\(\int \frac {\cot ^3(c+d x) (B \tan (c+d x)+C \tan ^2(c+d x))}{(a+b \tan (c+d x))^2} \, dx\) [37]

Optimal result

Integrand size = 40, antiderivative size = 192 \[ \int \frac {\cot ^3(c+d x) \left (B \tan (c+d x)+C \tan ^2(c+d x)\right )}{(a+b \tan (c+d x))^2} \, dx=-\frac {\left (a^2 B-b^2 B+2 a b C\right ) x}{\left (a^2+b^2\right )^2}-\frac {(2 b B-a C) \log (\sin (c+d x))}{a^3 d}+\frac {b^2 \left (4 a^2 b B+2 b^3 B-3 a^3 C-a b^2 C\right ) \log (a \cos (c+d x)+b \sin (c+d x))}{a^3 \left (a^2+b^2\right )^2 d}-\frac {b \left (a^2 B+2 b^2 B-a b C\right )}{a^2 \left (a^2+b^2\right ) d (a+b \tan (c+d x))}-\frac {B \cot (c+d x)}{a d (a+b \tan (c+d x))} \]

[Out]

-(B*a^2-B*b^2+2*C*a*b)*x/(a^2+b^2)^2-(2*B*b-C*a)*ln(sin(d*x+c))/a^3/d+b^2*(4*B*a^2*b+2*B*b^3-3*C*a^3-C*a*b^2)*
ln(a*cos(d*x+c)+b*sin(d*x+c))/a^3/(a^2+b^2)^2/d-b*(B*a^2+2*B*b^2-C*a*b)/a^2/(a^2+b^2)/d/(a+b*tan(d*x+c))-B*cot
(d*x+c)/a/d/(a+b*tan(d*x+c))

Rubi [A] (verified)

Time = 0.72 (sec) , antiderivative size = 192, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.150, Rules used = {3713, 3690, 3730, 3732, 3611, 3556} \[ \int \frac {\cot ^3(c+d x) \left (B \tan (c+d x)+C \tan ^2(c+d x)\right )}{(a+b \tan (c+d x))^2} \, dx=-\frac {(2 b B-a C) \log (\sin (c+d x))}{a^3 d}-\frac {b \left (a^2 B-a b C+2 b^2 B\right )}{a^2 d \left (a^2+b^2\right ) (a+b \tan (c+d x))}-\frac {x \left (a^2 B+2 a b C-b^2 B\right )}{\left (a^2+b^2\right )^2}+\frac {b^2 \left (-3 a^3 C+4 a^2 b B-a b^2 C+2 b^3 B\right ) \log (a \cos (c+d x)+b \sin (c+d x))}{a^3 d \left (a^2+b^2\right )^2}-\frac {B \cot (c+d x)}{a d (a+b \tan (c+d x))} \]

[In]

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

[Out]

-(((a^2*B - b^2*B + 2*a*b*C)*x)/(a^2 + b^2)^2) - ((2*b*B - a*C)*Log[Sin[c + d*x]])/(a^3*d) + (b^2*(4*a^2*b*B +
2*b^3*B - 3*a^3*C - a*b^2*C)*Log[a*Cos[c + d*x] + b*Sin[c + d*x]])/(a^3*(a^2 + b^2)^2*d) - (b*(a^2*B + 2*b^2*
B - a*b*C))/(a^2*(a^2 + b^2)*d*(a + b*Tan[c + d*x])) - (B*Cot[c + d*x])/(a*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 3690

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

Rule 3713

Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_.)*((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)])^(n_.)*((A_.) + (B_.)
*tan[(e_.) + (f_.)*(x_)] + (C_.)*tan[(e_.) + (f_.)*(x_)]^2), x_Symbol] :> Dist[1/b^2, Int[(a + b*Tan[e + f*x])
^(m + 1)*(c + d*Tan[e + f*x])^n*(b*B - a*C + b*C*Tan[e + f*x]), x], x] /; FreeQ[{a, b, c, d, e, f, A, B, C, m,
n}, x] && NeQ[b*c - a*d, 0] && EqQ[A*b^2 - a*b*B + a^2*C, 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}& = \int \frac {\cot ^2(c+d x) (B+C \tan (c+d x))}{(a+b \tan (c+d x))^2} \, dx \\ & = -\frac {B \cot (c+d x)}{a d (a+b \tan (c+d x))}-\frac {\int \frac {\cot (c+d x) \left (2 b B-a C+a B \tan (c+d x)+2 b B \tan ^2(c+d x)\right )}{(a+b \tan (c+d x))^2} \, dx}{a} \\ & = -\frac {b \left (a^2 B+2 b^2 B-a b C\right )}{a^2 \left (a^2+b^2\right ) d (a+b \tan (c+d x))}-\frac {B \cot (c+d x)}{a d (a+b \tan (c+d x))}-\frac {\int \frac {\cot (c+d x) \left (\left (a^2+b^2\right ) (2 b B-a C)+a^2 (a B+b C) \tan (c+d x)+b \left (a^2 B+2 b^2 B-a b C\right ) \tan ^2(c+d x)\right )}{a+b \tan (c+d x)} \, dx}{a^2 \left (a^2+b^2\right )} \\ & = -\frac {\left (a^2 B-b^2 B+2 a b C\right ) x}{\left (a^2+b^2\right )^2}-\frac {b \left (a^2 B+2 b^2 B-a b C\right )}{a^2 \left (a^2+b^2\right ) d (a+b \tan (c+d x))}-\frac {B \cot (c+d x)}{a d (a+b \tan (c+d x))}-\frac {(2 b B-a C) \int \cot (c+d x) \, dx}{a^3}+\frac {\left (b^2 \left (4 a^2 b B+2 b^3 B-3 a^3 C-a b^2 C\right )\right ) \int \frac {b-a \tan (c+d x)}{a+b \tan (c+d x)} \, dx}{a^3 \left (a^2+b^2\right )^2} \\ & = -\frac {\left (a^2 B-b^2 B+2 a b C\right ) x}{\left (a^2+b^2\right )^2}-\frac {(2 b B-a C) \log (\sin (c+d x))}{a^3 d}+\frac {b^2 \left (4 a^2 b B+2 b^3 B-3 a^3 C-a b^2 C\right ) \log (a \cos (c+d x)+b \sin (c+d x))}{a^3 \left (a^2+b^2\right )^2 d}-\frac {b \left (a^2 B+2 b^2 B-a b C\right )}{a^2 \left (a^2+b^2\right ) d (a+b \tan (c+d x))}-\frac {B \cot (c+d x)}{a d (a+b \tan (c+d x))} \\ \end{align*}

