Optimal. Leaf size=268 \[ -\frac {\cosh \left (\frac {2 a}{b}\right ) \text {Chi}\left (\frac {2 \left (a+b \sinh ^{-1}(c x)\right )}{b}\right )}{32 b c^3}+\frac {\cosh \left (\frac {4 a}{b}\right ) \text {Chi}\left (\frac {4 \left (a+b \sinh ^{-1}(c x)\right )}{b}\right )}{32 b c^3}+\frac {\cosh \left (\frac {6 a}{b}\right ) \text {Chi}\left (\frac {6 \left (a+b \sinh ^{-1}(c x)\right )}{b}\right )}{32 b c^3}+\frac {\cosh \left (\frac {8 a}{b}\right ) \text {Chi}\left (\frac {8 \left (a+b \sinh ^{-1}(c x)\right )}{b}\right )}{128 b c^3}+\frac {\sinh \left (\frac {2 a}{b}\right ) \text {Shi}\left (\frac {2 \left (a+b \sinh ^{-1}(c x)\right )}{b}\right )}{32 b c^3}-\frac {\sinh \left (\frac {4 a}{b}\right ) \text {Shi}\left (\frac {4 \left (a+b \sinh ^{-1}(c x)\right )}{b}\right )}{32 b c^3}-\frac {\sinh \left (\frac {6 a}{b}\right ) \text {Shi}\left (\frac {6 \left (a+b \sinh ^{-1}(c x)\right )}{b}\right )}{32 b c^3}-\frac {\sinh \left (\frac {8 a}{b}\right ) \text {Shi}\left (\frac {8 \left (a+b \sinh ^{-1}(c x)\right )}{b}\right )}{128 b c^3}-\frac {5 \log \left (a+b \sinh ^{-1}(c x)\right )}{128 b c^3} \]
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Rubi [A] time = 0.62, antiderivative size = 268, normalized size of antiderivative = 1.00, number of steps used = 15, number of rules used = 5, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.185, Rules used = {5779, 5448, 3303, 3298, 3301} \[ -\frac {\cosh \left (\frac {2 a}{b}\right ) \text {Chi}\left (\frac {2 a}{b}+2 \sinh ^{-1}(c x)\right )}{32 b c^3}+\frac {\cosh \left (\frac {4 a}{b}\right ) \text {Chi}\left (\frac {4 a}{b}+4 \sinh ^{-1}(c x)\right )}{32 b c^3}+\frac {\cosh \left (\frac {6 a}{b}\right ) \text {Chi}\left (\frac {6 a}{b}+6 \sinh ^{-1}(c x)\right )}{32 b c^3}+\frac {\cosh \left (\frac {8 a}{b}\right ) \text {Chi}\left (\frac {8 a}{b}+8 \sinh ^{-1}(c x)\right )}{128 b c^3}+\frac {\sinh \left (\frac {2 a}{b}\right ) \text {Shi}\left (\frac {2 a}{b}+2 \sinh ^{-1}(c x)\right )}{32 b c^3}-\frac {\sinh \left (\frac {4 a}{b}\right ) \text {Shi}\left (\frac {4 a}{b}+4 \sinh ^{-1}(c x)\right )}{32 b c^3}-\frac {\sinh \left (\frac {6 a}{b}\right ) \text {Shi}\left (\frac {6 a}{b}+6 \sinh ^{-1}(c x)\right )}{32 b c^3}-\frac {\sinh \left (\frac {8 a}{b}\right ) \text {Shi}\left (\frac {8 a}{b}+8 \sinh ^{-1}(c x)\right )}{128 b c^3}-\frac {5 \log \left (a+b \sinh ^{-1}(c x)\right )}{128 b c^3} \]
Antiderivative was successfully verified.
