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

Optimal. Leaf size=168 \[ \frac{(A-i B) \left (a^2+i a^2 \tan (c+d x)\right )^2}{d}+\frac{4 a^4 (B+i A) \tan (c+d x)}{d}-\frac{8 a^4 (A-i B) \log (\cos (c+d x))}{d}-8 a^4 x (B+i A)+\frac{a (A-i B) (a+i a \tan (c+d x))^3}{3 d}+\frac{A (a+i a \tan (c+d x))^4}{4 d}-\frac{i B (a+i a \tan (c+d x))^5}{5 a d} \]

[Out]

-8*a^4*(I*A + B)*x - (8*a^4*(A - I*B)*Log[Cos[c + d*x]])/d + (4*a^4*(I*A + B)*Tan[c + d*x])/d + (a*(A - I*B)*(
a + I*a*Tan[c + d*x])^3)/(3*d) + (A*(a + I*a*Tan[c + d*x])^4)/(4*d) - ((I/5)*B*(a + I*a*Tan[c + d*x])^5)/(a*d)
+ ((A - I*B)*(a^2 + I*a^2*Tan[c + d*x])^2)/d

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Rubi [A]  time = 0.157747, antiderivative size = 168, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 32, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.156, Rules used = {3592, 3527, 3478, 3477, 3475} \[ \frac{(A-i B) \left (a^2+i a^2 \tan (c+d x)\right )^2}{d}+\frac{4 a^4 (B+i A) \tan (c+d x)}{d}-\frac{8 a^4 (A-i B) \log (\cos (c+d x))}{d}-8 a^4 x (B+i A)+\frac{a (A-i B) (a+i a \tan (c+d x))^3}{3 d}+\frac{A (a+i a \tan (c+d x))^4}{4 d}-\frac{i B (a+i a \tan (c+d x))^5}{5 a d} \]

Antiderivative was successfully verified.

[In]

Int[Tan[c + d*x]*(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*(A - I*B)*Log[Cos[c + d*x]])/d + (4*a^4*(I*A + B)*Tan[c + d*x])/d + (a*(A - I*B)*(
a + I*a*Tan[c + d*x])^3)/(3*d) + (A*(a + I*a*Tan[c + d*x])^4)/(4*d) - ((I/5)*B*(a + I*a*Tan[c + d*x])^5)/(a*d)
+ ((A - I*B)*(a^2 + I*a^2*Tan[c + d*x])^2)/d

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 3527

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

Rule 3478

Int[((a_) + (b_.)*tan[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[(b*(a + b*Tan[c + d*x])^(n - 1))/(d*(n - 1)
), x] + Dist[2*a, Int[(a + b*Tan[c + d*x])^(n - 1), x], x] /; FreeQ[{a, b, c, d}, x] && EqQ[a^2 + b^2, 0] && G
tQ[n, 1]

