\(\int (3+b \sin (e+f x))^3 \sqrt {c+d \sin (e+f x)} \, dx\) [739]

Optimal result

Integrand size = 27, antiderivative size = 359 \[ \int (3+b \sin (e+f x))^3 \sqrt {c+d \sin (e+f x)} \, dx=\frac {2 b \left (126 b c d-945 d^2-b^2 \left (8 c^2+25 d^2\right )\right ) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{105 d^2 f}+\frac {8 b^2 (b c-12 d) \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{35 d^2 f}-\frac {2 b^2 \cos (e+f x) (3+b \sin (e+f x)) (c+d \sin (e+f x))^{3/2}}{7 d f}+\frac {2 \left (945 b c d^2+2835 d^3-63 b^2 d \left (2 c^2-9 d^2\right )+b^3 \left (8 c^3+19 c d^2\right )\right ) E\left (\frac {1}{2} \left (e-\frac {\pi }{2}+f x\right )|\frac {2 d}{c+d}\right ) \sqrt {c+d \sin (e+f x)}}{105 d^3 f \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}+\frac {2 b \left (c^2-d^2\right ) \left (126 b c d-945 d^2-b^2 \left (8 c^2+25 d^2\right )\right ) \operatorname {EllipticF}\left (\frac {1}{2} \left (e-\frac {\pi }{2}+f x\right ),\frac {2 d}{c+d}\right ) \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}{105 d^3 f \sqrt {c+d \sin (e+f x)}} \]

[Out]

8/35*b^2*(-4*a*d+b*c)*cos(f*x+e)*(c+d*sin(f*x+e))^(3/2)/d^2/f-2/7*b^2*cos(f*x+e)*(a+b*sin(f*x+e))*(c+d*sin(f*x
+e))^(3/2)/d/f+2/105*b*(42*a*b*c*d-105*a^2*d^2-b^2*(8*c^2+25*d^2))*cos(f*x+e)*(c+d*sin(f*x+e))^(1/2)/d^2/f-2/1
05*(105*a^2*b*c*d^2+105*a^3*d^3-21*a*b^2*d*(2*c^2-9*d^2)+b^3*(8*c^3+19*c*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)*EllipticE(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)-2/105*b*(c^2-d^2)*(42*a*b*c*d-105*a^2*d^2-b^2*(8*c^2+25*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)*EllipticF(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)

Rubi [A] (verified)

Time = 0.46 (sec) , antiderivative size = 375, normalized size of antiderivative = 1.04, number of steps used = 8, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.296, Rules used = {2872, 3102, 2832, 2831, 2742, 2740, 2734, 2732} \[ \int (3+b \sin (e+f x))^3 \sqrt {c+d \sin (e+f x)} \, dx=\frac {2 b \left (-105 a^2 d^2+42 a b c d-\left (b^2 \left (8 c^2+25 d^2\right )\right )\right ) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{105 d^2 f}+\frac {2 b \left (c^2-d^2\right ) \left (-105 a^2 d^2+42 a b c d-\left (b^2 \left (8 c^2+25 d^2\right )\right )\right ) \sqrt {\frac {c+d \sin (e+f x)}{c+d}} \operatorname {EllipticF}\left (\frac {1}{2} \left (e+f x-\frac {\pi }{2}\right ),\frac {2 d}{c+d}\right )}{105 d^3 f \sqrt {c+d \sin (e+f x)}}+\frac {2 \left (105 a^3 d^3+105 a^2 b c d^2-21 a b^2 d \left (2 c^2-9 d^2\right )+b^3 \left (8 c^3+19 c d^2\right )\right ) \sqrt {c+d \sin (e+f x)} E\left (\frac {1}{2} \left (e+f x-\frac {\pi }{2}\right )|\frac {2 d}{c+d}\right )}{105 d^3 f \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}+\frac {8 b^2 (b c-4 a d) \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{35 d^2 f}-\frac {2 b^2 \cos (e+f x) (a+b \sin (e+f x)) (c+d \sin (e+f x))^{3/2}}{7 d f} \]

[In]

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

[Out]

