\(\int (a \cos (c+d x)+i a \sin (c+d x))^4 \, dx\) [249]

Optimal result

Integrand size = 22, antiderivative size = 31 \[ \int (a \cos (c+d x)+i a \sin (c+d x))^4 \, dx=-\frac {i (a \cos (c+d x)+i a \sin (c+d x))^4}{4 d} \]

[Out]

-1/4*I*(a*cos(d*x+c)+I*a*sin(d*x+c))^4/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} \[ \int (a \cos (c+d x)+i a \sin (c+d x))^4 \, dx=-\frac {i (a \cos (c+d x)+i a \sin (c+d x))^4}{4 d} \]

[In]

Int[(a*Cos[c + d*x] + I*a*Sin[c + d*x])^4,x]

[Out]

((-1/4*I)*(a*Cos[c + d*x] + I*a*Sin[c + d*x])^4)/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))^4}{4 d} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.52 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.00 \[ \int (a \cos (c+d x)+i a \sin (c+d x))^4 \, dx=-\frac {i (a \cos (c+d x)+i a \sin (c+d x))^4}{4 d} \]

[In]

Integrate[(a*Cos[c + d*x] + I*a*Sin[c + d*x])^4,x]

[Out]

((-1/4*I)*(a*Cos[c + d*x] + I*a*Sin[c + d*x])^4)/d

Maple [A] (verified)

Time = 1.95 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.61

method	result	size
risch	\(-\frac {i a^{4} {\mathrm e}^{4 i \left (d x +c \right )}}{4 d}\)	\(19\)
parallelrisch	\(-\frac {a^{4} \left (i \cos \left (4 d x +4 c \right )-i-\sin \left (4 d x +4 c \right )\right )}{4 d}\)	\(35\)
derivativedivides	\(\frac {a^{4} \left (-\frac {\left (\sin \left (d x +c \right )^{3}+\frac {3 \sin \left (d x +c \right )}{2}\right ) \cos \left (d x +c \right )}{4}+\frac {3 d x}{8}+\frac {3 c}{8}\right )-i a^{4} \sin \left (d x +c \right )^{4}-6 a^{4} \left (-\frac {\sin \left (d x +c \right ) \cos \left (d x +c \right )^{3}}{4}+\frac {\sin \left (d x +c \right ) \cos \left (d x +c \right )}{8}+\frac {d x}{8}+\frac {c}{8}\right )-i a^{4} \cos \left (d x +c \right )^{4}+a^{4} \left (\frac {\left (\cos \left (d x +c \right )^{3}+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{4}+\frac {3 d x}{8}+\frac {3 c}{8}\right )}{d}\)	\(151\)
default	\(\frac {a^{4} \left (-\frac {\left (\sin \left (d x +c \right )^{3}+\frac {3 \sin \left (d x +c \right )}{2}\right ) \cos \left (d x +c \right )}{4}+\frac {3 d x}{8}+\frac {3 c}{8}\right )-i a^{4} \sin \left (d x +c \right )^{4}-6 a^{4} \left (-\frac {\sin \left (d x +c \right ) \cos \left (d x +c \right )^{3}}{4}+\frac {\sin \left (d x +c \right ) \cos \left (d x +c \right )}{8}+\frac {d x}{8}+\frac {c}{8}\right )-i a^{4} \cos \left (d x +c \right )^{4}+a^{4} \left (\frac {\left (\cos \left (d x +c \right )^{3}+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{4}+\frac {3 d x}{8}+\frac {3 c}{8}\right )}{d}\)	\(151\)
norman	\(\frac {\frac {2 a^{4} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{d}-\frac {14 a^{4} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{d}+\frac {14 a^{4} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}{d}-\frac {2 a^{4} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{7}}{d}-\frac {16 i a^{4} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}}{d}+\frac {8 i a^{4} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}{d}+\frac {8 i a^{4} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{6}}{d}}{\left (1+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right )^{4}}\)	\(152\)
parts	\(\frac {a^{4} \left (-\frac {\left (\sin \left (d x +c \right )^{3}+\frac {3 \sin \left (d x +c \right )}{2}\right ) \cos \left (d x +c \right )}{4}+\frac {3 d x}{8}+\frac {3 c}{8}\right )}{d}+\frac {a^{4} \left (\frac {\left (\cos \left (d x +c \right )^{3}+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{4}+\frac {3 d x}{8}+\frac {3 c}{8}\right )}{d}-\frac {i a^{4} \sin \left (d x +c \right )^{4}}{d}-\frac {i a^{4} \cos \left (d x +c \right )^{4}}{d}-\frac {6 a^{4} \left (-\frac {\sin \left (d x +c \right ) \cos \left (d x +c \right )^{3}}{4}+\frac {\sin \left (d x +c \right ) \cos \left (d x +c \right )}{8}+\frac {d x}{8}+\frac {c}{8}\right )}{d}\)	\(162\)

