\(\int \frac {\csc ^3(a+b x)}{(c+d x)^2} \, dx\) [37]

Optimal result

Integrand size = 16, antiderivative size = 16 \[ \int \frac {\csc ^3(a+b x)}{(c+d x)^2} \, dx=\text {Int}\left (\frac {\csc ^3(a+b x)}{(c+d x)^2},x\right ) \]

[Out]

Unintegrable(csc(b*x+a)^3/(d*x+c)^2,x)

Rubi [N/A]

Not integrable

Time = 0.03 (sec) , antiderivative size = 16, 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 = {} \[ \int \frac {\csc ^3(a+b x)}{(c+d x)^2} \, dx=\int \frac {\csc ^3(a+b x)}{(c+d x)^2} \, dx \]

[In]

Int[Csc[a + b*x]^3/(c + d*x)^2,x]

[Out]

Defer[Int][Csc[a + b*x]^3/(c + d*x)^2, x]

Rubi steps \begin{align*} \text {integral}& = \int \frac {\csc ^3(a+b x)}{(c+d x)^2} \, dx \\ \end{align*}

Mathematica [N/A]

Not integrable

Time = 25.49 (sec) , antiderivative size = 18, normalized size of antiderivative = 1.12 \[ \int \frac {\csc ^3(a+b x)}{(c+d x)^2} \, dx=\int \frac {\csc ^3(a+b x)}{(c+d x)^2} \, dx \]

[In]

Integrate[Csc[a + b*x]^3/(c + d*x)^2,x]

[Out]

Integrate[Csc[a + b*x]^3/(c + d*x)^2, x]

Maple [N/A] (verified)

Not integrable

Time = 0.07 (sec) , antiderivative size = 16, normalized size of antiderivative = 1.00

\[\int \frac {\csc ^{3}\left (b x +a \right )}{\left (d x +c \right )^{2}}d x\]

[In]

int(csc(b*x+a)^3/(d*x+c)^2,x)

[Out]

int(csc(b*x+a)^3/(d*x+c)^2,x)

Fricas [N/A]

Not integrable

Time = 0.33 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.81 \[ \int \frac {\csc ^3(a+b x)}{(c+d x)^2} \, dx=\int { \frac {\csc \left (b x + a\right )^{3}}{{\left (d x + c\right )}^{2}} \,d x } \]

[In]

integrate(csc(b*x+a)^3/(d*x+c)^2,x, algorithm="fricas")

[Out]

integral(csc(b*x + a)^3/(d^2*x^2 + 2*c*d*x + c^2), x)

Sympy [N/A]

Not integrable

Time = 0.59 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.94 \[ \int \frac {\csc ^3(a+b x)}{(c+d x)^2} \, dx=\int \frac {\csc ^{3}{\left (a + b x \right )}}{\left (c + d x\right )^{2}}\, dx \]

[In]

integrate(csc(b*x+a)**3/(d*x+c)**2,x)

[Out]

Integral(csc(a + b*x)**3/(c + d*x)**2, x)

Maxima [N/A]

Not integrable

Time = 8.30 (sec) , antiderivative size = 2287, normalized size of antiderivative = 142.94 \[ \int \frac {\csc ^3(a+b x)}{(c+d x)^2} \, dx=\int { \frac {\csc \left (b x + a\right )^{3}}{{\left (d x + c\right )}^{2}} \,d x } \]

[In]

integrate(csc(b*x+a)^3/(d*x+c)^2,x, algorithm="maxima")

[Out]

