3.26 \(\int \tan ^2(c+d x) (a+i a \tan (c+d x))^4 (A+B \tan (c+d x)) \, dx\)

Optimal. Leaf size=225 \[ -\frac {a^4 (92 A-93 i B) \tan ^3(c+d x)}{60 d}-\frac {(12 A-13 i B) \tan ^3(c+d x) \left (a^4+i a^4 \tan (c+d x)\right )}{20 d}+\frac {4 a^4 (B+i A) \tan ^2(c+d x)}{d}+\frac {8 a^4 (A-i B) \tan (c+d x)}{d}+\frac {8 a^4 (B+i A) \log (\cos (c+d x))}{d}-8 a^4 x (A-i B)-\frac {(2 A-3 i B) \tan ^3(c+d x) \left (a^2+i a^2 \tan (c+d x)\right )^2}{10 d}+\frac {i a B \tan ^3(c+d x) (a+i a \tan (c+d x))^3}{6 d} \]

[Out]

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

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Rubi [A]  time = 0.64, antiderivative size = 225, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 5, integrand size = 34, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.147, Rules used = {3594, 3592, 3528, 3525, 3475} \[ -\frac {a^4 (92 A-93 i B) \tan ^3(c+d x)}{60 d}+\frac {4 a^4 (B+i A) \tan ^2(c+d x)}{d}-\frac {(2 A-3 i B) \tan ^3(c+d x) \left (a^2+i a^2 \tan (c+d x)\right )^2}{10 d}-\frac {(12 A-13 i B) \tan ^3(c+d x) \left (a^4+i a^4 \tan (c+d x)\right )}{20 d}+\frac {8 a^4 (A-i B) \tan (c+d x)}{d}+\frac {8 a^4 (B+i A) \log (\cos (c+d x))}{d}-8 a^4 x (A-i B)+\frac {i a B \tan ^3(c+d x) (a+i a \tan (c+d x))^3}{6 d} \]

Antiderivative was successfully verified.

[In]

Int[Tan[c + d*x]^2*(a + I*a*Tan[c + d*x])^4*(A + B*Tan[c + d*x]),x]

[Out]

-8*a^4*(A - I*B)*x + (8*a^4*(I*A + B)*Log[Cos[c + d*x]])/d + (8*a^4*(A - I*B)*Tan[c + d*x])/d + (4*a^4*(I*A +
B)*Tan[c + d*x]^2)/d - (a^4*(92*A - (93*I)*B)*Tan[c + d*x]^3)/(60*d) + ((I/6)*a*B*Tan[c + d*x]^3*(a + I*a*Tan[
c + d*x])^3)/d - ((2*A - (3*I)*B)*Tan[c + d*x]^3*(a^2 + I*a^2*Tan[c + d*x])^2)/(10*d) - ((12*A - (13*I)*B)*Tan
[c + d*x]^3*(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 3525

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

Rule 3528

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

Rule 3592

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

Rule 3594

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*B*(a + b*Tan[e + f*x])^(m - 1)*(c + d*Tan[e + f*x])^(n + 1))/(d*f
*(m + n)), x] + Dist[1/(d*(m + n)), Int[(a + b*Tan[e + f*x])^(m - 1)*(c + d*Tan[e + f*x])^n*Simp[a*A*d*(m + n)
+ B*(a*c*(m - 1) - b*d*(n + 1)) - (B*(b*c - a*d)*(m - 1) - d*(A*b + a*B)*(m + n))*Tan[e + f*x], x], x], x] /;
FreeQ[{a, b, c, d, e, f, A, B, n}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 + b^2, 0] && GtQ[m, 1] &&  !LtQ[n, -1]

