3.93 \(\int \frac{A+B \cot (c+d x)}{(a+b \cot (c+d x))^2} \, dx\)

Optimal. Leaf size=111 \[ \frac{A b-a B}{d \left (a^2+b^2\right ) (a+b \cot (c+d x))}-\frac{\left (a^2 (-B)+2 a A b+b^2 B\right ) \log (a \sin (c+d x)+b \cos (c+d x))}{d \left (a^2+b^2\right )^2}+\frac{x \left (a^2 A+2 a b B-A b^2\right )}{\left (a^2+b^2\right )^2} \]

[Out]

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

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Rubi [A]  time = 0.149454, antiderivative size = 111, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.13, Rules used = {3529, 3531, 3530} \[ \frac{A b-a B}{d \left (a^2+b^2\right ) (a+b \cot (c+d x))}-\frac{\left (a^2 (-B)+2 a A b+b^2 B\right ) \log (a \sin (c+d x)+b \cos (c+d x))}{d \left (a^2+b^2\right )^2}+\frac{x \left (a^2 A+2 a b B-A b^2\right )}{\left (a^2+b^2\right )^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((a^2*A - A*b^2 + 2*a*b*B)*x)/(a^2 + b^2)^2 + (A*b - a*B)/((a^2 + b^2)*d*(a + b*Cot[c + d*x])) - ((2*a*A*b - a
^2*B + b^2*B)*Log[b*Cos[c + d*x] + a*Sin[c + d*x]])/((a^2 + b^2)^2*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))^2} \, dx &=\frac{A b-a B}{\left (a^2+b^2\right ) d (a+b \cot (c+d x))}+\frac{\int \frac{a A+b B-(A b-a B) \cot (c+d x)}{a+b \cot (c+d x)} \, dx}{a^2+b^2}\\ &=\frac{\left (a^2 A-A b^2+2 a b B\right ) x}{\left (a^2+b^2\right )^2}+\frac{A b-a B}{\left (a^2+b^2\right ) d (a+b \cot (c+d x))}-\frac{\left (2 a A b-a^2 B+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 )^2}\\ &=\frac{\left (a^2 A-A b^2+2 a b B\right ) x}{\left (a^2+b^2\right )^2}+\frac{A b-a B}{\left (a^2+b^2\right ) d (a+b \cot (c+d x))}-\frac{\left (2 a A b-a^2 B+b^2 B\right ) \log (b \cos (c+d x)+a \sin (c+d x))}{\left (a^2+b^2\right )^2 d}\\ \end{align*}

Mathematica [C]  time = 1.79675, size = 144, normalized size = 1.3 \[ \frac{\frac{2 b (a B-A b)}{a \left (a^2+b^2\right ) (a \tan (c+d x)+b)}+\frac{2 \left (a^2 B-2 a A b-b^2 B\right ) \log (a \tan (c+d x)+b)}{\left (a^2+b^2\right )^2}-\frac{(B+i A) \log (-\tan (c+d x)+i)}{(a-i b)^2}+\frac{i (A+i B) \log (\tan (c+d x)+i)}{(a+i b)^2}}{2 d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [B]  time = 0.031, size = 356, normalized size = 3.2 \begin{align*}{\frac{Ab}{d \left ({a}^{2}+{b}^{2} \right ) \left ( a+b\cot \left ( dx+c \right ) \right ) }}-{\frac{Ba}{d \left ({a}^{2}+{b}^{2} \right ) \left ( a+b\cot \left ( dx+c \right ) \right ) }}-2\,{\frac{\ln \left ( a+b\cot \left ( dx+c \right ) \right ) Aab}{d \left ({a}^{2}+{b}^{2} \right ) ^{2}}}+{\frac{\ln \left ( a+b\cot \left ( dx+c \right ) \right ) B{a}^{2}}{d \left ({a}^{2}+{b}^{2} \right ) ^{2}}}-{\frac{\ln \left ( a+b\cot \left ( dx+c \right ) \right ) B{b}^{2}}{d \left ({a}^{2}+{b}^{2} \right ) ^{2}}}+{\frac{\ln \left ( \left ( \cot \left ( dx+c \right ) \right ) ^{2}+1 \right ) Aab}{d \left ({a}^{2}+{b}^{2} \right ) ^{2}}}-{\frac{\ln \left ( \left ( \cot \left ( dx+c \right ) \right ) ^{2}+1 \right ) B{a}^{2}}{2\,d \left ({a}^{2}+{b}^{2} \right ) ^{2}}}+{\frac{\ln \left ( \left ( \cot \left ( dx+c \right ) \right ) ^{2}+1 \right ) B{b}^{2}}{2\,d \left ({a}^{2}+{b}^{2} \right ) ^{2}}}-{\frac{A\pi \,{a}^{2}}{2\,d \left ({a}^{2}+{b}^{2} \right ) ^{2}}}+{\frac{A\pi \,{b}^{2}}{2\,d \left ({a}^{2}+{b}^{2} \right ) ^{2}}}-{\frac{B\pi \,ab}{d \left ({a}^{2}+{b}^{2} \right ) ^{2}}}+{\frac{A{\rm arccot} \left (\cot \left ( dx+c \right ) \right ){a}^{2}}{d \left ({a}^{2}+{b}^{2} \right ) ^{2}}}-{\frac{A{\rm arccot} \left (\cot \left ( dx+c \right ) \right ){b}^{2}}{d \left ({a}^{2}+{b}^{2} \right ) ^{2}}}+2\,{\frac{B{\rm arccot} \left (\cot \left ( dx+c \right ) \right )ab}{d \left ({a}^{2}+{b}^{2} \right ) ^{2}}} \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))^2,x)

