Integrand size = 27, antiderivative size = 236 \[ \int (e \cos (c+d x))^{3/2} \sqrt {a+a \sin (c+d x)} \, dx=-\frac {a (e \cos (c+d x))^{5/2}}{2 d e \sqrt {a+a \sin (c+d x)}}+\frac {3 e \sqrt {e \cos (c+d x)} \sqrt {a+a \sin (c+d x)}}{4 d}-\frac {3 e^{3/2} \text {arcsinh}\left (\frac {\sqrt {e \cos (c+d x)}}{\sqrt {e}}\right ) \sqrt {1+\cos (c+d x)} \sqrt {a+a \sin (c+d x)}}{4 d (1+\cos (c+d x)+\sin (c+d x))}+\frac {3 e^{3/2} \arctan \left (\frac {\sqrt {e} \sin (c+d x)}{\sqrt {e \cos (c+d x)} \sqrt {1+\cos (c+d x)}}\right ) \sqrt {1+\cos (c+d x)} \sqrt {a+a \sin (c+d x)}}{4 d (1+\cos (c+d x)+\sin (c+d x))} \]
-1/2*a*(e*cos(d*x+c))^(5/2)/d/e/(a+a*sin(d*x+c))^(1/2)+3/4*e*(e*cos(d*x+c) )^(1/2)*(a+a*sin(d*x+c))^(1/2)/d-3/4*e^(3/2)*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))+3/4*e^(3/2)*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))
Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.
Time = 0.22 (sec) , antiderivative size = 77, normalized size of antiderivative = 0.33 \[ \int (e \cos (c+d x))^{3/2} \sqrt {a+a \sin (c+d x)} \, dx=-\frac {4\ 2^{3/4} (e \cos (c+d x))^{5/2} \operatorname {Hypergeometric2F1}\left (-\frac {3}{4},\frac {5}{4},\frac {9}{4},\frac {1}{2} (1-\sin (c+d x))\right ) \sqrt {a (1+\sin (c+d x))}}{5 d e (1+\sin (c+d x))^{7/4}} \]
(-4*2^(3/4)*(e*Cos[c + d*x])^(5/2)*Hypergeometric2F1[-3/4, 5/4, 9/4, (1 - Sin[c + d*x])/2]*Sqrt[a*(1 + Sin[c + d*x])])/(5*d*e*(1 + Sin[c + d*x])^(7/ 4))
Time = 0.98 (sec) , antiderivative size = 249, normalized size of antiderivative = 1.06, number of steps used = 14, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.481, Rules used = {3042, 3157, 3042, 3164, 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.
| \(\displaystyle \int \sqrt {a \sin (c+d x)+a} (e \cos (c+d x))^{3/2} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \sqrt {a \sin (c+d x)+a} (e \cos (c+d x))^{3/2}dx\) |
| \(\Big \downarrow \) 3157 |
| \(\displaystyle \frac {3}{4} a \int \frac {(e \cos (c+d x))^{3/2}}{\sqrt {\sin (c+d x) a+a}}dx-\frac {a (e \cos (c+d x))^{5/2}}{2 d e \sqrt {a \sin (c+d x)+a}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {3}{4} a \int \frac {(e \cos (c+d x))^{3/2}}{\sqrt {\sin (c+d x) a+a}}dx-\frac {a (e \cos (c+d x))^{5/2}}{2 d e \sqrt {a \sin (c+d x)+a}}\) |
| \(\Big \downarrow \) 3164 |
| \(\displaystyle \frac {3}{4} a \left (\frac {e^2 \int \frac {\sqrt {\sin (c+d x) a+a}}{\sqrt {e \cos (c+d x)}}dx}{2 a}+\frac {e \sqrt {a \sin (c+d x)+a} \sqrt {e \cos (c+d x)}}{a d}\right )-\frac {a (e \cos (c+d x))^{5/2}}{2 d e \sqrt {a \sin (c+d x)+a}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {3}{4} a \left (\frac {e^2 \int \frac {\sqrt {\sin (c+d x) a+a}}{\sqrt {e \cos (c+d x)}}dx}{2 a}+\frac {e \sqrt {a \sin (c+d x)+a} \sqrt {e \cos (c+d x)}}{a d}\right )-\frac {a (e \cos (c+d x))^{5/2}}{2 d e \sqrt {a \sin (c+d x)+a}}\) |
