3.33 \(\int \sin ^3(c+d x) \sqrt{a+a \sin (c+d x)} \, dx\)

Optimal. Leaf size=122 \[ -\frac{2 a \sin ^3(c+d x) \cos (c+d x)}{7 d \sqrt{a \sin (c+d x)+a}}-\frac{12 \cos (c+d x) (a \sin (c+d x)+a)^{3/2}}{35 a d}+\frac{8 \cos (c+d x) \sqrt{a \sin (c+d x)+a}}{35 d}-\frac{4 a \cos (c+d x)}{5 d \sqrt{a \sin (c+d x)+a}} \]

[Out]

(-4*a*Cos[c + d*x])/(5*d*Sqrt[a + a*Sin[c + d*x]]) - (2*a*Cos[c + d*x]*Sin[c + d*x]^3)/(7*d*Sqrt[a + a*Sin[c +
d*x]]) + (8*Cos[c + d*x]*Sqrt[a + a*Sin[c + d*x]])/(35*d) - (12*Cos[c + d*x]*(a + a*Sin[c + d*x])^(3/2))/(35*
a*d)

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Rubi [A]  time = 0.16932, antiderivative size = 122, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.174, Rules used = {2770, 2759, 2751, 2646} \[ -\frac{2 a \sin ^3(c+d x) \cos (c+d x)}{7 d \sqrt{a \sin (c+d x)+a}}-\frac{12 \cos (c+d x) (a \sin (c+d x)+a)^{3/2}}{35 a d}+\frac{8 \cos (c+d x) \sqrt{a \sin (c+d x)+a}}{35 d}-\frac{4 a \cos (c+d x)}{5 d \sqrt{a \sin (c+d x)+a}} \]

Antiderivative was successfully verified.

[In]

Int[Sin[c + d*x]^3*Sqrt[a + a*Sin[c + d*x]],x]

[Out]

(-4*a*Cos[c + d*x])/(5*d*Sqrt[a + a*Sin[c + d*x]]) - (2*a*Cos[c + d*x]*Sin[c + d*x]^3)/(7*d*Sqrt[a + a*Sin[c +
d*x]]) + (8*Cos[c + d*x]*Sqrt[a + a*Sin[c + d*x]])/(35*d) - (12*Cos[c + d*x]*(a + a*Sin[c + d*x])^(3/2))/(35*
a*d)

Rule 2770

Int[Sqrt[(a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]]*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Simp
[(-2*b*Cos[e + f*x]*(c + d*Sin[e + f*x])^n)/(f*(2*n + 1)*Sqrt[a + b*Sin[e + f*x]]), x] + Dist[(2*n*(b*c + a*d)
)/(b*(2*n + 1)), Int[Sqrt[a + b*Sin[e + f*x]]*(c + d*Sin[e + f*x])^(n - 1), x], x] /; FreeQ[{a, b, c, d, e, f}
, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && GtQ[n, 0] && IntegerQ[2*n]

Rule 2759

Int[sin[(e_.) + (f_.)*(x_)]^2*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_), x_Symbol] :> -Simp[(Cos[e + f*x]*(a
+ b*Sin[e + f*x])^(m + 1))/(b*f*(m + 2)), x] + Dist[1/(b*(m + 2)), Int[(a + b*Sin[e + f*x])^m*(b*(m + 1) - a*
Sin[e + f*x]), x], x] /; FreeQ[{a, b, e, f, m}, x] && EqQ[a^2 - b^2, 0] &&  !LtQ[m, -2^(-1)]

Rule 2751

Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)]), x_Symbol] :> -Simp[(d
*Cos[e + f*x]*(a + b*Sin[e + f*x])^m)/(f*(m + 1)), x] + Dist[(a*d*m + b*c*(m + 1))/(b*(m + 1)), Int[(a + b*Sin
[e + f*x])^m, x], x] /; FreeQ[{a, b, c, d, e, f, m}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 - b^2, 0] &&  !LtQ[m,
-2^(-1)]

Rule 2646

Int[Sqrt[(a_) + (b_.)*sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[(-2*b*Cos[c + d*x])/(d*Sqrt[a + b*Sin[c + d*
x]]), x] /; FreeQ[{a, b, c, d}, x] && EqQ[a^2 - b^2, 0]

Rubi steps

\begin{align*} \int \sin ^3(c+d x) \sqrt{a+a \sin (c+d x)} \, dx &=-\frac{2 a \cos (c+d x) \sin ^3(c+d x)}{7 d \sqrt{a+a \sin (c+d x)}}+\frac{6}{7} \int \sin ^2(c+d x) \sqrt{a+a \sin (c+d x)} \, dx\\ &=-\frac{2 a \cos (c+d x) \sin ^3(c+d x)}{7 d \sqrt{a+a \sin (c+d x)}}-\frac{12 \cos (c+d x) (a+a \sin (c+d x))^{3/2}}{35 a d}+\frac{12 \int \left (\frac{3 a}{2}-a \sin (c+d x)\right ) \sqrt{a+a \sin (c+d x)} \, dx}{35 a}\\ &=-\frac{2 a \cos (c+d x) \sin ^3(c+d x)}{7 d \sqrt{a+a \sin (c+d x)}}+\frac{8 \cos (c+d x) \sqrt{a+a \sin (c+d x)}}{35 d}-\frac{12 \cos (c+d x) (a+a \sin (c+d x))^{3/2}}{35 a d}+\frac{2}{5} \int \sqrt{a+a \sin (c+d x)} \, dx\\ &=-\frac{4 a \cos (c+d x)}{5 d \sqrt{a+a \sin (c+d x)}}-\frac{2 a \cos (c+d x) \sin ^3(c+d x)}{7 d \sqrt{a+a \sin (c+d x)}}+\frac{8 \cos (c+d x) \sqrt{a+a \sin (c+d x)}}{35 d}-\frac{12 \cos (c+d x) (a+a \sin (c+d x))^{3/2}}{35 a d}\\ \end{align*}

