\(\int \frac {(1+c^2 x^2)^{5/2}}{x (a+b \text {arcsinh}(c x))} \, dx\) [376]

Optimal result

Integrand size = 27, antiderivative size = 27 \[ \int \frac {\left (1+c^2 x^2\right )^{5/2}}{x (a+b \text {arcsinh}(c x))} \, dx=-\frac {11 \text {Chi}\left (\frac {a+b \text {arcsinh}(c x)}{b}\right ) \sinh \left (\frac {a}{b}\right )}{8 b}-\frac {7 \text {Chi}\left (\frac {3 (a+b \text {arcsinh}(c x))}{b}\right ) \sinh \left (\frac {3 a}{b}\right )}{16 b}-\frac {\text {Chi}\left (\frac {5 (a+b \text {arcsinh}(c x))}{b}\right ) \sinh \left (\frac {5 a}{b}\right )}{16 b}+\frac {11 \cosh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a+b \text {arcsinh}(c x)}{b}\right )}{8 b}+\frac {7 \cosh \left (\frac {3 a}{b}\right ) \text {Shi}\left (\frac {3 (a+b \text {arcsinh}(c x))}{b}\right )}{16 b}+\frac {\cosh \left (\frac {5 a}{b}\right ) \text {Shi}\left (\frac {5 (a+b \text {arcsinh}(c x))}{b}\right )}{16 b}+\text {Int}\left (\frac {1}{x \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))},x\right ) \]

[Out]

11/8*cosh(a/b)*Shi((a+b*arcsinh(c*x))/b)/b+7/16*cosh(3*a/b)*Shi(3*(a+b*arcsinh(c*x))/b)/b+1/16*cosh(5*a/b)*Shi
(5*(a+b*arcsinh(c*x))/b)/b-11/8*Chi((a+b*arcsinh(c*x))/b)*sinh(a/b)/b-7/16*Chi(3*(a+b*arcsinh(c*x))/b)*sinh(3*
a/b)/b-1/16*Chi(5*(a+b*arcsinh(c*x))/b)*sinh(5*a/b)/b+Unintegrable(1/x/(a+b*arcsinh(c*x))/(c^2*x^2+1)^(1/2),x)

Rubi [N/A]

Not integrable

Time = 0.81 (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 = {} \[ \int \frac {\left (1+c^2 x^2\right )^{5/2}}{x (a+b \text {arcsinh}(c x))} \, dx=\int \frac {\left (1+c^2 x^2\right )^{5/2}}{x (a+b \text {arcsinh}(c x))} \, dx \]

[In]

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

[Out]

(-11*CoshIntegral[(a + b*ArcSinh[c*x])/b]*Sinh[a/b])/(8*b) - (7*CoshIntegral[(3*(a + b*ArcSinh[c*x]))/b]*Sinh[
(3*a)/b])/(16*b) - (CoshIntegral[(5*(a + b*ArcSinh[c*x]))/b]*Sinh[(5*a)/b])/(16*b) + (11*Cosh[a/b]*SinhIntegra
l[(a + b*ArcSinh[c*x])/b])/(8*b) + (7*Cosh[(3*a)/b]*SinhIntegral[(3*(a + b*ArcSinh[c*x]))/b])/(16*b) + (Cosh[(
5*a)/b]*SinhIntegral[(5*(a + b*ArcSinh[c*x]))/b])/(16*b) + Defer[Int][1/(x*Sqrt[1 + c^2*x^2]*(a + b*ArcSinh[c*
x])), x]

Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {1}{x \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))}+\frac {3 c^2 x}{\sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))}+\frac {3 c^4 x^3}{\sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))}+\frac {c^6 x^5}{\sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))}\right ) \, dx \\ & = \left (3 c^2\right ) \int \frac {x}{\sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))} \, dx+\left (3 c^4\right ) \int \frac {x^3}{\sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))} \, dx+c^6 \int \frac {x^5}{\sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))} \, dx+\int \frac {1}{x \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))} \, dx \\ & = -\frac {\text {Subst}\left (\int \frac {\sinh ^5\left (\frac {a}{b}-\frac {x}{b}\right )}{x} \, dx,x,a+b \text {arcsinh}(c x)\right )}{b}-\frac {3 \text {Subst}\left (\int \frac {\sinh \left (\frac {a}{b}-\frac {x}{b}\right )}{x} \, dx,x,a+b \text {arcsinh}(c x)\right )}{b}-\frac {3 \text {Subst}\left (\int \frac {\sinh ^3\left (\frac {a}{b}-\frac {x}{b}\right )}{x} \, dx,x,a+b \text {arcsinh}(c x)\right )}{b}+\int \frac {1}{x \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))} \, dx \\ & = \frac {i \text {Subst}\left (\int \left (\frac {i \sinh \left (\frac {5 a}{b}-\frac {5 x}{b}\right )}{16 x}-\frac {5 i \sinh \left (\frac {3 a}{b}-\frac {3 x}{b}\right )}{16 x}+\frac {5 i \sinh \left (\frac {a}{b}-\frac {x}{b}\right )}{8 x}\right ) \, dx,x,a+b \text {arcsinh}(c x)\right )}{b}-\frac {(3 i) \text {Subst}\left (\int \left (-\frac {i \sinh \left (\frac {3 a}{b}-\frac {3 x}{b}\right )}{4 x}+\frac {3 i \sinh \left (\frac {a}{b}-\frac {x}{b}\right )}{4 x}\right ) \, dx,x,a+b \text {arcsinh}(c x)\right )}{b}+\frac {\left (3 \cosh \left (\frac {a}{b}\right )\right ) \text {Subst}\left (\int \frac {\sinh \left (\frac {x}{b}\right )}{x} \, dx,x,a+b \text {arcsinh}(c x)\right )}{b}-\frac {\left (3 \sinh \left (\frac {a}{b}\right )\right ) \text {Subst}\left (\int \frac {\cosh \left (\frac {x}{b}\right )}{x} \, dx,x,a+b \text {arcsinh}(c x)\right )}{b}+\int \frac {1}{x \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))} \, dx \\ & = -\frac {3 \text {Chi}\left (\frac {a+b \text {arcsinh}(c x)}{b}\right ) \sinh \left (\frac {a}{b}\right )}{b}+\frac {3 \cosh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a+b \text {arcsinh}(c x)}{b}\right )}{b}-\frac {\text {Subst}\left (\int \frac {\sinh \left (\frac {5 a}{b}-\frac {5 x}{b}\right )}{x} \, dx,x,a+b \text {arcsinh}(c x)\right )}{16 b}+\frac {5 \text {Subst}\left (\int \frac {\sinh \left (\frac {3 a}{b}-\frac {3 x}{b}\right )}{x} \, dx,x,a+b \text {arcsinh}(c x)\right )}{16 b}-\frac {5 \text {Subst}\left (\int \frac {\sinh \left (\frac {a}{b}-\frac {x}{b}\right )}{x} \, dx,x,a+b \text {arcsinh}(c x)\right )}{8 b}-\frac {3 \text {Subst}\left (\int \frac {\sinh \left (\frac {3 a}{b}-\frac {3 x}{b}\right )}{x} \, dx,x,a+b \text {arcsinh}(c x)\right )}{4 b}+\frac {9 \text {Subst}\left (\int \frac {\sinh \left (\frac {a}{b}-\frac {x}{b}\right )}{x} \, dx,x,a+b \text {arcsinh}(c x)\right )}{4 b}+\int \frac {1}{x \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))} \, dx \\ & = -\frac {3 \text {Chi}\left (\frac {a+b \text {arcsinh}(c x)}{b}\right ) \sinh \left (\frac {a}{b}\right )}{b}+\frac {3 \cosh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a+b \text {arcsinh}(c x)}{b}\right )}{b}+\frac {\left (5 \cosh \left (\frac {a}{b}\right )\right ) \text {Subst}\left (\int \frac {\sinh \left (\frac {x}{b}\right )}{x} \, dx,x,a+b \text {arcsinh}(c x)\right )}{8 b}-\frac {\left (9 \cosh \left (\frac {a}{b}\right )\right ) \text {Subst}\left (\int \frac {\sinh \left (\frac {x}{b}\right )}{x} \, dx,x,a+b \text {arcsinh}(c x)\right )}{4 b}-\frac {\left (5 \cosh \left (\frac {3 a}{b}\right )\right ) \text {Subst}\left (\int \frac {\sinh \left (\frac {3 x}{b}\right )}{x} \, dx,x,a+b \text {arcsinh}(c x)\right )}{16 b}+\frac {\left (3 \cosh \left (\frac {3 a}{b}\right )\right ) \text {Subst}\left (\int \frac {\sinh \left (\frac {3 x}{b}\right )}{x} \, dx,x,a+b \text {arcsinh}(c x)\right )}{4 b}+\frac {\cosh \left (\frac {5 a}{b}\right ) \text {Subst}\left (\int \frac {\sinh \left (\frac {5 x}{b}\right )}{x} \, dx,x,a+b \text {arcsinh}(c x)\right )}{16 b}-\frac {\left (5 \sinh \left (\frac {a}{b}\right )\right ) \text {Subst}\left (\int \frac {\cosh \left (\frac {x}{b}\right )}{x} \, dx,x,a+b \text {arcsinh}(c x)\right )}{8 b}+\frac {\left (9 \sinh \left (\frac {a}{b}\right )\right ) \text {Subst}\left (\int \frac {\cosh \left (\frac {x}{b}\right )}{x} \, dx,x,a+b \text {arcsinh}(c x)\right )}{4 b}+\frac {\left (5 \sinh \left (\frac {3 a}{b}\right )\right ) \text {Subst}\left (\int \frac {\cosh \left (\frac {3 x}{b}\right )}{x} \, dx,x,a+b \text {arcsinh}(c x)\right )}{16 b}-\frac {\left (3 \sinh \left (\frac {3 a}{b}\right )\right ) \text {Subst}\left (\int \frac {\cosh \left (\frac {3 x}{b}\right )}{x} \, dx,x,a+b \text {arcsinh}(c x)\right )}{4 b}-\frac {\sinh \left (\frac {5 a}{b}\right ) \text {Subst}\left (\int \frac {\cosh \left (\frac {5 x}{b}\right )}{x} \, dx,x,a+b \text {arcsinh}(c x)\right )}{16 b}+\int \frac {1}{x \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))} \, dx \\ & = -\frac {11 \text {Chi}\left (\frac {a+b \text {arcsinh}(c x)}{b}\right ) \sinh \left (\frac {a}{b}\right )}{8 b}-\frac {7 \text {Chi}\left (\frac {3 (a+b \text {arcsinh}(c x))}{b}\right ) \sinh \left (\frac {3 a}{b}\right )}{16 b}-\frac {\text {Chi}\left (\frac {5 (a+b \text {arcsinh}(c x))}{b}\right ) \sinh \left (\frac {5 a}{b}\right )}{16 b}+\frac {11 \cosh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a+b \text {arcsinh}(c x)}{b}\right )}{8 b}+\frac {7 \cosh \left (\frac {3 a}{b}\right ) \text {Shi}\left (\frac {3 (a+b \text {arcsinh}(c x))}{b}\right )}{16 b}+\frac {\cosh \left (\frac {5 a}{b}\right ) \text {Shi}\left (\frac {5 (a+b \text {arcsinh}(c x))}{b}\right )}{16 b}+\int \frac {1}{x \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))} \, dx \\ \end{align*}

