3.499 \(\int \frac{(a+a \sin (e+f x))^3}{\sqrt{c+d \sin (e+f x)}} \, dx\)

Optimal. Leaf size=258 \[ -\frac{4 a^3 (c-d) \left (4 c^2-11 c d+15 d^2\right ) \sqrt{\frac{c+d \sin (e+f x)}{c+d}} F\left (\frac{1}{2} \left (e+f x-\frac{\pi }{2}\right )|\frac{2 d}{c+d}\right )}{15 d^3 f \sqrt{c+d \sin (e+f x)}}+\frac{4 a^3 \left (4 c^2-15 c d+27 d^2\right ) \sqrt{c+d \sin (e+f x)} E\left (\frac{1}{2} \left (e+f x-\frac{\pi }{2}\right )|\frac{2 d}{c+d}\right )}{15 d^3 f \sqrt{\frac{c+d \sin (e+f x)}{c+d}}}+\frac{8 a^3 (c-3 d) \cos (e+f x) \sqrt{c+d \sin (e+f x)}}{15 d^2 f}-\frac{2 \cos (e+f x) \left (a^3 \sin (e+f x)+a^3\right ) \sqrt{c+d \sin (e+f x)}}{5 d f} \]

[Out]

(8*a^3*(c - 3*d)*Cos[e + f*x]*Sqrt[c + d*Sin[e + f*x]])/(15*d^2*f) - (2*Cos[e + f*x]*(a^3 + a^3*Sin[e + f*x])*
Sqrt[c + d*Sin[e + f*x]])/(5*d*f) + (4*a^3*(4*c^2 - 15*c*d + 27*d^2)*EllipticE[(e - Pi/2 + f*x)/2, (2*d)/(c +
d)]*Sqrt[c + d*Sin[e + f*x]])/(15*d^3*f*Sqrt[(c + d*Sin[e + f*x])/(c + d)]) - (4*a^3*(c - d)*(4*c^2 - 11*c*d +
15*d^2)*EllipticF[(e - Pi/2 + f*x)/2, (2*d)/(c + d)]*Sqrt[(c + d*Sin[e + f*x])/(c + d)])/(15*d^3*f*Sqrt[c + d
*Sin[e + f*x]])

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Rubi [A]  time = 0.476269, antiderivative size = 258, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 8, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.296, Rules used = {2763, 2968, 3023, 2752, 2663, 2661, 2655, 2653} \[ -\frac{4 a^3 (c-d) \left (4 c^2-11 c d+15 d^2\right ) \sqrt{\frac{c+d \sin (e+f x)}{c+d}} F\left (\frac{1}{2} \left (e+f x-\frac{\pi }{2}\right )|\frac{2 d}{c+d}\right )}{15 d^3 f \sqrt{c+d \sin (e+f x)}}+\frac{4 a^3 \left (4 c^2-15 c d+27 d^2\right ) \sqrt{c+d \sin (e+f x)} E\left (\frac{1}{2} \left (e+f x-\frac{\pi }{2}\right )|\frac{2 d}{c+d}\right )}{15 d^3 f \sqrt{\frac{c+d \sin (e+f x)}{c+d}}}+\frac{8 a^3 (c-3 d) \cos (e+f x) \sqrt{c+d \sin (e+f x)}}{15 d^2 f}-\frac{2 \cos (e+f x) \left (a^3 \sin (e+f x)+a^3\right ) \sqrt{c+d \sin (e+f x)}}{5 d f} \]

Antiderivative was successfully verified.

