3.1081 \(\int \frac{(c+d \tan (e+f x))^3}{(a+i a \tan (e+f x))^2} \, dx\)

Optimal. Leaf size=136 \[ \frac{x \left (-3 i c^2 d+c^3-3 c d^2-3 i d^3\right )}{4 a^2}+\frac{(c+i d)^2 (3 d+i c)}{4 a^2 f (1+i \tan (e+f x))}+\frac{d^3 \log (\cos (e+f x))}{a^2 f}+\frac{(-d+i c) (c+d \tan (e+f x))^2}{4 f (a+i a \tan (e+f x))^2} \]

[Out]

((c^3 - (3*I)*c^2*d - 3*c*d^2 - (3*I)*d^3)*x)/(4*a^2) + (d^3*Log[Cos[e + f*x]])/(a^2*f) + ((c + I*d)^2*(I*c +
3*d))/(4*a^2*f*(1 + I*Tan[e + f*x])) + ((I*c - d)*(c + d*Tan[e + f*x])^2)/(4*f*(a + I*a*Tan[e + f*x])^2)

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Rubi [A]  time = 0.302602, antiderivative size = 136, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.179, Rules used = {3558, 3589, 3475, 3526, 8} \[ \frac{x \left (-3 i c^2 d+c^3-3 c d^2-3 i d^3\right )}{4 a^2}+\frac{(c+i d)^2 (3 d+i c)}{4 a^2 f (1+i \tan (e+f x))}+\frac{d^3 \log (\cos (e+f x))}{a^2 f}+\frac{(-d+i c) (c+d \tan (e+f x))^2}{4 f (a+i a \tan (e+f x))^2} \]

Antiderivative was successfully verified.

[In]

Int[(c + d*Tan[e + f*x])^3/(a + I*a*Tan[e + f*x])^2,x]

[Out]

((c^3 - (3*I)*c^2*d - 3*c*d^2 - (3*I)*d^3)*x)/(4*a^2) + (d^3*Log[Cos[e + f*x]])/(a^2*f) + ((c + I*d)^2*(I*c +
3*d))/(4*a^2*f*(1 + I*Tan[e + f*x])) + ((I*c - d)*(c + d*Tan[e + f*x])^2)/(4*f*(a + I*a*Tan[e + f*x])^2)

Rule 3558

Int[((a_) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> -Si
mp[((b*c - a*d)*(a + b*Tan[e + f*x])^m*(c + d*Tan[e + f*x])^(n - 1))/(2*a*f*m), x] + Dist[1/(2*a^2*m), Int[(a
+ b*Tan[e + f*x])^(m + 1)*(c + d*Tan[e + f*x])^(n - 2)*Simp[c*(a*c*m + b*d*(n - 1)) - d*(b*c*m + a*d*(n - 1))
- d*(b*d*(m - n + 1) - a*c*(m + n - 1))*Tan[e + f*x], x], x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c -
a*d, 0] && EqQ[a^2 + b^2, 0] && NeQ[c^2 + d^2, 0] && LtQ[m, 0] && GtQ[n, 1] && (IntegerQ[m] || IntegersQ[2*m,
2*n])

Rule 3589

Int[(((A_.) + (B_.)*tan[(e_.) + (f_.)*(x_)])*((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)]))/((a_.) + (b_.)*tan[(e_.)
+ (f_.)*(x_)]), x_Symbol] :> Dist[(B*d)/b, Int[Tan[e + f*x], x], x] + Dist[1/b, Int[Simp[A*b*c + (A*b*d + B*(
b*c - a*d))*Tan[e + f*x], x]/(a + b*Tan[e + f*x]), x], x] /; FreeQ[{a, b, c, d, e, f, A, B}, x] && NeQ[b*c - a
*d, 0]

Rule 3475

Int[tan[(c_.) + (d_.)*(x_)], x_Symbol] :> -Simp[Log[RemoveContent[Cos[c + d*x], x]]/d, x] /; FreeQ[{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 8

