Integrand size = 20, antiderivative size = 109 \[ \int x \left (c+a^2 c x^2\right )^{3/2} \arctan (a x) \, dx=-\frac {3 c x \sqrt {c+a^2 c x^2}}{40 a}-\frac {x \left (c+a^2 c x^2\right )^{3/2}}{20 a}+\frac {\left (c+a^2 c x^2\right )^{5/2} \arctan (a x)}{5 a^2 c}-\frac {3 c^{3/2} \text {arctanh}\left (\frac {a \sqrt {c} x}{\sqrt {c+a^2 c x^2}}\right )}{40 a^2} \]
-1/20*x*(a^2*c*x^2+c)^(3/2)/a+1/5*(a^2*c*x^2+c)^(5/2)*arctan(a*x)/a^2/c-3/ 40*c^(3/2)*arctanh(a*x*c^(1/2)/(a^2*c*x^2+c)^(1/2))/a^2-3/40*c*x*(a^2*c*x^ 2+c)^(1/2)/a
Time = 0.16 (sec) , antiderivative size = 101, normalized size of antiderivative = 0.93 \[ \int x \left (c+a^2 c x^2\right )^{3/2} \arctan (a x) \, dx=-\frac {a c x \left (5+2 a^2 x^2\right ) \sqrt {c+a^2 c x^2}-8 c \left (1+a^2 x^2\right )^2 \sqrt {c+a^2 c x^2} \arctan (a x)+3 c^{3/2} \log \left (a c x+\sqrt {c} \sqrt {c+a^2 c x^2}\right )}{40 a^2} \]
-1/40*(a*c*x*(5 + 2*a^2*x^2)*Sqrt[c + a^2*c*x^2] - 8*c*(1 + a^2*x^2)^2*Sqr t[c + a^2*c*x^2]*ArcTan[a*x] + 3*c^(3/2)*Log[a*c*x + Sqrt[c]*Sqrt[c + a^2* c*x^2]])/a^2
Time = 0.26 (sec) , antiderivative size = 116, normalized size of antiderivative = 1.06, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {5465, 211, 211, 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.
\(\displaystyle \int x \arctan (a x) \left (a^2 c x^2+c\right )^{3/2} \, dx\) |
\(\Big \downarrow \) 5465 |
\(\displaystyle \frac {\arctan (a x) \left (a^2 c x^2+c\right )^{5/2}}{5 a^2 c}-\frac {\int \left (a^2 c x^2+c\right )^{3/2}dx}{5 a}\) |
\(\Big \downarrow \) 211 |
\(\displaystyle \frac {\arctan (a x) \left (a^2 c x^2+c\right )^{5/2}}{5 a^2 c}-\frac {\frac {3}{4} c \int \sqrt {a^2 c x^2+c}dx+\frac {1}{4} x \left (a^2 c x^2+c\right )^{3/2}}{5 a}\) |
\(\Big \downarrow \) 211 |
\(\displaystyle \frac {\arctan (a x) \left (a^2 c x^2+c\right )^{5/2}}{5 a^2 c}-\frac {\frac {3}{4} c \left (\frac {1}{2} c \int \frac {1}{\sqrt {a^2 c x^2+c}}dx+\frac {1}{2} x \sqrt {a^2 c x^2+c}\right )+\frac {1}{4} x \left (a^2 c x^2+c\right )^{3/2}}{5 a}\) |
\(\Big \downarrow \) 224 |
\(\displaystyle \frac {\arctan (a x) \left (a^2 c x^2+c\right )^{5/2}}{5 a^2 c}-\frac {\frac {3}{4} c \left (\frac {1}{2} c \int \frac {1}{1-\frac {a^2 c x^2}{a^2 c x^2+c}}d\frac {x}{\sqrt {a^2 c x^2+c}}+\frac {1}{2} x \sqrt {a^2 c x^2+c}\right )+\frac {1}{4} x \left (a^2 c x^2+c\right )^{3/2}}{5 a}\) |
\(\Big \downarrow \) 219 |
\(\displaystyle \frac {\arctan (a x) \left (a^2 c x^2+c\right )^{5/2}}{5 a^2 c}-\frac {\frac {3}{4} c \left (\frac {\sqrt {c} \text {arctanh}\left (\frac {a \sqrt {c} x}{\sqrt {a^2 c x^2+c}}\right )}{2 a}+\frac {1}{2} x \sqrt {a^2 c x^2+c}\right )+\frac {1}{4} x \left (a^2 c x^2+c\right )^{3/2}}{5 a}\) |
((c + a^2*c*x^2)^(5/2)*ArcTan[a*x])/(5*a^2*c) - ((x*(c + a^2*c*x^2)^(3/2)) /4 + (3*c*((x*Sqrt[c + a^2*c*x^2])/2 + (Sqrt[c]*ArcTanh[(a*Sqrt[c]*x)/Sqrt [c + a^2*c*x^2]])/(2*a)))/4)/(5*a)
3.3.10.3.1 Defintions of rubi rules used
Int[((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[x*((a + b*x^2)^p/(2*p + 1 )), x] + Simp[2*a*(p/(2*p + 1)) Int[(a + b*x^2)^(p - 1), x], x] /; FreeQ[ {a, b}, x] && GtQ[p, 0] && (IntegerQ[4*p] || IntegerQ[6*p])
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[((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]
Result contains complex when optimal does not.
