Integrand size = 27, antiderivative size = 161 \[ \int \frac {\sqrt {a+a \sin (c+d x)}}{\sqrt {e \cos (c+d x)}} \, dx=-\frac {2 \text {arcsinh}\left (\frac {\sqrt {e \cos (c+d x)}}{\sqrt {e}}\right ) \sqrt {1+\cos (c+d x)} \sqrt {a+a \sin (c+d x)}}{d \sqrt {e} (1+\cos (c+d x)+\sin (c+d x))}+\frac {2 \arctan \left (\frac {\sqrt {e} \sin (c+d x)}{\sqrt {e \cos (c+d x)} \sqrt {1+\cos (c+d x)}}\right ) \sqrt {1+\cos (c+d x)} \sqrt {a+a \sin (c+d x)}}{d \sqrt {e} (1+\cos (c+d x)+\sin (c+d x))} \]
-2*arcsinh((e*cos(d*x+c))^(1/2)/e^(1/2))*(1+cos(d*x+c))^(1/2)*(a+a*sin(d*x +c))^(1/2)/d/(1+cos(d*x+c)+sin(d*x+c))/e^(1/2)+2*arctan(sin(d*x+c)*e^(1/2) /(e*cos(d*x+c))^(1/2)/(1+cos(d*x+c))^(1/2))*(1+cos(d*x+c))^(1/2)*(a+a*sin( d*x+c))^(1/2)/d/(1+cos(d*x+c)+sin(d*x+c))/e^(1/2)
Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.
Time = 0.14 (sec) , antiderivative size = 75, normalized size of antiderivative = 0.47 \[ \int \frac {\sqrt {a+a \sin (c+d x)}}{\sqrt {e \cos (c+d x)}} \, dx=-\frac {2\ 2^{3/4} \sqrt {e \cos (c+d x)} \operatorname {Hypergeometric2F1}\left (\frac {1}{4},\frac {1}{4},\frac {5}{4},\frac {1}{2} (1-\sin (c+d x))\right ) \sqrt {a (1+\sin (c+d x))}}{d e (1+\sin (c+d x))^{3/4}} \]
(-2*2^(3/4)*Sqrt[e*Cos[c + d*x]]*Hypergeometric2F1[1/4, 1/4, 5/4, (1 - Sin [c + d*x])/2]*Sqrt[a*(1 + Sin[c + d*x])])/(d*e*(1 + Sin[c + d*x])^(3/4))
Time = 0.60 (sec) , antiderivative size = 161, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {3042, 3156, 3042, 25, 3254, 216, 3312, 63, 222}
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.
| \(\displaystyle \int \frac {\sqrt {a \sin (c+d x)+a}}{\sqrt {e \cos (c+d x)}} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\sqrt {a \sin (c+d x)+a}}{\sqrt {e \cos (c+d x)}}dx\) |
| \(\Big \downarrow \) 3156 |
| \(\displaystyle \frac {\sqrt {\cos (c+d x)+1} \sqrt {a \sin (c+d x)+a} \int \frac {\sqrt {\cos (c+d x)+1}}{\sqrt {e \cos (c+d x)}}dx}{\sin (c+d x)+\cos (c+d x)+1}+\frac {\sqrt {\cos (c+d x)+1} \sqrt {a \sin (c+d x)+a} \int \frac {\sin (c+d x)}{\sqrt {e \cos (c+d x)} \sqrt {\cos (c+d x)+1}}dx}{\sin (c+d x)+\cos (c+d x)+1}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\sqrt {\cos (c+d x)+1} \sqrt {a \sin (c+d x)+a} \int -\frac {\cos \left (c+d x+\frac {\pi }{2}\right )}{\sqrt {e \sin \left (c+d x+\frac {\pi }{2}\right )} \sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )+1}}dx}{\sin (c+d x)+\cos (c+d x)+1}+\frac {\sqrt {\cos (c+d x)+1} \sqrt {a \sin (c+d x)+a} \int \frac {\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )+1}}{\sqrt {e \sin \left (c+d x+\frac {\pi }{2}\right )}}dx}{\sin (c+d x)+\cos (c+d x)+1}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\sqrt {\cos (c+d x)+1} \sqrt {a \sin (c+d x)+a} \int \frac {\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )+1}}{\sqrt {e \sin \left (c+d x+\frac {\pi }{2}\right )}}dx}{\sin (c+d x)+\cos (c+d x)+1}-\frac {\sqrt {\cos (c+d x)+1} \sqrt {a \sin (c+d x)+a} \int \frac {\cos \left (\frac {1}{2} (2 c+\pi )+d x\right )}{\sqrt {e \sin \left (\frac {1}{2} (2 c+\pi )+d x\right )} \sqrt {\sin \left (\frac {1}{2} (2 c+\pi )+d x\right )+1}}dx}{\sin (c+d x)+\cos (c+d x)+1}\) |
