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]

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

________________________________________________________________________________________

Rubi [A]  time = 0.04, antiderivative size = 24, normalized size of antiderivative = 1.00, 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 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]

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]

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 = 0.02, size = 24, normalized size = 1.00 \[ \frac {2 (a+b \sin (c+d x))^{5/2}}{5 b d} \]

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)

________________________________________________________________________________________

fricas [B]  time = 0.94, size = 53, normalized size = 2.21 \[ -\frac {2 \, {\left (b^{2} \cos \left (d x + c\right )^{2} - 2 \, a b \sin \left (d x + c\right ) - a^{2} - b^{2}\right )} \sqrt {b \sin \left (d x + c\right ) + a}}{5 \, b d} \]

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)

________________________________________________________________________________________

giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (b \sin \left (d x + c\right ) + a\right )}^{\frac {3}{2}} \cos \left (d x + c\right )\,{d x} \]

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)

________________________________________________________________________________________

maple [A]  time = 0.04, size = 21, normalized size = 0.88 \[ \frac {2 \left (a +b \sin \left (d x +c \right )\right )^{\frac {5}{2}}}{5 b d} \]

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

________________________________________________________________________________________

maxima [A]  time = 0.32, size = 20, normalized size = 0.83 \[ \frac {2 \, {\left (b \sin \left (d x + c\right ) + a\right )}^{\frac {5}{2}}}{5 \, b d} \]

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)

________________________________________________________________________________________

mupad [B]  time = 5.41, size = 20, normalized size = 0.83 \[ \frac {2\,{\left (a+b\,\sin \left (c+d\,x\right )\right )}^{5/2}}{5\,b\,d} \]

Verification of antiderivative is not currently implemented for this CAS.

[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)

________________________________________________________________________________________

sympy [A]  time = 26.32, size = 116, normalized size = 4.83 \[ \begin {cases} a^{\frac {3}{2}} x \cos {\relax (c )} & \text {for}\: b = 0 \wedge d = 0 \\\frac {a^{\frac {3}{2}} \sin {\left (c + d x \right )}}{d} & \text {for}\: b = 0 \\x \left (a + b \sin {\relax (c )}\right )^{\frac {3}{2}} \cos {\relax (c )} & \text {for}\: d = 0 \\\frac {2 a^{2} \sqrt {a + b \sin {\left (c + d x \right )}}}{5 b d} + \frac {4 a \sqrt {a + b \sin {\left (c + d x \right )}} \sin {\left (c + d x \right )}}{5 d} + \frac {2 b \sqrt {a + b \sin {\left (c + d x \right )}} \sin ^{2}{\left (c + d x \right )}}{5 d} & \text {otherwise} \end {cases} \]

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)), (a**(3/2)*sin(c + d*x)/d, Eq(b, 0)), (x*(a + b*sin(c))**(3
/2)*cos(c), Eq(d, 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))

________________________________________________________________________________________