Integrand size = 27, antiderivative size = 244 \[ \int \frac {(e \cos (c+d x))^{5/2}}{\sqrt {a+a \sin (c+d x)}} \, dx=-\frac {a (e \cos (c+d x))^{7/2}}{2 d e (a+a \sin (c+d x))^{3/2}}+\frac {e (e \cos (c+d x))^{3/2}}{4 d \sqrt {a+a \sin (c+d x)}}+\frac {3 e^{5/2} \text {arcsinh}\left (\frac {\sqrt {e \cos (c+d x)}}{\sqrt {e}}\right ) \sqrt {1+\cos (c+d x)} \sqrt {a+a \sin (c+d x)}}{4 d (a+a \cos (c+d x)+a \sin (c+d x))}+\frac {3 e^{5/2} \arctan \left (\frac {\sqrt {e} \sin (c+d x)}{\sqrt {e \cos (c+d x)} \sqrt {1+\cos (c+d x)}}\right ) \sqrt {1+\cos (c+d x)} \sqrt {a+a \sin (c+d x)}}{4 d (a+a \cos (c+d x)+a \sin (c+d x))} \]
-1/2*a*(e*cos(d*x+c))^(7/2)/d/e/(a+a*sin(d*x+c))^(3/2)+1/4*e*(e*cos(d*x+c) )^(3/2)/d/(a+a*sin(d*x+c))^(1/2)+3/4*e^(5/2)*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/(a+a*cos(d*x+c)+a*s in(d*x+c))+3/4*e^(5/2)*arctan(sin(d*x+c)*e^(1/2)/(e*cos(d*x+c))^(1/2)/(1+c os(d*x+c))^(1/2))*(1+cos(d*x+c))^(1/2)*(a+a*sin(d*x+c))^(1/2)/d/(a+a*cos(d *x+c)+a*sin(d*x+c))
Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.
Time = 0.10 (sec) , antiderivative size = 77, normalized size of antiderivative = 0.32 \[ \int \frac {(e \cos (c+d x))^{5/2}}{\sqrt {a+a \sin (c+d x)}} \, dx=-\frac {4 \sqrt [4]{2} (e \cos (c+d x))^{7/2} \operatorname {Hypergeometric2F1}\left (-\frac {1}{4},\frac {7}{4},\frac {11}{4},\frac {1}{2} (1-\sin (c+d x))\right )}{7 d e (1+\sin (c+d x))^{5/4} \sqrt {a (1+\sin (c+d x))}} \]
(-4*2^(1/4)*(e*Cos[c + d*x])^(7/2)*Hypergeometric2F1[-1/4, 7/4, 11/4, (1 - Sin[c + d*x])/2])/(7*d*e*(1 + Sin[c + d*x])^(5/4)*Sqrt[a*(1 + Sin[c + d*x ])])
Time = 0.99 (sec) , antiderivative size = 257, normalized size of antiderivative = 1.05, number of steps used = 14, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.481, Rules used = {3042, 3165, 3042, 3158, 3042, 3163, 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.