Rubi steps

\begin {align*} \int \tan ^2(c+d x) (a+i a \tan (c+d x))^4 (A+B \tan (c+d x)) \, dx &=\frac {i a B \tan ^3(c+d x) (a+i a \tan (c+d x))^3}{6 d}+\frac {1}{6} \int \tan ^2(c+d x) (a+i a \tan (c+d x))^3 (3 a (2 A-i B)+3 a (2 i A+3 B) \tan (c+d x)) \, dx\\ &=\frac {i a B \tan ^3(c+d x) (a+i a \tan (c+d x))^3}{6 d}-\frac {(2 A-3 i B) \tan ^3(c+d x) \left (a^2+i a^2 \tan (c+d x)\right )^2}{10 d}+\frac {1}{30} \int \tan ^2(c+d x) (a+i a \tan (c+d x))^2 \left (6 a^2 (8 A-7 i B)+6 a^2 (12 i A+13 B) \tan (c+d x)\right ) \, dx\\ &=\frac {i a B \tan ^3(c+d x) (a+i a \tan (c+d x))^3}{6 d}-\frac {(2 A-3 i B) \tan ^3(c+d x) \left (a^2+i a^2 \tan (c+d x)\right )^2}{10 d}-\frac {(12 A-13 i B) \tan ^3(c+d x) \left (a^4+i a^4 \tan (c+d x)\right )}{20 d}+\frac {1}{120} \int \tan ^2(c+d x) (a+i a \tan (c+d x)) \left (6 a^3 (68 A-67 i B)+6 a^3 (92 i A+93 B) \tan (c+d x)\right ) \, dx\\ &=-\frac {a^4 (92 A-93 i B) \tan ^3(c+d x)}{60 d}+\frac {i a B \tan ^3(c+d x) (a+i a \tan (c+d x))^3}{6 d}-\frac {(2 A-3 i B) \tan ^3(c+d x) \left (a^2+i a^2 \tan (c+d x)\right )^2}{10 d}-\frac {(12 A-13 i B) \tan ^3(c+d x) \left (a^4+i a^4 \tan (c+d x)\right )}{20 d}+\frac {1}{120} \int \tan ^2(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 (i A+B) \tan ^2(c+d x)}{d}-\frac {a^4 (92 A-93 i B) \tan ^3(c+d x)}{60 d}+\frac {i a B \tan ^3(c+d x) (a+i a \tan (c+d x))^3}{6 d}-\frac {(2 A-3 i B) \tan ^3(c+d x) \left (a^2+i a^2 \tan (c+d x)\right )^2}{10 d}-\frac {(12 A-13 i B) \tan ^3(c+d x) \left (a^4+i a^4 \tan (c+d x)\right )}{20 d}+\frac {1}{120} \int \tan (c+d x) \left (-960 a^4 (i A+B)+960 a^4 (A-i B) \tan (c+d x)\right ) \, dx\\ &=-8 a^4 (A-i B) x+\frac {8 a^4 (A-i B) \tan (c+d x)}{d}+\frac {4 a^4 (i A+B) \tan ^2(c+d x)}{d}-\frac {a^4 (92 A-93 i B) \tan ^3(c+d x)}{60 d}+\frac {i a B \tan ^3(c+d x) (a+i a \tan (c+d x))^3}{6 d}-\frac {(2 A-3 i B) \tan ^3(c+d x) \left (a^2+i a^2 \tan (c+d x)\right )^2}{10 d}-\frac {(12 A-13 i B) \tan ^3(c+d x) \left (a^4+i a^4 \tan (c+d x)\right )}{20 d}-\left (8 a^4 (i A+B)\right ) \int \tan (c+d x) \, dx\\ &=-8 a^4 (A-i B) x+\frac {8 a^4 (i A+B) \log (\cos (c+d x))}{d}+\frac {8 a^4 (A-i B) \tan (c+d x)}{d}+\frac {4 a^4 (i A+B) \tan ^2(c+d x)}{d}-\frac {a^4 (92 A-93 i B) \tan ^3(c+d x)}{60 d}+\frac {i