\(\int (a \cos (c+d x)+i a \sin (c+d x))^3 \, dx\) [250]

Optimal result

Integrand size = 22, antiderivative size = 31 \[ \int (a \cos (c+d x)+i a \sin (c+d x))^3 \, dx=-\frac {i (a \cos (c+d x)+i a \sin (c+d x))^3}{3 d} \]

[Out]

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

Rubi [A] (verified)

Time = 0.02 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.045, Rules used = {3150} \[ \int (a \cos (c+d x)+i a \sin (c+d x))^3 \, dx=-\frac {i (a \cos (c+d x)+i a \sin (c+d x))^3}{3 d} \]

[In]

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

[Out]

((-1/3*I)*(a*Cos[c + d*x] + I*a*Sin[c + d*x])^3)/d

Rule 3150

Int[(cos[(c_.) + (d_.)*(x_)]*(a_.) + (b_.)*sin[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[a*((a*Cos[c + d*x]
+ b*Sin[c + d*x])^n/(b*d*n)), x] /; FreeQ[{a, b, c, d, n}, x] && EqQ[a^2 + b^2, 0]

Rubi steps \begin{align*} \text {integral}& = -\frac {i (a \cos (c+d x)+i a \sin (c+d x))^3}{3 d} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.27 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.00 \[ \int (a \cos (c+d x)+i a \sin (c+d x))^3 \, dx=-\frac {i (a \cos (c+d x)+i a \sin (c+d x))^3}{3 d} \]

[In]

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

[Out]

((-1/3*I)*(a*Cos[c + d*x] + I*a*Sin[c + d*x])^3)/d

Maple [A] (verified)

Time = 1.93 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.61

method	result	size
risch	\(-\frac {i a^{3} {\mathrm e}^{3 i \left (d x +c \right )}}{3 d}\)	\(19\)
parallelrisch	\(-\frac {a^{3} \left (i \cos \left (3 d x +3 c \right )+i-\sin \left (3 d x +3 c \right )\right )}{3 d}\)	\(35\)
derivativedivides	\(\frac {\frac {i a^{3} \left (2+\sin \left (d x +c \right )^{2}\right ) \cos \left (d x +c \right )}{3}-a^{3} \sin \left (d x +c \right )^{3}-i a^{3} \cos \left (d x +c \right )^{3}+\frac {a^{3} \left (2+\cos \left (d x +c \right )^{2}\right ) \sin \left (d x +c \right )}{3}}{d}\)	\(76\)
default	\(\frac {\frac {i a^{3} \left (2+\sin \left (d x +c \right )^{2}\right ) \cos \left (d x +c \right )}{3}-a^{3} \sin \left (d x +c \right )^{3}-i a^{3} \cos \left (d x +c \right )^{3}+\frac {a^{3} \left (2+\cos \left (d x +c \right )^{2}\right ) \sin \left (d x +c \right )}{3}}{d}\)	\(76\)
parts	\(\frac {a^{3} \left (2+\cos \left (d x +c \right )^{2}\right ) \sin \left (d x +c \right )}{3 d}+\frac {i a^{3} \left (2+\sin \left (d x +c \right )^{2}\right ) \cos \left (d x +c \right )}{3 d}-\frac {i a^{3} \cos \left (d x +c \right )^{3}}{d}-\frac {a^{3} \sin \left (d x +c \right )^{3}}{d}\)	\(84\)
norman	\(\frac {\frac {4 i a^{3} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}{d}-\frac {2 i a^{3}}{3 d}+\frac {2 a^{3} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{d}-\frac {20 a^{3} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{3 d}+\frac {2 a^{3} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}{d}-\frac {6 i a^{3} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}}{d}}{\left (1+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right )^{3}}\)	\(122\)

[In]

int((cos(d*x+c)*a+I*a*sin(d*x+c))^3,x,method=_RETURNVERBOSE)

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.55 \[ \int (a \cos (c+d x)+i a \sin (c+d x))^3 \, dx=-\frac {i \, a^{3} e^{\left (3 i \, d x + 3 i \, c\right )}}{3 \, d} \]

[In]

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

[Out]

-1/3*I*a^3*e^(3*I*d*x + 3*I*c)/d

Sympy [A] (verification not implemented)

Time = 0.08 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.16 \[ \int (a \cos (c+d x)+i a \sin (c+d x))^3 \, dx=\begin {cases} - \frac {i a^{3} e^{3 i c} e^{3 i d x}}{3 d} & \text {for}\: d \neq 0 \\a^{3} x e^{3 i c} & \text {otherwise} \end {cases} \]

[In]

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

[Out]

Piecewise((-I*a**3*exp(3*I*c)*exp(3*I*d*x)/(3*d), Ne(d, 0)), (a**3*x*exp(3*I*c), True))

Maxima [B] (verification not implemented)

Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 83 vs. \(2 (25) = 50\).

Time = 0.23 (sec) , antiderivative size = 83, normalized size of antiderivative = 2.68 \[ \int (a \cos (c+d x)+i a \sin (c+d x))^3 \, dx=-\frac {i \, a^{3} \cos \left (d x + c\right )^{3}}{d} - \frac {a^{3} \sin \left (d x + c\right )^{3}}{d} - \frac {i \, {\left (\cos \left (d x + c\right )^{3} - 3 \, \cos \left (d x + c\right )\right )} a^{3}}{3 \, d} - \frac {{\left (\sin \left (d x + c\right )^{3} - 3 \, \sin \left (d x + c\right )\right )} a^{3}}{3 \, d} \]

[In]

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

[Out]

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

Giac [B] (verification not implemented)

Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 52 vs. \(2 (25) = 50\).

Time = 0.28 (sec) , antiderivative size = 52, normalized size of antiderivative = 1.68 \[ \int (a \cos (c+d x)+i a \sin (c+d x))^3 \, dx=-\frac {i \, a^{3} e^{\left (3 i \, d x + 3 i \, c\right )}}{6 \, d} - \frac {i \, a^{3} e^{\left (-3 i \, d x - 3 i \, c\right )}}{6 \, d} + \frac {a^{3} \sin \left (3 \, d x + 3 \, c\right )}{3 \, d} \]

[In]

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

[Out]

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

Mupad [B] (verification not implemented)

Time = 26.30 (sec) , antiderivative size = 66, normalized size of antiderivative = 2.13 \[ \int (a \cos (c+d x)+i a \sin (c+d x))^3 \, dx=-\frac {2\,a^3\,\left (3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2-1\right )}{3\,d\,\left (-{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3-{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2\,3{}\mathrm {i}+3\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )+1{}\mathrm {i}\right )} \]

[In]

int((a*cos(c + d*x) + a*sin(c + d*x)*1i)^3,x)

[Out]

-(2*a^3*(3*tan(c/2 + (d*x)/2)^2 - 1))/(3*d*(3*tan(c/2 + (d*x)/2) - tan(c/2 + (d*x)/2)^2*3i - tan(c/2 + (d*x)/2
)^3 + 1i))