3.364 \(\int \frac{(f x)^m \sqrt{1-c^2 x^2}}{(a+b \cosh ^{-1}(c x))^2} \, dx\)

Optimal. Leaf size=32 \[ \text{Unintegrable}\left (\frac{\sqrt{1-c^2 x^2} (f x)^m}{\left (a+b \cosh ^{-1}(c x)\right )^2},x\right ) \]

[Out]

Unintegrable[((f*x)^m*Sqrt[1 - c^2*x^2])/(a + b*ArcCosh[c*x])^2, x]

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Rubi [A]  time = 0.450238, 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 = {} \[ \int \frac{(f x)^m \sqrt{1-c^2 x^2}}{\left (a+b \cosh ^{-1}(c x)\right )^2} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

Mathematica [A]  time = 0.185017, size = 0, normalized size = 0. \[ \int \frac{(f x)^m \sqrt{1-c^2 x^2}}{\left (a+b \cosh ^{-1}(c x)\right )^2} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

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Maple [A]  time = 0.865, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( fx \right ) ^{m}}{ \left ( a+b{\rm arccosh} \left (cx\right ) \right ) ^{2}}\sqrt{-{c}^{2}{x}^{2}+1}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 0., size = 0, normalized size = 0. \begin{align*} -\frac{{\left ({\left (c^{2} f^{m} x^{2} - f^{m}\right )}{\left (c x + 1\right )} \sqrt{c x - 1} x^{m} +{\left (c^{3} f^{m} x^{3} - c f^{m} x\right )} \sqrt{c x + 1} x^{m}\right )} \sqrt{-c x + 1}}{a b c^{3} x^{2} + \sqrt{c x + 1} \sqrt{c x - 1} a b c^{2} x - a b c +{\left (b^{2} c^{3} x^{2} + \sqrt{c x + 1} \sqrt{c x - 1} b^{2} c^{2} x - b^{2} c\right )} \log \left (c x + \sqrt{c x + 1} \sqrt{c x - 1}\right )} + \int \frac{{\left ({\left (c^{3} f^{m}{\left (m + 2\right )} x^{3} - c f^{m}{\left (m - 1\right )} x\right )}{\left (c x + 1\right )}^{\frac{3}{2}}{\left (c x - 1\right )} x^{m} +{\left (2 \, c^{4} f^{m}{\left (m + 2\right )} x^{4} - c^{2} f^{m}{\left (3 \, m + 2\right )} x^{2} + f^{m} m\right )}{\left (c x + 1\right )} \sqrt{c x - 1} x^{m} +{\left (c^{5} f^{m}{\left (m + 2\right )} x^{5} - c^{3} f^{m}{\left (2 \, m + 3\right )} x^{3} + c f^{m}{\left (m + 1\right )} x\right )} \sqrt{c x + 1} x^{m}\right )} \sqrt{-c x + 1}}{a b c^{5} x^{5} +{\left (c x + 1\right )}{\left (c x - 1\right )} a b c^{3} x^{3} - 2 \, a b c^{3} x^{3} + a b c x + 2 \,{\left (a b c^{4} x^{4} - a b c^{2} x^{2}\right )} \sqrt{c x + 1} \sqrt{c x - 1} +{\left (b^{2} c^{5} x^{5} +{\left (c x + 1\right )}{\left (c x - 1\right )} b^{2} c^{3} x^{3} - 2 \, b^{2} c^{3} x^{3} + b^{2} c x + 2 \,{\left (b^{2} c^{4} x^{4} - b^{2} c^{2} x^{2}\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} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-((c^2*f^m*x^2 - f^m)*(c*x + 1)*sqrt(c*x - 1)*x^m + (c^3*f^m*x^3 - c*f^m*x)*sqrt(c*x + 1)*x^m)*sqrt(-c*x + 1)/
(a*b*c^3*x^2 + sqrt(c*x + 1)*sqrt(c*x - 1)*a*b*c^2*x - a*b*c + (b^2*c^3*x^2 + sqrt(c*x + 1)*sqrt(c*x - 1)*b^2*
c^2*x - b^2*c)*log(c*x + sqrt(c*x + 1)*sqrt(c*x - 1))) + integrate(((c^3*f^m*(m + 2)*x^3 - c*f^m*(m - 1)*x)*(c
*x + 1)^(3/2)*(c*x - 1)*x^m + (2*c^4*f^m*(m + 2)*x^4 - c^2*f^m*(3*m + 2)*x^2 + f^m*m)*(c*x + 1)*sqrt(c*x - 1)*
x^m + (c^5*f^m*(m + 2)*x^5 - c^3*f^m*(2*m + 3)*x^3 + c*f^m*(m + 1)*x)*sqrt(c*x + 1)*x^m)*sqrt(-c*x + 1)/(a*b*c
^5*x^5 + (c*x + 1)*(c*x - 1)*a*b*c^3*x^3 - 2*a*b*c^3*x^3 + a*b*c*x + 2*(a*b*c^4*x^4 - a*b*c^2*x^2)*sqrt(c*x +
1)*sqrt(c*x - 1) + (b^2*c^5*x^5 + (c*x + 1)*(c*x - 1)*b^2*c^3*x^3 - 2*b^2*c^3*x^3 + b^2*c*x + 2*(b^2*c^4*x^4 -
b^2*c^2*x^2)*sqrt(c*x + 1)*sqrt(c*x - 1))*log(c*x + 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} \left (f x\right )^{m}}{b^{2} \operatorname{arcosh}\left (c x\right )^{2} + 2 \, a b \operatorname{arcosh}\left (c x\right ) + a^{2}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(sqrt(-c^2*x^2 + 1)*(f*x)^m/(b^2*arccosh(c*x)^2 + 2*a*b*arccosh(c*x) + a^2), x)

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Sympy [A]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\left (f x\right )^{m} \sqrt{- \left (c x - 1\right ) \left (c x + 1\right )}}{\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((f*x)**m*(-c**2*x**2+1)**(1/2)/(a+b*acosh(c*x))**2,x)

[Out]

Integral((f*x)**m*sqrt(-(c*x - 1)*(c*x + 1))/(a + b*acosh(c*x))**2, x)

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Giac [A]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt{-c^{2} x^{2} + 1} \left (f x\right )^{m}}{{\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((f*x)^m*(-c^2*x^2+1)^(1/2)/(a+b*arccosh(c*x))^2,x, algorithm="giac")

[Out]

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