\(\int \frac {\csc ^3(c+d x)}{(e+f x)^2 (a+a \sin (c+d x))} \, dx\) [214]

Optimal result

Integrand size = 28, antiderivative size = 28 \[ \int \frac {\csc ^3(c+d x)}{(e+f x)^2 (a+a \sin (c+d x))} \, dx=\text {Int}\left (\frac {\csc ^3(c+d x)}{(e+f x)^2 (a+a \sin (c+d x))},x\right ) \]

[Out]

Unintegrable(csc(d*x+c)^3/(f*x+e)^2/(a+a*sin(d*x+c)),x)

Rubi [N/A]

Not integrable

Time = 0.05 (sec) , antiderivative size = 28, 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 = {} \[ \int \frac {\csc ^3(c+d x)}{(e+f x)^2 (a+a \sin (c+d x))} \, dx=\int \frac {\csc ^3(c+d x)}{(e+f x)^2 (a+a \sin (c+d x))} \, dx \]

[In]

Int[Csc[c + d*x]^3/((e + f*x)^2*(a + a*Sin[c + d*x])),x]

[Out]

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

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

Mathematica [F(-1)]

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

[In]

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

[Out]

$Aborted

Maple [N/A] (verified)

Not integrable

Time = 0.19 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.00

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

[In]

int(csc(d*x+c)^3/(f*x+e)^2/(a+a*sin(d*x+c)),x)

[Out]

int(csc(d*x+c)^3/(f*x+e)^2/(a+a*sin(d*x+c)),x)

Fricas [N/A]

Not integrable

Time = 0.34 (sec) , antiderivative size = 60, normalized size of antiderivative = 2.14 \[ \int \frac {\csc ^3(c+d x)}{(e+f x)^2 (a+a \sin (c+d x))} \, dx=\int { \frac {\csc \left (d x + c\right )^{3}}{{\left (f x + e\right )}^{2} {\left (a \sin \left (d x + c\right ) + a\right )}} \,d x } \]

[In]

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

[Out]

integral(csc(d*x + c)^3/(a*f^2*x^2 + 2*a*e*f*x + a*e^2 + (a*f^2*x^2 + 2*a*e*f*x + a*e^2)*sin(d*x + c)), x)

Sympy [N/A]

Not integrable

Time = 4.47 (sec) , antiderivative size = 65, normalized size of antiderivative = 2.32 \[ \int \frac {\csc ^3(c+d x)}{(e+f x)^2 (a+a \sin (c+d x))} \, dx=\frac {\int \frac {\csc ^{3}{\left (c + d x \right )}}{e^{2} \sin {\left (c + d x \right )} + e^{2} + 2 e f x \sin {\left (c + d x \right )} + 2 e f x + f^{2} x^{2} \sin {\left (c + d x \right )} + f^{2} x^{2}}\, dx}{a} \]

[In]

integrate(csc(d*x+c)**3/(f*x+e)**2/(a+a*sin(d*x+c)),x)

[Out]

Integral(csc(c + d*x)**3/(e**2*sin(c + d*x) + e**2 + 2*e*f*x*sin(c + d*x) + 2*e*f*x + f**2*x**2*sin(c + d*x) +
f**2*x**2), x)/a

Maxima [N/A]

Not integrable

Time = 24.34 (sec) , antiderivative size = 9726, normalized size of antiderivative = 347.36 \[ \int \frac {\csc ^3(c+d x)}{(e+f x)^2 (a+a \sin (c+d x))} \, dx=\int { \frac {\csc \left (d x + c\right )^{3}}{{\left (f x + e\right )}^{2} {\left (a \sin \left (d x + c\right ) + a\right )}} \,d x } \]

[In]

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

[Out]