(((b*d*x + b*c)*cos(3*b*x + 3*a) + (b*d*x + b*c)*cos(b*x + a) - 2*d*sin(3*b*x + 3*a) + 2*d*sin(b*x + a))*cos(4
*b*x + 4*a) + (b*d*x + b*c - 2*(b*d*x + b*c)*cos(2*b*x + 2*a) - 4*d*sin(2*b*x + 2*a))*cos(3*b*x + 3*a) - 2*((b
*d*x + b*c)*cos(b*x + a) + 2*d*sin(b*x + a))*cos(2*b*x + 2*a) + (b*d*x + b*c)*cos(b*x + a) + (b^2*d^3*x^3 + 3*
b^2*c*d^2*x^2 + 3*b^2*c^2*d*x + b^2*c^3 + (b^2*d^3*x^3 + 3*b^2*c*d^2*x^2 + 3*b^2*c^2*d*x + b^2*c^3)*cos(4*b*x
+ 4*a)^2 + 4*(b^2*d^3*x^3 + 3*b^2*c*d^2*x^2 + 3*b^2*c^2*d*x + b^2*c^3)*cos(2*b*x + 2*a)^2 + (b^2*d^3*x^3 + 3*b
^2*c*d^2*x^2 + 3*b^2*c^2*d*x + b^2*c^3)*sin(4*b*x + 4*a)^2 - 4*(b^2*d^3*x^3 + 3*b^2*c*d^2*x^2 + 3*b^2*c^2*d*x
+ b^2*c^3)*sin(4*b*x + 4*a)*sin(2*b*x + 2*a) + 4*(b^2*d^3*x^3 + 3*b^2*c*d^2*x^2 + 3*b^2*c^2*d*x + b^2*c^3)*sin
(2*b*x + 2*a)^2 + 2*(b^2*d^3*x^3 + 3*b^2*c*d^2*x^2 + 3*b^2*c^2*d*x + b^2*c^3 - 2*(b^2*d^3*x^3 + 3*b^2*c*d^2*x^
2 + 3*b^2*c^2*d*x + b^2*c^3)*cos(2*b*x + 2*a))*cos(4*b*x + 4*a) - 4*(b^2*d^3*x^3 + 3*b^2*c*d^2*x^2 + 3*b^2*c^2
*d*x + b^2*c^3)*cos(2*b*x + 2*a))*integrate(1/2*(b^2*d^2*x^2 + 2*b^2*c*d*x + b^2*c^2 + 6*d^2)*sin(b*x + a)/(b^
2*d^4*x^4 + 4*b^2*c*d^3*x^3 + 6*b^2*c^2*d^2*x^2 + 4*b^2*c^3*d*x + b^2*c^4 + (b^2*d^4*x^4 + 4*b^2*c*d^3*x^3 + 6
*b^2*c^2*d^2*x^2 + 4*b^2*c^3*d*x + b^2*c^4)*cos(b*x + a)^2 + (b^2*d^4*x^4 + 4*b^2*c*d^3*x^3 + 6*b^2*c^2*d^2*x^
2 + 4*b^2*c^3*d*x + b^2*c^4)*sin(b*x + a)^2 + 2*(b^2*d^4*x^4 + 4*b^2*c*d^3*x^3 + 6*b^2*c^2*d^2*x^2 + 4*b^2*c^3
*d*x + b^2*c^4)*cos(b*x + a)), x) + (b^2*d^3*x^3 + 3*b^2*c*d^2*x^2 + 3*b^2*c^2*d*x + b^2*c^3 + (b^2*d^3*x^3 +
3*b^2*c*d^2*x^2 + 3*b^2*c^2*d*x + b^2*c^3)*cos(4*b*x + 4*a)^2 + 4*(b^2*d^3*x^3 + 3*b^2*c*d^2*x^2 + 3*b^2*c^2*d
*x + b^2*c^3)*cos(2*b*x + 2*a)^2 + (b^2*d^3*x^3 + 3*b^2*c*d^2*x^2 + 3*b^2*c^2*d*x + b^2*c^3)*sin(4*b*x + 4*a)^
2 - 4*(b^2*d^3*x^3 + 3*b^2*c*d^2*x^2 + 3*b^2*c^2*d*x + b^2*c^3)*sin(4*b*x + 4*a)*sin(2*b*x + 2*a) + 4*(b^2*d^3
*x^3 + 3*b^2*c*d^2*x^2 + 3*b^2*c^2*d*x + b^2*c^3)*sin(2*b*x + 2*a)^2 + 2*(b^2*d^3*x^3 + 3*b^2*c*d^2*x^2 + 3*b^
2*c^2*d*x + b^2*c^3 - 2*(b^2*d^3*x^3 + 3*b^2*c*d^2*x^2 + 3*b^2*c^2*d*x + b^2*c^3)*cos(2*b*x + 2*a))*cos(4*b*x