| \(\Big \downarrow \) 3156 |
| \(\displaystyle \frac {3}{4} a \left (\frac {e^2 \left (\frac {\sqrt {\cos (c+d x)+1} \sqrt {a \sin (c+d x)+a} \int \frac {\sqrt {\cos (c+d x)+1}}{\sqrt {e \cos (c+d x)}}dx}{\sin (c+d x)+\cos (c+d x)+1}+\frac {\sqrt {\cos (c+d x)+1} \sqrt {a \sin (c+d x)+a} \int \frac {\sin (c+d x)}{\sqrt {e \cos (c+d x)} \sqrt {\cos (c+d x)+1}}dx}{\sin (c+d x)+\cos (c+d x)+1}\right )}{2 a}+\frac {e \sqrt {a \sin (c+d x)+a} \sqrt {e \cos (c+d x)}}{a d}\right )-\frac {a (e \cos (c+d x))^{5/2}}{2 d e \sqrt {a \sin (c+d x)+a}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {3}{4} a \left (\frac {e^2 \left (\frac {\sqrt {\cos (c+d x)+1} \sqrt {a \sin (c+d x)+a} \int -\frac {\cos \left (c+d x+\frac {\pi }{2}\right )}{\sqrt {e \sin \left (c+d x+\frac {\pi }{2}\right )} \sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )+1}}dx}{\sin (c+d x)+\cos (c+d x)+1}+\frac {\sqrt {\cos (c+d x)+1} \sqrt {a \sin (c+d x)+a} \int \frac {\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )+1}}{\sqrt {e \sin \left (c+d x+\frac {\pi }{2}\right )}}dx}{\sin (c+d x)+\cos (c+d x)+1}\right )}{2 a}+\frac {e \sqrt {a \sin (c+d x)+a} \sqrt {e \cos (c+d x)}}{a d}\right )-\frac {a (e \cos (c+d x))^{5/2}}{2 d e \sqrt {a \sin (c+d x)+a}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {3}{4} a \left (\frac {e^2 \left (\frac {\sqrt {\cos (c+d x)+1} \sqrt {a \sin (c+d x)+a} \int \frac {\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )+1}}{\sqrt {e \sin \left (c+d x+\frac {\pi }{2}\right )}}dx}{\sin (c+d x)+\cos (c+d x)+1}-\frac {\sqrt {\cos (c+d x)+1} \sqrt {a \sin (c+d x)+a} \int \frac {\cos \left (\frac {1}{2} (2 c+\pi )+d x\right )}{\sqrt {e \sin \left (\frac {1}{2} (2 c+\pi )+d x\right )} \sqrt {\sin \left (\frac {1}{2} (2 c+\pi )+d x\right )+1}}dx}{\sin (c+d x)+\cos (c+d x)+1}\right )}{2 a}+\frac {e \sqrt {a \sin (c+d x)+a} \sqrt {e \cos (c+d x)}}{a d}\right )-\frac {a (e \cos (c+d x))^{5/2}}{2 d e \sqrt {a \sin (c+d x)+a}}\) |
| \(\Big \downarrow \) 3254 |
| \(\displaystyle \frac {3}{4} a \left (\frac {e^2 \left (-\frac {\sqrt {\cos (c+d x)+1} \sqrt {a \sin (c+d x)+a} \int \frac {\cos \left (\frac {1}{2} (2 c+\pi )+d x\right )}{\sqrt {e \sin \left (\frac {1}{2} (2 c+\pi )+d x\right )} \sqrt {\sin \left (\frac {1}{2} (2 c+\pi )+d x\right )+1}}dx}{\sin (c+d x)+\cos (c+d x)+1}-\frac {2 \sqrt {\cos (c+d x)+1} \sqrt {a \sin (c+d x)+a} \int \frac {1}{\frac {\sin (c+d x) \tan (c+d x)}{\cos (c+d x)+1}+1}d\left (-\frac {\sin (c+d x)}{\sqrt {e \cos (c+d x)} \sqrt {\cos (c+d x)+1}}\right )}{d (\sin (c+d x)+\cos (c+d x)+1)}\right )}{2 a}+\frac {e \sqrt {a \sin (c+d x)+a} \sqrt {e \cos (c+d x)}}{a d}\right )-\frac {a (e \cos (c+d x))^{5/2}}{2 d e \sqrt {a \sin (c+d x)+a}}\) |
