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

Optimal. Leaf size=81 \[ \text{Unintegrable}\left (\frac{\cot (a+b x)}{c+d x},x\right )-\frac{\sin \left (2 a-\frac{2 b c}{d}\right ) \text{CosIntegral}\left (\frac{2 b c}{d}+2 b x\right )}{2 d}-\frac{\cos \left (2 a-\frac{2 b c}{d}\right ) \text{Si}\left (\frac{2 b c}{d}+2 b x\right )}{2 d} \]

[Out]

-(CosIntegral[(2*b*c)/d + 2*b*x]*Sin[2*a - (2*b*c)/d])/(2*d) - (Cos[2*a - (2*b*c)/d]*SinIntegral[(2*b*c)/d + 2
*b*x])/(2*d) + Unintegrable[Cot[a + b*x]/(c + d*x), x]

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Rubi [A]  time = 0.144313, antiderivative size = 0, normalized size of antiderivative = 0., number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0., Rules used = {} \[ \int \frac{\cos ^2(a+b x) \cot (a+b x)}{c+d x} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

Rubi steps

\begin{align*} \int \frac{\cos ^2(a+b x) \cot (a+b x)}{c+d x} \, dx &=\int \frac{\cot (a+b x)}{c+d x} \, dx-\int \frac{\cos (a+b x) \sin (a+b x)}{c+d x} \, dx\\ &=\int \frac{\cot (a+b x)}{c+d x} \, dx-\int \frac{\sin (2 a+2 b x)}{2 (c+d x)} \, dx\\ &=-\left (\frac{1}{2} \int \frac{\sin (2 a+2 b x)}{c+d x} \, dx\right )+\int \frac{\cot (a+b x)}{c+d x} \, dx\\ &=-\left (\frac{1}{2} \cos \left (2 a-\frac{2 b c}{d}\right ) \int \frac{\sin \left (\frac{2 b c}{d}+2 b x\right )}{c+d x} \, dx\right )-\frac{1}{2} \sin \left (2 a-\frac{2 b c}{d}\right ) \int \frac{\cos \left (\frac{2 b c}{d}+2 b x\right )}{c+d x} \, dx+\int \frac{\cot (a+b x)}{c+d x} \, dx\\ &=-\frac{\text{Ci}\left (\frac{2 b c}{d}+2 b x\right ) \sin \left (2 a-\frac{2 b c}{d}\right )}{2 d}-\frac{\cos \left (2 a-\frac{2 b c}{d}\right ) \text{Si}\left (\frac{2 b c}{d}+2 b x\right )}{2 d}+\int \frac{\cot (a+b x)}{c+d x} \, dx\\ \end{align*}

Mathematica [A]  time = 0.794424, size = 0, normalized size = 0. \[ \int \frac{\cos ^2(a+b x) \cot (a+b x)}{c+d x} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

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Maple [A]  time = 0.355, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( \cos \left ( bx+a \right ) \right ) ^{2}\cot \left ( bx+a \right ) }{dx+c}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 0., size = 0, normalized size = 0. \begin{align*} -\frac{{\left (-i \, E_{1}\left (\frac{2 i \, b d x + 2 i \, b c}{d}\right ) + i \, E_{1}\left (-\frac{2 i \, b d x + 2 i \, b c}{d}\right )\right )} \cos \left (-\frac{2 \,{\left (b c - a d\right )}}{d}\right ) + 4 \, d \int \frac{\sin \left (b x + a\right )}{{\left (d x + c\right )}{\left (\cos \left (b x + a\right )^{2} + \sin \left (b x + a\right )^{2} + 2 \, \cos \left (b x + a\right ) + 1\right )}}\,{d x} - 4 \, d \int \frac{\sin \left (b x + a\right )}{{\left (d x + c\right )}{\left (\cos \left (b x + a\right )^{2} + \sin \left (b x + a\right )^{2} - 2 \, \cos \left (b x + a\right ) + 1\right )}}\,{d x} -{\left (E_{1}\left (\frac{2 i \, b d x + 2 i \, b c}{d}\right ) + E_{1}\left (-\frac{2 i \, b d x + 2 i \, b c}{d}\right )\right )} \sin \left (-\frac{2 \,{\left (b c - a d\right )}}{d}\right )}{4 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{\cos \left (b x + a\right )^{2} \cot \left (b x + a\right )}{d x + c}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\cos ^{2}{\left (a + b x \right )} \cot{\left (a + b x \right )}}{c + d x}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\cos \left (b x + a\right )^{2} \cot \left (b x + a\right )}{d x + c}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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