\(\int \frac {x^4}{(a+b \text {sech}(c+d x^2))^2} \, dx\) [24]

Optimal result

Integrand size = 18, antiderivative size = 18 \[ \int \frac {x^4}{\left (a+b \text {sech}\left (c+d x^2\right )\right )^2} \, dx=\text {Int}\left (\frac {x^4}{\left (a+b \text {sech}\left (c+d x^2\right )\right )^2},x\right ) \]

[Out]

Unintegrable(x^4/(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 = {} \[ \int \frac {x^4}{\left (a+b \text {sech}\left (c+d x^2\right )\right )^2} \, dx=\int \frac {x^4}{\left (a+b \text {sech}\left (c+d x^2\right )\right )^2} \, dx \]

[In]

Int[x^4/(a + b*Sech[c + d*x^2])^2,x]

[Out]

Defer[Int][x^4/(a + b*Sech[c + d*x^2])^2, x]

Rubi steps \begin{align*} \text {integral}& = \int \frac {x^4}{\left (a+b \text {sech}\left (c+d x^2\right )\right )^2} \, dx \\ \end{align*}

Mathematica [N/A]

Not integrable

Time = 20.98 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.11 \[ \int \frac {x^4}{\left (a+b \text {sech}\left (c+d x^2\right )\right )^2} \, dx=\int \frac {x^4}{\left (a+b \text {sech}\left (c+d x^2\right )\right )^2} \, dx \]

[In]

Integrate[x^4/(a + b*Sech[c + d*x^2])^2,x]

[Out]

Integrate[x^4/(a + b*Sech[c + d*x^2])^2, x]

Maple [N/A] (verified)

Not integrable

Time = 0.06 (sec) , antiderivative size = 18, normalized size of antiderivative = 1.00

\[\int \frac {x^{4}}{{\left (a +b \,\operatorname {sech}\left (d \,x^{2}+c \right )\right )}^{2}}d x\]

[In]

int(x^4/(a+b*sech(d*x^2+c))^2,x)

[Out]

int(x^4/(a+b*sech(d*x^2+c))^2,x)

Fricas [N/A]

Not integrable

Time = 0.27 (sec) , antiderivative size = 38, normalized size of antiderivative = 2.11 \[ \int \frac {x^4}{\left (a+b \text {sech}\left (c+d x^2\right )\right )^2} \, dx=\int { \frac {x^{4}}{{\left (b \operatorname {sech}\left (d x^{2} + c\right ) + a\right )}^{2}} \,d x } \]

[In]

integrate(x^4/(a+b*sech(d*x^2+c))^2,x, algorithm="fricas")

[Out]

integral(x^4/(b^2*sech(d*x^2 + c)^2 + 2*a*b*sech(d*x^2 + c) + a^2), x)

Sympy [N/A]

Not integrable

Time = 1.00 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.94 \[ \int \frac {x^4}{\left (a+b \text {sech}\left (c+d x^2\right )\right )^2} \, dx=\int \frac {x^{4}}{\left (a + b \operatorname {sech}{\left (c + d x^{2} \right )}\right )^{2}}\, dx \]

[In]

integrate(x**4/(a+b*sech(d*x**2+c))**2,x)

[Out]

Integral(x**4/(a + b*sech(c + d*x**2))**2, x)

Maxima [N/A]

Not integrable

Time = 0.42 (sec) , antiderivative size = 312, normalized size of antiderivative = 17.33 \[ \int \frac {x^4}{\left (a+b \text {sech}\left (c+d x^2\right )\right )^2} \, dx=\int { \frac {x^{4}}{{\left (b \operatorname {sech}\left (d x^{2} + c\right ) + a\right )}^{2}} \,d x } \]

[In]

integrate(x^4/(a+b*sech(d*x^2+c))^2,x, algorithm="maxima")

[Out]

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

Giac [N/A]

Not integrable

Time = 0.40 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.11 \[ \int \frac {x^4}{\left (a+b \text {sech}\left (c+d x^2\right )\right )^2} \, dx=\int { \frac {x^{4}}{{\left (b \operatorname {sech}\left (d x^{2} + c\right ) + a\right )}^{2}} \,d x } \]

[In]

integrate(x^4/(a+b*sech(d*x^2+c))^2,x, algorithm="giac")

[Out]

integrate(x^4/(b*sech(d*x^2 + c) + a)^2, x)

Mupad [N/A]

Not integrable

Time = 2.06 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.22 \[ \int \frac {x^4}{\left (a+b \text {sech}\left (c+d x^2\right )\right )^2} \, dx=\int \frac {x^4}{{\left (a+\frac {b}{\mathrm {cosh}\left (d\,x^2+c\right )}\right )}^2} \,d x \]

[In]

int(x^4/(a + b/cosh(c + d*x^2))^2,x)

[Out]

int(x^4/(a + b/cosh(c + d*x^2))^2, x)