Time = 0.41 (sec) , antiderivative size = 179, normalized size of antiderivative = 1.64
method | result | size |
default | \(\frac {c \sqrt {c \left (a x -i\right ) \left (a x +i\right )}\, \left (8 \arctan \left (a x \right ) a^{4} x^{4}-2 a^{3} x^{3}+16 a^{2} \arctan \left (a x \right ) x^{2}-5 a x +8 \arctan \left (a x \right )\right )}{40 a^{2}}-\frac {3 c \sqrt {c \left (a x -i\right ) \left (a x +i\right )}\, \ln \left (\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}+i\right )}{40 a^{2} \sqrt {a^{2} x^{2}+1}}+\frac {3 c \sqrt {c \left (a x -i\right ) \left (a x +i\right )}\, \ln \left (\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}-i\right )}{40 a^{2} \sqrt {a^{2} x^{2}+1}}\) | \(179\) |
1/40*c/a^2*(c*(a*x-I)*(I+a*x))^(1/2)*(8*arctan(a*x)*a^4*x^4-2*a^3*x^3+16*a ^2*arctan(a*x)*x^2-5*a*x+8*arctan(a*x))-3/40*c/a^2*(c*(a*x-I)*(I+a*x))^(1/ 2)*ln((1+I*a*x)/(a^2*x^2+1)^(1/2)+I)/(a^2*x^2+1)^(1/2)+3/40*c/a^2*(c*(a*x- I)*(I+a*x))^(1/2)*ln((1+I*a*x)/(a^2*x^2+1)^(1/2)-I)/(a^2*x^2+1)^(1/2)
Time = 0.27 (sec) , antiderivative size = 98, normalized size of antiderivative = 0.90 \[ \int x \left (c+a^2 c x^2\right )^{3/2} \arctan (a x) \, dx=\frac {3 \, c^{\frac {3}{2}} \log \left (-2 \, a^{2} c x^{2} + 2 \, \sqrt {a^{2} c x^{2} + c} a \sqrt {c} x - c\right ) - 2 \, {\left (2 \, a^{3} c x^{3} + 5 \, a c x - 8 \, {\left (a^{4} c x^{4} + 2 \, a^{2} c x^{2} + c\right )} \arctan \left (a x\right )\right )} \sqrt {a^{2} c x^{2} + c}}{80 \, a^{2}} \]
1/80*(3*c^(3/2)*log(-2*a^2*c*x^2 + 2*sqrt(a^2*c*x^2 + c)*a*sqrt(c)*x - c) - 2*(2*a^3*c*x^3 + 5*a*c*x - 8*(a^4*c*x^4 + 2*a^2*c*x^2 + c)*arctan(a*x))* sqrt(a^2*c*x^2 + c))/a^2
\[ \int x \left (c+a^2 c x^2\right )^{3/2} \arctan (a x) \, dx=\int x \left (c \left (a^{2} x^{2} + 1\right )\right )^{\frac {3}{2}} \operatorname {atan}{\left (a x \right )}\, dx \]
Leaf count of result is larger than twice the leaf count of optimal. 406 vs. \(2 (89) = 178\).
