3.6.50 \(\int \frac {1}{(d+e x^2)^{3/2} (a+b \cosh ^{-1}(c x))^2} \, dx\) [550]

Optimal. Leaf size=25 \[ \text {Int}\left (\frac {1}{\left (d+e x^2\right )^{3/2} \left (a+b \cosh ^{-1}(c x)\right )^2},x\right ) \]

[Out]

Unintegrable(1/(e*x^2+d)^(3/2)/(a+b*arccosh(c*x))^2,x)

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Rubi [A]
time = 0.03, 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 {1}{\left (d+e x^2\right )^{3/2} \left (a+b \cosh ^{-1}(c x)\right )^2} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[1/((d + e*x^2)^(3/2)*(a + b*ArcCosh[c*x])^2),x]

[Out]

Defer[Int][1/((d + e*x^2)^(3/2)*(a + b*ArcCosh[c*x])^2), x]

Rubi steps

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

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Mathematica [A]
time = 165.38, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{\left (d+e x^2\right )^{3/2} \left (a+b \cosh ^{-1}(c x)\right )^2} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Integrate[1/((d + e*x^2)^(3/2)*(a + b*ArcCosh[c*x])^2),x]

[Out]

Integrate[1/((d + e*x^2)^(3/2)*(a + b*ArcCosh[c*x])^2), x]

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Maple [A]
time = 180.00, size = 0, normalized size = 0.00 \[\int \frac {1}{\left (e \,x^{2}+d \right )^{\frac {3}{2}} \left (a +b \,\mathrm {arccosh}\left (c x \right )\right )^{2}}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(e*x^2+d)^(3/2)/(a+b*arccosh(c*x))^2,x)

[Out]

int(1/(e*x^2+d)^(3/2)/(a+b*arccosh(c*x))^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(1/(e*x^2+d)^(3/2)/(a+b*arccosh(c*x))^2,x, algorithm="maxima")

[Out]

-(c^3*x^3 + (c^2*x^2 - 1)*sqrt(c*x + 1)*sqrt(c*x - 1) - c*x)/((b^2*c^3*x^4*e - b^2*c*d + (b^2*c^3*d - b^2*c*e)
*x^2 + (b^2*c^2*x^3*e + b^2*c^2*d*x)*sqrt(c*x + 1)*sqrt(c*x - 1))*sqrt(x^2*e + d)*log(c*x + sqrt(c*x + 1)*sqrt
(c*x - 1)) + (a*b*c^3*x^4*e - a*b*c*d + (a*b*c^3*d - a*b*c*e)*x^2 + (a*b*c^2*x^3*e + a*b*c^2*d*x)*sqrt(c*x + 1
)*sqrt(c*x - 1))*sqrt(x^2*e + d)) - integrate((2*c^5*x^6*e - (c^5*d + 4*c^3*e)*x^4 + (2*c^3*x^4*e - (c^3*d + 4
*c*e)*x^2 - c*d)*(c*x + 1)*(c*x - 1) + 2*(c^3*d + c*e)*x^2 + (4*c^4*x^5*e - 2*(c^4*d + 4*c^2*e)*x^3 + (c^2*d +
3*e)*x)*sqrt(c*x + 1)*sqrt(c*x - 1) - c*d)/((b^2*c^5*x^8*e^2 + 2*(b^2*c^5*d*e - b^2*c^3*e^2)*x^6 + b^2*c*d^2
+ (b^2*c^5*d^2 - 4*b^2*c^3*d*e + b^2*c*e^2)*x^4 + (b^2*c^3*x^6*e^2 + 2*b^2*c^3*d*x^4*e + b^2*c^3*d^2*x^2)*(c*x
+ 1)*(c*x - 1) - 2*(b^2*c^3*d^2 - b^2*c*d*e)*x^2 + 2*(b^2*c^4*x^7*e^2 - b^2*c^2*d^2*x + (2*b^2*c^4*d*e - b^2*
c^2*e^2)*x^5 + (b^2*c^4*d^2 - 2*b^2*c^2*d*e)*x^3)*sqrt(c*x + 1)*sqrt(c*x - 1))*sqrt(x^2*e + d)*log(c*x + sqrt(
c*x + 1)*sqrt(c*x - 1)) + (a*b*c^5*x^8*e^2 + 2*(a*b*c^5*d*e - a*b*c^3*e^2)*x^6 + a*b*c*d^2 + (a*b*c^5*d^2 - 4*
a*b*c^3*d*e + a*b*c*e^2)*x^4 + (a*b*c^3*x^6*e^2 + 2*a*b*c^3*d*x^4*e + a*b*c^3*d^2*x^2)*(c*x + 1)*(c*x - 1) - 2
*(a*b*c^3*d^2 - a*b*c*d*e)*x^2 + 2*(a*b*c^4*x^7*e^2 - a*b*c^2*d^2*x + (2*a*b*c^4*d*e - a*b*c^2*e^2)*x^5 + (a*b
*c^4*d^2 - 2*a*b*c^2*d*e)*x^3)*sqrt(c*x + 1)*sqrt(c*x - 1))*sqrt(x^2*e + 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(1/(e*x^2+d)^(3/2)/(a+b*arccosh(c*x))^2,x, algorithm="fricas")

[Out]

integral(sqrt(x^2*e + d)/(a^2*x^4*e^2 + 2*a^2*d*x^2*e + a^2*d^2 + (b^2*x^4*e^2 + 2*b^2*d*x^2*e + b^2*d^2)*arcc
osh(c*x)^2 + 2*(a*b*x^4*e^2 + 2*a*b*d*x^2*e + a*b*d^2)*arccosh(c*x)), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(e*x**2+d)**(3/2)/(a+b*acosh(c*x))**2,x)

[Out]

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

[Out]

integrate(1/((e*x^2 + d)^(3/2)*(b*arccosh(c*x) + a)^2), x)

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Mupad [A]
time = 0.00, size = -1, normalized size = -0.04 \begin {gather*} \int \frac {1}{{\left (a+b\,\mathrm {acosh}\left (c\,x\right )\right )}^2\,{\left (e\,x^2+d\right )}^{3/2}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/((a + b*acosh(c*x))^2*(d + e*x^2)^(3/2)),x)

[Out]

int(1/((a + b*acosh(c*x))^2*(d + e*x^2)^(3/2)), x)

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