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

Optimal. Leaf size=385 \[ \frac{\sqrt{1-c x} \text{Unintegrable}\left (\frac{\left (c^2 x^2-1\right )^2}{x^2 \left (a+b \cosh ^{-1}(c x)\right )},x\right )}{b c \sqrt{c x-1}}-\frac{25 \sqrt{1-c x} \sinh \left (\frac{a}{b}\right ) \text{Chi}\left (\frac{a+b \cosh ^{-1}(c x)}{b}\right )}{8 b^2 \sqrt{c x-1}}+\frac{25 \sqrt{1-c x} \sinh \left (\frac{3 a}{b}\right ) \text{Chi}\left (\frac{3 \left (a+b \cosh ^{-1}(c x)\right )}{b}\right )}{16 b^2 \sqrt{c x-1}}-\frac{5 \sqrt{1-c x} \sinh \left (\frac{5 a}{b}\right ) \text{Chi}\left (\frac{5 \left (a+b \cosh ^{-1}(c x)\right )}{b}\right )}{16 b^2 \sqrt{c x-1}}+\frac{25 \sqrt{1-c x} \cosh \left (\frac{a}{b}\right ) \text{Shi}\left (\frac{a+b \cosh ^{-1}(c x)}{b}\right )}{8 b^2 \sqrt{c x-1}}-\frac{25 \sqrt{1-c x} \cosh \left (\frac{3 a}{b}\right ) \text{Shi}\left (\frac{3 \left (a+b \cosh ^{-1}(c x)\right )}{b}\right )}{16 b^2 \sqrt{c x-1}}+\frac{5 \sqrt{1-c x} \cosh \left (\frac{5 a}{b}\right ) \text{Shi}\left (\frac{5 \left (a+b \cosh ^{-1}(c x)\right )}{b}\right )}{16 b^2 \sqrt{c x-1}}-\frac{\sqrt{c x-1} \sqrt{c x+1} \left (1-c^2 x^2\right )^{5/2}}{b c x \left (a+b \cosh ^{-1}(c x)\right )} \]

[Out]

-((Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(1 - c^2*x^2)^(5/2))/(b*c*x*(a + b*ArcCosh[c*x]))) - (25*Sqrt[1 - c*x]*CoshInt
egral[(a + b*ArcCosh[c*x])/b]*Sinh[a/b])/(8*b^2*Sqrt[-1 + c*x]) + (25*Sqrt[1 - c*x]*CoshIntegral[(3*(a + b*Arc
Cosh[c*x]))/b]*Sinh[(3*a)/b])/(16*b^2*Sqrt[-1 + c*x]) - (5*Sqrt[1 - c*x]*CoshIntegral[(5*(a + b*ArcCosh[c*x]))
/b]*Sinh[(5*a)/b])/(16*b^2*Sqrt[-1 + c*x]) + (25*Sqrt[1 - c*x]*Cosh[a/b]*SinhIntegral[(a + b*ArcCosh[c*x])/b])
/(8*b^2*Sqrt[-1 + c*x]) - (25*Sqrt[1 - c*x]*Cosh[(3*a)/b]*SinhIntegral[(3*(a + b*ArcCosh[c*x]))/b])/(16*b^2*Sq
rt[-1 + c*x]) + (5*Sqrt[1 - c*x]*Cosh[(5*a)/b]*SinhIntegral[(5*(a + b*ArcCosh[c*x]))/b])/(16*b^2*Sqrt[-1 + c*x
]) + (Sqrt[1 - c*x]*Unintegrable[(-1 + c^2*x^2)^2/(x^2*(a + b*ArcCosh[c*x])), x])/(b*c*Sqrt[-1 + c*x])

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Rubi [A]  time = 0.984997, 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 = {} \[ \int \frac{\left (1-c^2 x^2\right )^{5/2}}{x \left (a+b \cosh ^{-1}(c x)\right )^2} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