Rule 3477

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

Rule 3475

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

Rubi steps

\begin{align*} \int \tan (c+d x) (a+i a \tan (c+d x))^4 (A+B \tan (c+d x)) \, dx &=-\frac{i B (a+i a \tan (c+d x))^5}{5 a d}+\int (a+i a \tan (c+d x))^4 (-B+A \tan (c+d x)) \, dx\\ &=\frac{A (a+i a \tan (c+d x))^4}{4 d}-\frac{i B (a+i a \tan (c+d x))^5}{5 a d}-(i A+B) \int (a+i a \tan (c+d x))^4 \, dx\\ &=\frac{a (A-i B) (a+i a \tan (c+d x))^3}{3 d}+\frac{A (a+i a \tan (c+d x))^4}{4 d}-\frac{i B (a+i a \tan (c+d x))^5}{5 a d}-(2 a (i A+B)) \int (a+i a \tan (c+d x))^3 \, dx\\ &=\frac{a (A-i B) (a+i a \tan (c+d x))^3}{3 d}+\frac{A (a+i a \tan (c+d x))^4}{4 d}-\frac{i B (a+i a \tan (c+d x))^5}{5 a d}+\frac{(A-i B) \left (a^2+i a^2 \tan (c+d x)\right )^2}{d}-\left (4 a^2 (i A+B)\right ) \int (a+i a \tan (c+d x))^2 \, dx\\ &=-8 a^4 (i A+B) x+\frac{4 a^4 (i A+B) \tan (c+d x)}{d}+\frac{a (A-i B) (a+i a \tan (c+d x))^3}{3 d}+\frac{A (a+i a \tan (c+d x))^4}{4 d}-\frac{i B (a+i a \tan (c+d x))^5}{5 a d}+\frac{(A-i B) \left (a^2+i a^2 \tan (c+d x)\right )^2}{d}+\left (8 a^4 (A-i B)\right ) \int \tan (c+d x) \, dx\\ &=-8 a^4 (i A+B) x-\frac{8 a^4 (A-i B) \log (\cos (c+d x))}{d}+\frac{4 a^4 (i A+B) \tan (c+d x)}{d}+\frac{a (A-i B) (a+i a \tan (c+d x))^3}{3 d}+\frac{A (a+i a \tan (c+d x))^4}{4 d}-\frac{i B (a+i a \tan (c+d x))^5}{5 a d}+\frac{(A-i B) \left (a^2+i a^2 \tan (c+d x)\right )^2}{d}\\ \end{align*}

Mathematica [B]  time = 4.52209, size = 589, normalized size = 3.51 \[ \frac{a^4 \sec (c) \sec ^5(c+d x) \left (-15 i \cos (d x) \left (-10 i (A-i B) \log \left (\cos ^2(c+d x)\right )+20 A d x-11 i A-20 i B d x-14 B\right )-15 i \cos (2 c+d x) \left (-10 i (A-i B) \log \left (\cos ^2(c+d x)\right )+20 A d x-11 i A-20 i B d x-14 B\right )-300 i A \sin (2 c+d x)+260 i A \sin (2 c+3 d x)-90 i A \sin (4 c+3 d x)+70 i A \sin (4 c+5 d x)-60 A \cos (2 c+3 d x)-150 i A d x \cos (2 c+3 d x)-60 A \cos (4 c+3 d x)-150 i A d x \cos (4 c+3 d x)-30 i A d x \cos (4 c+5 d x)-30 i A d x \cos (6 c+5 d x)-75 A \cos (2 c+3 d x) \log \left (\cos ^2(c+d x)\right )-75 A \cos (4 c+3 d x) \log \left (\cos ^2(c+d x)\right )-15 A \cos (4 c+5 d x) \log \left (\cos ^2(c+d x)\right )-15 A \cos (6 c+5 d x) \log \left (\cos ^2(c+d x)\right )+400 i A \sin (d x)-345 B \sin (2 c+d x)+275 B \sin (2 c+3 d x)-120 B \sin (4 c+3 d x)+79 B \sin (4 c+5 d x)+90 i B \cos (2 c+3 d x)-150 B d x \cos (2 c+3 d x)+90 i B \cos (4 c+3 d x)-150 B d x \cos (4 c+3 d x)-30 B d x \cos (4 c+5 d x)-30 B d x \cos (6 c+5 d x)+75 i B \cos (2 c+3 d x) \log \left (\cos ^2(c+d x)\right )+75 i B \cos (4 c+3 d x) \log \left (\cos ^2(c+d x)\right )+15 i B \cos (4 c+5 d x) \log \left (\cos ^2(c+d x)\right )+15 i B \cos (6 c+5 d x) \log \left (\cos ^2(c+d x)\right )+445 B \sin (d x)\right )}{120 d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(a^4*Sec[c]*Sec[c + d*x]^5*(-60*A*Cos[2*c + 3*d*x] + (90*I)*B*Cos[2*c + 3*d*x] - (150*I)*A*d*x*Cos[2*c + 3*d*x
] - 150*B*d*x*Cos[2*c + 3*d*x] - 60*A*Cos[4*c + 3*d*x] + (90*I)*B*Cos[4*c + 3*d*x] - (150*I)*A*d*x*Cos[4*c + 3
*d*x] - 150*B*d*x*Cos[4*c + 3*d*x] - (30*I)*A*d*x*Cos[4*c + 5*d*x] - 30*B*d*x*Cos[4*c + 5*d*x] - (30*I)*A*d*x*
Cos[6*c + 5*d*x] - 30*B*d*x*Cos[6*c + 5*d*x] - 75*A*Cos[2*c + 3*d*x]*Log[Cos[c + d*x]^2] + (75*I)*B*Cos[2*c +
3*d*x]*Log[Cos[c + d*x]^2] - 75*A*Cos[4*c + 3*d*x]*Log[Cos[c + d*x]^2] + (75*I)*B*Cos[4*c + 3*d*x]*Log[Cos[c +
d*x]^2] - 15*A*Cos[4*c + 5*d*x]*Log[Cos[c + d*x]^2] + (15*I)*B*Cos[4*c + 5*d*x]*Log[Cos[c + d*x]^2] - 15*A*Co
s[6*c + 5*d*x]*Log[Cos[c + d*x]^2] + (15*I)*B*Cos[6*c + 5*d*x]*Log[Cos[c + d*x]^2] - (15*I)*Cos[d*x]*((-11*I)*
A - 14*B + 20*A*d*x - (20*I)*B*d*x - (10*I)*(A - I*B)*Log[Cos[c + d*x]^2]) - (15*I)*Cos[2*c + d*x]*((-11*I)*A
- 14*B + 20*A*d*x - (20*I)*B*d*x - (10*I)*(A - I*B)*Log[Cos[c + d*x]^2]) + (400*I)*A*Sin[d*x] + 445*B*Sin[d*x]
- (300*I)*A*Sin[2*c + d*x] - 345*B*Sin[2*c + d*x] + (260*I)*A*Sin[2*c + 3*d*x] + 275*B*Sin[2*c + 3*d*x] - (90
*I)*A*Sin[4*c + 3*d*x] - 120*B*Sin[4*c + 3*d*x] + (70*I)*A*Sin[4*c + 5*d*x] + 79*B*Sin[4*c + 5*d*x]))/(120*d)

