3.5.84 \(\int (a+a \sin (e+f x)) \sqrt {c+d \sin (e+f x)} \, dx\) [484]

Optimal. Leaf size=179 \[ -\frac {2 a \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{3 f}+\frac {2 a (c+3 d) E\left (\frac {1}{2} \left (e-\frac {\pi }{2}+f x\right )|\frac {2 d}{c+d}\right ) \sqrt {c+d \sin (e+f x)}}{3 d f \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}-\frac {2 a \left (c^2-d^2\right ) F\left (\frac {1}{2} \left (e-\frac {\pi }{2}+f x\right )|\frac {2 d}{c+d}\right ) \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}{3 d f \sqrt {c+d \sin (e+f x)}} \]

[Out]

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

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Rubi [A]
time = 0.13, antiderivative size = 179, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.240, Rules used = {2832, 2831, 2742, 2740, 2734, 2732} \begin {gather*} -\frac {2 a \left (c^2-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 )}{3 d f \sqrt {c+d \sin (e+f x)}}-\frac {2 a \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{3 f}+\frac {2 a (c+3 d) \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 )}{3 d f \sqrt {\frac {c+d \sin (e+f x)}{c+d}}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rubi steps

\begin {align*} \int (a+a \sin (e+f x)) \sqrt {c+d \sin (e+f x)} \, dx &=-\frac {2 a \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{3 f}+\frac {2}{3} \int \frac {\frac {1}{2} a (3 c+d)+\frac {1}{2} a (c+3 d) \sin (e+f x)}{\sqrt {c+d \sin (e+f x)}} \, dx\\ &=-\frac {2 a \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{3 f}+\frac {(a (c+3 d)) \int \sqrt {c+d \sin (e+f x)} \, dx}{3 d}-\frac {\left (a \left (c^2-d^2\right )\right ) \int \frac {1}{\sqrt {c+d \sin (e+f x)}} \, dx}{3 d}\\ &=-\frac {2 a \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{3 f}+\frac {\left (a (c+3 d) \sqrt {c+d \sin (e+f x)}\right ) \int \sqrt {\frac {c}{c+d}+\frac {d \sin (e+f x)}{c+d}} \, dx}{3 d \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}-\frac {\left (a \left (c^2-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}{3 d \sqrt {c+d \sin (e+f x)}}\\ &=-\frac {2 a \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{3 f}+\frac {2 a (c+3 d) 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)}}{3 d f \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}-\frac {2 a \left (c^2-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}}}{3 d f \sqrt {c+d \sin (e+f x)}}\\ \end {align*}

