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

Optimal. Leaf size=386 \[ \frac {\sqrt {1-c x} \text {Int}\left (\frac {\left (c^2 x^2-1\right )^2}{x^2 \left (a+b \cosh ^{-1}(c x)\right )},x\right )}{b c \sqrt {c x-1}}-\frac {25 \sqrt {1-c x} \sinh \left (\frac {a}{b}\right ) \text {Chi}\left (\frac {a+b \cosh ^{-1}(c x)}{b}\right )}{8 b^2 \sqrt {c x-1}}+\frac {25 \sqrt {1-c x} \sinh \left (\frac {3 a}{b}\right ) \text {Chi}\left (\frac {3 \left (a+b \cosh ^{-1}(c x)\right )}{b}\right )}{16 b^2 \sqrt {c x-1}}-\frac {5 \sqrt {1-c x} \sinh \left (\frac {5 a}{b}\right ) \text {Chi}\left (\frac {5 \left (a+b \cosh ^{-1}(c x)\right )}{b}\right )}{16 b^2 \sqrt {c x-1}}+\frac {25 \sqrt {1-c x} \cosh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a+b \cosh ^{-1}(c x)}{b}\right )}{8 b^2 \sqrt {c x-1}}-\frac {25 \sqrt {1-c x} \cosh \left (\frac {3 a}{b}\right ) \text {Shi}\left (\frac {3 \left (a+b \cosh ^{-1}(c x)\right )}{b}\right )}{16 b^2 \sqrt {c x-1}}+\frac {5 \sqrt {1-c x} \cosh \left (\frac {5 a}{b}\right ) \text {Shi}\left (\frac {5 \left (a+b \cosh ^{-1}(c x)\right )}{b}\right )}{16 b^2 \sqrt {c x-1}}-\frac {\sqrt {c x-1} \sqrt {c x+1} \left (1-c^2 x^2\right )^{5/2}}{b c x \left (a+b \cosh ^{-1}(c x)\right )} \]

[Out]

25/8*cosh(a/b)*Shi((a+b*arccosh(c*x))/b)*(-c*x+1)^(1/2)/b^2/(c*x-1)^(1/2)-25/16*cosh(3*a/b)*Shi(3*(a+b*arccosh
(c*x))/b)*(-c*x+1)^(1/2)/b^2/(c*x-1)^(1/2)+5/16*cosh(5*a/b)*Shi(5*(a+b*arccosh(c*x))/b)*(-c*x+1)^(1/2)/b^2/(c*
x-1)^(1/2)-25/8*Chi((a+b*arccosh(c*x))/b)*sinh(a/b)*(-c*x+1)^(1/2)/b^2/(c*x-1)^(1/2)+25/16*Chi(3*(a+b*arccosh(
c*x))/b)*sinh(3*a/b)*(-c*x+1)^(1/2)/b^2/(c*x-1)^(1/2)-5/16*Chi(5*(a+b*arccosh(c*x))/b)*sinh(5*a/b)*(-c*x+1)^(1
/2)/b^2/(c*x-1)^(1/2)-(-c^2*x^2+1)^(5/2)*(c*x-1)^(1/2)*(c*x+1)^(1/2)/b/c/x/(a+b*arccosh(c*x))+(-c*x+1)^(1/2)*U
nintegrable((c^2*x^2-1)^2/x^2/(a+b*arccosh(c*x)),x)/b/c/(c*x-1)^(1/2)

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Rubi [A]  time = 0.98, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, 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*}

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Mathematica [A]  time = 9.00, size = 0, normalized size = 0.00 \[ \int \frac {\left (1-c^2 x^2\right )^{5/2}}{x \left (a+b \cosh ^{-1}(c x)\right )^2} \, dx \]

