Optimal. Leaf size=364 \[ -\frac {13}{6} a^2 c^{5/2} \tanh ^{-1}\left (\frac {a \sqrt {c} x}{\sqrt {a^2 c x^2+c}}\right )+\frac {5 i a^2 c^3 \sqrt {a^2 x^2+1} \text {Li}_2\left (-\frac {\sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{2 \sqrt {a^2 c x^2+c}}-\frac {5 i a^2 c^3 \sqrt {a^2 x^2+1} \text {Li}_2\left (\frac {\sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{2 \sqrt {a^2 c x^2+c}}-\frac {5 a^2 c^3 \sqrt {a^2 x^2+1} \tan ^{-1}(a x) \tanh ^{-1}\left (\frac {\sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{\sqrt {a^2 c x^2+c}}-\frac {a c^2 \sqrt {a^2 c x^2+c}}{2 x}+2 a^2 c^2 \sqrt {a^2 c x^2+c} \tan ^{-1}(a x)-\frac {c^2 \sqrt {a^2 c x^2+c} \tan ^{-1}(a x)}{2 x^2}+\frac {1}{3} a^2 c \left (a^2 c x^2+c\right )^{3/2} \tan ^{-1}(a x)-\frac {1}{6} a^3 c^2 x \sqrt {a^2 c x^2+c} \]
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Rubi [A] time = 1.14, antiderivative size = 364, normalized size of antiderivative = 1.00, number of steps used = 23, number of rules used = 10, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.454, Rules used = {4950, 4946, 4962, 264, 4958, 4954, 217, 206, 4930, 195} \[ \frac {5 i a^2 c^3 \sqrt {a^2 x^2+1} \text {PolyLog}\left (2,-\frac {\sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{2 \sqrt {a^2 c x^2+c}}-\frac {5 i a^2 c^3 \sqrt {a^2 x^2+1} \text {PolyLog}\left (2,\frac {\sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{2 \sqrt {a^2 c x^2+c}}-\frac {1}{6} a^3 c^2 x \sqrt {a^2 c x^2+c}-\frac {a c^2 \sqrt {a^2 c x^2+c}}{2 x}+2 a^2 c^2 \sqrt {a^2 c x^2+c} \tan ^{-1}(a x)-\frac {c^2 \sqrt {a^2 c x^2+c} \tan ^{-1}(a x)}{2 x^2}-\frac {13}{6} a^2 c^{5/2} \tanh ^{-1}\left (\frac {a \sqrt {c} x}{\sqrt {a^2 c x^2+c}}\right )-\frac {5 a^2 c^3 \sqrt {a^2 x^2+1} \tan ^{-1}(a x) \tanh ^{-1}\left (\frac {\sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{\sqrt {a^2 c x^2+c}}+\frac {1}{3} a^2 c \left (a^2 c x^2+c\right )^{3/2} \tan ^{-1}(a x) \]
Antiderivative was successfully verified.
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Rule 195
Rule 206
Rule 217
Rule 264
Rule 4930
Rule 4946
Rule 4950
Rule 4954
Rule 4958
Rule 4962
Rubi steps
\begin {align*} \int \frac {\left (c+a^2 c x^2\right )^{5/2} \tan ^{-1}(a x)}{x^3} \, dx &=c \int \frac {\left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)}{x^3} \, dx+\left (a^2 c\right ) \int \frac {\left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)}{x} \, dx\\ &=c^2 \int \frac {\sqrt {c+a^2 c x^2} \tan ^{-1}(a x)}{x^3} \, dx+2 \left (\left (a^2 c^2\right ) \int \frac {\sqrt {c+a^2 c x^2} \tan ^{-1}(a x)}{x} \, dx\right )+\left (a^4 c^2\right ) \int x \sqrt {c+a^2 c x^2} \tan ^{-1}(a x) \, dx\\ &=-\frac {c^2 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)}{x^2}+\frac {1}{3} a^2 c \left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)-\frac {1}{3} \left (a^3 c^2\right ) \int \sqrt {c+a^2 c x^2} \, dx-c^3 \int \frac {\tan ^{-1}(a x)}{x^3 \sqrt {c+a^2 c x^2}} \, dx+\left (a c^3\right ) \int \frac {1}{x^2 \sqrt {c+a^2 c x^2}} \, dx+2 \left (a^2 c^2 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)+\left (a^2 c^3\right ) \int \frac {\tan ^{-1}(a x)}{x \sqrt {c+a^2 c x^2}} \, dx-\left (a^3 c^3\right ) \int \frac {1}{\sqrt {c+a^2 c x^2}} \, dx\right )\\ &=-\frac {a c^2 \sqrt {c+a^2 c x^2}}{x}-\frac {1}{6} a^3 c^2 x \sqrt {c+a^2 c x^2}-\frac {c^2 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)}{2 x^2}+\frac {1}{3} a^2 c \left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)-\frac {1}{2} \left (a c^3\right ) \int \frac {1}{x^2 \sqrt {c+a^2 c x^2}} \, dx+\frac {1}{2} \left (a^2 c^3\right ) \int \frac {\tan ^{-1}(a x)}{x \sqrt {c+a^2 c x^2}} \, dx-\frac {1}{6} \left (a^3 c^3\right ) \int \frac {1}{\sqrt {c+a^2 c x^2}} \, dx+2 \left (a^2 c^2 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)-\left (a^3 c^3\right ) \operatorname {Subst}\left (\int \frac {1}{1-a^2 c x^2} \, dx,x,\frac {x}{\sqrt {c+a^2 c x^2}}\right )+\frac {\left (a^2 c^3 \sqrt {1+a^2 x^2}\right ) \int \frac {\tan ^{-1}(a x)}{x \sqrt {1+a^2 x^2}} \, dx}{\sqrt {c+a^2 c x^2}}\right )\\ &=-\frac {a c^2 \sqrt {c+a^2 c x^2}}{2 x}-\frac {1}{6} a^3 c^2 x \sqrt {c+a^2 c x^2}-\frac {c^2 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)}{2 x^2}+\frac {1}{3} a^2 c \left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)+2 \left (a^2 c^2 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)-\frac {2 a^2 c^3 \sqrt {1+a^2 x^2} \tan ^{-1}(a x) \tanh ^{-1}\left (\frac {\sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{\sqrt {c+a^2 c x^2}}-a^2 c^{5/2} \tanh ^{-1}\left (\frac {a \sqrt {c} x}{\sqrt {c+a^2 c x^2}}\right )+\frac {i a^2 c^3 \sqrt {1+a^2 x^2} \text {Li}_2\left (-\frac {\sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{\sqrt {c+a^2 c x^2}}-\frac {i a^2 c^3 \sqrt {1+a^2 x^2} \text {Li}_2\left (\frac {\sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{\sqrt {c+a^2 c x^2}}\right )-\frac {1}{6} \left (a^3 c^3\right ) \operatorname {Subst}\left (\int \frac {1}{1-a^2 c x^2} \, dx,x,\frac {x}{\sqrt {c+a^2 c x^2}}\right )+\frac {\left (a^2 c^3 \sqrt {1+a^2 x^2}\right ) \int \frac {\tan ^{-1}(a x)}{x \sqrt {1+a^2 x^2}} \, dx}{2 \sqrt {c+a^2 c x^2}}\\ &=-\frac {a c^2 \sqrt {c+a^2 c x^2}}{2 x}-\frac {1}{6} a^3 c^2 x \sqrt {c+a^2 c x^2}-\frac {c^2 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)}{2 x^2}+\frac {1}{3} a^2 c \left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)-\frac {a^2 c^3 \sqrt {1+a^2 x^2} \tan ^{-1}(a x) \tanh ^{-1}\left (\frac {\sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{\sqrt {c+a^2 c x^2}}-\frac {1}{6} a^2 c^{5/2} \tanh ^{-1}\left (\frac {a \sqrt {c} x}{\sqrt {c+a^2 c x^2}}\right )+\frac {i a^2 c^3 \sqrt {1+a^2 x^2} \text {Li}_2\left (-\frac {\sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{2 \sqrt {c+a^2 c x^2}}-\frac {i a^2 c^3 \sqrt {1+a^2 x^2} \text {Li}_2\left (\frac {\sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{2 \sqrt {c+a^2 c x^2}}+2 \left (a^2 c^2 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)-\frac {2 a^2 c^3 \sqrt {1+a^2 x^2} \tan ^{-1}(a x) \tanh ^{-1}\left (\frac {\sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{\sqrt {c+a^2 c x^2}}-a^2 c^{5/2} \tanh ^{-1}\left (\frac {a \sqrt {c} x}{\sqrt {c+a^2 c x^2}}\right )+\frac {i a^2 c^3 \sqrt {1+a^2 x^2} \text {Li}_2\left (-\frac {\sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{\sqrt {c+a^2 c x^2}}-\frac {i a^2 c^3 \sqrt {1+a^2 x^2} \text {Li}_2\left (\frac {\sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{\sqrt {c+a^2 c x^2}}\right )\\ \end {align*}
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Mathematica [A] time = 2.08, size = 361, normalized size = 0.99 \[ \frac {a^2 c^2 \sqrt {a^2 c x^2+c} \tan \left (\frac {1}{2} \tan ^{-1}(a x)\right ) \left (4 a^3 x^3 \tan ^{-1}(a x) \csc ^2\left (\frac {1}{2} \tan ^{-1}(a x)\right )-2 a^2 x^2 \csc ^2\left (\frac {1}{2} \tan ^{-1}(a x)\right )+60 i \text {Li}_2\left (-e^{i \tan ^{-1}(a x)}\right ) \cot \left (\frac {1}{2} \tan ^{-1}(a x)\right )-60 i \text {Li}_2\left (e^{i \tan ^{-1}(a x)}\right ) \cot \left (\frac {1}{2} \tan ^{-1}(a x)\right )-6 \cot ^2\left (\frac {1}{2} \tan ^{-1}(a x)\right )+28 a x \tan ^{-1}(a x) \csc ^2\left (\frac {1}{2} \tan ^{-1}(a x)\right )+60 \tan ^{-1}(a x) \log \left (1-e^{i \tan ^{-1}(a x)}\right ) \cot \left (\frac {1}{2} \tan ^{-1}(a x)\right )-60 \tan ^{-1}(a x) \log \left (1+e^{i \tan ^{-1}(a x)}\right ) \cot \left (\frac {1}{2} \tan ^{-1}(a x)\right )-3 \tan ^{-1}(a x) \cot \left (\frac {1}{2} \tan ^{-1}(a x)\right ) \csc ^2\left (\frac {1}{2} \tan ^{-1}(a x)\right )-4 \sinh ^{-1}(a x) \cot \left (\frac {1}{2} \tan ^{-1}(a x)\right )+3 \tan ^{-1}(a x) \csc \left (\frac {1}{2} \tan ^{-1}(a x)\right ) \sec \left (\frac {1}{2} \tan ^{-1}(a x)\right )+48 \cot \left (\frac {1}{2} \tan ^{-1}(a x)\right ) \log \left (\cos \left (\frac {1}{2} \tan ^{-1}(a x)\right )-\sin \left (\frac {1}{2} \tan ^{-1}(a x)\right )\right )-48 \cot \left (\frac {1}{2} \tan ^{-1}(a x)\right ) \log \left (\sin \left (\frac {1}{2} \tan ^{-1}(a x)\right )+\cos \left (\frac {1}{2} \tan ^{-1}(a x)\right )\right )-6\right )}{24 \sqrt {a^2 x^2+1}} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 1.05, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (a^{4} c^{2} x^{4} + 2 \, a^{2} c^{2} x^{2} + c^{2}\right )} \sqrt {a^{2} c x^{2} + c} \arctan \left (a x\right )}{x^{3}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 1.08, size = 204, normalized size = 0.56 \[ \frac {c^{2} \sqrt {c \left (a x -i\right ) \left (a x +i\right )}\, \left (2 \arctan \left (a x \right ) x^{4} a^{4}-a^{3} x^{3}+14 \arctan \left (a x \right ) x^{2} a^{2}-3 a x -3 \arctan \left (a x \right )\right )}{6 x^{2}}-\frac {\sqrt {c \left (a x -i\right ) \left (a x +i\right )}\, \left (-26 i \arctan \left (\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )-15 i \dilog \left (1+\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )-15 i \dilog \left (\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )+15 \arctan \left (a x \right ) \ln \left (1+\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )\right ) c^{2} a^{2}}{6 \sqrt {a^{2} x^{2}+1}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \frac {1}{3} \, {\left (a^{4} c^{2} x^{2} + a^{2} c^{2}\right )} \sqrt {a^{2} x^{2} + 1} \sqrt {c} \arctan \left (a x\right ) - \frac {1}{6} \, {\left (a^{4} x^{4} + 10 \, a^{2} x^{2} + 9\right )}^{\frac {1}{4}} {\left (a^{3} c^{2} x \cos \left (\frac {1}{2} \, \arctan \left (4 \, a x, -a^{2} x^{2} + 3\right )\right ) + 2 \, a^{2} c^{2} \sin \left (\frac {1}{2} \, \arctan \left (4 \, a x, -a^{2} x^{2} + 3\right )\right )\right )} \sqrt {c} + \frac {1}{12} \, {\left (a^{2} c^{2} \arctan \left ({\left (a^{4} x^{4} + 10 \, a^{2} x^{2} + 9\right )}^{\frac {1}{4}} \sin \left (\frac {1}{2} \, \arctan \left (4 \, a x, a^{2} x^{2} - 3\right )\right ) + 2, a x + {\left (a^{4} x^{4} + 10 \, a^{2} x^{2} + 9\right )}^{\frac {1}{4}} \cos \left (\frac {1}{2} \, \arctan \left (4 \, a x, a^{2} x^{2} - 3\right )\right )\right ) + a^{2} c^{2} \arctan \left ({\left (a^{4} x^{4} + 10 \, a^{2} x^{2} + 9\right )}^{\frac {1}{4}} \sin \left (\frac {1}{2} \, \arctan \left (4 \, a x, a^{2} x^{2} - 3\right )\right ) - 2, -a x + {\left (a^{4} x^{4} + 10 \, a^{2} x^{2} + 9\right )}^{\frac {1}{4}} \cos \left (\frac {1}{2} \, \arctan \left (4 \, a x, a^{2} x^{2} - 3\right )\right )\right ) + 24 \, a^{2} c^{2} \int \frac {\sqrt {a^{2} x^{2} + 1} \arctan \left (a x\right )}{x}\,{d x} + 12 \, c^{2} \int \frac {\sqrt {a^{2} x^{2} + 1} \arctan \left (a x\right )}{x^{3}}\,{d x}\right )} \sqrt {c} \]
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 \[ \int \frac {\mathrm {atan}\left (a\,x\right )\,{\left (c\,a^2\,x^2+c\right )}^{5/2}}{x^3} \,d x \]
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 \[ \int \frac {\left (c \left (a^{2} x^{2} + 1\right )\right )^{\frac {5}{2}} \operatorname {atan}{\left (a x \right )}}{x^{3}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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