3.767 \(\int \sqrt{a+b \sin (e+f x)} \sqrt{c+d \sin (e+f x)} \, dx\)

Optimal. Leaf size=628 \[ -\frac{b \cos (e+f x) \sqrt{c+d \sin (e+f x)}}{f \sqrt{a+b \sin (e+f x)}}+\frac{(a+b)^{3/2} \sec (e+f x) (c+d \sin (e+f x)) \sqrt{\frac{(b c-a d) (1-\sin (e+f x))}{(a+b) (c+d \sin (e+f x))}} \sqrt{-\frac{(b c-a d) (\sin (e+f x)+1)}{(a-b) (c+d \sin (e+f x))}} F\left (\sin ^{-1}\left (\frac{\sqrt{c+d} \sqrt{a+b \sin (e+f x)}}{\sqrt{a+b} \sqrt{c+d \sin (e+f x)}}\right )|\frac{(a+b) (c-d)}{(a-b) (c+d)}\right )}{b f \sqrt{c+d}}+\frac{\sqrt{a+b} (c-d) \sqrt{c+d} \sec (e+f x) (a+b \sin (e+f x)) \sqrt{-\frac{(b c-a d) (1-\sin (e+f x))}{(c+d) (a+b \sin (e+f x))}} \sqrt{\frac{(b c-a d) (\sin (e+f x)+1)}{(c-d) (a+b \sin (e+f x))}} E\left (\sin ^{-1}\left (\frac{\sqrt{a+b} \sqrt{c+d \sin (e+f x)}}{\sqrt{c+d} \sqrt{a+b \sin (e+f x)}}\right )|\frac{(a-b) (c+d)}{(a+b) (c-d)}\right )}{f (b c-a d)}+\frac{\sqrt{c+d} (a d+b c) \sec (e+f x) (a+b \sin (e+f x)) \sqrt{-\frac{(b c-a d) (1-\sin (e+f x))}{(c+d) (a+b \sin (e+f x))}} \sqrt{\frac{(b c-a d) (\sin (e+f x)+1)}{(c-d) (a+b \sin (e+f x))}} \Pi \left (\frac{b (c+d)}{(a+b) d};\sin ^{-1}\left (\frac{\sqrt{a+b} \sqrt{c+d \sin (e+f x)}}{\sqrt{c+d} \sqrt{a+b \sin (e+f x)}}\right )|\frac{(a-b) (c+d)}{(a+b) (c-d)}\right )}{b d f \sqrt{a+b}} \]

[Out]

(Sqrt[a + b]*(c - d)*Sqrt[c + d]*EllipticE[ArcSin[(Sqrt[a + b]*Sqrt[c + d*Sin[e + f*x]])/(Sqrt[c + d]*Sqrt[a +
b*Sin[e + f*x]])], ((a - b)*(c + d))/((a + b)*(c - d))]*Sec[e + f*x]*Sqrt[-(((b*c - a*d)*(1 - Sin[e + f*x]))/
((c + d)*(a + b*Sin[e + f*x])))]*Sqrt[((b*c - a*d)*(1 + Sin[e + f*x]))/((c - d)*(a + b*Sin[e + f*x]))]*(a + b*
Sin[e + f*x]))/((b*c - a*d)*f) + (Sqrt[c + d]*(b*c + a*d)*EllipticPi[(b*(c + d))/((a + b)*d), ArcSin[(Sqrt[a +
b]*Sqrt[c + d*Sin[e + f*x]])/(Sqrt[c + d]*Sqrt[a + b*Sin[e + f*x]])], ((a - b)*(c + d))/((a + b)*(c - d))]*Se
c[e + f*x]*Sqrt[-(((b*c - a*d)*(1 - Sin[e + f*x]))/((c + d)*(a + b*Sin[e + f*x])))]*Sqrt[((b*c - a*d)*(1 + Sin
[e + f*x]))/((c - d)*(a + b*Sin[e + f*x]))]*(a + b*Sin[e + f*x]))/(b*Sqrt[a + b]*d*f) - (b*Cos[e + f*x]*Sqrt[c
+ d*Sin[e + f*x]])/(f*Sqrt[a + b*Sin[e + f*x]]) + ((a + b)^(3/2)*EllipticF[ArcSin[(Sqrt[c + d]*Sqrt[a + b*Sin
[e + f*x]])/(Sqrt[a + b]*Sqrt[c + d*Sin[e + f*x]])], ((a + b)*(c - d))/((a - b)*(c + d))]*Sec[e + f*x]*Sqrt[((
b*c - a*d)*(1 - Sin[e + f*x]))/((a + b)*(c + d*Sin[e + f*x]))]*Sqrt[-(((b*c - a*d)*(1 + Sin[e + f*x]))/((a - b
)*(c + d*Sin[e + f*x])))]*(c + d*Sin[e + f*x]))/(b*Sqrt[c + d]*f)

