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

Optimal. Leaf size=223 \[ \frac {a^4 (92 B+93 i A) \cot ^3(c+d x)}{60 d}-\frac {4 a^4 (A-i B) \cot ^2(c+d x)}{d}-\frac {8 a^4 (B+i A) \cot (c+d x)}{d}-\frac {8 a^4 (A-i B) \log (\sin (c+d x))}{d}+\frac {(13 A-12 i B) \cot ^4(c+d x) \left (a^4+i a^4 \tan (c+d x)\right )}{20 d}-8 a^4 x (B+i A)-\frac {(2 B+3 i A) \cot ^5(c+d x) \left (a^2+i a^2 \tan (c+d x)\right )^2}{10 d}-\frac {a A \cot ^6(c+d x) (a+i a \tan (c+d x))^3}{6 d} \]

[Out]

-8*a^4*(I*A+B)*x-8*a^4*(I*A+B)*cot(d*x+c)/d-4*a^4*(A-I*B)*cot(d*x+c)^2/d+1/60*a^4*(93*I*A+92*B)*cot(d*x+c)^3/d
-8*a^4*(A-I*B)*ln(sin(d*x+c))/d-1/6*a*A*cot(d*x+c)^6*(a+I*a*tan(d*x+c))^3/d-1/10*(3*I*A+2*B)*cot(d*x+c)^5*(a^2
+I*a^2*tan(d*x+c))^2/d+1/20*(13*A-12*I*B)*cot(d*x+c)^4*(a^4+I*a^4*tan(d*x+c))/d

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Rubi [A]  time = 0.65, antiderivative size = 223, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 5, integrand size = 34, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.147, Rules used = {3593, 3591, 3529, 3531, 3475} \[ \frac {a^4 (92 B+93 i A) \cot ^3(c+d x)}{60 d}-\frac {4 a^4 (A-i B) \cot ^2(c+d x)}{d}-\frac {8 a^4 (B+i A) \cot (c+d x)}{d}-\frac {8 a^4 (A-i B) \log (\sin (c+d x))}{d}-\frac {(2 B+3 i A) \cot ^5(c+d x) \left (a^2+i a^2 \tan (c+d x)\right )^2}{10 d}+\frac {(13 A-12 i B) \cot ^4(c+d x) \left (a^4+i a^4 \tan (c+d x)\right )}{20 d}-8 a^4 x (B+i A)-\frac {a A \cot ^6(c+d x) (a+i a \tan (c+d x))^3}{6 d} \]

Antiderivative was successfully verified.

[In]

Int[Cot[c + d*x]^7*(a + I*a*Tan[c + d*x])^4*(A + B*Tan[c + d*x]),x]

[Out]

-8*a^4*(I*A + B)*x - (8*a^4*(I*A + B)*Cot[c + d*x])/d - (4*a^4*(A - I*B)*Cot[c + d*x]^2)/d + (a^4*((93*I)*A +
92*B)*Cot[c + d*x]^3)/(60*d) - (8*a^4*(A - I*B)*Log[Sin[c + d*x]])/d - (a*A*Cot[c + d*x]^6*(a + I*a*Tan[c + d*
x])^3)/(6*d) - (((3*I)*A + 2*B)*Cot[c + d*x]^5*(a^2 + I*a^2*Tan[c + d*x])^2)/(10*d) + ((13*A - (12*I)*B)*Cot[c
+ d*x]^4*(a^4 + I*a^4*Tan[c + d*x]))/(20*d)

Rule 3475

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

Rule 3529

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 3531

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 3591

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]

