3.94 \(\int \frac{A+B \cot (c+d x)}{(a+b \cot (c+d x))^3} \, dx\)

Optimal. Leaf size=175 \[ \frac{A b-a B}{2 d \left (a^2+b^2\right ) (a+b \cot (c+d x))^2}+\frac{a^2 (-B)+2 a A b+b^2 B}{d \left (a^2+b^2\right )^2 (a+b \cot (c+d x))}-\frac{\left (3 a^2 A b+a^3 (-B)+3 a b^2 B-A b^3\right ) \log (a \sin (c+d x)+b \cos (c+d x))}{d \left (a^2+b^2\right )^3}+\frac{x \left (a^3 A+3 a^2 b B-3 a A b^2-b^3 B\right )}{\left (a^2+b^2\right )^3} \]

[Out]

((a^3*A - 3*a*A*b^2 + 3*a^2*b*B - b^3*B)*x)/(a^2 + b^2)^3 + (A*b - a*B)/(2*(a^2 + b^2)*d*(a + b*Cot[c + d*x])^
2) + (2*a*A*b - a^2*B + b^2*B)/((a^2 + b^2)^2*d*(a + b*Cot[c + d*x])) - ((3*a^2*A*b - A*b^3 - a^3*B + 3*a*b^2*
B)*Log[b*Cos[c + d*x] + a*Sin[c + d*x]])/((a^2 + b^2)^3*d)

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Rubi [A]  time = 0.275609, antiderivative size = 175, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.13, Rules used = {3529, 3531, 3530} \[ \frac{A b-a B}{2 d \left (a^2+b^2\right ) (a+b \cot (c+d x))^2}+\frac{a^2 (-B)+2 a A b+b^2 B}{d \left (a^2+b^2\right )^2 (a+b \cot (c+d x))}-\frac{\left (3 a^2 A b+a^3 (-B)+3 a b^2 B-A b^3\right ) \log (a \sin (c+d x)+b \cos (c+d x))}{d \left (a^2+b^2\right )^3}+\frac{x \left (a^3 A+3 a^2 b B-3 a A b^2-b^3 B\right )}{\left (a^2+b^2\right )^3} \]

Antiderivative was successfully verified.

[In]

Int[(A + B*Cot[c + d*x])/(a + b*Cot[c + d*x])^3,x]

[Out]

((a^3*A - 3*a*A*b^2 + 3*a^2*b*B - b^3*B)*x)/(a^2 + b^2)^3 + (A*b - a*B)/(2*(a^2 + b^2)*d*(a + b*Cot[c + d*x])^
2) + (2*a*A*b - a^2*B + b^2*B)/((a^2 + b^2)^2*d*(a + b*Cot[c + d*x])) - ((3*a^2*A*b - A*b^3 - a^3*B + 3*a*b^2*
B)*Log[b*Cos[c + d*x] + a*Sin[c + d*x]])/((a^2 + b^2)^3*d)

Rule 3529

Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[((
b*c - a*d)*(a + b*Tan[e + f*x])^(m + 1))/(f*(m + 1)*(a^2 + b^2)), x] + Dist[1/(a^2 + b^2), Int[(a + b*Tan[e +
f*x])^(m + 1)*Simp[a*c + b*d - (b*c - a*d)*Tan[e + f*x], x], x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c
- a*d, 0] && NeQ[a^2 + b^2, 0] && LtQ[m, -1]

Rule 3531

Int[((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)])/((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[((a*c +
b*d)*x)/(a^2 + b^2), x] + Dist[(b*c - a*d)/(a^2 + b^2), Int[(b - a*Tan[e + f*x])/(a + b*Tan[e + f*x]), x], x]
/; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 + b^2, 0] && NeQ[a*c + b*d, 0]

Rule 3530

Int[((c_) + (d_.)*tan[(e_.) + (f_.)*(x_)])/((a_) + (b_.)*tan[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[(c*Log[Re
moveContent[a*Cos[e + f*x] + b*Sin[e + f*x], x]])/(b*f), x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d,
0] && NeQ[a^2 + b^2, 0] && EqQ[a*c + b*d, 0]

