3.57 \(\int \frac{\log (h (f+g x)^m)}{\sqrt{1+c^2 x^2} (a+b \sinh ^{-1}(c x))} \, dx\)

Optimal. Leaf size=36 \[ \text{Unintegrable}\left (\frac{\log \left (h (f+g x)^m\right )}{\sqrt{c^2 x^2+1} \left (a+b \sinh ^{-1}(c x)\right )},x\right ) \]

[Out]

Unintegrable[Log[h*(f + g*x)^m]/(Sqrt[1 + c^2*x^2]*(a + b*ArcSinh[c*x])), x]

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Rubi [A]  time = 0.194929, antiderivative size = 0, normalized size of antiderivative = 0., number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0., Rules used = {} \[ \int \frac{\log \left (h (f+g x)^m\right )}{\sqrt{1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )} \, dx \]

Verification is Not applicable to the result.

[In]

Int[Log[h*(f + g*x)^m]/(Sqrt[1 + c^2*x^2]*(a + b*ArcSinh[c*x])),x]

[Out]

Defer[Int][Log[h*(f + g*x)^m]/(Sqrt[1 + c^2*x^2]*(a + b*ArcSinh[c*x])), x]

Rubi steps

\begin{align*} \int \frac{\log \left (h (f+g x)^m\right )}{\sqrt{1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )} \, dx &=\int \frac{\log \left (h (f+g x)^m\right )}{\sqrt{1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )} \, dx\\ \end{align*}

Mathematica [A]  time = 0.266206, size = 0, normalized size = 0. \[ \int \frac{\log \left (h (f+g x)^m\right )}{\sqrt{1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )} \, dx \]

Verification is Not applicable to the result.

[In]

Integrate[Log[h*(f + g*x)^m]/(Sqrt[1 + c^2*x^2]*(a + b*ArcSinh[c*x])),x]

[Out]

Integrate[Log[h*(f + g*x)^m]/(Sqrt[1 + c^2*x^2]*(a + b*ArcSinh[c*x])), x]

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Maple [A]  time = 0.843, size = 0, normalized size = 0. \begin{align*} \int{\frac{\ln \left ( h \left ( gx+f \right ) ^{m} \right ) }{a+b{\it Arcsinh} \left ( cx \right ) }{\frac{1}{\sqrt{{c}^{2}{x}^{2}+1}}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(ln(h*(g*x+f)^m)/(a+b*arcsinh(c*x))/(c^2*x^2+1)^(1/2),x)

[Out]

int(ln(h*(g*x+f)^m)/(a+b*arcsinh(c*x))/(c^2*x^2+1)^(1/2),x)

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Maxima [A]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\log \left ({\left (g x + f\right )}^{m} h\right )}{\sqrt{c^{2} x^{2} + 1}{\left (b \operatorname{arsinh}\left (c x\right ) + a\right )}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(log(h*(g*x+f)^m)/(a+b*arcsinh(c*x))/(c^2*x^2+1)^(1/2),x, algorithm="maxima")

[Out]

integrate(log((g*x + f)^m*h)/(sqrt(c^2*x^2 + 1)*(b*arcsinh(c*x) + a)), x)

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Fricas [A]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{\sqrt{c^{2} x^{2} + 1} \log \left ({\left (g x + f\right )}^{m} h\right )}{a c^{2} x^{2} +{\left (b c^{2} x^{2} + b\right )} \operatorname{arsinh}\left (c x\right ) + a}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(log(h*(g*x+f)^m)/(a+b*arcsinh(c*x))/(c^2*x^2+1)^(1/2),x, algorithm="fricas")

[Out]

integral(sqrt(c^2*x^2 + 1)*log((g*x + f)^m*h)/(a*c^2*x^2 + (b*c^2*x^2 + b)*arcsinh(c*x) + a), x)

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Sympy [A]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\log{\left (h \left (f + g x\right )^{m} \right )}}{\left (a + b \operatorname{asinh}{\left (c x \right )}\right ) \sqrt{c^{2} x^{2} + 1}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(ln(h*(g*x+f)**m)/(a+b*asinh(c*x))/(c**2*x**2+1)**(1/2),x)

[Out]

Integral(log(h*(f + g*x)**m)/((a + b*asinh(c*x))*sqrt(c**2*x**2 + 1)), x)

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Giac [A]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\log \left ({\left (g x + f\right )}^{m} h\right )}{\sqrt{c^{2} x^{2} + 1}{\left (b \operatorname{arsinh}\left (c x\right ) + a\right )}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(log(h*(g*x+f)^m)/(a+b*arcsinh(c*x))/(c^2*x^2+1)^(1/2),x, algorithm="giac")

[Out]

integrate(log((g*x + f)^m*h)/(sqrt(c^2*x^2 + 1)*(b*arcsinh(c*x) + a)), x)