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

Optimal. Leaf size=309 \[ \text{Unintegrable}\left (\frac{1}{x \sqrt{1-c^2 x^2} \left (a+b \cosh ^{-1}(c x)\right )},x\right )-\frac{11 \sqrt{c x-1} \cosh \left (\frac{a}{b}\right ) \text{Chi}\left (\frac{a+b \cosh ^{-1}(c x)}{b}\right )}{8 b \sqrt{1-c x}}+\frac{7 \sqrt{c x-1} \cosh \left (\frac{3 a}{b}\right ) \text{Chi}\left (\frac{3 \left (a+b \cosh ^{-1}(c x)\right )}{b}\right )}{16 b \sqrt{1-c x}}-\frac{\sqrt{c x-1} \cosh \left (\frac{5 a}{b}\right ) \text{Chi}\left (\frac{5 \left (a+b \cosh ^{-1}(c x)\right )}{b}\right )}{16 b \sqrt{1-c x}}+\frac{11 \sqrt{c x-1} \sinh \left (\frac{a}{b}\right ) \text{Shi}\left (\frac{a+b \cosh ^{-1}(c x)}{b}\right )}{8 b \sqrt{1-c x}}-\frac{7 \sqrt{c x-1} \sinh \left (\frac{3 a}{b}\right ) \text{Shi}\left (\frac{3 \left (a+b \cosh ^{-1}(c x)\right )}{b}\right )}{16 b \sqrt{1-c x}}+\frac{\sqrt{c x-1} \sinh \left (\frac{5 a}{b}\right ) \text{Shi}\left (\frac{5 \left (a+b \cosh ^{-1}(c x)\right )}{b}\right )}{16 b \sqrt{1-c x}} \]

[Out]

(-11*Sqrt[-1 + c*x]*Cosh[a/b]*CoshIntegral[(a + b*ArcCosh[c*x])/b])/(8*b*Sqrt[1 - c*x]) + (7*Sqrt[-1 + c*x]*Co
sh[(3*a)/b]*CoshIntegral[(3*(a + b*ArcCosh[c*x]))/b])/(16*b*Sqrt[1 - c*x]) - (Sqrt[-1 + c*x]*Cosh[(5*a)/b]*Cos
hIntegral[(5*(a + b*ArcCosh[c*x]))/b])/(16*b*Sqrt[1 - c*x]) + (11*Sqrt[-1 + c*x]*Sinh[a/b]*SinhIntegral[(a + b
*ArcCosh[c*x])/b])/(8*b*Sqrt[1 - c*x]) - (7*Sqrt[-1 + c*x]*Sinh[(3*a)/b]*SinhIntegral[(3*(a + b*ArcCosh[c*x]))
/b])/(16*b*Sqrt[1 - c*x]) + (Sqrt[-1 + c*x]*Sinh[(5*a)/b]*SinhIntegral[(5*(a + b*ArcCosh[c*x]))/b])/(16*b*Sqrt
[1 - c*x]) + Unintegrable[1/(x*Sqrt[1 - c^2*x^2]*(a + b*ArcCosh[c*x])), x]

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Rubi [A]  time = 2.37614, 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 = {} \[ \int \frac{\left (1-c^2 x^2\right )^{5/2}}{x \left (a+b \cosh ^{-1}(c x)\right )} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