(2*b*(42*a*b*c*d - 105*a^2*d^2 - b^2*(8*c^2 + 25*d^2))*Cos[e + f*x]*Sqrt[c + d*Sin[e + f*x]])/(105*d^2*f) + (8
*b^2*(b*c - 4*a*d)*Cos[e + f*x]*(c + d*Sin[e + f*x])^(3/2))/(35*d^2*f) - (2*b^2*Cos[e + f*x]*(a + b*Sin[e + f*
x])*(c + d*Sin[e + f*x])^(3/2))/(7*d*f) + (2*(105*a^2*b*c*d^2 + 105*a^3*d^3 - 21*a*b^2*d*(2*c^2 - 9*d^2) + b^3
*(8*c^3 + 19*c*d^2))*EllipticE[(e - Pi/2 + f*x)/2, (2*d)/(c + d)]*Sqrt[c + d*Sin[e + f*x]])/(105*d^3*f*Sqrt[(c
+ d*Sin[e + f*x])/(c + d)]) + (2*b*(c^2 - d^2)*(42*a*b*c*d - 105*a^2*d^2 - b^2*(8*c^2 + 25*d^2))*EllipticF[(e
- Pi/2 + f*x)/2, (2*d)/(c + d)]*Sqrt[(c + d*Sin[e + f*x])/(c + d)])/(105*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 2832

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[1/(m + 1), Int[(a + b*Sin[e + f*x])^(m - 1)*Sim
p[b*d*m + a*c*(m + 1) + (a*d*m + b*c*(m + 1))*Sin[e + f*x], x], x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[
b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && GtQ[m, 0] && IntegerQ[2*m]