Time = 0.40 (sec) , antiderivative size = 406, normalized size of antiderivative = 3.72 \[ \int x \left (c+a^2 c x^2\right )^{3/2} \arctan (a x) \, dx=\frac {40 \, {\left (a^{2} c x^{2} + c\right )} \sqrt {a^{2} x^{2} + 1} \sqrt {c} \arctan \left (a x\right ) - 20 \, {\left (a^{4} x^{4} + 10 \, a^{2} x^{2} + 9\right )}^{\frac {1}{4}} {\left (a c x \cos \left (\frac {1}{2} \, \arctan \left (4 \, a x, -a^{2} x^{2} + 3\right )\right ) + 2 \, c \sin \left (\frac {1}{2} \, \arctan \left (4 \, a x, -a^{2} x^{2} + 3\right )\right )\right )} \sqrt {c} - {\left ({\left (a {\left (\frac {3 \, {\left (\frac {2 \, {\left (a^{2} x^{2} + 1\right )}^{\frac {3}{2}} x}{a^{2}} - \frac {\sqrt {a^{2} x^{2} + 1} x}{a^{2}} - \frac {\operatorname {arsinh}\left (a x\right )}{a^{3}}\right )}}{a^{2}} - \frac {8 \, {\left (\sqrt {a^{2} x^{2} + 1} x + \frac {\operatorname {arsinh}\left (a x\right )}{a}\right )}}{a^{4}}\right )} - 8 \, {\left (\frac {3 \, {\left (a^{2} x^{2} + 1\right )}^{\frac {3}{2}} x^{2}}{a^{2}} - \frac {2 \, {\left (a^{2} x^{2} + 1\right )}^{\frac {3}{2}}}{a^{4}}\right )} \arctan \left (a x\right )\right )} a^{4} c - 10 \, c \arctan \left ({\left (a^{4} x^{4} + 10 \, a^{2} x^{2} + 9\right )}^{\frac {1}{4}} \sin \left (\frac {1}{2} \, \arctan \left (4 \, a x, a^{2} x^{2} - 3\right )\right ) + 2, a x + {\left (a^{4} x^{4} + 10 \, a^{2} x^{2} + 9\right )}^{\frac {1}{4}} \cos \left (\frac {1}{2} \, \arctan \left (4 \, a x, a^{2} x^{2} - 3\right )\right )\right ) - 10 \, c \arctan \left ({\left (a^{4} x^{4} + 10 \, a^{2} x^{2} + 9\right )}^{\frac {1}{4}} \sin \left (\frac {1}{2} \, \arctan \left (4 \, a x, a^{2} x^{2} - 3\right )\right ) - 2, -a x + {\left (a^{4} x^{4} + 10 \, a^{2} x^{2} + 9\right )}^{\frac {1}{4}} \cos \left (\frac {1}{2} \, \arctan \left (4 \, a x, a^{2} x^{2} - 3\right )\right )\right )\right )} \sqrt {c}}{120 \, a^{2}} \]
1/120*(40*(a^2*c*x^2 + c)*sqrt(a^2*x^2 + 1)*sqrt(c)*arctan(a*x) - 20*(a^4* x^4 + 10*a^2*x^2 + 9)^(1/4)*(a*c*x*cos(1/2*arctan2(4*a*x, -a^2*x^2 + 3)) + 2*c*sin(1/2*arctan2(4*a*x, -a^2*x^2 + 3)))*sqrt(c) - ((a*(3*(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 - 8*(sqr t(a^2*x^2 + 1)*x + arcsinh(a*x)/a)/a^4) - 8*(3*(a^2*x^2 + 1)^(3/2)*x^2/a^2 - 2*(a^2*x^2 + 1)^(3/2)/a^4)*arctan(a*x))*a^4*c - 10*c*arctan2((a^4*x^4 + 10*a^2*x^2 + 9)^(1/4)*sin(1/2*arctan2(4*a*x, a^2*x^2 - 3)) + 2, a*x + (a^ 4*x^4 + 10*a^2*x^2 + 9)^(1/4)*cos(1/2*arctan2(4*a*x, a^2*x^2 - 3))) - 10*c *arctan2((a^4*x^4 + 10*a^2*x^2 + 9)^(1/4)*sin(1/2*arctan2(4*a*x, a^2*x^2 - 3)) - 2, -a*x + (a^4*x^4 + 10*a^2*x^2 + 9)^(1/4)*cos(1/2*arctan2(4*a*x, a ^2*x^2 - 3))))*sqrt(c))/a^2
Exception generated. \[ \int x \left (c+a^2 c x^2\right )^{3/2} \arctan (a x) \, dx=\text {Exception raised: TypeError} \]
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. \[ \int x \left (c+a^2 c x^2\right )^{3/2} \arctan (a x) \, dx=\int x\,\mathrm {atan}\left (a\,x\right )\,{\left (c\,a^2\,x^2+c\right )}^{3/2} \,d x \]