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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fricas [A]  time = 0.49, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (c^{4} x^{4} - 2 \, c^{2} x^{2} + 1\right )} \sqrt {-c^{2} x^{2} + 1}}{b^{2} x \operatorname {arcosh}\left (c x\right )^{2} + 2 \, a b x \operatorname {arcosh}\left (c x\right ) + a^{2} x}, x\right ) \]

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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giac [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]

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]

Exception raised: TypeError >> An error occurred running a Giac command:INPUT:sage2:=int(sage0,x):;OUTPUT:sym2
poly/r2sym(const gen & e,const index_m & i,const vecteur & l) Error: Bad Argument Value

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maple [A]  time = 0.74, size = 0, normalized size = 0.00 \[ \int \frac {\left (-c^{2} x^{2}+1\right )^{\frac {5}{2}}}{x \left (a +b \,\mathrm {arccosh}\left (c x \right )\right )^{2}}\, dx \]

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 = 0.00, size = 0, normalized size = 0.00 \[ -\frac {{\left ({\left (c^{6} x^{6} - 3 \, c^{4} x^{4} + 3 \, c^{2} x^{2} - 1\right )} {\left (c x + 1\right )} \sqrt {c x - 1} + {\left (c^{7} x^{7} - 3 \, c^{5} x^{5} + 3 \, c^{3} x^{3} - c x\right )} \sqrt {c x + 1}\right )} \sqrt {-c x + 1}}{a b c^{3} x^{3} + \sqrt {c x + 1} \sqrt {c x - 1} a b c^{2} x^{2} - a b c x + {\left (b^{2} c^{3} x^{3} + \sqrt {c x + 1} \sqrt {c x - 1} b^{2} c^{2} x^{2} - b^{2} c x\right )} \log \left (c x + \sqrt {c x + 1} \sqrt {c x - 1}\right )} + \int \frac {{\left ({\left (5 \, c^{7} x^{7} - 8 \, c^{5} x^{5} + c^{3} x^{3} + 2 \, c x\right )} {\left (c x + 1\right )}^{\frac {3}{2}} {\left (c x - 1\right )} + {\left (10 \, c^{8} x^{8} - 23 \, c^{6} x^{6} + 15 \, c^{4} x^{4} - c^{2} x^{2} - 1\right )} {\left (c x + 1\right )} \sqrt {c x - 1} + 5 \, {\left (c^{9} x^{9} - 3 \, c^{7} x^{7} + 3 \, c^{5} x^{5} - c^{3} x^{3}\right )} \sqrt {c x + 1}\right )} \sqrt {-c x + 1}}{a b c^{5} x^{6} + {\left (c x + 1\right )} {\left (c x - 1\right )} a b c^{3} x^{4} - 2 \, a b c^{3} x^{4} + a b c x^{2} + 2 \, {\left (a b c^{4} x^{5} - a b c^{2} x^{3}\right )} \sqrt {c x + 1} \sqrt {c x - 1} + {\left (b^{2} c^{5} x^{6} + {\left (c x + 1\right )} {\left (c x - 1\right )} b^{2} c^{3} x^{4} - 2 \, b^{2} c^{3} x^{4} + b^{2} c x^{2} + 2 \, {\left (b^{2} c^{4} x^{5} - b^{2} c^{2} x^{3}\right )} \sqrt {c x + 1} \sqrt {c x - 1}\right )} \log \left (c x + \sqrt {c x + 1} \sqrt {c x - 1}\right )}\,{d x} \]

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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mupad [A]  time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {{\left (1-c^2\,x^2\right )}^{5/2}}{x\,{\left (a+b\,\mathrm {acosh}\left (c\,x\right )\right )}^2} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [A]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\left (- \left (c x - 1\right ) \left (c x + 1\right )\right )^{\frac {5}{2}}}{x \left (a + b \operatorname {acosh}{\left (c x \right )}\right )^{2}}\, dx \]

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]

Integral((-(c*x - 1)*(c*x + 1))**(5/2)/(x*(a + b*acosh(c*x))**2), x)

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