Optimal. Leaf size=363 \[ -\frac {2 \sqrt {a^2 x^2+1} \sinh ^{-1}(a x) \text {Li}_2\left (-e^{2 \sinh ^{-1}(a x)}\right )}{a c^2 \sqrt {a^2 c x^2+c}}+\frac {\sqrt {a^2 x^2+1} \text {Li}_3\left (-e^{2 \sinh ^{-1}(a x)}\right )}{a c^2 \sqrt {a^2 c x^2+c}}+\frac {\sqrt {a^2 x^2+1} \log \left (a^2 x^2+1\right )}{2 a c^2 \sqrt {a^2 c x^2+c}}+\frac {2 x \sinh ^{-1}(a x)^3}{3 c^2 \sqrt {a^2 c x^2+c}}+\frac {2 \sqrt {a^2 x^2+1} \sinh ^{-1}(a x)^3}{3 a c^2 \sqrt {a^2 c x^2+c}}+\frac {\sinh ^{-1}(a x)^2}{2 a c^2 \sqrt {a^2 x^2+1} \sqrt {a^2 c x^2+c}}-\frac {x \sinh ^{-1}(a x)}{c^2 \sqrt {a^2 c x^2+c}}-\frac {2 \sqrt {a^2 x^2+1} \sinh ^{-1}(a x)^2 \log \left (e^{2 \sinh ^{-1}(a x)}+1\right )}{a c^2 \sqrt {a^2 c x^2+c}}+\frac {x \sinh ^{-1}(a x)^3}{3 c \left (a^2 c x^2+c\right )^{3/2}} \]
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Rubi [A] time = 0.33, antiderivative size = 363, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 10, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.476, Rules used = {5690, 5687, 5714, 3718, 2190, 2531, 2282, 6589, 5717, 260} \[ -\frac {2 \sqrt {a^2 x^2+1} \sinh ^{-1}(a x) \text {PolyLog}\left (2,-e^{2 \sinh ^{-1}(a x)}\right )}{a c^2 \sqrt {a^2 c x^2+c}}+\frac {\sqrt {a^2 x^2+1} \text {PolyLog}\left (3,-e^{2 \sinh ^{-1}(a x)}\right )}{a c^2 \sqrt {a^2 c x^2+c}}+\frac {\sqrt {a^2 x^2+1} \log \left (a^2 x^2+1\right )}{2 a c^2 \sqrt {a^2 c x^2+c}}+\frac {2 x \sinh ^{-1}(a x)^3}{3 c^2 \sqrt {a^2 c x^2+c}}+\frac {2 \sqrt {a^2 x^2+1} \sinh ^{-1}(a x)^3}{3 a c^2 \sqrt {a^2 c x^2+c}}+\frac {\sinh ^{-1}(a x)^2}{2 a c^2 \sqrt {a^2 x^2+1} \sqrt {a^2 c x^2+c}}-\frac {x \sinh ^{-1}(a x)}{c^2 \sqrt {a^2 c x^2+c}}-\frac {2 \sqrt {a^2 x^2+1} \sinh ^{-1}(a x)^2 \log \left (e^{2 \sinh ^{-1}(a x)}+1\right )}{a c^2 \sqrt {a^2 c x^2+c}}+\frac {x \sinh ^{-1}(a x)^3}{3 c \left (a^2 c x^2+c\right )^{3/2}} \]
Antiderivative was successfully verified.
