3.1.93 \(\int \frac {A+B \cot (c+d x)}{(a+b \cot (c+d x))^2} \, dx\) [93]

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

[Out]

(A*a^2-A*b^2+2*B*a*b)*x/(a^2+b^2)^2+(A*b-B*a)/(a^2+b^2)/d/(a+b*cot(d*x+c))-(2*A*a*b-B*a^2+B*b^2)*ln(b*cos(d*x+
c)+a*sin(d*x+c))/(a^2+b^2)^2/d

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Rubi [A]
time = 0.11, antiderivative size = 111, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.130, Rules used = {3610, 3612, 3611} \begin {gather*} \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} \end {gather*}

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 3610

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 3611

Int[((c_) + (d_.)*tan[(e_.) + (f_.)*(x_)])/((a_) + (b_.)*tan[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[(c/(b*f))
*Log[RemoveContent[a*Cos[e + f*x] + b*Sin[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] && EqQ[a*c + b*d, 0]

Rule 3612

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]

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*}

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Mathematica [C] Result contains complex when optimal does not.
time = 2.02, size = 144, normalized size = 1.30 \begin {gather*} \frac {-\frac {(i A+B) \log (i-\tan (c+d x))}{(a-i b)^2}+\frac {i (A+i B) \log (i+\tan (c+d x))}{(a+i b)^2}+\frac {2 \left (-2 a A b+a^2 B-b^2 B\right ) \log (b+a \tan (c+d x))}{\left (a^2+b^2\right )^2}+\frac {2 b (-A b+a B)}{a \left (a^2+b^2\right ) (b+a \tan (c+d x))}}{2 d} \end {gather*}

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 [A]
time = 0.35, size = 147, normalized size = 1.32

