\(\int \cot ^4(c+d x) (a+i a \tan (c+d x)) (A+B \tan (c+d x)) \, dx\) [7]

Optimal result

Integrand size = 32, antiderivative size = 89 \[ \int \cot ^4(c+d x) (a+i a \tan (c+d x)) (A+B \tan (c+d x)) \, dx=a (A-i B) x+\frac {a (A-i B) \cot (c+d x)}{d}-\frac {a (i A+B) \cot ^2(c+d x)}{2 d}-\frac {a A \cot ^3(c+d x)}{3 d}-\frac {a (i A+B) \log (\sin (c+d x))}{d} \]

[Out]

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

Rubi [A] (verified)

Time = 0.21 (sec) , antiderivative size = 89, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {3672, 3610, 3612, 3556} \[ \int \cot ^4(c+d x) (a+i a \tan (c+d x)) (A+B \tan (c+d x)) \, dx=-\frac {a (B+i A) \cot ^2(c+d x)}{2 d}+\frac {a (A-i B) \cot (c+d x)}{d}-\frac {a (B+i A) \log (\sin (c+d x))}{d}+a x (A-i B)-\frac {a A \cot ^3(c+d x)}{3 d} \]

[In]

Int[Cot[c + d*x]^4*(a + I*a*Tan[c + d*x])*(A + B*Tan[c + d*x]),x]

[Out]

a*(A - I*B)*x + (a*(A - I*B)*Cot[c + d*x])/d - (a*(I*A + B)*Cot[c + d*x]^2)/(2*d) - (a*A*Cot[c + d*x]^3)/(3*d)
- (a*(I*A + 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 ^3(c+d x)}{3 d}+\int \cot ^3(c+d x) (a (i A+B)-a (A-i B) \tan (c+d x)) \, dx \\ & = -\frac {a (i A+B) \cot ^2(c+d x)}{2 d}-\frac {a A \cot ^3(c+d x)}{3 d}+\int \cot ^2(c+d x) (-a (A-i B)-a (i A+B) \tan (c+d x)) \, dx \\ & = \frac {a (A-i B) \cot (c+d x)}{d}-\frac {a (i A+B) \cot ^2(c+d x)}{2 d}-\frac {a A \cot ^3(c+d x)}{3 d}+\int \cot (c+d x) (-a (i A+B)+a (A-i B) \tan (c+d x)) \, dx \\ & = a (A-i B) x+\frac {a (A-i B) \cot (c+d x)}{d}-\frac {a (i A+B) \cot ^2(c+d x)}{2 d}-\frac {a A \cot ^3(c+d x)}{3 d}-(a (i A+B)) \int \cot (c+d x) \, dx \\ & = a (A-i B) x+\frac {a (A-i B) \cot (c+d x)}{d}-\frac {a (i A+B) \cot ^2(c+d x)}{2 d}-\frac {a A \cot ^3(c+d x)}{3 d}-\frac {a (i A+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 = 0.79 (sec) , antiderivative size = 102, normalized size of antiderivative = 1.15 \[ \int \cot ^4(c+d x) (a+i a \tan (c+d x)) (A+B \tan (c+d x)) \, dx=-\frac {a \left (2 A \cot ^3(c+d x) \operatorname {Hypergeometric2F1}\left (-\frac {3}{2},1,-\frac {1}{2},-\tan ^2(c+d x)\right )+6 i B \cot (c+d x) \operatorname {Hypergeometric2F1}\left (-\frac {1}{2},1,\frac {1}{2},-\tan ^2(c+d x)\right )+3 (i A+B) \left (\cot ^2(c+d x)+2 (\log (\cos (c+d x))+\log (\tan (c+d x)))\right )\right )}{6 d} \]

[In]

Integrate[Cot[c + d*x]^4*(a + I*a*Tan[c + d*x])*(A + B*Tan[c + d*x]),x]

[Out]

-1/6*(a*(2*A*Cot[c + d*x]^3*Hypergeometric2F1[-3/2, 1, -1/2, -Tan[c + d*x]^2] + (6*I)*B*Cot[c + d*x]*Hypergeom
etric2F1[-1/2, 1, 1/2, -Tan[c + d*x]^2] + 3*(I*A + B)*(Cot[c + d*x]^2 + 2*(Log[Cos[c + d*x]] + Log[Tan[c + d*x
]]))))/d

