\(\int \frac {\cot (a+b x) \csc (a+b x)}{(c+d x)^2} \, dx\) [44]

Optimal result

Integrand size = 20, antiderivative size = 20 \[ \int \frac {\cot (a+b x) \csc (a+b x)}{(c+d x)^2} \, dx=\text {Int}\left (\frac {\cot (a+b x) \csc (a+b x)}{(c+d x)^2},x\right ) \]

[Out]

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

Rubi [N/A]

Not integrable

Time = 0.15 (sec) , antiderivative size = 20, 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 = {} \[ \int \frac {\cot (a+b x) \csc (a+b x)}{(c+d x)^2} \, dx=\int \frac {\cot (a+b x) \csc (a+b x)}{(c+d x)^2} \, dx \]

[In]

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

[Out]

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

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

Mathematica [N/A]

Not integrable

Time = 22.94 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.10 \[ \int \frac {\cot (a+b x) \csc (a+b x)}{(c+d x)^2} \, dx=\int \frac {\cot (a+b x) \csc (a+b x)}{(c+d x)^2} \, dx \]

[In]

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

[Out]

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

Maple [N/A] (verified)

Not integrable

Time = 0.54 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.10

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

[In]

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

[Out]

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

Fricas [N/A]

Not integrable

Time = 0.25 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.75 \[ \int \frac {\cot (a+b x) \csc (a+b x)}{(c+d x)^2} \, dx=\int { \frac {\cos \left (b x + a\right ) \csc \left (b x + a\right )^{2}}{{\left (d x + c\right )}^{2}} \,d x } \]

[In]

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

[Out]

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

Sympy [N/A]

Not integrable

Time = 1.37 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.10 \[ \int \frac {\cot (a+b x) \csc (a+b x)}{(c+d x)^2} \, dx=\int \frac {\cos {\left (a + b x \right )} \csc ^{2}{\left (a + b x \right )}}{\left (c + d x\right )^{2}}\, dx \]

[In]

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

[Out]

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

Maxima [N/A]

Not integrable

Time = 1.28 (sec) , antiderivative size = 745, normalized size of antiderivative = 37.25 \[ \int \frac {\cot (a+b x) \csc (a+b x)}{(c+d x)^2} \, dx=\int { \frac {\cos \left (b x + a\right ) \csc \left (b x + a\right )^{2}}{{\left (d x + c\right )}^{2}} \,d x } \]

[In]

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

[Out]

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

Giac [N/A]

Not integrable

Time = 2.88 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.20 \[ \int \frac {\cot (a+b x) \csc (a+b x)}{(c+d x)^2} \, dx=\int { \frac {\cos \left (b x + a\right ) \csc \left (b x + a\right )^{2}}{{\left (d x + c\right )}^{2}} \,d x } \]

[In]

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

[Out]

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

Mupad [N/A]

Not integrable

Time = 23.87 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.20 \[ \int \frac {\cot (a+b x) \csc (a+b x)}{(c+d x)^2} \, dx=\int \frac {\cos \left (a+b\,x\right )}{{\sin \left (a+b\,x\right )}^2\,{\left (c+d\,x\right )}^2} \,d x \]

[In]

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

[Out]

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