method	result	size
derivativedivides	\(\frac {\frac {A b -B a}{\left (a^{2}+b^{2}\right ) \left (a +b \cot \left (d x +c \right )\right )}-\frac {\left (2 A a b -B \,a^{2}+B \,b^{2}\right ) \ln \left (a +b \cot \left (d x +c \right )\right )}{\left (a^{2}+b^{2}\right )^{2}}+\frac {\frac {\left (2 A a b -B \,a^{2}+B \,b^{2}\right ) \ln \left (\cot ^{2}\left (d x +c \right )+1\right )}{2}+\left (-A \,a^{2}+A \,b^{2}-2 B a b \right ) \left (\frac {\pi }{2}-\mathrm {arccot}\left (\cot \left (d x +c \right )\right )\right )}{\left (a^{2}+b^{2}\right )^{2}}}{d}\)	\(147\)
default	\(\frac {\frac {A b -B a}{\left (a^{2}+b^{2}\right ) \left (a +b \cot \left (d x +c \right )\right )}-\frac {\left (2 A a b -B \,a^{2}+B \,b^{2}\right ) \ln \left (a +b \cot \left (d x +c \right )\right )}{\left (a^{2}+b^{2}\right )^{2}}+\frac {\frac {\left (2 A a b -B \,a^{2}+B \,b^{2}\right ) \ln \left (\cot ^{2}\left (d x +c \right )+1\right )}{2}+\left (-A \,a^{2}+A \,b^{2}-2 B a b \right ) \left (\frac {\pi }{2}-\mathrm {arccot}\left (\cot \left (d x +c \right )\right )\right )}{\left (a^{2}+b^{2}\right )^{2}}}{d}\)	\(147\)
norman	\(\frac {\frac {b \left (A \,a^{2}-A \,b^{2}+2 B a b \right ) x}{a^{4}+2 a^{2} b^{2}+b^{4}}+\frac {a \left (A \,a^{2}-A \,b^{2}+2 B a b \right ) x \tan \left (d x +c \right )}{a^{4}+2 a^{2} b^{2}+b^{4}}-\frac {\left (A b -B a \right ) b}{a d \left (a^{2}+b^{2}\right )}}{a \tan \left (d x +c \right )+b}+\frac {\left (2 A a b -B \,a^{2}+B \,b^{2}\right ) \ln \left (1+\tan ^{2}\left (d x +c \right )\right )}{2 d \left (a^{4}+2 a^{2} b^{2}+b^{4}\right )}-\frac {\left (2 A a b -B \,a^{2}+B \,b^{2}\right ) \ln \left (a \tan \left (d x +c \right )+b \right )}{d \left (a^{4}+2 a^{2} b^{2}+b^{4}\right )}\)	\(222\)
risch	\(\frac {i x B}{2 i a b +a^{2}-b^{2}}+\frac {x A}{2 i a b +a^{2}-b^{2}}+\frac {4 i A a b x}{a^{4}+2 a^{2} b^{2}+b^{4}}-\frac {2 i B \,a^{2} x}{a^{4}+2 a^{2} b^{2}+b^{4}}+\frac {2 i B \,b^{2} x}{a^{4}+2 a^{2} b^{2}+b^{4}}+\frac {4 i A a b c}{d \left (a^{4}+2 a^{2} b^{2}+b^{4}\right )}-\frac {2 i B \,a^{2} c}{d \left (a^{4}+2 a^{2} b^{2}+b^{4}\right )}+\frac {2 i B \,b^{2} c}{d \left (a^{4}+2 a^{2} b^{2}+b^{4}\right )}+\frac {2 i b^{2} A}{\left (i a +b \right ) d \left (-i a +b \right )^{2} \left (b \,{\mathrm e}^{2 i \left (d x +c \right )}-i a \,{\mathrm e}^{2 i \left (d x +c \right )}+b +i a \right )}-\frac {2 i b B a}{\left (i a +b \right ) d \left (-i a +b \right )^{2} \left (b \,{\mathrm e}^{2 i \left (d x +c \right )}-i a \,{\mathrm e}^{2 i \left (d x +c \right )}+b +i a \right )}-\frac {2 \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}+\frac {i b -a}{i b +a}\right ) A a b}{d \left (a^{4}+2 a^{2} b^{2}+b^{4}\right )}+\frac {\ln \left ({\mathrm e}^{2 i \left (d x +c \right )}+\frac {i b -a}{i b +a}\right ) B \,a^{2}}{d \left (a^{4}+2 a^{2} b^{2}+b^{4}\right )}-\frac {\ln \left ({\mathrm e}^{2 i \left (d x +c \right )}+\frac {i b -a}{i b +a}\right ) B \,b^{2}}{d \left (a^{4}+2 a^{2} b^{2}+b^{4}\right )}\)	\(474\)

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,method=_RETURNVERBOSE)

[Out]

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

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Maxima [A]
time = 0.53, size = 185, normalized size = 1.67 \begin {gather*} \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 {gather*}

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] Leaf count of result is larger than twice the leaf count of optimal. 340 vs. \(2 (111) = 222\).
time = 3.32, size = 340, normalized size = 3.06 \begin {gather*} \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 {gather*}

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 [C] Result contains complex when optimal does not.
time = 1.59, size = 3966, normalized size = 35.73 \begin {gather*} \text {Too large to display} \end {gather*}

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]

