3.56 \(\int \frac {1}{(c+d x)^2 (a+b \cot (e+f x))} \, dx\)

Optimal. Leaf size=23 \[ \text {Int}\left (\frac {1}{(c+d x)^2 (a+b \cot (e+f x))},x\right ) \]

[Out]

Unintegrable(1/(d*x+c)^2/(a+b*cot(f*x+e)),x)

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Rubi [A]  time = 0.06, 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 = {} \[ \int \frac {1}{(c+d x)^2 (a+b \cot (e+f x))} \, dx \]

Verification is Not applicable to the result.

[In]

Int[1/((c + d*x)^2*(a + b*Cot[e + f*x])),x]

[Out]

Defer[Int][1/((c + d*x)^2*(a + b*Cot[e + f*x])), x]

Rubi steps

\begin {align*} \int \frac {1}{(c+d x)^2 (a+b \cot (e+f x))} \, dx &=\int \frac {1}{(c+d x)^2 (a+b \cot (e+f x))} \, dx\\ \end {align*}

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Mathematica [A]  time = 3.94, size = 0, normalized size = 0.00 \[ \int \frac {1}{(c+d x)^2 (a+b \cot (e+f x))} \, dx \]

Verification is Not applicable to the result.

[In]

Integrate[1/((c + d*x)^2*(a + b*Cot[e + f*x])),x]

[Out]

Integrate[1/((c + d*x)^2*(a + b*Cot[e + f*x])), x]

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fricas [A]  time = 0.64, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {1}{a d^{2} x^{2} + 2 \, a c d x + a c^{2} + {\left (b d^{2} x^{2} + 2 \, b c d x + b c^{2}\right )} \cot \left (f x + e\right )}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(d*x+c)^2/(a+b*cot(f*x+e)),x, algorithm="fricas")

[Out]

integral(1/(a*d^2*x^2 + 2*a*c*d*x + a*c^2 + (b*d^2*x^2 + 2*b*c*d*x + b*c^2)*cot(f*x + e)), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(d*x+c)^2/(a+b*cot(f*x+e)),x, algorithm="giac")

[Out]

integrate(1/((d*x + c)^2*(b*cot(f*x + e) + a)), x)

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maple [A]  time = 5.05, size = 0, normalized size = 0.00 \[ \int \frac {1}{\left (d x +c \right )^{2} \left (a +b \cot \left (f x +e \right )\right )}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(d*x+c)^2/(a+b*cot(f*x+e)),x)

[Out]

int(1/(d*x+c)^2/(a+b*cot(f*x+e)),x)

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maxima [A]  time = 0.00, size = 0, normalized size = 0.00 \[ \frac {-2 \, {\left ({\left (a^{2} b + b^{3}\right )} d^{2} x + {\left (a^{2} b + b^{3}\right )} c d\right )} \int \frac {2 \, a b \cos \left (2 \, f x + 2 \, e\right ) + {\left (a^{2} - b^{2}\right )} \sin \left (2 \, f x + 2 \, e\right )}{{\left (a^{4} + 2 \, a^{2} b^{2} + b^{4}\right )} d^{2} x^{2} + 2 \, {\left (a^{4} + 2 \, a^{2} b^{2} + b^{4}\right )} c d x + {\left (a^{4} + 2 \, a^{2} b^{2} + b^{4}\right )} c^{2} + {\left ({\left (a^{4} + 2 \, a^{2} b^{2} + b^{4}\right )} d^{2} x^{2} + 2 \, {\left (a^{4} + 2 \, a^{2} b^{2} + b^{4}\right )} c d x + {\left (a^{4} + 2 \, a^{2} b^{2} + b^{4}\right )} c^{2}\right )} \cos \left (2 \, f x + 2 \, e\right )^{2} + {\left ({\left (a^{4} + 2 \, a^{2} b^{2} + b^{4}\right )} d^{2} x^{2} + 2 \, {\left (a^{4} + 2 \, a^{2} b^{2} + b^{4}\right )} c d x + {\left (a^{4} + 2 \, a^{2} b^{2} + b^{4}\right )} c^{2}\right )} \sin \left (2 \, f x + 2 \, e\right )^{2} - 2 \, {\left ({\left (a^{4} - b^{4}\right )} d^{2} x^{2} + 2 \, {\left (a^{4} - b^{4}\right )} c d x + {\left (a^{4} - b^{4}\right )} c^{2}\right )} \cos \left (2 \, f x + 2 \, e\right ) + 4 \, {\left ({\left (a^{3} b + a b^{3}\right )} d^{2} x^{2} + 2 \, {\left (a^{3} b + a b^{3}\right )} c d x + {\left (a^{3} b + a b^{3}\right )} c^{2}\right )} \sin \left (2 \, f x + 2 \, e\right )}\,{d x} - a}{{\left (a^{2} + b^{2}\right )} d^{2} x + {\left (a^{2} + b^{2}\right )} c d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(d*x+c)^2/(a+b*cot(f*x+e)),x, algorithm="maxima")

[Out]

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

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mupad [A]  time = 0.00, size = -1, normalized size = -0.04 \[ \int \frac {1}{\left (a+b\,\mathrm {cot}\left (e+f\,x\right )\right )\,{\left (c+d\,x\right )}^2} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/((a + b*cot(e + f*x))*(c + d*x)^2),x)

[Out]

int(1/((a + b*cot(e + f*x))*(c + d*x)^2), x)

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sympy [A]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\left (a + b \cot {\left (e + f x \right )}\right ) \left (c + d x\right )^{2}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(d*x+c)**2/(a+b*cot(f*x+e)),x)

[Out]

Integral(1/((a + b*cot(e + f*x))*(c + d*x)**2), x)

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