\(\int \frac {\cot (c+d x) (A+B \tan (c+d x))}{(a+b \tan (c+d x))^3} \, dx\) [287]

Optimal result

Integrand size = 29, antiderivative size = 215 \[ \int \frac {\cot (c+d x) (A+B \tan (c+d x))}{(a+b \tan (c+d x))^3} \, dx=-\frac {\left (3 a^2 A b-A b^3-a^3 B+3 a b^2 B\right ) x}{\left (a^2+b^2\right )^3}+\frac {A \log (\sin (c+d x))}{a^3 d}-\frac {b \left (6 a^4 A b+3 a^2 A b^3+A b^5-3 a^5 B+a^3 b^2 B\right ) \log (a \cos (c+d x)+b \sin (c+d x))}{a^3 \left (a^2+b^2\right )^3 d}+\frac {b (A b-a B)}{2 a \left (a^2+b^2\right ) d (a+b \tan (c+d x))^2}+\frac {b \left (3 a^2 A b+A b^3-2 a^3 B\right )}{a^2 \left (a^2+b^2\right )^2 d (a+b \tan (c+d x))} \]

[Out]

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

Rubi [A] (verified)

Time = 0.66 (sec) , antiderivative size = 215, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.172, Rules used = {3690, 3730, 3732, 3611, 3556} \[ \int \frac {\cot (c+d x) (A+B \tan (c+d x))}{(a+b \tan (c+d x))^3} \, dx=\frac {A \log (\sin (c+d x))}{a^3 d}+\frac {b (A b-a B)}{2 a d \left (a^2+b^2\right ) (a+b \tan (c+d x))^2}+\frac {b \left (-2 a^3 B+3 a^2 A b+A b^3\right )}{a^2 d \left (a^2+b^2\right )^2 (a+b \tan (c+d x))}-\frac {x \left (a^3 (-B)+3 a^2 A b+3 a b^2 B-A b^3\right )}{\left (a^2+b^2\right )^3}-\frac {b \left (-3 a^5 B+6 a^4 A b+a^3 b^2 B+3 a^2 A b^3+A b^5\right ) \log (a \cos (c+d x)+b \sin (c+d x))}{a^3 d \left (a^2+b^2\right )^3} \]

[In]

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

[Out]

