\(\int \frac {\csc (c+d x)}{(e+f x)^2 (a+a \sin (c+d x))} \, dx\) [202]

Optimal result

Integrand size = 26, antiderivative size = 26 \[ \int \frac {\csc (c+d x)}{(e+f x)^2 (a+a \sin (c+d x))} \, dx=\text {Int}\left (\frac {\csc (c+d x)}{(e+f x)^2 (a+a \sin (c+d x))},x\right ) \]

[Out]

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

Rubi [N/A]

Not integrable

Time = 0.04 (sec) , antiderivative size = 26, 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 = {} \[ \int \frac {\csc (c+d x)}{(e+f x)^2 (a+a \sin (c+d x))} \, dx=\int \frac {\csc (c+d x)}{(e+f x)^2 (a+a \sin (c+d x))} \, dx \]

[In]

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

[Out]

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

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

Mathematica [N/A]

Not integrable

Time = 8.84 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.08 \[ \int \frac {\csc (c+d x)}{(e+f x)^2 (a+a \sin (c+d x))} \, dx=\int \frac {\csc (c+d x)}{(e+f x)^2 (a+a \sin (c+d x))} \, dx \]

[In]

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

[Out]

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

Maple [N/A] (verified)

Not integrable

Time = 0.15 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.00

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

[In]

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

[Out]

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

Fricas [N/A]

Not integrable

Time = 0.27 (sec) , antiderivative size = 58, normalized size of antiderivative = 2.23 \[ \int \frac {\csc (c+d x)}{(e+f x)^2 (a+a \sin (c+d x))} \, dx=\int { \frac {\csc \left (d x + c\right )}{{\left (f x + e\right )}^{2} {\left (a \sin \left (d x + c\right ) + a\right )}} \,d x } \]

[In]

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

[Out]

integral(csc(d*x + c)/(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 = 4.45 (sec) , antiderivative size = 63, normalized size of antiderivative = 2.42 \[ \int \frac {\csc (c+d x)}{(e+f x)^2 (a+a \sin (c+d x))} \, dx=\frac {\int \frac {\csc {\left (c + d x \right )}}{e^{2} \sin {\left (c + d x \right )} + e^{2} + 2 e f x \sin {\left (c + d x \right )} + 2 e f x + f^{2} x^{2} \sin {\left (c + d x \right )} + f^{2} x^{2}}\, dx}{a} \]

[In]

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

[Out]

Integral(csc(c + d*x)/(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 = 1.24 (sec) , antiderivative size = 915, normalized size of antiderivative = 35.19 \[ \int \frac {\csc (c+d x)}{(e+f x)^2 (a+a \sin (c+d x))} \, dx=\int { \frac {\csc \left (d x + c\right )}{{\left (f x + e\right )}^{2} {\left (a \sin \left (d x + c\right ) + a\right )}} \,d x } \]

[In]

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

[Out]

(4*(a*d*f^3*x^2 + 2*a*d*e*f^2*x + a*d*e^2*f + (a*d*f^3*x^2 + 2*a*d*e*f^2*x + a*d*e^2*f)*cos(d*x + c)^2 + (a*d*
f^3*x^2 + 2*a*d*e*f^2*x + a*d*e^2*f)*sin(d*x + c)^2 + 2*(a*d*f^3*x^2 + 2*a*d*e*f^2*x + a*d*e^2*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) + (a*d*f^2*x^2 + 2
*a*d*e*f*x + a*d*e^2 + (a*d*f^2*x^2 + 2*a*d*e*f*x + a*d*e^2)*cos(d*x + c)^2 + (a*d*f^2*x^2 + 2*a*d*e*f*x + a*d
*e^2)*sin(d*x + c)^2 + 2*(a*d*f^2*x^2 + 2*a*d*e*f*x + a*d*e^2)*sin(d*x + c))*integrate(sin(d*x + c)/(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)*cos(d*x + c)^2 + (a*f^2*x^2 + 2*a*e*f*x + a*e^2)*sin(d*
x + c)^2 + 2*(a*f^2*x^2 + 2*a*e*f*x + a*e^2)*cos(d*x + c)), x) + (a*d*f^2*x^2 + 2*a*d*e*f*x + a*d*e^2 + (a*d*f
^2*x^2 + 2*a*d*e*f*x + a*d*e^2)*cos(d*x + c)^2 + (a*d*f^2*x^2 + 2*a*d*e*f*x + a*d*e^2)*sin(d*x + c)^2 + 2*(a*d
*f^2*x^2 + 2*a*d*e*f*x + a*d*e^2)*sin(d*x + c))*integrate(sin(d*x + c)/(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)*cos(d*x + c)^2 + (a*f^2*x^2 + 2*a*e*f*x + a*e^2)*sin(d*x + c)^2 - 2*(a*f^2*x^2 + 2*a
*e*f*x + a*e^2)*cos(d*x + c)), x) + 2*cos(d*x + c))/(a*d*f^2*x^2 + 2*a*d*e*f*x + a*d*e^2 + (a*d*f^2*x^2 + 2*a*
d*e*f*x + a*d*e^2)*cos(d*x + c)^2 + (a*d*f^2*x^2 + 2*a*d*e*f*x + a*d*e^2)*sin(d*x + c)^2 + 2*(a*d*f^2*x^2 + 2*
a*d*e*f*x + a*d*e^2)*sin(d*x + c))

Giac [F(-1)]

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

[In]

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

[Out]

Timed out

Mupad [N/A]

Not integrable

Time = 1.44 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.15 \[ \int \frac {\csc (c+d x)}{(e+f x)^2 (a+a \sin (c+d x))} \, dx=\int \frac {1}{\sin \left (c+d\,x\right )\,{\left (e+f\,x\right )}^2\,\left (a+a\,\sin \left (c+d\,x\right )\right )} \,d x \]

[In]

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

[Out]

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