Optimal. Leaf size=256 \[ -\frac {135 \sqrt {\text {ArcTan}(a x)}}{2048 a^4 c^3}-\frac {15 x^4 \sqrt {\text {ArcTan}(a x)}}{256 c^3 \left (1+a^2 x^2\right )^2}+\frac {45 \sqrt {\text {ArcTan}(a x)}}{256 a^4 c^3 \left (1+a^2 x^2\right )}+\frac {5 x^3 \text {ArcTan}(a x)^{3/2}}{32 a c^3 \left (1+a^2 x^2\right )^2}+\frac {15 x \text {ArcTan}(a x)^{3/2}}{64 a^3 c^3 \left (1+a^2 x^2\right )}-\frac {3 \text {ArcTan}(a x)^{5/2}}{32 a^4 c^3}+\frac {x^4 \text {ArcTan}(a x)^{5/2}}{4 c^3 \left (1+a^2 x^2\right )^2}+\frac {15 \sqrt {\frac {\pi }{2}} \text {FresnelC}\left (2 \sqrt {\frac {2}{\pi }} \sqrt {\text {ArcTan}(a x)}\right )}{4096 a^4 c^3}-\frac {15 \sqrt {\pi } \text {FresnelC}\left (\frac {2 \sqrt {\text {ArcTan}(a x)}}{\sqrt {\pi }}\right )}{256 a^4 c^3} \]
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Rubi [A]
time = 0.36, antiderivative size = 256, normalized size of antiderivative = 1.00, number of steps
used = 16, number of rules used = 9, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.375, Rules used = {5064, 5060,
5056, 5050, 5024, 3393, 3385, 3433, 5090} \begin {gather*} \frac {15 \sqrt {\frac {\pi }{2}} \text {FresnelC}\left (2 \sqrt {\frac {2}{\pi }} \sqrt {\text {ArcTan}(a x)}\right )}{4096 a^4 c^3}-\frac {15 \sqrt {\pi } \text {FresnelC}\left (\frac {2 \sqrt {\text {ArcTan}(a x)}}{\sqrt {\pi }}\right )}{256 a^4 c^3}-\frac {3 \text {ArcTan}(a x)^{5/2}}{32 a^4 c^3}-\frac {135 \sqrt {\text {ArcTan}(a x)}}{2048 a^4 c^3}+\frac {x^4 \text {ArcTan}(a x)^{5/2}}{4 c^3 \left (a^2 x^2+1\right )^2}-\frac {15 x^4 \sqrt {\text {ArcTan}(a x)}}{256 c^3 \left (a^2 x^2+1\right )^2}+\frac {5 x^3 \text {ArcTan}(a x)^{3/2}}{32 a c^3 \left (a^2 x^2+1\right )^2}+\frac {45 \sqrt {\text {ArcTan}(a x)}}{256 a^4 c^3 \left (a^2 x^2+1\right )}+\frac {15 x \text {ArcTan}(a x)^{3/2}}{64 a^3 c^3 \left (a^2 x^2+1\right )} \end {gather*}
Antiderivative was successfully verified.