-(((3*a^2*A*b - A*b^3 - a^3*B + 3*a*b^2*B)*x)/(a^2 + b^2)^3) + (A*Log[Sin[c + d*x]])/(a^3*d) - (b*(6*a^4*A*b +
3*a^2*A*b^3 + A*b^5 - 3*a^5*B + a^3*b^2*B)*Log[a*Cos[c + d*x] + b*Sin[c + d*x]])/(a^3*(a^2 + b^2)^3*d) + (b*(
A*b - a*B))/(2*a*(a^2 + b^2)*d*(a + b*Tan[c + d*x])^2) + (b*(3*a^2*A*b + A*b^3 - 2*a^3*B))/(a^2*(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 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 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 {b (A b-a B)}{2 a \left (a^2+b^2\right ) d (a+b \tan (c+d x))^2}+\frac {\int \frac {\cot (c+d x) \left (2 A \left (a^2+b^2\right )-2 a (A b-a B) \tan (c+d x)+2 b (A b-a B) \tan ^2(c+d x)\right )}{(a+b \tan (c+d x))^2} \, dx}{2 a \left (a^2+b^2\right )} \\ & = \frac {b (A b-a B)}{2 a \left (a^2+b^2\right ) d (a+b \tan (c+d x))^2}+\frac {b \left (3 a^2 A b+A b^3-2 a^3 B\right )}{a^2 \left (a^2+b^2\right )^2 d (a+b \tan (c+d x))}+\frac {\int \frac {\cot (c+d x) \left (2 A \left (a^2+b^2\right )^2-2 a^2 \left (2 a A b-a^2 B+b^2 B\right ) \tan (c+d x)+2 b \left (3 a^2 A b+A b^3-2 a^3 B\right ) \tan ^2(c+d x)\right )}{a+b \tan (c+d x)} \, dx}{2 a^2 \left (a^2+b^2\right )^2} \\ & = -\frac {\left (3 a^2 A b-A b^3-a^3 B+3 a b^2 B\right ) x}{\left (a^2+b^2\right )^3}+\frac {b (A b-a B)}{2 a \left (a^2+b^2\right ) d (a+b \tan (c+d x))^2}+\frac {b \left (3 a^2 A b+A b^3-2 a^3 B\right )}{a^2 \left (a^2+b^2\right )^2 d (a+b \tan (c+d x))}+\frac {A \int \cot (c+d x) \, dx}{a^3}-\frac {\left (b \left (6 a^4 A b+3 a^2 A b^3+A b^5-3 a^5 B+a^3 b^2 B\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 )^3} \\ & = -\frac {\left (3 a^2 A b-A b^3-a^3 B+3 a b^2 B\right ) x}{\left (a^2+b^2\right )^3}+\frac {A \log (\sin (c+d x))}{a^3 d}-\frac {b \left (6 a^4 A b+3 a^2 A b^3+A b^5-3 a^5 B+a^3 b^2 B\right ) \log (a \cos (c+d x)+b \sin (c+d x))}{a^3 \left (a^2+b^2\right )^3 d}+\frac {b (A b-a B)}{2 a \left (a^2+b^2\right ) d (a+b \tan (c+d x))^2}+\frac {b \left (3 a^2 A b+A b^3-2 a^3 B\right )}{a^2 \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 = 3.88 (sec) , antiderivative size = 254, normalized size of antiderivative = 1.18 \[ \int \frac {\cot (c+d x) (A+B \tan (c+d x))}{(a+b \tan (c+d x))^3} \, dx=\frac {-\frac {a (a-i b) (A+i B) \log (i-\tan (c+d x))}{(a+i b)^2}+\frac {2 A \left (a^2+b^2\right ) \log (\tan (c+d x))}{a^2}-\frac {a (a+i b) (A-i B) \log (i+\tan (c+d x))}{(a-i b)^2}-\frac {2 b \left (6 a^4 A b+3 a^2 A b^3+A b^5-3 a^5 B+a^3 b^2 B\right ) \log (a+b \tan (c+d x))}{a^2 \left (a^2+b^2\right )^2}+\frac {b (A b-a B)}{(a+b \tan (c+d x))^2}+\frac {4 a b (A b-a B)}{\left (a^2+b^2\right ) (a+b \tan (c+d x))}+\frac {2 A b^2}{a^2+a b \tan (c+d x)}}{2 a \left (a^2+b^2\right ) d} \]

[In]

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

[Out]

(-((a*(a - I*b)*(A + I*B)*Log[I - Tan[c + d*x]])/(a + I*b)^2) + (2*A*(a^2 + b^2)*Log[Tan[c + d*x]])/a^2 - (a*(
a + I*b)*(A - I*B)*Log[I + Tan[c + d*x]])/(a - I*b)^2 - (2*b*(6*a^4*A*b + 3*a^2*A*b^3 + A*b^5 - 3*a^5*B + a^3*
b^2*B)*Log[a + b*Tan[c + d*x]])/(a^2*(a^2 + b^2)^2) + (b*(A*b - a*B))/(a + b*Tan[c + d*x])^2 + (4*a*b*(A*b - a
*B))/((a^2 + b^2)*(a + b*Tan[c + d*x])) + (2*A*b^2)/(a^2 + a*b*Tan[c + d*x]))/(2*a*(a^2 + b^2)*d)

Maple [A] (verified)