(2*f*cos(4*d*x + 4*c)^2 + 4*f*cos(3*d*x + 3*c)^2 + 4*f*cos(2*d*x + 2*c)^2 + 2*f*cos(d*x + c)^2 + 2*f*sin(4*d*x
+ 4*c)^2 + 4*f*sin(3*d*x + 3*c)^2 + 4*f*sin(2*d*x + 2*c)^2 + 2*f*sin(d*x + c)^2 + (4*d*f*x + 4*d*e + 3*(d*f*x
+ d*e)*cos(4*d*x + 4*c) - 2*f*cos(3*d*x + 3*c) - 5*(d*f*x + d*e)*cos(2*d*x + 2*c) + 2*f*cos(d*x + c) - 2*f*si
n(4*d*x + 4*c) - 3*(d*f*x + d*e)*sin(3*d*x + 3*c) + 2*f*sin(2*d*x + 2*c) + (d*f*x + d*e)*sin(d*x + c))*cos(5*d
*x + 5*c) - (3*(d*f*x + d*e)*cos(3*d*x + 3*c) + 6*f*cos(2*d*x + 2*c) - 2*(d*f*x + d*e)*cos(d*x + c) + 6*f*sin(
3*d*x + 3*c) - (d*f*x + d*e)*sin(2*d*x + 2*c) - 4*f*sin(d*x + c) - 2*f)*cos(4*d*x + 4*c) - (5*d*f*x + 5*d*e -
4*(d*f*x + d*e)*cos(2*d*x + 2*c) + 6*f*cos(d*x + c) + 8*f*sin(2*d*x + 2*c) - (d*f*x + d*e)*sin(d*x + c))*cos(3
*d*x + 3*c) - (3*(d*f*x + d*e)*cos(d*x + c) + 6*f*sin(d*x + c) + 2*f)*cos(2*d*x + 2*c) + 3*(d*f*x + d*e)*cos(d
*x + c) + (a*d^2*f^3*x^3 + 3*a*d^2*e*f^2*x^2 + 3*a*d^2*e^2*f*x + a*d^2*e^3 + (a*d^2*f^3*x^3 + 3*a*d^2*e*f^2*x^
2 + 3*a*d^2*e^2*f*x + a*d^2*e^3)*cos(5*d*x + 5*c)^2 + (a*d^2*f^3*x^3 + 3*a*d^2*e*f^2*x^2 + 3*a*d^2*e^2*f*x + a
*d^2*e^3)*cos(4*d*x + 4*c)^2 + 4*(a*d^2*f^3*x^3 + 3*a*d^2*e*f^2*x^2 + 3*a*d^2*e^2*f*x + a*d^2*e^3)*cos(3*d*x +
3*c)^2 + 4*(a*d^2*f^3*x^3 + 3*a*d^2*e*f^2*x^2 + 3*a*d^2*e^2*f*x + a*d^2*e^3)*cos(2*d*x + 2*c)^2 + (a*d^2*f^3*
x^3 + 3*a*d^2*e*f^2*x^2 + 3*a*d^2*e^2*f*x + a*d^2*e^3)*cos(d*x + c)^2 + (a*d^2*f^3*x^3 + 3*a*d^2*e*f^2*x^2 + 3
*a*d^2*e^2*f*x + a*d^2*e^3)*sin(5*d*x + 5*c)^2 + (a*d^2*f^3*x^3 + 3*a*d^2*e*f^2*x^2 + 3*a*d^2*e^2*f*x + a*d^2*
e^3)*sin(4*d*x + 4*c)^2 + 4*(a*d^2*f^3*x^3 + 3*a*d^2*e*f^2*x^2 + 3*a*d^2*e^2*f*x + a*d^2*e^3)*sin(3*d*x + 3*c)
^2 + 4*(a*d^2*f^3*x^3 + 3*a*d^2*e*f^2*x^2 + 3*a*d^2*e^2*f*x + a*d^2*e^3)*cos(d*x + c)*sin(2*d*x + 2*c) + 4*(a*
d^2*f^3*x^3 + 3*a*d^2*e*f^2*x^2 + 3*a*d^2*e^2*f*x + a*d^2*e^3)*sin(2*d*x + 2*c)^2 + (a*d^2*f^3*x^3 + 3*a*d^2*e
*f^2*x^2 + 3*a*d^2*e^2*f*x + a*d^2*e^3)*sin(d*x + c)^2 - 2*(2*(a*d^2*f^3*x^3 + 3*a*d^2*e*f^2*x^2 + 3*a*d^2*e^2
*f*x + a*d^2*e^3)*cos(3*d*x + 3*c) - (a*d^2*f^3*x^3 + 3*a*d^2*e*f^2*x^2 + 3*a*d^2*e^2*f*x + a*d^2*e^3)*cos(d*x
+ c) + (a*d^2*f^3*x^3 + 3*a*d^2*e*f^2*x^2 + 3*a*d^2*e^2*f*x + a*d^2*e^3)*sin(4*d*x + 4*c) - 2*(a*d^2*f^3*x^3
+ 3*a*d^2*e*f^2*x^2 + 3*a*d^2*e^2*f*x + a*d^2*e^3)*sin(2*d*x + 2*c))*cos(5*d*x + 5*c) + 2*(a*d^2*f^3*x^3 + 3*a