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Rubi [A]  time = 1.32657, antiderivative size = 628, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 7, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.241, Rules used = {2821, 3053, 2811, 12, 2801, 2818, 2996} \[ -\frac{b \cos (e+f x) \sqrt{c+d \sin (e+f x)}}{f \sqrt{a+b \sin (e+f x)}}+\frac{(a+b)^{3/2} \sec (e+f x) (c+d \sin (e+f x)) \sqrt{\frac{(b c-a d) (1-\sin (e+f x))}{(a+b) (c+d \sin (e+f x))}} \sqrt{-\frac{(b c-a d) (\sin (e+f x)+1)}{(a-b) (c+d \sin (e+f x))}} F\left (\sin ^{-1}\left (\frac{\sqrt{c+d} \sqrt{a+b \sin (e+f x)}}{\sqrt{a+b} \sqrt{c+d \sin (e+f x)}}\right )|\frac{(a+b) (c-d)}{(a-b) (c+d)}\right )}{b f \sqrt{c+d}}+\frac{\sqrt{a+b} (c-d) \sqrt{c+d} \sec (e+f x) (a+b \sin (e+f x)) \sqrt{-\frac{(b c-a d) (1-\sin (e+f x))}{(c+d) (a+b \sin (e+f x))}} \sqrt{\frac{(b c-a d) (\sin (e+f x)+1)}{(c-d) (a+b \sin (e+f x))}} E\left (\sin ^{-1}\left (\frac{\sqrt{a+b} \sqrt{c+d \sin (e+f x)}}{\sqrt{c+d} \sqrt{a+b \sin (e+f x)}}\right )|\frac{(a-b) (c+d)}{(a+b) (c-d)}\right )}{f (b c-a d)}+\frac{\sqrt{c+d} (a d+b c) \sec (e+f x) (a+b \sin (e+f x)) \sqrt{-\frac{(b c-a d) (1-\sin (e+f x))}{(c+d) (a+b \sin (e+f x))}} \sqrt{\frac{(b c-a d) (\sin (e+f x)+1)}{(c-d) (a+b \sin (e+f x))}} \Pi \left (\frac{b (c+d)}{(a+b) d};\sin ^{-1}\left (\frac{\sqrt{a+b} \sqrt{c+d \sin (e+f x)}}{\sqrt{c+d} \sqrt{a+b \sin (e+f x)}}\right )|\frac{(a-b) (c+d)}{(a+b) (c-d)}\right )}{b d f \sqrt{a+b}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(Sqrt[a + b]*(c - d)*Sqrt[c + d]*EllipticE[ArcSin[(Sqrt[a + b]*Sqrt[c + d*Sin[e + f*x]])/(Sqrt[c + d]*Sqrt[a +
b*Sin[e + f*x]])], ((a - b)*(c + d))/((a + b)*(c - d))]*Sec[e + f*x]*Sqrt[-(((b*c - a*d)*(1 - Sin[e + f*x]))/
((c + d)*(a + b*Sin[e + f*x])))]*Sqrt[((b*c - a*d)*(1 + Sin[e + f*x]))/((c - d)*(a + b*Sin[e + f*x]))]*(a + b*
Sin[e + f*x]))/((b*c - a*d)*f) + (Sqrt[c + d]*(b*c + a*d)*EllipticPi[(b*(c + d))/((a + b)*d), ArcSin[(Sqrt[a +
b]*Sqrt[c + d*Sin[e + f*x]])/(Sqrt[c + d]*Sqrt[a + b*Sin[e + f*x]])], ((a - b)*(c + d))/((a + b)*(c - d))]*Se
c[e + f*x]*Sqrt[-(((b*c - a*d)*(1 - Sin[e + f*x]))/((c + d)*(a + b*Sin[e + f*x])))]*Sqrt[((b*c - a*d)*(1 + Sin
[e + f*x]))/((c - d)*(a + b*Sin[e + f*x]))]*(a + b*Sin[e + f*x]))/(b*Sqrt[a + b]*d*f) - (b*Cos[e + f*x]*Sqrt[c
+ d*Sin[e + f*x]])/(f*Sqrt[a + b*Sin[e + f*x]]) + ((a + b)^(3/2)*EllipticF[ArcSin[(Sqrt[c + d]*Sqrt[a + b*Sin
[e + f*x]])/(Sqrt[a + b]*Sqrt[c + d*Sin[e + f*x]])], ((a + b)*(c - d))/((a - b)*(c + d))]*Sec[e + f*x]*Sqrt[((
b*c - a*d)*(1 - Sin[e + f*x]))/((a + b)*(c + d*Sin[e + f*x]))]*Sqrt[-(((b*c - a*d)*(1 + Sin[e + f*x]))/((a - b
)*(c + d*Sin[e + f*x])))]*(c + d*Sin[e + f*x]))/(b*Sqrt[c + d]*f)

