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

Optimal. Leaf size=106 \[ \frac{2 \sqrt{c x-1} \text{Unintegrable}\left (\frac{x}{\left (c^2 x^2-1\right )^2 \left (a+b \cosh ^{-1}(c x)\right )},x\right )}{b c \sqrt{1-c x}}-\frac{x^2 \sqrt{c x-1} \sqrt{c x+1}}{b c \left (1-c^2 x^2\right )^{3/2} \left (a+b \cosh ^{-1}(c x)\right )} \]

[Out]

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

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Rubi [A]  time = 0.64553, antiderivative size = 0, normalized size of antiderivative = 0., number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0., 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*}

Mathematica [A]  time = 5.78185, size = 0, normalized size = 0. \[ \int \frac{x^2}{\left (1-c^2 x^2\right )^{3/2} \left (a+b \cosh ^{-1}(c x)\right )^2} \, dx \]

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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Maple [A]  time = 0.152, size = 0, normalized size = 0. \begin{align*} \int{\frac{{x}^{2}}{ \left ( a+b{\rm arccosh} \left (cx\right ) \right ) ^{2}} \left ( -{c}^{2}{x}^{2}+1 \right ) ^{-{\frac{3}{2}}}}\, dx \end{align*}

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., size = 0, normalized size = 0. \begin{align*} \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} \end{align*}

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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Fricas [A]  time = 0., size = 0, normalized size = 0. \begin{align*}{\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 ) \end{align*}

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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Sympy [A]  time = 0., size = 0, normalized size = 0. \begin{align*} \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 \end{align*}

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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Giac [A]  time = 0., size = 0, normalized size = 0. \begin{align*} \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} \end{align*}

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)