Integrand size = 22, antiderivative size = 217 \[ \int x^3 \left (c+a^2 c x^2\right )^{3/2} \arctan (a x) \, dx=\frac {3 c x \sqrt {c+a^2 c x^2}}{112 a^3}-\frac {23 c x^3 \sqrt {c+a^2 c x^2}}{840 a}-\frac {1}{42} a c x^5 \sqrt {c+a^2 c x^2}-\frac {2 c \sqrt {c+a^2 c x^2} \arctan (a x)}{35 a^4}+\frac {c x^2 \sqrt {c+a^2 c x^2} \arctan (a x)}{35 a^2}+\frac {8}{35} c x^4 \sqrt {c+a^2 c x^2} \arctan (a x)+\frac {1}{7} a^2 c x^6 \sqrt {c+a^2 c x^2} \arctan (a x)+\frac {17 c^{3/2} \text {arctanh}\left (\frac {a \sqrt {c} x}{\sqrt {c+a^2 c x^2}}\right )}{560 a^4} \] Output:
3/112*c*x*(a^2*c*x^2+c)^(1/2)/a^3-23/840*c*x^3*(a^2*c*x^2+c)^(1/2)/a-1/42* a*c*x^5*(a^2*c*x^2+c)^(1/2)-2/35*c*(a^2*c*x^2+c)^(1/2)*arctan(a*x)/a^4+1/3 5*c*x^2*(a^2*c*x^2+c)^(1/2)*arctan(a*x)/a^2+8/35*c*x^4*(a^2*c*x^2+c)^(1/2) *arctan(a*x)+1/7*a^2*c*x^6*(a^2*c*x^2+c)^(1/2)*arctan(a*x)+17/560*c^(3/2)* arctanh(a*c^(1/2)*x/(a^2*c*x^2+c)^(1/2))/a^4
Time = 0.13 (sec) , antiderivative size = 119, normalized size of antiderivative = 0.55 \[ \int x^3 \left (c+a^2 c x^2\right )^{3/2} \arctan (a x) \, dx=\frac {a c x \sqrt {c+a^2 c x^2} \left (45-46 a^2 x^2-40 a^4 x^4\right )+48 c \left (1+a^2 x^2\right )^2 \left (-2+5 a^2 x^2\right ) \sqrt {c+a^2 c x^2} \arctan (a x)+51 c^{3/2} \log \left (a c x+\sqrt {c} \sqrt {c+a^2 c x^2}\right )}{1680 a^4} \] Input:
Integrate[x^3*(c + a^2*c*x^2)^(3/2)*ArcTan[a*x],x]
Output:
(a*c*x*Sqrt[c + a^2*c*x^2]*(45 - 46*a^2*x^2 - 40*a^4*x^4) + 48*c*(1 + a^2* x^2)^2*(-2 + 5*a^2*x^2)*Sqrt[c + a^2*c*x^2]*ArcTan[a*x] + 51*c^(3/2)*Log[a *c*x + Sqrt[c]*Sqrt[c + a^2*c*x^2]])/(1680*a^4)
Leaf count is larger than twice the leaf count of optimal. \(783\) vs. \(2(217)=434\).
Time = 2.48 (sec) , antiderivative size = 783, normalized size of antiderivative = 3.61, number of steps used = 27, number of rules used = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.182, Rules used = {5485, 5481, 262, 262, 224, 219, 262, 224, 219, 5487, 262, 224, 219, 262, 224, 219, 5465, 224, 219, 5487, 262, 224, 219, 5465, 224, 219}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int x^3 \arctan (a x) \left (a^2 c x^2+c\right )^{3/2} \, dx\) |
| \(\Big \downarrow \) 5485 |
| \(\displaystyle a^2 c \int x^5 \sqrt {a^2 c x^2+c} \arctan (a x)dx+c \int x^3 \sqrt {a^2 c x^2+c} \arctan (a x)dx\) |
| \(\Big \downarrow \) 5481 |
| \(\displaystyle a^2 c \left (\frac {1}{7} c \int \frac {x^5 \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx-\frac {1}{7} a c \int \frac {x^6}{\sqrt {a^2 c x^2+c}}dx+\frac {1}{7} x^6 \arctan (a x) \sqrt {a^2 c x^2+c}\right )+c \left (\frac {1}{5} c \int \frac {x^3 \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx-\frac {1}{5} a c \int \frac {x^4}{\sqrt {a^2 c x^2+c}}dx+\frac {1}{5} x^4 \arctan (a x) \sqrt {a^2 c x^2+c}\right )\) |
| \(\Big \downarrow \) 262 |
| \(\displaystyle c \left (\frac {1}{5} c \int \frac {x^3 \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx-\frac {1}{5} a c \left (\frac {x^3 \sqrt {a^2 c x^2+c}}{4 a^2 c}-\frac {3 \int \frac {x^2}{\sqrt {a^2 c x^2+c}}dx}{4 a^2}\right )+\frac {1}{5} x^4 \arctan (a x) \sqrt {a^2 c x^2+c}\right )+a^2 c \left (\frac {1}{7} c \int \frac {x^5 \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx-\frac {1}{7} a c \left (\frac {x^5 \sqrt {a^2 c x^2+c}}{6 a^2 c}-\frac {5 \int \frac {x^4}{\sqrt {a^2 c x^2+c}}dx}{6 a^2}\right )+\frac {1}{7} x^6 \arctan (a x) \sqrt {a^2 c x^2+c}\right )\) |
| \(\Big \downarrow \) 262 |
| \(\displaystyle c \left (\frac {1}{5} c \int \frac {x^3 \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx-\frac {1}{5} a c \left (\frac {x^3 \sqrt {a^2 c x^2+c}}{4 a^2 c}-\frac {3 \left (\frac {x \sqrt {a^2 c x^2+c}}{2 a^2 c}-\frac {\int \frac {1}{\sqrt {a^2 c x^2+c}}dx}{2 a^2}\right )}{4 a^2}\right )+\frac {1}{5} x^4 \arctan (a x) \sqrt {a^2 c x^2+c}\right )+a^2 c \left (\frac {1}{7} c \int \frac {x^5 \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx-\frac {1}{7} a c \left (\frac {x^5 \sqrt {a^2 c x^2+c}}{6 a^2 c}-\frac {5 \left (\frac {x^3 \sqrt {a^2 c x^2+c}}{4 a^2 c}-\frac {3 \int \frac {x^2}{\sqrt {a^2 c x^2+c}}dx}{4 a^2}\right )}{6 a^2}\right )+\frac {1}{7} x^6 \arctan (a x) \sqrt {a^2 c x^2+c}\right )\) |
| \(\Big \downarrow \) 224 |
| \(\displaystyle c \left (\frac {1}{5} c \int \frac {x^3 \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx-\frac {1}{5} a c \left (\frac {x^3 \sqrt {a^2 c x^2+c}}{4 a^2 c}-\frac {3 \left (\frac {x \sqrt {a^2 c x^2+c}}{2 a^2 c}-\frac {\int \frac {1}{1-\frac {a^2 c x^2}{a^2 c x^2+c}}d\frac {x}{\sqrt {a^2 c x^2+c}}}{2 a^2}\right )}{4 a^2}\right )+\frac {1}{5} x^4 \arctan (a x) \sqrt {a^2 c x^2+c}\right )+a^2 c \left (\frac {1}{7} c \int \frac {x^5 \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx-\frac {1}{7} a c \left (\frac {x^5 \sqrt {a^2 c x^2+c}}{6 a^2 c}-\frac {5 \left (\frac {x^3 \sqrt {a^2 c x^2+c}}{4 a^2 c}-\frac {3 \int \frac {x^2}{\sqrt {a^2 c x^2+c}}dx}{4 a^2}\right )}{6 a^2}\right )+\frac {1}{7} x^6 \arctan (a x) \sqrt {a^2 c x^2+c}\right )\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle a^2 c \left (\frac {1}{7} c \int \frac {x^5 \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx-\frac {1}{7} a c \left (\frac {x^5 \sqrt {a^2 c x^2+c}}{6 a^2 c}-\frac {5 \left (\frac {x^3 \sqrt {a^2 c x^2+c}}{4 a^2 c}-\frac {3 \int \frac {x^2}{\sqrt {a^2 c x^2+c}}dx}{4 a^2}\right )}{6 a^2}\right )+\frac {1}{7} x^6 \arctan (a x) \sqrt {a^2 c x^2+c}\right )+c \left (\frac {1}{5} c \int \frac {x^3 \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx+\frac {1}{5} x^4 \arctan (a x) \sqrt {a^2 c x^2+c}-\frac {1}{5} a c \left (\frac {x^3 \sqrt {a^2 c x^2+c}}{4 a^2 c}-\frac {3 \left (\frac {x \sqrt {a^2 c x^2+c}}{2 a^2 c}-\frac {\text {arctanh}\left (\frac {a \sqrt {c} x}{\sqrt {a^2 c x^2+c}}\right )}{2 a^3 \sqrt {c}}\right )}{4 a^2}\right )\right )\) |