a B \tan ^3(c+d x) (a+i a \tan (c+d x))^3}{6 d}-\frac {(2 A-3 i B) \tan ^3(c+d x) \left (a^2+i a^2 \tan (c+d x)\right )^2}{10 d}-\frac {(12 A-13 i B) \tan ^3(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 = 9.16, size = 951, normalized size = 4.23 \[ \frac {x \left (-4 A \cos ^4(c)+4 i B \cos ^4(c)+20 i A \sin (c) \cos ^3(c)+20 B \sin (c) \cos ^3(c)+40 A \sin ^2(c) \cos ^2(c)-40 i B \sin ^2(c) \cos ^2(c)+4 A \cos ^2(c)-4 i B \cos ^2(c)-40 i A \sin ^3(c) \cos (c)-40 B \sin ^3(c) \cos (c)-12 i A \sin (c) \cos (c)-12 B \sin (c) \cos (c)-20 A \sin ^4(c)+20 i B \sin ^4(c)-12 A \sin ^2(c)+12 i B \sin ^2(c)+4 i A \sin ^4(c) \tan (c)+4 B \sin ^4(c) \tan (c)+4 i A \sin ^2(c) \tan (c)+4 B \sin ^2(c) \tan (c)-i (A-i B) (8 \cos (4 c)-8 i \sin (4 c)) \tan (c)\right ) (i \tan (c+d x) a+a)^4 (A+B \tan (c+d x)) \cos ^5(c+d x)}{(\cos (d x)+i \sin (d x))^4 (A \cos (c+d x)+B \sin (c+d x))}+\frac {(i A \cos (2 c)+B \cos (2 c)+A \sin (2 c)-i B \sin (2 c)) \left (4 \cos (2 c) \log \left (\cos ^2(c+d x)\right )-4 i \log \left (\cos ^2(c+d x)\right ) \sin (2 c)\right ) (i \tan (c+d x) a+a)^4 (A+B \tan (c+d x)) \cos ^5(c+d x)}{d (\cos (d x)+i \sin (d x))^4 (A \cos (c+d x)+B \sin (c+d x))}+\frac {\sec (c) \sec (c+d x) \left (\frac {1}{240} \cos (4 c)-\frac {1}{240} i \sin (4 c)\right ) (420 i A \cos (c)+490 B \cos (c)-600 A d x \cos (c)+600 i B d x \cos (c)+300 i A \cos (c+2 d x)+345 B \cos (c+2 d x)-450 A d x \cos (c+2 d x)+450 i B d x \cos (c+2 d x)+300 i A \cos (3 c+2 d x)+345 B \cos (3 c+2 d x)-450 A d x \cos (3 c+2 d x)+450 i B d x \cos (3 c+2 d x)+90 i A \cos (3 c+4 d x)+120 B \cos (3 c+4 d x)-180 A d x \cos (3 c+4 d x)+180 i B d x \cos (3 c+4 d x)+90 i A \cos (5 c+4 d x)+120 B \cos (5 c+4 d x)-180 A d x \cos (5 c+4 d x)+180 i B d x \cos (5 c+4 d x)-30 A d x \cos (5 c+6 d x)+30 i B d x \cos (5 c+6 d x)-30 A d x \cos (7 c+6 d x)+30 i B d x \cos (7 c+6 d x)-790 A \sin (c)+860 i B \sin (c)+720 A \sin (c+2 d x)-780 i B \sin (c+2 d x)-465 A \sin (3 c+2 d x)+510 i B \sin (3 c+2 d x)+354 A \sin (3 c+4 d x)-366 i B \sin (3 c+4 d x)-120 A \sin (5 c+4 d x)+150 i B \sin (5 c+4 d x)+79 A \sin (5 c+6 d x)-86 i B \sin (5 c+6 d x)) (i \tan (c+d x) a+a)^4 (A+B \tan (c+d x))}{d (\cos (d x)+i \sin (d x))^4 (A \cos (c+d x)+B \sin (c+d x))} \]

