Optimal. Leaf size=78 \[ -\frac {\text {Chi}\left (\frac {a+b \sinh ^{-1}(c x)}{b}\right ) \sinh \left (\frac {a}{b}\right )}{b}+\frac {\cosh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a+b \sinh ^{-1}(c x)}{b}\right )}{b}+\text {Int}\left (\frac {1}{x \sqrt {1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )},x\right ) \]
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Rubi [A]
time = 0.27, 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 )} \, 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 )} \, dx &=\int \left (\frac {1}{x \sqrt {1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )}+\frac {c^2 x}{\sqrt {1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )}\right ) \, dx\\ &=c^2 \int \frac {x}{\sqrt {1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )} \, dx+\int \frac {1}{x \sqrt {1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )} \, dx\\ &=\int \frac {1}{x \sqrt {1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )} \, dx+\text {Subst}\left (\int \frac {\sinh (x)}{a+b x} \, dx,x,\sinh ^{-1}(c x)\right )\\ &=\cosh \left (\frac {a}{b}\right ) \text {Subst}\left (\int \frac {\sinh \left (\frac {a}{b}+x\right )}{a+b x} \, dx,x,\sinh ^{-1}(c x)\right )-\sinh \left (\frac {a}{b}\right ) \text {Subst}\left (\int \frac {\cosh \left (\frac {a}{b}+x\right )}{a+b x} \, dx,x,\sinh ^{-1}(c x)\right )+\int \frac {1}{x \sqrt {1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )} \, dx\\ &=-\frac {\text {Chi}\left (\frac {a}{b}+\sinh ^{-1}(c x)\right ) \sinh \left (\frac {a}{b}\right )}{b}+\frac {\cosh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a}{b}+\sinh ^{-1}(c x)\right )}{b}+\int \frac {1}{x \sqrt {1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )} \, dx\\ \end {align*}
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Mathematica [A]
time = 1.37, 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 )} \, 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 \[\int \frac {\sqrt {c^{2} x^{2}+1}}{x \left (a +b \arcsinh \left (c x \right )\right )}\, dx\]
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 )}\, 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 )} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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