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

Rubi steps

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

Mathematica [B]  time = 2.14423, size = 305, normalized size = 2.24 \[ -\frac{\sec ^2(e+f x) \left (\cos (2 (e+f x)) \left (3 c^2 d (-1-4 i f x)+c^3 (4 f x+i)-3 c d^2 (4 f x+i)+8 d^3 \log \left (\cos ^2(e+f x)\right )+d^3 (1+4 i f x)\right )+3 i c^2 d \sin (2 (e+f x))+12 c^2 d f x \sin (2 (e+f x))+4 i c^3 f x \sin (2 (e+f x))+c^3 \sin (2 (e+f x))+4 i c^3-3 c d^2 \sin (2 (e+f x))-12 i c d^2 f x \sin (2 (e+f x))+12 i c d^2-i d^3 \sin (2 (e+f x))-4 d^3 f x \sin (2 (e+f x))+8 i d^3 \sin (2 (e+f x)) \log \left (\cos ^2(e+f x)\right )+16 d^3 \tan ^{-1}(\tan (f x)) (\sin (2 (e+f x))-i \cos (2 (e+f x)))-8 d^3\right )}{16 a^2 f (\tan (e+f x)-i)^2} \]

Antiderivative was successfully verified.

[In]

Integrate[(c + d*Tan[e + f*x])^3/(a + I*a*Tan[e + f*x])^2,x]

[Out]

-(Sec[e + f*x]^2*((4*I)*c^3 + (12*I)*c*d^2 - 8*d^3 + Cos[2*(e + f*x)]*(3*c^2*d*(-1 - (4*I)*f*x) + d^3*(1 + (4*
I)*f*x) + c^3*(I + 4*f*x) - 3*c*d^2*(I + 4*f*x) + 8*d^3*Log[Cos[e + f*x]^2]) + c^3*Sin[2*(e + f*x)] + (3*I)*c^
2*d*Sin[2*(e + f*x)] - 3*c*d^2*Sin[2*(e + f*x)] - I*d^3*Sin[2*(e + f*x)] + (4*I)*c^3*f*x*Sin[2*(e + f*x)] + 12
*c^2*d*f*x*Sin[2*(e + f*x)] - (12*I)*c*d^2*f*x*Sin[2*(e + f*x)] - 4*d^3*f*x*Sin[2*(e + f*x)] + (8*I)*d^3*Log[C
os[e + f*x]^2]*Sin[2*(e + f*x)] + 16*d^3*ArcTan[Tan[f*x]]*((-I)*Cos[2*(e + f*x)] + Sin[2*(e + f*x)])))/(16*a^2
*f*(-I + Tan[e + f*x])^2)

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Maple [B]  time = 0.034, size = 362, normalized size = 2.7 \begin{align*}{\frac{3\,{c}^{2}d}{4\,f{a}^{2} \left ( \tan \left ( fx+e \right ) -i \right ) ^{2}}}-{\frac{{d}^{3}}{4\,f{a}^{2} \left ( \tan \left ( fx+e \right ) -i \right ) ^{2}}}-{\frac{{\frac{3\,i}{8}}\ln \left ( \tan \left ( fx+e \right ) +i \right ) c{d}^{2}}{f{a}^{2}}}-{\frac{{\frac{i}{4}}{c}^{3}}{f{a}^{2} \left ( \tan \left ( fx+e \right ) -i \right ) ^{2}}}+{\frac{{\frac{3\,i}{8}}\ln \left ( \tan \left ( fx+e \right ) -i \right ) c{d}^{2}}{f{a}^{2}}}-{\frac{{\frac{i}{8}}\ln \left ( \tan \left ( fx+e \right ) -i \right ){c}^{3}}{f{a}^{2}}}-{\frac{3\,\ln \left ( \tan \left ( fx+e \right ) -i \right ){c}^{2}d}{8\,f{a}^{2}}}-{\frac{7\,\ln \left ( \tan \left ( fx+e \right ) -i \right ){d}^{3}}{8\,f{a}^{2}}}-{\frac{{\frac{3\,i}{4}}{c}^{2}d}{f{a}^{2} \left ( \tan \left ( fx+e \right ) -i \right ) }}+{\frac{{\frac{3\,i}{4}}c{d}^{2}}{f{a}^{2} \left ( \tan \left ( fx+e \right ) -i \right ) ^{2}}}+{\frac{{c}^{3}}{4\,f{a}^{2} \left ( \tan \left ( fx+e \right ) -i \right ) }}+{\frac{9\,c{d}^{2}}{4\,f{a}^{2} \left ( \tan \left ( fx+e \right ) -i \right ) }}+{\frac{3\,\ln \left ( \tan \left ( fx+e \right ) +i \right ){c}^{2}d}{8\,f{a}^{2}}}-{\frac{\ln \left ( \tan \left ( fx+e \right ) +i \right ){d}^{3}}{8\,f{a}^{2}}}+{\frac{{\frac{5\,i}{4}}{d}^{3}}{f{a}^{2} \left ( \tan \left ( fx+e \right ) -i \right ) }}+{\frac{{\frac{i}{8}}\ln \left ( \tan \left ( fx+e \right ) +i \right ){c}^{3}}{f{a}^{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c+d*tan(f*x+e))^3/(a+I*a*tan(f*x+e))^2,x)