Maple [A] (verified)

Time = 0.18 (sec) , antiderivative size = 101, normalized size of antiderivative = 1.13

method	result	size
derivativedivides	\(\frac {a \left (\frac {\left (i A +B \right ) \ln \left (1+\tan ^{2}\left (d x +c \right )\right )}{2}+\left (-i B +A \right ) \arctan \left (\tan \left (d x +c \right )\right )-\frac {A}{3 \tan \left (d x +c \right )^{3}}-\frac {i B -A}{\tan \left (d x +c \right )}+\left (-i A -B \right ) \ln \left (\tan \left (d x +c \right )\right )-\frac {i A +B}{2 \tan \left (d x +c \right )^{2}}\right )}{d}\)	\(101\)
default	\(\frac {a \left (\frac {\left (i A +B \right ) \ln \left (1+\tan ^{2}\left (d x +c \right )\right )}{2}+\left (-i B +A \right ) \arctan \left (\tan \left (d x +c \right )\right )-\frac {A}{3 \tan \left (d x +c \right )^{3}}-\frac {i B -A}{\tan \left (d x +c \right )}+\left (-i A -B \right ) \ln \left (\tan \left (d x +c \right )\right )-\frac {i A +B}{2 \tan \left (d x +c \right )^{2}}\right )}{d}\)	\(101\)
norman	\(\frac {\frac {\left (-i a B +a A \right ) \left (\tan ^{2}\left (d x +c \right )\right )}{d}+\left (-i a B +a A \right ) x \left (\tan ^{3}\left (d x +c \right )\right )-\frac {a A}{3 d}-\frac {\left (i a A +B a \right ) \tan \left (d x +c \right )}{2 d}}{\tan \left (d x +c \right )^{3}}-\frac {\left (i a A +B a \right ) \ln \left (\tan \left (d x +c \right )\right )}{d}+\frac {\left (i a A +B a \right ) \ln \left (1+\tan ^{2}\left (d x +c \right )\right )}{2 d}\)	\(125\)
parallelrisch	\(-\frac {a \left (6 i B d x +6 i A \ln \left (\tan \left (d x +c \right )\right )-3 i A \ln \left (\sec ^{2}\left (d x +c \right )\right )-6 A d x +3 i A \left (\cot ^{2}\left (d x +c \right )\right )+6 i B \cot \left (d x +c \right )+6 B \ln \left (\tan \left (d x +c \right )\right )-3 B \ln \left (\sec ^{2}\left (d x +c \right )\right )+8 A \left (\cot ^{3}\left (d x +c \right )\right )+3 B \left (\cot ^{2}\left (d x +c \right )\right )-6 A \cot \left (d x +c \right ) \left (\csc ^{2}\left (d x +c \right )\right )\right )}{6 d}\)	\(126\)
risch	\(\frac {2 i a B c}{d}-\frac {2 a A c}{d}+\frac {2 a \left (9 i A \,{\mathrm e}^{4 i \left (d x +c \right )}+6 B \,{\mathrm e}^{4 i \left (d x +c \right )}-9 i A \,{\mathrm e}^{2 i \left (d x +c \right )}-9 B \,{\mathrm e}^{2 i \left (d x +c \right )}+4 i A +3 B \right )}{3 d \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )^{3}}-\frac {a \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right ) B}{d}-\frac {i a \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right ) A}{d}\)	\(135\)

[In]

int(cot(d*x+c)^4*(a+I*a*tan(d*x+c))*(A+B*tan(d*x+c)),x,method=_RETURNVERBOSE)

[Out]

a/d*(1/2*(I*A+B)*ln(1+tan(d*x+c)^2)+(A-I*B)*arctan(tan(d*x+c))-1/3*A/tan(d*x+c)^3-(-A+I*B)/tan(d*x+c)+(-I*A-B)
*ln(tan(d*x+c))-1/2*(I*A+B)/tan(d*x+c)^2)

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. 166 vs. \(2 (77) = 154\).