Rubi steps

\begin{align*} \int \frac{A+B \cot (c+d x)}{(a+b \cot (c+d x))^3} \, dx &=\frac{A b-a B}{2 \left (a^2+b^2\right ) d (a+b \cot (c+d x))^2}+\frac{\int \frac{a A+b B-(A b-a B) \cot (c+d x)}{(a+b \cot (c+d x))^2} \, dx}{a^2+b^2}\\ &=\frac{A b-a B}{2 \left (a^2+b^2\right ) d (a+b \cot (c+d x))^2}+\frac{2 a A b-a^2 B+b^2 B}{\left (a^2+b^2\right )^2 d (a+b \cot (c+d x))}+\frac{\int \frac{a^2 A-A b^2+2 a b B-\left (2 a A b-a^2 B+b^2 B\right ) \cot (c+d x)}{a+b \cot (c+d x)} \, dx}{\left (a^2+b^2\right )^2}\\ &=\frac{\left (a^3 A-3 a A b^2+3 a^2 b B-b^3 B\right ) x}{\left (a^2+b^2\right )^3}+\frac{A b-a B}{2 \left (a^2+b^2\right ) d (a+b \cot (c+d x))^2}+\frac{2 a A b-a^2 B+b^2 B}{\left (a^2+b^2\right )^2 d (a+b \cot (c+d x))}-\frac{\left (3 a^2 A b-A b^3-a^3 B+3 a b^2 B\right ) \int \frac{-b+a \cot (c+d x)}{a+b \cot (c+d x)} \, dx}{\left (a^2+b^2\right )^3}\\ &=\frac{\left (a^3 A-3 a A b^2+3 a^2 b B-b^3 B\right ) x}{\left (a^2+b^2\right )^3}+\frac{A b-a B}{2 \left (a^2+b^2\right ) d (a+b \cot (c+d x))^2}+\frac{2 a A b-a^2 B+b^2 B}{\left (a^2+b^2\right )^2 d (a+b \cot (c+d x))}-\frac{\left (3 a^2 A b-A b^3-a^3 B+3 a b^2 B\right ) \log (b \cos (c+d x)+a \sin (c+d x))}{\left (a^2+b^2\right )^3 d}\\ \end{align*}

Mathematica [C]  time = 4.7199, size = 202, normalized size = 1.15 \[ \frac{\frac{2 \left (-3 a^2 A b+a^3 B-3 a b^2 B+A b^3\right ) \log (a \tan (c+d x)+b)-\frac{b \left (a^2+b^2\right ) \left (\left (6 a^3 A b-4 a^4 B+2 a A b^3\right ) \tan (c+d x)+b \left (5 a^2 A b-3 a^3 B+a b^2 B+A b^3\right )\right )}{a^2 (a \tan (c+d x)+b)^2}}{\left (a^2+b^2\right )^3}-\frac{i (A-i B) \log (-\tan (c+d x)+i)}{(a-i b)^3}+\frac{i (A+i B) \log (\tan (c+d x)+i)}{(a+i b)^3}}{2 d} \]

Antiderivative was successfully verified.

[In]

Integrate[(A + B*Cot[c + d*x])/(a + b*Cot[c + d*x])^3,x]

[Out]

(((-I)*(A - I*B)*Log[I - Tan[c + d*x]])/(a - I*b)^3 + (I*(A + I*B)*Log[I + Tan[c + d*x]])/(a + I*b)^3 + (2*(-3
*a^2*A*b + A*b^3 + a^3*B - 3*a*b^2*B)*Log[b + a*Tan[c + d*x]] - (b*(a^2 + b^2)*(b*(5*a^2*A*b + A*b^3 - 3*a^3*B
+ a*b^2*B) + (6*a^3*A*b + 2*a*A*b^3 - 4*a^4*B)*Tan[c + d*x]))/(a^2*(b + a*Tan[c + d*x])^2))/(a^2 + b^2)^3)/(2
*d)

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Maple [B]  time = 0.037, size = 559, normalized size = 3.2 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((A+B*cot(d*x+c))/(a+b*cot(d*x+c))^3,x)

[Out]

