\(\int \frac {1}{x^2 (a+b \csc (c+d x^2))^2} \, dx\) [29]

Optimal result

Integrand size = 18, antiderivative size = 18 \[ \int \frac {1}{x^2 \left (a+b \csc \left (c+d x^2\right )\right )^2} \, dx=\text {Int}\left (\frac {1}{x^2 \left (a+b \csc \left (c+d x^2\right )\right )^2},x\right ) \]

[Out]

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

[In]

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

[Out]

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

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

Mathematica [N/A]

Not integrable

Time = 14.00 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.11 \[ \int \frac {1}{x^2 \left (a+b \csc \left (c+d x^2\right )\right )^2} \, dx=\int \frac {1}{x^2 \left (a+b \csc \left (c+d x^2\right )\right )^2} \, dx \]

[In]

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

[Out]

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

Maple [N/A] (verified)

Not integrable

Time = 0.14 (sec) , antiderivative size = 18, normalized size of antiderivative = 1.00

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

[In]

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

[Out]

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

Fricas [N/A]

Not integrable

Time = 0.26 (sec) , antiderivative size = 44, normalized size of antiderivative = 2.44 \[ \int \frac {1}{x^2 \left (a+b \csc \left (c+d x^2\right )\right )^2} \, dx=\int { \frac {1}{{\left (b \csc \left (d x^{2} + c\right ) + a\right )}^{2} x^{2}} \,d x } \]

[In]

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

[Out]

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

Sympy [N/A]

Not integrable

Time = 1.03 (sec) , antiderivative size = 19, normalized size of antiderivative = 1.06 \[ \int \frac {1}{x^2 \left (a+b \csc \left (c+d x^2\right )\right )^2} \, dx=\int \frac {1}{x^{2} \left (a + b \csc {\left (c + d x^{2} \right )}\right )^{2}}\, dx \]

[In]

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

[Out]

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

Maxima [N/A]

Not integrable

Time = 5.41 (sec) , antiderivative size = 4560, normalized size of antiderivative = 253.33 \[ \int \frac {1}{x^2 \left (a+b \csc \left (c+d x^2\right )\right )^2} \, dx=\int { \frac {1}{{\left (b \csc \left (d x^{2} + c\right ) + a\right )}^{2} x^{2}} \,d x } \]

[In]

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

[Out]

