3.7 \(\int \cos ^2(c+d x) (a+i a \tan (c+d x)) \, dx\)

Optimal. Leaf size=45 \[ -\frac{i a \cos ^2(c+d x)}{2 d}+\frac{a \sin (c+d x) \cos (c+d x)}{2 d}+\frac{a x}{2} \]

[Out]

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

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Rubi [A]  time = 0.0310949, antiderivative size = 45, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.136, Rules used = {3486, 2635, 8} \[ -\frac{i a \cos ^2(c+d x)}{2 d}+\frac{a \sin (c+d x) \cos (c+d x)}{2 d}+\frac{a x}{2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(a*x)/2 - ((I/2)*a*Cos[c + d*x]^2)/d + (a*Cos[c + d*x]*Sin[c + d*x])/(2*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 2635

Int[((b_.)*sin[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> -Simp[(b*Cos[c + d*x]*(b*Sin[c + d*x])^(n - 1))/(d*n),
x] + Dist[(b^2*(n - 1))/n, Int[(b*Sin[c + d*x])^(n - 2), x], x] /; FreeQ[{b, c, d}, x] && GtQ[n, 1] && Integer
Q[2*n]

Rule 8

Int[a_, x_Symbol] :> Simp[a*x, x] /; FreeQ[a, x]

Rubi steps

\begin{align*} \int \cos ^2(c+d x) (a+i a \tan (c+d x)) \, dx &=-\frac{i a \cos ^2(c+d x)}{2 d}+a \int \cos ^2(c+d x) \, dx\\ &=-\frac{i a \cos ^2(c+d x)}{2 d}+\frac{a \cos (c+d x) \sin (c+d x)}{2 d}+\frac{1}{2} a \int 1 \, dx\\ &=\frac{a x}{2}-\frac{i a \cos ^2(c+d x)}{2 d}+\frac{a \cos (c+d x) \sin (c+d x)}{2 d}\\ \end{align*}

Mathematica [A]  time = 0.0560812, size = 48, normalized size = 1.07 \[ \frac{a (c+d x)}{2 d}+\frac{a \sin (2 (c+d x))}{4 d}-\frac{i a \cos ^2(c+d x)}{2 d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(a*(c + d*x))/(2*d) - ((I/2)*a*Cos[c + d*x]^2)/d + (a*Sin[2*(c + d*x)])/(4*d)

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Maple [A]  time = 0.044, size = 42, normalized size = 0.9 \begin{align*}{\frac{1}{d} \left ( -{\frac{i}{2}}a \left ( \cos \left ( dx+c \right ) \right ) ^{2}+a \left ({\frac{\cos \left ( dx+c \right ) \sin \left ( dx+c \right ) }{2}}+{\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.66952, size = 51, normalized size = 1.13 \begin{align*} \frac{{\left (d x + c\right )} a + \frac{a \tan \left (d x + c\right ) - i \, a}{\tan \left (d x + c\right )^{2} + 1}}{2 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 1.10973, size = 58, normalized size = 1.29 \begin{align*} \frac{2 \, a d x - i \, a e^{\left (2 i \, d x + 2 i \, c\right )}}{4 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]  time = 0.227572, size = 41, normalized size = 0.91 \begin{align*} \frac{a x}{2} + \begin{cases} - \frac{i a e^{2 i c} e^{2 i d x}}{4 d} & \text{for}\: 4 d \neq 0 \\\frac{a x e^{2 i c}}{2} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

a*x/2 + Piecewise((-I*a*exp(2*I*c)*exp(2*I*d*x)/(4*d), Ne(4*d, 0)), (a*x*exp(2*I*c)/2, True))

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Giac [A]  time = 1.12101, size = 31, normalized size = 0.69 \begin{align*} \frac{2 \, a d x - i \, a e^{\left (2 i \, d x + 2 i \, c\right )}}{4 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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