Time = 0.25 (sec) , antiderivative size = 166, normalized size of antiderivative = 1.87 \[ \int \cot ^4(c+d x) (a+i a \tan (c+d x)) (A+B \tan (c+d x)) \, dx=-\frac {6 \, {\left (-3 i \, A - 2 \, B\right )} a e^{\left (4 i \, d x + 4 i \, c\right )} + 18 \, {\left (i \, A + B\right )} a e^{\left (2 i \, d x + 2 i \, c\right )} + 2 \, {\left (-4 i \, A - 3 \, B\right )} a + 3 \, {\left ({\left (i \, A + B\right )} a e^{\left (6 i \, d x + 6 i \, c\right )} + 3 \, {\left (-i \, A - B\right )} a e^{\left (4 i \, d x + 4 i \, c\right )} + 3 \, {\left (i \, A + B\right )} a e^{\left (2 i \, d x + 2 i \, c\right )} + {\left (-i \, A - B\right )} a\right )} \log \left (e^{\left (2 i \, d x + 2 i \, c\right )} - 1\right )}{3 \, {\left (d e^{\left (6 i \, d x + 6 i \, c\right )} - 3 \, d e^{\left (4 i \, d x + 4 i \, c\right )} + 3 \, d e^{\left (2 i \, d x + 2 i \, c\right )} - d\right )}} \]

[In]

integrate(cot(d*x+c)^4*(a+I*a*tan(d*x+c))*(A+B*tan(d*x+c)),x, algorithm="fricas")

[Out]

-1/3*(6*(-3*I*A - 2*B)*a*e^(4*I*d*x + 4*I*c) + 18*(I*A + B)*a*e^(2*I*d*x + 2*I*c) + 2*(-4*I*A - 3*B)*a + 3*((I
*A + B)*a*e^(6*I*d*x + 6*I*c) + 3*(-I*A - B)*a*e^(4*I*d*x + 4*I*c) + 3*(I*A + B)*a*e^(2*I*d*x + 2*I*c) + (-I*A
- B)*a)*log(e^(2*I*d*x + 2*I*c) - 1))/(d*e^(6*I*d*x + 6*I*c) - 3*d*e^(4*I*d*x + 4*I*c) + 3*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. 168 vs. \(2 (73) = 146\).

Time = 0.37 (sec) , antiderivative size = 168, normalized size of antiderivative = 1.89 \[ \int \cot ^4(c+d x) (a+i a \tan (c+d x)) (A+B \tan (c+d x)) \, dx=- \frac {i a \left (A - i B\right ) \log {\left (e^{2 i d x} - e^{- 2 i c} \right )}}{d} + \frac {8 i A a + 6 B a + \left (- 18 i A a e^{2 i c} - 18 B a e^{2 i c}\right ) e^{2 i d x} + \left (18 i A a e^{4 i c} + 12 B a e^{4 i c}\right ) e^{4 i d x}}{3 d e^{6 i c} e^{6 i d x} - 9 d e^{4 i c} e^{4 i d x} + 9 d e^{2 i c} e^{2 i d x} - 3 d} \]

[In]

integrate(cot(d*x+c)**4*(a+I*a*tan(d*x+c))*(A+B*tan(d*x+c)),x)

[Out]

-I*a*(A - I*B)*log(exp(2*I*d*x) - exp(-2*I*c))/d + (8*I*A*a + 6*B*a + (-18*I*A*a*exp(2*I*c) - 18*B*a*exp(2*I*c
))*exp(2*I*d*x) + (18*I*A*a*exp(4*I*c) + 12*B*a*exp(4*I*c))*exp(4*I*d*x))/(3*d*exp(6*I*c)*exp(6*I*d*x) - 9*d*e
xp(4*I*c)*exp(4*I*d*x) + 9*d*exp(2*I*c)*exp(2*I*d*x) - 3*d)

