\(\int \frac {\cot ^2(a+b x) \csc (a+b x)}{(c+d x)^2} \, dx\) [117]

Optimal result

Integrand size = 22, antiderivative size = 22 \[ \int \frac {\cot ^2(a+b x) \csc (a+b x)}{(c+d x)^2} \, dx=-\text {Int}\left (\frac {\csc (a+b x)}{(c+d x)^2},x\right )+\text {Int}\left (\frac {\csc ^3(a+b x)}{(c+d x)^2},x\right ) \]

[Out]

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

Rubi [N/A]

Not integrable

Time = 0.10 (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 = {} \[ \int \frac {\cot ^2(a+b x) \csc (a+b x)}{(c+d x)^2} \, dx=\int \frac {\cot ^2(a+b x) \csc (a+b x)}{(c+d x)^2} \, dx \]

[In]

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

[Out]

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

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

Mathematica [N/A]

Not integrable

Time = 45.77 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.09 \[ \int \frac {\cot ^2(a+b x) \csc (a+b x)}{(c+d x)^2} \, dx=\int \frac {\cot ^2(a+b x) \csc (a+b x)}{(c+d x)^2} \, dx \]

[In]

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

[Out]

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

Maple [N/A] (verified)

Not integrable

Time = 0.50 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.00

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

[In]

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

[Out]

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

Fricas [N/A]

Not integrable

Time = 0.26 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.59 \[ \int \frac {\cot ^2(a+b x) \csc (a+b x)}{(c+d x)^2} \, dx=\int { \frac {\cot \left (b x + a\right )^{2} \csc \left (b x + a\right )}{{\left (d x + c\right )}^{2}} \,d x } \]

[In]

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

[Out]

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

Sympy [N/A]

Not integrable

Time = 3.86 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.00 \[ \int \frac {\cot ^2(a+b x) \csc (a+b x)}{(c+d x)^2} \, dx=\int \frac {\cot ^{2}{\left (a + b x \right )} \csc {\left (a + b x \right )}}{\left (c + d x\right )^{2}}\, dx \]

[In]

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

[Out]

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

Maxima [N/A]

Not integrable

Time = 9.88 (sec) , antiderivative size = 2289, normalized size of antiderivative = 104.05 \[ \int \frac {\cot ^2(a+b x) \csc (a+b x)}{(c+d x)^2} \, dx=\int { \frac {\cot \left (b x + a\right )^{2} \csc \left (b x + a\right )}{{\left (d x + c\right )}^{2}} \,d x } \]

[In]

integrate(cot(b*x+a)^2*csc(b*x+a)/(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(-2)]

Exception generated. \[ \int \frac {\cot ^2(a+b x) \csc (a+b x)}{(c+d x)^2} \, dx=\text {Exception raised: AttributeError} \]

[In]

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

[Out]

Exception raised: AttributeError >> type

Mupad [N/A]

Not integrable

Time = 24.38 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.18 \[ \int \frac {\cot ^2(a+b x) \csc (a+b x)}{(c+d x)^2} \, dx=\int \frac {{\mathrm {cot}\left (a+b\,x\right )}^2}{\sin \left (a+b\,x\right )\,{\left (c+d\,x\right )}^2} \,d x \]

[In]

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

[Out]

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