-((a^6 - a^4*b^2)*d*x^2*cos(2*d*x^2 + 2*c)^2 + (a^6 - a^4*b^2)*d*x^2*sin(2*d*x^2 + 2*c)^2 + (a^6 - 2*a^4*b^2 +
a^2*b^4)*d*x^2 - (a^2*b^4*sin(2*c) - (a^4*b^2 - a^2*b^4)*d*x^2*cos(2*c))*cos(2*d*x^2) + (a^3*b^3*cos(d*x^2 +
c) - (a^4*b^2 - a^2*b^4)*d*x^2*cos(2*d*x^2)*cos(2*c) - 2*(a^5*b - 2*a^3*b^3 + a*b^5)*d*x^2*cos(c)*sin(d*x^2) +
(a^4*b^2 - a^2*b^4)*d*x^2*sin(2*d*x^2)*sin(2*c) - 2*(a^5*b - 2*a^3*b^3 + a*b^5)*d*x^2*cos(d*x^2)*sin(c) - 2*(
a^5*b - a^3*b^3)*d*x^2*sin(d*x^2 + c) - (2*a^6 - 3*a^4*b^2 + a^2*b^4)*d*x^2)*cos(2*d*x^2 + 2*c) - (a^3*b^3 - a
*b^5 + (a*b^5*cos(2*c) + 2*(a^3*b^3 - a*b^5)*d*x^2*sin(2*c))*cos(2*d*x^2) - 2*(2*(a^4*b^2 - 2*a^2*b^4 + b^6)*d
*x^2*cos(c) - (a^2*b^4 - b^6)*sin(c))*cos(d*x^2) - (a*b^5*sin(2*c) - 2*(a^3*b^3 - a*b^5)*d*x^2*cos(2*c))*sin(2
*d*x^2) + 2*(2*(a^4*b^2 - 2*a^2*b^4 + b^6)*d*x^2*sin(c) + (a^2*b^4 - b^6)*cos(c))*sin(d*x^2))*cos(d*x^2 + c) +
2*((a^5*b - 2*a^3*b^3 + a*b^5)*d*x^2*sin(c) + (a^3*b^3 - a*b^5)*cos(c))*cos(d*x^2) + (a^8*d*x^3*cos(2*d*x^2 +
2*c)^2 + a^8*d*x^3*sin(2*d*x^2 + 2*c)^2 + (a^4*b^4*cos(2*c)^2 + a^4*b^4*sin(2*c)^2)*d*x^3*cos(2*d*x^2)^2 + 4*
((a^6*b^2 - 2*a^4*b^4 + a^2*b^6)*cos(c)^2 + (a^6*b^2 - 2*a^4*b^4 + a^2*b^6)*sin(c)^2)*d*x^3*cos(d*x^2)^2 + (a^
4*b^4*cos(2*c)^2 + a^4*b^4*sin(2*c)^2)*d*x^3*sin(2*d*x^2)^2 + 4*(a^7*b - 2*a^5*b^3 + a^3*b^5)*d*x^3*cos(c)*sin
(d*x^2) + 4*((a^6*b^2 - 2*a^4*b^4 + a^2*b^6)*cos(c)^2 + (a^6*b^2 - 2*a^4*b^4 + a^2*b^6)*sin(c)^2)*d*x^3*sin(d*
x^2)^2 + 4*(a^7*b - 2*a^5*b^3 + a^3*b^5)*d*x^3*cos(d*x^2)*sin(c) + (a^8 - 2*a^6*b^2 + a^4*b^4)*d*x^3 - 2*(2*((
a^5*b^3 - a^3*b^5)*cos(c)*sin(2*c) - (a^5*b^3 - a^3*b^5)*cos(2*c)*sin(c))*d*x^3*cos(d*x^2) - (a^6*b^2 - a^4*b^
4)*d*x^3*cos(2*c) - 2*((a^5*b^3 - a^3*b^5)*cos(2*c)*cos(c) + (a^5*b^3 - a^3*b^5)*sin(2*c)*sin(c))*d*x^3*sin(d*
x^2))*cos(2*d*x^2) - 2*(a^6*b^2*d*x^3*cos(2*d*x^2)*cos(2*c) - a^6*b^2*d*x^3*sin(2*d*x^2)*sin(2*c) + 2*(a^7*b -
a^5*b^3)*d*x^3*cos(c)*sin(d*x^2) + 2*(a^7*b - a^5*b^3)*d*x^3*cos(d*x^2)*sin(c) + (a^8 - a^6*b^2)*d*x^3)*cos(2