*d^2*e*f^2*x^2 + 3*a*d^2*e^2*f*x + a*d^2*e^3 - 2*(a*d^2*f^3*x^3 + 3*a*d^2*e*f^2*x^2 + 3*a*d^2*e^2*f*x + a*d^2*
e^3)*cos(2*d*x + 2*c) - 2*(a*d^2*f^3*x^3 + 3*a*d^2*e*f^2*x^2 + 3*a*d^2*e^2*f*x + a*d^2*e^3)*sin(3*d*x + 3*c) +
(a*d^2*f^3*x^3 + 3*a*d^2*e*f^2*x^2 + 3*a*d^2*e^2*f*x + a*d^2*e^3)*sin(d*x + c))*cos(4*d*x + 4*c) - 4*((a*d^2*
f^3*x^3 + 3*a*d^2*e*f^2*x^2 + 3*a*d^2*e^2*f*x + a*d^2*e^3)*cos(d*x + c) + 2*(a*d^2*f^3*x^3 + 3*a*d^2*e*f^2*x^2
+ 3*a*d^2*e^2*f*x + a*d^2*e^3)*sin(2*d*x + 2*c))*cos(3*d*x + 3*c) - 4*(a*d^2*f^3*x^3 + 3*a*d^2*e*f^2*x^2 + 3*
a*d^2*e^2*f*x + a*d^2*e^3 + (a*d^2*f^3*x^3 + 3*a*d^2*e*f^2*x^2 + 3*a*d^2*e^2*f*x + a*d^2*e^3)*sin(d*x + c))*co
s(2*d*x + 2*c) + 2*(a*d^2*f^3*x^3 + 3*a*d^2*e*f^2*x^2 + 3*a*d^2*e^2*f*x + a*d^2*e^3 + (a*d^2*f^3*x^3 + 3*a*d^2
*e*f^2*x^2 + 3*a*d^2*e^2*f*x + a*d^2*e^3)*cos(4*d*x + 4*c) - 2*(a*d^2*f^3*x^3 + 3*a*d^2*e*f^2*x^2 + 3*a*d^2*e^
2*f*x + a*d^2*e^3)*cos(2*d*x + 2*c) - 2*(a*d^2*f^3*x^3 + 3*a*d^2*e*f^2*x^2 + 3*a*d^2*e^2*f*x + a*d^2*e^3)*sin(
3*d*x + 3*c) + (a*d^2*f^3*x^3 + 3*a*d^2*e*f^2*x^2 + 3*a*d^2*e^2*f*x + a*d^2*e^3)*sin(d*x + c))*sin(5*d*x + 5*c
) + 2*(2*(a*d^2*f^3*x^3 + 3*a*d^2*e*f^2*x^2 + 3*a*d^2*e^2*f*x + a*d^2*e^3)*cos(3*d*x + 3*c) - (a*d^2*f^3*x^3 +
3*a*d^2*e*f^2*x^2 + 3*a*d^2*e^2*f*x + a*d^2*e^3)*cos(d*x + c) - 2*(a*d^2*f^3*x^3 + 3*a*d^2*e*f^2*x^2 + 3*a*d^
2*e^2*f*x + a*d^2*e^3)*sin(2*d*x + 2*c))*sin(4*d*x + 4*c) - 4*(a*d^2*f^3*x^3 + 3*a*d^2*e*f^2*x^2 + 3*a*d^2*e^2
*f*x + a*d^2*e^3 - 2*(a*d^2*f^3*x^3 + 3*a*d^2*e*f^2*x^2 + 3*a*d^2*e^2*f*x + a*d^2*e^3)*cos(2*d*x + 2*c) + (a*d
^2*f^3*x^3 + 3*a*d^2*e*f^2*x^2 + 3*a*d^2*e^2*f*x + a*d^2*e^3)*sin(d*x + c))*sin(3*d*x + 3*c) + 2*(a*d^2*f^3*x^
3 + 3*a*d^2*e*f^2*x^2 + 3*a*d^2*e^2*f*x + a*d^2*e^3)*sin(d*x + c))*integrate(1/2*(3*d^2*f^2*x^2 + 3*d^2*e^2 +
4*d*e*f + 6*f^2 + 2*(3*d^2*e*f + 2*d*f^2)*x)*sin(d*x + c)/(a*d^2*f^4*x^4 + 4*a*d^2*e*f^3*x^3 + 6*a*d^2*e^2*f^2
*x^2 + 4*a*d^2*e^3*f*x + a*d^2*e^4 + (a*d^2*f^4*x^4 + 4*a*d^2*e*f^3*x^3 + 6*a*d^2*e^2*f^2*x^2 + 4*a*d^2*e^3*f*
x + a*d^2*e^4)*cos(d*x + c)^2 + (a*d^2*f^4*x^4 + 4*a*d^2*e*f^3*x^3 + 6*a*d^2*e^2*f^2*x^2 + 4*a*d^2*e^3*f*x + a
*d^2*e^4)*sin(d*x + c)^2 - 2*(a*d^2*f^4*x^4 + 4*a*d^2*e*f^3*x^3 + 6*a*d^2*e^2*f^2*x^2 + 4*a*d^2*e^3*f*x + a*d^
2*e^4)*cos(d*x + c)), x) + (a*d^2*f^3*x^3 + 3*a*d^2*e*f^2*x^2 + 3*a*d^2*e^2*f*x + a*d^2*e^3 + (a*d^2*f^3*x^3 +
