3.3.91 \(\int \frac {(1-c^2 x^2)^{5/2}}{x^4 (a+b \cosh ^{-1}(c x))} \, dx\) [291]

Optimal. Leaf size=31 \[ \text {Int}\left (\frac {\left (1-c^2 x^2\right )^{5/2}}{x^4 \left (a+b \cosh ^{-1}(c x)\right )},x\right ) \]

[Out]

Unintegrable((-c^2*x^2+1)^(5/2)/x^4/(a+b*arccosh(c*x)),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 = {} \begin {gather*} \int \frac {\left (1-c^2 x^2\right )^{5/2}}{x^4 \left (a+b \cosh ^{-1}(c x)\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(1 - c^2*x^2)^(5/2)/(x^4*(a + b*ArcCosh[c*x])),x]

[Out]

Defer[Int][(1 - c^2*x^2)^(5/2)/(x^4*(a + b*ArcCosh[c*x])), x]

Rubi steps

\begin {align*} \int \frac {\left (1-c^2 x^2\right )^{5/2}}{x^4 \left (a+b \cosh ^{-1}(c x)\right )} \, dx &=\frac {\sqrt {1-c^2 x^2} \int \frac {(-1+c x)^{5/2} (1+c x)^{5/2}}{x^4 \left (a+b \cosh ^{-1}(c x)\right )} \, dx}{\sqrt {-1+c x} \sqrt {1+c x}}\\ \end {align*}

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Mathematica [A]
time = 0.73, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (1-c^2 x^2\right )^{5/2}}{x^4 \left (a+b \cosh ^{-1}(c x)\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Integrate[(1 - c^2*x^2)^(5/2)/(x^4*(a + b*ArcCosh[c*x])),x]

[Out]

Integrate[(1 - c^2*x^2)^(5/2)/(x^4*(a + b*ArcCosh[c*x])), x]

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Maple [A]
time = 180.00, size = 0, normalized size = 0.00 \[\int \frac {\left (-c^{2} x^{2}+1\right )^{\frac {5}{2}}}{x^{4} \left (a +b \,\mathrm {arccosh}\left (c x \right )\right )}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-c^2*x^2+1)^(5/2)/x^4/(a+b*arccosh(c*x)),x)

[Out]

int((-c^2*x^2+1)^(5/2)/x^4/(a+b*arccosh(c*x)),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((-c^2*x^2+1)^(5/2)/x^4/(a+b*arccosh(c*x)),x, algorithm="maxima")

[Out]

integrate((-c^2*x^2 + 1)^(5/2)/((b*arccosh(c*x) + a)*x^4), 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((-c^2*x^2+1)^(5/2)/x^4/(a+b*arccosh(c*x)),x, algorithm="fricas")

[Out]

integral((c^4*x^4 - 2*c^2*x^2 + 1)*sqrt(-c^2*x^2 + 1)/(b*x^4*arccosh(c*x) + a*x^4), x)

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Sympy [A]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (- \left (c x - 1\right ) \left (c x + 1\right )\right )^{\frac {5}{2}}}{x^{4} \left (a + b \operatorname {acosh}{\left (c x \right )}\right )}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-c**2*x**2+1)**(5/2)/x**4/(a+b*acosh(c*x)),x)

[Out]

Integral((-(c*x - 1)*(c*x + 1))**(5/2)/(x**4*(a + b*acosh(c*x))), x)

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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((-c^2*x^2+1)^(5/2)/x^4/(a+b*arccosh(c*x)),x, algorithm="giac")

[Out]

integrate((-c^2*x^2 + 1)^(5/2)/((b*arccosh(c*x) + a)*x^4), x)

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Mupad [A]
time = 0.00, size = -1, normalized size = -0.03 \begin {gather*} \int \frac {{\left (1-c^2\,x^2\right )}^{5/2}}{x^4\,\left (a+b\,\mathrm {acosh}\left (c\,x\right )\right )} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((1 - c^2*x^2)^(5/2)/(x^4*(a + b*acosh(c*x))),x)

[Out]

int((1 - c^2*x^2)^(5/2)/(x^4*(a + b*acosh(c*x))), x)

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