Optimal. Leaf size=36 \[ -\frac {1+(a+b x)^2}{b \sinh ^{-1}(a+b x)}+\frac {\text {Shi}\left (2 \sinh ^{-1}(a+b x)\right )}{b} \]
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Rubi [A]
time = 0.08, antiderivative size = 36, normalized size of antiderivative = 1.00, number of steps
used = 6, number of rules used = 6, integrand size = 30, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {5860, 5790,
5780, 5556, 12, 3379} \begin {gather*} \frac {\text {Shi}\left (2 \sinh ^{-1}(a+b x)\right )}{b}-\frac {(a+b x)^2+1}{b \sinh ^{-1}(a+b x)} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 3379
Rule 5556
Rule 5780
Rule 5790
Rule 5860
Rubi steps
\begin {align*} \int \frac {\sqrt {1+a^2+2 a b x+b^2 x^2}}{\sinh ^{-1}(a+b x)^2} \, dx &=\frac {\text {Subst}\left (\int \frac {\sqrt {1+x^2}}{\sinh ^{-1}(x)^2} \, dx,x,a+b x\right )}{b}\\ &=-\frac {1+(a+b x)^2}{b \sinh ^{-1}(a+b x)}+\frac {2 \text {Subst}\left (\int \frac {x}{\sinh ^{-1}(x)} \, dx,x,a+b x\right )}{b}\\ &=-\frac {1+(a+b x)^2}{b \sinh ^{-1}(a+b x)}+\frac {2 \text {Subst}\left (\int \frac {\cosh (x) \sinh (x)}{x} \, dx,x,\sinh ^{-1}(a+b x)\right )}{b}\\ &=-\frac {1+(a+b x)^2}{b \sinh ^{-1}(a+b x)}+\frac {2 \text {Subst}\left (\int \frac {\sinh (2 x)}{2 x} \, dx,x,\sinh ^{-1}(a+b x)\right )}{b}\\ &=-\frac {1+(a+b x)^2}{b \sinh ^{-1}(a+b x)}+\frac {\text {Subst}\left (\int \frac {\sinh (2 x)}{x} \, dx,x,\sinh ^{-1}(a+b x)\right )}{b}\\ &=-\frac {1+(a+b x)^2}{b \sinh ^{-1}(a+b x)}+\frac {\text {Shi}\left (2 \sinh ^{-1}(a+b x)\right )}{b}\\ \end {align*}
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Mathematica [A]
time = 0.05, size = 47, normalized size = 1.31 \begin {gather*} -\frac {1+a^2+2 a b x+b^2 x^2-\sinh ^{-1}(a+b x) \text {Shi}\left (2 \sinh ^{-1}(a+b x)\right )}{b \sinh ^{-1}(a+b x)} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 4.95, size = 44, normalized size = 1.22
method | result | size |
default | \(\frac {2 \hyperbolicSineIntegral \left (2 \arcsinh \left (b x +a \right )\right ) \arcsinh \left (b x +a \right )-\cosh \left (2 \arcsinh \left (b x +a \right )\right )-1}{2 b \arcsinh \left (b x +a \right )}\) | \(44\) |
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 {\sqrt {a^{2} + 2 a b x + b^{2} x^{2} + 1}}{\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 = -0.03 \begin {gather*} \int \frac {\sqrt {a^2+2\,a\,b\,x+b^2\,x^2+1}}{{\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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