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

Optimal. Leaf size=74 \[ \text{CannotIntegrate}\left (\frac{\cot (a+b x) \csc (a+b x)}{c+d x},x\right )-\frac{\cos \left (a-\frac{b c}{d}\right ) \text{CosIntegral}\left (\frac{b c}{d}+b x\right )}{d}+\frac{\sin \left (a-\frac{b c}{d}\right ) \text{Si}\left (\frac{b c}{d}+b x\right )}{d} \]

[Out]

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

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Rubi [A]  time = 0.213677, 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 = {} \[ \int \frac{\cos (a+b x) \cot ^2(a+b x)}{c+d x} \, dx \]

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*}

Mathematica [A]  time = 3.70494, size = 0, normalized size = 0. \[ \int \frac{\cos (a+b x) \cot ^2(a+b x)}{c+d x} \, dx \]

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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Maple [A]  time = 0.346, size = 0, normalized size = 0. \begin{align*} \int{\frac{\cos \left ( bx+a \right ) \left ( \cot \left ( bx+a \right ) \right ) ^{2}}{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)^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 [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)^2/(d*x+c),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 )^{2}}{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)^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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Sympy [A]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\cos{\left (a + b x \right )} \cot ^{2}{\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)**2/(d*x+c),x)

[Out]

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

[Out]

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