Time = 0.44 (sec) , antiderivative size = 243, normalized size of antiderivative = 1.13

method	result	size
derivativedivides	\(\frac {\frac {\frac {\left (-A \,a^{3}+3 A a \,b^{2}-3 B \,a^{2} b +B \,b^{3}\right ) \ln \left (1+\tan ^{2}\left (d x +c \right )\right )}{2}+\left (-3 A \,a^{2} b +A \,b^{3}+B \,a^{3}-3 B a \,b^{2}\right ) \arctan \left (\tan \left (d x +c \right )\right )}{\left (a^{2}+b^{2}\right )^{3}}+\frac {A \ln \left (\tan \left (d x +c \right )\right )}{a^{3}}+\frac {b \left (3 A \,a^{2} b +A \,b^{3}-2 B \,a^{3}\right )}{\left (a^{2}+b^{2}\right )^{2} a^{2} \left (a +b \tan \left (d x +c \right )\right )}-\frac {b \left (6 A \,a^{4} b +3 A \,a^{2} b^{3}+A \,b^{5}-3 B \,a^{5}+B \,a^{3} b^{2}\right ) \ln \left (a +b \tan \left (d x +c \right )\right )}{\left (a^{2}+b^{2}\right )^{3} a^{3}}+\frac {\left (A b -B a \right ) b}{2 \left (a^{2}+b^{2}\right ) a \left (a +b \tan \left (d x +c \right )\right )^{2}}}{d}\)	\(243\)
default	\(\frac {\frac {\frac {\left (-A \,a^{3}+3 A a \,b^{2}-3 B \,a^{2} b +B \,b^{3}\right ) \ln \left (1+\tan ^{2}\left (d x +c \right )\right )}{2}+\left (-3 A \,a^{2} b +A \,b^{3}+B \,a^{3}-3 B a \,b^{2}\right ) \arctan \left (\tan \left (d x +c \right )\right )}{\left (a^{2}+b^{2}\right )^{3}}+\frac {A \ln \left (\tan \left (d x +c \right )\right )}{a^{3}}+\frac {b \left (3 A \,a^{2} b +A \,b^{3}-2 B \,a^{3}\right )}{\left (a^{2}+b^{2}\right )^{2} a^{2} \left (a +b \tan \left (d x +c \right )\right )}-\frac {b \left (6 A \,a^{4} b +3 A \,a^{2} b^{3}+A \,b^{5}-3 B \,a^{5}+B \,a^{3} b^{2}\right ) \ln \left (a +b \tan \left (d x +c \right )\right )}{\left (a^{2}+b^{2}\right )^{3} a^{3}}+\frac {\left (A b -B a \right ) b}{2 \left (a^{2}+b^{2}\right ) a \left (a +b \tan \left (d x +c \right )\right )^{2}}}{d}\)	\(243\)
parallelrisch	\(\frac {-12 b \left (a +b \tan \left (d x +c \right )\right )^{2} \left (A \,a^{4} b +\frac {1}{2} A \,a^{2} b^{3}+\frac {1}{6} A \,b^{5}-\frac {1}{2} B \,a^{5}+\frac {1}{6} B \,a^{3} b^{2}\right ) \ln \left (a +b \tan \left (d x +c \right )\right )-a^{3} \left (a +b \tan \left (d x +c \right )\right )^{2} \left (A \,a^{3}-3 A a \,b^{2}+3 B \,a^{2} b -B \,b^{3}\right ) \ln \left (\sec ^{2}\left (d x +c \right )\right )+2 A \left (a^{2}+b^{2}\right )^{3} \left (a +b \tan \left (d x +c \right )\right )^{2} \ln \left (\tan \left (d x +c \right )\right )-6 b^{2} \left (-\frac {B \,a^{6} d x}{3}+b \left (A d x -\frac {B}{3}\right ) a^{5}+\frac {b^{2} \left (2 B d x +A \right ) a^{4}}{2}-\frac {b^{3} \left (A d x +B \right ) a^{3}}{3}+\frac {2 A \,a^{2} b^{4}}{3}+\frac {A \,b^{6}}{6}\right ) \left (\tan ^{2}\left (d x +c \right )\right )-12 d \left (A \,a^{2} b -\frac {1}{3} A \,b^{3}-\frac {1}{3} B \,a^{3}+B a \,b^{2}\right ) a^{4} x b \tan \left (d x +c \right )-6 a^{2} \left (-\frac {B \,a^{6} d x}{3}+b \left (A d x +\frac {B}{2}\right ) a^{5}-\frac {2 \left (-\frac {3 B d x}{2}+A \right ) b^{2} a^{4}}{3}-\frac {b^{3} \left (A d x -2 B \right ) a^{3}}{3}-A \,a^{2} b^{4}+\frac {B a \,b^{5}}{6}-\frac {A \,b^{6}}{3}\right )}{2 \left (a^{2}+b^{2}\right )^{3} a^{3} d \left (a +b \tan \left (d x +c \right )\right )^{2}}\)	\(375\)
norman	\(\frac {-\frac {\left (3 A \,a^{2} b -A \,b^{3}-B \,a^{3}+3 B a \,b^{2}\right ) a^{2} x}{\left (a^{4}+2 a^{2} b^{2}+b^{4}\right ) \left (a^{2}+b^{2}\right )}-\frac {b^{2} \left (3 A \,a^{2} b -A \,b^{3}-B \,a^{3}+3 B a \,b^{2}\right ) x \left (\tan ^{2}\left (d x +c \right )\right )}{\left (a^{4}+2 a^{2} b^{2}+b^{4}\right ) \left (a^{2}+b^{2}\right )}-\frac {b \left (4 A \,a^{2} b^{2}+2 A \,b^{4}-3 B \,a^{3} b -B a \,b^{3}\right ) \tan \left (d x +c \right )}{d \,a^{2} \left (a^{4}+2 a^{2} b^{2}+b^{4}\right )}-\frac {b^{2} \left (7 A \,a^{2} b^{2}+3 A \,b^{4}-5 B \,a^{3} b -B a \,b^{3}\right ) \left (\tan ^{2}\left (d x +c \right )\right )}{2 d \,a^{3} \left (a^{4}+2 a^{2} b^{2}+b^{4}\right )}-\frac {2 b \left (3 A \,a^{2} b -A \,b^{3}-B \,a^{3}+3 B a \,b^{2}\right ) a x \tan \left (d x +c \right )}{\left (a^{4}+2 a^{2} b^{2}+b^{4}\right ) \left (a^{2}+b^{2}\right )}}{\left (a +b \tan \left (d x +c \right )\right )^{2}}+\frac {A \ln \left (\tan \left (d x +c \right )\right )}{a^{3} d}-\frac {\left (A \,a^{3}-3 A a \,b^{2}+3 B \,a^{2} b -B \,b^{3}\right ) \ln \left (1+\tan ^{2}\left (d x +c \right )\right )}{2 d \left (a^{6}+3 a^{4} b^{2}+3 a^{2} b^{4}+b^{6}\right )}-\frac {b \left (6 A \,a^{4} b +3 A \,a^{2} b^{3}+A \,b^{5}-3 B \,a^{5}+B \,a^{3} b^{2}\right ) \ln \left (a +b \tan \left (d x +c \right )\right )}{\left (a^{6}+3 a^{4} b^{2}+3 a^{2} b^{4}+b^{6}\right ) a^{3} d}\)	\(497\)