Rule 3593

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

Rubi steps

\begin {align*} \int \cot ^7(c+d x) (a+i a \tan (c+d x))^4 (A+B \tan (c+d x)) \, dx &=-\frac {a A \cot ^6(c+d x) (a+i a \tan (c+d x))^3}{6 d}+\frac {1}{6} \int \cot ^6(c+d x) (a+i a \tan (c+d x))^3 (3 a (3 i A+2 B)-3 a (A-2 i B) \tan (c+d x)) \, dx\\ &=-\frac {a A \cot ^6(c+d x) (a+i a \tan (c+d x))^3}{6 d}-\frac {(3 i A+2 B) \cot ^5(c+d x) \left (a^2+i a^2 \tan (c+d x)\right )^2}{10 d}+\frac {1}{30} \int \cot ^5(c+d x) (a+i a \tan (c+d x))^2 \left (-6 a^2 (13 A-12 i B)-6 a^2 (7 i A+8 B) \tan (c+d x)\right ) \, dx\\ &=-\frac {a A \cot ^6(c+d x) (a+i a \tan (c+d x))^3}{6 d}-\frac {(3 i A+2 B) \cot ^5(c+d x) \left (a^2+i a^2 \tan (c+d x)\right )^2}{10 d}+\frac {(13 A-12 i B) \cot ^4(c+d x) \left (a^4+i a^4 \tan (c+d x)\right )}{20 d}+\frac {1}{120} \int \cot ^4(c+d x) (a+i a \tan (c+d x)) \left (-6 a^3 (93 i A+92 B)+6 a^3 (67 A-68 i B) \tan (c+d x)\right ) \, dx\\ &=\frac {a^4 (93 i A+92 B) \cot ^3(c+d x)}{60 d}-\frac {a A \cot ^6(c+d x) (a+i a \tan (c+d x))^3}{6 d}-\frac {(3 i A+2 B) \cot ^5(c+d x) \left (a^2+i a^2 \tan (c+d x)\right )^2}{10 d}+\frac {(13 A-12 i B) \cot ^4(c+d x) \left (a^4+i a^4 \tan (c+d x)\right )}{20 d}+\frac {1}{120} \int \cot ^3(c+d x) \left (960 a^4 (A-i B)+960 a^4 (i A+B) \tan (c+d x)\right ) \, dx\\ &=-\frac {4 a^4 (A-i B) \cot ^2(c+d x)}{d}+\frac {a^4 (93 i A+92 B) \cot ^3(c+d x)}{60 d}-\frac {a A \cot ^6(c+d x) (a+i a \tan (c+d x))^3}{6 d}-\frac {(3 i A+2 B) \cot ^5(c+d x) \left (a^2+i a^2 \tan (c+d x)\right )^2}{10 d}+\frac {(13 A-12 i B) \cot ^4(c+d x) \left (a^4+i a^4 \tan (c+d x)\right )}{20 d}+\frac {1}{120} \int \cot ^2(c+d x) \left (960 a^4 (i A+B)-960 a^4 (A-i B) \tan (c+d x)\right ) \, dx\\ &=-\frac {8 a^4 (i A+B) \cot (c+d x)}{d}-\frac {4 a^4 (A-i B) \cot ^2(c+d x)}{d}+\frac {a^4 (93 i A+92 B) \cot ^3(c+d x)}{60 d}-\frac {a A \cot ^6(c+d x) (a+i a \tan (c+d x))^3}{6 d}-\frac {(3 i A+2 B) \cot ^5(c+d x) \left (a^2+i a^2 \tan (c+d x)\right )^2}{10 d}+\frac {(13 A-12 i B) \cot ^4(c+d x) \left (a^4+i a^4 \tan (c+d x)\right )}{20 d}+\frac {1}{120} \int \cot (c+d x) \left (-960 a^4 (A-i B)-960 a^4 (i A+B) \tan (c+d x)\right ) \, dx\\ &=-8 a^4 (i A+B) x-\frac {8 a^4 (i A+B) \cot (c+d x)}{d}-\frac {4 a^4 (A-i B) \cot ^2(c+d x)}{d}+\frac {a^4 (93 i A+92 B) \cot ^3(c+d x)}{60 d}-\frac {a A \cot ^6(c+d x) (a+i a \tan (c+d x))^3}{6 d}-\frac {(3 i A+2 B) \cot ^5(c+d x) \left (a^2+i a^2 \tan (c+d x)\right )^2}{10 d}+\frac {(13 A-12 i B) \cot ^4(c+d x) \left (a^4+i a^4 \tan (c+d x)\right )}{20 d}-\left (8 a^4 (A-i B)\right ) \int \cot (c+d x) \, dx\\ &=-8 a^4 (i A+B) x-\frac {8 a^4 (i A+B) \cot (c+d x)}{d}-\frac {4 a^4 (A-i B) \cot ^2(c+d x)}{d}+\frac {a^4 (93 i A+92 B) \cot ^3(c+d x)}{60 d}-\frac {8 a^4 (A-i B) \log (\sin (c+d x))}{d}-\frac {a A \cot ^6(c+d x) (a+i a \tan (c+d x))^3}{6 d}-\frac {(3 i A+2 B) \cot ^5(c+d x) \left (a^2+i a^2 \tan (c+d x)\right )^2}{10 d}+\frac {(13 A-12 i B) \cot ^4(c+d x) \left (a^4+i a^4 \tan (c+d x)\right )}{20 d}\\ \end {align*}