[Out]

3/4/f/a^2/(tan(f*x+e)-I)^2*c^2*d-1/4/f/a^2/(tan(f*x+e)-I)^2*d^3-3/8*I/f/a^2*ln(tan(f*x+e)+I)*c*d^2-1/4*I/f/a^2
/(tan(f*x+e)-I)^2*c^3+3/8*I/f/a^2*ln(tan(f*x+e)-I)*c*d^2-1/8*I/f/a^2*ln(tan(f*x+e)-I)*c^3-3/8/f/a^2*ln(tan(f*x
+e)-I)*c^2*d-7/8/f/a^2*ln(tan(f*x+e)-I)*d^3-3/4*I/f/a^2/(tan(f*x+e)-I)*c^2*d+3/4*I/f/a^2/(tan(f*x+e)-I)^2*c*d^
2+1/4/f*c^3/a^2/(tan(f*x+e)-I)+9/4/f/a^2/(tan(f*x+e)-I)*c*d^2+3/8/f/a^2*ln(tan(f*x+e)+I)*c^2*d-1/8/f/a^2*ln(ta
n(f*x+e)+I)*d^3+5/4*I/f/a^2/(tan(f*x+e)-I)*d^3+1/8*I/f/a^2*ln(tan(f*x+e)+I)*c^3

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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((c+d*tan(f*x+e))^3/(a+I*a*tan(f*x+e))^2,x, algorithm="maxima")

[Out]

Exception raised: RuntimeError

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Fricas [A]  time = 1.64889, size = 339, normalized size = 2.49 \begin{align*} \frac{{\left (16 \, d^{3} e^{\left (4 i \, f x + 4 i \, e\right )} \log \left (e^{\left (2 i \, f x + 2 i \, e\right )} + 1\right ) +{\left (4 \, c^{3} - 12 i \, c^{2} d - 12 \, c d^{2} - 28 i \, d^{3}\right )} f x e^{\left (4 i \, f x + 4 i \, e\right )} + i \, c^{3} - 3 \, c^{2} d - 3 i \, c d^{2} + d^{3} +{\left (4 i \, c^{3} + 12 i \, c d^{2} - 8 \, d^{3}\right )} e^{\left (2 i \, f x + 2 i \, e\right )}\right )} e^{\left (-4 i \, f x - 4 i \, e\right )}}{16 \, a^{2} f} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c+d*tan(f*x+e))^3/(a+I*a*tan(f*x+e))^2,x, algorithm="fricas")

[Out]

1/16*(16*d^3*e^(4*I*f*x + 4*I*e)*log(e^(2*I*f*x + 2*I*e) + 1) + (4*c^3 - 12*I*c^2*d - 12*c*d^2 - 28*I*d^3)*f*x
*e^(4*I*f*x + 4*I*e) + I*c^3 - 3*c^2*d - 3*I*c*d^2 + d^3 + (4*I*c^3 + 12*I*c*d^2 - 8*d^3)*e^(2*I*f*x + 2*I*e))
*e^(-4*I*f*x - 4*I*e)/(a^2*f)

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Sympy [A]  time = 3.51069, size = 394, normalized size = 2.9 \begin{align*} \begin{cases} \frac{\left (\left (16 i a^{2} c^{3} f e^{4 i e} + 48 i a^{2} c d^{2} f e^{4 i e} - 32 a^{2} d^{3} f e^{4 i e}\right ) e^{- 2 i f x} + \left (4 i a^{2} c^{3} f e^{2 i e} - 12 a^{2} c^{2} d f e^{2 i e} - 12 i a^{2} c d^{2} f e^{2 i e} + 4 a^{2} d^{3} f e^{2 i e}\right ) e^{- 4 i f x}\right ) e^{- 6 i e}}{64 a^{4} f^{2}} & \text{for}\: 64 a^{4} f^{2} e^{6 i e} \neq 0 \\x \left (- \frac{c^{3} - 3 i c^{2} d - 3 c d^{2} - 7 i d^{3}}{4 a^{2}} + \frac{\left (c^{3} e^{4 i e} + 2 c^{3} e^{2 i e} + c^{3} - 3 i c^{2} d e^{4 i e} + 3 i c^{2} d - 3 c d^{2} e^{4 i e} + 6 c d^{2} e^{2 i e} - 3 c d^{2} - 7 i d^{3} e^{4 i e} + 4 i d^{3} e^{2 i e} - i d^{3}\right ) e^{- 4 i e}}{4 a^{2}}\right ) & \text{otherwise} \end{cases} + \frac{d^{3} \log{\left (e^{2 i f x} + e^{- 2 i e} \right )}}{a^{2} f} + \frac{x \left (c^{3} - 3 i c^{2} d - 3 c d^{2} - 7 i d^{3}\right )}{4 a^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c+d*tan(f*x+e))**3/(a+I*a*tan(f*x+e))**2,x)

