Integrand size = 25, antiderivative size = 371 \[ \int \sqrt {\sin (c+d x)} \sqrt {a+b \sin (c+d x)} \, dx=-\frac {\cos (c+d x) \sqrt {a+b \sin (c+d x)}}{d \sqrt {\sin (c+d x)}}+\frac {(a-b) \sqrt {a+b} \sqrt {\frac {a (1-\csc (c+d x))}{a+b}} \sqrt {\frac {a (1+\csc (c+d x))}{a-b}} E\left (\arcsin \left (\frac {\sqrt {a+b \sin (c+d x)}}{\sqrt {a+b} \sqrt {\sin (c+d x)}}\right )|-\frac {a+b}{a-b}\right ) \tan (c+d x)}{a d}-\frac {\sqrt {a+b} \sqrt {\frac {a (1-\csc (c+d x))}{a+b}} \sqrt {\frac {a (1+\csc (c+d x))}{a-b}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {a+b \sin (c+d x)}}{\sqrt {a+b} \sqrt {\sin (c+d x)}}\right ),-\frac {a+b}{a-b}\right ) \tan (c+d x)}{d}+\frac {a \sqrt {a+b} \sqrt {\frac {a (1-\csc (c+d x))}{a+b}} \sqrt {\frac {a (1+\csc (c+d x))}{a-b}} \operatorname {EllipticPi}\left (\frac {a+b}{b},\arcsin \left (\frac {\sqrt {a+b \sin (c+d x)}}{\sqrt {a+b} \sqrt {\sin (c+d x)}}\right ),-\frac {a+b}{a-b}\right ) \tan (c+d x)}{b d} \]
-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)*tan( 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-c sc(d*x+c))/(a+b))^(1/2)*(a*(1+csc(d*x+c))/(a-b))^(1/2)*tan(d*x+c)/b/d
Result contains complex when optimal does not.
Time = 27.53 (sec) , antiderivative size = 10847, normalized size of antiderivative = 29.24 \[ \int \sqrt {\sin (c+d x)} \sqrt {a+b \sin (c+d x)} \, dx=\text {Result too large to show} \]
Time = 1.37 (sec) , antiderivative size = 385, normalized size of antiderivative = 1.04, number of steps used = 12, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.480, Rules used = {3042, 3300, 27, 3042, 3533, 27, 3042, 3280, 3042, 3288, 3295, 3473}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
\(\displaystyle \int \sqrt {\sin (c+d x)} \sqrt {a+b \sin (c+d x)} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \sqrt {\sin (c+d x)} \sqrt {a+b \sin (c+d x)}dx\) |
\(\Big \downarrow \) 3300 |
\(\displaystyle \frac {\int -\frac {a b-a b \sin ^2(c+d x)}{2 \sin ^{\frac {3}{2}}(c+d x) \sqrt {a+b \sin (c+d x)}}dx}{b}-\frac {\cos (c+d x) \sqrt {a+b \sin (c+d x)}}{d \sqrt {\sin (c+d x)}}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle -\frac {\int \frac {a b-a b \sin ^2(c+d x)}{\sin ^{\frac {3}{2}}(c+d x) \sqrt {a+b \sin (c+d x)}}dx}{2 b}-\frac {\cos (c+d x) \sqrt {a+b \sin (c+d x)}}{d \sqrt {\sin (c+d x)}}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle -\frac {\int \frac {a b-a b \sin (c+d x)^2}{\sin (c+d x)^{3/2} \sqrt {a+b \sin (c+d x)}}dx}{2 b}-\frac {\cos (c+d x) \sqrt {a+b \sin (c+d x)}}{d \sqrt {\sin (c+d x)}}\) |
\(\Big \downarrow \) 3533 |
\(\displaystyle -\frac {\int \frac {a b}{\sin ^{\frac {3}{2}}(c+d x) \sqrt {a+b \sin (c+d x)}}dx-a b \int \frac {\sqrt {\sin (c+d x)}}{\sqrt {a+b \sin (c+d x)}}dx}{2 b}-\frac {\cos (c+d x) \sqrt {a+b \sin (c+d x)}}{d \sqrt {\sin (c+d x)}}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle -\frac {a b \int \frac {1}{\sin ^{\frac {3}{2}}(c+d x) \sqrt {a+b \sin (c+d x)}}dx-a b \int \frac {\sqrt {\sin (c+d x)}}{\sqrt {a+b \sin (c+d x)}}dx}{2 b}-\frac {\cos (c+d x) \sqrt {a+b \sin (c+d x)}}{d \sqrt {\sin (c+d x)}}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle -\frac {a b \int \frac {1}{\sin (c+d x)^{3/2} \sqrt {a+b \sin (c+d x)}}dx-a b \int \frac {\sqrt {\sin (c+d x)}}{\sqrt {a+b \sin (c+d x)}}dx}{2 b}-\frac {\cos (c+d x) \sqrt {a+b \sin (c+d x)}}{d \sqrt {\sin (c+d x)}}\) |