3*a*d^2*e*f^2*x^2 + 3*a*d^2*e^2*f*x + a*d^2*e^3)*cos(5*d*x + 5*c)^2 + (a*d^2*f^3*x^3 + 3*a*d^2*e*f^2*x^2 + 3*
a*d^2*e^2*f*x + a*d^2*e^3)*cos(4*d*x + 4*c)^2 + 4*(a*d^2*f^3*x^3 + 3*a*d^2*e*f^2*x^2 + 3*a*d^2*e^2*f*x + a*d^2
*e^3)*cos(3*d*x + 3*c)^2 + 4*(a*d^2*f^3*x^3 + 3*a*d^2*e*f^2*x^2 + 3*a*d^2*e^2*f*x + a*d^2*e^3)*cos(2*d*x + 2*c
)^2 + (a*d^2*f^3*x^3 + 3*a*d^2*e*f^2*x^2 + 3*a*d^2*e^2*f*x + a*d^2*e^3)*cos(d*x + c)^2 + (a*d^2*f^3*x^3 + 3*a*
d^2*e*f^2*x^2 + 3*a*d^2*e^2*f*x + a*d^2*e^3)*sin(5*d*x + 5*c)^2 + (a*d^2*f^3*x^3 + 3*a*d^2*e*f^2*x^2 + 3*a*d^2
*e^2*f*x + a*d^2*e^3)*sin(4*d*x + 4*c)^2 + 4*(a*d^2*f^3*x^3 + 3*a*d^2*e*f^2*x^2 + 3*a*d^2*e^2*f*x + a*d^2*e^3)
*sin(3*d*x + 3*c)^2 + 4*(a*d^2*f^3*x^3 + 3*a*d^2*e*f^2*x^2 + 3*a*d^2*e^2*f*x + a*d^2*e^3)*cos(d*x + c)*sin(2*d
*x + 2*c) + 4*(a*d^2*f^3*x^3 + 3*a*d^2*e*f^2*x^2 + 3*a*d^2*e^2*f*x + a*d^2*e^3)*sin(2*d*x + 2*c)^2 + (a*d^2*f^
3*x^3 + 3*a*d^2*e*f^2*x^2 + 3*a*d^2*e^2*f*x + a*d^2*e^3)*sin(d*x + c)^2 - 2*(2*(a*d^2*f^3*x^3 + 3*a*d^2*e*f^2*
x^2 + 3*a*d^2*e^2*f*x + a*d^2*e^3)*cos(3*d*x + 3*c) - (a*d^2*f^3*x^3 + 3*a*d^2*e*f^2*x^2 + 3*a*d^2*e^2*f*x + a
*d^2*e^3)*cos(d*x + c) + (a*d^2*f^3*x^3 + 3*a*d^2*e*f^2*x^2 + 3*a*d^2*e^2*f*x + a*d^2*e^3)*sin(4*d*x + 4*c) -
2*(a*d^2*f^3*x^3 + 3*a*d^2*e*f^2*x^2 + 3*a*d^2*e^2*f*x + a*d^2*e^3)*sin(2*d*x + 2*c))*cos(5*d*x + 5*c) + 2*(a*
d^2*f^3*x^3 + 3*a*d^2*e*f^2*x^2 + 3*a*d^2*e^2*f*x + a*d^2*e^3 - 2*(a*d^2*f^3*x^3 + 3*a*d^2*e*f^2*x^2 + 3*a*d^2
*e^2*f*x + a*d^2*e^3)*cos(2*d*x + 2*c) - 2*(a*d^2*f^3*x^3 + 3*a*d^2*e*f^2*x^2 + 3*a*d^2*e^2*f*x + a*d^2*e^3)*s
in(3*d*x + 3*c) + (a*d^2*f^3*x^3 + 3*a*d^2*e*f^2*x^2 + 3*a*d^2*e^2*f*x + a*d^2*e^3)*sin(d*x + c))*cos(4*d*x +
4*c) - 4*((a*d^2*f^3*x^3 + 3*a*d^2*e*f^2*x^2 + 3*a*d^2*e^2*f*x + a*d^2*e^3)*cos(d*x + c) + 2*(a*d^2*f^3*x^3 +
3*a*d^2*e*f^2*x^2 + 3*a*d^2*e^2*f*x + a*d^2*e^3)*sin(2*d*x + 2*c))*cos(3*d*x + 3*c) - 4*(a*d^2*f^3*x^3 + 3*a*d
^2*e*f^2*x^2 + 3*a*d^2*e^2*f*x + a*d^2*e^3 + (a*d^2*f^3*x^3 + 3*a*d^2*e*f^2*x^2 + 3*a*d^2*e^2*f*x + a*d^2*e^3)
*sin(d*x + c))*cos(2*d*x + 2*c) + 2*(a*d^2*f^3*x^3 + 3*a*d^2*e*f^2*x^2 + 3*a*d^2*e^2*f*x + a*d^2*e^3 + (a*d^2*
f^3*x^3 + 3*a*d^2*e*f^2*x^2 + 3*a*d^2*e^2*f*x + a*d^2*e^3)*cos(4*d*x + 4*c) - 2*(a*d^2*f^3*x^3 + 3*a*d^2*e*f^2
*x^2 + 3*a*d^2*e^2*f*x + a*d^2*e^3)*cos(2*d*x + 2*c) - 2*(a*d^2*f^3*x^3 + 3*a*d^2*e*f^2*x^2 + 3*a*d^2*e^2*f*x
