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(d*x+c)*(a+b*sin(d*x+c))^(1/2)/d/sin(d*x+c)^(1/2)+(a-b)*EllipticE((a+b*sin(d*x+c))^(1/2)/(a+b)^(1/2)/sin(d
*x+c)^(1/2),((-a-b)/(a-b))^(1/2))*(a+b)^(1/2)*(a*(1-csc(d*x+c))/(a+b))^(1/2)*(a*(1+csc(d*x+c))/(a-b))^(1/2)*ta
n(d*x+c)/a/d-EllipticF((a+b*sin(d*x+c))^(1/2)/(a+b)^(1/2)/sin(d*x+c)^(1/2),((-a-b)/(a-b))^(1/2))*(a+b)^(1/2)*(
a*(1-csc(d*x+c))/(a+b))^(1/2)*(a*(1+csc(d*x+c))/(a-b))^(1/2)*tan(d*x+c)/d+a*EllipticPi((a+b*sin(d*x+c))^(1/2)/
(a+b)^(1/2)/sin(d*x+c)^(1/2),(a+b)/b,((-a-b)/(a-b))^(1/2))*(a+b)^(1/2)*(a*(1-csc(d*x+c))/(a+b))^(1/2)*(a*(1+cs
c(d*x+c))/(a-b))^(1/2)*tan(d*x+c)/b/d

________________________________________________________________________________________

Rubi [A]  time = 0.56, antiderivative size = 371, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 7, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.280, 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 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 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 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 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 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]

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]

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 = 26.83, 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

________________________________________________________________________________________

fricas [F]  time = 2.70, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\sqrt {b \sin \left (d x + c\right ) + a} \sqrt {\sin \left (d x + c\right )}, x\right ) \]

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)

________________________________________________________________________________________

giac [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]

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 >> An error occurred running a Giac command:INPUT:sage2:=int(sage0,x):;OUTPUT:Simp
lification assuming c near 0Unable to check sign: (2*pi/x/2)>(-2*pi/x/2)Unable to check sign: (2*pi/x/2)>(-2*p
i/x/2)Unable to check sign: (2*pi/x/2)>(-2*pi/x/2)Unable to check sign: (2*pi/x/2)>(-2*pi/x/2)Simplification a
ssuming c near 0Unable to check sign: (2*pi/x/2)>(-2*pi/x/2)Unable to check sign: (2*pi/x/2)>(-2*pi/x/2)Unable
to check sign: (2*pi/x/2)>(-2*pi/x/2)ext_reduce Error: Bad Argument TypeSimplification assuming c near 0Unabl
e to check sign: (2*pi/x/2)>(-2*pi/x/2)Unable to check sign: (2*pi/x/2)>(-2*pi/x/2)Unable to check sign: (2*pi
/x/2)>(-2*pi/x/2)Unable to check sign: (2*pi/x/2)>(-2*pi/x/2)Simplification assuming c near 0Unable to check s
ign: (2*pi/x/2)>(-2*pi/x/2)Unable to check sign: (2*pi/x/2)>(-2*pi/x/2)Unable to check sign: (2*pi/x/2)>(-2*pi
/x/2)ext_reduce Error: Bad Argument TypeDone

________________________________________________________________________________________

maple [C]  time = 0.76, size = 9823, normalized size = 26.48 \[ \text {output too large to display} \]

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

________________________________________________________________________________________

maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \sqrt {b \sin \left (d x + c\right ) + a} \sqrt {\sin \left (d x + c\right )}\,{d x} \]

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)

________________________________________________________________________________________

mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \[ \int \sqrt {\sin \left (c+d\,x\right )}\,\sqrt {a+b\,\sin \left (c+d\,x\right )} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \sqrt {a + b \sin {\left (c + d x \right )}} \sqrt {\sin {\left (c + d x \right )}}\, dx \]

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)

________________________________________________________________________________________