| \(\Big \downarrow \) 262 |
| \(\displaystyle a^2 c \left (\frac {1}{7} c \int \frac {x^5 \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx-\frac {1}{7} a c \left (\frac {x^5 \sqrt {a^2 c x^2+c}}{6 a^2 c}-\frac {5 \left (\frac {x^3 \sqrt {a^2 c x^2+c}}{4 a^2 c}-\frac {3 \left (\frac {x \sqrt {a^2 c x^2+c}}{2 a^2 c}-\frac {\int \frac {1}{\sqrt {a^2 c x^2+c}}dx}{2 a^2}\right )}{4 a^2}\right )}{6 a^2}\right )+\frac {1}{7} x^6 \arctan (a x) \sqrt {a^2 c x^2+c}\right )+c \left (\frac {1}{5} c \int \frac {x^3 \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx+\frac {1}{5} x^4 \arctan (a x) \sqrt {a^2 c x^2+c}-\frac {1}{5} a c \left (\frac {x^3 \sqrt {a^2 c x^2+c}}{4 a^2 c}-\frac {3 \left (\frac {x \sqrt {a^2 c x^2+c}}{2 a^2 c}-\frac {\text {arctanh}\left (\frac {a \sqrt {c} x}{\sqrt {a^2 c x^2+c}}\right )}{2 a^3 \sqrt {c}}\right )}{4 a^2}\right )\right )\) |
| \(\Big \downarrow \) 224 |
| \(\displaystyle a^2 c \left (\frac {1}{7} c \int \frac {x^5 \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx-\frac {1}{7} a c \left (\frac {x^5 \sqrt {a^2 c x^2+c}}{6 a^2 c}-\frac {5 \left (\frac {x^3 \sqrt {a^2 c x^2+c}}{4 a^2 c}-\frac {3 \left (\frac {x \sqrt {a^2 c x^2+c}}{2 a^2 c}-\frac {\int \frac {1}{1-\frac {a^2 c x^2}{a^2 c x^2+c}}d\frac {x}{\sqrt {a^2 c x^2+c}}}{2 a^2}\right )}{4 a^2}\right )}{6 a^2}\right )+\frac {1}{7} x^6 \arctan (a x) \sqrt {a^2 c x^2+c}\right )+c \left (\frac {1}{5} c \int \frac {x^3 \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx+\frac {1}{5} x^4 \arctan (a x) \sqrt {a^2 c x^2+c}-\frac {1}{5} a c \left (\frac {x^3 \sqrt {a^2 c x^2+c}}{4 a^2 c}-\frac {3 \left (\frac {x \sqrt {a^2 c x^2+c}}{2 a^2 c}-\frac {\text {arctanh}\left (\frac {a \sqrt {c} x}{\sqrt {a^2 c x^2+c}}\right )}{2 a^3 \sqrt {c}}\right )}{4 a^2}\right )\right )\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle c \left (\frac {1}{5} c \int \frac {x^3 \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx+\frac {1}{5} x^4 \arctan (a x) \sqrt {a^2 c x^2+c}-\frac {1}{5} a c \left (\frac {x^3 \sqrt {a^2 c x^2+c}}{4 a^2 c}-\frac {3 \left (\frac {x \sqrt {a^2 c x^2+c}}{2 a^2 c}-\frac {\text {arctanh}\left (\frac {a \sqrt {c} x}{\sqrt {a^2 c x^2+c}}\right )}{2 a^3 \sqrt {c}}\right )}{4 a^2}\right )\right )+a^2 c \left (\frac {1}{7} c \int \frac {x^5 \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx+\frac {1}{7} x^6 \arctan (a x) \sqrt {a^2 c x^2+c}-\frac {1}{7} a c \left (\frac {x^5 \sqrt {a^2 c x^2+c}}{6 a^2 c}-\frac {5 \left (\frac {x^3 \sqrt {a^2 c x^2+c}}{4 a^2 c}-\frac {3 \left (\frac {x \sqrt {a^2 c x^2+c}}{2 a^2 c}-\frac {\text {arctanh}\left (\frac {a \sqrt {c} x}{\sqrt {a^2 c x^2+c}}\right )}{2 a^3 \sqrt {c}}\right )}{4 a^2}\right )}{6 a^2}\right )\right )\) |
| \(\Big \downarrow \) 5487 |
| \(\displaystyle c \left (\frac {1}{5} c \left (-\frac {2 \int \frac {x \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{3 a^2}-\frac {\int \frac {x^2}{\sqrt {a^2 c x^2+c}}dx}{3 a}+\frac {x^2 \arctan (a x) \sqrt {a^2 c x^2+c}}{3 a^2 c}\right )+\frac {1}{5} x^4 \arctan (a x) \sqrt {a^2 c x^2+c}-\frac {1}{5} a c \left (\frac {x^3 \sqrt {a^2 c x^2+c}}{4 a^2 c}-\frac {3 \left (\frac {x \sqrt {a^2 c x^2+c}}{2 a^2 c}-\frac {\text {arctanh}\left (\frac {a \sqrt {c} x}{\sqrt {a^2 c x^2+c}}\right )}{2 a^3 \sqrt {c}}\right )}{4 a^2}\right )\right )+a^2 c \left (\frac {1}{7} c \left (-\frac {4 \int \frac {x^3 \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{5 a^2}-\frac {\int \frac {x^4}{\sqrt {a^2 c x^2+c}}dx}{5 a}+\frac {x^4 \arctan (a x) \sqrt {a^2 c x^2+c}}{5 a^2 c}\right )+\frac {1}{7} x^6 \arctan (a x) \sqrt {a^2 c x^2+c}-\frac {1}{7} a c \left (\frac {x^5 \sqrt {a^2 c x^2+c}}{6 a^2 c}-\frac {5 \left (\frac {x^3 \sqrt {a^2 c x^2+c}}{4 a^2 c}-\frac {3 \left (\frac {x \sqrt {a^2 c x^2+c}}{2 a^2 c}-\frac {\text {arctanh}\left (\frac {a \sqrt {c} x}{\sqrt {a^2 c x^2+c}}\right )}{2 a^3 \sqrt {c}}\right )}{4 a^2}\right )}{6 a^2}\right )\right )\) |
| \(\Big \downarrow \) 262 |
| \(\displaystyle c \left (\frac {1}{5} c \left (-\frac {2 \int \frac {x \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{3 a^2}-\frac {\frac {x \sqrt {a^2 c x^2+c}}{2 a^2 c}-\frac {\int \frac {1}{\sqrt {a^2 c x^2+c}}dx}{2 a^2}}{3 a}+\frac {x^2 \arctan (a x) \sqrt {a^2 c x^2+c}}{3 a^2 c}\right )+\frac {1}{5} x^4 \arctan (a x) \sqrt {a^2 c x^2+c}-\frac {1}{5} a c \left (\frac {x^3 \sqrt {a^2 c x^2+c}}{4 a^2 c}-\frac {3 \left (\frac {x \sqrt {a^2 c x^2+c}}{2 a^2 c}-\frac {\text {arctanh}\left (\frac {a \sqrt {c} x}{\sqrt {a^2 c x^2+c}}\right )}{2 a^3 \sqrt {c}}\right )}{4 a^2}\right )\right )+a^2 c \left (\frac {1}{7} c \left (-\frac {4 \int \frac {x^3 \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{5 a^2}-\frac {\frac {x^3 \sqrt {a^2 c x^2+c}}{4 a^2 c}-\frac {3 \int \frac {x^2}{\sqrt {a^2 c x^2+c}}dx}{4 a^2}}{5 a}+\frac {x^4 \arctan (a x) \sqrt {a^2 c x^2+c}}{5 a^2 c}\right )+\frac {1}{7} x^6 \arctan (a x) \sqrt {a^2 c x^2+c}-\frac {1}{7} a c \left (\frac {x^5 \sqrt {a^2 c x^2+c}}{6 a^2 c}-\frac {5 \left (\frac {x^3 \sqrt {a^2 c x^2+c}}{4 a^2 c}-\frac {3 \left (\frac {x \sqrt {a^2 c x^2+c}}{2 a^2 c}-\frac {\text {arctanh}\left (\frac {a \sqrt {c} x}{\sqrt {a^2 c x^2+c}}\right )}{2 a^3 \sqrt {c}}\right )}{4 a^2}\right )}{6 a^2}\right )\right )\) |
| \(\Big \downarrow \) 224 |