-3/d/(a^2+b^2)^3*ln(a+b*cot(d*x+c))*A*a^2*b+1/d/(a^2+b^2)^3*ln(a+b*cot(d*x+c))*A*b^3+1/d/(a^2+b^2)^3*ln(a+b*co
t(d*x+c))*B*a^3-3/d/(a^2+b^2)^3*ln(a+b*cot(d*x+c))*B*a*b^2+1/2/d/(a^2+b^2)/(a+b*cot(d*x+c))^2*A*b-1/2/d/(a^2+b
^2)/(a+b*cot(d*x+c))^2*B*a+2/d/(a^2+b^2)^2/(a+b*cot(d*x+c))*A*a*b-1/d/(a^2+b^2)^2/(a+b*cot(d*x+c))*B*a^2+1/d/(
a^2+b^2)^2/(a+b*cot(d*x+c))*B*b^2+3/2/d/(a^2+b^2)^3*ln(cot(d*x+c)^2+1)*A*a^2*b-1/2/d/(a^2+b^2)^3*ln(cot(d*x+c)
^2+1)*A*b^3-1/2/d/(a^2+b^2)^3*ln(cot(d*x+c)^2+1)*B*a^3+3/2/d/(a^2+b^2)^3*ln(cot(d*x+c)^2+1)*B*a*b^2-1/2/d/(a^2
+b^2)^3*A*Pi*a^3+3/2/d/(a^2+b^2)^3*A*Pi*a*b^2-3/2/d/(a^2+b^2)^3*B*Pi*a^2*b+1/2/d/(a^2+b^2)^3*B*Pi*b^3+1/d/(a^2
+b^2)^3*A*arccot(cot(d*x+c))*a^3-3/d/(a^2+b^2)^3*A*arccot(cot(d*x+c))*a*b^2+3/d/(a^2+b^2)^3*B*arccot(cot(d*x+c
))*a^2*b-1/d/(a^2+b^2)^3*B*arccot(cot(d*x+c))*b^3

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Maxima [A]  time = 1.81501, size = 455, normalized size = 2.6 \begin{align*} \frac{\frac{2 \,{\left (A a^{3} + 3 \, B a^{2} b - 3 \, A a b^{2} - B b^{3}\right )}{\left (d x + c\right )}}{a^{6} + 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} + b^{6}} + \frac{2 \,{\left (B a^{3} - 3 \, A a^{2} b - 3 \, B a b^{2} + A b^{3}\right )} \log \left (a \tan \left (d x + c\right ) + b\right )}{a^{6} + 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} + b^{6}} - \frac{{\left (B a^{3} - 3 \, A a^{2} b - 3 \, B a b^{2} + A b^{3}\right )} \log \left (\tan \left (d x + c\right )^{2} + 1\right )}{a^{6} + 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} + b^{6}} + \frac{3 \, B a^{3} b^{2} - 5 \, A a^{2} b^{3} - B a b^{4} - A b^{5} + 2 \,{\left (2 \, B a^{4} b - 3 \, A a^{3} b^{2} - A a b^{4}\right )} \tan \left (d x + c\right )}{a^{6} b^{2} + 2 \, a^{4} b^{4} + a^{2} b^{6} +{\left (a^{8} + 2 \, a^{6} b^{2} + a^{4} b^{4}\right )} \tan \left (d x + c\right )^{2} + 2 \,{\left (a^{7} b + 2 \, a^{5} b^{3} + a^{3} b^{5}\right )} \tan \left (d x + c\right )}}{2 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((A+B*cot(d*x+c))/(a+b*cot(d*x+c))^3,x, algorithm="maxima")

[Out]

1/2*(2*(A*a^3 + 3*B*a^2*b - 3*A*a*b^2 - B*b^3)*(d*x + c)/(a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6) + 2*(B*a^3 - 3*A*
a^2*b - 3*B*a*b^2 + A*b^3)*log(a*tan(d*x + c) + b)/(a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6) - (B*a^3 - 3*A*a^2*b -
3*B*a*b^2 + A*b^3)*log(tan(d*x + c)^2 + 1)/(a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6) + (3*B*a^3*b^2 - 5*A*a^2*b^3 -
B*a*b^4 - A*b^5 + 2*(2*B*a^4*b - 3*A*a^3*b^2 - A*a*b^4)*tan(d*x + c))/(a^6*b^2 + 2*a^4*b^4 + a^2*b^6 + (a^8 +
2*a^6*b^2 + a^4*b^4)*tan(d*x + c)^2 + 2*(a^7*b + 2*a^5*b^3 + a^3*b^5)*tan(d*x + c)))/d