Antiderivative was successfully verified.

[In]

Integrate[Tan[c + d*x]^2*(a + I*a*Tan[c + d*x])^4*(A + B*Tan[c + d*x]),x]

[Out]

(Cos[c + d*x]^5*(I*A*Cos[2*c] + B*Cos[2*c] + A*Sin[2*c] - I*B*Sin[2*c])*(4*Cos[2*c]*Log[Cos[c + d*x]^2] - (4*I
)*Log[Cos[c + d*x]^2]*Sin[2*c])*(a + I*a*Tan[c + d*x])^4*(A + B*Tan[c + d*x]))/(d*(Cos[d*x] + I*Sin[d*x])^4*(A
*Cos[c + d*x] + B*Sin[c + d*x])) + (Sec[c]*Sec[c + d*x]*(Cos[4*c]/240 - (I/240)*Sin[4*c])*((420*I)*A*Cos[c] +
490*B*Cos[c] - 600*A*d*x*Cos[c] + (600*I)*B*d*x*Cos[c] + (300*I)*A*Cos[c + 2*d*x] + 345*B*Cos[c + 2*d*x] - 450
*A*d*x*Cos[c + 2*d*x] + (450*I)*B*d*x*Cos[c + 2*d*x] + (300*I)*A*Cos[3*c + 2*d*x] + 345*B*Cos[3*c + 2*d*x] - 4
50*A*d*x*Cos[3*c + 2*d*x] + (450*I)*B*d*x*Cos[3*c + 2*d*x] + (90*I)*A*Cos[3*c + 4*d*x] + 120*B*Cos[3*c + 4*d*x
] - 180*A*d*x*Cos[3*c + 4*d*x] + (180*I)*B*d*x*Cos[3*c + 4*d*x] + (90*I)*A*Cos[5*c + 4*d*x] + 120*B*Cos[5*c +
4*d*x] - 180*A*d*x*Cos[5*c + 4*d*x] + (180*I)*B*d*x*Cos[5*c + 4*d*x] - 30*A*d*x*Cos[5*c + 6*d*x] + (30*I)*B*d*
x*Cos[5*c + 6*d*x] - 30*A*d*x*Cos[7*c + 6*d*x] + (30*I)*B*d*x*Cos[7*c + 6*d*x] - 790*A*Sin[c] + (860*I)*B*Sin[
c] + 720*A*Sin[c + 2*d*x] - (780*I)*B*Sin[c + 2*d*x] - 465*A*Sin[3*c + 2*d*x] + (510*I)*B*Sin[3*c + 2*d*x] + 3
54*A*Sin[3*c + 4*d*x] - (366*I)*B*Sin[3*c + 4*d*x] - 120*A*Sin[5*c + 4*d*x] + (150*I)*B*Sin[5*c + 4*d*x] + 79*
A*Sin[5*c + 6*d*x] - (86*I)*B*Sin[5*c + 6*d*x])*(a + I*a*Tan[c + d*x])^4*(A + B*Tan[c + d*x]))/(d*(Cos[d*x] +
I*Sin[d*x])^4*(A*Cos[c + d*x] + B*Sin[c + d*x])) + (x*Cos[c + d*x]^5*(4*A*Cos[c]^2 - (4*I)*B*Cos[c]^2 - 4*A*Co
s[c]^4 + (4*I)*B*Cos[c]^4 - (12*I)*A*Cos[c]*Sin[c] - 12*B*Cos[c]*Sin[c] + (20*I)*A*Cos[c]^3*Sin[c] + 20*B*Cos[
c]^3*Sin[c] - 12*A*Sin[c]^2 + (12*I)*B*Sin[c]^2 + 40*A*Cos[c]^2*Sin[c]^2 - (40*I)*B*Cos[c]^2*Sin[c]^2 - (40*I)
*A*Cos[c]*Sin[c]^3 - 40*B*Cos[c]*Sin[c]^3 - 20*A*Sin[c]^4 + (20*I)*B*Sin[c]^4 + (4*I)*A*Sin[c]^2*Tan[c] + 4*B*
