\(\int \frac {1}{x (1+c^2 x^2)^{3/2} (a+b \text {arcsinh}(c x))} \, dx\) [399]

Optimal result

Integrand size = 27, antiderivative size = 27 \[ \int \frac {1}{x \left (1+c^2 x^2\right )^{3/2} (a+b \text {arcsinh}(c x))} \, dx=\text {Int}\left (\frac {1}{x \left (1+c^2 x^2\right )^{3/2} (a+b \text {arcsinh}(c x))},x\right ) \]

[Out]

Unintegrable(1/x/(c^2*x^2+1)^(3/2)/(a+b*arcsinh(c*x)),x)

Rubi [N/A]

Not integrable

Time = 0.10 (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 = {} \[ \int \frac {1}{x \left (1+c^2 x^2\right )^{3/2} (a+b \text {arcsinh}(c x))} \, dx=\int \frac {1}{x \left (1+c^2 x^2\right )^{3/2} (a+b \text {arcsinh}(c x))} \, dx \]

[In]

Int[1/(x*(1 + c^2*x^2)^(3/2)*(a + b*ArcSinh[c*x])),x]

[Out]

Defer[Int][1/(x*(1 + c^2*x^2)^(3/2)*(a + b*ArcSinh[c*x])), x]

Rubi steps \begin{align*} \text {integral}& = \int \frac {1}{x \left (1+c^2 x^2\right )^{3/2} (a+b \text {arcsinh}(c x))} \, dx \\ \end{align*}

Mathematica [N/A]

Not integrable

Time = 2.32 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.07 \[ \int \frac {1}{x \left (1+c^2 x^2\right )^{3/2} (a+b \text {arcsinh}(c x))} \, dx=\int \frac {1}{x \left (1+c^2 x^2\right )^{3/2} (a+b \text {arcsinh}(c x))} \, dx \]

[In]

Integrate[1/(x*(1 + c^2*x^2)^(3/2)*(a + b*ArcSinh[c*x])),x]

[Out]

Integrate[1/(x*(1 + c^2*x^2)^(3/2)*(a + b*ArcSinh[c*x])), x]

Maple [N/A] (verified)

Not integrable

Time = 0.16 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.93

\[\int \frac {1}{x \left (c^{2} x^{2}+1\right )^{\frac {3}{2}} \left (a +b \,\operatorname {arcsinh}\left (c x \right )\right )}d x\]

[In]

int(1/x/(c^2*x^2+1)^(3/2)/(a+b*arcsinh(c*x)),x)

[Out]

int(1/x/(c^2*x^2+1)^(3/2)/(a+b*arcsinh(c*x)),x)

Fricas [N/A]

Not integrable

Time = 0.25 (sec) , antiderivative size = 63, normalized size of antiderivative = 2.33 \[ \int \frac {1}{x \left (1+c^2 x^2\right )^{3/2} (a+b \text {arcsinh}(c x))} \, dx=\int { \frac {1}{{\left (c^{2} x^{2} + 1\right )}^{\frac {3}{2}} {\left (b \operatorname {arsinh}\left (c x\right ) + a\right )} x} \,d x } \]

[In]

integrate(1/x/(c^2*x^2+1)^(3/2)/(a+b*arcsinh(c*x)),x, algorithm="fricas")

[Out]

integral(sqrt(c^2*x^2 + 1)/(a*c^4*x^5 + 2*a*c^2*x^3 + a*x + (b*c^4*x^5 + 2*b*c^2*x^3 + b*x)*arcsinh(c*x)), x)

Sympy [N/A]

Not integrable

Time = 2.75 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.89 \[ \int \frac {1}{x \left (1+c^2 x^2\right )^{3/2} (a+b \text {arcsinh}(c x))} \, dx=\int \frac {1}{x \left (a + b \operatorname {asinh}{\left (c x \right )}\right ) \left (c^{2} x^{2} + 1\right )^{\frac {3}{2}}}\, dx \]

[In]

integrate(1/x/(c**2*x**2+1)**(3/2)/(a+b*asinh(c*x)),x)

[Out]

Integral(1/(x*(a + b*asinh(c*x))*(c**2*x**2 + 1)**(3/2)), x)

Maxima [N/A]

Not integrable

Time = 0.28 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.00 \[ \int \frac {1}{x \left (1+c^2 x^2\right )^{3/2} (a+b \text {arcsinh}(c x))} \, dx=\int { \frac {1}{{\left (c^{2} x^{2} + 1\right )}^{\frac {3}{2}} {\left (b \operatorname {arsinh}\left (c x\right ) + a\right )} x} \,d x } \]

[In]

integrate(1/x/(c^2*x^2+1)^(3/2)/(a+b*arcsinh(c*x)),x, algorithm="maxima")

[Out]

integrate(1/((c^2*x^2 + 1)^(3/2)*(b*arcsinh(c*x) + a)*x), x)

Giac [F(-2)]

Exception generated. \[ \int \frac {1}{x \left (1+c^2 x^2\right )^{3/2} (a+b \text {arcsinh}(c x))} \, dx=\text {Exception raised: TypeError} \]

[In]

integrate(1/x/(c^2*x^2+1)^(3/2)/(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 = 2.62 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.00 \[ \int \frac {1}{x \left (1+c^2 x^2\right )^{3/2} (a+b \text {arcsinh}(c x))} \, dx=\int \frac {1}{x\,\left (a+b\,\mathrm {asinh}\left (c\,x\right )\right )\,{\left (c^2\,x^2+1\right )}^{3/2}} \,d x \]

[In]

int(1/(x*(a + b*asinh(c*x))*(c^2*x^2 + 1)^(3/2)),x)

[Out]

int(1/(x*(a + b*asinh(c*x))*(c^2*x^2 + 1)^(3/2)), x)