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 3.81 (sec) , antiderivative size = 193, normalized size of antiderivative = 1.01 \[ \int \frac {\cot ^3(c+d x) \left (B \tan (c+d x)+C \tan ^2(c+d x)\right )}{(a+b \tan (c+d x))^2} \, dx=\frac {-\frac {2 B \cot (c+d x)}{a^2}+\frac {i (B+i C) \log (i-\tan (c+d x))}{(a+i b)^2}+\frac {2 (-2 b B+a C) \log (\tan (c+d x))}{a^3}-\frac {(i B+C) \log (i+\tan (c+d x))}{(a-i b)^2}-\frac {2 b^2 \left (-4 a^2 b B-2 b^3 B+3 a^3 C+a b^2 C\right ) \log (a+b \tan (c+d x))}{a^3 \left (a^2+b^2\right )^2}+\frac {2 b^2 (-b B+a C)}{a^2 \left (a^2+b^2\right ) (a+b \tan (c+d x))}}{2 d} \]

[In]

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

[Out]

((-2*B*Cot[c + d*x])/a^2 + (I*(B + I*C)*Log[I - Tan[c + d*x]])/(a + I*b)^2 + (2*(-2*b*B + a*C)*Log[Tan[c + d*x
]])/a^3 - ((I*B + C)*Log[I + Tan[c + d*x]])/(a - I*b)^2 - (2*b^2*(-4*a^2*b*B - 2*b^3*B + 3*a^3*C + a*b^2*C)*Lo
g[a + b*Tan[c + d*x]])/(a^3*(a^2 + b^2)^2) + (2*b^2*(-(b*B) + a*C))/(a^2*(a^2 + b^2)*(a + b*Tan[c + d*x])))/(2
*d)

Maple [A] (verified)

Time = 0.54 (sec) , antiderivative size = 196, normalized size of antiderivative = 1.02

method	result	size
derivativedivides	\(\frac {-\frac {B}{a^{2} \tan \left (d x +c \right )}+\frac {\left (-2 B b +C a \right ) \ln \left (\tan \left (d x +c \right )\right )}{a^{3}}+\frac {\frac {\left (2 B a b -C \,a^{2}+C \,b^{2}\right ) \ln \left (1+\tan \left (d x +c \right )^{2}\right )}{2}+\left (-B \,a^{2}+B \,b^{2}-2 C a b \right ) \arctan \left (\tan \left (d x +c \right )\right )}{\left (a^{2}+b^{2}\right )^{2}}+\frac {b^{2} \left (4 B \,a^{2} b +2 B \,b^{3}-3 C \,a^{3}-C a \,b^{2}\right ) \ln \left (a +b \tan \left (d x +c \right )\right )}{\left (a^{2}+b^{2}\right )^{2} a^{3}}-\frac {\left (B b -C a \right ) b^{2}}{\left (a^{2}+b^{2}\right ) a^{2} \left (a +b \tan \left (d x +c \right )\right )}}{d}\)	\(196\)
default	\(\frac {-\frac {B}{a^{2} \tan \left (d x +c \right )}+\frac {\left (-2 B b +C a \right ) \ln \left (\tan \left (d x +c \right )\right )}{a^{3}}+\frac {\frac {\left (2 B a b -C \,a^{2}+C \,b^{2}\right ) \ln \left (1+\tan \left (d x +c \right )^{2}\right )}{2}+\left (-B \,a^{2}+B \,b^{2}-2 C a b \right ) \arctan \left (\tan \left (d x +c \right )\right )}{\left (a^{2}+b^{2}\right )^{2}}+\frac {b^{2} \left (4 B \,a^{2} b +2 B \,b^{3}-3 C \,a^{3}-C a \,b^{2}\right ) \ln \left (a +b \tan \left (d x +c \right )\right )}{\left (a^{2}+b^{2}\right )^{2} a^{3}}-\frac {\left (B b -C a \right ) b^{2}}{\left (a^{2}+b^{2}\right ) a^{2} \left (a +b \tan \left (d x +c \right )\right )}}{d}\)	\(196\)
parallelrisch	\(\frac {4 b^{2} \left (a +b \tan \left (d x +c \right )\right ) \left (B \,a^{2} b +\frac {1}{2} B \,b^{3}-\frac {3}{4} C \,a^{3}-\frac {1}{4} C a \,b^{2}\right ) \ln \left (a +b \tan \left (d x +c \right )\right )+a^{3} \left (B a b -\frac {1}{2} C \,a^{2}+\frac {1}{2} C \,b^{2}\right ) \left (a +b \tan \left (d x +c \right )\right ) \ln \left (\sec \left (d x +c \right )^{2}\right )-2 \left (a^{2}+b^{2}\right )^{2} \left (a +b \tan \left (d x +c \right )\right ) \left (B b -\frac {C a}{2}\right ) \ln \left (\tan \left (d x +c \right )\right )-b \left (-2 B \,b^{5}+C a \,b^{4}-3 B \,a^{2} b^{3}-\left (B d x -C \right ) a^{3} b^{2}-a^{4} \left (-2 C d x +B \right ) b +B \,a^{5} d x \right ) \tan \left (d x +c \right )-a^{2} \left (B \left (a^{2}+b^{2}\right )^{2} \cot \left (d x +c \right )+a^{2} d x \left (B \,a^{2}-B \,b^{2}+2 C a b \right )\right )}{\left (a^{2}+b^{2}\right )^{2} a^{3} d \left (a +b \tan \left (d x +c \right )\right )}\)	\(271\)
norman	\(\frac {\frac {\left (B \,a^{2} b +2 B \,b^{3}-C a \,b^{2}\right ) b \tan \left (d x +c \right )^{3}}{d \,a^{3} \left (a^{2}+b^{2}\right )}-\frac {B \tan \left (d x +c \right )}{a d}-\frac {a \left (B \,a^{2}-B \,b^{2}+2 C a b \right ) x \tan \left (d x +c \right )^{2}}{a^{4}+2 a^{2} b^{2}+b^{4}}-\frac {b \left (B \,a^{2}-B \,b^{2}+2 C a b \right ) x \tan \left (d x +c \right )^{3}}{a^{4}+2 a^{2} b^{2}+b^{4}}}{\tan \left (d x +c \right )^{2} \left (a +b \tan \left (d x +c \right )\right )}+\frac {b^{2} \left (4 B \,a^{2} b +2 B \,b^{3}-3 C \,a^{3}-C a \,b^{2}\right ) \ln \left (a +b \tan \left (d x +c \right )\right )}{\left (a^{4}+2 a^{2} b^{2}+b^{4}\right ) a^{3} d}-\frac {\left (2 B b -C a \right ) \ln \left (\tan \left (d x +c \right )\right )}{a^{3} d}+\frac {\left (2 B a b -C \,a^{2}+C \,b^{2}\right ) \ln \left (1+\tan \left (d x +c \right )^{2}\right )}{2 d \left (a^{4}+2 a^{2} b^{2}+b^{4}\right )}\)	\(315\)
risch	\(\frac {x B}{2 i b a -a^{2}+b^{2}}-\frac {4 i b^{5} B c}{a^{3} d \left (a^{4}+2 a^{2} b^{2}+b^{4}\right )}+\frac {2 i b^{4} C x}{a^{2} \left (a^{4}+2 a^{2} b^{2}+b^{4}\right )}-\frac {4 i b^{5} B x}{a^{3} \left (a^{4}+2 a^{2} b^{2}+b^{4}\right )}-\frac {i x C}{2 i b a -a^{2}+b^{2}}-\frac {2 i \left (-2 i B \,a^{3} b \,{\mathrm e}^{2 i \left (d x +c \right )}-2 i B a \,b^{3} {\mathrm e}^{2 i \left (d x +c \right )}+B \,a^{4} {\mathrm e}^{2 i \left (d x +c \right )}-2 B \,b^{4} {\mathrm e}^{2 i \left (d x +c \right )}+B \,a^{4}+2 B \,a^{2} b^{2}+2 B \,b^{4}+C a \,b^{3} {\mathrm e}^{2 i \left (d x +c \right )}-C a \,b^{3}\right )}{\left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right ) \left (i b +a \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 ) a^{2} d}+\frac {4 i B b x}{a^{3}}+\frac {4 i B b c}{a^{3} d}-\frac {8 i b^{3} B c}{a d \left (a^{4}+2 a^{2} b^{2}+b^{4}\right )}-\frac {2 i C c}{a^{2} d}-\frac {2 i C x}{a^{2}}+\frac {6 i C \,b^{2} c}{d \left (a^{4}+2 a^{2} b^{2}+b^{4}\right )}+\frac {6 i C \,b^{2} x}{a^{4}+2 a^{2} b^{2}+b^{4}}-\frac {8 i b^{3} B x}{a \left (a^{4}+2 a^{2} b^{2}+b^{4}\right )}+\frac {2 i b^{4} C c}{a^{2} d \left (a^{4}+2 a^{2} b^{2}+b^{4}\right )}-\frac {2 \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right ) B b}{a^{3} d}+\frac {\ln \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right ) C}{a^{2} d}+\frac {4 b^{3} \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}-\frac {i b +a}{i b -a}\right ) B}{a d \left (a^{4}+2 a^{2} b^{2}+b^{4}\right )}+\frac {2 b^{5} \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^{4}+2 a^{2} b^{2}+b^{4}\right )}-\frac {3 \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}-\frac {i b +a}{i b -a}\right ) C \,b^{2}}{d \left (a^{4}+2 a^{2} b^{2}+b^{4}\right )}-\frac {b^{4} \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}-\frac {i b +a}{i b -a}\right ) C}{a^{2} d \left (a^{4}+2 a^{2} b^{2}+b^{4}\right )}\)	\(759\)