[In]

Int[(a + a*Sin[e + f*x])^3/Sqrt[c + d*Sin[e + f*x]],x]

[Out]

(8*a^3*(c - 3*d)*Cos[e + f*x]*Sqrt[c + d*Sin[e + f*x]])/(15*d^2*f) - (2*Cos[e + f*x]*(a^3 + a^3*Sin[e + f*x])*
Sqrt[c + d*Sin[e + f*x]])/(5*d*f) + (4*a^3*(4*c^2 - 15*c*d + 27*d^2)*EllipticE[(e - Pi/2 + f*x)/2, (2*d)/(c +
d)]*Sqrt[c + d*Sin[e + f*x]])/(15*d^3*f*Sqrt[(c + d*Sin[e + f*x])/(c + d)]) - (4*a^3*(c - d)*(4*c^2 - 11*c*d +
15*d^2)*EllipticF[(e - Pi/2 + f*x)/2, (2*d)/(c + d)]*Sqrt[(c + d*Sin[e + f*x])/(c + d)])/(15*d^3*f*Sqrt[c + d
*Sin[e + f*x]])

Rule 2763

Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> -Si
mp[(b^2*Cos[e + f*x]*(a + b*Sin[e + f*x])^(m - 2)*(c + d*Sin[e + f*x])^(n + 1))/(d*f*(m + n)), x] + Dist[1/(d*
(m + n)), Int[(a + b*Sin[e + f*x])^(m - 2)*(c + d*Sin[e + f*x])^n*Simp[a*b*c*(m - 2) + b^2*d*(n + 1) + a^2*d*(
m + n) - b*(b*c*(m - 1) - a*d*(3*m + 2*n - 2))*Sin[e + f*x], x], x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] &&
NeQ[b*c - a*d, 0] && EqQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && GtQ[m, 1] &&  !LtQ[n, -1] && (IntegersQ[2*m, 2*
n] || IntegerQ[m + 1/2] || (IntegerQ[m] && EqQ[c, 0]))

Rule 2968

Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)])*((c_.) + (d_.)*sin[(
e_.) + (f_.)*(x_)]), x_Symbol] :> Int[(a + b*Sin[e + f*x])^m*(A*c + (B*c + A*d)*Sin[e + f*x] + B*d*Sin[e + f*x
]^2), x] /; FreeQ[{a, b, c, d, e, f, A, B, m}, x] && NeQ[b*c - a*d, 0]

Rule 3023

Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)] + (C_.)*sin[(e_.) + (
f_.)*(x_)]^2), x_Symbol] :> -Simp[(C*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*Simp[A*b*(m + 2) + b*C*(m + 1) + (b*B*(m + 2) - a*C)*Sin[e + f*x], x], x]
, x] /; FreeQ[{a, b, e, f, A, B, C, m}, x] &&  !LtQ[m, -1]

Rule 2752

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

Rule 2663

Int[1/Sqrt[(a_) + (b_.)*sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Dist[Sqrt[(a + b*Sin[c + d*x])/(a + b)]/Sqrt[a
+ b*Sin[c + d*x]], Int[1/Sqrt[a/(a + b) + (b*Sin[c + d*x])/(a + b)], x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[a
^2 - b^2, 0] &&  !GtQ[a + b, 0]

Rule 2661

Int[1/Sqrt[(a_) + (b_.)*sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[(2*EllipticF[(1*(c - Pi/2 + d*x))/2, (2*b)
/(a + b)])/(d*Sqrt[a + b]), x] /; FreeQ[{a, b, c, d}, x] && NeQ[a^2 - b^2, 0] && GtQ[a + b, 0]

Rule 2655

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

Rule 2653

Int[Sqrt[(a_) + (b_.)*sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[(2*Sqrt[a + b]*EllipticE[(1*(c - Pi/2 + d*x)
)/2, (2*b)/(a + b)])/d, x] /; FreeQ[{a, b, c, d}, x] && NeQ[a^2 - b^2, 0] && GtQ[a + b, 0]