+ 4*a) - 4*(b^2*d^3*x^3 + 3*b^2*c*d^2*x^2 + 3*b^2*c^2*d*x + b^2*c^3)*cos(2*b*x + 2*a))*integrate(1/2*(b^2*d^2*
x^2 + 2*b^2*c*d*x + b^2*c^2 + 6*d^2)*sin(b*x + a)/(b^2*d^4*x^4 + 4*b^2*c*d^3*x^3 + 6*b^2*c^2*d^2*x^2 + 4*b^2*c
^3*d*x + b^2*c^4 + (b^2*d^4*x^4 + 4*b^2*c*d^3*x^3 + 6*b^2*c^2*d^2*x^2 + 4*b^2*c^3*d*x + b^2*c^4)*cos(b*x + a)^
2 + (b^2*d^4*x^4 + 4*b^2*c*d^3*x^3 + 6*b^2*c^2*d^2*x^2 + 4*b^2*c^3*d*x + b^2*c^4)*sin(b*x + a)^2 - 2*(b^2*d^4*
x^4 + 4*b^2*c*d^3*x^3 + 6*b^2*c^2*d^2*x^2 + 4*b^2*c^3*d*x + b^2*c^4)*cos(b*x + a)), x) + (2*d*cos(3*b*x + 3*a)
- 2*d*cos(b*x + a) + (b*d*x + b*c)*sin(3*b*x + 3*a) + (b*d*x + b*c)*sin(b*x + a))*sin(4*b*x + 4*a) + 2*(2*d*c
os(2*b*x + 2*a) - (b*d*x + b*c)*sin(2*b*x + 2*a) - d)*sin(3*b*x + 3*a) + 2*(2*d*cos(b*x + a) - (b*d*x + b*c)*s
in(b*x + a))*sin(2*b*x + 2*a) + 2*d*sin(b*x + a))/(b^2*d^3*x^3 + 3*b^2*c*d^2*x^2 + 3*b^2*c^2*d*x + b^2*c^3 + (
b^2*d^3*x^3 + 3*b^2*c*d^2*x^2 + 3*b^2*c^2*d*x + b^2*c^3)*cos(4*b*x + 4*a)^2 + 4*(b^2*d^3*x^3 + 3*b^2*c*d^2*x^2
+ 3*b^2*c^2*d*x + b^2*c^3)*cos(2*b*x + 2*a)^2 + (b^2*d^3*x^3 + 3*b^2*c*d^2*x^2 + 3*b^2*c^2*d*x + b^2*c^3)*sin
(4*b*x + 4*a)^2 - 4*(b^2*d^3*x^3 + 3*b^2*c*d^2*x^2 + 3*b^2*c^2*d*x + b^2*c^3)*sin(4*b*x + 4*a)*sin(2*b*x + 2*a
) + 4*(b^2*d^3*x^3 + 3*b^2*c*d^2*x^2 + 3*b^2*c^2*d*x + b^2*c^3)*sin(2*b*x + 2*a)^2 + 2*(b^2*d^3*x^3 + 3*b^2*c*
d^2*x^2 + 3*b^2*c^2*d*x + b^2*c^3 - 2*(b^2*d^3*x^3 + 3*b^2*c*d^2*x^2 + 3*b^2*c^2*d*x + b^2*c^3)*cos(2*b*x + 2*
a))*cos(4*b*x + 4*a) - 4*(b^2*d^3*x^3 + 3*b^2*c*d^2*x^2 + 3*b^2*c^2*d*x + b^2*c^3)*cos(2*b*x + 2*a))

Giac [F(-1)]

Timed out. \[ \int \frac {\csc ^3(a+b x)}{(c+d x)^2} \, dx=\text {Timed out} \]

[In]

integrate(csc(b*x+a)^3/(d*x+c)^2,x, algorithm="giac")

[Out]

Timed out

Mupad [N/A]

Not integrable

Time = 0.50 (sec) , antiderivative size = 18, normalized size of antiderivative = 1.12 \[ \int \frac {\csc ^3(a+b x)}{(c+d x)^2} \, dx=\int \frac {1}{{\sin \left (a+b\,x\right )}^3\,{\left (c+d\,x\right )}^2} \,d x \]

[In]

int(1/(sin(a + b*x)^3*(c + d*x)^2),x)

[Out]

int(1/(sin(a + b*x)^3*(c + d*x)^2), x)