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Mathematica [B]  time = 10.15, size = 1009, normalized size = 4.52 \[ a^4 \left (\frac {(\cot (c+d x)+i)^4 (B+A \cot (c+d x)) (A \cos (2 c)-i B \cos (2 c)-i A \sin (2 c)-B \sin (2 c)) \left (8 i \tan ^{-1}(\tan (5 c+d x)) \cos (2 c)+8 \tan ^{-1}(\tan (5 c+d x)) \sin (2 c)\right ) \sin ^5(c+d x)}{d (\cos (d x)+i \sin (d x))^4 (A \cos (c+d x)+B \sin (c+d x))}+\frac {(\cot (c+d x)+i)^4 (B+A \cot (c+d x)) (A \cos (2 c)-i B \cos (2 c)-i A \sin (2 c)-B \sin (2 c)) \left (4 i \log \left (\sin ^2(c+d x)\right ) \sin (2 c)-4 \cos (2 c) \log \left (\sin ^2(c+d x)\right )\right ) \sin ^5(c+d x)}{d (\cos (d x)+i \sin (d x))^4 (A \cos (c+d x)+B \sin (c+d x))}+\frac {x (\cot (c+d x)+i)^4 (B+A \cot (c+d x)) \left (-40 i A \cos ^4(c)-40 B \cos ^4(c)+8 A \cot (c) \cos ^4(c)-8 i B \cot (c) \cos ^4(c)-80 A \sin (c) \cos ^3(c)+80 i B \sin (c) \cos ^3(c)+80 i A \sin ^2(c) \cos ^2(c)+80 B \sin ^2(c) \cos ^2(c)+40 A \sin ^3(c) \cos (c)-40 i B \sin ^3(c) \cos (c)-8 i A \sin ^4(c)-8 B \sin ^4(c)+(A-i B) \cot (c) (8 i \sin (4 c)-8 \cos (4 c))\right ) \sin ^5(c+d x)}{(\cos (d x)+i \sin (d x))^4 (A \cos (c+d x)+B \sin (c+d x))}+\frac {(\cot (c+d x)+i)^4 (B+A \cot (c+d x)) \csc (c) \csc (c+d x) \left (\frac {1}{240} \cos (4 c)-\frac {1}{240} i \sin (4 c)\right ) (860 i A \cos (c)+790 B \cos (c)-780 i A \cos (c+2 d x)-720 B \cos (c+2 d x)-510 i A \cos (3 c+2 d x)-465 B \cos (3 c+2 d x)+366 i A \cos (3 c+4 d x)+354 B \cos (3 c+4 d x)+150 i A \cos (5 c+4 d x)+120 B \cos (5 c+4 d x)-86 i A \cos (5 c+6 d x)-79 B \cos (5 c+6 d x)-490 A \sin (c)+420 i B \sin (c)-600 i A d x \sin (c)-600 B d x \sin (c)-345 A \sin (c+2 d x)+300 i B \sin (c+2 d x)-450 i A d x \sin (c+2 d x)-450 B d x \sin (c+2 d x)+345 A \sin (3 c+2 d x)-300 i B \sin (3 c+2 d x)+450 i A d x \sin (3 c+2 d x)+450 B d x \sin (3 c+2 d x)+120 A \sin (3 c+4 d x)-90 i B \sin (3 c+4 d x)+180 i A d x \sin (3 c+4 d x)+180 B d x \sin (3 c+4 d x)-120 A \sin (5 c+4 d x)+90 i B \sin (5 c+4 d x)-180 i A d x \sin (5 c+4 d x)-180 B d x \sin (5 c+4 d x)-30 i A d x \sin (5 c+6 d x)-30 B d x \sin (5 c+6 d x)+30 i A d x \sin (7 c+6 d x)+30 B d x \sin (7 c+6 d x))}{d (\cos (d x)+i \sin (d x))^4 (A \cos (c+d x)+B \sin (c+d x))}\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[Cot[c + d*x]^7*(a + I*a*Tan[c + d*x])^4*(A + B*Tan[c + d*x]),x]

