\(\int \frac {\cot ^2(a+b x) \csc (a+b x)}{c+d x} \, dx\) [116]
Optimal result
Integrand size = 22, antiderivative size = 22 \[
\int \frac {\cot ^2(a+b x) \csc (a+b x)}{c+d x} \, dx=-\text {Int}\left (\frac {\csc (a+b x)}{c+d x},x\right )+\text {Int}\left (\frac {\csc ^3(a+b x)}{c+d x},x\right )
\]
[Out]
-Unintegrable(csc(b*x+a)/(d*x+c),x)+Unintegrable(csc(b*x+a)^3/(d*x+c),x)
Rubi [N/A]
Not integrable
Time = 0.11 (sec) , antiderivative size = 22, 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 = {} \[
\int \frac {\cot ^2(a+b x) \csc (a+b x)}{c+d x} \, dx=\int \frac {\cot ^2(a+b x) \csc (a+b x)}{c+d x} \, dx
\]
[In]
Int[(Cot[a + b*x]^2*Csc[a + b*x])/(c + d*x),x]
[Out]
-Defer[Int][Csc[a + b*x]/(c + d*x), x] + Defer[Int][Csc[a + b*x]^3/(c + d*x), x]
Rubi steps \begin{align*}
\text {integral}& = -\int \frac {\csc (a+b x)}{c+d x} \, dx+\int \frac {\csc ^3(a+b x)}{c+d x} \, dx \\
\end{align*}
Mathematica [N/A]
Not integrable
Time = 40.32 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.09
\[
\int \frac {\cot ^2(a+b x) \csc (a+b x)}{c+d x} \, dx=\int \frac {\cot ^2(a+b x) \csc (a+b x)}{c+d x} \, dx
\]
[In]
Integrate[(Cot[a + b*x]^2*Csc[a + b*x])/(c + d*x),x]
[Out]
Integrate[(Cot[a + b*x]^2*Csc[a + b*x])/(c + d*x), x]
Maple [N/A] (verified)
Not integrable
Time = 0.52 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.00
\[\int \frac {\cot \left (x b +a \right )^{2} \csc \left (x b +a \right )}{d x +c}d x\]
[In]
int(cot(b*x+a)^2*csc(b*x+a)/(d*x+c),x)
[Out]
int(cot(b*x+a)^2*csc(b*x+a)/(d*x+c),x)
Fricas [N/A]
Not integrable
Time = 0.25 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.09
\[
\int \frac {\cot ^2(a+b x) \csc (a+b x)}{c+d x} \, dx=\int { \frac {\cot \left (b x + a\right )^{2} \csc \left (b x + a\right )}{d x + c} \,d x }
\]
[In]
integrate(cot(b*x+a)^2*csc(b*x+a)/(d*x+c),x, algorithm="fricas")
[Out]
integral(cot(b*x + a)^2*csc(b*x + a)/(d*x + c), x)
Sympy [N/A]
Not integrable
Time = 1.03 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.91
\[
\int \frac {\cot ^2(a+b x) \csc (a+b x)}{c+d x} \, dx=\int \frac {\cot ^{2}{\left (a + b x \right )} \csc {\left (a + b x \right )}}{c + d x}\, dx
\]
[In]
integrate(cot(b*x+a)**2*csc(b*x+a)/(d*x+c),x)
[Out]
Integral(cot(a + b*x)**2*csc(a + b*x)/(c + d*x), x)
Maxima [N/A]
Not integrable
Time = 2.82 (sec) , antiderivative size = 1793, normalized size of antiderivative = 81.50
\[
\int \frac {\cot ^2(a+b x) \csc (a+b x)}{c+d x} \, dx=\int { \frac {\cot \left (b x + a\right )^{2} \csc \left (b x + a\right )}{d x + c} \,d x }
\]
[In]
integrate(cot(b*x+a)^2*csc(b*x+a)/(d*x+c),x, algorithm="maxima")
[Out]
(((b*d*x + b*c)*cos(3*b*x + 3*a) + (b*d*x + b*c)*cos(b*x + a) - d*sin(3*b*x + 3*a) + 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) - 2*d*sin(2*b*x + 2*a))*cos(3*b*x + 3*a) - 2*((b*d*x
+ b*c)*cos(b*x + a) + d*sin(b*x + a))*cos(2*b*x + 2*a) + (b*d*x + b*c)*cos(b*x + a) - (b^2*d^2*x^2 + 2*b^2*c*
d*x + b^2*c^2 + (b^2*d^2*x^2 + 2*b^2*c*d*x + b^2*c^2)*cos(4*b*x + 4*a)^2 + 4*(b^2*d^2*x^2 + 2*b^2*c*d*x + b^2*