+ a*d^2*e^3)*sin(3*d*x + 3*c) + (a*d^2*f^3*x^3 + 3*a*d^2*e*f^2*x^2 + 3*a*d^2*e^2*f*x + a*d^2*e^3)*sin(d*x + c)
)*sin(5*d*x + 5*c) + 2*(2*(a*d^2*f^3*x^3 + 3*a*d^2*e*f^2*x^2 + 3*a*d^2*e^2*f*x + a*d^2*e^3)*cos(3*d*x + 3*c) -
(a*d^2*f^3*x^3 + 3*a*d^2*e*f^2*x^2 + 3*a*d^2*e^2*f*x + a*d^2*e^3)*cos(d*x + c) - 2*(a*d^2*f^3*x^3 + 3*a*d^2*e
*f^2*x^2 + 3*a*d^2*e^2*f*x + a*d^2*e^3)*sin(2*d*x + 2*c))*sin(4*d*x + 4*c) - 4*(a*d^2*f^3*x^3 + 3*a*d^2*e*f^2*
x^2 + 3*a*d^2*e^2*f*x + a*d^2*e^3 - 2*(a*d^2*f^3*x^3 + 3*a*d^2*e*f^2*x^2 + 3*a*d^2*e^2*f*x + a*d^2*e^3)*cos(2*
d*x + 2*c) + (a*d^2*f^3*x^3 + 3*a*d^2*e*f^2*x^2 + 3*a*d^2*e^2*f*x + a*d^2*e^3)*sin(d*x + c))*sin(3*d*x + 3*c)
+ 2*(a*d^2*f^3*x^3 + 3*a*d^2*e*f^2*x^2 + 3*a*d^2*e^2*f*x + a*d^2*e^3)*sin(d*x + c))*integrate(1/2*(3*d^2*f^2*x
^2 + 3*d^2*e^2 - 4*d*e*f + 6*f^2 + 2*(3*d^2*e*f - 2*d*f^2)*x)*sin(d*x + c)/(a*d^2*f^4*x^4 + 4*a*d^2*e*f^3*x^3
+ 6*a*d^2*e^2*f^2*x^2 + 4*a*d^2*e^3*f*x + a*d^2*e^4 + (a*d^2*f^4*x^4 + 4*a*d^2*e*f^3*x^3 + 6*a*d^2*e^2*f^2*x^2
+ 4*a*d^2*e^3*f*x + a*d^2*e^4)*cos(d*x + c)^2 + (a*d^2*f^4*x^4 + 4*a*d^2*e*f^3*x^3 + 6*a*d^2*e^2*f^2*x^2 + 4*
a*d^2*e^3*f*x + a*d^2*e^4)*sin(d*x + c)^2 + 2*(a*d^2*f^4*x^4 + 4*a*d^2*e*f^3*x^3 + 6*a*d^2*e^2*f^2*x^2 + 4*a*d
^2*e^3*f*x + a*d^2*e^4)*cos(d*x + c)), x) + 4*(a*d^2*f^4*x^3 + 3*a*d^2*e*f^3*x^2 + 3*a*d^2*e^2*f^2*x + a*d^2*e
^3*f + (a*d^2*f^4*x^3 + 3*a*d^2*e*f^3*x^2 + 3*a*d^2*e^2*f^2*x + a*d^2*e^3*f)*cos(5*d*x + 5*c)^2 + (a*d^2*f^4*x
^3 + 3*a*d^2*e*f^3*x^2 + 3*a*d^2*e^2*f^2*x + a*d^2*e^3*f)*cos(4*d*x + 4*c)^2 + 4*(a*d^2*f^4*x^3 + 3*a*d^2*e*f^
3*x^2 + 3*a*d^2*e^2*f^2*x + a*d^2*e^3*f)*cos(3*d*x + 3*c)^2 + 4*(a*d^2*f^4*x^3 + 3*a*d^2*e*f^3*x^2 + 3*a*d^2*e
^2*f^2*x + a*d^2*e^3*f)*cos(2*d*x + 2*c)^2 + (a*d^2*f^4*x^3 + 3*a*d^2*e*f^3*x^2 + 3*a*d^2*e^2*f^2*x + a*d^2*e^
3*f)*cos(d*x + c)^2 + (a*d^2*f^4*x^3 + 3*a*d^2*e*f^3*x^2 + 3*a*d^2*e^2*f^2*x + a*d^2*e^3*f)*sin(5*d*x + 5*c)^2
+ (a*d^2*f^4*x^3 + 3*a*d^2*e*f^3*x^2 + 3*a*d^2*e^2*f^2*x + a*d^2*e^3*f)*sin(4*d*x + 4*c)^2 + 4*(a*d^2*f^4*x^3
+ 3*a*d^2*e*f^3*x^2 + 3*a*d^2*e^2*f^2*x + a*d^2*e^3*f)*sin(3*d*x + 3*c)^2 + 4*(a*d^2*f^4*x^3 + 3*a*d^2*e*f^3*
x^2 + 3*a*d^2*e^2*f^2*x + a*d^2*e^3*f)*cos(d*x + c)*sin(2*d*x + 2*c) + 4*(a*d^2*f^4*x^3 + 3*a*d^2*e*f^3*x^2 +
3*a*d^2*e^2*f^2*x + a*d^2*e^3*f)*sin(2*d*x + 2*c)^2 + (a*d^2*f^4*x^3 + 3*a*d^2*e*f^3*x^2 + 3*a*d^2*e^2*f^2*x +
