3.71 \(\int \frac{\tan (c+d x)}{(a+i a \tan (c+d x))^3} \, dx\)

Optimal. Leaf size=84 \[ \frac{1}{8 d \left (a^3+i a^3 \tan (c+d x)\right )}-\frac{i x}{8 a^3}+\frac{1}{8 a d (a+i a \tan (c+d x))^2}-\frac{1}{6 d (a+i a \tan (c+d x))^3} \]

[Out]

((-I/8)*x)/a^3 - 1/(6*d*(a + I*a*Tan[c + d*x])^3) + 1/(8*a*d*(a + I*a*Tan[c + d*x])^2) + 1/(8*d*(a^3 + I*a^3*T
an[c + d*x]))

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Rubi [A]  time = 0.0600328, antiderivative size = 84, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.136, Rules used = {3526, 3479, 8} \[ \frac{1}{8 d \left (a^3+i a^3 \tan (c+d x)\right )}-\frac{i x}{8 a^3}+\frac{1}{8 a d (a+i a \tan (c+d x))^2}-\frac{1}{6 d (a+i a \tan (c+d x))^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((-I/8)*x)/a^3 - 1/(6*d*(a + I*a*Tan[c + d*x])^3) + 1/(8*a*d*(a + I*a*Tan[c + d*x])^2) + 1/(8*d*(a^3 + I*a^3*T
an[c + d*x]))

Rule 3526

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)/(2*a*f*m), x] + Dist[(b*c + a*d)/(2*a*b), Int[(a + b*Tan[e + f*x])^(m + 1),
x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 + b^2, 0] && LtQ[m, 0]

Rule 3479

Int[((a_) + (b_.)*tan[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[(a*(a + b*Tan[c + d*x])^n)/(2*b*d*n), x] +
Dist[1/(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] && LtQ[n
, 0]

Rule 8

Int[a_, x_Symbol] :> Simp[a*x, x] /; FreeQ[a, x]

Rubi steps

\begin{align*} \int \frac{\tan (c+d x)}{(a+i a \tan (c+d x))^3} \, dx &=-\frac{1}{6 d (a+i a \tan (c+d x))^3}-\frac{i \int \frac{1}{(a+i a \tan (c+d x))^2} \, dx}{2 a}\\ &=-\frac{1}{6 d (a+i a \tan (c+d x))^3}+\frac{1}{8 a d (a+i a \tan (c+d x))^2}-\frac{i \int \frac{1}{a+i a \tan (c+d x)} \, dx}{4 a^2}\\ &=-\frac{1}{6 d (a+i a \tan (c+d x))^3}+\frac{1}{8 a d (a+i a \tan (c+d x))^2}+\frac{1}{8 d \left (a^3+i a^3 \tan (c+d x)\right )}-\frac{i \int 1 \, dx}{8 a^3}\\ &=-\frac{i x}{8 a^3}-\frac{1}{6 d (a+i a \tan (c+d x))^3}+\frac{1}{8 a d (a+i a \tan (c+d x))^2}+\frac{1}{8 d \left (a^3+i a^3 \tan (c+d x)\right )}\\ \end{align*}

Mathematica [A]  time = 0.387834, size = 91, normalized size = 1.08 \[ \frac{\sec ^3(c+d x) (-9 \sin (c+d x)+12 i d x \sin (3 (c+d x))-2 \sin (3 (c+d x))+3 i \cos (c+d x)+2 (6 d x-i) \cos (3 (c+d x)))}{96 a^3 d (\tan (c+d x)-i)^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(Sec[c + d*x]^3*((3*I)*Cos[c + d*x] + 2*(-I + 6*d*x)*Cos[3*(c + d*x)] - 9*Sin[c + d*x] - 2*Sin[3*(c + d*x)] +
(12*I)*d*x*Sin[3*(c + d*x)]))/(96*a^3*d*(-I + Tan[c + d*x])^3)