risch	\(-\frac {x B}{3 i a^{2} b -i b^{3}-a^{3}+3 a \,b^{2}}-\frac {6 i a^{2} b B x}{a^{6}+3 a^{4} b^{2}+3 a^{2} b^{4}+b^{6}}+\frac {2 i B \,b^{3} c}{d \left (a^{6}+3 a^{4} b^{2}+3 a^{2} b^{4}+b^{6}\right )}+\frac {2 i b^{6} A x}{a^{3} \left (a^{6}+3 a^{4} b^{2}+3 a^{2} b^{4}+b^{6}\right )}-\frac {6 i a^{2} b B c}{d \left (a^{6}+3 a^{4} b^{2}+3 a^{2} b^{4}+b^{6}\right )}+\frac {12 i a \,b^{2} A c}{d \left (a^{6}+3 a^{4} b^{2}+3 a^{2} b^{4}+b^{6}\right )}-\frac {2 i x A}{a^{3}}+\frac {2 i B \,b^{3} x}{a^{6}+3 a^{4} b^{2}+3 a^{2} b^{4}+b^{6}}+\frac {2 i b^{2} \left (3 i A \,a^{2} b^{2} {\mathrm e}^{2 i \left (d x +c \right )}+i A \,b^{4} {\mathrm e}^{2 i \left (d x +c \right )}-2 i B \,a^{3} b \,{\mathrm e}^{2 i \left (d x +c \right )}-4 A \,a^{3} b \,{\mathrm e}^{2 i \left (d x +c \right )}-2 A a \,b^{3} {\mathrm e}^{2 i \left (d x +c \right )}+3 B \,a^{4} {\mathrm e}^{2 i \left (d x +c \right )}+B \,a^{2} b^{2} {\mathrm e}^{2 i \left (d x +c \right )}-4 i A \,a^{2} b^{2}-i A \,b^{4}+3 i B \,a^{3} b -4 A \,a^{3} b -A a \,b^{3}+3 B \,a^{4}\right )}{a^{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 )^{2} d \left (-i b +a \right )^{3}}+\frac {6 i b^{4} A x}{a \left (a^{6}+3 a^{4} b^{2}+3 a^{2} b^{4}+b^{6}\right )}+\frac {6 i b^{4} A c}{a d \left (a^{6}+3 a^{4} b^{2}+3 a^{2} b^{4}+b^{6}\right )}-\frac {i x A}{3 i a^{2} b -i b^{3}-a^{3}+3 a \,b^{2}}+\frac {2 i b^{6} A c}{a^{3} d \left (a^{6}+3 a^{4} b^{2}+3 a^{2} b^{4}+b^{6}\right )}-\frac {2 i A c}{a^{3} d}+\frac {12 i a \,b^{2} A x}{a^{6}+3 a^{4} b^{2}+3 a^{2} b^{4}+b^{6}}-\frac {6 a \,b^{2} \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}-\frac {i b +a}{i b -a}\right ) A}{d \left (a^{6}+3 a^{4} b^{2}+3 a^{2} b^{4}+b^{6}\right )}-\frac {3 b^{4} \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}-\frac {i b +a}{i b -a}\right ) A}{a d \left (a^{6}+3 a^{4} b^{2}+3 a^{2} b^{4}+b^{6}\right )}-\frac {b^{6} \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}-\frac {i b +a}{i b -a}\right ) A}{a^{3} d \left (a^{6}+3 a^{4} b^{2}+3 a^{2} b^{4}+b^{6}\right )}+\frac {3 a^{2} b \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}-\frac {i b +a}{i b -a}\right ) 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 ) B \,b^{3}}{d \left (a^{6}+3 a^{4} b^{2}+3 a^{2} b^{4}+b^{6}\right )}+\frac {A \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )}{a^{3} d}\)	\(1010\)

