\(\int \cot ^5(c+d x) (a+i a \tan (c+d x)) (A+B \tan (c+d x)) \, dx\) [8]

Optimal result

Integrand size = 32, antiderivative size = 111 \[ \int \cot ^5(c+d x) (a+i a \tan (c+d x)) (A+B \tan (c+d x)) \, dx=a (i A+B) x+\frac {a (i A+B) \cot (c+d x)}{d}+\frac {a (A-i B) \cot ^2(c+d x)}{2 d}-\frac {a (i A+B) \cot ^3(c+d x)}{3 d}-\frac {a A \cot ^4(c+d x)}{4 d}+\frac {a (A-i B) \log (\sin (c+d x))}{d} \]

[Out]

a*(I*A+B)*x+a*(I*A+B)*cot(d*x+c)/d+1/2*a*(A-I*B)*cot(d*x+c)^2/d-1/3*a*(I*A+B)*cot(d*x+c)^3/d-1/4*a*A*cot(d*x+c
)^4/d+a*(A-I*B)*ln(sin(d*x+c))/d

Rubi [A] (verified)

Time = 0.30 (sec) , antiderivative size = 111, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {3672, 3610, 3612, 3556} \[ \int \cot ^5(c+d x) (a+i a \tan (c+d x)) (A+B \tan (c+d x)) \, dx=-\frac {a (B+i A) \cot ^3(c+d x)}{3 d}+\frac {a (A-i B) \cot ^2(c+d x)}{2 d}+\frac {a (B+i A) \cot (c+d x)}{d}+\frac {a (A-i B) \log (\sin (c+d x))}{d}+a x (B+i A)-\frac {a A \cot ^4(c+d x)}{4 d} \]

[In]

Int[Cot[c + d*x]^5*(a + I*a*Tan[c + d*x])*(A + B*Tan[c + d*x]),x]

[Out]

a*(I*A + B)*x + (a*(I*A + B)*Cot[c + d*x])/d + (a*(A - I*B)*Cot[c + d*x]^2)/(2*d) - (a*(I*A + B)*Cot[c + d*x]^
3)/(3*d) - (a*A*Cot[c + d*x]^4)/(4*d) + (a*(A - I*B)*Log[Sin[c + d*x]])/d

Rule 3556

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

Rule 3610

Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[(b
*c - a*d)*((a + b*Tan[e + f*x])^(m + 1)/(f*(m + 1)*(a^2 + b^2))), x] + Dist[1/(a^2 + b^2), Int[(a + b*Tan[e +
f*x])^(m + 1)*Simp[a*c + b*d - (b*c - a*d)*Tan[e + f*x], x], x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c
- a*d, 0] && NeQ[a^2 + b^2, 0] && LtQ[m, -1]

Rule 3612

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

Rule 3672

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

Rubi steps \begin{align*} \text {integral}& = -\frac {a A \cot ^4(c+d x)}{4 d}+\int \cot ^4(c+d x) (a (i A+B)-a (A-i B) \tan (c+d x)) \, dx \\ & = -\frac {a (i A+B) \cot ^3(c+d x)}{3 d}-\frac {a A \cot ^4(c+d x)}{4 d}+\int \cot ^3(c+d x) (-a (A-i B)-a (i A+B) \tan (c+d x)) \, dx \\ & = \frac {a (A-i B) \cot ^2(c+d x)}{2 d}-\frac {a (i A+B) \cot ^3(c+d x)}{3 d}-\frac {a A \cot ^4(c+d x)}{4 d}+\int \cot ^2(c+d x) (-a (i A+B)+a (A-i B) \tan (c+d x)) \, dx \\ & = \frac {a (i A+B) \cot (c+d x)}{d}+\frac {a (A-i B) \cot ^2(c+d x)}{2 d}-\frac {a (i A+B) \cot ^3(c+d x)}{3 d}-\frac {a A \cot ^4(c+d x)}{4 d}+\int \cot (c+d x) (a (A-i B)+a (i A+B) \tan (c+d x)) \, dx \\ & = a (i A+B) x+\frac {a (i A+B) \cot (c+d x)}{d}+\frac {a (A-i B) \cot ^2(c+d x)}{2 d}-\frac {a (i A+B) \cot ^3(c+d x)}{3 d}-\frac {a A \cot ^4(c+d x)}{4 d}+(a (A-i B)) \int \cot (c+d x) \, dx \\ & = a (i A+B) x+\frac {a (i A+B) \cot (c+d x)}{d}+\frac {a (A-i B) \cot ^2(c+d x)}{2 d}-\frac {a (i A+B) \cot ^3(c+d x)}{3 d}-\frac {a A \cot ^4(c+d x)}{4 d}+\frac {a (A-i B) \log (\sin (c+d x))}{d} \\ \end{align*}

Mathematica [C] (verified)

Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.

