\(\int \frac {(1-c^2 x^2)^{5/2}}{x^2 (a+b \text {arccosh}(c x))} \, dx\) [289]

Optimal result

Integrand size = 28, antiderivative size = 28 \[ \int \frac {\left (1-c^2 x^2\right )^{5/2}}{x^2 (a+b \text {arccosh}(c x))} \, dx=\frac {c \sqrt {-1+c x} \cosh \left (\frac {2 a}{b}\right ) \text {Chi}\left (\frac {2 (a+b \text {arccosh}(c x))}{b}\right )}{b \sqrt {1-c x}}-\frac {c \sqrt {-1+c x} \cosh \left (\frac {4 a}{b}\right ) \text {Chi}\left (\frac {4 (a+b \text {arccosh}(c x))}{b}\right )}{8 b \sqrt {1-c x}}-\frac {15 c \sqrt {-1+c x} \log (a+b \text {arccosh}(c x))}{8 b \sqrt {1-c x}}-\frac {c \sqrt {-1+c x} \sinh \left (\frac {2 a}{b}\right ) \text {Shi}\left (\frac {2 (a+b \text {arccosh}(c x))}{b}\right )}{b \sqrt {1-c x}}+\frac {c \sqrt {-1+c x} \sinh \left (\frac {4 a}{b}\right ) \text {Shi}\left (\frac {4 (a+b \text {arccosh}(c x))}{b}\right )}{8 b \sqrt {1-c x}}+\text {Int}\left (\frac {1}{x^2 \sqrt {1-c^2 x^2} (a+b \text {arccosh}(c x))},x\right ) \]

[Out]

c*Chi(2*(a+b*arccosh(c*x))/b)*cosh(2*a/b)*(c*x-1)^(1/2)/b/(-c*x+1)^(1/2)-1/8*c*Chi(4*(a+b*arccosh(c*x))/b)*cos
h(4*a/b)*(c*x-1)^(1/2)/b/(-c*x+1)^(1/2)-15/8*c*ln(a+b*arccosh(c*x))*(c*x-1)^(1/2)/b/(-c*x+1)^(1/2)-c*Shi(2*(a+
b*arccosh(c*x))/b)*sinh(2*a/b)*(c*x-1)^(1/2)/b/(-c*x+1)^(1/2)+1/8*c*Shi(4*(a+b*arccosh(c*x))/b)*sinh(4*a/b)*(c
*x-1)^(1/2)/b/(-c*x+1)^(1/2)+Unintegrable(1/x^2/(a+b*arccosh(c*x))/(-c^2*x^2+1)^(1/2),x)

Rubi [N/A]

Not integrable

Time = 0.70 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.00, number of steps used = 0, number of rules used = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \[ \int \frac {\left (1-c^2 x^2\right )^{5/2}}{x^2 (a+b \text {arccosh}(c x))} \, dx=\int \frac {\left (1-c^2 x^2\right )^{5/2}}{x^2 (a+b \text {arccosh}(c x))} \, dx \]

[In]

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

[Out]

(c*Sqrt[-1 + c*x]*Cosh[(2*a)/b]*CoshIntegral[(2*(a + b*ArcCosh[c*x]))/b])/(b*Sqrt[1 - c*x]) - (c*Sqrt[-1 + c*x
]*Cosh[(4*a)/b]*CoshIntegral[(4*(a + b*ArcCosh[c*x]))/b])/(8*b*Sqrt[1 - c*x]) - (15*c*Sqrt[-1 + c*x]*Log[a + b
*ArcCosh[c*x]])/(8*b*Sqrt[1 - c*x]) - (c*Sqrt[-1 + c*x]*Sinh[(2*a)/b]*SinhIntegral[(2*(a + b*ArcCosh[c*x]))/b]
)/(b*Sqrt[1 - c*x]) + (c*Sqrt[-1 + c*x]*Sinh[(4*a)/b]*SinhIntegral[(4*(a + b*ArcCosh[c*x]))/b])/(8*b*Sqrt[1 -
c*x]) + Defer[Int][1/(x^2*Sqrt[1 - c^2*x^2]*(a + b*ArcCosh[c*x])), x]