Maxima [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 103, normalized size of antiderivative = 1.16 \[ \int \cot ^4(c+d x) (a+i a \tan (c+d x)) (A+B \tan (c+d x)) \, dx=\frac {6 \, {\left (d x + c\right )} {\left (A - i \, B\right )} a - 3 \, {\left (-i \, A - B\right )} a \log \left (\tan \left (d x + c\right )^{2} + 1\right ) + 6 \, {\left (-i \, A - B\right )} a \log \left (\tan \left (d x + c\right )\right ) + \frac {6 \, {\left (A - i \, B\right )} a \tan \left (d x + c\right )^{2} + 3 \, {\left (-i \, A - B\right )} a \tan \left (d x + c\right ) - 2 \, A a}{\tan \left (d x + c\right )^{3}}}{6 \, d} \]

[In]

integrate(cot(d*x+c)^4*(a+I*a*tan(d*x+c))*(A+B*tan(d*x+c)),x, algorithm="maxima")

[Out]

1/6*(6*(d*x + c)*(A - I*B)*a - 3*(-I*A - B)*a*log(tan(d*x + c)^2 + 1) + 6*(-I*A - B)*a*log(tan(d*x + c)) + (6*
(A - I*B)*a*tan(d*x + c)^2 + 3*(-I*A - B)*a*tan(d*x + c) - 2*A*a)/tan(d*x + c)^3)/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. 221 vs. \(2 (77) = 154\).

Time = 0.71 (sec) , antiderivative size = 221, normalized size of antiderivative = 2.48 \[ \int \cot ^4(c+d x) (a+i a \tan (c+d x)) (A+B \tan (c+d x)) \, dx=\frac {A a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 3 i \, A a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 3 \, B a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 15 \, A a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 12 i \, B a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 48 \, {\left (i \, A a + B a\right )} \log \left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + i\right ) - 24 \, {\left (i \, A a + B a\right )} \log \left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right ) - \frac {-44 i \, A a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 44 \, B a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 15 \, A a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 12 i \, B a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 3 i \, A a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 3 \, B a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + A a}{\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3}}}{24 \, d} \]

[In]

integrate(cot(d*x+c)^4*(a+I*a*tan(d*x+c))*(A+B*tan(d*x+c)),x, algorithm="giac")

[Out]

1/24*(A*a*tan(1/2*d*x + 1/2*c)^3 - 3*I*A*a*tan(1/2*d*x + 1/2*c)^2 - 3*B*a*tan(1/2*d*x + 1/2*c)^2 - 15*A*a*tan(
1/2*d*x + 1/2*c) + 12*I*B*a*tan(1/2*d*x + 1/2*c) + 48*(I*A*a + B*a)*log(tan(1/2*d*x + 1/2*c) + I) - 24*(I*A*a
+ B*a)*log(tan(1/2*d*x + 1/2*c)) - (-44*I*A*a*tan(1/2*d*x + 1/2*c)^3 - 44*B*a*tan(1/2*d*x + 1/2*c)^3 - 15*A*a*
tan(1/2*d*x + 1/2*c)^2 + 12*I*B*a*tan(1/2*d*x + 1/2*c)^2 + 3*I*A*a*tan(1/2*d*x + 1/2*c) + 3*B*a*tan(1/2*d*x +
1/2*c) + A*a)/tan(1/2*d*x + 1/2*c)^3)/d

Mupad [B] (verification not implemented)

Time = 7.57 (sec) , antiderivative size = 80, normalized size of antiderivative = 0.90 \[ \int \cot ^4(c+d x) (a+i a \tan (c+d x)) (A+B \tan (c+d x)) \, dx=-\frac {\left (-A\,a+B\,a\,1{}\mathrm {i}\right )\,{\mathrm {tan}\left (c+d\,x\right )}^2+\left (\frac {B\,a}{2}+\frac {A\,a\,1{}\mathrm {i}}{2}\right )\,\mathrm {tan}\left (c+d\,x\right )+\frac {A\,a}{3}}{d\,{\mathrm {tan}\left (c+d\,x\right )}^3}-\frac {a\,\mathrm {atan}\left (2\,\mathrm {tan}\left (c+d\,x\right )+1{}\mathrm {i}\right )\,\left (B+A\,1{}\mathrm {i}\right )\,2{}\mathrm {i}}{d} \]

[In]

int(cot(c + d*x)^4*(A + B*tan(c + d*x))*(a + a*tan(c + d*x)*1i),x)

[Out]

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