\(\int \frac {\cot ^2(c+d x)}{(a+b \tan (c+d x))^4} \, dx\) [494]

Optimal result

Integrand size = 21, antiderivative size = 278 \[ \int \frac {\cot ^2(c+d x)}{(a+b \tan (c+d x))^4} \, dx=-\frac {\left (a^4-6 a^2 b^2+b^4\right ) x}{\left (a^2+b^2\right )^4}-\frac {4 b \log (\sin (c+d x))}{a^5 d}+\frac {4 b^3 \left (5 a^6+6 a^4 b^2+4 a^2 b^4+b^6\right ) \log (a \cos (c+d x)+b \sin (c+d x))}{a^5 \left (a^2+b^2\right )^4 d}-\frac {b \left (3 a^2+4 b^2\right )}{3 a^2 \left (a^2+b^2\right ) d (a+b \tan (c+d x))^3}-\frac {\cot (c+d x)}{a d (a+b \tan (c+d x))^3}-\frac {b \left (a^4+4 a^2 b^2+2 b^4\right )}{a^3 \left (a^2+b^2\right )^2 d (a+b \tan (c+d x))^2}-\frac {b \left (a^6+13 a^4 b^2+12 a^2 b^4+4 b^6\right )}{a^4 \left (a^2+b^2\right )^3 d (a+b \tan (c+d x))} \]

[Out]

-(a^4-6*a^2*b^2+b^4)*x/(a^2+b^2)^4-4*b*ln(sin(d*x+c))/a^5/d+4*b^3*(5*a^6+6*a^4*b^2+4*a^2*b^4+b^6)*ln(a*cos(d*x
+c)+b*sin(d*x+c))/a^5/(a^2+b^2)^4/d-1/3*b*(3*a^2+4*b^2)/a^2/(a^2+b^2)/d/(a+b*tan(d*x+c))^3-cot(d*x+c)/a/d/(a+b
*tan(d*x+c))^3-b*(a^4+4*a^2*b^2+2*b^4)/a^3/(a^2+b^2)^2/d/(a+b*tan(d*x+c))^2-b*(a^6+13*a^4*b^2+12*a^2*b^4+4*b^6
)/a^4/(a^2+b^2)^3/d/(a+b*tan(d*x+c))

Rubi [A] (verified)

Time = 1.03 (sec) , antiderivative size = 278, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.238, Rules used = {3650, 3730, 3732, 3611, 3556} \[ \int \frac {\cot ^2(c+d x)}{(a+b \tan (c+d x))^4} \, dx=-\frac {4 b \log (\sin (c+d x))}{a^5 d}-\frac {b \left (3 a^2+4 b^2\right )}{3 a^2 d \left (a^2+b^2\right ) (a+b \tan (c+d x))^3}-\frac {x \left (a^4-6 a^2 b^2+b^4\right )}{\left (a^2+b^2\right )^4}-\frac {b \left (a^6+13 a^4 b^2+12 a^2 b^4+4 b^6\right )}{a^4 d \left (a^2+b^2\right )^3 (a+b \tan (c+d x))}-\frac {b \left (a^4+4 a^2 b^2+2 b^4\right )}{a^3 d \left (a^2+b^2\right )^2 (a+b \tan (c+d x))^2}+\frac {4 b^3 \left (5 a^6+6 a^4 b^2+4 a^2 b^4+b^6\right ) \log (a \cos (c+d x)+b \sin (c+d x))}{a^5 d \left (a^2+b^2\right )^4}-\frac {\cot (c+d x)}{a d (a+b \tan (c+d x))^3} \]

[In]

Int[Cot[c + d*x]^2/(a + b*Tan[c + d*x])^4,x]

[Out]