[In]

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

[Out]

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 683 vs. \(2 (213) = 426\).

Time = 0.33 (sec) , antiderivative size = 683, normalized size of antiderivative = 3.18 \[ \int \frac {\cot (c+d x) (A+B \tan (c+d x))}{(a+b \tan (c+d x))^3} \, dx=-\frac {7 \, B a^{5} b^{3} - 9 \, A a^{4} b^{4} + B a^{3} b^{5} - 3 \, A a^{2} b^{6} - 2 \, {\left (B a^{8} - 3 \, A a^{7} b - 3 \, B a^{6} b^{2} + A a^{5} b^{3}\right )} d x - {\left (5 \, B a^{5} b^{3} - 7 \, A a^{4} b^{4} - B a^{3} b^{5} - A a^{2} b^{6} + 2 \, {\left (B a^{6} b^{2} - 3 \, A a^{5} b^{3} - 3 \, B a^{4} b^{4} + A a^{3} b^{5}\right )} d x\right )} \tan \left (d x + c\right )^{2} - {\left (A a^{8} + 3 \, A a^{6} b^{2} + 3 \, A a^{4} b^{4} + A a^{2} b^{6} + {\left (A a^{6} b^{2} + 3 \, A a^{4} b^{4} + 3 \, A a^{2} b^{6} + A b^{8}\right )} \tan \left (d x + c\right )^{2} + 2 \, {\left (A a^{7} b + 3 \, A a^{5} b^{3} + 3 \, A a^{3} b^{5} + A a 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 (3 \, B a^{7} b - 6 \, A a^{6} b^{2} - B a^{5} b^{3} - 3 \, A a^{4} b^{4} - A a^{2} b^{6} + {\left (3 \, B a^{5} b^{3} - 6 \, A a^{4} b^{4} - B a^{3} b^{5} - 3 \, A a^{2} b^{6} - A b^{8}\right )} \tan \left (d x + c\right )^{2} + 2 \, {\left (3 \, B a^{6} b^{2} - 6 \, A a^{5} b^{3} - B a^{4} b^{4} - 3 \, A a^{3} b^{5} - A a 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 ) - 2 \, {\left (3 \, B a^{6} b^{2} - 4 \, A a^{5} b^{3} - 3 \, B a^{4} b^{4} + 3 \, A a^{3} b^{5} + A a b^{7} + 2 \, {\left (B a^{7} b - 3 \, A a^{6} b^{2} - 3 \, B a^{5} b^{3} + A a^{4} b^{4}\right )} d x\right )} \tan \left (d x + c\right )}{2 \, {\left ({\left (a^{9} b^{2} + 3 \, a^{7} b^{4} + 3 \, a^{5} b^{6} + a^{3} b^{8}\right )} d \tan \left (d x + c\right )^{2} + 2 \, {\left (a^{10} b + 3 \, a^{8} b^{3} + 3 \, a^{6} b^{5} + a^{4} b^{7}\right )} d \tan \left (d x + c\right ) + {\left (a^{11} + 3 \, a^{9} b^{2} + 3 \, a^{7} b^{4} + a^{5} b^{6}\right )} d\right )}} \]

[In]

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

[Out]

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

Sympy [F(-2)]

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

[In]

integrate(cot(d*x+c)*(A+B*tan(d*x+c))/(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 = 0.31 (sec) , antiderivative size = 372, normalized size of antiderivative = 1.73 \[ \int \frac {\cot (c+d x) (A+B \tan (c+d x))}{(a+b \tan (c+d x))^3} \, dx=\frac {\frac {2 \, {\left (B a^{3} - 3 \, A a^{2} b - 3 \, B a b^{2} + A b^{3}\right )} {\left (d x + c\right )}}{a^{6} + 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} + b^{6}} + \frac {2 \, {\left (3 \, B a^{5} b - 6 \, A a^{4} b^{2} - B a^{3} b^{3} - 3 \, A a^{2} b^{4} - A b^{6}\right )} \log \left (b \tan \left (d x + c\right ) + a\right )}{a^{9} + 3 \, a^{7} b^{2} + 3 \, a^{5} b^{4} + a^{3} b^{6}} - \frac {{\left (A a^{3} + 3 \, B a^{2} b - 3 \, A a b^{2} - B b^{3}\right )} \log \left (\tan \left (d x + c\right )^{2} + 1\right )}{a^{6} + 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} + b^{6}} - \frac {5 \, B a^{4} b - 7 \, A a^{3} b^{2} + B a^{2} b^{3} - 3 \, A a b^{4} + 2 \, {\left (2 \, B a^{3} b^{2} - 3 \, A a^{2} b^{3} - A b^{5}\right )} \tan \left (d x + c\right )}{a^{8} + 2 \, a^{6} b^{2} + a^{4} b^{4} + {\left (a^{6} b^{2} + 2 \, a^{4} b^{4} + a^{2} b^{6}\right )} \tan \left (d x + c\right )^{2} + 2 \, {\left (a^{7} b + 2 \, a^{5} b^{3} + a^{3} b^{5}\right )} \tan \left (d x + c\right )} + \frac {2 \, A \log \left (\tan \left (d x + c\right )\right )}{a^{3}}}{2 \, d} \]

[In]

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

[Out]

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

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 479 vs. \(2 (213) = 426\).

