3.5.99 \(\int \frac {(a+a \sin (e+f x))^3}{\sqrt {c+d \sin (e+f x)}} \, dx\) [499]

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

[Out]

8/15*a^3*(c-3*d)*cos(f*x+e)*(c+d*sin(f*x+e))^(1/2)/d^2/f-2/5*cos(f*x+e)*(a^3+a^3*sin(f*x+e))*(c+d*sin(f*x+e))^
(1/2)/d/f-4/15*a^3*(4*c^2-15*c*d+27*d^2)*(sin(1/2*e+1/4*Pi+1/2*f*x)^2)^(1/2)/sin(1/2*e+1/4*Pi+1/2*f*x)*Ellipti
cE(cos(1/2*e+1/4*Pi+1/2*f*x),2^(1/2)*(d/(c+d))^(1/2))*(c+d*sin(f*x+e))^(1/2)/d^3/f/((c+d*sin(f*x+e))/(c+d))^(1
/2)+4/15*a^3*(c-d)*(4*c^2-11*c*d+15*d^2)*(sin(1/2*e+1/4*Pi+1/2*f*x)^2)^(1/2)/sin(1/2*e+1/4*Pi+1/2*f*x)*Ellipti
cF(cos(1/2*e+1/4*Pi+1/2*f*x),2^(1/2)*(d/(c+d))^(1/2))*((c+d*sin(f*x+e))/(c+d))^(1/2)/d^3/f/(c+d*sin(f*x+e))^(1
/2)

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Rubi [A]
time = 0.31, antiderivative size = 258, normalized size of antiderivative = 1.00, 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 = {2842, 3047, 3102, 2831, 2742, 2740, 2734, 2732} \begin {gather*} -\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} \end {gather*}

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 2732

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

Rule 2734

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/(a + b))*Sin[c + d*x]], x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[a^2 -
b^2, 0] &&  !GtQ[a + b, 0]

Rule 2740

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

Rule 2742

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/(a + b))*Sin[c + d*x]], x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[a
^2 - b^2, 0] &&  !GtQ[a + b, 0]

Rule 2831

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 2842

Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Sim
p[(-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 3047

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 3102

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]

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*}

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Mathematica [A]
time = 1.10, size = 246, normalized size = 0.95 \begin {gather*} -\frac {a^3 (1+\sin (e+f x))^3 \left (4 \left (4 c^3-11 c^2 d+12 c d^2+27 d^3\right ) E\left (\frac {1}{4} (-2 e+\pi -2 f x)|\frac {2 d}{c+d}\right ) \sqrt {\frac {c+d \sin (e+f x)}{c+d}}-4 \left (4 c^3-15 c^2 d+26 c d^2-15 d^3\right ) F\left (\frac {1}{4} (-2 e+\pi -2 f x)|\frac {2 d}{c+d}\right ) \sqrt {\frac {c+d \sin (e+f x)}{c+d}}-d \cos (e+f x) \left (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)\right )\right )}{15 d^3 f \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )^6 \sqrt {c+d \sin (e+f x)}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/15*(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])))/(d^3*f*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^6*Sqrt[c + d*
Sin[e + f*x]])

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(1034\) vs. \(2(304)=608\).
time = 5.10, size = 1035, normalized size = 4.01

