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

Optimal. Leaf size=31 \[ -\frac{i (a \cos (c+d x)+i a \sin (c+d x))^4}{4 d} \]

[Out]

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

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Rubi [A]  time = 0.0181858, antiderivative size = 31, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.045, Rules used = {3071} \[ -\frac{i (a \cos (c+d x)+i a \sin (c+d x))^4}{4 d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 3071

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*} \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}\\ \end{align*}

Mathematica [A]  time = 0.113335, size = 31, normalized size = 1. \[ -\frac{i (a \cos (c+d x)+i a \sin (c+d x))^4}{4 d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [B]  time = 0.06, size = 151, normalized size = 4.9 \begin{align*}{\frac{1}{d} \left ({a}^{4} \left ( -{\frac{\cos \left ( dx+c \right ) }{4} \left ( \left ( \sin \left ( dx+c \right ) \right ) ^{3}+{\frac{3\,\sin \left ( dx+c \right ) }{2}} \right ) }+{\frac{3\,dx}{8}}+{\frac{3\,c}{8}} \right ) -i{a}^{4} \left ( \sin \left ( dx+c \right ) \right ) ^{4}-6\,{a}^{4} \left ( -1/4\,\sin \left ( dx+c \right ) \left ( \cos \left ( dx+c \right ) \right ) ^{3}+1/8\,\sin \left ( dx+c \right ) \cos \left ( dx+c \right ) +1/8\,dx+c/8 \right ) -i{a}^{4} \left ( \cos \left ( dx+c \right ) \right ) ^{4}+{a}^{4} \left ({\frac{\sin \left ( dx+c \right ) }{4} \left ( \left ( \cos \left ( dx+c \right ) \right ) ^{3}+{\frac{3\,\cos \left ( dx+c \right ) }{2}} \right ) }+{\frac{3\,dx}{8}}+{\frac{3\,c}{8}} \right ) \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/d*(a^4*(-1/4*(sin(d*x+c)^3+3/2*sin(d*x+c))*cos(d*x+c)+3/8*d*x+3/8*c)-I*a^4*sin(d*x+c)^4-6*a^4*(-1/4*sin(d*x+
c)*cos(d*x+c)^3+1/8*sin(d*x+c)*cos(d*x+c)+1/8*d*x+1/8*c)-I*a^4*cos(d*x+c)^4+a^4*(1/4*(cos(d*x+c)^3+3/2*cos(d*x
+c))*sin(d*x+c)+3/8*d*x+3/8*c))

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Maxima [B]  time = 1.00341, size = 178, normalized size = 5.74 \begin{align*} -\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} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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

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Fricas [A]  time = 2.02668, size = 46, normalized size = 1.48 \begin{align*} -\frac{i \, a^{4} e^{\left (4 i \, d x + 4 i \, c\right )}}{4 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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

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Sympy [A]  time = 0.171032, size = 37, normalized size = 1.19 \begin{align*} \begin{cases} - \frac{i a^{4} e^{4 i c} e^{4 i d x}}{4 d} & \text{for}\: 4 d \neq 0 \\a^{4} x e^{4 i c} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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(4*d, 0)), (a**4*x*exp(4*I*c), True))

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Giac [B]  time = 1.15405, size = 70, normalized size = 2.26 \begin{align*} -\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} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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