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

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 = 0.05, 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 = {} \[ \int \frac {1}{\left (d+e x^2\right )^{3/2} \left (a+b \cosh ^{-1}(c x)\right )^2} \, dx \]

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 [F]  time = 180.00, size = 0, normalized size = 0.00 \[ \text {\$Aborted} \]

Verification is Not applicable to the result.

[In]

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

[Out]

$Aborted

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fricas [A]  time = 0.72, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {e x^{2} + d}}{a^{2} e^{2} x^{4} + 2 \, a^{2} d e x^{2} + a^{2} d^{2} + {\left (b^{2} e^{2} x^{4} + 2 \, b^{2} d e x^{2} + b^{2} d^{2}\right )} \operatorname {arcosh}\left (c x\right )^{2} + 2 \, {\left (a b e^{2} x^{4} + 2 \, a b d e x^{2} + a b d^{2}\right )} \operatorname {arcosh}\left (c x\right )}, x\right ) \]

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

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giac [A]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{{\left (e x^{2} + d\right )}^{\frac {3}{2}} {\left (b \operatorname {arcosh}\left (c x\right ) + a\right )}^{2}}\,{d x} \]

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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maple [A]  time = 0.37, 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 \[ -\frac {c^{3} x^{3} + {\left (c^{2} x^{2} - 1\right )} \sqrt {c x + 1} \sqrt {c x - 1} - c x}{{\left (b^{2} c^{3} e x^{4} + {\left (c^{3} d - c e\right )} b^{2} x^{2} - b^{2} c d + {\left (b^{2} c^{2} e x^{3} + b^{2} c^{2} d x\right )} \sqrt {c x + 1} \sqrt {c x - 1}\right )} \sqrt {e x^{2} + d} \log \left (c x + \sqrt {c x + 1} \sqrt {c x - 1}\right ) + {\left (a b c^{3} e x^{4} + {\left (c^{3} d - c e\right )} a b x^{2} - a b c d + {\left (a b c^{2} e x^{3} + a b c^{2} d x\right )} \sqrt {c x + 1} \sqrt {c x - 1}\right )} \sqrt {e x^{2} + d}} - \int \frac {2 \, c^{5} e x^{6} - {\left (c^{5} d + 4 \, c^{3} e\right )} x^{4} + {\left (2 \, c^{3} e x^{4} - {\left (c^{3} d + 4 \, c e\right )} x^{2} - c d\right )} {\left (c x + 1\right )} {\left (c x - 1\right )} + 2 \, {\left (c^{3} d + c e\right )} x^{2} + {\left (4 \, c^{4} e x^{5} - 2 \, {\left (c^{4} d + 4 \, c^{2} e\right )} x^{3} + {\left (c^{2} d + 3 \, e\right )} x\right )} \sqrt {c x + 1} \sqrt {c x - 1} - c d}{{\left (b^{2} c^{5} e^{2} x^{8} + 2 \, {\left (c^{5} d e - c^{3} e^{2}\right )} b^{2} x^{6} + {\left (c^{5} d^{2} - 4 \, c^{3} d e + c e^{2}\right )} b^{2} x^{4} + b^{2} c d^{2} - 2 \, {\left (c^{3} d^{2} - c d e\right )} b^{2} x^{2} + {\left (b^{2} c^{3} e^{2} x^{6} + 2 \, b^{2} c^{3} d e x^{4} + b^{2} c^{3} d^{2} x^{2}\right )} {\left (c x + 1\right )} {\left (c x - 1\right )} + 2 \, {\left (b^{2} c^{4} e^{2} x^{7} + {\left (2 \, c^{4} d e - c^{2} e^{2}\right )} b^{2} x^{5} - b^{2} c^{2} d^{2} x + {\left (c^{4} d^{2} - 2 \, c^{2} d e\right )} b^{2} x^{3}\right )} \sqrt {c x + 1} \sqrt {c x - 1}\right )} \sqrt {e x^{2} + d} \log \left (c x + \sqrt {c x + 1} \sqrt {c x - 1}\right ) + {\left (a b c^{5} e^{2} x^{8} + 2 \, {\left (c^{5} d e - c^{3} e^{2}\right )} a b x^{6} + {\left (c^{5} d^{2} - 4 \, c^{3} d e + c e^{2}\right )} a b x^{4} + a b c d^{2} - 2 \, {\left (c^{3} d^{2} - c d e\right )} a b x^{2} + {\left (a b c^{3} e^{2} x^{6} + 2 \, a b c^{3} d e x^{4} + a b c^{3} d^{2} x^{2}\right )} {\left (c x + 1\right )} {\left (c x - 1\right )} + 2 \, {\left (a b c^{4} e^{2} x^{7} + {\left (2 \, c^{4} d e - c^{2} e^{2}\right )} a b x^{5} - a b c^{2} d^{2} x + {\left (c^{4} d^{2} - 2 \, c^{2} d e\right )} a b x^{3}\right )} \sqrt {c x + 1} \sqrt {c x - 1}\right )} \sqrt {e x^{2} + d}}\,{d x} \]

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

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

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

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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