Rule 2872

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 - 3)*(c + d*Sin[e + f*x])^n*Simp[a^3*d*(m + n) + b^2*(b*c*(m - 2) + a
*d*(n + 1)) - b*(a*b*c - b^2*d*(m + n - 1) - 3*a^2*d*(m + n))*Sin[e + f*x] - b^2*(b*c*(m - 1) - a*d*(3*m + 2*n
- 2))*Sin[e + f*x]^2, x], x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0]
&& NeQ[c^2 - d^2, 0] && GtQ[m, 2] && (IntegerQ[m] || IntegersQ[2*m, 2*n]) &&  !(IGtQ[n, 2] && ( !IntegerQ[m]
|| (EqQ[a, 0] && NeQ[c, 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*} \text {integral}& = -\frac {2 b^2 \cos (e+f x) (a+b \sin (e+f x)) (c+d \sin (e+f x))^{3/2}}{7 d f}+\frac {2 \int \sqrt {c+d \sin (e+f x)} \left (\frac {1}{2} \left (2 b^3 c+7 a^3 d+3 a b^2 d\right )-\frac {1}{2} b \left (2 a b c-21 a^2 d-5 b^2 d\right ) \sin (e+f x)-2 b^2 (b c-4 a d) \sin ^2(e+f x)\right ) \, dx}{7 d} \\ & = \frac {8 b^2 (b c-4 a d) \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{35 d^2 f}-\frac {2 b^2 \cos (e+f x) (a+b \sin (e+f x)) (c+d \sin (e+f x))^{3/2}}{7 d f}+\frac {4 \int \sqrt {c+d \sin (e+f x)} \left (-\frac {1}{4} d \left (2 b^3 c-35 a^3 d-63 a b^2 d\right )-\frac {1}{4} b \left (42 a b c d-105 a^2 d^2-b^2 \left (8 c^2+25 d^2\right )\right ) \sin (e+f x)\right ) \, dx}{35 d^2} \\ & = \frac {2 b \left (42 a b c d-105 a^2 d^2-b^2 \left (8 c^2+25 d^2\right )\right ) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{105 d^2 f}+\frac {8 b^2 (b c-4 a d) \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{35 d^2 f}-\frac {2 b^2 \cos (e+f x) (a+b \sin (e+f x)) (c+d \sin (e+f x))^{3/2}}{7 d f}+\frac {8 \int \frac {\frac {1}{8} d \left (105 a^3 c d+147 a b^2 c d+105 a^2 b d^2+b^3 \left (2 c^2+25 d^2\right )\right )+\frac {1}{8} \left (105 a^2 b c d^2+105 a^3 d^3-21 a b^2 d \left (2 c^2-9 d^2\right )+b^3 \left (8 c^3+19 c d^2\right )\right ) \sin (e+f x)}{\sqrt {c+d \sin (e+f x)}} \, dx}{105 d^2} \\ & = \frac {2 b \left (42 a b c d-105 a^2 d^2-b^2 \left (8 c^2+25 d^2\right )\right ) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{105 d^2 f}+\frac {8 b^2 (b c-4 a d) \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{35 d^2 f}-\frac {2 b^2 \cos (e+f x) (a+b \sin (e+f x)) (c+d \sin (e+f x))^{3/2}}{7 d f}+\frac {\left (b \left (c^2-d^2\right ) \left (42 a b c d-105 a^2 d^2-b^2 \left (8 c^2+25 d^2\right )\right )\right ) \int \frac {1}{\sqrt {c+d \sin (e+f x)}} \, dx}{105 d^3}+\frac {\left (105 a^2 b c d^2+105 a^3 d^3-21 a b^2 d \left (2 c^2-9 d^2\right )+b^3 \left (8 c^3+19 c d^2\right )\right ) \int \sqrt {c+d \sin (e+f x)} \, dx}{105 d^3} \\ & = \frac {2 b \left (42 a b c d-105 a^2 d^2-b^2 \left (8 c^2+25 d^2\right )\right ) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{105 d^2 f}+\frac {8 b^2 (b c-4 a d) \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{35 d^2 f}-\frac {2 b^2 \cos (e+f x) (a+b \sin (e+f x)) (c+d \sin (e+f x))^{3/2}}{7 d f}+\frac {\left (\left (105 a^2 b c d^2+105 a^3 d^3-21 a b^2 d \left (2 c^2-9 d^2\right )+b^3 \left (8 c^3+19 c d^2\right )\right ) \sqrt {c+d \sin (e+f x)}\right ) \int \sqrt {\frac {c}{c+d}+\frac {d \sin (e+f x)}{c+d}} \, dx}{105 d^3 \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}+\frac {\left (b \left (c^2-d^2\right ) \left (42 a b c d-105 a^2 d^2-b^2 \left (8 c^2+25 d^2\right )\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}{105 d^3 \sqrt {c+d \sin (e+f x)}} \\ & = \frac {2 b \left (42 a b c d-105 a^2 d^2-b^2 \left (8 c^2+25 d^2\right )\right ) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{105 d^2 f}+\frac {8 b^2 (b c-4 a d) \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{35 d^2 f}-\frac {2 b^2 \cos (e+f x) (a+b \sin (e+f x)) (c+d \sin (e+f x))^{3/2}}{7 d f}+\frac {2 \left (105 a^2 b c d^2+105 a^3 d^3-21 a b^2 d \left (2 c^2-9 d^2\right )+b^3 \left (8 c^3+19 c d^2\right )\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)}}{105 d^3 f \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}+\frac {2 b \left (c^2-d^2\right ) \left (42 a b c d-105 a^2 d^2-b^2 \left (8 c^2+25 d^2\right )\right ) \operatorname {EllipticF}\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}}}{105 d^3 f \sqrt {c+d \sin (e+f x)}} \\ \end{align*}

Mathematica [A] (verified)

Time = 3.67 (sec) , antiderivative size = 287, normalized size of antiderivative = 0.80 \[ \int (3+b \sin (e+f x))^3 \sqrt {c+d \sin (e+f x)} \, dx=\frac {-4 \left (d^2 \left (2835 c d+441 b^2 c d+945 b d^2+b^3 \left (2 c^2+25 d^2\right )\right ) \operatorname {EllipticF}\left (\frac {1}{4} (-2 e+\pi -2 f x),\frac {2 d}{c+d}\right )+\left (945 b c d^2+2835 d^3+b^3 \left (8 c^3+19 c d^2\right )-63 b^2 \left (2 c^2 d-9 d^3\right )\right ) \left ((c+d) E\left (\frac {1}{4} (-2 e+\pi -2 f x)|\frac {2 d}{c+d}\right )-c \operatorname {EllipticF}\left (\frac {1}{4} (-2 e+\pi -2 f x),\frac {2 d}{c+d}\right )\right )\right ) \sqrt {\frac {c+d \sin (e+f x)}{c+d}}+b d (c+d \sin (e+f x)) \left (\left (-252 b c d-3780 d^2+b^2 \left (16 c^2-115 d^2\right )\right ) \cos (e+f x)+3 b d (5 b d \cos (3 (e+f x))-2 (b c+63 d) \sin (2 (e+f x)))\right )}{210 d^3 f \sqrt {c+d \sin (e+f x)}} \]

