Integrand size = 22, antiderivative size = 175 \[ \int \frac {x}{\left (c+a^2 c x^2\right )^{5/2} \arctan (a x)^3} \, dx=-\frac {x}{2 a c \left (c+a^2 c x^2\right )^{3/2} \arctan (a x)^2}-\frac {3}{2 a^2 c \left (c+a^2 c x^2\right )^{3/2} \arctan (a x)}+\frac {1}{a^2 c^2 \sqrt {c+a^2 c x^2} \arctan (a x)}-\frac {\sqrt {1+a^2 x^2} \text {Si}(\arctan (a x))}{8 a^2 c^2 \sqrt {c+a^2 c x^2}}-\frac {9 \sqrt {1+a^2 x^2} \text {Si}(3 \arctan (a x))}{8 a^2 c^2 \sqrt {c+a^2 c x^2}} \]
[Out]
Time = 0.66 (sec) , antiderivative size = 175, normalized size of antiderivative = 1.00, number of steps used = 20, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.318, Rules used = {5088, 5084, 5022, 5091, 5090, 3380, 4491} \[ \int \frac {x}{\left (c+a^2 c x^2\right )^{5/2} \arctan (a x)^3} \, dx=-\frac {\sqrt {a^2 x^2+1} \text {Si}(\arctan (a x))}{8 a^2 c^2 \sqrt {a^2 c x^2+c}}-\frac {9 \sqrt {a^2 x^2+1} \text {Si}(3 \arctan (a x))}{8 a^2 c^2 \sqrt {a^2 c x^2+c}}+\frac {1}{a^2 c^2 \arctan (a x) \sqrt {a^2 c x^2+c}}-\frac {x}{2 a c \arctan (a x)^2 \left (a^2 c x^2+c\right )^{3/2}}-\frac {3}{2 a^2 c \arctan (a x) \left (a^2 c x^2+c\right )^{3/2}} \]
[In]
[Out]
Rule 3380
Rule 4491
Rule 5022
Rule 5084
Rule 5088
Rule 5090
Rule 5091
Rubi steps \begin{align*} \text {integral}& = -\frac {x}{2 a c \left (c+a^2 c x^2\right )^{3/2} \arctan (a x)^2}+\frac {\int \frac {1}{\left (c+a^2 c x^2\right )^{5/2} \arctan (a x)^2} \, dx}{2 a}-a \int \frac {x^2}{\left (c+a^2 c x^2\right )^{5/2} \arctan (a x)^2} \, dx \\ & = -\frac {x}{2 a c \left (c+a^2 c x^2\right )^{3/2} \arctan (a x)^2}-\frac {1}{2 a^2 c \left (c+a^2 c x^2\right )^{3/2} \arctan (a x)}-\frac {3}{2} \int \frac {x}{\left (c+a^2 c x^2\right )^{5/2} \arctan (a x)} \, dx+\frac {\int \frac {1}{\left (c+a^2 c x^2\right )^{5/2} \arctan (a x)^2} \, dx}{a}-\frac {\int \frac {1}{\left (c+a^2 c x^2\right )^{3/2} \arctan (a x)^2} \, dx}{a c} \\ & = -\frac {x}{2 a c \left (c+a^2 c x^2\right )^{3/2} \arctan (a x)^2}-\frac {3}{2 a^2 c \left (c+a^2 c x^2\right )^{3/2} \arctan (a x)}+\frac {1}{a^2 c^2 \sqrt {c+a^2 c x^2} \arctan (a x)}-3 \int \frac {x}{\left (c+a^2 c x^2\right )^{5/2} \arctan (a x)} \, dx+\frac {\int \frac {x}{\left (c+a^2 c x^2\right )^{3/2} \arctan (a x)} \, dx}{c}-\frac {\left (3 \sqrt {1+a^2 x^2}\right ) \int \frac {x}{\left (1+a^2 x^2\right )^{5/2} \arctan (a x)} \, dx}{2 c^2 \sqrt {c+a^2 c x^2}} \\ & = -\frac {x}{2 a c \left (c+a^2 c x^2\right )^{3/2} \arctan (a x)^2}-\frac {3}{2 a^2 c \left (c+a^2 c x^2\right )^{3/2} \arctan (a x)}+\frac {1}{a^2 c^2 \sqrt {c+a^2 c x^2} \arctan (a x)}+\frac {\sqrt {1+a^2 x^2} \int \frac {x}{\left (1+a^2 x^2\right )^{3/2} \arctan (a x)} \, dx}{c^2 \sqrt {c+a^2 c x^2}}-\frac {\left (3 \sqrt {1+a^2 x^2}\right ) \int \frac {x}{\left (1+a^2 x^2\right )^{5/2} \arctan (a x)} \, dx}{c^2 \sqrt {c+a^2 c x^2}}-\frac {\left (3 \sqrt {1+a^2 x^2}\right ) \text {Subst}\left (\int \frac {\cos ^2(x) \sin (x)}{x} \, dx,x,\arctan (a x)\right )}{2 a^2 c^2 \sqrt {c+a^2 c x^2}} \\ & = -\frac {x}{2 a c \left (c+a^2 c x^2\right )^{3/2} \arctan (a x)^2}-\frac {3}{2 a^2 c \left (c+a^2 c x^2\right )^{3/2} \arctan (a x)}+\frac {1}{a^2 c^2 \sqrt {c+a^2 c x^2} \arctan (a x)}+\frac {\sqrt {1+a^2 x^2} \text {Subst}\left (\int \frac {\sin (x)}{x} \, dx,x,\arctan (a x)\right )}{a^2 c^2 \sqrt {c+a^2 c x^2}}-\frac {\left (3 \sqrt {1+a^2 x^2}\right ) \text {Subst}\left (\int \left (\frac {\sin (x)}{4 x}+\frac {\sin (3 x)}{4 x}\right ) \, dx,x,\arctan (a x)\right )}{2 a^2 c^2 \sqrt {c+a^2 c x^2}}-\frac {\left (3 \sqrt {1+a^2 x^2}\right ) \text {Subst}\left (\int \frac {\cos ^2(x) \sin (x)}{x} \, dx,x,\arctan (a x)\right )}{a^2 c^2 \sqrt {c+a^2 c x^2}} \\ & = -\frac {x}{2 a c \left (c+a^2 c x^2\right )^{3/2} \arctan (a x)^2}-\frac {3}{2 a^2 c \left (c+a^2 c x^2\right )^{3/2} \arctan (a x)}+\frac {1}{a^2 c^2 \sqrt {c+a^2 c x^2} \arctan (a x)}+\frac {\sqrt {1+a^2 x^2} \text {Si}(\arctan (a x))}{a^2 c^2 \sqrt {c+a^2 c x^2}}-\frac {\left (3 \sqrt {1+a^2 x^2}\right ) \text {Subst}\left (\int \frac {\sin (x)}{x} \, dx,x,\arctan (a x)\right )}{8 a^2 c^2 \sqrt {c+a^2 c x^2}}-\frac {\left (3 \sqrt {1+a^2 x^2}\right ) \text {Subst}\left (\int \frac {\sin (3 x)}{x} \, dx,x,\arctan (a x)\right )}{8 a^2 c^2 \sqrt {c+a^2 c x^2}}-\frac {\left (3 \sqrt {1+a^2 x^2}\right ) \text {Subst}\left (\int \left (\frac {\sin (x)}{4 x}+\frac {\sin (3 x)}{4 x}\right ) \, dx,x,\arctan (a x)\right )}{a^2 c^2 \sqrt {c+a^2 c x^2}} \\ & = -\frac {x}{2 a c \left (c+a^2 c x^2\right )^{3/2} \arctan (a x)^2}-\frac {3}{2 a^2 c \left (c+a^2 c x^2\right )^{3/2} \arctan (a x)}+\frac {1}{a^2 c^2 \sqrt {c+a^2 c x^2} \arctan (a x)}+\frac {5 \sqrt {1+a^2 x^2} \text {Si}(\arctan (a x))}{8 a^2 c^2 \sqrt {c+a^2 c x^2}}-\frac {3 \sqrt {1+a^2 x^2} \text {Si}(3 \arctan (a x))}{8 a^2 c^2 \sqrt {c+a^2 c x^2}}-\frac {\left (3 \sqrt {1+a^2 x^2}\right ) \text {Subst}\left (\int \frac {\sin (x)}{x} \, dx,x,\arctan (a x)\right )}{4 a^2 c^2 \sqrt {c+a^2 c x^2}}-\frac {\left (3 \sqrt {1+a^2 x^2}\right ) \text {Subst}\left (\int \frac {\sin (3 x)}{x} \, dx,x,\arctan (a x)\right )}{4 a^2 c^2 \sqrt {c+a^2 c x^2}} \\ & = -\frac {x}{2 a c \left (c+a^2 c x^2\right )^{3/2} \arctan (a x)^2}-\frac {3}{2 a^2 c \left (c+a^2 c x^2\right )^{3/2} \arctan (a x)}+\frac {1}{a^2 c^2 \sqrt {c+a^2 c x^2} \arctan (a x)}-\frac {\sqrt {1+a^2 x^2} \text {Si}(\arctan (a x))}{8 a^2 c^2 \sqrt {c+a^2 c x^2}}-\frac {9 \sqrt {1+a^2 x^2} \text {Si}(3 \arctan (a x))}{8 a^2 c^2 \sqrt {c+a^2 c x^2}} \\ \end{align*}
Time = 0.35 (sec) , antiderivative size = 118, normalized size of antiderivative = 0.67 \[ \int \frac {x}{\left (c+a^2 c x^2\right )^{5/2} \arctan (a x)^3} \, dx=\frac {-4 a x-4 \arctan (a x)+8 a^2 x^2 \arctan (a x)-\left (1+a^2 x^2\right )^{3/2} \arctan (a x)^2 \text {Si}(\arctan (a x))-9 \left (1+a^2 x^2\right )^{3/2} \arctan (a x)^2 \text {Si}(3 \arctan (a x))}{8 a^2 c^2 \left (1+a^2 x^2\right ) \sqrt {c+a^2 c x^2} \arctan (a x)^2} \]
[In]
[Out]
Result contains complex when optimal does not.
