Optimal. Leaf size=219 \[ -\frac {45 \sqrt {\text {ArcTan}(a x)}}{256 a c^3}+\frac {3 \sqrt {\text {ArcTan}(a x)}}{32 a c^3 \left (1+a^2 x^2\right )^2}+\frac {9 \sqrt {\text {ArcTan}(a x)}}{32 a c^3 \left (1+a^2 x^2\right )}+\frac {x \text {ArcTan}(a x)^{3/2}}{4 c^3 \left (1+a^2 x^2\right )^2}+\frac {3 x \text {ArcTan}(a x)^{3/2}}{8 c^3 \left (1+a^2 x^2\right )}+\frac {3 \text {ArcTan}(a x)^{5/2}}{20 a c^3}-\frac {3 \sqrt {\frac {\pi }{2}} \text {FresnelC}\left (2 \sqrt {\frac {2}{\pi }} \sqrt {\text {ArcTan}(a x)}\right )}{512 a c^3}-\frac {3 \sqrt {\pi } \text {FresnelC}\left (\frac {2 \sqrt {\text {ArcTan}(a x)}}{\sqrt {\pi }}\right )}{32 a c^3} \]
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Rubi [A]
time = 0.22, antiderivative size = 219, normalized size of antiderivative = 1.00, number of steps
used = 15, number of rules used = 7, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {5020, 5012,
5050, 5024, 3393, 3385, 3433} \begin {gather*} \frac {3 x \text {ArcTan}(a x)^{3/2}}{8 c^3 \left (a^2 x^2+1\right )}+\frac {x \text {ArcTan}(a x)^{3/2}}{4 c^3 \left (a^2 x^2+1\right )^2}+\frac {9 \sqrt {\text {ArcTan}(a x)}}{32 a c^3 \left (a^2 x^2+1\right )}+\frac {3 \sqrt {\text {ArcTan}(a x)}}{32 a c^3 \left (a^2 x^2+1\right )^2}-\frac {3 \sqrt {\frac {\pi }{2}} \text {FresnelC}\left (2 \sqrt {\frac {2}{\pi }} \sqrt {\text {ArcTan}(a x)}\right )}{512 a c^3}-\frac {3 \sqrt {\pi } \text {FresnelC}\left (\frac {2 \sqrt {\text {ArcTan}(a x)}}{\sqrt {\pi }}\right )}{32 a c^3}+\frac {3 \text {ArcTan}(a x)^{5/2}}{20 a c^3}-\frac {45 \sqrt {\text {ArcTan}(a x)}}{256 a c^3} \end {gather*}
Antiderivative was successfully verified.
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Rule 3385
Rule 3393
Rule 3433
Rule 5012
Rule 5020
Rule 5024
Rule 5050
Rubi steps
\begin {align*} \int \frac {\tan ^{-1}(a x)^{3/2}}{\left (c+a^2 c x^2\right )^3} \, dx &=\frac {3 \sqrt {\tan ^{-1}(a x)}}{32 a c^3 \left (1+a^2 x^2\right )^2}+\frac {x \tan ^{-1}(a x)^{3/2}}{4 c^3 \left (1+a^2 x^2\right )^2}-\frac {3}{64} \int \frac {1}{\left (c+a^2 c x^2\right )^3 \sqrt {\tan ^{-1}(a x)}} \, dx+\frac {3 \int \frac {\tan ^{-1}(a x)^{3/2}}{\left (c+a^2 c x^2\right )^2} \, dx}{4 c}\\ &=\frac {3 \sqrt {\tan ^{-1}(a x)}}{32 a c^3 \left (1+a^2 x^2\right )^2}+\frac {x \tan ^{-1}(a x)^{3/2}}{4 c^3 \left (1+a^2 x^2\right )^2}+\frac {3 x \tan ^{-1}(a x)^{3/2}}{8 c^3 \left (1+a^2 x^2\right )}+\frac {3 \tan ^{-1}(a x)^{5/2}}{20 a c^3}-\frac {3 \text {Subst}\left (\int \frac {\cos ^4(x)}{\sqrt {x}} \, dx,x,\tan ^{-1}(a x)\right )}{64 a c^3}-\frac {(9 a) \int \frac {x \sqrt {\tan ^{-1}(a x)}}{\left (c+a^2 c x^2\right )^2} \, dx}{16 c}\\ &=\frac {3 \sqrt {\tan ^{-1}(a x)}}{32 a c^3 \left (1+a^2 x^2\right )^2}+\frac {9 \sqrt {\tan ^{-1}(a x)}}{32 a c^3 \left (1+a^2 x^2\right )}+\frac {x \tan ^{-1}(a x)^{3/2}}{4 c^3 \left (1+a^2 