\(\int \frac {\cot ^2(c+d x)}{(a+b \tan (c+d x))^3} \, dx\) [485]

Optimal result

Integrand size = 21, antiderivative size = 211 \[ \int \frac {\cot ^2(c+d x)}{(a+b \tan (c+d x))^3} \, dx=-\frac {a \left (a^2-3 b^2\right ) x}{\left (a^2+b^2\right )^3}-\frac {3 b \log (\sin (c+d x))}{a^4 d}+\frac {b^3 \left (10 a^4+9 a^2 b^2+3 b^4\right ) \log (a \cos (c+d x)+b \sin (c+d x))}{a^4 \left (a^2+b^2\right )^3 d}-\frac {b \left (2 a^2+3 b^2\right )}{2 a^2 \left (a^2+b^2\right ) d (a+b \tan (c+d x))^2}-\frac {\cot (c+d x)}{a d (a+b \tan (c+d x))^2}-\frac {b \left (a^4+6 a^2 b^2+3 b^4\right )}{a^3 \left (a^2+b^2\right )^2 d (a+b \tan (c+d x))} \]

[Out]

-a*(a^2-3*b^2)*x/(a^2+b^2)^3-3*b*ln(sin(d*x+c))/a^4/d+b^3*(10*a^4+9*a^2*b^2+3*b^4)*ln(a*cos(d*x+c)+b*sin(d*x+c
))/a^4/(a^2+b^2)^3/d-1/2*b*(2*a^2+3*b^2)/a^2/(a^2+b^2)/d/(a+b*tan(d*x+c))^2-cot(d*x+c)/a/d/(a+b*tan(d*x+c))^2-
b*(a^4+6*a^2*b^2+3*b^4)/a^3/(a^2+b^2)^2/d/(a+b*tan(d*x+c))

Rubi [A] (verified)

Time = 0.70 (sec) , antiderivative size = 211, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.238, Rules used = {3650, 3730, 3732, 3611, 3556} \[ \int \frac {\cot ^2(c+d x)}{(a+b \tan (c+d x))^3} \, dx=-\frac {3 b \log (\sin (c+d x))}{a^4 d}-\frac {b \left (2 a^2+3 b^2\right )}{2 a^2 d \left (a^2+b^2\right ) (a+b \tan (c+d x))^2}-\frac {a x \left (a^2-3 b^2\right )}{\left (a^2+b^2\right )^3}+\frac {b^3 \left (10 a^4+9 a^2 b^2+3 b^4\right ) \log (a \cos (c+d x)+b \sin (c+d x))}{a^4 d \left (a^2+b^2\right )^3}-\frac {b \left (a^4+6 a^2 b^2+3 b^4\right )}{a^3 d \left (a^2+b^2\right )^2 (a+b \tan (c+d x))}-\frac {\cot (c+d x)}{a d (a+b \tan (c+d x))^2} \]

[In]

Int[Cot[c + d*x]^2/(a + b*Tan[c + d*x])^3,x]

[Out]

-((a*(a^2 - 3*b^2)*x)/(a^2 + b^2)^3) - (3*b*Log[Sin[c + d*x]])/(a^4*d) + (b^3*(10*a^4 + 9*a^2*b^2 + 3*b^4)*Log
[a*Cos[c + d*x] + b*Sin[c + d*x]])/(a^4*(a^2 + b^2)^3*d) - (b*(2*a^2 + 3*b^2))/(2*a^2*(a^2 + b^2)*d*(a + b*Tan
[c + d*x])^2) - Cot[c + d*x]/(a*d*(a + b*Tan[c + d*x])^2) - (b*(a^4 + 6*a^2*b^2 + 3*b^4))/(a^3*(a^2 + b^2)^2*d
*(a + b*Tan[c + d*x]))

Rule 3556

Int[tan[(c_.) + (d_.)*(x_)], x_Symbol] :> Simp[-Log[RemoveContent[Cos[c + d*x], x]]/d, x] /; FreeQ[{c, d}, x]

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 3650

Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Si
mp[b^2*(a + b*Tan[e + f*x])^(m + 1)*((c + d*Tan[e + f*x])^(n + 1)/(f*(m + 1)*(a^2 + b^2)*(b*c - a*d))), x] + D
ist[1/((m + 1)*(a^2 + b^2)*(b*c - a*d)), Int[(a + b*Tan[e + f*x])^(m + 1)*(c + d*Tan[e + f*x])^n*Simp[a*(b*c -
a*d)*(m + 1) - b^2*d*(m + n + 2) - b*(b*c - a*d)*(m + 1)*Tan[e + f*x] - b^2*d*(m + n + 2)*Tan[e + f*x]^2, x],
x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 + b^2, 0] && NeQ[c^2 + d^2, 0] && I
ntegerQ[2*m] && LtQ[m, -1] && (LtQ[n, 0] || IntegerQ[m]) &&  !(ILtQ[n, -1] && ( !IntegerQ[m] || (EqQ[c, 0] &&
NeQ[a, 0])))