Rule 2821

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

Rule 3053

Int[((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)] + (C_.)*sin[(e_.) + (f_.)*(x_)]^2)/(((a_.) + (b_.)*sin[(e_.) + (f_.
)*(x_)])^(3/2)*Sqrt[(c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)]]), x_Symbol] :> Dist[C/b^2, Int[Sqrt[a + b*Sin[e + f
*x]]/Sqrt[c + d*Sin[e + f*x]], x], x] + Dist[1/b^2, Int[(A*b^2 - a^2*C + b*(b*B - 2*a*C)*Sin[e + f*x])/((a + b
*Sin[e + f*x])^(3/2)*Sqrt[c + d*Sin[e + f*x]]), x], x] /; FreeQ[{a, b, c, d, e, f, A, B, C}, x] && NeQ[b*c - a
*d, 0] && NeQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0]

Rule 2811

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

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 2801

Int[1/(((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(3/2)*Sqrt[(c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)]]), x_Symbol] :
> Dist[1/(a - b), Int[1/(Sqrt[a + b*Sin[e + f*x]]*Sqrt[c + d*Sin[e + f*x]]), x], x] - Dist[b/(a - b), Int[(1 +
Sin[e + f*x])/((a + b*Sin[e + f*x])^(3/2)*Sqrt[c + d*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] && NeQ[c^2 - d^2, 0]

Rule 2818

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

Rule 2996

Int[((A_) + (B_.)*sin[(e_.) + (f_.)*(x_)])/(((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(3/2)*Sqrt[(c_) + (d_.)*sin
[(e_.) + (f_.)*(x_)]]), x_Symbol] :> Simp[(-2*A*(c - d)*(a + b*Sin[e + f*x])*Sqrt[((b*c - a*d)*(1 + Sin[e + f*
x]))/((c - d)*(a + b*Sin[e + f*x]))]*Sqrt[-(((b*c - a*d)*(1 - Sin[e + f*x]))/((c + d)*(a + b*Sin[e + f*x])))]*
EllipticE[ArcSin[(Rt[(a + b)/(c + d), 2]*Sqrt[c + d*Sin[e + f*x]])/Sqrt[a + b*Sin[e + f*x]]], ((a - b)*(c + d)
)/((a + b)*(c - d))])/(f*(b*c - a*d)^2*Rt[(a + b)/(c + d), 2]*Cos[e + f*x]), x] /; FreeQ[{a, b, c, d, e, f, A,
B}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && EqQ[A, B] && PosQ[(a + b)/(c + d)]

Rubi steps

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

Mathematica [C]  time = 31.6832, size = 228392, normalized size = 363.68 \[ \text{Result too large to show} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

Result too large to show

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Maple [C]  time = 1.348, size = 146613, normalized size = 233.5 \begin{align*} \text{output too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

result too large to display

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \sqrt{b \sin \left (f x + e\right ) + a} \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+b*sin(f*x+e))^(1/2)*(c+d*sin(f*x+e))^(1/2),x, algorithm="maxima")

[Out]

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

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

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \sqrt{b \sin \left (f x + e\right ) + a} \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+b*sin(f*x+e))^(1/2)*(c+d*sin(f*x+e))^(1/2),x, algorithm="giac")

[Out]

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