\(\Big \downarrow \) 3280 |
\(\displaystyle -\frac {a b \left (\int \frac {\sin (c+d x)+1}{\sin ^{\frac {3}{2}}(c+d x) \sqrt {a+b \sin (c+d x)}}dx-\int \frac {1}{\sqrt {\sin (c+d x)} \sqrt {a+b \sin (c+d x)}}dx\right )-a b \int \frac {\sqrt {\sin (c+d x)}}{\sqrt {a+b \sin (c+d x)}}dx}{2 b}-\frac {\cos (c+d x) \sqrt {a+b \sin (c+d x)}}{d \sqrt {\sin (c+d x)}}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle -\frac {a b \left (\int \frac {\sin (c+d x)+1}{\sin (c+d x)^{3/2} \sqrt {a+b \sin (c+d x)}}dx-\int \frac {1}{\sqrt {\sin (c+d x)} \sqrt {a+b \sin (c+d x)}}dx\right )-a b \int \frac {\sqrt {\sin (c+d x)}}{\sqrt {a+b \sin (c+d x)}}dx}{2 b}-\frac {\cos (c+d x) \sqrt {a+b \sin (c+d x)}}{d \sqrt {\sin (c+d x)}}\) |
\(\Big \downarrow \) 3288 |
\(\displaystyle -\frac {a b \left (\int \frac {\sin (c+d x)+1}{\sin (c+d x)^{3/2} \sqrt {a+b \sin (c+d x)}}dx-\int \frac {1}{\sqrt {\sin (c+d x)} \sqrt {a+b \sin (c+d x)}}dx\right )-\frac {2 a \sqrt {a+b} \tan (c+d x) \sqrt {\frac {a (1-\csc (c+d x))}{a+b}} \sqrt {\frac {a (\csc (c+d x)+1)}{a-b}} \operatorname {EllipticPi}\left (\frac {a+b}{b},\arcsin \left (\frac {\sqrt {a+b \sin (c+d x)}}{\sqrt {a+b} \sqrt {\sin (c+d x)}}\right ),-\frac {a+b}{a-b}\right )}{d}}{2 b}-\frac {\cos (c+d x) \sqrt {a+b \sin (c+d x)}}{d \sqrt {\sin (c+d x)}}\) |
\(\Big \downarrow \) 3295 |
\(\displaystyle -\frac {a b \left (\int \frac {\sin (c+d x)+1}{\sin (c+d x)^{3/2} \sqrt {a+b \sin (c+d x)}}dx+\frac {2 \sqrt {a+b} \tan (c+d x) \sqrt {\frac {a (1-\csc (c+d x))}{a+b}} \sqrt {\frac {a (\csc (c+d x)+1)}{a-b}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {a+b \sin (c+d x)}}{\sqrt {a+b} \sqrt {\sin (c+d x)}}\right ),-\frac {a+b}{a-b}\right )}{a d}\right )-\frac {2 a \sqrt {a+b} \tan (c+d x) \sqrt {\frac {a (1-\csc (c+d x))}{a+b}} \sqrt {\frac {a (\csc (c+d x)+1)}{a-b}} \operatorname {EllipticPi}\left (\frac {a+b}{b},\arcsin \left (\frac {\sqrt {a+b \sin (c+d x)}}{\sqrt {a+b} \sqrt {\sin (c+d x)}}\right ),-\frac {a+b}{a-b}\right )}{d}}{2 b}-\frac {\cos (c+d x) \sqrt {a+b \sin (c+d x)}}{d \sqrt {\sin (c+d x)}}\) |
\(\Big \downarrow \) 3473 |
\(\displaystyle -\frac {a b \left (\frac {2 \sqrt {a+b} \tan (c+d x) \sqrt {\frac {a (1-\csc (c+d x))}{a+b}} \sqrt {\frac {a (\csc (c+d x)+1)}{a-b}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {a+b \sin (c+d x)}}{\sqrt {a+b} \sqrt {\sin (c+d x)}}\right ),-\frac {a+b}{a-b}\right )}{a d}-\frac {2 (a-b) \sqrt {a+b} \tan (c+d x) \sqrt {\frac {a (1-\csc (c+d x))}{a+b}} \sqrt {\frac {a (\csc (c+d x)+1)}{a-b}} E\left (\arcsin \left (\frac {\sqrt {a+b \sin (c+d x)}}{\sqrt {a+b} \sqrt {\sin (c+d x)}}\right )|-\frac {a+b}{a-b}\right )}{a^2 d}\right )-\frac {2 a \sqrt {a+b} \tan (c+d x) \sqrt {\frac {a (1-\csc (c+d x))}{a+b}} \sqrt {\frac {a (\csc (c+d x)+1)}{a-b}} \operatorname {EllipticPi}\left (\frac {a+b}{b},\arcsin \left (\frac {\sqrt {a+b \sin (c+d x)}}{\sqrt {a+b} \sqrt {\sin (c+d x)}}\right ),-\frac {a+b}{a-b}\right )}{d}}{2 b}-\frac {\cos (c+d x) \sqrt {a+b \sin (c+d x)}}{d \sqrt {\sin (c+d x)}}\) |