-(((a^4 - 6*a^2*b^2 + b^4)*x)/(a^2 + b^2)^4) - (4*b*Log[Sin[c + d*x]])/(a^5*d) + (4*b^3*(5*a^6 + 6*a^4*b^2 + 4
*a^2*b^4 + b^6)*Log[a*Cos[c + d*x] + b*Sin[c + d*x]])/(a^5*(a^2 + b^2)^4*d) - (b*(3*a^2 + 4*b^2))/(3*a^2*(a^2
+ b^2)*d*(a + b*Tan[c + d*x])^3) - Cot[c + d*x]/(a*d*(a + b*Tan[c + d*x])^3) - (b*(a^4 + 4*a^2*b^2 + 2*b^4))/(
a^3*(a^2 + b^2)^2*d*(a + b*Tan[c + d*x])^2) - (b*(a^6 + 13*a^4*b^2 + 12*a^2*b^4 + 4*b^6))/(a^4*(a^2 + b^2)^3*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))^3}-\frac {\int \frac {\cot (c+d x) \left (4 b+a \tan (c+d x)+4 b \tan ^2(c+d x)\right )}{(a+b \tan (c+d x))^4} \, dx}{a} \\ & = -\frac {b \left (3 a^2+4 b^2\right )}{3 a^2 \left (a^2+b^2\right ) d (a+b \tan (c+d x))^3}-\frac {\cot (c+d x)}{a d (a+b \tan (c+d x))^3}-\frac {\int \frac {\cot (c+d x) \left (12 b \left (a^2+b^2\right )+3 a^3 \tan (c+d x)+3 b \left (3 a^2+4 b^2\right ) \tan ^2(c+d x)\right )}{(a+b \tan (c+d x))^3} \, dx}{3 a^2 \left (a^2+b^2\right )} \\ & = -\frac {b \left (3 a^2+4 b^2\right )}{3 a^2 \left (a^2+b^2\right ) d (a+b \tan (c+d x))^3}-\frac {\cot (c+d x)}{a d (a+b \tan (c+d x))^3}-\frac {b \left (a^4+4 a^2 b^2+2 b^4\right )}{a^3 \left (a^2+b^2\right )^2 d (a+b \tan (c+d x))^2}-\frac {\int \frac {\cot (c+d x) \left (24 b \left (a^2+b^2\right )^2+6 a^3 \left (a^2-b^2\right ) \tan (c+d x)+12 b \left (a^4+4 a^2 b^2+2 b^4\right ) \tan ^2(c+d x)\right )}{(a+b \tan (c+d x))^2} \, dx}{6 a^3 \left (a^2+b^2\right )^2} \\ & = -\frac {b \left (3 a^2+4 b^2\right )}{3 a^2 \left (a^2+b^2\right ) d (a+b \tan (c+d x))^3}-\frac {\cot (c+d x)}{a d (a+b \tan (c+d x))^3}-\frac {b \left (a^4+4 a^2 b^2+2 b^4\right )}{a^3 \left (a^2+b^2\right )^2 d (a+b \tan (c+d x))^2}-\frac {b \left (a^6+13 a^4 b^2+12 a^2 b^4+4 b^6\right )}{a^4 \left (a^2+b^2\right )^3 d (a+b \tan (c+d x))}-\frac {\int \frac {\cot (c+d x) \left (24 b \left (a^2+b^2\right )^3+6 a^5 \left (a^2-3 b^2\right ) \tan (c+d x)+6 b \left (a^6+13 a^4 b^2+12 a^2 b^4+4 b^6\right ) \tan ^2(c+d x)\right )}{a+b \tan (c+d x)} \, dx}{6 a^4 \left (a^2+b^2\right )^3} \\ & = -\frac {\left (a^4-6 a^2 b^2+b^4\right ) x}{\left (a^2+b^2\right )^4}-\frac {b \left (3 a^2+4 b^2\right )}{3 a^2 \left (a^2+b^2\right ) d (a+b \tan (c+d x))^3}-\frac {\cot (c+d x)}{a d (a+b \tan (c+d x))^3}-\frac {b \left (a^4+4 a^2 b^2+2 b^4\right )}{a^3 \left (a^2+b^2\right )^2 d (a+b \tan (c+d x))^2}-\frac {b \left (a^6+13 a^4 b^2+12 a^2 b^4+4 b^6\right )}{a^4 \left (a^2+b^2\right )^3 d (a+b \tan (c+d x))}-\frac {(4 b) \int \cot (c+d x) \, dx}{a^5}+\frac {\left (4 b^3 \left (5 a^6+6 a^4 b^2+4 a^2 b^4+b^6\right )\right ) \int \frac {b-a \tan (c+d x)}{a+b \tan (c+d x)} \, dx}{a^5 \left (a^2+b^2\right )^4} \\ & = -\frac {\left (a^4-6 a^2 b^2+b^4\right ) x}{\left (a^2+b^2\right )^4}-\frac {4 b \log (\sin (c+d x))}{a^5 d}+\frac {4 b^3 \left (5 a^6+6 a^4 b^2+4 a^2 b^4+b^6\right ) \log (a \cos (c+d x)+b \sin (c+d x))}{a^5 \left (a^2+b^2\right )^4 d}-\frac {b \left (3 a^2+4 b^2\right )}{3 a^2 \left (a^2+b^2\right ) d (a+b \tan (c+d x))^3}-\frac {\cot (c+d x)}{a d (a+b \tan (c+d x))^3}-\frac {b \left (a^4+4 a^2 b^2+2 b^4\right )}{a^3 \left (a^2+b^2\right )^2 d (a+b \tan (c+d x))^2}-\frac {b \left (a^6+13 a^4 b^2+12 a^2 b^4+4 b^6\right )}{a^4 \left (a^2+b^2\right )^3 d (a+b \tan (c+d x))} \\ \end{align*}

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 3.71 (sec) , antiderivative size = 241, normalized size of antiderivative = 0.87 \[ \int \frac {\cot ^2(c+d x)}{(a+b \tan (c+d x))^4} \, dx=-\frac {\frac {\cot (c+d x)}{a^4}-\frac {b^6}{3 a^5 \left (a^2+b^2\right ) (b+a \cot (c+d x))^3}+\frac {b^5 \left (3 a^2+2 b^2\right )}{a^5 \left (a^2+b^2\right )^2 (b+a \cot (c+d x))^2}-\frac {b^4 \left (15 a^4+17 a^2 b^2+6 b^4\right )}{a^5 \left (a^2+b^2\right )^3 (b+a \cot (c+d x))}+\frac {i \log (i-\cot (c+d x))}{2 (a-i b)^4}-\frac {i \log (i+\cot (c+d x))}{2 (a+i b)^4}-\frac {4 b^3 \left (5 a^6+6 a^4 b^2+4 a^2 b^4+b^6\right ) \log (b+a \cot (c+d x))}{a^5 \left (a^2+b^2\right )^4}}{d} \]

[In]

Integrate[Cot[c + d*x]^2/(a + b*Tan[c + d*x])^4,x]

[Out]

