3.367 \(\int \frac{(f x)^m}{(1-c^2 x^2)^{5/2} (a+b \cosh ^{-1}(c x))^2} \, dx\)

Optimal. Leaf size=32 \[ \text{Unintegrable}\left (\frac{(f x)^m}{\left (1-c^2 x^2\right )^{5/2} \left (a+b \cosh ^{-1}(c x)\right )^2},x\right ) \]

[Out]

Unintegrable[(f*x)^m/((1 - c^2*x^2)^(5/2)*(a + b*ArcCosh[c*x])^2), x]

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Rubi [A]  time = 0.551211, 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 = {} \[ \int \frac{(f x)^m}{\left (1-c^2 x^2\right )^{5/2} \left (a+b \cosh ^{-1}(c x)\right )^2} \, dx \]

Verification is Not applicable to the result.

[In]

Int[(f*x)^m/((1 - c^2*x^2)^(5/2)*(a + b*ArcCosh[c*x])^2),x]

[Out]

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

Rubi steps

\begin{align*} \int \frac{(f x)^m}{\left (1-c^2 x^2\right )^{5/2} \left (a+b \cosh ^{-1}(c x)\right )^2} \, dx &=\frac{\left (\sqrt{-1+c x} \sqrt{1+c x}\right ) \int \frac{(f x)^m}{(-1+c x)^{5/2} (1+c x)^{5/2} \left (a+b \cosh ^{-1}(c x)\right )^2} \, dx}{\sqrt{1-c^2 x^2}}\\ \end{align*}

Mathematica [A]  time = 1.68472, size = 0, normalized size = 0. \[ \int \frac{(f x)^m}{\left (1-c^2 x^2\right )^{5/2} \left (a+b \cosh ^{-1}(c x)\right )^2} \, dx \]

Verification is Not applicable to the result.

[In]

Integrate[(f*x)^m/((1 - c^2*x^2)^(5/2)*(a + b*ArcCosh[c*x])^2),x]

[Out]

Integrate[(f*x)^m/((1 - c^2*x^2)^(5/2)*(a + b*ArcCosh[c*x])^2), x]

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Maple [A]  time = 0.509, 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{5}{2}}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((f*x)^m/(-c^2*x^2+1)^(5/2)/(a+b*arccosh(c*x))^2,x)

[Out]

int((f*x)^m/(-c^2*x^2+1)^(5/2)/(a+b*arccosh(c*x))^2,x)

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Maxima [A]  time = 0., size = 0, normalized size = 0. \begin{align*} -\frac{c f^{m} x x^{m} + \sqrt{c x + 1} \sqrt{c x - 1} f^{m} x^{m}}{{\left ({\left (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^{5} x^{4} - 2 \, 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^{4} x^{3} - a b c^{2} x\right )}{\left (c x + 1\right )} \sqrt{c x - 1} +{\left (a b c^{5} x^{4} - 2 \, a b c^{3} x^{2} + a b c\right )} \sqrt{c x + 1}\right )} \sqrt{-c x + 1}} + \int \frac{{\left (c^{3} f^{m}{\left (m - 4\right )} x^{3} - c f^{m}{\left (m - 1\right )} x\right )}{\left (c x + 1\right )}{\left (c x - 1\right )} x^{m} +{\left (2 \, c^{4} f^{m}{\left (m - 4\right )} x^{4} - c^{2} f^{m}{\left (3 \, m - 4\right )} x^{2} + f^{m} m\right )} \sqrt{c x + 1} \sqrt{c x - 1} x^{m} +{\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 )} x^{m}}{{\left ({\left (b^{2} c^{7} x^{7} - 2 \, b^{2} c^{5} x^{5} + b^{2} c^{3} x^{3}\right )}{\left (c x + 1\right )}^{\frac{3}{2}}{\left (c x - 1\right )} + 2 \,{\left (b^{2} c^{8} x^{8} - 3 \, b^{2} c^{6} x^{6} + 3 \, b^{2} c^{4} x^{4} - b^{2} c^{2} x^{2}\right )}{\left (c x + 1\right )} \sqrt{c x - 1} +{\left (b^{2} c^{9} x^{9} - 4 \, b^{2} c^{7} x^{7} + 6 \, b^{2} c^{5} x^{5} - 4 \, b^{2} c^{3} x^{3} + b^{2} c x\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^{7} x^{7} - 2 \, a b c^{5} x^{5} + a b c^{3} x^{3}\right )}{\left (c x + 1\right )}^{\frac{3}{2}}{\left (c x - 1\right )} + 2 \,{\left (a b c^{8} x^{8} - 3 \, a b c^{6} x^{6} + 3 \, a b c^{4} x^{4} - a b c^{2} x^{2}\right )}{\left (c x + 1\right )} \sqrt{c x - 1} +{\left (a b c^{9} x^{9} - 4 \, a b c^{7} x^{7} + 6 \, a b c^{5} x^{5} - 4 \, a b c^{3} x^{3} + a b c x\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((f*x)^m/(-c^2*x^2+1)^(5/2)/(a+b*arccosh(c*x))^2,x, algorithm="maxima")

