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

Optimal. Leaf size=91 \[ \frac{f m \sqrt{c x-1} \text{Unintegrable}\left (\frac{(f x)^{m-1}}{a+b \cosh ^{-1}(c x)},x\right )}{b c \sqrt{1-c x}}-\frac{\sqrt{c x-1} (f x)^m}{b c \sqrt{1-c x} \left (a+b \cosh ^{-1}(c x)\right )} \]

[Out]

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

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Rubi [A]  time = 0.52198, 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]

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

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

Mathematica [A]  time = 0.635352, 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 = 0.358, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( fx \right ) ^{m}}{ \left ( a+b{\rm arccosh} \left (cx\right ) \right ) ^{2}}{\frac{1}{\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 (c^{2} f^{m} x^{2} - f^{m}\right )} \sqrt{c x + 1} \sqrt{c x - 1} x^{m} +{\left (c^{3} f^{m} x^{3} - c f^{m} x\right )} x^{m}}{{\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{{\left (c^{3} f^{m} m 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} m x^{4} - 3 \, c^{2} f^{m} m x^{2} + f^{m} m\right )} \sqrt{c x + 1} \sqrt{c x - 1} x^{m} +{\left (c^{5} f^{m} m x^{5} - c^{3} f^{m}{\left (2 \, m + 1\right )} x^{3} + c f^{m}{\left (m + 1\right )} x\right )} x^{m}}{{\left ({\left (c x + 1\right )}^{\frac{3}{2}}{\left (c x - 1\right )} b^{2} c^{3} x^{3} + 2 \,{\left (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^{5} x^{5} - 2 \, 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 (c x + 1\right )}^{\frac{3}{2}}{\left (c x - 1\right )} a b c^{3} x^{3} + 2 \,{\left (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^{5} x^{5} - 2 \, 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)^(1/2)/(a+b*arccosh(c*x))^2,x, algorithm="maxima")

[Out]

-((c^2*f^m*x^2 - f^m)*sqrt(c*x + 1)*sqrt(c*x - 1)*x^m + (c^3*f^m*x^3 - c*f^m*x)*x^m)/(((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(((c^3*f^m*m*
x^3 - c*f^m*(m - 1)*x)*(c*x + 1)*(c*x - 1)*x^m + (2*c^4*f^m*m*x^4 - 3*c^2*f^m*m*x^2 + f^m*m)*sqrt(c*x + 1)*sqr
t(c*x - 1)*x^m + (c^5*f^m*m*x^5 - c^3*f^m*(2*m + 1)*x^3 + c*f^m*(m + 1)*x)*x^m)/(((c*x + 1)^(3/2)*(c*x - 1)*b^
2*c^3*x^3 + 2*(b^2*c^4*x^4 - b^2*c^2*x^2)*(c*x + 1)*sqrt(c*x - 1) + (b^2*c^5*x^5 - 2*b^2*c^3*x^3 + b^2*c*x)*sq
rt(c*x + 1))*sqrt(-c*x + 1)*log(c*x + sqrt(c*x + 1)*sqrt(c*x - 1)) + ((c*x + 1)^(3/2)*(c*x - 1)*a*b*c^3*x^3 +
2*(a*b*c^4*x^4 - a*b*c^2*x^2)*(c*x + 1)*sqrt(c*x - 1) + (a*b*c^5*x^5 - 2*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^{2} x^{2} +{\left (b^{2} c^{2} x^{2} - b^{2}\right )} \operatorname{arcosh}\left (c x\right )^{2} - a^{2} + 2 \,{\left (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)^(1/2)/(a+b*arccosh(c*x))^2,x, algorithm="fricas")

[Out]

integral(-sqrt(-c^2*x^2 + 1)*(f*x)^m/(a^2*c^2*x^2 + (b^2*c^2*x^2 - b^2)*arccosh(c*x)^2 - a^2 + 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{\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{\left (f x\right )^{m}}{\sqrt{-c^{2} x^{2} + 1}{\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((f*x)^m/(sqrt(-c^2*x^2 + 1)*(b*arccosh(c*x) + a)^2), x)