3.1.94 \(\int \frac {A+B \cot (c+d x)}{(a+b \cot (c+d x))^3} \, dx\) [94]

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

[Out]

(A*a^3-3*A*a*b^2+3*B*a^2*b-B*b^3)*x/(a^2+b^2)^3+1/2*(A*b-B*a)/(a^2+b^2)/d/(a+b*cot(d*x+c))^2+(2*A*a*b-B*a^2+B*
b^2)/(a^2+b^2)^2/d/(a+b*cot(d*x+c))-(3*A*a^2*b-A*b^3-B*a^3+3*B*a*b^2)*ln(b*cos(d*x+c)+a*sin(d*x+c))/(a^2+b^2)^
3/d

________________________________________________________________________________________

Rubi [A]
time = 0.19, antiderivative size = 175, normalized size of antiderivative = 1.00, number of steps used = 4, 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}{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 (a^3 (-B)+3 a^2 A 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} \end {gather*}

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 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))^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] Result contains complex when optimal does not.
time = 5.21, size = 202, normalized size = 1.15 \begin {gather*} \frac {-\frac {i (A-i B) \log (i-\tan (c+d x))}{(a-i b)^3}+\frac {i (A+i B) \log (i+\tan (c+d x))}{(a+i b)^3}+\frac {2 \left (-3 a^2 A b+A b^3+a^3 B-3 a b^2 B\right ) \log (b+a \tan (c+d x))-\frac {b \left (a^2+b^2\right ) \left (b \left (5 a^2 A b+A b^3-3 a^3 B+a b^2 B\right )+\left (6 a^3 A b+2 a A b^3-4 a^4 B\right ) \tan (c+d x)\right )}{a^2 (b+a \tan (c+d x))^2}}{\left (a^2+b^2\right )^3}}{2 d} \end {gather*}

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)