-((Cot[c + d*x]/a^4 - b^6/(3*a^5*(a^2 + b^2)*(b + a*Cot[c + d*x])^3) + (b^5*(3*a^2 + 2*b^2))/(a^5*(a^2 + b^2)^
2*(b + a*Cot[c + d*x])^2) - (b^4*(15*a^4 + 17*a^2*b^2 + 6*b^4))/(a^5*(a^2 + b^2)^3*(b + a*Cot[c + d*x])) + ((I
/2)*Log[I - Cot[c + d*x]])/(a - I*b)^4 - ((I/2)*Log[I + Cot[c + d*x]])/(a + I*b)^4 - (4*b^3*(5*a^6 + 6*a^4*b^2
+ 4*a^2*b^4 + b^6)*Log[b + a*Cot[c + d*x]])/(a^5*(a^2 + b^2)^4))/d)

Maple [A] (verified)

Time = 1.47 (sec) , antiderivative size = 264, normalized size of antiderivative = 0.95

method	result	size
derivativedivides	\(\frac {\frac {\frac {\left (4 a^{3} b -4 a \,b^{3}\right ) \ln \left (1+\tan ^{2}\left (d x +c \right )\right )}{2}+\left (-a^{4}+6 a^{2} b^{2}-b^{4}\right ) \arctan \left (\tan \left (d x +c \right )\right )}{\left (a^{2}+b^{2}\right )^{4}}-\frac {b^{3}}{3 a^{2} \left (a^{2}+b^{2}\right ) \left (a +b \tan \left (d x +c \right )\right )^{3}}-\frac {b^{3} \left (10 a^{4}+9 a^{2} b^{2}+3 b^{4}\right )}{a^{4} \left (a^{2}+b^{2}\right )^{3} \left (a +b \tan \left (d x +c \right )\right )}-\frac {b^{3} \left (2 a^{2}+b^{2}\right )}{a^{3} \left (a^{2}+b^{2}\right )^{2} \left (a +b \tan \left (d x +c \right )\right )^{2}}+\frac {4 b^{3} \left (5 a^{6}+6 a^{4} b^{2}+4 a^{2} b^{4}+b^{6}\right ) \ln \left (a +b \tan \left (d x +c \right )\right )}{a^{5} \left (a^{2}+b^{2}\right )^{4}}-\frac {1}{a^{4} \tan \left (d x +c \right )}-\frac {4 b \ln \left (\tan \left (d x +c \right )\right )}{a^{5}}}{d}\)	\(264\)
default	\(\frac {\frac {\frac {\left (4 a^{3} b -4 a \,b^{3}\right ) \ln \left (1+\tan ^{2}\left (d x +c \right )\right )}{2}+\left (-a^{4}+6 a^{2} b^{2}-b^{4}\right ) \arctan \left (\tan \left (d x +c \right )\right )}{\left (a^{2}+b^{2}\right )^{4}}-\frac {b^{3}}{3 a^{2} \left (a^{2}+b^{2}\right ) \left (a +b \tan \left (d x +c \right )\right )^{3}}-\frac {b^{3} \left (10 a^{4}+9 a^{2} b^{2}+3 b^{4}\right )}{a^{4} \left (a^{2}+b^{2}\right )^{3} \left (a +b \tan \left (d x +c \right )\right )}-\frac {b^{3} \left (2 a^{2}+b^{2}\right )}{a^{3} \left (a^{2}+b^{2}\right )^{2} \left (a +b \tan \left (d x +c \right )\right )^{2}}+\frac {4 b^{3} \left (5 a^{6}+6 a^{4} b^{2}+4 a^{2} b^{4}+b^{6}\right ) \ln \left (a +b \tan \left (d x +c \right )\right )}{a^{5} \left (a^{2}+b^{2}\right )^{4}}-\frac {1}{a^{4} \tan \left (d x +c \right )}-\frac {4 b \ln \left (\tan \left (d x +c \right )\right )}{a^{5}}}{d}\)	\(264\)
parallelrisch	\(\frac {20 \left (a +b \tan \left (d x +c \right )\right )^{3} b^{3} \left (a^{6}+\frac {6}{5} a^{4} b^{2}+\frac {4}{5} a^{2} b^{4}+\frac {1}{5} b^{6}\right ) \ln \left (a +b \tan \left (d x +c \right )\right )+2 b \,a^{6} \left (a -b \right ) \left (a +b \right ) \left (a +b \tan \left (d x +c \right )\right )^{3} \ln \left (\sec ^{2}\left (d x +c \right )\right )-4 b \left (a^{2}+b^{2}\right )^{4} \left (a +b \tan \left (d x +c \right )\right )^{3} \ln \left (\tan \left (d x +c \right )\right )-b^{3} \left (a^{9} d x -6 x \,a^{7} b^{2} d +a^{5} b^{4} d x -3 a^{8} b -\frac {73}{3} a^{6} b^{3}-43 a^{4} b^{5}-29 a^{2} b^{7}-\frac {22}{3} b^{9}\right ) \left (\tan ^{3}\left (d x +c \right )\right )-3 a \,b^{2} \left (a^{9} d x -6 x \,a^{7} b^{2} d +a^{5} b^{4} d x -\frac {8}{3} a^{8} b -\frac {59}{3} a^{6} b^{3}-\frac {104}{3} a^{4} b^{5}-\frac {71}{3} a^{2} b^{7}-6 b^{9}\right ) \left (\tan ^{2}\left (d x +c \right )\right )-3 a^{2} \left (a^{9} d x -6 x \,a^{7} b^{2} d +a^{5} b^{4} d x -2 a^{8} b -13 a^{6} b^{3}-\frac {68}{3} a^{4} b^{5}-\frac {47}{3} a^{2} b^{7}-4 b^{9}\right ) b \tan \left (d x +c \right )-a^{4} \left (\left (a^{2}+b^{2}\right )^{4} \cot \left (d x +c \right )+a^{8} d x -6 a^{6} b^{2} d x +a^{4} b^{4} d x \right )}{d \,a^{5} \left (a^{2}+b^{2}\right )^{4} \left (a +b \tan \left (d x +c \right )\right )^{3}}\)	\(418\)
norman	\(\frac {\frac {b \left (6 a^{6} b +33 a^{4} b^{3}+35 b^{5} a^{2}+12 b^{7}\right ) \left (\tan ^{2}\left (d x +c \right )\right )}{d \,a^{3} \left (a^{6}+3 a^{4} b^{2}+3 a^{2} b^{4}+b^{6}\right )}+\frac {b^{2} \left (8 a^{6} b +51 a^{4} b^{3}+53 b^{5} a^{2}+18 b^{7}\right ) \left (\tan ^{3}\left (d x +c \right )\right )}{d \,a^{4} \left (a^{6}+3 a^{4} b^{2}+3 a^{2} b^{4}+b^{6}\right )}-\frac {1}{a d}-\frac {b^{3} \left (a^{4}-6 a^{2} b^{2}+b^{4}\right ) x \left (\tan ^{4}\left (d x +c \right )\right )}{\left (a^{6}+3 a^{4} b^{2}+3 a^{2} b^{4}+b^{6}\right ) \left (a^{2}+b^{2}\right )}+\frac {b^{3} \left (9 a^{6} b +64 a^{4} b^{3}+65 b^{5} a^{2}+22 b^{7}\right ) \left (\tan ^{4}\left (d x +c \right )\right )}{3 d \,a^{5} \left (a^{6}+3 a^{4} b^{2}+3 a^{2} b^{4}+b^{6}\right )}-\frac {\left (a^{4}-6 a^{2} b^{2}+b^{4}\right ) a^{3} x \tan \left (d x +c \right )}{\left (a^{6}+3 a^{4} b^{2}+3 a^{2} b^{4}+b^{6}\right ) \left (a^{2}+b^{2}\right )}-\frac {3 b \left (a^{4}-6 a^{2} b^{2}+b^{4}\right ) a^{2} x \left (\tan ^{2}\left (d x +c \right )\right )}{\left (a^{6}+3 a^{4} b^{2}+3 a^{2} b^{4}+b^{6}\right ) \left (a^{2}+b^{2}\right )}-\frac {3 b^{2} \left (a^{4}-6 a^{2} b^{2}+b^{4}\right ) a x \left (\tan ^{3}\left (d x +c \right )\right )}{\left (a^{6}+3 a^{4} b^{2}+3 a^{2} b^{4}+b^{6}\right ) \left (a^{2}+b^{2}\right )}}{\tan \left (d x +c \right ) \left (a +b \tan \left (d x +c \right )\right )^{3}}-\frac {4 b \ln \left (\tan \left (d x +c \right )\right )}{a^{5} d}+\frac {2 a b \left (a^{2}-b^{2}\right ) \ln \left (1+\tan ^{2}\left (d x +c \right )\right )}{d \left (a^{8}+4 a^{6} b^{2}+6 a^{4} b^{4}+4 b^{6} a^{2}+b^{8}\right )}+\frac {4 b^{3} \left (5 a^{6}+6 a^{4} b^{2}+4 a^{2} b^{4}+b^{6}\right ) \ln \left (a +b \tan \left (d x +c \right )\right )}{d \,a^{5} \left (a^{8}+4 a^{6} b^{2}+6 a^{4} b^{4}+4 b^{6} a^{2}+b^{8}\right )}\)	\(652\)
risch	\(\text {Expression too large to display}\)	\(1370\)

