3.257 \(\int (a \cos (c+d x)+i a \sin (c+d x))^{5/2} \, dx\)

Optimal. Leaf size=33 \[ -\frac{2 i (a \cos (c+d x)+i a \sin (c+d x))^{5/2}}{5 d} \]

[Out]

(((-2*I)/5)*(a*Cos[c + d*x] + I*a*Sin[c + d*x])^(5/2))/d

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Rubi [A]  time = 0.015664, 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} \[ -\frac{2 i (a \cos (c+d x)+i a \sin (c+d x))^{5/2}}{5 d} \]

Antiderivative was successfully verified.

[In]

Int[(a*Cos[c + d*x] + I*a*Sin[c + d*x])^(5/2),x]

[Out]

(((-2*I)/5)*(a*Cos[c + d*x] + I*a*Sin[c + d*x])^(5/2))/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))^{5/2} \, dx &=-\frac{2 i (a \cos (c+d x)+i a \sin (c+d x))^{5/2}}{5 d}\\ \end{align*}

Mathematica [A]  time = 0.0304076, size = 32, normalized size = 0.97 \[ -\frac{2 i (a (\cos (c+d x)+i \sin (c+d x)))^{5/2}}{5 d} \]

Antiderivative was successfully verified.

[In]

Integrate[(a*Cos[c + d*x] + I*a*Sin[c + d*x])^(5/2),x]

[Out]

(((-2*I)/5)*(a*(Cos[c + d*x] + I*Sin[c + d*x]))^(5/2))/d

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Maple [A]  time = 0.055, size = 28, normalized size = 0.9 \begin{align*}{\frac{-{\frac{2\,i}{5}}}{d} \left ( a\cos \left ( dx+c \right ) +ia\sin \left ( dx+c \right ) \right ) ^{{\frac{5}{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))^(5/2),x)

[Out]

-2/5*I*(a*cos(d*x+c)+I*a*sin(d*x+c))^(5/2)/d

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Maxima [B]  time = 1.57343, size = 69, normalized size = 2.09 \begin{align*} -\frac{2 i \, a^{\frac{5}{2}}{\left (-\frac{\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + i\right )}^{\frac{5}{2}}}{5 \, d{\left (\frac{\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + i\right )}^{\frac{5}{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))^(5/2),x, algorithm="maxima")

[Out]

-2/5*I*a^(5/2)*(-sin(d*x + c)/(cos(d*x + c) + 1) + I)^(5/2)/(d*(sin(d*x + c)/(cos(d*x + c) + 1) + I)^(5/2))

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Fricas [A]  time = 2.00607, size = 57, normalized size = 1.73 \begin{align*} -\frac{2 i \, a^{\frac{5}{2}} e^{\left (\frac{5}{2} i \, d x + \frac{5}{2} i \, c\right )}}{5 \, 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))^(5/2),x, algorithm="fricas")

[Out]

-2/5*I*a^(5/2)*e^(5/2*I*d*x + 5/2*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))**(5/2),x)

[Out]

Timed out

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Giac [A]  time = 2.14632, size = 23, normalized size = 0.7 \begin{align*} -\frac{2 i \, a^{\frac{5}{2}} e^{\left (\frac{5}{2} i \, d x + \frac{5}{2} i \, c\right )}}{5 \, 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))^(5/2),x, algorithm="giac")

[Out]

-2/5*I*a^(5/2)*e^(5/2*I*d*x + 5/2*I*c)/d