\(\int \frac {1}{x (1+c^2 x^2)^{3/2} (a+b \text {arcsinh}(c x))^2} \, dx\) [447]

Optimal result

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

[Out]

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

[In]

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

[Out]

Defer[Int][1/(x*(1 + c^2*x^2)^(3/2)*(a + b*ArcSinh[c*x])^2), 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))^2} \, dx \\ \end{align*}

Mathematica [N/A]

Not integrable

Time = 35.98 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.07 \[ \int \frac {1}{x \left (1+c^2 x^2\right )^{3/2} (a+b \text {arcsinh}(c x))^2} \, dx=\int \frac {1}{x \left (1+c^2 x^2\right )^{3/2} (a+b \text {arcsinh}(c x))^2} \, dx \]

[In]

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

[Out]

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

Maple [N/A] (verified)

Not integrable

Time = 0.20 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.93

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

[In]

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

[Out]

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

Fricas [N/A]

Not integrable

Time = 0.25 (sec) , antiderivative size = 107, normalized size of antiderivative = 3.96 \[ \int \frac {1}{x \left (1+c^2 x^2\right )^{3/2} (a+b \text {arcsinh}(c x))^2} \, dx=\int { \frac {1}{{\left (c^{2} x^{2} + 1\right )}^{\frac {3}{2}} {\left (b \operatorname {arsinh}\left (c x\right ) + a\right )}^{2} x} \,d x } \]

[In]

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

[Out]

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

Sympy [N/A]

Not integrable

Time = 4.46 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.96 \[ \int \frac {1}{x \left (1+c^2 x^2\right )^{3/2} (a+b \text {arcsinh}(c x))^2} \, dx=\int \frac {1}{x \left (a + b \operatorname {asinh}{\left (c x \right )}\right )^{2} \left (c^{2} x^{2} + 1\right )^{\frac {3}{2}}}\, dx \]

[In]

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

[Out]

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

Maxima [N/A]

Not integrable

Time = 0.56 (sec) , antiderivative size = 477, normalized size of antiderivative = 17.67 \[ \int \frac {1}{x \left (1+c^2 x^2\right )^{3/2} (a+b \text {arcsinh}(c x))^2} \, dx=\int { \frac {1}{{\left (c^{2} x^{2} + 1\right )}^{\frac {3}{2}} {\left (b \operatorname {arsinh}\left (c x\right ) + a\right )}^{2} x} \,d x } \]

[In]

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

[Out]

-(c*x + sqrt(c^2*x^2 + 1))/((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)*s
qrt(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)) - integrate((3*c^5
*x^5 + 3*c^3*x^3 + (3*c^3*x^3 + 2*c*x)*(c^2*x^2 + 1) + (6*c^4*x^4 + 5*c^2*x^2 + 1)*sqrt(c^2*x^2 + 1))/((a*b*c^
5*x^6 + a*b*c^3*x^4)*(c^2*x^2 + 1)^(3/2) + 2*(a*b*c^6*x^7 + 2*a*b*c^4*x^5 + a*b*c^2*x^3)*(c^2*x^2 + 1) + ((b^2
*c^5*x^6 + b^2*c^3*x^4)*(c^2*x^2 + 1)^(3/2) + 2*(b^2*c^6*x^7 + 2*b^2*c^4*x^5 + b^2*c^2*x^3)*(c^2*x^2 + 1) + (b
^2*c^7*x^8 + 3*b^2*c^5*x^6 + 3*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^7*x^8 + 3*a*b*c^5*x^6 + 3*a*b*c^3*x^4 + a*b*c*x^2)*sqrt(c^2*x^2 + 1)), x)

Giac [F(-2)]

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

[In]

integrate(1/x/(c^2*x^2+1)^(3/2)/(a+b*arcsinh(c*x))^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 = 2.56 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.00 \[ \int \frac {1}{x \left (1+c^2 x^2\right )^{3/2} (a+b \text {arcsinh}(c x))^2} \, dx=\int \frac {1}{x\,{\left (a+b\,\mathrm {asinh}\left (c\,x\right )\right )}^2\,{\left (c^2\,x^2+1\right )}^{3/2}} \,d x \]

[In]

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

[Out]

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