Optimal. Leaf size=124 \[ \frac {x \tan ^{-1}(a x)^{3/2}}{2 c^2 \left (a^2 x^2+1\right )}+\frac {3 \sqrt {\tan ^{-1}(a x)}}{8 a c^2 \left (a^2 x^2+1\right )}-\frac {3 \sqrt {\pi } C\left (\frac {2 \sqrt {\tan ^{-1}(a x)}}{\sqrt {\pi }}\right )}{32 a c^2}+\frac {\tan ^{-1}(a x)^{5/2}}{5 a c^2}-\frac {3 \sqrt {\tan ^{-1}(a x)}}{16 a c^2} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.15, antiderivative size = 124, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {4892, 4930, 4904, 3312, 3304, 3352} \[ \frac {x \tan ^{-1}(a x)^{3/2}}{2 c^2 \left (a^2 x^2+1\right )}+\frac {3 \sqrt {\tan ^{-1}(a x)}}{8 a c^2 \left (a^2 x^2+1\right )}-\frac {3 \sqrt {\pi } \text {FresnelC}\left (\frac {2 \sqrt {\tan ^{-1}(a x)}}{\sqrt {\pi }}\right )}{32 a c^2}+\frac {\tan ^{-1}(a x)^{5/2}}{5 a c^2}-\frac {3 \sqrt {\tan ^{-1}(a x)}}{16 a c^2} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 3304
Rule 3312
Rule 3352
Rule 4892
Rule 4904
Rule 4930
Rubi steps
\begin {align*} \int \frac {\tan ^{-1}(a x)^{3/2}}{\left (c+a^2 c x^2\right )^2} \, dx &=\frac {x \tan ^{-1}(a x)^{3/2}}{2 c^2 \left (1+a^2 x^2\right )}+\frac {\tan ^{-1}(a x)^{5/2}}{5 a c^2}-\frac {1}{4} (3 a) \int \frac {x \sqrt {\tan ^{-1}(a x)}}{\left (c+a^2 c x^2\right )^2} \, dx\\ &=\frac {3 \sqrt {\tan ^{-1}(a x)}}{8 a c^2 \left (1+a^2 x^2\right )}+\frac {x \tan ^{-1}(a x)^{3/2}}{2 c^2 \left (1+a^2 x^2\right )}+\frac {\tan ^{-1}(a x)^{5/2}}{5 a c^2}-\frac {3}{16} \int \frac {1}{\left (c+a^2 c x^2\right )^2 \sqrt {\tan ^{-1}(a x)}} \, dx\\ &=\frac {3 \sqrt {\tan ^{-1}(a x)}}{8 a c^2 \left (1+a^2 x^2\right )}+\frac {x \tan ^{-1}(a x)^{3/2}}{2 c^2 \left (1+a^2 x^2\right )}+\frac {\tan ^{-1}(a x)^{5/2}}{5 a c^2}-\frac {3 \operatorname {Subst}\left (\int \frac {\cos ^2(x)}{\sqrt {x}} \, dx,x,\tan ^{-1}(a x)\right )}{16 a c^2}\\ &=\frac {3 \sqrt {\tan ^{-1}(a x)}}{8 a c^2 \left (1+a^2 x^2\right )}+\frac {x \tan ^{-1}(a x)^{3/2}}{2 c^2 \left (1+a^2 x^2\right )}+\frac {\tan ^{-1}(a x)^{5/2}}{5 a c^2}-\frac {3 \operatorname {Subst}\left (\int \left (\frac {1}{2 \sqrt {x}}+\frac {\cos (2 x)}{2 \sqrt {x}}\right ) \, dx,x,\tan ^{-1}(a x)\right )}{16 a c^2}\\ &=-\frac {3 \sqrt {\tan ^{-1}(a x)}}{16 a c^2}+\frac {3 \sqrt {\tan ^{-1}(a x)}}{8 a c^2 \left (1+a^2 x^2\right )}+\frac {x \tan ^{-1}(a x)^{3/2}}{2 c^2 \left (1+a^2 x^2\right )}+\frac {\tan ^{-1}(a x)^{5/2}}{5 a c^2}-\frac {3 \operatorname {Subst}\left (\int \frac {\cos (2 x)}{\sqrt {x}} \, dx,x,\tan ^{-1}(a x)\right )}{32 a c^2}\\ &=-\frac {3 \sqrt {\tan ^{-1}(a x)}}{16 a c^2}+\frac {3 \sqrt {\tan ^{-1}(a x)}}{8 a c^2 \left (1+a^2 x^2\right )}+\frac {x \tan ^{-1}(a x)^{3/2}}{2 c^2 \left (1+a^2 x^2\right )}+\frac {\tan ^{-1}(a x)^{5/2}}{5 a c^2}-\frac {3 \operatorname {Subst}\left (\int \cos \left (2 x^2\right ) \, dx,x,\sqrt {\tan ^{-1}(a x)}\right )}{16 a c^2}\\ &=-\frac {3 \sqrt {\tan ^{-1}(a x)}}{16 a c^2}+\frac {3 \sqrt {\tan ^{-1}(a x)}}{8 a c^2 \left (1+a^2 x^2\right )}+\frac {x \tan ^{-1}(a x)^{3/2}}{2 c^2 \left (1+a^2 x^2\right )}+\frac {\tan ^{-1}(a x)^{5/2}}{5 a c^2}-\frac {3 \sqrt {\pi } C\left (\frac {2 \sqrt {\tan ^{-1}(a x)}}{\sqrt {\pi }}\right )}{32 a c^2}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.18, size = 90, normalized size = 0.73 \[ \frac {\frac {2 \sqrt {\tan ^{-1}(a x)} \left (-15 a^2 x^2+16 \left (a^2 x^2+1\right ) \tan ^{-1}(a x)^2+40 a x \tan ^{-1}(a x)+15\right )}{a^2 x^2+1}-15 \sqrt {\pi } C\left (\frac {2 \sqrt {\tan ^{-1}(a x)}}{\sqrt {\pi }}\right )}{160 a c^2} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \mathit {sage}_{0} x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.53, size = 75, normalized size = 0.60 \[ \frac {32 \arctan \left (a x \right )^{3}+40 \arctan \left (a x \right )^{2} \sin \left (2 \arctan \left (a x \right )\right )+30 \cos \left (2 \arctan \left (a x \right )\right ) \arctan \left (a x \right )-15 \sqrt {\arctan \left (a x \right )}\, \sqrt {\pi }\, \FresnelC \left (\frac {2 \sqrt {\arctan \left (a x \right )}}{\sqrt {\pi }}\right )}{160 a \,c^{2} \sqrt {\arctan \left (a x \right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: RuntimeError} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {{\mathrm {atan}\left (a\,x\right )}^{3/2}}{{\left (c\,a^2\,x^2+c\right )}^2} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \frac {\int \frac {\operatorname {atan}^{\frac {3}{2}}{\left (a x \right )}}{a^{4} x^{4} + 2 a^{2} x^{2} + 1}\, dx}{c^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________