\(\int \frac {(a+b \arccos (c x))^n \log (h (f+g x)^m)}{\sqrt {1-c^2 x^2}} \, dx\) [19]

Optimal result

Integrand size = 35, antiderivative size = 35 \[ \int \frac {(a+b \arccos (c x))^n \log \left (h (f+g x)^m\right )}{\sqrt {1-c^2 x^2}} \, dx=\text {Int}\left (\frac {(a+b \arccos (c x))^n \log \left (h (f+g x)^m\right )}{\sqrt {1-c^2 x^2}},x\right ) \]

[Out]

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

Rubi [N/A]

Not integrable

Time = 0.12 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.00, number of steps used = 0, number of rules used = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \[ \int \frac {(a+b \arccos (c x))^n \log \left (h (f+g x)^m\right )}{\sqrt {1-c^2 x^2}} \, dx=\int \frac {(a+b \arccos (c x))^n \log \left (h (f+g x)^m\right )}{\sqrt {1-c^2 x^2}} \, dx \]

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

Mathematica [N/A]

Not integrable

Time = 0.18 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.06 \[ \int \frac {(a+b \arccos (c x))^n \log \left (h (f+g x)^m\right )}{\sqrt {1-c^2 x^2}} \, dx=\int \frac {(a+b \arccos (c x))^n \log \left (h (f+g x)^m\right )}{\sqrt {1-c^2 x^2}} \, dx \]

[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]

Maple [N/A] (verified)

Not integrable

Time = 22.42 (sec) , antiderivative size = 33, normalized size of antiderivative = 0.94

\[\int \frac {\left (a +b \arccos \left (c x \right )\right )^{n} \ln \left (h \left (g x +f \right )^{m}\right )}{\sqrt {-c^{2} x^{2}+1}}d x\]

[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)

Fricas [N/A]

Not integrable

Time = 0.27 (sec) , antiderivative size = 47, normalized size of antiderivative = 1.34 \[ \int \frac {(a+b \arccos (c x))^n \log \left (h (f+g x)^m\right )}{\sqrt {1-c^2 x^2}} \, dx=\int { \frac {{\left (b \arccos \left (c x\right ) + a\right )}^{n} \log \left ({\left (g x + f\right )}^{m} h\right )}{\sqrt {-c^{2} x^{2} + 1}} \,d x } \]

[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)

Sympy [F(-1)]

Timed out. \[ \int \frac {(a+b \arccos (c x))^n \log \left (h (f+g x)^m\right )}{\sqrt {1-c^2 x^2}} \, dx=\text {Timed out} \]

[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

Maxima [N/A]

Not integrable

Time = 1.99 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.00 \[ \int \frac {(a+b \arccos (c x))^n \log \left (h (f+g x)^m\right )}{\sqrt {1-c^2 x^2}} \, dx=\int { \frac {{\left (b \arccos \left (c x\right ) + a\right )}^{n} \log \left ({\left (g x + f\right )}^{m} h\right )}{\sqrt {-c^{2} x^{2} + 1}} \,d x } \]

[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)

Giac [N/A]

Not integrable

Time = 0.43 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.00 \[ \int \frac {(a+b \arccos (c x))^n \log \left (h (f+g x)^m\right )}{\sqrt {1-c^2 x^2}} \, dx=\int { \frac {{\left (b \arccos \left (c x\right ) + a\right )}^{n} \log \left ({\left (g x + f\right )}^{m} h\right )}{\sqrt {-c^{2} x^{2} + 1}} \,d x } \]

[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)

Mupad [N/A]

Not integrable

Time = 0.30 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.00 \[ \int \frac {(a+b \arccos (c x))^n \log \left (h (f+g x)^m\right )}{\sqrt {1-c^2 x^2}} \, dx=\int \frac {\ln \left (h\,{\left (f+g\,x\right )}^m\right )\,{\left (a+b\,\mathrm {acos}\left (c\,x\right )\right )}^n}{\sqrt {1-c^2\,x^2}} \,d x \]

[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)