Piecewise((zoo*x*(A + B*cot(c))/cot(c)**2, Eq(a, 0) & Eq(b, 0) & Eq(d, 0)), ((-A*x + A*tan(c + d*x)/d + B*log(
tan(c + d*x)**2 + 1)/(2*d))/b**2, Eq(a, 0)), (-A*d*x*cot(c + d*x)**2/(4*b**2*d*cot(c + d*x)**2 - 8*I*b**2*d*co
t(c + d*x) - 4*b**2*d) + 2*I*A*d*x*cot(c + d*x)/(4*b**2*d*cot(c + d*x)**2 - 8*I*b**2*d*cot(c + d*x) - 4*b**2*d
) + A*d*x/(4*b**2*d*cot(c + d*x)**2 - 8*I*b**2*d*cot(c + d*x) - 4*b**2*d) + A*cot(c + d*x)/(4*b**2*d*cot(c + d
*x)**2 - 8*I*b**2*d*cot(c + d*x) - 4*b**2*d) - 2*I*A/(4*b**2*d*cot(c + d*x)**2 - 8*I*b**2*d*cot(c + d*x) - 4*b
**2*d) + I*B*d*x*cot(c + d*x)**2/(4*b**2*d*cot(c + d*x)**2 - 8*I*b**2*d*cot(c + d*x) - 4*b**2*d) + 2*B*d*x*cot
(c + d*x)/(4*b**2*d*cot(c + d*x)**2 - 8*I*b**2*d*cot(c + d*x) - 4*b**2*d) - I*B*d*x/(4*b**2*d*cot(c + d*x)**2
- 8*I*b**2*d*cot(c + d*x) - 4*b**2*d) - I*B*cot(c + d*x)/(4*b**2*d*cot(c + d*x)**2 - 8*I*b**2*d*cot(c + d*x) -
4*b**2*d), Eq(a, -I*b)), (-A*d*x*cot(c + d*x)**2/(4*b**2*d*cot(c + d*x)**2 + 8*I*b**2*d*cot(c + d*x) - 4*b**2
*d) - 2*I*A*d*x*cot(c + d*x)/(4*b**2*d*cot(c + d*x)**2 + 8*I*b**2*d*cot(c + d*x) - 4*b**2*d) + A*d*x/(4*b**2*d
*cot(c + d*x)**2 + 8*I*b**2*d*cot(c + d*x) - 4*b**2*d) + A*cot(c + d*x)/(4*b**2*d*cot(c + d*x)**2 + 8*I*b**2*d
*cot(c + d*x) - 4*b**2*d) + 2*I*A/(4*b**2*d*cot(c + d*x)**2 + 8*I*b**2*d*cot(c + d*x) - 4*b**2*d) - I*B*d*x*co
t(c + d*x)**2/(4*b**2*d*cot(c + d*x)**2 + 8*I*b**2*d*cot(c + d*x) - 4*b**2*d) + 2*B*d*x*cot(c + d*x)/(4*b**2*d
*cot(c + d*x)**2 + 8*I*b**2*d*cot(c + d*x) - 4*b**2*d) + I*B*d*x/(4*b**2*d*cot(c + d*x)**2 + 8*I*b**2*d*cot(c
+ d*x) - 4*b**2*d) + I*B*cot(c + d*x)/(4*b**2*d*cot(c + d*x)**2 + 8*I*b**2*d*cot(c + d*x) - 4*b**2*d), Eq(a, I
*b)), (A*d*x*tan(c + d*x)**2*cot(c + d*x)**2/(8*b**2*d*tan(c + d*x)**2*cot(c + d*x)**2 - 16*b**2*d*tan(c + d*x
)*cot(c + d*x) + 8*b**2*d) + 3*A*d*x*tan(c + d*x)**2/(8*b**2*d*tan(c + d*x)**2*cot(c + d*x)**2 - 16*b**2*d*tan
(c + d*x)*cot(c + d*x) + 8*b**2*d) + 4*A*d*x*tan(c + d*x)*cot(c + d*x)/(8*b**2*d*tan(c + d*x)**2*cot(c + d*x)*
*2 - 16*b**2*d*tan(c + d*x)*cot(c + d*x) + 8*b**2*d) + 3*A*d*x*cot(c + d*x)**2/(8*b**2*d*tan(c + d*x)**2*cot(c
+ d*x)**2 - 16*b**2*d*tan(c + d*x)*cot(c + d*x) + 8*b**2*d) + A*d*x/(8*b**2*d*tan(c + d*x)**2*cot(c + d*x)**2
- 16*b**2*d*tan(c + d*x)*cot(c + d*x) + 8*b**2*d) + A*tan(c + d*x)*cot(c + d*x)**2/(8*b**2*d*tan(c + d*x)**2*
cot(c + d*x)**2 - 16*b**2*d*tan(c + d*x)*cot(c + d*x) + 8*b**2*d) - 5*A*tan(c + d*x)/(8*b**2*d*tan(c + d*x)**2
*cot(c + d*x)**2 - 16*b**2*d*tan(c + d*x)*cot(c + d*x) + 8*b**2*d) - 4*A*cot(c + d*x)/(8*b**2*d*tan(c + d*x)**