Rule 3730

Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)])^(n_)*((A_.) + (B_.)*t
an[(e_.) + (f_.)*(x_)] + (C_.)*tan[(e_.) + (f_.)*(x_)]^2), x_Symbol] :> Simp[(A*b^2 - a*(b*B - a*C))*(a + b*Ta
n[e + f*x])^(m + 1)*((c + d*Tan[e + f*x])^(n + 1)/(f*(m + 1)*(b*c - a*d)*(a^2 + b^2))), x] + Dist[1/((m + 1)*(
b*c - a*d)*(a^2 + b^2)), Int[(a + b*Tan[e + f*x])^(m + 1)*(c + d*Tan[e + f*x])^n*Simp[A*(a*(b*c - a*d)*(m + 1)
- b^2*d*(m + n + 2)) + (b*B - a*C)*(b*c*(m + 1) + a*d*(n + 1)) - (m + 1)*(b*c - a*d)*(A*b - a*B - b*C)*Tan[e
+ f*x] - d*(A*b^2 - a*(b*B - a*C))*(m + n + 2)*Tan[e + f*x]^2, x], x], x] /; FreeQ[{a, b, c, d, e, f, A, B, C,
n}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 + b^2, 0] && NeQ[c^2 + d^2, 0] && LtQ[m, -1] &&  !(ILtQ[n, -1] && ( !I
ntegerQ[m] || (EqQ[c, 0] && NeQ[a, 0])))

Rule 3732

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

Rubi steps \begin{align*} \text {integral}& = -\frac {\cot (c+d x)}{a d (a+b \tan (c+d x))^2}-\frac {\int \frac {\cot (c+d x) \left (3 b+a \tan (c+d x)+3 b \tan ^2(c+d x)\right )}{(a+b \tan (c+d x))^3} \, dx}{a} \\ & = -\frac {b \left (2 a^2+3 b^2\right )}{2 a^2 \left (a^2+b^2\right ) d (a+b \tan (c+d x))^2}-\frac {\cot (c+d x)}{a d (a+b \tan (c+d x))^2}-\frac {\int \frac {\cot (c+d x) \left (6 b \left (a^2+b^2\right )+2 a^3 \tan (c+d x)+2 b \left (2 a^2+3 b^2\right ) \tan ^2(c+d x)\right )}{(a+b \tan (c+d x))^2} \, dx}{2 a^2 \left (a^2+b^2\right )} \\ & = -\frac {b \left (2 a^2+3 b^2\right )}{2 a^2 \left (a^2+b^2\right ) d (a+b \tan (c+d x))^2}-\frac {\cot (c+d x)}{a d (a+b \tan (c+d x))^2}-\frac {b \left (a^4+6 a^2 b^2+3 b^4\right )}{a^3 \left (a^2+b^2\right )^2 d (a+b \tan (c+d x))}-\frac {\int \frac {\cot (c+d x) \left (6 b \left (a^2+b^2\right )^2+2 a^3 \left (a^2-b^2\right ) \tan (c+d x)+2 b \left (a^4+6 a^2 b^2+3 b^4\right ) \tan ^2(c+d x)\right )}{a+b \tan (c+d x)} \, dx}{2 a^3 \left (a^2+b^2\right )^2} \\ & = -\frac {a \left (a^2-3 b^2\right ) x}{\left (a^2+b^2\right )^3}-\frac {b \left (2 a^2+3 b^2\right )}{2 a^2 \left (a^2+b^2\right ) d (a+b \tan (c+d x))^2}-\frac {\cot (c+d x)}{a d (a+b \tan (c+d x))^2}-\frac {b \left (a^4+6 a^2 b^2+3 b^4\right )}{a^3 \left (a^2+b^2\right )^2 d (a+b \tan (c+d x))}-\frac {(3 b) \int \cot (c+d x) \, dx}{a^4}+\frac {\left (b^3 \left (10 a^4+9 a^2 b^2+3 b^4\right )\right ) \int \frac {b-a \tan (c+d x)}{a+b \tan (c+d x)} \, dx}{a^4 \left (a^2+b^2\right )^3} \\ & = -\frac {a \left (a^2-3 b^2\right ) x}{\left (a^2+b^2\right )^3}-\frac {3 b \log (\sin (c+d x))}{a^4 d}+\frac {b^3 \left (10 a^4+9 a^2 b^2+3 b^4\right ) \log (a \cos (c+d x)+b \sin (c+d x))}{a^4 \left (a^2+b^2\right )^3 d}-\frac {b \left (2 a^2+3 b^2\right )}{2 a^2 \left (a^2+b^2\right ) d (a+b \tan (c+d x))^2}-\frac {\cot (c+d x)}{a d (a+b \tan (c+d x))^2}-\frac {b \left (a^4+6 a^2 b^2+3 b^4\right )}{a^3 \left (a^2+b^2\right )^2 d (a+b \tan (c+d x))} \\ \end{align*}