method	result	size
default	\(\frac {2 a^{3} \left (8 \sqrt {\frac {c +d \sin \left (f x +e \right )}{c -d}}\, \sqrt {-\frac {\left (\sin \left (f x +e \right )-1\right ) d}{c +d}}\, \sqrt {-\frac {d \left (1+\sin \left (f x +e \right )\right )}{c -d}}\, \EllipticF \left (\sqrt {\frac {c +d \sin \left (f x +e \right )}{c -d}}, \sqrt {\frac {c -d}{c +d}}\right ) c^{3} d -36 c^{2} \sqrt {\frac {c +d \sin \left (f x +e \right )}{c -d}}\, \sqrt {-\frac {\left (\sin \left (f x +e \right )-1\right ) d}{c +d}}\, \sqrt {-\frac {d \left (1+\sin \left (f x +e \right )\right )}{c -d}}\, \EllipticF \left (\sqrt {\frac {c +d \sin \left (f x +e \right )}{c -d}}, \sqrt {\frac {c -d}{c +d}}\right ) d^{2}+112 c \sqrt {\frac {c +d \sin \left (f x +e \right )}{c -d}}\, \sqrt {-\frac {\left (\sin \left (f x +e \right )-1\right ) d}{c +d}}\, \sqrt {-\frac {d \left (1+\sin \left (f x +e \right )\right )}{c -d}}\, \EllipticF \left (\sqrt {\frac {c +d \sin \left (f x +e \right )}{c -d}}, \sqrt {\frac {c -d}{c +d}}\right ) d^{3}-84 \sqrt {\frac {c +d \sin \left (f x +e \right )}{c -d}}\, \sqrt {-\frac {\left (\sin \left (f x +e \right )-1\right ) d}{c +d}}\, \sqrt {-\frac {d \left (1+\sin \left (f x +e \right )\right )}{c -d}}\, \EllipticF \left (\sqrt {\frac {c +d \sin \left (f x +e \right )}{c -d}}, \sqrt {\frac {c -d}{c +d}}\right ) d^{4}-8 \sqrt {\frac {c +d \sin \left (f x +e \right )}{c -d}}\, \sqrt {-\frac {\left (\sin \left (f x +e \right )-1\right ) d}{c +d}}\, \sqrt {-\frac {d \left (1+\sin \left (f x +e \right )\right )}{c -d}}\, \EllipticE \left (\sqrt {\frac {c +d \sin \left (f x +e \right )}{c -d}}, \sqrt {\frac {c -d}{c +d}}\right ) c^{4}+30 \sqrt {\frac {c +d \sin \left (f x +e \right )}{c -d}}\, \sqrt {-\frac {\left (\sin \left (f x +e \right )-1\right ) d}{c +d}}\, \sqrt {-\frac {d \left (1+\sin \left (f x +e \right )\right )}{c -d}}\, \EllipticE \left (\sqrt {\frac {c +d \sin \left (f x +e \right )}{c -d}}, \sqrt {\frac {c -d}{c +d}}\right ) c^{3} d -46 \sqrt {\frac {c +d \sin \left (f x +e \right )}{c -d}}\, \sqrt {-\frac {\left (\sin \left (f x +e \right )-1\right ) d}{c +d}}\, \sqrt {-\frac {d \left (1+\sin \left (f x +e \right )\right )}{c -d}}\, \EllipticE \left (\sqrt {\frac {c +d \sin \left (f x +e \right )}{c -d}}, \sqrt {\frac {c -d}{c +d}}\right ) c^{2} d^{2}-30 \sqrt {\frac {c +d \sin \left (f x +e \right )}{c -d}}\, \sqrt {-\frac {\left (\sin \left (f x +e \right )-1\right ) d}{c +d}}\, \sqrt {-\frac {d \left (1+\sin \left (f x +e \right )\right )}{c -d}}\, \EllipticE \left (\sqrt {\frac {c +d \sin \left (f x +e \right )}{c -d}}, \sqrt {\frac {c -d}{c +d}}\right ) c \,d^{3}+54 \sqrt {\frac {c +d \sin \left (f x +e \right )}{c -d}}\, \sqrt {-\frac {\left (\sin \left (f x +e \right )-1\right ) d}{c +d}}\, \sqrt {-\frac {d \left (1+\sin \left (f x +e \right )\right )}{c -d}}\, \EllipticE \left (\sqrt {\frac {c +d \sin \left (f x +e \right )}{c -d}}, \sqrt {\frac {c -d}{c +d}}\right ) d^{4}+3 d^{4} \left (\sin ^{4}\left (f x +e \right )\right )-c \,d^{3} \left (\sin ^{3}\left (f x +e \right )\right )+15 d^{4} \left (\sin ^{3}\left (f x +e \right )\right )-4 c^{2} d^{2} \left (\sin ^{2}\left (f x +e \right )\right )+15 c \,d^{3} \left (\sin ^{2}\left (f x +e \right )\right )-3 d^{4} \left (\sin ^{2}\left (f x +e \right )\right )+c \,d^{3} \sin \left (f x +e \right )-15 d^{4} \sin \left (f x +e \right )+4 c^{2} d^{2}-15 d^{3} c \right )}{15 d^{4} \cos \left (f x +e \right ) \sqrt {c +d \sin \left (f x +e \right )}\, f}\)	\(1035\)