2*cot(c + d*x)**2 - 16*b**2*d*tan(c + d*x)*cot(c + d*x) + 8*b**2*d) + 2*B*d*x*tan(c + d*x)**2*cot(c + d*x)/(8*
b**2*d*tan(c + d*x)**2*cot(c + d*x)**2 - 16*b**2*d*tan(c + d*x)*cot(c + d*x) + 8*b**2*d) + 2*B*d*x*tan(c + d*x
)*cot(c + d*x)**2/(8*b**2*d*tan(c + d*x)**2*cot(c + d*x)**2 - 16*b**2*d*tan(c + d*x)*cot(c + d*x) + 8*b**2*d)
- 2*B*d*x*tan(c + d*x)/(8*b**2*d*tan(c + d*x)**2*cot(c + d*x)**2 - 16*b**2*d*tan(c + d*x)*cot(c + d*x) + 8*b**
2*d) - 2*B*d*x*cot(c + d*x)/(8*b**2*d*tan(c + d*x)**2*cot(c + d*x)**2 - 16*b**2*d*tan(c + d*x)*cot(c + d*x) +
8*b**2*d) + 3*B*tan(c + d*x)**2/(8*b**2*d*tan(c + d*x)**2*cot(c + d*x)**2 - 16*b**2*d*tan(c + d*x)*cot(c + d*x
) + 8*b**2*d) + B*cot(c + d*x)**2/(8*b**2*d*tan(c + d*x)**2*cot(c + d*x)**2 - 16*b**2*d*tan(c + d*x)*cot(c + d
*x) + 8*b**2*d) + 2*B/(8*b**2*d*tan(c + d*x)**2*cot(c + d*x)**2 - 16*b**2*d*tan(c + d*x)*cot(c + d*x) + 8*b**2
*d), Eq(a, -b/tan(c + d*x))), (x*(A + B*cot(c))/(a + b*cot(c))**2, Eq(d, 0)), (2*A*a**4*d*x*tan(c + d*x)/(2*a*
*6*d*tan(c + d*x) + 2*a**5*b*d + 4*a**4*b**2*d*tan(c + d*x) + 4*a**3*b**3*d + 2*a**2*b**4*d*tan(c + d*x) + 2*a
*b**5*d) + 2*A*a**3*b*d*x/(2*a**6*d*tan(c + d*x) + 2*a**5*b*d + 4*a**4*b**2*d*tan(c + d*x) + 4*a**3*b**3*d + 2
*a**2*b**4*d*tan(c + d*x) + 2*a*b**5*d) - 4*A*a**3*b*log(tan(c + d*x) + b/a)*tan(c + d*x)/(2*a**6*d*tan(c + d*
x) + 2*a**5*b*d + 4*a**4*b**2*d*tan(c + d*x) + 4*a**3*b**3*d + 2*a**2*b**4*d*tan(c + d*x) + 2*a*b**5*d) + 2*A*
a**3*b*log(tan(c + d*x)**2 + 1)*tan(c + d*x)/(2*a**6*d*tan(c + d*x) + 2*a**5*b*d + 4*a**4*b**2*d*tan(c + d*x)
+ 4*a**3*b**3*d + 2*a**2*b**4*d*tan(c + d*x) + 2*a*b**5*d) - 2*A*a**2*b**2*d*x*tan(c + d*x)/(2*a**6*d*tan(c +
d*x) + 2*a**5*b*d + 4*a**4*b**2*d*tan(c + d*x) + 4*a**3*b**3*d + 2*a**2*b**4*d*tan(c + d*x) + 2*a*b**5*d) - 4*
A*a**2*b**2*log(tan(c + d*x) + b/a)/(2*a**6*d*tan(c + d*x) + 2*a**5*b*d + 4*a**4*b**2*d*tan(c + d*x) + 4*a**3*
b**3*d + 2*a**2*b**4*d*tan(c + d*x) + 2*a*b**5*d) + 2*A*a**2*b**2*log(tan(c + d*x)**2 + 1)/(2*a**6*d*tan(c + d
*x) + 2*a**5*b*d + 4*a**4*b**2*d*tan(c + d*x) + 4*a**3*b**3*d + 2*a**2*b**4*d*tan(c + d*x) + 2*a*b**5*d) - 2*A
*a**2*b**2/(2*a**6*d*tan(c + d*x) + 2*a**5*b*d + 4*a**4*b**2*d*tan(c + d*x) + 4*a**3*b**3*d + 2*a**2*b**4*d*ta
n(c + d*x) + 2*a*b**5*d) - 2*A*a*b**3*d*x/(2*a**6*d*tan(c + d*x) + 2*a**5*b*d + 4*a**4*b**2*d*tan(c + d*x) + 4
*a**3*b**3*d + 2*a**2*b**4*d*tan(c + d*x) + 2*a*b**5*d) - 2*A*b**4/(2*a**6*d*tan(c + d*x) + 2*a**5*b*d + 4*a**
4*b**2*d*tan(c + d*x) + 4*a**3*b**3*d + 2*a**2*...

