3.37 \(\int (a+i a \tan (c+d x))^4 \, dx\)

Optimal. Leaf size=89 \[ -\frac{4 a^4 \tan (c+d x)}{d}+\frac{i \left (a^2+i a^2 \tan (c+d x)\right )^2}{d}-\frac{8 i a^4 \log (\cos (c+d x))}{d}+8 a^4 x+\frac{i a (a+i a \tan (c+d x))^3}{3 d} \]

[Out]

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

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Rubi [A]  time = 0.0514113, antiderivative size = 89, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {3478, 3477, 3475} \[ -\frac{4 a^4 \tan (c+d x)}{d}+\frac{i \left (a^2+i a^2 \tan (c+d x)\right )^2}{d}-\frac{8 i a^4 \log (\cos (c+d x))}{d}+8 a^4 x+\frac{i a (a+i a \tan (c+d x))^3}{3 d} \]

Antiderivative was successfully verified.

[In]

Int[(a + I*a*Tan[c + d*x])^4,x]

[Out]

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

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

Mathematica [A]  time = 1.09216, size = 176, normalized size = 1.98 \[ \frac{a^4 \sec (c) \sec ^3(c+d x) \left (12 \sin (2 c+d x)-11 \sin (2 c+3 d x)+6 d x \cos (2 c+3 d x)+6 d x \cos (4 c+3 d x)-3 i \cos (2 c+3 d x) \log \left (\cos ^2(c+d x)\right )+3 \cos (d x) \left (-3 i \log \left (\cos ^2(c+d x)\right )+6 d x-2 i\right )+3 \cos (2 c+d x) \left (-3 i \log \left (\cos ^2(c+d x)\right )+6 d x-2 i\right )-3 i \cos (4 c+3 d x) \log \left (\cos ^2(c+d x)\right )-21 \sin (d x)\right )}{6 d} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + I*a*Tan[c + d*x])^4,x]

[Out]

(a^4*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] + 3*Cos[d*x]*(-2*I + 6*d*x - (3*I)
*Log[Cos[c + d*x]^2]) + 3*Cos[2*c + d*x]*(-2*I + 6*d*x - (3*I)*Log[Cos[c + d*x]^2]) - (3*I)*Cos[2*c + 3*d*x]*L
og[Cos[c + d*x]^2] - (3*I)*Cos[4*c + 3*d*x]*Log[Cos[c + d*x]^2] - 21*Sin[d*x] + 12*Sin[2*c + d*x] - 11*Sin[2*c
+ 3*d*x]))/(6*d)

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Maple [A]  time = 0.003, size = 84, normalized size = 0.9 \begin{align*} -7\,{\frac{{a}^{4}\tan \left ( dx+c \right ) }{d}}+{\frac{{a}^{4} \left ( \tan \left ( dx+c \right ) \right ) ^{3}}{3\,d}}-{\frac{2\,i{a}^{4} \left ( \tan \left ( dx+c \right ) \right ) ^{2}}{d}}+{\frac{4\,i{a}^{4}\ln \left ( 1+ \left ( \tan \left ( dx+c \right ) \right ) ^{2} \right ) }{d}}+8\,{\frac{{a}^{4}\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,x)

[Out]

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

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Maxima [A]  time = 1.68964, size = 146, normalized size = 1.64 \begin{align*} a^{4} x + \frac{{\left (\tan \left (d x + c\right )^{3} + 3 \, d x + 3 \, c - 3 \, \tan \left (d x + c\right )\right )} a^{4}}{3 \, d} + \frac{6 \,{\left (d x + c - \tan \left (d x + c\right )\right )} a^{4}}{d} + \frac{2 i \, a^{4}{\left (\frac{1}{\sin \left (d x + c\right )^{2} - 1} - \log \left (\sin \left (d x + c\right )^{2} - 1\right )\right )}}{d} + \frac{4 i \, a^{4} \log \left (\sec \left (d x + c\right )\right )}{d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

a^4*x + 1/3*(tan(d*x + c)^3 + 3*d*x + 3*c - 3*tan(d*x + c))*a^4/d + 6*(d*x + c - tan(d*x + c))*a^4/d + 2*I*a^4
*(1/(sin(d*x + c)^2 - 1) - log(sin(d*x + c)^2 - 1))/d + 4*I*a^4*log(sec(d*x + c))/d

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Fricas [A]  time = 2.28897, size = 400, normalized size = 4.49 \begin{align*} \frac{-72 i \, a^{4} e^{\left (4 i \, d x + 4 i \, c\right )} - 108 i \, a^{4} e^{\left (2 i \, d x + 2 i \, c\right )} - 44 i \, a^{4} +{\left (-24 i \, a^{4} e^{\left (6 i \, d x + 6 i \, c\right )} - 72 i \, a^{4} e^{\left (4 i \, d x + 4 i \, c\right )} - 72 i \, a^{4} e^{\left (2 i \, d x + 2 i \, c\right )} - 24 i \, a^{4}\right )} \log \left (e^{\left (2 i \, d x + 2 i \, c\right )} + 1\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((a+I*a*tan(d*x+c))^4,x, algorithm="fricas")

[Out]

1/3*(-72*I*a^4*e^(4*I*d*x + 4*I*c) - 108*I*a^4*e^(2*I*d*x + 2*I*c) - 44*I*a^4 + (-24*I*a^4*e^(6*I*d*x + 6*I*c)
- 72*I*a^4*e^(4*I*d*x + 4*I*c) - 72*I*a^4*e^(2*I*d*x + 2*I*c) - 24*I*a^4)*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 = 3.3214, size = 143, normalized size = 1.61 \begin{align*} - \frac{8 i a^{4} \log{\left (e^{2 i d x} + e^{- 2 i c} \right )}}{d} + \frac{- \frac{24 i a^{4} e^{- 2 i c} e^{4 i d x}}{d} - \frac{36 i a^{4} e^{- 4 i c} e^{2 i d x}}{d} - \frac{44 i a^{4} 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((a+I*a*tan(d*x+c))**4,x)

[Out]

-8*I*a**4*log(exp(2*I*d*x) + exp(-2*I*c))/d + (-24*I*a**4*exp(-2*I*c)*exp(4*I*d*x)/d - 36*I*a**4*exp(-4*I*c)*e
xp(2*I*d*x)/d - 44*I*a**4*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 = 1.27527, size = 230, normalized size = 2.58 \begin{align*} \frac{-24 i \, 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 ) - 72 i \, 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 ) - 72 i \, 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^{4} e^{\left (4 i \, d x + 4 i \, c\right )} - 108 i \, a^{4} e^{\left (2 i \, d x + 2 i \, c\right )} - 24 i \, a^{4} \log \left (e^{\left (2 i \, d x + 2 i \, c\right )} + 1\right ) - 44 i \, a^{4}}{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((a+I*a*tan(d*x+c))^4,x, algorithm="giac")

[Out]

1/3*(-24*I*a^4*e^(6*I*d*x + 6*I*c)*log(e^(2*I*d*x + 2*I*c) + 1) - 72*I*a^4*e^(4*I*d*x + 4*I*c)*log(e^(2*I*d*x
+ 2*I*c) + 1) - 72*I*a^4*e^(2*I*d*x + 2*I*c)*log(e^(2*I*d*x + 2*I*c) + 1) - 72*I*a^4*e^(4*I*d*x + 4*I*c) - 108
*I*a^4*e^(2*I*d*x + 2*I*c) - 24*I*a^4*log(e^(2*I*d*x + 2*I*c) + 1) - 44*I*a^4)/(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)