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

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

[Out]

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

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Rubi [A]  time = 0.05, 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 = {} \[ \int \frac {(a+b \cot (e+f x))^2}{(c+d x)^2} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

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Mathematica [A]  time = 16.28, size = 0, normalized size = 0.00 \[ \int \frac {(a+b \cot (e+f x))^2}{(c+d x)^2} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

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fricas [A]  time = 0.46, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {b^{2} \cot \left (f x + e\right )^{2} + 2 \, a b \cot \left (f x + e\right ) + a^{2}}{d^{2} x^{2} + 2 \, c d x + c^{2}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [A]  time = 7.24, size = 0, normalized size = 0.00 \[ \int \frac {\left (a +b \cot \left (f x +e \right )\right )^{2}}{\left (d x +c \right )^{2}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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