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

Optimal. Leaf size=24 \[ \frac{2 (a+b \sin (c+d x))^{5/2}}{5 b d} \]

[Out]

(2*(a + b*Sin[c + d*x])^(5/2))/(5*b*d)

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Rubi [A]  time = 0.0387544, antiderivative size = 24, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.095, Rules used = {2668, 32} \[ \frac{2 (a+b \sin (c+d x))^{5/2}}{5 b d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*(a + b*Sin[c + d*x])^(5/2))/(5*b*d)

Rule 2668

Int[cos[(e_.) + (f_.)*(x_)]^(p_.)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.), x_Symbol] :> Dist[1/(b^p*f), S
ubst[Int[(a + x)^m*(b^2 - x^2)^((p - 1)/2), x], x, b*Sin[e + f*x]], x] /; FreeQ[{a, b, e, f, m}, x] && Integer
Q[(p - 1)/2] && NeQ[a^2 - b^2, 0]

Rule 32

Int[((a_.) + (b_.)*(x_))^(m_), x_Symbol] :> Simp[(a + b*x)^(m + 1)/(b*(m + 1)), x] /; FreeQ[{a, b, m}, x] && N
eQ[m, -1]

Rubi steps

\begin{align*} \int \cos (c+d x) (a+b \sin (c+d x))^{3/2} \, dx &=\frac{\operatorname{Subst}\left (\int (a+x)^{3/2} \, dx,x,b \sin (c+d x)\right )}{b d}\\ &=\frac{2 (a+b \sin (c+d x))^{5/2}}{5 b d}\\ \end{align*}

Mathematica [A]  time = 0.0212864, size = 24, normalized size = 1. \[ \frac{2 (a+b \sin (c+d x))^{5/2}}{5 b d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*(a + b*Sin[c + d*x])^(5/2))/(5*b*d)

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Maple [A]  time = 0.006, size = 21, normalized size = 0.9 \begin{align*}{\frac{2}{5\,bd} \left ( a+b\sin \left ( dx+c \right ) \right ) ^{{\frac{5}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(d*x+c)*(a+b*sin(d*x+c))^(3/2),x)

[Out]

2/5*(a+b*sin(d*x+c))^(5/2)/b/d

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Maxima [A]  time = 0.949259, size = 27, normalized size = 1.12 \begin{align*} \frac{2 \,{\left (b \sin \left (d x + c\right ) + a\right )}^{\frac{5}{2}}}{5 \, b d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)*(a+b*sin(d*x+c))^(3/2),x, algorithm="maxima")

[Out]

2/5*(b*sin(d*x + c) + a)^(5/2)/(b*d)

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Fricas [B]  time = 2.35848, size = 123, normalized size = 5.12 \begin{align*} -\frac{2 \,{\left (b^{2} \cos \left (d x + c\right )^{2} - 2 \, a b \sin \left (d x + c\right ) - a^{2} - b^{2}\right )} \sqrt{b \sin \left (d x + c\right ) + a}}{5 \, b d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)*(a+b*sin(d*x+c))^(3/2),x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 27.4081, size = 116, normalized size = 4.83 \begin{align*} \begin{cases} a^{\frac{3}{2}} x \cos{\left (c \right )} & \text{for}\: b = 0 \wedge d = 0 \\x \left (a + b \sin{\left (c \right )}\right )^{\frac{3}{2}} \cos{\left (c \right )} & \text{for}\: d = 0 \\\frac{a^{\frac{3}{2}} \sin{\left (c + d x \right )}}{d} & \text{for}\: b = 0 \\\frac{2 a^{2} \sqrt{a + b \sin{\left (c + d x \right )}}}{5 b d} + \frac{4 a \sqrt{a + b \sin{\left (c + d x \right )}} \sin{\left (c + d x \right )}}{5 d} + \frac{2 b \sqrt{a + b \sin{\left (c + d x \right )}} \sin ^{2}{\left (c + d x \right )}}{5 d} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)*(a+b*sin(d*x+c))**(3/2),x)

[Out]

Piecewise((a**(3/2)*x*cos(c), Eq(b, 0) & Eq(d, 0)), (x*(a + b*sin(c))**(3/2)*cos(c), Eq(d, 0)), (a**(3/2)*sin(
c + d*x)/d, Eq(b, 0)), (2*a**2*sqrt(a + b*sin(c + d*x))/(5*b*d) + 4*a*sqrt(a + b*sin(c + d*x))*sin(c + d*x)/(5
*d) + 2*b*sqrt(a + b*sin(c + d*x))*sin(c + d*x)**2/(5*d), True))

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (b \sin \left (d x + c\right ) + a\right )}^{\frac{3}{2}} \cos \left (d x + c\right )\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)*(a+b*sin(d*x+c))^(3/2),x, algorithm="giac")

[Out]

integrate((b*sin(d*x + c) + a)^(3/2)*cos(d*x + c), x)