3.258 \(\int (a \cos (c+d x)+i a \sin (c+d x))^{3/2} \, dx\)

Optimal. Leaf size=33 \[ -\frac{2 i (a \cos (c+d x)+i a \sin (c+d x))^{3/2}}{3 d} \]

[Out]

(((-2*I)/3)*(a*Cos[c + d*x] + I*a*Sin[c + d*x])^(3/2))/d

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Rubi [A]  time = 0.0160221, antiderivative size = 33, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.042, Rules used = {3071} \[ -\frac{2 i (a \cos (c+d x)+i a \sin (c+d x))^{3/2}}{3 d} \]

Antiderivative was successfully verified.

[In]

Int[(a*Cos[c + d*x] + I*a*Sin[c + d*x])^(3/2),x]

[Out]

(((-2*I)/3)*(a*Cos[c + d*x] + I*a*Sin[c + d*x])^(3/2))/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))^{3/2} \, dx &=-\frac{2 i (a \cos (c+d x)+i a \sin (c+d x))^{3/2}}{3 d}\\ \end{align*}

Mathematica [A]  time = 0.0294467, size = 32, normalized size = 0.97 \[ -\frac{2 i (a (\cos (c+d x)+i \sin (c+d x)))^{3/2}}{3 d} \]

Antiderivative was successfully verified.

[In]

Integrate[(a*Cos[c + d*x] + I*a*Sin[c + d*x])^(3/2),x]

[Out]

(((-2*I)/3)*(a*(Cos[c + d*x] + I*Sin[c + d*x]))^(3/2))/d

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Maple [A]  time = 0.03, size = 28, normalized size = 0.9 \begin{align*}{\frac{-{\frac{2\,i}{3}}}{d} \left ( a\cos \left ( dx+c \right ) +ia\sin \left ( dx+c \right ) \right ) ^{{\frac{3}{2}}}} \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))^(3/2),x)

[Out]

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

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Maxima [B]  time = 1.54804, size = 69, normalized size = 2.09 \begin{align*} -\frac{2 i \, a^{\frac{3}{2}}{\left (-\frac{\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + i\right )}^{\frac{3}{2}}}{3 \, d{\left (\frac{\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + i\right )}^{\frac{3}{2}}} \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))^(3/2),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 1.9168, size = 57, normalized size = 1.73 \begin{align*} -\frac{2 i \, a^{\frac{3}{2}} e^{\left (\frac{3}{2} i \, d x + \frac{3}{2} i \, c\right )}}{3 \, 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))^(3/2),x, algorithm="fricas")

[Out]

-2/3*I*a^(3/2)*e^(3/2*I*d*x + 3/2*I*c)/d

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \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))**(3/2),x)

[Out]

Timed out

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Giac [A]  time = 1.15549, size = 34, normalized size = 1.03 \begin{align*} -\frac{2 i \,{\left (a \cos \left (d x + c\right ) + i \, a \sin \left (d x + c\right )\right )}^{\frac{3}{2}}}{3 \, 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))^(3/2),x, algorithm="giac")

[Out]

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