Time = 0.91 (sec) , antiderivative size = 96, normalized size of antiderivative = 0.86 \[ \int \cot ^5(c+d x) (a+i a \tan (c+d x)) (A+B \tan (c+d x)) \, dx=-\frac {a \left (-6 (A-i B) \cot ^2(c+d x)+3 A \cot ^4(c+d x)+4 (i A+B) \cot ^3(c+d x) \operatorname {Hypergeometric2F1}\left (-\frac {3}{2},1,-\frac {1}{2},-\tan ^2(c+d x)\right )-12 (A-i B) (\log (\cos (c+d x))+\log (\tan (c+d x)))\right )}{12 d} \]

[In]

Integrate[Cot[c + d*x]^5*(a + I*a*Tan[c + d*x])*(A + B*Tan[c + d*x]),x]

[Out]

-1/12*(a*(-6*(A - I*B)*Cot[c + d*x]^2 + 3*A*Cot[c + d*x]^4 + 4*(I*A + B)*Cot[c + d*x]^3*Hypergeometric2F1[-3/2
, 1, -1/2, -Tan[c + d*x]^2] - 12*(A - I*B)*(Log[Cos[c + d*x]] + Log[Tan[c + d*x]])))/d

Maple [A] (verified)

Time = 0.20 (sec) , antiderivative size = 106, normalized size of antiderivative = 0.95

method	result	size
parallelrisch	\(\frac {a \left (\left (-\frac {A}{2}+\frac {i B}{2}\right ) \ln \left (\sec ^{2}\left (d x +c \right )\right )+\left (-i B +A \right ) \ln \left (\tan \left (d x +c \right )\right )-\frac {A \left (\cot ^{4}\left (d x +c \right )\right )}{4}+\left (\cot ^{3}\left (d x +c \right )\right ) \left (-\frac {i A}{3}-\frac {B}{3}\right )+\left (\cot ^{2}\left (d x +c \right )\right ) \left (\frac {A}{2}-\frac {i B}{2}\right )+\cot \left (d x +c \right ) \left (i A +B \right )+\left (i A +B \right ) x d \right )}{d}\)	\(106\)
derivativedivides	\(\frac {a \left (\frac {\left (i B -A \right ) \ln \left (1+\tan ^{2}\left (d x +c \right )\right )}{2}+\left (i A +B \right ) \arctan \left (\tan \left (d x +c \right )\right )-\frac {A}{4 \tan \left (d x +c \right )^{4}}-\frac {i B -A}{2 \tan \left (d x +c \right )^{2}}+\left (-i B +A \right ) \ln \left (\tan \left (d x +c \right )\right )-\frac {-i A -B}{\tan \left (d x +c \right )}-\frac {i A +B}{3 \tan \left (d x +c \right )^{3}}\right )}{d}\)	\(119\)
default	\(\frac {a \left (\frac {\left (i B -A \right ) \ln \left (1+\tan ^{2}\left (d x +c \right )\right )}{2}+\left (i A +B \right ) \arctan \left (\tan \left (d x +c \right )\right )-\frac {A}{4 \tan \left (d x +c \right )^{4}}-\frac {i B -A}{2 \tan \left (d x +c \right )^{2}}+\left (-i B +A \right ) \ln \left (\tan \left (d x +c \right )\right )-\frac {-i A -B}{\tan \left (d x +c \right )}-\frac {i A +B}{3 \tan \left (d x +c \right )^{3}}\right )}{d}\)	\(119\)
norman	\(\frac {\frac {\left (i a A +B a \right ) \left (\tan ^{3}\left (d x +c \right )\right )}{d}+\left (i a A +B a \right ) x \left (\tan ^{4}\left (d x +c \right )\right )-\frac {a A}{4 d}+\frac {\left (-i a B +a A \right ) \left (\tan ^{2}\left (d x +c \right )\right )}{2 d}-\frac {\left (i a A +B a \right ) \tan \left (d x +c \right )}{3 d}}{\tan \left (d x +c \right )^{4}}+\frac {\left (-i a B +a A \right ) \ln \left (\tan \left (d x +c \right )\right )}{d}-\frac {\left (-i a B +a A \right ) \ln \left (1+\tan ^{2}\left (d x +c \right )\right )}{2 d}\)	\(146\)
risch	\(-\frac {2 a B c}{d}-\frac {2 i a A c}{d}+\frac {2 i a \left (12 i A \,{\mathrm e}^{6 i \left (d x +c \right )}+9 B \,{\mathrm e}^{6 i \left (d x +c \right )}-18 i A \,{\mathrm e}^{4 i \left (d x +c \right )}-18 B \,{\mathrm e}^{4 i \left (d x +c \right )}+16 i A \,{\mathrm e}^{2 i \left (d x +c \right )}+13 B \,{\mathrm e}^{2 i \left (d x +c \right )}-4 i A -4 B \right )}{3 d \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )^{4}}-\frac {i a \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right ) B}{d}+\frac {a A \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )}{d}\)	\(160\)