[In]

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

[Out]

(-4*(d^2*(2835*c*d + 441*b^2*c*d + 945*b*d^2 + b^3*(2*c^2 + 25*d^2))*EllipticF[(-2*e + Pi - 2*f*x)/4, (2*d)/(c
+ d)] + (945*b*c*d^2 + 2835*d^3 + b^3*(8*c^3 + 19*c*d^2) - 63*b^2*(2*c^2*d - 9*d^3))*((c + d)*EllipticE[(-2*e
+ Pi - 2*f*x)/4, (2*d)/(c + d)] - c*EllipticF[(-2*e + Pi - 2*f*x)/4, (2*d)/(c + d)]))*Sqrt[(c + d*Sin[e + f*x
])/(c + d)] + b*d*(c + d*Sin[e + f*x])*((-252*b*c*d - 3780*d^2 + b^2*(16*c^2 - 115*d^2))*Cos[e + f*x] + 3*b*d*
(5*b*d*Cos[3*(e + f*x)] - 2*(b*c + 63*d)*Sin[2*(e + f*x)])))/(210*d^3*f*Sqrt[c + d*Sin[e + f*x]])

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(1560\) vs. \(2(417)=834\).

Time = 14.36 (sec) , antiderivative size = 1561, normalized size of antiderivative = 4.35

method	result	size
default	\(\text {Expression too large to display}\)	\(1561\)
parts	\(\text {Expression too large to display}\)	\(2445\)

[In]

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

[Out]