| \(\displaystyle c \left (\frac {1}{5} c \left (-\frac {2 \int \frac {x \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{3 a^2}-\frac {\frac {x \sqrt {a^2 c x^2+c}}{2 a^2 c}-\frac {\int \frac {1}{1-\frac {a^2 c x^2}{a^2 c x^2+c}}d\frac {x}{\sqrt {a^2 c x^2+c}}}{2 a^2}}{3 a}+\frac {x^2 \arctan (a x) \sqrt {a^2 c x^2+c}}{3 a^2 c}\right )+\frac {1}{5} x^4 \arctan (a x) \sqrt {a^2 c x^2+c}-\frac {1}{5} a c \left (\frac {x^3 \sqrt {a^2 c x^2+c}}{4 a^2 c}-\frac {3 \left (\frac {x \sqrt {a^2 c x^2+c}}{2 a^2 c}-\frac {\text {arctanh}\left (\frac {a \sqrt {c} x}{\sqrt {a^2 c x^2+c}}\right )}{2 a^3 \sqrt {c}}\right )}{4 a^2}\right )\right )+a^2 c \left (\frac {1}{7} c \left (-\frac {4 \int \frac {x^3 \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{5 a^2}-\frac {\frac {x^3 \sqrt {a^2 c x^2+c}}{4 a^2 c}-\frac {3 \int \frac {x^2}{\sqrt {a^2 c x^2+c}}dx}{4 a^2}}{5 a}+\frac {x^4 \arctan (a x) \sqrt {a^2 c x^2+c}}{5 a^2 c}\right )+\frac {1}{7} x^6 \arctan (a x) \sqrt {a^2 c x^2+c}-\frac {1}{7} a c \left (\frac {x^5 \sqrt {a^2 c x^2+c}}{6 a^2 c}-\frac {5 \left (\frac {x^3 \sqrt {a^2 c x^2+c}}{4 a^2 c}-\frac {3 \left (\frac {x \sqrt {a^2 c x^2+c}}{2 a^2 c}-\frac {\text {arctanh}\left (\frac {a \sqrt {c} x}{\sqrt {a^2 c x^2+c}}\right )}{2 a^3 \sqrt {c}}\right )}{4 a^2}\right )}{6 a^2}\right )\right )\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle c \left (\frac {1}{5} c \left (-\frac {2 \int \frac {x \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{3 a^2}+\frac {x^2 \arctan (a x) \sqrt {a^2 c x^2+c}}{3 a^2 c}-\frac {\frac {x \sqrt {a^2 c x^2+c}}{2 a^2 c}-\frac {\text {arctanh}\left (\frac {a \sqrt {c} x}{\sqrt {a^2 c x^2+c}}\right )}{2 a^3 \sqrt {c}}}{3 a}\right )+\frac {1}{5} x^4 \arctan (a x) \sqrt {a^2 c x^2+c}-\frac {1}{5} a c \left (\frac {x^3 \sqrt {a^2 c x^2+c}}{4 a^2 c}-\frac {3 \left (\frac {x \sqrt {a^2 c x^2+c}}{2 a^2 c}-\frac {\text {arctanh}\left (\frac {a \sqrt {c} x}{\sqrt {a^2 c x^2+c}}\right )}{2 a^3 \sqrt {c}}\right )}{4 a^2}\right )\right )+a^2 c \left (\frac {1}{7} c \left (-\frac {4 \int \frac {x^3 \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{5 a^2}-\frac {\frac {x^3 \sqrt {a^2 c x^2+c}}{4 a^2 c}-\frac {3 \int \frac {x^2}{\sqrt {a^2 c x^2+c}}dx}{4 a^2}}{5 a}+\frac {x^4 \arctan (a x) \sqrt {a^2 c x^2+c}}{5 a^2 c}\right )+\frac {1}{7} x^6 \arctan (a x) \sqrt {a^2 c x^2+c}-\frac {1}{7} a c \left (\frac {x^5 \sqrt {a^2 c x^2+c}}{6 a^2 c}-\frac {5 \left (\frac {x^3 \sqrt {a^2 c x^2+c}}{4 a^2 c}-\frac {3 \left (\frac {x \sqrt {a^2 c x^2+c}}{2 a^2 c}-\frac {\text {arctanh}\left (\frac {a \sqrt {c} x}{\sqrt {a^2 c x^2+c}}\right )}{2 a^3 \sqrt {c}}\right )}{4 a^2}\right )}{6 a^2}\right )\right )\) |
| \(\Big \downarrow \) 262 |
| \(\displaystyle c \left (\frac {1}{5} c \left (-\frac {2 \int \frac {x \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{3 a^2}+\frac {x^2 \arctan (a x) \sqrt {a^2 c x^2+c}}{3 a^2 c}-\frac {\frac {x \sqrt {a^2 c x^2+c}}{2 a^2 c}-\frac {\text {arctanh}\left (\frac {a \sqrt {c} x}{\sqrt {a^2 c x^2+c}}\right )}{2 a^3 \sqrt {c}}}{3 a}\right )+\frac {1}{5} x^4 \arctan (a x) \sqrt {a^2 c x^2+c}-\frac {1}{5} a c \left (\frac {x^3 \sqrt {a^2 c x^2+c}}{4 a^2 c}-\frac {3 \left (\frac {x \sqrt {a^2 c x^2+c}}{2 a^2 c}-\frac {\text {arctanh}\left (\frac {a \sqrt {c} x}{\sqrt {a^2 c x^2+c}}\right )}{2 a^3 \sqrt {c}}\right )}{4 a^2}\right )\right )+a^2 c \left (\frac {1}{7} c \left (-\frac {4 \int \frac {x^3 \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{5 a^2}-\frac {\frac {x^3 \sqrt {a^2 c x^2+c}}{4 a^2 c}-\frac {3 \left (\frac {x \sqrt {a^2 c x^2+c}}{2 a^2 c}-\frac {\int \frac {1}{\sqrt {a^2 c x^2+c}}dx}{2 a^2}\right )}{4 a^2}}{5 a}+\frac {x^4 \arctan (a x) \sqrt {a^2 c x^2+c}}{5 a^2 c}\right )+\frac {1}{7} x^6 \arctan (a x) \sqrt {a^2 c x^2+c}-\frac {1}{7} a c \left (\frac {x^5 \sqrt {a^2 c x^2+c}}{6 a^2 c}-\frac {5 \left (\frac {x^3 \sqrt {a^2 c x^2+c}}{4 a^2 c}-\frac {3 \left (\frac {x \sqrt {a^2 c x^2+c}}{2 a^2 c}-\frac {\text {arctanh}\left (\frac {a \sqrt {c} x}{\sqrt {a^2 c x^2+c}}\right )}{2 a^3 \sqrt {c}}\right )}{4 a^2}\right )}{6 a^2}\right )\right )\) |
| \(\Big \downarrow \) 224 |
| \(\displaystyle c \left (\frac {1}{5} c \left (-\frac {2 \int \frac {x \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{3 a^2}+\frac {x^2 \arctan (a x) \sqrt {a^2 c x^2+c}}{3 a^2 c}-\frac {\frac {x \sqrt {a^2 c x^2+c}}{2 a^2 c}-\frac {\text {arctanh}\left (\frac {a \sqrt {c} x}{\sqrt {a^2 c x^2+c}}\right )}{2 a^3 \sqrt {c}}}{3 a}\right )+\frac {1}{5} x^4 \arctan (a x) \sqrt {a^2 c x^2+c}-\frac {1}{5} a c \left (\frac {x^3 \sqrt {a^2 c x^2+c}}{4 a^2 c}-\frac {3 \left (\frac {x \sqrt {a^2 c x^2+c}}{2 a^2 c}-\frac {\text {arctanh}\left (\frac {a \sqrt {c} x}{\sqrt {a^2 c x^2+c}}\right )}{2 a^3 \sqrt {c}}\right )}{4 a^2}\right )\right )+a^2 c \left (\frac {1}{7} c \left (-\frac {4 \int \frac {x^3 \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{5 a^2}-\frac {\frac {x^3 \sqrt {a^2 c x^2+c}}{4 a^2 c}-\frac {3 \left (\frac {x \sqrt {a^2 c x^2+c}}{2 a^2 c}-\frac {\int \frac {1}{1-\frac {a^2 c x^2}{a^2 c x^2+c}}d\frac {x}{\sqrt {a^2 c x^2+c}}}{2 a^2}\right )}{4 a^2}}{5 a}+\frac {x^4 \arctan (a x) \sqrt {a^2 c x^2+c}}{5 a^2 c}\right )+\frac {1}{7} x^6 \arctan (a x) \sqrt {a^2 c x^2+c}-\frac {1}{7} a c \left (\frac {x^5 \sqrt {a^2 c x^2+c}}{6 a^2 c}-\frac {5 \left (\frac {x^3 \sqrt {a^2 c x^2+c}}{4 a^2 c}-\frac {3 \left (\frac {x \sqrt {a^2 c x^2+c}}{2 a^2 c}-\frac {\text {arctanh}\left (\frac {a \sqrt {c} x}{\sqrt {a^2 c x^2+c}}\right )}{2 a^3 \sqrt {c}}\right )}{4 a^2}\right )}{6 a^2}\right )\right )\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle c \left (\frac {1}{5} c \left (-\frac {2 \int \frac {x \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{3 a^2}+\frac {x^2 \arctan (a x) \sqrt {a^2 c x^2+c}}{3 a^2 c}-\frac {\frac {x \sqrt {a^2 c x^2+c}}{2 a^2 c}-\frac {\text {arctanh}\left (\frac {a \sqrt {c} x}{\sqrt {a^2 c x^2+c}}\right )}{2 a^3 \sqrt {c}}}{3 a}\right )+\frac {1}{5} x^4 \arctan (a x) \sqrt {a^2 c x^2+c}-\frac {1}{5} a c \left (\frac {x^3 \sqrt {a^2 c x^2+c}}{4 a^2 c}-\frac {3 \left (\frac {x \sqrt {a^2 c x^2+c}}{2 a^2 c}-\frac {\text {arctanh}\left (\frac {a \sqrt {c} x}{\sqrt {a^2 c x^2+c}}\right )}{2 a^3 \sqrt {c}}\right )}{4 a^2}\right )\right )+a^2 c \left (\frac {1}{7} c \left (-\frac {4 \int \frac {x^3 \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{5 a^2}+\frac {x^4 \arctan (a x) \sqrt {a^2 c x^2+c}}{5 a^2 c}-\frac {\frac {x^3 \sqrt {a^2 c x^2+c}}{4 a^2 c}-\frac {3 \left (\frac {x \sqrt {a^2 c x^2+c}}{2 a^2 c}-\frac {\text {arctanh}\left (\frac {a \sqrt {c} x}{\sqrt {a^2 c x^2+c}}\right )}{2 a^3 \sqrt {c}}\right )}{4 a^2}}{5 a}\right )+\frac {1}{7} x^6 \arctan (a x) \sqrt {a^2 c x^2+c}-\frac {1}{7} a c \left (\frac {x^5 \sqrt {a^2 c x^2+c}}{6 a^2 c}-\frac {5 \left (\frac {x^3 \sqrt {a^2 c x^2+c}}{4 a^2 c}-\frac {3 \left (\frac {x \sqrt {a^2 c x^2+c}}{2 a^2 c}-\frac {\text {arctanh}\left (\frac {a \sqrt {c} x}{\sqrt {a^2 c x^2+c}}\right )}{2 a^3 \sqrt {c}}\right )}{4 a^2}\right )}{6 a^2}\right )\right )\) |
| \(\Big \downarrow \) 5465 |
| \(\displaystyle c \left (\frac {1}{5} c \left (-\frac {2 \left (\frac {\arctan (a x) \sqrt {a^2 c x^2+c}}{a^2 c}-\frac {\int \frac {1}{\sqrt {a^2 c x^2+c}}dx}{a}\right )}{3 a^2}+\frac {x^2 \arctan (a x) \sqrt {a^2 c x^2+c}}{3 a^2 c}-\frac {\frac {x \sqrt {a^2 c x^2+c}}{2 a^2 c}-\frac {\text {arctanh}\left (\frac {a \sqrt {c} x}{\sqrt {a^2 c x^2+c}}\right )}{2 a^3 \sqrt {c}}}{3 a}\right )+\frac {1}{5} x^4 \arctan (a x) \sqrt {a^2 c x^2+c}-\frac {1}{5} a c \left (\frac {x^3 \sqrt {a^2 c x^2+c}}{4 a^2 c}-\frac {3 \left (\frac {x \sqrt {a^2 c x^2+c}}{2 a^2 c}-\frac {\text {arctanh}\left (\frac {a \sqrt {c} x}{\sqrt {a^2 c x^2+c}}\right )}{2 a^3 \sqrt {c}}\right )}{4 a^2}\right )\right )+a^2 c \left (\frac {1}{7} c \left (-\frac {4 \int \frac {x^3 \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{5 a^2}+\frac {x^4 \arctan (a x) \sqrt {a^2 c x^2+c}}{5 a^2 c}-\frac {\frac {x^3 \sqrt {a^2 c x^2+c}}{4 a^2 c}-\frac {3 \left (\frac {x \sqrt {a^2 c x^2+c}}{2 a^2 c}-\frac {\text {arctanh}\left (\frac {a \sqrt {c} x}{\sqrt {a^2 c x^2+c}}\right )}{2 a^3 \sqrt {c}}\right )}{4 a^2}}{5 a}\right )+\frac {1}{7} x^6 \arctan (a x) \sqrt {a^2 c x^2+c}-\frac {1}{7} a c \left (\frac {x^5 \sqrt {a^2 c x^2+c}}{6 a^2 c}-\frac {5 \left (\frac {x^3 \sqrt {a^2 c x^2+c}}{4 a^2 c}-\frac {3 \left (\frac {x \sqrt {a^2 c x^2+c}}{2 a^2 c}-\frac {\text {arctanh}\left (\frac {a \sqrt {c} x}{\sqrt {a^2 c x^2+c}}\right )}{2 a^3 \sqrt {c}}\right )}{4 a^2}\right )}{6 a^2}\right )\right )\) |
| \(\Big \downarrow \) 224 |
| \(\displaystyle c \left (\frac {1}{5} c \left (-\frac {2 \left (\frac {\arctan (a x) \sqrt {a^2 c x^2+c}}{a^2 c}-\frac {\int \frac {1}{1-\frac {a^2 c x^2}{a^2 c x^2+c}}d\frac {x}{\sqrt {a^2 c x^2+c}}}{a}\right )}{3 a^2}+\frac {x^2 \arctan (a x) \sqrt {a^2 c x^2+c}}{3 a^2 c}-\frac {\frac {x \sqrt {a^2 c x^2+c}}{2 a^2 c}-\frac {\text {arctanh}\left (\frac {a \sqrt {c} x}{\sqrt {a^2 c x^2+c}}\right )}{2 a^3 \sqrt {c}}}{3 a}\right )+\frac {1}{5} x^4 \arctan (a x) \sqrt {a^2 c x^2+c}-\frac {1}{5} a c \left (\frac {x^3 \sqrt {a^2 c x^2+c}}{4 a^2 c}-\frac {3 \left (\frac {x \sqrt {a^2 c x^2+c}}{2 a^2 c}-\frac {\text {arctanh}\left (\frac {a \sqrt {c} x}{\sqrt {a^2 c x^2+c}}\right )}{2 a^3 \sqrt {c}}\right )}{4 a^2}\right )\right )+a^2 c \left (\frac {1}{7} c \left (-\frac {4 \int \frac {x^3 \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{5 a^2}+\frac {x^4 \arctan (a x) \sqrt {a^2 c x^2+c}}{5 a^2 c}-\frac {\frac {x^3 \sqrt {a^2 c x^2+c}}{4 a^2 c}-\frac {3 \left (\frac {x \sqrt {a^2 c x^2+c}}{2 a^2 c}-\frac {\text {arctanh}\left (\frac {a \sqrt {c} x}{\sqrt {a^2 c x^2+c}}\right )}{2 a^3 \sqrt {c}}\right )}{4 a^2}}{5 a}\right )+\frac {1}{7} x^6 \arctan (a x) \sqrt {a^2 c x^2+c}-\frac {1}{7} a c \left (\frac {x^5 \sqrt {a^2 c x^2+c}}{6 a^2 c}-\frac {5 \left (\frac {x^3 \sqrt {a^2 c x^2+c}}{4 a^2 c}-\frac {3 \left (\frac {x \sqrt {a^2 c x^2+c}}{2 a^2 c}-\frac {\text {arctanh}\left (\frac {a \sqrt {c} x}{\sqrt {a^2 c x^2+c}}\right )}{2 a^3 \sqrt {c}}\right )}{4 a^2}\right )}{6 a^2}\right )\right )\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle a^2 c \left (\frac {1}{7} c \left (-\frac {4 \int \frac {x^3 \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{5 a^2}+\frac {x^4 \arctan (a x) \sqrt {a^2 c x^2+c}}{5 a^2 c}-\frac {\frac {x^3 \sqrt {a^2 c x^2+c}}{4 a^2 c}-\frac {3 \left (\frac {x \sqrt {a^2 c x^2+c}}{2 a^2 c}-\frac {\text {arctanh}\left (\frac {a \sqrt {c} x}{\sqrt {a^2 c x^2+c}}\right )}{2 a^3 \sqrt {c}}\right )}{4 a^2}}{5 a}\right )+\frac {1}{7} x^6 \arctan (a x) \sqrt {a^2 c x^2+c}-\frac {1}{7} a c \left (\frac {x^5 \sqrt {a^2 c x^2+c}}{6 a^2 c}-\frac {5 \left (\frac {x^3 \sqrt {a^2 c x^2+c}}{4 a^2 c}-\frac {3 \left (\frac {x \sqrt {a^2 c x^2+c}}{2 a^2 c}-\frac {\text {arctanh}\left (\frac {a \sqrt {c} x}{\sqrt {a^2 c x^2+c}}\right )}{2 a^3 \sqrt {c}}\right )}{4 a^2}\right )}{6 a^2}\right )\right )+c \left (\frac {1}{5} x^4 \arctan (a x) \sqrt {a^2 c x^2+c}+\frac {1}{5} c \left (-\frac {2 \left (\frac {\arctan (a x) \sqrt {a^2 c x^2+c}}{a^2 c}-\frac {\text {arctanh}\left (\frac {a \sqrt {c} x}{\sqrt {a^2 c x^2+c}}\right )}{a^2 \sqrt {c}}\right )}{3 a^2}+\frac {x^2 \arctan (a x) \sqrt {a^2 c x^2+c}}{3 a^2 c}-\frac {\frac {x \sqrt {a^2 c x^2+c}}{2 a^2 c}-\frac {\text {arctanh}\left (\frac {a \sqrt {c} x}{\sqrt {a^2 c x^2+c}}\right )}{2 a^3 \sqrt {c}}}{3 a}\right )-\frac {1}{5} a c \left (\frac {x^3 \sqrt {a^2 c x^2+c}}{4 a^2 c}-\frac {3 \left (\frac {x \sqrt {a^2 c x^2+c}}{2 a^2 c}-\frac {\text {arctanh}\left (\frac {a \sqrt {c} x}{\sqrt {a^2 c x^2+c}}\right )}{2 a^3 \sqrt {c}}\right )}{4 a^2}\right )\right )\) |
