\(\int \frac {\cos (a+b x) \cot (a+b x)}{(c+d x)^2} \, dx\) [103]

Optimal result

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

[Out]

-b*Ci(b*c/d+b*x)*cos(a-b*c/d)/d^2+b*Si(b*c/d+b*x)*sin(a-b*c/d)/d^2+sin(b*x+a)/d/(d*x+c)+Unintegrable(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 = {} \[ \int \frac {\cos (a+b x) \cot (a+b x)}{(c+d x)^2} \, dx=\int \frac {\cos (a+b x) \cot (a+b x)}{(c+d x)^2} \, dx \]

[In]

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

[Out]

-((b*Cos[a - (b*c)/d]*CosIntegral[(b*c)/d + b*x])/d^2) + Sin[a + b*x]/(d*(c + d*x)) + (b*Sin[a - (b*c)/d]*SinI
ntegral[(b*c)/d + b*x])/d^2 + Defer[Int][Csc[a + b*x]/(c + d*x)^2, x]

Rubi steps \begin{align*} \text {integral}& = \int \frac {\csc (a+b x)}{(c+d x)^2} \, dx-\int \frac {\sin (a+b x)}{(c+d x)^2} \, dx \\ & = \frac {\sin (a+b x)}{d (c+d x)}-\frac {b \int \frac {\cos (a+b x)}{c+d x} \, dx}{d}+\int \frac {\csc (a+b x)}{(c+d x)^2} \, dx \\ & = \frac {\sin (a+b x)}{d (c+d x)}-\frac {\left (b \cos \left (a-\frac {b c}{d}\right )\right ) \int \frac {\cos \left (\frac {b c}{d}+b x\right )}{c+d x} \, dx}{d}+\frac {\left (b \sin \left (a-\frac {b c}{d}\right )\right ) \int \frac {\sin \left (\frac {b c}{d}+b x\right )}{c+d x} \, dx}{d}+\int \frac {\csc (a+b x)}{(c+d x)^2} \, dx \\ & = -\frac {b \cos \left (a-\frac {b c}{d}\right ) \operatorname {CosIntegral}\left (\frac {b c}{d}+b x\right )}{d^2}+\frac {\sin (a+b x)}{d (c+d x)}+\frac {b \sin \left (a-\frac {b c}{d}\right ) \text {Si}\left (\frac {b c}{d}+b x\right )}{d^2}+\int \frac {\csc (a+b x)}{(c+d x)^2} \, dx \\ \end{align*}

Mathematica [N/A]

Not integrable

Time = 4.66 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.10 \[ \int \frac {\cos (a+b x) \cot (a+b x)}{(c+d x)^2} \, dx=\int \frac {\cos (a+b x) \cot (a+b x)}{(c+d x)^2} \, dx \]

[In]

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

[Out]

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

Maple [N/A] (verified)

Not integrable

Time = 0.83 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.00

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

[In]

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

[Out]

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

Fricas [N/A]

Not integrable

Time = 0.25 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.65 \[ \int \frac {\cos (a+b x) \cot (a+b x)}{(c+d x)^2} \, dx=\int { \frac {\cos \left (b x + a\right ) \cot \left (b x + a\right )}{{\left (d x + c\right )}^{2}} \,d x } \]

[In]

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

[Out]

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

Sympy [N/A]

Not integrable

Time = 0.71 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.00 \[ \int \frac {\cos (a+b x) \cot (a+b x)}{(c+d x)^2} \, dx=\int \frac {\cos {\left (a + b x \right )} \cot {\left (a + b x \right )}}{\left (c + d x\right )^{2}}\, dx \]

[In]

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

[Out]

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

Maxima [N/A]

Not integrable

Time = 0.89 (sec) , antiderivative size = 340, normalized size of antiderivative = 17.00 \[ \int \frac {\cos (a+b x) \cot (a+b x)}{(c+d x)^2} \, dx=\int { \frac {\cos \left (b x + a\right ) \cot \left (b x + a\right )}{{\left (d x + c\right )}^{2}} \,d x } \]

[In]

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

[Out]

1/2*((I*exp_integral_e(2, (I*b*d*x + I*b*c)/d) - I*exp_integral_e(2, -(I*b*d*x + I*b*c)/d))*cos(-(b*c - a*d)/d
) + 2*(d^2*x + c*d)*integrate(sin(b*x + a)/(d^2*x^2 + 2*c*d*x + (d^2*x^2 + 2*c*d*x + c^2)*cos(b*x + a)^2 + (d^
2*x^2 + 2*c*d*x + c^2)*sin(b*x + a)^2 + c^2 + 2*(d^2*x^2 + 2*c*d*x + c^2)*cos(b*x + a)), x) + 2*(d^2*x + c*d)*
integrate(sin(b*x + a)/(d^2*x^2 + 2*c*d*x + (d^2*x^2 + 2*c*d*x + c^2)*cos(b*x + a)^2 + (d^2*x^2 + 2*c*d*x + c^
2)*sin(b*x + a)^2 + c^2 - 2*(d^2*x^2 + 2*c*d*x + c^2)*cos(b*x + a)), x) + (exp_integral_e(2, (I*b*d*x + I*b*c)
/d) + exp_integral_e(2, -(I*b*d*x + I*b*c)/d))*sin(-(b*c - a*d)/d))/(d^2*x + c*d)

Giac [N/A]

Not integrable

Time = 2.96 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.10 \[ \int \frac {\cos (a+b x) \cot (a+b x)}{(c+d x)^2} \, dx=\int { \frac {\cos \left (b x + a\right ) \cot \left (b x + a\right )}{{\left (d x + c\right )}^{2}} \,d x } \]

[In]

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

[Out]

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

Mupad [N/A]

Not integrable

Time = 22.99 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.10 \[ \int \frac {\cos (a+b x) \cot (a+b x)}{(c+d x)^2} \, dx=\int \frac {\cos \left (a+b\,x\right )\,\mathrm {cot}\left (a+b\,x\right )}{{\left (c+d\,x\right )}^2} \,d x \]

[In]

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

[Out]

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