Rubi steps

\begin{align*} \int \frac{(a+a \sin (e+f x))^3}{\sqrt{c+d \sin (e+f x)}} \, dx &=-\frac{2 \cos (e+f x) \left (a^3+a^3 \sin (e+f x)\right ) \sqrt{c+d \sin (e+f x)}}{5 d f}+\frac{2 \int \frac{(a+a \sin (e+f x)) \left (a^2 (c+3 d)-2 a^2 (c-3 d) \sin (e+f x)\right )}{\sqrt{c+d \sin (e+f x)}} \, dx}{5 d}\\ &=-\frac{2 \cos (e+f x) \left (a^3+a^3 \sin (e+f x)\right ) \sqrt{c+d \sin (e+f x)}}{5 d f}+\frac{2 \int \frac{a^3 (c+3 d)+\left (-2 a^3 (c-3 d)+a^3 (c+3 d)\right ) \sin (e+f x)-2 a^3 (c-3 d) \sin ^2(e+f x)}{\sqrt{c+d \sin (e+f x)}} \, dx}{5 d}\\ &=\frac{8 a^3 (c-3 d) \cos (e+f x) \sqrt{c+d \sin (e+f x)}}{15 d^2 f}-\frac{2 \cos (e+f x) \left (a^3+a^3 \sin (e+f x)\right ) \sqrt{c+d \sin (e+f x)}}{5 d f}+\frac{4 \int \frac{\frac{1}{2} a^3 d (c+15 d)+\frac{1}{2} a^3 \left (4 c^2-15 c d+27 d^2\right ) \sin (e+f x)}{\sqrt{c+d \sin (e+f x)}} \, dx}{15 d^2}\\ &=\frac{8 a^3 (c-3 d) \cos (e+f x) \sqrt{c+d \sin (e+f x)}}{15 d^2 f}-\frac{2 \cos (e+f x) \left (a^3+a^3 \sin (e+f x)\right ) \sqrt{c+d \sin (e+f x)}}{5 d f}-\frac{\left (2 a^3 (c-d) \left (4 c^2-11 c d+15 d^2\right )\right ) \int \frac{1}{\sqrt{c+d \sin (e+f x)}} \, dx}{15 d^3}+\frac{\left (2 a^3 \left (4 c^2-15 c d+27 d^2\right )\right ) \int \sqrt{c+d \sin (e+f x)} \, dx}{15 d^3}\\ &=\frac{8 a^3 (c-3 d) \cos (e+f x) \sqrt{c+d \sin (e+f x)}}{15 d^2 f}-\frac{2 \cos (e+f x) \left (a^3+a^3 \sin (e+f x)\right ) \sqrt{c+d \sin (e+f x)}}{5 d f}+\frac{\left (2 a^3 \left (4 c^2-15 c d+27 d^2\right ) \sqrt{c+d \sin (e+f x)}\right ) \int \sqrt{\frac{c}{c+d}+\frac{d \sin (e+f x)}{c+d}} \, dx}{15 d^3 \sqrt{\frac{c+d \sin (e+f x)}{c+d}}}-\frac{\left (2 a^3 (c-d) \left (4 c^2-11 c d+15 d^2\right ) \sqrt{\frac{c+d \sin (e+f x)}{c+d}}\right ) \int \frac{1}{\sqrt{\frac{c}{c+d}+\frac{d \sin (e+f x)}{c+d}}} \, dx}{15 d^3 \sqrt{c+d \sin (e+f x)}}\\ &=\frac{8 a^3 (c-3 d) \cos (e+f x) \sqrt{c+d \sin (e+f x)}}{15 d^2 f}-\frac{2 \cos (e+f x) \left (a^3+a^3 \sin (e+f x)\right ) \sqrt{c+d \sin (e+f x)}}{5 d f}+\frac{4 a^3 \left (4 c^2-15 c d+27 d^2\right ) E\left (\frac{1}{2} \left (e-\frac{\pi }{2}+f x\right )|\frac{2 d}{c+d}\right ) \sqrt{c+d \sin (e+f x)}}{15 d^3 f \sqrt{\frac{c+d \sin (e+f x)}{c+d}}}-\frac{4 a^3 (c-d) \left (4 c^2-11 c d+15 d^2\right ) F\left (\frac{1}{2} \left (e-\frac{\pi }{2}+f x\right )|\frac{2 d}{c+d}\right ) \sqrt{\frac{c+d \sin (e+f x)}{c+d}}}{15 d^3 f \sqrt{c+d \sin (e+f x)}}\\ \end{align*}