[Out]

-(c*f^m*x*x^m + sqrt(c*x + 1)*sqrt(c*x - 1)*f^m*x^m)/(((b^2*c^4*x^3 - b^2*c^2*x)*(c*x + 1)*sqrt(c*x - 1) + (b^
2*c^5*x^4 - 2*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^4*x^3 - a*b*c^2*x)*(c*x + 1)*sqrt(c*x - 1) + (a*b*c^5*x^4 - 2*a*b*c^3*x^2 + a*b*c)*sqrt(c*x + 1))*sqrt(-c*
x + 1)) + integrate(((c^3*f^m*(m - 4)*x^3 - c*f^m*(m - 1)*x)*(c*x + 1)*(c*x - 1)*x^m + (2*c^4*f^m*(m - 4)*x^4
- c^2*f^m*(3*m - 4)*x^2 + f^m*m)*sqrt(c*x + 1)*sqrt(c*x - 1)*x^m + (c^5*f^m*(m - 4)*x^5 - c^3*f^m*(2*m - 3)*x^
3 + c*f^m*(m + 1)*x)*x^m)/(((b^2*c^7*x^7 - 2*b^2*c^5*x^5 + b^2*c^3*x^3)*(c*x + 1)^(3/2)*(c*x - 1) + 2*(b^2*c^8
*x^8 - 3*b^2*c^6*x^6 + 3*b^2*c^4*x^4 - b^2*c^2*x^2)*(c*x + 1)*sqrt(c*x - 1) + (b^2*c^9*x^9 - 4*b^2*c^7*x^7 + 6
*b^2*c^5*x^5 - 4*b^2*c^3*x^3 + b^2*c*x)*sqrt(c*x + 1))*sqrt(-c*x + 1)*log(c*x + sqrt(c*x + 1)*sqrt(c*x - 1)) +
((a*b*c^7*x^7 - 2*a*b*c^5*x^5 + a*b*c^3*x^3)*(c*x + 1)^(3/2)*(c*x - 1) + 2*(a*b*c^8*x^8 - 3*a*b*c^6*x^6 + 3*a
*b*c^4*x^4 - a*b*c^2*x^2)*(c*x + 1)*sqrt(c*x - 1) + (a*b*c^9*x^9 - 4*a*b*c^7*x^7 + 6*a*b*c^5*x^5 - 4*a*b*c^3*x
^3 + a*b*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}}{a^{2} c^{6} x^{6} - 3 \, a^{2} c^{4} x^{4} + 3 \, a^{2} c^{2} x^{2} +{\left (b^{2} c^{6} x^{6} - 3 \, b^{2} c^{4} x^{4} + 3 \, b^{2} c^{2} x^{2} - b^{2}\right )} \operatorname{arcosh}\left (c x\right )^{2} - a^{2} + 2 \,{\left (a b c^{6} x^{6} - 3 \, a b c^{4} x^{4} + 3 \, 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((f*x)^m/(-c^2*x^2+1)^(5/2)/(a+b*arccosh(c*x))^2,x, algorithm="fricas")

[Out]

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

[Out]

integrate((f*x)^m/((-c^2*x^2 + 1)^(5/2)*(b*arccosh(c*x) + a)^2), x)