Optimal. Leaf size=37 \[ \text {Int}\left (\frac {\left (a+b \sinh ^{-1}(c x)\right )^n \log \left (h (f+g x)^m\right )}{\sqrt {1+c^2 x^2}},x\right ) \]
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Rubi [A]
time = 0.13, 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 {\left (a+b \sinh ^{-1}(c x)\right )^n \log \left (h (f+g x)^m\right )}{\sqrt {1+c^2 x^2}} \, dx \end {gather*}
Verification is not applicable to the result.
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Rubi steps
\begin {align*} \int \frac {\left (a+b \sinh ^{-1}(c x)\right )^n \log \left (h (f+g x)^m\right )}{\sqrt {1+c^2 x^2}} \, dx &=\int \frac {\left (a+b \sinh ^{-1}(c x)\right )^n \log \left (h (f+g x)^m\right )}{\sqrt {1+c^2 x^2}} \, dx\\ \end {align*}
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Mathematica [A]
time = 0.10, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (a+b \sinh ^{-1}(c x)\right )^n \log \left (h (f+g x)^m\right )}{\sqrt {1+c^2 x^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 \[\int \frac {\left (a +b \arcsinh \left (c x \right )\right )^{n} \ln \left (h \left (g x +f \right )^{m}\right )}{\sqrt {c^{2} x^{2}+1}}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: ValueError} \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 [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [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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Mupad [A]
time = 0.00, size = -1, normalized size = -0.03 \begin {gather*} \int \frac {\ln \left (h\,{\left (f+g\,x\right )}^m\right )\,{\left (a+b\,\mathrm {asinh}\left (c\,x\right )\right )}^n}{\sqrt {c^2\,x^2+1}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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