3.136 \(\int \frac{\sec ^2(c+d x)}{(a+i a \tan (c+d x))^3} \, dx\)

Optimal. Leaf size=27 \[ \frac{i}{2 a d (a+i a \tan (c+d x))^2} \]

[Out]

(I/2)/(a*d*(a + I*a*Tan[c + d*x])^2)

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Rubi [A]  time = 0.0531279, 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} \[ \frac{i}{2 a d (a+i a \tan (c+d x))^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(I/2)/(a*d*(a + I*a*Tan[c + d*x])^2)

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

Mathematica [A]  time = 0.0702105, size = 42, normalized size = 1.56 \[ -\frac{i (\tan (c+d x)-3 i) \sec ^2(c+d x)}{8 a^3 d (\tan (c+d x)-i)^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.038, size = 24, normalized size = 0.9 \begin{align*}{\frac{{\frac{i}{2}}}{ad \left ( a+ia\tan \left ( dx+c \right ) \right ) ^{2}}} \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))^3,x)

[Out]

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

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Maxima [A]  time = 0.946087, size = 28, normalized size = 1.04 \begin{align*} \frac{i}{2 \,{\left (i \, a \tan \left (d x + c\right ) + a\right )}^{2} 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))^3,x, algorithm="maxima")

[Out]

1/2*I/((I*a*tan(d*x + c) + a)^2*a*d)

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Fricas [A]  time = 2.22497, size = 86, normalized size = 3.19 \begin{align*} \frac{{\left (2 i \, e^{\left (2 i \, d x + 2 i \, c\right )} + i\right )} e^{\left (-4 i \, d x - 4 i \, c\right )}}{8 \, a^{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))^3,x, algorithm="fricas")

[Out]

1/8*(2*I*e^(2*I*d*x + 2*I*c) + I)*e^(-4*I*d*x - 4*I*c)/(a^3*d)

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Sympy [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: AttributeError} \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))**3,x)

[Out]

Exception raised: AttributeError

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Giac [B]  time = 1.16061, size = 77, normalized size = 2.85 \begin{align*} -\frac{2 \,{\left (\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} - i \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} - \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )\right )}}{a^{3} d{\left (\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) - i\right )}^{4}} \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))^3,x, algorithm="giac")

[Out]

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