[In]

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

[Out]

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 465 vs. \(2 (190) = 380\).

Time = 0.32 (sec) , antiderivative size = 465, normalized size of antiderivative = 2.42 \[ \int \frac {\cot ^3(c+d x) \left (B \tan (c+d x)+C \tan ^2(c+d x)\right )}{(a+b \tan (c+d x))^2} \, dx=-\frac {2 \, B a^{6} + 4 \, B a^{4} b^{2} + 2 \, B a^{2} b^{4} + 2 \, {\left (C a^{3} b^{3} - B a^{2} b^{4} + {\left (B a^{5} b + 2 \, C a^{4} b^{2} - B a^{3} b^{3}\right )} d x\right )} \tan \left (d x + c\right )^{2} - {\left ({\left (C a^{5} b - 2 \, B a^{4} b^{2} + 2 \, C a^{3} b^{3} - 4 \, B a^{2} b^{4} + C a b^{5} - 2 \, B b^{6}\right )} \tan \left (d x + c\right )^{2} + {\left (C a^{6} - 2 \, B a^{5} b + 2 \, C a^{4} b^{2} - 4 \, B a^{3} b^{3} + C a^{2} b^{4} - 2 \, B a b^{5}\right )} \tan \left (d x + c\right )\right )} \log \left (\frac {\tan \left (d x + c\right )^{2}}{\tan \left (d x + c\right )^{2} + 1}\right ) + {\left ({\left (3 \, C a^{3} b^{3} - 4 \, B a^{2} b^{4} + C a b^{5} - 2 \, B b^{6}\right )} \tan \left (d x + c\right )^{2} + {\left (3 \, C a^{4} b^{2} - 4 \, B a^{3} b^{3} + C a^{2} b^{4} - 2 \, B a b^{5}\right )} \tan \left (d x + c\right )\right )} \log \left (\frac {b^{2} \tan \left (d x + c\right )^{2} + 2 \, a b \tan \left (d x + c\right ) + a^{2}}{\tan \left (d x + c\right )^{2} + 1}\right ) + 2 \, {\left (B a^{5} b + 2 \, B a^{3} b^{3} - C a^{2} b^{4} + 2 \, B a b^{5} + {\left (B a^{6} + 2 \, C a^{5} b - B a^{4} b^{2}\right )} d x\right )} \tan \left (d x + c\right )}{2 \, {\left ({\left (a^{7} b + 2 \, a^{5} b^{3} + a^{3} b^{5}\right )} d \tan \left (d x + c\right )^{2} + {\left (a^{8} + 2 \, a^{6} b^{2} + a^{4} b^{4}\right )} d \tan \left (d x + c\right )\right )}} \]

[In]

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

[Out]

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

Sympy [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 6.03 (sec) , antiderivative size = 8143, normalized size of antiderivative = 42.41 \[ \int \frac {\cot ^3(c+d x) \left (B \tan (c+d x)+C \tan ^2(c+d x)\right )}{(a+b \tan (c+d x))^2} \, dx=\text {Too large to display} \]

[In]

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

[Out]