Time = 1.00 (sec) , antiderivative size = 479, normalized size of antiderivative = 2.23 \[ \int \frac {\cot (c+d x) (A+B \tan (c+d x))}{(a+b \tan (c+d x))^3} \, dx=\frac {\frac {2 \, {\left (B a^{3} - 3 \, A a^{2} b - 3 \, B a b^{2} + A b^{3}\right )} {\left (d x + c\right )}}{a^{6} + 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} + b^{6}} - \frac {{\left (A a^{3} + 3 \, B a^{2} b - 3 \, A a b^{2} - B b^{3}\right )} \log \left (\tan \left (d x + c\right )^{2} + 1\right )}{a^{6} + 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} + b^{6}} + \frac {2 \, {\left (3 \, B a^{5} b^{2} - 6 \, A a^{4} b^{3} - B a^{3} b^{4} - 3 \, A a^{2} b^{5} - A b^{7}\right )} \log \left ({\left | b \tan \left (d x + c\right ) + a \right |}\right )}{a^{9} b + 3 \, a^{7} b^{3} + 3 \, a^{5} b^{5} + a^{3} b^{7}} + \frac {2 \, A \log \left ({\left | \tan \left (d x + c\right ) \right |}\right )}{a^{3}} - \frac {9 \, B a^{5} b^{3} \tan \left (d x + c\right )^{2} - 18 \, A a^{4} b^{4} \tan \left (d x + c\right )^{2} - 3 \, B a^{3} b^{5} \tan \left (d x + c\right )^{2} - 9 \, A a^{2} b^{6} \tan \left (d x + c\right )^{2} - 3 \, A b^{8} \tan \left (d x + c\right )^{2} + 22 \, B a^{6} b^{2} \tan \left (d x + c\right ) - 42 \, A a^{5} b^{3} \tan \left (d x + c\right ) - 2 \, B a^{4} b^{4} \tan \left (d x + c\right ) - 26 \, A a^{3} b^{5} \tan \left (d x + c\right ) - 8 \, A a b^{7} \tan \left (d x + c\right ) + 14 \, B a^{7} b - 25 \, A a^{6} b^{2} + 3 \, B a^{5} b^{3} - 19 \, A a^{4} b^{4} + B a^{3} b^{5} - 6 \, A a^{2} b^{6}}{{\left (a^{9} + 3 \, a^{7} b^{2} + 3 \, a^{5} b^{4} + a^{3} b^{6}\right )} {\left (b \tan \left (d x + c\right ) + a\right )}^{2}}}{2 \, d} \]

[In]

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

[Out]

1/2*(2*(B*a^3 - 3*A*a^2*b - 3*B*a*b^2 + A*b^3)*(d*x + c)/(a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6) - (A*a^3 + 3*B*a^
2*b - 3*A*a*b^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*(3*B*a^5*b^2 - 6*A*a^
4*b^3 - B*a^3*b^4 - 3*A*a^2*b^5 - A*b^7)*log(abs(b*tan(d*x + c) + a))/(a^9*b + 3*a^7*b^3 + 3*a^5*b^5 + a^3*b^7
) + 2*A*log(abs(tan(d*x + c)))/a^3 - (9*B*a^5*b^3*tan(d*x + c)^2 - 18*A*a^4*b^4*tan(d*x + c)^2 - 3*B*a^3*b^5*t
an(d*x + c)^2 - 9*A*a^2*b^6*tan(d*x + c)^2 - 3*A*b^8*tan(d*x + c)^2 + 22*B*a^6*b^2*tan(d*x + c) - 42*A*a^5*b^3
*tan(d*x + c) - 2*B*a^4*b^4*tan(d*x + c) - 26*A*a^3*b^5*tan(d*x + c) - 8*A*a*b^7*tan(d*x + c) + 14*B*a^7*b - 2
5*A*a^6*b^2 + 3*B*a^5*b^3 - 19*A*a^4*b^4 + B*a^3*b^5 - 6*A*a^2*b^6)/((a^9 + 3*a^7*b^2 + 3*a^5*b^4 + a^3*b^6)*(
b*tan(d*x + c) + a)^2))/d

