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

Optimal. Leaf size=75 \[ \text {Int}\left (\frac {\cot (a+b x) \csc (a+b x)}{c+d x},x\right )-\frac {\cos \left (a-\frac {b c}{d}\right ) \text {Ci}\left (\frac {b c}{d}+b x\right )}{d}+\frac {\sin \left (a-\frac {b c}{d}\right ) \text {Si}\left (\frac {b c}{d}+b x\right )}{d} \]

[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

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Rubi [A]  time = 0.21, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \[ \int \frac {\cos (a+b x) \cot ^2(a+b x)}{c+d x} \, dx \]

Verification is Not applicable to the result.

[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*} \int \frac {\cos (a+b x) \cot ^2(a+b x)}{c+d x} \, dx &=-\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 ) \text {Ci}\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*}

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Mathematica [A]  time = 3.72, size = 0, normalized size = 0.00 \[ \int \frac {\cos (a+b x) \cot ^2(a+b x)}{c+d x} \, dx \]

Verification is Not applicable to the result.

[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]

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fricas [A]  time = 0.44, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\cos \left (b x + a\right ) \cot \left (b x + a\right )^{2}}{d x + c}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[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)

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giac [A]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\cos \left (b x + a\right ) \cot \left (b x + a\right )^{2}}{d x + c}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[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)

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maple [A]  time = 0.27, size = 0, normalized size = 0.00 \[ \int \frac {\cos \left (b x +a \right ) \left (\cot ^{2}\left (b x +a \right )\right )}{d x +c}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[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)

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maxima [A]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[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*(-2*I*exp_integral_e(1, (I*b*d*x + I*b*c)/d) + 2*I*exp_integral_e(1, -(I*b*d*x + I*b*c)/d))*sin(-(b*c - a*d
)/d) + (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*(-2*I*exp_integral_e(1, (I*b*d*x + I*b*c)/d) + 2*I*exp_integral_e(1, -(I*b*d*x + I*b*c)/d))*sin(-
(b*c - a*d)/d))*x - 4*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))*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) - 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))

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mupad [A]  time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {\cos \left (a+b\,x\right )\,{\mathrm {cot}\left (a+b\,x\right )}^2}{c+d\,x} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[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)

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sympy [A]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\cos {\left (a + b x \right )} \cot ^{2}{\left (a + b x \right )}}{c + d x}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[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)

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