3.16 \(\int \cos ^3(c+d x) (a+i a \tan (c+d x)) \, dx\)

Optimal. Leaf size=46 \[ -\frac{a \sin ^3(c+d x)}{3 d}+\frac{a \sin (c+d x)}{d}-\frac{i a \cos ^3(c+d x)}{3 d} \]

[Out]

((-I/3)*a*Cos[c + d*x]^3)/d + (a*Sin[c + d*x])/d - (a*Sin[c + d*x]^3)/(3*d)

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Rubi [A]  time = 0.0323892, antiderivative size = 46, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.091, Rules used = {3486, 2633} \[ -\frac{a \sin ^3(c+d x)}{3 d}+\frac{a \sin (c+d x)}{d}-\frac{i a \cos ^3(c+d x)}{3 d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((-I/3)*a*Cos[c + d*x]^3)/d + (a*Sin[c + d*x])/d - (a*Sin[c + d*x]^3)/(3*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 2633

Int[sin[(c_.) + (d_.)*(x_)]^(n_), x_Symbol] :> -Dist[d^(-1), Subst[Int[Expand[(1 - x^2)^((n - 1)/2), x], x], x
, Cos[c + d*x]], x] /; FreeQ[{c, d}, x] && IGtQ[(n - 1)/2, 0]

Rubi steps

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

Mathematica [A]  time = 0.0095743, size = 46, normalized size = 1. \[ -\frac{a \sin ^3(c+d x)}{3 d}+\frac{a \sin (c+d x)}{d}-\frac{i a \cos ^3(c+d x)}{3 d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((-I/3)*a*Cos[c + d*x]^3)/d + (a*Sin[c + d*x])/d - (a*Sin[c + d*x]^3)/(3*d)

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Maple [A]  time = 0.078, size = 37, normalized size = 0.8 \begin{align*}{\frac{1}{d} \left ( -{\frac{i}{3}}a \left ( \cos \left ( dx+c \right ) \right ) ^{3}+{\frac{a \left ( 2+ \left ( \cos \left ( dx+c \right ) \right ) ^{2} \right ) \sin \left ( dx+c \right ) }{3}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(d*x+c)^3*(a+I*a*tan(d*x+c)),x)

[Out]

1/d*(-1/3*I*a*cos(d*x+c)^3+1/3*a*(2+cos(d*x+c)^2)*sin(d*x+c))

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Maxima [A]  time = 1.12166, size = 49, normalized size = 1.07 \begin{align*} -\frac{i \, a \cos \left (d x + c\right )^{3} +{\left (\sin \left (d x + c\right )^{3} - 3 \, \sin \left (d x + c\right )\right )} a}{3 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^3*(a+I*a*tan(d*x+c)),x, algorithm="maxima")

[Out]

-1/3*(I*a*cos(d*x + c)^3 + (sin(d*x + c)^3 - 3*sin(d*x + c))*a)/d

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Fricas [A]  time = 1.10006, size = 119, normalized size = 2.59 \begin{align*} \frac{{\left (-i \, a e^{\left (4 i \, d x + 4 i \, c\right )} - 6 i \, a e^{\left (2 i \, d x + 2 i \, c\right )} + 3 i \, a\right )} e^{\left (-i \, d x - i \, c\right )}}{12 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^3*(a+I*a*tan(d*x+c)),x, algorithm="fricas")

[Out]

1/12*(-I*a*e^(4*I*d*x + 4*I*c) - 6*I*a*e^(2*I*d*x + 2*I*c) + 3*I*a)*e^(-I*d*x - I*c)/d

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Sympy [A]  time = 0.558693, size = 107, normalized size = 2.33 \begin{align*} \begin{cases} \frac{\left (- 8 i a d^{2} e^{4 i c} e^{3 i d x} - 48 i a d^{2} e^{2 i c} e^{i d x} + 24 i a d^{2} e^{- i d x}\right ) e^{- i c}}{96 d^{3}} & \text{for}\: 96 d^{3} e^{i c} \neq 0 \\\frac{x \left (a e^{4 i c} + 2 a e^{2 i c} + a\right ) e^{- i c}}{4} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)**3*(a+I*a*tan(d*x+c)),x)

[Out]

Piecewise(((-8*I*a*d**2*exp(4*I*c)*exp(3*I*d*x) - 48*I*a*d**2*exp(2*I*c)*exp(I*d*x) + 24*I*a*d**2*exp(-I*d*x))
*exp(-I*c)/(96*d**3), Ne(96*d**3*exp(I*c), 0)), (x*(a*exp(4*I*c) + 2*a*exp(2*I*c) + a)*exp(-I*c)/4, True))

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Giac [B]  time = 1.14382, size = 265, normalized size = 5.76 \begin{align*} -\frac{{\left (9 \, a e^{\left (i \, d x + i \, c\right )} \log \left (i \, e^{\left (i \, d x + i \, c\right )} + 1\right ) + 6 \, a e^{\left (i \, d x + i \, c\right )} \log \left (i \, e^{\left (i \, d x + i \, c\right )} - 1\right ) - 9 \, a e^{\left (i \, d x + i \, c\right )} \log \left (-i \, e^{\left (i \, d x + i \, c\right )} + 1\right ) - 6 \, a e^{\left (i \, d x + i \, c\right )} \log \left (-i \, e^{\left (i \, d x + i \, c\right )} - 1\right ) - 3 \, a e^{\left (i \, d x + i \, c\right )} \log \left (i \, e^{\left (i \, d x\right )} + e^{\left (-i \, c\right )}\right ) + 3 \, a e^{\left (i \, d x + i \, c\right )} \log \left (-i \, e^{\left (i \, d x\right )} + e^{\left (-i \, c\right )}\right ) + 4 i \, a e^{\left (4 i \, d x + 4 i \, c\right )} + 24 i \, a e^{\left (2 i \, d x + 2 i \, c\right )} - 12 i \, a\right )} e^{\left (-i \, d x - i \, c\right )}}{48 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^3*(a+I*a*tan(d*x+c)),x, algorithm="giac")

[Out]

-1/48*(9*a*e^(I*d*x + I*c)*log(I*e^(I*d*x + I*c) + 1) + 6*a*e^(I*d*x + I*c)*log(I*e^(I*d*x + I*c) - 1) - 9*a*e
^(I*d*x + I*c)*log(-I*e^(I*d*x + I*c) + 1) - 6*a*e^(I*d*x + I*c)*log(-I*e^(I*d*x + I*c) - 1) - 3*a*e^(I*d*x +
I*c)*log(I*e^(I*d*x) + e^(-I*c)) + 3*a*e^(I*d*x + I*c)*log(-I*e^(I*d*x) + e^(-I*c)) + 4*I*a*e^(4*I*d*x + 4*I*c
) + 24*I*a*e^(2*I*d*x + 2*I*c) - 12*I*a)*e^(-I*d*x - I*c)/d