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Maple [A]  time = 0.006, size = 229, normalized size = 1.4 \begin{align*}{\frac{-i{a}^{4}B \left ( \tan \left ( dx+c \right ) \right ) ^{4}}{d}}+{\frac{{a}^{4}B \left ( \tan \left ( dx+c \right ) \right ) ^{5}}{5\,d}}-{\frac{{\frac{4\,i}{3}}{a}^{4}A \left ( \tan \left ( dx+c \right ) \right ) ^{3}}{d}}+{\frac{{a}^{4}A \left ( \tan \left ( dx+c \right ) \right ) ^{4}}{4\,d}}+{\frac{4\,i{a}^{4}B \left ( \tan \left ( dx+c \right ) \right ) ^{2}}{d}}-{\frac{7\,{a}^{4}B \left ( \tan \left ( dx+c \right ) \right ) ^{3}}{3\,d}}+{\frac{8\,i{a}^{4}A\tan \left ( dx+c \right ) }{d}}-{\frac{7\,{a}^{4}A \left ( \tan \left ( dx+c \right ) \right ) ^{2}}{2\,d}}+8\,{\frac{{a}^{4}B\tan \left ( dx+c \right ) }{d}}-{\frac{4\,i{a}^{4}B\ln \left ( 1+ \left ( \tan \left ( dx+c \right ) \right ) ^{2} \right ) }{d}}+4\,{\frac{{a}^{4}A\ln \left ( 1+ \left ( \tan \left ( dx+c \right ) \right ) ^{2} \right ) }{d}}-{\frac{8\,i{a}^{4}A\arctan \left ( \tan \left ( dx+c \right ) \right ) }{d}}-8\,{\frac{{a}^{4}B\arctan \left ( \tan \left ( dx+c \right ) \right ) }{d}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 2.20945, size = 182, normalized size = 1.08 \begin{align*} \frac{12 \, B a^{4} \tan \left (d x + c\right )^{5} +{\left (15 \, A - 60 i \, B\right )} a^{4} \tan \left (d x + c\right )^{4} - 20 \,{\left (4 i \, A + 7 \, B\right )} a^{4} \tan \left (d x + c\right )^{3} -{\left (210 \, A - 240 i \, B\right )} a^{4} \tan \left (d x + c\right )^{2} - 480 \,{\left (d x + c\right )}{\left (i \, A + B\right )} a^{4} + 60 \,{\left (4 \, A - 4 i \, B\right )} a^{4} \log \left (\tan \left (d x + c\right )^{2} + 1\right ) - 480 \,{\left (-i \, A - B\right )} a^{4} \tan \left (d x + c\right )}{60 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/60*(12*B*a^4*tan(d*x + c)^5 + (15*A - 60*I*B)*a^4*tan(d*x + c)^4 - 20*(4*I*A + 7*B)*a^4*tan(d*x + c)^3 - (21
0*A - 240*I*B)*a^4*tan(d*x + c)^2 - 480*(d*x + c)*(I*A + B)*a^4 + 60*(4*A - 4*I*B)*a^4*log(tan(d*x + c)^2 + 1)
- 480*(-I*A - B)*a^4*tan(d*x + c))/d

