\(\int \frac {\cot (a+b x) \csc ^2(a+b x)}{c+d x} \, dx\) [50]

Optimal result

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

[Out]

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

Rubi [N/A]

Not integrable

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

[In]

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

[Out]

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

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

Mathematica [N/A]

Not integrable

Time = 13.14 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.09 \[ \int \frac {\cot (a+b x) \csc ^2(a+b x)}{c+d x} \, dx=\int \frac {\cot (a+b x) \csc ^2(a+b x)}{c+d x} \, dx \]

[In]

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

[Out]

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

Maple [N/A] (verified)

Not integrable

Time = 0.57 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.00

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

[In]

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

[Out]

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

Fricas [N/A]

Not integrable

Time = 0.25 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.09 \[ \int \frac {\cot (a+b x) \csc ^2(a+b x)}{c+d x} \, dx=\int { \frac {\cos \left (b x + a\right ) \csc \left (b x + a\right )^{3}}{d x + c} \,d x } \]

[In]

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

[Out]

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

Sympy [N/A]

Not integrable

Time = 2.07 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.91 \[ \int \frac {\cot (a+b x) \csc ^2(a+b x)}{c+d x} \, dx=\int \frac {\cos {\left (a + b x \right )} \csc ^{3}{\left (a + b x \right )}}{c + d x}\, dx \]

[In]

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

[Out]

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

Maxima [N/A]

Not integrable

Time = 1.15 (sec) , antiderivative size = 1664, normalized size of antiderivative = 75.64 \[ \int \frac {\cot (a+b x) \csc ^2(a+b x)}{c+d x} \, dx=\int { \frac {\cos \left (b x + a\right ) \csc \left (b x + a\right )^{3}}{d x + c} \,d x } \]

[In]

integrate(cos(b*x+a)*csc(b*x+a)^3/(d*x+c),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) + (b^2*d^4*x^2 + 2*b^2*c*d^3*x + b^2
*c^2*d^2 + (b^2*d^4*x^2 + 2*b^2*c*d^3*x + b^2*c^2*d^2)*cos(4*b*x + 4*a)^2 + 4*(b^2*d^4*x^2 + 2*b^2*c*d^3*x + b
^2*c^2*d^2)*cos(2*b*x + 2*a)^2 + (b^2*d^4*x^2 + 2*b^2*c*d^3*x + b^2*c^2*d^2)*sin(4*b*x + 4*a)^2 - 4*(b^2*d^4*x
^2 + 2*b^2*c*d^3*x + b^2*c^2*d^2)*sin(4*b*x + 4*a)*sin(2*b*x + 2*a) + 4*(b^2*d^4*x^2 + 2*b^2*c*d^3*x + b^2*c^2
*d^2)*sin(2*b*x + 2*a)^2 + 2*(b^2*d^4*x^2 + 2*b^2*c*d^3*x + b^2*c^2*d^2 - 2*(b^2*d^4*x^2 + 2*b^2*c*d^3*x + b^2
*c^2*d^2)*cos(2*b*x + 2*a))*cos(4*b*x + 4*a) - 4*(b^2*d^4*x^2 + 2*b^2*c*d^3*x + b^2*c^2*d^2)*cos(2*b*x + 2*a))
*integrate(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^4*x^2 + 2*
b^2*c*d^3*x + b^2*c^2*d^2 + (b^2*d^4*x^2 + 2*b^2*c*d^3*x + b^2*c^2*d^2)*cos(4*b*x + 4*a)^2 + 4*(b^2*d^4*x^2 +
2*b^2*c*d^3*x + b^2*c^2*d^2)*cos(2*b*x + 2*a)^2 + (b^2*d^4*x^2 + 2*b^2*c*d^3*x + b^2*c^2*d^2)*sin(4*b*x + 4*a)
^2 - 4*(b^2*d^4*x^2 + 2*b^2*c*d^3*x + b^2*c^2*d^2)*sin(4*b*x + 4*a)*sin(2*b*x + 2*a) + 4*(b^2*d^4*x^2 + 2*b^2*
c*d^3*x + b^2*c^2*d^2)*sin(2*b*x + 2*a)^2 + 2*(b^2*d^4*x^2 + 2*b^2*c*d^3*x + b^2*c^2*d^2 - 2*(b^2*d^4*x^2 + 2*
b^2*c*d^3*x + b^2*c^2*d^2)*cos(2*b*x + 2*a))*cos(4*b*x + 4*a) - 4*(b^2*d^4*x^2 + 2*b^2*c*d^3*x + b^2*c^2*d^2)*
cos(2*b*x + 2*a))*integrate(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(2*b*x + 2*a) + 2*(b*d*x + b*c)*sin(2*b*x + 2*a) - d)*sin(4*b*x + 4*a) - d*sin(2*b*x + 2*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 = 0.33 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.09 \[ \int \frac {\cot (a+b x) \csc ^2(a+b x)}{c+d x} \, dx=\int { \frac {\cos \left (b x + a\right ) \csc \left (b x + a\right )^{3}}{d x + c} \,d x } \]

[In]

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

[Out]

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

Mupad [N/A]

Not integrable

Time = 23.32 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.09 \[ \int \frac {\cot (a+b x) \csc ^2(a+b x)}{c+d x} \, dx=\int \frac {\cos \left (a+b\,x\right )}{{\sin \left (a+b\,x\right )}^3\,\left (c+d\,x\right )} \,d x \]

[In]

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

[Out]

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