Mathematica [N/A]

Not integrable

Time = 2.40 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.07 \[ \int \frac {\left (1+c^2 x^2\right )^{5/2}}{x (a+b \text {arcsinh}(c x))} \, dx=\int \frac {\left (1+c^2 x^2\right )^{5/2}}{x (a+b \text {arcsinh}(c x))} \, dx \]

[In]

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

[Out]

Integrate[(1 + c^2*x^2)^(5/2)/(x*(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

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

[In]

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

[Out]

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

Fricas [N/A]

Not integrable

Time = 0.25 (sec) , antiderivative size = 44, normalized size of antiderivative = 1.63 \[ \int \frac {\left (1+c^2 x^2\right )^{5/2}}{x (a+b \text {arcsinh}(c x))} \, dx=\int { \frac {{\left (c^{2} x^{2} + 1\right )}^{\frac {5}{2}}}{{\left (b \operatorname {arsinh}\left (c x\right ) + a\right )} x} \,d x } \]

[In]

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

Sympy [N/A]

Not integrable

Time = 5.23 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.81 \[ \int \frac {\left (1+c^2 x^2\right )^{5/2}}{x (a+b \text {arcsinh}(c x))} \, dx=\int \frac {\left (c^{2} x^{2} + 1\right )^{\frac {5}{2}}}{x \left (a + b \operatorname {asinh}{\left (c x \right )}\right )}\, dx \]

[In]

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

[Out]

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

Maxima [N/A]

Not integrable

Time = 0.26 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.00 \[ \int \frac {\left (1+c^2 x^2\right )^{5/2}}{x (a+b \text {arcsinh}(c x))} \, dx=\int { \frac {{\left (c^{2} x^{2} + 1\right )}^{\frac {5}{2}}}{{\left (b \operatorname {arsinh}\left (c x\right ) + a\right )} x} \,d x } \]

[In]

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

[Out]

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

Giac [F(-2)]

Exception generated. \[ \int \frac {\left (1+c^2 x^2\right )^{5/2}}{x (a+b \text {arcsinh}(c x))} \, dx=\text {Exception raised: TypeError} \]

[In]

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

[Out]

Exception raised: TypeError >> an error occurred running a Giac command:INPUT:sage2:=int(sage0,sageVARx):;OUTP
UT:sym2poly/r2sym(const gen & e,const index_m & i,const vecteur & l) Error: Bad Argument Value

Mupad [N/A]

Not integrable

Time = 2.67 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.00 \[ \int \frac {\left (1+c^2 x^2\right )^{5/2}}{x (a+b \text {arcsinh}(c x))} \, dx=\int \frac {{\left (c^2\,x^2+1\right )}^{5/2}}{x\,\left (a+b\,\mathrm {asinh}\left (c\,x\right )\right )} \,d x \]

[In]

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

[Out]

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