Optimal. Leaf size=133 \[ \frac {\text {ArcTan}(a x)^{7/2}}{28 a^3 c^3}-\frac {5 \text {ArcTan}(a x)^{3/2} \cos (4 \text {ArcTan}(a x))}{256 a^3 c^3}-\frac {15 \sqrt {\frac {\pi }{2}} S\left (2 \sqrt {\frac {2}{\pi }} \sqrt {\text {ArcTan}(a x)}\right )}{4096 a^3 c^3}+\frac {15 \sqrt {\text {ArcTan}(a x)} \sin (4 \text {ArcTan}(a x))}{2048 a^3 c^3}-\frac {\text {ArcTan}(a x)^{5/2} \sin (4 \text {ArcTan}(a x))}{32 a^3 c^3} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.14, antiderivative size = 133, normalized size of antiderivative = 1.00, number of steps
used = 8, number of rules used = 5, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.208, Rules used = {5090, 4491,
3377, 3386, 3432} \begin {gather*} -\frac {15 \sqrt {\frac {\pi }{2}} S\left (2 \sqrt {\frac {2}{\pi }} \sqrt {\text {ArcTan}(a x)}\right )}{4096 a^3 c^3}+\frac {\text {ArcTan}(a x)^{7/2}}{28 a^3 c^3}-\frac {\text {ArcTan}(a x)^{5/2} \sin (4 \text {ArcTan}(a x))}{32 a^3 c^3}+\frac {15 \sqrt {\text {ArcTan}(a x)} \sin (4 \text {ArcTan}(a x))}{2048 a^3 c^3}-\frac {5 \text {ArcTan}(a x)^{3/2} \cos (4 \text {ArcTan}(a x))}{256 a^3 c^3} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 3377
Rule 3386
Rule 3432
Rule 4491
Rule 5090
Rubi steps
\begin {align*} \int \frac {x^2 \tan ^{-1}(a x)^{5/2}}{\left (c+a^2 c x^2\right )^3} \, dx &=\frac {\text {Subst}\left (\int x^{5/2} \cos ^2(x) \sin ^2(x) \, dx,x,\tan ^{-1}(a x)\right )}{a^3 c^3}\\ &=\frac {\text {Subst}\left (\int \left (\frac {x^{5/2}}{8}-\frac {1}{8} x^{5/2} \cos (4 x)\right ) \, dx,x,\tan ^{-1}(a x)\right )}{a^3 c^3}\\ &=\frac {\tan ^{-1}(a x)^{7/2}}{28 a^3 c^3}-\frac {\text {Subst}\left (\int x^{5/2} \cos (4 x) \, dx,x,\tan ^{-1}(a x)\right )}{8 a^3 c^3}\\ &=\frac {\tan ^{-1}(a x)^{7/2}}{28 a^3 c^3}-\frac {\tan ^{-1}(a x)^{5/2} \sin \left (4 \tan ^{-1}(a x)\right )}{32 a^3 c^3}+\frac {5 \text {Subst}\left (\int x^{3/2} \sin (4 x) \, dx,x,\tan ^{-1}(a x)\right )}{64 a^3 c^3}\\ &=\frac {\tan ^{-1}(a x)^{7/2}}{28 a^3 c^3}-\frac {5 \tan ^{-1}(a x)^{3/2} \cos \left (4 \tan ^{-1}(a x)\right )}{256 a^3 c^3}-\frac {\tan ^{-1}(a x)^{5/2} \sin \left (4 \tan ^{-1}(a x)\right )}{32 a^3 c^3}+\frac {15 \text {Subst}\left (\int \sqrt {x} \cos (4 x) \, dx,x,\tan ^{-1}(a x)\right )}{512 a^3 c^3}\\ &=\frac {\tan ^{-1}(a x)^{7/2}}{28 a^3 c^3}-\frac {5 \tan ^{-1}(a x)^{3/2} \cos \left (4 \tan ^{-1}(a x)\right )}{256 a^3 c^3}+\frac {15 \sqrt {\tan ^{-1}(a x)} \sin \left (4 \tan ^{-1}(a x)\right )}{2048 a^3 c^3}-\frac {\tan ^{-1}(a x)^{5/2} \sin \left (4 \tan ^{-1}(a x)\right )}{32 a^3 c^3}-\frac {15 \text {Subst}\left (\int \frac {\sin (4 x)}{\sqrt {x}} \, dx,x,\tan ^{-1}(a x)\right )}{4096 a^3 c^3}\\ &=\frac {\tan ^{-1}(a x)^{7/2}}{28 a^3 c^3}-\frac {5 \tan ^{-1}(a x)^{3/2} \cos \left (4 \tan ^{-1}(a x)\right )}{256 a^3 c^3}+\frac {15 \sqrt {\tan ^{-1}(a x)} \sin \left (4 \tan ^{-1}(a x)\right )}{2048 a^3 c^3}-\frac {\tan ^{-1}(a x)^{5/2} \sin \left (4 \tan ^{-1}(a x)\right )}{32 a^3 c^3}-\frac {15 \text {Subst}\left (\int \sin \left (4 x^2\right ) \, dx,x,\sqrt {\tan ^{-1}(a x)}\right )}{2048 a^3 c^3}\\ &=\frac {\tan ^{-1}(a x)^{7/2}}{28 a^3 c^3}-\frac {5 \tan ^{-1}(a x)^{3/2} \cos \left (4 \tan ^{-1}(a x)\right )}{256 a^3 c^3}-\frac {15 \sqrt {\frac {\pi }{2}} S\left (2 \sqrt {\frac {2}{\pi }} \sqrt {\tan ^{-1}(a x)}\right )}{4096 a^3 c^3}+\frac {15 \sqrt {\tan ^{-1}(a x)} \sin \left (4 \tan ^{-1}(a x)\right )}{2048 a^3 c^3}-\frac {\tan ^{-1}(a x)^{5/2} \sin \left (4 \tan ^{-1}(a x)\right )}{32 a^3 c^3}\\ \end {align*}
________________________________________________________________________________________
Mathematica [C] Result contains complex when optimal does not.
time = 0.18, size = 185, normalized size = 1.39 \begin {gather*} \frac {32 \text {ArcTan}(a x) \left (-105 a x \left (-1+a^2 x^2\right )-70 \left (1-6 a^2 x^2+a^4 x^4\right ) \text {ArcTan}(a x)+448 a x \left (-1+a^2 x^2\right ) \text {ArcTan}(a x)^2+128 \left (1+a^2 x^2\right )^2 \text {ArcTan}(a x)^3\right )+105 \left (1+a^2 x^2\right )^2 \sqrt {-i \text {ArcTan}(a x)} \text {Gamma}\left (\frac {1}{2},-4 i \text {ArcTan}(a x)\right )+105 \left (1+a^2 x^2\right )^2 \sqrt {i \text {ArcTan}(a x)} \text {Gamma}\left (\frac {1}{2},4 i \text {ArcTan}(a x)\right )}{114688 a^3 c^3 \left (1+a^2 x^2\right )^2 \sqrt {\text {ArcTan}(a x)}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A]
time = 0.34, size = 96, normalized size = 0.72
method | result | size |
default | \(\frac {2048 \arctan \left (a x \right )^{4}-1792 \arctan \left (a x \right )^{3} \sin \left (4 \arctan \left (a x \right )\right )-105 \,\mathrm {S}\left (\frac {2 \sqrt {2}\, \sqrt {\arctan \left (a x \right )}}{\sqrt {\pi }}\right ) \sqrt {2}\, \sqrt {\arctan \left (a x \right )}\, \sqrt {\pi }-1120 \arctan \left (a x \right )^{2} \cos \left (4 \arctan \left (a x \right )\right )+420 \sin \left (4 \arctan \left (a x \right )\right ) \arctan \left (a x \right )}{57344 c^{3} a^{3} \sqrt {\arctan \left (a x \right )}}\) | \(96\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: RuntimeError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \frac {\int \frac {x^{2} \operatorname {atan}^{\frac {5}{2}}{\left (a x \right )}}{a^{6} x^{6} + 3 a^{4} x^{4} + 3 a^{2} x^{2} + 1}\, dx}{c^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {x^2\,{\mathrm {atan}\left (a\,x\right )}^{5/2}}{{\left (c\,a^2\,x^2+c\right )}^3} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________