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 4.62 (sec) , antiderivative size = 178, normalized size of antiderivative = 0.84 \[ \int \frac {\cot ^2(c+d x)}{(a+b \tan (c+d x))^3} \, dx=-\frac {\frac {2 \cot (c+d x)}{a^3}+\frac {b^5}{a^4 \left (a^2+b^2\right ) (b+a \cot (c+d x))^2}-\frac {2 b^4 \left (5 a^2+3 b^2\right )}{a^4 \left (a^2+b^2\right )^2 (b+a \cot (c+d x))}+\frac {\log (i-\cot (c+d x))}{(i a+b)^3}+\frac {\log (i+\cot (c+d x))}{(-i a+b)^3}-\frac {2 b^3 \left (10 a^4+9 a^2 b^2+3 b^4\right ) \log (b+a \cot (c+d x))}{a^4 \left (a^2+b^2\right )^3}}{2 d} \]

[In]

Integrate[Cot[c + d*x]^2/(a + b*Tan[c + d*x])^3,x]

[Out]

-1/2*((2*Cot[c + d*x])/a^3 + b^5/(a^4*(a^2 + b^2)*(b + a*Cot[c + d*x])^2) - (2*b^4*(5*a^2 + 3*b^2))/(a^4*(a^2
+ b^2)^2*(b + a*Cot[c + d*x])) + Log[I - Cot[c + d*x]]/(I*a + b)^3 + Log[I + Cot[c + d*x]]/((-I)*a + b)^3 - (2
*b^3*(10*a^4 + 9*a^2*b^2 + 3*b^4)*Log[b + a*Cot[c + d*x]])/(a^4*(a^2 + b^2)^3))/d

Maple [A] (verified)

Time = 1.07 (sec) , antiderivative size = 201, normalized size of antiderivative = 0.95

