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

Optimal. Leaf size=32 \[ \text{Unintegrable}\left (\frac{\left (1-c^2 x^2\right )^{3/2} (f x)^m}{\left (a+b \cosh ^{-1}(c x)\right )^2},x\right ) \]

[Out]

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

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Rubi [A]  time = 0.53223, 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 = {} \[ \int \frac{(f x)^m \left (1-c^2 x^2\right )^{3/2}}{\left (a+b \cosh ^{-1}(c x)\right )^2} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

Mathematica [A]  time = 1.11353, size = 0, normalized size = 0. \[ \int \frac{(f x)^m \left (1-c^2 x^2\right )^{3/2}}{\left (a+b \cosh ^{-1}(c x)\right )^2} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

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Maple [A]  time = 0.875, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( fx \right ) ^{m}}{ \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((f*x)^m*(-c^2*x^2+1)^(3/2)/(a+b*arccosh(c*x))^2,x)

[Out]

int((f*x)^m*(-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{{\left ({\left (c^{4} f^{m} x^{4} - 2 \, c^{2} f^{m} x^{2} + f^{m}\right )}{\left (c x + 1\right )} \sqrt{c x - 1} x^{m} +{\left (c^{5} f^{m} x^{5} - 2 \, 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^{5} f^{m}{\left (m + 4\right )} x^{5} - c^{3} f^{m}{\left (2 \, m + 3\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^{6} f^{m}{\left (m + 4\right )} x^{6} - c^{4} f^{m}{\left (5 \, m + 12\right )} x^{4} + 4 \, c^{2} f^{m}{\left (m + 1\right )} x^{2} - f^{m} m\right )}{\left (c x + 1\right )} \sqrt{c x - 1} x^{m} +{\left (c^{7} f^{m}{\left (m + 4\right )} x^{7} - 3 \, c^{5} f^{m}{\left (m + 3\right )} x^{5} + 3 \, c^{3} f^{m}{\left (m + 2\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)^(3/2)/(a+b*arccosh(c*x))^2,x, algorithm="maxima")

[Out]

((c^4*f^m*x^4 - 2*c^2*f^m*x^2 + f^m)*(c*x + 1)*sqrt(c*x - 1)*x^m + (c^5*f^m*x^5 - 2*c^3*f^m*x^3 + c*f^m*x)*sqr
t(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 + s
qrt(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^5*f^m*(m
+ 4)*x^5 - c^3*f^m*(2*m + 3)*x^3 + c*f^m*(m - 1)*x)*(c*x + 1)^(3/2)*(c*x - 1)*x^m + (2*c^6*f^m*(m + 4)*x^6 -
c^4*f^m*(5*m + 12)*x^4 + 4*c^2*f^m*(m + 1)*x^2 - f^m*m)*(c*x + 1)*sqrt(c*x - 1)*x^m + (c^7*f^m*(m + 4)*x^7 - 3
*c^5*f^m*(m + 3)*x^5 + 3*c^3*f^m*(m + 2)*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)*sqr
t(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{{\left (c^{2} x^{2} - 1\right )} \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)^(3/2)/(a+b*arccosh(c*x))^2,x, algorithm="fricas")

[Out]

integral(-(c^2*x^2 - 1)*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 [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Giac [A]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (-c^{2} x^{2} + 1\right )}^{\frac{3}{2}} \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)^(3/2)/(a+b*arccosh(c*x))^2,x, algorithm="giac")

[Out]

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