[Out]

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

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Maxima [A]  time = 1.75696, size = 250, normalized size = 2.25 \begin{align*} \frac{\frac{2 \,{\left (A a^{2} + 2 \, B a b - A b^{2}\right )}{\left (d x + c\right )}}{a^{4} + 2 \, a^{2} b^{2} + b^{4}} + \frac{2 \,{\left (B a^{2} - 2 \, A a b - B b^{2}\right )} \log \left (a \tan \left (d x + c\right ) + b\right )}{a^{4} + 2 \, a^{2} b^{2} + b^{4}} - \frac{{\left (B a^{2} - 2 \, A a b - B b^{2}\right )} \log \left (\tan \left (d x + c\right )^{2} + 1\right )}{a^{4} + 2 \, a^{2} b^{2} + b^{4}} + \frac{2 \,{\left (B a b - A b^{2}\right )}}{a^{3} b + a b^{3} +{\left (a^{4} + a^{2} b^{2}\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))^2,x, algorithm="maxima")

[Out]

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

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Fricas [B]  time = 1.69335, size = 748, normalized size = 6.74 \begin{align*} \frac{2 \, B a^{2} b - 2 \, A a b^{2} + 2 \,{\left (A a^{2} b + 2 \, B a b^{2} - A b^{3}\right )} d x + 2 \,{\left (B a^{2} b - A a b^{2} +{\left (A a^{2} b + 2 \, B a b^{2} - A b^{3}\right )} d x\right )} \cos \left (2 \, d x + 2 \, c\right ) +{\left (B a^{2} b - 2 \, A a b^{2} - B b^{3} +{\left (B a^{2} b - 2 \, A a b^{2} - B b^{3}\right )} \cos \left (2 \, d x + 2 \, c\right ) +{\left (B a^{3} - 2 \, A a^{2} b - B a b^{2}\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 (B a b^{2} - A b^{3} -{\left (A a^{3} + 2 \, B a^{2} b - A a b^{2}\right )} d x\right )} \sin \left (2 \, d x + 2 \, c\right )}{2 \,{\left ({\left (a^{4} b + 2 \, a^{2} b^{3} + b^{5}\right )} d \cos \left (2 \, d x + 2 \, c\right ) +{\left (a^{5} + 2 \, a^{3} b^{2} + a b^{4}\right )} d \sin \left (2 \, d x + 2 \, c\right ) +{\left (a^{4} b + 2 \, a^{2} b^{3} + b^{5}\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))^2,x, algorithm="fricas")

[Out]

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

[Out]

Exception raised: AttributeError

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Giac [B]  time = 1.38256, size = 325, normalized size = 2.93 \begin{align*} \frac{\frac{2 \,{\left (A a^{2} + 2 \, B a b - A b^{2}\right )}{\left (d x + c\right )}}{a^{4} + 2 \, a^{2} b^{2} + b^{4}} - \frac{{\left (B a^{2} - 2 \, A a b - B b^{2}\right )} \log \left (\tan \left (d x + c\right )^{2} + 1\right )}{a^{4} + 2 \, a^{2} b^{2} + b^{4}} + \frac{2 \,{\left (B a^{3} - 2 \, A a^{2} b - B a b^{2}\right )} \log \left ({\left | a \tan \left (d x + c\right ) + b \right |}\right )}{a^{5} + 2 \, a^{3} b^{2} + a b^{4}} - \frac{2 \,{\left (B a^{4} \tan \left (d x + c\right ) - 2 \, A a^{3} b \tan \left (d x + c\right ) - B a^{2} b^{2} \tan \left (d x + c\right ) - A a^{2} b^{2} - 2 \, B a b^{3} + A b^{4}\right )}}{{\left (a^{5} + 2 \, a^{3} b^{2} + a b^{4}\right )}{\left (a \tan \left (d x + c\right ) + b\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))^2,x, algorithm="giac")

[Out]

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