a*d^2*e^3*f)*sin(d*x + c)^2 - 2*(2*(a*d^2*f^4*x^3 + 3*a*d^2*e*f^3*x^2 + 3*a*d^2*e^2*f^2*x + a*d^2*e^3*f)*cos(
3*d*x + 3*c) - (a*d^2*f^4*x^3 + 3*a*d^2*e*f^3*x^2 + 3*a*d^2*e^2*f^2*x + a*d^2*e^3*f)*cos(d*x + c) + (a*d^2*f^4
*x^3 + 3*a*d^2*e*f^3*x^2 + 3*a*d^2*e^2*f^2*x + a*d^2*e^3*f)*sin(4*d*x + 4*c) - 2*(a*d^2*f^4*x^3 + 3*a*d^2*e*f^
3*x^2 + 3*a*d^2*e^2*f^2*x + a*d^2*e^3*f)*sin(2*d*x + 2*c))*cos(5*d*x + 5*c) + 2*(a*d^2*f^4*x^3 + 3*a*d^2*e*f^3
*x^2 + 3*a*d^2*e^2*f^2*x + a*d^2*e^3*f - 2*(a*d^2*f^4*x^3 + 3*a*d^2*e*f^3*x^2 + 3*a*d^2*e^2*f^2*x + a*d^2*e^3*
f)*cos(2*d*x + 2*c) - 2*(a*d^2*f^4*x^3 + 3*a*d^2*e*f^3*x^2 + 3*a*d^2*e^2*f^2*x + a*d^2*e^3*f)*sin(3*d*x + 3*c)
+ (a*d^2*f^4*x^3 + 3*a*d^2*e*f^3*x^2 + 3*a*d^2*e^2*f^2*x + a*d^2*e^3*f)*sin(d*x + c))*cos(4*d*x + 4*c) - 4*((
a*d^2*f^4*x^3 + 3*a*d^2*e*f^3*x^2 + 3*a*d^2*e^2*f^2*x + a*d^2*e^3*f)*cos(d*x + c) + 2*(a*d^2*f^4*x^3 + 3*a*d^2
*e*f^3*x^2 + 3*a*d^2*e^2*f^2*x + a*d^2*e^3*f)*sin(2*d*x + 2*c))*cos(3*d*x + 3*c) - 4*(a*d^2*f^4*x^3 + 3*a*d^2*
e*f^3*x^2 + 3*a*d^2*e^2*f^2*x + a*d^2*e^3*f + (a*d^2*f^4*x^3 + 3*a*d^2*e*f^3*x^2 + 3*a*d^2*e^2*f^2*x + a*d^2*e
^3*f)*sin(d*x + c))*cos(2*d*x + 2*c) + 2*(a*d^2*f^4*x^3 + 3*a*d^2*e*f^3*x^2 + 3*a*d^2*e^2*f^2*x + a*d^2*e^3*f
+ (a*d^2*f^4*x^3 + 3*a*d^2*e*f^3*x^2 + 3*a*d^2*e^2*f^2*x + a*d^2*e^3*f)*cos(4*d*x + 4*c) - 2*(a*d^2*f^4*x^3 +
3*a*d^2*e*f^3*x^2 + 3*a*d^2*e^2*f^2*x + a*d^2*e^3*f)*cos(2*d*x + 2*c) - 2*(a*d^2*f^4*x^3 + 3*a*d^2*e*f^3*x^2 +
3*a*d^2*e^2*f^2*x + a*d^2*e^3*f)*sin(3*d*x + 3*c) + (a*d^2*f^4*x^3 + 3*a*d^2*e*f^3*x^2 + 3*a*d^2*e^2*f^2*x +
a*d^2*e^3*f)*sin(d*x + c))*sin(5*d*x + 5*c) + 2*(2*(a*d^2*f^4*x^3 + 3*a*d^2*e*f^3*x^2 + 3*a*d^2*e^2*f^2*x + a*
d^2*e^3*f)*cos(3*d*x + 3*c) - (a*d^2*f^4*x^3 + 3*a*d^2*e*f^3*x^2 + 3*a*d^2*e^2*f^2*x + a*d^2*e^3*f)*cos(d*x +
c) - 2*(a*d^2*f^4*x^3 + 3*a*d^2*e*f^3*x^2 + 3*a*d^2*e^2*f^2*x + a*d^2*e^3*f)*sin(2*d*x + 2*c))*sin(4*d*x + 4*c
) - 4*(a*d^2*f^4*x^3 + 3*a*d^2*e*f^3*x^2 + 3*a*d^2*e^2*f^2*x + a*d^2*e^3*f - 2*(a*d^2*f^4*x^3 + 3*a*d^2*e*f^3*
x^2 + 3*a*d^2*e^2*f^2*x + a*d^2*e^3*f)*cos(2*d*x + 2*c) + (a*d^2*f^4*x^3 + 3*a*d^2*e*f^3*x^2 + 3*a*d^2*e^2*f^2
*x + a*d^2*e^3*f)*sin(d*x + c))*sin(3*d*x + 3*c) + 2*(a*d^2*f^4*x^3 + 3*a*d^2*e*f^3*x^2 + 3*a*d^2*e^2*f^2*x +
a*d^2*e^3*f)*sin(d*x + c))*integrate(cos(d*x + c)/(a*d*f^3*x^3 + 3*a*d*e*f^2*x^2 + 3*a*d*e^2*f*x + a*d*e^3 + (