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,method=_RETURNVERBOSE)

[Out]

2/15*a^3*(8*((c+d*sin(f*x+e))/(c-d))^(1/2)*(-(sin(f*x+e)-1)*d/(c+d))^(1/2)*(-d*(1+sin(f*x+e))/(c-d))^(1/2)*Ell
ipticF(((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)*(-(sin(
f*x+e)-1)*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)*(-(sin(f*x+e)-1)*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)*(-(sin
(f*x+e)-1)*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)*(-(sin(f*x+e)-1)*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)*(-(sin(f*
x+e)-1)*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)*(-(sin(f*x+e)-1)*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)*(-(si
n(f*x+e)-1)*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)*(-(sin(f*x+e)-1)*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+1
5*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)-15*d^4*sin
(f*x+e)+4*c^2*d^2-15*d^3*c)/d^4/cos(f*x+e)/(c+d*sin(f*x+e))^(1/2)/f

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

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 [C] Result contains higher order function than in optimal. Order 9 vs. order 4.
time = 0.20, size = 556, normalized size = 2.16 \begin {gather*} -\frac {2 \, {\left (\sqrt {2} {\left (8 \, a^{3} c^{3} - 30 \, a^{3} c^{2} d + 51 \, a^{3} c d^{2} - 45 \, a^{3} d^{3}\right )} \sqrt {i \, d} {\rm weierstrassPInverse}\left (-\frac {4 \, {\left (4 \, c^{2} - 3 \, d^{2}\right )}}{3 \, d^{2}}, -\frac {8 \, {\left (8 i \, c^{3} - 9 i \, c d^{2}\right )}}{27 \, d^{3}}, \frac {3 \, d \cos \left (f x + e\right ) - 3 i \, d \sin \left (f x + e\right ) - 2 i \, c}{3 \, d}\right ) + \sqrt {2} {\left (8 \, a^{3} c^{3} - 30 \, a^{3} c^{2} d + 51 \, a^{3} c d^{2} - 45 \, a^{3} d^{3}\right )} \sqrt {-i \, d} {\rm weierstrassPInverse}\left (-\frac {4 \, {\left (4 \, c^{2} - 3 \, d^{2}\right )}}{3 \, d^{2}}, -\frac {8 \, {\left (-8 i \, c^{3} + 9 i \, c d^{2}\right )}}{27 \, d^{3}}, \frac {3 \, d \cos \left (f x + e\right ) + 3 i \, d \sin \left (f x + e\right ) + 2 i \, c}{3 \, d}\right ) + 3 \, \sqrt {2} {\left (4 i \, a^{3} c^{2} d - 15 i \, a^{3} c d^{2} + 27 i \, a^{3} d^{3}\right )} \sqrt {i \, d} {\rm weierstrassZeta}\left (-\frac {4 \, {\left (4 \, c^{2} - 3 \, d^{2}\right )}}{3 \, d^{2}}, -\frac {8 \, {\left (8 i \, c^{3} - 9 i \, c d^{2}\right )}}{27 \, d^{3}}, {\rm weierstrassPInverse}\left (-\frac {4 \, {\left (4 \, c^{2} - 3 \, d^{2}\right )}}{3 \, d^{2}}, -\frac {8 \, {\left (8 i \, c^{3} - 9 i \, c d^{2}\right )}}{27 \, d^{3}}, \frac {3 \, d \cos \left (f x + e\right ) - 3 i \, d \sin \left (f x + e\right ) - 2 i \, c}{3 \, d}\right )\right ) + 3 \, \sqrt {2} {\left (-4 i \, a^{3} c^{2} d + 15 i \, a^{3} c d^{2} - 27 i \, a^{3} d^{3}\right )} \sqrt {-i \, d} {\rm weierstrassZeta}\left (-\frac {4 \, {\left (4 \, c^{2} - 3 \, d^{2}\right )}}{3 \, d^{2}}, -\frac {8 \, {\left (-8 i \, c^{3} + 9 i \, c d^{2}\right )}}{27 \, d^{3}}, {\rm weierstrassPInverse}\left (-\frac {4 \, {\left (4 \, c^{2} - 3 \, d^{2}\right )}}{3 \, d^{2}}, -\frac {8 \, {\left (-8 i \, c^{3} + 9 i \, c d^{2}\right )}}{27 \, d^{3}}, \frac {3 \, d \cos \left (f x + e\right ) + 3 i \, d \sin \left (f x + e\right ) + 2 i \, c}{3 \, d}\right )\right ) + 3 \, {\left (3 \, a^{3} d^{3} \cos \left (f x + e\right ) \sin \left (f x + e\right ) - {\left (4 \, a^{3} c d^{2} - 15 \, a^{3} d^{3}\right )} \cos \left (f x + e\right )\right )} \sqrt {d \sin \left (f x + e\right ) + c}\right )}}{45 \, d^{4} f} \end {gather*}

