\(\int \frac {\cos (a+b x) \cot ^2(a+b x)}{c+d x} \, dx\) [175]

Optimal result

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

[Out]

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

Rubi [N/A]

Not integrable

Time = 0.21 (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 = {} \[ \int \frac {\cos (a+b x) \cot ^2(a+b x)}{c+d x} \, dx=\int \frac {\cos (a+b x) \cot ^2(a+b x)}{c+d x} \, dx \]

[In]

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

[Out]

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

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

Mathematica [N/A]

Not integrable

Time = 3.79 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.09 \[ \int \frac {\cos (a+b x) \cot ^2(a+b x)}{c+d x} \, dx=\int \frac {\cos (a+b x) \cot ^2(a+b x)}{c+d x} \, dx \]

[In]

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

[Out]

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

Maple [N/A] (verified)

Not integrable

Time = 0.41 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.00

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

[In]

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

[Out]

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

Fricas [N/A]

Not integrable

Time = 0.26 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.09 \[ \int \frac {\cos (a+b x) \cot ^2(a+b x)}{c+d x} \, dx=\int { \frac {\cos \left (b x + a\right ) \cot \left (b x + a\right )^{2}}{d x + c} \,d x } \]

[In]

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

[Out]

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

Sympy [N/A]

Not integrable

Time = 0.63 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.91 \[ \int \frac {\cos (a+b x) \cot ^2(a+b x)}{c+d x} \, dx=\int \frac {\cos {\left (a + b x \right )} \cot ^{2}{\left (a + b x \right )}}{c + d x}\, dx \]

[In]

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

[Out]

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

Maxima [N/A]

Not integrable

Time = 0.66 (sec) , antiderivative size = 1438, normalized size of antiderivative = 65.36 \[ \int \frac {\cos (a+b x) \cot ^2(a+b x)}{c+d x} \, dx=\int { \frac {\cos \left (b x + a\right ) \cot \left (b x + a\right )^{2}}{d x + c} \,d x } \]

[In]

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

[Out]

1/2*(b*c*(exp_integral_e(1, (I*b*d*x + I*b*c)/d) + exp_integral_e(1, -(I*b*d*x + I*b*c)/d))*cos(-(b*c - a*d)/d
) - b*c*(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))*sin(-(b*c - a*d
)/d) + (b*c*(exp_integral_e(1, (I*b*d*x + I*b*c)/d) + exp_integral_e(1, -(I*b*d*x + I*b*c)/d))*cos(-(b*c - a*d
)/d) - b*c*(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))*sin(-(b*c -
a*d)/d) + (b*d*(exp_integral_e(1, (I*b*d*x + I*b*c)/d) + exp_integral_e(1, -(I*b*d*x + I*b*c)/d))*cos(-(b*c -
a*d)/d) - b*d*(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))*sin(-(b*c
- a*d)/d))*x)*cos(2*b*x + 2*a)^2 - 4*d*cos(b*x + a)*sin(2*b*x + 2*a) + (b*c*(exp_integral_e(1, (I*b*d*x + I*b
*c)/d) + exp_integral_e(1, -(I*b*d*x + I*b*c)/d))*cos(-(b*c - a*d)/d) - b*c*(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))*sin(-(b*c - a*d)/d) + (b*d*(exp_integral_e(1, (I*b*d*x +
I*b*c)/d) + exp_integral_e(1, -(I*b*d*x + I*b*c)/d))*cos(-(b*c - a*d)/d) - b*d*(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))*sin(-(b*c - a*d)/d))*x)*sin(2*b*x + 2*a)^2 + (b*d*(exp
_integral_e(1, (I*b*d*x + I*b*c)/d) + exp_integral_e(1, -(I*b*d*x + I*b*c)/d))*cos(-(b*c - a*d)/d) - b*d*(I*ex
p_integral_e(1, (I*b*d*x + I*b*c)/d) - I*exp_integral_e(1, -(I*b*d*x + I*b*c)/d))*sin(-(b*c - a*d)/d))*x - 2*(
b*c*(exp_integral_e(1, (I*b*d*x + I*b*c)/d) + exp_integral_e(1, -(I*b*d*x + I*b*c)/d))*cos(-(b*c - a*d)/d) - b
*c*(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))*sin(-(b*c - a*d)/d)
+ (b*d*(exp_integral_e(1, (I*b*d*x + I*b*c)/d) + exp_integral_e(1, -(I*b*d*x + I*b*c)/d))*cos(-(b*c - a*d)/d)
- b*d*(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))*sin(-(b*c - a*d)/
d))*x - 2*d*sin(b*x + a))*cos(2*b*x + 2*a) - 2*(b*d^3*x + b*c*d^2 + (b*d^3*x + b*c*d^2)*cos(2*b*x + 2*a)^2 + (
b*d^3*x + b*c*d^2)*sin(2*b*x + 2*a)^2 - 2*(b*d^3*x + b*c*d^2)*cos(2*b*x + 2*a))*integrate(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(b*x + a)^2 + (b*d^2*x^2 + 2*b*c*d*x + b*c^2)*sin
(b*x + a)^2 + 2*(b*d^2*x^2 + 2*b*c*d*x + b*c^2)*cos(b*x + a)), x) - 2*(b*d^3*x + b*c*d^2 + (b*d^3*x + b*c*d^2)
*cos(2*b*x + 2*a)^2 + (b*d^3*x + b*c*d^2)*sin(2*b*x + 2*a)^2 - 2*(b*d^3*x + b*c*d^2)*cos(2*b*x + 2*a))*integra
te(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(b*x + a)^2 + (b*d^2*x^2 +
2*b*c*d*x + b*c^2)*sin(b*x + a)^2 - 2*(b*d^2*x^2 + 2*b*c*d*x + b*c^2)*cos(b*x + a)), x) - 4*d*sin(b*x + a))/(
b*d^2*x + b*c*d + (b*d^2*x + b*c*d)*cos(2*b*x + 2*a)^2 + (b*d^2*x + b*c*d)*sin(2*b*x + 2*a)^2 - 2*(b*d^2*x + b
*c*d)*cos(2*b*x + 2*a))

Giac [N/A]

Not integrable

Time = 1.33 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.09 \[ \int \frac {\cos (a+b x) \cot ^2(a+b x)}{c+d x} \, dx=\int { \frac {\cos \left (b x + a\right ) \cot \left (b x + a\right )^{2}}{d x + c} \,d x } \]

[In]

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

[Out]

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

Mupad [N/A]

Not integrable

Time = 26.13 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.09 \[ \int \frac {\cos (a+b x) \cot ^2(a+b x)}{c+d x} \, dx=\int \frac {\cos \left (a+b\,x\right )\,{\mathrm {cot}\left (a+b\,x\right )}^2}{c+d\,x} \,d x \]

[In]

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

[Out]

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