*d*x^2 + 2*c) - 2*(2*((a^5*b^3 - a^3*b^5)*cos(2*c)*cos(c) + (a^5*b^3 - a^3*b^5)*sin(2*c)*sin(c))*d*x^3*cos(d*x
^2) + 2*((a^5*b^3 - a^3*b^5)*cos(c)*sin(2*c) - (a^5*b^3 - a^3*b^5)*cos(2*c)*sin(c))*d*x^3*sin(d*x^2) + (a^6*b^
2 - a^4*b^4)*d*x^3*sin(2*c))*sin(2*d*x^2) - 2*(a^6*b^2*d*x^3*cos(2*c)*sin(2*d*x^2) + a^6*b^2*d*x^3*cos(2*d*x^2
)*sin(2*c) - 2*(a^7*b - a^5*b^3)*d*x^3*cos(d*x^2)*cos(c) + 2*(a^7*b - a^5*b^3)*d*x^3*sin(d*x^2)*sin(c))*sin(2*
d*x^2 + 2*c))*integrate(-(3*a^2*b^4*cos(2*c)*sin(2*d*x^2) + 3*a^2*b^4*cos(2*d*x^2)*sin(2*c) - 6*(a^3*b^3 - a*b
^5)*cos(d*x^2)*cos(c) + 6*(a^3*b^3 - a*b^5)*sin(d*x^2)*sin(c) - (3*a^3*b^3*cos(d*x^2 + c) - 2*(2*a^5*b - a^3*b
^3)*d*x^2*sin(d*x^2 + c))*cos(2*d*x^2 + 2*c) + (3*a^3*b^3 - 3*a*b^5 + (3*a*b^5*cos(2*c) + 2*(2*a^3*b^3 - a*b^5
)*d*x^2*sin(2*c))*cos(2*d*x^2) - 2*(2*(2*a^4*b^2 - 3*a^2*b^4 + b^6)*d*x^2*cos(c) - 3*(a^2*b^4 - b^6)*sin(c))*c
os(d*x^2) - (3*a*b^5*sin(2*c) - 2*(2*a^3*b^3 - a*b^5)*d*x^2*cos(2*c))*sin(2*d*x^2) + 2*(2*(2*a^4*b^2 - 3*a^2*b
^4 + b^6)*d*x^2*sin(c) + 3*(a^2*b^4 - b^6)*cos(c))*sin(d*x^2))*cos(d*x^2 + c) - (3*a^3*b^3*sin(d*x^2 + c) + 3*
a^4*b^2 + 2*(2*a^5*b - a^3*b^3)*d*x^2*cos(d*x^2 + c))*sin(2*d*x^2 + 2*c) - (2*(2*a^5*b - 3*a^3*b^3 + a*b^5)*d*
x^2 - (3*a*b^5*sin(2*c) - 2*(2*a^3*b^3 - a*b^5)*d*x^2*cos(2*c))*cos(2*d*x^2) + 2*(2*(2*a^4*b^2 - 3*a^2*b^4 + b
^6)*d*x^2*sin(c) + 3*(a^2*b^4 - b^6)*cos(c))*cos(d*x^2) - (3*a*b^5*cos(2*c) + 2*(2*a^3*b^3 - a*b^5)*d*x^2*sin(
2*c))*sin(2*d*x^2) + 2*(2*(2*a^4*b^2 - 3*a^2*b^4 + b^6)*d*x^2*cos(c) - 3*(a^2*b^4 - b^6)*sin(c))*sin(d*x^2))*s
in(d*x^2 + c))/(a^8*d*x^4*cos(2*d*x^2 + 2*c)^2 + a^8*d*x^4*sin(2*d*x^2 + 2*c)^2 + (a^4*b^4*cos(2*c)^2 + a^4*b^
4*sin(2*c)^2)*d*x^4*cos(2*d*x^2)^2 + 4*((a^6*b^2 - 2*a^4*b^4 + a^2*b^6)*cos(c)^2 + (a^6*b^2 - 2*a^4*b^4 + a^2*
b^6)*sin(c)^2)*d*x^4*cos(d*x^2)^2 + (a^4*b^4*cos(2*c)^2 + a^4*b^4*sin(2*c)^2)*d*x^4*sin(2*d*x^2)^2 + 4*(a^7*b
- 2*a^5*b^3 + a^3*b^5)*d*x^4*cos(c)*sin(d*x^2) + 4*((a^6*b^2 - 2*a^4*b^4 + a^2*b^6)*cos(c)^2 + (a^6*b^2 - 2*a^
4*b^4 + a^2*b^6)*sin(c)^2)*d*x^4*sin(d*x^2)^2 + 4*(a^7*b - 2*a^5*b^3 + a^3*b^5)*d*x^4*cos(d*x^2)*sin(c) + (a^8