| \(\Big \downarrow \) 216 |
| \(\displaystyle \frac {3}{4} a \left (\frac {e^2 \left (\frac {2 \sqrt {\cos (c+d x)+1} \sqrt {a \sin (c+d x)+a} \arctan \left (\frac {\sqrt {e} \sin (c+d x)}{\sqrt {\cos (c+d x)+1} \sqrt {e \cos (c+d x)}}\right )}{d \sqrt {e} (\sin (c+d x)+\cos (c+d x)+1)}-\frac {\sqrt {\cos (c+d x)+1} \sqrt {a \sin (c+d x)+a} \int \frac {\cos \left (\frac {1}{2} (2 c+\pi )+d x\right )}{\sqrt {e \sin \left (\frac {1}{2} (2 c+\pi )+d x\right )} \sqrt {\sin \left (\frac {1}{2} (2 c+\pi )+d x\right )+1}}dx}{\sin (c+d x)+\cos (c+d x)+1}\right )}{2 a}+\frac {e \sqrt {a \sin (c+d x)+a} \sqrt {e \cos (c+d x)}}{a d}\right )-\frac {a (e \cos (c+d x))^{5/2}}{2 d e \sqrt {a \sin (c+d x)+a}}\) |
| \(\Big \downarrow \) 3312 |
| \(\displaystyle \frac {3}{4} a \left (\frac {e^2 \left (\frac {2 \sqrt {\cos (c+d x)+1} \sqrt {a \sin (c+d x)+a} \arctan \left (\frac {\sqrt {e} \sin (c+d x)}{\sqrt {\cos (c+d x)+1} \sqrt {e \cos (c+d x)}}\right )}{d \sqrt {e} (\sin (c+d x)+\cos (c+d x)+1)}-\frac {\sqrt {\cos (c+d x)+1} \sqrt {a \sin (c+d x)+a} \int \frac {1}{\sqrt {e \cos (c+d x)} \sqrt {\cos (c+d x)+1}}d\cos (c+d x)}{d (\sin (c+d x)+\cos (c+d x)+1)}\right )}{2 a}+\frac {e \sqrt {a \sin (c+d x)+a} \sqrt {e \cos (c+d x)}}{a d}\right )-\frac {a (e \cos (c+d x))^{5/2}}{2 d e \sqrt {a \sin (c+d x)+a}}\) |
| \(\Big \downarrow \) 63 |
| \(\displaystyle \frac {3}{4} a \left (\frac {e^2 \left (\frac {2 \sqrt {\cos (c+d x)+1} \sqrt {a \sin (c+d x)+a} \arctan \left (\frac {\sqrt {e} \sin (c+d x)}{\sqrt {\cos (c+d x)+1} \sqrt {e \cos (c+d x)}}\right )}{d \sqrt {e} (\sin (c+d x)+\cos (c+d x)+1)}-\frac {2 \sqrt {\cos (c+d x)+1} \sqrt {a \sin (c+d x)+a} \int \frac {1}{\sqrt {\cos (c+d x)+1}}d\sqrt {e \cos (c+d x)}}{d e (\sin (c+d x)+\cos (c+d x)+1)}\right )}{2 a}+\frac {e \sqrt {a \sin (c+d x)+a} \sqrt {e \cos (c+d x)}}{a d}\right )-\frac {a (e \cos (c+d x))^{5/2}}{2 d e \sqrt {a \sin (c+d x)+a}}\) |
| \(\Big \downarrow \) 222 |
| \(\displaystyle \frac {3}{4} a \left (\frac {e^2 \left (\frac {2 \sqrt {\cos (c+d x)+1} \sqrt {a \sin (c+d x)+a} \arctan \left (\frac {\sqrt {e} \sin (c+d x)}{\sqrt {\cos (c+d x)+1} \sqrt {e \cos (c+d x)}}\right )}{d \sqrt {e} (\sin (c+d x)+\cos (c+d x)+1)}-\frac {2 \sqrt {\cos (c+d x)+1} \sqrt {a \sin (c+d x)+a} \text {arcsinh}\left (\frac {\sqrt {e \cos (c+d x)}}{\sqrt {e}}\right )}{d \sqrt {e} (\sin (c+d x)+\cos (c+d x)+1)}\right )}{2 a}+\frac {e \sqrt {a \sin (c+d x)+a} \sqrt {e \cos (c+d x)}}{a d}\right )-\frac {a (e \cos (c+d x))^{5/2}}{2 d e \sqrt {a \sin (c+d x)+a}}\) |
-1/2*(a*(e*Cos[c + d*x])^(5/2))/(d*e*Sqrt[a + a*Sin[c + d*x]]) + (3*a*((e* Sqrt[e*Cos[c + d*x]]*Sqrt[a + a*Sin[c + d*x]])/(a*d) + (e^2*((-2*ArcSinh[S qrt[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]])]*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*a)))/4