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Mathematica [C] Result contains higher order function than in optimal. Order 6 vs. order 4 in optimal.
time = 6.21, size = 1736, normalized size = 9.70 \begin {gather*} a \left (\frac {c \sec (e) (1+\sin (e+f x)) \left (-\frac {F_1\left (-\frac {1}{2};-\frac {1}{2},-\frac {1}{2};\frac {1}{2};-\frac {\csc (e) \left (c+d \cos \left (f x-\tan ^{-1}(\cot (e))\right ) \sqrt {1+\cot ^2(e)} \sin (e)\right )}{d \sqrt {1+\cot ^2(e)} \left (1-\frac {c \csc (e)}{d \sqrt {1+\cot ^2(e)}}\right )},-\frac {\csc (e) \left (c+d \cos \left (f x-\tan ^{-1}(\cot (e))\right ) \sqrt {1+\cot ^2(e)} \sin (e)\right )}{d \sqrt {1+\cot ^2(e)} \left (-1-\frac {c \csc (e)}{d \sqrt {1+\cot ^2(e)}}\right )}\right ) \cot (e) \sin \left (f x-\tan ^{-1}(\cot (e))\right )}{\sqrt {1+\cot ^2(e)} \sqrt {\frac {d \sqrt {1+\cot ^2(e)}+d \cos \left (f x-\tan ^{-1}(\cot (e))\right ) \sqrt {1+\cot ^2(e)}}{d \sqrt {1+\cot ^2(e)}-c \csc (e)}} \sqrt {\frac {d \sqrt {1+\cot ^2(e)}-d \cos \left (f x-\tan ^{-1}(\cot (e))\right ) \sqrt {1+\cot ^2(e)}}{d \sqrt {1+\cot ^2(e)}+c \csc (e)}} \sqrt {c+d \cos \left (f x-\tan ^{-1}(\cot (e))\right ) \sqrt {1+\cot ^2(e)} \sin (e)}}-\frac {\frac {2 d \sin (e) \left (c+d \cos \left (f x-\tan ^{-1}(\cot (e))\right ) \sqrt {1+\cot ^2(e)} \sin (e)\right )}{d^2 \cos ^2(e)+d^2 \sin ^2(e)}-\frac {\cot (e) \sin \left (f x-\tan ^{-1}(\cot (e))\right )}{\sqrt {1+\cot ^2(e)}}}{\sqrt {c+d \cos \left (f x-\tan ^{-1}(\cot (e))\right ) \sqrt {1+\cot ^2(e)} \sin (e)}}\right )}{3 f \left (\cos \left (\frac {e}{2}+\frac {f x}{2}\right )+\sin \left (\frac {e}{2}+\frac {f x}{2}\right )\right )^2}+\frac {d \sec (e) (1+\sin (e+f x)) \left (-\frac {F_1\left (-\frac {1}{2};-\frac {1}{2},-\frac {1}{2};\frac {1}{2};-\frac {\csc (e) \left (c+d \cos \left (f x-\tan ^{-1}(\cot (e))\right ) \sqrt {1+\cot ^2(e)} \sin (e)\right )}{d \sqrt {1+\cot ^2(e)} \left (1-\frac {c \csc (e)}{d \sqrt {1+\cot ^2(e)}}\right )},-\frac {\csc (e) \left (c+d \cos \left (f x-\tan ^{-1}(\cot (e))\right ) \sqrt {1+\cot ^2(e)} \sin (e)\right )}{d \sqrt {1+\cot ^2(e)} \left (-1-\frac {c \csc (e)}{d \sqrt {1+\cot ^2(e)}}\right )}\right ) \cot (e) \sin \left (f x-\tan ^{-1}(\cot (e))\right )}{\sqrt {1+\cot ^2(e)} \sqrt {\frac {d \sqrt {1+\cot ^2(e)}+d \cos \left (f x-\tan ^{-1}(\cot (e))\right ) \sqrt {1+\cot ^2(e)}}{d \sqrt {1+\cot ^2(e)}-c \csc (e)}} \sqrt {\frac {d \sqrt {1+\cot ^2(e)}-d \cos \left (f x-\tan ^{-1}(\cot (e))\right ) \sqrt {1+\cot ^2(e)}}{d \sqrt {1+\cot ^2(e)}+c \csc (e)}} \sqrt {c+d \cos \left (f x-\tan ^{-1}(\cot (e))\right ) \sqrt {1+\cot ^2(e)} \sin (e)}}-\frac {\frac {2 d \sin (e) \left (c+d \cos \left (f x-\tan ^{-1}(\cot (e))\right ) \sqrt {1+\cot ^2(e)} \sin (e)\right )}{d^2 \cos ^2(e)+d^2 \sin ^2(e)}-\frac {\cot (e) \sin \left (f x-\tan ^{-1}(\cot (e))\right )}{\sqrt {1+\cot ^2(e)}}}{\sqrt {c+d \cos \left (f x-\tan ^{-1}(\cot (e))\right ) \sqrt {1+\cot ^2(e)} \sin (e)}}\right )}{f \left (\cos \left (\frac {e}{2}+\frac {f x}{2}\right )+\sin \left (\frac {e}{2}+\frac {f x}{2}\right )\right )^2}+\frac {(1+\sin (e+f x)) \sqrt {c+d \sin (e+f x)} \left (-\frac {2 \cos (e) \cos (f x)}{3 f}+\frac {2 \sin (e) \sin (f x)}{3 f}+\frac {2 (c+3 d) \tan (e)}{3 d f}\right )}{\left (\cos \left (\frac {e}{2}+\frac {f x}{2}\right )+\sin \left (\frac {e}{2}+\frac {f x}{2}\right )\right )^2}+\frac {2 F_1\left (\frac {1}{2};\frac {1}{2},\frac {1}{2};\frac {3}{2};-\frac {\sec (e) \left (c+d \cos (e) \sin \left (f x+\tan ^{-1}(\tan (e))\right ) \sqrt {1+\tan ^2(e)}\right )}{d \sqrt {1+\tan ^2(e)} \left (1-\frac {c \sec (e)}{d \sqrt {1+\tan ^2(e)}}\right )},-\frac {\sec (e) \left (c+d \cos (e) \sin \left (f x+\tan ^{-1}(\tan (e))\right ) \sqrt {1+\tan ^2(e)}\right )}{d \sqrt {1+\tan ^2(e)} \left (-1-\frac {c \sec (e)}{d \sqrt {1+\tan ^2(e)}}\right )}\right ) \sec (e) \sec \left (f x+\tan ^{-1}(\tan (e))\right ) (1+\sin (e+f x)) \sqrt {\frac {d \sqrt {1+\tan ^2(e)}-d \sin \left (f x+\tan ^{-1}(\tan (e))\right ) \sqrt {1+\tan ^2(e)}}{c \sec (e)+d \sqrt {1+\tan ^2(e)}}} \sqrt {\frac {d \sqrt {1+\tan ^2(e)}+d \sin \left (f x+\tan ^{-1}(\tan (e))\right ) \sqrt {1+\tan ^2(e)}}{-c \sec (e)+d \sqrt {1+\tan ^2(e)}}} \sqrt {c+d \cos (e) \sin \left (f x+\tan ^{-1}(\tan (e))\right ) \sqrt {1+\tan ^2(e)}}}{3 f \left (\cos \left (\frac {e}{2}+\frac {f x}{2}\right )+\sin \left (\frac {e}{2}+\frac {f x}{2}\right )\right )^2 \sqrt {1+\tan ^2(e)}}+\frac {2 c F_1\left (\frac {1}{2};\frac {1}{2},\frac {1}{2};\frac {3}{2};-\frac {\sec (e) \left (c+d \cos (e) \sin \left (f x+\tan ^{-1}(\tan (e))\right ) \sqrt {1+\tan ^2(e)}\right )}{d \sqrt {1+\tan ^2(e)} \left (1-\frac {c \sec (e)}{d \sqrt {1+\tan ^2(e)}}\right )},-\frac {\sec (e) \left (c+d \cos (e) \sin \left (f x+\tan ^{-1}(\tan (e))\right ) \sqrt {1+\tan ^2(e)}\right )}{d \sqrt {1+\tan ^2(e)} \left (-1-\frac {c \sec (e)}{d \sqrt {1+\tan ^2(e)}}\right )}\right ) \sec (e) \sec \left (f x+\tan ^{-1}(\tan (e))\right ) (1+\sin (e+f x)) \sqrt {\frac {d \sqrt {1+\tan ^2(e)}-d \sin \left (f x+\tan ^{-1}(\tan (e))\right ) \sqrt {1+\tan ^2(e)}}{c \sec (e)+d \sqrt {1+\tan ^2(e)}}} \sqrt {\frac {d \sqrt {1+\tan ^2(e)}+d \sin \left (f x+\tan ^{-1}(\tan (e))\right ) \sqrt {1+\tan ^2(e)}}{-c \sec (e)+d \sqrt {1+\tan ^2(e)}}} \sqrt {c+d \cos (e) \sin \left (f x+\tan ^{-1}(\tan (e))\right ) \sqrt {1+\tan ^2(e)}}}{d f \left (\cos \left (\frac {e}{2}+\frac {f x}{2}\right )+\sin \left (\frac {e}{2}+\frac {f x}{2}\right )\right )^2 \sqrt {1+\tan ^2(e)}}\right ) \end {gather*}

