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

Optimal. Leaf size=26 \[ \frac{i}{d \left (a^2+i a^2 \tan (c+d x)\right )} \]

[Out]

I/(d*(a^2 + I*a^2*Tan[c + d*x]))

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Rubi [A]  time = 0.0432143, antiderivative size = 26, 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} \[ \frac{i}{d \left (a^2+i a^2 \tan (c+d x)\right )} \]

Antiderivative was successfully verified.

[In]

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

[Out]

I/(d*(a^2 + I*a^2*Tan[c + d*x]))

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

Mathematica [A]  time = 0.0401591, size = 32, normalized size = 1.23 \[ \frac{i \sec ^2(c+d x)}{2 d (a+i a \tan (c+d x))^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.032, size = 24, normalized size = 0.9 \begin{align*}{\frac{i}{ad \left ( a+ia\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]

I/a/d/(a+I*a*tan(d*x+c))

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Maxima [A]  time = 0.955199, size = 28, normalized size = 1.08 \begin{align*} \frac{i}{{\left (i \, a \tan \left (d x + c\right ) + a\right )} 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]

I/((I*a*tan(d*x + c) + a)*a*d)

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Fricas [A]  time = 2.26377, size = 49, normalized size = 1.88 \begin{align*} \frac{i \, e^{\left (-2 i \, d x - 2 i \, c\right )}}{2 \, a^{2} 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="fricas")

[Out]

1/2*I*e^(-2*I*d*x - 2*I*c)/(a^2*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))**2,x)

[Out]

Exception raised: AttributeError

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Giac [A]  time = 1.19236, size = 41, normalized size = 1.58 \begin{align*} -\frac{2 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )}{a^{2} d{\left (\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) - i\right )}^{2}} \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]

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