a*d*f^3*x^3 + 3*a*d*e*f^2*x^2 + 3*a*d*e^2*f*x + a*d*e^3)*cos(d*x + c)^2 + (a*d*f^3*x^3 + 3*a*d*e*f^2*x^2 + 3*a
*d*e^2*f*x + a*d*e^3)*sin(d*x + c)^2 + 2*(a*d*f^3*x^3 + 3*a*d*e*f^2*x^2 + 3*a*d*e^2*f*x + a*d*e^3)*sin(d*x + c
)), x) + (2*f*cos(4*d*x + 4*c) + 3*(d*f*x + d*e)*cos(3*d*x + 3*c) - 2*f*cos(2*d*x + 2*c) - (d*f*x + d*e)*cos(d
*x + c) + 3*(d*f*x + d*e)*sin(4*d*x + 4*c) - 2*f*sin(3*d*x + 3*c) - 5*(d*f*x + d*e)*sin(2*d*x + 2*c) + 2*f*sin
(d*x + c))*sin(5*d*x + 5*c) - (d*f*x + d*e - 6*f*cos(3*d*x + 3*c) + (d*f*x + d*e)*cos(2*d*x + 2*c) + 4*f*cos(d
*x + c) + 3*(d*f*x + d*e)*sin(3*d*x + 3*c) + 6*f*sin(2*d*x + 2*c) - 2*(d*f*x + d*e)*sin(d*x + c))*sin(4*d*x +
4*c) + (8*f*cos(2*d*x + 2*c) - (d*f*x + d*e)*cos(d*x + c) + 4*(d*f*x + d*e)*sin(2*d*x + 2*c) - 6*f*sin(d*x + c
) - 2*f)*sin(3*d*x + 3*c) + 3*(d*f*x + d*e + 2*f*cos(d*x + c) - (d*f*x + d*e)*sin(d*x + c))*sin(2*d*x + 2*c) +
2*f*sin(d*x + c))/(a*d^2*f^3*x^3 + 3*a*d^2*e*f^2*x^2 + 3*a*d^2*e^2*f*x + a*d^2*e^3 + (a*d^2*f^3*x^3 + 3*a*d^2
*e*f^2*x^2 + 3*a*d^2*e^2*f*x + a*d^2*e^3)*cos(5*d*x + 5*c)^2 + (a*d^2*f^3*x^3 + 3*a*d^2*e*f^2*x^2 + 3*a*d^2*e^
2*f*x + a*d^2*e^3)*cos(4*d*x + 4*c)^2 + 4*(a*d^2*f^3*x^3 + 3*a*d^2*e*f^2*x^2 + 3*a*d^2*e^2*f*x + a*d^2*e^3)*co
s(3*d*x + 3*c)^2 + 4*(a*d^2*f^3*x^3 + 3*a*d^2*e*f^2*x^2 + 3*a*d^2*e^2*f*x + a*d^2*e^3)*cos(2*d*x + 2*c)^2 + (a
*d^2*f^3*x^3 + 3*a*d^2*e*f^2*x^2 + 3*a*d^2*e^2*f*x + a*d^2*e^3)*cos(d*x + c)^2 + (a*d^2*f^3*x^3 + 3*a*d^2*e*f^
2*x^2 + 3*a*d^2*e^2*f*x + a*d^2*e^3)*sin(5*d*x + 5*c)^2 + (a*d^2*f^3*x^3 + 3*a*d^2*e*f^2*x^2 + 3*a*d^2*e^2*f*x
+ a*d^2*e^3)*sin(4*d*x + 4*c)^2 + 4*(a*d^2*f^3*x^3 + 3*a*d^2*e*f^2*x^2 + 3*a*d^2*e^2*f*x + a*d^2*e^3)*sin(3*d
*x + 3*c)^2 + 4*(a*d^2*f^3*x^3 + 3*a*d^2*e*f^2*x^2 + 3*a*d^2*e^2*f*x + a*d^2*e^3)*cos(d*x + c)*sin(2*d*x + 2*c
) + 4*(a*d^2*f^3*x^3 + 3*a*d^2*e*f^2*x^2 + 3*a*d^2*e^2*f*x + a*d^2*e^3)*sin(2*d*x + 2*c)^2 + (a*d^2*f^3*x^3 +
3*a*d^2*e*f^2*x^2 + 3*a*d^2*e^2*f*x + a*d^2*e^3)*sin(d*x + c)^2 - 2*(2*(a*d^2*f^3*x^3 + 3*a*d^2*e*f^2*x^2 + 3*
a*d^2*e^2*f*x + a*d^2*e^3)*cos(3*d*x + 3*c) - (a*d^2*f^3*x^3 + 3*a*d^2*e*f^2*x^2 + 3*a*d^2*e^2*f*x + a*d^2*e^3
)*cos(d*x + c) + (a*d^2*f^3*x^3 + 3*a*d^2*e*f^2*x^2 + 3*a*d^2*e^2*f*x + a*d^2*e^3)*sin(4*d*x + 4*c) - 2*(a*d^2
*f^3*x^3 + 3*a*d^2*e*f^2*x^2 + 3*a*d^2*e^2*f*x + a*d^2*e^3)*sin(2*d*x + 2*c))*cos(5*d*x + 5*c) + 2*(a*d^2*f^3*
