\(\int \frac {1}{x^2 \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))^2} \, dx\) [442]

Optimal result

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

[Out]

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

Rubi [N/A]

Not integrable

Time = 0.12 (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 = {} \[ \int \frac {1}{x^2 \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))^2} \, dx=\int \frac {1}{x^2 \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))^2} \, dx \]

[In]

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

[Out]

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

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

Mathematica [N/A]

Not integrable

Time = 2.02 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.07 \[ \int \frac {1}{x^2 \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))^2} \, dx=\int \frac {1}{x^2 \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))^2} \, dx \]

[In]

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

[Out]

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

Maple [N/A] (verified)

Not integrable

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

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

[In]

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

[Out]

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

Fricas [N/A]

Not integrable

Time = 0.25 (sec) , antiderivative size = 81, normalized size of antiderivative = 3.00 \[ \int \frac {1}{x^2 \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))^2} \, dx=\int { \frac {1}{\sqrt {c^{2} x^{2} + 1} {\left (b \operatorname {arsinh}\left (c x\right ) + a\right )}^{2} x^{2}} \,d x } \]

[In]

integrate(1/x^2/(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^4 + a^2*x^2 + (b^2*c^2*x^4 + b^2*x^2)*arcsinh(c*x)^2 + 2*(a*b*c^2*x^4 +
a*b*x^2)*arcsinh(c*x)), x)

Sympy [N/A]

Not integrable

Time = 1.48 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.00 \[ \int \frac {1}{x^2 \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))^2} \, dx=\int \frac {1}{x^{2} \left (a + b \operatorname {asinh}{\left (c x \right )}\right )^{2} \sqrt {c^{2} x^{2} + 1}}\, dx \]

[In]

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

[Out]

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

Maxima [N/A]

Not integrable

Time = 0.49 (sec) , antiderivative size = 427, normalized size of antiderivative = 15.81 \[ \int \frac {1}{x^2 \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))^2} \, dx=\int { \frac {1}{\sqrt {c^{2} x^{2} + 1} {\left (b \operatorname {arsinh}\left (c x\right ) + a\right )}^{2} x^{2}} \,d x } \]

[In]

integrate(1/x^2/(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^3 + ((c^2*x^2 + 1)*b^2*c^2*x^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^3*x^4 + a*b*c*x^2)*sqrt(c^2*x^2 + 1)) -
integrate((2*c^5*x^5 + 3*c^3*x^3 + (2*c^3*x^3 + 3*c*x)*(c^2*x^2 + 1) + c*x + 2*(2*c^4*x^4 + 3*c^2*x^2 + 1)*sqr
t(c^2*x^2 + 1))/((c^2*x^2 + 1)^(3/2)*a*b*c^3*x^5 + 2*(a*b*c^4*x^6 + a*b*c^2*x^4)*(c^2*x^2 + 1) + ((c^2*x^2 + 1
)^(3/2)*b^2*c^3*x^5 + 2*(b^2*c^4*x^6 + b^2*c^2*x^4)*(c^2*x^2 + 1) + (b^2*c^5*x^7 + 2*b^2*c^3*x^5 + b^2*c*x^3)*
sqrt(c^2*x^2 + 1))*log(c*x + sqrt(c^2*x^2 + 1)) + (a*b*c^5*x^7 + 2*a*b*c^3*x^5 + a*b*c*x^3)*sqrt(c^2*x^2 + 1))
, x)

Giac [N/A]

Not integrable

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

[In]

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

[Out]

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

Mupad [N/A]

Not integrable

Time = 2.72 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.00 \[ \int \frac {1}{x^2 \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))^2} \, dx=\int \frac {1}{x^2\,{\left (a+b\,\mathrm {asinh}\left (c\,x\right )\right )}^2\,\sqrt {c^2\,x^2+1}} \,d x \]

[In]

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

[Out]

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