Piecewise(((-B*x - B/(d*tan(c + d*x)) - C*log(tan(c + d*x)**2 + 1)/(2*d) + C*log(tan(c + d*x))/d)/a**2, Eq(b,
0)), ((B*x + B/(d*tan(c + d*x)) - B/(3*d*tan(c + d*x)**3) + C*log(tan(c + d*x)**2 + 1)/(2*d) - C*log(tan(c + d
*x))/d - C/(2*d*tan(c + d*x)**2))/b**2, Eq(a, 0)), (-9*B*d*x*tan(c + d*x)**3/(4*a**2*d*tan(c + d*x)**3 + 8*I*a
**2*d*tan(c + d*x)**2 - 4*a**2*d*tan(c + d*x)) - 18*I*B*d*x*tan(c + d*x)**2/(4*a**2*d*tan(c + d*x)**3 + 8*I*a*
*2*d*tan(c + d*x)**2 - 4*a**2*d*tan(c + d*x)) + 9*B*d*x*tan(c + d*x)/(4*a**2*d*tan(c + d*x)**3 + 8*I*a**2*d*ta
n(c + d*x)**2 - 4*a**2*d*tan(c + d*x)) - 4*I*B*log(tan(c + d*x)**2 + 1)*tan(c + d*x)**3/(4*a**2*d*tan(c + d*x)
**3 + 8*I*a**2*d*tan(c + d*x)**2 - 4*a**2*d*tan(c + d*x)) + 8*B*log(tan(c + d*x)**2 + 1)*tan(c + d*x)**2/(4*a*
*2*d*tan(c + d*x)**3 + 8*I*a**2*d*tan(c + d*x)**2 - 4*a**2*d*tan(c + d*x)) + 4*I*B*log(tan(c + d*x)**2 + 1)*ta
n(c + d*x)/(4*a**2*d*tan(c + d*x)**3 + 8*I*a**2*d*tan(c + d*x)**2 - 4*a**2*d*tan(c + d*x)) + 8*I*B*log(tan(c +
d*x))*tan(c + d*x)**3/(4*a**2*d*tan(c + d*x)**3 + 8*I*a**2*d*tan(c + d*x)**2 - 4*a**2*d*tan(c + d*x)) - 16*B*
log(tan(c + d*x))*tan(c + d*x)**2/(4*a**2*d*tan(c + d*x)**3 + 8*I*a**2*d*tan(c + d*x)**2 - 4*a**2*d*tan(c + d*
x)) - 8*I*B*log(tan(c + d*x))*tan(c + d*x)/(4*a**2*d*tan(c + d*x)**3 + 8*I*a**2*d*tan(c + d*x)**2 - 4*a**2*d*t
an(c + d*x)) - 9*B*tan(c + d*x)**2/(4*a**2*d*tan(c + d*x)**3 + 8*I*a**2*d*tan(c + d*x)**2 - 4*a**2*d*tan(c + d
*x)) - 14*I*B*tan(c + d*x)/(4*a**2*d*tan(c + d*x)**3 + 8*I*a**2*d*tan(c + d*x)**2 - 4*a**2*d*tan(c + d*x)) + 4
*B/(4*a**2*d*tan(c + d*x)**3 + 8*I*a**2*d*tan(c + d*x)**2 - 4*a**2*d*tan(c + d*x)) + 3*I*C*d*x*tan(c + d*x)**3
/(4*a**2*d*tan(c + d*x)**3 + 8*I*a**2*d*tan(c + d*x)**2 - 4*a**2*d*tan(c + d*x)) - 6*C*d*x*tan(c + d*x)**2/(4*
a**2*d*tan(c + d*x)**3 + 8*I*a**2*d*tan(c + d*x)**2 - 4*a**2*d*tan(c + d*x)) - 3*I*C*d*x*tan(c + d*x)/(4*a**2*
d*tan(c + d*x)**3 + 8*I*a**2*d*tan(c + d*x)**2 - 4*a**2*d*tan(c + d*x)) - 2*C*log(tan(c + d*x)**2 + 1)*tan(c +
d*x)**3/(4*a**2*d*tan(c + d*x)**3 + 8*I*a**2*d*tan(c + d*x)**2 - 4*a**2*d*tan(c + d*x)) - 4*I*C*log(tan(c + d
*x)**2 + 1)*tan(c + d*x)**2/(4*a**2*d*tan(c + d*x)**3 + 8*I*a**2*d*tan(c + d*x)**2 - 4*a**2*d*tan(c + d*x)) +
2*C*log(tan(c + d*x)**2 + 1)*tan(c + d*x)/(4*a**2*d*tan(c + d*x)**3 + 8*I*a**2*d*tan(c + d*x)**2 - 4*a**2*d*ta
n(c + d*x)) + 4*C*log(tan(c + d*x))*tan(c + d*x)**3/(4*a**2*d*tan(c + d*x)**3 + 8*I*a**2*d*tan(c + d*x)**2 - 4
*a**2*d*tan(c + d*x)) + 8*I*C*log(tan(c + d*x))*tan(c + d*x)**2/(4*a**2*d*tan(c + d*x)**3 + 8*I*a**2*d*tan(c +
d*x)**2 - 4*a**2*d*tan(c + d*x)) - 4*C*log(tan(c + d*x))*tan(c + d*x)/(4*a**2*d*tan(c + d*x)**3 + 8*I*a**2*d*
tan(c + d*x)**2 - 4*a**2*d*tan(c + d*x)) + 3*I*C*tan(c + d*x)**2/(4*a**2*d*tan(c + d*x)**3 + 8*I*a**2*d*tan(c
+ d*x)**2 - 4*a**2*d*tan(c + d*x)) - 4*C*tan(c + d*x)/(4*a**2*d*tan(c + d*x)**3 + 8*I*a**2*d*tan(c + d*x)**2 -
4*a**2*d*tan(c + d*x)), Eq(b, -I*a)), (-9*B*d*x*tan(c + d*x)**3/(4*a**2*d*tan(c + d*x)**3 - 8*I*a**2*d*tan(c
+ d*x)**2 - 4*a**2*d*tan(c + d*x)) + 18*I*B*d*x*tan(c + d*x)**2/(4*a**2*d*tan(c + d*x)**3 - 8*I*a**2*d*tan(c +
d*x)**2 - 4*a**2*d*tan(c + d*x)) + 9*B*d*x*tan(c + d*x)/(4*a**2*d*tan(c + d*x)**3 - 8*I*a**2*d*tan(c + d*x)**
2 - 4*a**2*d*tan(c + d*x)) + 4*I*B*log(tan(c + d*x)**2 + 1)*tan(c + d*x)**3/(4*a**2*d*tan(c + d*x)**3 - 8*I*a*
*2*d*tan(c + d*x)**2 - 4*a**2*d*tan(c + d*x)) + 8*B*log(tan(c + d*x)**2 + 1)*tan(c + d*x)**2/(4*a**2*d*tan(c +
d*x)**3 - 8*I*a**2*d*tan(c + d*x)**2 - 4*a**2*d*tan(c + d*x)) - 4*I*B*log(tan(c + d*x)**2 + 1)*tan(c + d*x)/(
4*a**2*d*tan(c + d*x)**3 - 8*I*a**2*d*tan(c + d*x)**2 - 4*a**2*d*tan(c + d*x)) - 8*I*B*log(tan(c + d*x))*tan(c
+ d*x)**3/(4*a**2*d*tan(c + d*x)**3 - 8*I*a**2*d*tan(c + d*x)**2 - 4*a**2*d*tan(c + d*x)) - 16*B*log(tan(c +
d*x))*tan(c + d*x)**2/(4*a**2*d*tan(c + d*x)**3 - 8*I*a**2*d*tan(c + d*x)**2 - 4*a**2*d*tan(c + d*x)) + 8*I*B*
log(tan(c + d*x))*tan(c + d*x)/(4*a**2*d*tan(c + d*x)**3 - 8*I*a**2*d*tan(c + d*x)**2 - 4*a**2*d*tan(c + d*x))
- 9*B*tan(c + d*x)**2/(4*a**2*d*tan(c + d*x)**3 - 8*I*a**2*d*tan(c + d*x)**2 - 4*a**2*d*tan(c + d*x)) + 14*I*
B*tan(c + d*x)/(4*a**2*d*tan(c + d*x)**3 - 8*I*a**2*d*tan(c + d*x)**2 - 4*a**2*d*tan(c + d*x)) + 4*B/(4*a**2*d
*tan(c + d*x)**3 - 8*I*a**2*d*tan(c + d*x)**2 - 4*a**2*d*tan(c + d*x)) - 3*I*C*d*x*tan(c + d*x)**3/(4*a**2*d*t
an(c + d*x)**3 - 8*I*a**2*d*tan(c + d*x)**2 - 4*a**2*d*tan(c + d*x)) - 6*C*d*x*tan(c + d*x)**2/(4*a**2*d*tan(c
+ d*x)**3 - 8*I*a**2*d*tan(c + d*x)**2 - 4*a**2*d*tan(c + d*x)) + 3*I*C*d*x*tan(c + d*x)/(4*a**2*d*tan(c + d*
x)**3 - 8*I*a**2*d*tan(c + d*x)**2 - 4*a**2*d*tan(c + d*x)) - 2*C*log(tan(c + d*x)**2 + 1)*tan(c + d*x)**3/(4*
a**2*d*tan(c + d*x)**3 - 8*I*a**2*d*tan(c + d*x)**2 - 4*a**2*d*tan(c + d*x)) + 4*I*C*log(tan(c + d*x)**2 + 1)*
tan(c + d*x)**2/(4*a**2*d*tan(c + d*x)**3 - 8*I*a**2*d*tan(c + d*x)**2 - 4*a**2*d*tan(c + d*x)) + 2*C*log(tan(
