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

Optimal. Leaf size=87 \[ \text{Unintegrable}\left (\frac{\csc (a+b x)}{(c+d x)^2},x\right )-\frac{b \cos \left (a-\frac{b c}{d}\right ) \text{CosIntegral}\left (\frac{b c}{d}+b x\right )}{d^2}+\frac{b \sin \left (a-\frac{b c}{d}\right ) \text{Si}\left (\frac{b c}{d}+b x\right )}{d^2}+\frac{\sin (a+b x)}{d (c+d x)} \]

[Out]

-((b*Cos[a - (b*c)/d]*CosIntegral[(b*c)/d + b*x])/d^2) + Sin[a + b*x]/(d*(c + d*x)) + (b*Sin[a - (b*c)/d]*SinI
ntegral[(b*c)/d + b*x])/d^2 + Unintegrable[Csc[a + b*x]/(c + d*x)^2, x]

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Rubi [A]  time = 0.130614, 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 = {} \[ \int \frac{\cos (a+b x) \cot (a+b x)}{(c+d x)^2} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

Mathematica [A]  time = 4.12491, size = 0, normalized size = 0. \[ \int \frac{\cos (a+b x) \cot (a+b x)}{(c+d x)^2} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

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Maple [A]  time = 0.63, size = 0, normalized size = 0. \begin{align*} \int{\frac{\cos \left ( bx+a \right ) \cot \left ( bx+a \right ) }{ \left ( dx+c \right ) ^{2}}}\, 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)^2,x)

[Out]

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

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Maxima [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \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)^2,x, algorithm="maxima")

[Out]

Timed out

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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^{2} x^{2} + 2 \, c d x + c^{2}}, 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)^2,x, algorithm="fricas")

[Out]

integral(cos(b*x + a)*cot(b*x + a)/(d^2*x^2 + 2*c*d*x + c^2), 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 )}}{\left (c + d x\right )^{2}}\, 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)**2,x)

[Out]

Integral(cos(a + b*x)*cot(a + b*x)/(c + d*x)**2, 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 )}{{\left (d x + c\right )}^{2}}\,{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)^2,x, algorithm="giac")

[Out]

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