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Fricas [B]  time = 1.85088, size = 1191, normalized size = 6.81 \begin{align*} \frac{2 \, B a^{3} b^{2} - 2 \, A a^{2} b^{3} + 2 \, B a b^{4} - 2 \, A b^{5} - 2 \,{\left (A a^{5} + 3 \, B a^{4} b - 2 \, A a^{3} b^{2} + 2 \, B a^{2} b^{3} - 3 \, A a b^{4} - B b^{5}\right )} d x - 2 \,{\left (4 \, B a^{3} b^{2} - 6 \, A a^{2} b^{3} - 2 \, B a b^{4} -{\left (A a^{5} + 3 \, B a^{4} b - 4 \, A a^{3} b^{2} - 4 \, B a^{2} b^{3} + 3 \, A a b^{4} + B b^{5}\right )} d x\right )} \cos \left (2 \, d x + 2 \, c\right ) -{\left (B a^{5} - 3 \, A a^{4} b - 2 \, B a^{3} b^{2} - 2 \, A a^{2} b^{3} - 3 \, B a b^{4} + A b^{5} -{\left (B a^{5} - 3 \, A a^{4} b - 4 \, B a^{3} b^{2} + 4 \, A a^{2} b^{3} + 3 \, B a b^{4} - A b^{5}\right )} \cos \left (2 \, d x + 2 \, c\right ) + 2 \,{\left (B a^{4} b - 3 \, A a^{3} b^{2} - 3 \, B a^{2} b^{3} + A a b^{4}\right )} \sin \left (2 \, d x + 2 \, c\right )\right )} \log \left (a b \sin \left (2 \, d x + 2 \, c\right ) + \frac{1}{2} \, a^{2} + \frac{1}{2} \, b^{2} - \frac{1}{2} \,{\left (a^{2} - b^{2}\right )} \cos \left (2 \, d x + 2 \, c\right )\right ) - 2 \,{\left (2 \, B a^{4} b - 3 \, A a^{3} b^{2} - 3 \, B a^{2} b^{3} + 3 \, A a b^{4} + B b^{5} + 2 \,{\left (A a^{4} b + 3 \, B a^{3} b^{2} - 3 \, A a^{2} b^{3} - B a b^{4}\right )} d x\right )} \sin \left (2 \, d x + 2 \, c\right )}{2 \,{\left ({\left (a^{8} + 2 \, a^{6} b^{2} - 2 \, a^{2} b^{6} - b^{8}\right )} d \cos \left (2 \, d x + 2 \, c\right ) - 2 \,{\left (a^{7} b + 3 \, a^{5} b^{3} + 3 \, a^{3} b^{5} + a b^{7}\right )} d \sin \left (2 \, d x + 2 \, c\right ) -{\left (a^{8} + 4 \, a^{6} b^{2} + 6 \, a^{4} b^{4} + 4 \, a^{2} b^{6} + b^{8}\right )} d\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((A+B*cot(d*x+c))/(a+b*cot(d*x+c))^3,x, algorithm="fricas")

[Out]

1/2*(2*B*a^3*b^2 - 2*A*a^2*b^3 + 2*B*a*b^4 - 2*A*b^5 - 2*(A*a^5 + 3*B*a^4*b - 2*A*a^3*b^2 + 2*B*a^2*b^3 - 3*A*
a*b^4 - B*b^5)*d*x - 2*(4*B*a^3*b^2 - 6*A*a^2*b^3 - 2*B*a*b^4 - (A*a^5 + 3*B*a^4*b - 4*A*a^3*b^2 - 4*B*a^2*b^3
+ 3*A*a*b^4 + B*b^5)*d*x)*cos(2*d*x + 2*c) - (B*a^5 - 3*A*a^4*b - 2*B*a^3*b^2 - 2*A*a^2*b^3 - 3*B*a*b^4 + A*b
^5 - (B*a^5 - 3*A*a^4*b - 4*B*a^3*b^2 + 4*A*a^2*b^3 + 3*B*a*b^4 - A*b^5)*cos(2*d*x + 2*c) + 2*(B*a^4*b - 3*A*a
^3*b^2 - 3*B*a^2*b^3 + A*a*b^4)*sin(2*d*x + 2*c))*log(a*b*sin(2*d*x + 2*c) + 1/2*a^2 + 1/2*b^2 - 1/2*(a^2 - b^
2)*cos(2*d*x + 2*c)) - 2*(2*B*a^4*b - 3*A*a^3*b^2 - 3*B*a^2*b^3 + 3*A*a*b^4 + B*b^5 + 2*(A*a^4*b + 3*B*a^3*b^2
- 3*A*a^2*b^3 - B*a*b^4)*d*x)*sin(2*d*x + 2*c))/((a^8 + 2*a^6*b^2 - 2*a^2*b^6 - b^8)*d*cos(2*d*x + 2*c) - 2*(
a^7*b + 3*a^5*b^3 + 3*a^3*b^5 + a*b^7)*d*sin(2*d*x + 2*c) - (a^8 + 4*a^6*b^2 + 6*a^4*b^4 + 4*a^2*b^6 + b^8)*d)

