3.51 \(\int \frac{(a+b \cot (e+f x))^3}{(c+d x)^2} \, dx\)
Optimal. Leaf size=22 \[ \text{Unintegrable}\left (\frac{(a+b \cot (e+f x))^3}{(c+d x)^2},x\right ) \]
[Out]
Unintegrable[(a + b*Cot[e + f*x])^3/(c + d*x)^2, x]
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Rubi [A] time = 0.0510741, 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 = {}
\[ \int \frac{(a+b \cot (e+f x))^3}{(c+d x)^2} \, dx \]
Verification is Not applicable to the result.
[In]
Int[(a + b*Cot[e + f*x])^3/(c + d*x)^2,x]
[Out]
Defer[Int][(a + b*Cot[e + f*x])^3/(c + d*x)^2, x]
Rubi steps
\begin{align*} \int \frac{(a+b \cot (e+f x))^3}{(c+d x)^2} \, dx &=\int \frac{(a+b \cot (e+f x))^3}{(c+d x)^2} \, dx\\ \end{align*}
Mathematica [A] time = 11.8618, size = 0, normalized size = 0. \[ \int \frac{(a+b \cot (e+f x))^3}{(c+d x)^2} \, dx \]
Verification is Not applicable to the result.
[In]
Integrate[(a + b*Cot[e + f*x])^3/(c + d*x)^2,x]
[Out]
Integrate[(a + b*Cot[e + f*x])^3/(c + d*x)^2, x]
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Maple [A] time = 4.519, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( a+b\cot \left ( fx+e \right ) \right ) ^{3}}{ \left ( dx+c \right ) ^{2}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((a+b*cot(f*x+e))^3/(d*x+c)^2,x)
[Out]
int((a+b*cot(f*x+e))^3/(d*x+c)^2,x)
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Maxima [A] time = 0., size = 0, normalized size = 0. \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((a+b*cot(f*x+e))^3/(d*x+c)^2,x, algorithm="maxima")
[Out]
-((a^3 - 3*a*b^2)*d^2*f^2*x^2 + 2*(a^3 - 3*a*b^2)*c*d*f^2*x + (a^3 - 3*a*b^2)*c^2*f^2 + ((a^3 - 3*a*b^2)*d^2*f
^2*x^2 + 2*(a^3 - 3*a*b^2)*c*d*f^2*x + (a^3 - 3*a*b^2)*c^2*f^2)*cos(4*f*x + 4*e)^2 + 4*((a^3 - 3*a*b^2)*d^2*f^
2*x^2 + b^3*c*d*f + (a^3 - 3*a*b^2)*c^2*f^2 + (b^3*d^2*f + 2*(a^3 - 3*a*b^2)*c*d*f^2)*x)*cos(2*f*x + 2*e)^2 +
((a^3 - 3*a*b^2)*d^2*f^2*x^2 + 2*(a^3 - 3*a*b^2)*c*d*f^2*x + (a^3 - 3*a*b^2)*c^2*f^2)*sin(4*f*x + 4*e)^2 + 4*(
(a^3 - 3*a*b^2)*d^2*f^2*x^2 + b^3*c*d*f + (a^3 - 3*a*b^2)*c^2*f^2 + (b^3*d^2*f + 2*(a^3 - 3*a*b^2)*c*d*f^2)*x)
*sin(2*f*x + 2*e)^2 + 2*((a^3 - 3*a*b^2)*d^2*f^2*x^2 + 2*(a^3 - 3*a*b^2)*c*d*f^2*x + (a^3 - 3*a*b^2)*c^2*f^2 -
(2*(a^3 - 3*a*b^2)*d^2*f^2*x^2 + b^3*c*d*f + 2*(a^3 - 3*a*b^2)*c^2*f^2 + (b^3*d^2*f + 4*(a^3 - 3*a*b^2)*c*d*f
^2)*x)*cos(2*f*x + 2*e) - (3*a*b^2*d^2*f*x + 3*a*b^2*c*d*f - b^3*d^2)*sin(2*f*x + 2*e))*cos(4*f*x + 4*e) - 2*(