Mathematica [A]  time = 1.63038, size = 246, normalized size = 0.95 \[ -\frac{a^3 (\sin (e+f x)+1)^3 \left (-d \cos (e+f x) \left (8 c^2+2 d (c-15 d) \sin (e+f x)-30 c d+3 d^2 \cos (2 (e+f x))-3 d^2\right )-4 \left (-15 c^2 d+4 c^3+26 c d^2-15 d^3\right ) \sqrt{\frac{c+d \sin (e+f x)}{c+d}} F\left (\frac{1}{4} (-2 e-2 f x+\pi )|\frac{2 d}{c+d}\right )+4 \left (-11 c^2 d+4 c^3+12 c d^2+27 d^3\right ) \sqrt{\frac{c+d \sin (e+f x)}{c+d}} E\left (\frac{1}{4} (-2 e-2 f x+\pi )|\frac{2 d}{c+d}\right )\right )}{15 d^3 f \left (\sin \left (\frac{1}{2} (e+f x)\right )+\cos \left (\frac{1}{2} (e+f x)\right )\right )^6 \sqrt{c+d \sin (e+f x)}} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + a*Sin[e + f*x])^3/Sqrt[c + d*Sin[e + f*x]],x]

[Out]

-(a^3*(1 + Sin[e + f*x])^3*(4*(4*c^3 - 11*c^2*d + 12*c*d^2 + 27*d^3)*EllipticE[(-2*e + Pi - 2*f*x)/4, (2*d)/(c
+ d)]*Sqrt[(c + d*Sin[e + f*x])/(c + d)] - 4*(4*c^3 - 15*c^2*d + 26*c*d^2 - 15*d^3)*EllipticF[(-2*e + Pi - 2*
f*x)/4, (2*d)/(c + d)]*Sqrt[(c + d*Sin[e + f*x])/(c + d)] - d*Cos[e + f*x]*(8*c^2 - 30*c*d - 3*d^2 + 3*d^2*Cos
[2*(e + f*x)] + 2*(c - 15*d)*d*Sin[e + f*x])))/(15*d^3*f*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^6*Sqrt[c + d*Si
n[e + f*x]])