c + d*x)**2 + 1)*tan(c + d*x)/(4*a**2*d*tan(c + d*x)**3 - 8*I*a**2*d*tan(c + d*x)**2 - 4*a**2*d*tan(c + d*x))
+ 4*C*log(tan(c + d*x))*tan(c + d*x)**3/(4*a**2*d*tan(c + d*x)**3 - 8*I*a**2*d*tan(c + d*x)**2 - 4*a**2*d*tan(
c + d*x)) - 8*I*C*log(tan(c + d*x))*tan(c + d*x)**2/(4*a**2*d*tan(c + d*x)**3 - 8*I*a**2*d*tan(c + d*x)**2 - 4
*a**2*d*tan(c + d*x)) - 4*C*log(tan(c + d*x))*tan(c + d*x)/(4*a**2*d*tan(c + d*x)**3 - 8*I*a**2*d*tan(c + d*x)
**2 - 4*a**2*d*tan(c + d*x)) - 3*I*C*tan(c + d*x)**2/(4*a**2*d*tan(c + d*x)**3 - 8*I*a**2*d*tan(c + d*x)**2 -
4*a**2*d*tan(c + d*x)) - 4*C*tan(c + d*x)/(4*a**2*d*tan(c + d*x)**3 - 8*I*a**2*d*tan(c + d*x)**2 - 4*a**2*d*ta
n(c + d*x)), Eq(b, I*a)), (zoo*(-B*x - B/(d*tan(c + d*x)) - C*log(tan(c + d*x)**2 + 1)/(2*d) + C*log(tan(c + d
*x))/d), Eq(b, -a/tan(c + d*x))), (nan, Eq(c, -d*x)), (x*(B*tan(c) + C*tan(c)**2)*cot(c)**3/(a + b*tan(c))**2,
Eq(d, 0)), (-2*B*a**6*d*x*tan(c + d*x)/(2*a**8*d*tan(c + d*x) + 2*a**7*b*d*tan(c + d*x)**2 + 4*a**6*b**2*d*ta
n(c + d*x) + 4*a**5*b**3*d*tan(c + d*x)**2 + 2*a**4*b**4*d*tan(c + d*x) + 2*a**3*b**5*d*tan(c + d*x)**2) - 2*B
*a**6/(2*a**8*d*tan(c + d*x) + 2*a**7*b*d*tan(c + d*x)**2 + 4*a**6*b**2*d*tan(c + d*x) + 4*a**5*b**3*d*tan(c +
d*x)**2 + 2*a**4*b**4*d*tan(c + d*x) + 2*a**3*b**5*d*tan(c + d*x)**2) - 2*B*a**5*b*d*x*tan(c + d*x)**2/(2*a**
8*d*tan(c + d*x) + 2*a**7*b*d*tan(c + d*x)**2 + 4*a**6*b**2*d*tan(c + d*x) + 4*a**5*b**3*d*tan(c + d*x)**2 + 2
*a**4*b**4*d*tan(c + d*x) + 2*a**3*b**5*d*tan(c + d*x)**2) + 2*B*a**5*b*log(tan(c + d*x)**2 + 1)*tan(c + d*x)/
(2*a**8*d*tan(c + d*x) + 2*a**7*b*d*tan(c + d*x)**2 + 4*a**6*b**2*d*tan(c + d*x) + 4*a**5*b**3*d*tan(c + d*x)*
*2 + 2*a**4*b**4*d*tan(c + d*x) + 2*a**3*b**5*d*tan(c + d*x)**2) - 4*B*a**5*b*log(tan(c + d*x))*tan(c + d*x)/(
2*a**8*d*tan(c + d*x) + 2*a**7*b*d*tan(c + d*x)**2 + 4*a**6*b**2*d*tan(c + d*x) + 4*a**5*b**3*d*tan(c + d*x)**
2 + 2*a**4*b**4*d*tan(c + d*x) + 2*a**3*b**5*d*tan(c + d*x)**2) - 2*B*a**5*b*tan(c + d*x)/(2*a**8*d*tan(c + d*
x) + 2*a**7*b*d*tan(c + d*x)**2 + 4*a**6*b**2*d*tan(c + d*x) + 4*a**5*b**3*d*tan(c + d*x)**2 + 2*a**4*b**4*d*t
an(c + d*x) + 2*a**3*b**5*d*tan(c + d*x)**2) + 2*B*a**4*b**2*d*x*tan(c + d*x)/(2*a**8*d*tan(c + d*x) + 2*a**7*
b*d*tan(c + d*x)**2 + 4*a**6*b**2*d*tan(c + d*x) + 4*a**5*b**3*d*tan(c + d*x)**2 + 2*a**4*b**4*d*tan(c + d*x)
+ 2*a**3*b**5*d*tan(c + d*x)**2) + 2*B*a**4*b**2*log(tan(c + d*x)**2 + 1)*tan(c + d*x)**2/(2*a**8*d*tan(c + d*
x) + 2*a**7*b*d*tan(c + d*x)**2 + 4*a**6*b**2*d*tan(c + d*x) + 4*a**5*b**3*d*tan(c + d*x)**2 + 2*a**4*b**4*d*t
an(c + d*x) + 2*a**3*b**5*d*tan(c + d*x)**2) - 4*B*a**4*b**2*log(tan(c + d*x))*tan(c + d*x)**2/(2*a**8*d*tan(c
+ d*x) + 2*a**7*b*d*tan(c + d*x)**2 + 4*a**6*b**2*d*tan(c + d*x) + 4*a**5*b**3*d*tan(c + d*x)**2 + 2*a**4*b**
4*d*tan(c + d*x) + 2*a**3*b**5*d*tan(c + d*x)**2) - 4*B*a**4*b**2/(2*a**8*d*tan(c + d*x) + 2*a**7*b*d*tan(c +
d*x)**2 + 4*a**6*b**2*d*tan(c + d*x) + 4*a**5*b**3*d*tan(c + d*x)**2 + 2*a**4*b**4*d*tan(c + d*x) + 2*a**3*b**
5*d*tan(c + d*x)**2) + 2*B*a**3*b**3*d*x*tan(c + d*x)**2/(2*a**8*d*tan(c + d*x) + 2*a**7*b*d*tan(c + d*x)**2 +
4*a**6*b**2*d*tan(c + d*x) + 4*a**5*b**3*d*tan(c + d*x)**2 + 2*a**4*b**4*d*tan(c + d*x) + 2*a**3*b**5*d*tan(c
+ d*x)**2) + 8*B*a**3*b**3*log(a/b + tan(c + d*x))*tan(c + d*x)/(2*a**8*d*tan(c + d*x) + 2*a**7*b*d*tan(c + d
*x)**2 + 4*a**6*b**2*d*tan(c + d*x) + 4*a**5*b**3*d*tan(c + d*x)**2 + 2*a**4*b**4*d*tan(c + d*x) + 2*a**3*b**5
*d*tan(c + d*x)**2) - 8*B*a**3*b**3*log(tan(c + d*x))*tan(c + d*x)/(2*a**8*d*tan(c + d*x) + 2*a**7*b*d*tan(c +
d*x)**2 + 4*a**6*b**2*d*tan(c + d*x) + 4*a**5*b**3*d*tan(c + d*x)**2 + 2*a**4*b**4*d*tan(c + d*x) + 2*a**3*b*
*5*d*tan(c + d*x)**2) - 6*B*a**3*b**3*tan(c + d*x)/(2*a**8*d*tan(c + d*x) + 2*a**7*b*d*tan(c + d*x)**2 + 4*a**
6*b**2*d*tan(c + d*x) + 4*a**5*b**3*d*tan(c + d*x)**2 + 2*a**4*b**4*d*tan(c + d*x) + 2*a**3*b**5*d*tan(c + d*x
)**2) + 8*B*a**2*b**4*log(a/b + tan(c + d*x))*tan(c + d*x)**2/(2*a**8*d*tan(c + d*x) + 2*a**7*b*d*tan(c + d*x)
**2 + 4*a**6*b**2*d*tan(c + d*x) + 4*a**5*b**3*d*tan(c + d*x)**2 + 2*a**4*b**4*d*tan(c + d*x) + 2*a**3*b**5*d*
tan(c + d*x)**2) - 8*B*a**2*b**4*log(tan(c + d*x))*tan(c + d*x)**2/(2*a**8*d*tan(c + d*x) + 2*a**7*b*d*tan(c +
d*x)**2 + 4*a**6*b**2*d*tan(c + d*x) + 4*a**5*b**3*d*tan(c + d*x)**2 + 2*a**4*b**4*d*tan(c + d*x) + 2*a**3*b*
*5*d*tan(c + d*x)**2) - 2*B*a**2*b**4/(2*a**8*d*tan(c + d*x) + 2*a**7*b*d*tan(c + d*x)**2 + 4*a**6*b**2*d*tan(
c + d*x) + 4*a**5*b**3*d*tan(c + d*x)**2 + 2*a**4*b**4*d*tan(c + d*x) + 2*a**3*b**5*d*tan(c + d*x)**2) + 4*B*a
*b**5*log(a/b + tan(c + d*x))*tan(c + d*x)/(2*a**8*d*tan(c + d*x) + 2*a**7*b*d*tan(c + d*x)**2 + 4*a**6*b**2*d
*tan(c + d*x) + 4*a**5*b**3*d*tan(c + d*x)**2 + 2*a**4*b**4*d*tan(c + d*x) + 2*a**3*b**5*d*tan(c + d*x)**2) -
