Integrand size = 24, antiderivative size = 493 \[ \int \frac {\arctan (a x)^3}{x^2 \left (c+a^2 c x^2\right )^{5/2}} \, dx=\frac {2 a}{27 c \left (c+a^2 c x^2\right )^{3/2}}+\frac {94 a}{9 c^2 \sqrt {c+a^2 c x^2}}+\frac {2 a^2 x \arctan (a x)}{9 c \left (c+a^2 c x^2\right )^{3/2}}+\frac {94 a^2 x \arctan (a x)}{9 c^2 \sqrt {c+a^2 c x^2}}-\frac {a \arctan (a x)^2}{3 c \left (c+a^2 c x^2\right )^{3/2}}-\frac {5 a \arctan (a x)^2}{c^2 \sqrt {c+a^2 c x^2}}-\frac {a^2 x \arctan (a x)^3}{3 c \left (c+a^2 c x^2\right )^{3/2}}-\frac {5 a^2 x \arctan (a x)^3}{3 c^2 \sqrt {c+a^2 c x^2}}-\frac {\sqrt {c+a^2 c x^2} \arctan (a x)^3}{c^3 x}-\frac {6 a \sqrt {1+a^2 x^2} \arctan (a x)^2 \text {arctanh}\left (e^{i \arctan (a x)}\right )}{c^2 \sqrt {c+a^2 c x^2}}+\frac {6 i a \sqrt {1+a^2 x^2} \arctan (a x) \operatorname {PolyLog}\left (2,-e^{i \arctan (a x)}\right )}{c^2 \sqrt {c+a^2 c x^2}}-\frac {6 i a \sqrt {1+a^2 x^2} \arctan (a x) \operatorname {PolyLog}\left (2,e^{i \arctan (a x)}\right )}{c^2 \sqrt {c+a^2 c x^2}}-\frac {6 a \sqrt {1+a^2 x^2} \operatorname {PolyLog}\left (3,-e^{i \arctan (a x)}\right )}{c^2 \sqrt {c+a^2 c x^2}}+\frac {6 a \sqrt {1+a^2 x^2} \operatorname {PolyLog}\left (3,e^{i \arctan (a x)}\right )}{c^2 \sqrt {c+a^2 c x^2}} \] Output:
2/27*a/c/(a^2*c*x^2+c)^(3/2)+94/9*a/c^2/(a^2*c*x^2+c)^(1/2)+2/9*a^2*x*arct an(a*x)/c/(a^2*c*x^2+c)^(3/2)+94/9*a^2*x*arctan(a*x)/c^2/(a^2*c*x^2+c)^(1/ 2)-1/3*a*arctan(a*x)^2/c/(a^2*c*x^2+c)^(3/2)-5*a*arctan(a*x)^2/c^2/(a^2*c* x^2+c)^(1/2)-1/3*a^2*x*arctan(a*x)^3/c/(a^2*c*x^2+c)^(3/2)-5/3*a^2*x*arcta n(a*x)^3/c^2/(a^2*c*x^2+c)^(1/2)-(a^2*c*x^2+c)^(1/2)*arctan(a*x)^3/c^3/x-6 *a*(a^2*x^2+1)^(1/2)*arctan(a*x)^2*arctanh((1+I*a*x)/(a^2*x^2+1)^(1/2))/c^ 2/(a^2*c*x^2+c)^(1/2)+6*I*a*(a^2*x^2+1)^(1/2)*arctan(a*x)*polylog(2,-(1+I* a*x)/(a^2*x^2+1)^(1/2))/c^2/(a^2*c*x^2+c)^(1/2)-6*I*a*(a^2*x^2+1)^(1/2)*ar ctan(a*x)*polylog(2,(1+I*a*x)/(a^2*x^2+1)^(1/2))/c^2/(a^2*c*x^2+c)^(1/2)-6 *a*(a^2*x^2+1)^(1/2)*polylog(3,-(1+I*a*x)/(a^2*x^2+1)^(1/2))/c^2/(a^2*c*x^ 2+c)^(1/2)+6*a*(a^2*x^2+1)^(1/2)*polylog(3,(1+I*a*x)/(a^2*x^2+1)^(1/2))/c^ 2/(a^2*c*x^2+c)^(1/2)
Time = 1.70 (sec) , antiderivative size = 399, normalized size of antiderivative = 0.81 \[ \int \frac {\arctan (a x)^3}{x^2 \left (c+a^2 c x^2\right )^{5/2}} \, dx=-\frac {a \left (-1134-1134 a x \arctan (a x)+567 \arctan (a x)^2+189 a x \arctan (a x)^3-2 \sqrt {1+a^2 x^2} \cos (3 \arctan (a x))+9 \sqrt {1+a^2 x^2} \arctan (a x)^2 \cos (3 \arctan (a x))+27 a x \arctan (a x)^3 \csc ^2\left (\frac {1}{2} \arctan (a x)\right )-324 \sqrt {1+a^2 x^2} \arctan (a x)^2 \log \left (1-e^{i \arctan (a x)}\right )+324 \sqrt {1+a^2 x^2} \arctan (a x)^2 \log \left (1+e^{i \arctan (a x)}\right )-648 i \sqrt {1+a^2 x^2} \arctan (a x) \operatorname {PolyLog}\left (2,-e^{i \arctan (a x)}\right )+648 i \sqrt {1+a^2 x^2} \arctan (a x) \operatorname {PolyLog}\left (2,e^{i \arctan (a x)}\right )+648 \sqrt {1+a^2 x^2} \operatorname {PolyLog}\left (3,-e^{i \arctan (a x)}\right )-648 \sqrt {1+a^2 x^2} \operatorname {PolyLog}\left (3,e^{i \arctan (a x)}\right )-6 \sqrt {1+a^2 x^2} \arctan (a x) \sin (3 \arctan (a x))+9 \sqrt {1+a^2 x^2} \arctan (a x)^3 \sin (3 \arctan (a x))+54 \sqrt {1+a^2 x^2} \arctan (a x)^3 \tan \left (\frac {1}{2} \arctan (a x)\right )\right )}{108 c^2 \sqrt {c+a^2 c x^2}} \] Input:
Integrate[ArcTan[a*x]^3/(x^2*(c + a^2*c*x^2)^(5/2)),x]
Output:
-1/108*(a*(-1134 - 1134*a*x*ArcTan[a*x] + 567*ArcTan[a*x]^2 + 189*a*x*ArcT an[a*x]^3 - 2*Sqrt[1 + a^2*x^2]*Cos[3*ArcTan[a*x]] + 9*Sqrt[1 + a^2*x^2]*A rcTan[a*x]^2*Cos[3*ArcTan[a*x]] + 27*a*x*ArcTan[a*x]^3*Csc[ArcTan[a*x]/2]^ 2 - 324*Sqrt[1 + a^2*x^2]*ArcTan[a*x]^2*Log[1 - E^(I*ArcTan[a*x])] + 324*S qrt[1 + a^2*x^2]*ArcTan[a*x]^2*Log[1 + E^(I*ArcTan[a*x])] - (648*I)*Sqrt[1 + a^2*x^2]*ArcTan[a*x]*PolyLog[2, -E^(I*ArcTan[a*x])] + (648*I)*Sqrt[1 + a^2*x^2]*ArcTan[a*x]*PolyLog[2, E^(I*ArcTan[a*x])] + 648*Sqrt[1 + a^2*x^2] *PolyLog[3, -E^(I*ArcTan[a*x])] - 648*Sqrt[1 + a^2*x^2]*PolyLog[3, E^(I*Ar cTan[a*x])] - 6*Sqrt[1 + a^2*x^2]*ArcTan[a*x]*Sin[3*ArcTan[a*x]] + 9*Sqrt[ 1 + a^2*x^2]*ArcTan[a*x]^3*Sin[3*ArcTan[a*x]] + 54*Sqrt[1 + a^2*x^2]*ArcTa n[a*x]^3*Tan[ArcTan[a*x]/2]))/(c^2*Sqrt[c + a^2*c*x^2])
Time = 4.29 (sec) , antiderivative size = 552, normalized size of antiderivative = 1.12, number of steps used = 18, number of rules used = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.708, Rules used = {5501, 5435, 5431, 5429, 5433, 5429, 5501, 5433, 5429, 5479, 5493, 5491, 3042, 4671, 3011, 2720, 7143}
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.