[Out]

a^4*(((I + Cot[c + d*x])^4*(B + A*Cot[c + d*x])*(A*Cos[2*c] - I*B*Cos[2*c] - I*A*Sin[2*c] - B*Sin[2*c])*((8*I)
*ArcTan[Tan[5*c + d*x]]*Cos[2*c] + 8*ArcTan[Tan[5*c + d*x]]*Sin[2*c])*Sin[c + d*x]^5)/(d*(Cos[d*x] + I*Sin[d*x
])^4*(A*Cos[c + d*x] + B*Sin[c + d*x])) + ((I + Cot[c + d*x])^4*(B + A*Cot[c + d*x])*(A*Cos[2*c] - I*B*Cos[2*c
] - I*A*Sin[2*c] - B*Sin[2*c])*(-4*Cos[2*c]*Log[Sin[c + d*x]^2] + (4*I)*Log[Sin[c + d*x]^2]*Sin[2*c])*Sin[c +
d*x]^5)/(d*(Cos[d*x] + I*Sin[d*x])^4*(A*Cos[c + d*x] + B*Sin[c + d*x])) + (x*(I + Cot[c + d*x])^4*(B + A*Cot[c
+ d*x])*((-40*I)*A*Cos[c]^4 - 40*B*Cos[c]^4 + 8*A*Cos[c]^4*Cot[c] - (8*I)*B*Cos[c]^4*Cot[c] - 80*A*Cos[c]^3*S
in[c] + (80*I)*B*Cos[c]^3*Sin[c] + (80*I)*A*Cos[c]^2*Sin[c]^2 + 80*B*Cos[c]^2*Sin[c]^2 + 40*A*Cos[c]*Sin[c]^3
- (40*I)*B*Cos[c]*Sin[c]^3 - (8*I)*A*Sin[c]^4 - 8*B*Sin[c]^4 + (A - I*B)*Cot[c]*(-8*Cos[4*c] + (8*I)*Sin[4*c])
)*Sin[c + d*x]^5)/((Cos[d*x] + I*Sin[d*x])^4*(A*Cos[c + d*x] + B*Sin[c + d*x])) + ((I + Cot[c + d*x])^4*(B + A
*Cot[c + d*x])*Csc[c]*Csc[c + d*x]*(Cos[4*c]/240 - (I/240)*Sin[4*c])*((860*I)*A*Cos[c] + 790*B*Cos[c] - (780*I
)*A*Cos[c + 2*d*x] - 720*B*Cos[c + 2*d*x] - (510*I)*A*Cos[3*c + 2*d*x] - 465*B*Cos[3*c + 2*d*x] + (366*I)*A*Co
s[3*c + 4*d*x] + 354*B*Cos[3*c + 4*d*x] + (150*I)*A*Cos[5*c + 4*d*x] + 120*B*Cos[5*c + 4*d*x] - (86*I)*A*Cos[5
*c + 6*d*x] - 79*B*Cos[5*c + 6*d*x] - 490*A*Sin[c] + (420*I)*B*Sin[c] - (600*I)*A*d*x*Sin[c] - 600*B*d*x*Sin[c
] - 345*A*Sin[c + 2*d*x] + (300*I)*B*Sin[c + 2*d*x] - (450*I)*A*d*x*Sin[c + 2*d*x] - 450*B*d*x*Sin[c + 2*d*x]
+ 345*A*Sin[3*c + 2*d*x] - (300*I)*B*Sin[3*c + 2*d*x] + (450*I)*A*d*x*Sin[3*c + 2*d*x] + 450*B*d*x*Sin[3*c + 2
*d*x] + 120*A*Sin[3*c + 4*d*x] - (90*I)*B*Sin[3*c + 4*d*x] + (180*I)*A*d*x*Sin[3*c + 4*d*x] + 180*B*d*x*Sin[3*
c + 4*d*x] - 120*A*Sin[5*c + 4*d*x] + (90*I)*B*Sin[5*c + 4*d*x] - (180*I)*A*d*x*Sin[5*c + 4*d*x] - 180*B*d*x*S
in[5*c + 4*d*x] - (30*I)*A*d*x*Sin[5*c + 6*d*x] - 30*B*d*x*Sin[5*c + 6*d*x] + (30*I)*A*d*x*Sin[7*c + 6*d*x] +
30*B*d*x*Sin[7*c + 6*d*x]))/(d*(Cos[d*x] + I*Sin[d*x])^4*(A*Cos[c + d*x] + B*Sin[c + d*x])))