4*B*a*b**5*log(tan(c + d*x))*tan(c + d*x)/(2*a**8*d*tan(c + d*x) + 2*a**7*b*d*tan(c + d*x)**2 + 4*a**6*b**2*d*
tan(c + d*x) + 4*a**5*b**3*d*tan(c + d*x)**2 + 2*a**4*b**4*d*tan(c + d*x) + 2*a**3*b**5*d*tan(c + d*x)**2) - 4
*B*a*b**5*tan(c + d*x)/(2*a**8*d*tan(c + d*x) + 2*a**7*b*d*tan(c + d*x)**2 + 4*a**6*b**2*d*tan(c + d*x) + 4*a*
*5*b**3*d*tan(c + d*x)**2 + 2*a**4*b**4*d*tan(c + d*x) + 2*a**3*b**5*d*tan(c + d*x)**2) + 4*B*b**6*log(a/b + t
an(c + d*x))*tan(c + d*x)**2/(2*a**8*d*tan(c + d*x) + 2*a**7*b*d*tan(c + d*x)**2 + 4*a**6*b**2*d*tan(c + d*x)
+ 4*a**5*b**3*d*tan(c + d*x)**2 + 2*a**4*b**4*d*tan(c + d*x) + 2*a**3*b**5*d*tan(c + d*x)**2) - 4*B*b**6*log(t
an(c + d*x))*tan(c + d*x)**2/(2*a**8*d*tan(c + d*x) + 2*a**7*b*d*tan(c + d*x)**2 + 4*a**6*b**2*d*tan(c + d*x)
+ 4*a**5*b**3*d*tan(c + d*x)**2 + 2*a**4*b**4*d*tan(c + d*x) + 2*a**3*b**5*d*tan(c + d*x)**2) - C*a**6*log(tan
(c + d*x)**2 + 1)*tan(c + d*x)/(2*a**8*d*tan(c + d*x) + 2*a**7*b*d*tan(c + d*x)**2 + 4*a**6*b**2*d*tan(c + d*x
) + 4*a**5*b**3*d*tan(c + d*x)**2 + 2*a**4*b**4*d*tan(c + d*x) + 2*a**3*b**5*d*tan(c + d*x)**2) + 2*C*a**6*log
(tan(c + d*x))*tan(c + d*x)/(2*a**8*d*tan(c + d*x) + 2*a**7*b*d*tan(c + d*x)**2 + 4*a**6*b**2*d*tan(c + d*x) +
4*a**5*b**3*d*tan(c + d*x)**2 + 2*a**4*b**4*d*tan(c + d*x) + 2*a**3*b**5*d*tan(c + d*x)**2) - 4*C*a**5*b*d*x*
tan(c + d*x)/(2*a**8*d*tan(c + d*x) + 2*a**7*b*d*tan(c + d*x)**2 + 4*a**6*b**2*d*tan(c + d*x) + 4*a**5*b**3*d*
tan(c + d*x)**2 + 2*a**4*b**4*d*tan(c + d*x) + 2*a**3*b**5*d*tan(c + d*x)**2) - C*a**5*b*log(tan(c + d*x)**2 +
1)*tan(c + d*x)**2/(2*a**8*d*tan(c + d*x) + 2*a**7*b*d*tan(c + d*x)**2 + 4*a**6*b**2*d*tan(c + d*x) + 4*a**5*
b**3*d*tan(c + d*x)**2 + 2*a**4*b**4*d*tan(c + d*x) + 2*a**3*b**5*d*tan(c + d*x)**2) + 2*C*a**5*b*log(tan(c +
d*x))*tan(c + d*x)**2/(2*a**8*d*tan(c + d*x) + 2*a**7*b*d*tan(c + d*x)**2 + 4*a**6*b**2*d*tan(c + d*x) + 4*a**
5*b**3*d*tan(c + d*x)**2 + 2*a**4*b**4*d*tan(c + d*x) + 2*a**3*b**5*d*tan(c + d*x)**2) - 4*C*a**4*b**2*d*x*tan
(c + d*x)**2/(2*a**8*d*tan(c + d*x) + 2*a**7*b*d*tan(c + d*x)**2 + 4*a**6*b**2*d*tan(c + d*x) + 4*a**5*b**3*d*
tan(c + d*x)**2 + 2*a**4*b**4*d*tan(c + d*x) + 2*a**3*b**5*d*tan(c + d*x)**2) - 6*C*a**4*b**2*log(a/b + tan(c
+ d*x))*tan(c + d*x)/(2*a**8*d*tan(c + d*x) + 2*a**7*b*d*tan(c + d*x)**2 + 4*a**6*b**2*d*tan(c + d*x) + 4*a**5
*b**3*d*tan(c + d*x)**2 + 2*a**4*b**4*d*tan(c + d*x) + 2*a**3*b**5*d*tan(c + d*x)**2) + C*a**4*b**2*log(tan(c
+ d*x)**2 + 1)*tan(c + d*x)/(2*a**8*d*tan(c + d*x) + 2*a**7*b*d*tan(c + d*x)**2 + 4*a**6*b**2*d*tan(c + d*x) +
4*a**5*b**3*d*tan(c + d*x)**2 + 2*a**4*b**4*d*tan(c + d*x) + 2*a**3*b**5*d*tan(c + d*x)**2) + 4*C*a**4*b**2*l
og(tan(c + d*x))*tan(c + d*x)/(2*a**8*d*tan(c + d*x) + 2*a**7*b*d*tan(c + d*x)**2 + 4*a**6*b**2*d*tan(c + d*x)
+ 4*a**5*b**3*d*tan(c + d*x)**2 + 2*a**4*b**4*d*tan(c + d*x) + 2*a**3*b**5*d*tan(c + d*x)**2) + 2*C*a**4*b**2
*tan(c + d*x)/(2*a**8*d*tan(c + d*x) + 2*a**7*b*d*tan(c + d*x)**2 + 4*a**6*b**2*d*tan(c + d*x) + 4*a**5*b**3*d
*tan(c + d*x)**2 + 2*a**4*b**4*d*tan(c + d*x) + 2*a**3*b**5*d*tan(c + d*x)**2) - 6*C*a**3*b**3*log(a/b + tan(c
+ d*x))*tan(c + d*x)**2/(2*a**8*d*tan(c + d*x) + 2*a**7*b*d*tan(c + d*x)**2 + 4*a**6*b**2*d*tan(c + d*x) + 4*
a**5*b**3*d*tan(c + d*x)**2 + 2*a**4*b**4*d*tan(c + d*x) + 2*a**3*b**5*d*tan(c + d*x)**2) + C*a**3*b**3*log(ta
n(c + d*x)**2 + 1)*tan(c + d*x)**2/(2*a**8*d*tan(c + d*x) + 2*a**7*b*d*tan(c + d*x)**2 + 4*a**6*b**2*d*tan(c +
d*x) + 4*a**5*b**3*d*tan(c + d*x)**2 + 2*a**4*b**4*d*tan(c + d*x) + 2*a**3*b**5*d*tan(c + d*x)**2) + 4*C*a**3
*b**3*log(tan(c + d*x))*tan(c + d*x)**2/(2*a**8*d*tan(c + d*x) + 2*a**7*b*d*tan(c + d*x)**2 + 4*a**6*b**2*d*ta
n(c + d*x) + 4*a**5*b**3*d*tan(c + d*x)**2 + 2*a**4*b**4*d*tan(c + d*x) + 2*a**3*b**5*d*tan(c + d*x)**2) - 2*C
*a**2*b**4*log(a/b + tan(c + d*x))*tan(c + d*x)/(2*a**8*d*tan(c + d*x) + 2*a**7*b*d*tan(c + d*x)**2 + 4*a**6*b
**2*d*tan(c + d*x) + 4*a**5*b**3*d*tan(c + d*x)**2 + 2*a**4*b**4*d*tan(c + d*x) + 2*a**3*b**5*d*tan(c + d*x)**
2) + 2*C*a**2*b**4*log(tan(c + d*x))*tan(c + d*x)/(2*a**8*d*tan(c + d*x) + 2*a**7*b*d*tan(c + d*x)**2 + 4*a**6
*b**2*d*tan(c + d*x) + 4*a**5*b**3*d*tan(c + d*x)**2 + 2*a**4*b**4*d*tan(c + d*x) + 2*a**3*b**5*d*tan(c + d*x)
**2) + 2*C*a**2*b**4*tan(c + d*x)/(2*a**8*d*tan(c + d*x) + 2*a**7*b*d*tan(c + d*x)**2 + 4*a**6*b**2*d*tan(c +
d*x) + 4*a**5*b**3*d*tan(c + d*x)**2 + 2*a**4*b**4*d*tan(c + d*x) + 2*a**3*b**5*d*tan(c + d*x)**2) - 2*C*a*b**
5*log(a/b + tan(c + d*x))*tan(c + d*x)**2/(2*a**8*d*tan(c + d*x) + 2*a**7*b*d*tan(c + d*x)**2 + 4*a**6*b**2*d*
tan(c + d*x) + 4*a**5*b**3*d*tan(c + d*x)**2 + 2*a**4*b**4*d*tan(c + d*x) + 2*a**3*b**5*d*tan(c + d*x)**2) + 2
*C*a*b**5*log(tan(c + d*x))*tan(c + d*x)**2/(2*a**8*d*tan(c + d*x) + 2*a**7*b*d*tan(c + d*x)**2 + 4*a**6*b**2*
d*tan(c + d*x) + 4*a**5*b**3*d*tan(c + d*x)**2 + 2*a**4*b**4*d*tan(c + d*x) + 2*a**3*b**5*d*tan(c + d*x)**2),
True))