[In]

int(cot(d*x+c)^5*(a+I*a*tan(d*x+c))*(A+B*tan(d*x+c)),x,method=_RETURNVERBOSE)

[Out]

a*((-1/2*A+1/2*I*B)*ln(sec(d*x+c)^2)+(A-I*B)*ln(tan(d*x+c))-1/4*A*cot(d*x+c)^4+cot(d*x+c)^3*(-1/3*I*A-1/3*B)+c
ot(d*x+c)^2*(1/2*A-1/2*I*B)+cot(d*x+c)*(I*A+B)+(I*A+B)*x*d)/d

Fricas [B] (verification not implemented)

Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 206 vs. \(2 (95) = 190\).

Time = 0.25 (sec) , antiderivative size = 206, normalized size of antiderivative = 1.86 \[ \int \cot ^5(c+d x) (a+i a \tan (c+d x)) (A+B \tan (c+d x)) \, dx=-\frac {6 \, {\left (4 \, A - 3 i \, B\right )} a e^{\left (6 i \, d x + 6 i \, c\right )} - 36 \, {\left (A - i \, B\right )} a e^{\left (4 i \, d x + 4 i \, c\right )} + 2 \, {\left (16 \, A - 13 i \, B\right )} a e^{\left (2 i \, d x + 2 i \, c\right )} - 8 \, {\left (A - i \, B\right )} a - 3 \, {\left ({\left (A - i \, B\right )} a e^{\left (8 i \, d x + 8 i \, c\right )} - 4 \, {\left (A - i \, B\right )} a e^{\left (6 i \, d x + 6 i \, c\right )} + 6 \, {\left (A - i \, B\right )} a e^{\left (4 i \, d x + 4 i \, c\right )} - 4 \, {\left (A - i \, B\right )} a e^{\left (2 i \, d x + 2 i \, c\right )} + {\left (A - i \, B\right )} a\right )} \log \left (e^{\left (2 i \, d x + 2 i \, c\right )} - 1\right )}{3 \, {\left (d e^{\left (8 i \, d x + 8 i \, c\right )} - 4 \, d e^{\left (6 i \, d x + 6 i \, c\right )} + 6 \, d e^{\left (4 i \, d x + 4 i \, c\right )} - 4 \, d e^{\left (2 i \, d x + 2 i \, c\right )} + d\right )}} \]

[In]

integrate(cot(d*x+c)^5*(a+I*a*tan(d*x+c))*(A+B*tan(d*x+c)),x, algorithm="fricas")

[Out]