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Sympy [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: AttributeError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((A+B*cot(d*x+c))/(a+b*cot(d*x+c))**3,x)

[Out]

Exception raised: AttributeError

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Giac [B]  time = 1.42441, size = 556, normalized size = 3.18 \begin{align*} \frac{\frac{2 \,{\left (A a^{3} + 3 \, B a^{2} b - 3 \, A a b^{2} - B b^{3}\right )}{\left (d x + c\right )}}{a^{6} + 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} + b^{6}} - \frac{{\left (B a^{3} - 3 \, A a^{2} b - 3 \, B a b^{2} + A b^{3}\right )} \log \left (\tan \left (d x + c\right )^{2} + 1\right )}{a^{6} + 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} + b^{6}} + \frac{2 \,{\left (B a^{4} - 3 \, A a^{3} b - 3 \, B a^{2} b^{2} + A a b^{3}\right )} \log \left ({\left | a \tan \left (d x + c\right ) + b \right |}\right )}{a^{7} + 3 \, a^{5} b^{2} + 3 \, a^{3} b^{4} + a b^{6}} - \frac{3 \, B a^{7} \tan \left (d x + c\right )^{2} - 9 \, A a^{6} b \tan \left (d x + c\right )^{2} - 9 \, B a^{5} b^{2} \tan \left (d x + c\right )^{2} + 3 \, A a^{4} b^{3} \tan \left (d x + c\right )^{2} + 2 \, B a^{6} b \tan \left (d x + c\right ) - 12 \, A a^{5} b^{2} \tan \left (d x + c\right ) - 22 \, B a^{4} b^{3} \tan \left (d x + c\right ) + 14 \, A a^{3} b^{4} \tan \left (d x + c\right ) + 2 \, A a b^{6} \tan \left (d x + c\right ) - 4 \, A a^{4} b^{3} - 11 \, B a^{3} b^{4} + 9 \, A a^{2} b^{5} + B a b^{6} + A b^{7}}{{\left (a^{8} + 3 \, a^{6} b^{2} + 3 \, a^{4} b^{4} + a^{2} b^{6}\right )}{\left (a \tan \left (d x + c\right ) + b\right )}^{2}}}{2 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((A+B*cot(d*x+c))/(a+b*cot(d*x+c))^3,x, algorithm="giac")

[Out]

1/2*(2*(A*a^3 + 3*B*a^2*b - 3*A*a*b^2 - B*b^3)*(d*x + c)/(a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6) - (B*a^3 - 3*A*a^
2*b - 3*B*a*b^2 + A*b^3)*log(tan(d*x + c)^2 + 1)/(a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6) + 2*(B*a^4 - 3*A*a^3*b -
3*B*a^2*b^2 + A*a*b^3)*log(abs(a*tan(d*x + c) + b))/(a^7 + 3*a^5*b^2 + 3*a^3*b^4 + a*b^6) - (3*B*a^7*tan(d*x +
c)^2 - 9*A*a^6*b*tan(d*x + c)^2 - 9*B*a^5*b^2*tan(d*x + c)^2 + 3*A*a^4*b^3*tan(d*x + c)^2 + 2*B*a^6*b*tan(d*x
+ c) - 12*A*a^5*b^2*tan(d*x + c) - 22*B*a^4*b^3*tan(d*x + c) + 14*A*a^3*b^4*tan(d*x + c) + 2*A*a*b^6*tan(d*x
+ c) - 4*A*a^4*b^3 - 11*B*a^3*b^4 + 9*A*a^2*b^5 + B*a*b^6 + A*b^7)/((a^8 + 3*a^6*b^2 + 3*a^4*b^4 + a^2*b^6)*(a
*tan(d*x + c) + b)^2))/d