3.34 \(\int \cot ^6(c+d x) (a+i a \tan (c+d x))^4 (A+B \tan (c+d x)) \, dx\)

Optimal. Leaf size=200 \[ \frac {a^4 (145 B+148 i A) \cot ^2(c+d x)}{60 d}-\frac {8 a^4 (A-i B) \cot (c+d x)}{d}+\frac {8 a^4 (B+i A) \log (\sin (c+d x))}{d}+\frac {(28 A-25 i B) \cot ^3(c+d x) \left (a^4+i a^4 \tan (c+d x)\right )}{30 d}-8 a^4 x (A-i B)-\frac {(5 B+8 i A) \cot ^4(c+d x) \left (a^2+i a^2 \tan (c+d x)\right )^2}{20 d}-\frac {a A \cot ^5(c+d x) (a+i a \tan (c+d x))^3}{5 d} \]

[Out]

-8*a^4*(A-I*B)*x-8*a^4*(A-I*B)*cot(d*x+c)/d+1/60*a^4*(148*I*A+145*B)*cot(d*x+c)^2/d+8*a^4*(I*A+B)*ln(sin(d*x+c
))/d-1/5*a*A*cot(d*x+c)^5*(a+I*a*tan(d*x+c))^3/d-1/20*(8*I*A+5*B)*cot(d*x+c)^4*(a^2+I*a^2*tan(d*x+c))^2/d+1/30
*(28*A-25*I*B)*cot(d*x+c)^3*(a^4+I*a^4*tan(d*x+c))/d

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Rubi [A]  time = 0.59, antiderivative size = 200, 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 = {3593, 3591, 3529, 3531, 3475} \[ \frac {a^4 (145 B+148 i A) \cot ^2(c+d x)}{60 d}-\frac {8 a^4 (A-i B) \cot (c+d x)}{d}+\frac {8 a^4 (B+i A) \log (\sin (c+d x))}{d}-\frac {(5 B+8 i A) \cot ^4(c+d x) \left (a^2+i a^2 \tan (c+d x)\right )^2}{20 d}+\frac {(28 A-25 i B) \cot ^3(c+d x) \left (a^4+i a^4 \tan (c+d x)\right )}{30 d}-8 a^4 x (A-i B)-\frac {a A \cot ^5(c+d x) (a+i a \tan (c+d x))^3}{5 d} \]

Antiderivative was successfully verified.

[In]

