3.1.15 \(\int \cos (c+d x) (a+i a \tan (c+d x)) \, dx\) [15]

Optimal. Leaf size=26 \[ -\frac {i a \cos (c+d x)}{d}+\frac {a \sin (c+d x)}{d} \]

[Out]

-I*a*cos(d*x+c)/d+a*sin(d*x+c)/d

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Rubi [A]
time = 0.02, antiderivative size = 26, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {3567, 2717} \begin {gather*} \frac {a \sin (c+d x)}{d}-\frac {i a \cos (c+d x)}{d} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((-I)*a*Cos[c + d*x])/d + (a*Sin[c + d*x])/d

Rule 2717

Int[sin[Pi/2 + (c_.) + (d_.)*(x_)], x_Symbol] :> Simp[Sin[c + d*x]/d, x] /; FreeQ[{c, d}, x]

Rule 3567

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])

Rubi steps

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

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Mathematica [A]
time = 0.03, size = 51, normalized size = 1.96 \begin {gather*} -\frac {i a \cos (c) \cos (d x)}{d}+\frac {a \cos (d x) \sin (c)}{d}+\frac {a \cos (c) \sin (d x)}{d}+\frac {i a \sin (c) \sin (d x)}{d} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((-I)*a*Cos[c]*Cos[d*x])/d + (a*Cos[d*x]*Sin[c])/d + (a*Cos[c]*Sin[d*x])/d + (I*a*Sin[c]*Sin[d*x])/d

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Maple [A]
time = 0.20, size = 24, normalized size = 0.92

method	result	size
risch	\(-\frac {i a \,{\mathrm e}^{i \left (d x +c \right )}}{d}\)	\(17\)
derivativedivides	\(\frac {-i a \cos \left (d x +c \right )+a \sin \left (d x +c \right )}{d}\)	\(24\)
default	\(\frac {-i a \cos \left (d x +c \right )+a \sin \left (d x +c \right )}{d}\)	\(24\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(d*x+c)*(a+I*a*tan(d*x+c)),x,method=_RETURNVERBOSE)

[Out]

1/d*(-I*a*cos(d*x+c)+a*sin(d*x+c))

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Maxima [A]
time = 0.28, size = 22, normalized size = 0.85 \begin {gather*} \frac {-i \, a \cos \left (d x + c\right ) + a \sin \left (d x + c\right )}{d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(-I*a*cos(d*x + c) + a*sin(d*x + c))/d

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Fricas [A]
time = 0.37, size = 15, normalized size = 0.58 \begin {gather*} -\frac {i \, a e^{\left (i \, d x + i \, c\right )}}{d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-I*a*e^(I*d*x + I*c)/d

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Sympy [A]
time = 0.06, size = 26, normalized size = 1.00 \begin {gather*} \begin {cases} - \frac {i a e^{i c} e^{i d x}}{d} & \text {for}\: d \neq 0 \\a x e^{i c} & \text {otherwise} \end {cases} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((-I*a*exp(I*c)*exp(I*d*x)/d, Ne(d, 0)), (a*x*exp(I*c), True))

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Giac [B] Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 84 vs. \(2 (24) = 48\).
time = 0.44, size = 84, normalized size = 3.23 \begin {gather*} -\frac {4 i \, a e^{\left (i \, d x + i \, c\right )} + a \log \left (i \, e^{\left (i \, d x + i \, c\right )} + 1\right ) + a \log \left (i \, e^{\left (i \, d x + i \, c\right )} - 1\right ) - a \log \left (-i \, e^{\left (i \, d x + i \, c\right )} + 1\right ) - a \log \left (-i \, e^{\left (i \, d x + i \, c\right )} - 1\right )}{4 \, d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/4*(4*I*a*e^(I*d*x + I*c) + a*log(I*e^(I*d*x + I*c) + 1) + a*log(I*e^(I*d*x + I*c) - 1) - a*log(-I*e^(I*d*x
+ I*c) + 1) - a*log(-I*e^(I*d*x + I*c) - 1))/d

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Mupad [B]
time = 3.28, size = 20, normalized size = 0.77 \begin {gather*} \frac {2\,a}{d\,\left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )+1{}\mathrm {i}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(c + d*x)*(a + a*tan(c + d*x)*1i),x)

[Out]

(2*a)/(d*(tan(c/2 + (d*x)/2) + 1i))

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