x^2\right )^2}+\frac {3 x \tan ^{-1}(a x)^{3/2}}{8 c^3 \left (1+a^2 x^2\right )}+\frac {3 \tan ^{-1}(a x)^{5/2}}{20 a c^3}-\frac {3 \text {Subst}\left (\int \left (\frac {3}{8 \sqrt {x}}+\frac {\cos (2 x)}{2 \sqrt {x}}+\frac {\cos (4 x)}{8 \sqrt {x}}\right ) \, dx,x,\tan ^{-1}(a x)\right )}{64 a c^3}-\frac {9 \int \frac {1}{\left (c+a^2 c x^2\right )^2 \sqrt {\tan ^{-1}(a x)}} \, dx}{64 c}\\ &=-\frac {9 \sqrt {\tan ^{-1}(a x)}}{256 a c^3}+\frac {3 \sqrt {\tan ^{-1}(a x)}}{32 a c^3 \left (1+a^2 x^2\right )^2}+\frac {9 \sqrt {\tan ^{-1}(a x)}}{32 a c^3 \left (1+a^2 x^2\right )}+\frac {x \tan ^{-1}(a x)^{3/2}}{4 c^3 \left (1+a^2 x^2\right )^2}+\frac {3 x \tan ^{-1}(a x)^{3/2}}{8 c^3 \left (1+a^2 x^2\right )}+\frac {3 \tan ^{-1}(a x)^{5/2}}{20 a c^3}-\frac {3 \text {Subst}\left (\int \frac {\cos (4 x)}{\sqrt {x}} \, dx,x,\tan ^{-1}(a x)\right )}{512 a c^3}-\frac {3 \text {Subst}\left (\int \frac {\cos (2 x)}{\sqrt {x}} \, dx,x,\tan ^{-1}(a x)\right )}{128 a c^3}-\frac {9 \text {Subst}\left (\int \frac {\cos ^2(x)}{\sqrt {x}} \, dx,x,\tan ^{-1}(a x)\right )}{64 a c^3}\\ &=-\frac {9 \sqrt {\tan ^{-1}(a x)}}{256 a c^3}+\frac {3 \sqrt {\tan ^{-1}(a x)}}{32 a c^3 \left (1+a^2 x^2\right )^2}+\frac {9 \sqrt {\tan ^{-1}(a x)}}{32 a c^3 \left (1+a^2 x^2\right )}+\frac {x \tan ^{-1}(a x)^{3/2}}{4 c^3 \left (1+a^2 x^2\right )^2}+\frac {3 x \tan ^{-1}(a x)^{3/2}}{8 c^3 \left (1+a^2 x^2\right )}+\frac {3 \tan ^{-1}(a x)^{5/2}}{20 a c^3}-\frac {3 \text {Subst}\left (\int \cos \left (4 x^2\right ) \, dx,x,\sqrt {\tan ^{-1}(a x)}\right )}{256 a c^3}-\frac {3 \text {Subst}\left (\int \cos \left (2 x^2\right ) \, dx,x,\sqrt {\tan ^{-1}(a x)}\right )}{64 a c^3}-\frac {9 \text {Subst}\left (\int \left (\frac {1}{2 \sqrt {x}}+\frac {\cos (2 x)}{2 \sqrt {x}}\right ) \, dx,x,\tan ^{-1}(a x)\right )}{64 a c^3}\\ &=-\frac {45 \sqrt {\tan ^{-1}(a x)}}{256 a c^3}+\frac {3 \sqrt {\tan ^{-1}(a x)}}{32 a c^3 \left (1+a^2 x^2\right )^2}+\frac {9 \sqrt {\tan ^{-1}(a x)}}{32 a c^3 \left (1+a^2 x^2\right )}+\frac {x \tan ^{-1}(a x)^{3/2}}{4 c^3 \left (1+a^2 x^2\right )^2}+\frac {3 x \tan ^{-1}(a x)^{3/2}}{8 c^3 \left (1+a^2 x^2\right )}+\frac {3 \tan ^{-1}(a x)^{5/2}}{20 a c^3}-\frac {3 \sqrt {\frac {\pi }{2}} C\left (2 \sqrt {\frac {2}{\pi }} \sqrt {\tan ^{-1}(a x)}\right )}{512 a c^3}-\frac {3 \sqrt {\pi } C\left (\frac {2 \sqrt {\tan ^{-1}(a x)}}{\sqrt {\pi }}\right )}{128 a c^3}-\frac {9 \text {Subst}\left (\int \frac {\cos (2 x)}{\sqrt {x}} \, dx,x,\tan ^{-1}(a x)\right )}{128 a c^3}\\ &=-\frac {45 \sqrt {\tan ^{-1}(a x)}}{256 a c^3}+\frac {3 \sqrt {\tan ^{-1}(a x)}}{32 a c^3 \left (1+a^2 x^2\right )^2}+\frac {9 \sqrt {\tan ^{-1}(a x)}}{32 a c^3 \left (1+a^2 x^2\right )}+\frac {x \tan ^{-1}(a x)^{3/2}}{4 c^3 \left (1+a^2 x^2\right )^2}+\frac {3 x \tan ^{-1}(a x)^{3/2}}{8 c^3 \left (1+a^2 x^2\right )}+\frac {3 \tan ^{-1}(a