Optimal. Leaf size=206 \[ -\frac {\cosh \left (\frac {2 a}{b}\right ) \text {Chi}\left (\frac {2 \left (a+b \sinh ^{-1}(c x)\right )}{b}\right )}{32 b c^3}+\frac {\cosh \left (\frac {4 a}{b}\right ) \text {Chi}\left (\frac {4 \left (a+b \sinh ^{-1}(c x)\right )}{b}\right )}{16 b c^3}+\frac {\cosh \left (\frac {6 a}{b}\right ) \text {Chi}\left (\frac {6 \left (a+b \sinh ^{-1}(c x)\right )}{b}\right )}{32 b c^3}-\frac {\log \left (a+b \sinh ^{-1}(c x)\right )}{16 b c^3}+\frac {\sinh \left (\frac {2 a}{b}\right ) \text {Shi}\left (\frac {2 \left (a+b \sinh ^{-1}(c x)\right )}{b}\right )}{32 b c^3}-\frac {\sinh \left (\frac {4 a}{b}\right ) \text {Shi}\left (\frac {4 \left (a+b \sinh ^{-1}(c x)\right )}{b}\right )}{16 b c^3}-\frac {\sinh \left (\frac {6 a}{b}\right ) \text {Shi}\left (\frac {6 \left (a+b \sinh ^{-1}(c x)\right )}{b}\right )}{32 b c^3} \]
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Rubi [A]
time = 0.28, antiderivative size = 206, normalized size of antiderivative = 1.00, number of steps
used = 12, number of rules used = 5, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.185, Rules used = {5819, 5556,
3384, 3379, 3382} \begin {gather*} -\frac {\cosh \left (\frac {2 a}{b}\right ) \text {Chi}\left (\frac {2 \left (a+b \sinh ^{-1}(c x)\right )}{b}\right )}{32 b c^3}+\frac {\cosh \left (\frac {4 a}{b}\right ) \text {Chi}\left (\frac {4 \left (a+b \sinh ^{-1}(c x)\right )}{b}\right )}{16 b c^3}+\frac {\cosh \left (\frac {6 a}{b}\right ) \text {Chi}\left (\frac {6 \left (a+b \sinh ^{-1}(c x)\right )}{b}\right )}{32 b c^3}+\frac {\sinh \left (\frac {2 a}{b}\right ) \text {Shi}\left (\frac {2 \left (a+b \sinh ^{-1}(c x)\right )}{b}\right )}{32 b c^3}-\frac {\sinh \left (\frac {4 a}{b}\right ) \text {Shi}\left (\frac {4 \left (a+b \sinh ^{-1}(c x)\right )}{b}\right )}{16 b c^3}-\frac {\sinh \left (\frac {6 a}{b}\right ) \text {Shi}\left (\frac {6 \left (a+b \sinh ^{-1}(c x)\right )}{b}\right )}{32 b c^3}-\frac {\log \left (a+b \sinh ^{-1}(c x)\right )}{16 b c^3} \end {gather*}
Antiderivative was successfully verified.
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Rule 3379
Rule 3382
Rule 3384
Rule 5556
Rule 5819
Rubi steps
\begin {align*} \int \frac {x^2 \left (1+c^2 x^2\right )^{3/2}}{a+b \sinh ^{-1}(c x)} \, dx &=\frac {\text {Subst}\left (\int \frac {\cosh ^4(x) \sinh ^2(x)}{a+b x} \, dx,x,\sinh ^{-1}(c x)\right )}{c^3}\\ &=\frac {\text {Subst}\left (\int \left (-\frac {1}{16 (a+b x)}-\frac {\cosh (2 x)}{32 (a+b x)}+\frac {\cosh (4 x)}{16 (a+b x)}+\frac {\cosh (6 x)}{32 (a+b x)}\right ) \, dx,x,\sinh ^{-1}(c x)\right )}{c^3}\\ &=-\frac {\log \left (a+b \sinh ^{-1}(c x)\right )}{16 b c^3}-\frac {\text {Subst}\left (\int \frac {\cosh (2 x)}{a+b x} \, dx,x,\sinh ^{-1}(c x)\right )}{32 c^3}+\frac {\text {Subst}\left (\int \frac {\cosh (6 x)}{a+b x} \, dx,x,\sinh ^{-1}(c x)\right )}{32 c^3}+\frac {\text {Subst}\left (\int \frac {\cosh (4 x)}{a+b x} \, dx,x,\sinh ^{-1}(c x)\right )}{16 c^3}\\ &=-\frac {\log \left (a+b \sinh ^{-1}(c x)\right )}{16 b c^3}-\frac {\cosh \left (\frac {2 a}{b}\right ) \text {Subst}\left (\int \frac {\cosh \left (\frac {2 a}{b}+2 x\right )}{a+b x} \, dx,x,\sinh ^{-1}(c x)\right )}{32 c^3}+\frac {\cosh \left (\frac {4 a}{b}\right ) \text {Subst}\left (\int \frac {\cosh \left (\frac {4 a}{b}+4 x\right )}{a+b x} \, dx,x,\sinh ^{-1}(c x)\right )}{16 c^3}+\frac {\cosh \left (\frac {6 a}{b}\right ) \text {Subst}\left (\int \frac {\cosh \left (\frac {6 a}{b}+6 x\right )}{a+b x} \, dx,x,\sinh ^{-1}(c x)\right )}{32 c^3}+\frac {\sinh \left (\frac {2 a}{b}\right ) \text {Subst}\left (\int \frac {\sinh \left (\frac {2 a}{b}+2 x\right )}{a+b x} \, dx,x,\sinh ^{-1}(c x)\right )}{32 c^3}-\frac {\sinh \left (\frac {4 a}{b}\right ) \text {Subst}\left (\int \frac {\sinh \left (\frac {4 a}{b}+4 x\right )}{a+b x} \, dx,x,\sinh ^{-1}(c x)\right )}{16 c^3}-\frac {\sinh \left (\frac {6 a}{b}\right ) \text {Subst}\left (\int \frac {\sinh \left (\frac {6 a}{b}+6 x\right )}{a+b x} \, dx,x,\sinh ^{-1}(c x)\right )}{32 c^3}\\ &=-\frac {\cosh \left (\frac {2 a}{b}\right ) \text {Chi}\left (\frac {2 a}{b}+2 \sinh ^{-1}(c x)\right )}{32 b c^3}+\frac {\cosh \left (\frac {4 a}{b}\right ) \text {Chi}\left (\frac {4 a}{b}+4 \sinh ^{-1}(c x)\right )}{16 b c^3}+\frac {\cosh \left (\frac {6 a}{b}\right ) \text {Chi}\left (\frac {6 a}{b}+6 \sinh ^{-1}(c x)\right )}{32 b c^3}-\frac {\log \left (a+b \sinh ^{-1}(c x)\right )}{16 b c^3}+\frac {\sinh \left (\frac {2 a}{b}\right ) \text {Shi}\left (\frac {2 a}{b}+2 \sinh ^{-1}(c x)\right )}{32 b c^3}-\frac {\sinh \left (\frac {4 a}{b}\right ) \text {Shi}\left (\frac {4 a}{b}+4 \sinh ^{-1}(c x)\right )}{16 b c^3}-\frac {\sinh \left (\frac {6 a}{b}\right ) \text {Shi}\left (\frac {6 a}{b}+6 \sinh ^{-1}(c x)\right )}{32 b c^3}\\ \end {align*}
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Mathematica [A]
time = 0.35, size = 152, normalized size = 0.74 \begin {gather*} \frac {-\cosh \left (\frac {2 a}{b}\right ) \text {Chi}\left (2 \left (\frac {a}{b}+\sinh ^{-1}(c x)\right )\right )+2 \cosh \left (\frac {4 a}{b}\right ) \text {Chi}\left (4 \left (\frac {a}{b}+\sinh ^{-1}(c x)\right )\right )+\cosh \left (\frac {6 a}{b}\right ) \text {Chi}\left (6 \left (\frac {a}{b}+\sinh ^{-1}(c x)\right )\right )-2 \log \left (a+b \sinh ^{-1}(c x)\right )+\sinh \left (\frac {2 a}{b}\right ) \text {Shi}\left (2 \left (\frac {a}{b}+\sinh ^{-1}(c x)\right )\right )-2 \sinh \left (\frac {4 a}{b}\right ) \text {Shi}\left (4 \left (\frac {a}{b}+\sinh ^{-1}(c x)\right )\right )-\sinh \left (\frac {6 a}{b}\right ) \text {Shi}\left (6 \left (\frac {a}{b}+\sinh ^{-1}(c x)\right )\right )}{32 b c^3} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 8.98, size = 199, normalized size = 0.97
method | result | size |
default | \(-\frac {\ln \left (a +b \arcsinh \left (c x \right )\right )}{16 b \,c^{3}}-\frac {{\mathrm e}^{\frac {6 a}{b}} \expIntegral \left (1, 6 \arcsinh \left (c x \right )+\frac {6 a}{b}\right )}{64 c^{3} b}-\frac {{\mathrm e}^{\frac {4 a}{b}} \expIntegral \left (1, 4 \arcsinh \left (c x \right )+\frac {4 a}{b}\right )}{32 c^{3} b}+\frac {{\mathrm e}^{\frac {2 a}{b}} \expIntegral \left (1, 2 \arcsinh \left (c x \right )+\frac {2 a}{b}\right )}{64 c^{3} b}+\frac {{\mathrm e}^{-\frac {2 a}{b}} \expIntegral \left (1, -2 \arcsinh \left (c x \right )-\frac {2 a}{b}\right )}{64 c^{3} b}-\frac {{\mathrm e}^{-\frac {4 a}{b}} \expIntegral \left (1, -4 \arcsinh \left (c x \right )-\frac {4 a}{b}\right )}{32 c^{3} b}-\frac {{\mathrm e}^{-\frac {6 a}{b}} \expIntegral \left (1, -6 \arcsinh \left (c x \right )-\frac {6 a}{b}\right )}{64 c^{3} b}\) | \(199\) |
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 {x^{2} \left (c^{2} x^{2} + 1\right )^{\frac {3}{2}}}{a + b \operatorname {asinh}{\left (c x \right )}}\, dx \end {gather*}
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 \begin {gather*} \text {could not integrate} \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 {x^2\,{\left (c^2\,x^2+1\right )}^{3/2}}{a+b\,\mathrm {asinh}\left (c\,x\right )} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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