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 241 vs. \(2 (111) = 222\).
time = 0.52, size = 241, normalized size = 2.17 \begin {gather*} \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 {gather*}

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

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Mupad [B]
time = 1.46, size = 268, normalized size = 2.41 \begin {gather*} \ln \left (a+b\,\mathrm {cot}\left (c+d\,x\right )\right )\,\left (\frac {B}{d\,\left (a^2+b^2\right )}-\frac {2\,B\,b^2}{d\,{\left (a^2+b^2\right )}^2}\right )+\frac {A\,\ln \left (\mathrm {cot}\left (c+d\,x\right )-\mathrm {i}\right )}{2\,\left (-1{}\mathrm {i}\,d\,a^2+2\,d\,a\,b+1{}\mathrm {i}\,d\,b^2\right )}-\frac {B\,\ln \left (\mathrm {cot}\left (c+d\,x\right )-\mathrm {i}\right )}{2\,\left (d\,a^2+2{}\mathrm {i}\,d\,a\,b-d\,b^2\right )}+\frac {A\,b}{\left (a\,d+b\,d\,\mathrm {cot}\left (c+d\,x\right )\right )\,\left (a^2+b^2\right )}-\frac {B\,a}{\left (a\,d+b\,d\,\mathrm {cot}\left (c+d\,x\right )\right )\,\left (a^2+b^2\right )}-\frac {2\,A\,a\,b\,\ln \left (a+b\,\mathrm {cot}\left (c+d\,x\right )\right )}{d\,{\left (a^2+b^2\right )}^2}+\frac {A\,\ln \left (\mathrm {cot}\left (c+d\,x\right )+1{}\mathrm {i}\right )\,1{}\mathrm {i}}{2\,\left (-d\,a^2+2{}\mathrm {i}\,d\,a\,b+d\,b^2\right )}-\frac {B\,\ln \left (\mathrm {cot}\left (c+d\,x\right )+1{}\mathrm {i}\right )\,1{}\mathrm {i}}{2\,\left (1{}\mathrm {i}\,d\,a^2+2\,d\,a\,b-1{}\mathrm {i}\,d\,b^2\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((A + B*cot(c + d*x))/(a + b*cot(c + d*x))^2,x)

[Out]

log(a + b*cot(c + d*x))*(B/(d*(a^2 + b^2)) - (2*B*b^2)/(d*(a^2 + b^2)^2)) + (A*log(cot(c + d*x) + 1i)*1i)/(2*(
b^2*d - a^2*d + a*b*d*2i)) + (A*log(cot(c + d*x) - 1i))/(2*(b^2*d*1i - a^2*d*1i + 2*a*b*d)) - (B*log(cot(c + d
*x) - 1i))/(2*(a^2*d - b^2*d + a*b*d*2i)) - (B*log(cot(c + d*x) + 1i)*1i)/(2*(a^2*d*1i - b^2*d*1i + 2*a*b*d))
+ (A*b)/((a*d + b*d*cot(c + d*x))*(a^2 + b^2)) - (B*a)/((a*d + b*d*cot(c + d*x))*(a^2 + b^2)) - (2*A*a*b*log(a
+ b*cot(c + d*x)))/(d*(a^2 + b^2)^2)

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