Sin[c]^2*Tan[c] + (4*I)*A*Sin[c]^4*Tan[c] + 4*B*Sin[c]^4*Tan[c] - I*(A - I*B)*(8*Cos[4*c] - (8*I)*Sin[4*c])*Ta
n[c])*(a + I*a*Tan[c + d*x])^4*(A + B*Tan[c + d*x]))/((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 = 0.48, size = 334, normalized size = 1.48 \[ \frac {{\left (840 i \, A + 1080 \, B\right )} a^{4} e^{\left (10 i \, d x + 10 i \, c\right )} + {\left (3060 i \, A + 3420 \, B\right )} a^{4} e^{\left (8 i \, d x + 8 i \, c\right )} + {\left (4840 i \, A + 5400 \, B\right )} a^{4} e^{\left (6 i \, d x + 6 i \, c\right )} + {\left (4080 i \, A + 4500 \, B\right )} a^{4} e^{\left (4 i \, d x + 4 i \, c\right )} + {\left (1776 i \, A + 1944 \, B\right )} a^{4} e^{\left (2 i \, d x + 2 i \, c\right )} + {\left (316 i \, A + 344 \, B\right )} a^{4} + {\left ({\left (120 i \, A + 120 \, B\right )} a^{4} e^{\left (12 i \, d x + 12 i \, c\right )} + {\left (720 i \, A + 720 \, B\right )} a^{4} e^{\left (10 i \, d x + 10 i \, c\right )} + {\left (1800 i \, A + 1800 \, B\right )} a^{4} e^{\left (8 i \, d x + 8 i \, c\right )} + {\left (2400 i \, A + 2400 \, B\right )} a^{4} e^{\left (6 i \, d x + 6 i \, c\right )} + {\left (1800 i \, A + 1800 \, B\right )} a^{4} e^{\left (4 i \, d x + 4 i \, c\right )} + {\left (720 i \, A + 720 \, B\right )} a^{4} e^{\left (2 i \, d x + 2 i \, c\right )} + {\left (120 i \, A + 120 \, B\right )} a^{4}\right )} \log \left (e^{\left (2 i \, d x + 2 i \, c\right )} + 1\right )}{15 \, {\left (d e^{\left (12 i \, d x + 12 i \, c\right )} + 6 \, d e^{\left (10 i \, d x + 10 i \, c\right )} + 15 \, d e^{\left (8 i \, d x + 8 i \, c\right )} + 20 \, d e^{\left (6 i \, d x + 6 i \, c\right )} + 15 \, d e^{\left (4 i \, d x + 4 i \, c\right )} + 6 \, d e^{\left (2 i \, d x + 2 i \, c\right )} + d\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(tan(d*x+c)^2*(a+I*a*tan(d*x+c))^4*(A+B*tan(d*x+c)),x, algorithm="fricas")

