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]

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

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Rubi [A]  time = 0.02, antiderivative size = 31, normalized size of antiderivative = 1.00, 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*}

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Mathematica [A]  time = 0.11, size = 31, normalized size = 1.00 \[ -\frac {i (a \cos (c+d x)+i a \sin (c+d x))^4}{4 d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [A]  time = 1.83, size = 17, normalized size = 0.55 \[ -\frac {i \, a^{4} e^{\left (4 i \, d x + 4 i \, c\right )}}{4 \, d} \]

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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giac [B]  time = 0.19, size = 52, normalized size = 1.68 \[ -\frac {i \, a^{4} e^{\left (4 i \, d x + 4 i \, c\right )}}{8 \, d} - \frac {i \, a^{4} e^{\left (-4 i \, d x - 4 i \, c\right )}}{8 \, d} + \frac {a^{4} \sin \left (4 \, d x + 4 \, c\right )}{4 \, d} \]

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

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maple [B]  time = 0.29, size = 151, normalized size = 4.87 \[ \frac {a^{4} \left (-\frac {\left (\sin ^{3}\left (d x +c \right )+\frac {3 \sin \left (d x +c \right )}{2}\right ) \cos \left (d x +c \right )}{4}+\frac {3 d x}{8}+\frac {3 c}{8}\right )-i a^{4} \left (\sin ^{4}\left (d x +c \right )\right )-6 a^{4} \left (-\frac {\sin \left (d x +c \right ) \left (\cos ^{3}\left (d x +c \right )\right )}{4}+\frac {\sin \left (d x +c \right ) \cos \left (d x +c \right )}{8}+\frac {d x}{8}+\frac {c}{8}\right )-i a^{4} \left (\cos ^{4}\left (d x +c \right )\right )+a^{4} \left (\frac {\left (\cos ^{3}\left (d x +c \right )+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{4}+\frac {3 d x}{8}+\frac {3 c}{8}\right )}{d} \]

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 = 0.46, size = 132, normalized size = 4.26 \[ -\frac {i \, a^{4} \cos \left (d x + c\right )^{4}}{d} - \frac {i \, a^{4} \sin \left (d x + c\right )^{4}}{d} + \frac {{\left (12 \, d x + 12 \, c + \sin \left (4 \, d x + 4 \, c\right ) + 8 \, \sin \left (2 \, d x + 2 \, c\right )\right )} a^{4}}{32 \, d} + \frac {{\left (12 \, d x + 12 \, c + \sin \left (4 \, d x + 4 \, c\right ) - 8 \, \sin \left (2 \, d x + 2 \, c\right )\right )} a^{4}}{32 \, d} - \frac {3 \, {\left (4 \, d x + 4 \, c - \sin \left (4 \, d x + 4 \, c\right )\right )} a^{4}}{16 \, d} \]

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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mupad [B]  time = 2.55, size = 84, normalized size = 2.71 \[ -\frac {2\,a^4\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2-1\right )}{d\,\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3\,4{}\mathrm {i}-6\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2-\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,4{}\mathrm {i}+1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[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))

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sympy [A]  time = 0.14, size = 37, normalized size = 1.19 \[ \begin {cases} - \frac {i a^{4} e^{4 i c} e^{4 i d x}}{4 d} & \text {for}\: 4 d \neq 0 \\a^{4} x e^{4 i c} & \text {otherwise} \end {cases} \]

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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