Int[Cot[c + d*x]^6*(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*(A - I*B)*Cot[c + d*x])/d + (a^4*((148*I)*A + 145*B)*Cot[c + d*x]^2)/(60*d) + (8*a
^4*(I*A + B)*Log[Sin[c + d*x]])/d - (a*A*Cot[c + d*x]^5*(a + I*a*Tan[c + d*x])^3)/(5*d) - (((8*I)*A + 5*B)*Cot
[c + d*x]^4*(a^2 + I*a^2*Tan[c + d*x])^2)/(20*d) + ((28*A - (25*I)*B)*Cot[c + d*x]^3*(a^4 + I*a^4*Tan[c + d*x]
))/(30*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 ^6(c+d x) (a+i a \tan (c+d x))^4 (A+B \tan (c+d x)) \, dx &=-\frac {a A \cot ^5(c+d x) (a+i a \tan (c+d x))^3}{5 d}+\frac {1}{5} \int \cot ^5(c+d x) (a+i a \tan (c+d x))^3 (a (8 i A+5 B)-a (2 A-5 i B) \tan (c+d x)) \, dx\\ &=-\frac {a A \cot ^5(c+d x) (a+i a \tan (c+d x))^3}{5 d}-\frac {(8 i A+5 B) \cot ^4(c+d x) \left (a^2+i a^2 \tan (c+d x)\right )^2}{20 d}+\frac {1}{20} \int \cot ^4(c+d x) (a+i a \tan (c+d x))^2 \left (-2 a^2 (28 A-25 i B)-6 a^2 (4 i A+5 B) \tan (c+d x)\right ) \, dx\\ &=-\frac {a A \cot ^5(c+d x) (a+i a \tan (c+d x))^3}{5 d}-\frac {(8 i A+5 B) \cot ^4(c+d x) \left (a^2+i a^2 \tan (c+d x)\right )^2}{20 d}+\frac {(28 A-25 i B) \cot ^3(c+d x) \left (a^4+i a^4 \tan (c+d x)\right )}{30 d}+\frac {1}{60} \int \cot ^3(c+d x) (a+i a \tan (c+d x)) \left (-2 a^3 (148 i A+145 B)+2 a^3 (92 A-95 i B) \tan (c+d x)\right ) \, dx\\ &=\frac {a^4 (148 i A+145 B) \cot ^2(c+d x)}{60 d}-\frac {a A \cot ^5(c+d x) (a+i a \tan (c+d x))^3}{5 d}-\frac {(8 i A+5 B) \cot ^4(c+d x) \left (a^2+i a^2 \tan (c+d x)\right )^2}{20 d}+\frac {(28 A-25 i B) \cot ^3(c+d x) \left (a^4+i a^4 \tan (c+d x)\right )}{30 d}+\frac {1}{60} \int \cot ^2(c+d x) \left (480 a^4 (A-i B)+480 a^4 (i A+B) \tan (c+d x)\right ) \, dx\\ &=-\frac {8 a^4 (A-i B) \cot (c+d x)}{d}+\frac {a^4 (148 i A+145 B) \cot ^2(c+d x)}{60 d}-\frac {a A \cot ^5(c+d x) (a+i a \tan (c+d x))^3}{5 d}-\frac {(8 i A+5 B) \cot ^4(c+d x) \left (a^2+i a^2 \tan (c+d x)\right )^2}{20 d}+\frac {(28 A-25 i B) \cot ^3(c+d x) \left (a^4+i a^4 \tan (c+d x)\right )}{30 d}+\frac {1}{60} \int \cot (c+d x) \left (480 a^4 (i A+B)-480 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) \cot (c+d x)}{d}+\frac {a^4 (148 i A+145 B) \cot ^2(c+d x)}{60 d}-\frac {a A \cot ^5(c+d x) (a+i a \tan (c+d x))^3}{5 d}-\frac {(8 i A+5 B) \cot ^4(c+d x) \left (a^2+i a^2 \tan (c+d x)\right )^2}{20 d}+\frac {(28 A-25 i B) \cot ^3(c+d x) \left (a^4+i a^4 \tan (c+d x)\right )}{30 d}+\left (8 a^4 (i A+B)\right ) \int \cot (c+d x) \, dx\\ &=-8 a^4 (A-i B) x-\frac {8 a^4 (A-i B) \cot (c+d x)}{d}+\frac {a^4 (148 i A+145 B) \cot ^2(c+d x)}{60 d}+\frac {8 a^4 (i A+B) \log (\sin (c+d x))}{d}-\frac {a A \cot ^5(c+d x) (a+i a \tan (c+d x))^3}{5 d}-\frac {(8 i A+5 B) \cot ^4(c+d x) \left (a^2+i a^2 \tan (c+d x)\right )^2}{20 d}+\frac {(28 A-25 i B) \cot ^3(c+d x) \left (a^4+i a^4 \tan (c+d x)\right )}{30 d}\\ \end {align*}