| \(\Big \downarrow \) 5487 |
| \(\displaystyle a^2 c \left (\frac {1}{7} c \left (-\frac {4 \left (-\frac {2 \int \frac {x \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{3 a^2}-\frac {\int \frac {x^2}{\sqrt {a^2 c x^2+c}}dx}{3 a}+\frac {x^2 \arctan (a x) \sqrt {a^2 c x^2+c}}{3 a^2 c}\right )}{5 a^2}+\frac {x^4 \arctan (a x) \sqrt {a^2 c x^2+c}}{5 a^2 c}-\frac {\frac {x^3 \sqrt {a^2 c x^2+c}}{4 a^2 c}-\frac {3 \left (\frac {x \sqrt {a^2 c x^2+c}}{2 a^2 c}-\frac {\text {arctanh}\left (\frac {a \sqrt {c} x}{\sqrt {a^2 c x^2+c}}\right )}{2 a^3 \sqrt {c}}\right )}{4 a^2}}{5 a}\right )+\frac {1}{7} x^6 \arctan (a x) \sqrt {a^2 c x^2+c}-\frac {1}{7} a c \left (\frac {x^5 \sqrt {a^2 c x^2+c}}{6 a^2 c}-\frac {5 \left (\frac {x^3 \sqrt {a^2 c x^2+c}}{4 a^2 c}-\frac {3 \left (\frac {x \sqrt {a^2 c x^2+c}}{2 a^2 c}-\frac {\text {arctanh}\left (\frac {a \sqrt {c} x}{\sqrt {a^2 c x^2+c}}\right )}{2 a^3 \sqrt {c}}\right )}{4 a^2}\right )}{6 a^2}\right )\right )+c \left (\frac {1}{5} x^4 \arctan (a x) \sqrt {a^2 c x^2+c}+\frac {1}{5} c \left (-\frac {2 \left (\frac {\arctan (a x) \sqrt {a^2 c x^2+c}}{a^2 c}-\frac {\text {arctanh}\left (\frac {a \sqrt {c} x}{\sqrt {a^2 c x^2+c}}\right )}{a^2 \sqrt {c}}\right )}{3 a^2}+\frac {x^2 \arctan (a x) \sqrt {a^2 c x^2+c}}{3 a^2 c}-\frac {\frac {x \sqrt {a^2 c x^2+c}}{2 a^2 c}-\frac {\text {arctanh}\left (\frac {a \sqrt {c} x}{\sqrt {a^2 c x^2+c}}\right )}{2 a^3 \sqrt {c}}}{3 a}\right )-\frac {1}{5} a c \left (\frac {x^3 \sqrt {a^2 c x^2+c}}{4 a^2 c}-\frac {3 \left (\frac {x \sqrt {a^2 c x^2+c}}{2 a^2 c}-\frac {\text {arctanh}\left (\frac {a \sqrt {c} x}{\sqrt {a^2 c x^2+c}}\right )}{2 a^3 \sqrt {c}}\right )}{4 a^2}\right )\right )\) |
| \(\Big \downarrow \) 262 |
| \(\displaystyle a^2 c \left (\frac {1}{7} c \left (-\frac {4 \left (-\frac {2 \int \frac {x \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{3 a^2}-\frac {\frac {x \sqrt {a^2 c x^2+c}}{2 a^2 c}-\frac {\int \frac {1}{\sqrt {a^2 c x^2+c}}dx}{2 a^2}}{3 a}+\frac {x^2 \arctan (a x) \sqrt {a^2 c x^2+c}}{3 a^2 c}\right )}{5 a^2}+\frac {x^4 \arctan (a x) \sqrt {a^2 c x^2+c}}{5 a^2 c}-\frac {\frac {x^3 \sqrt {a^2 c x^2+c}}{4 a^2 c}-\frac {3 \left (\frac {x \sqrt {a^2 c x^2+c}}{2 a^2 c}-\frac {\text {arctanh}\left (\frac {a \sqrt {c} x}{\sqrt {a^2 c x^2+c}}\right )}{2 a^3 \sqrt {c}}\right )}{4 a^2}}{5 a}\right )+\frac {1}{7} x^6 \arctan (a x) \sqrt {a^2 c x^2+c}-\frac {1}{7} a c \left (\frac {x^5 \sqrt {a^2 c x^2+c}}{6 a^2 c}-\frac {5 \left (\frac {x^3 \sqrt {a^2 c x^2+c}}{4 a^2 c}-\frac {3 \left (\frac {x \sqrt {a^2 c x^2+c}}{2 a^2 c}-\frac {\text {arctanh}\left (\frac {a \sqrt {c} x}{\sqrt {a^2 c x^2+c}}\right )}{2 a^3 \sqrt {c}}\right )}{4 a^2}\right )}{6 a^2}\right )\right )+c \left (\frac {1}{5} x^4 \arctan (a x) \sqrt {a^2 c x^2+c}+\frac {1}{5} c \left (-\frac {2 \left (\frac {\arctan (a x) \sqrt {a^2 c x^2+c}}{a^2 c}-\frac {\text {arctanh}\left (\frac {a \sqrt {c} x}{\sqrt {a^2 c x^2+c}}\right )}{a^2 \sqrt {c}}\right )}{3 a^2}+\frac {x^2 \arctan (a x) \sqrt {a^2 c x^2+c}}{3 a^2 c}-\frac {\frac {x \sqrt {a^2 c x^2+c}}{2 a^2 c}-\frac {\text {arctanh}\left (\frac {a \sqrt {c} x}{\sqrt {a^2 c x^2+c}}\right )}{2 a^3 \sqrt {c}}}{3 a}\right )-\frac {1}{5} a c \left (\frac {x^3 \sqrt {a^2 c x^2+c}}{4 a^2 c}-\frac {3 \left (\frac {x \sqrt {a^2 c x^2+c}}{2 a^2 c}-\frac {\text {arctanh}\left (\frac {a \sqrt {c} x}{\sqrt {a^2 c x^2+c}}\right )}{2 a^3 \sqrt {c}}\right )}{4 a^2}\right )\right )\) |
| \(\Big \downarrow \) 224 |
| \(\displaystyle a^2 c \left (\frac {1}{7} c \left (-\frac {4 \left (-\frac {2 \int \frac {x \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{3 a^2}-\frac {\frac {x \sqrt {a^2 c x^2+c}}{2 a^2 c}-\frac {\int \frac {1}{1-\frac {a^2 c x^2}{a^2 c x^2+c}}d\frac {x}{\sqrt {a^2 c x^2+c}}}{2 a^2}}{3 a}+\frac {x^2 \arctan (a x) \sqrt {a^2 c x^2+c}}{3 a^2 c}\right )}{5 a^2}+\frac {x^4 \arctan (a x) \sqrt {a^2 c x^2+c}}{5 a^2 c}-\frac {\frac {x^3 \sqrt {a^2 c x^2+c}}{4 a^2 c}-\frac {3 \left (\frac {x \sqrt {a^2 c x^2+c}}{2 a^2 c}-\frac {\text {arctanh}\left (\frac {a \sqrt {c} x}{\sqrt {a^2 c x^2+c}}\right )}{2 a^3 \sqrt {c}}\right )}{4 a^2}}{5 a}\right )+\frac {1}{7} x^6 \arctan (a x) \sqrt {a^2 c x^2+c}-\frac {1}{7} a c \left (\frac {x^5 \sqrt {a^2 c x^2+c}}{6 a^2 c}-\frac {5 \left (\frac {x^3 \sqrt {a^2 c x^2+c}}{4 a^2 c}-\frac {3 \left (\frac {x \sqrt {a^2 c x^2+c}}{2 a^2 c}-\frac {\text {arctanh}\left (\frac {a \sqrt {c} x}{\sqrt {a^2 c x^2+c}}\right )}{2 a^3 \sqrt {c}}\right )}{4 a^2}\right )}{6 a^2}\right )\right )+c \left (\frac {1}{5} x^4 \arctan (a x) \sqrt {a^2 c x^2+c}+\frac {1}{5} c \left (-\frac {2 \left (\frac {\arctan (a x) \sqrt {a^2 c x^2+c}}{a^2 c}-\frac {\text {arctanh}\left (\frac {a \sqrt {c} x}{\sqrt {a^2 c x^2+c}}\right )}{a^2 \sqrt {c}}\right )}{3 a^2}+\frac {x^2 \arctan (a x) \sqrt {a^2 c x^2+c}}{3 a^2 c}-\frac {\frac {x \sqrt {a^2 c x^2+c}}{2 a^2 c}-\frac {\text {arctanh}\left (\frac {a \sqrt {c} x}{\sqrt {a^2 c x^2+c}}\right )}{2 a^3 \sqrt {c}}}{3 a}\right )-\frac {1}{5} a c \left (\frac {x^3 \sqrt {a^2 c x^2+c}}{4 a^2 c}-\frac {3 \left (\frac {x \sqrt {a^2 c x^2+c}}{2 a^2 