((1 - c*x)^3*(1 + c*x)^(5/2)*Sqrt[1 - c^2*x^2])/(b*c*x*Sqrt[-1 + c*x]*(a + b*ArcCosh[c*x])) - (25*Sqrt[1 - c^2
*x^2]*CoshIntegral[a/b + ArcCosh[c*x]]*Sinh[a/b])/(8*b^2*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + (25*Sqrt[1 - c^2*x^2]
*CoshIntegral[(3*a)/b + 3*ArcCosh[c*x]]*Sinh[(3*a)/b])/(16*b^2*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) - (5*Sqrt[1 - c^2
*x^2]*CoshIntegral[(5*a)/b + 5*ArcCosh[c*x]]*Sinh[(5*a)/b])/(16*b^2*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + (25*Sqrt[1
- c^2*x^2]*Cosh[a/b]*SinhIntegral[a/b + ArcCosh[c*x]])/(8*b^2*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) - (25*Sqrt[1 - c^
2*x^2]*Cosh[(3*a)/b]*SinhIntegral[(3*a)/b + 3*ArcCosh[c*x]])/(16*b^2*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + (5*Sqrt[1
- c^2*x^2]*Cosh[(5*a)/b]*SinhIntegral[(5*a)/b + 5*ArcCosh[c*x]])/(16*b^2*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + (Sqr
t[1 - c^2*x^2]*Defer[Int][(-1 + c^2*x^2)^2/(x^2*(a + b*ArcCosh[c*x])), x])/(b*c*Sqrt[-1 + c*x]*Sqrt[1 + c*x])