- 2*a^6*b^2 + a^4*b^4)*d*x^4 - 2*(2*((a^5*b^3 - a^3*b^5)*cos(c)*sin(2*c) - (a^5*b^3 - a^3*b^5)*cos(2*c)*sin(c
))*d*x^4*cos(d*x^2) - (a^6*b^2 - a^4*b^4)*d*x^4*cos(2*c) - 2*((a^5*b^3 - a^3*b^5)*cos(2*c)*cos(c) + (a^5*b^3 -
a^3*b^5)*sin(2*c)*sin(c))*d*x^4*sin(d*x^2))*cos(2*d*x^2) - 2*(a^6*b^2*d*x^4*cos(2*d*x^2)*cos(2*c) - a^6*b^2*d
*x^4*sin(2*d*x^2)*sin(2*c) + 2*(a^7*b - a^5*b^3)*d*x^4*cos(c)*sin(d*x^2) + 2*(a^7*b - a^5*b^3)*d*x^4*cos(d*x^2
)*sin(c) + (a^8 - a^6*b^2)*d*x^4)*cos(2*d*x^2 + 2*c) - 2*(2*((a^5*b^3 - a^3*b^5)*cos(2*c)*cos(c) + (a^5*b^3 -
a^3*b^5)*sin(2*c)*sin(c))*d*x^4*cos(d*x^2) + 2*((a^5*b^3 - a^3*b^5)*cos(c)*sin(2*c) - (a^5*b^3 - a^3*b^5)*cos(
2*c)*sin(c))*d*x^4*sin(d*x^2) + (a^6*b^2 - a^4*b^4)*d*x^4*sin(2*c))*sin(2*d*x^2) - 2*(a^6*b^2*d*x^4*cos(2*c)*s
in(2*d*x^2) + a^6*b^2*d*x^4*cos(2*d*x^2)*sin(2*c) - 2*(a^7*b - a^5*b^3)*d*x^4*cos(d*x^2)*cos(c) + 2*(a^7*b - a
^5*b^3)*d*x^4*sin(d*x^2)*sin(c))*sin(2*d*x^2 + 2*c)), x) - (a^2*b^4*cos(2*c) + (a^4*b^2 - a^2*b^4)*d*x^2*sin(2
*c))*sin(2*d*x^2) + (a^3*b^3*sin(d*x^2 + c) + a^4*b^2 + 2*(a^5*b - 2*a^3*b^3 + a*b^5)*d*x^2*cos(d*x^2)*cos(c)
- (a^4*b^2 - a^2*b^4)*d*x^2*cos(2*c)*sin(2*d*x^2) - (a^4*b^2 - a^2*b^4)*d*x^2*cos(2*d*x^2)*sin(2*c) - 2*(a^5*b
- 2*a^3*b^3 + a*b^5)*d*x^2*sin(d*x^2)*sin(c) + 2*(a^5*b - a^3*b^3)*d*x^2*cos(d*x^2 + c))*sin(2*d*x^2 + 2*c) +
(2*(a^5*b - 2*a^3*b^3 + a*b^5)*d*x^2 - (a*b^5*sin(2*c) - 2*(a^3*b^3 - a*b^5)*d*x^2*cos(2*c))*cos(2*d*x^2) + 2
*(2*(a^4*b^2 - 2*a^2*b^4 + b^6)*d*x^2*sin(c) + (a^2*b^4 - b^6)*cos(c))*cos(d*x^2) - (a*b^5*cos(2*c) + 2*(a^3*b
^3 - a*b^5)*d*x^2*sin(2*c))*sin(2*d*x^2) + 2*(2*(a^4*b^2 - 2*a^2*b^4 + b^6)*d*x^2*cos(c) - (a^2*b^4 - b^6)*sin
(c))*sin(d*x^2))*sin(d*x^2 + c) + 2*((a^5*b - 2*a^3*b^3 + a*b^5)*d*x^2*cos(c) - (a^3*b^3 - a*b^5)*sin(c))*sin(
d*x^2))/(a^8*d*x^3*cos(2*d*x^2 + 2*c)^2 + a^8*d*x^3*sin(2*d*x^2 + 2*c)^2 + (a^4*b^4*cos(2*c)^2 + a^4*b^4*sin(2
*c)^2)*d*x^3*cos(2*d*x^2)^2 + 4*((a^6*b^2 - 2*a^4*b^4 + a^2*b^6)*cos(c)^2 + (a^6*b^2 - 2*a^4*b^4 + a^2*b^6)*si