Rubi steps \begin{align*} \text {integral}& = \int \left (-\frac {3 c^2}{\sqrt {1-c^2 x^2} (a+b \text {arccosh}(c x))}+\frac {1}{x^2 \sqrt {1-c^2 x^2} (a+b \text {arccosh}(c x))}+\frac {3 c^4 x^2}{\sqrt {1-c^2 x^2} (a+b \text {arccosh}(c x))}-\frac {c^6 x^4}{\sqrt {1-c^2 x^2} (a+b \text {arccosh}(c x))}\right ) \, dx \\ & = -\left (\left (3 c^2\right ) \int \frac {1}{\sqrt {1-c^2 x^2} (a+b \text {arccosh}(c x))} \, dx\right )+\left (3 c^4\right ) \int \frac {x^2}{\sqrt {1-c^2 x^2} (a+b \text {arccosh}(c x))} \, dx-c^6 \int \frac {x^4}{\sqrt {1-c^2 x^2} (a+b \text {arccosh}(c x))} \, dx+\int \frac {1}{x^2 \sqrt {1-c^2 x^2} (a+b \text {arccosh}(c x))} \, dx \\ & = -\frac {3 c \sqrt {-1+c x} \log (a+b \text {arccosh}(c x))}{b \sqrt {1-c x}}-\frac {\left (c \sqrt {-1+c x}\right ) \text {Subst}\left (\int \frac {\cosh ^4\left (\frac {a}{b}-\frac {x}{b}\right )}{x} \, dx,x,a+b \text {arccosh}(c x)\right )}{b \sqrt {1-c x}}+\frac {\left (3 c \sqrt {-1+c x}\right ) \text {Subst}\left (\int \frac {\cosh ^2\left (\frac {a}{b}-\frac {x}{b}\right )}{x} \, dx,x,a+b \text {arccosh}(c x)\right )}{b \sqrt {1-c x}}+\int \frac {1}{x^2 \sqrt {1-c^2 x^2} (a+b \text {arccosh}(c x))} \, dx \\ & = -\frac {3 c \sqrt {-1+c x} \log (a+b \text {arccosh}(c x))}{b \sqrt {1-c x}}-\frac {\left (c \sqrt {-1+c x}\right ) \text {Subst}\left (\int \left (\frac {3}{8 x}+\frac {\cosh \left (\frac {4 a}{b}-\frac {4 x}{b}\right )}{8 x}+\frac {\cosh \left (\frac {2 a}{b}-\frac {2 x}{b}\right )}{2 x}\right ) \, dx,x,a+b \text {arccosh}(c x)\right )}{b \sqrt {1-c x}}+\frac {\left (3 c \sqrt {-1+c x}\right ) \text {Subst}\left (\int \left (\frac {1}{2 x}+\frac {\cosh \left (\frac {2 a}{b}-\frac {2 x}{b}\right )}{2 x}\right ) \, dx,x,a+b \text {arccosh}(c x)\right )}{b \sqrt {1-c x}}+\int \frac {1}{x^2 \sqrt {1-c^2 x^2} (a+b \text {arccosh}(c x))} \, dx \\ & = -\frac {15 c \sqrt {-1+c x} \log (a+b \text {arccosh}(c x))}{8 b \sqrt {1-c x}}-\frac {\left (c \sqrt {-1+c x}\right ) \text {Subst}\left (\int \frac {\cosh \left (\frac {4 a}{b}-\frac {4 x}{b}\right )}{x} \, dx,x,a+b \text {arccosh}(c x)\right )}{8 b \sqrt {1-c x}}-\frac {\left (c \sqrt {-1+c x}\right ) \text {Subst}\left (\int \frac {\cosh \left (\frac {2 a}{b}-\frac {2 x}{b}\right )}{x} \, dx,x,a+b \text {arccosh}(c x)\right )}{2 b \sqrt {1-c x}}+\frac {\left (3 c \sqrt {-1+c x}\right ) \text {Subst}\left (\int \frac {\cosh \left (\frac {2 a}{b}-\frac {2 x}{b}\right )}{x} \, dx,x,a+b \text {arccosh}(c x)\right )}{2 b \sqrt {1-c x}}+\int \frac {1}{x^2 \sqrt {1-c^2 x^2} (a+b \text {arccosh}(c x))} \, dx \\ & = -\frac {15 c \sqrt {-1+c x} \log (a+b \text {arccosh}(c x))}{8 b \sqrt {1-c x}}-\frac {\left (c \sqrt {-1+c x} \cosh \left (\frac {2 a}{b}\right )\right ) \text {Subst}\left (\int \frac {\cosh \left (\frac {2 x}{b}\right )}{x} \, dx,x,a+b \text {arccosh}(c x)\right )}{2 b \sqrt {1-c x}}+\frac {\left (3 c \sqrt {-1+c x} \cosh \left (\frac {2 a}{b}\right )\right ) \text {Subst}\left (\int \frac {\cosh \left (\frac {2 x}{b}\right )}{x} \, dx,x,a+b \text {arccosh}(c x)\right )}{2 b \sqrt {1-c x}}-\frac {\left (c \sqrt {-1+c x} \cosh \left (\frac {4 a}{b}\right )\right ) \text {Subst}\left (\int \frac {\cosh \left (\frac {4 x}{b}\right )}{x} \, dx,x,a+b \text {arccosh}(c x)\right )}{8 b \sqrt {1-c x}}+\frac {\left (c \sqrt {-1+c x} \sinh \left (\frac {2 a}{b}\right )\right ) \text {Subst}\left (\int \frac {\sinh \left (\frac {2 x}{b}\right )}{x} \, dx,x,a+b \text {arccosh}(c x)\right )}{2 b \sqrt {1-c x}}-\frac {\left (3 c \sqrt {-1+c x} \sinh \left (\frac {2 a}{b}\right )\right ) \text {Subst}\left (\int \frac {\sinh \left (\frac {2 x}{b}\right )}{x} \, dx,x,a+b \text {arccosh}(c x)\right )}{2 b \sqrt {1-c x}}+\frac {\left (c \sqrt {-1+c x} \sinh \left (\frac {4 a}{b}\right )\right ) \text {Subst}\left (\int \frac {\sinh \left (\frac {4 x}{b}\right )}{x} \, dx,x,a+b \text {arccosh}(c x)\right )}{8 b \sqrt {1-c x}}+\int \frac {1}{x^2 \sqrt {1-c^2 x^2} (a+b \text {arccosh}(c x))} \, dx \\ & = \frac {c \sqrt {-1+c x} \cosh \left (\frac {2 a}{b}\right ) \text {Chi}\left (\frac {2 (a+b \text {arccosh}(c x))}{b}\right )}{b \sqrt {1-c x}}-\frac {c \sqrt {-1+c x} \cosh \left (\frac {4 a}{b}\right ) \text {Chi}\left (\frac {4 (a+b \text {arccosh}(c x))}{b}\right )}{8 b \sqrt {1-c x}}-\frac {15 c \sqrt {-1+c x} \log (a+b \text {arccosh}(c x))}{8 b \sqrt {1-c x}}-\frac {c \sqrt {-1+c x} \sinh \left (\frac {2 a}{b}\right ) \text {Shi}\left (\frac {2 (a+b \text {arccosh}(c x))}{b}\right )}{b \sqrt {1-c x}}+\frac {c \sqrt {-1+c x} \sinh \left (\frac {4 a}{b}\right ) \text {Shi}\left (\frac {4 (a+b \text {arccosh}(c x))}{b}\right )}{8 b \sqrt {1-c x}}+\int \frac {1}{x^2 \sqrt {1-c^2 x^2} (a+b \text {arccosh}(c x))} \, dx \\ \end{align*}