| \(\Big \downarrow \) 3254 |
| \(\displaystyle -\frac {\sqrt {\cos (c+d x)+1} \sqrt {a \sin (c+d x)+a} \int \frac {\cos \left (\frac {1}{2} (2 c+\pi )+d x\right )}{\sqrt {e \sin \left (\frac {1}{2} (2 c+\pi )+d x\right )} \sqrt {\sin \left (\frac {1}{2} (2 c+\pi )+d x\right )+1}}dx}{\sin (c+d x)+\cos (c+d x)+1}-\frac {2 \sqrt {\cos (c+d x)+1} \sqrt {a \sin (c+d x)+a} \int \frac {1}{\frac {\sin (c+d x) \tan (c+d x)}{\cos (c+d x)+1}+1}d\left (-\frac {\sin (c+d x)}{\sqrt {e \cos (c+d x)} \sqrt {\cos (c+d x)+1}}\right )}{d (\sin (c+d x)+\cos (c+d x)+1)}\) |
| \(\Big \downarrow \) 216 |
| \(\displaystyle \frac {2 \sqrt {\cos (c+d x)+1} \sqrt {a \sin (c+d x)+a} \arctan \left (\frac {\sqrt {e} \sin (c+d x)}{\sqrt {\cos (c+d x)+1} \sqrt {e \cos (c+d x)}}\right )}{d \sqrt {e} (\sin (c+d x)+\cos (c+d x)+1)}-\frac {\sqrt {\cos (c+d x)+1} \sqrt {a \sin (c+d x)+a} \int \frac {\cos \left (\frac {1}{2} (2 c+\pi )+d x\right )}{\sqrt {e \sin \left (\frac {1}{2} (2 c+\pi )+d x\right )} \sqrt {\sin \left (\frac {1}{2} (2 c+\pi )+d x\right )+1}}dx}{\sin (c+d x)+\cos (c+d x)+1}\) |
| \(\Big \downarrow \) 3312 |
| \(\displaystyle \frac {2 \sqrt {\cos (c+d x)+1} \sqrt {a \sin (c+d x)+a} \arctan \left (\frac {\sqrt {e} \sin (c+d x)}{\sqrt {\cos (c+d x)+1} \sqrt {e \cos (c+d x)}}\right )}{d \sqrt {e} (\sin (c+d x)+\cos (c+d x)+1)}-\frac {\sqrt {\cos (c+d x)+1} \sqrt {a \sin (c+d x)+a} \int \frac {1}{\sqrt {e \cos (c+d x)} \sqrt {\cos (c+d x)+1}}d\cos (c+d x)}{d (\sin (c+d x)+\cos (c+d x)+1)}\) |
| \(\Big \downarrow \) 63 |
| \(\displaystyle \frac {2 \sqrt {\cos (c+d x)+1} \sqrt {a \sin (c+d x)+a} \arctan \left (\frac {\sqrt {e} \sin (c+d x)}{\sqrt {\cos (c+d x)+1} \sqrt {e \cos (c+d x)}}\right )}{d \sqrt {e} (\sin (c+d x)+\cos (c+d x)+1)}-\frac {2 \sqrt {\cos (c+d x)+1} \sqrt {a \sin (c+d x)+a} \int \frac {1}{\sqrt {\cos (c+d x)+1}}d\sqrt {e \cos (c+d x)}}{d e (\sin (c+d x)+\cos (c+d x)+1)}\) |
| \(\Big \downarrow \) 222 |
| \(\displaystyle \frac {2 \sqrt {\cos (c+d x)+1} \sqrt {a \sin (c+d x)+a} \arctan \left (\frac {\sqrt {e} \sin (c+d x)}{\sqrt {\cos (c+d x)+1} \sqrt {e \cos (c+d x)}}\right )}{d \sqrt {e} (\sin (c+d x)+\cos (c+d x)+1)}-\frac {2 \sqrt {\cos (c+d x)+1} \sqrt {a \sin (c+d x)+a} \text {arcsinh}\left (\frac {\sqrt {e \cos (c+d x)}}{\sqrt {e}}\right )}{d \sqrt {e} (\sin (c+d x)+\cos (c+d x)+1)}\) |