3.3.73.3.1 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[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x _)])^(m_), x_Symbol] :> Simp[(-b)*(g*Cos[e + f*x])^(p + 1)*((a + b*Sin[e + f*x])^(m - 1)/(f*g*(m + p))), x] + Simp[a*((2*m + p - 1)/(m + p)) Int[(g* Cos[e + f*x])^p*(a + b*Sin[e + f*x])^(m - 1), x], x] /; FreeQ[{a, b, e, f, g, m, p}, x] && EqQ[a^2 - b^2, 0] && GtQ[m, 0] && NeQ[m + p, 0] && Integers Q[2*m, 2*p]
Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(3/2)/Sqrt[(a_) + (b_.)*sin[(e_.) + (f_ .)*(x_)]], x_Symbol] :> Simp[g*Sqrt[g*Cos[e + f*x]]*(Sqrt[a + b*Sin[e + f*x ]]/(b*f)), x] + Simp[g^2/(2*a) Int[Sqrt[a + b*Sin[e + f*x]]/Sqrt[g*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 = 6.39 (sec) , antiderivative size = 303, normalized size of antiderivative = 1.28
| method | result | size |
| default | \(\frac {\sqrt {e \cos \left (d x +c \right )}\, \sqrt {a \left (1+\sin \left (d x +c \right )\right )}\, e \left (3 \sqrt {-\frac {\cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\, \operatorname {arctanh}\left (\frac {\sin \left (d x +c \right )}{\left (1+\cos \left (d x +c \right )\right ) \sqrt {-\frac {\cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}}\right )+3 \sqrt {-\frac {\cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\, \arctan \left (\sqrt {-\frac {\cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\right )-2 \left (\cos ^{2}\left (d x +c \right )\right )+2 \cos \left (d x +c \right ) \sin \left (d x +c \right )+3 \sec \left (d x +c \right ) \sqrt {-\frac {\cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\, \operatorname {arctanh}\left (\frac {\sin \left (d x +c \right )}{\left (1+\cos \left (d x +c \right )\right ) \sqrt {-\frac {\cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}}\right )+3 \sec \left (d x +c \right ) \sqrt {-\frac {\cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\, \arctan \left (\sqrt {-\frac {\cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\right )+\cos \left (d x +c \right )+3 \sin \left (d x +c \right )+3\right )}{4 d \left (1+\cos \left (d x +c \right )+\sin \left (d x +c \right )\right )}\) | \(303\) |
1/4/d*(e*cos(d*x+c))^(1/2)*(a*(1+sin(d*x+c)))^(1/2)*e/(1+cos(d*x+c)+sin(d* x+c))*(3*(-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))+3*(-cos(d*x+c)/(1+cos(d*x+c)))^(1/ 2)*arctan((-cos(d*x+c)/(1+cos(d*x+c)))^(1/2))-2*cos(d*x+c)^2+2*cos(d*x+c)* sin(d*x+c)+3*sec(d*x+c)*(-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))+3*sec(d*x+c)*(-cos( d*x+c)/(1+cos(d*x+c)))^(1/2)*arctan((-cos(d*x+c)/(1+cos(d*x+c)))^(1/2))+co s(d*x+c)+3*sin(d*x+c)+3)
Result contains complex when optimal does not.