-1/3*(6*(4*A - 3*I*B)*a*e^(6*I*d*x + 6*I*c) - 36*(A - I*B)*a*e^(4*I*d*x + 4*I*c) + 2*(16*A - 13*I*B)*a*e^(2*I*
d*x + 2*I*c) - 8*(A - I*B)*a - 3*((A - I*B)*a*e^(8*I*d*x + 8*I*c) - 4*(A - I*B)*a*e^(6*I*d*x + 6*I*c) + 6*(A -
I*B)*a*e^(4*I*d*x + 4*I*c) - 4*(A - I*B)*a*e^(2*I*d*x + 2*I*c) + (A - I*B)*a)*log(e^(2*I*d*x + 2*I*c) - 1))/(
d*e^(8*I*d*x + 8*I*c) - 4*d*e^(6*I*d*x + 6*I*c) + 6*d*e^(4*I*d*x + 4*I*c) - 4*d*e^(2*I*d*x + 2*I*c) + d)

Sympy [B] (verification not implemented)

Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 218 vs. \(2 (92) = 184\).

Time = 0.72 (sec) , antiderivative size = 218, normalized size of antiderivative = 1.96 \[ \int \cot ^5(c+d x) (a+i a \tan (c+d x)) (A+B \tan (c+d x)) \, dx=\frac {a \left (A - i B\right ) \log {\left (e^{2 i d x} - e^{- 2 i c} \right )}}{d} + \frac {8 A a - 8 i B a + \left (- 32 A a e^{2 i c} + 26 i B a e^{2 i c}\right ) e^{2 i d x} + \left (36 A a e^{4 i c} - 36 i B a e^{4 i c}\right ) e^{4 i d x} + \left (- 24 A a e^{6 i c} + 18 i B a e^{6 i c}\right ) e^{6 i d x}}{3 d e^{8 i c} e^{8 i d x} - 12 d e^{6 i c} e^{6 i d x} + 18 d e^{4 i c} e^{4 i d x} - 12 d e^{2 i c} e^{2 i d x} + 3 d} \]

[In]

integrate(cot(d*x+c)**5*(a+I*a*tan(d*x+c))*(A+B*tan(d*x+c)),x)

[Out]

a*(A - I*B)*log(exp(2*I*d*x) - exp(-2*I*c))/d + (8*A*a - 8*I*B*a + (-32*A*a*exp(2*I*c) + 26*I*B*a*exp(2*I*c))*
exp(2*I*d*x) + (36*A*a*exp(4*I*c) - 36*I*B*a*exp(4*I*c))*exp(4*I*d*x) + (-24*A*a*exp(6*I*c) + 18*I*B*a*exp(6*I
*c))*exp(6*I*d*x))/(3*d*exp(8*I*c)*exp(8*I*d*x) - 12*d*exp(6*I*c)*exp(6*I*d*x) + 18*d*exp(4*I*c)*exp(4*I*d*x)
- 12*d*exp(2*I*c)*exp(2*I*d*x) + 3*d)

Maxima [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 116, normalized size of antiderivative = 1.05 \[ \int \cot ^5(c+d x) (a+i a \tan (c+d x)) (A+B \tan (c+d x)) \, dx=\frac {12 \, {\left (d x + c\right )} {\left (i \, A + B\right )} a - 6 \, {\left (A - i \, B\right )} a \log \left (\tan \left (d x + c\right )^{2} + 1\right ) + 12 \, {\left (A - i \, B\right )} a \log \left (\tan \left (d x + c\right )\right ) - \frac {12 \, {\left (-i \, A - B\right )} a \tan \left (d x + c\right )^{3} - 6 \, {\left (A - i \, B\right )} a \tan \left (d x + c\right )^{2} + 4 \, {\left (i \, A + B\right )} a \tan \left (d x + c\right ) + 3 \, A a}{\tan \left (d x + c\right )^{4}}}{12 \, d} \]

[In]

integrate(cot(d*x+c)^5*(a+I*a*tan(d*x+c))*(A+B*tan(d*x+c)),x, algorithm="maxima")

[Out]

1/12*(12*(d*x + c)*(I*A + B)*a - 6*(A - I*B)*a*log(tan(d*x + c)^2 + 1) + 12*(A - I*B)*a*log(tan(d*x + c)) - (1
2*(-I*A - B)*a*tan(d*x + c)^3 - 6*(A - I*B)*a*tan(d*x + c)^2 + 4*(I*A + B)*a*tan(d*x + c) + 3*A*a)/tan(d*x + c
)^4)/d

Giac [B] (verification not implemented)

Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 282 vs. \(2 (95) = 190\).

