3.2.48 \(\int \frac {(f x)^m (a+b \cosh ^{-1}(c x))}{d-c^2 d x^2} \, dx\) [148]

Optimal. Leaf size=30 \[ \text {Int}\left (\frac {(f x)^m \left (a+b \cosh ^{-1}(c x)\right )}{d-c^2 d x^2},x\right ) \]

[Out]

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

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Rubi [A]
time = 0.05, 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 {(f x)^m \left (a+b \cosh ^{-1}(c x)\right )}{d-c^2 d x^2} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

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Mathematica [A]
time = 3.12, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {(f x)^m \left (a+b \cosh ^{-1}(c x)\right )}{d-c^2 d x^2} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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Maple [A]
time = 180.00, size = 0, normalized size = 0.00 \[\int \frac {\left (f x \right )^{m} \left (a +b \,\mathrm {arccosh}\left (c x \right )\right )}{-c^{2} d \,x^{2}+d}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

[Out]

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

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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