Optimal. Leaf size=183 \[ -\frac {2}{3 a c \left (c+a^2 c x^2\right )^{3/2} \text {ArcTan}(a x)^{3/2}}+\frac {4 x}{c \left (c+a^2 c x^2\right )^{3/2} \sqrt {\text {ArcTan}(a x)}}-\frac {\sqrt {2 \pi } \sqrt {1+a^2 x^2} \text {FresnelC}\left (\sqrt {\frac {2}{\pi }} \sqrt {\text {ArcTan}(a x)}\right )}{a c^2 \sqrt {c+a^2 c x^2}}-\frac {\sqrt {6 \pi } \sqrt {1+a^2 x^2} \text {FresnelC}\left (\sqrt {\frac {6}{\pi }} \sqrt {\text {ArcTan}(a x)}\right )}{a c^2 \sqrt {c+a^2 c x^2}} \]
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Rubi [A]
time = 0.43, antiderivative size = 183, normalized size of antiderivative = 1.00, number of steps
used = 18, number of rules used = 10, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.435, Rules used = {5022, 5088,
5091, 5090, 4491, 3385, 3433, 5025, 5024, 3393} \begin {gather*} -\frac {\sqrt {2 \pi } \sqrt {a^2 x^2+1} \text {FresnelC}\left (\sqrt {\frac {2}{\pi }} \sqrt {\text {ArcTan}(a x)}\right )}{a c^2 \sqrt {a^2 c x^2+c}}-\frac {\sqrt {6 \pi } \sqrt {a^2 x^2+1} \text {FresnelC}\left (\sqrt {\frac {6}{\pi }} \sqrt {\text {ArcTan}(a x)}\right )}{a c^2 \sqrt {a^2 c x^2+c}}+\frac {4 x}{c \sqrt {\text {ArcTan}(a x)} \left (a^2 c x^2+c\right )^{3/2}}-\frac {2}{3 a c \text {ArcTan}(a x)^{3/2} \left (a^2 c x^2+c\right )^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 3385
Rule 3393
Rule 3433
Rule 4491
Rule 5022
Rule 5024
Rule 5025
Rule 5088
Rule 5090
Rule 5091
Rubi steps
\begin {align*} \int \frac {1}{\left (c+a^2 c x^2\right )^{5/2} \tan ^{-1}(a x)^{5/2}} \, dx &=-\frac {2}{3 a c \left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)^{3/2}}-(2 a) \int \frac {x}{\left (c+a^2 c x^2\right )^{5/2} \tan ^{-1}(a x)^{3/2}} \, dx\\ &=-\frac {2}{3 a c \left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)^{3/2}}+\frac {4 x}{c \left (c+a^2 c x^2\right )^{3/2} \sqrt {\tan ^{-1}(a x)}}-4 \int \frac {1}{\left (c+a^2 c x^2\right )^{5/2} \sqrt {\tan ^{-1}(a x)}} \, dx+\left (8 a^2\right ) \int \frac {x^2}{\left (c+a^2 c x^2\right )^{5/2} \sqrt {\tan ^{-1}(a x)}} \, dx\\ &=-\frac {2}{3 a c \left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)^{3/2}}+\frac {4 x}{c \left (c+a^2 c x^2\right )^{3/2} \sqrt {\tan ^{-1}(a x)}}-\frac {\left (4 \sqrt {1+a^2 x^2}\right ) \int \frac {1}{\left (1+a^2 x^2\right )^{5/2} \sqrt {\tan ^{-1}(a x)}} \, dx}{c^2 \sqrt {c+a^2 c x^2}}+\frac {\left (8 a^2 \sqrt {1+a^2 x^2}\right ) \int \frac {x^2}{\left (1+a^2 x^2\right )^{5/2} \sqrt {\tan ^{-1}(a x)}} \, dx}{c^2 \sqrt {c+a^2 c x^2}}\\ &=-\frac {2}{3 a c \left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)^{3/2}}+\frac {4 x}{c \left (c+a^2 c x^2\right )^{3/2} \sqrt {\tan ^{-1}(a x)}}-\frac {\left (4 \sqrt {1+a^2 x^2}\right ) \text {Subst}\left (\int \frac {\cos ^3(x)}{\sqrt {x}} \, dx,x,\tan ^{-1}(a x)\right )}{a c^2 \sqrt {c+a^2 c x^2}}+\frac {\left (8 \sqrt {1+a^2 x^2}\right ) \text {Subst}\left (\int \frac {\cos (x) \sin ^2(x)}{\sqrt {x}} \, dx,x,\tan ^{-1}(a x)\right )}{a c^2 \sqrt {c+a^2 c x^2}}\\ &=-\frac {2}{3 a c \left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)^{3/2}}+\frac {4 x}{c \left (c+a^2 c x^2\right )^{3/2} \sqrt {\tan ^{-1}(a x)}}-\frac {\left (4 \sqrt {1+a^2 x^2}\right ) \text {Subst}\left (\int \left (\frac {3 \cos (x)}{4 \sqrt {x}}+\frac {\cos (3 x)}{4 \sqrt {x}}\right ) \, dx,x,\tan ^{-1}(a x)\right )}{a c^2 \sqrt {c+a^2 c x^2}}+\frac {\left (8 \sqrt {1+a^2 x^2}\right ) \text {Subst}\left (\int \left (\frac {\cos (x)}{4 \sqrt {x}}-\frac {\cos (3 x)}{4 \sqrt {x}}\right ) \, dx,x,\tan ^{-1}(a x)\right )}{a c^2 \sqrt {c+a^2 c x^2}}\\ &=-\frac {2}{3 a c \left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)^{3/2}}+\frac {4 x}{c \left (c+a^2 c x^2\right )^{3/2} \sqrt {\tan ^{-1}(a x)}}-\frac {\sqrt {1+a^2 x^2} \text {Subst}\left (\int \frac {\cos (3 x)}{\sqrt {x}} \, dx,x,\tan ^{-1}(a x)\right )}{a c^2 \sqrt {c+a^2 c x^2}}+\frac {\left (2 \sqrt {1+a^2 x^2}\right ) \text {Subst}\left (\int \frac {\cos (x)}{\sqrt {x}} \, dx,x,\tan ^{-1}(a x)\right )}{a c^2 \sqrt {c+a^2 c x^2}}-\frac {\left (2 \sqrt {1+a^2 x^2}\right ) \text {Subst}\left (\int \frac {\cos (3 x)}{\sqrt {x}} \, dx,x,\tan ^{-1}(a x)\right )}{a c^2 \sqrt {c+a^2 c x^2}}-\frac {\left (3 \sqrt {1+a^2 x^2}\right ) \text {Subst}\left (\int \frac {\cos (x)}{\sqrt {x}} \, dx,x,\tan ^{-1}(a x)\right )}{a c^2 \sqrt {c+a^2 c x^2}}\\ &=-\frac {2}{3 a c \left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)^{3/2}}+\frac {4 x}{c \left (c+a^2 c x^2\right )^{3/2} \sqrt {\tan ^{-1}(a x)}}-\frac {\left (2 \sqrt {1+a^2 x^2}\right ) \text {Subst}\left (\int \cos \left (3 x^2\right ) \, dx,x,\sqrt {\tan ^{-1}(a x)}\right )}{a c^2 \sqrt {c+a^2 c x^2}}+\frac {\left (4 \sqrt {1+a^2 x^2}\right ) \text {Subst}\left (\int \cos \left (x^2\right ) \, dx,x,\sqrt {\tan ^{-1}(a x)}\right )}{a c^2 \sqrt {c+a^2 c x^2}}-\frac {\left (4 \sqrt {1+a^2 x^2}\right ) \text {Subst}\left (\int \cos \left (3 x^2\right ) \, dx,x,\sqrt {\tan ^{-1}(a x)}\right )}{a c^2 \sqrt {c+a^2 c x^2}}-\frac {\left (6 \sqrt {1+a^2 x^2}\right ) \text {Subst}\left (\int \cos \left (x^2\right ) \, dx,x,\sqrt {\tan ^{-1}(a x)}\right )}{a c^2 \sqrt {c+a^2 c x^2}}\\ &=-\frac {2}{3 a c \left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)^{3/2}}+\frac {4 x}{c \left (c+a^2 c x^2\right )^{3/2} \sqrt {\tan ^{-1}(a x)}}-\frac {\sqrt {2 \pi } \sqrt {1+a^2 x^2} C\left (\sqrt {\frac {2}{\pi }} \sqrt {\tan ^{-1}(a x)}\right )}{a c^2 \sqrt {c+a^2 c x^2}}-\frac {\sqrt {6 \pi } \sqrt {1+a^2 x^2} C\left (\sqrt {\frac {6}{\pi }} \sqrt {\tan ^{-1}(a x)}\right )}{a c^2 \sqrt {c+a^2 c x^2}}\\ \end {align*}
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Mathematica [C] Result contains complex when optimal does not.
time = 0.37, size = 300, normalized size = 1.64 \begin {gather*} \frac {-4+24 a x \text {ArcTan}(a x)-3 \left (1+a^2 x^2\right )^{3/2} (-i \text {ArcTan}(a x))^{3/2} \text {Gamma}\left (\frac {1}{2},-i \text {ArcTan}(a x)\right )-3 \left (1+a^2 x^2\right )^{3/2} (i \text {ArcTan}(a x))^{3/2} \text {Gamma}\left (\frac {1}{2},i \text {ArcTan}(a x)\right )-3 \sqrt {3+3 a^2 x^2} (-i \text {ArcTan}(a x))^{3/2} \text {Gamma}\left (\frac {1}{2},-3 i \text {ArcTan}(a x)\right )-3 a^2 x^2 \sqrt {3+3 a^2 x^2} (-i \text {ArcTan}(a x))^{3/2} \text {Gamma}\left (\frac {1}{2},-3 i \text {ArcTan}(a x)\right )-3 \sqrt {3+3 a^2 x^2} (i \text {ArcTan}(a x))^{3/2} \text {Gamma}\left (\frac {1}{2},3 i \text {ArcTan}(a x)\right )-3 a^2 x^2 \sqrt {3+3 a^2 x^2} (i \text {ArcTan}(a x))^{3/2} \text {Gamma}\left (\frac {1}{2},3 i \text {ArcTan}(a x)\right )}{6 c^2 \left (a+a^3 x^2\right ) \sqrt {c+a^2 c x^2} \text {ArcTan}(a x)^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.82, size = 0, normalized size = 0.00 \[\int \frac {1}{\left (a^{2} c \,x^{2}+c \right )^{\frac {5}{2}} \arctan \left (a x \right )^{\frac {5}{2}}}\, dx\]
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(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: SystemError} \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 {1}{{\mathrm {atan}\left (a\,x\right )}^{5/2}\,{\left (c\,a^2\,x^2+c\right )}^{5/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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