[In]

int(cot(d*x+c)^2/(a+b*tan(d*x+c))^4,x,method=_RETURNVERBOSE)

[Out]

1/d*(1/(a^2+b^2)^4*(1/2*(4*a^3*b-4*a*b^3)*ln(1+tan(d*x+c)^2)+(-a^4+6*a^2*b^2-b^4)*arctan(tan(d*x+c)))-1/3*b^3/
a^2/(a^2+b^2)/(a+b*tan(d*x+c))^3-b^3*(10*a^4+9*a^2*b^2+3*b^4)/a^4/(a^2+b^2)^3/(a+b*tan(d*x+c))-b^3*(2*a^2+b^2)
/a^3/(a^2+b^2)^2/(a+b*tan(d*x+c))^2+4*b^3*(5*a^6+6*a^4*b^2+4*a^2*b^4+b^6)/a^5/(a^2+b^2)^4*ln(a+b*tan(d*x+c))-1
/a^4/tan(d*x+c)-4/a^5*b*ln(tan(d*x+c)))

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 925 vs. \(2 (276) = 552\).

Time = 0.33 (sec) , antiderivative size = 925, normalized size of antiderivative = 3.33 \[ \int \frac {\cot ^2(c+d x)}{(a+b \tan (c+d x))^4} \, dx=-\frac {3 \, a^{12} + 12 \, a^{10} b^{2} + 18 \, a^{8} b^{4} + 12 \, a^{6} b^{6} + 3 \, a^{4} b^{8} - {\left (37 \, a^{6} b^{6} + 21 \, a^{4} b^{8} + 6 \, a^{2} b^{10} - 3 \, {\left (a^{9} b^{3} - 6 \, a^{7} b^{5} + a^{5} b^{7}\right )} d x\right )} \tan \left (d x + c\right )^{4} + 3 \, {\left (a^{9} b^{3} - 23 \, a^{7} b^{5} + 4 \, a^{5} b^{7} + 10 \, a^{3} b^{9} + 4 \, a b^{11} + 3 \, {\left (a^{10} b^{2} - 6 \, a^{8} b^{4} + a^{6} b^{6}\right )} d x\right )} \tan \left (d x + c\right )^{3} + 3 \, {\left (3 \, a^{10} b^{2} - 3 \, a^{8} b^{4} + 40 \, a^{6} b^{6} + 34 \, a^{4} b^{8} + 10 \, a^{2} b^{10} + 3 \, {\left (a^{11} b - 6 \, a^{9} b^{3} + a^{7} b^{5}\right )} d x\right )} \tan \left (d x + c\right )^{2} + 6 \, {\left ({\left (a^{8} b^{4} + 4 \, a^{6} b^{6} + 6 \, a^{4} b^{8} + 4 \, a^{2} b^{10} + b^{12}\right )} \tan \left (d x + c\right )^{4} + 3 \, {\left (a^{9} b^{3} + 4 \, a^{7} b^{5} + 6 \, a^{5} b^{7} + 4 \, a^{3} b^{9} + a b^{11}\right )} \tan \left (d x + c\right )^{3} + 3 \, {\left (a^{10} b^{2} + 4 \, a^{8} b^{4} + 6 \, a^{6} b^{6} + 4 \, a^{4} b^{8} + a^{2} b^{10}\right )} \tan \left (d x + c\right )^{2} + {\left (a^{11} b + 4 \, a^{9} b^{3} + 6 \, a^{7} b^{5} + 4 \, a^{5} b^{7} + a^{3} b^{9}\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 ) - 6 \, {\left ({\left (5 \, a^{6} b^{6} + 6 \, a^{4} b^{8} + 4 \, a^{2} b^{10} + b^{12}\right )} \tan \left (d x + c\right )^{4} + 3 \, {\left (5 \, a^{7} b^{5} + 6 \, a^{5} b^{7} + 4 \, a^{3} b^{9} + a b^{11}\right )} \tan \left (d x + c\right )^{3} + 3 \, {\left (5 \, a^{8} b^{4} + 6 \, a^{6} b^{6} + 4 \, a^{4} b^{8} + a^{2} b^{10}\right )} \tan \left (d x + c\right )^{2} + {\left (5 \, a^{9} b^{3} + 6 \, a^{7} b^{5} + 4 \, a^{5} b^{7} + a^{3} b^{9}\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 (9 \, a^{11} b + 36 \, a^{9} b^{3} + 108 \, a^{7} b^{5} + 81 \, a^{5} b^{7} + 22 \, a^{3} b^{9} + 3 \, {\left (a^{12} - 6 \, a^{10} b^{2} + a^{8} b^{4}\right )} d x\right )} \tan \left (d x + c\right )}{3 \, {\left ({\left (a^{13} b^{3} + 4 \, a^{11} b^{5} + 6 \, a^{9} b^{7} + 4 \, a^{7} b^{9} + a^{5} b^{11}\right )} d \tan \left (d x + c\right )^{4} + 3 \, {\left (a^{14} b^{2} + 4 \, a^{12} b^{4} + 6 \, a^{10} b^{6} + 4 \, a^{8} b^{8} + a^{6} b^{10}\right )} d \tan \left (d x + c\right )^{3} + 3 \, {\left (a^{15} b + 4 \, a^{13} b^{3} + 6 \, a^{11} b^{5} + 4 \, a^{9} b^{7} + a^{7} b^{9}\right )} d \tan \left (d x + c\right )^{2} + {\left (a^{16} + 4 \, a^{14} b^{2} + 6 \, a^{12} b^{4} + 4 \, a^{10} b^{6} + a^{8} b^{8}\right )} d \tan \left (d x + c\right )\right )}} \]

[In]

integrate(cot(d*x+c)^2/(a+b*tan(d*x+c))^4,x, algorithm="fricas")

[Out]