[Out]

1/15*((840*I*A + 1080*B)*a^4*e^(10*I*d*x + 10*I*c) + (3060*I*A + 3420*B)*a^4*e^(8*I*d*x + 8*I*c) + (4840*I*A +
5400*B)*a^4*e^(6*I*d*x + 6*I*c) + (4080*I*A + 4500*B)*a^4*e^(4*I*d*x + 4*I*c) + (1776*I*A + 1944*B)*a^4*e^(2*
I*d*x + 2*I*c) + (316*I*A + 344*B)*a^4 + ((120*I*A + 120*B)*a^4*e^(12*I*d*x + 12*I*c) + (720*I*A + 720*B)*a^4*
e^(10*I*d*x + 10*I*c) + (1800*I*A + 1800*B)*a^4*e^(8*I*d*x + 8*I*c) + (2400*I*A + 2400*B)*a^4*e^(6*I*d*x + 6*I
*c) + (1800*I*A + 1800*B)*a^4*e^(4*I*d*x + 4*I*c) + (720*I*A + 720*B)*a^4*e^(2*I*d*x + 2*I*c) + (120*I*A + 120
*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 = 1.32, size = 600, normalized size = 2.67 \[ \frac {120 i \, A a^{4} e^{\left (12 i \, d x + 12 i \, c\right )} \log \left (e^{\left (2 i \, d x + 2 i \, c\right )} + 1\right ) + 120 \, B a^{4} e^{\left (12 i \, d x + 12 i \, c\right )} \log \left (e^{\left (2 i \, d x + 2 i \, c\right )} + 1\right ) + 720 i \, A a^{4} e^{\left (10 i \, d x + 10 i \, c\right )} \log \left (e^{\left (2 i \, d x + 2 i \, c\right )} + 1\right ) + 720 \, B a^{4} e^{\left (10 i \, d x + 10 i \, c\right )} \log \left (e^{\left (2 i \, d x + 2 i \, c\right )} + 1\right ) + 1800 i \, A a^{4} e^{\left (8 i \, d x + 8 i \, c\right )} \log \left (e^{\left (2 i \, d x + 2 i \, c\right )} + 1\right ) + 1800 \, B a^{4} e^{\left (8 i \, d x + 8 i \, c\right )} \log \left (e^{\left (2 i \, d x + 2 i \, c\right )} + 1\right ) + 2400 i \, A a^{4} e^{\left (6 i \, d x + 6 i \, c\right )} \log \left (e^{\left (2 i \, d x + 2 i \, c\right )} + 1\right ) + 2400 \, B a^{4} e^{\left (6 i \, d x + 6 i \, c\right )} \log \left (e^{\left (2 i \, d x + 2 i \, c\right )} + 1\right ) + 1800 i \, A a^{4} e^{\left (4 i \, d x + 4 i \, c\right )} \log \left (e^{\left (2 i \, d x + 2 i \, c\right )} + 1\right ) + 1800 \, B a^{4} e^{\left (4 i \, d x + 4 i \, c\right )} \log \left (e^{\left (2 i \, d x + 2 i \, c\right )} + 1\right ) + 720 i \, A a^{4} e^{\left (2 i \, d x + 2 i \, c\right )} \log \left (e^{\left (2 i \, d x + 2 i \, c\right )} + 1\right ) + 720 \, B a^{4} e^{\left (2 i \, d x + 2 i \, c\right )} \log \left (e^{\left (2 i \, d x + 2 i \, c\right )} + 1\right ) + 840 i \, A a^{4} e^{\left (10 i \, d x + 10 i \, c\right )} + 1080 \, B a^{4} e^{\left (10 i \, d x + 10 i \, c\right )} + 3060 i \, A a^{4} e^{\left (8 i \, d x + 8 i \, c\right )} + 3420 \, B a^{4} e^{\left (8 i \, d x + 8 i \, c\right )} + 4840 i \, A a^{4} e^{\left (6 i \, d x + 6 i \, c\right )} + 5400 \, B a^{4} e^{\left (6 i \, d x + 6 i \, c\right )} + 4080 i \, A a^{4} e^{\left (4 i \, d x + 4 i \, c\right )} + 4500 \, B a^{4} e^{\left (4 i \, d x + 4 i \, c\right )} + 1776 i \, A a^{4} e^{\left (2 i \, d x + 2 i \, c\right )} + 1944 \, B a^{4} e^{\left (2 i \, d x + 2 i \, c\right )} + 120 i \, A a^{4} \log \left (e^{\left (2 i \, d x + 2 i \, c\right )} + 1\right ) + 120 \, B a^{4} \log \left (e^{\left (2 i \, d x + 2 i \, c\right )} + 1\right ) + 316 i \, A a^{4} + 344 \, B a^{4}}{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(tan(d*x+c)^2*(a+I*a*tan(d*x+c))^4*(A+B*tan(d*x+c)),x, algorithm="giac")

