\(\int x^4 (a+b \csc (c+d x^2))^2 \, dx\) [9]

Optimal result

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

[Out]

Unintegrable(x^4*(a+b*csc(d*x^2+c))^2,x)

Rubi [N/A]

Not integrable

Time = 0.03 (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 = {} \[ \int x^4 \left (a+b \csc \left (c+d x^2\right )\right )^2 \, dx=\int x^4 \left (a+b \csc \left (c+d x^2\right )\right )^2 \, dx \]

[In]

Int[x^4*(a + b*Csc[c + d*x^2])^2,x]

[Out]

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

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

Mathematica [N/A]

Not integrable

Time = 22.32 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.11 \[ \int x^4 \left (a+b \csc \left (c+d x^2\right )\right )^2 \, dx=\int x^4 \left (a+b \csc \left (c+d x^2\right )\right )^2 \, dx \]

[In]

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

[Out]

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

Maple [N/A] (verified)

Not integrable

Time = 0.20 (sec) , antiderivative size = 18, normalized size of antiderivative = 1.00

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

[In]

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

[Out]

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

Fricas [N/A]

Not integrable

Time = 0.24 (sec) , antiderivative size = 42, normalized size of antiderivative = 2.33 \[ \int x^4 \left (a+b \csc \left (c+d x^2\right )\right )^2 \, dx=\int { {\left (b \csc \left (d x^{2} + c\right ) + a\right )}^{2} x^{4} \,d x } \]

[In]

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

[Out]

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

Sympy [N/A]

Not integrable

Time = 2.49 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.94 \[ \int x^4 \left (a+b \csc \left (c+d x^2\right )\right )^2 \, dx=\int x^{4} \left (a + b \csc {\left (c + d x^{2} \right )}\right )^{2}\, dx \]

[In]

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

[Out]

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

Maxima [N/A]

Not integrable

Time = 0.40 (sec) , antiderivative size = 310, normalized size of antiderivative = 17.22 \[ \int x^4 \left (a+b \csc \left (c+d x^2\right )\right )^2 \, dx=\int { {\left (b \csc \left (d x^{2} + c\right ) + a\right )}^{2} x^{4} \,d x } \]

[In]

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

[Out]

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

Giac [N/A]

Not integrable

Time = 1.02 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.11 \[ \int x^4 \left (a+b \csc \left (c+d x^2\right )\right )^2 \, dx=\int { {\left (b \csc \left (d x^{2} + c\right ) + a\right )}^{2} x^{4} \,d x } \]

[In]

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

[Out]

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

Mupad [N/A]

Not integrable

Time = 17.67 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.22 \[ \int x^4 \left (a+b \csc \left (c+d x^2\right )\right )^2 \, dx=\int x^4\,{\left (a+\frac {b}{\sin \left (d\,x^2+c\right )}\right )}^2 \,d x \]

[In]

int(x^4*(a + b/sin(c + d*x^2))^2,x)

[Out]

int(x^4*(a + b/sin(c + d*x^2))^2, x)