-1/3*(3*a^12 + 12*a^10*b^2 + 18*a^8*b^4 + 12*a^6*b^6 + 3*a^4*b^8 - (37*a^6*b^6 + 21*a^4*b^8 + 6*a^2*b^10 - 3*(
a^9*b^3 - 6*a^7*b^5 + a^5*b^7)*d*x)*tan(d*x + c)^4 + 3*(a^9*b^3 - 23*a^7*b^5 + 4*a^5*b^7 + 10*a^3*b^9 + 4*a*b^
11 + 3*(a^10*b^2 - 6*a^8*b^4 + a^6*b^6)*d*x)*tan(d*x + c)^3 + 3*(3*a^10*b^2 - 3*a^8*b^4 + 40*a^6*b^6 + 34*a^4*
b^8 + 10*a^2*b^10 + 3*(a^11*b - 6*a^9*b^3 + a^7*b^5)*d*x)*tan(d*x + c)^2 + 6*((a^8*b^4 + 4*a^6*b^6 + 6*a^4*b^8
+ 4*a^2*b^10 + b^12)*tan(d*x + c)^4 + 3*(a^9*b^3 + 4*a^7*b^5 + 6*a^5*b^7 + 4*a^3*b^9 + a*b^11)*tan(d*x + c)^3
+ 3*(a^10*b^2 + 4*a^8*b^4 + 6*a^6*b^6 + 4*a^4*b^8 + a^2*b^10)*tan(d*x + c)^2 + (a^11*b + 4*a^9*b^3 + 6*a^7*b^
5 + 4*a^5*b^7 + a^3*b^9)*tan(d*x + c))*log(tan(d*x + c)^2/(tan(d*x + c)^2 + 1)) - 6*((5*a^6*b^6 + 6*a^4*b^8 +
4*a^2*b^10 + b^12)*tan(d*x + c)^4 + 3*(5*a^7*b^5 + 6*a^5*b^7 + 4*a^3*b^9 + a*b^11)*tan(d*x + c)^3 + 3*(5*a^8*b
^4 + 6*a^6*b^6 + 4*a^4*b^8 + a^2*b^10)*tan(d*x + c)^2 + (5*a^9*b^3 + 6*a^7*b^5 + 4*a^5*b^7 + a^3*b^9)*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)) + (9*a^11*b + 36*a^9*b^3 + 108
*a^7*b^5 + 81*a^5*b^7 + 22*a^3*b^9 + 3*(a^12 - 6*a^10*b^2 + a^8*b^4)*d*x)*tan(d*x + c))/((a^13*b^3 + 4*a^11*b^
5 + 6*a^9*b^7 + 4*a^7*b^9 + a^5*b^11)*d*tan(d*x + c)^4 + 3*(a^14*b^2 + 4*a^12*b^4 + 6*a^10*b^6 + 4*a^8*b^8 + a
^6*b^10)*d*tan(d*x + c)^3 + 3*(a^15*b + 4*a^13*b^3 + 6*a^11*b^5 + 4*a^9*b^7 + a^7*b^9)*d*tan(d*x + c)^2 + (a^1
6 + 4*a^14*b^2 + 6*a^12*b^4 + 4*a^10*b^6 + a^8*b^8)*d*tan(d*x + c))

Sympy [F(-2)]

Exception generated. \[ \int \frac {\cot ^2(c+d x)}{(a+b \tan (c+d x))^4} \, dx=\text {Exception raised: AttributeError} \]

[In]

integrate(cot(d*x+c)**2/(a+b*tan(d*x+c))**4,x)

[Out]

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

Maxima [A] (verification not implemented)

none

Time = 0.62 (sec) , antiderivative size = 516, normalized size of antiderivative = 1.86 \[ \int \frac {\cot ^2(c+d x)}{(a+b \tan (c+d x))^4} \, dx=-\frac {\frac {3 \, {\left (a^{4} - 6 \, a^{2} b^{2} + b^{4}\right )} {\left (d x + c\right )}}{a^{8} + 4 \, a^{6} b^{2} + 6 \, a^{4} b^{4} + 4 \, a^{2} b^{6} + b^{8}} - \frac {12 \, {\left (5 \, a^{6} b^{3} + 6 \, a^{4} b^{5} + 4 \, a^{2} b^{7} + b^{9}\right )} \log \left (b \tan \left (d x + c\right ) + a\right )}{a^{13} + 4 \, a^{11} b^{2} + 6 \, a^{9} b^{4} + 4 \, a^{7} b^{6} + a^{5} b^{8}} - \frac {6 \, {\left (a^{3} b - a b^{3}\right )} \log \left (\tan \left (d x + c\right )^{2} + 1\right )}{a^{8} + 4 \, a^{6} b^{2} + 6 \, a^{4} b^{4} + 4 \, a^{2} b^{6} + b^{8}} + \frac {3 \, a^{9} + 9 \, a^{7} b^{2} + 9 \, a^{5} b^{4} + 3 \, a^{3} b^{6} + 3 \, {\left (a^{6} b^{3} + 13 \, a^{4} b^{5} + 12 \, a^{2} b^{7} + 4 \, b^{9}\right )} \tan \left (d x + c\right )^{3} + 3 \, {\left (3 \, a^{7} b^{2} + 31 \, a^{5} b^{4} + 30 \, a^{3} b^{6} + 10 \, a b^{8}\right )} \tan \left (d x + c\right )^{2} + {\left (9 \, a^{8} b + 64 \, a^{6} b^{3} + 65 \, a^{4} b^{5} + 22 \, a^{2} b^{7}\right )} \tan \left (d x + c\right )}{{\left (a^{10} b^{3} + 3 \, a^{8} b^{5} + 3 \, a^{6} b^{7} + a^{4} b^{9}\right )} \tan \left (d x + c\right )^{4} + 3 \, {\left (a^{11} b^{2} + 3 \, a^{9} b^{4} + 3 \, a^{7} b^{6} + a^{5} b^{8}\right )} \tan \left (d x + c\right )^{3} + 3 \, {\left (a^{12} b + 3 \, a^{10} b^{3} + 3 \, a^{8} b^{5} + a^{6} b^{7}\right )} \tan \left (d x + c\right )^{2} + {\left (a^{13} + 3 \, a^{11} b^{2} + 3 \, a^{9} b^{4} + a^{7} b^{6}\right )} \tan \left (d x + c\right )} + \frac {12 \, b \log \left (\tan \left (d x + c\right )\right )}{a^{5}}}{3 \, d} \]

[In]

integrate(cot(d*x+c)^2/(a+b*tan(d*x+c))^4,x, algorithm="maxima")

[Out]

