Optimal. Leaf size=160 \[ \frac{4 \sqrt{2 \pi } S\left (2 \sqrt{\frac{2}{\pi }} \sqrt{\tan ^{-1}(a x)}\right )}{3 a^4 c^3}-\frac{4 \sqrt{\pi } S\left (\frac{2 \sqrt{\tan ^{-1}(a x)}}{\sqrt{\pi }}\right )}{3 a^4 c^3}+\frac{4 x^4}{3 c^3 \left (a^2 x^2+1\right )^2 \sqrt{\tan ^{-1}(a x)}}-\frac{2 x^3}{3 a c^3 \left (a^2 x^2+1\right )^2 \tan ^{-1}(a x)^{3/2}}-\frac{4 x^2}{a^2 c^3 \left (a^2 x^2+1\right )^2 \sqrt{\tan ^{-1}(a x)}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.590159, antiderivative size = 160, normalized size of antiderivative = 1., number of steps used = 24, number of rules used = 6, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {4968, 4942, 4970, 4406, 3305, 3351} \[ \frac{4 \sqrt{2 \pi } S\left (2 \sqrt{\frac{2}{\pi }} \sqrt{\tan ^{-1}(a x)}\right )}{3 a^4 c^3}-\frac{4 \sqrt{\pi } S\left (\frac{2 \sqrt{\tan ^{-1}(a x)}}{\sqrt{\pi }}\right )}{3 a^4 c^3}+\frac{4 x^4}{3 c^3 \left (a^2 x^2+1\right )^2 \sqrt{\tan ^{-1}(a x)}}-\frac{2 x^3}{3 a c^3 \left (a^2 x^2+1\right )^2 \tan ^{-1}(a x)^{3/2}}-\frac{4 x^2}{a^2 c^3 \left (a^2 x^2+1\right )^2 \sqrt{\tan ^{-1}(a x)}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 4968
Rule 4942
Rule 4970
Rule 4406
Rule 3305
Rule 3351
Rubi steps
\begin{align*} \int \frac{x^3}{\left (c+a^2 c x^2\right )^3 \tan ^{-1}(a x)^{5/2}} \, dx &=-\frac{2 x^3}{3 a c^3 \left (1+a^2 x^2\right )^2 \tan ^{-1}(a x)^{3/2}}+\frac{2 \int \frac{x^2}{\left (c+a^2 c x^2\right )^3 \tan ^{-1}(a x)^{3/2}} \, dx}{a}-\frac{1}{3} (2 a) \int \frac{x^4}{\left (c+a^2 c x^2\right )^3 \tan ^{-1}(a x)^{3/2}} \, dx\\ &=-\frac{2 x^3}{3 a c^3 \left (1+a^2 x^2\right )^2 \tan ^{-1}(a x)^{3/2}}-\frac{4 x^2}{a^2 c^3 \left (1+a^2 x^2\right )^2 \sqrt{\tan ^{-1}(a x)}}+\frac{4 x^4}{3 c^3 \left (1+a^2 x^2\right )^2 \sqrt{\tan ^{-1}(a x)}}-\frac{16}{3} \int \frac{x^3}{\left (c+a^2 c x^2\right )^3 \sqrt{\tan ^{-1}(a x)}} \, dx-8 \int \frac{x^3}{\left (c+a^2 c x^2\right )^3 \sqrt{\tan ^{-1}(a x)}} \, dx+\frac{8 \int \frac{x}{\left (c+a^2 c x^2\right )^3 \sqrt{\tan ^{-1}(a x)}} \, dx}{a^2}\\ &=-\frac{2 x^3}{3 a c^3 \left (1+a^2 x^2\right )^2 \tan ^{-1}(a x)^{3/2}}-\frac{4 x^2}{a^2 c^3 \left (1+a^2 x^2\right )^2 \sqrt{\tan ^{-1}(a x)}}+\frac{4 x^4}{3 c^3 \left (1+a^2 x^2\right )^2 \sqrt{\tan ^{-1}(a x)}}-\frac{16 \operatorname{Subst}\left (\int \frac{\cos (x) \sin ^3(x)}{\sqrt{x}} \, dx,x,\tan ^{-1}(a x)\right )}{3 a^4 c^3}+\frac{8 \operatorname{Subst}\left (\int \frac{\cos ^3(x) \sin (x)}{\sqrt{x}} \, dx,x,\tan ^{-1}(a x)\right )}{a^4 c^3}-\frac{8 \operatorname{Subst}\left (\int \frac{\cos (x) \sin ^3(x)}{\sqrt{x}} \, dx,x,\tan ^{-1}(a x)\right )}{a^4 c^3}\\ &=-\frac{2 x^3}{3 a c^3 \left (1+a^2 x^2\right )^2 \tan ^{-1}(a x)^{3/2}}-\frac{4 x^2}{a^2 c^3 \left (1+a^2 x^2\right )^2 \sqrt{\tan ^{-1}(a x)}}+\frac{4 x^4}{3 c^3 \left (1+a^2 x^2\right )^2 \sqrt{\tan ^{-1}(a x)}}-\frac{16 \operatorname{Subst}\left (\int \left (\frac{\sin (2 x)}{4 \sqrt{x}}-\frac{\sin (4 x)}{8 \sqrt{x}}\right ) \, dx,x,\tan ^{-1}(a x)\right )}{3 a^4 c^3}-\frac{8 \operatorname{Subst}\left (\int \left (\frac{\sin (2 x)}{4 \sqrt{x}}-\frac{\sin (4 x)}{8 \sqrt{x}}\right ) \, dx,x,\tan ^{-1}(a x)\right )}{a^4 c^3}+\frac{8 \operatorname{Subst}\left (\int \left (\frac{\sin (2 x)}{4 \sqrt{x}}+\frac{\sin (4 x)}{8 \sqrt{x}}\right ) \, dx,x,\tan ^{-1}(a x)\right )}{a^4 c^3}\\ &=-\frac{2 x^3}{3 a c^3 \left (1+a^2 x^2\right )^2 \tan ^{-1}(a x)^{3/2}}-\frac{4 x^2}{a^2 c^3 \left (1+a^2 x^2\right )^2 \sqrt{\tan ^{-1}(a x)}}+\frac{4 x^4}{3 c^3 \left (1+a^2 x^2\right )^2 \sqrt{\tan ^{-1}(a x)}}+\frac{2 \operatorname{Subst}\left (\int \frac{\sin (4 x)}{\sqrt{x}} \, dx,x,\tan ^{-1}(a x)\right )}{3 a^4 c^3}+2 \frac{\operatorname{Subst}\left (\int \frac{\sin (4 x)}{\sqrt{x}} \, dx,x,\tan ^{-1}(a x)\right )}{a^4 c^3}-\frac{4 \operatorname{Subst}\left (\int \frac{\sin (2 x)}{\sqrt{x}} \, dx,x,\tan ^{-1}(a x)\right )}{3 a^4 c^3}\\ &=-\frac{2 x^3}{3 a c^3 \left (1+a^2 x^2\right )^2 \tan ^{-1}(a x)^{3/2}}-\frac{4 x^2}{a^2 c^3 \left (1+a^2 x^2\right )^2 \sqrt{\tan ^{-1}(a x)}}+\frac{4 x^4}{3 c^3 \left (1+a^2 x^2\right )^2 \sqrt{\tan ^{-1}(a x)}}+\frac{4 \operatorname{Subst}\left (\int \sin \left (4 x^2\right ) \, dx,x,\sqrt{\tan ^{-1}(a x)}\right )}{3 a^4 c^3}+2 \frac{2 \operatorname{Subst}\left (\int \sin \left (4 x^2\right ) \, dx,x,\sqrt{\tan ^{-1}(a x)}\right )}{a^4 c^3}-\frac{8 \operatorname{Subst}\left (\int \sin \left (2 x^2\right ) \, dx,x,\sqrt{\tan ^{-1}(a x)}\right )}{3 a^4 c^3}\\ &=-\frac{2 x^3}{3 a c^3 \left (1+a^2 x^2\right )^2 \tan ^{-1}(a x)^{3/2}}-\frac{4 x^2}{a^2 c^3 \left (1+a^2 x^2\right )^2 \sqrt{\tan ^{-1}(a x)}}+\frac{4 x^4}{3 c^3 \left (1+a^2 x^2\right )^2 \sqrt{\tan ^{-1}(a x)}}+\frac{4 \sqrt{2 \pi } S\left (2 \sqrt{\frac{2}{\pi }} \sqrt{\tan ^{-1}(a x)}\right )}{3 a^4 c^3}-\frac{4 \sqrt{\pi } S\left (\frac{2 \sqrt{\tan ^{-1}(a x)}}{\sqrt{\pi }}\right )}{3 a^4 c^3}\\ \end{align*}