(-2*ArcSinh[Sqrt[e*Cos[c + d*x]]/Sqrt[e]]*Sqrt[1 + Cos[c + d*x]]*Sqrt[a + a*Sin[c + d*x]])/(d*Sqrt[e]*(1 + Cos[c + d*x] + Sin[c + d*x])) + (2*ArcTan [(Sqrt[e]*Sin[c + d*x])/(Sqrt[e*Cos[c + d*x]]*Sqrt[1 + Cos[c + d*x]])]*Sqr t[1 + Cos[c + d*x]]*Sqrt[a + a*Sin[c + d*x]])/(d*Sqrt[e]*(1 + Cos[c + d*x] + Sin[c + d*x]))
3.3.75.3.1 Defintions of rubi rules used
Int[1/(Sqrt[(b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_)]), x_Symbol] :> Simp[2/b S ubst[Int[1/Sqrt[c + d*(x^2/b)], x], x, Sqrt[b*x]], x] /; FreeQ[{b, c, d}, x ] && GtQ[c, 0]
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[b, 2]))*A rcTan[Rt[b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a , 0] || GtQ[b, 0])
Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Simp[ArcSinh[Rt[b, 2]*(x/Sqrt [a])]/Rt[b, 2], x] /; FreeQ[{a, b}, x] && GtQ[a, 0] && PosQ[b]
Int[Sqrt[(a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]]/Sqrt[cos[(e_.) + (f_.)*(x_)] *(g_.)], x_Symbol] :> Simp[a*Sqrt[1 + Cos[e + f*x]]*(Sqrt[a + b*Sin[e + f*x ]]/(a + a*Cos[e + f*x] + b*Sin[e + f*x])) Int[Sqrt[1 + Cos[e + f*x]]/Sqrt [g*Cos[e + f*x]], x], x] + Simp[b*Sqrt[1 + Cos[e + f*x]]*(Sqrt[a + b*Sin[e + f*x]]/(a + a*Cos[e + f*x] + b*Sin[e + f*x])) Int[Sin[e + f*x]/(Sqrt[g*C os[e + f*x]]*Sqrt[1 + Cos[e + f*x]]), x], x] /; FreeQ[{a, b, e, f, g}, x] & & EqQ[a^2 - b^2, 0]
Int[Sqrt[(a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]]/Sqrt[(c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)]], x_Symbol] :> Simp[-2*(b/f) Subst[Int[1/(b + d*x^2), x], x, b*(Cos[e + f*x]/(Sqrt[a + b*Sin[e + f*x]]*Sqrt[c + d*Sin[e + f*x]]))], x ] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0]
Int[cos[(e_.) + (f_.)*(x_)]*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*(( c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_.), x_Symbol] :> Simp[1/(b*f) Su bst[Int[(a + x)^m*(c + (d/b)*x)^n, x], x, b*Sin[e + f*x]], x] /; FreeQ[{a, b, c, d, e, f, m, n}, x]
Time = 2.00 (sec) , antiderivative size = 134, normalized size of antiderivative = 0.83
| method | result | size |
| default | \(-\frac {\sqrt {a \left (1+\sin \left (d x +c \right )\right )}\, \left (\arctan \left (\sqrt {-\frac {\cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\right )+\operatorname {arctanh}\left (\frac {\sin \left (d x +c \right )}{\left (1+\cos \left (d x +c \right )\right ) \sqrt {-\frac {\cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}}\right )\right ) \left (\cos \left (d x +c \right )-\sin \left (d x +c \right )+1\right )}{d \left (1+\cos \left (d x +c \right )\right ) \sqrt {-\frac {\cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\, \sqrt {e \cos \left (d x +c \right )}}\) | \(134\) |
-1/d*(a*(1+sin(d*x+c)))^(1/2)*(arctan((-cos(d*x+c)/(1+cos(d*x+c)))^(1/2))+ arctanh(sin(d*x+c)/(1+cos(d*x+c))/(-cos(d*x+c)/(1+cos(d*x+c)))^(1/2)))*(co s(d*x+c)-sin(d*x+c)+1)/(1+cos(d*x+c))/(-cos(d*x+c)/(1+cos(d*x+c)))^(1/2)/( e*cos(d*x+c))^(1/2)
Result contains complex when optimal does not.
