3.26 \(\int \tan (c+d x) (a+i a \tan (c+d x))^3 \, dx\)

Optimal. Leaf size=85 \[ \frac{2 i a^3 \tan (c+d x)}{d}-\frac{4 a^3 \log (\cos (c+d x))}{d}-4 i a^3 x+\frac{a (a+i a \tan (c+d x))^2}{2 d}+\frac{(a+i a \tan (c+d x))^3}{3 d} \]

[Out]

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

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Rubi [A]  time = 0.0576672, antiderivative size = 85, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182, Rules used = {3527, 3478, 3477, 3475} \[ \frac{2 i a^3 \tan (c+d x)}{d}-\frac{4 a^3 \log (\cos (c+d x))}{d}-4 i a^3 x+\frac{a (a+i a \tan (c+d x))^2}{2 d}+\frac{(a+i a \tan (c+d x))^3}{3 d} \]

Antiderivative was successfully verified.

[In]

Int[Tan[c + d*x]*(a + I*a*Tan[c + d*x])^3,x]

[Out]

(-4*I)*a^3*x - (4*a^3*Log[Cos[c + d*x]])/d + ((2*I)*a^3*Tan[c + d*x])/d + (a*(a + I*a*Tan[c + d*x])^2)/(2*d) +
(a + I*a*Tan[c + d*x])^3/(3*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 \tan (c+d x) (a+i a \tan (c+d x))^3 \, dx &=\frac{(a+i a \tan (c+d x))^3}{3 d}-i \int (a+i a \tan (c+d x))^3 \, dx\\ &=\frac{a (a+i a \tan (c+d x))^2}{2 d}+\frac{(a+i a \tan (c+d x))^3}{3 d}-(2 i a) \int (a+i a \tan (c+d x))^2 \, dx\\ &=-4 i a^3 x+\frac{2 i a^3 \tan (c+d x)}{d}+\frac{a (a+i a \tan (c+d x))^2}{2 d}+\frac{(a+i a \tan (c+d x))^3}{3 d}+\left (4 a^3\right ) \int \tan (c+d x) \, dx\\ &=-4 i a^3 x-\frac{4 a^3 \log (\cos (c+d x))}{d}+\frac{2 i a^3 \tan (c+d x)}{d}+\frac{a (a+i a \tan (c+d x))^2}{2 d}+\frac{(a+i a \tan (c+d x))^3}{3 d}\\ \end{align*}

Mathematica [B]  time = 1.13465, size = 178, normalized size = 2.09 \[ -\frac{i a^3 \sec (c) \sec ^3(c+d x) \left (15 \sin (2 c+d x)-13 \sin (2 c+3 d x)+6 d x \cos (2 c+3 d x)+6 d x \cos (4 c+3 d x)-3 i \cos (2 c+3 d x) \log \left (\cos ^2(c+d x)\right )+9 \cos (d x) \left (-i \log \left (\cos ^2(c+d x)\right )+2 d x-i\right )+9 \cos (2 c+d x) \left (-i \log \left (\cos ^2(c+d x)\right )+2 d x-i\right )-3 i \cos (4 c+3 d x) \log \left (\cos ^2(c+d x)\right )-24 \sin (d x)\right )}{12 d} \]

Antiderivative was successfully verified.

[In]

Integrate[Tan[c + d*x]*(a + I*a*Tan[c + d*x])^3,x]

[Out]

((-I/12)*a^3*Sec[c]*Sec[c + d*x]^3*(6*d*x*Cos[2*c + 3*d*x] + 6*d*x*Cos[4*c + 3*d*x] + 9*Cos[d*x]*(-I + 2*d*x -
I*Log[Cos[c + d*x]^2]) + 9*Cos[2*c + d*x]*(-I + 2*d*x - I*Log[Cos[c + d*x]^2]) - (3*I)*Cos[2*c + 3*d*x]*Log[C
os[c + d*x]^2] - (3*I)*Cos[4*c + 3*d*x]*Log[Cos[c + d*x]^2] - 24*Sin[d*x] + 15*Sin[2*c + d*x] - 13*Sin[2*c + 3
*d*x]))/d

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Maple [A]  time = 0.003, size = 85, normalized size = 1. \begin{align*}{\frac{4\,i{a}^{3}\tan \left ( dx+c \right ) }{d}}-{\frac{{\frac{i}{3}}{a}^{3} \left ( \tan \left ( dx+c \right ) \right ) ^{3}}{d}}-{\frac{3\,{a}^{3} \left ( \tan \left ( dx+c \right ) \right ) ^{2}}{2\,d}}+2\,{\frac{{a}^{3}\ln \left ( 1+ \left ( \tan \left ( dx+c \right ) \right ) ^{2} \right ) }{d}}-{\frac{4\,i{a}^{3}\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))^3,x)

[Out]

4*I/d*a^3*tan(d*x+c)-1/3*I/d*a^3*tan(d*x+c)^3-3/2*a^3*tan(d*x+c)^2/d+2/d*a^3*ln(1+tan(d*x+c)^2)-4*I/d*a^3*arct
an(tan(d*x+c))

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Maxima [A]  time = 2.00956, size = 93, normalized size = 1.09 \begin{align*} -\frac{2 i \, a^{3} \tan \left (d x + c\right )^{3} + 9 \, a^{3} \tan \left (d x + c\right )^{2} + 24 i \,{\left (d x + c\right )} a^{3} - 12 \, a^{3} \log \left (\tan \left (d x + c\right )^{2} + 1\right ) - 24 i \, a^{3} \tan \left (d x + c\right )}{6 \, 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))^3,x, algorithm="maxima")