________________________________________________________________________________________

Maple [A]
time = 0.40, size = 216, normalized size = 1.23

method	result	size
derivativedivides	\(\frac {\frac {\frac {\left (3 A \,a^{2} b -A \,b^{3}-B \,a^{3}+3 B a \,b^{2}\right ) \ln \left (\cot ^{2}\left (d x +c \right )+1\right )}{2}+\left (-A \,a^{3}+3 A a \,b^{2}-3 B \,a^{2} b +B \,b^{3}\right ) \left (\frac {\pi }{2}-\mathrm {arccot}\left (\cot \left (d x +c \right )\right )\right )}{\left (a^{2}+b^{2}\right )^{3}}-\frac {\left (3 A \,a^{2} b -A \,b^{3}-B \,a^{3}+3 B a \,b^{2}\right ) \ln \left (a +b \cot \left (d x +c \right )\right )}{\left (a^{2}+b^{2}\right )^{3}}+\frac {A b -B a}{2 \left (a^{2}+b^{2}\right ) \left (a +b \cot \left (d x +c \right )\right )^{2}}+\frac {2 A a b -B \,a^{2}+B \,b^{2}}{\left (a^{2}+b^{2}\right )^{2} \left (a +b \cot \left (d x +c \right )\right )}}{d}\)	\(216\)
default	\(\frac {\frac {\frac {\left (3 A \,a^{2} b -A \,b^{3}-B \,a^{3}+3 B a \,b^{2}\right ) \ln \left (\cot ^{2}\left (d x +c \right )+1\right )}{2}+\left (-A \,a^{3}+3 A a \,b^{2}-3 B \,a^{2} b +B \,b^{3}\right ) \left (\frac {\pi }{2}-\mathrm {arccot}\left (\cot \left (d x +c \right )\right )\right )}{\left (a^{2}+b^{2}\right )^{3}}-\frac {\left (3 A \,a^{2} b -A \,b^{3}-B \,a^{3}+3 B a \,b^{2}\right ) \ln \left (a +b \cot \left (d x +c \right )\right )}{\left (a^{2}+b^{2}\right )^{3}}+\frac {A b -B a}{2 \left (a^{2}+b^{2}\right ) \left (a +b \cot \left (d x +c \right )\right )^{2}}+\frac {2 A a b -B \,a^{2}+B \,b^{2}}{\left (a^{2}+b^{2}\right )^{2} \left (a +b \cot \left (d x +c \right )\right )}}{d}\)	\(216\)
norman	\(\frac {\frac {b^{2} \left (A \,a^{3}-3 A a \,b^{2}+3 B \,a^{2} b -B \,b^{3}\right ) x}{\left (a^{4}+2 a^{2} b^{2}+b^{4}\right ) \left (a^{2}+b^{2}\right )}+\frac {\left (A \,a^{3}-3 A a \,b^{2}+3 B \,a^{2} b -B \,b^{3}\right ) a^{2} x \left (\tan ^{2}\left (d x +c \right )\right )}{\left (a^{4}+2 a^{2} b^{2}+b^{4}\right ) \left (a^{2}+b^{2}\right )}-\frac {b^{2} \left (5 A \,a^{2} b +A \,b^{3}-3 B \,a^{3}+B a \,b^{2}\right )}{2 d \,a^{2} \left (a^{4}+2 a^{2} b^{2}+b^{4}\right )}-\frac {b \left (3 A \,a^{2} b +A \,b^{3}-2 B \,a^{3}\right ) \tan \left (d x +c \right )}{d a \left (a^{4}+2 a^{2} b^{2}+b^{4}\right )}+\frac {2 a b \left (A \,a^{3}-3 A a \,b^{2}+3 B \,a^{2} b -B \,b^{3}\right ) x \tan \left (d x +c \right )}{\left (a^{4}+2 a^{2} b^{2}+b^{4}\right ) \left (a^{2}+b^{2}\right )}}{\left (a \tan \left (d x +c \right )+b \right )^{2}}+\frac {\left (3 A \,a^{2} b -A \,b^{3}-B \,a^{3}+3 B a \,b^{2}\right ) \ln \left (1+\tan ^{2}\left (d x +c \right )\right )}{2 d \left (a^{6}+3 a^{4} b^{2}+3 a^{2} b^{4}+b^{6}\right )}-\frac {\left (3 A \,a^{2} b -A \,b^{3}-B \,a^{3}+3 B a \,b^{2}\right ) \ln \left (a \tan \left (d x +c \right )+b \right )}{d \left (a^{6}+3 a^{4} b^{2}+3 a^{2} b^{4}+b^{6}\right )}\)	\(441\)
risch	\(\frac {i x B}{3 i a^{2} b -i b^{3}+a^{3}-3 a \,b^{2}}+\frac {x A}{3 i a^{2} b -i b^{3}+a^{3}-3 a \,b^{2}}+\frac {6 i A \,a^{2} b x}{a^{6}+3 a^{4} b^{2}+3 a^{2} b^{4}+b^{6}}-\frac {2 i A \,b^{3} x}{a^{6}+3 a^{4} b^{2}+3 a^{2} b^{4}+b^{6}}-\frac {2 i B \,a^{3} x}{a^{6}+3 a^{4} b^{2}+3 a^{2} b^{4}+b^{6}}+\frac {6 i B a \,b^{2} x}{a^{6}+3 a^{4} b^{2}+3 a^{2} b^{4}+b^{6}}+\frac {6 i A \,a^{2} b c}{d \left (a^{6}+3 a^{4} b^{2}+3 a^{2} b^{4}+b^{6}\right )}-\frac {2 i A \,b^{3} c}{d \left (a^{6}+3 a^{4} b^{2}+3 a^{2} b^{4}+b^{6}\right )}-\frac {2 i B \,a^{3} c}{d \left (a^{6}+3 a^{4} b^{2}+3 a^{2} b^{4}+b^{6}\right )}+\frac {6 i B a \,b^{2} c}{d \left (a^{6}+3 a^{4} b^{2}+3 a^{2} b^{4}+b^{6}\right )}+\frac {2 i \left (-2 i A a \,b^{3} {\mathrm e}^{2 i \left (d x +c \right )}+i B \,a^{2} b^{2} {\mathrm e}^{2 i \left (d x +c \right )}-A \,b^{4} {\mathrm e}^{2 i \left (d x +c \right )}-3 i A a \,b^{3}+2 i B \,a^{2} b^{2}+B a \,b^{3}-i B \,b^{4} {\mathrm e}^{2 i \left (d x +c \right )}-3 A \,a^{2} b^{2} {\mathrm e}^{2 i \left (d x +c \right )}+2 B \,a^{3} b \,{\mathrm e}^{2 i \left (d x +c \right )}-i B \,b^{4}+3 A \,a^{2} b^{2}-2 B \,a^{3} b \right )}{\left (-i b +a \right )^{2} \left (i {\mathrm e}^{2 i \left (d x +c \right )} b +a \,{\mathrm e}^{2 i \left (d x +c \right )}+i b -a \right )^{2} d \left (i b +a \right )^{3}}-\frac {3 \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}+\frac {i b -a}{i b +a}\right ) A \,a^{2} b}{d \left (a^{6}+3 a^{4} b^{2}+3 a^{2} b^{4}+b^{6}\right )}+\frac {\ln \left ({\mathrm e}^{2 i \left (d x +c \right )}+\frac {i b -a}{i b +a}\right ) A \,b^{3}}{d \left (a^{6}+3 a^{4} b^{2}+3 a^{2} b^{4}+b^{6}\right )}+\frac {\ln \left ({\mathrm e}^{2 i \left (d x +c \right )}+\frac {i b -a}{i b +a}\right ) B \,a^{3}}{d \left (a^{6}+3 a^{4} b^{2}+3 a^{2} b^{4}+b^{6}\right )}-\frac {3 \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}+\frac {i b -a}{i b +a}\right ) B a \,b^{2}}{d \left (a^{6}+3 a^{4} b^{2}+3 a^{2} b^{4}+b^{6}\right )}\)	\(793\)

