Optimal. Leaf size=372 \[ -\frac {a c^2 \sqrt {a^2 c x^2+c}}{6 x^2}-\frac {2 a^2 c^2 \sqrt {a^2 c x^2+c} \tan ^{-1}(a x)}{x}-\frac {c \left (a^2 c x^2+c\right )^{3/2} \tan ^{-1}(a x)}{3 x^3}+\frac {1}{2} a^4 c^2 x \sqrt {a^2 c x^2+c} \tan ^{-1}(a x)-\frac {13}{6} a^3 c^{5/2} \tanh ^{-1}\left (\frac {\sqrt {a^2 c x^2+c}}{\sqrt {c}}\right )+\frac {5 i a^3 c^3 \sqrt {a^2 x^2+1} \text {Li}_2\left (-\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{2 \sqrt {a^2 c x^2+c}}-\frac {5 i a^3 c^3 \sqrt {a^2 x^2+1} \text {Li}_2\left (\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{2 \sqrt {a^2 c x^2+c}}-\frac {5 i a^3 c^3 \sqrt {a^2 x^2+1} \tan ^{-1}(a x) \tan ^{-1}\left (\frac {\sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{\sqrt {a^2 c x^2+c}}-\frac {1}{2} a^3 c^2 \sqrt {a^2 c x^2+c} \]
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Rubi [A] time = 0.98, antiderivative size = 372, normalized size of antiderivative = 1.00, number of steps used = 25, number of rules used = 9, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.409, Rules used = {4950, 4944, 266, 47, 63, 208, 4890, 4886, 4878} \[ \frac {5 i a^3 c^3 \sqrt {a^2 x^2+1} \text {PolyLog}\left (2,-\frac {i \sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{2 \sqrt {a^2 c x^2+c}}-\frac {5 i a^3 c^3 \sqrt {a^2 x^2+1} \text {PolyLog}\left (2,\frac {i \sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{2 \sqrt {a^2 c x^2+c}}-\frac {1}{2} a^3 c^2 \sqrt {a^2 c x^2+c}-\frac {a c^2 \sqrt {a^2 c x^2+c}}{6 x^2}+\frac {1}{2} a^4 c^2 x \sqrt {a^2 c x^2+c} \tan ^{-1}(a x)-\frac {5 i a^3 c^3 \sqrt {a^2 x^2+1} \tan ^{-1}(a x) \tan ^{-1}\left (\frac {\sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{\sqrt {a^2 c x^2+c}}-\frac {2 a^2 c^2 \sqrt {a^2 c x^2+c} \tan ^{-1}(a x)}{x}-\frac {13}{6} a^3 c^{5/2} \tanh ^{-1}\left (\frac {\sqrt {a^2 c x^2+c}}{\sqrt {c}}\right )-\frac {c \left (a^2 c x^2+c\right )^{3/2} \tan ^{-1}(a x)}{3 x^3} \]
Antiderivative was successfully verified.
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Rule 47
Rule 63
Rule 208
Rule 266
Rule 4878
Rule 4886
Rule 4890
Rule 4944
Rule 4950
Rubi steps
\begin {align*} \int \frac {\left (c+a^2 c x^2\right )^{5/2} \tan ^{-1}(a x)}{x^4} \, dx &=c \int \frac {\left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)}{x^4} \, dx+\left (a^2 c\right ) \int \frac {\left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)}{x^2} \, dx\\ &=c^2 \int \frac {\sqrt {c+a^2 c x^2} \tan ^{-1}(a x)}{x^4} \, dx+2 \left (\left (a^2 c^2\right ) \int \frac {\sqrt {c+a^2 c x^2} \tan ^{-1}(a x)}{x^2} \, dx\right )+\left (a^4 c^2\right ) \int \sqrt {c+a^2 c x^2} \tan ^{-1}(a x) \, dx\\ &=-\frac {1}{2} a^3 c^2 \sqrt {c+a^2 c x^2}+\frac {1}{2} a^4 c^2 x \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)-\frac {c \left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)}{3 x^3}+\frac {1}{3} \left (a c^2\right ) \int \frac {\sqrt {c+a^2 c x^2}}{x^3} \, dx+\frac {1}{2} \left (a^4 c^3\right ) \int \frac {\tan ^{-1}(a x)}{\sqrt {c+a^2 c x^2}} \, dx+2 \left (\left (a^2 c^3\right ) \int \frac {\tan ^{-1}(a x)}{x^2 \sqrt {c+a^2 c x^2}} \, dx+\left (a^4 c^3\right ) \int \frac {\tan ^{-1}(a x)}{\sqrt {c+a^2 c x^2}} \, dx\right )\\ &=-\frac {1}{2} a^3 c^2 \sqrt {c+a^2 c x^2}+\frac {1}{2} a^4 c^2 x \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)-\frac {c \left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)}{3 x^3}+\frac {1}{6} \left (a c^2\right ) \operatorname {Subst}\left (\int \frac {\sqrt {c+a^2 c x}}{x^2} \, dx,x,x^2\right )+\frac {\left (a^4 c^3 \sqrt {1+a^2 x^2}\right ) \int \frac {\tan ^{-1}(a x)}{\sqrt {1+a^2 x^2}} \, dx}{2 \sqrt {c+a^2 c x^2}}+2 \left (-\frac {a^2 c^2 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)}{x}+\left (a^3 c^3\right ) \int \frac {1}{x \sqrt {c+a^2 c x^2}} \, dx+\frac {\left (a^4 c^3 \sqrt {1+a^2 x^2}\right ) \int \frac {\tan ^{-1}(a x)}{\sqrt {1+a^2 x^2}} \, dx}{\sqrt {c+a^2 c x^2}}\right )\\ &=-\frac {1}{2} a^3 c^2 \sqrt {c+a^2 c x^2}-\frac {a c^2 \sqrt {c+a^2 c x^2}}{6 x^2}+\frac {1}{2} a^4 c^2 x \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)-\frac {c \left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)}{3 x^3}-\frac {i a^3 c^3 \sqrt {1+a^2 x^2} \tan ^{-1}(a x) \tan ^{-1}\left (\frac {\sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{\sqrt {c+a^2 c x^2}}+\frac {i a^3 c^3 \sqrt {1+a^2 x^2} \text {Li}_2\left (-\frac {i \sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{2 \sqrt {c+a^2 c x^2}}-\frac {i a^3 c^3 \sqrt {1+a^2 x^2} \text {Li}_2\left (\frac {i \sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{2 \sqrt {c+a^2 c x^2}}+\frac {1}{12} \left (a^3 c^3\right ) \operatorname {Subst}\left (\int \frac {1}{x \sqrt {c+a^2 c x}} \, dx,x,x^2\right )+2 \left (-\frac {a^2 c^2 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)}{x}-\frac {2 i a^3 c^3 \sqrt {1+a^2 x^2} \tan ^{-1}(a x) \tan ^{-1}\left (\frac {\sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{\sqrt {c+a^2 c x^2}}+\frac {i a^3 c^3 \sqrt {1+a^2 x^2} \text {Li}_2\left (-\frac {i \sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{\sqrt {c+a^2 c x^2}}-\frac {i a^3 c^3 \sqrt {1+a^2 x^2} \text {Li}_2\left (\frac {i \sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{\sqrt {c+a^2 c x^2}}+\frac {1}{2} \left (a^3 c^3\right ) \operatorname {Subst}\left (\int \frac {1}{x \sqrt {c+a^2 c x}} \, dx,x,x^2\right )\right )\\ &=-\frac {1}{2} a^3 c^2 \sqrt {c+a^2 c x^2}-\frac {a c^2 \sqrt {c+a^2 c x^2}}{6 x^2}+\frac {1}{2} a^4 c^2 x \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)-\frac {c \left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)}{3 x^3}-\frac {i a^3 c^3 \sqrt {1+a^2 x^2} \tan ^{-1}(a x) \tan ^{-1}\left (\frac {\sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{\sqrt {c+a^2 c x^2}}+\frac {i a^3 c^3 \sqrt {1+a^2 x^2} \text {Li}_2\left (-\frac {i \sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{2 \sqrt {c+a^2 c x^2}}-\frac {i a^3 c^3 \sqrt {1+a^2 x^2} \text {Li}_2\left (\frac {i \sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{2 \sqrt {c+a^2 c x^2}}+\frac {1}{6} \left (a c^2\right ) \operatorname {Subst}\left (\int \frac {1}{-\frac {1}{a^2}+\frac {x^2}{a^2 c}} \, dx,x,\sqrt {c+a^2 c x^2}\right )+2 \left (-\frac {a^2 c^2 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)}{x}-\frac {2 i a^3 c^3 \sqrt {1+a^2 x^2} \tan ^{-1}(a x) \tan ^{-1}\left (\frac {\sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{\sqrt {c+a^2 c x^2}}+\frac {i a^3 c^3 \sqrt {1+a^2 x^2} \text {Li}_2\left (-\frac {i \sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{\sqrt {c+a^2 c x^2}}-\frac {i a^3 c^3 \sqrt {1+a^2 x^2} \text {Li}_2\left (\frac {i \sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{\sqrt {c+a^2 c x^2}}+\left (a c^2\right ) \operatorname {Subst}\left (\int \frac {1}{-\frac {1}{a^2}+\frac {x^2}{a^2 c}} \, dx,x,\sqrt {c+a^2 c x^2}\right )\right )\\ &=-\frac {1}{2} a^3 c^2 \sqrt {c+a^2 c x^2}-\frac {a c^2 \sqrt {c+a^2 c x^2}}{6 x^2}+\frac {1}{2} a^4 c^2 x \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)-\frac {c \left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)}{3 x^3}-\frac {i a^3 c^3 \sqrt {1+a^2 x^2} \tan ^{-1}(a x) \tan ^{-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^3 c^{5/2} \tanh ^{-1}\left (\frac {\sqrt {c+a^2 c x^2}}{\sqrt {c}}\right )+\frac {i a^3 c^3 \sqrt {1+a^2 x^2} \text {Li}_2\left (-\frac {i \sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{2 \sqrt {c+a^2 c x^2}}-\frac {i a^3 c^3 \sqrt {1+a^2 x^2} \text {Li}_2\left (\frac {i \sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{2 \sqrt {c+a^2 c x^2}}+2 \left (-\frac {a^2 c^2 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)}{x}-\frac {2 i a^3 c^3 \sqrt {1+a^2 x^2} \tan ^{-1}(a x) \tan ^{-1}\left (\frac {\sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{\sqrt {c+a^2 c x^2}}-a^3 c^{5/2} \tanh ^{-1}\left (\frac {\sqrt {c+a^2 c x^2}}{\sqrt {c}}\right )+\frac {i a^3 c^3 \sqrt {1+a^2 x^2} \text {Li}_2\left (-\frac {i \sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{\sqrt {c+a^2 c x^2}}-\frac {i a^3 c^3 \sqrt {1+a^2 x^2} \text {Li}_2\left (\frac {i \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 = 1.04, size = 313, normalized size = 0.84 \[ \frac {c^2 \sqrt {a^2 c x^2+c} \left (15 i a^3 x^3 \text {Li}_2\left (-i e^{i \tan ^{-1}(a x)}\right )-15 i a^3 x^3 \text {Li}_2\left (i e^{i \tan ^{-1}(a x)}\right )+15 a^3 x^3 \tan ^{-1}(a x) \log \left (1-i e^{i \tan ^{-1}(a x)}\right )-15 a^3 x^3 \tan ^{-1}(a x) \log \left (1+i e^{i \tan ^{-1}(a x)}\right )+12 a^3 x^3 \log \left (\sin \left (\frac {1}{2} \tan ^{-1}(a x)\right )\right )-12 a^3 x^3 \log \left (\cos \left (\frac {1}{2} \tan ^{-1}(a x)\right )\right )-a x \sqrt {a^2 x^2+1}-14 a^2 x^2 \sqrt {a^2 x^2+1} \tan ^{-1}(a x)-2 \sqrt {a^2 x^2+1} \tan ^{-1}(a x)+3 a^4 x^4 \sqrt {a^2 x^2+1} \tan ^{-1}(a x)-3 a^3 x^3 \sqrt {a^2 x^2+1}-a^3 x^3 \tanh ^{-1}\left (\sqrt {a^2 x^2+1}\right )\right )}{6 x^3 \sqrt {a^2 x^2+1}} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.60, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (a^{4} c^{2} x^{4} + 2 \, a^{2} c^{2} x^{2} + c^{2}\right )} \sqrt {a^{2} c x^{2} + c} \arctan \left (a x\right )}{x^{4}}, x\right ) \]
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 = 1.42, size = 270, normalized size = 0.73 \[ \frac {c^{2} \sqrt {c \left (a x -i\right ) \left (a x +i\right )}\, \left (3 \arctan \left (a x \right ) x^{4} a^{4}-3 a^{3} x^{3}-14 \arctan \left (a x \right ) x^{2} a^{2}-a x -2 \arctan \left (a x \right )\right )}{6 x^{3}}+\frac {i a^{3} c^{2} \sqrt {c \left (a x -i\right ) \left (a x +i\right )}\, \left (15 i \arctan \left (a x \right ) \ln \left (1+\frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )-15 i \arctan \left (a x \right ) \ln \left (1-\frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )+13 i \ln \left (1+\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )-13 i \ln \left (\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}-1\right )+15 \dilog \left (1+\frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )-15 \dilog \left (1-\frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )\right )}{6 \sqrt {a^{2} x^{2}+1}} \]
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 \[ \int \frac {{\left (a^{2} c x^{2} + c\right )}^{\frac {5}{2}} \arctan \left (a x\right )}{x^{4}}\,{d x} \]
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 \[ \int \frac {\mathrm {atan}\left (a\,x\right )\,{\left (c\,a^2\,x^2+c\right )}^{5/2}}{x^4} \,d x \]
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 \[ \int \frac {\left (c \left (a^{2} x^{2} + 1\right )\right )^{\frac {5}{2}} \operatorname {atan}{\left (a x \right )}}{x^{4}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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