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

Optimal. Leaf size=21 \[ \text {Int}\left (\frac {a+b \cot (e+f x)}{c+d x},x\right ) \]

[Out]

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

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Rubi [A]  time = 0.03, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \[ \int \frac {a+b \cot (e+f x)}{c+d x} \, dx \]

Verification is Not applicable to the result.

[In]

Int[(a + b*Cot[e + f*x])/(c + d*x),x]

[Out]

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

Rubi steps

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

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Mathematica [A]  time = 2.23, size = 0, normalized size = 0.00 \[ \int \frac {a+b \cot (e+f x)}{c+d x} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

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fricas [A]  time = 0.65, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {b \cot \left (f x + e\right ) + a}{d x + c}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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giac [A]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {b \cot \left (f x + e\right ) + a}{d x + c}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [A]  time = 1.78, size = 0, normalized size = 0.00 \[ \int \frac {a +b \cot \left (f x +e \right )}{d x +c}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 0.00, size = 0, normalized size = 0.00 \[ -\frac {b d \int \frac {\sin \left (f x + e\right )}{{\left (d x + c\right )} {\left (\cos \left (f x + e\right )^{2} + \sin \left (f x + e\right )^{2} + 2 \, \cos \left (f x + e\right ) + 1\right )}}\,{d x} - b d \int \frac {\sin \left (f x + e\right )}{{\left (d x + c\right )} {\left (\cos \left (f x + e\right )^{2} + \sin \left (f x + e\right )^{2} - 2 \, \cos \left (f x + e\right ) + 1\right )}}\,{d x} - a \log \left (d x + c\right )}{d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [A]  time = 0.00, size = -1, normalized size = -0.05 \[ \int \frac {a+b\,\mathrm {cot}\left (e+f\,x\right )}{c+d\,x} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*cot(e + f*x))/(c + d*x),x)

[Out]

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

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sympy [A]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {a + b \cot {\left (e + f x \right )}}{c + d x}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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