3.5.47 \(\int \frac {x (a+b \cosh ^{-1}(c x))^n}{(d-c^2 d x^2)^{3/2}} \, dx\) [447]

Optimal. Leaf size=30 \[ \text {Int}\left (\frac {x \left (a+b \cosh ^{-1}(c x)\right )^n}{\left (d-c^2 d x^2\right )^{3/2}},x\right ) \]

[Out]

Unintegrable(x*(a+b*arccosh(c*x))^n/(-c^2*d*x^2+d)^(3/2),x)

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Rubi [A]
time = 0.08, 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 {x \left (a+b \cosh ^{-1}(c x)\right )^n}{\left (d-c^2 d x^2\right )^{3/2}} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(x*(a + b*ArcCosh[c*x])^n)/(d - c^2*d*x^2)^(3/2),x]

[Out]

Defer[Int][(x*(a + b*ArcCosh[c*x])^n)/(d - c^2*d*x^2)^(3/2), x]

Rubi steps

\begin {align*} \int \frac {x \left (a+b \cosh ^{-1}(c x)\right )^n}{\left (d-c^2 d x^2\right )^{3/2}} \, dx &=-\frac {\left (\sqrt {-1+c x} \sqrt {1+c x}\right ) \int \frac {x \left (a+b \cosh ^{-1}(c x)\right )^n}{(-1+c x)^{3/2} (1+c x)^{3/2}} \, dx}{d \sqrt {d-c^2 d x^2}}\\ \end {align*}

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Mathematica [A]
time = 0.46, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x \left (a+b \cosh ^{-1}(c x)\right )^n}{\left (d-c^2 d x^2\right )^{3/2}} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Integrate[(x*(a + b*ArcCosh[c*x])^n)/(d - c^2*d*x^2)^(3/2),x]

[Out]

Integrate[(x*(a + b*ArcCosh[c*x])^n)/(d - c^2*d*x^2)^(3/2), x]

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Maple [A]
time = 180.00, size = 0, normalized size = 0.00 \[\int \frac {x \left (a +b \,\mathrm {arccosh}\left (c x \right )\right )^{n}}{\left (-c^{2} d \,x^{2}+d \right )^{\frac {3}{2}}}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x*(a+b*arccosh(c*x))^n/(-c^2*d*x^2+d)^(3/2),x)

[Out]

int(x*(a+b*arccosh(c*x))^n/(-c^2*d*x^2+d)^(3/2),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(x*(a+b*arccosh(c*x))^n/(-c^2*d*x^2+d)^(3/2),x, algorithm="maxima")

[Out]

integrate((b*arccosh(c*x) + a)^n*x/(-c^2*d*x^2 + d)^(3/2), 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(x*(a+b*arccosh(c*x))^n/(-c^2*d*x^2+d)^(3/2),x, algorithm="fricas")

[Out]

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

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Sympy [A]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x \left (a + b \operatorname {acosh}{\left (c x \right )}\right )^{n}}{\left (- d \left (c x - 1\right ) \left (c x + 1\right )\right )^{\frac {3}{2}}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*(a+b*acosh(c*x))**n/(-c**2*d*x**2+d)**(3/2),x)

[Out]

Integral(x*(a + b*acosh(c*x))**n/(-d*(c*x - 1)*(c*x + 1))**(3/2), 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(x*(a+b*arccosh(c*x))^n/(-c^2*d*x^2+d)^(3/2),x, algorithm="giac")

[Out]

integrate((b*arccosh(c*x) + a)^n*x/(-c^2*d*x^2 + d)^(3/2), x)

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Mupad [A]
time = 0.00, size = -1, normalized size = -0.03 \begin {gather*} \int \frac {x\,{\left (a+b\,\mathrm {acosh}\left (c\,x\right )\right )}^n}{{\left (d-c^2\,d\,x^2\right )}^{3/2}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x*(a + b*acosh(c*x))^n)/(d - c^2*d*x^2)^(3/2),x)

[Out]

int((x*(a + b*acosh(c*x))^n)/(d - c^2*d*x^2)^(3/2), x)

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