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,method=_RETURNVERBOSE)

[Out]

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

________________________________________________________________________________________

Maxima [A]
time = 0.54, size = 337, normalized size = 1.93 \begin {gather*} \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 {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))^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

________________________________________________________________________________________

Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 549 vs. \(2 (171) = 342\).
time = 2.86, size = 549, normalized size = 3.14 \begin {gather*} \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 {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))^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)

________________________________________________________________________________________

Sympy [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: AttributeError} \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))**3,x)

[Out]

Exception raised: AttributeError >> 'NoneType' object has no attribute 'primitive'

________________________________________________________________________________________

Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 412 vs. \(2 (171) = 342\).
time = 0.58, size = 412, normalized size = 2.35 \begin {gather*} \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 {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))^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

________________________________________________________________________________________

Mupad [B]
time = 2.60, size = 481, normalized size = 2.75 \begin {gather*} \frac {\frac {5\,A\,a^2\,b+A\,b^3}{2\,\left (a^4+2\,a^2\,b^2+b^4\right )}+\frac {2\,A\,a\,b^2\,\mathrm {cot}\left (c+d\,x\right )}{a^4+2\,a^2\,b^2+b^4}}{d\,a^2+2\,d\,a\,b\,\mathrm {cot}\left (c+d\,x\right )+d\,b^2\,{\mathrm {cot}\left (c+d\,x\right )}^2}-\ln \left (a+b\,\mathrm {cot}\left (c+d\,x\right )\right )\,\left (\frac {3\,A\,b}{d\,{\left (a^2+b^2\right )}^2}-\frac {4\,A\,b^3}{d\,{\left (a^2+b^2\right )}^3}\right )-\frac {\frac {3\,B\,a^3-B\,a\,b^2}{2\,\left (a^4+2\,a^2\,b^2+b^4\right )}-\frac {\mathrm {cot}\left (c+d\,x\right )\,\left (B\,b^3-B\,a^2\,b\right )}{a^4+2\,a^2\,b^2+b^4}}{d\,a^2+2\,d\,a\,b\,\mathrm {cot}\left (c+d\,x\right )+d\,b^2\,{\mathrm {cot}\left (c+d\,x\right )}^2}+\ln \left (a+b\,\mathrm {cot}\left (c+d\,x\right )\right )\,\left (\frac {B\,a}{d\,{\left (a^2+b^2\right )}^2}-\frac {4\,B\,a\,b^2}{d\,{\left (a^2+b^2\right )}^3}\right )+\frac {A\,\ln \left (\mathrm {cot}\left (c+d\,x\right )-\mathrm {i}\right )\,1{}\mathrm {i}}{2\,\left (d\,a^3+3{}\mathrm {i}\,d\,a^2\,b-3\,d\,a\,b^2-1{}\mathrm {i}\,d\,b^3\right )}+\frac {A\,\ln \left (\mathrm {cot}\left (c+d\,x\right )+1{}\mathrm {i}\right )}{2\,\left (1{}\mathrm {i}\,d\,a^3+3\,d\,a^2\,b-3{}\mathrm {i}\,d\,a\,b^2-d\,b^3\right )}-\frac {B\,\ln \left (\mathrm {cot}\left (c+d\,x\right )-\mathrm {i}\right )\,1{}\mathrm {i}}{2\,\left (1{}\mathrm {i}\,d\,a^3-3\,d\,a^2\,b-3{}\mathrm {i}\,d\,a\,b^2+d\,b^3\right )}-\frac {B\,\ln \left (\mathrm {cot}\left (c+d\,x\right )+1{}\mathrm {i}\right )}{2\,\left (d\,a^3-3{}\mathrm {i}\,d\,a^2\,b-3\,d\,a\,b^2+1{}\mathrm {i}\,d\,b^3\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))^3,x)

[Out]

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

________________________________________________________________________________________