Time = 0.42 (sec) , antiderivative size = 885, normalized size of antiderivative = 5.50 \[ \int \frac {\sqrt {a+a \sin (c+d x)}}{\sqrt {e \cos (c+d x)}} \, dx=\text {Too large to display} \]
1/2*I*(-a^2/(d^4*e^2))^(1/4)*log(-(2*sqrt(e*cos(d*x + c))*(a*sin(d*x + c) + (d^2*e*cos(d*x + c) + d^2*e)*sqrt(-a^2/(d^4*e^2)))*sqrt(a*sin(d*x + c) + a) - (I*d^3*e^2*cos(d*x + c) + I*d^3*e^2 + (2*I*d^3*e^2*cos(d*x + c) + I* d^3*e^2)*sin(d*x + c))*(-a^2/(d^4*e^2))^(3/4) - (-2*I*a*d*e*cos(d*x + c)^2 - I*a*d*e*cos(d*x + c) + I*a*d*e*sin(d*x + c) + I*a*d*e)*(-a^2/(d^4*e^2)) ^(1/4))/(cos(d*x + c) + sin(d*x + c) + 1)) - 1/2*I*(-a^2/(d^4*e^2))^(1/4)* log(-(2*sqrt(e*cos(d*x + c))*(a*sin(d*x + c) + (d^2*e*cos(d*x + c) + d^2*e )*sqrt(-a^2/(d^4*e^2)))*sqrt(a*sin(d*x + c) + a) - (-I*d^3*e^2*cos(d*x + c ) - I*d^3*e^2 + (-2*I*d^3*e^2*cos(d*x + c) - I*d^3*e^2)*sin(d*x + c))*(-a^ 2/(d^4*e^2))^(3/4) - (2*I*a*d*e*cos(d*x + c)^2 + I*a*d*e*cos(d*x + c) - I* a*d*e*sin(d*x + c) - I*a*d*e)*(-a^2/(d^4*e^2))^(1/4))/(cos(d*x + c) + sin( d*x + c) + 1)) + 1/2*(-a^2/(d^4*e^2))^(1/4)*log(-(2*sqrt(e*cos(d*x + c))*( a*sin(d*x + c) - (d^2*e*cos(d*x + c) + d^2*e)*sqrt(-a^2/(d^4*e^2)))*sqrt(a *sin(d*x + c) + a) + (d^3*e^2*cos(d*x + c) + d^3*e^2 + (2*d^3*e^2*cos(d*x + c) + d^3*e^2)*sin(d*x + c))*(-a^2/(d^4*e^2))^(3/4) + (2*a*d*e*cos(d*x + c)^2 + a*d*e*cos(d*x + c) - a*d*e*sin(d*x + c) - a*d*e)*(-a^2/(d^4*e^2))^( 1/4))/(cos(d*x + c) + sin(d*x + c) + 1)) - 1/2*(-a^2/(d^4*e^2))^(1/4)*log( -(2*sqrt(e*cos(d*x + c))*(a*sin(d*x + c) - (d^2*e*cos(d*x + c) + d^2*e)*sq rt(-a^2/(d^4*e^2)))*sqrt(a*sin(d*x + c) + a) - (d^3*e^2*cos(d*x + c) + d^3 *e^2 + (2*d^3*e^2*cos(d*x + c) + d^3*e^2)*sin(d*x + c))*(-a^2/(d^4*e^2)...
\[ \int \frac {\sqrt {a+a \sin (c+d x)}}{\sqrt {e \cos (c+d x)}} \, dx=\int \frac {\sqrt {a \left (\sin {\left (c + d x \right )} + 1\right )}}{\sqrt {e \cos {\left (c + d x \right )}}}\, dx \]
\[ \int \frac {\sqrt {a+a \sin (c+d x)}}{\sqrt {e \cos (c+d x)}} \, dx=\int { \frac {\sqrt {a \sin \left (d x + c\right ) + a}}{\sqrt {e \cos \left (d x + c\right )}} \,d x } \]
\[ \int \frac {\sqrt {a+a \sin (c+d x)}}{\sqrt {e \cos (c+d x)}} \, dx=\int { \frac {\sqrt {a \sin \left (d x + c\right ) + a}}{\sqrt {e \cos \left (d x + c\right )}} \,d x } \]
Timed out. \[ \int \frac {\sqrt {a+a \sin (c+d x)}}{\sqrt {e \cos (c+d x)}} \, dx=\int \frac {\sqrt {a+a\,\sin \left (c+d\,x\right )}}{\sqrt {e\,\cos \left (c+d\,x\right )}} \,d x \]