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

Optimal. Leaf size=38 \[ \text {Int}\left (\frac {\left (a+b \cos ^{-1}(c x)\right )^n \log \left (h (f+g x)^m\right )}{\sqrt {1-c^2 x^2}},x\right ) \]

[Out]

Unintegrable((a+b*arccos(c*x))^n*ln(h*(g*x+f)^m)/(-c^2*x^2+1)^(1/2),x)

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Rubi [A]  time = 0.19, 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 = {} \[ \int \frac {\left (a+b \cos ^{-1}(c x)\right )^n \log \left (h (f+g x)^m\right )}{\sqrt {1-c^2 x^2}} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

Rubi steps

\begin {align*} \int \frac {\left (a+b \cos ^{-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 \cos ^{-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 = 0.16, size = 0, normalized size = 0.00 \[ \int \frac {\left (a+b \cos ^{-1}(c x)\right )^n \log \left (h (f+g x)^m\right )}{\sqrt {1-c^2 x^2}} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

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fricas [A]  time = 0.52, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-\frac {\sqrt {-c^{2} x^{2} + 1} {\left (b \arccos \left (c x\right ) + a\right )}^{n} \log \left ({\left (g x + f\right )}^{m} h\right )}{c^{2} x^{2} - 1}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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giac [A]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (b \arccos \left (c x\right ) + a\right )}^{n} \log \left ({\left (g x + f\right )}^{m} h\right )}{\sqrt {-c^{2} x^{2} + 1}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [A]  time = 12.11, size = 0, normalized size = 0.00 \[ \int \frac {\left (a +b \arccos \left (c x \right )\right )^{n} \ln \left (h \left (g x +f \right )^{m}\right )}{\sqrt {-c^{2} x^{2}+1}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (b \arccos \left (c x\right ) + a\right )}^{n} \log \left ({\left (g x + f\right )}^{m} h\right )}{\sqrt {-c^{2} x^{2} + 1}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [A]  time = 0.00, size = -1, normalized size = -0.03 \[ \int \frac {\ln \left (h\,{\left (f+g\,x\right )}^m\right )\,{\left (a+b\,\mathrm {acos}\left (c\,x\right )\right )}^n}{\sqrt {1-c^2\,x^2}} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((log(h*(f + g*x)^m)*(a + b*acos(c*x))^n)/(1 - c^2*x^2)^(1/2),x)

[Out]

int((log(h*(f + g*x)^m)*(a + b*acos(c*x))^n)/(1 - c^2*x^2)^(1/2), x)

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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