Mathematica [A]  time = 0.292307, size = 141, normalized size = 1.16 \[ \frac{\sqrt{a (\sin (c+d x)+1)} \left (105 \sin \left (\frac{1}{2} (c+d x)\right )-35 \sin \left (\frac{3}{2} (c+d x)\right )-7 \sin \left (\frac{5}{2} (c+d x)\right )+5 \sin \left (\frac{7}{2} (c+d x)\right )-105 \cos \left (\frac{1}{2} (c+d x)\right )-35 \cos \left (\frac{3}{2} (c+d x)\right )+7 \cos \left (\frac{5}{2} (c+d x)\right )+5 \cos \left (\frac{7}{2} (c+d x)\right )\right )}{140 d \left (\sin \left (\frac{1}{2} (c+d x)\right )+\cos \left (\frac{1}{2} (c+d x)\right )\right )} \]

Antiderivative was successfully verified.

[In]

Integrate[Sin[c + d*x]^3*Sqrt[a + a*Sin[c + d*x]],x]

[Out]

(Sqrt[a*(1 + Sin[c + d*x])]*(-105*Cos[(c + d*x)/2] - 35*Cos[(3*(c + d*x))/2] + 7*Cos[(5*(c + d*x))/2] + 5*Cos[
(7*(c + d*x))/2] + 105*Sin[(c + d*x)/2] - 35*Sin[(3*(c + d*x))/2] - 7*Sin[(5*(c + d*x))/2] + 5*Sin[(7*(c + d*x
))/2]))/(140*d*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2]))

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Maple [A]  time = 0.539, size = 73, normalized size = 0.6 \begin{align*}{\frac{ \left ( 2+2\,\sin \left ( dx+c \right ) \right ) a \left ( \sin \left ( dx+c \right ) -1 \right ) \left ( 5\, \left ( \sin \left ( dx+c \right ) \right ) ^{3}+6\, \left ( \sin \left ( dx+c \right ) \right ) ^{2}+8\,\sin \left ( dx+c \right ) +16 \right ) }{35\,d\cos \left ( dx+c \right ) }{\frac{1}{\sqrt{a+a\sin \left ( dx+c \right ) }}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sin(d*x+c)^3*(a+a*sin(d*x+c))^(1/2),x)

[Out]

2/35*(1+sin(d*x+c))*a*(sin(d*x+c)-1)*(5*sin(d*x+c)^3+6*sin(d*x+c)^2+8*sin(d*x+c)+16)/cos(d*x+c)/(a+a*sin(d*x+c
))^(1/2)/d

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sin(d*x+c)^3*(a+a*sin(d*x+c))^(1/2),x, algorithm="maxima")

[Out]

integrate(sqrt(a*sin(d*x + c) + a)*sin(d*x + c)^3, x)

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Fricas [A]  time = 1.37575, size = 300, normalized size = 2.46 \begin{align*} \frac{2 \,{\left (5 \, \cos \left (d x + c\right )^{4} + 6 \, \cos \left (d x + c\right )^{3} - 12 \, \cos \left (d x + c\right )^{2} +{\left (5 \, \cos \left (d x + c\right )^{3} - \cos \left (d x + c\right )^{2} - 13 \, \cos \left (d x + c\right ) + 9\right )} \sin \left (d x + c\right ) - 22 \, \cos \left (d x + c\right ) - 9\right )} \sqrt{a \sin \left (d x + c\right ) + a}}{35 \,{\left (d \cos \left (d x + c\right ) + d \sin \left (d x + c\right ) + d\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sin(d*x+c)^3*(a+a*sin(d*x+c))^(1/2),x, algorithm="fricas")

[Out]

2/35*(5*cos(d*x + c)^4 + 6*cos(d*x + c)^3 - 12*cos(d*x + c)^2 + (5*cos(d*x + c)^3 - cos(d*x + c)^2 - 13*cos(d*
x + c) + 9)*sin(d*x + c) - 22*cos(d*x + c) - 9)*sqrt(a*sin(d*x + c) + a)/(d*cos(d*x + c) + d*sin(d*x + 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(sin(d*x+c)**3*(a+a*sin(d*x+c))**(1/2),x)

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sin(d*x+c)^3*(a+a*sin(d*x+c))^(1/2),x, algorithm="giac")

[Out]

integrate(sqrt(a*sin(d*x + c) + a)*sin(d*x + c)^3, x)