-((Cos[c + d*x]*Sqrt[a + b*Sin[c + d*x]])/(d*Sqrt[Sin[c + d*x]])) - ((-2*a *Sqrt[a + b]*Sqrt[(a*(1 - Csc[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Csc[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])/d + a*b*((-2 *(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 ]*Sqrt[Sin[c + d*x]])], -((a + b)/(a - b))]*Tan[c + d*x])/(a^2*d) + (2*Sqr t[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])/(a*d)))/(2*b)
3.3.11.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[1/(((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(3/2)*Sqrt[(c_.) + (d_.)*sin [(e_.) + (f_.)*(x_)]]), x_Symbol] :> Simp[1/(a - b) Int[1/(Sqrt[a + b*Sin [e + f*x]]*Sqrt[c + d*Sin[e + f*x]]), x], x] - Simp[b/(a - b) Int[(1 + Si n[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]
Int[Sqrt[(b_.)*sin[(e_.) + (f_.)*(x_)]]/Sqrt[(c_) + (d_.)*sin[(e_.) + (f_.) *(x_)]], x_Symbol] :> Simp[2*b*(Tan[e + f*x]/(d*f))*Rt[(c + d)/b, 2]*Sqrt[c *((1 + Csc[e + f*x])/(c - d))]*Sqrt[c*((1 - Csc[e + f*x])/(c + d))]*Ellipti cPi[(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)], x] /; FreeQ[{b, c, d, e, f}, x] && NeQ[c^2 - d^2, 0] && PosQ[(c + d)/b]
Int[1/(Sqrt[(d_.)*sin[(e_.) + (f_.)*(x_)]]*Sqrt[(a_) + (b_.)*sin[(e_.) + (f _.)*(x_)]]), x_Symbol] :> Simp[-2*(Tan[e + f*x]/(a*f))*Rt[(a + b)/d, 2]*Sqr t[a*((1 - Csc[e + f*x])/(a + b))]*Sqrt[a*((1 + Csc[e + f*x])/(a - b))]*Elli pticF[ArcSin[Sqrt[a + b*Sin[e + f*x]]/Sqrt[d*Sin[e + f*x]]/Rt[(a + b)/d, 2] ], -(a + b)/(a - b)], x] /; FreeQ[{a, b, d, e, f}, x] && NeQ[a^2 - b^2, 0] && PosQ[(a + b)/d]
Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Simp[(-b)*Cos[e + f*x]*(a + b*Sin[e + f*x] )^(m - 1)*((c + d*Sin[e + f*x])^n/(f*(m + n))), x] + Simp[1/(d*(m + n)) I nt[(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))*Sin[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])
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]/(f*b*c^2))*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)], x] /; FreeQ[{b, c, d, e, f, A, B}, x] && NeQ[c^2 - d^2, 0] && EqQ[A, B] && PosQ[(c + d)/b]
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] : > Simp[C/b^2 Int[Sqrt[a + b*Sin[e + f*x]]/Sqrt[c + d*Sin[e + f*x]], x], x ] + Simp[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]
Result contains complex when optimal does not.
Time = 5.26 (sec) , antiderivative size = 8415, normalized size of antiderivative = 22.68
Timed out. \[ \int \sqrt {\sin (c+d x)} \sqrt {a+b \sin (c+d x)} \, dx=\text {Timed out} \]
\[ \int \sqrt {\sin (c+d x)} \sqrt {a+b \sin (c+d x)} \, dx=\int \sqrt {a + b \sin {\left (c + d x \right )}} \sqrt {\sin {\left (c + d x \right )}}\, dx \]
\[ \int \sqrt {\sin (c+d x)} \sqrt {a+b \sin (c+d x)} \, dx=\int { \sqrt {b \sin \left (d x + c\right ) + a} \sqrt {\sin \left (d x + c\right )} \,d x } \]
Exception generated. \[ \int \sqrt {\sin (c+d x)} \sqrt {a+b \sin (c+d x)} \, dx=\text {Exception raised: TypeError} \]
Exception raised: TypeError >> an error occurred running a Giac command:IN PUT:sage2:=int(sage0,sageVARx):;OUTPUT:The choice was done assuming 0=[0,0 ,0]ext_reduce Error: Bad Argument TypeThe choice was done assuming 0=[0,0, 0]ext_red
Timed out. \[ \int \sqrt {\sin (c+d x)} \sqrt {a+b \sin (c+d x)} \, dx=\int \sqrt {\sin \left (c+d\,x\right )}\,\sqrt {a+b\,\sin \left (c+d\,x\right )} \,d x \]