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Maple [B]  time = 1.255, size = 1035, normalized size = 4. \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/15*a^3*(8*((c+d*sin(f*x+e))/(c-d))^(1/2)*(-(-1+sin(f*x+e))*d/(c+d))^(1/2)*(-d*(1+sin(f*x+e))/(c-d))^(1/2)*El
lipticF(((c+d*sin(f*x+e))/(c-d))^(1/2),((c-d)/(c+d))^(1/2))*c^3*d-36*c^2*((c+d*sin(f*x+e))/(c-d))^(1/2)*(-(-1+
sin(f*x+e))*d/(c+d))^(1/2)*(-d*(1+sin(f*x+e))/(c-d))^(1/2)*EllipticF(((c+d*sin(f*x+e))/(c-d))^(1/2),((c-d)/(c+
d))^(1/2))*d^2+112*c*((c+d*sin(f*x+e))/(c-d))^(1/2)*(-(-1+sin(f*x+e))*d/(c+d))^(1/2)*(-d*(1+sin(f*x+e))/(c-d))
^(1/2)*EllipticF(((c+d*sin(f*x+e))/(c-d))^(1/2),((c-d)/(c+d))^(1/2))*d^3-84*((c+d*sin(f*x+e))/(c-d))^(1/2)*(-(
-1+sin(f*x+e))*d/(c+d))^(1/2)*(-d*(1+sin(f*x+e))/(c-d))^(1/2)*EllipticF(((c+d*sin(f*x+e))/(c-d))^(1/2),((c-d)/
(c+d))^(1/2))*d^4-8*((c+d*sin(f*x+e))/(c-d))^(1/2)*(-(-1+sin(f*x+e))*d/(c+d))^(1/2)*(-d*(1+sin(f*x+e))/(c-d))^
(1/2)*EllipticE(((c+d*sin(f*x+e))/(c-d))^(1/2),((c-d)/(c+d))^(1/2))*c^4+30*((c+d*sin(f*x+e))/(c-d))^(1/2)*(-(-
1+sin(f*x+e))*d/(c+d))^(1/2)*(-d*(1+sin(f*x+e))/(c-d))^(1/2)*EllipticE(((c+d*sin(f*x+e))/(c-d))^(1/2),((c-d)/(
c+d))^(1/2))*c^3*d-46*((c+d*sin(f*x+e))/(c-d))^(1/2)*(-(-1+sin(f*x+e))*d/(c+d))^(1/2)*(-d*(1+sin(f*x+e))/(c-d)
)^(1/2)*EllipticE(((c+d*sin(f*x+e))/(c-d))^(1/2),((c-d)/(c+d))^(1/2))*c^2*d^2-30*((c+d*sin(f*x+e))/(c-d))^(1/2
)*(-(-1+sin(f*x+e))*d/(c+d))^(1/2)*(-d*(1+sin(f*x+e))/(c-d))^(1/2)*EllipticE(((c+d*sin(f*x+e))/(c-d))^(1/2),((
c-d)/(c+d))^(1/2))*c*d^3+54*((c+d*sin(f*x+e))/(c-d))^(1/2)*(-(-1+sin(f*x+e))*d/(c+d))^(1/2)*(-d*(1+sin(f*x+e))
/(c-d))^(1/2)*EllipticE(((c+d*sin(f*x+e))/(c-d))^(1/2),((c-d)/(c+d))^(1/2))*d^4+3*d^4*sin(f*x+e)^4-c*d^3*sin(f
*x+e)^3+15*d^4*sin(f*x+e)^3-4*c^2*d^2*sin(f*x+e)^2+15*c*d^3*sin(f*x+e)^2-3*d^4*sin(f*x+e)^2+c*d^3*sin(f*x+e)-1
5*d^4*sin(f*x+e)+4*c^2*d^2-15*c*d^3)/d^4/cos(f*x+e)/(c+d*sin(f*x+e))^(1/2)/f

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (-\frac{3 \, a^{3} \cos \left (f x + e\right )^{2} - 4 \, a^{3} +{\left (a^{3} \cos \left (f x + e\right )^{2} - 4 \, a^{3}\right )} \sin \left (f x + e\right )}{\sqrt{d \sin \left (f x + e\right ) + c}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(-(3*a^3*cos(f*x + e)^2 - 4*a^3 + (a^3*cos(f*x + e)^2 - 4*a^3)*sin(f*x + e))/sqrt(d*sin(f*x + e) + c),
x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} a^{3} \left (\int \frac{3 \sin{\left (e + f x \right )}}{\sqrt{c + d \sin{\left (e + f x \right )}}}\, dx + \int \frac{3 \sin ^{2}{\left (e + f x \right )}}{\sqrt{c + d \sin{\left (e + f x \right )}}}\, dx + \int \frac{\sin ^{3}{\left (e + f x \right )}}{\sqrt{c + d \sin{\left (e + f x \right )}}}\, dx + \int \frac{1}{\sqrt{c + d \sin{\left (e + f x \right )}}}\, dx\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+a*sin(f*x+e))**3/(c+d*sin(f*x+e))**(1/2),x)

[Out]

a**3*(Integral(3*sin(e + f*x)/sqrt(c + d*sin(e + f*x)), x) + Integral(3*sin(e + f*x)**2/sqrt(c + d*sin(e + f*x
)), x) + Integral(sin(e + f*x)**3/sqrt(c + d*sin(e + f*x)), x) + Integral(1/sqrt(c + d*sin(e + f*x)), x))

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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