x)^{5/2}}{20 a c^3}-\frac {3 \sqrt {\frac {\pi }{2}} C\left (2 \sqrt {\frac {2}{\pi }} \sqrt {\tan ^{-1}(a x)}\right )}{512 a c^3}-\frac {3 \sqrt {\pi } C\left (\frac {2 \sqrt {\tan ^{-1}(a x)}}{\sqrt {\pi }}\right )}{128 a c^3}-\frac {9 \text {Subst}\left (\int \cos \left (2 x^2\right ) \, dx,x,\sqrt {\tan ^{-1}(a x)}\right )}{64 a c^3}\\ &=-\frac {45 \sqrt {\tan ^{-1}(a x)}}{256 a c^3}+\frac {3 \sqrt {\tan ^{-1}(a x)}}{32 a c^3 \left (1+a^2 x^2\right )^2}+\frac {9 \sqrt {\tan ^{-1}(a x)}}{32 a c^3 \left (1+a^2 x^2\right )}+\frac {x \tan ^{-1}(a x)^{3/2}}{4 c^3 \left (1+a^2 x^2\right )^2}+\frac {3 x \tan ^{-1}(a x)^{3/2}}{8 c^3 \left (1+a^2 x^2\right )}+\frac {3 \tan ^{-1}(a x)^{5/2}}{20 a c^3}-\frac {3 \sqrt {\frac {\pi }{2}} C\left (2 \sqrt {\frac {2}{\pi }} \sqrt {\tan ^{-1}(a x)}\right )}{512 a c^3}-\frac {3 \sqrt {\pi } C\left (\frac {2 \sqrt {\tan ^{-1}(a x)}}{\sqrt {\pi }}\right )}{32 a c^3}\\ \end {align*}
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Mathematica [A]
time = 0.15, size = 142, normalized size = 0.65 \begin {gather*} \frac {\frac {4 \sqrt {\text {ArcTan}(a x)} \left (-15 \left (-17+6 a^2 x^2+15 a^4 x^4\right )+160 a x \left (5+3 a^2 x^2\right ) \text {ArcTan}(a x)+192 \left (1+a^2 x^2\right )^2 \text {ArcTan}(a x)^2\right )}{\left (1+a^2 x^2\right )^2}-15 \sqrt {2 \pi } \text {FresnelC}\left (2 \sqrt {\frac {2}{\pi }} \sqrt {\text {ArcTan}(a x)}\right )-480 \sqrt {\pi } \text {FresnelC}\left (\frac {2 \sqrt {\text {ArcTan}(a x)}}{\sqrt {\pi }}\right )}{5120 a c^3} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.32, size = 132, normalized size = 0.60
method | result | size |
default | \(\frac {768 \arctan \left (a x \right )^{3}+1280 \arctan \left (a x \right )^{2} \sin \left (2 \arctan \left (a x \right )\right )+160 \arctan \left (a x \right )^{2} \sin \left (4 \arctan \left (a x \right )\right )-15 \sqrt {2}\, \sqrt {\arctan \left (a x \right )}\, \sqrt {\pi }\, \FresnelC \left (\frac {2 \sqrt {2}\, \sqrt {\arctan \left (a x \right )}}{\sqrt {\pi }}\right )+960 \cos \left (2 \arctan \left (a x \right )\right ) \arctan \left (a x \right )-480 \sqrt {\arctan \left (a x \right )}\, \sqrt {\pi }\, \FresnelC \left (\frac {2 \sqrt {\arctan \left (a x \right )}}{\sqrt {\pi }}\right )+60 \cos \left (4 \arctan \left (a x \right )\right ) \arctan \left (a x \right )}{5120 c^{3} a \sqrt {\arctan \left (a x \right )}}\) | \(132\) |
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 {\operatorname {atan}^{\frac {3}{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.
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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 = -0.00 \begin {gather*} \int \frac {{\mathrm {atan}\left (a\,x\right )}^{3/2}}{{\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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