Optimal. Leaf size=103 \[ -\frac {1+c^2 x^2}{b c x \left (a+b \sinh ^{-1}(c x)\right )}+\frac {\cosh \left (\frac {a}{b}\right ) \text {Chi}\left (\frac {a+b \sinh ^{-1}(c x)}{b}\right )}{b^2}-\frac {\sinh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a+b \sinh ^{-1}(c x)}{b}\right )}{b^2}-\frac {\text {Int}\left (\frac {1}{x^2 \left (a+b \sinh ^{-1}(c x)\right )},x\right )}{b c} \]
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Rubi [A]
time = 0.15, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps
used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {}
\begin {gather*} \int \frac {\sqrt {1+c^2 x^2}}{x \left (a+b \sinh ^{-1}(c x)\right )^2} \, dx \end {gather*}
Verification is not applicable to the result.
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Rubi steps
\begin {align*} \int \frac {\sqrt {1+c^2 x^2}}{x \left (a+b \sinh ^{-1}(c x)\right )^2} \, dx &=-\frac {1+c^2 x^2}{b c x \left (a+b \sinh ^{-1}(c x)\right )}-\frac {\int \frac {1}{x^2 \left (a+b \sinh ^{-1}(c x)\right )} \, dx}{b c}+\frac {c \int \frac {1}{a+b \sinh ^{-1}(c x)} \, dx}{b}\\ &=-\frac {1+c^2 x^2}{b c x \left (a+b \sinh ^{-1}(c x)\right )}+\frac {\text {Subst}\left (\int \frac {\cosh \left (\frac {a}{b}-\frac {x}{b}\right )}{x} \, dx,x,a+b \sinh ^{-1}(c x)\right )}{b^2}-\frac {\int \frac {1}{x^2 \left (a+b \sinh ^{-1}(c x)\right )} \, dx}{b c}\\ &=-\frac {1+c^2 x^2}{b c x \left (a+b \sinh ^{-1}(c x)\right )}-\frac {\int \frac {1}{x^2 \left (a+b \sinh ^{-1}(c x)\right )} \, dx}{b c}+\frac {\cosh \left (\frac {a}{b}\right ) \text {Subst}\left (\int \frac {\cosh \left (\frac {x}{b}\right )}{x} \, dx,x,a+b \sinh ^{-1}(c x)\right )}{b^2}-\frac {\sinh \left (\frac {a}{b}\right ) \text {Subst}\left (\int \frac {\sinh \left (\frac {x}{b}\right )}{x} \, dx,x,a+b \sinh ^{-1}(c x)\right )}{b^2}\\ &=-\frac {1+c^2 x^2}{b c x \left (a+b \sinh ^{-1}(c x)\right )}+\frac {\cosh \left (\frac {a}{b}\right ) \text {Chi}\left (\frac {a+b \sinh ^{-1}(c x)}{b}\right )}{b^2}-\frac {\sinh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a+b \sinh ^{-1}(c x)}{b}\right )}{b^2}-\frac {\int \frac {1}{x^2 \left (a+b \sinh ^{-1}(c x)\right )} \, dx}{b c}\\ \end {align*}
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Mathematica [A]
time = 7.36, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt {1+c^2 x^2}}{x \left (a+b \sinh ^{-1}(c x)\right )^2} \, dx \end {gather*}
Verification is not applicable to the result.
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Maple [A]
time = 180.00, size = 0, normalized size = 0.00 \[\int \frac {\sqrt {c^{2} x^{2}+1}}{x \left (a +b \arcsinh \left (c x \right )\right )^{2}}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
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 [A]
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 [A]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt {c^{2} x^{2} + 1}}{x \left (a + b \operatorname {asinh}{\left (c x \right )}\right )^{2}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [A]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {\sqrt {c^2\,x^2+1}}{x\,{\left (a+b\,\mathrm {asinh}\left (c\,x\right )\right )}^2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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