2*(a^3 - 3*a*b^2)*d^2*f^2*x^2 + b^3*c*d*f + 2*(a^3 - 3*a*b^2)*c^2*f^2 + (b^3*d^2*f + 4*(a^3 - 3*a*b^2)*c*d*f^2
)*x)*cos(2*f*x + 2*e) - (d^4*f^2*x^3 + 3*c*d^3*f^2*x^2 + 3*c^2*d^2*f^2*x + c^3*d*f^2 + (d^4*f^2*x^3 + 3*c*d^3*
f^2*x^2 + 3*c^2*d^2*f^2*x + c^3*d*f^2)*cos(4*f*x + 4*e)^2 + 4*(d^4*f^2*x^3 + 3*c*d^3*f^2*x^2 + 3*c^2*d^2*f^2*x
+ c^3*d*f^2)*cos(2*f*x + 2*e)^2 + (d^4*f^2*x^3 + 3*c*d^3*f^2*x^2 + 3*c^2*d^2*f^2*x + c^3*d*f^2)*sin(4*f*x + 4
*e)^2 - 4*(d^4*f^2*x^3 + 3*c*d^3*f^2*x^2 + 3*c^2*d^2*f^2*x + c^3*d*f^2)*sin(4*f*x + 4*e)*sin(2*f*x + 2*e) + 4*
(d^4*f^2*x^3 + 3*c*d^3*f^2*x^2 + 3*c^2*d^2*f^2*x + c^3*d*f^2)*sin(2*f*x + 2*e)^2 + 2*(d^4*f^2*x^3 + 3*c*d^3*f^
2*x^2 + 3*c^2*d^2*f^2*x + c^3*d*f^2 - 2*(d^4*f^2*x^3 + 3*c*d^3*f^2*x^2 + 3*c^2*d^2*f^2*x + c^3*d*f^2)*cos(2*f*
x + 2*e))*cos(4*f*x + 4*e) - 4*(d^4*f^2*x^3 + 3*c*d^3*f^2*x^2 + 3*c^2*d^2*f^2*x + c^3*d*f^2)*cos(2*f*x + 2*e))
*integrate(-((3*a^2*b - b^3)*d^2*f^2*x^2 - 6*a*b^2*c*d*f + 3*b^3*d^2 + (3*a^2*b - b^3)*c^2*f^2 - 2*(3*a*b^2*d^
2*f - (3*a^2*b - b^3)*c*d*f^2)*x)*sin(f*x + e)/(d^4*f^2*x^4 + 4*c*d^3*f^2*x^3 + 6*c^2*d^2*f^2*x^2 + 4*c^3*d*f^
2*x + c^4*f^2 + (d^4*f^2*x^4 + 4*c*d^3*f^2*x^3 + 6*c^2*d^2*f^2*x^2 + 4*c^3*d*f^2*x + c^4*f^2)*cos(f*x + e)^2 +
(d^4*f^2*x^4 + 4*c*d^3*f^2*x^3 + 6*c^2*d^2*f^2*x^2 + 4*c^3*d*f^2*x + c^4*f^2)*sin(f*x + e)^2 + 2*(d^4*f^2*x^4
+ 4*c*d^3*f^2*x^3 + 6*c^2*d^2*f^2*x^2 + 4*c^3*d*f^2*x + c^4*f^2)*cos(f*x + e)), x) + (d^4*f^2*x^3 + 3*c*d^3*f
^2*x^2 + 3*c^2*d^2*f^2*x + c^3*d*f^2 + (d^4*f^2*x^3 + 3*c*d^3*f^2*x^2 + 3*c^2*d^2*f^2*x + c^3*d*f^2)*cos(4*f*x
+ 4*e)^2 + 4*(d^4*f^2*x^3 + 3*c*d^3*f^2*x^2 + 3*c^2*d^2*f^2*x + c^3*d*f^2)*cos(2*f*x + 2*e)^2 + (d^4*f^2*x^3
+ 3*c*d^3*f^2*x^2 + 3*c^2*d^2*f^2*x + c^3*d*f^2)*sin(4*f*x + 4*e)^2 - 4*(d^4*f^2*x^3 + 3*c*d^3*f^2*x^2 + 3*c^2
*d^2*f^2*x + c^3*d*f^2)*sin(4*f*x + 4*e)*sin(2*f*x + 2*e) + 4*(d^4*f^2*x^3 + 3*c*d^3*f^2*x^2 + 3*c^2*d^2*f^2*x
+ c^3*d*f^2)*sin(2*f*x + 2*e)^2 + 2*(d^4*f^2*x^3 + 3*c*d^3*f^2*x^2 + 3*c^2*d^2*f^2*x + c^3*d*f^2 - 2*(d^4*f^2
*x^3 + 3*c*d^3*f^2*x^2 + 3*c^2*d^2*f^2*x + c^3*d*f^2)*cos(2*f*x + 2*e))*cos(4*f*x + 4*e) - 4*(d^4*f^2*x^3 + 3*
c*d^3*f^2*x^2 + 3*c^2*d^2*f^2*x + c^3*d*f^2)*cos(2*f*x + 2*e))*integrate(-((3*a^2*b - b^3)*d^2*f^2*x^2 - 6*a*b
^2*c*d*f + 3*b^3*d^2 + (3*a^2*b - b^3)*c^2*f^2 - 2*(3*a*b^2*d^2*f - (3*a^2*b - b^3)*c*d*f^2)*x)*sin(f*x + e)/(
d^4*f^2*x^4 + 4*c*d^3*f^2*x^3 + 6*c^2*d^2*f^2*x^2 + 4*c^3*d*f^2*x + c^4*f^2 + (d^4*f^2*x^4 + 4*c*d^3*f^2*x^3 +
6*c^2*d^2*f^2*x^2 + 4*c^3*d*f^2*x + c^4*f^2)*cos(f*x + e)^2 + (d^4*f^2*x^4 + 4*c*d^3*f^2*x^3 + 6*c^2*d^2*f^2*
