\(\int \frac {\cos (a+b x) \cot ^2(a+b x)}{(c+d x)^2} \, dx\) [176]

Optimal result

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

[Out]

CannotIntegrate(cot(b*x+a)*csc(b*x+a)/(d*x+c)^2,x)+cos(b*x+a)/d/(d*x+c)+b*cos(a-b*c/d)*Si(b*c/d+b*x)/d^2+b*Ci(
b*c/d+b*x)*sin(a-b*c/d)/d^2

Rubi [N/A]

Not integrable

Time = 0.27 (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 = {} \[ \int \frac {\cos (a+b x) \cot ^2(a+b x)}{(c+d x)^2} \, dx=\int \frac {\cos (a+b x) \cot ^2(a+b x)}{(c+d x)^2} \, dx \]

[In]

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

[Out]

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

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

Mathematica [N/A]

Not integrable

Time = 4.05 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.09 \[ \int \frac {\cos (a+b x) \cot ^2(a+b x)}{(c+d x)^2} \, dx=\int \frac {\cos (a+b x) \cot ^2(a+b x)}{(c+d x)^2} \, dx \]

[In]

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

[Out]

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

Maple [N/A] (verified)

Not integrable

Time = 0.44 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.00

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

[In]

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

[Out]

int(cos(b*x+a)*cot(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 = 1.59 \[ \int \frac {\cos (a+b x) \cot ^2(a+b x)}{(c+d x)^2} \, dx=\int { \frac {\cos \left (b x + a\right ) \cot \left (b x + a\right )^{2}}{{\left (d x + c\right )}^{2}} \,d x } \]

[In]

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

[Out]

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

Sympy [N/A]

Not integrable

Time = 0.96 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.00 \[ \int \frac {\cos (a+b x) \cot ^2(a+b x)}{(c+d x)^2} \, dx=\int \frac {\cos {\left (a + b x \right )} \cot ^{2}{\left (a + b x \right )}}{\left (c + d x\right )^{2}}\, dx \]

[In]

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

[Out]

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

Maxima [N/A]

Not integrable

Time = 1.43 (sec) , antiderivative size = 1678, normalized size of antiderivative = 76.27 \[ \int \frac {\cos (a+b x) \cot ^2(a+b x)}{(c+d x)^2} \, dx=\int { \frac {\cos \left (b x + a\right ) \cot \left (b x + a\right )^{2}}{{\left (d x + c\right )}^{2}} \,d x } \]

[In]

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

[Out]

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

Giac [N/A]

Not integrable

Time = 6.95 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.09 \[ \int \frac {\cos (a+b x) \cot ^2(a+b x)}{(c+d x)^2} \, dx=\int { \frac {\cos \left (b x + a\right ) \cot \left (b x + a\right )^{2}}{{\left (d x + c\right )}^{2}} \,d x } \]

[In]

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

[Out]

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

Mupad [N/A]

Not integrable

Time = 25.79 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.09 \[ \int \frac {\cos (a+b x) \cot ^2(a+b x)}{(c+d x)^2} \, dx=\int \frac {\cos \left (a+b\,x\right )\,{\mathrm {cot}\left (a+b\,x\right )}^2}{{\left (c+d\,x\right )}^2} \,d x \]

[In]

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

[Out]

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