3.455 \(\int \frac {(f x)^m (a+b \cosh ^{-1}(c x))^n}{d-c^2 d x^2} \, dx\)

Optimal. Leaf size=32 \[ \text {Int}\left (\frac {(f x)^m \left (a+b \cosh ^{-1}(c x)\right )^n}{d-c^2 d x^2},x\right ) \]

[Out]

Unintegrable((f*x)^m*(a+b*arccosh(c*x))^n/(-c^2*d*x^2+d),x)

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Rubi [A]  time = 0.10, 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 = {} \[ \int \frac {(f x)^m \left (a+b \cosh ^{-1}(c x)\right )^n}{d-c^2 d x^2} \, dx \]

Verification is Not applicable to the result.

[In]

Int[((f*x)^m*(a + b*ArcCosh[c*x])^n)/(d - c^2*d*x^2),x]

[Out]

Defer[Int][((f*x)^m*(a + b*ArcCosh[c*x])^n)/(d - c^2*d*x^2), x]

Rubi steps

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

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Mathematica [A]  time = 0.65, size = 0, normalized size = 0.00 \[ \int \frac {(f x)^m \left (a+b \cosh ^{-1}(c x)\right )^n}{d-c^2 d x^2} \, dx \]

Verification is Not applicable to the result.

[In]

Integrate[((f*x)^m*(a + b*ArcCosh[c*x])^n)/(d - c^2*d*x^2),x]

[Out]

Integrate[((f*x)^m*(a + b*ArcCosh[c*x])^n)/(d - c^2*d*x^2), x]

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fricas [A]  time = 0.59, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-\frac {\left (f x\right )^{m} {\left (b \operatorname {arcosh}\left (c x\right ) + a\right )}^{n}}{c^{2} d x^{2} - d}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((f*x)^m*(a+b*arccosh(c*x))^n/(-c^2*d*x^2+d),x, algorithm="fricas")

[Out]

integral(-(f*x)^m*(b*arccosh(c*x) + a)^n/(c^2*d*x^2 - d), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(-(f*x)^m*(b*arccosh(c*x) + a)^n/(c^2*d*x^2 - d), x)

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maple [A]  time = 0.28, size = 0, normalized size = 0.00 \[ \int \frac {\left (f x \right )^{m} \left (a +b \,\mathrm {arccosh}\left (c x \right )\right )^{n}}{-c^{2} d \,x^{2}+d}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((f*x)^m*(a+b*arccosh(c*x))^n/(-c^2*d*x^2+d),x)

[Out]

int((f*x)^m*(a+b*arccosh(c*x))^n/(-c^2*d*x^2+d),x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-integrate((f*x)^m*(b*arccosh(c*x) + a)^n/(c^2*d*x^2 - d), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((a + b*acosh(c*x))^n*(f*x)^m)/(d - c^2*d*x^2),x)

[Out]

int(((a + b*acosh(c*x))^n*(f*x)^m)/(d - c^2*d*x^2), x)

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sympy [A]  time = 0.00, size = 0, normalized size = 0.00 \[ - \frac {\int \frac {\left (f x\right )^{m} \left (a + b \operatorname {acosh}{\left (c x \right )}\right )^{n}}{c^{2} x^{2} - 1}\, dx}{d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((f*x)**m*(a+b*acosh(c*x))**n/(-c**2*d*x**2+d),x)

[Out]

-Integral((f*x)**m*(a + b*acosh(c*x))**n/(c**2*x**2 - 1), x)/d

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