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

Optimal. Leaf size=140 \[ -\frac{4 a^4 (A-i B) \tan (c+d x)}{d}+\frac{(B+i A) \left (a^2+i a^2 \tan (c+d x)\right )^2}{d}-\frac{8 a^4 (B+i A) \log (\cos (c+d x))}{d}+8 a^4 x (A-i B)+\frac{a (B+i A) (a+i a \tan (c+d x))^3}{3 d}+\frac{B (a+i a \tan (c+d x))^4}{4 d} \]

[Out]

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

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Rubi [A]  time = 0.114768, antiderivative size = 140, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154, Rules used = {3527, 3478, 3477, 3475} \[ -\frac{4 a^4 (A-i B) \tan (c+d x)}{d}+\frac{(B+i A) \left (a^2+i a^2 \tan (c+d x)\right )^2}{d}-\frac{8 a^4 (B+i A) \log (\cos (c+d x))}{d}+8 a^4 x (A-i B)+\frac{a (B+i A) (a+i a \tan (c+d x))^3}{3 d}+\frac{B (a+i a \tan (c+d x))^4}{4 d} \]

Antiderivative was successfully verified.

[In]

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

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

Mathematica [B]  time = 3.22407, size = 448, normalized size = 3.2 \[ \frac{a^4 \sec (c) \sec ^4(c+d x) \left (3 \cos (c) \left ((-6 B-6 i A) \log \left (\cos ^2(c+d x)\right )+12 A d x-4 i A-12 i B d x-7 B\right )+6 \cos (c+2 d x) \left ((-2 B-2 i A) \log \left (\cos ^2(c+d x)\right )+4 A d x-i A-4 i B d x-2 B\right )-32 A \sin (c+2 d x)+12 A \sin (3 c+2 d x)-11 A \sin (3 c+4 d x)-6 i A \cos (3 c+2 d x)+24 A d x \cos (3 c+2 d x)+6 A d x \cos (3 c+4 d x)+6 A d x \cos (5 c+4 d x)-12 i A \cos (3 c+2 d x) \log \left (\cos ^2(c+d x)\right )-3 i A \cos (3 c+4 d x) \log \left (\cos ^2(c+d x)\right )-3 i A \cos (5 c+4 d x) \log \left (\cos ^2(c+d x)\right )+33 A \sin (c)+38 i B \sin (c+2 d x)-18 i B \sin (3 c+2 d x)+14 i B \sin (3 c+4 d x)-12 B \cos (3 c+2 d x)-24 i B d x \cos (3 c+2 d x)-6 i B d x \cos (3 c+4 d x)-6 i B d x \cos (5 c+4 d x)-12 B \cos (3 c+2 d x) \log \left (\cos ^2(c+d x)\right )-3 B \cos (3 c+4 d x) \log \left (\cos ^2(c+d x)\right )-3 B \cos (5 c+4 d x) \log \left (\cos ^2(c+d x)\right )-42 i B \sin (c)\right )}{12 d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(a^4*Sec[c]*Sec[c + d*x]^4*((-6*I)*A*Cos[3*c + 2*d*x] - 12*B*Cos[3*c + 2*d*x] + 24*A*d*x*Cos[3*c + 2*d*x] - (2
4*I)*B*d*x*Cos[3*c + 2*d*x] + 6*A*d*x*Cos[3*c + 4*d*x] - (6*I)*B*d*x*Cos[3*c + 4*d*x] + 6*A*d*x*Cos[5*c + 4*d*
x] - (6*I)*B*d*x*Cos[5*c + 4*d*x] - (12*I)*A*Cos[3*c + 2*d*x]*Log[Cos[c + d*x]^2] - 12*B*Cos[3*c + 2*d*x]*Log[
Cos[c + d*x]^2] - (3*I)*A*Cos[3*c + 4*d*x]*Log[Cos[c + d*x]^2] - 3*B*Cos[3*c + 4*d*x]*Log[Cos[c + d*x]^2] - (3
*I)*A*Cos[5*c + 4*d*x]*Log[Cos[c + d*x]^2] - 3*B*Cos[5*c + 4*d*x]*Log[Cos[c + d*x]^2] + 3*Cos[c]*((-4*I)*A - 7
*B + 12*A*d*x - (12*I)*B*d*x + ((-6*I)*A - 6*B)*Log[Cos[c + d*x]^2]) + 6*Cos[c + 2*d*x]*((-I)*A - 2*B + 4*A*d*
x - (4*I)*B*d*x + ((-2*I)*A - 2*B)*Log[Cos[c + d*x]^2]) + 33*A*Sin[c] - (42*I)*B*Sin[c] - 32*A*Sin[c + 2*d*x]
+ (38*I)*B*Sin[c + 2*d*x] + 12*A*Sin[3*c + 2*d*x] - (18*I)*B*Sin[3*c + 2*d*x] - 11*A*Sin[3*c + 4*d*x] + (14*I)
*B*Sin[3*c + 4*d*x]))/(12*d)

