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

Optimal. Leaf size=69 \[ \text{Unintegrable}\left (\frac{\csc (a+b x)}{c+d x},x\right )-\frac{\sin \left (a-\frac{b c}{d}\right ) \text{CosIntegral}\left (\frac{b c}{d}+b x\right )}{d}-\frac{\cos \left (a-\frac{b c}{d}\right ) \text{Si}\left (\frac{b c}{d}+b x\right )}{d} \]

[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 + Uninteg
rable[Csc[a + b*x]/(c + d*x), x]

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Rubi [A]  time = 0.106625, 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 = {} \[ \int \frac{\cos (a+b x) \cot (a+b x)}{c+d x} \, dx \]

Verification is Not applicable to the result.

[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*} \int \frac{\cos (a+b x) \cot (a+b x)}{c+d x} \, dx &=\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{\text{Ci}\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 [A]  time = 8.49259, size = 0, normalized size = 0. \[ \int \frac{\cos (a+b x) \cot (a+b x)}{c+d x} \, dx \]

Verification is Not applicable to the result.

[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]

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Maple [A]  time = 0.302, size = 0, normalized size = 0. \begin{align*} \int{\frac{\cos \left ( bx+a \right ) \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)*cot(b*x+a)/(d*x+c),x)

[Out]

int(cos(b*x+a)*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{i \, b d x + i \, b c}{d}\right ) - i \, E_{1}\left (-\frac{i \, b d x + i \, b c}{d}\right )\right )} \cos \left (-\frac{b c - a d}{d}\right ) + 2 \, 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} + 2 \, 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{i \, b d x + i \, b c}{d}\right ) + E_{1}\left (-\frac{i \, b d x + i \, b c}{d}\right )\right )} \sin \left (-\frac{b c - a d}{d}\right )}{2 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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

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

[Out]

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

[Out]

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

[Out]

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