\(\displaystyle \int \frac {\arctan (a x)^3}{x^2 \left (a^2 c x^2+c\right )^{5/2}} \, dx\) |
\(\Big \downarrow \) 5501 |
\(\displaystyle \frac {\int \frac {\arctan (a x)^3}{x^2 \left (a^2 c x^2+c\right )^{3/2}}dx}{c}-a^2 \int \frac {\arctan (a x)^3}{\left (a^2 c x^2+c\right )^{5/2}}dx\) |
\(\Big \downarrow \) 5435 |
\(\displaystyle \frac {\int \frac {\arctan (a x)^3}{x^2 \left (a^2 c x^2+c\right )^{3/2}}dx}{c}-a^2 \left (-\frac {2}{3} \int \frac {\arctan (a x)}{\left (a^2 c x^2+c\right )^{5/2}}dx+\frac {2 \int \frac {\arctan (a x)^3}{\left (a^2 c x^2+c\right )^{3/2}}dx}{3 c}+\frac {x \arctan (a x)^3}{3 c \left (a^2 c x^2+c\right )^{3/2}}+\frac {\arctan (a x)^2}{3 a c \left (a^2 c x^2+c\right )^{3/2}}\right )\) |
\(\Big \downarrow \) 5431 |
\(\displaystyle \frac {\int \frac {\arctan (a x)^3}{x^2 \left (a^2 c x^2+c\right )^{3/2}}dx}{c}-a^2 \left (-\frac {2}{3} \left (\frac {2 \int \frac {\arctan (a x)}{\left (a^2 c x^2+c\right )^{3/2}}dx}{3 c}+\frac {x \arctan (a x)}{3 c \left (a^2 c x^2+c\right )^{3/2}}+\frac {1}{9 a c \left (a^2 c x^2+c\right )^{3/2}}\right )+\frac {2 \int \frac {\arctan (a x)^3}{\left (a^2 c x^2+c\right )^{3/2}}dx}{3 c}+\frac {x \arctan (a x)^3}{3 c \left (a^2 c x^2+c\right )^{3/2}}+\frac {\arctan (a x)^2}{3 a c \left (a^2 c x^2+c\right )^{3/2}}\right )\) |
\(\Big \downarrow \) 5429 |
\(\displaystyle \frac {\int \frac {\arctan (a x)^3}{x^2 \left (a^2 c x^2+c\right )^{3/2}}dx}{c}-a^2 \left (\frac {2 \int \frac {\arctan (a x)^3}{\left (a^2 c x^2+c\right )^{3/2}}dx}{3 c}+\frac {x \arctan (a x)^3}{3 c \left (a^2 c x^2+c\right )^{3/2}}+\frac {\arctan (a x)^2}{3 a c \left (a^2 c x^2+c\right )^{3/2}}-\frac {2}{3} \left (\frac {x \arctan (a x)}{3 c \left (a^2 c x^2+c\right )^{3/2}}+\frac {2 \left (\frac {x \arctan (a x)}{c \sqrt {a^2 c x^2+c}}+\frac {1}{a c \sqrt {a^2 c x^2+c}}\right )}{3 c}+\frac {1}{9 a c \left (a^2 c x^2+c\right )^{3/2}}\right )\right )\) |
\(\Big \downarrow \) 5433 |
\(\displaystyle \frac {\int \frac {\arctan (a x)^3}{x^2 \left (a^2 c x^2+c\right )^{3/2}}dx}{c}-a^2 \left (\frac {2 \left (-6 \int \frac {\arctan (a x)}{\left (a^2 c x^2+c\right )^{3/2}}dx+\frac {x \arctan (a x)^3}{c \sqrt {a^2 c x^2+c}}+\frac {3 \arctan (a x)^2}{a c \sqrt {a^2 c x^2+c}}\right )}{3 c}+\frac {x \arctan (a x)^3}{3 c \left (a^2 c x^2+c\right )^{3/2}}+\frac {\arctan (a x)^2}{3 a c \left (a^2 c x^2+c\right )^{3/2}}-\frac {2}{3} \left (\frac {x \arctan (a x)}{3 c \left (a^2 c x^2+c\right )^{3/2}}+\frac {2 \left (\frac {x \arctan (a x)}{c \sqrt {a^2 c x^2+c}}+\frac {1}{a c \sqrt {a^2 c x^2+c}}\right )}{3 c}+\frac {1}{9 a c \left (a^2 c x^2+c\right )^{3/2}}\right )\right )\) |
\(\Big \downarrow \) 5429 |
\(\displaystyle \frac {\int \frac {\arctan (a x)^3}{x^2 \left (a^2 c x^2+c\right )^{3/2}}dx}{c}-a^2 \left (\frac {x \arctan (a x)^3}{3 c \left (a^2 c x^2+c\right )^{3/2}}+\frac {\arctan (a x)^2}{3 a c \left (a^2 c x^2+c\right )^{3/2}}+\frac {2 \left (\frac {x \arctan (a x)^3}{c \sqrt {a^2 c x^2+c}}+\frac {3 \arctan (a x)^2}{a c \sqrt {a^2 c x^2+c}}-6 \left (\frac {x \arctan (a x)}{c \sqrt {a^2 c x^2+c}}+\frac {1}{a c \sqrt {a^2 c x^2+c}}\right )\right )}{3 c}-\frac {2}{3} \left (\frac {x \arctan (a x)}{3 c \left (a^2 c x^2+c\right )^{3/2}}+\frac {2 \left (\frac {x \arctan (a x)}{c \sqrt {a^2 c x^2+c}}+\frac {1}{a c \sqrt {a^2 c x^2+c}}\right )}{3 c}+\frac {1}{9 a c \left (a^2 c x^2+c\right )^{3/2}}\right )\right )\) |
\(\Big \downarrow \) 5501 |