Mupad [B] (verification not implemented)

Time = 10.32 (sec) , antiderivative size = 315, normalized size of antiderivative = 1.47 \[ \int \frac {\cot (c+d x) (A+B \tan (c+d x))}{(a+b \tan (c+d x))^3} \, dx=\frac {\frac {-5\,B\,a^3\,b+7\,A\,a^2\,b^2-B\,a\,b^3+3\,A\,b^4}{2\,a\,\left (a^4+2\,a^2\,b^2+b^4\right )}+\frac {\mathrm {tan}\left (c+d\,x\right )\,\left (-2\,B\,a^3\,b^2+3\,A\,a^2\,b^3+A\,b^5\right )}{a^2\,\left (a^4+2\,a^2\,b^2+b^4\right )}}{d\,\left (a^2+2\,a\,b\,\mathrm {tan}\left (c+d\,x\right )+b^2\,{\mathrm {tan}\left (c+d\,x\right )}^2\right )}+\frac {A\,\ln \left (\mathrm {tan}\left (c+d\,x\right )\right )}{a^3\,d}+\frac {\ln \left (\mathrm {tan}\left (c+d\,x\right )-\mathrm {i}\right )\,\left (-B+A\,1{}\mathrm {i}\right )}{2\,d\,\left (-a^3\,1{}\mathrm {i}+3\,a^2\,b+a\,b^2\,3{}\mathrm {i}-b^3\right )}+\frac {\ln \left (\mathrm {tan}\left (c+d\,x\right )+1{}\mathrm {i}\right )\,\left (A-B\,1{}\mathrm {i}\right )}{2\,d\,\left (-a^3+a^2\,b\,3{}\mathrm {i}+3\,a\,b^2-b^3\,1{}\mathrm {i}\right )}-\frac {b\,\ln \left (a+b\,\mathrm {tan}\left (c+d\,x\right )\right )\,\left (-3\,B\,a^5+6\,A\,a^4\,b+B\,a^3\,b^2+3\,A\,a^2\,b^3+A\,b^5\right )}{a^3\,d\,{\left (a^2+b^2\right )}^3} \]

[In]

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

[Out]

((3*A*b^4 + 7*A*a^2*b^2 - B*a*b^3 - 5*B*a^3*b)/(2*a*(a^4 + b^4 + 2*a^2*b^2)) + (tan(c + d*x)*(A*b^5 + 3*A*a^2*
b^3 - 2*B*a^3*b^2))/(a^2*(a^4 + b^4 + 2*a^2*b^2)))/(d*(a^2 + b^2*tan(c + d*x)^2 + 2*a*b*tan(c + d*x))) + (A*lo
g(tan(c + d*x)))/(a^3*d) + (log(tan(c + d*x) - 1i)*(A*1i - B))/(2*d*(a*b^2*3i + 3*a^2*b - a^3*1i - b^3)) + (lo
g(tan(c + d*x) + 1i)*(A - B*1i))/(2*d*(3*a*b^2 + a^2*b*3i - a^3 - b^3*1i)) - (b*log(a + b*tan(c + d*x))*(A*b^5
- 3*B*a^5 + 3*A*a^2*b^3 + B*a^3*b^2 + 6*A*a^4*b))/(a^3*d*(a^2 + b^2)^3)