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

Optimal. Leaf size=202 \[ \frac{d (c+d x) e^{-2 i e-2 i f x}}{4 a^2 f^2}+\frac{d (c+d x) e^{-4 i e-4 i f x}}{32 a^2 f^2}+\frac{i (c+d x)^2 e^{-2 i e-2 i f x}}{4 a^2 f}+\frac{i (c+d x)^2 e^{-4 i e-4 i f x}}{16 a^2 f}+\frac{(c+d x)^3}{12 a^2 d}-\frac{i d^2 e^{-2 i e-2 i f x}}{8 a^2 f^3}-\frac{i d^2 e^{-4 i e-4 i f x}}{128 a^2 f^3} \]

[Out]

((-I/8)*d^2*E^((-2*I)*e - (2*I)*f*x))/(a^2*f^3) - ((I/128)*d^2*E^((-4*I)*e - (4*I)*f*x))/(a^2*f^3) + (d*E^((-2
*I)*e - (2*I)*f*x)*(c + d*x))/(4*a^2*f^2) + (d*E^((-4*I)*e - (4*I)*f*x)*(c + d*x))/(32*a^2*f^2) + ((I/4)*E^((-
2*I)*e - (2*I)*f*x)*(c + d*x)^2)/(a^2*f) + ((I/16)*E^((-4*I)*e - (4*I)*f*x)*(c + d*x)^2)/(a^2*f) + (c + d*x)^3
/(12*a^2*d)

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Rubi [A]  time = 0.208059, antiderivative size = 202, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 3, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.13, Rules used = {3729, 2176, 2194} \[ \frac{d (c+d x) e^{-2 i e-2 i f x}}{4 a^2 f^2}+\frac{d (c+d x) e^{-4 i e-4 i f x}}{32 a^2 f^2}+\frac{i (c+d x)^2 e^{-2 i e-2 i f x}}{4 a^2 f}+\frac{i (c+d x)^2 e^{-4 i e-4 i f x}}{16 a^2 f}+\frac{(c+d x)^3}{12 a^2 d}-\frac{i d^2 e^{-2 i e-2 i f x}}{8 a^2 f^3}-\frac{i d^2 e^{-4 i e-4 i f x}}{128 a^2 f^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((-I/8)*d^2*E^((-2*I)*e - (2*I)*f*x))/(a^2*f^3) - ((I/128)*d^2*E^((-4*I)*e - (4*I)*f*x))/(a^2*f^3) + (d*E^((-2
*I)*e - (2*I)*f*x)*(c + d*x))/(4*a^2*f^2) + (d*E^((-4*I)*e - (4*I)*f*x)*(c + d*x))/(32*a^2*f^2) + ((I/4)*E^((-
2*I)*e - (2*I)*f*x)*(c + d*x)^2)/(a^2*f) + ((I/16)*E^((-4*I)*e - (4*I)*f*x)*(c + d*x)^2)/(a^2*f) + (c + d*x)^3
/(12*a^2*d)

Rule 3729

Int[((c_.) + (d_.)*(x_))^(m_)*((a_) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Int[ExpandIntegrand[(c
+ d*x)^m, (1/(2*a) + E^((2*a*(e + f*x))/b)/(2*a))^(-n), x], x] /; FreeQ[{a, b, c, d, e, f, m}, x] && EqQ[a^2
+ b^2, 0] && ILtQ[n, 0]

Rule 2176

Int[((b_.)*(F_)^((g_.)*((e_.) + (f_.)*(x_))))^(n_.)*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> Simp[((c + d*x)^m
*(b*F^(g*(e + f*x)))^n)/(f*g*n*Log[F]), x] - Dist[(d*m)/(f*g*n*Log[F]), Int[(c + d*x)^(m - 1)*(b*F^(g*(e + f*x
)))^n, x], x] /; FreeQ[{F, b, c, d, e, f, g, n}, x] && GtQ[m, 0] && IntegerQ[2*m] &&  !$UseGamma === True

Rule 2194

Int[((F_)^((c_.)*((a_.) + (b_.)*(x_))))^(n_.), x_Symbol] :> Simp[(F^(c*(a + b*x)))^n/(b*c*n*Log[F]), x] /; Fre
eQ[{F, a, b, c, n}, x]