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Rule 3298
Rule 3301
Rule 3303
Rule 5448
Rule 5779
Rubi steps
\begin {align*} \int \frac {x^2 \left (1+c^2 x^2\right )^{5/2}}{a+b \sinh ^{-1}(c x)} \, dx &=\frac {\operatorname {Subst}\left (\int \frac {\cosh ^6(x) \sinh ^2(x)}{a+b x} \, dx,x,\sinh ^{-1}(c x)\right )}{c^3}\\ &=\frac {\operatorname {Subst}\left (\int \left (-\frac {5}{128 (a+b x)}-\frac {\cosh (2 x)}{32 (a+b x)}+\frac {\cosh (4 x)}{32 (a+b x)}+\frac {\cosh (6 x)}{32 (a+b x)}+\frac {\cosh (8 x)}{128 (a+b x)}\right ) \, dx,x,\sinh ^{-1}(c x)\right )}{c^3}\\ &=-\frac {5 \log \left (a+b \sinh ^{-1}(c x)\right )}{128 b c^3}+\frac {\operatorname {Subst}\left (\int \frac {\cosh (8 x)}{a+b x} \, dx,x,\sinh ^{-1}(c x)\right )}{128 c^3}-\frac {\operatorname {Subst}\left (\int \frac {\cosh (2 x)}{a+b x} \, dx,x,\sinh ^{-1}(c x)\right )}{32 c^3}+\frac {\operatorname {Subst}\left (\int \frac {\cosh (4 x)}{a+b x} \, dx,x,\sinh ^{-1}(c x)\right )}{32 c^3}+\frac {\operatorname {Subst}\left (\int \frac {\cosh (6 x)}{a+b x} \, dx,x,\sinh ^{-1}(c x)\right )}{32 c^3}\\ &=-\frac {5 \log \left (a+b \sinh ^{-1}(c x)\right )}{128 b c^3}-\frac {\cosh \left (\frac {2 a}{b}\right ) \operatorname {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 ) \operatorname {Subst}\left (\int \frac {\cosh \left (\frac {4 a}{b}+4 x\right )}{a+b x} \, dx,x,\sinh ^{-1}(c x)\right )}{32 c^3}+\frac {\cosh \left (\frac {6 a}{b}\right ) \operatorname {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 {\cosh \left (\frac {8 a}{b}\right ) \operatorname {Subst}\left (\int \frac {\cosh \left (\frac {8 a}{b}+8 x\right )}{a+b x} \, dx,x,\sinh ^{-1}(c x)\right )}{128 c^3}+\frac {\sinh \left (\frac {2 a}{b}\right ) \operatorname {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 ) \operatorname {Subst}\left (\int \frac {\sinh \left (\frac {4 a}{b}+4 x\right )}{a+b x} \, dx,x,\sinh ^{-1}(c x)\right )}{32 c^3}-\frac {\sinh \left (\frac {6 a}{b}\right ) \operatorname {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 {\sinh \left (\frac {8 a}{b}\right ) \operatorname {Subst}\left (\int \frac {\sinh \left (\frac {8 a}{b}+8 x\right )}{a+b x} \, dx,x,\sinh ^{-1}(c x)\right )}{128 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 )}{32 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 {\cosh \left (\frac {8 a}{b}\right ) \text {Chi}\left (\frac {8 a}{b}+8 \sinh ^{-1}(c x)\right )}{128 b c^3}-\frac {5 \log \left (a+b \sinh ^{-1}(c x)\right )}{128 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 )}{32 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}-\frac {\sinh \left (\frac {8 a}{b}\right ) \text {Shi}\left (\frac {8 a}{b}+8 \sinh ^{-1}(c x)\right )}{128 b c^3}\\ \end {align*}