Mathematica [N/A]

Not integrable

Time = 2.59 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.07 \[ \int \frac {\left (1-c^2 x^2\right )^{5/2}}{x^2 (a+b \text {arccosh}(c x))} \, dx=\int \frac {\left (1-c^2 x^2\right )^{5/2}}{x^2 (a+b \text {arccosh}(c x))} \, dx \]

[In]

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

[Out]

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

Maple [N/A] (verified)

Not integrable

Time = 0.98 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.93

\[\int \frac {\left (-c^{2} x^{2}+1\right )^{\frac {5}{2}}}{x^{2} \left (a +b \,\operatorname {arccosh}\left (c x \right )\right )}d x\]

[In]

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

[Out]

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

Fricas [N/A]

Not integrable

Time = 0.25 (sec) , antiderivative size = 49, normalized size of antiderivative = 1.75 \[ \int \frac {\left (1-c^2 x^2\right )^{5/2}}{x^2 (a+b \text {arccosh}(c x))} \, dx=\int { \frac {{\left (-c^{2} x^{2} + 1\right )}^{\frac {5}{2}}}{{\left (b \operatorname {arcosh}\left (c x\right ) + a\right )} x^{2}} \,d x } \]

[In]

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

Sympy [F(-1)]

Timed out. \[ \int \frac {\left (1-c^2 x^2\right )^{5/2}}{x^2 (a+b \text {arccosh}(c x))} \, dx=\text {Timed out} \]

[In]

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

[Out]

Timed out

Maxima [N/A]

Not integrable

Time = 0.31 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.00 \[ \int \frac {\left (1-c^2 x^2\right )^{5/2}}{x^2 (a+b \text {arccosh}(c x))} \, dx=\int { \frac {{\left (-c^{2} x^{2} + 1\right )}^{\frac {5}{2}}}{{\left (b \operatorname {arcosh}\left (c x\right ) + a\right )} x^{2}} \,d x } \]

[In]

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

[Out]

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

Giac [N/A]

Not integrable

Time = 0.31 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.00 \[ \int \frac {\left (1-c^2 x^2\right )^{5/2}}{x^2 (a+b \text {arccosh}(c x))} \, dx=\int { \frac {{\left (-c^{2} x^{2} + 1\right )}^{\frac {5}{2}}}{{\left (b \operatorname {arcosh}\left (c x\right ) + a\right )} x^{2}} \,d x } \]

[In]

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

[Out]

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

Mupad [N/A]

Not integrable

Time = 2.76 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.00 \[ \int \frac {\left (1-c^2 x^2\right )^{5/2}}{x^2 (a+b \text {arccosh}(c x))} \, dx=\int \frac {{\left (1-c^2\,x^2\right )}^{5/2}}{x^2\,\left (a+b\,\mathrm {acosh}\left (c\,x\right )\right )} \,d x \]

[In]

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

[Out]

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