(11*Sqrt[1 - c^2*x^2]*Cosh[a/b]*CoshIntegral[a/b + ArcCosh[c*x]])/(8*b*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) - (7*Sqrt
[1 - c^2*x^2]*Cosh[(3*a)/b]*CoshIntegral[(3*a)/b + 3*ArcCosh[c*x]])/(16*b*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + (Sqr
t[1 - c^2*x^2]*Cosh[(5*a)/b]*CoshIntegral[(5*a)/b + 5*ArcCosh[c*x]])/(16*b*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) - (11
*Sqrt[1 - c^2*x^2]*Sinh[a/b]*SinhIntegral[a/b + ArcCosh[c*x]])/(8*b*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + (7*Sqrt[1
- c^2*x^2]*Sinh[(3*a)/b]*SinhIntegral[(3*a)/b + 3*ArcCosh[c*x]])/(16*b*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) - (Sqrt[1
- c^2*x^2]*Sinh[(5*a)/b]*SinhIntegral[(5*a)/b + 5*ArcCosh[c*x]])/(16*b*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) - (Sqrt[
1 - c^2*x^2]*Defer[Int][1/(x*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(a + b*ArcCosh[c*x])), x])/(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 )} \, 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 )} \, dx}{\sqrt{-1+c x} \sqrt{1+c x}}\\ &=\frac{\sqrt{1-c^2 x^2} \int \left (-\frac{1}{x \sqrt{-1+c x} \sqrt{1+c x} \left (a+b \cosh ^{-1}(c x)\right )}+\frac{3 c^2 x}{\sqrt{-1+c x} \sqrt{1+c x} \left (a+b \cosh ^{-1}(c x)\right )}-\frac{3 c^4 x^3}{\sqrt{-1+c x} \sqrt{1+c x} \left (a+b \cosh ^{-1}(c x)\right )}+\frac{c^6 x^5}{\sqrt{-1+c x} \sqrt{1+c x} \left (a+b \cosh ^{-1}(c x)\right )}\right ) \, dx}{\sqrt{-1+c x} \sqrt{1+c x}}\\ &=-\frac{\sqrt{1-c^2 x^2} \int \frac{1}{x \sqrt{-1+c x} \sqrt{1+c x} \left (a+b \cosh ^{-1}(c x)\right )} \, dx}{\sqrt{-1+c x} \sqrt{1+c x}}+\frac{\left (3 c^2 \sqrt{1-c^2 x^2}\right ) \int \frac{x}{\sqrt{-1+c x} \sqrt{1+c x} \left (a+b \cosh ^{-1}(c x)\right )} \, dx}{\sqrt{-1+c x} \sqrt{1+c x}}-\frac{\left (3 c^4 \sqrt{1-c^2 x^2}\right ) \int \frac{x^3}{\sqrt{-1+c x} \sqrt{1+c x} \left (a+b \cosh ^{-1}(c x)\right )} \, dx}{\sqrt{-1+c x} \sqrt{1+c x}}+\frac{\left (c^6 \sqrt{1-c^2 x^2}\right ) \int \frac{x^5}{\sqrt{-1+c x} \sqrt{1+c x} \left (a+b \cosh ^{-1}(c x)\right )} \, dx}{\sqrt{-1+c x} \sqrt{1+c x}}\\ &=-\frac{\sqrt{1-c^2 x^2} \int \frac{1}{x \sqrt{-1+c x} \sqrt{1+c x} \left (a+b \cosh ^{-1}(c x)\right )} \, dx}{\sqrt{-1+c x} \sqrt{1+c x}}+\frac{\sqrt{1-c^2 x^2} \operatorname{Subst}\left (\int \frac{\cosh ^5(x)}{a+b x} \, dx,x,\cosh ^{-1}(c x)\right )}{\sqrt{-1+c x} \sqrt{1+c x}}+\frac{\left (3 \sqrt{1-c^2 x^2}\right ) \operatorname{Subst}\left (\int \frac{\cosh (x)}{a+b x} \, dx,x,\cosh ^{-1}(c x)\right )}{\sqrt{-1+c x} \sqrt{1+c x}}-\frac{\left (3 \sqrt{1-c^2 x^2}\right ) \operatorname{Subst}\left (\int \frac{\cosh ^3(x)}{a+b x} \, dx,x,\cosh ^{-1}(c x)\right )}{\sqrt{-1+c x} \sqrt{1+c x}}\\ &=-\frac{\sqrt{1-c^2 x^2} \int \frac{1}{x \sqrt{-1+c x} \sqrt{1+c x} \left (a+b \cosh ^{-1}(c x)\right )} \, dx}{\sqrt{-1+c x} \sqrt{1+c x}}+\frac{\sqrt{1-c^2 x^2} \operatorname{Subst}\left (\int \left (\frac{5 \cosh (x)}{8 (a+b x)}+\frac{5 \cosh (3 x)}{16 (a+b x)}+\frac{\cosh (5 x)}{16 (a+b x)}\right ) \, dx,x,\cosh ^{-1}(c x)\right )}{\sqrt{-1+c x} \sqrt{1+c x}}-\frac{\left (3 \sqrt{1-c^2 x^2}\right ) \operatorname{Subst}\left (\int \left (\frac{3 \cosh (x)}{4 (a+b x)}+\frac{\cosh (3 x)}{4 (a+b x)}\right ) \, dx,x,\cosh ^{-1}(c x)\right )}{\sqrt{-1+c x} \sqrt{1+c x}}+\frac{\left (3 \sqrt{1-c^2 x^2} \cosh \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 )}{\sqrt{-1+c x} \sqrt{1+c x}}-\frac{\left (3 \sqrt{1-c^2 x^2} \sinh \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 )}{\sqrt{-1+c x} \sqrt{1+c x}}\\ &=\frac{3 \sqrt{1-c^2 x^2} \cosh \left (\frac{a}{b}\right ) \text{Chi}\left (\frac{a}{b}+\cosh ^{-1}(c x)\right )}{b \sqrt{-1+c x} \sqrt{1+c x}}-\frac{3 \sqrt{1-c^2 x^2} \sinh \left (\frac{a}{b}\right ) \text{Shi}\left (\frac{a}{b}+\cosh ^{-1}(c x)\right )}{b \sqrt{-1+c x} \sqrt{1+c x}}+\frac{\sqrt{1-c^2 x^2} \operatorname{Subst}\left (\int \frac{\cosh (5 x)}{a+b x} \, dx,x,\cosh ^{-1}(c x)\right )}{16 \sqrt{-1+c x} \sqrt{1+c x}}+\frac{\left (5 \sqrt{1-c^2 x^2}\right ) \operatorname{Subst}\left (\int \frac{\cosh (3 x)}{a+b x} \, dx,x,\cosh ^{-1}(c x)\right )}{16 \sqrt{-1+c x} \sqrt{1+c x}}+\frac{\left (5 \sqrt{1-c^2 x^2}\right ) \operatorname{Subst}\left (\int \frac{\cosh (x)}{a+b x} \, dx,x,\cosh ^{-1}(c x)\right )}{8 \sqrt{-1+c x} \sqrt{1+c x}}-\frac{\left (3 \sqrt{1-c^2 x^2}\right ) \operatorname{Subst}\left (\int \frac{\cosh (3 x)}{a+b x} \, dx,x,\cosh ^{-1}(c x)\right )}{4 \sqrt{-1+c x} \sqrt{1+c x}}-\frac{\sqrt{1-c^2 x^2} \int \frac{1}{x \sqrt{-1+c x} \sqrt{1+c x} \left (a+b \cosh ^{-1}(c x)\right )} \, dx}{\sqrt{-1+c x} \sqrt{1+c x}}-\frac{\left (9 \sqrt{1-c^2 x^2}\right ) \operatorname{Subst}\left (\int \frac{\cosh (x)}{a+b x} \, dx,x,\cosh ^{-1}(c x)\right )}{4 \sqrt{-1+c x} \sqrt{1+c x}}\\ &=\frac{3 \sqrt{1-c^2 x^2} \cosh \left (\frac{a}{b}\right ) \text{Chi}\left (\frac{a}{b}+\cosh ^{-1}(c x)\right )}{b \sqrt{-1+c x} \sqrt{1+c x}}-\frac{3 \sqrt{1-c^2 x^2} \sinh \left (\frac{a}{b}\right ) \text{Shi}\left (\frac{a}{b}+\cosh ^{-1}(c x)\right )}{b \sqrt{-1+c x} \sqrt{1+c x}}-\frac{\sqrt{1-c^2 x^2} \int \frac{1}{x \sqrt{-1+c x} \sqrt{1+c x} \left (a+b \cosh ^{-1}(c x)\right )} \, dx}{\sqrt{-1+c x} \sqrt{1+c x}}+\frac{\left (5 \sqrt{1-c^2 x^2} \cosh \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 \sqrt{-1+c x} \sqrt{1+c x}}-\frac{\left (9 \sqrt{1-c^2 x^2} \cosh \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 )}{4 \sqrt{-1+c x} \sqrt{1+c x}}+\frac{\left (5 \sqrt{1-c^2 x^2} \cosh \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 \sqrt{-1+c x} \sqrt{1+c x}}-\frac{\left (3 \sqrt{1-c^2 x^2} \cosh \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 )}{4 \sqrt{-1+c x} \sqrt{1+c x}}+\frac{\left (\sqrt{1-c^2 x^2} \cosh \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 \sqrt{-1+c x} \sqrt{1+c x}}-\frac{\left (5 \sqrt{1-c^2 x^2} \sinh \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 \sqrt{-1+c x} \sqrt{1+c x}}+\frac{\left (9 \sqrt{1-c^2 x^2} \sinh \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 )}{4 \sqrt{-1+c x} \sqrt{1+c x}}-\frac{\left (5 \sqrt{1-c^2 x^2} \sinh \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 \sqrt{-1+c x} \sqrt{1+c x}}+\frac{\left (3 \sqrt{1-c^2 x^2} \sinh \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 )}{4 \sqrt{-1+c x} \sqrt{1+c x}}-\frac{\left (\sqrt{1-c^2 x^2} \sinh \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 \sqrt{-1+c x} \sqrt{1+c x}}\\ &=\frac{11 \sqrt{1-c^2 x^2} \cosh \left (\frac{a}{b}\right ) \text{Chi}\left (\frac{a}{b}+\cosh ^{-1}(c x)\right )}{8 b \sqrt{-1+c x} \sqrt{1+c x}}-\frac{7 \sqrt{1-c^2 x^2} \cosh \left (\frac{3 a}{b}\right ) \text{Chi}\left (\frac{3 a}{b}+3 \cosh ^{-1}(c x)\right )}{16 b \sqrt{-1+c x} \sqrt{1+c x}}+\frac{\sqrt{1-c^2 x^2} \cosh \left (\frac{5 a}{b}\right ) \text{Chi}\left (\frac{5 a}{b}+5 \cosh ^{-1}(c x)\right )}{16 b \sqrt{-1+c x} \sqrt{1+c x}}-\frac{11 \sqrt{1-c^2 x^2} \sinh \left (\frac{a}{b}\right ) \text{Shi}\left (\frac{a}{b}+\cosh ^{-1}(c x)\right )}{8 b \sqrt{-1+c x} \sqrt{1+c x}}+\frac{7 \sqrt{1-c^2 x^2} \sinh \left (\frac{3 a}{b}\right ) \text{Shi}\left (\frac{3 a}{b}+3 \cosh ^{-1}(c x)\right )}{16 b \sqrt{-1+c x} \sqrt{1+c x}}-\frac{\sqrt{1-c^2 x^2} \sinh \left (\frac{5 a}{b}\right ) \text{Shi}\left (\frac{5 a}{b}+5 \cosh ^{-1}(c x)\right )}{16 b \sqrt{-1+c x} \sqrt{1+c x}}-\frac{\sqrt{1-c^2 x^2} \int \frac{1}{x \sqrt{-1+c x} \sqrt{1+c x} \left (a+b \cosh ^{-1}(c x)\right )} \, dx}{\sqrt{-1+c x} \sqrt{1+c x}}\\ \end{align*}

Mathematica [A]  time = 1.26989, size = 0, normalized size = 0. \[ \int \frac{\left (1-c^2 x^2\right )^{5/2}}{x \left (a+b \cosh ^{-1}(c x)\right )} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

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Maple [A]  time = 0.268, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{x \left ( a+b{\rm arccosh} \left (cx\right ) \right ) } \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)),x)

[Out]

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

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Maxima [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 )} 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)),x, algorithm="maxima")

[Out]

integrate((-c^2*x^2 + 1)^(5/2)/((b*arccosh(c*x) + a)*x), 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 x \operatorname{arcosh}\left (c x\right ) + a 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)),x, algorithm="fricas")

[Out]

integral((c^4*x^4 - 2*c^2*x^2 + 1)*sqrt(-c^2*x^2 + 1)/(b*x*arccosh(c*x) + a*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)),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 )} 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)),x, algorithm="giac")

[Out]

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