[In]

int((cos(d*x+c)*a+I*a*sin(d*x+c))^4,x,method=_RETURNVERBOSE)

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.55 \[ \int (a \cos (c+d x)+i a \sin (c+d x))^4 \, dx=-\frac {i \, a^{4} e^{\left (4 i \, d x + 4 i \, c\right )}}{4 \, d} \]

[In]

integrate((a*cos(d*x+c)+I*a*sin(d*x+c))^4,x, algorithm="fricas")

[Out]

-1/4*I*a^4*e^(4*I*d*x + 4*I*c)/d

Sympy [A] (verification not implemented)

Time = 0.09 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.16 \[ \int (a \cos (c+d x)+i a \sin (c+d x))^4 \, dx=\begin {cases} - \frac {i a^{4} e^{4 i c} e^{4 i d x}}{4 d} & \text {for}\: d \neq 0 \\a^{4} x e^{4 i c} & \text {otherwise} \end {cases} \]

[In]

integrate((a*cos(d*x+c)+I*a*sin(d*x+c))**4,x)

[Out]

Piecewise((-I*a**4*exp(4*I*c)*exp(4*I*d*x)/(4*d), Ne(d, 0)), (a**4*x*exp(4*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. 132 vs. \(2 (25) = 50\).

Time = 0.23 (sec) , antiderivative size = 132, normalized size of antiderivative = 4.26 \[ \int (a \cos (c+d x)+i a \sin (c+d x))^4 \, dx=-\frac {i \, a^{4} \cos \left (d x + c\right )^{4}}{d} - \frac {i \, a^{4} \sin \left (d x + c\right )^{4}}{d} + \frac {{\left (12 \, d x + 12 \, c + \sin \left (4 \, d x + 4 \, c\right ) + 8 \, \sin \left (2 \, d x + 2 \, c\right )\right )} a^{4}}{32 \, d} + \frac {{\left (12 \, d x + 12 \, c + \sin \left (4 \, d x + 4 \, c\right ) - 8 \, \sin \left (2 \, d x + 2 \, c\right )\right )} a^{4}}{32 \, d} - \frac {3 \, {\left (4 \, d x + 4 \, c - \sin \left (4 \, d x + 4 \, c\right )\right )} a^{4}}{16 \, d} \]

[In]

integrate((a*cos(d*x+c)+I*a*sin(d*x+c))^4,x, algorithm="maxima")

[Out]

-I*a^4*cos(d*x + c)^4/d - I*a^4*sin(d*x + c)^4/d + 1/32*(12*d*x + 12*c + sin(4*d*x + 4*c) + 8*sin(2*d*x + 2*c)
)*a^4/d + 1/32*(12*d*x + 12*c + sin(4*d*x + 4*c) - 8*sin(2*d*x + 2*c))*a^4/d - 3/16*(4*d*x + 4*c - sin(4*d*x +
4*c))*a^4/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 = 0.31 (sec) , antiderivative size = 52, normalized size of antiderivative = 1.68 \[ \int (a \cos (c+d x)+i a \sin (c+d x))^4 \, dx=-\frac {i \, a^{4} e^{\left (4 i \, d x + 4 i \, c\right )}}{8 \, d} - \frac {i \, a^{4} e^{\left (-4 i \, d x - 4 i \, c\right )}}{8 \, d} + \frac {a^{4} \sin \left (4 \, d x + 4 \, c\right )}{4 \, d} \]

[In]

integrate((a*cos(d*x+c)+I*a*sin(d*x+c))^4,x, algorithm="giac")

[Out]

-1/8*I*a^4*e^(4*I*d*x + 4*I*c)/d - 1/8*I*a^4*e^(-4*I*d*x - 4*I*c)/d + 1/4*a^4*sin(4*d*x + 4*c)/d

Mupad [B] (verification not implemented)

Time = 27.28 (sec) , antiderivative size = 84, normalized size of antiderivative = 2.71 \[ \int (a \cos (c+d x)+i a \sin (c+d x))^4 \, dx=-\frac {2\,a^4\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2-1\right )}{d\,\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3\,4{}\mathrm {i}-6\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2-\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,4{}\mathrm {i}+1\right )} \]

[In]

int((a*cos(c + d*x) + a*sin(c + d*x)*1i)^4,x)

[Out]

-(2*a^4*tan(c/2 + (d*x)/2)*(tan(c/2 + (d*x)/2)^2 - 1))/(d*(tan(c/2 + (d*x)/2)^3*4i - 6*tan(c/2 + (d*x)/2)^2 -
tan(c/2 + (d*x)/2)*4i + tan(c/2 + (d*x)/2)^4 + 1))