method	result	size
derivativedivides	\(\frac {\frac {\frac {\left (3 a^{2} b -b^{3}\right ) \ln \left (1+\tan ^{2}\left (d x +c \right )\right )}{2}+\left (-a^{3}+3 a \,b^{2}\right ) \arctan \left (\tan \left (d x +c \right )\right )}{\left (a^{2}+b^{2}\right )^{3}}-\frac {b^{3}}{2 a^{2} \left (a^{2}+b^{2}\right ) \left (a +b \tan \left (d x +c \right )\right )^{2}}+\frac {b^{3} \left (10 a^{4}+9 a^{2} b^{2}+3 b^{4}\right ) \ln \left (a +b \tan \left (d x +c \right )\right )}{a^{4} \left (a^{2}+b^{2}\right )^{3}}-\frac {2 b^{3} \left (2 a^{2}+b^{2}\right )}{a^{3} \left (a^{2}+b^{2}\right )^{2} \left (a +b \tan \left (d x +c \right )\right )}-\frac {1}{a^{3} \tan \left (d x +c \right )}-\frac {3 b \ln \left (\tan \left (d x +c \right )\right )}{a^{4}}}{d}\)	\(201\)
default	\(\frac {\frac {\frac {\left (3 a^{2} b -b^{3}\right ) \ln \left (1+\tan ^{2}\left (d x +c \right )\right )}{2}+\left (-a^{3}+3 a \,b^{2}\right ) \arctan \left (\tan \left (d x +c \right )\right )}{\left (a^{2}+b^{2}\right )^{3}}-\frac {b^{3}}{2 a^{2} \left (a^{2}+b^{2}\right ) \left (a +b \tan \left (d x +c \right )\right )^{2}}+\frac {b^{3} \left (10 a^{4}+9 a^{2} b^{2}+3 b^{4}\right ) \ln \left (a +b \tan \left (d x +c \right )\right )}{a^{4} \left (a^{2}+b^{2}\right )^{3}}-\frac {2 b^{3} \left (2 a^{2}+b^{2}\right )}{a^{3} \left (a^{2}+b^{2}\right )^{2} \left (a +b \tan \left (d x +c \right )\right )}-\frac {1}{a^{3} \tan \left (d x +c \right )}-\frac {3 b \ln \left (\tan \left (d x +c \right )\right )}{a^{4}}}{d}\)	\(201\)
parallelrisch	\(\frac {20 \left (a^{4}+\frac {9}{10} a^{2} b^{2}+\frac {3}{10} b^{4}\right ) \left (a +b \tan \left (d x +c \right )\right )^{2} b^{3} \ln \left (a +b \tan \left (d x +c \right )\right )+3 a^{4} \left (a^{2}-\frac {b^{2}}{3}\right ) \left (a +b \tan \left (d x +c \right )\right )^{2} b \ln \left (\sec ^{2}\left (d x +c \right )\right )-6 b \left (a^{2}+b^{2}\right )^{3} \left (a +b \tan \left (d x +c \right )\right )^{2} \ln \left (\tan \left (d x +c \right )\right )+\left (-2 x \,a^{7} b^{2} d +6 x \,a^{5} b^{4} d +4 a^{6} b^{3}+21 a^{4} b^{5}+26 a^{2} b^{7}+9 b^{9}\right ) \left (\tan ^{2}\left (d x +c \right )\right )+\left (-4 x \,a^{8} b d +12 x \,a^{6} b^{3} d +6 b^{2} a^{7}+28 b^{4} a^{5}+34 b^{6} a^{3}+12 a \,b^{8}\right ) \tan \left (d x +c \right )-2 \left (\left (a^{2}+b^{2}\right )^{3} \cot \left (d x +c \right )+a^{4} d x \left (a^{2}-3 b^{2}\right )\right ) a^{3}}{2 d \,a^{4} \left (a^{2}+b^{2}\right )^{3} \left (a +b \tan \left (d x +c \right )\right )^{2}}\)	\(295\)
norman	\(\frac {\frac {b \left (3 a^{4} b +11 a^{2} b^{3}+6 b^{5}\right ) \left (\tan ^{2}\left (d x +c \right )\right )}{d \,a^{3} \left (a^{4}+2 a^{2} b^{2}+b^{4}\right )}-\frac {1}{a d}+\frac {b^{2} \left (4 a^{4} b +17 a^{2} b^{3}+9 b^{5}\right ) \left (\tan ^{3}\left (d x +c \right )\right )}{2 d \,a^{4} \left (a^{4}+2 a^{2} b^{2}+b^{4}\right )}-\frac {\left (a^{2}-3 b^{2}\right ) a^{3} x \tan \left (d x +c \right )}{\left (a^{4}+2 a^{2} b^{2}+b^{4}\right ) \left (a^{2}+b^{2}\right )}-\frac {2 b \left (a^{2}-3 b^{2}\right ) a^{2} x \left (\tan ^{2}\left (d x +c \right )\right )}{\left (a^{4}+2 a^{2} b^{2}+b^{4}\right ) \left (a^{2}+b^{2}\right )}-\frac {b^{2} \left (a^{2}-3 b^{2}\right ) a x \left (\tan ^{3}\left (d x +c \right )\right )}{\left (a^{4}+2 a^{2} b^{2}+b^{4}\right ) \left (a^{2}+b^{2}\right )}}{\tan \left (d x +c \right ) \left (a +b \tan \left (d x +c \right )\right )^{2}}+\frac {b^{3} \left (10 a^{4}+9 a^{2} b^{2}+3 b^{4}\right ) \ln \left (a +b \tan \left (d x +c \right )\right )}{d \,a^{4} \left (a^{6}+3 a^{4} b^{2}+3 a^{2} b^{4}+b^{6}\right )}-\frac {3 b \ln \left (\tan \left (d x +c \right )\right )}{a^{4} d}+\frac {b \left (3 a^{2}-b^{2}\right ) \ln \left (1+\tan ^{2}\left (d x +c \right )\right )}{2 d \left (a^{6}+3 a^{4} b^{2}+3 a^{2} b^{4}+b^{6}\right )}\)	\(422\)
risch	\(\frac {x}{3 i b \,a^{2}-i b^{3}-a^{3}+3 a \,b^{2}}+\frac {6 i b x}{a^{4}}+\frac {6 i b c}{d \,a^{4}}-\frac {20 i b^{3} x}{a^{6}+3 a^{4} b^{2}+3 a^{2} b^{4}+b^{6}}-\frac {20 i b^{3} c}{d \left (a^{6}+3 a^{4} b^{2}+3 a^{2} b^{4}+b^{6}\right )}-\frac {18 i b^{5} x}{a^{2} \left (a^{6}+3 a^{4} b^{2}+3 a^{2} b^{4}+b^{6}\right )}-\frac {18 i b^{5} c}{d \,a^{2} \left (a^{6}+3 a^{4} b^{2}+3 a^{2} b^{4}+b^{6}\right )}-\frac {6 i b^{7} x}{a^{4} \left (a^{6}+3 a^{4} b^{2}+3 a^{2} b^{4}+b^{6}\right )}-\frac {6 i b^{7} c}{d \,a^{4} \left (a^{6}+3 a^{4} b^{2}+3 a^{2} b^{4}+b^{6}\right )}-\frac {2 i \left (-15 i a^{2} b^{5} {\mathrm e}^{2 i \left (d x +c \right )}+a^{7}+3 i a^{4} b^{3}-6 i a^{4} b^{3} {\mathrm e}^{2 i \left (d x +c \right )}+3 i a^{2} b^{5} {\mathrm e}^{4 i \left (d x +c \right )}+6 a^{5} b^{2} {\mathrm e}^{2 i \left (d x +c \right )}+i a^{6} b +3 a^{5} b^{2}+a^{7} {\mathrm e}^{4 i \left (d x +c \right )}+2 a^{7} {\mathrm e}^{2 i \left (d x +c \right )}+3 a \,b^{6}+3 a \,b^{6} {\mathrm e}^{2 i \left (d x +c \right )}-a^{5} b^{2} {\mathrm e}^{4 i \left (d x +c \right )}-10 a^{3} b^{4} {\mathrm e}^{4 i \left (d x +c \right )}-6 a \,b^{6} {\mathrm e}^{4 i \left (d x +c \right )}+6 a^{3} b^{4} {\mathrm e}^{2 i \left (d x +c \right )}-6 i b^{7} {\mathrm e}^{2 i \left (d x +c \right )}+8 a^{3} b^{4}+3 i b^{7} {\mathrm e}^{4 i \left (d x +c \right )}+8 i a^{2} b^{5}-5 i a^{4} b^{3} {\mathrm e}^{4 i \left (d x +c \right )}-3 i a^{6} b \,{\mathrm e}^{4 i \left (d x +c \right )}-2 i a^{6} b \,{\mathrm e}^{2 i \left (d x +c \right )}+3 i b^{7}\right )}{\left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right ) \left (i b +a \right )^{2} \left (-i b \,{\mathrm e}^{2 i \left (d x +c \right )}+a \,{\mathrm e}^{2 i \left (d x +c \right )}+i b +a \right )^{2} \left (-i b +a \right )^{3} a^{3} d}-\frac {3 b \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )}{a^{4} d}+\frac {10 b^{3} \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}-\frac {i b +a}{i b -a}\right )}{d \left (a^{6}+3 a^{4} b^{2}+3 a^{2} b^{4}+b^{6}\right )}+\frac {9 b^{5} \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}-\frac {i b +a}{i b -a}\right )}{d \,a^{2} \left (a^{6}+3 a^{4} b^{2}+3 a^{2} b^{4}+b^{6}\right )}+\frac {3 b^{7} \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}-\frac {i b +a}{i b -a}\right )}{d \,a^{4} \left (a^{6}+3 a^{4} b^{2}+3 a^{2} b^{4}+b^{6}\right )}\)	\(862\)

