\(\int x^2 (a+b \text {sech}(c+d x^2))^2 \, dx\) [11]

Optimal result

Integrand size = 18, antiderivative size = 18 \[ \int x^2 \left (a+b \text {sech}\left (c+d x^2\right )\right )^2 \, dx=\text {Int}\left (x^2 \left (a+b \text {sech}\left (c+d x^2\right )\right )^2,x\right ) \]

[Out]

Unintegrable(x^2*(a+b*sech(d*x^2+c))^2,x)

Rubi [N/A]

Not integrable

Time = 0.02 (sec) , antiderivative size = 18, 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 = {} \[ \int x^2 \left (a+b \text {sech}\left (c+d x^2\right )\right )^2 \, dx=\int x^2 \left (a+b \text {sech}\left (c+d x^2\right )\right )^2 \, dx \]

[In]

Int[x^2*(a + b*Sech[c + d*x^2])^2,x]

[Out]

Defer[Int][x^2*(a + b*Sech[c + d*x^2])^2, x]

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

Mathematica [N/A]

Not integrable

Time = 9.29 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.11 \[ \int x^2 \left (a+b \text {sech}\left (c+d x^2\right )\right )^2 \, dx=\int x^2 \left (a+b \text {sech}\left (c+d x^2\right )\right )^2 \, dx \]

[In]

Integrate[x^2*(a + b*Sech[c + d*x^2])^2,x]

[Out]

Integrate[x^2*(a + b*Sech[c + d*x^2])^2, x]

Maple [N/A] (verified)

Not integrable

Time = 0.07 (sec) , antiderivative size = 18, normalized size of antiderivative = 1.00

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

[In]

int(x^2*(a+b*sech(d*x^2+c))^2,x)

[Out]

int(x^2*(a+b*sech(d*x^2+c))^2,x)

Fricas [N/A]

Not integrable

Time = 0.26 (sec) , antiderivative size = 42, normalized size of antiderivative = 2.33 \[ \int x^2 \left (a+b \text {sech}\left (c+d x^2\right )\right )^2 \, dx=\int { {\left (b \operatorname {sech}\left (d x^{2} + c\right ) + a\right )}^{2} x^{2} \,d x } \]

[In]

integrate(x^2*(a+b*sech(d*x^2+c))^2,x, algorithm="fricas")

[Out]

integral(b^2*x^2*sech(d*x^2 + c)^2 + 2*a*b*x^2*sech(d*x^2 + c) + a^2*x^2, x)

Sympy [N/A]

Not integrable

Time = 1.01 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.94 \[ \int x^2 \left (a+b \text {sech}\left (c+d x^2\right )\right )^2 \, dx=\int x^{2} \left (a + b \operatorname {sech}{\left (c + d x^{2} \right )}\right )^{2}\, dx \]

[In]

integrate(x**2*(a+b*sech(d*x**2+c))**2,x)

[Out]

Integral(x**2*(a + b*sech(c + d*x**2))**2, x)

Maxima [N/A]

Not integrable

Time = 0.40 (sec) , antiderivative size = 72, normalized size of antiderivative = 4.00 \[ \int x^2 \left (a+b \text {sech}\left (c+d x^2\right )\right )^2 \, dx=\int { {\left (b \operatorname {sech}\left (d x^{2} + c\right ) + a\right )}^{2} x^{2} \,d x } \]

[In]

integrate(x^2*(a+b*sech(d*x^2+c))^2,x, algorithm="maxima")

[Out]

1/3*a^2*x^3 - b^2*x/(d*e^(2*d*x^2 + 2*c) + d) + integrate((4*a*b*d*x^2*e^(d*x^2 + c) + b^2)/(d*e^(2*d*x^2 + 2*
c) + d), x)

Giac [N/A]

Not integrable

Time = 0.67 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.11 \[ \int x^2 \left (a+b \text {sech}\left (c+d x^2\right )\right )^2 \, dx=\int { {\left (b \operatorname {sech}\left (d x^{2} + c\right ) + a\right )}^{2} x^{2} \,d x } \]

[In]

integrate(x^2*(a+b*sech(d*x^2+c))^2,x, algorithm="giac")

[Out]

integrate((b*sech(d*x^2 + c) + a)^2*x^2, x)

Mupad [N/A]

Not integrable

Time = 1.99 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.22 \[ \int x^2 \left (a+b \text {sech}\left (c+d x^2\right )\right )^2 \, dx=\int x^2\,{\left (a+\frac {b}{\mathrm {cosh}\left (d\,x^2+c\right )}\right )}^2 \,d x \]

[In]

int(x^2*(a + b/cosh(c + d*x^2))^2,x)

[Out]

int(x^2*(a + b/cosh(c + d*x^2))^2, x)