Time = 11.43 (sec) , antiderivative size = 371, normalized size of antiderivative = 2.12
method | result | size |
default | \(-\frac {i \left (\arctan \left (a x \right )^{2} \operatorname {Ei}_{1}\left (-i \arctan \left (a x \right )\right ) a^{4} x^{4}-\arctan \left (a x \right )^{2} \operatorname {Ei}_{1}\left (i \arctan \left (a x \right )\right ) a^{4} x^{4}+9 \arctan \left (a x \right )^{2} \operatorname {Ei}_{1}\left (-3 i \arctan \left (a x \right )\right ) a^{4} x^{4}-9 \arctan \left (a x \right )^{2} \operatorname {Ei}_{1}\left (3 i \arctan \left (a x \right )\right ) a^{4} x^{4}+2 \arctan \left (a x \right )^{2} \operatorname {Ei}_{1}\left (-i \arctan \left (a x \right )\right ) a^{2} x^{2}-2 \arctan \left (a x \right )^{2} \operatorname {Ei}_{1}\left (i \arctan \left (a x \right )\right ) a^{2} x^{2}+18 \arctan \left (a x \right )^{2} \operatorname {Ei}_{1}\left (-3 i \arctan \left (a x \right )\right ) a^{2} x^{2}-18 \arctan \left (a x \right )^{2} \operatorname {Ei}_{1}\left (3 i \arctan \left (a x \right )\right ) a^{2} x^{2}+16 i \arctan \left (a x \right ) \sqrt {a^{2} x^{2}+1}\, a^{2} x^{2}+\operatorname {Ei}_{1}\left (-i \arctan \left (a x \right )\right ) \arctan \left (a x \right )^{2}-\operatorname {Ei}_{1}\left (i \arctan \left (a x \right )\right ) \arctan \left (a x \right )^{2}+9 \,\operatorname {Ei}_{1}\left (-3 i \arctan \left (a x \right )\right ) \arctan \left (a x \right )^{2}-9 \,\operatorname {Ei}_{1}\left (3 i \arctan \left (a x \right )\right ) \arctan \left (a x \right )^{2}-8 i \sqrt {a^{2} x^{2}+1}\, a x -8 i \sqrt {a^{2} x^{2}+1}\, \arctan \left (a x \right )\right ) \sqrt {c \left (a x -i\right ) \left (a x +i\right )}}{16 \sqrt {a^{2} x^{2}+1}\, \arctan \left (a x \right )^{2} a^{2} c^{3} \left (a^{4} x^{4}+2 a^{2} x^{2}+1\right )}\) | \(371\) |
[In]
[Out]
\[ \int \frac {x}{\left (c+a^2 c x^2\right )^{5/2} \arctan (a x)^3} \, dx=\int { \frac {x}{{\left (a^{2} c x^{2} + c\right )}^{\frac {5}{2}} \arctan \left (a x\right )^{3}} \,d x } \]
[In]
[Out]
\[ \int \frac {x}{\left (c+a^2 c x^2\right )^{5/2} \arctan (a x)^3} \, dx=\int \frac {x}{\left (c \left (a^{2} x^{2} + 1\right )\right )^{\frac {5}{2}} \operatorname {atan}^{3}{\left (a x \right )}}\, dx \]
[In]
[Out]
\[ \int \frac {x}{\left (c+a^2 c x^2\right )^{5/2} \arctan (a x)^3} \, dx=\int { \frac {x}{{\left (a^{2} c x^{2} + c\right )}^{\frac {5}{2}} \arctan \left (a x\right )^{3}} \,d x } \]
[In]
[Out]
Exception generated. \[ \int \frac {x}{\left (c+a^2 c x^2\right )^{5/2} \arctan (a x)^3} \, dx=\text {Exception raised: TypeError} \]
[In]
[Out]
Timed out. \[ \int \frac {x}{\left (c+a^2 c x^2\right )^{5/2} \arctan (a x)^3} \, dx=\int \frac {x}{{\mathrm {atan}\left (a\,x\right )}^3\,{\left (c\,a^2\,x^2+c\right )}^{5/2}} \,d x \]
[In]
[Out]