[In]

int(cot(d*x+c)^2/(a+b*tan(d*x+c))^3,x,method=_RETURNVERBOSE)

[Out]

1/d*(1/(a^2+b^2)^3*(1/2*(3*a^2*b-b^3)*ln(1+tan(d*x+c)^2)+(-a^3+3*a*b^2)*arctan(tan(d*x+c)))-1/2*b^3/a^2/(a^2+b
^2)/(a+b*tan(d*x+c))^2+b^3*(10*a^4+9*a^2*b^2+3*b^4)/a^4/(a^2+b^2)^3*ln(a+b*tan(d*x+c))-2*b^3*(2*a^2+b^2)/a^3/(
a^2+b^2)^2/(a+b*tan(d*x+c))-1/a^3/tan(d*x+c)-3/a^4*b*ln(tan(d*x+c)))

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 585 vs. \(2 (209) = 418\).

Time = 0.30 (sec) , antiderivative size = 585, normalized size of antiderivative = 2.77 \[ \int \frac {\cot ^2(c+d x)}{(a+b \tan (c+d x))^3} \, dx=-\frac {2 \, a^{9} + 6 \, a^{7} b^{2} + 6 \, a^{5} b^{4} + 2 \, a^{3} b^{6} - {\left (9 \, a^{4} b^{5} + 3 \, a^{2} b^{7} - 2 \, {\left (a^{7} b^{2} - 3 \, a^{5} b^{4}\right )} d x\right )} \tan \left (d x + c\right )^{3} + 2 \, {\left (a^{7} b^{2} - 2 \, a^{5} b^{4} + 6 \, a^{3} b^{6} + 3 \, a b^{8} + 2 \, {\left (a^{8} b - 3 \, a^{6} b^{3}\right )} d x\right )} \tan \left (d x + c\right )^{2} + 3 \, {\left ({\left (a^{6} b^{3} + 3 \, a^{4} b^{5} + 3 \, a^{2} b^{7} + b^{9}\right )} \tan \left (d x + c\right )^{3} + 2 \, {\left (a^{7} b^{2} + 3 \, a^{5} b^{4} + 3 \, a^{3} b^{6} + a b^{8}\right )} \tan \left (d x + c\right )^{2} + {\left (a^{8} b + 3 \, a^{6} b^{3} + 3 \, a^{4} b^{5} + a^{2} b^{7}\right )} \tan \left (d x + c\right )\right )} \log \left (\frac {\tan \left (d x + c\right )^{2}}{\tan \left (d x + c\right )^{2} + 1}\right ) - {\left ({\left (10 \, a^{4} b^{5} + 9 \, a^{2} b^{7} + 3 \, b^{9}\right )} \tan \left (d x + c\right )^{3} + 2 \, {\left (10 \, a^{5} b^{4} + 9 \, a^{3} b^{6} + 3 \, a b^{8}\right )} \tan \left (d x + c\right )^{2} + {\left (10 \, a^{6} b^{3} + 9 \, a^{4} b^{5} + 3 \, a^{2} b^{7}\right )} \tan \left (d x + c\right )\right )} \log \left (\frac {b^{2} \tan \left (d x + c\right )^{2} + 2 \, a b \tan \left (d x + c\right ) + a^{2}}{\tan \left (d x + c\right )^{2} + 1}\right ) + {\left (4 \, a^{8} b + 12 \, a^{6} b^{3} + 23 \, a^{4} b^{5} + 9 \, a^{2} b^{7} + 2 \, {\left (a^{9} - 3 \, a^{7} b^{2}\right )} d x\right )} \tan \left (d x + c\right )}{2 \, {\left ({\left (a^{10} b^{2} + 3 \, a^{8} b^{4} + 3 \, a^{6} b^{6} + a^{4} b^{8}\right )} d \tan \left (d x + c\right )^{3} + 2 \, {\left (a^{11} b + 3 \, a^{9} b^{3} + 3 \, a^{7} b^{5} + a^{5} b^{7}\right )} d \tan \left (d x + c\right )^{2} + {\left (a^{12} + 3 \, a^{10} b^{2} + 3 \, a^{8} b^{4} + a^{6} b^{6}\right )} d \tan \left (d x + c\right )\right )}} \]

[In]

integrate(cot(d*x+c)^2/(a+b*tan(d*x+c))^3,x, algorithm="fricas")

[Out]

-1/2*(2*a^9 + 6*a^7*b^2 + 6*a^5*b^4 + 2*a^3*b^6 - (9*a^4*b^5 + 3*a^2*b^7 - 2*(a^7*b^2 - 3*a^5*b^4)*d*x)*tan(d*
x + c)^3 + 2*(a^7*b^2 - 2*a^5*b^4 + 6*a^3*b^6 + 3*a*b^8 + 2*(a^8*b - 3*a^6*b^3)*d*x)*tan(d*x + c)^2 + 3*((a^6*
b^3 + 3*a^4*b^5 + 3*a^2*b^7 + b^9)*tan(d*x + c)^3 + 2*(a^7*b^2 + 3*a^5*b^4 + 3*a^3*b^6 + a*b^8)*tan(d*x + c)^2
+ (a^8*b + 3*a^6*b^3 + 3*a^4*b^5 + a^2*b^7)*tan(d*x + c))*log(tan(d*x + c)^2/(tan(d*x + c)^2 + 1)) - ((10*a^4
*b^5 + 9*a^2*b^7 + 3*b^9)*tan(d*x + c)^3 + 2*(10*a^5*b^4 + 9*a^3*b^6 + 3*a*b^8)*tan(d*x + c)^2 + (10*a^6*b^3 +
9*a^4*b^5 + 3*a^2*b^7)*tan(d*x + c))*log((b^2*tan(d*x + c)^2 + 2*a*b*tan(d*x + c) + a^2)/(tan(d*x + c)^2 + 1)
) + (4*a^8*b + 12*a^6*b^3 + 23*a^4*b^5 + 9*a^2*b^7 + 2*(a^9 - 3*a^7*b^2)*d*x)*tan(d*x + c))/((a^10*b^2 + 3*a^8
*b^4 + 3*a^6*b^6 + a^4*b^8)*d*tan(d*x + c)^3 + 2*(a^11*b + 3*a^9*b^3 + 3*a^7*b^5 + a^5*b^7)*d*tan(d*x + c)^2 +
(a^12 + 3*a^10*b^2 + 3*a^8*b^4 + a^6*b^6)*d*tan(d*x + c))

Sympy [F(-2)]

Exception generated. \[ \int \frac {\cot ^2(c+d x)}{(a+b \tan (c+d x))^3} \, dx=\text {Exception raised: AttributeError} \]

[In]

integrate(cot(d*x+c)**2/(a+b*tan(d*x+c))**3,x)

[Out]