c}-\frac {\text {arctanh}\left (\frac {a \sqrt {c} x}{\sqrt {a^2 c x^2+c}}\right )}{2 a^3 \sqrt {c}}\right )}{4 a^2}\right )\right )\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle a^2 c \left (\frac {1}{7} c \left (-\frac {4 \left (-\frac {2 \int \frac {x \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{3 a^2}+\frac {x^2 \arctan (a x) \sqrt {a^2 c x^2+c}}{3 a^2 c}-\frac {\frac {x \sqrt {a^2 c x^2+c}}{2 a^2 c}-\frac {\text {arctanh}\left (\frac {a \sqrt {c} x}{\sqrt {a^2 c x^2+c}}\right )}{2 a^3 \sqrt {c}}}{3 a}\right )}{5 a^2}+\frac {x^4 \arctan (a x) \sqrt {a^2 c x^2+c}}{5 a^2 c}-\frac {\frac {x^3 \sqrt {a^2 c x^2+c}}{4 a^2 c}-\frac {3 \left (\frac {x \sqrt {a^2 c x^2+c}}{2 a^2 c}-\frac {\text {arctanh}\left (\frac {a \sqrt {c} x}{\sqrt {a^2 c x^2+c}}\right )}{2 a^3 \sqrt {c}}\right )}{4 a^2}}{5 a}\right )+\frac {1}{7} x^6 \arctan (a x) \sqrt {a^2 c x^2+c}-\frac {1}{7} a c \left (\frac {x^5 \sqrt {a^2 c x^2+c}}{6 a^2 c}-\frac {5 \left (\frac {x^3 \sqrt {a^2 c x^2+c}}{4 a^2 c}-\frac {3 \left (\frac {x \sqrt {a^2 c x^2+c}}{2 a^2 c}-\frac {\text {arctanh}\left (\frac {a \sqrt {c} x}{\sqrt {a^2 c x^2+c}}\right )}{2 a^3 \sqrt {c}}\right )}{4 a^2}\right )}{6 a^2}\right )\right )+c \left (\frac {1}{5} x^4 \arctan (a x) \sqrt {a^2 c x^2+c}+\frac {1}{5} c \left (-\frac {2 \left (\frac {\arctan (a x) \sqrt {a^2 c x^2+c}}{a^2 c}-\frac {\text {arctanh}\left (\frac {a \sqrt {c} x}{\sqrt {a^2 c x^2+c}}\right )}{a^2 \sqrt {c}}\right )}{3 a^2}+\frac {x^2 \arctan (a x) \sqrt {a^2 c x^2+c}}{3 a^2 c}-\frac {\frac {x \sqrt {a^2 c x^2+c}}{2 a^2 c}-\frac {\text {arctanh}\left (\frac {a \sqrt {c} x}{\sqrt {a^2 c x^2+c}}\right )}{2 a^3 \sqrt {c}}}{3 a}\right )-\frac {1}{5} a c \left (\frac {x^3 \sqrt {a^2 c x^2+c}}{4 a^2 c}-\frac {3 \left (\frac {x \sqrt {a^2 c x^2+c}}{2 a^2 c}-\frac {\text {arctanh}\left (\frac {a \sqrt {c} x}{\sqrt {a^2 c x^2+c}}\right )}{2 a^3 \sqrt {c}}\right )}{4 a^2}\right )\right )\) |
| \(\Big \downarrow \) 5465 |
| \(\displaystyle a^2 c \left (\frac {1}{7} c \left (-\frac {4 \left (-\frac {2 \left (\frac {\arctan (a x) \sqrt {a^2 c x^2+c}}{a^2 c}-\frac {\int \frac {1}{\sqrt {a^2 c x^2+c}}dx}{a}\right )}{3 a^2}+\frac {x^2 \arctan (a x) \sqrt {a^2 c x^2+c}}{3 a^2 c}-\frac {\frac {x \sqrt {a^2 c x^2+c}}{2 a^2 c}-\frac {\text {arctanh}\left (\frac {a \sqrt {c} x}{\sqrt {a^2 c x^2+c}}\right )}{2 a^3 \sqrt {c}}}{3 a}\right )}{5 a^2}+\frac {x^4 \arctan (a x) \sqrt {a^2 c x^2+c}}{5 a^2 c}-\frac {\frac {x^3 \sqrt {a^2 c x^2+c}}{4 a^2 c}-\frac {3 \left (\frac {x \sqrt {a^2 c x^2+c}}{2 a^2 c}-\frac {\text {arctanh}\left (\frac {a \sqrt {c} x}{\sqrt {a^2 c x^2+c}}\right )}{2 a^3 \sqrt {c}}\right )}{4 a^2}}{5 a}\right )+\frac {1}{7} x^6 \arctan (a x) \sqrt {a^2 c x^2+c}-\frac {1}{7} a c \left (\frac {x^5 \sqrt {a^2 c x^2+c}}{6 a^2 c}-\frac {5 \left (\frac {x^3 \sqrt {a^2 c x^2+c}}{4 a^2 c}-\frac {3 \left (\frac {x \sqrt {a^2 c x^2+c}}{2 a^2 c}-\frac {\text {arctanh}\left (\frac {a \sqrt {c} x}{\sqrt {a^2 c x^2+c}}\right )}{2 a^3 \sqrt {c}}\right )}{4 a^2}\right )}{6 a^2}\right )\right )+c \left (\frac {1}{5} x^4 \arctan (a x) \sqrt {a^2 c x^2+c}+\frac {1}{5} c \left (-\frac {2 \left (\frac {\arctan (a x) \sqrt {a^2 c x^2+c}}{a^2 c}-\frac {\text {arctanh}\left (\frac {a \sqrt {c} x}{\sqrt {a^2 c x^2+c}}\right )}{a^2 \sqrt {c}}\right )}{3 a^2}+\frac {x^2 \arctan (a x) \sqrt {a^2 c x^2+c}}{3 a^2 c}-\frac {\frac {x \sqrt {a^2 c x^2+c}}{2 a^2 c}-\frac {\text {arctanh}\left (\frac {a \sqrt {c} x}{\sqrt {a^2 c x^2+c}}\right )}{2 a^3 \sqrt {c}}}{3 a}\right )-\frac {1}{5} a c \left (\frac {x^3 \sqrt {a^2 c x^2+c}}{4 a^2 c}-\frac {3 \left (\frac {x \sqrt {a^2 c x^2+c}}{2 a^2 c}-\frac {\text {arctanh}\left (\frac {a \sqrt {c} x}{\sqrt {a^2 c x^2+c}}\right )}{2 a^3 \sqrt {c}}\right )}{4 a^2}\right )\right )\) |
| \(\Big \downarrow \) 224 |
| \(\displaystyle c \left (\frac {1}{7} \sqrt {a^2 c x^2+c} \arctan (a x) x^6-\frac {1}{7} a c \left (\frac {x^5 \sqrt {a^2 c x^2+c}}{6 a^2 c}-\frac {5 \left (\frac {x^3 \sqrt {a^2 c x^2+c}}{4 a^2 c}-\frac {3 \left (\frac {x \sqrt {a^2 c x^2+c}}{2 a^2 c}-\frac {\text {arctanh}\left (\frac {a \sqrt {c} x}{\sqrt {a^2 c x^2+c}}\right )}{2 a^3 \sqrt {c}}\right )}{4 a^2}\right )}{6 a^2}\right )+\frac {1}{7} c \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x) x^4}{5 a^2 c}-\frac {\frac {x^3 \sqrt {a^2 c x^2+c}}{4 a^2 c}-\frac {3 \left (\frac {x \sqrt {a^2 c x^2+c}}{2 a^2 c}-\frac {\text {arctanh}\left (\frac {a \sqrt {c} x}{\sqrt {a^2 c x^2+c}}\right )}{2 a^3 \sqrt {c}}\right )}{4 a^2}}{5 a}-\frac {4 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x) x^2}{3 a^2 c}-\frac {\frac {x \sqrt {a^2 c x^2+c}}{2 a^2 c}-\frac {\text {arctanh}\left (\frac {a \sqrt {c} x}{\sqrt {a^2 c x^2+c}}\right )}{2 a^3 \sqrt {c}}}{3 a}-\frac {2 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)}{a^2 c}-\frac {\int \frac {1}{1-\frac {a^2 c x^2}{a^2 c x^2+c}}d\frac {x}{\sqrt {a^2 c x^2+c}}}{a}\right )}{3 a^2}\right )}{5 a^2}\right )\right ) a^2+c \left (\frac {1}{5} \sqrt {a^2 c x^2+c} \arctan (a x) x^4-\frac {1}{5} a c \left (\frac {x^3 \sqrt {a^2 c x^2+c}}{4 a^2 c}-\frac {3 \left (\frac {x \sqrt {a^2 c x^2+c}}{2 a^2 c}-\frac {\text {arctanh}\left (\frac {a \sqrt {c} x}{\sqrt {a^2 c x^2+c}}\right )}{2 a^3 \sqrt {c}}\right )}{4 a^2}\right )+\frac {1}{5} c \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x) x^2}{3 a^2 c}-\frac {\frac {x \sqrt {a^2 c x^2+c}}{2 a^2 c}-\frac {\text {arctanh}\left (\frac {a \sqrt {c} x}{\sqrt {a^2 c x^2+c}}\right )}{2 a^3 \sqrt {c}}}{3 a}-\frac {2 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)}{a^2 c}-\frac {\text {arctanh}\left (\frac {a \sqrt {c} x}{\sqrt {a^2 c x^2+c}}\right )}{a^2 \sqrt {c}}\right )}{3 a^2}\right )\right )\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle c \left (\frac {1}{5} x^4 \arctan (a x) \sqrt {a^2 c x^2+c}+\frac {1}{5} c \left (-\frac {2 \left (\frac {\arctan (a x) \sqrt {a^2 c x^2+c}}{a^2 c}-\frac {\text {arctanh}\left (\frac {a \sqrt {c} x}{\sqrt {a^2 c x^2+c}}\right )}{a^2 \sqrt {c}}\right )}{3 a^2}+\frac {x^2 \arctan (a x) \sqrt {a^2 c x^2+c}}{3 a^2 c}-\frac {\frac {x \sqrt {a^2 c x^2+c}}{2 a^2 c}-\frac {\text {arctanh}\left (\frac {a \sqrt {c} x}{\sqrt {a^2 c x^2+c}}\right )}{2 a^3 \sqrt {c}}}{3 a}\right )-\frac {1}{5} a c \left (\frac {x^3 \sqrt {a^2 c x^2+c}}{4 a^2 c}-\frac {3 \left (\frac {x \sqrt {a^2 c x^2+c}}{2 a^2 c}-\frac {\text {arctanh}\left (\frac {a \sqrt {c} x}{\sqrt {a^2 c x^2+c}}\right )}{2 a^3 \sqrt {c}}\right )}{4 a^2}\right )\right )+a^2 c \left (\frac {1}{7} x^6 \arctan (a x) \sqrt {a^2 c x^2+c}+\frac {1}{7} c \left (\frac {x^4 \arctan (a x) \sqrt {a^2 c x^2+c}}{5 a^2 c}-\frac {4 \left (-\frac {2 \left (\frac {\arctan (a x) \sqrt {a^2 c x^2+c}}{a^2 c}-\frac {\text {arctanh}\left (\frac {a \sqrt {c} x}{\sqrt {a^2 c x^2+c}}\right )}{a^2 \sqrt {c}}\right )}{3 a^2}+\frac {x^2 \arctan (a x) \sqrt {a^2 c x^2+c}}{3 a^2 c}-\frac {\frac {x \sqrt {a^2 c x^2+c}}{2 a^2 c}-\frac {\text {arctanh}\left (\frac {a \sqrt {c} x}{\sqrt {a^2 c x^2+c}}\right )}{2 a^3 \sqrt {c}}}{3 a}\right )}{5 a^2}-\frac {\frac {x^3 \sqrt {a^2 c x^2+c}}{4 a^2 c}-\frac {3 \left (\frac {x \sqrt {a^2 c x^2+c}}{2 a^2 c}-\frac {\text {arctanh}\left (\frac {a \sqrt {c} x}{\sqrt {a^2 c x^2+c}}\right )}{2 a^3 \sqrt {c}}\right )}{4 a^2}}{5 a}\right )-\frac {1}{7} a c \left (\frac {x^5 \sqrt {a^2 c x^2+c}}{6 a^2 c}-\frac {5 \left (\frac {x^3 \sqrt {a^2 c x^2+c}}{4 a^2 c}-\frac {3 \left (\frac {x \sqrt {a^2 c x^2+c}}{2 a^2 c}-\frac {\text {arctanh}\left (\frac {a \sqrt {c} x}{\sqrt {a^2 c x^2+c}}\right )}{2 a^3 \sqrt {c}}\right )}{4 a^2}\right )}{6 a^2}\right )\right )\) |
Input:
Int[x^3*(c + a^2*c*x^2)^(3/2)*ArcTan[a*x],x]
Output:
c*((x^4*Sqrt[c + a^2*c*x^2]*ArcTan[a*x])/5 - (a*c*((x^3*Sqrt[c + a^2*c*x^2 ])/(4*a^2*c) - (3*((x*Sqrt[c + a^2*c*x^2])/(2*a^2*c) - ArcTanh[(a*Sqrt[c]* x)/Sqrt[c + a^2*c*x^2]]/(2*a^3*Sqrt[c])))/(4*a^2)))/5 + (c*((x^2*Sqrt[c + a^2*c*x^2]*ArcTan[a*x])/(3*a^2*c) - ((x*Sqrt[c + a^2*c*x^2])/(2*a^2*c) - A rcTanh[(a*Sqrt[c]*x)/Sqrt[c + a^2*c*x^2]]/(2*a^3*Sqrt[c]))/(3*a) - (2*((Sq rt[c + a^2*c*x^2]*ArcTan[a*x])/(a^2*c) - ArcTanh[(a*Sqrt[c]*x)/Sqrt[c + a^ 2*c*x^2]]/(a^2*Sqrt[c])))/(3*a^2)))/5) + a^2*c*((x^6*Sqrt[c + a^2*c*x^2]*A rcTan[a*x])/7 - (a*c*((x^5*Sqrt[c + a^2*c*x^2])/(6*a^2*c) - (5*((x^3*Sqrt[ c + a^2*c*x^2])/(4*a^2*c) - (3*((x*Sqrt[c + a^2*c*x^2])/(2*a^2*c) - ArcTan h[(a*Sqrt[c]*x)/Sqrt[c + a^2*c*x^2]]/(2*a^3*Sqrt[c])))/(4*a^2)))/(6*a^2))) /7 + (c*((x^4*Sqrt[c + a^2*c*x^2]*ArcTan[a*x])/(5*a^2*c) - ((x^3*Sqrt[c + a^2*c*x^2])/(4*a^2*c) - (3*((x*Sqrt[c + a^2*c*x^2])/(2*a^2*c) - ArcTanh[(a *Sqrt[c]*x)/Sqrt[c + a^2*c*x^2]]/(2*a^3*Sqrt[c])))/(4*a^2))/(5*a) - (4*((x ^2*Sqrt[c + a^2*c*x^2]*ArcTan[a*x])/(3*a^2*c) - ((x*Sqrt[c + a^2*c*x^2])/( 2*a^2*c) - ArcTanh[(a*Sqrt[c]*x)/Sqrt[c + a^2*c*x^2]]/(2*a^3*Sqrt[c]))/(3* a) - (2*((Sqrt[c + a^2*c*x^2]*ArcTan[a*x])/(a^2*c) - ArcTanh[(a*Sqrt[c]*x) /Sqrt[c + a^2*c*x^2]]/(a^2*Sqrt[c])))/(3*a^2)))/(5*a^2)))/7)
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))* ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (Gt Q[a, 0] || LtQ[b, 0])
Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Subst[Int[1/(1 - b*x^2), x], x, x/Sqrt[a + b*x^2]] /; FreeQ[{a, b}, x] && !GtQ[a, 0]
Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[c*(c*x) ^(m - 1)*((a + b*x^2)^(p + 1)/(b*(m + 2*p + 1))), x] - Simp[a*c^2*((m - 1)/ (b*(m + 2*p + 1))) Int[(c*x)^(m - 2)*(a + b*x^2)^p, x], x] /; FreeQ[{a, b , c, p}, x] && GtQ[m, 2 - 1] && NeQ[m + 2*p + 1, 0] && IntBinomialQ[a, b, c , 2, m, p, x]
Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.)*(x_)*((d_) + (e_.)*(x_)^2)^(q_ .), x_Symbol] :> Simp[(d + e*x^2)^(q + 1)*((a + b*ArcTan[c*x])^p/(2*e*(q + 1))), x] - Simp[b*(p/(2*c*(q + 1))) Int[(d + e*x^2)^q*(a + b*ArcTan[c*x]) ^(p - 1), x], x] /; FreeQ[{a, b, c, d, e, q}, x] && EqQ[e, c^2*d] && GtQ[p, 0] && NeQ[q, -1]
Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))*((f_.)*(x_))^(m_)*Sqrt[(d_) + (e_.)* (x_)^2], x_Symbol] :> Simp[(f*x)^(m + 1)*Sqrt[d + e*x^2]*((a + b*ArcTan[c*x ])/(f*(m + 2))), x] + (Simp[d/(m + 2) Int[(f*x)^m*((a + b*ArcTan[c*x])/Sq rt[d + e*x^2]), x], x] - Simp[b*c*(d/(f*(m + 2))) Int[(f*x)^(m + 1)/Sqrt[ d + e*x^2], x], x]) /; FreeQ[{a, b, c, d, e, f, m}, x] && EqQ[e, c^2*d] && NeQ[m, -2]
Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.)*((f_.)*(x_))^(m_)*((d_) + (e_. )*(x_)^2)^(q_.), x_Symbol] :> Simp[d Int[(f*x)^m*(d + e*x^2)^(q - 1)*(a + b*ArcTan[c*x])^p, x], x] + Simp[c^2*(d/f^2) Int[(f*x)^(m + 2)*(d + e*x^2 )^(q - 1)*(a + b*ArcTan[c*x])^p, x], x] /; FreeQ[{a, b, c, d, e, f, m}, x] && EqQ[e, c^2*d] && GtQ[q, 0] && IGtQ[p, 0] && (RationalQ[m] || (EqQ[p, 1] && IntegerQ[q]))
Int[(((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.)*((f_.)*(x_))^(m_))/Sqrt[(d_) + (e_.)*(x_)^2], x_Symbol] :> Simp[f*(f*x)^(m - 1)*Sqrt[d + e*x^2]*((a + b* ArcTan[c*x])^p/(c^2*d*m)), x] + (-Simp[b*f*(p/(c*m)) Int[(f*x)^(m - 1)*(( a + b*ArcTan[c*x])^(p - 1)/Sqrt[d + e*x^2]), x], x] - Simp[f^2*((m - 1)/(c^ 2*m)) Int[(f*x)^(m - 2)*((a + b*ArcTan[c*x])^p/Sqrt[d + e*x^2]), x], x]) /; FreeQ[{a, b, c, d, e, f}, x] && EqQ[e, c^2*d] && GtQ[p, 0] && GtQ[m, 1]
Result contains complex when optimal does not.