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Maple [A]  time = 0.003, size = 194, normalized size = 1.4 \begin{align*}{\frac{-{\frac{4\,i}{3}}{a}^{4}B \left ( \tan \left ( dx+c \right ) \right ) ^{3}}{d}}+{\frac{{a}^{4}B \left ( \tan \left ( dx+c \right ) \right ) ^{4}}{4\,d}}-{\frac{2\,i{a}^{4}A \left ( \tan \left ( dx+c \right ) \right ) ^{2}}{d}}+{\frac{{a}^{4}A \left ( \tan \left ( dx+c \right ) \right ) ^{3}}{3\,d}}+{\frac{8\,i{a}^{4}B\tan \left ( dx+c \right ) }{d}}-{\frac{7\,{a}^{4}B \left ( \tan \left ( dx+c \right ) \right ) ^{2}}{2\,d}}-7\,{\frac{{a}^{4}A\tan \left ( dx+c \right ) }{d}}+{\frac{4\,i{a}^{4}A\ln \left ( 1+ \left ( \tan \left ( dx+c \right ) \right ) ^{2} \right ) }{d}}+4\,{\frac{{a}^{4}B\ln \left ( 1+ \left ( \tan \left ( dx+c \right ) \right ) ^{2} \right ) }{d}}-{\frac{8\,i{a}^{4}B\arctan \left ( \tan \left ( dx+c \right ) \right ) }{d}}+8\,{\frac{{a}^{4}A\arctan \left ( \tan \left ( dx+c \right ) \right ) }{d}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-4/3*I/d*a^4*B*tan(d*x+c)^3+1/4/d*a^4*B*tan(d*x+c)^4-2*I/d*a^4*A*tan(d*x+c)^2+1/3/d*a^4*A*tan(d*x+c)^3+8*I/d*a
^4*B*tan(d*x+c)-7/2/d*a^4*B*tan(d*x+c)^2-7/d*a^4*A*tan(d*x+c)+4*I/d*a^4*A*ln(1+tan(d*x+c)^2)+4/d*a^4*B*ln(1+ta
n(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 = 2.06552, size = 158, normalized size = 1.13 \begin{align*} \frac{3 \, B a^{4} \tan \left (d x + c\right )^{4} +{\left (4 \, A - 16 i \, B\right )} a^{4} \tan \left (d x + c\right )^{3} - 6 \,{\left (4 i \, A + 7 \, B\right )} a^{4} \tan \left (d x + c\right )^{2} + 12 \,{\left (d x + c\right )}{\left (8 \, A - 8 i \, B\right )} a^{4} - 48 \,{\left (-i \, A - B\right )} a^{4} \log \left (\tan \left (d x + c\right )^{2} + 1\right ) -{\left (84 \, A - 96 i \, B\right )} a^{4} \tan \left (d x + c\right )}{12 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/12*(3*B*a^4*tan(d*x + c)^4 + (4*A - 16*I*B)*a^4*tan(d*x + c)^3 - 6*(4*I*A + 7*B)*a^4*tan(d*x + c)^2 + 12*(d*
x + c)*(8*A - 8*I*B)*a^4 - 48*(-I*A - B)*a^4*log(tan(d*x + c)^2 + 1) - (84*A - 96*I*B)*a^4*tan(d*x + c))/d

