3.55 \(\int \frac{1}{(c+d x) (a+b \cot (e+f x))} \, dx\)

Optimal. Leaf size=22 \[ \text{Unintegrable}\left (\frac{1}{(c+d x) (a+b \cot (e+f x))},x\right ) \]

[Out]

Unintegrable[1/((c + d*x)*(a + b*Cot[e + f*x])), x]

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Rubi [A]  time = 0.0638416, 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 = {} \[ \int \frac{1}{(c+d x) (a+b \cot (e+f x))} \, dx \]

Verification is Not applicable to the result.

[In]

Int[1/((c + d*x)*(a + b*Cot[e + f*x])),x]

[Out]

Defer[Int][1/((c + d*x)*(a + b*Cot[e + f*x])), x]

Rubi steps

\begin{align*} \int \frac{1}{(c+d x) (a+b \cot (e+f x))} \, dx &=\int \frac{1}{(c+d x) (a+b \cot (e+f x))} \, dx\\ \end{align*}

Mathematica [A]  time = 1.79597, size = 0, normalized size = 0. \[ \int \frac{1}{(c+d x) (a+b \cot (e+f x))} \, dx \]

Verification is Not applicable to the result.

[In]

Integrate[1/((c + d*x)*(a + b*Cot[e + f*x])),x]

[Out]

Integrate[1/((c + d*x)*(a + b*Cot[e + f*x])), x]

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Maple [A]  time = 0.837, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{ \left ( dx+c \right ) \left ( a+b\cot \left ( fx+e \right ) \right ) }}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(d*x+c)/(a+b*cot(f*x+e)),x)

[Out]

int(1/(d*x+c)/(a+b*cot(f*x+e)),x)

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Maxima [A]  time = 0., size = 0, normalized size = 0. \begin{align*} \frac{-2 \,{\left (a^{2} b + b^{3}\right )} d \int \frac{2 \, a b \cos \left (2 \, f x + 2 \, e\right ) +{\left (a^{2} - b^{2}\right )} \sin \left (2 \, f x + 2 \, e\right )}{{\left (a^{4} + 2 \, a^{2} b^{2} + b^{4}\right )} d x +{\left ({\left (a^{4} + 2 \, a^{2} b^{2} + b^{4}\right )} d x +{\left (a^{4} + 2 \, a^{2} b^{2} + b^{4}\right )} c\right )} \cos \left (2 \, f x + 2 \, e\right )^{2} +{\left ({\left (a^{4} + 2 \, a^{2} b^{2} + b^{4}\right )} d x +{\left (a^{4} + 2 \, a^{2} b^{2} + b^{4}\right )} c\right )} \sin \left (2 \, f x + 2 \, e\right )^{2} +{\left (a^{4} + 2 \, a^{2} b^{2} + b^{4}\right )} c - 2 \,{\left ({\left (a^{4} - b^{4}\right )} d x +{\left (a^{4} - b^{4}\right )} c\right )} \cos \left (2 \, f x + 2 \, e\right ) + 4 \,{\left ({\left (a^{3} b + a b^{3}\right )} d x +{\left (a^{3} b + a b^{3}\right )} c\right )} \sin \left (2 \, f x + 2 \, e\right )}\,{d x} + a \log \left (d x + c\right )}{{\left (a^{2} + b^{2}\right )} d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(d*x+c)/(a+b*cot(f*x+e)),x, algorithm="maxima")

[Out]

(2*(a^2*b + b^3)*d*integrate(-(2*a*b*cos(2*f*x + 2*e) + (a^2 - b^2)*sin(2*f*x + 2*e))/((a^4 + 2*a^2*b^2 + b^4)
*d*x + ((a^4 + 2*a^2*b^2 + b^4)*d*x + (a^4 + 2*a^2*b^2 + b^4)*c)*cos(2*f*x + 2*e)^2 + ((a^4 + 2*a^2*b^2 + b^4)
*d*x + (a^4 + 2*a^2*b^2 + b^4)*c)*sin(2*f*x + 2*e)^2 + (a^4 + 2*a^2*b^2 + b^4)*c - 2*((a^4 - b^4)*d*x + (a^4 -
b^4)*c)*cos(2*f*x + 2*e) + 4*((a^3*b + a*b^3)*d*x + (a^3*b + a*b^3)*c)*sin(2*f*x + 2*e)), x) + a*log(d*x + c)
)/((a^2 + b^2)*d)

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Fricas [A]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{1}{a d x + a c +{\left (b d x + b c\right )} \cot \left (f x + e\right )}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(d*x+c)/(a+b*cot(f*x+e)),x, algorithm="fricas")

[Out]

integral(1/(a*d*x + a*c + (b*d*x + b*c)*cot(f*x + e)), x)

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Sympy [A]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\left (a + b \cot{\left (e + f x \right )}\right ) \left (c + d x\right )}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(d*x+c)/(a+b*cot(f*x+e)),x)

[Out]

Integral(1/((a + b*cot(e + f*x))*(c + d*x)), x)

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Giac [A]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{{\left (d x + c\right )}{\left (b \cot \left (f x + e\right ) + a\right )}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(d*x+c)/(a+b*cot(f*x+e)),x, algorithm="giac")

[Out]

integrate(1/((d*x + c)*(b*cot(f*x + e) + a)), x)