x^3 + 3*a*d^2*e*f^2*x^2 + 3*a*d^2*e^2*f*x + a*d^2*e^3 - 2*(a*d^2*f^3*x^3 + 3*a*d^2*e*f^2*x^2 + 3*a*d^2*e^2*f*x
+ a*d^2*e^3)*cos(2*d*x + 2*c) - 2*(a*d^2*f^3*x^3 + 3*a*d^2*e*f^2*x^2 + 3*a*d^2*e^2*f*x + a*d^2*e^3)*sin(3*d*x
+ 3*c) + (a*d^2*f^3*x^3 + 3*a*d^2*e*f^2*x^2 + 3*a*d^2*e^2*f*x + a*d^2*e^3)*sin(d*x + c))*cos(4*d*x + 4*c) - 4
*((a*d^2*f^3*x^3 + 3*a*d^2*e*f^2*x^2 + 3*a*d^2*e^2*f*x + a*d^2*e^3)*cos(d*x + c) + 2*(a*d^2*f^3*x^3 + 3*a*d^2*
e*f^2*x^2 + 3*a*d^2*e^2*f*x + a*d^2*e^3)*sin(2*d*x + 2*c))*cos(3*d*x + 3*c) - 4*(a*d^2*f^3*x^3 + 3*a*d^2*e*f^2
*x^2 + 3*a*d^2*e^2*f*x + a*d^2*e^3 + (a*d^2*f^3*x^3 + 3*a*d^2*e*f^2*x^2 + 3*a*d^2*e^2*f*x + a*d^2*e^3)*sin(d*x
+ c))*cos(2*d*x + 2*c) + 2*(a*d^2*f^3*x^3 + 3*a*d^2*e*f^2*x^2 + 3*a*d^2*e^2*f*x + a*d^2*e^3 + (a*d^2*f^3*x^3
+ 3*a*d^2*e*f^2*x^2 + 3*a*d^2*e^2*f*x + a*d^2*e^3)*cos(4*d*x + 4*c) - 2*(a*d^2*f^3*x^3 + 3*a*d^2*e*f^2*x^2 + 3
*a*d^2*e^2*f*x + a*d^2*e^3)*cos(2*d*x + 2*c) - 2*(a*d^2*f^3*x^3 + 3*a*d^2*e*f^2*x^2 + 3*a*d^2*e^2*f*x + a*d^2*
e^3)*sin(3*d*x + 3*c) + (a*d^2*f^3*x^3 + 3*a*d^2*e*f^2*x^2 + 3*a*d^2*e^2*f*x + a*d^2*e^3)*sin(d*x + c))*sin(5*
d*x + 5*c) + 2*(2*(a*d^2*f^3*x^3 + 3*a*d^2*e*f^2*x^2 + 3*a*d^2*e^2*f*x + a*d^2*e^3)*cos(3*d*x + 3*c) - (a*d^2*
f^3*x^3 + 3*a*d^2*e*f^2*x^2 + 3*a*d^2*e^2*f*x + a*d^2*e^3)*cos(d*x + c) - 2*(a*d^2*f^3*x^3 + 3*a*d^2*e*f^2*x^2
+ 3*a*d^2*e^2*f*x + a*d^2*e^3)*sin(2*d*x + 2*c))*sin(4*d*x + 4*c) - 4*(a*d^2*f^3*x^3 + 3*a*d^2*e*f^2*x^2 + 3*
a*d^2*e^2*f*x + a*d^2*e^3 - 2*(a*d^2*f^3*x^3 + 3*a*d^2*e*f^2*x^2 + 3*a*d^2*e^2*f*x + a*d^2*e^3)*cos(2*d*x + 2*
c) + (a*d^2*f^3*x^3 + 3*a*d^2*e*f^2*x^2 + 3*a*d^2*e^2*f*x + a*d^2*e^3)*sin(d*x + c))*sin(3*d*x + 3*c) + 2*(a*d
^2*f^3*x^3 + 3*a*d^2*e*f^2*x^2 + 3*a*d^2*e^2*f*x + a*d^2*e^3)*sin(d*x + c))

Giac [F(-1)]

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

[In]

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

[Out]

Timed out

Mupad [N/A]

Not integrable

Time = 2.76 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.07 \[ \int \frac {\csc ^3(c+d x)}{(e+f x)^2 (a+a \sin (c+d x))} \, dx=\int \frac {1}{{\sin \left (c+d\,x\right )}^3\,{\left (e+f\,x\right )}^2\,\left (a+a\,\sin \left (c+d\,x\right )\right )} \,d x \]

[In]

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

[Out]

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