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Mathematica [B]  time = 8.90, size = 542, normalized size = 2.71 \[ \frac {a^4 (\cot (c+d x)+i)^4 (A \cot (c+d x)+B) \left (-8 d x (A-i B) (\cos (4 c)-i \sin (4 c)) \sin ^5(c+d x)+4 (A-i B) (\sin (4 c)+i \cos (4 c)) \sin ^5(c+d x) \log \left (\sin ^2(c+d x)\right )+8 (A-i B) (\cos (4 c)-i \sin (4 c)) \sin ^5(c+d x) \tan ^{-1}(\tan (5 c+d x))+\frac {1}{120} \csc (c) (\cos (4 c)-i \sin (4 c)) (15 (20 A d x-14 i A-20 i B d x-11 B) \cos (2 c+d x)+15 \cos (d x) (A (-20 d x+14 i)+B (11+20 i d x))+345 A \sin (2 c+d x)-275 A \sin (2 c+3 d x)-120 A \sin (4 c+3 d x)+79 A \sin (4 c+5 d x)-90 i A \cos (2 c+3 d x)+150 A d x \cos (2 c+3 d x)+90 i A \cos (4 c+3 d x)-150 A d x \cos (4 c+3 d x)-30 A d x \cos (4 c+5 d x)+30 A d x \cos (6 c+5 d x)+445 A \sin (d x)-300 i B \sin (2 c+d x)+260 i B \sin (2 c+3 d x)+90 i B \sin (4 c+3 d x)-70 i B \sin (4 c+5 d x)-60 B \cos (2 c+3 d x)-150 i B d x \cos (2 c+3 d x)+60 B \cos (4 c+3 d x)+150 i B d x \cos (4 c+3 d x)+30 i B d x \cos (4 c+5 d x)-30 i B d x \cos (6 c+5 d x)-400 i B \sin (d x))\right )}{d (\cos (d x)+i \sin (d x))^4 (A \cos (c+d x)+B \sin (c+d x))} \]

Antiderivative was successfully verified.

[In]

Integrate[Cot[c + d*x]^6*(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])*(-8*(A - I*B)*d*x*(Cos[4*c] - I*Sin[4*c])*Sin[c + d*x]^5 + 8*(A
- I*B)*ArcTan[Tan[5*c + d*x]]*(Cos[4*c] - I*Sin[4*c])*Sin[c + d*x]^5 + 4*(A - I*B)*Log[Sin[c + d*x]^2]*(I*Cos
[4*c] + Sin[4*c])*Sin[c + d*x]^5 + (Csc[c]*(Cos[4*c] - I*Sin[4*c])*(15*(A*(14*I - 20*d*x) + B*(11 + (20*I)*d*x
))*Cos[d*x] + 15*((-14*I)*A - 11*B + 20*A*d*x - (20*I)*B*d*x)*Cos[2*c + d*x] - (90*I)*A*Cos[2*c + 3*d*x] - 60*
B*Cos[2*c + 3*d*x] + 150*A*d*x*Cos[2*c + 3*d*x] - (150*I)*B*d*x*Cos[2*c + 3*d*x] + (90*I)*A*Cos[4*c + 3*d*x] +
60*B*Cos[4*c + 3*d*x] - 150*A*d*x*Cos[4*c + 3*d*x] + (150*I)*B*d*x*Cos[4*c + 3*d*x] - 30*A*d*x*Cos[4*c + 5*d*
x] + (30*I)*B*d*x*Cos[4*c + 5*d*x] + 30*A*d*x*Cos[6*c + 5*d*x] - (30*I)*B*d*x*Cos[6*c + 5*d*x] + 445*A*Sin[d*x
] - (400*I)*B*Sin[d*x] + 345*A*Sin[2*c + d*x] - (300*I)*B*Sin[2*c + d*x] - 275*A*Sin[2*c + 3*d*x] + (260*I)*B*
Sin[2*c + 3*d*x] - 120*A*Sin[4*c + 3*d*x] + (90*I)*B*Sin[4*c + 3*d*x] + 79*A*Sin[4*c + 5*d*x] - (70*I)*B*Sin[4
*c + 5*d*x]))/120))/(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 = 1.16, size = 284, normalized size = 1.42 \[ \frac {{\left (-840 i \, A - 600 \, B\right )} a^{4} e^{\left (8 i \, d x + 8 i \, c\right )} + {\left (2220 i \, A + 1860 \, B\right )} a^{4} e^{\left (6 i \, d x + 6 i \, c\right )} + {\left (-2620 i \, A - 2260 \, B\right )} a^{4} e^{\left (4 i \, d x + 4 i \, c\right )} + {\left (1460 i \, A + 1280 \, B\right )} a^{4} e^{\left (2 i \, d x + 2 i \, c\right )} + {\left (-316 i \, A - 280 \, B\right )} a^{4} + {\left ({\left (120 i \, A + 120 \, B\right )} a^{4} e^{\left (10 i \, d x + 10 i \, c\right )} + {\left (-600 i \, A - 600 \, B\right )} a^{4} e^{\left (8 i \, d x + 8 i \, c\right )} + {\left (1200 i \, A + 1200 \, B\right )} a^{4} e^{\left (6 i \, d x + 6 i \, c\right )} + {\left (-1200 i \, A - 1200 \, B\right )} a^{4} e^{\left (4 i \, d x + 4 i \, c\right )} + {\left (600 i \, A + 600 \, B\right )} a^{4} e^{\left (2 i \, d x + 2 i \, c\right )} + {\left (-120 i \, A - 120 \, B\right )} a^{4}\right )} \log \left (e^{\left (2 i \, d x + 2 i \, c\right )} - 1\right )}{15 \, {\left (d e^{\left (10 i \, d x + 10 i \, c\right )} - 5 \, d e^{\left (8 i \, d x + 8 i \, c\right )} + 10 \, d e^{\left (6 i \, d x + 6 i \, c\right )} - 10 \, d e^{\left (4 i \, d x + 4 i \, c\right )} + 5 \, d e^{\left (2 i \, d x + 2 i \, c\right )} - d\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(d*x+c)^6*(a+I*a*tan(d*x+c))^4*(A+B*tan(d*x+c)),x, algorithm="fricas")