n(c)^2)*d*x^3*cos(d*x^2)^2 + (a^4*b^4*cos(2*c)^2 + a^4*b^4*sin(2*c)^2)*d*x^3*sin(2*d*x^2)^2 + 4*(a^7*b - 2*a^5
*b^3 + a^3*b^5)*d*x^3*cos(c)*sin(d*x^2) + 4*((a^6*b^2 - 2*a^4*b^4 + a^2*b^6)*cos(c)^2 + (a^6*b^2 - 2*a^4*b^4 +
a^2*b^6)*sin(c)^2)*d*x^3*sin(d*x^2)^2 + 4*(a^7*b - 2*a^5*b^3 + a^3*b^5)*d*x^3*cos(d*x^2)*sin(c) + (a^8 - 2*a^
6*b^2 + a^4*b^4)*d*x^3 - 2*(2*((a^5*b^3 - a^3*b^5)*cos(c)*sin(2*c) - (a^5*b^3 - a^3*b^5)*cos(2*c)*sin(c))*d*x^
3*cos(d*x^2) - (a^6*b^2 - a^4*b^4)*d*x^3*cos(2*c) - 2*((a^5*b^3 - a^3*b^5)*cos(2*c)*cos(c) + (a^5*b^3 - a^3*b^
5)*sin(2*c)*sin(c))*d*x^3*sin(d*x^2))*cos(2*d*x^2) - 2*(a^6*b^2*d*x^3*cos(2*d*x^2)*cos(2*c) - a^6*b^2*d*x^3*si
n(2*d*x^2)*sin(2*c) + 2*(a^7*b - a^5*b^3)*d*x^3*cos(c)*sin(d*x^2) + 2*(a^7*b - a^5*b^3)*d*x^3*cos(d*x^2)*sin(c
) + (a^8 - a^6*b^2)*d*x^3)*cos(2*d*x^2 + 2*c) - 2*(2*((a^5*b^3 - a^3*b^5)*cos(2*c)*cos(c) + (a^5*b^3 - a^3*b^5
)*sin(2*c)*sin(c))*d*x^3*cos(d*x^2) + 2*((a^5*b^3 - a^3*b^5)*cos(c)*sin(2*c) - (a^5*b^3 - a^3*b^5)*cos(2*c)*si
n(c))*d*x^3*sin(d*x^2) + (a^6*b^2 - a^4*b^4)*d*x^3*sin(2*c))*sin(2*d*x^2) - 2*(a^6*b^2*d*x^3*cos(2*c)*sin(2*d*
x^2) + a^6*b^2*d*x^3*cos(2*d*x^2)*sin(2*c) - 2*(a^7*b - a^5*b^3)*d*x^3*cos(d*x^2)*cos(c) + 2*(a^7*b - a^5*b^3)
*d*x^3*sin(d*x^2)*sin(c))*sin(2*d*x^2 + 2*c))

Giac [N/A]

Not integrable

Time = 0.39 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.11 \[ \int \frac {1}{x^2 \left (a+b \csc \left (c+d x^2\right )\right )^2} \, dx=\int { \frac {1}{{\left (b \csc \left (d x^{2} + c\right ) + a\right )}^{2} x^{2}} \,d x } \]

[In]

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

[Out]

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

Mupad [N/A]

Not integrable

Time = 17.97 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.22 \[ \int \frac {1}{x^2 \left (a+b \csc \left (c+d x^2\right )\right )^2} \, dx=\int \frac {1}{x^2\,{\left (a+\frac {b}{\sin \left (d\,x^2+c\right )}\right )}^2} \,d x \]

[In]

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

[Out]

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