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

Maxima [A] (verification not implemented)

none

Time = 0.41 (sec) , antiderivative size = 349, normalized size of antiderivative = 1.65 \[ \int \frac {\cot ^2(c+d x)}{(a+b \tan (c+d x))^3} \, dx=-\frac {\frac {2 \, {\left (a^{3} - 3 \, a b^{2}\right )} {\left (d x + c\right )}}{a^{6} + 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} + b^{6}} - \frac {2 \, {\left (10 \, a^{4} b^{3} + 9 \, a^{2} b^{5} + 3 \, b^{7}\right )} \log \left (b \tan \left (d x + c\right ) + a\right )}{a^{10} + 3 \, a^{8} b^{2} + 3 \, a^{6} b^{4} + a^{4} b^{6}} - \frac {{\left (3 \, a^{2} b - b^{3}\right )} \log \left (\tan \left (d x + c\right )^{2} + 1\right )}{a^{6} + 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} + b^{6}} + \frac {2 \, a^{6} + 4 \, a^{4} b^{2} + 2 \, a^{2} b^{4} + 2 \, {\left (a^{4} b^{2} + 6 \, a^{2} b^{4} + 3 \, b^{6}\right )} \tan \left (d x + c\right )^{2} + {\left (4 \, a^{5} b + 17 \, a^{3} b^{3} + 9 \, a b^{5}\right )} \tan \left (d x + c\right )}{{\left (a^{7} b^{2} + 2 \, a^{5} b^{4} + a^{3} b^{6}\right )} \tan \left (d x + c\right )^{3} + 2 \, {\left (a^{8} b + 2 \, a^{6} b^{3} + a^{4} b^{5}\right )} \tan \left (d x + c\right )^{2} + {\left (a^{9} + 2 \, a^{7} b^{2} + a^{5} b^{4}\right )} \tan \left (d x + c\right )} + \frac {6 \, b \log \left (\tan \left (d x + c\right )\right )}{a^{4}}}{2 \, d} \]

[In]

integrate(cot(d*x+c)^2/(a+b*tan(d*x+c))^3,x, algorithm="maxima")

[Out]

-1/2*(2*(a^3 - 3*a*b^2)*(d*x + c)/(a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6) - 2*(10*a^4*b^3 + 9*a^2*b^5 + 3*b^7)*log
(b*tan(d*x + c) + a)/(a^10 + 3*a^8*b^2 + 3*a^6*b^4 + a^4*b^6) - (3*a^2*b - b^3)*log(tan(d*x + c)^2 + 1)/(a^6 +
3*a^4*b^2 + 3*a^2*b^4 + b^6) + (2*a^6 + 4*a^4*b^2 + 2*a^2*b^4 + 2*(a^4*b^2 + 6*a^2*b^4 + 3*b^6)*tan(d*x + c)^
2 + (4*a^5*b + 17*a^3*b^3 + 9*a*b^5)*tan(d*x + c))/((a^7*b^2 + 2*a^5*b^4 + a^3*b^6)*tan(d*x + c)^3 + 2*(a^8*b
+ 2*a^6*b^3 + a^4*b^5)*tan(d*x + c)^2 + (a^9 + 2*a^7*b^2 + a^5*b^4)*tan(d*x + c)) + 6*b*log(tan(d*x + c))/a^4)
/d

Giac [A] (verification not implemented)

none

Time = 0.86 (sec) , antiderivative size = 357, normalized size of antiderivative = 1.69 \[ \int \frac {\cot ^2(c+d x)}{(a+b \tan (c+d x))^3} \, dx=-\frac {\frac {2 \, {\left (a^{3} - 3 \, a b^{2}\right )} {\left (d x + c\right )}}{a^{6} + 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} + b^{6}} - \frac {{\left (3 \, a^{2} b - b^{3}\right )} \log \left (\tan \left (d x + c\right )^{2} + 1\right )}{a^{6} + 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} + b^{6}} - \frac {2 \, {\left (10 \, a^{4} b^{4} + 9 \, a^{2} b^{6} + 3 \, b^{8}\right )} \log \left ({\left | b \tan \left (d x + c\right ) + a \right |}\right )}{a^{10} b + 3 \, a^{8} b^{3} + 3 \, a^{6} b^{5} + a^{4} b^{7}} + \frac {30 \, a^{4} b^{5} \tan \left (d x + c\right )^{2} + 27 \, a^{2} b^{7} \tan \left (d x + c\right )^{2} + 9 \, b^{9} \tan \left (d x + c\right )^{2} + 68 \, a^{5} b^{4} \tan \left (d x + c\right ) + 66 \, a^{3} b^{6} \tan \left (d x + c\right ) + 22 \, a b^{8} \tan \left (d x + c\right ) + 39 \, a^{6} b^{3} + 41 \, a^{4} b^{5} + 14 \, a^{2} b^{7}}{{\left (a^{10} + 3 \, a^{8} b^{2} + 3 \, a^{6} b^{4} + a^{4} b^{6}\right )} {\left (b \tan \left (d x + c\right ) + a\right )}^{2}} + \frac {6 \, b \log \left ({\left | \tan \left (d x + c\right ) \right |}\right )}{a^{4}} - \frac {2 \, {\left (3 \, b \tan \left (d x + c\right ) - a\right )}}{a^{4} \tan \left (d x + c\right )}}{2 \, d} \]