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Rule 260
Rule 2190
Rule 2282
Rule 2531
Rule 3718
Rule 5687
Rule 5690
Rule 5714
Rule 5717
Rule 6589
Rubi steps
\begin {align*} \int \frac {\sinh ^{-1}(a x)^3}{\left (c+a^2 c x^2\right )^{5/2}} \, dx &=\frac {x \sinh ^{-1}(a x)^3}{3 c \left (c+a^2 c x^2\right )^{3/2}}+\frac {2 \int \frac {\sinh ^{-1}(a x)^3}{\left (c+a^2 c x^2\right )^{3/2}} \, dx}{3 c}-\frac {\left (a \sqrt {1+a^2 x^2}\right ) \int \frac {x \sinh ^{-1}(a x)^2}{\left (1+a^2 x^2\right )^2} \, dx}{c^2 \sqrt {c+a^2 c x^2}}\\ &=\frac {\sinh ^{-1}(a x)^2}{2 a c^2 \sqrt {1+a^2 x^2} \sqrt {c+a^2 c x^2}}+\frac {x \sinh ^{-1}(a x)^3}{3 c \left (c+a^2 c x^2\right )^{3/2}}+\frac {2 x \sinh ^{-1}(a x)^3}{3 c^2 \sqrt {c+a^2 c x^2}}-\frac {\sqrt {1+a^2 x^2} \int \frac {\sinh ^{-1}(a x)}{\left (1+a^2 x^2\right )^{3/2}} \, dx}{c^2 \sqrt {c+a^2 c x^2}}-\frac {\left (2 a \sqrt {1+a^2 x^2}\right ) \int \frac {x \sinh ^{-1}(a x)^2}{1+a^2 x^2} \, dx}{c^2 \sqrt {c+a^2 c x^2}}\\ &=-\frac {x \sinh ^{-1}(a x)}{c^2 \sqrt {c+a^2 c x^2}}+\frac {\sinh ^{-1}(a x)^2}{2 a c^2 \sqrt {1+a^2 x^2} \sqrt {c+a^2 c x^2}}+\frac {x \sinh ^{-1}(a x)^3}{3 c \left (c+a^2 c x^2\right )^{3/2}}+\frac {2 x \sinh ^{-1}(a x)^3}{3 c^2 \sqrt {c+a^2 c x^2}}-\frac {\left (2 \sqrt {1+a^2 x^2}\right ) \operatorname {Subst}\left (\int x^2 \tanh (x) \, dx,x,\sinh ^{-1}(a x)\right )}{a c^2 \sqrt {c+a^2 c x^2}}+\frac {\left (a \sqrt {1+a^2 x^2}\right ) \int \frac {x}{1+a^2 x^2} \, dx}{c^2 \sqrt {c+a^2 c x^2}}\\ &=-\frac {x \sinh ^{-1}(a x)}{c^2 \sqrt {c+a^2 c x^2}}+\frac {\sinh ^{-1}(a x)^2}{2 a c^2 \sqrt {1+a^2 x^2} \sqrt {c+a^2 c x^2}}+\frac {x \sinh ^{-1}(a x)^3}{3 c \left (c+a^2 c x^2\right )^{3/2}}+\frac {2 x \sinh ^{-1}(a x)^3}{3 c^2 \sqrt {c+a^2 c x^2}}+\frac {2 \sqrt {1+a^2 x^2} \sinh ^{-1}(a x)^3}{3 a c^2 \sqrt {c+a^2 c x^2}}+\frac {\sqrt {1+a^2 x^2} \log \left (1+a^2 x^2\right )}{2 a c^2 \sqrt {c+a^2 c x^2}}-\frac {\left (4 \sqrt {1+a^2 x^2}\right ) \operatorname {Subst}\left (\int \frac {e^{2 x} x^2}{1+e^{2 x}} \, dx,x,\sinh ^{-1}(a x)\right )}{a c^2 \sqrt {c+a^2 c x^2}}\\ &=-\frac {x \sinh ^{-1}(a x)}{c^2 \sqrt {c+a^2 c x^2}}+\frac {\sinh ^{-1}(a x)^2}{2 a c^2 \sqrt {1+a^2 x^2} \sqrt {c+a^2 c x^2}}+\frac {x \sinh ^{-1}(a x)^3}{3 c \left (c+a^2 c x^2\right )^{3/2}}+\frac {2 x \sinh ^{-1}(a x)^3}{3 c^2 \sqrt {c+a^2 c x^2}}+\frac {2 \sqrt {1+a^2 x^2} \sinh ^{-1}(a x)^3}{3 a c^2 \sqrt {c+a^2 c x^2}}-\frac {2 \sqrt {1+a^2 x^2} \sinh ^{-1}(a x)^2 \log \left (1+e^{2 \sinh ^{-1}(a x)}\right )}{a c^2 \sqrt {c+a^2 c x^2}}+\frac {\sqrt {1+a^2 x^2} \log \left (1+a^2 x^2\right )}{2 a c^2 \sqrt {c+a^2 c x^2}}+\frac {\left (4 \sqrt {1+a^2 x^2}\right ) \operatorname {Subst}\left (\int x \log \left (1+e^{2 x}\right ) \, dx,x,\sinh ^{-1}(a x)\right )}{a c^2 \sqrt {c+a^2 c x^2}}\\ &=-\frac {x \sinh ^{-1}(a x)}{c^2 \sqrt {c+a^2 