\(\int \frac {(1+c^2 x^2)^{5/2}}{x^2 (a+b \text {arcsinh}(c x))} \, dx\) [377]

Optimal result

Integrand size = 27, antiderivative size = 27 \[ \int \frac {\left (1+c^2 x^2\right )^{5/2}}{x^2 (a+b \text {arcsinh}(c x))} \, dx=\frac {c \cosh \left (\frac {2 a}{b}\right ) \text {Chi}\left (\frac {2 (a+b \text {arcsinh}(c x))}{b}\right )}{b}+\frac {c \cosh \left (\frac {4 a}{b}\right ) \text {Chi}\left (\frac {4 (a+b \text {arcsinh}(c x))}{b}\right )}{8 b}+\frac {15 c \log (a+b \text {arcsinh}(c x))}{8 b}-\frac {c \sinh \left (\frac {2 a}{b}\right ) \text {Shi}\left (\frac {2 (a+b \text {arcsinh}(c x))}{b}\right )}{b}-\frac {c \sinh \left (\frac {4 a}{b}\right ) \text {Shi}\left (\frac {4 (a+b \text {arcsinh}(c x))}{b}\right )}{8 b}+\text {Int}\left (\frac {1}{x^2 \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))},x\right ) \]

[Out]

c*Chi(2*(a+b*arcsinh(c*x))/b)*cosh(2*a/b)/b+1/8*c*Chi(4*(a+b*arcsinh(c*x))/b)*cosh(4*a/b)/b+15/8*c*ln(a+b*arcs
inh(c*x))/b-c*Shi(2*(a+b*arcsinh(c*x))/b)*sinh(2*a/b)/b-1/8*c*Shi(4*(a+b*arcsinh(c*x))/b)*sinh(4*a/b)/b+Uninte
grable(1/x^2/(a+b*arcsinh(c*x))/(c^2*x^2+1)^(1/2),x)

Rubi [N/A]

Not integrable

Time = 0.64 (sec) , antiderivative size = 27, 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 = {} \[ \int \frac {\left (1+c^2 x^2\right )^{5/2}}{x^2 (a+b \text {arcsinh}(c x))} \, dx=\int \frac {\left (1+c^2 x^2\right )^{5/2}}{x^2 (a+b \text {arcsinh}(c x))} \, dx \]

[In]

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

[Out]

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

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

Mathematica [N/A]

Not integrable

Time = 2.41 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.07 \[ \int \frac {\left (1+c^2 x^2\right )^{5/2}}{x^2 (a+b \text {arcsinh}(c x))} \, dx=\int \frac {\left (1+c^2 x^2\right )^{5/2}}{x^2 (a+b \text {arcsinh}(c x))} \, dx \]

[In]

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

[Out]

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

Maple [N/A] (verified)

Not integrable

Time = 0.08 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.93

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

[In]

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

[Out]

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

Fricas [N/A]

Not integrable

Time = 0.25 (sec) , antiderivative size = 48, normalized size of antiderivative = 1.78 \[ \int \frac {\left (1+c^2 x^2\right )^{5/2}}{x^2 (a+b \text {arcsinh}(c x))} \, dx=\int { \frac {{\left (c^{2} x^{2} + 1\right )}^{\frac {5}{2}}}{{\left (b \operatorname {arsinh}\left (c x\right ) + a\right )} x^{2}} \,d x } \]

[In]

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

Sympy [N/A]

Not integrable

Time = 3.88 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.89 \[ \int \frac {\left (1+c^2 x^2\right )^{5/2}}{x^2 (a+b \text {arcsinh}(c x))} \, dx=\int \frac {\left (c^{2} x^{2} + 1\right )^{\frac {5}{2}}}{x^{2} \left (a + b \operatorname {asinh}{\left (c x \right )}\right )}\, dx \]

[In]

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

[Out]

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

Maxima [N/A]

Not integrable

Time = 0.27 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.00 \[ \int \frac {\left (1+c^2 x^2\right )^{5/2}}{x^2 (a+b \text {arcsinh}(c x))} \, dx=\int { \frac {{\left (c^{2} x^{2} + 1\right )}^{\frac {5}{2}}}{{\left (b \operatorname {arsinh}\left (c x\right ) + a\right )} x^{2}} \,d x } \]

[In]

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

[Out]

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

Giac [N/A]

Not integrable

Time = 0.32 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.00 \[ \int \frac {\left (1+c^2 x^2\right )^{5/2}}{x^2 (a+b \text {arcsinh}(c x))} \, dx=\int { \frac {{\left (c^{2} x^{2} + 1\right )}^{\frac {5}{2}}}{{\left (b \operatorname {arsinh}\left (c x\right ) + a\right )} x^{2}} \,d x } \]

[In]

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

[Out]

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

Mupad [N/A]

Not integrable

Time = 2.68 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.00 \[ \int \frac {\left (1+c^2 x^2\right )^{5/2}}{x^2 (a+b \text {arcsinh}(c x))} \, dx=\int \frac {{\left (c^2\,x^2+1\right )}^{5/2}}{x^2\,\left (a+b\,\mathrm {asinh}\left (c\,x\right )\right )} \,d x \]

[In]

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

[Out]

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