[Out]

-1/6*(2*I*a^3*tan(d*x + c)^3 + 9*a^3*tan(d*x + c)^2 + 24*I*(d*x + c)*a^3 - 12*a^3*log(tan(d*x + c)^2 + 1) - 24
*I*a^3*tan(d*x + c))/d

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Fricas [A]  time = 2.22603, size = 370, normalized size = 4.35 \begin{align*} -\frac{2 \,{\left (24 \, a^{3} e^{\left (4 i \, d x + 4 i \, c\right )} + 33 \, a^{3} e^{\left (2 i \, d x + 2 i \, c\right )} + 13 \, a^{3} + 6 \,{\left (a^{3} e^{\left (6 i \, d x + 6 i \, c\right )} + 3 \, a^{3} e^{\left (4 i \, d x + 4 i \, c\right )} + 3 \, a^{3} e^{\left (2 i \, d x + 2 i \, c\right )} + a^{3}\right )} \log \left (e^{\left (2 i \, d x + 2 i \, c\right )} + 1\right )\right )}}{3 \,{\left (d e^{\left (6 i \, d x + 6 i \, c\right )} + 3 \, d e^{\left (4 i \, d x + 4 i \, c\right )} + 3 \, 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))^3,x, algorithm="fricas")

[Out]

-2/3*(24*a^3*e^(4*I*d*x + 4*I*c) + 33*a^3*e^(2*I*d*x + 2*I*c) + 13*a^3 + 6*(a^3*e^(6*I*d*x + 6*I*c) + 3*a^3*e^
(4*I*d*x + 4*I*c) + 3*a^3*e^(2*I*d*x + 2*I*c) + a^3)*log(e^(2*I*d*x + 2*I*c) + 1))/(d*e^(6*I*d*x + 6*I*c) + 3*
d*e^(4*I*d*x + 4*I*c) + 3*d*e^(2*I*d*x + 2*I*c) + d)

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Sympy [A]  time = 5.55981, size = 136, normalized size = 1.6 \begin{align*} - \frac{4 a^{3} \log{\left (e^{2 i d x} + e^{- 2 i c} \right )}}{d} + \frac{- \frac{16 a^{3} e^{- 2 i c} e^{4 i d x}}{d} - \frac{22 a^{3} e^{- 4 i c} e^{2 i d x}}{d} - \frac{26 a^{3} e^{- 6 i c}}{3 d}}{e^{6 i d x} + 3 e^{- 2 i c} e^{4 i d x} + 3 e^{- 4 i c} e^{2 i d x} + e^{- 6 i c}} \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))**3,x)

[Out]

-4*a**3*log(exp(2*I*d*x) + exp(-2*I*c))/d + (-16*a**3*exp(-2*I*c)*exp(4*I*d*x)/d - 22*a**3*exp(-4*I*c)*exp(2*I
*d*x)/d - 26*a**3*exp(-6*I*c)/(3*d))/(exp(6*I*d*x) + 3*exp(-2*I*c)*exp(4*I*d*x) + 3*exp(-4*I*c)*exp(2*I*d*x) +
exp(-6*I*c))

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Giac [B]  time = 1.38674, size = 230, normalized size = 2.71 \begin{align*} -\frac{2 \,{\left (6 \, a^{3} e^{\left (6 i \, d x + 6 i \, c\right )} \log \left (e^{\left (2 i \, d x + 2 i \, c\right )} + 1\right ) + 18 \, a^{3} e^{\left (4 i \, d x + 4 i \, c\right )} \log \left (e^{\left (2 i \, d x + 2 i \, c\right )} + 1\right ) + 18 \, a^{3} e^{\left (2 i \, d x + 2 i \, c\right )} \log \left (e^{\left (2 i \, d x + 2 i \, c\right )} + 1\right ) + 24 \, a^{3} e^{\left (4 i \, d x + 4 i \, c\right )} + 33 \, a^{3} e^{\left (2 i \, d x + 2 i \, c\right )} + 6 \, a^{3} \log \left (e^{\left (2 i \, d x + 2 i \, c\right )} + 1\right ) + 13 \, a^{3}\right )}}{3 \,{\left (d e^{\left (6 i \, d x + 6 i \, c\right )} + 3 \, d e^{\left (4 i \, d x + 4 i \, c\right )} + 3 \, 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))^3,x, algorithm="giac")

[Out]

-2/3*(6*a^3*e^(6*I*d*x + 6*I*c)*log(e^(2*I*d*x + 2*I*c) + 1) + 18*a^3*e^(4*I*d*x + 4*I*c)*log(e^(2*I*d*x + 2*I
*c) + 1) + 18*a^3*e^(2*I*d*x + 2*I*c)*log(e^(2*I*d*x + 2*I*c) + 1) + 24*a^3*e^(4*I*d*x + 4*I*c) + 33*a^3*e^(2*
I*d*x + 2*I*c) + 6*a^3*log(e^(2*I*d*x + 2*I*c) + 1) + 13*a^3)/(d*e^(6*I*d*x + 6*I*c) + 3*d*e^(4*I*d*x + 4*I*c)
+ 3*d*e^(2*I*d*x + 2*I*c) + d)