c x^2}}+\frac {\sinh ^{-1}(a x)^2}{2 a c^2 \sqrt {1+a^2 x^2} \sqrt {c+a^2 c x^2}}+\frac {x \sinh ^{-1}(a x)^3}{3 c \left (c+a^2 c x^2\right )^{3/2}}+\frac {2 x \sinh ^{-1}(a x)^3}{3 c^2 \sqrt {c+a^2 c x^2}}+\frac {2 \sqrt {1+a^2 x^2} \sinh ^{-1}(a x)^3}{3 a c^2 \sqrt {c+a^2 c x^2}}-\frac {2 \sqrt {1+a^2 x^2} \sinh ^{-1}(a x)^2 \log \left (1+e^{2 \sinh ^{-1}(a x)}\right )}{a c^2 \sqrt {c+a^2 c x^2}}+\frac {\sqrt {1+a^2 x^2} \log \left (1+a^2 x^2\right )}{2 a c^2 \sqrt {c+a^2 c x^2}}-\frac {2 \sqrt {1+a^2 x^2} \sinh ^{-1}(a x) \text {Li}_2\left (-e^{2 \sinh ^{-1}(a x)}\right )}{a c^2 \sqrt {c+a^2 c x^2}}+\frac {\left (2 \sqrt {1+a^2 x^2}\right ) \operatorname {Subst}\left (\int \text {Li}_2\left (-e^{2 x}\right ) \, dx,x,\sinh ^{-1}(a x)\right )}{a c^2 \sqrt {c+a^2 c x^2}}\\ &=-\frac {x \sinh ^{-1}(a x)}{c^2 \sqrt {c+a^2 c x^2}}+\frac {\sinh ^{-1}(a x)^2}{2 a c^2 \sqrt {1+a^2 x^2} \sqrt {c+a^2 c x^2}}+\frac {x \sinh ^{-1}(a x)^3}{3 c \left (c+a^2 c x^2\right )^{3/2}}+\frac {2 x \sinh ^{-1}(a x)^3}{3 c^2 \sqrt {c+a^2 c x^2}}+\frac {2 \sqrt {1+a^2 x^2} \sinh ^{-1}(a x)^3}{3 a c^2 \sqrt {c+a^2 c x^2}}-\frac {2 \sqrt {1+a^2 x^2} \sinh ^{-1}(a x)^2 \log \left (1+e^{2 \sinh ^{-1}(a x)}\right )}{a c^2 \sqrt {c+a^2 c x^2}}+\frac {\sqrt {1+a^2 x^2} \log \left (1+a^2 x^2\right )}{2 a c^2 \sqrt {c+a^2 c x^2}}-\frac {2 \sqrt {1+a^2 x^2} \sinh ^{-1}(a x) \text {Li}_2\left (-e^{2 \sinh ^{-1}(a x)}\right )}{a c^2 \sqrt {c+a^2 c x^2}}+\frac {\sqrt {1+a^2 x^2} \operatorname {Subst}\left (\int \frac {\text {Li}_2(-x)}{x} \, dx,x,e^{2 \sinh ^{-1}(a x)}\right )}{a c^2 \sqrt {c+a^2 c x^2}}\\ &=-\frac {x \sinh ^{-1}(a x)}{c^2 \sqrt {c+a^2 c x^2}}+\frac {\sinh ^{-1}(a x)^2}{2 a c^2 \sqrt {1+a^2 x^2} \sqrt {c+a^2 c x^2}}+\frac {x \sinh ^{-1}(a x)^3}{3 c \left (c+a^2 c x^2\right )^{3/2}}+\frac {2 x \sinh ^{-1}(a x)^3}{3 c^2 \sqrt {c+a^2 c x^2}}+\frac {2 \sqrt {1+a^2 x^2} \sinh ^{-1}(a x)^3}{3 a c^2 \sqrt {c+a^2 c x^2}}-\frac {2 \sqrt {1+a^2 x^2} \sinh ^{-1}(a x)^2 \log \left (1+e^{2 \sinh ^{-1}(a x)}\right )}{a c^2 \sqrt {c+a^2 c x^2}}+\frac {\sqrt {1+a^2 x^2} \log \left (1+a^2 x^2\right )}{2 a c^2 \sqrt {c+a^2 c x^2}}-\frac {2 \sqrt {1+a^2 x^2} \sinh ^{-1}(a x) \text {Li}_2\left (-e^{2 \sinh ^{-1}(a x)}\right )}{a c^2 \sqrt {c+a^2 c x^2}}+\frac {\sqrt {1+a^2 x^2} \text {Li}_3\left (-e^{2 \sinh ^{-1}(a x)}\right )}{a c^2 \sqrt {c+a^2 c x^2}}\\ \end {align*}
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Mathematica [A] time = 0.55, size = 195, normalized size = 0.54 \[ \frac {\left (a^2 x^2+1\right )^{3/2} \left (3 \log \left (a^2 x^2+1\right )+\frac {4 a x \sinh ^{-1}(a x)^3}{\sqrt {a^2 x^2+1}}+\frac {2 a x \sinh ^{-1}(a x)^3}{\left (a^2 x^2+1\right )^{3/2}}+\frac {3 \sinh ^{-1}(a x)^2}{a^2 x^2+1}-\frac {6 a x \sinh ^{-1}(a x)}{\sqrt {a^2 x^2+1}}+12 \sinh ^{-1}(a x) \text {Li}_2\left (-e^{-2 \sinh ^{-1}(a x)}\right )+6 \text {Li}_3\left (-e^{-2 \sinh ^{-1}(a x)}\right )-4 \sinh ^{-1}(a x)^3-12 \sinh ^{-1}(a x)^2 \log \left (e^{-2 \sinh ^{-1}(a x)}+1\right )\right )}{6 a c \left (a^2 c x^2+c\right )^{3/2}} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.48, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {a^{2} c x^{2} + c} \operatorname {arsinh}\left (a x\right )^{3}}{a^{6} c^{3} x^{6} + 3 \, a^{4} c^{3} x^{4} + 3 \, a^{2} c^{3} x^{2} + c^{3}}, x\right ) \]
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 = 0.27, size = 550, normalized size = 1.52 \[ \frac {\sqrt {c \left (a^{2} x^{2}+1\right )}\, \left (2 x^{3} a^{3}-2 \sqrt {a^{2} x^{2}+1}\, x^{2} a^{2}+3 a x -2 \sqrt {a^{2} x^{2}+1}\right ) \arcsinh \left (a x \right ) \left (-6 \arcsinh \left (a x \right ) x^{4} a^{4}-6 \arcsinh \left (a x \right ) \sqrt {a^{2} x^{2}+1}\, a^{3} x^{3}-6 x^{4} a^{4}-6 \sqrt {a^{2} x^{2}+1}\, x^{3} a^{3}+6 \arcsinh \left (a x \right )^{2} a^{2} x^{2}-12 \arcsinh \left (a x \right ) x^{2} a^{2}-9 \arcsinh \left (a x \right ) \sqrt {a^{2} x^{2}+1}\, a x -18 a^{2} x^{2}-6 \sqrt {a^{2} x^{2}+1}\, x a +8 \arcsinh \left (a x \right )^{2}-6 \arcsinh \left (a x \right )-12\right )}{6 \left (3 x^{6} a^{6}+10 x^{4} a^{4}+11 a^{2} x^{2}+4\right ) a \,c^{3}}-\frac {2 \sqrt {c \left (a^{2} x^{2}+1\right )}\, \ln \left (a x +\sqrt {a^{2} x^{2}+1}\right )}{\sqrt {a^{2} x^{2}+1}\, a \,c^{3}}+\frac {\sqrt {c \left (a^{2} x^{2}+1\right )}\, \ln \left (1+\left (a x +\sqrt {a^{2} x^{2}+1}\right )^{2}\right )}{\sqrt {a^{2} x^{2}+1}\, a \,c^{3}}+\frac {4 \sqrt {c \left (a^{2} x^{2}+1\right )}\, \arcsinh \left (a x \right )^{3}}{3 \sqrt {a^{2} x^{2}+1}\, a \,c^{3}}-\frac {2 \sqrt {c \left (a^{2} x^{2}+1\right )}\, \arcsinh \left (a x \right )^{2} \ln \left (1+\left (a x +\sqrt {a^{2} x^{2}+1}\right )^{2}\right )}{\sqrt {a^{2} x^{2}+1}\, a \,c^{3}}-\frac {2 \sqrt {c \left (a^{2} x^{2}+1\right )}\, \arcsinh \left (a x \right ) \polylog \left (2, -\left (a x +\sqrt {a^{2} x^{2}+1}\right )^{2}\right )}{\sqrt {a^{2} x^{2}+1}\, a \,c^{3}}+\frac {\sqrt {c \left (a^{2} x^{2}+1\right )}\, \polylog \left (3, -\left (a x +\sqrt {a^{2} x^{2}+1}\right )^{2}\right )}{\sqrt {a^{2} x^{2}+1}\, a \,c^{3}} \]
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 \[ \int \frac {\operatorname {arsinh}\left (a x\right )^{3}}{{\left (a^{2} c x^{2} + c\right )}^{\frac {5}{2}}}\,{d x} \]
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 \[ \int \frac {{\mathrm {asinh}\left (a\,x\right )}^3}{{\left (c\,a^2\,x^2+c\right )}^{5/2}} \,d x \]
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 \[ \int \frac {\operatorname {asinh}^{3}{\left (a x \right )}}{\left (c \left (a^{2} x^{2} + 1\right )\right )^{\frac {5}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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