Optimal. Leaf size=71 \[ \frac {\sqrt {\frac {\pi }{2}} S\left (2 \sqrt {\frac {2}{\pi }} \sqrt {\text {ArcTan}(a x)}\right )}{8 a^2 c^3}+\frac {\sqrt {\pi } S\left (\frac {2 \sqrt {\text {ArcTan}(a x)}}{\sqrt {\pi }}\right )}{4 a^2 c^3} \]
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Rubi [A]
time = 0.08, antiderivative size = 71, normalized size of antiderivative = 1.00, number of steps
used = 7, number of rules used = 4, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {5090, 4491,
3386, 3432} \begin {gather*} \frac {\sqrt {\frac {\pi }{2}} S\left (2 \sqrt {\frac {2}{\pi }} \sqrt {\text {ArcTan}(a x)}\right )}{8 a^2 c^3}+\frac {\sqrt {\pi } S\left (\frac {2 \sqrt {\text {ArcTan}(a x)}}{\sqrt {\pi }}\right )}{4 a^2 c^3} \end {gather*}
Antiderivative was successfully verified.
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Rule 3386
Rule 3432
Rule 4491
Rule 5090
Rubi steps
\begin {align*} \int \frac {x}{\left (c+a^2 c x^2\right )^3 \sqrt {\tan ^{-1}(a x)}} \, dx &=\frac {\text {Subst}\left (\int \frac {\cos ^3(x) \sin (x)}{\sqrt {x}} \, dx,x,\tan ^{-1}(a x)\right )}{a^2 c^3}\\ &=\frac {\text {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^2 c^3}\\ &=\frac {\text {Subst}\left (\int \frac {\sin (4 x)}{\sqrt {x}} \, dx,x,\tan ^{-1}(a x)\right )}{8 a^2 c^3}+\frac {\text {Subst}\left (\int \frac {\sin (2 x)}{\sqrt {x}} \, dx,x,\tan ^{-1}(a x)\right )}{4 a^2 c^3}\\ &=\frac {\text {Subst}\left (\int \sin \left (4 x^2\right ) \, dx,x,\sqrt {\tan ^{-1}(a x)}\right )}{4 a^2 c^3}+\frac {\text {Subst}\left (\int \sin \left (2 x^2\right ) \, dx,x,\sqrt {\tan ^{-1}(a x)}\right )}{2 a^2 c^3}\\ &=\frac {\sqrt {\frac {\pi }{2}} S\left (2 \sqrt {\frac {2}{\pi }} \sqrt {\tan ^{-1}(a x)}\right )}{8 a^2 c^3}+\frac {\sqrt {\pi } S\left (\frac {2 \sqrt {\tan ^{-1}(a x)}}{\sqrt {\pi }}\right )}{4 a^2 c^3}\\ \end {align*}
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Mathematica [C] Result contains complex when optimal does not.
time = 0.08, size = 133, normalized size = 1.87 \begin {gather*} \frac {-2 \sqrt {2} \sqrt {-i \text {ArcTan}(a x)} \text {Gamma}\left (\frac {1}{2},-2 i \text {ArcTan}(a x)\right )-2 \sqrt {2} \sqrt {i \text {ArcTan}(a x)} \text {Gamma}\left (\frac {1}{2},2 i \text {ArcTan}(a x)\right )-\sqrt {-i \text {ArcTan}(a x)} \text {Gamma}\left (\frac {1}{2},-4 i \text {ArcTan}(a x)\right )-\sqrt {i \text {ArcTan}(a x)} \text {Gamma}\left (\frac {1}{2},4 i \text {ArcTan}(a x)\right )}{32 a^2 c^3 \sqrt {\text {ArcTan}(a x)}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.19, size = 46, normalized size = 0.65
method | result | size |
default | \(\frac {\sqrt {\pi }\, \left (\sqrt {2}\, \mathrm {S}\left (\frac {2 \sqrt {2}\, \sqrt {\arctan \left (a x \right )}}{\sqrt {\pi }}\right )+4 \,\mathrm {S}\left (\frac {2 \sqrt {\arctan \left (a x \right )}}{\sqrt {\pi }}\right )\right )}{16 c^{3} a^{2}}\) | \(46\) |
Verification of antiderivative is not currently implemented for this CAS.
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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.
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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.
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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \frac {\int \frac {x}{a^{6} x^{6} \sqrt {\operatorname {atan}{\left (a x \right )}} + 3 a^{4} x^{4} \sqrt {\operatorname {atan}{\left (a x \right )}} + 3 a^{2} x^{2} \sqrt {\operatorname {atan}{\left (a x \right )}} + \sqrt {\operatorname {atan}{\left (a x \right )}}}\, dx}{c^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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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.
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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {x}{\sqrt {\mathrm {atan}\left (a\,x\right )}\,{\left (c\,a^2\,x^2+c\right )}^3} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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