Rubi steps

\begin{align*} \int \frac{\left (1-c^2 x^2\right )^{5/2}}{x \left (a+b \cosh ^{-1}(c x)\right )^2} \, dx &=\frac{\sqrt{1-c^2 x^2} \int \frac{(-1+c x)^{5/2} (1+c x)^{5/2}}{x \left (a+b \cosh ^{-1}(c x)\right )^2} \, dx}{\sqrt{-1+c x} \sqrt{1+c x}}\\ &=\frac{(1-c x)^3 (1+c x)^{5/2} \sqrt{1-c^2 x^2}}{b c x \sqrt{-1+c x} \left (a+b \cosh ^{-1}(c x)\right )}+\frac{\sqrt{1-c^2 x^2} \int \frac{\left (-1+c^2 x^2\right )^2}{x^2 \left (a+b \cosh ^{-1}(c x)\right )} \, dx}{b c \sqrt{-1+c x} \sqrt{1+c x}}+\frac{\left (5 c \sqrt{1-c^2 x^2}\right ) \int \frac{\left (-1+c^2 x^2\right )^2}{a+b \cosh ^{-1}(c x)} \, dx}{b \sqrt{-1+c x} \sqrt{1+c x}}\\ &=\frac{(1-c x)^3 (1+c x)^{5/2} \sqrt{1-c^2 x^2}}{b c x \sqrt{-1+c x} \left (a+b \cosh ^{-1}(c x)\right )}+\frac{\left (5 \sqrt{1-c^2 x^2}\right ) \operatorname{Subst}\left (\int \frac{\sinh ^5(x)}{a+b x} \, dx,x,\cosh ^{-1}(c x)\right )}{b \sqrt{-1+c x} \sqrt{1+c x}}+\frac{\sqrt{1-c^2 x^2} \int \frac{\left (-1+c^2 x^2\right )^2}{x^2 \left (a+b \cosh ^{-1}(c x)\right )} \, dx}{b c \sqrt{-1+c x} \sqrt{1+c x}}\\ &=\frac{(1-c x)^3 (1+c x)^{5/2} \sqrt{1-c^2 x^2}}{b c x \sqrt{-1+c x} \left (a+b \cosh ^{-1}(c x)\right )}-\frac{\left (5 i \sqrt{1-c^2 x^2}\right ) \operatorname{Subst}\left (\int \left (\frac{5 i \sinh (x)}{8 (a+b x)}-\frac{5 i \sinh (3 x)}{16 (a+b x)}+\frac{i \sinh (5 x)}{16 (a+b x)}\right ) \, dx,x,\cosh ^{-1}(c x)\right )}{b \sqrt{-1+c x} \sqrt{1+c x}}+\frac{\sqrt{1-c^2 x^2} \int \frac{\left (-1+c^2 x^2\right )^2}{x^2 \left (a+b \cosh ^{-1}(c x)\right )} \, dx}{b c \sqrt{-1+c x} \sqrt{1+c x}}\\ &=\frac{(1-c x)^3 (1+c x)^{5/2} \sqrt{1-c^2 x^2}}{b c x \sqrt{-1+c x} \left (a+b \cosh ^{-1}(c x)\right )}+\frac{\left (5 \sqrt{1-c^2 x^2}\right ) \operatorname{Subst}\left (\int \frac{\sinh (5 x)}{a+b x} \, dx,x,\cosh ^{-1}(c x)\right )}{16 b \sqrt{-1+c x} \sqrt{1+c x}}-\frac{\left (25 \sqrt{1-c^2 x^2}\right ) \operatorname{Subst}\left (\int \frac{\sinh (3 x)}{a+b x} \, dx,x,\cosh ^{-1}(c x)\right )}{16 b \sqrt{-1+c x} \sqrt{1+c x}}+\frac{\left (25 \sqrt{1-c^2 x^2}\right ) \operatorname{Subst}\left (\int \frac{\sinh (x)}{a+b x} \, dx,x,\cosh ^{-1}(c x)\right )}{8 b \sqrt{-1+c x} \sqrt{1+c x}}+\frac{\sqrt{1-c^2 x^2} \int \frac{\left (-1+c^2 x^2\right )^2}{x^2 \left (a+b \cosh ^{-1}(c x)\right )} \, dx}{b c \sqrt{-1+c x} \sqrt{1+c x}}\\ &=\frac{(1-c x)^3 (1+c x)^{5/2} \sqrt{1-c^2 x^2}}{b c x \sqrt{-1+c x} \left (a+b \cosh ^{-1}(c x)\right )}+\frac{\sqrt{1-c^2 x^2} \int \frac{\left (-1+c^2 x^2\right )^2}{x^2 \left (a+b \cosh ^{-1}(c x)\right )} \, dx}{b c \sqrt{-1+c