Optimal. Leaf size=38 \[ -\frac {\sqrt {1+(a+b x)^2}}{b \sinh ^{-1}(a+b x)}+\frac {\text {Shi}\left (\sinh ^{-1}(a+b x)\right )}{b} \]
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Rubi [A]
time = 0.05, antiderivative size = 38, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 4, integrand size = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {5858, 5773,
5819, 3379} \begin {gather*} \frac {\text {Shi}\left (\sinh ^{-1}(a+b x)\right )}{b}-\frac {\sqrt {(a+b x)^2+1}}{b \sinh ^{-1}(a+b x)} \end {gather*}
Antiderivative was successfully verified.
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Rule 3379
Rule 5773
Rule 5819
Rule 5858
Rubi steps
\begin {align*} \int \frac {1}{\sinh ^{-1}(a+b x)^2} \, dx &=\frac {\text {Subst}\left (\int \frac {1}{\sinh ^{-1}(x)^2} \, dx,x,a+b x\right )}{b}\\ &=-\frac {\sqrt {1+(a+b x)^2}}{b \sinh ^{-1}(a+b x)}+\frac {\text {Subst}\left (\int \frac {x}{\sqrt {1+x^2} \sinh ^{-1}(x)} \, dx,x,a+b x\right )}{b}\\ &=-\frac {\sqrt {1+(a+b x)^2}}{b \sinh ^{-1}(a+b x)}+\frac {\text {Subst}\left (\int \frac {\sinh (x)}{x} \, dx,x,\sinh ^{-1}(a+b x)\right )}{b}\\ &=-\frac {\sqrt {1+(a+b x)^2}}{b \sinh ^{-1}(a+b x)}+\frac {\text {Shi}\left (\sinh ^{-1}(a+b x)\right )}{b}\\ \end {align*}
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Mathematica [A]
time = 0.06, size = 42, normalized size = 1.11 \begin {gather*} \frac {-\sqrt {1+(a+b x)^2}+\sinh ^{-1}(a+b x) \text {Shi}\left (\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 = 1.92, size = 34, normalized size = 0.89
method | result | size |
derivativedivides | \(\frac {-\frac {\sqrt {1+\left (b x +a \right )^{2}}}{\arcsinh \left (b x +a \right )}+\hyperbolicSineIntegral \left (\arcsinh \left (b x +a \right )\right )}{b}\) | \(34\) |
default | \(\frac {-\frac {\sqrt {1+\left (b x +a \right )^{2}}}{\arcsinh \left (b x +a \right )}+\hyperbolicSineIntegral \left (\arcsinh \left (b x +a \right )\right )}{b}\) | \(34\) |
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 {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 {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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