Mathematica [C] time = 0.442479, size = 227, normalized size = 1.42 \[ \frac{i \sqrt{2} \left (a^2 x^2+1\right )^2 \left (-i \tan ^{-1}(a x)\right )^{3/2} \text{Gamma}\left (\frac{1}{2},-2 i \tan ^{-1}(a x)\right )+\sqrt{2} \left (a^2 x^2+1\right )^2 \sqrt{i \tan ^{-1}(a x)} \tan ^{-1}(a x) \text{Gamma}\left (\frac{1}{2},2 i \tan ^{-1}(a x)\right )-2 \left (i \left (a^2 x^2+1\right )^2 \left (-i \tan ^{-1}(a x)\right )^{3/2} \text{Gamma}\left (\frac{1}{2},-4 i \tan ^{-1}(a x)\right )+\left (a^2 x^2+1\right )^2 \sqrt{i \tan ^{-1}(a x)} \tan ^{-1}(a x) \text{Gamma}\left (\frac{1}{2},4 i \tan ^{-1}(a x)\right )+a^2 x^2 \left (\left (6-2 a^2 x^2\right ) \tan ^{-1}(a x)+a x\right )\right )}{3 a^4 c^3 \left (a^2 x^2+1\right )^2 \tan ^{-1}(a x)^{3/2}} \]
Warning: Unable to verify antiderivative.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.119, size = 112, normalized size = 0.7 \begin{align*} -{\frac{1}{12\,{c}^{3}{a}^{4}} \left ( -16\,\sqrt{2}\sqrt{\pi }{\it FresnelS} \left ( 2\,{\frac{\sqrt{2}\sqrt{\arctan \left ( ax \right ) }}{\sqrt{\pi }}} \right ) \left ( \arctan \left ( ax \right ) \right ) ^{3/2}+16\,\sqrt{\pi }{\it FresnelS} \left ( 2\,{\frac{\sqrt{\arctan \left ( ax \right ) }}{\sqrt{\pi }}} \right ) \left ( \arctan \left ( ax \right ) \right ) ^{3/2}+8\,\cos \left ( 2\,\arctan \left ( ax \right ) \right ) \arctan \left ( ax \right ) -8\,\cos \left ( 4\,\arctan \left ( ax \right ) \right ) \arctan \left ( ax \right ) +2\,\sin \left ( 2\,\arctan \left ( ax \right ) \right ) -\sin \left ( 4\,\arctan \left ( ax \right ) \right ) \right ) \left ( \arctan \left ( ax \right ) \right ) ^{-{\frac{3}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: RuntimeError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: UnboundLocalError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{3}}{{\left (a^{2} c x^{2} + c\right )}^{3} \arctan \left (a x\right )^{\frac{5}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]