3.1.78 \(\int \frac {(a+b \cosh ^{-1}(c x))^n \log (h (f+g x)^m)}{\sqrt {1-c^2 x^2}} \, dx\) [78]

Optimal. Leaf size=75 \[ \frac {\sqrt {-1+c x} \sqrt {1+c x} \text {Int}\left (\frac {\left (a+b \cosh ^{-1}(c x)\right )^n \log \left (h (f+g x)^m\right )}{\sqrt {-1+c x} \sqrt {1+c x}},x\right )}{\sqrt {1-c^2 x^2}} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]
time = 0.43, 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 = {} \begin {gather*} \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 \end {gather*}

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 = 0.12, size = 0, normalized size = 0.00 \begin {gather*} \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 \end {gather*}

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 = 180.00, size = 0, normalized size = 0.00 \[\int \frac {\left (a +b \,\mathrm {arccosh}\left (c x \right )\right )^{n} \ln \left (h \left (g x +f \right )^{m}\right )}{\sqrt {-c^{2} x^{2}+1}}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+b*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 = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

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 = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

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)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

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 [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}

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]

Exception raised: TypeError >> An error occurred running a Giac command:INPUT:sage2:=int(sage0,sageVARx):;OUTP
UT:Simplification assuming sageVARc near 0Simplification assuming sageVARc near 0Simplification assuming sageV
ARc near 0S

________________________________________________________________________________________

Mupad [A]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {\ln \left (h\,{\left (f+g\,x\right )}^m\right )\,{\left (a+b\,\mathrm {acosh}\left (c\,x\right )\right )}^n}{\sqrt {1-c^2\,x^2}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________