[Out]

1/15*((-840*I*A - 600*B)*a^4*e^(8*I*d*x + 8*I*c) + (2220*I*A + 1860*B)*a^4*e^(6*I*d*x + 6*I*c) + (-2620*I*A -
2260*B)*a^4*e^(4*I*d*x + 4*I*c) + (1460*I*A + 1280*B)*a^4*e^(2*I*d*x + 2*I*c) + (-316*I*A - 280*B)*a^4 + ((120
*I*A + 120*B)*a^4*e^(10*I*d*x + 10*I*c) + (-600*I*A - 600*B)*a^4*e^(8*I*d*x + 8*I*c) + (1200*I*A + 1200*B)*a^4
*e^(6*I*d*x + 6*I*c) + (-1200*I*A - 1200*B)*a^4*e^(4*I*d*x + 4*I*c) + (600*I*A + 600*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^(10*I*d*x + 10*I*c) - 5*d*e^(8*I*d*x + 8*I*c) +
10*d*e^(6*I*d*x + 6*I*c) - 10*d*e^(4*I*d*x + 4*I*c) + 5*d*e^(2*I*d*x + 2*I*c) - d)

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giac [B]  time = 9.17, size = 392, normalized size = 1.96 \[ \frac {6 \, A a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 60 i \, A a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 15 \, B a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 310 \, A a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 160 i \, B a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 1200 i \, A a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 900 \, B a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 4740 \, A a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 4320 i \, B a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1920 \, {\left (-8 i \, A a^{4} - 8 \, B a^{4}\right )} \log \left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + i\right ) + 1920 \, {\left (4 i \, A a^{4} + 4 \, B a^{4}\right )} \log \left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right ) + \frac {-17536 i \, A a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 17536 \, B a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 4740 \, A a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 4320 i \, B a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 1200 i \, A a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 900 \, B a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 310 \, A a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 160 i \, B a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 60 i \, A a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 15 \, B a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 6 \, A a^{4}}{\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5}}}{960 \, d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(d*x+c)^6*(a+I*a*tan(d*x+c))^4*(A+B*tan(d*x+c)),x, algorithm="giac")

[Out]