x} \sqrt{1+c x}}+\frac{\left (25 \sqrt{1-c^2 x^2} \cosh \left (\frac{a}{b}\right )\right ) \operatorname{Subst}\left (\int \frac{\sinh \left (\frac{a}{b}+x\right )}{a+b x} \, dx,x,\cosh ^{-1}(c x)\right )}{8 b \sqrt{-1+c x} \sqrt{1+c x}}-\frac{\left (25 \sqrt{1-c^2 x^2} \cosh \left (\frac{3 a}{b}\right )\right ) \operatorname{Subst}\left (\int \frac{\sinh \left (\frac{3 a}{b}+3 x\right )}{a+b x} \, dx,x,\cosh ^{-1}(c x)\right )}{16 b \sqrt{-1+c x} \sqrt{1+c x}}+\frac{\left (5 \sqrt{1-c^2 x^2} \cosh \left (\frac{5 a}{b}\right )\right ) \operatorname{Subst}\left (\int \frac{\sinh \left (\frac{5 a}{b}+5 x\right )}{a+b x} \, dx,x,\cosh ^{-1}(c x)\right )}{16 b \sqrt{-1+c x} \sqrt{1+c x}}-\frac{\left (25 \sqrt{1-c^2 x^2} \sinh \left (\frac{a}{b}\right )\right ) \operatorname{Subst}\left (\int \frac{\cosh \left (\frac{a}{b}+x\right )}{a+b x} \, dx,x,\cosh ^{-1}(c x)\right )}{8 b \sqrt{-1+c x} \sqrt{1+c x}}+\frac{\left (25 \sqrt{1-c^2 x^2} \sinh \left (\frac{3 a}{b}\right )\right ) \operatorname{Subst}\left (\int \frac{\cosh \left (\frac{3 a}{b}+3 x\right )}{a+b x} \, dx,x,\cosh ^{-1}(c x)\right )}{16 b \sqrt{-1+c x} \sqrt{1+c x}}-\frac{\left (5 \sqrt{1-c^2 x^2} \sinh \left (\frac{5 a}{b}\right )\right ) \operatorname{Subst}\left (\int \frac{\cosh \left (\frac{5 a}{b}+5 x\right )}{a+b x} \, dx,x,\cosh ^{-1}(c x)\right )}{16 b \sqrt{-1+c x} \sqrt{1+c x}}\\ &=\frac{(1-c x)^3 (1+c x)^{5/2} \sqrt{1-c^2 x^2}}{b c x \sqrt{-1+c x} \left (a+b \cosh ^{-1}(c x)\right )}-\frac{25 \sqrt{1-c^2 x^2} \text{Chi}\left (\frac{a}{b}+\cosh ^{-1}(c x)\right ) \sinh \left (\frac{a}{b}\right )}{8 b^2 \sqrt{-1+c x} \sqrt{1+c x}}+\frac{25 \sqrt{1-c^2 x^2} \text{Chi}\left (\frac{3 a}{b}+3 \cosh ^{-1}(c x)\right ) \sinh \left (\frac{3 a}{b}\right )}{16 b^2 \sqrt{-1+c x} \sqrt{1+c x}}-\frac{5 \sqrt{1-c^2 x^2} \text{Chi}\left (\frac{5 a}{b}+5 \cosh ^{-1}(c x)\right ) \sinh \left (\frac{5 a}{b}\right )}{16 b^2 \sqrt{-1+c x} \sqrt{1+c x}}+\frac{25 \sqrt{1-c^2 x^2} \cosh \left (\frac{a}{b}\right ) \text{Shi}\left (\frac{a}{b}+\cosh ^{-1}(c x)\right )}{8 b^2 \sqrt{-1+c x} \sqrt{1+c x}}-\frac{25 \sqrt{1-c^2 x^2} \cosh \left (\frac{3 a}{b}\right ) \text{Shi}\left (\frac{3 a}{b}+3 \cosh ^{-1}(c x)\right )}{16 b^2 \sqrt{-1+c x} \sqrt{1+c x}}+\frac{5 \sqrt{1-c^2 x^2} \cosh \left (\frac{5 a}{b}\right ) \text{Shi}\left (\frac{5 a}{b}+5 \cosh ^{-1}(c x)\right )}{16 b^2 \sqrt{-1+c x} \sqrt{1+c x}}+\frac{\sqrt{1-c^2 x^2} \int \frac{\left (-1+c^2 x^2\right )^2}{x^2 \left (a+b \cosh ^{-1}(c x)\right )} \, dx}{b c \sqrt{-1+c x} \sqrt{1+c x}}\\ \end{align*}