Time = 1.40 (sec) , antiderivative size = 199, normalized size of antiderivative = 0.92
| method | result | size |
| default | \(\frac {c \sqrt {c \left (a x -i\right ) \left (a x +i\right )}\, \left (240 a^{6} \arctan \left (a x \right ) x^{6}-40 a^{5} x^{5}+384 x^{4} \arctan \left (a x \right ) a^{4}-46 a^{3} x^{3}+48 x^{2} a^{2} \arctan \left (a x \right )+45 a x -96 \arctan \left (a x \right )\right )}{1680 a^{4}}+\frac {17 c \sqrt {c \left (a x -i\right ) \left (a x +i\right )}\, \ln \left (\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}+i\right )}{560 a^{4} \sqrt {a^{2} x^{2}+1}}-\frac {17 c \sqrt {c \left (a x -i\right ) \left (a x +i\right )}\, \ln \left (\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}-i\right )}{560 a^{4} \sqrt {a^{2} x^{2}+1}}\) | \(199\) |
Input:
int(x^3*(a^2*c*x^2+c)^(3/2)*arctan(a*x),x,method=_RETURNVERBOSE)
Output:
1/1680*c/a^4*(c*(a*x-I)*(a*x+I))^(1/2)*(240*a^6*arctan(a*x)*x^6-40*a^5*x^5 +384*x^4*arctan(a*x)*a^4-46*a^3*x^3+48*x^2*a^2*arctan(a*x)+45*a*x-96*arcta n(a*x))+17/560*c/a^4*(c*(a*x-I)*(a*x+I))^(1/2)*ln((1+I*a*x)/(a^2*x^2+1)^(1 /2)+I)/(a^2*x^2+1)^(1/2)-17/560*c/a^4*(c*(a*x-I)*(a*x+I))^(1/2)*ln((1+I*a* x)/(a^2*x^2+1)^(1/2)-I)/(a^2*x^2+1)^(1/2)
Time = 0.18 (sec) , antiderivative size = 118, normalized size of antiderivative = 0.54 \[ \int x^3 \left (c+a^2 c x^2\right )^{3/2} \arctan (a x) \, dx=\frac {51 \, c^{\frac {3}{2}} \log \left (-2 \, a^{2} c x^{2} - 2 \, \sqrt {a^{2} c x^{2} + c} a \sqrt {c} x - c\right ) - 2 \, {\left (40 \, a^{5} c x^{5} + 46 \, a^{3} c x^{3} - 45 \, a c x - 48 \, {\left (5 \, a^{6} c x^{6} + 8 \, a^{4} c x^{4} + a^{2} c x^{2} - 2 \, c\right )} \arctan \left (a x\right )\right )} \sqrt {a^{2} c x^{2} + c}}{3360 \, a^{4}} \] Input:
integrate(x^3*(a^2*c*x^2+c)^(3/2)*arctan(a*x),x, algorithm="fricas")
Output:
1/3360*(51*c^(3/2)*log(-2*a^2*c*x^2 - 2*sqrt(a^2*c*x^2 + c)*a*sqrt(c)*x - c) - 2*(40*a^5*c*x^5 + 46*a^3*c*x^3 - 45*a*c*x - 48*(5*a^6*c*x^6 + 8*a^4*c *x^4 + a^2*c*x^2 - 2*c)*arctan(a*x))*sqrt(a^2*c*x^2 + c))/a^4
\[ \int x^3 \left (c+a^2 c x^2\right )^{3/2} \arctan (a x) \, dx=\int x^{3} \left (c \left (a^{2} x^{2} + 1\right )\right )^{\frac {3}{2}} \operatorname {atan}{\left (a x \right )}\, dx \] Input:
integrate(x**3*(a**2*c*x**2+c)**(3/2)*atan(a*x),x)
Output:
Integral(x**3*(c*(a**2*x**2 + 1))**(3/2)*atan(a*x), x)
Time = 0.18 (sec) , antiderivative size = 214, normalized size of antiderivative = 0.99 \[ \int x^3 \left (c+a^2 c x^2\right )^{3/2} \arctan (a x) \, dx=-\frac {1}{1680} \, {\left ({\left (5 \, {\left (\frac {8 \, {\left (a^{2} x^{2} + 1\right )}^{\frac {3}{2}} x^{3}}{a^{2}} - \frac {6 \, {\left (a^{2} x^{2} + 1\right )}^{\frac {3}{2}} x}{a^{4}} + \frac {3 \, \sqrt {a^{2} x^{2} + 1} x}{a^{4}} + \frac {3 \, \operatorname {arsinh}\left (a x\right )}{a^{5}}\right )} c + \frac {18 \, c {\left (\frac {2 \, {\left (a^{2} x^{2} + 1\right )}^{\frac {3}{2}} x}{a^{2}} - \frac {\sqrt {a^{2} x^{2} + 1} x}{a^{2}} - \frac {\operatorname {arsinh}\left (a x\right )}{a^{3}}\right )}}{a^{2}} - \frac {48 \, {\left (\sqrt {a^{2} x^{2} + 1} x + \frac {\operatorname {arsinh}\left (a x\right )}{a}\right )} c}{a^{4}}\right )} a - 48 \, {\left (5 \, {\left (a^{2} x^{2} + 1\right )}^{\frac {3}{2}} c x^{4} + \frac {3 \, {\left (a^{2} x^{2} + 1\right )}^{\frac {3}{2}} c x^{2}}{a^{2}} - \frac {2 \, {\left (a^{2} x^{2} + 1\right )}^{\frac {3}{2}} c}{a^{4}}\right )} \arctan \left (a x\right )\right )} \sqrt {c} \] Input:
integrate(x^3*(a^2*c*x^2+c)^(3/2)*arctan(a*x),x, algorithm="maxima")
Output:
-1/1680*((5*(8*(a^2*x^2 + 1)^(3/2)*x^3/a^2 - 6*(a^2*x^2 + 1)^(3/2)*x/a^4 + 3*sqrt(a^2*x^2 + 1)*x/a^4 + 3*arcsinh(a*x)/a^5)*c + 18*c*(2*(a^2*x^2 + 1) ^(3/2)*x/a^2 - sqrt(a^2*x^2 + 1)*x/a^2 - arcsinh(a*x)/a^3)/a^2 - 48*(sqrt( a^2*x^2 + 1)*x + arcsinh(a*x)/a)*c/a^4)*a - 48*(5*(a^2*x^2 + 1)^(3/2)*c*x^ 4 + 3*(a^2*x^2 + 1)^(3/2)*c*x^2/a^2 - 2*(a^2*x^2 + 1)^(3/2)*c/a^4)*arctan( a*x))*sqrt(c)
Exception generated. \[ \int x^3 \left (c+a^2 c x^2\right )^{3/2} \arctan (a x) \, dx=\text {Exception raised: TypeError} \] Input:
integrate(x^3*(a^2*c*x^2+c)^(3/2)*arctan(a*x),x, algorithm="giac")
Output:
Exception raised: TypeError >> an error occurred running a Giac command:IN PUT:sage2:=int(sage0,sageVARx):;OUTPUT:sym2poly/r2sym(const gen & e,const index_m & i,const vecteur & l) Error: Bad Argument Value
Timed out. \[ \int x^3 \left (c+a^2 c x^2\right )^{3/2} \arctan (a x) \, dx=\int x^3\,\mathrm {atan}\left (a\,x\right )\,{\left (c\,a^2\,x^2+c\right )}^{3/2} \,d x \] Input:
int(x^3*atan(a*x)*(c + a^2*c*x^2)^(3/2),x)
Output:
int(x^3*atan(a*x)*(c + a^2*c*x^2)^(3/2), x)
Time = 0.20 (sec) , antiderivative size = 158, normalized size of antiderivative = 0.73 \[ \int x^3 \left (c+a^2 c x^2\right )^{3/2} \arctan (a x) \, dx=\frac {\sqrt {c}\, c \left (240 \sqrt {a^{2} x^{2}+1}\, \mathit {atan} \left (a x \right ) a^{6} x^{6}+384 \sqrt {a^{2} x^{2}+1}\, \mathit {atan} \left (a x \right ) a^{4} x^{4}+48 \sqrt {a^{2} x^{2}+1}\, \mathit {atan} \left (a x \right ) a^{2} x^{2}-96 \sqrt {a^{2} x^{2}+1}\, \mathit {atan} \left (a x \right )-40 \sqrt {a^{2} x^{2}+1}\, a^{5} x^{5}-46 \sqrt {a^{2} x^{2}+1}\, a^{3} x^{3}+45 \sqrt {a^{2} x^{2}+1}\, a x +51 \,\mathrm {log}\left (\sqrt {a^{2} x^{2}+1}+a x \right )\right )}{1680 a^{4}} \] Input:
int(x^3*(a^2*c*x^2+c)^(3/2)*atan(a*x),x)
Output:
(sqrt(c)*c*(240*sqrt(a**2*x**2 + 1)*atan(a*x)*a**6*x**6 + 384*sqrt(a**2*x* *2 + 1)*atan(a*x)*a**4*x**4 + 48*sqrt(a**2*x**2 + 1)*atan(a*x)*a**2*x**2 - 96*sqrt(a**2*x**2 + 1)*atan(a*x) - 40*sqrt(a**2*x**2 + 1)*a**5*x**5 - 46* sqrt(a**2*x**2 + 1)*a**3*x**3 + 45*sqrt(a**2*x**2 + 1)*a*x + 51*log(sqrt(a **2*x**2 + 1) + a*x)))/(1680*a**4)