| \(\displaystyle \int \frac {(e \cos (c+d x))^{5/2}}{\sqrt {a \sin (c+d x)+a}} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {(e \cos (c+d x))^{5/2}}{\sqrt {a \sin (c+d x)+a}}dx\) |
| \(\Big \downarrow \) 3165 |
| \(\displaystyle \frac {1}{4} a \int \frac {(e \cos (c+d x))^{5/2}}{(\sin (c+d x) a+a)^{3/2}}dx-\frac {a (e \cos (c+d x))^{7/2}}{2 d e (a \sin (c+d x)+a)^{3/2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{4} a \int \frac {(e \cos (c+d x))^{5/2}}{(\sin (c+d x) a+a)^{3/2}}dx-\frac {a (e \cos (c+d x))^{7/2}}{2 d e (a \sin (c+d x)+a)^{3/2}}\) |
| \(\Big \downarrow \) 3158 |
| \(\displaystyle \frac {1}{4} a \left (\frac {3 e^2 \int \frac {\sqrt {e \cos (c+d x)}}{\sqrt {\sin (c+d x) a+a}}dx}{2 a}+\frac {e (e \cos (c+d x))^{3/2}}{a d \sqrt {a \sin (c+d x)+a}}\right )-\frac {a (e \cos (c+d x))^{7/2}}{2 d e (a \sin (c+d x)+a)^{3/2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{4} a \left (\frac {3 e^2 \int \frac {\sqrt {e \cos (c+d x)}}{\sqrt {\sin (c+d x) a+a}}dx}{2 a}+\frac {e (e \cos (c+d x))^{3/2}}{a d \sqrt {a \sin (c+d x)+a}}\right )-\frac {a (e \cos (c+d x))^{7/2}}{2 d e (a \sin (c+d x)+a)^{3/2}}\) |
| \(\Big \downarrow \) 3163 |
| \(\displaystyle \frac {1}{4} a \left (\frac {3 e^2 \left (\frac {e \sqrt {\cos (c+d x)+1} \sqrt {a \sin (c+d x)+a} \int \frac {\sqrt {\cos (c+d x)+1}}{\sqrt {e \cos (c+d x)}}dx}{a \sin (c+d x)+a \cos (c+d x)+a}-\frac {e \sqrt {\cos (c+d x)+1} \sqrt {a \sin (c+d x)+a} \int \frac {\sin (c+d x)}{\sqrt {e \cos (c+d x)} \sqrt {\cos (c+d x)+1}}dx}{a \sin (c+d x)+a \cos (c+d x)+a}\right )}{2 a}+\frac {e (e \cos (c+d x))^{3/2}}{a d \sqrt {a \sin (c+d x)+a}}\right )-\frac {a (e \cos (c+d x))^{7/2}}{2 d e (a \sin (c+d x)+a)^{3/2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{4} a \left (\frac {3 e^2 \left (\frac {e \sqrt {\cos (c+d x)+1} \sqrt {a \sin (c+d x)+a} \int \frac {\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )+1}}{\sqrt {e \sin \left (c+d x+\frac {\pi }{2}\right )}}dx}{a \sin (c+d x)+a \cos (c+d x)+a}-\frac {e \sqrt {\cos (c+d x)+1} \sqrt {a \sin (c+d x)+a} \int -\frac {\cos \left (c+d x+\frac {\pi }{2}\right )}{\sqrt {e \sin \left (c+d x+\frac {\pi }{2}\right )} \sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )+1}}dx}{a \sin (c+d x)+a \cos (c+d x)+a}\right )}{2 a}+\frac {e (e \cos (c+d x))^{3/2}}{a d \sqrt {a \sin (c+d x)+a}}\right )-\frac {a (e \cos (c+d x))^{7/2}}{2 d e (a \sin (c+d x)+a)^{3/2}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {1}{4} a \left (\frac {3 e^2 \left (\frac {e \sqrt {\cos (c+d x)+1} \sqrt {a \sin (c+d