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

Optimal. Leaf size=371 \[ -\frac{\cos (c+d x) \sqrt{a+b \sin (c+d x)}}{d \sqrt{\sin (c+d x)}}-\frac{\sqrt{a+b} \tan (c+d x) \sqrt{\frac{a (1-\csc (c+d x))}{a+b}} \sqrt{\frac{a (\csc (c+d x)+1)}{a-b}} F\left (\sin ^{-1}\left (\frac{\sqrt{a+b \sin (c+d x)}}{\sqrt{a+b} \sqrt{\sin (c+d x)}}\right )|-\frac{a+b}{a-b}\right )}{d}+\frac{(a-b) \sqrt{a+b} \tan (c+d x) \sqrt{\frac{a (1-\csc (c+d x))}{a+b}} \sqrt{\frac{a (\csc (c+d x)+1)}{a-b}} E\left (\sin ^{-1}\left (\frac{\sqrt{a+b \sin (c+d x)}}{\sqrt{a+b} \sqrt{\sin (c+d x)}}\right )|-\frac{a+b}{a-b}\right )}{a d}+\frac{a \sqrt{a+b} \tan (c+d x) \sqrt{\frac{a (1-\csc (c+d x))}{a+b}} \sqrt{\frac{a (\csc (c+d x)+1)}{a-b}} \Pi \left (\frac{a+b}{b};\sin ^{-1}\left (\frac{\sqrt{a+b \sin (c+d x)}}{\sqrt{a+b} \sqrt{\sin (c+d x)}}\right )|-\frac{a+b}{a-b}\right )}{b d} \]

[Out]

-((Cos[c + d*x]*Sqrt[a + b*Sin[c + d*x]])/(d*Sqrt[Sin[c + d*x]])) + ((a - b)*Sqrt[a + b]*Sqrt[(a*(1 - Csc[c +
d*x]))/(a + b)]*Sqrt[(a*(1 + Csc[c + d*x]))/(a - b)]*EllipticE[ArcSin[Sqrt[a + b*Sin[c + d*x]]/(Sqrt[a + b]*Sq
rt[Sin[c + d*x]])], -((a + b)/(a - b))]*Tan[c + d*x])/(a*d) - (Sqrt[a + b]*Sqrt[(a*(1 - Csc[c + d*x]))/(a + b)
]*Sqrt[(a*(1 + Csc[c + d*x]))/(a - b)]*EllipticF[ArcSin[Sqrt[a + b*Sin[c + d*x]]/(Sqrt[a + b]*Sqrt[Sin[c + d*x
]])], -((a + b)/(a - b))]*Tan[c + d*x])/d + (a*Sqrt[a + b]*Sqrt[(a*(1 - Csc[c + d*x]))/(a + b)]*Sqrt[(a*(1 + C
sc[c + d*x]))/(a - b)]*EllipticPi[(a + b)/b, ArcSin[Sqrt[a + b*Sin[c + d*x]]/(Sqrt[a + b]*Sqrt[Sin[c + d*x]])]
, -((a + b)/(a - b))]*Tan[c + d*x])/(b*d)

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Rubi [A]  time = 0.563021, antiderivative size = 371, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 7, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.28, Rules used = {2821, 3054, 2809, 12, 2801, 2816, 2994} \[ -\frac{\cos (c+d x) \sqrt{a+b \sin (c+d x)}}{d \sqrt{\sin (c+d x)}}-\frac{\sqrt{a+b} \tan (c+d x) \sqrt{\frac{a (1-\csc (c+d x))}{a+b}} \sqrt{\frac{a (\csc (c+d x)+1)}{a-b}} F\left (\sin ^{-1}\left (\frac{\sqrt{a+b \sin (c+d x)}}{\sqrt{a+b} \sqrt{\sin (c+d x)}}\right )|-\frac{a+b}{a-b}\right )}{d}+\frac{(a-b) \sqrt{a+b} \tan (c+d x) \sqrt{\frac{a (1-\csc (c+d x))}{a+b}} \sqrt{\frac{a (\csc (c+d x)+1)}{a-b}} E\left (\sin ^{-1}\left (\frac{\sqrt{a+b \sin (c+d x)}}{\sqrt{a+b} \sqrt{\sin (c+d x)}}\right )|-\frac{a+b}{a-b}\right )}{a d}+\frac{a \sqrt{a+b} \tan (c+d x) \sqrt{\frac{a (1-\csc (c+d x))}{a+b}} \sqrt{\frac{a (\csc (c+d x)+1)}{a-b}} \Pi \left (\frac{a+b}{b};\sin ^{-1}\left (\frac{\sqrt{a+b \sin (c+d x)}}{\sqrt{a+b} \sqrt{\sin (c+d x)}}\right )|-\frac{a+b}{a-b}\right )}{b d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-((Cos[c + d*x]*Sqrt[a + b*Sin[c + d*x]])/(d*Sqrt[Sin[c + d*x]])) + ((a - b)*Sqrt[a + b]*Sqrt[(a*(1 - Csc[c +
d*x]))/(a + b)]*Sqrt[(a*(1 + Csc[c + d*x]))/(a - b)]*EllipticE[ArcSin[Sqrt[a + b*Sin[c + d*x]]/(Sqrt[a + b]*Sq
rt[Sin[c + d*x]])], -((a + b)/(a - b))]*Tan[c + d*x])/(a*d) - (Sqrt[a + b]*Sqrt[(a*(1 - Csc[c + d*x]))/(a + b)
]*Sqrt[(a*(1 + Csc[c + d*x]))/(a - b)]*EllipticF[ArcSin[Sqrt[a + b*Sin[c + d*x]]/(Sqrt[a + b]*Sqrt[Sin[c + d*x
]])], -((a + b)/(a - b))]*Tan[c + d*x])/d + (a*Sqrt[a + b]*Sqrt[(a*(1 - Csc[c + d*x]))/(a + b)]*Sqrt[(a*(1 + C
sc[c + d*x]))/(a - b)]*EllipticPi[(a + b)/b, ArcSin[Sqrt[a + b*Sin[c + d*x]]/(Sqrt[a + b]*Sqrt[Sin[c + d*x]])]
, -((a + b)/(a - b))]*Tan[c + d*x])/(b*d)

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 3054

Int[((A_.) + (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 - 2*a*b*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, C}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && NeQ[c^2 - d^
2, 0]

Rule 2809

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

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 2816

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

Rule 2994

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

Rubi steps

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

Mathematica [C]  time = 26.7561, size = 10847, normalized size = 29.24 \[ \text{Result too large to show} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

Result too large to show

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Maple [C]  time = 0.568, size = 9567, normalized size = 25.8 \begin{align*} \text{output too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sin(d*x+c)^(1/2)*(a+b*sin(d*x+c))^(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 (d x + c\right ) + a} \sqrt{\sin \left (d x + c\right )}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\sqrt{b \sin \left (d x + c\right ) + a} \sqrt{\sin \left (d x + c\right )}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(sqrt(b*sin(d*x + c) + a)*sqrt(sin(d*x + c)), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: TypeError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: TypeError