[Out]

1/15*(120*I*A*a^4*e^(12*I*d*x + 12*I*c)*log(e^(2*I*d*x + 2*I*c) + 1) + 120*B*a^4*e^(12*I*d*x + 12*I*c)*log(e^(
2*I*d*x + 2*I*c) + 1) + 720*I*A*a^4*e^(10*I*d*x + 10*I*c)*log(e^(2*I*d*x + 2*I*c) + 1) + 720*B*a^4*e^(10*I*d*x
+ 10*I*c)*log(e^(2*I*d*x + 2*I*c) + 1) + 1800*I*A*a^4*e^(8*I*d*x + 8*I*c)*log(e^(2*I*d*x + 2*I*c) + 1) + 1800
*B*a^4*e^(8*I*d*x + 8*I*c)*log(e^(2*I*d*x + 2*I*c) + 1) + 2400*I*A*a^4*e^(6*I*d*x + 6*I*c)*log(e^(2*I*d*x + 2*
I*c) + 1) + 2400*B*a^4*e^(6*I*d*x + 6*I*c)*log(e^(2*I*d*x + 2*I*c) + 1) + 1800*I*A*a^4*e^(4*I*d*x + 4*I*c)*log
(e^(2*I*d*x + 2*I*c) + 1) + 1800*B*a^4*e^(4*I*d*x + 4*I*c)*log(e^(2*I*d*x + 2*I*c) + 1) + 720*I*A*a^4*e^(2*I*d
*x + 2*I*c)*log(e^(2*I*d*x + 2*I*c) + 1) + 720*B*a^4*e^(2*I*d*x + 2*I*c)*log(e^(2*I*d*x + 2*I*c) + 1) + 840*I*
A*a^4*e^(10*I*d*x + 10*I*c) + 1080*B*a^4*e^(10*I*d*x + 10*I*c) + 3060*I*A*a^4*e^(8*I*d*x + 8*I*c) + 3420*B*a^4
*e^(8*I*d*x + 8*I*c) + 4840*I*A*a^4*e^(6*I*d*x + 6*I*c) + 5400*B*a^4*e^(6*I*d*x + 6*I*c) + 4080*I*A*a^4*e^(4*I
*d*x + 4*I*c) + 4500*B*a^4*e^(4*I*d*x + 4*I*c) + 1776*I*A*a^4*e^(2*I*d*x + 2*I*c) + 1944*B*a^4*e^(2*I*d*x + 2*
I*c) + 120*I*A*a^4*log(e^(2*I*d*x + 2*I*c) + 1) + 120*B*a^4*log(e^(2*I*d*x + 2*I*c) + 1) + 316*I*A*a^4 + 344*B
*a^4)/(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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maple [A]  time = 0.03, size = 264, normalized size = 1.17 \[ -\frac {4 i a^{4} B \left (\tan ^{5}\left (d x +c \right )\right )}{5 d}+\frac {a^{4} B \left (\tan ^{6}\left (d x +c \right )\right )}{6 d}-\frac {i a^{4} A \left (\tan ^{4}\left (d x +c \right )\right )}{d}+\frac {a^{4} A \left (\tan ^{5}\left (d x +c \right )\right )}{5 d}+\frac {8 i a^{4} B \left (\tan ^{3}\left (d x +c \right )\right )}{3 d}-\frac {7 a^{4} B \left (\tan ^{4}\left (d x +c \right )\right )}{4 d}+\frac {4 i a^{4} A \left (\tan ^{2}\left (d x +c \right )\right )}{d}-\frac {7 a^{4} A \left (\tan ^{3}\left (d x +c \right )\right )}{3 d}-\frac {8 i a^{4} B \tan \left (d x +c \right )}{d}+\frac {4 a^{4} B \left (\tan ^{2}\left (d x +c \right )\right )}{d}+\frac {8 A \,a^{4} \tan \left (d x +c \right )}{d}-\frac {4 i a^{4} A \ln \left (1+\tan ^{2}\left (d x +c \right )\right )}{d}-\frac {4 a^{4} B \ln \left (1+\tan ^{2}\left (d x +c \right )\right )}{d}+\frac {8 i a^{4} B \arctan \left (\tan \left (d x +c \right )\right )}{d}-\frac {8 a^{4} A \arctan \left (\tan \left (d x +c \right )\right )}{d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(tan(d*x+c)^2*(a+I*a*tan(d*x+c))^4*(A+B*tan(d*x+c)),x)

[Out]

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

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maxima [A]  time = 0.66, size = 150, normalized size = 0.67 \[ \frac {10 \, B a^{4} \tan \left (d x + c\right )^{6} + 12 \, {\left (A - 4 i \, B\right )} a^{4} \tan \left (d x + c\right )^{5} + {\left (-60 i \, A - 105 \, B\right )} a^{4} \tan \left (d x + c\right )^{4} - 20 \, {\left (7 \, A - 8 i \, B\right )} a^{4} \tan \left (d x + c\right )^{3} + {\left (240 i \, A + 240 \, B\right )} a^{4} \tan \left (d x + c\right )^{2} - 480 \, {\left (d x + c\right )} {\left (A - i \, B\right )} a^{4} + 60 \, {\left (-4 i \, A - 4 \, B\right )} a^{4} \log \left (\tan \left (d x + c\right )^{2} + 1\right ) + 480 \, {\left (A - i \, B\right )} a^{4} \tan \left (d x + c\right )}{60 \, d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(tan(d*x+c)^2*(a+I*a*tan(d*x+c))^4*(A+B*tan(d*x+c)),x, algorithm="maxima")

[Out]

1/60*(10*B*a^4*tan(d*x + c)^6 + 12*(A - 4*I*B)*a^4*tan(d*x + c)^5 + (-60*I*A - 105*B)*a^4*tan(d*x + c)^4 - 20*
(7*A - 8*I*B)*a^4*tan(d*x + c)^3 + (240*I*A + 240*B)*a^4*tan(d*x + c)^2 - 480*(d*x + c)*(A - I*B)*a^4 + 60*(-4
*I*A - 4*B)*a^4*log(tan(d*x + c)^2 + 1) + 480*(A - I*B)*a^4*tan(d*x + c))/d