x)+a} \int \frac {\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )+1}}{\sqrt {e \sin \left (c+d x+\frac {\pi }{2}\right )}}dx}{a \sin (c+d x)+a \cos (c+d x)+a}+\frac {e \sqrt {\cos (c+d x)+1} \sqrt {a \sin (c+d x)+a} \int \frac {\cos \left (\frac {1}{2} (2 c+\pi )+d x\right )}{\sqrt {e \sin \left (\frac {1}{2} (2 c+\pi )+d x\right )} \sqrt {\sin \left (\frac {1}{2} (2 c+\pi )+d x\right )+1}}dx}{a \sin (c+d x)+a \cos (c+d x)+a}\right )}{2 a}+\frac {e (e \cos (c+d x))^{3/2}}{a d \sqrt {a \sin (c+d x)+a}}\right )-\frac {a (e \cos (c+d x))^{7/2}}{2 d e (a \sin (c+d x)+a)^{3/2}}\) |
| \(\Big \downarrow \) 3254 |
| \(\displaystyle \frac {1}{4} a \left (\frac {3 e^2 \left (\frac {e \sqrt {\cos (c+d x)+1} \sqrt {a \sin (c+d x)+a} \int \frac {\cos \left (\frac {1}{2} (2 c+\pi )+d x\right )}{\sqrt {e \sin \left (\frac {1}{2} (2 c+\pi )+d x\right )} \sqrt {\sin \left (\frac {1}{2} (2 c+\pi )+d x\right )+1}}dx}{a \sin (c+d x)+a \cos (c+d x)+a}-\frac {2 e \sqrt {\cos (c+d x)+1} \sqrt {a \sin (c+d x)+a} \int \frac {1}{\frac {\sin (c+d x) \tan (c+d x)}{\cos (c+d x)+1}+1}d\left (-\frac {\sin (c+d x)}{\sqrt {e \cos (c+d x)} \sqrt {\cos (c+d x)+1}}\right )}{d (a \sin (c+d x)+a \cos (c+d x)+a)}\right )}{2 a}+\frac {e (e \cos (c+d x))^{3/2}}{a d \sqrt {a \sin (c+d x)+a}}\right )-\frac {a (e \cos (c+d x))^{7/2}}{2 d e (a \sin (c+d x)+a)^{3/2}}\) |
| \(\Big \downarrow \) 216 |
| \(\displaystyle \frac {1}{4} a \left (\frac {3 e^2 \left (\frac {e \sqrt {\cos (c+d x)+1} \sqrt {a \sin (c+d x)+a} \int \frac {\cos \left (\frac {1}{2} (2 c+\pi )+d x\right )}{\sqrt {e \sin \left (\frac {1}{2} (2 c+\pi )+d x\right )} \sqrt {\sin \left (\frac {1}{2} (2 c+\pi )+d x\right )+1}}dx}{a \sin (c+d x)+a \cos (c+d x)+a}+\frac {2 \sqrt {e} \sqrt {\cos (c+d x)+1} \sqrt {a \sin (c+d x)+a} \arctan \left (\frac {\sqrt {e} \sin (c+d x)}{\sqrt {\cos (c+d x)+1} \sqrt {e \cos (c+d x)}}\right )}{d (a \sin (c+d x)+a \cos (c+d x)+a)}\right )}{2 a}+\frac {e (e \cos (c+d x))^{3/2}}{a d \sqrt {a \sin (c+d x)+a}}\right )-\frac {a (e \cos (c+d x))^{7/2}}{2 d e (a \sin (c+d x)+a)^{3/2}}\) |
| \(\Big \downarrow \) 3312 |
| \(\displaystyle \frac {1}{4} a \left (\frac {3 e^2 \left (\frac {e \sqrt {\cos (c+d x)+1} \sqrt {a \sin (c+d x)+a} \int \frac {1}{\sqrt {e \cos (c+d x)} \sqrt {\cos (c+d x)+1}}d\cos (c+d x)}{d (a \sin (c+d x)+a \cos (c+d x)+a)}+\frac {2 \sqrt {e} \sqrt {\cos (c+d x)+1} \sqrt {a \sin (c+d x)+a} \arctan \left (\frac {\sqrt {e} \sin (c+d x)}{\sqrt {\cos (c+d x)+1} \sqrt {e \cos (c+d x)}}\right )}{d (a \sin (c+d x)+a \cos (c+d x)+a)}\right )}{2 a}+\frac {e (e \cos (c+d x))^{3/2}}{a d \sqrt {a \sin (c+d x)+a}}\right )-\frac {a (e \cos (c+d x))^{7/2}}{2 d e (a \sin (c+d x)+a)^{3/2}}\) |