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Maple [A]  time = 0.024, size = 97, normalized size = 1.2 \begin{align*}{\frac{-{\frac{i}{6}}}{d{a}^{3} \left ( \tan \left ( dx+c \right ) -i \right ) ^{3}}}-{\frac{{\frac{i}{8}}}{d{a}^{3} \left ( \tan \left ( dx+c \right ) -i \right ) }}-{\frac{1}{8\,d{a}^{3} \left ( \tan \left ( dx+c \right ) -i \right ) ^{2}}}-{\frac{\ln \left ( \tan \left ( dx+c \right ) -i \right ) }{16\,d{a}^{3}}}+{\frac{\ln \left ( \tan \left ( dx+c \right ) +i \right ) }{16\,d{a}^{3}}} \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]

-1/6*I/d/a^3/(tan(d*x+c)-I)^3-1/8*I/d/a^3/(tan(d*x+c)-I)-1/8/d/a^3/(tan(d*x+c)-I)^2-1/16/d/a^3*ln(tan(d*x+c)-I
)+1/16/d/a^3*ln(tan(d*x+c)+I)

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: RuntimeError} \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]

Exception raised: RuntimeError

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Fricas [A]  time = 2.22399, size = 161, normalized size = 1.92 \begin{align*} \frac{{\left (-12 i \, d x e^{\left (6 i \, d x + 6 i \, c\right )} + 6 \, e^{\left (4 i \, d x + 4 i \, c\right )} - 3 \, e^{\left (2 i \, d x + 2 i \, c\right )} - 2\right )} e^{\left (-6 i \, d x - 6 i \, c\right )}}{96 \, a^{3} 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="fricas")

[Out]

1/96*(-12*I*d*x*e^(6*I*d*x + 6*I*c) + 6*e^(4*I*d*x + 4*I*c) - 3*e^(2*I*d*x + 2*I*c) - 2)*e^(-6*I*d*x - 6*I*c)/
(a^3*d)

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Sympy [A]  time = 0.861223, size = 155, normalized size = 1.85 \begin{align*} \begin{cases} \frac{\left (1536 a^{6} d^{2} e^{10 i c} e^{- 2 i d x} - 768 a^{6} d^{2} e^{8 i c} e^{- 4 i d x} - 512 a^{6} d^{2} e^{6 i c} e^{- 6 i d x}\right ) e^{- 12 i c}}{24576 a^{9} d^{3}} & \text{for}\: 24576 a^{9} d^{3} e^{12 i c} \neq 0 \\x \left (- \frac{\left (i e^{6 i c} + i e^{4 i c} - i e^{2 i c} - i\right ) e^{- 6 i c}}{8 a^{3}} + \frac{i}{8 a^{3}}\right ) & \text{otherwise} \end{cases} - \frac{i x}{8 a^{3}} \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]

Piecewise(((1536*a**6*d**2*exp(10*I*c)*exp(-2*I*d*x) - 768*a**6*d**2*exp(8*I*c)*exp(-4*I*d*x) - 512*a**6*d**2*
exp(6*I*c)*exp(-6*I*d*x))*exp(-12*I*c)/(24576*a**9*d**3), Ne(24576*a**9*d**3*exp(12*I*c), 0)), (x*(-(I*exp(6*I
*c) + I*exp(4*I*c) - I*exp(2*I*c) - I)*exp(-6*I*c)/(8*a**3) + I/(8*a**3)), True)) - I*x/(8*a**3)

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Giac [A]  time = 1.3192, size = 109, normalized size = 1.3 \begin{align*} -\frac{\frac{6 \, \log \left (\tan \left (d x + c\right ) - i\right )}{a^{3}} - \frac{6 \, \log \left (i \, \tan \left (d x + c\right ) - 1\right )}{a^{3}} - \frac{11 \, \tan \left (d x + c\right )^{3} - 45 i \, \tan \left (d x + c\right )^{2} - 69 \, \tan \left (d x + c\right ) + 19 i}{a^{3}{\left (\tan \left (d x + c\right ) - i\right )}^{3}}}{96 \, 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="giac")

[Out]

-1/96*(6*log(tan(d*x + c) - I)/a^3 - 6*log(I*tan(d*x + c) - 1)/a^3 - (11*tan(d*x + c)^3 - 45*I*tan(d*x + c)^2
- 69*tan(d*x + c) + 19*I)/(a^3*(tan(d*x + c) - I)^3))/d