(-(-d*sin(f*x+e)-c)*cos(f*x+e)^2)^(1/2)*(2*a^3*c*(c/d-1)*((c+d*sin(f*x+e))/(c-d))^(1/2)*(d*(1-sin(f*x+e))/(c+d
))^(1/2)*(1/(c-d)*(-sin(f*x+e)-1)*d)^(1/2)/(-(-d*sin(f*x+e)-c)*cos(f*x+e)^2)^(1/2)*EllipticF(((c+d*sin(f*x+e))
/(c-d))^(1/2),((c-d)/(c+d))^(1/2))+b^3*d*(-2/7/d*sin(f*x+e)^2*(-(-d*sin(f*x+e)-c)*cos(f*x+e)^2)^(1/2)+12/35*c/
d^2*sin(f*x+e)*(-(-d*sin(f*x+e)-c)*cos(f*x+e)^2)^(1/2)-2/3*(5/7+24/35*c^2/d^2)/d*(-(-d*sin(f*x+e)-c)*cos(f*x+e
)^2)^(1/2)+2*(-4/35*c^2/d^2+5/21)*(c/d-1)*((c+d*sin(f*x+e))/(c-d))^(1/2)*(d*(1-sin(f*x+e))/(c+d))^(1/2)*(1/(c-
d)*(-sin(f*x+e)-1)*d)^(1/2)/(-(-d*sin(f*x+e)-c)*cos(f*x+e)^2)^(1/2)*EllipticF(((c+d*sin(f*x+e))/(c-d))^(1/2),(
(c-d)/(c+d))^(1/2))+2/105*(-48*c^3-44*c*d^2)/d^3*(c/d-1)*((c+d*sin(f*x+e))/(c-d))^(1/2)*(d*(1-sin(f*x+e))/(c+d
))^(1/2)*(1/(c-d)*(-sin(f*x+e)-1)*d)^(1/2)/(-(-d*sin(f*x+e)-c)*cos(f*x+e)^2)^(1/2)*((-c/d-1)*EllipticE(((c+d*s
in(f*x+e))/(c-d))^(1/2),((c-d)/(c+d))^(1/2))+EllipticF(((c+d*sin(f*x+e))/(c-d))^(1/2),((c-d)/(c+d))^(1/2))))+(
3*a*b^2*d+b^3*c)*(-2/5/d*sin(f*x+e)*(-(-d*sin(f*x+e)-c)*cos(f*x+e)^2)^(1/2)+8/15*c/d^2*(-(-d*sin(f*x+e)-c)*cos
(f*x+e)^2)^(1/2)+4/15*c/d*(c/d-1)*((c+d*sin(f*x+e))/(c-d))^(1/2)*(d*(1-sin(f*x+e))/(c+d))^(1/2)*(1/(c-d)*(-sin
(f*x+e)-1)*d)^(1/2)/(-(-d*sin(f*x+e)-c)*cos(f*x+e)^2)^(1/2)*EllipticF(((c+d*sin(f*x+e))/(c-d))^(1/2),((c-d)/(c
+d))^(1/2))+2*(3/5+8/15*c^2/d^2)*(c/d-1)*((c+d*sin(f*x+e))/(c-d))^(1/2)*(d*(1-sin(f*x+e))/(c+d))^(1/2)*(1/(c-d
)*(-sin(f*x+e)-1)*d)^(1/2)/(-(-d*sin(f*x+e)-c)*cos(f*x+e)^2)^(1/2)*((-c/d-1)*EllipticE(((c+d*sin(f*x+e))/(c-d)
)^(1/2),((c-d)/(c+d))^(1/2))+EllipticF(((c+d*sin(f*x+e))/(c-d))^(1/2),((c-d)/(c+d))^(1/2))))+2*(a^3*d+3*a^2*b*
c)*(c/d-1)*((c+d*sin(f*x+e))/(c-d))^(1/2)*(d*(1-sin(f*x+e))/(c+d))^(1/2)*(1/(c-d)*(-sin(f*x+e)-1)*d)^(1/2)/(-(
-d*sin(f*x+e)-c)*cos(f*x+e)^2)^(1/2)*((-c/d-1)*EllipticE(((c+d*sin(f*x+e))/(c-d))^(1/2),((c-d)/(c+d))^(1/2))+E
llipticF(((c+d*sin(f*x+e))/(c-d))^(1/2),((c-d)/(c+d))^(1/2)))+(3*a^2*b*d+3*a*b^2*c)*(-2/3/d*(-(-d*sin(f*x+e)-c
)*cos(f*x+e)^2)^(1/2)+2/3*(c/d-1)*((c+d*sin(f*x+e))/(c-d))^(1/2)*(d*(1-sin(f*x+e))/(c+d))^(1/2)*(1/(c-d)*(-sin
(f*x+e)-1)*d)^(1/2)/(-(-d*sin(f*x+e)-c)*cos(f*x+e)^2)^(1/2)*EllipticF(((c+d*sin(f*x+e))/(c-d))^(1/2),((c-d)/(c
+d))^(1/2))-4/3*c/d*(c/d-1)*((c+d*sin(f*x+e))/(c-d))^(1/2)*(d*(1-sin(f*x+e))/(c+d))^(1/2)*(1/(c-d)*(-sin(f*x+e
)-1)*d)^(1/2)/(-(-d*sin(f*x+e)-c)*cos(f*x+e)^2)^(1/2)*((-c/d-1)*EllipticE(((c+d*sin(f*x+e))/(c-d))^(1/2),((c-d
)/(c+d))^(1/2))+EllipticF(((c+d*sin(f*x+e))/(c-d))^(1/2),((c-d)/(c+d))^(1/2)))))/cos(f*x+e)/(c+d*sin(f*x+e))^(
1/2)/f

Fricas [C] (verification not implemented)

Result contains higher order function than in optimal. Order 9 vs. order 4.