[Out]

Piecewise((((16*I*a**2*c**3*f*exp(4*I*e) + 48*I*a**2*c*d**2*f*exp(4*I*e) - 32*a**2*d**3*f*exp(4*I*e))*exp(-2*I
*f*x) + (4*I*a**2*c**3*f*exp(2*I*e) - 12*a**2*c**2*d*f*exp(2*I*e) - 12*I*a**2*c*d**2*f*exp(2*I*e) + 4*a**2*d**
3*f*exp(2*I*e))*exp(-4*I*f*x))*exp(-6*I*e)/(64*a**4*f**2), Ne(64*a**4*f**2*exp(6*I*e), 0)), (x*(-(c**3 - 3*I*c
**2*d - 3*c*d**2 - 7*I*d**3)/(4*a**2) + (c**3*exp(4*I*e) + 2*c**3*exp(2*I*e) + c**3 - 3*I*c**2*d*exp(4*I*e) +
3*I*c**2*d - 3*c*d**2*exp(4*I*e) + 6*c*d**2*exp(2*I*e) - 3*c*d**2 - 7*I*d**3*exp(4*I*e) + 4*I*d**3*exp(2*I*e)
- I*d**3)*exp(-4*I*e)/(4*a**2)), True)) + d**3*log(exp(2*I*f*x) + exp(-2*I*e))/(a**2*f) + x*(c**3 - 3*I*c**2*d
- 3*c*d**2 - 7*I*d**3)/(4*a**2)

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Giac [A]  time = 1.9154, size = 305, normalized size = 2.24 \begin{align*} -\frac{\frac{2 \,{\left (-i \, c^{3} - 3 \, c^{2} d + 3 i \, c d^{2} + d^{3}\right )} \log \left (\tan \left (f x + e\right ) + i\right )}{a^{2}} + \frac{2 \,{\left (i \, c^{3} + 3 \, c^{2} d - 3 i \, c d^{2} + 7 \, d^{3}\right )} \log \left (\tan \left (f x + e\right ) - i\right )}{a^{2}} + \frac{-3 i \, c^{3} \tan \left (f x + e\right )^{2} - 9 \, c^{2} d \tan \left (f x + e\right )^{2} + 9 i \, c d^{2} \tan \left (f x + e\right )^{2} - 21 \, d^{3} \tan \left (f x + e\right )^{2} - 10 \, c^{3} \tan \left (f x + e\right ) + 30 i \, c^{2} d \tan \left (f x + e\right ) - 18 \, c d^{2} \tan \left (f x + e\right ) + 22 i \, d^{3} \tan \left (f x + e\right ) + 11 i \, c^{3} + 9 \, c^{2} d + 15 i \, c d^{2} + 5 \, d^{3}}{a^{2}{\left (\tan \left (f x + e\right ) - i\right )}^{2}}}{16 \, f} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c+d*tan(f*x+e))^3/(a+I*a*tan(f*x+e))^2,x, algorithm="giac")

[Out]

-1/16*(2*(-I*c^3 - 3*c^2*d + 3*I*c*d^2 + d^3)*log(tan(f*x + e) + I)/a^2 + 2*(I*c^3 + 3*c^2*d - 3*I*c*d^2 + 7*d
^3)*log(tan(f*x + e) - I)/a^2 + (-3*I*c^3*tan(f*x + e)^2 - 9*c^2*d*tan(f*x + e)^2 + 9*I*c*d^2*tan(f*x + e)^2 -
21*d^3*tan(f*x + e)^2 - 10*c^3*tan(f*x + e) + 30*I*c^2*d*tan(f*x + e) - 18*c*d^2*tan(f*x + e) + 22*I*d^3*tan(
f*x + e) + 11*I*c^3 + 9*c^2*d + 15*I*c*d^2 + 5*d^3)/(a^2*(tan(f*x + e) - I)^2))/f