Time = 0.43 (sec) , antiderivative size = 917, normalized size of antiderivative = 3.89 \[ \int (e \cos (c+d x))^{3/2} \sqrt {a+a \sin (c+d x)} \, dx=\text {Too large to display} \]
-1/16*(3*I*(-a^2*e^6/d^4)^(1/4)*d*log(-27/2*(2*(a*e^4*sin(d*x + c) + sqrt( -a^2*e^6/d^4)*(d^2*e*cos(d*x + c) + d^2*e))*sqrt(e*cos(d*x + c))*sqrt(a*si n(d*x + c) + a) + (-a^2*e^6/d^4)^(3/4)*(I*d^3*cos(d*x + c) + I*d^3 + (2*I* d^3*cos(d*x + c) + I*d^3)*sin(d*x + c)) + (-2*I*a*d*e^3*cos(d*x + c)^2 - I *a*d*e^3*cos(d*x + c) + I*a*d*e^3*sin(d*x + c) + I*a*d*e^3)*(-a^2*e^6/d^4) ^(1/4))/(cos(d*x + c) + sin(d*x + c) + 1)) - 3*I*(-a^2*e^6/d^4)^(1/4)*d*lo g(-27/2*(2*(a*e^4*sin(d*x + c) + sqrt(-a^2*e^6/d^4)*(d^2*e*cos(d*x + c) + d^2*e))*sqrt(e*cos(d*x + c))*sqrt(a*sin(d*x + c) + a) + (-a^2*e^6/d^4)^(3/ 4)*(-I*d^3*cos(d*x + c) - I*d^3 + (-2*I*d^3*cos(d*x + c) - I*d^3)*sin(d*x + c)) + (2*I*a*d*e^3*cos(d*x + c)^2 + I*a*d*e^3*cos(d*x + c) - I*a*d*e^3*s in(d*x + c) - I*a*d*e^3)*(-a^2*e^6/d^4)^(1/4))/(cos(d*x + c) + sin(d*x + c ) + 1)) - 3*(-a^2*e^6/d^4)^(1/4)*d*log(-27/2*(2*(a*e^4*sin(d*x + c) - sqrt (-a^2*e^6/d^4)*(d^2*e*cos(d*x + c) + d^2*e))*sqrt(e*cos(d*x + c))*sqrt(a*s in(d*x + c) + a) + (-a^2*e^6/d^4)^(3/4)*(d^3*cos(d*x + c) + d^3 + (2*d^3*c os(d*x + c) + d^3)*sin(d*x + c)) + (2*a*d*e^3*cos(d*x + c)^2 + a*d*e^3*cos (d*x + c) - a*d*e^3*sin(d*x + c) - a*d*e^3)*(-a^2*e^6/d^4)^(1/4))/(cos(d*x + c) + sin(d*x + c) + 1)) + 3*(-a^2*e^6/d^4)^(1/4)*d*log(-27/2*(2*(a*e^4* sin(d*x + c) - sqrt(-a^2*e^6/d^4)*(d^2*e*cos(d*x + c) + d^2*e))*sqrt(e*cos (d*x + c))*sqrt(a*sin(d*x + c) + a) - (-a^2*e^6/d^4)^(3/4)*(d^3*cos(d*x + c) + d^3 + (2*d^3*cos(d*x + c) + d^3)*sin(d*x + c)) - (2*a*d*e^3*cos(d*...
\[ \int (e \cos (c+d x))^{3/2} \sqrt {a+a \sin (c+d x)} \, dx=\int \sqrt {a \left (\sin {\left (c + d x \right )} + 1\right )} \left (e \cos {\left (c + d x \right )}\right )^{\frac {3}{2}}\, dx \]
\[ \int (e \cos (c+d x))^{3/2} \sqrt {a+a \sin (c+d x)} \, dx=\int { \left (e \cos \left (d x + c\right )\right )^{\frac {3}{2}} \sqrt {a \sin \left (d x + c\right ) + a} \,d x } \]
\[ \int (e \cos (c+d x))^{3/2} \sqrt {a+a \sin (c+d x)} \, dx=\int { \left (e \cos \left (d x + c\right )\right )^{\frac {3}{2}} \sqrt {a \sin \left (d x + c\right ) + a} \,d x } \]
Timed out. \[ \int (e \cos (c+d x))^{3/2} \sqrt {a+a \sin (c+d x)} \, dx=\int {\left (e\,\cos \left (c+d\,x\right )\right )}^{3/2}\,\sqrt {a+a\,\sin \left (c+d\,x\right )} \,d x \]