Rubi steps

\begin{align*} \int \frac{(c+d x)^2}{(a+i a \tan (e+f x))^2} \, dx &=\int \left (\frac{(c+d x)^2}{4 a^2}+\frac{e^{-2 i e-2 i f x} (c+d x)^2}{2 a^2}+\frac{e^{-4 i e-4 i f x} (c+d x)^2}{4 a^2}\right ) \, dx\\ &=\frac{(c+d x)^3}{12 a^2 d}+\frac{\int e^{-4 i e-4 i f x} (c+d x)^2 \, dx}{4 a^2}+\frac{\int e^{-2 i e-2 i f x} (c+d x)^2 \, dx}{2 a^2}\\ &=\frac{i e^{-2 i e-2 i f x} (c+d x)^2}{4 a^2 f}+\frac{i e^{-4 i e-4 i f x} (c+d x)^2}{16 a^2 f}+\frac{(c+d x)^3}{12 a^2 d}-\frac{(i d) \int e^{-4 i e-4 i f x} (c+d x) \, dx}{8 a^2 f}-\frac{(i d) \int e^{-2 i e-2 i f x} (c+d x) \, dx}{2 a^2 f}\\ &=\frac{d e^{-2 i e-2 i f x} (c+d x)}{4 a^2 f^2}+\frac{d e^{-4 i e-4 i f x} (c+d x)}{32 a^2 f^2}+\frac{i e^{-2 i e-2 i f x} (c+d x)^2}{4 a^2 f}+\frac{i e^{-4 i e-4 i f x} (c+d x)^2}{16 a^2 f}+\frac{(c+d x)^3}{12 a^2 d}-\frac{d^2 \int e^{-4 i e-4 i f x} \, dx}{32 a^2 f^2}-\frac{d^2 \int e^{-2 i e-2 i f x} \, dx}{4 a^2 f^2}\\ &=-\frac{i d^2 e^{-2 i e-2 i f x}}{8 a^2 f^3}-\frac{i d^2 e^{-4 i e-4 i f x}}{128 a^2 f^3}+\frac{d e^{-2 i e-2 i f x} (c+d x)}{4 a^2 f^2}+\frac{d e^{-4 i e-4 i f x} (c+d x)}{32 a^2 f^2}+\frac{i e^{-2 i e-2 i f x} (c+d x)^2}{4 a^2 f}+\frac{i e^{-4 i e-4 i f x} (c+d x)^2}{16 a^2 f}+\frac{(c+d x)^3}{12 a^2 d}\\ \end{align*}

Mathematica [A]  time = 0.668001, size = 282, normalized size = 1.4 \[ \frac{\sec ^2(e+f x) (\cos (f x)+i \sin (f x))^2 \left (\frac{2}{3} f^3 x \left (3 c^2+3 c d x+d^2 x^2\right ) (\cos (2 e)+i \sin (2 e))+\frac{1}{16} (\cos (2 e)-i \sin (2 e)) \cos (4 f x) ((2+2 i) c f+(2+2 i) d f x+d) ((2+2 i) c f+d ((2+2 i) f x-i))-\frac{1}{16} i (\cos (2 e)-i \sin (2 e)) \sin (4 f x) ((2+2 i) c f+(2+2 i) d f x+d) ((2+2 i) c f+d ((2+2 i) f x-i))-i \sin (2 f x) ((1+i) c f+(1+i) d f x+d) ((1+i) c f+d ((1+i) f x-i))+\cos (2 f x) ((1+i) c f+(1+i) d f x+d) ((1+i) c f+d ((1+i) f x-i))\right )}{8 f^3 (a+i a \tan (e+f x))^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.252, size = 162, normalized size = 0.8 \begin{align*}{\frac{{d}^{2}{x}^{3}}{12\,{a}^{2}}}+{\frac{cd{x}^{2}}{4\,{a}^{2}}}+{\frac{{c}^{2}x}{4\,{a}^{2}}}+{\frac{{\frac{i}{8}} \left ( 2\,{d}^{2}{x}^{2}{f}^{2}-2\,i{d}^{2}fx+4\,cd{f}^{2}x-2\,icdf+2\,{c}^{2}{f}^{2}-{d}^{2} \right ){{\rm e}^{-2\,i \left ( fx+e \right ) }}}{{a}^{2}{f}^{3}}}+{\frac{{\frac{i}{128}} \left ( 8\,{d}^{2}{x}^{2}{f}^{2}-4\,i{d}^{2}fx+16\,cd{f}^{2}x-4\,icdf+8\,{c}^{2}{f}^{2}-{d}^{2} \right ){{\rm e}^{-4\,i \left ( fx+e \right ) }}}{{a}^{2}{f}^{3}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/12/a^2*d^2*x^3+1/4/a^2*c*d*x^2+1/4/a^2*c^2*x+1/8*I*(2*d^2*x^2*f^2-2*I*d^2*f*x+4*c*d*f^2*x-2*I*c*d*f+2*c^2*f^
2-d^2)/a^2/f^3*exp(-2*I*(f*x+e))+1/128*I*(8*d^2*x^2*f^2-4*I*d^2*f*x+16*c*d*f^2*x-4*I*c*d*f+8*c^2*f^2-d^2)/a^2/
f^3*exp(-4*I*(f*x+e))