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Fricas [A]  time = 1.76039, size = 790, normalized size = 4.7 \begin{align*} -\frac{4 \,{\left (30 \,{\left (5 \, A - 7 i \, B\right )} a^{4} e^{\left (8 i \, d x + 8 i \, c\right )} + 15 \,{\left (31 \, A - 37 i \, B\right )} a^{4} e^{\left (6 i \, d x + 6 i \, c\right )} + 5 \,{\left (113 \, A - 131 i \, B\right )} a^{4} e^{\left (4 i \, d x + 4 i \, c\right )} + 5 \,{\left (64 \, A - 73 i \, B\right )} a^{4} e^{\left (2 i \, d x + 2 i \, c\right )} +{\left (70 \, A - 79 i \, B\right )} a^{4} + 30 \,{\left ({\left (A - i \, B\right )} a^{4} e^{\left (10 i \, d x + 10 i \, c\right )} + 5 \,{\left (A - i \, B\right )} a^{4} e^{\left (8 i \, d x + 8 i \, c\right )} + 10 \,{\left (A - i \, B\right )} a^{4} e^{\left (6 i \, d x + 6 i \, c\right )} + 10 \,{\left (A - i \, B\right )} a^{4} e^{\left (4 i \, d x + 4 i \, c\right )} + 5 \,{\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 (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 )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-4/15*(30*(5*A - 7*I*B)*a^4*e^(8*I*d*x + 8*I*c) + 15*(31*A - 37*I*B)*a^4*e^(6*I*d*x + 6*I*c) + 5*(113*A - 131*
I*B)*a^4*e^(4*I*d*x + 4*I*c) + 5*(64*A - 73*I*B)*a^4*e^(2*I*d*x + 2*I*c) + (70*A - 79*I*B)*a^4 + 30*((A - I*B)
*a^4*e^(10*I*d*x + 10*I*c) + 5*(A - I*B)*a^4*e^(8*I*d*x + 8*I*c) + 10*(A - I*B)*a^4*e^(6*I*d*x + 6*I*c) + 10*(
A - I*B)*a^4*e^(4*I*d*x + 4*I*c) + 5*(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^(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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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Giac [B]  time = 1.62291, size = 680, normalized size = 4.05 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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