\(\displaystyle \frac {\frac {\int \frac {\arctan (a x)^3}{x^2 \sqrt {a^2 c x^2+c}}dx}{c}-a^2 \int \frac {\arctan (a x)^3}{\left (a^2 c x^2+c\right )^{3/2}}dx}{c}-a^2 \left (\frac {x \arctan (a x)^3}{3 c \left (a^2 c x^2+c\right )^{3/2}}+\frac {\arctan (a x)^2}{3 a c \left (a^2 c x^2+c\right )^{3/2}}+\frac {2 \left (\frac {x \arctan (a x)^3}{c \sqrt {a^2 c x^2+c}}+\frac {3 \arctan (a x)^2}{a c \sqrt {a^2 c x^2+c}}-6 \left (\frac {x \arctan (a x)}{c \sqrt {a^2 c x^2+c}}+\frac {1}{a c \sqrt {a^2 c x^2+c}}\right )\right )}{3 c}-\frac {2}{3} \left (\frac {x \arctan (a x)}{3 c \left (a^2 c x^2+c\right )^{3/2}}+\frac {2 \left (\frac {x \arctan (a x)}{c \sqrt {a^2 c x^2+c}}+\frac {1}{a c \sqrt {a^2 c x^2+c}}\right )}{3 c}+\frac {1}{9 a c \left (a^2 c x^2+c\right )^{3/2}}\right )\right )\) |
\(\Big \downarrow \) 5433 |
\(\displaystyle \frac {\frac {\int \frac {\arctan (a x)^3}{x^2 \sqrt {a^2 c x^2+c}}dx}{c}-a^2 \left (-6 \int \frac {\arctan (a x)}{\left (a^2 c x^2+c\right )^{3/2}}dx+\frac {x \arctan (a x)^3}{c \sqrt {a^2 c x^2+c}}+\frac {3 \arctan (a x)^2}{a c \sqrt {a^2 c x^2+c}}\right )}{c}-a^2 \left (\frac {x \arctan (a x)^3}{3 c \left (a^2 c x^2+c\right )^{3/2}}+\frac {\arctan (a x)^2}{3 a c \left (a^2 c x^2+c\right )^{3/2}}+\frac {2 \left (\frac {x \arctan (a x)^3}{c \sqrt {a^2 c x^2+c}}+\frac {3 \arctan (a x)^2}{a c \sqrt {a^2 c x^2+c}}-6 \left (\frac {x \arctan (a x)}{c \sqrt {a^2 c x^2+c}}+\frac {1}{a c \sqrt {a^2 c x^2+c}}\right )\right )}{3 c}-\frac {2}{3} \left (\frac {x \arctan (a x)}{3 c \left (a^2 c x^2+c\right )^{3/2}}+\frac {2 \left (\frac {x \arctan (a x)}{c \sqrt {a^2 c x^2+c}}+\frac {1}{a c \sqrt {a^2 c x^2+c}}\right )}{3 c}+\frac {1}{9 a c \left (a^2 c x^2+c\right )^{3/2}}\right )\right )\) |
\(\Big \downarrow \) 5429 |
\(\displaystyle \frac {\frac {\int \frac {\arctan (a x)^3}{x^2 \sqrt {a^2 c x^2+c}}dx}{c}-a^2 \left (\frac {x \arctan (a x)^3}{c \sqrt {a^2 c x^2+c}}+\frac {3 \arctan (a x)^2}{a c \sqrt {a^2 c x^2+c}}-6 \left (\frac {x \arctan (a x)}{c \sqrt {a^2 c x^2+c}}+\frac {1}{a c \sqrt {a^2 c x^2+c}}\right )\right )}{c}-a^2 \left (\frac {x \arctan (a x)^3}{3 c \left (a^2 c x^2+c\right )^{3/2}}+\frac {\arctan (a x)^2}{3 a c \left (a^2 c x^2+c\right )^{3/2}}+\frac {2 \left (\frac {x \arctan (a x)^3}{c \sqrt {a^2 c x^2+c}}+\frac {3 \arctan (a x)^2}{a c \sqrt {a^2 c x^2+c}}-6 \left (\frac {x \arctan (a x)}{c \sqrt {a^2 c x^2+c}}+\frac {1}{a c \sqrt {a^2 c x^2+c}}\right )\right )}{3 c}-\frac {2}{3} \left (\frac {x \arctan (a x)}{3 c \left (a^2 c x^2+c\right )^{3/2}}+\frac {2 \left (\frac {x \arctan (a x)}{c \sqrt {a^2 c x^2+c}}+\frac {1}{a c \sqrt {a^2 c x^2+c}}\right )}{3 c}+\frac {1}{9 a c \left (a^2 c x^2+c\right )^{3/2}}\right )\right )\) |
\(\Big \downarrow \) 5479 |
\(\displaystyle \frac {\frac {3 a \int \frac {\arctan (a x)^2}{x \sqrt {a^2 c x^2+c}}dx-\frac {\arctan (a x)^3 \sqrt {a^2 c x^2+c}}{c x}}{c}-a^2 \left (\frac {x \arctan (a x)^3}{c \sqrt {a^2 c x^2+c}}+\frac {3 \arctan (a x)^2}{a c \sqrt {a^2 c x^2+c}}-6 \left (\frac {x \arctan (a x)}{c \sqrt {a^2 c x^2+c}}+\frac {1}{a c \sqrt {a^2 c x^2+c}}\right )\right )}{c}-a^2 \left (\frac {x \arctan (a x)^3}{3 c \left (a^2 c x^2+c\right )^{3/2}}+\frac {\arctan (a x)^2}{3 a c \left (a^2 c x^2+c\right )^{3/2}}+\frac {2 \left (\frac {x \arctan (a x)^3}{c \sqrt {a^2 c x^2+c}}+\frac {3 \arctan (a x)^2}{a c \sqrt {a^2 c x^2+c}}-6 \left (\frac {x \arctan (a x)}{c \sqrt {a^2 c x^2+c}}+\frac {1}{a c \sqrt {a^2 c x^2+c}}\right )\right )}{3 c}-\frac {2}{3} \left (\frac {x \arctan (a x)}{3 c \left (a^2 c x^2+c\right )^{3/2}}+\frac {2 \left (\frac {x \arctan (a x)}{c \sqrt {a^2 c x^2+c}}+\frac {1}{a c \sqrt {a^2 c x^2+c}}\right )}{3 c}+\frac {1}{9 a c \left (a^2 c x^2+c\right )^{3/2}}\right )\right )\) |
\(\Big \downarrow \) 5493 |
