3.22 \(\int \sec ^2(c+d x) (a+i a \tan (c+d x))^2 \, dx\)

Optimal. Leaf size=27 \[ -\frac{i (a+i a \tan (c+d x))^3}{3 a d} \]

[Out]

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

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Rubi [A]  time = 0.0375375, antiderivative size = 27, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.083, Rules used = {3487, 32} \[ -\frac{i (a+i a \tan (c+d x))^3}{3 a d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 3487

Int[sec[(e_.) + (f_.)*(x_)]^(m_)*((a_) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Dist[1/(a^(m - 2)*b
*f), Subst[Int[(a - x)^(m/2 - 1)*(a + x)^(n + m/2 - 1), x], x, b*Tan[e + f*x]], x] /; FreeQ[{a, b, e, f, n}, x
] && EqQ[a^2 + b^2, 0] && IntegerQ[m/2]

Rule 32

Int[((a_.) + (b_.)*(x_))^(m_), x_Symbol] :> Simp[(a + b*x)^(m + 1)/(b*(m + 1)), x] /; FreeQ[{a, b, m}, x] && N
eQ[m, -1]

Rubi steps

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

Mathematica [B]  time = 0.327802, size = 68, normalized size = 2.52 \[ \frac{a^2 \sec (c) \sec ^3(c+d x) (-3 \sin (2 c+d x)+2 \sin (2 c+3 d x)+3 i \cos (2 c+d x)+3 \sin (d x)+3 i \cos (d x))}{6 d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [B]  time = 0.051, size = 51, normalized size = 1.9 \begin{align*}{\frac{1}{d} \left ( -{\frac{{a}^{2} \left ( \sin \left ( dx+c \right ) \right ) ^{3}}{3\, \left ( \cos \left ( dx+c \right ) \right ) ^{3}}}+{\frac{i{a}^{2}}{ \left ( \cos \left ( dx+c \right ) \right ) ^{2}}}+{a}^{2}\tan \left ( dx+c \right ) \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sec(d*x+c)^2*(a+I*a*tan(d*x+c))^2,x)

[Out]

1/d*(-1/3*a^2*sin(d*x+c)^3/cos(d*x+c)^3+I*a^2/cos(d*x+c)^2+a^2*tan(d*x+c))

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Maxima [A]  time = 1.11372, size = 28, normalized size = 1.04 \begin{align*} -\frac{i \,{\left (i \, a \tan \left (d x + c\right ) + a\right )}^{3}}{3 \, a d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sec(d*x+c)^2*(a+I*a*tan(d*x+c))^2,x, algorithm="maxima")

[Out]

-1/3*I*(I*a*tan(d*x + c) + a)^3/(a*d)

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Fricas [B]  time = 1.12419, size = 212, normalized size = 7.85 \begin{align*} \frac{24 i \, a^{2} e^{\left (4 i \, d x + 4 i \, c\right )} + 24 i \, a^{2} e^{\left (2 i \, d x + 2 i \, c\right )} + 8 i \, a^{2}}{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(sec(d*x+c)^2*(a+I*a*tan(d*x+c))^2,x, algorithm="fricas")

[Out]

1/3*(24*I*a^2*e^(4*I*d*x + 4*I*c) + 24*I*a^2*e^(2*I*d*x + 2*I*c) + 8*I*a^2)/(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 [F]  time = 0., size = 0, normalized size = 0. \begin{align*} a^{2} \left (\int - \tan ^{2}{\left (c + d x \right )} \sec ^{2}{\left (c + d x \right )}\, dx + \int 2 i \tan{\left (c + d x \right )} \sec ^{2}{\left (c + d x \right )}\, dx + \int \sec ^{2}{\left (c + d x \right )}\, dx\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sec(d*x+c)**2*(a+I*a*tan(d*x+c))**2,x)

[Out]

a**2*(Integral(-tan(c + d*x)**2*sec(c + d*x)**2, x) + Integral(2*I*tan(c + d*x)*sec(c + d*x)**2, x) + Integral
(sec(c + d*x)**2, x))

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Giac [B]  time = 1.18351, size = 57, normalized size = 2.11 \begin{align*} -\frac{a^{2} \tan \left (d x + c\right )^{3} - 3 i \, a^{2} \tan \left (d x + c\right )^{2} - 3 \, a^{2} \tan \left (d x + c\right )}{3 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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