\(\int \frac {1}{(d+e x^2) (a+b \text {arcsinh}(c x))^2} \, dx\) [627]

Optimal result

Integrand size = 20, antiderivative size = 20 \[ \int \frac {1}{\left (d+e x^2\right ) (a+b \text {arcsinh}(c x))^2} \, dx=\text {Int}\left (\frac {1}{\left (d+e x^2\right ) (a+b \text {arcsinh}(c x))^2},x\right ) \]

[Out]

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

Rubi [N/A]

Not integrable

Time = 0.03 (sec) , antiderivative size = 20, 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 = {} \[ \int \frac {1}{\left (d+e x^2\right ) (a+b \text {arcsinh}(c x))^2} \, dx=\int \frac {1}{\left (d+e x^2\right ) (a+b \text {arcsinh}(c x))^2} \, dx \]

[In]

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

[Out]

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

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

Mathematica [N/A]

Not integrable

Time = 12.00 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.10 \[ \int \frac {1}{\left (d+e x^2\right ) (a+b \text {arcsinh}(c x))^2} \, dx=\int \frac {1}{\left (d+e x^2\right ) (a+b \text {arcsinh}(c x))^2} \, dx \]

[In]

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

[Out]

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

Maple [N/A] (verified)

Not integrable

Time = 0.48 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.00

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

[In]

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

[Out]

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

Fricas [N/A]

Not integrable

Time = 0.26 (sec) , antiderivative size = 57, normalized size of antiderivative = 2.85 \[ \int \frac {1}{\left (d+e x^2\right ) (a+b \text {arcsinh}(c x))^2} \, dx=\int { \frac {1}{{\left (e x^{2} + d\right )} {\left (b \operatorname {arsinh}\left (c x\right ) + a\right )}^{2}} \,d x } \]

[In]

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

[Out]

integral(1/(a^2*e*x^2 + a^2*d + (b^2*e*x^2 + b^2*d)*arcsinh(c*x)^2 + 2*(a*b*e*x^2 + a*b*d)*arcsinh(c*x)), x)

Sympy [N/A]

Not integrable

Time = 33.78 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.95 \[ \int \frac {1}{\left (d+e x^2\right ) (a+b \text {arcsinh}(c x))^2} \, dx=\int \frac {1}{\left (a + b \operatorname {asinh}{\left (c x \right )}\right )^{2} \left (d + e x^{2}\right )}\, dx \]

[In]

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

[Out]

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

Maxima [N/A]

Not integrable

Time = 0.91 (sec) , antiderivative size = 766, normalized size of antiderivative = 38.30 \[ \int \frac {1}{\left (d+e x^2\right ) (a+b \text {arcsinh}(c x))^2} \, dx=\int { \frac {1}{{\left (e x^{2} + d\right )} {\left (b \operatorname {arsinh}\left (c x\right ) + a\right )}^{2}} \,d x } \]

[In]

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

[Out]

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

Giac [N/A]

Not integrable

Time = 0.29 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.10 \[ \int \frac {1}{\left (d+e x^2\right ) (a+b \text {arcsinh}(c x))^2} \, dx=\int { \frac {1}{{\left (e x^{2} + d\right )} {\left (b \operatorname {arsinh}\left (c x\right ) + a\right )}^{2}} \,d x } \]

[In]

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

[Out]

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

Mupad [N/A]

Not integrable

Time = 2.66 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.10 \[ \int \frac {1}{\left (d+e x^2\right ) (a+b \text {arcsinh}(c x))^2} \, dx=\int \frac {1}{{\left (a+b\,\mathrm {asinh}\left (c\,x\right )\right )}^2\,\left (e\,x^2+d\right )} \,d x \]

[In]

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

[Out]

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