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Rule 3385
Rule 3393
Rule 3433
Rule 5024
Rule 5050
Rule 5056
Rule 5060
Rule 5064
Rule 5090
Rubi steps
\begin {align*} \int \frac {x^3 \tan ^{-1}(a x)^{5/2}}{\left (c+a^2 c x^2\right )^3} \, dx &=\frac {x^4 \tan ^{-1}(a x)^{5/2}}{4 c^3 \left (1+a^2 x^2\right )^2}-\frac {1}{8} (5 a) \int \frac {x^4 \tan ^{-1}(a x)^{3/2}}{\left (c+a^2 c x^2\right )^3} \, dx\\ &=-\frac {15 x^4 \sqrt {\tan ^{-1}(a x)}}{256 c^3 \left (1+a^2 x^2\right )^2}+\frac {5 x^3 \tan ^{-1}(a x)^{3/2}}{32 a c^3 \left (1+a^2 x^2\right )^2}+\frac {x^4 \tan ^{-1}(a x)^{5/2}}{4 c^3 \left (1+a^2 x^2\right )^2}+\frac {1}{512} (15 a) \int \frac {x^4}{\left (c+a^2 c x^2\right )^3 \sqrt {\tan ^{-1}(a x)}} \, dx-\frac {15 \int \frac {x^2 \tan ^{-1}(a x)^{3/2}}{\left (c+a^2 c x^2\right )^2} \, dx}{32 a c}\\ &=-\frac {15 x^4 \sqrt {\tan ^{-1}(a x)}}{256 c^3 \left (1+a^2 x^2\right )^2}+\frac {5 x^3 \tan ^{-1}(a x)^{3/2}}{32 a c^3 \left (1+a^2 x^2\right )^2}+\frac {15 x \tan ^{-1}(a x)^{3/2}}{64 a^3 c^3 \left (1+a^2 x^2\right )}-\frac {3 \tan ^{-1}(a x)^{5/2}}{32 a^4 c^3}+\frac {x^4 \tan ^{-1}(a x)^{5/2}}{4 c^3 \left (1+a^2 x^2\right )^2}+\frac {15 \text {Subst}\left (\int \frac {\sin ^4(x)}{\sqrt {x}} \, dx,x,\tan ^{-1}(a x)\right )}{512 a^4 c^3}-\frac {45 \int \frac {x \sqrt {\tan ^{-1}(a x)}}{\left (c+a^2 c x^2\right )^2} \, dx}{128 a^2 c}\\ &=-\frac {15 x^4 \sqrt {\tan ^{-1}(a x)}}{256 c^3 \left (1+a^2 x^2\right )^2}+\frac {45 \sqrt {\tan ^{-1}(a x)}}{256 a^4 c^3 \left (1+a^2 x^2\right )}+\frac {5 x^3 \tan ^{-1}(a x)^{3/2}}{32 a c^3 \left (1+a^2 x^2\right )^2}+\frac {15 x \tan ^{-1}(a x)^{3/2}}{64 a^3 c^3 \left (1+a^2 x^2\right )}-\frac {3 \tan ^{-1}(a x)^{5/2}}{32 a^4 c^3}+\frac {x^4 \tan ^{-1}(a x)^{5/2}}{4 c^3 \left (1+a^2 x^2\right )^2}+\frac {15 \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 )}{512 a^4 c^3}-\frac {45 \int \frac {1}{\left (c+a^2 c x^2\right )^2 \sqrt {\tan ^{-1}(a x)}} \, dx}{512 a^3 c}\\ &=\frac {45 \sqrt {\tan ^{-1}(a x)}}{2048 a^4 c^3}-\frac {15 x^4 \sqrt {\tan ^{-1}(a x)}}{256 c^3 \left (1+a^2 x^2\right )^2}+\frac {45 \sqrt {\tan ^{-1}(a x)}}{256 a^4 c^3 \left (1+a^2 x^2\right )}+\frac {5 x^3 \tan ^{-1}(a x)^{3/2}}{32 a c^3 \left (1+a^2 x^2\right )^2}+\frac {15 x \tan ^{-1}(a x)^{3/2}}{64 a^3 c^3 \left (1+a^2 x^2\right )}-\frac {3 \tan ^{-1}(a x)^{5/2}}{32 a^4 c^3}+\frac {x^4 \tan ^{-1}(a x)^{5/2}}{4 c^3 \left (1+a^2 x^2\right )^2}+\frac {15 \text {Subst}\left (\int \frac {\cos (4 x)}{\sqrt {x}} \, dx,x,\tan ^{-1}(a x)\right )}{4096 a^4 c^3}-\frac {15 \text {Subst}\left (\int \frac {\cos (2 x)}{\sqrt {x}} \, dx,x,\tan ^{-1}(a x)\right )}{1024 a^4 c^3}-\frac {45 \text {Subst}\left (\int \frac {\cos ^2(x)}{\sqrt {x}} \, dx,x,\tan ^{-1}(a x)\right )}{512 a^4 c^3}\\ &=\frac {45 \sqrt {\tan ^{-1}(a x)}}{2048 a^4 c^3}-\frac {15 x^4 \sqrt {\tan ^{-1}(a x)}}{256 c^3 \left (1+a^2 x^2\right )^2}+\frac {45 \sqrt {\tan ^{-1}(a x)}}{256 a^4 c^3 \left (1+a^2 x^2\right )}+\frac {5 x^3 \tan ^{-1}(a x)^{3/2}}{32 a c^3 \left (1+a^2 x^2\right )^2}+\frac {15 x \tan ^{-1}(a x)^{3/2}}{64 a^3 c^3 \left (1+a^2 x^2\right )}-\frac {3 \tan ^{-1}(a x)^{5/2}}{32 a^4 c^3}+\frac {x^4 \tan ^{-1}(a x)^{5/2}}{4 c^3 \left (1+a^2 x^2\right )^2}+\frac {15 \text {Subst}\left (\int \cos \left (4 x^2\right ) \, dx,x,\sqrt {\tan ^{-1}(a x)}\right )}{2048 a^4 c^3}-\frac {15 \text {Subst}\left (\int \cos \left (2 x^2\right ) \, dx,x,\sqrt {\tan ^{-1}(a x)}\right )}{512 a^4 c^3}-\frac {45 \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 )}{512 a^4 c^3}\\ &=-\frac {135 \sqrt {\tan ^{-1}(a x)}}{2048 a^4 c^3}-\frac {15 x^4 \sqrt {\tan ^{-1}(a x)}}{256 c^3 \left (1+a^2 x^2\right )^2}+\frac {45 \sqrt {\tan ^{-1}(a x)}}{256 a^4 c^3 \left (1+a^2 x^2\right )}+\frac {5 x^3 \tan ^{-1}(a x)^{3/2}}{32 a c^3 \left (1+a^2 x^2\right )^2}+\frac {15 x \tan ^{-1}(a x)^{3/2}}{64 a^3 c^3 \left (1+a^2 x^2\right )}-\frac {3 \tan ^{-1}(a x)^{5/2}}{32 a^4 c^3}+\frac {x^4 \tan ^{-1}(a x)^{5/2}}{4 c^3 \left (1+a^2 x^2\right )^2}+\frac {15 \sqrt {\frac {\pi }{2}} C\left (2 \sqrt {\frac {2}{\pi }} \sqrt {\tan ^{-1}(a x)}\right )}{4096 a^4 c^3}-\frac {15 \sqrt {\pi } C\left (\frac {2 \sqrt {\tan ^{-1}(a x)}}{\sqrt {\pi }}\right )}{1024 a^4 c^3}-\frac {45 \text {Subst}\left (\int \frac {\cos (2 x)}{\sqrt {x}} \, dx,x,\tan ^{-1}(a x)\right )}{1024 a^4 c^3}\\ &=-\frac {135 \sqrt {\tan ^{-1}(a x)}}{2048 a^4 c^3}-\frac {15 x^4 \sqrt {\tan ^{-1}(a x)}}{256 c^3 \left (1+a^2 x^2\right )^2}+\frac {45 \sqrt {\tan ^{-1}(a x)}}{256 a^4 c^3 \left (1+a^2 x^2\right )}+\frac {5 x^3 \tan ^{-1}(a x)^{3/2}}{32 a c^3 \left (1+a^2 x^2\right )^2}+\frac {15 x \tan ^{-1}(a x)^{3/2}}{64 a^3 c^3 \left (1+a^2 x^2\right )}-\frac {3 \tan ^{-1}(a x)^{5/2}}{32 a^4 c^3}+\frac {x^4 \tan ^{-1}(a x)^{5/2}}{4 c^3 \left (1+a^2 x^2\right )^2}+\frac {15 \sqrt {\frac {\pi }{2}} C\left (2 \sqrt {\frac {2}{\pi }} \sqrt {\tan ^{-1}(a x)}\right )}{4096 a^4 c^3}-\frac {15 \sqrt {\pi } C\left (\frac {2 \sqrt {\tan ^{-1}(a x)}}{\sqrt {\pi }}\right )}{1024 a^4 c^3}-\frac {45 \text {Subst}\left (\int \cos \left (2 x^2\right ) \, dx,x,\sqrt {\tan ^{-1}(a x)}\right )}{512 a^4 c^3}\\ &=-\frac {135 \sqrt {\tan ^{-1}(a x)}}{2048 a^4 c^3}-\frac {15 x^4 \sqrt {\tan ^{-1}(a x)}}{256 c^3 \left (1+a^2 x^2\right )^2}+\frac {45 \sqrt {\tan ^{-1}(a x)}}{256 a^4 c^3 \left (1+a^2 x^2\right )}+\frac {5 x^3 \tan ^{-1}(a x)^{3/2}}{32 a c^3 \left (1+a^2 x^2\right )^2}+\frac {15 x \tan ^{-1}(a x)^{3/2}}{64 a^3 c^3 \left (1+a^2 x^2\right )}-\frac {3 \tan ^{-1}(a x)^{5/2}}{32 a^4 c^3}+\frac {x^4 \tan ^{-1}(a x)^{5/2}}{4 c^3 \left (1+a^2 x^2\right )^2}+\frac {15 \sqrt {\frac {\pi }{2}} C\left (2 \sqrt {\frac {2}{\pi }} \sqrt {\tan ^{-1}(a x)}\right )}{4096 a^4 c^3}-\frac {15 \sqrt {\pi } C\left (\frac {2 \sqrt {\tan ^{-1}(a x)}}{\sqrt {\pi }}\right )}{256 a^4 c^3}\\ \end {align*}
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Mathematica [C] Result contains complex when optimal does not.
