Optimal. Leaf size=174 \[ -\frac {3 i \sinh ^{-1}(a x)^2 \text {Li}_2\left (-i e^{\sinh ^{-1}(a x)}\right )}{a c}+\frac {3 i \sinh ^{-1}(a x)^2 \text {Li}_2\left (i e^{\sinh ^{-1}(a x)}\right )}{a c}+\frac {6 i \sinh ^{-1}(a x) \text {Li}_3\left (-i e^{\sinh ^{-1}(a x)}\right )}{a c}-\frac {6 i \sinh ^{-1}(a x) \text {Li}_3\left (i e^{\sinh ^{-1}(a x)}\right )}{a c}-\frac {6 i \text {Li}_4\left (-i e^{\sinh ^{-1}(a x)}\right )}{a c}+\frac {6 i \text {Li}_4\left (i e^{\sinh ^{-1}(a x)}\right )}{a c}+\frac {2 \sinh ^{-1}(a x)^3 \tan ^{-1}\left (e^{\sinh ^{-1}(a x)}\right )}{a c} \]
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Rubi [A] time = 0.13, antiderivative size = 174, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 6, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.316, Rules used = {5693, 4180, 2531, 6609, 2282, 6589} \[ -\frac {3 i \sinh ^{-1}(a x)^2 \text {PolyLog}\left (2,-i e^{\sinh ^{-1}(a x)}\right )}{a c}+\frac {3 i \sinh ^{-1}(a x)^2 \text {PolyLog}\left (2,i e^{\sinh ^{-1}(a x)}\right )}{a c}+\frac {6 i \sinh ^{-1}(a x) \text {PolyLog}\left (3,-i e^{\sinh ^{-1}(a x)}\right )}{a c}-\frac {6 i \sinh ^{-1}(a x) \text {PolyLog}\left (3,i e^{\sinh ^{-1}(a x)}\right )}{a c}-\frac {6 i \text {PolyLog}\left (4,-i e^{\sinh ^{-1}(a x)}\right )}{a c}+\frac {6 i \text {PolyLog}\left (4,i e^{\sinh ^{-1}(a x)}\right )}{a c}+\frac {2 \sinh ^{-1}(a x)^3 \tan ^{-1}\left (e^{\sinh ^{-1}(a x)}\right )}{a c} \]
Antiderivative was successfully verified.
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Rule 2282
Rule 2531
Rule 4180
Rule 5693
Rule 6589
Rule 6609
Rubi steps
\begin {align*} \int \frac {\sinh ^{-1}(a x)^3}{c+a^2 c x^2} \, dx &=\frac {\operatorname {Subst}\left (\int x^3 \text {sech}(x) \, dx,x,\sinh ^{-1}(a x)\right )}{a c}\\ &=\frac {2 \sinh ^{-1}(a x)^3 \tan ^{-1}\left (e^{\sinh ^{-1}(a x)}\right )}{a c}-\frac {(3 i) \operatorname {Subst}\left (\int x^2 \log \left (1-i e^x\right ) \, dx,x,\sinh ^{-1}(a x)\right )}{a c}+\frac {(3 i) \operatorname {Subst}\left (\int x^2 \log \left (1+i e^x\right ) \, dx,x,\sinh ^{-1}(a x)\right )}{a c}\\ &=\frac {2 \sinh ^{-1}(a x)^3 \tan ^{-1}\left (e^{\sinh ^{-1}(a x)}\right )}{a c}-\frac {3 i \sinh ^{-1}(a x)^2 \text {Li}_2\left (-i e^{\sinh ^{-1}(a x)}\right )}{a c}+\frac {3 i \sinh ^{-1}(a x)^2 \text {Li}_2\left (i e^{\sinh ^{-1}(a x)}\right )}{a c}+\frac {(6 i) \operatorname {Subst}\left (\int x \text {Li}_2\left (-i e^x\right ) \, dx,x,\sinh ^{-1}(a x)\right )}{a c}-\frac {(6 i) \operatorname {Subst}\left (\int x \text {Li}_2\left (i e^x\right ) \, dx,x,\sinh ^{-1}(a x)\right )}{a c}\\ &=\frac {2 \sinh ^{-1}(a x)^3 \tan ^{-1}\left (e^{\sinh ^{-1}(a x)}\right )}{a c}-\frac {3 i \sinh ^{-1}(a x)^2 \text {Li}_2\left (-i e^{\sinh ^{-1}(a x)}\right )}{a c}+\frac {3 i \sinh ^{-1}(a x)^2 \text {Li}_2\left (i e^{\sinh ^{-1}(a x)}\right )}{a c}+\frac {6 i \sinh ^{-1}(a x) \text {Li}_3\left (-i e^{\sinh ^{-1}(a x)}\right )}{a c}-\frac {6 i \sinh ^{-1}(a x) \text {Li}_3\left (i e^{\sinh ^{-1}(a x)}\right )}{a c}-\frac {(6 i) \operatorname {Subst}\left (\int \text {Li}_3\left (-i e^x\right ) \, dx,x,\sinh ^{-1}(a x)\right )}{a c}+\frac {(6 i) \operatorname {Subst}\left (\int \text {Li}_3\left (i e^x\right ) \, dx,x,\sinh ^{-1}(a x)\right )}{a c}\\ &=\frac {2 \sinh ^{-1}(a x)^3 \tan ^{-1}\left (e^{\sinh ^{-1}(a x)}\right )}{a c}-\frac {3 i \sinh ^{-1}(a x)^2 \text {Li}_2\left (-i e^{\sinh ^{-1}(a x)}\right )}{a c}+\frac {3 i \sinh ^{-1}(a x)^2 \text {Li}_2\left (i e^{\sinh ^{-1}(a x)}\right )}{a c}+\frac {6 i \sinh ^{-1}(a x) \text {Li}_3\left (-i e^{\sinh ^{-1}(a x)}\right )}{a c}-\frac {6 i \sinh ^{-1}(a x) \text {Li}_3\left (i e^{\sinh ^{-1}(a x)}\right )}{a c}-\frac {(6 i) \operatorname {Subst}\left (\int \frac {\text {Li}_3(-i x)}{x} \, dx,x,e^{\sinh ^{-1}(a x)}\right )}{a c}+\frac {(6 i) \operatorname {Subst}\left (\int \frac {\text {Li}_3(i x)}{x} \, dx,x,e^{\sinh ^{-1}(a x)}\right )}{a c}\\ &=\frac {2 \sinh ^{-1}(a x)^3 \tan ^{-1}\left (e^{\sinh ^{-1}(a x)}\right )}{a c}-\frac {3 i \sinh ^{-1}(a x)^2 \text {Li}_2\left (-i e^{\sinh ^{-1}(a x)}\right )}{a c}+\frac {3 i \sinh ^{-1}(a x)^2 \text {Li}_2\left (i e^{\sinh ^{-1}(a x)}\right )}{a c}+\frac {6 i \sinh ^{-1}(a x) \text {Li}_3\left (-i e^{\sinh ^{-1}(a x)}\right )}{a c}-\frac {6 i \sinh ^{-1}(a x) \text {Li}_3\left (i e^{\sinh ^{-1}(a x)}\right )}{a c}-\frac {6 i \text {Li}_4\left (-i e^{\sinh ^{-1}(a x)}\right )}{a c}+\frac {6 i \text {Li}_4\left (i e^{\sinh ^{-1}(a x)}\right )}{a c}\\ \end {align*}
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Mathematica [B] time = 0.23, size = 454, normalized size = 2.61 \[ -\frac {i \left (192 \sinh ^{-1}(a x)^2 \text {Li}_2\left (-i e^{\sinh ^{-1}(a x)}\right )+192 i \pi \sinh ^{-1}(a x) \text {Li}_2\left (i e^{\sinh ^{-1}(a x)}\right )+384 \sinh ^{-1}(a x) \text {Li}_3\left (-i e^{-\sinh ^{-1}(a x)}\right )-384 \sinh ^{-1}(a x) \text {Li}_3\left (-i e^{\sinh ^{-1}(a x)}\right )-48 \left (\pi -2 i \sinh ^{-1}(a x)\right )^2 \text {Li}_2\left (-i e^{-\sinh ^{-1}(a x)}\right )-48 \pi ^2 \text {Li}_2\left (i e^{\sinh ^{-1}(a x)}\right )+192 i \pi \text {Li}_3\left (-i e^{-\sinh ^{-1}(a x)}\right )-192 i \pi \text {Li}_3\left (i e^{\sinh ^{-1}(a x)}\right )+384 \text {Li}_4\left (-i e^{-\sinh ^{-1}(a x)}\right )+384 \text {Li}_4\left (-i e^{\sinh ^{-1}(a x)}\right )-16 \sinh ^{-1}(a x)^4-32 i \pi \sinh ^{-1}(a x)^3+24 \pi ^2 \sinh ^{-1}(a x)^2+8 i \pi ^3 \sinh ^{-1}(a x)-64 \sinh ^{-1}(a x)^3 \log \left (1+i e^{-\sinh ^{-1}(a x)}\right )+64 \sinh ^{-1}(a x)^3 \log \left (1+i e^{\sinh ^{-1}(a x)}\right )-96 i \pi \sinh ^{-1}(a x)^2 \log \left (1+i e^{-\sinh ^{-1}(a x)}\right )+96 i \pi \sinh ^{-1}(a x)^2 \log \left (1-i e^{\sinh ^{-1}(a x)}\right )+48 \pi ^2 \sinh ^{-1}(a x) \log \left (1+i e^{-\sinh ^{-1}(a x)}\right )-48 \pi ^2 \sinh ^{-1}(a x) \log \left (1-i e^{\sinh ^{-1}(a x)}\right )+8 i \pi ^3 \log \left (1+i e^{-\sinh ^{-1}(a x)}\right )-8 i \pi ^3 \log \left (1+i e^{\sinh ^{-1}(a x)}\right )+8 i \pi ^3 \log \left (\tan \left (\frac {1}{4} \left (\pi +2 i \sinh ^{-1}(a x)\right )\right )\right )+7 \pi ^4\right )}{64 a c} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.43, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\operatorname {arsinh}\left (a x\right )^{3}}{a^{2} c x^{2} + c}, x\right ) \]
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 \[ \int \frac {\operatorname {arsinh}\left (a x\right )^{3}}{a^{2} c x^{2} + c}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.09, size = 0, normalized size = 0.00 \[ \int \frac {\arcsinh \left (a x \right )^{3}}{a^{2} c \,x^{2}+c}\, dx \]
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 \[ \int \frac {\operatorname {arsinh}\left (a x\right )^{3}}{a^{2} c x^{2} + c}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {{\mathrm {asinh}\left (a\,x\right )}^3}{c\,a^2\,x^2+c} \,d x \]
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 \[ \frac {\int \frac {\operatorname {asinh}^{3}{\left (a x \right )}}{a^{2} x^{2} + 1}\, dx}{c} \]
Verification of antiderivative is not currently implemented for this CAS.
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