Time = 0.85 (sec) , antiderivative size = 282, normalized size of antiderivative = 2.54 \[ \int \cot ^5(c+d x) (a+i a \tan (c+d x)) (A+B \tan (c+d x)) \, dx=-\frac {3 \, A a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 8 i \, A a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 8 \, B a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 36 \, A a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 24 i \, B a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 120 i \, A a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 120 \, B a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 384 \, {\left (A a - i \, B a\right )} \log \left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + i\right ) - 192 \, {\left (A a - i \, B a\right )} \log \left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right ) + \frac {400 \, A a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 400 i \, B a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 120 i \, A a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 120 \, B a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 36 \, A a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 24 i \, B a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 8 i \, A a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 8 \, B a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 3 \, A a}{\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4}}}{192 \, d} \]

[In]

integrate(cot(d*x+c)^5*(a+I*a*tan(d*x+c))*(A+B*tan(d*x+c)),x, algorithm="giac")

[Out]

-1/192*(3*A*a*tan(1/2*d*x + 1/2*c)^4 - 8*I*A*a*tan(1/2*d*x + 1/2*c)^3 - 8*B*a*tan(1/2*d*x + 1/2*c)^3 - 36*A*a*
tan(1/2*d*x + 1/2*c)^2 + 24*I*B*a*tan(1/2*d*x + 1/2*c)^2 + 120*I*A*a*tan(1/2*d*x + 1/2*c) + 120*B*a*tan(1/2*d*
x + 1/2*c) + 384*(A*a - I*B*a)*log(tan(1/2*d*x + 1/2*c) + I) - 192*(A*a - I*B*a)*log(tan(1/2*d*x + 1/2*c)) + (
400*A*a*tan(1/2*d*x + 1/2*c)^4 - 400*I*B*a*tan(1/2*d*x + 1/2*c)^4 - 120*I*A*a*tan(1/2*d*x + 1/2*c)^3 - 120*B*a
*tan(1/2*d*x + 1/2*c)^3 - 36*A*a*tan(1/2*d*x + 1/2*c)^2 + 24*I*B*a*tan(1/2*d*x + 1/2*c)^2 + 8*I*A*a*tan(1/2*d*
x + 1/2*c) + 8*B*a*tan(1/2*d*x + 1/2*c) + 3*A*a)/tan(1/2*d*x + 1/2*c)^4)/d

Mupad [B] (verification not implemented)

Time = 8.46 (sec) , antiderivative size = 100, normalized size of antiderivative = 0.90 \[ \int \cot ^5(c+d x) (a+i a \tan (c+d x)) (A+B \tan (c+d x)) \, dx=-\frac {\left (-B\,a-A\,a\,1{}\mathrm {i}\right )\,{\mathrm {tan}\left (c+d\,x\right )}^3+\left (-\frac {A\,a}{2}+\frac {B\,a\,1{}\mathrm {i}}{2}\right )\,{\mathrm {tan}\left (c+d\,x\right )}^2+\left (\frac {B\,a}{3}+\frac {A\,a\,1{}\mathrm {i}}{3}\right )\,\mathrm {tan}\left (c+d\,x\right )+\frac {A\,a}{4}}{d\,{\mathrm {tan}\left (c+d\,x\right )}^4}+\frac {a\,\mathrm {atan}\left (2\,\mathrm {tan}\left (c+d\,x\right )+1{}\mathrm {i}\right )\,\left (A-B\,1{}\mathrm {i}\right )\,2{}\mathrm {i}}{d} \]

[In]

int(cot(c + d*x)^5*(A + B*tan(c + d*x))*(a + a*tan(c + d*x)*1i),x)

[Out]

(a*atan(2*tan(c + d*x) + 1i)*(A - B*1i)*2i)/d - ((A*a)/4 + tan(c + d*x)*((A*a*1i)/3 + (B*a)/3) - tan(c + d*x)^
3*(A*a*1i + B*a) - tan(c + d*x)^2*((A*a)/2 - (B*a*1i)/2))/(d*tan(c + d*x)^4)