\(\int \frac {1}{x \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))^2} \, dx\) [441]

Optimal result

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

[Out]

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

Rubi [N/A]

Not integrable

Time = 0.11 (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 = {} \[ \int \frac {1}{x \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))^2} \, dx=\int \frac {1}{x \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))^2} \, dx \]

[In]

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

[Out]

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

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

Mathematica [N/A]

Not integrable

Time = 6.48 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.07 \[ \int \frac {1}{x \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))^2} \, dx=\int \frac {1}{x \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))^2} \, dx \]

[In]

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

[Out]

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

Maple [N/A] (verified)

Not integrable

Time = 0.15 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.93

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

[In]

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

[Out]

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

Fricas [N/A]

Not integrable

Time = 0.26 (sec) , antiderivative size = 75, normalized size of antiderivative = 2.78 \[ \int \frac {1}{x \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))^2} \, dx=\int { \frac {1}{\sqrt {c^{2} x^{2} + 1} {\left (b \operatorname {arsinh}\left (c x\right ) + a\right )}^{2} x} \,d x } \]

[In]

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

[Out]

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

Sympy [N/A]

Not integrable

Time = 1.70 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.96 \[ \int \frac {1}{x \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))^2} \, dx=\int \frac {1}{x \left (a + b \operatorname {asinh}{\left (c x \right )}\right )^{2} \sqrt {c^{2} x^{2} + 1}}\, dx \]

[In]

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

[Out]

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

Maxima [N/A]

Not integrable

Time = 0.49 (sec) , antiderivative size = 416, normalized size of antiderivative = 15.41 \[ \int \frac {1}{x \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))^2} \, dx=\int { \frac {1}{\sqrt {c^{2} x^{2} + 1} {\left (b \operatorname {arsinh}\left (c x\right ) + a\right )}^{2} x} \,d x } \]

[In]

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

[Out]

-(c^3*x^3 + c*x + (c^2*x^2 + 1)^(3/2))/((c^2*x^2 + 1)*a*b*c^2*x^2 + ((c^2*x^2 + 1)*b^2*c^2*x^2 + (b^2*c^3*x^3
+ b^2*c*x)*sqrt(c^2*x^2 + 1))*log(c*x + sqrt(c^2*x^2 + 1)) + (a*b*c^3*x^3 + a*b*c*x)*sqrt(c^2*x^2 + 1)) - inte
grate((c^5*x^5 + c^3*x^3 + (c^3*x^3 + 2*c*x)*(c^2*x^2 + 1) + (2*c^4*x^4 + 3*c^2*x^2 + 1)*sqrt(c^2*x^2 + 1))/((
c^2*x^2 + 1)^(3/2)*a*b*c^3*x^4 + 2*(a*b*c^4*x^5 + a*b*c^2*x^3)*(c^2*x^2 + 1) + ((c^2*x^2 + 1)^(3/2)*b^2*c^3*x^
4 + 2*(b^2*c^4*x^5 + b^2*c^2*x^3)*(c^2*x^2 + 1) + (b^2*c^5*x^6 + 2*b^2*c^3*x^4 + b^2*c*x^2)*sqrt(c^2*x^2 + 1))
*log(c*x + sqrt(c^2*x^2 + 1)) + (a*b*c^5*x^6 + 2*a*b*c^3*x^4 + a*b*c*x^2)*sqrt(c^2*x^2 + 1)), x)

Giac [F(-2)]

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

[In]

integrate(1/x/(a+b*arcsinh(c*x))^2/(c^2*x^2+1)^(1/2),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 = 2.81 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.00 \[ \int \frac {1}{x \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))^2} \, dx=\int \frac {1}{x\,{\left (a+b\,\mathrm {asinh}\left (c\,x\right )\right )}^2\,\sqrt {c^2\,x^2+1}} \,d x \]

[In]

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

[Out]

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