x^2 + 4*c^3*d*f^2*x + c^4*f^2)*sin(f*x + e)^2 - 2*(d^4*f^2*x^4 + 4*c*d^3*f^2*x^3 + 6*c^2*d^2*f^2*x^2 + 4*c^3*d
*f^2*x + c^4*f^2)*cos(f*x + e)), x) - 2*(3*a*b^2*d^2*f*x + 3*a*b^2*c*d*f - b^3*d^2 - (3*a*b^2*d^2*f*x + 3*a*b^
2*c*d*f - b^3*d^2)*cos(2*f*x + 2*e) + (2*(a^3 - 3*a*b^2)*d^2*f^2*x^2 + b^3*c*d*f + 2*(a^3 - 3*a*b^2)*c^2*f^2 +
(b^3*d^2*f + 4*(a^3 - 3*a*b^2)*c*d*f^2)*x)*sin(2*f*x + 2*e))*sin(4*f*x + 4*e) + 2*(3*a*b^2*d^2*f*x + 3*a*b^2*
c*d*f - b^3*d^2)*sin(2*f*x + 2*e))/(d^4*f^2*x^3 + 3*c*d^3*f^2*x^2 + 3*c^2*d^2*f^2*x + c^3*d*f^2 + (d^4*f^2*x^3
+ 3*c*d^3*f^2*x^2 + 3*c^2*d^2*f^2*x + c^3*d*f^2)*cos(4*f*x + 4*e)^2 + 4*(d^4*f^2*x^3 + 3*c*d^3*f^2*x^2 + 3*c^
2*d^2*f^2*x + c^3*d*f^2)*cos(2*f*x + 2*e)^2 + (d^4*f^2*x^3 + 3*c*d^3*f^2*x^2 + 3*c^2*d^2*f^2*x + c^3*d*f^2)*si
n(4*f*x + 4*e)^2 - 4*(d^4*f^2*x^3 + 3*c*d^3*f^2*x^2 + 3*c^2*d^2*f^2*x + c^3*d*f^2)*sin(4*f*x + 4*e)*sin(2*f*x
+ 2*e) + 4*(d^4*f^2*x^3 + 3*c*d^3*f^2*x^2 + 3*c^2*d^2*f^2*x + c^3*d*f^2)*sin(2*f*x + 2*e)^2 + 2*(d^4*f^2*x^3 +
3*c*d^3*f^2*x^2 + 3*c^2*d^2*f^2*x + c^3*d*f^2 - 2*(d^4*f^2*x^3 + 3*c*d^3*f^2*x^2 + 3*c^2*d^2*f^2*x + c^3*d*f^
2)*cos(2*f*x + 2*e))*cos(4*f*x + 4*e) - 4*(d^4*f^2*x^3 + 3*c*d^3*f^2*x^2 + 3*c^2*d^2*f^2*x + c^3*d*f^2)*cos(2*
f*x + 2*e))
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Fricas [A] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{b^{3} \cot \left (f x + e\right )^{3} + 3 \, a b^{2} \cot \left (f x + e\right )^{2} + 3 \, a^{2} b \cot \left (f x + e\right ) + a^{3}}{d^{2} x^{2} + 2 \, c d x + c^{2}}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((a+b*cot(f*x+e))^3/(d*x+c)^2,x, algorithm="fricas")
[Out]
integral((b^3*cot(f*x + e)^3 + 3*a*b^2*cot(f*x + e)^2 + 3*a^2*b*cot(f*x + e) + a^3)/(d^2*x^2 + 2*c*d*x + c^2),
x)
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Sympy [A] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\left (a + b \cot{\left (e + f x \right )}\right )^{3}}{\left (c + d x\right )^{2}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((a+b*cot(f*x+e))**3/(d*x+c)**2,x)
[Out]
Integral((a + b*cot(e + f*x))**3/(c + d*x)**2, x)
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Giac [A] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (b \cot \left (f x + e\right ) + a\right )}^{3}}{{\left (d x + c\right )}^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((a+b*cot(f*x+e))^3/(d*x+c)^2,x, algorithm="giac")
[Out]
integrate((b*cot(f*x + e) + a)^3/(d*x + c)^2, x)