\(\int \frac {1}{(d+e x^2)^2 (a+b \text {arcsinh}(c x))^2} \, dx\) [628]

Optimal result

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

[Out]

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

[In]

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

[Out]

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

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

Mathematica [N/A]

Not integrable

Time = 25.14 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.10 \[ \int \frac {1}{\left (d+e x^2\right )^2 (a+b \text {arcsinh}(c x))^2} \, dx=\int \frac {1}{\left (d+e x^2\right )^2 (a+b \text {arcsinh}(c x))^2} \, dx \]

[In]

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

[Out]

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

Maple [N/A] (verified)

Not integrable

Time = 0.75 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.00

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

[In]

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

[Out]

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

Fricas [N/A]

Not integrable

Time = 0.25 (sec) , antiderivative size = 98, normalized size of antiderivative = 4.90 \[ \int \frac {1}{\left (d+e x^2\right )^2 (a+b \text {arcsinh}(c x))^2} \, dx=\int { \frac {1}{{\left (e x^{2} + d\right )}^{2} {\left (b \operatorname {arsinh}\left (c x\right ) + a\right )}^{2}} \,d x } \]

[In]

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

[Out]

integral(1/(a^2*e^2*x^4 + 2*a^2*d*e*x^2 + a^2*d^2 + (b^2*e^2*x^4 + 2*b^2*d*e*x^2 + b^2*d^2)*arcsinh(c*x)^2 + 2
*(a*b*e^2*x^4 + 2*a*b*d*e*x^2 + a*b*d^2)*arcsinh(c*x)), x)

Sympy [F(-1)]

Timed out. \[ \int \frac {1}{\left (d+e x^2\right )^2 (a+b \text {arcsinh}(c x))^2} \, dx=\text {Timed out} \]

[In]

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

[Out]

Timed out

Maxima [N/A]

Not integrable

Time = 1.18 (sec) , antiderivative size = 1027, normalized size of antiderivative = 51.35 \[ \int \frac {1}{\left (d+e x^2\right )^2 (a+b \text {arcsinh}(c x))^2} \, dx=\int { \frac {1}{{\left (e x^{2} + d\right )}^{2} {\left (b \operatorname {arsinh}\left (c x\right ) + a\right )}^{2}} \,d x } \]

[In]

integrate(1/(e*x^2+d)^2/(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^2*x^6 + (2*c^3*d*e + c*e^2)*a*b*x^4 + a*b*c*d^2 + (c^3*d^2 +
2*c*d*e)*a*b*x^2 + (b^2*c^3*e^2*x^6 + (2*c^3*d*e + c*e^2)*b^2*x^4 + b^2*c*d^2 + (c^3*d^2 + 2*c*d*e)*b^2*x^2 +
(b^2*c^2*e^2*x^5 + 2*b^2*c^2*d*e*x^3 + b^2*c^2*d^2*x)*sqrt(c^2*x^2 + 1))*log(c*x + sqrt(c^2*x^2 + 1)) + (a*b*
c^2*e^2*x^5 + 2*a*b*c^2*d*e*x^3 + a*b*c^2*d^2*x)*sqrt(c^2*x^2 + 1)) - integrate((3*c^5*e*x^6 - (c^5*d - 6*c^3*
e)*x^4 - (2*c^3*d - 3*c*e)*x^2 + (3*c^3*e*x^4 - (c^3*d - 5*c*e)*x^2 + c*d)*(c^2*x^2 + 1) - c*d + (6*c^4*e*x^5
- (2*c^4*d - 11*c^2*e)*x^3 - (c^2*d - 4*e)*x)*sqrt(c^2*x^2 + 1))/(a*b*c^5*e^3*x^10 + (3*c^5*d*e^2 + 2*c^3*e^3)
*a*b*x^8 + (3*c^5*d^2*e + 6*c^3*d*e^2 + c*e^3)*a*b*x^6 + (c^5*d^3 + 6*c^3*d^2*e + 3*c*d*e^2)*a*b*x^4 + a*b*c*d
^3 + (2*c^3*d^3 + 3*c*d^2*e)*a*b*x^2 + (a*b*c^3*e^3*x^8 + 3*a*b*c^3*d*e^2*x^6 + 3*a*b*c^3*d^2*e*x^4 + a*b*c^3*
d^3*x^2)*(c^2*x^2 + 1) + (b^2*c^5*e^3*x^10 + (3*c^5*d*e^2 + 2*c^3*e^3)*b^2*x^8 + (3*c^5*d^2*e + 6*c^3*d*e^2 +
c*e^3)*b^2*x^6 + (c^5*d^3 + 6*c^3*d^2*e + 3*c*d*e^2)*b^2*x^4 + b^2*c*d^3 + (2*c^3*d^3 + 3*c*d^2*e)*b^2*x^2 + (
b^2*c^3*e^3*x^8 + 3*b^2*c^3*d*e^2*x^6 + 3*b^2*c^3*d^2*e*x^4 + b^2*c^3*d^3*x^2)*(c^2*x^2 + 1) + 2*(b^2*c^4*e^3*
x^9 + (3*c^4*d*e^2 + c^2*e^3)*b^2*x^7 + b^2*c^2*d^3*x + 3*(c^4*d^2*e + c^2*d*e^2)*b^2*x^5 + (c^4*d^3 + 3*c^2*d
^2*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^3*x^9 + (3*c^4*d*e^2 + c^2*e^3)*
a*b*x^7 + a*b*c^2*d^3*x + 3*(c^4*d^2*e + c^2*d*e^2)*a*b*x^5 + (c^4*d^3 + 3*c^2*d^2*e)*a*b*x^3)*sqrt(c^2*x^2 +
1)), x)

Giac [N/A]

Not integrable

Time = 0.30 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.10 \[ \int \frac {1}{\left (d+e x^2\right )^2 (a+b \text {arcsinh}(c x))^2} \, dx=\int { \frac {1}{{\left (e x^{2} + d\right )}^{2} {\left (b \operatorname {arsinh}\left (c x\right ) + a\right )}^{2}} \,d x } \]

[In]

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

[Out]

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

Mupad [N/A]

Not integrable

Time = 2.55 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.10 \[ \int \frac {1}{\left (d+e x^2\right )^2 (a+b \text {arcsinh}(c x))^2} \, dx=\int \frac {1}{{\left (a+b\,\mathrm {asinh}\left (c\,x\right )\right )}^2\,{\left (e\,x^2+d\right )}^2} \,d x \]

[In]

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

[Out]

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