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Fricas [A]  time = 1.71488, size = 670, normalized size = 4.79 \begin{align*} \frac{{\left (-72 i \, A - 120 \, B\right )} a^{4} e^{\left (6 i \, d x + 6 i \, c\right )} +{\left (-180 i \, A - 252 \, B\right )} a^{4} e^{\left (4 i \, d x + 4 i \, c\right )} +{\left (-152 i \, A - 200 \, B\right )} a^{4} e^{\left (2 i \, d x + 2 i \, c\right )} +{\left (-44 i \, A - 56 \, B\right )} a^{4} +{\left ({\left (-24 i \, A - 24 \, B\right )} a^{4} e^{\left (8 i \, d x + 8 i \, c\right )} +{\left (-96 i \, A - 96 \, B\right )} a^{4} e^{\left (6 i \, d x + 6 i \, c\right )} +{\left (-144 i \, A - 144 \, B\right )} a^{4} e^{\left (4 i \, d x + 4 i \, c\right )} +{\left (-96 i \, A - 96 \, B\right )} a^{4} e^{\left (2 i \, d x + 2 i \, c\right )} +{\left (-24 i \, A - 24 \, B\right )} a^{4}\right )} \log \left (e^{\left (2 i \, d x + 2 i \, c\right )} + 1\right )}{3 \,{\left (d e^{\left (8 i \, d x + 8 i \, c\right )} + 4 \, d e^{\left (6 i \, d x + 6 i \, c\right )} + 6 \, d e^{\left (4 i \, d x + 4 i \, c\right )} + 4 \, 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((a+I*a*tan(d*x+c))^4*(A+B*tan(d*x+c)),x, algorithm="fricas")

[Out]

1/3*((-72*I*A - 120*B)*a^4*e^(6*I*d*x + 6*I*c) + (-180*I*A - 252*B)*a^4*e^(4*I*d*x + 4*I*c) + (-152*I*A - 200*
B)*a^4*e^(2*I*d*x + 2*I*c) + (-44*I*A - 56*B)*a^4 + ((-24*I*A - 24*B)*a^4*e^(8*I*d*x + 8*I*c) + (-96*I*A - 96*
B)*a^4*e^(6*I*d*x + 6*I*c) + (-144*I*A - 144*B)*a^4*e^(4*I*d*x + 4*I*c) + (-96*I*A - 96*B)*a^4*e^(2*I*d*x + 2*
I*c) + (-24*I*A - 24*B)*a^4)*log(e^(2*I*d*x + 2*I*c) + 1))/(d*e^(8*I*d*x + 8*I*c) + 4*d*e^(6*I*d*x + 6*I*c) +
6*d*e^(4*I*d*x + 4*I*c) + 4*d*e^(2*I*d*x + 2*I*c) + d)

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Sympy [A]  time = 48.7133, size = 223, normalized size = 1.59 \begin{align*} - \frac{8 a^{4} \left (i A + B\right ) \log{\left (e^{2 i d x} + e^{- 2 i c} \right )}}{d} + \frac{- \frac{\left (24 i A a^{4} + 40 B a^{4}\right ) e^{- 2 i c} e^{6 i d x}}{d} - \frac{\left (44 i A a^{4} + 56 B a^{4}\right ) e^{- 8 i c}}{3 d} - \frac{\left (60 i A a^{4} + 84 B a^{4}\right ) e^{- 4 i c} e^{4 i d x}}{d} - \frac{\left (152 i A a^{4} + 200 B a^{4}\right ) e^{- 6 i c} e^{2 i d x}}{3 d}}{e^{8 i d x} + 4 e^{- 2 i c} e^{6 i d x} + 6 e^{- 4 i c} e^{4 i d x} + 4 e^{- 6 i c} e^{2 i d x} + e^{- 8 i c}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-8*a**4*(I*A + B)*log(exp(2*I*d*x) + exp(-2*I*c))/d + (-(24*I*A*a**4 + 40*B*a**4)*exp(-2*I*c)*exp(6*I*d*x)/d -
(44*I*A*a**4 + 56*B*a**4)*exp(-8*I*c)/(3*d) - (60*I*A*a**4 + 84*B*a**4)*exp(-4*I*c)*exp(4*I*d*x)/d - (152*I*A
*a**4 + 200*B*a**4)*exp(-6*I*c)*exp(2*I*d*x)/(3*d))/(exp(8*I*d*x) + 4*exp(-2*I*c)*exp(6*I*d*x) + 6*exp(-4*I*c)
*exp(4*I*d*x) + 4*exp(-6*I*c)*exp(2*I*d*x) + exp(-8*I*c))