\(\displaystyle \frac {\frac {\frac {3 a \sqrt {a^2 x^2+1} \int \frac {\arctan (a x)^2}{x \sqrt {a^2 x^2+1}}dx}{\sqrt {a^2 c x^2+c}}-\frac {\arctan (a x)^3 \sqrt {a^2 c x^2+c}}{c x}}{c}-a^2 \left (\frac {x \arctan (a x)^3}{c \sqrt {a^2 c x^2+c}}+\frac {3 \arctan (a x)^2}{a c \sqrt {a^2 c x^2+c}}-6 \left (\frac {x \arctan (a x)}{c \sqrt {a^2 c x^2+c}}+\frac {1}{a c \sqrt {a^2 c x^2+c}}\right )\right )}{c}-a^2 \left (\frac {x \arctan (a x)^3}{3 c \left (a^2 c x^2+c\right )^{3/2}}+\frac {\arctan (a x)^2}{3 a c \left (a^2 c x^2+c\right )^{3/2}}+\frac {2 \left (\frac {x \arctan (a x)^3}{c \sqrt {a^2 c x^2+c}}+\frac {3 \arctan (a x)^2}{a c \sqrt {a^2 c x^2+c}}-6 \left (\frac {x \arctan (a x)}{c \sqrt {a^2 c x^2+c}}+\frac {1}{a c \sqrt {a^2 c x^2+c}}\right )\right )}{3 c}-\frac {2}{3} \left (\frac {x \arctan (a x)}{3 c \left (a^2 c x^2+c\right )^{3/2}}+\frac {2 \left (\frac {x \arctan (a x)}{c \sqrt {a^2 c x^2+c}}+\frac {1}{a c \sqrt {a^2 c x^2+c}}\right )}{3 c}+\frac {1}{9 a c \left (a^2 c x^2+c\right )^{3/2}}\right )\right )\) |
\(\Big \downarrow \) 5491 |
\(\displaystyle \frac {\frac {\frac {3 a \sqrt {a^2 x^2+1} \int \frac {\sqrt {a^2 x^2+1} \arctan (a x)^2}{a x}d\arctan (a x)}{\sqrt {a^2 c x^2+c}}-\frac {\arctan (a x)^3 \sqrt {a^2 c x^2+c}}{c x}}{c}-a^2 \left (\frac {x \arctan (a x)^3}{c \sqrt {a^2 c x^2+c}}+\frac {3 \arctan (a x)^2}{a c \sqrt {a^2 c x^2+c}}-6 \left (\frac {x \arctan (a x)}{c \sqrt {a^2 c x^2+c}}+\frac {1}{a c \sqrt {a^2 c x^2+c}}\right )\right )}{c}-a^2 \left (\frac {x \arctan (a x)^3}{3 c \left (a^2 c x^2+c\right )^{3/2}}+\frac {\arctan (a x)^2}{3 a c \left (a^2 c x^2+c\right )^{3/2}}+\frac {2 \left (\frac {x \arctan (a x)^3}{c \sqrt {a^2 c x^2+c}}+\frac {3 \arctan (a x)^2}{a c \sqrt {a^2 c x^2+c}}-6 \left (\frac {x \arctan (a x)}{c \sqrt {a^2 c x^2+c}}+\frac {1}{a c \sqrt {a^2 c x^2+c}}\right )\right )}{3 c}-\frac {2}{3} \left (\frac {x \arctan (a x)}{3 c \left (a^2 c x^2+c\right )^{3/2}}+\frac {2 \left (\frac {x \arctan (a x)}{c \sqrt {a^2 c x^2+c}}+\frac {1}{a c \sqrt {a^2 c x^2+c}}\right )}{3 c}+\frac {1}{9 a c \left (a^2 c x^2+c\right )^{3/2}}\right )\right )\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {\frac {\frac {3 a \sqrt {a^2 x^2+1} \int \arctan (a x)^2 \csc (\arctan (a x))d\arctan (a x)}{\sqrt {a^2 c x^2+c}}-\frac {\arctan (a x)^3 \sqrt {a^2 c x^2+c}}{c x}}{c}-a^2 \left (\frac {x \arctan (a x)^3}{c \sqrt {a^2 c x^2+c}}+\frac {3 \arctan (a x)^2}{a c \sqrt {a^2 c x^2+c}}-6 \left (\frac {x \arctan (a x)}{c \sqrt {a^2 c x^2+c}}+\frac {1}{a c \sqrt {a^2 c x^2+c}}\right )\right )}{c}-a^2 \left (\frac {x \arctan (a x)^3}{3 c \left (a^2 c x^2+c\right )^{3/2}}+\frac {\arctan (a x)^2}{3 a c \left (a^2 c x^2+c\right )^{3/2}}+\frac {2 \left (\frac {x \arctan (a x)^3}{c \sqrt {a^2 c x^2+c}}+\frac {3 \arctan (a x)^2}{a c \sqrt {a^2 c x^2+c}}-6 \left (\frac {x \arctan (a x)}{c \sqrt {a^2 c x^2+c}}+\frac {1}{a c \sqrt {a^2 c x^2+c}}\right )\right )}{3 c}-\frac {2}{3} \left (\frac {x \arctan (a x)}{3 c \left (a^2 c x^2+c\right )^{3/2}}+\frac {2 \left (\frac {x \arctan (a x)}{c \sqrt {a^2 c x^2+c}}+\frac {1}{a c \sqrt {a^2 c x^2+c}}\right )}{3 c}+\frac {1}{9 a c \left (a^2 c x^2+c\right )^{3/2}}\right )\right )\) |
\(\Big \downarrow \) 4671 |
\(\displaystyle -a^2 \left (\frac {x \arctan (a x)^3}{3 c \left (a^2 c x^2+c\right )^{3/2}}+\frac {\arctan (a x)^2}{3 a c \left (a^2 c x^2+c\right )^{3/2}}+\frac {2 \left (\frac {x \arctan (a x)^3}{c \sqrt {a^2 c x^2+c}}+\frac {3 \arctan (a x)^2}{a c \sqrt {a^2 c x^2+c}}-6 \left (\frac {x \arctan (a x)}{c \sqrt {a^2 c x^2+c}}+\frac {1}{a c \sqrt {a^2 c x^2+c}}\right )\right )}{3 c}-\frac {2}{3} \left (\frac {x \arctan (a x)}{3 c \left (a^2 c x^2+c\right )^{3/2}}+\frac {2 \left (\frac {x \arctan (a x)}{c \sqrt {a^2 c x^2+c}}+\frac {1}{a c \sqrt {a^2 c x^2+c}}\right )}{3 c}+\frac {1}{9 a c \left (a^2 c x^2+c\right )^{3/2}}\right )\right )+\frac {-a^2 \left (\frac {x \arctan (a x)^3}{c \sqrt {a^2 c x^2+c}}+\frac {3 \arctan (a x)^2}{a c \sqrt {a^2 c x^2+c}}-6 \left (\frac {x \arctan (a x)}{c \sqrt {a^2 c x^2+c}}+\frac {1}{a c \sqrt {a^2 c x^2+c}}\right )\right )+\frac {-\frac {\arctan (a x)^3 \sqrt {a^2 c x^2+c}}{c x}+\frac {3 a \sqrt {a^2 x^2+1} \left (-2 \int \arctan (a x) \log \left (1-e^{i \arctan (a x)}\right )d\arctan (a x)+2 \int \arctan (a x) \log \left (1+e^{i \arctan (a x)}\right )d\arctan (a x)-2 \arctan (a x)^2 \text {arctanh}\left (e^{i \arctan (a x)}\right )\right )}{\sqrt {a^2 c x^2+c}}}{c}}{c}\) |
