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

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

[Out]

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

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Rubi [A]  time = 0.65, 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 = {} \[ \int \frac {x^2}{\left (1-c^2 x^2\right )^{3/2} \left (a+b \cosh ^{-1}(c x)\right )^2} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

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Mathematica [A]  time = 5.99, size = 0, normalized size = 0.00 \[ \int \frac {x^2}{\left (1-c^2 x^2\right )^{3/2} \left (a+b \cosh ^{-1}(c x)\right )^2} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

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fricas [A]  time = 0.52, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {-c^{2} x^{2} + 1} x^{2}}{a^{2} c^{4} x^{4} - 2 \, a^{2} c^{2} x^{2} + {\left (b^{2} c^{4} x^{4} - 2 \, b^{2} c^{2} x^{2} + b^{2}\right )} \operatorname {arcosh}\left (c x\right )^{2} + a^{2} + 2 \, {\left (a b c^{4} x^{4} - 2 \, a b c^{2} x^{2} + a b\right )} \operatorname {arcosh}\left (c x\right )}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [A]  time = 0.59, size = 0, normalized size = 0.00 \[ \int \frac {x^{2}}{\left (-c^{2} x^{2}+1\right )^{\frac {3}{2}} \left (a +b \,\mathrm {arccosh}\left (c x \right )\right )^{2}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

int(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 \[ \int \frac {x^{2}}{\left (- \left (c x - 1\right ) \left (c x + 1\right )\right )^{\frac {3}{2}} \left (a + b \operatorname {acosh}{\left (c x \right )}\right )^{2}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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