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

Optimal. Leaf size=37 \[ \text{Unintegrable}\left (\frac{\log \left (h (f+g x)^m\right )}{\sqrt{1-c^2 x^2} \left (a+b \cos ^{-1}(c x)\right )},x\right ) \]

[Out]

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

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Rubi [A]  time = 0.200123, 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 = {} \[ \int \frac{\log \left (h (f+g x)^m\right )}{\sqrt{1-c^2 x^2} \left (a+b \cos ^{-1}(c x)\right )} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

Defer[Int][Log[h*(f + g*x)^m]/(Sqrt[1 - c^2*x^2]*(a + b*ArcCos[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 \cos ^{-1}(c x)\right )} \, dx &=\int \frac{\log \left (h (f+g x)^m\right )}{\sqrt{1-c^2 x^2} \left (a+b \cos ^{-1}(c x)\right )} \, dx\\ \end{align*}

Mathematica [A]  time = 0.199132, size = 0, normalized size = 0. \[ \int \frac{\log \left (h (f+g x)^m\right )}{\sqrt{1-c^2 x^2} \left (a+b \cos ^{-1}(c x)\right )} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

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Maple [A]  time = 2.447, size = 0, normalized size = 0. \begin{align*} \int{\frac{\ln \left ( h \left ( gx+f \right ) ^{m} \right ) }{a+b\arccos \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*arccos(c*x))/(-c^2*x^2+1)^(1/2),x)

[Out]

int(ln(h*(g*x+f)^m)/(a+b*arccos(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 \arccos \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*arccos(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*arccos(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 )} \arccos \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*arccos(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)*arccos(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 )}}{\sqrt{- \left (c x - 1\right ) \left (c x + 1\right )} \left (a + b \operatorname{acos}{\left (c x \right )}\right )}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(log(h*(f + g*x)**m)/(sqrt(-(c*x - 1)*(c*x + 1))*(a + b*acos(c*x))), 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 \arccos \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*arccos(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*arccos(c*x) + a)), x)