1/960*(6*A*a^4*tan(1/2*d*x + 1/2*c)^5 - 60*I*A*a^4*tan(1/2*d*x + 1/2*c)^4 - 15*B*a^4*tan(1/2*d*x + 1/2*c)^4 -
310*A*a^4*tan(1/2*d*x + 1/2*c)^3 + 160*I*B*a^4*tan(1/2*d*x + 1/2*c)^3 + 1200*I*A*a^4*tan(1/2*d*x + 1/2*c)^2 +
900*B*a^4*tan(1/2*d*x + 1/2*c)^2 + 4740*A*a^4*tan(1/2*d*x + 1/2*c) - 4320*I*B*a^4*tan(1/2*d*x + 1/2*c) + 1920*
(-8*I*A*a^4 - 8*B*a^4)*log(tan(1/2*d*x + 1/2*c) + I) + 1920*(4*I*A*a^4 + 4*B*a^4)*log(tan(1/2*d*x + 1/2*c)) +
(-17536*I*A*a^4*tan(1/2*d*x + 1/2*c)^5 - 17536*B*a^4*tan(1/2*d*x + 1/2*c)^5 - 4740*A*a^4*tan(1/2*d*x + 1/2*c)^
4 + 4320*I*B*a^4*tan(1/2*d*x + 1/2*c)^4 + 1200*I*A*a^4*tan(1/2*d*x + 1/2*c)^3 + 900*B*a^4*tan(1/2*d*x + 1/2*c)
^3 + 310*A*a^4*tan(1/2*d*x + 1/2*c)^2 - 160*I*B*a^4*tan(1/2*d*x + 1/2*c)^2 - 60*I*A*a^4*tan(1/2*d*x + 1/2*c) -
15*B*a^4*tan(1/2*d*x + 1/2*c) - 6*A*a^4)/tan(1/2*d*x + 1/2*c)^5)/d

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maple [A]  time = 0.46, size = 224, normalized size = 1.12 \[ \frac {8 i B \,a^{4} c}{d}+8 i B x \,a^{4}-\frac {4 i B \,a^{4} \left (\cot ^{3}\left (d x +c \right )\right )}{3 d}-\frac {8 A \,a^{4} c}{d}-\frac {8 A \cot \left (d x +c \right ) a^{4}}{d}-8 A \,a^{4} x +\frac {8 i A \,a^{4} \ln \left (\sin \left (d x +c \right )\right )}{d}-\frac {A \,a^{4} \left (\cot ^{5}\left (d x +c \right )\right )}{5 d}+\frac {4 i A \,a^{4} \left (\cot ^{2}\left (d x +c \right )\right )}{d}+\frac {8 a^{4} B \ln \left (\sin \left (d x +c \right )\right )}{d}-\frac {a^{4} B \left (\cot ^{4}\left (d x +c \right )\right )}{4 d}+\frac {8 i B \cot \left (d x +c \right ) a^{4}}{d}+\frac {7 A \,a^{4} \left (\cot ^{3}\left (d x +c \right )\right )}{3 d}+\frac {7 a^{4} B \left (\cot ^{2}\left (d x +c \right )\right )}{2 d}-\frac {i A \,a^{4} \left (\cot ^{4}\left (d x +c \right )\right )}{d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cot(d*x+c)^6*(a+I*a*tan(d*x+c))^4*(A+B*tan(d*x+c)),x)

[Out]

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

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maxima [A]  time = 0.77, size = 153, normalized size = 0.76 \[ -\frac {480 \, {\left (d x + c\right )} {\left (A - i \, B\right )} a^{4} - 60 \, {\left (-4 i \, A - 4 \, B\right )} a^{4} \log \left (\tan \left (d x + c\right )^{2} + 1\right ) - 60 \, {\left (8 i \, A + 8 \, B\right )} a^{4} \log \left (\tan \left (d x + c\right )\right ) + \frac {480 \, {\left (A - i \, B\right )} a^{4} \tan \left (d x + c\right )^{4} + {\left (-240 i \, A - 210 \, B\right )} a^{4} \tan \left (d x + c\right )^{3} - 20 \, {\left (7 \, A - 4 i \, B\right )} a^{4} \tan \left (d x + c\right )^{2} + {\left (60 i \, A + 15 \, B\right )} a^{4} \tan \left (d x + c\right ) + 12 \, A a^{4}}{\tan \left (d x + c\right )^{5}}}{60 \, d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(d*x+c)^6*(a+I*a*tan(d*x+c))^4*(A+B*tan(d*x+c)),x, algorithm="maxima")