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]

-2/45*(sqrt(2)*(8*a^3*c^3 - 30*a^3*c^2*d + 51*a^3*c*d^2 - 45*a^3*d^3)*sqrt(I*d)*weierstrassPInverse(-4/3*(4*c^
2 - 3*d^2)/d^2, -8/27*(8*I*c^3 - 9*I*c*d^2)/d^3, 1/3*(3*d*cos(f*x + e) - 3*I*d*sin(f*x + e) - 2*I*c)/d) + sqrt
(2)*(8*a^3*c^3 - 30*a^3*c^2*d + 51*a^3*c*d^2 - 45*a^3*d^3)*sqrt(-I*d)*weierstrassPInverse(-4/3*(4*c^2 - 3*d^2)
/d^2, -8/27*(-8*I*c^3 + 9*I*c*d^2)/d^3, 1/3*(3*d*cos(f*x + e) + 3*I*d*sin(f*x + e) + 2*I*c)/d) + 3*sqrt(2)*(4*
I*a^3*c^2*d - 15*I*a^3*c*d^2 + 27*I*a^3*d^3)*sqrt(I*d)*weierstrassZeta(-4/3*(4*c^2 - 3*d^2)/d^2, -8/27*(8*I*c^
3 - 9*I*c*d^2)/d^3, weierstrassPInverse(-4/3*(4*c^2 - 3*d^2)/d^2, -8/27*(8*I*c^3 - 9*I*c*d^2)/d^3, 1/3*(3*d*co
s(f*x + e) - 3*I*d*sin(f*x + e) - 2*I*c)/d)) + 3*sqrt(2)*(-4*I*a^3*c^2*d + 15*I*a^3*c*d^2 - 27*I*a^3*d^3)*sqrt
(-I*d)*weierstrassZeta(-4/3*(4*c^2 - 3*d^2)/d^2, -8/27*(-8*I*c^3 + 9*I*c*d^2)/d^3, weierstrassPInverse(-4/3*(4
*c^2 - 3*d^2)/d^2, -8/27*(-8*I*c^3 + 9*I*c*d^2)/d^3, 1/3*(3*d*cos(f*x + e) + 3*I*d*sin(f*x + e) + 2*I*c)/d)) +
3*(3*a^3*d^3*cos(f*x + e)*sin(f*x + e) - (4*a^3*c*d^2 - 15*a^3*d^3)*cos(f*x + e))*sqrt(d*sin(f*x + e) + c))/(
d^4*f)

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} 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 {gather*}

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 = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

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)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {{\left (a+a\,\sin \left (e+f\,x\right )\right )}^3}{\sqrt {c+d\,\sin \left (e+f\,x\right )}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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