\(\Big \downarrow \) 3011 |
\(\displaystyle -a^2 \left (\frac {x \arctan (a x)^3}{3 c \left (a^2 c x^2+c\right )^{3/2}}+\frac {\arctan (a x)^2}{3 a c \left (a^2 c x^2+c\right )^{3/2}}+\frac {2 \left (\frac {x \arctan (a x)^3}{c \sqrt {a^2 c x^2+c}}+\frac {3 \arctan (a x)^2}{a c \sqrt {a^2 c x^2+c}}-6 \left (\frac {x \arctan (a x)}{c \sqrt {a^2 c x^2+c}}+\frac {1}{a c \sqrt {a^2 c x^2+c}}\right )\right )}{3 c}-\frac {2}{3} \left (\frac {x \arctan (a x)}{3 c \left (a^2 c x^2+c\right )^{3/2}}+\frac {2 \left (\frac {x \arctan (a x)}{c \sqrt {a^2 c x^2+c}}+\frac {1}{a c \sqrt {a^2 c x^2+c}}\right )}{3 c}+\frac {1}{9 a c \left (a^2 c x^2+c\right )^{3/2}}\right )\right )+\frac {-a^2 \left (\frac {x \arctan (a x)^3}{c \sqrt {a^2 c x^2+c}}+\frac {3 \arctan (a x)^2}{a c \sqrt {a^2 c x^2+c}}-6 \left (\frac {x \arctan (a x)}{c \sqrt {a^2 c x^2+c}}+\frac {1}{a c \sqrt {a^2 c x^2+c}}\right )\right )+\frac {-\frac {\arctan (a x)^3 \sqrt {a^2 c x^2+c}}{c x}+\frac {3 a \sqrt {a^2 x^2+1} \left (2 \left (i \arctan (a x) \operatorname {PolyLog}\left (2,-e^{i \arctan (a x)}\right )-i \int \operatorname {PolyLog}\left (2,-e^{i \arctan (a x)}\right )d\arctan (a x)\right )-2 \left (i \arctan (a x) \operatorname {PolyLog}\left (2,e^{i \arctan (a x)}\right )-i \int \operatorname {PolyLog}\left (2,e^{i \arctan (a x)}\right )d\arctan (a x)\right )-2 \arctan (a x)^2 \text {arctanh}\left (e^{i \arctan (a x)}\right )\right )}{\sqrt {a^2 c x^2+c}}}{c}}{c}\) |
\(\Big \downarrow \) 2720 |
\(\displaystyle -a^2 \left (\frac {x \arctan (a x)^3}{3 c \left (a^2 c x^2+c\right )^{3/2}}+\frac {\arctan (a x)^2}{3 a c \left (a^2 c x^2+c\right )^{3/2}}+\frac {2 \left (\frac {x \arctan (a x)^3}{c \sqrt {a^2 c x^2+c}}+\frac {3 \arctan (a x)^2}{a c \sqrt {a^2 c x^2+c}}-6 \left (\frac {x \arctan (a x)}{c \sqrt {a^2 c x^2+c}}+\frac {1}{a c \sqrt {a^2 c x^2+c}}\right )\right )}{3 c}-\frac {2}{3} \left (\frac {x \arctan (a x)}{3 c \left (a^2 c x^2+c\right )^{3/2}}+\frac {2 \left (\frac {x \arctan (a x)}{c \sqrt {a^2 c x^2+c}}+\frac {1}{a c \sqrt {a^2 c x^2+c}}\right )}{3 c}+\frac {1}{9 a c \left (a^2 c x^2+c\right )^{3/2}}\right )\right )+\frac {-a^2 \left (\frac {x \arctan (a x)^3}{c \sqrt {a^2 c x^2+c}}+\frac {3 \arctan (a x)^2}{a c \sqrt {a^2 c x^2+c}}-6 \left (\frac {x \arctan (a x)}{c \sqrt {a^2 c x^2+c}}+\frac {1}{a c \sqrt {a^2 c x^2+c}}\right )\right )+\frac {-\frac {\arctan (a x)^3 \sqrt {a^2 c x^2+c}}{c x}+\frac {3 a \sqrt {a^2 x^2+1} \left (2 \left (i \arctan (a x) \operatorname {PolyLog}\left (2,-e^{i \arctan (a x)}\right )-\int e^{-i \arctan (a x)} \operatorname {PolyLog}\left (2,-e^{i \arctan (a x)}\right )de^{i \arctan (a x)}\right )-2 \left (i \arctan (a x) \operatorname {PolyLog}\left (2,e^{i \arctan (a x)}\right )-\int e^{-i \arctan (a x)} \operatorname {PolyLog}\left (2,e^{i \arctan (a x)}\right )de^{i \arctan (a x)}\right )-2 \arctan (a x)^2 \text {arctanh}\left (e^{i \arctan (a x)}\right )\right )}{\sqrt {a^2 c x^2+c}}}{c}}{c}\) |
\(\Big \downarrow \) 7143 |