Maxima [A] (verification not implemented)

none

Time = 0.35 (sec) , antiderivative size = 262, normalized size of antiderivative = 1.36 \[ \int \frac {\cot ^3(c+d x) \left (B \tan (c+d x)+C \tan ^2(c+d x)\right )}{(a+b \tan (c+d x))^2} \, dx=-\frac {\frac {2 \, {\left (B a^{2} + 2 \, C a b - B b^{2}\right )} {\left (d x + c\right )}}{a^{4} + 2 \, a^{2} b^{2} + b^{4}} + \frac {2 \, {\left (3 \, C a^{3} b^{2} - 4 \, B a^{2} b^{3} + C a b^{4} - 2 \, B b^{5}\right )} \log \left (b \tan \left (d x + c\right ) + a\right )}{a^{7} + 2 \, a^{5} b^{2} + a^{3} b^{4}} + \frac {{\left (C a^{2} - 2 \, B a b - C b^{2}\right )} \log \left (\tan \left (d x + c\right )^{2} + 1\right )}{a^{4} + 2 \, a^{2} b^{2} + b^{4}} + \frac {2 \, {\left (B a^{3} + B a b^{2} + {\left (B a^{2} b - C a b^{2} + 2 \, B b^{3}\right )} \tan \left (d x + c\right )\right )}}{{\left (a^{4} b + a^{2} b^{3}\right )} \tan \left (d x + c\right )^{2} + {\left (a^{5} + a^{3} b^{2}\right )} \tan \left (d x + c\right )} - \frac {2 \, {\left (C a - 2 \, B b\right )} \log \left (\tan \left (d x + c\right )\right )}{a^{3}}}{2 \, d} \]