c^2)*cos(2*b*x + 2*a)^2 + (b^2*d^2*x^2 + 2*b^2*c*d*x + b^2*c^2)*sin(4*b*x + 4*a)^2 - 4*(b^2*d^2*x^2 + 2*b^2*c*
d*x + b^2*c^2)*sin(4*b*x + 4*a)*sin(2*b*x + 2*a) + 4*(b^2*d^2*x^2 + 2*b^2*c*d*x + b^2*c^2)*sin(2*b*x + 2*a)^2
+ 2*(b^2*d^2*x^2 + 2*b^2*c*d*x + b^2*c^2 - 2*(b^2*d^2*x^2 + 2*b^2*c*d*x + b^2*c^2)*cos(2*b*x + 2*a))*cos(4*b*x
+ 4*a) - 4*(b^2*d^2*x^2 + 2*b^2*c*d*x + b^2*c^2)*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 - 2*d^2)*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(b*x + 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(b*x + 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)*cos(b*x + a)), x) - (b^2*d^2*x
^2 + 2*b^2*c*d*x + b^2*c^2 + (b^2*d^2*x^2 + 2*b^2*c*d*x + b^2*c^2)*cos(4*b*x + 4*a)^2 + 4*(b^2*d^2*x^2 + 2*b^2
*c*d*x + b^2*c^2)*cos(2*b*x + 2*a)^2 + (b^2*d^2*x^2 + 2*b^2*c*d*x + b^2*c^2)*sin(4*b*x + 4*a)^2 - 4*(b^2*d^2*x
^2 + 2*b^2*c*d*x + b^2*c^2)*sin(4*b*x + 4*a)*sin(2*b*x + 2*a) + 4*(b^2*d^2*x^2 + 2*b^2*c*d*x + b^2*c^2)*sin(2*
b*x + 2*a)^2 + 2*(b^2*d^2*x^2 + 2*b^2*c*d*x + b^2*c^2 - 2*(b^2*d^2*x^2 + 2*b^2*c*d*x + b^2*c^2)*cos(2*b*x + 2*
a))*cos(4*b*x + 4*a) - 4*(b^2*d^2*x^2 + 2*b^2*c*d*x + b^2*c^2)*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 - 2*d^2)*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(b*x + 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(b*x + 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)*cos(b*x + a)), x)
+ (d*cos(3*b*x + 3*a) - 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*d*cos(2*b*x + 2*a) - 2*(b*d*x + b*c)*sin(2*b*x + 2*a) - d)*sin(3*b*x + 3*a) + 2*(d*cos(b*x + a)
- (b*d*x + b*c)*sin(b*x + a))*sin(2*b*x + 2*a) + d*sin(b*x + a))/(b^2*d^2*x^2 + 2*b^2*c*d*x + b^2*c^2 + (b^2*
d^2*x^2 + 2*b^2*c*d*x + b^2*c^2)*cos(4*b*x + 4*a)^2 + 4*(b^2*d^2*x^2 + 2*b^2*c*d*x + b^2*c^2)*cos(2*b*x + 2*a)
^2 + (b^2*d^2*x^2 + 2*b^2*c*d*x + b^2*c^2)*sin(4*b*x + 4*a)^2 - 4*(b^2*d^2*x^2 + 2*b^2*c*d*x + b^2*c^2)*sin(4*
b*x + 4*a)*sin(2*b*x + 2*a) + 4*(b^2*d^2*x^2 + 2*b^2*c*d*x + b^2*c^2)*sin(2*b*x + 2*a)^2 + 2*(b^2*d^2*x^2 + 2*
b^2*c*d*x + b^2*c^2 - 2*(b^2*d^2*x^2 + 2*b^2*c*d*x + b^2*c^2)*cos(2*b*x + 2*a))*cos(4*b*x + 4*a) - 4*(b^2*d^2*
x^2 + 2*b^2*c*d*x + b^2*c^2)*cos(2*b*x + 2*a))
Giac [N/A]
Not integrable
Time = 4.95 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.09
\[
\int \frac {\cot ^2(a+b x) \csc (a+b x)}{c+d x} \, dx=\int { \frac {\cot \left (b x + a\right )^{2} \csc \left (b x + a\right )}{d x + c} \,d x }
\]
[In]
integrate(cot(b*x+a)^2*csc(b*x+a)/(d*x+c),x, algorithm="giac")
[Out]
integrate(cot(b*x + a)^2*csc(b*x + a)/(d*x + c), x)
Mupad [N/A]
Not integrable
Time = 24.04 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.18
\[
\int \frac {\cot ^2(a+b x) \csc (a+b x)}{c+d x} \, dx=\int \frac {{\mathrm {cot}\left (a+b\,x\right )}^2}{\sin \left (a+b\,x\right )\,\left (c+d\,x\right )} \,d x
\]
[In]
int(cot(a + b*x)^2/(sin(a + b*x)*(c + d*x)),x)
[Out]
int(cot(a + b*x)^2/(sin(a + b*x)*(c + d*x)), x)