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Mathematica [A] time = 0.88, size = 197, normalized size = 0.74 \[ \frac {-4 \cosh \left (\frac {2 a}{b}\right ) \text {Chi}\left (2 \left (\frac {a}{b}+\sinh ^{-1}(c x)\right )\right )+4 \cosh \left (\frac {4 a}{b}\right ) \text {Chi}\left (4 \left (\frac {a}{b}+\sinh ^{-1}(c x)\right )\right )+4 \cosh \left (\frac {6 a}{b}\right ) \text {Chi}\left (6 \left (\frac {a}{b}+\sinh ^{-1}(c x)\right )\right )+\cosh \left (\frac {8 a}{b}\right ) \text {Chi}\left (8 \left (\frac {a}{b}+\sinh ^{-1}(c x)\right )\right )+4 \sinh \left (\frac {2 a}{b}\right ) \text {Shi}\left (2 \left (\frac {a}{b}+\sinh ^{-1}(c x)\right )\right )-4 \sinh \left (\frac {4 a}{b}\right ) \text {Shi}\left (4 \left (\frac {a}{b}+\sinh ^{-1}(c x)\right )\right )-4 \sinh \left (\frac {6 a}{b}\right ) \text {Shi}\left (6 \left (\frac {a}{b}+\sinh ^{-1}(c x)\right )\right )-\sinh \left (\frac {8 a}{b}\right ) \text {Shi}\left (8 \left (\frac {a}{b}+\sinh ^{-1}(c x)\right )\right )-5 \log \left (a+b \sinh ^{-1}(c x)\right )}{128 b c^3} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.49, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (c^{4} x^{6} + 2 \, c^{2} x^{4} + x^{2}\right )} \sqrt {c^{2} x^{2} + 1}}{b \operatorname {arsinh}\left (c x\right ) + a}, x\right ) \]
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 \[ \int \frac {{\left (c^{2} x^{2} + 1\right )}^{\frac {5}{2}} x^{2}}{b \operatorname {arsinh}\left (c x\right ) + a}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.60, size = 259, normalized size = 0.97 \[ -\frac {5 \ln \left (a +b \arcsinh \left (c x \right )\right )}{128 b \,c^{3}}-\frac {{\mathrm e}^{\frac {8 a}{b}} \Ei \left (1, 8 \arcsinh \left (c x \right )+\frac {8 a}{b}\right )}{256 c^{3} b}-\frac {{\mathrm e}^{\frac {6 a}{b}} \Ei \left (1, 6 \arcsinh \left (c x \right )+\frac {6 a}{b}\right )}{64 c^{3} b}-\frac {{\mathrm e}^{\frac {4 a}{b}} \Ei \left (1, 4 \arcsinh \left (c x \right )+\frac {4 a}{b}\right )}{64 c^{3} b}+\frac {{\mathrm e}^{\frac {2 a}{b}} \Ei \left (1, 2 \arcsinh \left (c x \right )+\frac {2 a}{b}\right )}{64 c^{3} b}+\frac {{\mathrm e}^{-\frac {2 a}{b}} \Ei \left (1, -2 \arcsinh \left (c x \right )-\frac {2 a}{b}\right )}{64 c^{3} b}-\frac {{\mathrm e}^{-\frac {4 a}{b}} \Ei \left (1, -4 \arcsinh \left (c x \right )-\frac {4 a}{b}\right )}{64 c^{3} b}-\frac {{\mathrm e}^{-\frac {6 a}{b}} \Ei \left (1, -6 \arcsinh \left (c x \right )-\frac {6 a}{b}\right )}{64 c^{3} b}-\frac {{\mathrm e}^{-\frac {8 a}{b}} \Ei \left (1, -8 \arcsinh \left (c x \right )-\frac {8 a}{b}\right )}{256 c^{3} b} \]
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 \[ \int \frac {{\left (c^{2} x^{2} + 1\right )}^{\frac {5}{2}} x^{2}}{b \operatorname {arsinh}\left (c x\right ) + a}\,{d x} \]
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 \[ \int \frac {x^2\,{\left (c^2\,x^2+1\right )}^{5/2}}{a+b\,\mathrm {asinh}\left (c\,x\right )} \,d x \]
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 \[ \int \frac {x^{2} \left (c^{2} x^{2} + 1\right )^{\frac {5}{2}}}{a + b \operatorname {asinh}{\left (c x \right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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