[In]

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

[Out]

-1/2*(2*(B*a^2 + 2*C*a*b - B*b^2)*(d*x + c)/(a^4 + 2*a^2*b^2 + b^4) + 2*(3*C*a^3*b^2 - 4*B*a^2*b^3 + C*a*b^4 -
2*B*b^5)*log(b*tan(d*x + c) + a)/(a^7 + 2*a^5*b^2 + a^3*b^4) + (C*a^2 - 2*B*a*b - C*b^2)*log(tan(d*x + c)^2 +
1)/(a^4 + 2*a^2*b^2 + b^4) + 2*(B*a^3 + B*a*b^2 + (B*a^2*b - C*a*b^2 + 2*B*b^3)*tan(d*x + c))/((a^4*b + a^2*b
^3)*tan(d*x + c)^2 + (a^5 + a^3*b^2)*tan(d*x + c)) - 2*(C*a - 2*B*b)*log(tan(d*x + c))/a^3)/d

Giac [A] (verification not implemented)

none

Time = 1.23 (sec) , antiderivative size = 362, normalized size of antiderivative = 1.89 \[ \int \frac {\cot ^3(c+d x) \left (B \tan (c+d x)+C \tan ^2(c+d x)\right )}{(a+b \tan (c+d x))^2} \, dx=-\frac {\frac {2 \, {\left (B a^{2} + 2 \, C a b - B b^{2}\right )} {\left (d x + c\right )}}{a^{4} + 2 \, a^{2} b^{2} + b^{4}} + \frac {{\left (C a^{2} - 2 \, B a b - C b^{2}\right )} \log \left (\tan \left (d x + c\right )^{2} + 1\right )}{a^{4} + 2 \, a^{2} b^{2} + b^{4}} + \frac {2 \, {\left (3 \, C a^{3} b^{3} - 4 \, B a^{2} b^{4} + C a b^{5} - 2 \, B b^{6}\right )} \log \left ({\left | b \tan \left (d x + c\right ) + a \right |}\right )}{a^{7} b + 2 \, a^{5} b^{3} + a^{3} b^{5}} + \frac {C a^{4} b \tan \left (d x + c\right )^{2} - 2 \, B a^{3} b^{2} \tan \left (d x + c\right )^{2} - C a^{2} b^{3} \tan \left (d x + c\right )^{2} + C a^{5} \tan \left (d x + c\right ) - 3 \, C a^{3} b^{2} \tan \left (d x + c\right ) + 6 \, B a^{2} b^{3} \tan \left (d x + c\right ) - 2 \, C a b^{4} \tan \left (d x + c\right ) + 4 \, B b^{5} \tan \left (d x + c\right ) + 2 \, B a^{5} + 4 \, B a^{3} b^{2} + 2 \, B a b^{4}}{{\left (a^{6} + 2 \, a^{4} b^{2} + a^{2} b^{4}\right )} {\left (b \tan \left (d x + c\right )^{2} + a \tan \left (d x + c\right )\right )}} - \frac {2 \, {\left (C a - 2 \, B b\right )} \log \left ({\left | \tan \left (d x + c\right ) \right |}\right )}{a^{3}}}{2 \, d} \]

[In]

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

[Out]

-1/2*(2*(B*a^2 + 2*C*a*b - B*b^2)*(d*x + c)/(a^4 + 2*a^2*b^2 + b^4) + (C*a^2 - 2*B*a*b - C*b^2)*log(tan(d*x +
c)^2 + 1)/(a^4 + 2*a^2*b^2 + b^4) + 2*(3*C*a^3*b^3 - 4*B*a^2*b^4 + C*a*b^5 - 2*B*b^6)*log(abs(b*tan(d*x + c) +
a))/(a^7*b + 2*a^5*b^3 + a^3*b^5) + (C*a^4*b*tan(d*x + c)^2 - 2*B*a^3*b^2*tan(d*x + c)^2 - C*a^2*b^3*tan(d*x
+ c)^2 + C*a^5*tan(d*x + c) - 3*C*a^3*b^2*tan(d*x + c) + 6*B*a^2*b^3*tan(d*x + c) - 2*C*a*b^4*tan(d*x + c) + 4
*B*b^5*tan(d*x + c) + 2*B*a^5 + 4*B*a^3*b^2 + 2*B*a*b^4)/((a^6 + 2*a^4*b^2 + a^2*b^4)*(b*tan(d*x + c)^2 + a*ta
n(d*x + c))) - 2*(C*a - 2*B*b)*log(abs(tan(d*x + c)))/a^3)/d

Mupad [B] (verification not implemented)

Time = 11.13 (sec) , antiderivative size = 230, normalized size of antiderivative = 1.20 \[ \int \frac {\cot ^3(c+d x) \left (B \tan (c+d x)+C \tan ^2(c+d x)\right )}{(a+b \tan (c+d x))^2} \, dx=\frac {b^2\,\ln \left (a+b\,\mathrm {tan}\left (c+d\,x\right )\right )\,\left (-3\,C\,a^3+4\,B\,a^2\,b-C\,a\,b^2+2\,B\,b^3\right )}{a^3\,d\,{\left (a^2+b^2\right )}^2}-\frac {\ln \left (\mathrm {tan}\left (c+d\,x\right )\right )\,\left (2\,B\,b-C\,a\right )}{a^3\,d}+\frac {\ln \left (\mathrm {tan}\left (c+d\,x\right )+1{}\mathrm {i}\right )\,\left (C+B\,1{}\mathrm {i}\right )}{2\,d\,\left (-a^2+a\,b\,2{}\mathrm {i}+b^2\right )}+\frac {\ln \left (\mathrm {tan}\left (c+d\,x\right )-\mathrm {i}\right )\,\left (B+C\,1{}\mathrm {i}\right )}{2\,d\,\left (-a^2\,1{}\mathrm {i}+2\,a\,b+b^2\,1{}\mathrm {i}\right )}-\frac {\frac {B}{a}+\frac {\mathrm {tan}\left (c+d\,x\right )\,\left (B\,a^2\,b-C\,a\,b^2+2\,B\,b^3\right )}{a^2\,\left (a^2+b^2\right )}}{d\,\left (b\,{\mathrm {tan}\left (c+d\,x\right )}^2+a\,\mathrm {tan}\left (c+d\,x\right )\right )} \]

[In]

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

[Out]

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