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

Optimal. Leaf size=31 \[ \text {Int}\left (\frac {1}{x^2 \left (1-c^2 x^2\right )^{3/2} \left (a+b \cosh ^{-1}(c x)\right )^2},x\right ) \]

[Out]

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

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Rubi [A]  time = 0.56, 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 = {} \[ \int \frac {1}{x^2 \left (1-c^2 x^2\right )^{3/2} \left (a+b \cosh ^{-1}(c x)\right )^2} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

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Mathematica [A]  time = 21.04, size = 0, normalized size = 0.00 \[ \int \frac {1}{x^2 \left (1-c^2 x^2\right )^{3/2} \left (a+b \cosh ^{-1}(c x)\right )^2} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

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fricas [A]  time = 0.55, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {-c^{2} x^{2} + 1}}{a^{2} c^{4} x^{6} - 2 \, a^{2} c^{2} x^{4} + a^{2} x^{2} + {\left (b^{2} c^{4} x^{6} - 2 \, b^{2} c^{2} x^{4} + b^{2} x^{2}\right )} \operatorname {arcosh}\left (c x\right )^{2} + 2 \, {\left (a b c^{4} x^{6} - 2 \, a b c^{2} x^{4} + a b x^{2}\right )} \operatorname {arcosh}\left (c x\right )}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^2/(-c^2*x^2+1)^(3/2)/(a+b*arccosh(c*x))^2,x, algorithm="fricas")

[Out]

integral(sqrt(-c^2*x^2 + 1)/(a^2*c^4*x^6 - 2*a^2*c^2*x^4 + a^2*x^2 + (b^2*c^4*x^6 - 2*b^2*c^2*x^4 + b^2*x^2)*a
rccosh(c*x)^2 + 2*(a*b*c^4*x^6 - 2*a*b*c^2*x^4 + a*b*x^2)*arccosh(c*x)), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [A]  time = 0.44, size = 0, normalized size = 0.00 \[ \int \frac {1}{x^{2} \left (-c^{2} x^{2}+1\right )^{\frac {3}{2}} \left (a +b \,\mathrm {arccosh}\left (c x \right )\right )^{2}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 0.00, size = 0, normalized size = 0.00 \[ \frac {c x + \sqrt {c x + 1} \sqrt {c x - 1}}{{\left ({\left (c x + 1\right )} \sqrt {c x - 1} b^{2} c^{2} x^{3} + {\left (b^{2} c^{3} x^{4} - b^{2} c x^{2}\right )} \sqrt {c x + 1}\right )} \sqrt {-c x + 1} \log \left (c x + \sqrt {c x + 1} \sqrt {c x - 1}\right ) + {\left ({\left (c x + 1\right )} \sqrt {c x - 1} a b c^{2} x^{3} + {\left (a b c^{3} x^{4} - a b c x^{2}\right )} \sqrt {c x + 1}\right )} \sqrt {-c x + 1}} + \int \frac {4 \, c^{5} x^{5} - 5 \, c^{3} x^{3} + {\left (4 \, c^{3} x^{3} - 3 \, c x\right )} {\left (c x + 1\right )} {\left (c x - 1\right )} + 2 \, {\left (4 \, c^{4} x^{4} - 4 \, c^{2} x^{2} + 1\right )} \sqrt {c x + 1} \sqrt {c x - 1} + c x}{{\left ({\left (b^{2} c^{5} x^{7} - b^{2} c^{3} x^{5}\right )} {\left (c x + 1\right )}^{\frac {3}{2}} {\left (c x - 1\right )} + 2 \, {\left (b^{2} c^{6} x^{8} - 2 \, b^{2} c^{4} x^{6} + b^{2} c^{2} x^{4}\right )} {\left (c x + 1\right )} \sqrt {c x - 1} + {\left (b^{2} c^{7} x^{9} - 3 \, b^{2} c^{5} x^{7} + 3 \, b^{2} c^{3} x^{5} - b^{2} c x^{3}\right )} \sqrt {c x + 1}\right )} \sqrt {-c x + 1} \log \left (c x + \sqrt {c x + 1} \sqrt {c x - 1}\right ) + {\left ({\left (a b c^{5} x^{7} - a b c^{3} x^{5}\right )} {\left (c x + 1\right )}^{\frac {3}{2}} {\left (c x - 1\right )} + 2 \, {\left (a b c^{6} x^{8} - 2 \, a b c^{4} x^{6} + a b c^{2} x^{4}\right )} {\left (c x + 1\right )} \sqrt {c x - 1} + {\left (a b c^{7} x^{9} - 3 \, a b c^{5} x^{7} + 3 \, a b c^{3} x^{5} - a b c x^{3}\right )} \sqrt {c x + 1}\right )} \sqrt {-c x + 1}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^2/(-c^2*x^2+1)^(3/2)/(a+b*arccosh(c*x))^2,x, algorithm="maxima")

[Out]

(c*x + sqrt(c*x + 1)*sqrt(c*x - 1))/(((c*x + 1)*sqrt(c*x - 1)*b^2*c^2*x^3 + (b^2*c^3*x^4 - b^2*c*x^2)*sqrt(c*x
+ 1))*sqrt(-c*x + 1)*log(c*x + sqrt(c*x + 1)*sqrt(c*x - 1)) + ((c*x + 1)*sqrt(c*x - 1)*a*b*c^2*x^3 + (a*b*c^3
*x^4 - a*b*c*x^2)*sqrt(c*x + 1))*sqrt(-c*x + 1)) + integrate((4*c^5*x^5 - 5*c^3*x^3 + (4*c^3*x^3 - 3*c*x)*(c*x
+ 1)*(c*x - 1) + 2*(4*c^4*x^4 - 4*c^2*x^2 + 1)*sqrt(c*x + 1)*sqrt(c*x - 1) + c*x)/(((b^2*c^5*x^7 - b^2*c^3*x^
5)*(c*x + 1)^(3/2)*(c*x - 1) + 2*(b^2*c^6*x^8 - 2*b^2*c^4*x^6 + b^2*c^2*x^4)*(c*x + 1)*sqrt(c*x - 1) + (b^2*c^
7*x^9 - 3*b^2*c^5*x^7 + 3*b^2*c^3*x^5 - b^2*c*x^3)*sqrt(c*x + 1))*sqrt(-c*x + 1)*log(c*x + sqrt(c*x + 1)*sqrt(
c*x - 1)) + ((a*b*c^5*x^7 - a*b*c^3*x^5)*(c*x + 1)^(3/2)*(c*x - 1) + 2*(a*b*c^6*x^8 - 2*a*b*c^4*x^6 + a*b*c^2*
x^4)*(c*x + 1)*sqrt(c*x - 1) + (a*b*c^7*x^9 - 3*a*b*c^5*x^7 + 3*a*b*c^3*x^5 - a*b*c*x^3)*sqrt(c*x + 1))*sqrt(-
c*x + 1)), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(1/(x**2*(-(c*x - 1)*(c*x + 1))**(3/2)*(a + b*acosh(c*x))**2), x)

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