Warning: Unable to verify antiderivative.

[In]

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

[Out]

a*((c*Sec[e]*(1 + Sin[e + f*x])*(-((AppellF1[-1/2, -1/2, -1/2, 1/2, -((Csc[e]*(c + d*Cos[f*x - ArcTan[Cot[e]]]
*Sqrt[1 + Cot[e]^2]*Sin[e]))/(d*Sqrt[1 + Cot[e]^2]*(1 - (c*Csc[e])/(d*Sqrt[1 + Cot[e]^2])))), -((Csc[e]*(c + d
*Cos[f*x - ArcTan[Cot[e]]]*Sqrt[1 + Cot[e]^2]*Sin[e]))/(d*Sqrt[1 + Cot[e]^2]*(-1 - (c*Csc[e])/(d*Sqrt[1 + Cot[
e]^2]))))]*Cot[e]*Sin[f*x - ArcTan[Cot[e]]])/(Sqrt[1 + Cot[e]^2]*Sqrt[(d*Sqrt[1 + Cot[e]^2] + d*Cos[f*x - ArcT
an[Cot[e]]]*Sqrt[1 + Cot[e]^2])/(d*Sqrt[1 + Cot[e]^2] - c*Csc[e])]*Sqrt[(d*Sqrt[1 + Cot[e]^2] - d*Cos[f*x - Ar
cTan[Cot[e]]]*Sqrt[1 + Cot[e]^2])/(d*Sqrt[1 + Cot[e]^2] + c*Csc[e])]*Sqrt[c + d*Cos[f*x - ArcTan[Cot[e]]]*Sqrt
[1 + Cot[e]^2]*Sin[e]])) - ((2*d*Sin[e]*(c + d*Cos[f*x - ArcTan[Cot[e]]]*Sqrt[1 + Cot[e]^2]*Sin[e]))/(d^2*Cos[
e]^2 + d^2*Sin[e]^2) - (Cot[e]*Sin[f*x - ArcTan[Cot[e]]])/Sqrt[1 + Cot[e]^2])/Sqrt[c + d*Cos[f*x - ArcTan[Cot[
e]]]*Sqrt[1 + Cot[e]^2]*Sin[e]]))/(3*f*(Cos[e/2 + (f*x)/2] + Sin[e/2 + (f*x)/2])^2) + (d*Sec[e]*(1 + Sin[e + f
*x])*(-((AppellF1[-1/2, -1/2, -1/2, 1/2, -((Csc[e]*(c + d*Cos[f*x - ArcTan[Cot[e]]]*Sqrt[1 + Cot[e]^2]*Sin[e])
)/(d*Sqrt[1 + Cot[e]^2]*(1 - (c*Csc[e])/(d*Sqrt[1 + Cot[e]^2])))), -((Csc[e]*(c + d*Cos[f*x - ArcTan[Cot[e]]]*
Sqrt[1 + Cot[e]^2]*Sin[e]))/(d*Sqrt[1 + Cot[e]^2]*(-1 - (c*Csc[e])/(d*Sqrt[1 + Cot[e]^2]))))]*Cot[e]*Sin[f*x -
ArcTan[Cot[e]]])/(Sqrt[1 + Cot[e]^2]*Sqrt[(d*Sqrt[1 + Cot[e]^2] + d*Cos[f*x - ArcTan[Cot[e]]]*Sqrt[1 + Cot[e]
^2])/(d*Sqrt[1 + Cot[e]^2] - c*Csc[e])]*Sqrt[(d*Sqrt[1 + Cot[e]^2] - d*Cos[f*x - ArcTan[Cot[e]]]*Sqrt[1 + Cot[
e]^2])/(d*Sqrt[1 + Cot[e]^2] + c*Csc[e])]*Sqrt[c + d*Cos[f*x - ArcTan[Cot[e]]]*Sqrt[1 + Cot[e]^2]*Sin[e]])) -
((2*d*Sin[e]*(c + d*Cos[f*x - ArcTan[Cot[e]]]*Sqrt[1 + Cot[e]^2]*Sin[e]))/(d^2*Cos[e]^2 + d^2*Sin[e]^2) - (Cot
[e]*Sin[f*x - ArcTan[Cot[e]]])/Sqrt[1 + Cot[e]^2])/Sqrt[c + d*Cos[f*x - ArcTan[Cot[e]]]*Sqrt[1 + Cot[e]^2]*Sin
[e]]))/(f*(Cos[e/2 + (f*x)/2] + Sin[e/2 + (f*x)/2])^2) + ((1 + Sin[e + f*x])*Sqrt[c + d*Sin[e + f*x]]*((-2*Cos
[e]*Cos[f*x])/(3*f) + (2*Sin[e]*Sin[f*x])/(3*f) + (2*(c + 3*d)*Tan[e])/(3*d*f)))/(Cos[e/2 + (f*x)/2] + Sin[e/2
+ (f*x)/2])^2 + (2*AppellF1[1/2, 1/2, 1/2, 3/2, -((Sec[e]*(c + d*Cos[e]*Sin[f*x + ArcTan[Tan[e]]]*Sqrt[1 + Ta
n[e]^2]))/(d*Sqrt[1 + Tan[e]^2]*(1 - (c*Sec[e])/(d*Sqrt[1 + Tan[e]^2])))), -((Sec[e]*(c + d*Cos[e]*Sin[f*x + A
rcTan[Tan[e]]]*Sqrt[1 + Tan[e]^2]))/(d*Sqrt[1 + Tan[e]^2]*(-1 - (c*Sec[e])/(d*Sqrt[1 + Tan[e]^2]))))]*Sec[e]*S
ec[f*x + ArcTan[Tan[e]]]*(1 + Sin[e + f*x])*Sqrt[(d*Sqrt[1 + Tan[e]^2] - d*Sin[f*x + ArcTan[Tan[e]]]*Sqrt[1 +
Tan[e]^2])/(c*Sec[e] + d*Sqrt[1 + Tan[e]^2])]*Sqrt[(d*Sqrt[1 + Tan[e]^2] + d*Sin[f*x + ArcTan[Tan[e]]]*Sqrt[1
+ Tan[e]^2])/(-(c*Sec[e]) + d*Sqrt[1 + Tan[e]^2])]*Sqrt[c + d*Cos[e]*Sin[f*x + ArcTan[Tan[e]]]*Sqrt[1 + Tan[e]
^2]])/(3*f*(Cos[e/2 + (f*x)/2] + Sin[e/2 + (f*x)/2])^2*Sqrt[1 + Tan[e]^2]) + (2*c*AppellF1[1/2, 1/2, 1/2, 3/2,
-((Sec[e]*(c + d*Cos[e]*Sin[f*x + ArcTan[Tan[e]]]*Sqrt[1 + Tan[e]^2]))/(d*Sqrt[1 + Tan[e]^2]*(1 - (c*Sec[e])/
(d*Sqrt[1 + Tan[e]^2])))), -((Sec[e]*(c + d*Cos[e]*Sin[f*x + ArcTan[Tan[e]]]*Sqrt[1 + Tan[e]^2]))/(d*Sqrt[1 +
Tan[e]^2]*(-1 - (c*Sec[e])/(d*Sqrt[1 + Tan[e]^2]))))]*Sec[e]*Sec[f*x + ArcTan[Tan[e]]]*(1 + Sin[e + f*x])*Sqrt
[(d*Sqrt[1 + Tan[e]^2] - d*Sin[f*x + ArcTan[Tan[e]]]*Sqrt[1 + Tan[e]^2])/(c*Sec[e] + d*Sqrt[1 + Tan[e]^2])]*Sq
rt[(d*Sqrt[1 + Tan[e]^2] + d*Sin[f*x + ArcTan[Tan[e]]]*Sqrt[1 + Tan[e]^2])/(-(c*Sec[e]) + d*Sqrt[1 + Tan[e]^2]
)]*Sqrt[c + d*Cos[e]*Sin[f*x + ArcTan[Tan[e]]]*Sqrt[1 + Tan[e]^2]])/(d*f*(Cos[e/2 + (f*x)/2] + Sin[e/2 + (f*x)
/2])^2*Sqrt[1 + Tan[e]^2]))