-1/3*(3*(a^4 - 6*a^2*b^2 + b^4)*(d*x + c)/(a^8 + 4*a^6*b^2 + 6*a^4*b^4 + 4*a^2*b^6 + b^8) - 12*(5*a^6*b^3 + 6*
a^4*b^5 + 4*a^2*b^7 + b^9)*log(b*tan(d*x + c) + a)/(a^13 + 4*a^11*b^2 + 6*a^9*b^4 + 4*a^7*b^6 + a^5*b^8) - 6*(
a^3*b - a*b^3)*log(tan(d*x + c)^2 + 1)/(a^8 + 4*a^6*b^2 + 6*a^4*b^4 + 4*a^2*b^6 + b^8) + (3*a^9 + 9*a^7*b^2 +
9*a^5*b^4 + 3*a^3*b^6 + 3*(a^6*b^3 + 13*a^4*b^5 + 12*a^2*b^7 + 4*b^9)*tan(d*x + c)^3 + 3*(3*a^7*b^2 + 31*a^5*b
^4 + 30*a^3*b^6 + 10*a*b^8)*tan(d*x + c)^2 + (9*a^8*b + 64*a^6*b^3 + 65*a^4*b^5 + 22*a^2*b^7)*tan(d*x + c))/((
a^10*b^3 + 3*a^8*b^5 + 3*a^6*b^7 + a^4*b^9)*tan(d*x + c)^4 + 3*(a^11*b^2 + 3*a^9*b^4 + 3*a^7*b^6 + a^5*b^8)*ta
n(d*x + c)^3 + 3*(a^12*b + 3*a^10*b^3 + 3*a^8*b^5 + a^6*b^7)*tan(d*x + c)^2 + (a^13 + 3*a^11*b^2 + 3*a^9*b^4 +
a^7*b^6)*tan(d*x + c)) + 12*b*log(tan(d*x + c))/a^5)/d

Giac [A] (verification not implemented)

none

Time = 1.06 (sec) , antiderivative size = 502, normalized size of antiderivative = 1.81 \[ \int \frac {\cot ^2(c+d x)}{(a+b \tan (c+d x))^4} \, dx=-\frac {\frac {3 \, {\left (a^{4} - 6 \, a^{2} b^{2} + b^{4}\right )} {\left (d x + c\right )}}{a^{8} + 4 \, a^{6} b^{2} + 6 \, a^{4} b^{4} + 4 \, a^{2} b^{6} + b^{8}} - \frac {6 \, {\left (a^{3} b - a b^{3}\right )} \log \left (\tan \left (d x + c\right )^{2} + 1\right )}{a^{8} + 4 \, a^{6} b^{2} + 6 \, a^{4} b^{4} + 4 \, a^{2} b^{6} + b^{8}} - \frac {12 \, {\left (5 \, a^{6} b^{4} + 6 \, a^{4} b^{6} + 4 \, a^{2} b^{8} + b^{10}\right )} \log \left ({\left | b \tan \left (d x + c\right ) + a \right |}\right )}{a^{13} b + 4 \, a^{11} b^{3} + 6 \, a^{9} b^{5} + 4 \, a^{7} b^{7} + a^{5} b^{9}} + \frac {110 \, a^{6} b^{6} \tan \left (d x + c\right )^{3} + 132 \, a^{4} b^{8} \tan \left (d x + c\right )^{3} + 88 \, a^{2} b^{10} \tan \left (d x + c\right )^{3} + 22 \, b^{12} \tan \left (d x + c\right )^{3} + 360 \, a^{7} b^{5} \tan \left (d x + c\right )^{2} + 453 \, a^{5} b^{7} \tan \left (d x + c\right )^{2} + 300 \, a^{3} b^{9} \tan \left (d x + c\right )^{2} + 75 \, a b^{11} \tan \left (d x + c\right )^{2} + 396 \, a^{8} b^{4} \tan \left (d x + c\right ) + 525 \, a^{6} b^{6} \tan \left (d x + c\right ) + 348 \, a^{4} b^{8} \tan \left (d x + c\right ) + 87 \, a^{2} b^{10} \tan \left (d x + c\right ) + 147 \, a^{9} b^{3} + 207 \, a^{7} b^{5} + 139 \, a^{5} b^{7} + 35 \, a^{3} b^{9}}{{\left (a^{13} + 4 \, a^{11} b^{2} + 6 \, a^{9} b^{4} + 4 \, a^{7} b^{6} + a^{5} b^{8}\right )} {\left (b \tan \left (d x + c\right ) + a\right )}^{3}} + \frac {12 \, b \log \left ({\left | \tan \left (d x + c\right ) \right |}\right )}{a^{5}} - \frac {3 \, {\left (4 \, b \tan \left (d x + c\right ) - a\right )}}{a^{5} \tan \left (d x + c\right )}}{3 \, d} \]

[In]

integrate(cot(d*x+c)^2/(a+b*tan(d*x+c))^4,x, algorithm="giac")

[Out]