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fricas [A]  time = 0.71, size = 332, normalized size = 1.49 \[ \frac {4 \, {\left (30 \, {\left (9 \, A - 7 i \, B\right )} a^{4} e^{\left (10 i \, d x + 10 i \, c\right )} - 45 \, {\left (19 \, A - 17 i \, B\right )} a^{4} e^{\left (8 i \, d x + 8 i \, c\right )} + 10 \, {\left (135 \, A - 121 i \, B\right )} a^{4} e^{\left (6 i \, d x + 6 i \, c\right )} - 15 \, {\left (75 \, A - 68 i \, B\right )} a^{4} e^{\left (4 i \, d x + 4 i \, c\right )} + 6 \, {\left (81 \, A - 74 i \, B\right )} a^{4} e^{\left (2 i \, d x + 2 i \, c\right )} - {\left (86 \, A - 79 i \, B\right )} a^{4} - 30 \, {\left ({\left (A - i \, B\right )} a^{4} e^{\left (12 i \, d x + 12 i \, c\right )} - 6 \, {\left (A - i \, B\right )} a^{4} e^{\left (10 i \, d x + 10 i \, c\right )} + 15 \, {\left (A - i \, B\right )} a^{4} e^{\left (8 i \, d x + 8 i \, c\right )} - 20 \, {\left (A - i \, B\right )} a^{4} e^{\left (6 i \, d x + 6 i \, c\right )} + 15 \, {\left (A - i \, B\right )} a^{4} e^{\left (4 i \, d x + 4 i \, c\right )} - 6 \, {\left (A - i \, B\right )} a^{4} e^{\left (2 i \, d x + 2 i \, c\right )} + {\left (A - i \, B\right )} a^{4}\right )} \log \left (e^{\left (2 i \, d x + 2 i \, c\right )} - 1\right )\right )}}{15 \, {\left (d e^{\left (12 i \, d x + 12 i \, c\right )} - 6 \, d e^{\left (10 i \, d x + 10 i \, c\right )} + 15 \, d e^{\left (8 i \, d x + 8 i \, c\right )} - 20 \, d e^{\left (6 i \, d x + 6 i \, c\right )} + 15 \, d e^{\left (4 i \, d x + 4 i \, c\right )} - 6 \, d e^{\left (2 i \, d x + 2 i \, c\right )} + d\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(d*x+c)^7*(a+I*a*tan(d*x+c))^4*(A+B*tan(d*x+c)),x, algorithm="fricas")

[Out]

4/15*(30*(9*A - 7*I*B)*a^4*e^(10*I*d*x + 10*I*c) - 45*(19*A - 17*I*B)*a^4*e^(8*I*d*x + 8*I*c) + 10*(135*A - 12
1*I*B)*a^4*e^(6*I*d*x + 6*I*c) - 15*(75*A - 68*I*B)*a^4*e^(4*I*d*x + 4*I*c) + 6*(81*A - 74*I*B)*a^4*e^(2*I*d*x
+ 2*I*c) - (86*A - 79*I*B)*a^4 - 30*((A - I*B)*a^4*e^(12*I*d*x + 12*I*c) - 6*(A - I*B)*a^4*e^(10*I*d*x + 10*I
*c) + 15*(A - I*B)*a^4*e^(8*I*d*x + 8*I*c) - 20*(A - I*B)*a^4*e^(6*I*d*x + 6*I*c) + 15*(A - I*B)*a^4*e^(4*I*d*
x + 4*I*c) - 6*(A - I*B)*a^4*e^(2*I*d*x + 2*I*c) + (A - I*B)*a^4)*log(e^(2*I*d*x + 2*I*c) - 1))/(d*e^(12*I*d*x
+ 12*I*c) - 6*d*e^(10*I*d*x + 10*I*c) + 15*d*e^(8*I*d*x + 8*I*c) - 20*d*e^(6*I*d*x + 6*I*c) + 15*d*e^(4*I*d*x
+ 4*I*c) - 6*d*e^(2*I*d*x + 2*I*c) + d)

