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

Optimal. Leaf size=74 \[ \frac{\sqrt{c x-1} \sqrt{c x+1} \text{Unintegrable}\left (\frac{\left (a+b \cosh ^{-1}(c x)\right )^n \log \left (h (f+g x)^m\right )}{\sqrt{c x-1} \sqrt{c x+1}},x\right )}{\sqrt{1-c^2 x^2}} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.955479, 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 = {} \[ \int \frac{\left (a+b \cosh ^{-1}(c x)\right )^n \log \left (h (f+g x)^m\right )}{\sqrt{1-c^2 x^2}} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

Mathematica [A]  time = 0.169582, size = 0, normalized size = 0. \[ \int \frac{\left (a+b \cosh ^{-1}(c x)\right )^n \log \left (h (f+g x)^m\right )}{\sqrt{1-c^2 x^2}} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

________________________________________________________________________________________

Maple [A]  time = 1.755, size = 0, normalized size = 0. \begin{align*} \int{ \left ( a+b{\rm arccosh} \left (cx\right ) \right ) ^{n}\ln \left ( h \left ( gx+f \right ) ^{m} \right ){\frac{1}{\sqrt{-{c}^{2}{x}^{2}+1}}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Maxima [A]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (b \operatorname{arcosh}\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} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

________________________________________________________________________________________

Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

________________________________________________________________________________________

Giac [A]  time = 0., size = 0, normalized size = 0. \begin{align*} \mathit{sage}_{0} x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

sage0*x