3.14 \(\int \sec (c+d x) (a+i a \tan (c+d x)) \, dx\)

Optimal. Leaf size=27 \[ \frac{a \tanh ^{-1}(\sin (c+d x))}{d}+\frac{i a \sec (c+d x)}{d} \]

[Out]

(a*ArcTanh[Sin[c + d*x]])/d + (I*a*Sec[c + d*x])/d

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Rubi [A]  time = 0.0158266, antiderivative size = 27, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.1, Rules used = {3486, 3770} \[ \frac{a \tanh ^{-1}(\sin (c+d x))}{d}+\frac{i a \sec (c+d x)}{d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(a*ArcTanh[Sin[c + d*x]])/d + (I*a*Sec[c + d*x])/d

Rule 3486

Int[((d_.)*sec[(e_.) + (f_.)*(x_)])^(m_.)*((a_) + (b_.)*tan[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[(b*(d*Sec[
e + f*x])^m)/(f*m), x] + Dist[a, Int[(d*Sec[e + f*x])^m, x], x] /; FreeQ[{a, b, d, e, f, m}, x] && (IntegerQ[2
*m] || NeQ[a^2 + b^2, 0])

Rule 3770

Int[csc[(c_.) + (d_.)*(x_)], x_Symbol] :> -Simp[ArcTanh[Cos[c + d*x]]/d, x] /; FreeQ[{c, d}, x]

Rubi steps

\begin{align*} \int \sec (c+d x) (a+i a \tan (c+d x)) \, dx &=\frac{i a \sec (c+d x)}{d}+a \int \sec (c+d x) \, dx\\ &=\frac{a \tanh ^{-1}(\sin (c+d x))}{d}+\frac{i a \sec (c+d x)}{d}\\ \end{align*}

Mathematica [A]  time = 0.0080077, size = 27, normalized size = 1. \[ \frac{a \tanh ^{-1}(\sin (c+d x))}{d}+\frac{i a \sec (c+d x)}{d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(a*ArcTanh[Sin[c + d*x]])/d + (I*a*Sec[c + d*x])/d

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Maple [A]  time = 0.014, size = 36, normalized size = 1.3 \begin{align*}{\frac{ia}{d\cos \left ( dx+c \right ) }}+{\frac{a\ln \left ( \sec \left ( dx+c \right ) +\tan \left ( dx+c \right ) \right ) }{d}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

I/d*a/cos(d*x+c)+1/d*a*ln(sec(d*x+c)+tan(d*x+c))

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Maxima [A]  time = 1.06567, size = 43, normalized size = 1.59 \begin{align*} \frac{a \log \left (\sec \left (d x + c\right ) + \tan \left (d x + c\right )\right ) + \frac{i \, a}{\cos \left (d x + c\right )}}{d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(a*log(sec(d*x + c) + tan(d*x + c)) + I*a/cos(d*x + c))/d

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Fricas [B]  time = 1.15591, size = 220, normalized size = 8.15 \begin{align*} \frac{2 i \, a e^{\left (i \, d x + i \, c\right )} +{\left (a e^{\left (2 i \, d x + 2 i \, c\right )} + a\right )} \log \left (e^{\left (i \, d x + i \, c\right )} + i\right ) -{\left (a e^{\left (2 i \, d x + 2 i \, c\right )} + a\right )} \log \left (e^{\left (i \, d x + i \, c\right )} - i\right )}{d e^{\left (2 i \, d x + 2 i \, c\right )} + d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(2*I*a*e^(I*d*x + I*c) + (a*e^(2*I*d*x + 2*I*c) + a)*log(e^(I*d*x + I*c) + I) - (a*e^(2*I*d*x + 2*I*c) + a)*lo
g(e^(I*d*x + I*c) - I))/(d*e^(2*I*d*x + 2*I*c) + d)

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Sympy [A]  time = 5.99679, size = 41, normalized size = 1.52 \begin{align*} \begin{cases} \frac{a \log{\left (\tan{\left (c + d x \right )} + \sec{\left (c + d x \right )} \right )} + i a \sec{\left (c + d x \right )}}{d} & \text{for}\: d \neq 0 \\x \left (i a \tan{\left (c \right )} + a\right ) \sec{\left (c \right )} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise(((a*log(tan(c + d*x) + sec(c + d*x)) + I*a*sec(c + d*x))/d, Ne(d, 0)), (x*(I*a*tan(c) + a)*sec(c), T
rue))

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Giac [B]  time = 1.18006, size = 73, normalized size = 2.7 \begin{align*} \frac{a \log \left ({\left | \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 1 \right |}\right ) - a \log \left ({\left | \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) - 1 \right |}\right ) - \frac{2 i \, a}{\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} - 1}}{d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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