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giac [B]  time = 10.24, size = 459, normalized size = 2.06 \[ -\frac {5 \, A a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} - 48 i \, A a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 12 \, B a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 240 \, A a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 120 i \, B a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 880 i \, A a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 620 \, B a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 2835 \, A a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 2400 i \, B a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 10080 i \, A a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 9480 \, B a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 30720 \, {\left (A a^{4} - i \, B a^{4}\right )} \log \left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + i\right ) + 15360 \, {\left (A a^{4} - i \, B a^{4}\right )} \log \left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right ) - \frac {37632 \, A a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} - 37632 i \, B a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} - 10080 i \, A a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 9480 \, B a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 2835 \, A a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 2400 i \, B a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 880 i \, A a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 620 \, B a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 240 \, A a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 120 i \, B a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 48 i \, A a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 12 \, B a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 5 \, A a^{4}}{\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6}}}{1920 \, d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(d*x+c)^7*(a+I*a*tan(d*x+c))^4*(A+B*tan(d*x+c)),x, algorithm="giac")

[Out]

-1/1920*(5*A*a^4*tan(1/2*d*x + 1/2*c)^6 - 48*I*A*a^4*tan(1/2*d*x + 1/2*c)^5 - 12*B*a^4*tan(1/2*d*x + 1/2*c)^5
- 240*A*a^4*tan(1/2*d*x + 1/2*c)^4 + 120*I*B*a^4*tan(1/2*d*x + 1/2*c)^4 + 880*I*A*a^4*tan(1/2*d*x + 1/2*c)^3 +
620*B*a^4*tan(1/2*d*x + 1/2*c)^3 + 2835*A*a^4*tan(1/2*d*x + 1/2*c)^2 - 2400*I*B*a^4*tan(1/2*d*x + 1/2*c)^2 -
10080*I*A*a^4*tan(1/2*d*x + 1/2*c) - 9480*B*a^4*tan(1/2*d*x + 1/2*c) - 30720*(A*a^4 - I*B*a^4)*log(tan(1/2*d*x
+ 1/2*c) + I) + 15360*(A*a^4 - I*B*a^4)*log(tan(1/2*d*x + 1/2*c)) - (37632*A*a^4*tan(1/2*d*x + 1/2*c)^6 - 376
32*I*B*a^4*tan(1/2*d*x + 1/2*c)^6 - 10080*I*A*a^4*tan(1/2*d*x + 1/2*c)^5 - 9480*B*a^4*tan(1/2*d*x + 1/2*c)^5 -
2835*A*a^4*tan(1/2*d*x + 1/2*c)^4 + 2400*I*B*a^4*tan(1/2*d*x + 1/2*c)^4 + 880*I*A*a^4*tan(1/2*d*x + 1/2*c)^3
+ 620*B*a^4*tan(1/2*d*x + 1/2*c)^3 + 240*A*a^4*tan(1/2*d*x + 1/2*c)^2 - 120*I*B*a^4*tan(1/2*d*x + 1/2*c)^2 - 4
8*I*A*a^4*tan(1/2*d*x + 1/2*c) - 12*B*a^4*tan(1/2*d*x + 1/2*c) - 5*A*a^4)/tan(1/2*d*x + 1/2*c)^6)/d