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

[Out]

Exception raised: RuntimeError

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Fricas [A]  time = 1.58838, size = 406, normalized size = 2.01 \begin{align*} \frac{{\left (24 i \, d^{2} f^{2} x^{2} + 24 i \, c^{2} f^{2} + 12 \, c d f - 3 i \, d^{2} +{\left (48 i \, c d f^{2} + 12 \, d^{2} f\right )} x + 32 \,{\left (d^{2} f^{3} x^{3} + 3 \, c d f^{3} x^{2} + 3 \, c^{2} f^{3} x\right )} e^{\left (4 i \, f x + 4 i \, e\right )} +{\left (96 i \, d^{2} f^{2} x^{2} + 96 i \, c^{2} f^{2} + 96 \, c d f - 48 i \, d^{2} +{\left (192 i \, c d f^{2} + 96 \, d^{2} f\right )} x\right )} e^{\left (2 i \, f x + 2 i \, e\right )}\right )} e^{\left (-4 i \, f x - 4 i \, e\right )}}{384 \, a^{2} f^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/384*(24*I*d^2*f^2*x^2 + 24*I*c^2*f^2 + 12*c*d*f - 3*I*d^2 + (48*I*c*d*f^2 + 12*d^2*f)*x + 32*(d^2*f^3*x^3 +
3*c*d*f^3*x^2 + 3*c^2*f^3*x)*e^(4*I*f*x + 4*I*e) + (96*I*d^2*f^2*x^2 + 96*I*c^2*f^2 + 96*c*d*f - 48*I*d^2 + (1
92*I*c*d*f^2 + 96*d^2*f)*x)*e^(2*I*f*x + 2*I*e))*e^(-4*I*f*x - 4*I*e)/(a^2*f^3)