\(\displaystyle -a^2 \left (\frac {x \arctan (a x)^3}{3 c \left (a^2 c x^2+c\right )^{3/2}}+\frac {\arctan (a x)^2}{3 a c \left (a^2 c x^2+c\right )^{3/2}}+\frac {2 \left (\frac {x \arctan (a x)^3}{c \sqrt {a^2 c x^2+c}}+\frac {3 \arctan (a x)^2}{a c \sqrt {a^2 c x^2+c}}-6 \left (\frac {x \arctan (a x)}{c \sqrt {a^2 c x^2+c}}+\frac {1}{a c \sqrt {a^2 c x^2+c}}\right )\right )}{3 c}-\frac {2}{3} \left (\frac {x \arctan (a x)}{3 c \left (a^2 c x^2+c\right )^{3/2}}+\frac {2 \left (\frac {x \arctan (a x)}{c \sqrt {a^2 c x^2+c}}+\frac {1}{a c \sqrt {a^2 c x^2+c}}\right )}{3 c}+\frac {1}{9 a c \left (a^2 c x^2+c\right )^{3/2}}\right )\right )+\frac {-a^2 \left (\frac {x \arctan (a x)^3}{c \sqrt {a^2 c x^2+c}}+\frac {3 \arctan (a x)^2}{a c \sqrt {a^2 c x^2+c}}-6 \left (\frac {x \arctan (a x)}{c \sqrt {a^2 c x^2+c}}+\frac {1}{a c \sqrt {a^2 c x^2+c}}\right )\right )+\frac {-\frac {\arctan (a x)^3 \sqrt {a^2 c x^2+c}}{c x}+\frac {3 a \sqrt {a^2 x^2+1} \left (-2 \arctan (a x)^2 \text {arctanh}\left (e^{i \arctan (a x)}\right )+2 \left (i \arctan (a x) \operatorname {PolyLog}\left (2,-e^{i \arctan (a x)}\right )-\operatorname {PolyLog}\left (3,-e^{i \arctan (a x)}\right )\right )-2 \left (i \arctan (a x) \operatorname {PolyLog}\left (2,e^{i \arctan (a x)}\right )-\operatorname {PolyLog}\left (3,e^{i \arctan (a x)}\right )\right )\right )}{\sqrt {a^2 c x^2+c}}}{c}}{c}\) |
Input:
Int[ArcTan[a*x]^3/(x^2*(c + a^2*c*x^2)^(5/2)),x]
Output:
-(a^2*(ArcTan[a*x]^2/(3*a*c*(c + a^2*c*x^2)^(3/2)) + (x*ArcTan[a*x]^3)/(3* c*(c + a^2*c*x^2)^(3/2)) + (2*((3*ArcTan[a*x]^2)/(a*c*Sqrt[c + a^2*c*x^2]) + (x*ArcTan[a*x]^3)/(c*Sqrt[c + a^2*c*x^2]) - 6*(1/(a*c*Sqrt[c + a^2*c*x^ 2]) + (x*ArcTan[a*x])/(c*Sqrt[c + a^2*c*x^2]))))/(3*c) - (2*(1/(9*a*c*(c + a^2*c*x^2)^(3/2)) + (x*ArcTan[a*x])/(3*c*(c + a^2*c*x^2)^(3/2)) + (2*(1/( a*c*Sqrt[c + a^2*c*x^2]) + (x*ArcTan[a*x])/(c*Sqrt[c + a^2*c*x^2])))/(3*c) ))/3)) + (-(a^2*((3*ArcTan[a*x]^2)/(a*c*Sqrt[c + a^2*c*x^2]) + (x*ArcTan[a *x]^3)/(c*Sqrt[c + a^2*c*x^2]) - 6*(1/(a*c*Sqrt[c + a^2*c*x^2]) + (x*ArcTa n[a*x])/(c*Sqrt[c + a^2*c*x^2])))) + (-((Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^3 )/(c*x)) + (3*a*Sqrt[1 + a^2*x^2]*(-2*ArcTan[a*x]^2*ArcTanh[E^(I*ArcTan[a* x])] + 2*(I*ArcTan[a*x]*PolyLog[2, -E^(I*ArcTan[a*x])] - PolyLog[3, -E^(I* ArcTan[a*x])]) - 2*(I*ArcTan[a*x]*PolyLog[2, E^(I*ArcTan[a*x])] - PolyLog[ 3, E^(I*ArcTan[a*x])])))/Sqrt[c + a^2*c*x^2])/c)/c
Int[u_, x_Symbol] :> With[{v = FunctionOfExponential[u, x]}, Simp[v/D[v, x] Subst[Int[FunctionOfExponentialFunction[u, x]/x, x], x, v], x]] /; Funct ionOfExponentialQ[u, x] && !MatchQ[u, (w_)*((a_.)*(v_)^(n_))^(m_) /; FreeQ [{a, m, n}, x] && IntegerQ[m*n]] && !MatchQ[u, E^((c_.)*((a_.) + (b_.)*x)) *(F_)[v_] /; FreeQ[{a, b, c}, x] && InverseFunctionQ[F[x]]]
Int[Log[1 + (e_.)*((F_)^((c_.)*((a_.) + (b_.)*(x_))))^(n_.)]*((f_.) + (g_.) *(x_))^(m_.), x_Symbol] :> Simp[(-(f + g*x)^m)*(PolyLog[2, (-e)*(F^(c*(a + b*x)))^n]/(b*c*n*Log[F])), x] + Simp[g*(m/(b*c*n*Log[F])) Int[(f + g*x)^( m - 1)*PolyLog[2, (-e)*(F^(c*(a + b*x)))^n], x], x] /; FreeQ[{F, a, b, c, e , f, g, n}, x] && GtQ[m, 0]
Int[csc[(e_.) + (f_.)*(x_)]*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> Simp[- 2*(c + d*x)^m*(ArcTanh[E^(I*(e + f*x))]/f), x] + (-Simp[d*(m/f) Int[(c + d*x)^(m - 1)*Log[1 - E^(I*(e + f*x))], x], x] + Simp[d*(m/f) Int[(c + d*x )^(m - 1)*Log[1 + E^(I*(e + f*x))], x], x]) /; FreeQ[{c, d, e, f}, x] && IG tQ[m, 0]
Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))/((d_) + (e_.)*(x_)^2)^(3/2), x_Symbo l] :> Simp[b/(c*d*Sqrt[d + e*x^2]), x] + Simp[x*((a + b*ArcTan[c*x])/(d*Sqr t[d + e*x^2])), x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[e, c^2*d]
Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))*((d_) + (e_.)*(x_)^2)^(q_), x_Symbol ] :> Simp[b*((d + e*x^2)^(q + 1)/(4*c*d*(q + 1)^2)), x] + (-Simp[x*(d + e*x ^2)^(q + 1)*((a + b*ArcTan[c*x])/(2*d*(q + 1))), x] + Simp[(2*q + 3)/(2*d*( q + 1)) Int[(d + e*x^2)^(q + 1)*(a + b*ArcTan[c*x]), x], x]) /; FreeQ[{a, b, c, d, e}, x] && EqQ[e, c^2*d] && LtQ[q, -1] && NeQ[q, -3/2]
Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_)/((d_) + (e_.)*(x_)^2)^(3/2), x_ Symbol] :> Simp[b*p*((a + b*ArcTan[c*x])^(p - 1)/(c*d*Sqrt[d + e*x^2])), x] + (Simp[x*((a + b*ArcTan[c*x])^p/(d*Sqrt[d + e*x^2])), x] - Simp[b^2*p*(p - 1) Int[(a + b*ArcTan[c*x])^(p - 2)/(d + e*x^2)^(3/2), x], x]) /; FreeQ[ {a, b, c, d, e}, x] && EqQ[e, c^2*d] && GtQ[p, 1]
Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_)*((d_) + (e_.)*(x_)^2)^(q_), x_S ymbol] :> Simp[b*p*(d + e*x^2)^(q + 1)*((a + b*ArcTan[c*x])^(p - 1)/(4*c*d* (q + 1)^2)), x] + (-Simp[x*(d + e*x^2)^(q + 1)*((a + b*ArcTan[c*x])^p/(2*d* (q + 1))), x] + Simp[(2*q + 3)/(2*d*(q + 1)) Int[(d + e*x^2)^(q + 1)*(a + b*ArcTan[c*x])^p, x], x] - Simp[b^2*p*((p - 1)/(4*(q + 1)^2)) Int[(d + e *x^2)^q*(a + b*ArcTan[c*x])^(p - 2), x], x]) /; FreeQ[{a, b, c, d, e}, x] & & EqQ[e, c^2*d] && LtQ[q, -1] && GtQ[p, 1] && NeQ[q, -3/2]
Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.)*((f_.)*(x_))^(m_.)*((d_) + (e_ .)*(x_)^2)^(q_.), x_Symbol] :> Simp[(f*x)^(m + 1)*(d + e*x^2)^(q + 1)*((a + b*ArcTan[c*x])^p/(d*f*(m + 1))), x] - Simp[b*c*(p/(f*(m + 1))) Int[(f*x) ^(m + 1)*(d + e*x^2)^q*(a + b*ArcTan[c*x])^(p - 1), x], x] /; FreeQ[{a, b, c, d, e, f, m, q}, x] && EqQ[e, c^2*d] && EqQ[m + 2*q + 3, 0] && GtQ[p, 0] && NeQ[m, -1]
Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_)/((x_)*Sqrt[(d_) + (e_.)*(x_)^2] ), x_Symbol] :> Simp[1/Sqrt[d] Subst[Int[(a + b*x)^p*Csc[x], x], x, ArcTa n[c*x]], x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[e, c^2*d] && IGtQ[p, 0] && GtQ[d, 0]
Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.)/((x_)*Sqrt[(d_) + (e_.)*(x_)^2 ]), x_Symbol] :> Simp[Sqrt[1 + c^2*x^2]/Sqrt[d + e*x^2] Int[(a + b*ArcTan [c*x])^p/(x*Sqrt[1 + c^2*x^2]), x], x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[ e, c^2*d] && IGtQ[p, 0] && !GtQ[d, 0]
Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.)*(x_)^(m_)*((d_) + (e_.)*(x_)^2 )^(q_), x_Symbol] :> Simp[1/d Int[x^m*(d + e*x^2)^(q + 1)*(a + b*ArcTan[c *x])^p, x], x] - Simp[e/d Int[x^(m + 2)*(d + e*x^2)^q*(a + b*ArcTan[c*x]) ^p, x], x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[e, c^2*d] && IntegersQ[p, 2* q] && LtQ[q, -1] && ILtQ[m, 0] && NeQ[p, -1]
Int[PolyLog[n_, (c_.)*((a_.) + (b_.)*(x_))^(p_.)]/((d_.) + (e_.)*(x_)), x_S ymbol] :> Simp[PolyLog[n + 1, c*(a + b*x)^p]/(e*p), x] /; FreeQ[{a, b, c, d , e, n, p}, x] && EqQ[b*d, a*e]
Time = 4.93 (sec) , antiderivative size = 528, normalized size of antiderivative = 1.07
method | result | size |
default | \(\frac {a \left (9 i \arctan \left (a x \right )^{2}+9 \arctan \left (a x \right )^{3}-2 i-6 \arctan \left (a x \right )\right ) \left (a^{3} x^{3}-3 i a^{2} x^{2}-3 a x +i\right ) \sqrt {c \left (a x -i\right ) \left (a x +i\right )}}{216 c^{3} \left (a^{2} x^{2}+1\right )^{2}}-\frac {7 a \left (\arctan \left (a x \right )^{3}-6 \arctan \left (a x \right )+3 i \arctan \left (a x \right )^{2}-6 i\right ) \left (a x -i\right ) \sqrt {c \left (a x -i\right ) \left (a x +i\right )}}{8 \left (a^{2} x^{2}+1\right ) c^{3}}-\frac {7 \sqrt {c \left (a x -i\right ) \left (a x +i\right )}\, \left (a x +i\right ) \left (\arctan \left (a x \right )^{3}-6 \arctan \left (a x \right )-3 i \arctan \left (a x \right )^{2}+6 i\right ) a}{8 \left (a^{2} x^{2}+1\right ) c^{3}}+\frac {\sqrt {c \left (a x -i\right ) \left (a x +i\right )}\, \left (a^{3} x^{3}+3 i a^{2} x^{2}-3 a x -i\right ) \left (-9 i \arctan \left (a x \right )^{2}+9 \arctan \left (a x \right )^{3}+2 i-6 \arctan \left (a x \right )\right ) a}{216 c^{3} \left (a^{4} x^{4}+2 a^{2} x^{2}+1\right )}-\frac {\arctan \left (a x \right )^{3} \sqrt {c \left (a x -i\right ) \left (a x +i\right )}}{c^{3} x}-\frac {3 a \left (\arctan \left (a x \right )^{2} \ln \left (1+\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )-2 i \arctan \left (a x \right ) \operatorname {polylog}\left (2, -\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )-\arctan \left (a x \right )^{2} \ln \left (1-\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )+2 i \arctan \left (a x \right ) \operatorname {polylog}\left (2, \frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )+2 \operatorname {polylog}\left (3, -\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )-2 \operatorname {polylog}\left (3, \frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )\right ) \sqrt {c \left (a x -i\right ) \left (a x +i\right )}}{\sqrt {a^{2} x^{2}+1}\, c^{3}}\) | \(528\) |
Input:
int(arctan(a*x)^3/x^2/(a^2*c*x^2+c)^(5/2),x,method=_RETURNVERBOSE)
Output:
1/216*a*(9*I*arctan(a*x)^2+9*arctan(a*x)^3-2*I-6*arctan(a*x))*(a^3*x^3-3*I *a^2*x^2-3*a*x+I)*(c*(a*x-I)*(a*x+I))^(1/2)/c^3/(a^2*x^2+1)^2-7/8*a*(arcta n(a*x)^3-6*arctan(a*x)+3*I*arctan(a*x)^2-6*I)*(a*x-I)*(c*(a*x-I)*(a*x+I))^ (1/2)/(a^2*x^2+1)/c^3-7/8*(c*(a*x-I)*(a*x+I))^(1/2)*(a*x+I)*(arctan(a*x)^3 -6*arctan(a*x)-3*I*arctan(a*x)^2+6*I)*a/(a^2*x^2+1)/c^3+1/216*(c*(a*x-I)*( a*x+I))^(1/2)*(a^3*x^3+3*I*a^2*x^2-3*a*x-I)*(-9*I*arctan(a*x)^2+9*arctan(a *x)^3+2*I-6*arctan(a*x))*a/c^3/(a^4*x^4+2*a^2*x^2+1)-arctan(a*x)^3*(c*(a*x -I)*(a*x+I))^(1/2)/c^3/x-3*a*(arctan(a*x)^2*ln(1+(1+I*a*x)/(a^2*x^2+1)^(1/ 2))-2*I*arctan(a*x)*polylog(2,-(1+I*a*x)/(a^2*x^2+1)^(1/2))-arctan(a*x)^2* ln(1-(1+I*a*x)/(a^2*x^2+1)^(1/2))+2*I*arctan(a*x)*polylog(2,(1+I*a*x)/(a^2 *x^2+1)^(1/2))+2*polylog(3,-(1+I*a*x)/(a^2*x^2+1)^(1/2))-2*polylog(3,(1+I* a*x)/(a^2*x^2+1)^(1/2)))*(c*(a*x-I)*(a*x+I))^(1/2)/(a^2*x^2+1)^(1/2)/c^3
\[ \int \frac {\arctan (a x)^3}{x^2 \left (c+a^2 c x^2\right )^{5/2}} \, dx=\int { \frac {\arctan \left (a x\right )^{3}}{{\left (a^{2} c x^{2} + c\right )}^{\frac {5}{2}} x^{2}} \,d x } \] Input:
integrate(arctan(a*x)^3/x^2/(a^2*c*x^2+c)^(5/2),x, algorithm="fricas")
Output:
integral(sqrt(a^2*c*x^2 + c)*arctan(a*x)^3/(a^6*c^3*x^8 + 3*a^4*c^3*x^6 + 3*a^2*c^3*x^4 + c^3*x^2), x)
\[ \int \frac {\arctan (a x)^3}{x^2 \left (c+a^2 c x^2\right )^{5/2}} \, dx=\int \frac {\operatorname {atan}^{3}{\left (a x \right )}}{x^{2} \left (c \left (a^{2} x^{2} + 1\right )\right )^{\frac {5}{2}}}\, dx \] Input:
integrate(atan(a*x)**3/x**2/(a**2*c*x**2+c)**(5/2),x)
Output:
Integral(atan(a*x)**3/(x**2*(c*(a**2*x**2 + 1))**(5/2)), x)
\[ \int \frac {\arctan (a x)^3}{x^2 \left (c+a^2 c x^2\right )^{5/2}} \, dx=\int { \frac {\arctan \left (a x\right )^{3}}{{\left (a^{2} c x^{2} + c\right )}^{\frac {5}{2}} x^{2}} \,d x } \] Input:
integrate(arctan(a*x)^3/x^2/(a^2*c*x^2+c)^(5/2),x, algorithm="maxima")
Output:
integrate(arctan(a*x)^3/((a^2*c*x^2 + c)^(5/2)*x^2), x)
Exception generated. \[ \int \frac {\arctan (a x)^3}{x^2 \left (c+a^2 c x^2\right )^{5/2}} \, dx=\text {Exception raised: TypeError} \] Input:
integrate(arctan(a*x)^3/x^2/(a^2*c*x^2+c)^(5/2),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. \[ \int \frac {\arctan (a x)^3}{x^2 \left (c+a^2 c x^2\right )^{5/2}} \, dx=\int \frac {{\mathrm {atan}\left (a\,x\right )}^3}{x^2\,{\left (c\,a^2\,x^2+c\right )}^{5/2}} \,d x \] Input:
int(atan(a*x)^3/(x^2*(c + a^2*c*x^2)^(5/2)),x)
Output:
int(atan(a*x)^3/(x^2*(c + a^2*c*x^2)^(5/2)), x)
\[ \int \frac {\arctan (a x)^3}{x^2 \left (c+a^2 c x^2\right )^{5/2}} \, dx=\frac {\int \frac {\mathit {atan} \left (a x \right )^{3}}{\sqrt {a^{2} x^{2}+1}\, a^{4} x^{6}+2 \sqrt {a^{2} x^{2}+1}\, a^{2} x^{4}+\sqrt {a^{2} x^{2}+1}\, x^{2}}d x}{\sqrt {c}\, c^{2}} \] Input:
int(atan(a*x)^3/x^2/(a^2*c*x^2+c)^(5/2),x)
Output:
int(atan(a*x)**3/(sqrt(a**2*x**2 + 1)*a**4*x**6 + 2*sqrt(a**2*x**2 + 1)*a* *2*x**4 + sqrt(a**2*x**2 + 1)*x**2),x)/(sqrt(c)*c**2)