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maple [A]  time = 0.50, size = 259, normalized size = 1.16 \[ -\frac {4 A \,a^{4} \left (\cot ^{2}\left (d x +c \right )\right )}{d}+\frac {7 a^{4} B \left (\cot ^{3}\left (d x +c \right )\right )}{3 d}-\frac {8 a^{4} B c}{d}-\frac {8 B \cot \left (d x +c \right ) a^{4}}{d}+\frac {7 A \,a^{4} \left (\cot ^{4}\left (d x +c \right )\right )}{4 d}-\frac {i B \,a^{4} \left (\cot ^{4}\left (d x +c \right )\right )}{d}-\frac {8 i A \cot \left (d x +c \right ) a^{4}}{d}+\frac {8 i B \,a^{4} \ln \left (\sin \left (d x +c \right )\right )}{d}+\frac {4 i B \,a^{4} \left (\cot ^{2}\left (d x +c \right )\right )}{d}+\frac {8 i A \,a^{4} \left (\cot ^{3}\left (d x +c \right )\right )}{3 d}-\frac {8 i A \,a^{4} c}{d}-\frac {4 i A \,a^{4} \left (\cot ^{5}\left (d x +c \right )\right )}{5 d}-8 i A x \,a^{4}-8 a^{4} B x -\frac {8 a^{4} A \ln \left (\sin \left (d x +c \right )\right )}{d}-\frac {A \,a^{4} \left (\cot ^{6}\left (d x +c \right )\right )}{6 d}-\frac {a^{4} B \left (\cot ^{5}\left (d x +c \right )\right )}{5 d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cot(d*x+c)^7*(a+I*a*tan(d*x+c))^4*(A+B*tan(d*x+c)),x)

[Out]

-4/d*A*a^4*cot(d*x+c)^2+7/3/d*a^4*B*cot(d*x+c)^3-8/d*a^4*B*c-8/d*B*cot(d*x+c)*a^4+7/4/d*A*a^4*cot(d*x+c)^4-I/d
*B*a^4*cot(d*x+c)^4-8*I/d*A*a^4*cot(d*x+c)+8*I/d*B*a^4*ln(sin(d*x+c))+4*I/d*B*a^4*cot(d*x+c)^2+8/3*I/d*A*a^4*c
ot(d*x+c)^3-8*I/d*A*a^4*c-4/5*I/d*A*a^4*cot(d*x+c)^5-8*I*A*x*a^4-8*a^4*B*x-8*a^4*A*ln(sin(d*x+c))/d-1/6/d*A*a^
4*cot(d*x+c)^6-1/5/d*a^4*B*cot(d*x+c)^5

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maxima [A]  time = 0.71, size = 171, normalized size = 0.77 \[ \frac {60 \, {\left (d x + c\right )} {\left (-8 i \, A - 8 \, B\right )} a^{4} + 240 \, {\left (A - i \, B\right )} a^{4} \log \left (\tan \left (d x + c\right )^{2} + 1\right ) - 480 \, {\left (A - i \, B\right )} a^{4} \log \left (\tan \left (d x + c\right )\right ) - \frac {{\left (480 i \, A + 480 \, B\right )} a^{4} \tan \left (d x + c\right )^{5} + 240 \, {\left (A - i \, B\right )} a^{4} \tan \left (d x + c\right )^{4} + {\left (-160 i \, A - 140 \, B\right )} a^{4} \tan \left (d x + c\right )^{3} - 15 \, {\left (7 \, A - 4 i \, B\right )} a^{4} \tan \left (d x + c\right )^{2} + {\left (48 i \, A + 12 \, B\right )} a^{4} \tan \left (d x + c\right ) + 10 \, A a^{4}}{\tan \left (d x + c\right )^{6}}}{60 \, d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(d*x+c)^7*(a+I*a*tan(d*x+c))^4*(A+B*tan(d*x+c)),x, algorithm="maxima")

[Out]

1/60*(60*(d*x + c)*(-8*I*A - 8*B)*a^4 + 240*(A - I*B)*a^4*log(tan(d*x + c)^2 + 1) - 480*(A - I*B)*a^4*log(tan(
d*x + c)) - ((480*I*A + 480*B)*a^4*tan(d*x + c)^5 + 240*(A - I*B)*a^4*tan(d*x + c)^4 + (-160*I*A - 140*B)*a^4*
tan(d*x + c)^3 - 15*(7*A - 4*I*B)*a^4*tan(d*x + c)^2 + (48*I*A + 12*B)*a^4*tan(d*x + c) + 10*A*a^4)/tan(d*x +
c)^6)/d

