\(\int \frac {\cos (a+b x) \cot (a+b x)}{c+d x} \, dx\) [102]

Optimal result

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

[Out]

-cos(a-b*c/d)*Si(b*c/d+b*x)/d-Ci(b*c/d+b*x)*sin(a-b*c/d)/d+Unintegrable(csc(b*x+a)/(d*x+c),x)

Rubi [N/A]

Not integrable

Time = 0.14 (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 = {} \[ \int \frac {\cos (a+b x) \cot (a+b x)}{c+d x} \, dx=\int \frac {\cos (a+b x) \cot (a+b x)}{c+d x} \, dx \]

[In]

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

[Out]

-((CosIntegral[(b*c)/d + b*x]*Sin[a - (b*c)/d])/d) - (Cos[a - (b*c)/d]*SinIntegral[(b*c)/d + b*x])/d + Defer[I
nt][Csc[a + b*x]/(c + d*x), x]

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

Mathematica [N/A]

Not integrable

Time = 9.75 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.10 \[ \int \frac {\cos (a+b x) \cot (a+b x)}{c+d x} \, dx=\int \frac {\cos (a+b x) \cot (a+b x)}{c+d x} \, dx \]

[In]

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

[Out]

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

Maple [N/A] (verified)

Not integrable

Time = 0.78 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.00

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

[In]

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

[Out]

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

Fricas [N/A]

Not integrable

Time = 0.25 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.10 \[ \int \frac {\cos (a+b x) \cot (a+b x)}{c+d x} \, dx=\int { \frac {\cos \left (b x + a\right ) \cot \left (b x + a\right )}{d x + c} \,d x } \]

[In]

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

[Out]

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

Sympy [N/A]

Not integrable

Time = 0.46 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.95 \[ \int \frac {\cos (a+b x) \cot (a+b x)}{c+d x} \, dx=\int \frac {\cos {\left (a + b x \right )} \cot {\left (a + b x \right )}}{c + d x}\, dx \]

[In]

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

[Out]

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

Maxima [N/A]

Not integrable

Time = 0.49 (sec) , antiderivative size = 228, normalized size of antiderivative = 11.40 \[ \int \frac {\cos (a+b x) \cot (a+b x)}{c+d x} \, dx=\int { \frac {\cos \left (b x + a\right ) \cot \left (b x + a\right )}{d x + c} \,d x } \]

[In]

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

[Out]

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

Giac [N/A]

Not integrable

Time = 0.52 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.10 \[ \int \frac {\cos (a+b x) \cot (a+b x)}{c+d x} \, dx=\int { \frac {\cos \left (b x + a\right ) \cot \left (b x + a\right )}{d x + c} \,d x } \]

[In]

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

[Out]

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

Mupad [N/A]

Not integrable

Time = 23.17 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.10 \[ \int \frac {\cos (a+b x) \cot (a+b x)}{c+d x} \, dx=\int \frac {\cos \left (a+b\,x\right )\,\mathrm {cot}\left (a+b\,x\right )}{c+d\,x} \,d x \]

[In]

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

[Out]

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