-1/3*(3*(a^4 - 6*a^2*b^2 + b^4)*(d*x + c)/(a^8 + 4*a^6*b^2 + 6*a^4*b^4 + 4*a^2*b^6 + b^8) - 6*(a^3*b - a*b^3)*
log(tan(d*x + c)^2 + 1)/(a^8 + 4*a^6*b^2 + 6*a^4*b^4 + 4*a^2*b^6 + b^8) - 12*(5*a^6*b^4 + 6*a^4*b^6 + 4*a^2*b^
8 + b^10)*log(abs(b*tan(d*x + c) + a))/(a^13*b + 4*a^11*b^3 + 6*a^9*b^5 + 4*a^7*b^7 + a^5*b^9) + (110*a^6*b^6*
tan(d*x + c)^3 + 132*a^4*b^8*tan(d*x + c)^3 + 88*a^2*b^10*tan(d*x + c)^3 + 22*b^12*tan(d*x + c)^3 + 360*a^7*b^
5*tan(d*x + c)^2 + 453*a^5*b^7*tan(d*x + c)^2 + 300*a^3*b^9*tan(d*x + c)^2 + 75*a*b^11*tan(d*x + c)^2 + 396*a^
8*b^4*tan(d*x + c) + 525*a^6*b^6*tan(d*x + c) + 348*a^4*b^8*tan(d*x + c) + 87*a^2*b^10*tan(d*x + c) + 147*a^9*
b^3 + 207*a^7*b^5 + 139*a^5*b^7 + 35*a^3*b^9)/((a^13 + 4*a^11*b^2 + 6*a^9*b^4 + 4*a^7*b^6 + a^5*b^8)*(b*tan(d*
x + c) + a)^3) + 12*b*log(abs(tan(d*x + c)))/a^5 - 3*(4*b*tan(d*x + c) - a)/(a^5*tan(d*x + c)))/d

Mupad [B] (verification not implemented)

Time = 7.10 (sec) , antiderivative size = 430, normalized size of antiderivative = 1.55 \[ \int \frac {\cot ^2(c+d x)}{(a+b \tan (c+d x))^4} \, dx=\frac {4\,b^3\,\ln \left (a+b\,\mathrm {tan}\left (c+d\,x\right )\right )\,\left (5\,a^6+6\,a^4\,b^2+4\,a^2\,b^4+b^6\right )}{a^5\,d\,{\left (a^2+b^2\right )}^4}-\frac {\frac {1}{a}+\frac {{\mathrm {tan}\left (c+d\,x\right )}^3\,\left (a^6\,b^3+13\,a^4\,b^5+12\,a^2\,b^7+4\,b^9\right )}{a^4\,\left (a^6+3\,a^4\,b^2+3\,a^2\,b^4+b^6\right )}+\frac {{\mathrm {tan}\left (c+d\,x\right )}^2\,\left (3\,a^6\,b^2+31\,a^4\,b^4+30\,a^2\,b^6+10\,b^8\right )}{a^3\,\left (a^6+3\,a^4\,b^2+3\,a^2\,b^4+b^6\right )}+\frac {\mathrm {tan}\left (c+d\,x\right )\,\left (9\,a^6\,b+64\,a^4\,b^3+65\,a^2\,b^5+22\,b^7\right )}{3\,a^2\,\left (a^6+3\,a^4\,b^2+3\,a^2\,b^4+b^6\right )}}{d\,\left (a^3\,\mathrm {tan}\left (c+d\,x\right )+3\,a^2\,b\,{\mathrm {tan}\left (c+d\,x\right )}^2+3\,a\,b^2\,{\mathrm {tan}\left (c+d\,x\right )}^3+b^3\,{\mathrm {tan}\left (c+d\,x\right )}^4\right )}-\frac {4\,b\,\ln \left (\mathrm {tan}\left (c+d\,x\right )\right )}{a^5\,d}-\frac {\ln \left (\mathrm {tan}\left (c+d\,x\right )-\mathrm {i}\right )}{2\,d\,\left (a^4\,1{}\mathrm {i}-4\,a^3\,b-a^2\,b^2\,6{}\mathrm {i}+4\,a\,b^3+b^4\,1{}\mathrm {i}\right )}-\frac {\ln \left (\mathrm {tan}\left (c+d\,x\right )+1{}\mathrm {i}\right )\,1{}\mathrm {i}}{2\,d\,\left (a^4-a^3\,b\,4{}\mathrm {i}-6\,a^2\,b^2+a\,b^3\,4{}\mathrm {i}+b^4\right )} \]

[In]

int(cot(c + d*x)^2/(a + b*tan(c + d*x))^4,x)

[Out]

(4*b^3*log(a + b*tan(c + d*x))*(5*a^6 + b^6 + 4*a^2*b^4 + 6*a^4*b^2))/(a^5*d*(a^2 + b^2)^4) - (log(tan(c + d*x
) + 1i)*1i)/(2*d*(a*b^3*4i - a^3*b*4i + a^4 + b^4 - 6*a^2*b^2)) - (1/a + (tan(c + d*x)^3*(4*b^9 + 12*a^2*b^7 +
13*a^4*b^5 + a^6*b^3))/(a^4*(a^6 + b^6 + 3*a^2*b^4 + 3*a^4*b^2)) + (tan(c + d*x)^2*(10*b^8 + 30*a^2*b^6 + 31*
a^4*b^4 + 3*a^6*b^2))/(a^3*(a^6 + b^6 + 3*a^2*b^4 + 3*a^4*b^2)) + (tan(c + d*x)*(9*a^6*b + 22*b^7 + 65*a^2*b^5
+ 64*a^4*b^3))/(3*a^2*(a^6 + b^6 + 3*a^2*b^4 + 3*a^4*b^2)))/(d*(a^3*tan(c + d*x) + b^3*tan(c + d*x)^4 + 3*a^2
*b*tan(c + d*x)^2 + 3*a*b^2*tan(c + d*x)^3)) - (4*b*log(tan(c + d*x)))/(a^5*d) - log(tan(c + d*x) - 1i)/(2*d*(
4*a*b^3 - 4*a^3*b + a^4*1i + b^4*1i - a^2*b^2*6i))