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mupad [B]  time = 7.59, size = 162, normalized size = 0.73 \[ -\frac {{\mathrm {tan}\left (c+d\,x\right )}^4\,\left (4\,A\,a^4-B\,a^4\,4{}\mathrm {i}\right )-{\mathrm {tan}\left (c+d\,x\right )}^2\,\left (\frac {7\,A\,a^4}{4}-B\,a^4\,1{}\mathrm {i}\right )+{\mathrm {tan}\left (c+d\,x\right )}^5\,\left (8\,B\,a^4+A\,a^4\,8{}\mathrm {i}\right )-{\mathrm {tan}\left (c+d\,x\right )}^3\,\left (\frac {7\,B\,a^4}{3}+\frac {A\,a^4\,8{}\mathrm {i}}{3}\right )+\frac {A\,a^4}{6}+\mathrm {tan}\left (c+d\,x\right )\,\left (\frac {B\,a^4}{5}+\frac {A\,a^4\,4{}\mathrm {i}}{5}\right )}{d\,{\mathrm {tan}\left (c+d\,x\right )}^6}-\frac {16\,a^4\,\mathrm {atan}\left (2\,\mathrm {tan}\left (c+d\,x\right )+1{}\mathrm {i}\right )\,\left (B+A\,1{}\mathrm {i}\right )}{d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cot(c + d*x)^7*(A + B*tan(c + d*x))*(a + a*tan(c + d*x)*1i)^4,x)

[Out]

- (tan(c + d*x)^4*(4*A*a^4 - B*a^4*4i) - tan(c + d*x)^2*((7*A*a^4)/4 - B*a^4*1i) + tan(c + d*x)^5*(A*a^4*8i +
8*B*a^4) - tan(c + d*x)^3*((A*a^4*8i)/3 + (7*B*a^4)/3) + (A*a^4)/6 + tan(c + d*x)*((A*a^4*4i)/5 + (B*a^4)/5))/
(d*tan(c + d*x)^6) - (16*a^4*atan(2*tan(c + d*x) + 1i)*(A*1i + B))/d

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sympy [A]  time = 4.93, size = 347, normalized size = 1.56 \[ - \frac {8 a^{4} \left (A - i B\right ) \log {\left (e^{2 i d x} - e^{- 2 i c} \right )}}{d} + \frac {344 A a^{4} - 316 i B a^{4} + \left (- 1944 A a^{4} e^{2 i c} + 1776 i B a^{4} e^{2 i c}\right ) e^{2 i d x} + \left (4500 A a^{4} e^{4 i c} - 4080 i B a^{4} e^{4 i c}\right ) e^{4 i d x} + \left (- 5400 A a^{4} e^{6 i c} + 4840 i B a^{4} e^{6 i c}\right ) e^{6 i d x} + \left (3420 A a^{4} e^{8 i c} - 3060 i B a^{4} e^{8 i c}\right ) e^{8 i d x} + \left (- 1080 A a^{4} e^{10 i c} + 840 i B a^{4} e^{10 i c}\right ) e^{10 i d x}}{- 15 d e^{12 i c} e^{12 i d x} + 90 d e^{10 i c} e^{10 i d x} - 225 d e^{8 i c} e^{8 i d x} + 300 d e^{6 i c} e^{6 i d x} - 225 d e^{4 i c} e^{4 i d x} + 90 d e^{2 i c} e^{2 i d x} - 15 d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(d*x+c)**7*(a+I*a*tan(d*x+c))**4*(A+B*tan(d*x+c)),x)

[Out]

-8*a**4*(A - I*B)*log(exp(2*I*d*x) - exp(-2*I*c))/d + (344*A*a**4 - 316*I*B*a**4 + (-1944*A*a**4*exp(2*I*c) +
1776*I*B*a**4*exp(2*I*c))*exp(2*I*d*x) + (4500*A*a**4*exp(4*I*c) - 4080*I*B*a**4*exp(4*I*c))*exp(4*I*d*x) + (-
5400*A*a**4*exp(6*I*c) + 4840*I*B*a**4*exp(6*I*c))*exp(6*I*d*x) + (3420*A*a**4*exp(8*I*c) - 3060*I*B*a**4*exp(
8*I*c))*exp(8*I*d*x) + (-1080*A*a**4*exp(10*I*c) + 840*I*B*a**4*exp(10*I*c))*exp(10*I*d*x))/(-15*d*exp(12*I*c)
*exp(12*I*d*x) + 90*d*exp(10*I*c)*exp(10*I*d*x) - 225*d*exp(8*I*c)*exp(8*I*d*x) + 300*d*exp(6*I*c)*exp(6*I*d*x
) - 225*d*exp(4*I*c)*exp(4*I*d*x) + 90*d*exp(2*I*c)*exp(2*I*d*x) - 15*d)

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