Optimal. Leaf size=363 \[ -\frac {x \sinh ^{-1}(a x)}{c^2 \sqrt {c+a^2 c x^2}}+\frac {\sinh ^{-1}(a x)^2}{2 a c^2 \sqrt {1+a^2 x^2} \sqrt {c+a^2 c x^2}}+\frac {x \sinh ^{-1}(a x)^3}{3 c \left (c+a^2 c x^2\right )^{3/2}}+\frac {2 x \sinh ^{-1}(a x)^3}{3 c^2 \sqrt {c+a^2 c x^2}}+\frac {2 \sqrt {1+a^2 x^2} \sinh ^{-1}(a x)^3}{3 a c^2 \sqrt {c+a^2 c x^2}}-\frac {2 \sqrt {1+a^2 x^2} \sinh ^{-1}(a x)^2 \log \left (1+e^{2 \sinh ^{-1}(a x)}\right )}{a c^2 \sqrt {c+a^2 c x^2}}+\frac {\sqrt {1+a^2 x^2} \log \left (1+a^2 x^2\right )}{2 a c^2 \sqrt {c+a^2 c x^2}}-\frac {2 \sqrt {1+a^2 x^2} \sinh ^{-1}(a x) \text {PolyLog}\left (2,-e^{2 \sinh ^{-1}(a x)}\right )}{a c^2 \sqrt {c+a^2 c x^2}}+\frac {\sqrt {1+a^2 x^2} \text {PolyLog}\left (3,-e^{2 \sinh ^{-1}(a x)}\right )}{a c^2 \sqrt {c+a^2 c x^2}} \]
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Rubi [A]
time = 0.23, 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 = {5788, 5787,
5797, 3799, 2221, 2611, 2320, 6724, 5798, 266} \begin {gather*} -\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}} \end {gather*}
Antiderivative was successfully verified.
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Rule 266
Rule 2221
Rule 2320
Rule 2611
Rule 3799
Rule 5787
Rule 5788
Rule 5797
Rule 5798
Rule 6724
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 ) \text {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 ) \text {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 ) \text {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 ) \text {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} \text {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 = 0.37, size = 195, normalized size = 0.54 \begin {gather*} \frac {\left (1+a^2 x^2\right )^{3/2} \left (-\frac {6 a x \sinh ^{-1}(a x)}{\sqrt {1+a^2 x^2}}+\frac {3 \sinh ^{-1}(a x)^2}{1+a^2 x^2}-4 \sinh ^{-1}(a x)^3+\frac {2 a x \sinh ^{-1}(a x)^3}{\left (1+a^2 x^2\right )^{3/2}}+\frac {4 a x \sinh ^{-1}(a x)^3}{\sqrt {1+a^2 x^2}}-12 \sinh ^{-1}(a x)^2 \log \left (1+e^{-2 \sinh ^{-1}(a x)}\right )+3 \log \left (1+a^2 x^2\right )+12 \sinh ^{-1}(a x) \text {PolyLog}\left (2,-e^{-2 \sinh ^{-1}(a x)}\right )+6 \text {PolyLog}\left (3,-e^{-2 \sinh ^{-1}(a x)}\right )\right )}{6 a c \left (c+a^2 c x^2\right )^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 2.87, size = 550, normalized size = 1.52
method | result | size |
default | \(\frac {\sqrt {c \left (a^{2} x^{2}+1\right )}\, \left (2 a^{3} x^{3}-2 \sqrt {a^{2} x^{2}+1}\, a^{2} x^{2}+3 a x -2 \sqrt {a^{2} x^{2}+1}\right ) \arcsinh \left (a x \right ) \left (-6 \arcsinh \left (a x \right ) a^{4} x^{4}-6 \sqrt {a^{2} x^{2}+1}\, \arcsinh \left (a x \right ) a^{3} x^{3}-6 a^{4} x^{4}-6 \sqrt {a^{2} x^{2}+1}\, a^{3} x^{3}+6 \arcsinh \left (a x \right )^{2} a^{2} x^{2}-12 x^{2} \arcsinh \left (a x \right ) a^{2}-9 \sqrt {a^{2} x^{2}+1}\, \arcsinh \left (a x \right ) a x -18 a^{2} x^{2}-6 \sqrt {a^{2} x^{2}+1}\, a x +8 \arcsinh \left (a x \right )^{2}-6 \arcsinh \left (a x \right )-12\right )}{6 \left (3 a^{6} x^{6}+10 a^{4} x^{4}+11 a^{2} x^{2}+4\right ) a \,c^{3}}+\frac {\sqrt {c \left (a^{2} x^{2}+1\right )}\, \ln \left (1+\left (a x +\sqrt {a^{2} x^{2}+1}\right )^{2}\right )}{\sqrt {a^{2} x^{2}+1}\, a \,c^{3}}-\frac {2 \sqrt {c \left (a^{2} x^{2}+1\right )}\, \ln \left (a x +\sqrt {a^{2} x^{2}+1}\right )}{\sqrt {a^{2} x^{2}+1}\, a \,c^{3}}+\frac {4 \sqrt {c \left (a^{2} x^{2}+1\right )}\, \arcsinh \left (a x \right )^{3}}{3 \sqrt {a^{2} x^{2}+1}\, a \,c^{3}}-\frac {2 \sqrt {c \left (a^{2} x^{2}+1\right )}\, \arcsinh \left (a x \right )^{2} \ln \left (1+\left (a x +\sqrt {a^{2} x^{2}+1}\right )^{2}\right )}{\sqrt {a^{2} x^{2}+1}\, a \,c^{3}}-\frac {2 \sqrt {c \left (a^{2} x^{2}+1\right )}\, \arcsinh \left (a x \right ) \polylog \left (2, -\left (a x +\sqrt {a^{2} x^{2}+1}\right )^{2}\right )}{\sqrt {a^{2} x^{2}+1}\, a \,c^{3}}+\frac {\sqrt {c \left (a^{2} x^{2}+1\right )}\, \polylog \left (3, -\left (a x +\sqrt {a^{2} x^{2}+1}\right )^{2}\right )}{\sqrt {a^{2} x^{2}+1}\, a \,c^{3}}\) | \(550\) |
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 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [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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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\operatorname {asinh}^{3}{\left (a x \right )}}{\left (c \left (a^{2} x^{2} + 1\right )\right )^{\frac {5}{2}}}\, dx \end {gather*}
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 \begin {gather*} \text {Exception raised: TypeError} \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 = -0.00 \begin {gather*} \int \frac {{\mathrm {asinh}\left (a\,x\right )}^3}{{\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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