\(\int \frac {\cot (a+b x) \csc ^2(a+b x)}{(c+d x)^2} \, dx\) [51]

Optimal result

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

[Out]

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

Rubi [N/A]

Not integrable

Time = 0.18 (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 = {} \[ \int \frac {\cot (a+b x) \csc ^2(a+b x)}{(c+d x)^2} \, dx=\int \frac {\cot (a+b x) \csc ^2(a+b x)}{(c+d x)^2} \, dx \]

[In]

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

[Out]

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

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

Mathematica [N/A]

Not integrable

Time = 12.40 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.09 \[ \int \frac {\cot (a+b x) \csc ^2(a+b x)}{(c+d x)^2} \, dx=\int \frac {\cot (a+b x) \csc ^2(a+b x)}{(c+d x)^2} \, dx \]

[In]

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

[Out]

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

Maple [N/A] (verified)

Not integrable

Time = 0.56 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.00

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

[In]

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

[Out]

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

Fricas [N/A]

Not integrable

Time = 0.26 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.59 \[ \int \frac {\cot (a+b x) \csc ^2(a+b x)}{(c+d x)^2} \, dx=\int { \frac {\cos \left (b x + a\right ) \csc \left (b x + a\right )^{3}}{{\left (d x + c\right )}^{2}} \,d x } \]

[In]

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

[Out]

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

Sympy [N/A]

Not integrable

Time = 3.14 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.00 \[ \int \frac {\cot (a+b x) \csc ^2(a+b x)}{(c+d x)^2} \, dx=\int \frac {\cos {\left (a + b x \right )} \csc ^{3}{\left (a + b x \right )}}{\left (c + d x\right )^{2}}\, dx \]

[In]

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

[Out]

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

Maxima [N/A]

Not integrable

Time = 2.66 (sec) , antiderivative size = 2153, normalized size of antiderivative = 97.86 \[ \int \frac {\cot (a+b x) \csc ^2(a+b x)}{(c+d x)^2} \, dx=\int { \frac {\cos \left (b x + a\right ) \csc \left (b x + a\right )^{3}}{{\left (d x + c\right )}^{2}} \,d x } \]

[In]

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

[Out]

-(4*(b*d*x + b*c)*cos(2*b*x + 2*a)^2 + 4*(b*d*x + b*c)*sin(2*b*x + 2*a)^2 - 2*((b*d*x + b*c)*cos(2*b*x + 2*a)
- d*sin(2*b*x + 2*a))*cos(4*b*x + 4*a) - 2*(b*d*x + b*c)*cos(2*b*x + 2*a) + 3*(b^2*d^5*x^3 + 3*b^2*c*d^4*x^2 +
3*b^2*c^2*d^3*x + b^2*c^3*d^2 + (b^2*d^5*x^3 + 3*b^2*c*d^4*x^2 + 3*b^2*c^2*d^3*x + b^2*c^3*d^2)*cos(4*b*x + 4
*a)^2 + 4*(b^2*d^5*x^3 + 3*b^2*c*d^4*x^2 + 3*b^2*c^2*d^3*x + b^2*c^3*d^2)*cos(2*b*x + 2*a)^2 + (b^2*d^5*x^3 +
3*b^2*c*d^4*x^2 + 3*b^2*c^2*d^3*x + b^2*c^3*d^2)*sin(4*b*x + 4*a)^2 - 4*(b^2*d^5*x^3 + 3*b^2*c*d^4*x^2 + 3*b^2
*c^2*d^3*x + b^2*c^3*d^2)*sin(4*b*x + 4*a)*sin(2*b*x + 2*a) + 4*(b^2*d^5*x^3 + 3*b^2*c*d^4*x^2 + 3*b^2*c^2*d^3
*x + b^2*c^3*d^2)*sin(2*b*x + 2*a)^2 + 2*(b^2*d^5*x^3 + 3*b^2*c*d^4*x^2 + 3*b^2*c^2*d^3*x + b^2*c^3*d^2 - 2*(b
^2*d^5*x^3 + 3*b^2*c*d^4*x^2 + 3*b^2*c^2*d^3*x + b^2*c^3*d^2)*cos(2*b*x + 2*a))*cos(4*b*x + 4*a) - 4*(b^2*d^5*
x^3 + 3*b^2*c*d^4*x^2 + 3*b^2*c^2*d^3*x + b^2*c^3*d^2)*cos(2*b*x + 2*a))*integrate(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) - 3*(b^2*d^5*x^3 + 3*b^2*c*d^4*x^2 + 3*b^2*c^2*d^3*x + b^2*c^3*d^2 + (b^2*d^5*x^3 + 3*b
^2*c*d^4*x^2 + 3*b^2*c^2*d^3*x + b^2*c^3*d^2)*cos(4*b*x + 4*a)^2 + 4*(b^2*d^5*x^3 + 3*b^2*c*d^4*x^2 + 3*b^2*c^
2*d^3*x + b^2*c^3*d^2)*cos(2*b*x + 2*a)^2 + (b^2*d^5*x^3 + 3*b^2*c*d^4*x^2 + 3*b^2*c^2*d^3*x + b^2*c^3*d^2)*si
n(4*b*x + 4*a)^2 - 4*(b^2*d^5*x^3 + 3*b^2*c*d^4*x^2 + 3*b^2*c^2*d^3*x + b^2*c^3*d^2)*sin(4*b*x + 4*a)*sin(2*b*
x + 2*a) + 4*(b^2*d^5*x^3 + 3*b^2*c*d^4*x^2 + 3*b^2*c^2*d^3*x + b^2*c^3*d^2)*sin(2*b*x + 2*a)^2 + 2*(b^2*d^5*x
^3 + 3*b^2*c*d^4*x^2 + 3*b^2*c^2*d^3*x + b^2*c^3*d^2 - 2*(b^2*d^5*x^3 + 3*b^2*c*d^4*x^2 + 3*b^2*c^2*d^3*x + b^
2*c^3*d^2)*cos(2*b*x + 2*a))*cos(4*b*x + 4*a) - 4*(b^2*d^5*x^3 + 3*b^2*c*d^4*x^2 + 3*b^2*c^2*d^3*x + b^2*c^3*d
^2)*cos(2*b*x + 2*a))*integrate(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(2*b*x + 2*a) + (b
*d*x + b*c)*sin(2*b*x + 2*a) - d)*sin(4*b*x + 4*a) - 2*d*sin(2*b*x + 2*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)*co
s(2*b*x + 2*a))

Giac [F(-1)]

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

[In]

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

[Out]

Timed out

Mupad [N/A]

Not integrable

Time = 23.27 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.09 \[ \int \frac {\cot (a+b x) \csc ^2(a+b x)}{(c+d x)^2} \, dx=\int \frac {\cos \left (a+b\,x\right )}{{\sin \left (a+b\,x\right )}^3\,{\left (c+d\,x\right )}^2} \,d x \]

[In]

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

[Out]

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