[In]

integrate(cot(d*x+c)^2/(a+b*tan(d*x+c))^3,x, algorithm="giac")

[Out]

-1/2*(2*(a^3 - 3*a*b^2)*(d*x + c)/(a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6) - (3*a^2*b - b^3)*log(tan(d*x + c)^2 + 1
)/(a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6) - 2*(10*a^4*b^4 + 9*a^2*b^6 + 3*b^8)*log(abs(b*tan(d*x + c) + a))/(a^10*
b + 3*a^8*b^3 + 3*a^6*b^5 + a^4*b^7) + (30*a^4*b^5*tan(d*x + c)^2 + 27*a^2*b^7*tan(d*x + c)^2 + 9*b^9*tan(d*x
+ c)^2 + 68*a^5*b^4*tan(d*x + c) + 66*a^3*b^6*tan(d*x + c) + 22*a*b^8*tan(d*x + c) + 39*a^6*b^3 + 41*a^4*b^5 +
14*a^2*b^7)/((a^10 + 3*a^8*b^2 + 3*a^6*b^4 + a^4*b^6)*(b*tan(d*x + c) + a)^2) + 6*b*log(abs(tan(d*x + c)))/a^
4 - 2*(3*b*tan(d*x + c) - a)/(a^4*tan(d*x + c)))/d

Mupad [B] (verification not implemented)

Time = 5.17 (sec) , antiderivative size = 293, normalized size of antiderivative = 1.39 \[ \int \frac {\cot ^2(c+d x)}{(a+b \tan (c+d x))^3} \, dx=\frac {b^3\,\ln \left (a+b\,\mathrm {tan}\left (c+d\,x\right )\right )\,\left (10\,a^4+9\,a^2\,b^2+3\,b^4\right )}{a^4\,d\,{\left (a^2+b^2\right )}^3}-\frac {\frac {1}{a}+\frac {{\mathrm {tan}\left (c+d\,x\right )}^2\,\left (a^4\,b^2+6\,a^2\,b^4+3\,b^6\right )}{a^3\,\left (a^4+2\,a^2\,b^2+b^4\right )}+\frac {\mathrm {tan}\left (c+d\,x\right )\,\left (4\,a^4\,b+17\,a^2\,b^3+9\,b^5\right )}{2\,a^2\,\left (a^4+2\,a^2\,b^2+b^4\right )}}{d\,\left (a^2\,\mathrm {tan}\left (c+d\,x\right )+2\,a\,b\,{\mathrm {tan}\left (c+d\,x\right )}^2+b^2\,{\mathrm {tan}\left (c+d\,x\right )}^3\right )}-\frac {3\,b\,\ln \left (\mathrm {tan}\left (c+d\,x\right )\right )}{a^4\,d}-\frac {\ln \left (\mathrm {tan}\left (c+d\,x\right )+1{}\mathrm {i}\right )}{2\,d\,\left (-a^3\,1{}\mathrm {i}-3\,a^2\,b+a\,b^2\,3{}\mathrm {i}+b^3\right )}-\frac {\ln \left (\mathrm {tan}\left (c+d\,x\right )-\mathrm {i}\right )\,1{}\mathrm {i}}{2\,d\,\left (-a^3-a^2\,b\,3{}\mathrm {i}+3\,a\,b^2+b^3\,1{}\mathrm {i}\right )} \]

[In]

int(cot(c + d*x)^2/(a + b*tan(c + d*x))^3,x)

[Out]

(b^3*log(a + b*tan(c + d*x))*(10*a^4 + 3*b^4 + 9*a^2*b^2))/(a^4*d*(a^2 + b^2)^3) - (log(tan(c + d*x) - 1i)*1i)
/(2*d*(3*a*b^2 - a^2*b*3i - a^3 + b^3*1i)) - (1/a + (tan(c + d*x)^2*(3*b^6 + 6*a^2*b^4 + a^4*b^2))/(a^3*(a^4 +
b^4 + 2*a^2*b^2)) + (tan(c + d*x)*(4*a^4*b + 9*b^5 + 17*a^2*b^3))/(2*a^2*(a^4 + b^4 + 2*a^2*b^2)))/(d*(a^2*ta
n(c + d*x) + b^2*tan(c + d*x)^3 + 2*a*b*tan(c + d*x)^2)) - (3*b*log(tan(c + d*x)))/(a^4*d) - log(tan(c + d*x)
+ 1i)/(2*d*(a*b^2*3i - 3*a^2*b - a^3*1i + b^3))