Time = 0.14 (sec) , antiderivative size = 730, normalized size of antiderivative = 2.03 \[ \int (3+b \sin (e+f x))^3 \sqrt {c+d \sin (e+f x)} \, dx=-\frac {\sqrt {2} {\left (16 \, b^{3} c^{4} - 84 \, a b^{2} c^{3} d + 2 \, {\left (105 \, a^{2} b + 16 \, b^{3}\right )} c^{2} d^{2} - 21 \, {\left (5 \, a^{3} + 3 \, a b^{2}\right )} c d^{3} - 15 \, {\left (21 \, a^{2} b + 5 \, b^{3}\right )} d^{4}\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 (16 \, b^{3} c^{4} - 84 \, a b^{2} c^{3} d + 2 \, {\left (105 \, a^{2} b + 16 \, b^{3}\right )} c^{2} d^{2} - 21 \, {\left (5 \, a^{3} + 3 \, a b^{2}\right )} c d^{3} - 15 \, {\left (21 \, a^{2} b + 5 \, b^{3}\right )} d^{4}\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 (8 i \, b^{3} c^{3} d - 42 i \, a b^{2} c^{2} d^{2} + i \, {\left (105 \, a^{2} b + 19 \, b^{3}\right )} c d^{3} + 21 i \, {\left (5 \, a^{3} + 9 \, a b^{2}\right )} d^{4}\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 (-8 i \, b^{3} c^{3} d + 42 i \, a b^{2} c^{2} d^{2} - i \, {\left (105 \, a^{2} b + 19 \, b^{3}\right )} c d^{3} - 21 i \, {\left (5 \, a^{3} + 9 \, a b^{2}\right )} d^{4}\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 ) - 6 \, {\left (15 \, b^{3} d^{4} \cos \left (f x + e\right )^{3} - 3 \, {\left (b^{3} c d^{3} + 21 \, a b^{2} d^{4}\right )} \cos \left (f x + e\right ) \sin \left (f x + e\right ) + {\left (4 \, b^{3} c^{2} d^{2} - 21 \, a b^{2} c d^{3} - 5 \, {\left (21 \, a^{2} b + 8 \, b^{3}\right )} d^{4}\right )} \cos \left (f x + e\right )\right )} \sqrt {d \sin \left (f x + e\right ) + c}}{315 \, d^{4} f} \]

[In]

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

[Out]

-1/315*(sqrt(2)*(16*b^3*c^4 - 84*a*b^2*c^3*d + 2*(105*a^2*b + 16*b^3)*c^2*d^2 - 21*(5*a^3 + 3*a*b^2)*c*d^3 - 1
5*(21*a^2*b + 5*b^3)*d^4)*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)*(16*b^3*c^4 - 84*a*b^2*c^3*d + 2*(105*a^
2*b + 16*b^3)*c^2*d^2 - 21*(5*a^3 + 3*a*b^2)*c*d^3 - 15*(21*a^2*b + 5*b^3)*d^4)*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)*(8*I*b^3*c^3*d - 42*I*a*b^2*c^2*d^2 + I*(105*a^2*b + 19*b^3)*c*d^3 + 21*I*(5*a^3 + 9*a*b^2)*
d^4)*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*sqrt(2)*(-8*I*b^3*c^3*d + 42*I*a*b^2*c^2*d^2 - I*(105*a^2*b + 19*b^3)*c*d^3 - 21*I*(5*a^3 + 9*a*b^2)*
d^4)*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, weierstrassPInvers
e(-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)) - 6*(15*b^3*d^4*cos(f*x + e)^3 - 3*(b^3*c*d^3 + 21*a*b^2*d^4)*cos(f*x + e)*sin(f*x + e) + (4*b^3*c^2*d
^2 - 21*a*b^2*c*d^3 - 5*(21*a^2*b + 8*b^3)*d^4)*cos(f*x + e))*sqrt(d*sin(f*x + e) + c))/(d^4*f)

Sympy [F]

\[ \int (3+b \sin (e+f x))^3 \sqrt {c+d \sin (e+f x)} \, dx=\int \left (a + b \sin {\left (e + f x \right )}\right )^{3} \sqrt {c + d \sin {\left (e + f x \right )}}\, dx \]

[In]

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

[Out]

Integral((a + b*sin(e + f*x))**3*sqrt(c + d*sin(e + f*x)), x)

Maxima [F]

\[ \int (3+b \sin (e+f x))^3 \sqrt {c+d \sin (e+f x)} \, dx=\int { {\left (b \sin \left (f x + e\right ) + a\right )}^{3} \sqrt {d \sin \left (f x + e\right ) + c} \,d x } \]

[In]

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

[Out]

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

Giac [F]

\[ \int (3+b \sin (e+f x))^3 \sqrt {c+d \sin (e+f x)} \, dx=\int { {\left (b \sin \left (f x + e\right ) + a\right )}^{3} \sqrt {d \sin \left (f x + e\right ) + c} \,d x } \]

[In]

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

[Out]

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

Mupad [F(-1)]

Timed out. \[ \int (3+b \sin (e+f x))^3 \sqrt {c+d \sin (e+f x)} \, dx=\int {\left (a+b\,\sin \left (e+f\,x\right )\right )}^3\,\sqrt {c+d\,\sin \left (e+f\,x\right )} \,d x \]

[In]

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

[Out]

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