Optimal. Leaf size=60 \[ \frac {a^2 \text {Chi}\left (\sinh ^{-1}(a+b x)\right )}{b^3}-\frac {\text {Chi}\left (\sinh ^{-1}(a+b x)\right )}{4 b^3}+\frac {\text {Chi}\left (3 \sinh ^{-1}(a+b x)\right )}{4 b^3}-\frac {a \text {Shi}\left (2 \sinh ^{-1}(a+b x)\right )}{b^3} \]
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Rubi [A] time = 0.53, antiderivative size = 60, normalized size of antiderivative = 1.00, number of steps used = 14, number of rules used = 8, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.667, Rules used = {5865, 5805, 6741, 12, 6742, 3301, 5448, 3298} \[ \frac {a^2 \text {Chi}\left (\sinh ^{-1}(a+b x)\right )}{b^3}-\frac {\text {Chi}\left (\sinh ^{-1}(a+b x)\right )}{4 b^3}+\frac {\text {Chi}\left (3 \sinh ^{-1}(a+b x)\right )}{4 b^3}-\frac {a \text {Shi}\left (2 \sinh ^{-1}(a+b x)\right )}{b^3} \]
Antiderivative was successfully verified.
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Rule 12
Rule 3298
Rule 3301
Rule 5448
Rule 5805
Rule 5865
Rule 6741
Rule 6742
Rubi steps
\begin {align*} \int \frac {x^2}{\sinh ^{-1}(a+b x)} \, dx &=\frac {\operatorname {Subst}\left (\int \frac {\left (-\frac {a}{b}+\frac {x}{b}\right )^2}{\sinh ^{-1}(x)} \, dx,x,a+b x\right )}{b}\\ &=\frac {\operatorname {Subst}\left (\int \frac {\cosh (x) \left (-\frac {a}{b}+\frac {\sinh (x)}{b}\right )^2}{x} \, dx,x,\sinh ^{-1}(a+b x)\right )}{b}\\ &=\frac {\operatorname {Subst}\left (\int \frac {\cosh (x) (a-\sinh (x))^2}{b^2 x} \, dx,x,\sinh ^{-1}(a+b x)\right )}{b}\\ &=\frac {\operatorname {Subst}\left (\int \frac {\cosh (x) (a-\sinh (x))^2}{x} \, dx,x,\sinh ^{-1}(a+b x)\right )}{b^3}\\ &=\frac {\operatorname {Subst}\left (\int \left (\frac {a^2 \cosh (x)}{x}-\frac {2 a \cosh (x) \sinh (x)}{x}+\frac {\cosh (x) \sinh ^2(x)}{x}\right ) \, dx,x,\sinh ^{-1}(a+b x)\right )}{b^3}\\ &=\frac {\operatorname {Subst}\left (\int \frac {\cosh (x) \sinh ^2(x)}{x} \, dx,x,\sinh ^{-1}(a+b x)\right )}{b^3}-\frac {(2 a) \operatorname {Subst}\left (\int \frac {\cosh (x) \sinh (x)}{x} \, dx,x,\sinh ^{-1}(a+b x)\right )}{b^3}+\frac {a^2 \operatorname {Subst}\left (\int \frac {\cosh (x)}{x} \, dx,x,\sinh ^{-1}(a+b x)\right )}{b^3}\\ &=\frac {a^2 \text {Chi}\left (\sinh ^{-1}(a+b x)\right )}{b^3}+\frac {\operatorname {Subst}\left (\int \left (-\frac {\cosh (x)}{4 x}+\frac {\cosh (3 x)}{4 x}\right ) \, dx,x,\sinh ^{-1}(a+b x)\right )}{b^3}-\frac {(2 a) \operatorname {Subst}\left (\int \frac {\sinh (2 x)}{2 x} \, dx,x,\sinh ^{-1}(a+b x)\right )}{b^3}\\ &=\frac {a^2 \text {Chi}\left (\sinh ^{-1}(a+b x)\right )}{b^3}-\frac {\operatorname {Subst}\left (\int \frac {\cosh (x)}{x} \, dx,x,\sinh ^{-1}(a+b x)\right )}{4 b^3}+\frac {\operatorname {Subst}\left (\int \frac {\cosh (3 x)}{x} \, dx,x,\sinh ^{-1}(a+b x)\right )}{4 b^3}-\frac {a \operatorname {Subst}\left (\int \frac {\sinh (2 x)}{x} \, dx,x,\sinh ^{-1}(a+b x)\right )}{b^3}\\ &=-\frac {\text {Chi}\left (\sinh ^{-1}(a+b x)\right )}{4 b^3}+\frac {a^2 \text {Chi}\left (\sinh ^{-1}(a+b x)\right )}{b^3}+\frac {\text {Chi}\left (3 \sinh ^{-1}(a+b x)\right )}{4 b^3}-\frac {a \text {Shi}\left (2 \sinh ^{-1}(a+b x)\right )}{b^3}\\ \end {align*}
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Mathematica [A] time = 0.15, size = 44, normalized size = 0.73 \[ \frac {\left (4 a^2-1\right ) \text {Chi}\left (\sinh ^{-1}(a+b x)\right )+\text {Chi}\left (3 \sinh ^{-1}(a+b x)\right )-4 a \text {Shi}\left (2 \sinh ^{-1}(a+b x)\right )}{4 b^3} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.48, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {x^{2}}{\operatorname {arsinh}\left (b x + a\right )}, x\right ) \]
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 \[ \int \frac {x^{2}}{\operatorname {arsinh}\left (b x + a\right )}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.12, size = 49, normalized size = 0.82 \[ \frac {-a \Shi \left (2 \arcsinh \left (b x +a \right )\right )-\frac {\Chi \left (\arcsinh \left (b x +a \right )\right )}{4}+\frac {\Chi \left (3 \arcsinh \left (b x +a \right )\right )}{4}+a^{2} \Chi \left (\arcsinh \left (b x +a \right )\right )}{b^{3}} \]
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 \[ \int \frac {x^{2}}{\operatorname {arsinh}\left (b x + a\right )}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.02 \[ \int \frac {x^2}{\mathrm {asinh}\left (a+b\,x\right )} \,d x \]
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 \[ \int \frac {x^{2}}{\operatorname {asinh}{\left (a + b x \right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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