Mathematica [A]  time = 8.35788, size = 0, normalized size = 0. \[ \int \frac{\left (1-c^2 x^2\right )^{5/2}}{x \left (a+b \cosh ^{-1}(c x)\right )^2} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

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Maple [A]  time = 0.553, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{x \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((-c^2*x^2+1)^(5/2)/x/(a+b*arccosh(c*x))^2,x)

[Out]

int((-c^2*x^2+1)^(5/2)/x/(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^{6} x^{6} - 3 \, c^{4} x^{4} + 3 \, c^{2} x^{2} - 1\right )}{\left (c x + 1\right )} \sqrt{c x - 1} +{\left (c^{7} x^{7} - 3 \, c^{5} x^{5} + 3 \, c^{3} x^{3} - c x\right )} \sqrt{c x + 1}\right )} \sqrt{-c x + 1}}{a b c^{3} x^{3} + \sqrt{c x + 1} \sqrt{c x - 1} a b c^{2} x^{2} - a b c x +{\left (b^{2} c^{3} x^{3} + \sqrt{c x + 1} \sqrt{c x - 1} b^{2} c^{2} x^{2} - b^{2} c x\right )} \log \left (c x + \sqrt{c x + 1} \sqrt{c x - 1}\right )} + \int \frac{{\left ({\left (5 \, c^{7} x^{7} - 8 \, c^{5} x^{5} + c^{3} x^{3} + 2 \, c x\right )}{\left (c x + 1\right )}^{\frac{3}{2}}{\left (c x - 1\right )} +{\left (10 \, c^{8} x^{8} - 23 \, c^{6} x^{6} + 15 \, c^{4} x^{4} - c^{2} x^{2} - 1\right )}{\left (c x + 1\right )} \sqrt{c x - 1} + 5 \,{\left (c^{9} x^{9} - 3 \, c^{7} x^{7} + 3 \, c^{5} x^{5} - c^{3} x^{3}\right )} \sqrt{c x + 1}\right )} \sqrt{-c x + 1}}{a b c^{5} x^{6} +{\left (c x + 1\right )}{\left (c x - 1\right )} a b c^{3} x^{4} - 2 \, a b c^{3} x^{4} + a b c x^{2} + 2 \,{\left (a b c^{4} x^{5} - a b c^{2} x^{3}\right )} \sqrt{c x + 1} \sqrt{c x - 1} +{\left (b^{2} c^{5} x^{6} +{\left (c x + 1\right )}{\left (c x - 1\right )} b^{2} c^{3} x^{4} - 2 \, b^{2} c^{3} x^{4} + b^{2} c x^{2} + 2 \,{\left (b^{2} c^{4} x^{5} - b^{2} c^{2} x^{3}\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((-c^2*x^2+1)^(5/2)/x/(a+b*arccosh(c*x))^2,x, algorithm="maxima")

[Out]

-((c^6*x^6 - 3*c^4*x^4 + 3*c^2*x^2 - 1)*(c*x + 1)*sqrt(c*x - 1) + (c^7*x^7 - 3*c^5*x^5 + 3*c^3*x^3 - c*x)*sqrt
(c*x + 1))*sqrt(-c*x + 1)/(a*b*c^3*x^3 + sqrt(c*x + 1)*sqrt(c*x - 1)*a*b*c^2*x^2 - a*b*c*x + (b^2*c^3*x^3 + sq
rt(c*x + 1)*sqrt(c*x - 1)*b^2*c^2*x^2 - b^2*c*x)*log(c*x + sqrt(c*x + 1)*sqrt(c*x - 1))) + integrate(((5*c^7*x
^7 - 8*c^5*x^5 + c^3*x^3 + 2*c*x)*(c*x + 1)^(3/2)*(c*x - 1) + (10*c^8*x^8 - 23*c^6*x^6 + 15*c^4*x^4 - c^2*x^2
- 1)*(c*x + 1)*sqrt(c*x - 1) + 5*(c^9*x^9 - 3*c^7*x^7 + 3*c^5*x^5 - c^3*x^3)*sqrt(c*x + 1))*sqrt(-c*x + 1)/(a*
b*c^5*x^6 + (c*x + 1)*(c*x - 1)*a*b*c^3*x^4 - 2*a*b*c^3*x^4 + a*b*c*x^2 + 2*(a*b*c^4*x^5 - a*b*c^2*x^3)*sqrt(c
*x + 1)*sqrt(c*x - 1) + (b^2*c^5*x^6 + (c*x + 1)*(c*x - 1)*b^2*c^3*x^4 - 2*b^2*c^3*x^4 + b^2*c*x^2 + 2*(b^2*c^
4*x^5 - b^2*c^2*x^3)*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^{4} x^{4} - 2 \, c^{2} x^{2} + 1\right )} \sqrt{-c^{2} x^{2} + 1}}{b^{2} x \operatorname{arcosh}\left (c x\right )^{2} + 2 \, a b x \operatorname{arcosh}\left (c x\right ) + a^{2} x}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-c^2*x^2+1)^(5/2)/x/(a+b*arccosh(c*x))^2,x, algorithm="fricas")

[Out]

integral((c^4*x^4 - 2*c^2*x^2 + 1)*sqrt(-c^2*x^2 + 1)/(b^2*x*arccosh(c*x)^2 + 2*a*b*x*arccosh(c*x) + a^2*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((-c**2*x**2+1)**(5/2)/x/(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{5}{2}}}{{\left (b \operatorname{arcosh}\left (c x\right ) + a\right )}^{2} x}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-c^2*x^2+1)^(5/2)/x/(a+b*arccosh(c*x))^2,x, algorithm="giac")

[Out]

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