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Giac [B]  time = 1.57072, size = 551, normalized size = 3.94 \begin{align*} \frac{-24 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 ) - 24 \, 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 ) - 96 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 ) - 96 \, 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 ) - 144 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 ) - 144 \, 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 ) - 96 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 ) - 96 \, 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 ) - 72 i \, A a^{4} e^{\left (6 i \, d x + 6 i \, c\right )} - 120 \, B a^{4} e^{\left (6 i \, d x + 6 i \, c\right )} - 180 i \, A a^{4} e^{\left (4 i \, d x + 4 i \, c\right )} - 252 \, B a^{4} e^{\left (4 i \, d x + 4 i \, c\right )} - 152 i \, A a^{4} e^{\left (2 i \, d x + 2 i \, c\right )} - 200 \, B a^{4} e^{\left (2 i \, d x + 2 i \, c\right )} - 24 i \, A a^{4} \log \left (e^{\left (2 i \, d x + 2 i \, c\right )} + 1\right ) - 24 \, B a^{4} \log \left (e^{\left (2 i \, d x + 2 i \, c\right )} + 1\right ) - 44 i \, A a^{4} - 56 \, B a^{4}}{3 \,{\left (d e^{\left (8 i \, d x + 8 i \, c\right )} + 4 \, d e^{\left (6 i \, d x + 6 i \, c\right )} + 6 \, d e^{\left (4 i \, d x + 4 i \, c\right )} + 4 \, 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((a+I*a*tan(d*x+c))^4*(A+B*tan(d*x+c)),x, algorithm="giac")

[Out]

1/3*(-24*I*A*a^4*e^(8*I*d*x + 8*I*c)*log(e^(2*I*d*x + 2*I*c) + 1) - 24*B*a^4*e^(8*I*d*x + 8*I*c)*log(e^(2*I*d*
x + 2*I*c) + 1) - 96*I*A*a^4*e^(6*I*d*x + 6*I*c)*log(e^(2*I*d*x + 2*I*c) + 1) - 96*B*a^4*e^(6*I*d*x + 6*I*c)*l
og(e^(2*I*d*x + 2*I*c) + 1) - 144*I*A*a^4*e^(4*I*d*x + 4*I*c)*log(e^(2*I*d*x + 2*I*c) + 1) - 144*B*a^4*e^(4*I*
d*x + 4*I*c)*log(e^(2*I*d*x + 2*I*c) + 1) - 96*I*A*a^4*e^(2*I*d*x + 2*I*c)*log(e^(2*I*d*x + 2*I*c) + 1) - 96*B
*a^4*e^(2*I*d*x + 2*I*c)*log(e^(2*I*d*x + 2*I*c) + 1) - 72*I*A*a^4*e^(6*I*d*x + 6*I*c) - 120*B*a^4*e^(6*I*d*x
+ 6*I*c) - 180*I*A*a^4*e^(4*I*d*x + 4*I*c) - 252*B*a^4*e^(4*I*d*x + 4*I*c) - 152*I*A*a^4*e^(2*I*d*x + 2*I*c) -
200*B*a^4*e^(2*I*d*x + 2*I*c) - 24*I*A*a^4*log(e^(2*I*d*x + 2*I*c) + 1) - 24*B*a^4*log(e^(2*I*d*x + 2*I*c) +
1) - 44*I*A*a^4 - 56*B*a^4)/(d*e^(8*I*d*x + 8*I*c) + 4*d*e^(6*I*d*x + 6*I*c) + 6*d*e^(4*I*d*x + 4*I*c) + 4*d*e
^(2*I*d*x + 2*I*c) + d)