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Sympy [A]  time = 1.2545, size = 420, normalized size = 2.08 \begin{align*} \begin{cases} \frac{\left (\left (64 i a^{10} c^{2} f^{11} e^{14 i e} + 128 i a^{10} c d f^{11} x e^{14 i e} + 32 a^{10} c d f^{10} e^{14 i e} + 64 i a^{10} d^{2} f^{11} x^{2} e^{14 i e} + 32 a^{10} d^{2} f^{10} x e^{14 i e} - 8 i a^{10} d^{2} f^{9} e^{14 i e}\right ) e^{- 4 i f x} + \left (256 i a^{10} c^{2} f^{11} e^{16 i e} + 512 i a^{10} c d f^{11} x e^{16 i e} + 256 a^{10} c d f^{10} e^{16 i e} + 256 i a^{10} d^{2} f^{11} x^{2} e^{16 i e} + 256 a^{10} d^{2} f^{10} x e^{16 i e} - 128 i a^{10} d^{2} f^{9} e^{16 i e}\right ) e^{- 2 i f x}\right ) e^{- 18 i e}}{1024 a^{12} f^{12}} & \text{for}\: 1024 a^{12} f^{12} e^{18 i e} \neq 0 \\\frac{x^{3} \left (2 d^{2} e^{2 i e} + d^{2}\right ) e^{- 4 i e}}{12 a^{2}} + \frac{x^{2} \left (2 c d e^{2 i e} + c d\right ) e^{- 4 i e}}{4 a^{2}} + \frac{x \left (2 c^{2} e^{2 i e} + c^{2}\right ) e^{- 4 i e}}{4 a^{2}} & \text{otherwise} \end{cases} + \frac{c^{2} x}{4 a^{2}} + \frac{c d x^{2}}{4 a^{2}} + \frac{d^{2} x^{3}}{12 a^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((((64*I*a**10*c**2*f**11*exp(14*I*e) + 128*I*a**10*c*d*f**11*x*exp(14*I*e) + 32*a**10*c*d*f**10*exp(
14*I*e) + 64*I*a**10*d**2*f**11*x**2*exp(14*I*e) + 32*a**10*d**2*f**10*x*exp(14*I*e) - 8*I*a**10*d**2*f**9*exp
(14*I*e))*exp(-4*I*f*x) + (256*I*a**10*c**2*f**11*exp(16*I*e) + 512*I*a**10*c*d*f**11*x*exp(16*I*e) + 256*a**1
0*c*d*f**10*exp(16*I*e) + 256*I*a**10*d**2*f**11*x**2*exp(16*I*e) + 256*a**10*d**2*f**10*x*exp(16*I*e) - 128*I
*a**10*d**2*f**9*exp(16*I*e))*exp(-2*I*f*x))*exp(-18*I*e)/(1024*a**12*f**12), Ne(1024*a**12*f**12*exp(18*I*e),
0)), (x**3*(2*d**2*exp(2*I*e) + d**2)*exp(-4*I*e)/(12*a**2) + x**2*(2*c*d*exp(2*I*e) + c*d)*exp(-4*I*e)/(4*a*
*2) + x*(2*c**2*exp(2*I*e) + c**2)*exp(-4*I*e)/(4*a**2), True)) + c**2*x/(4*a**2) + c*d*x**2/(4*a**2) + d**2*x
**3/(12*a**2)

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Giac [A]  time = 1.1848, size = 306, normalized size = 1.51 \begin{align*} \frac{{\left (32 \, d^{2} f^{3} x^{3} e^{\left (4 i \, f x + 4 i \, e\right )} + 96 \, c d f^{3} x^{2} e^{\left (4 i \, f x + 4 i \, e\right )} + 96 \, c^{2} f^{3} x e^{\left (4 i \, f x + 4 i \, e\right )} + 96 i \, d^{2} f^{2} x^{2} e^{\left (2 i \, f x + 2 i \, e\right )} + 24 i \, d^{2} f^{2} x^{2} + 192 i \, c d f^{2} x e^{\left (2 i \, f x + 2 i \, e\right )} + 48 i \, c d f^{2} x + 96 i \, c^{2} f^{2} e^{\left (2 i \, f x + 2 i \, e\right )} + 96 \, d^{2} f x e^{\left (2 i \, f x + 2 i \, e\right )} + 24 i \, c^{2} f^{2} + 12 \, d^{2} f x + 96 \, c d f e^{\left (2 i \, f x + 2 i \, e\right )} + 12 \, c d f - 48 i \, d^{2} e^{\left (2 i \, f x + 2 i \, e\right )} - 3 i \, d^{2}\right )} e^{\left (-4 i \, f x - 4 i \, e\right )}}{384 \, a^{2} f^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/384*(32*d^2*f^3*x^3*e^(4*I*f*x + 4*I*e) + 96*c*d*f^3*x^2*e^(4*I*f*x + 4*I*e) + 96*c^2*f^3*x*e^(4*I*f*x + 4*I
*e) + 96*I*d^2*f^2*x^2*e^(2*I*f*x + 2*I*e) + 24*I*d^2*f^2*x^2 + 192*I*c*d*f^2*x*e^(2*I*f*x + 2*I*e) + 48*I*c*d
*f^2*x + 96*I*c^2*f^2*e^(2*I*f*x + 2*I*e) + 96*d^2*f*x*e^(2*I*f*x + 2*I*e) + 24*I*c^2*f^2 + 12*d^2*f*x + 96*c*
d*f*e^(2*I*f*x + 2*I*e) + 12*c*d*f - 48*I*d^2*e^(2*I*f*x + 2*I*e) - 3*I*d^2)*e^(-4*I*f*x - 4*I*e)/(a^2*f^3)