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mupad [B]  time = 6.21, size = 308, normalized size = 1.37 \[ \frac {{\mathrm {tan}\left (c+d\,x\right )}^3\,\left (-a^4\,\left (A-B\,1{}\mathrm {i}\right )+\frac {a^4\,\left (B+A\,3{}\mathrm {i}\right )\,1{}\mathrm {i}}{3}+\frac {B\,a^4\,1{}\mathrm {i}}{3}+\frac {a^4\,\left (3\,B+A\,1{}\mathrm {i}\right )\,1{}\mathrm {i}}{3}\right )}{d}-\frac {\mathrm {tan}\left (c+d\,x\right )\,\left (-A\,a^4-3\,a^4\,\left (A-B\,1{}\mathrm {i}\right )+a^4\,\left (B+A\,3{}\mathrm {i}\right )\,1{}\mathrm {i}+B\,a^4\,1{}\mathrm {i}+a^4\,\left (3\,B+A\,1{}\mathrm {i}\right )\,1{}\mathrm {i}\right )}{d}-\frac {{\mathrm {tan}\left (c+d\,x\right )}^5\,\left (\frac {B\,a^4\,1{}\mathrm {i}}{5}+\frac {a^4\,\left (3\,B+A\,1{}\mathrm {i}\right )\,1{}\mathrm {i}}{5}\right )}{d}-\frac {\ln \left (\mathrm {tan}\left (c+d\,x\right )+1{}\mathrm {i}\right )\,\left (8\,B\,a^4+A\,a^4\,8{}\mathrm {i}\right )}{d}+\frac {{\mathrm {tan}\left (c+d\,x\right )}^2\,\left (\frac {A\,a^4\,1{}\mathrm {i}}{2}+\frac {a^4\,\left (A-B\,1{}\mathrm {i}\right )\,3{}\mathrm {i}}{2}+\frac {a^4\,\left (B+A\,3{}\mathrm {i}\right )}{2}+\frac {B\,a^4}{2}+\frac {a^4\,\left (3\,B+A\,1{}\mathrm {i}\right )}{2}\right )}{d}-\frac {{\mathrm {tan}\left (c+d\,x\right )}^4\,\left (\frac {a^4\,\left (A-B\,1{}\mathrm {i}\right )\,3{}\mathrm {i}}{4}+\frac {B\,a^4}{4}+\frac {a^4\,\left (3\,B+A\,1{}\mathrm {i}\right )}{4}\right )}{d}+\frac {B\,a^4\,{\mathrm {tan}\left (c+d\,x\right )}^6}{6\,d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(tan(c + d*x)^2*(A + B*tan(c + d*x))*(a + a*tan(c + d*x)*1i)^4,x)

[Out]

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

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sympy [A]  time = 3.20, size = 359, normalized size = 1.60 \[ \frac {8 i a^{4} \left (A - i B\right ) \log {\left (e^{2 i d x} + e^{- 2 i c} \right )}}{d} + \frac {- 316 i A a^{4} - 344 B a^{4} + \left (- 1776 i A a^{4} e^{2 i c} - 1944 B a^{4} e^{2 i c}\right ) e^{2 i d x} + \left (- 4080 i A a^{4} e^{4 i c} - 4500 B a^{4} e^{4 i c}\right ) e^{4 i d x} + \left (- 4840 i A a^{4} e^{6 i c} - 5400 B a^{4} e^{6 i c}\right ) e^{6 i d x} + \left (- 3060 i A a^{4} e^{8 i c} - 3420 B a^{4} e^{8 i c}\right ) e^{8 i d x} + \left (- 840 i A a^{4} e^{10 i c} - 1080 B a^{4} e^{10 i c}\right ) e^{10 i d x}}{- 15 d e^{12 i c} e^{12 i d x} - 90 d e^{10 i c} e^{10 i d x} - 225 d e^{8 i c} e^{8 i d x} - 300 d e^{6 i c} e^{6 i d x} - 225 d e^{4 i c} e^{4 i d x} - 90 d e^{2 i c} e^{2 i d x} - 15 d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(tan(d*x+c)**2*(a+I*a*tan(d*x+c))**4*(A+B*tan(d*x+c)),x)

[Out]

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