time = 0.49, size = 359, normalized size = 1.40 \begin {gather*} \frac {510 \sqrt {2 \pi } \text {FresnelC}\left (2 \sqrt {\frac {2}{\pi }} \sqrt {\text {ArcTan}(a x)}\right )+\frac {14400 \text {ArcTan}(a x)+5760 a^2 x^2 \text {ArcTan}(a x)-16320 a^4 x^4 \text {ArcTan}(a x)+30720 a x \text {ArcTan}(a x)^2+51200 a^3 x^3 \text {ArcTan}(a x)^2-12288 \text {ArcTan}(a x)^3-24576 a^2 x^2 \text {ArcTan}(a x)^3+20480 a^4 x^4 \text {ArcTan}(a x)^3-4080 \sqrt {\pi } \left (1+a^2 x^2\right )^2 \sqrt {\text {ArcTan}(a x)} \text {FresnelC}\left (\frac {2 \sqrt {\text {ArcTan}(a x)}}{\sqrt {\pi }}\right )+900 i \sqrt {2} \left (1+a^2 x^2\right )^2 \sqrt {-i \text {ArcTan}(a x)} \text {Gamma}\left (\frac {1}{2},-2 i \text {ArcTan}(a x)\right )-900 i \sqrt {2} \left (1+a^2 x^2\right )^2 \sqrt {i \text {ArcTan}(a x)} \text {Gamma}\left (\frac {1}{2},2 i \text {ArcTan}(a x)\right )+135 i \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 )-135 i \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 )}{\left (1+a^2 x^2\right )^2 \sqrt {\text {ArcTan}(a x)}}}{131072 a^4 c^3} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.35, size = 157, normalized size = 0.61
method | result | size |
default | \(\frac {-1024 \arctan \left (a x \right )^{\frac {5}{2}} \cos \left (2 \arctan \left (a x \right )\right ) \sqrt {\pi }+256 \arctan \left (a x \right )^{\frac {5}{2}} \sqrt {\pi }\, \cos \left (4 \arctan \left (a x \right )\right )+1280 \arctan \left (a x \right )^{\frac {3}{2}} \sin \left (2 \arctan \left (a x \right )\right ) \sqrt {\pi }-160 \arctan \left (a x \right )^{\frac {3}{2}} \sqrt {\pi }\, \sin \left (4 \arctan \left (a x \right )\right )+15 \pi \sqrt {2}\, \FresnelC \left (\frac {2 \sqrt {2}\, \sqrt {\arctan \left (a x \right )}}{\sqrt {\pi }}\right )+960 \sqrt {\arctan \left (a x \right )}\, \sqrt {\pi }\, \cos \left (2 \arctan \left (a x \right )\right )-60 \sqrt {\arctan \left (a x \right )}\, \sqrt {\pi }\, \cos \left (4 \arctan \left (a x \right )\right )-480 \pi \FresnelC \left (\frac {2 \sqrt {\arctan \left (a x \right )}}{\sqrt {\pi }}\right )}{8192 c^{3} a^{4} \sqrt {\pi }}\) | \(157\) |
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^{3} \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.
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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 {x^3\,{\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.
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