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(656\) vs. \(2(229)=458\).
time = 4.68, size = 657, normalized size = 3.67

method	result	size
default	\(\frac {2 a \left (4 \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^{2} d -4 \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}-\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}-3 \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 +\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^{2}+3 \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^{3}+d^{3} \left (\sin ^{3}\left (f x +e \right )\right )+c \,d^{2} \left (\sin ^{2}\left (f x +e \right )\right )-d^{3} \sin \left (f x +e \right )-c \,d^{2}\right )}{3 d^{2} \cos \left (f x +e \right ) \sqrt {c +d \sin \left (f x +e \right )}\, f}\)	\(657\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/3*a*(4*((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)*Ellipt
icF(((c+d*sin(f*x+e))/(c-d))^(1/2),((c-d)/(c+d))^(1/2))*c^2*d-4*((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-((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^3-3*((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
+((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*d^2+3*((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^3+d^3
*sin(f*x+e)^3+c*d^2*sin(f*x+e)^2-d^3*sin(f*x+e)-c*d^2)/d^2/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))*(c+d*sin(f*x+e))^(1/2),x, algorithm="maxima")

[Out]

integrate((a*sin(f*x + e) + a)*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 = 0.12, size = 453, normalized size = 2.53 \begin {gather*} -\frac {6 \, \sqrt {d \sin \left (f x + e\right ) + c} a d^{2} \cos \left (f x + e\right ) + \sqrt {2} {\left (2 \, a c^{2} - 3 \, a c d - 3 \, a d^{2}\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 (2 \, a c^{2} - 3 \, a c d - 3 \, a d^{2}\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 (i \, a c d + 3 i \, a d^{2}\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 (-i \, a c d - 3 i \, a d^{2}\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 )}{9 \, d^{2} f} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/9*(6*sqrt(d*sin(f*x + e) + c)*a*d^2*cos(f*x + e) + sqrt(2)*(2*a*c^2 - 3*a*c*d - 3*a*d^2)*sqrt(I*d)*weierstr
assPInverse(-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)*(2*a*c^2 - 3*a*c*d - 3*a*d^2)*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)*(I*a*c
*d + 3*I*a*d^2)*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, weierstra
ssPInverse(-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)*(-I*a*c*d - 3*I*a*d^2)*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)))/(d^2*f)

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} a \left (\int \sqrt {c + d \sin {\left (e + f x \right )}} \sin {\left (e + f x \right )}\, dx + \int \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))*(c+d*sin(f*x+e))**(1/2),x)

[Out]

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

[Out]

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

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \left (a+a\,\sin \left (e+f\,x\right )\right )\,\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))*(c + d*sin(e + f*x))^(1/2),x)

[Out]

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

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