Optimal. Leaf size=154 \[ -\frac {a^2 \sqrt {1+(a+b x)^2}}{b^3 \sinh ^{-1}(a+b x)}+\frac {2 a (a+b x) \sqrt {1+(a+b x)^2}}{b^3 \sinh ^{-1}(a+b x)}-\frac {(a+b x)^2 \sqrt {1+(a+b x)^2}}{b^3 \sinh ^{-1}(a+b x)}-\frac {2 a \text {Chi}\left (2 \sinh ^{-1}(a+b x)\right )}{b^3}-\frac {\text {Shi}\left (\sinh ^{-1}(a+b x)\right )}{4 b^3}+\frac {a^2 \text {Shi}\left (\sinh ^{-1}(a+b x)\right )}{b^3}+\frac {3 \text {Shi}\left (3 \sinh ^{-1}(a+b x)\right )}{4 b^3} \]
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Rubi [A]
time = 0.15, antiderivative size = 154, normalized size of antiderivative = 1.00, number of steps
used = 12, number of rules used = 7, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.583, Rules used = {5859, 5829,
5773, 5819, 3379, 5778, 3382} \begin {gather*} \frac {a^2 \text {Shi}\left (\sinh ^{-1}(a+b x)\right )}{b^3}-\frac {a^2 \sqrt {(a+b x)^2+1}}{b^3 \sinh ^{-1}(a+b x)}-\frac {2 a \text {Chi}\left (2 \sinh ^{-1}(a+b x)\right )}{b^3}-\frac {\text {Shi}\left (\sinh ^{-1}(a+b x)\right )}{4 b^3}+\frac {3 \text {Shi}\left (3 \sinh ^{-1}(a+b x)\right )}{4 b^3}+\frac {2 a (a+b x) \sqrt {(a+b x)^2+1}}{b^3 \sinh ^{-1}(a+b x)}-\frac {(a+b x)^2 \sqrt {(a+b x)^2+1}}{b^3 \sinh ^{-1}(a+b x)} \end {gather*}
Antiderivative was successfully verified.
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Rule 3379
Rule 3382
Rule 5773
Rule 5778
Rule 5819
Rule 5829
Rule 5859
Rubi steps
\begin {align*} \int \frac {x^2}{\sinh ^{-1}(a+b x)^2} \, dx &=\frac {\text {Subst}\left (\int \frac {\left (-\frac {a}{b}+\frac {x}{b}\right )^2}{\sinh ^{-1}(x)^2} \, dx,x,a+b x\right )}{b}\\ &=\frac {\text {Subst}\left (\int \left (\frac {a^2}{b^2 \sinh ^{-1}(x)^2}-\frac {2 a x}{b^2 \sinh ^{-1}(x)^2}+\frac {x^2}{b^2 \sinh ^{-1}(x)^2}\right ) \, dx,x,a+b x\right )}{b}\\ &=\frac {\text {Subst}\left (\int \frac {x^2}{\sinh ^{-1}(x)^2} \, dx,x,a+b x\right )}{b^3}-\frac {(2 a) \text {Subst}\left (\int \frac {x}{\sinh ^{-1}(x)^2} \, dx,x,a+b x\right )}{b^3}+\frac {a^2 \text {Subst}\left (\int \frac {1}{\sinh ^{-1}(x)^2} \, dx,x,a+b x\right )}{b^3}\\ &=-\frac {a^2 \sqrt {1+(a+b x)^2}}{b^3 \sinh ^{-1}(a+b x)}+\frac {2 a (a+b x) \sqrt {1+(a+b x)^2}}{b^3 \sinh ^{-1}(a+b x)}-\frac {(a+b x)^2 \sqrt {1+(a+b x)^2}}{b^3 \sinh ^{-1}(a+b x)}+\frac {\text {Subst}\left (\int \left (-\frac {\sinh (x)}{4 x}+\frac {3 \sinh (3 x)}{4 x}\right ) \, dx,x,\sinh ^{-1}(a+b x)\right )}{b^3}-\frac {(2 a) \text {Subst}\left (\int \frac {\cosh (2 x)}{x} \, dx,x,\sinh ^{-1}(a+b x)\right )}{b^3}+\frac {a^2 \text {Subst}\left (\int \frac {x}{\sqrt {1+x^2} \sinh ^{-1}(x)} \, dx,x,a+b x\right )}{b^3}\\ &=-\frac {a^2 \sqrt {1+(a+b x)^2}}{b^3 \sinh ^{-1}(a+b x)}+\frac {2 a (a+b x) \sqrt {1+(a+b x)^2}}{b^3 \sinh ^{-1}(a+b x)}-\frac {(a+b x)^2 \sqrt {1+(a+b x)^2}}{b^3 \sinh ^{-1}(a+b x)}-\frac {2 a \text {Chi}\left (2 \sinh ^{-1}(a+b x)\right )}{b^3}-\frac {\text {Subst}\left (\int \frac {\sinh (x)}{x} \, dx,x,\sinh ^{-1}(a+b x)\right )}{4 b^3}+\frac {3 \text {Subst}\left (\int \frac {\sinh (3 x)}{x} \, dx,x,\sinh ^{-1}(a+b x)\right )}{4 b^3}+\frac {a^2 \text {Subst}\left (\int \frac {\sinh (x)}{x} \, dx,x,\sinh ^{-1}(a+b x)\right )}{b^3}\\ &=-\frac {a^2 \sqrt {1+(a+b x)^2}}{b^3 \sinh ^{-1}(a+b x)}+\frac {2 a (a+b x) \sqrt {1+(a+b x)^2}}{b^3 \sinh ^{-1}(a+b x)}-\frac {(a+b x)^2 \sqrt {1+(a+b x)^2}}{b^3 \sinh ^{-1}(a+b x)}-\frac {2 a \text {Chi}\left (2 \sinh ^{-1}(a+b x)\right )}{b^3}-\frac {\text {Shi}\left (\sinh ^{-1}(a+b x)\right )}{4 b^3}+\frac {a^2 \text {Shi}\left (\sinh ^{-1}(a+b x)\right )}{b^3}+\frac {3 \text {Shi}\left (3 \sinh ^{-1}(a+b x)\right )}{4 b^3}\\ \end {align*}
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Mathematica [A]
time = 0.28, size = 83, normalized size = 0.54 \begin {gather*} \frac {-\frac {4 b^2 x^2 \sqrt {1+a^2+2 a b x+b^2 x^2}}{\sinh ^{-1}(a+b x)}-8 a \text {Chi}\left (2 \sinh ^{-1}(a+b x)\right )+\left (-1+4 a^2\right ) \text {Shi}\left (\sinh ^{-1}(a+b x)\right )+3 \text {Shi}\left (3 \sinh ^{-1}(a+b x)\right )}{4 b^3} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 4.12, size = 146, normalized size = 0.95
method | result | size |
derivativedivides | \(\frac {-\frac {a \left (2 \hyperbolicCosineIntegral \left (2 \arcsinh \left (b x +a \right )\right ) \arcsinh \left (b x +a \right )-\sinh \left (2 \arcsinh \left (b x +a \right )\right )\right )}{\arcsinh \left (b x +a \right )}+\frac {\sqrt {1+\left (b x +a \right )^{2}}}{4 \arcsinh \left (b x +a \right )}-\frac {\hyperbolicSineIntegral \left (\arcsinh \left (b x +a \right )\right )}{4}-\frac {\cosh \left (3 \arcsinh \left (b x +a \right )\right )}{4 \arcsinh \left (b x +a \right )}+\frac {3 \hyperbolicSineIntegral \left (3 \arcsinh \left (b x +a \right )\right )}{4}+\frac {a^{2} \left (\hyperbolicSineIntegral \left (\arcsinh \left (b x +a \right )\right ) \arcsinh \left (b x +a \right )-\sqrt {1+\left (b x +a \right )^{2}}\right )}{\arcsinh \left (b x +a \right )}}{b^{3}}\) | \(146\) |
default | \(\frac {-\frac {a \left (2 \hyperbolicCosineIntegral \left (2 \arcsinh \left (b x +a \right )\right ) \arcsinh \left (b x +a \right )-\sinh \left (2 \arcsinh \left (b x +a \right )\right )\right )}{\arcsinh \left (b x +a \right )}+\frac {\sqrt {1+\left (b x +a \right )^{2}}}{4 \arcsinh \left (b x +a \right )}-\frac {\hyperbolicSineIntegral \left (\arcsinh \left (b x +a \right )\right )}{4}-\frac {\cosh \left (3 \arcsinh \left (b x +a \right )\right )}{4 \arcsinh \left (b x +a \right )}+\frac {3 \hyperbolicSineIntegral \left (3 \arcsinh \left (b x +a \right )\right )}{4}+\frac {a^{2} \left (\hyperbolicSineIntegral \left (\arcsinh \left (b x +a \right )\right ) \arcsinh \left (b x +a \right )-\sqrt {1+\left (b x +a \right )^{2}}\right )}{\arcsinh \left (b x +a \right )}}{b^{3}}\) | \(146\) |
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}}{\operatorname {asinh}^{2}{\left (a + b 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 = -0.01 \begin {gather*} \int \frac {x^2}{{\mathrm {asinh}\left (a+b\,x\right )}^2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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