3.334 \(\int \frac {(1-c^2 x^2)^{3/2}}{x^4 (a+b \cosh ^{-1}(c x))^2} \, dx\)

Optimal. Leaf size=107 \[ -\frac {4 \sqrt {1-c x} \text {Int}\left (\frac {c^2 x^2-1}{x^5 \left (a+b \cosh ^{-1}(c x)\right )},x\right )}{b c \sqrt {c x-1}}-\frac {\sqrt {c x-1} \sqrt {c x+1} \left (1-c^2 x^2\right )^{3/2}}{b c x^4 \left (a+b \cosh ^{-1}(c x)\right )} \]

[Out]

-(-c^2*x^2+1)^(3/2)*(c*x-1)^(1/2)*(c*x+1)^(1/2)/b/c/x^4/(a+b*arccosh(c*x))-4*(-c*x+1)^(1/2)*Unintegrable((c^2*
x^2-1)/x^5/(a+b*arccosh(c*x)),x)/b/c/(c*x-1)^(1/2)

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Rubi [A]  time = 0.61, 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 = {} \[ \int \frac {\left (1-c^2 x^2\right )^{3/2}}{x^4 \left (a+b \cosh ^{-1}(c x)\right )^2} \, dx \]

Verification is Not applicable to the result.

[In]

Int[(1 - c^2*x^2)^(3/2)/(x^4*(a + b*ArcCosh[c*x])^2),x]

[Out]

((1 - c*x)^2*(1 + c*x)^(3/2)*Sqrt[1 - c^2*x^2])/(b*c*x^4*Sqrt[-1 + c*x]*(a + b*ArcCosh[c*x])) - (4*Sqrt[1 - c^
2*x^2]*Defer[Int][(-1 + c^2*x^2)/(x^5*(a + b*ArcCosh[c*x])), x])/(b*c*Sqrt[-1 + c*x]*Sqrt[1 + c*x])

Rubi steps

\begin {align*} \int \frac {\left (1-c^2 x^2\right )^{3/2}}{x^4 \left (a+b \cosh ^{-1}(c x)\right )^2} \, dx &=-\frac {\sqrt {1-c^2 x^2} \int \frac {(-1+c x)^{3/2} (1+c x)^{3/2}}{x^4 \left (a+b \cosh ^{-1}(c x)\right )^2} \, dx}{\sqrt {-1+c x} \sqrt {1+c x}}\\ &=\frac {(1-c x)^2 (1+c x)^{3/2} \sqrt {1-c^2 x^2}}{b c x^4 \sqrt {-1+c x} \left (a+b \cosh ^{-1}(c x)\right )}-\frac {\left (4 \sqrt {1-c^2 x^2}\right ) \int \frac {-1+c^2 x^2}{x^5 \left (a+b \cosh ^{-1}(c x)\right )} \, dx}{b c \sqrt {-1+c x} \sqrt {1+c x}}\\ \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 - c^2*x^2)^(3/2)/(x^4*(a + b*ArcCosh[c*x])^2),x]

[Out]

$Aborted

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fricas [A]  time = 1.01, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (-c^{2} x^{2} + 1\right )}^{\frac {3}{2}}}{b^{2} x^{4} \operatorname {arcosh}\left (c x\right )^{2} + 2 \, a b x^{4} \operatorname {arcosh}\left (c x\right ) + a^{2} x^{4}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-c^2*x^2+1)^(3/2)/x^4/(a+b*arccosh(c*x))^2,x, algorithm="fricas")

[Out]

integral((-c^2*x^2 + 1)^(3/2)/(b^2*x^4*arccosh(c*x)^2 + 2*a*b*x^4*arccosh(c*x) + a^2*x^4), x)

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giac [A]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (-c^{2} x^{2} + 1\right )}^{\frac {3}{2}}}{{\left (b \operatorname {arcosh}\left (c x\right ) + a\right )}^{2} x^{4}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-c^2*x^2+1)^(3/2)/x^4/(a+b*arccosh(c*x))^2,x, algorithm="giac")

[Out]

integrate((-c^2*x^2 + 1)^(3/2)/((b*arccosh(c*x) + a)^2*x^4), x)

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maple [A]  time = 0.83, size = 0, normalized size = 0.00 \[ \int \frac {\left (-c^{2} x^{2}+1\right )^{\frac {3}{2}}}{x^{4} \left (a +b \,\mathrm {arccosh}\left (c x \right )\right )^{2}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-c^2*x^2+1)^(3/2)/x^4/(a+b*arccosh(c*x))^2,x)

[Out]

int((-c^2*x^2+1)^(3/2)/x^4/(a+b*arccosh(c*x))^2,x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-c^2*x^2+1)^(3/2)/x^4/(a+b*arccosh(c*x))^2,x, algorithm="maxima")

[Out]

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

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mupad [A]  time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {{\left (1-c^2\,x^2\right )}^{3/2}}{x^4\,{\left (a+b\,\mathrm {acosh}\left (c\,x\right )\right )}^2} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((1 - c^2*x^2)^(3/2)/(x^4*(a + b*acosh(c*x))^2),x)

[Out]

int((1 - c^2*x^2)^(3/2)/(x^4*(a + b*acosh(c*x))^2), x)

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sympy [A]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\left (- \left (c x - 1\right ) \left (c x + 1\right )\right )^{\frac {3}{2}}}{x^{4} \left (a + b \operatorname {acosh}{\left (c x \right )}\right )^{2}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-c**2*x**2+1)**(3/2)/x**4/(a+b*acosh(c*x))**2,x)

[Out]

Integral((-(c*x - 1)*(c*x + 1))**(3/2)/(x**4*(a + b*acosh(c*x))**2), x)

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