[Out]

-1/60*(480*(d*x + c)*(A - I*B)*a^4 - 60*(-4*I*A - 4*B)*a^4*log(tan(d*x + c)^2 + 1) - 60*(8*I*A + 8*B)*a^4*log(
tan(d*x + c)) + (480*(A - I*B)*a^4*tan(d*x + c)^4 + (-240*I*A - 210*B)*a^4*tan(d*x + c)^3 - 20*(7*A - 4*I*B)*a
^4*tan(d*x + c)^2 + (60*I*A + 15*B)*a^4*tan(d*x + c) + 12*A*a^4)/tan(d*x + c)^5)/d

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mupad [B]  time = 6.85, size = 140, normalized size = 0.70 \[ -\frac {\frac {A\,a^4}{5}-{\mathrm {tan}\left (c+d\,x\right )}^2\,\left (\frac {7\,A\,a^4}{3}-\frac {B\,a^4\,4{}\mathrm {i}}{3}\right )+{\mathrm {tan}\left (c+d\,x\right )}^4\,\left (8\,A\,a^4-B\,a^4\,8{}\mathrm {i}\right )-{\mathrm {tan}\left (c+d\,x\right )}^3\,\left (\frac {7\,B\,a^4}{2}+A\,a^4\,4{}\mathrm {i}\right )+\mathrm {tan}\left (c+d\,x\right )\,\left (\frac {B\,a^4}{4}+A\,a^4\,1{}\mathrm {i}\right )}{d\,{\mathrm {tan}\left (c+d\,x\right )}^5}+\frac {a^4\,\mathrm {atan}\left (2\,\mathrm {tan}\left (c+d\,x\right )+1{}\mathrm {i}\right )\,\left (B+A\,1{}\mathrm {i}\right )\,16{}\mathrm {i}}{d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cot(c + d*x)^6*(A + B*tan(c + d*x))*(a + a*tan(c + d*x)*1i)^4,x)

[Out]

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

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sympy [A]  time = 17.31, size = 296, normalized size = 1.48 \[ \frac {8 i a^{4} \left (A - i B\right ) \log {\left (e^{2 i d x} - e^{- 2 i c} \right )}}{d} + \frac {- 316 i A a^{4} - 280 B a^{4} + \left (1460 i A a^{4} e^{2 i c} + 1280 B a^{4} e^{2 i c}\right ) e^{2 i d x} + \left (- 2620 i A a^{4} e^{4 i c} - 2260 B a^{4} e^{4 i c}\right ) e^{4 i d x} + \left (2220 i A a^{4} e^{6 i c} + 1860 B a^{4} e^{6 i c}\right ) e^{6 i d x} + \left (- 840 i A a^{4} e^{8 i c} - 600 B a^{4} e^{8 i c}\right ) e^{8 i d x}}{15 d e^{10 i c} e^{10 i d x} - 75 d e^{8 i c} e^{8 i d x} + 150 d e^{6 i c} e^{6 i d x} - 150 d e^{4 i c} e^{4 i d x} + 75 d e^{2 i c} e^{2 i d x} - 15 d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(d*x+c)**6*(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 - 280*B*a**4 + (1460*I*A*a**4*exp(2*I*c)
+ 1280*B*a**4*exp(2*I*c))*exp(2*I*d*x) + (-2620*I*A*a**4*exp(4*I*c) - 2260*B*a**4*exp(4*I*c))*exp(4*I*d*x) +
(2220*I*A*a**4*exp(6*I*c) + 1860*B*a**4*exp(6*I*c))*exp(6*I*d*x) + (-840*I*A*a**4*exp(8*I*c) - 600*B*a**4*exp(
8*I*c))*exp(8*I*d*x))/(15*d*exp(10*I*c)*exp(10*I*d*x) - 75*d*exp(8*I*c)*exp(8*I*d*x) + 150*d*exp(6*I*c)*exp(6*
I*d*x) - 150*d*exp(4*I*c)*exp(4*I*d*x) + 75*d*exp(2*I*c)*exp(2*I*d*x) - 15*d)

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