| \(\Big \downarrow \) 63 |
| \(\displaystyle \frac {1}{4} a \left (\frac {3 e^2 \left (\frac {2 \sqrt {\cos (c+d x)+1} \sqrt {a \sin (c+d x)+a} \int \frac {1}{\sqrt {\cos (c+d x)+1}}d\sqrt {e \cos (c+d x)}}{d (a \sin (c+d x)+a \cos (c+d x)+a)}+\frac {2 \sqrt {e} \sqrt {\cos (c+d x)+1} \sqrt {a \sin (c+d x)+a} \arctan \left (\frac {\sqrt {e} \sin (c+d x)}{\sqrt {\cos (c+d x)+1} \sqrt {e \cos (c+d x)}}\right )}{d (a \sin (c+d x)+a \cos (c+d x)+a)}\right )}{2 a}+\frac {e (e \cos (c+d x))^{3/2}}{a d \sqrt {a \sin (c+d x)+a}}\right )-\frac {a (e \cos (c+d x))^{7/2}}{2 d e (a \sin (c+d x)+a)^{3/2}}\) |
| \(\Big \downarrow \) 222 |
| \(\displaystyle \frac {1}{4} a \left (\frac {3 e^2 \left (\frac {2 \sqrt {e} \sqrt {\cos (c+d x)+1} \sqrt {a \sin (c+d x)+a} \text {arcsinh}\left (\frac {\sqrt {e \cos (c+d x)}}{\sqrt {e}}\right )}{d (a \sin (c+d x)+a \cos (c+d x)+a)}+\frac {2 \sqrt {e} \sqrt {\cos (c+d x)+1} \sqrt {a \sin (c+d x)+a} \arctan \left (\frac {\sqrt {e} \sin (c+d x)}{\sqrt {\cos (c+d x)+1} \sqrt {e \cos (c+d x)}}\right )}{d (a \sin (c+d x)+a \cos (c+d x)+a)}\right )}{2 a}+\frac {e (e \cos (c+d x))^{3/2}}{a d \sqrt {a \sin (c+d x)+a}}\right )-\frac {a (e \cos (c+d x))^{7/2}}{2 d e (a \sin (c+d x)+a)^{3/2}}\) |
-1/2*(a*(e*Cos[c + d*x])^(7/2))/(d*e*(a + a*Sin[c + d*x])^(3/2)) + (a*((e* (e*Cos[c + d*x])^(3/2))/(a*d*Sqrt[a + a*Sin[c + d*x]]) + (3*e^2*((2*Sqrt[e ]*ArcSinh[Sqrt[e*Cos[c + d*x]]/Sqrt[e]]*Sqrt[1 + Cos[c + d*x]]*Sqrt[a + a* Sin[c + d*x]])/(d*(a + a*Cos[c + d*x] + a*Sin[c + d*x])) + (2*Sqrt[e]*ArcT an[(Sqrt[e]*Sin[c + d*x])/(Sqrt[e*Cos[c + d*x]]*Sqrt[1 + Cos[c + d*x]])]*S qrt[1 + Cos[c + d*x]]*Sqrt[a + a*Sin[c + d*x]])/(d*(a + a*Cos[c + d*x] + a *Sin[c + d*x]))))/(2*a)))/4
3.3.98.3.1 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[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x _)])^(m_), x_Symbol] :> Simp[g*(g*Cos[e + f*x])^(p - 1)*((a + b*Sin[e + f*x ])^(m + 1)/(b*f*(m + p))), x] + Simp[g^2*((p - 1)/(a*(m + p))) Int[(g*Cos [e + f*x])^(p - 2)*(a + b*Sin[e + f*x])^(m + 1), x], x] /; FreeQ[{a, b, e, f, g}, x] && EqQ[a^2 - b^2, 0] && LtQ[m, -1] && GtQ[p, 1] && (GtQ[m, -2] || EqQ[2*m + p + 1, 0] || (EqQ[m, -2] && IntegerQ[p])) && NeQ[m + p, 0] && In tegersQ[2*m, 2*p]
Int[Sqrt[cos[(e_.) + (f_.)*(x_)]*(g_.)]/Sqrt[(a_) + (b_.)*sin[(e_.) + (f_.) *(x_)]], x_Symbol] :> Simp[g*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[g*Sqrt[1 + Cos[e + f*x]]*(Sqrt[a + b*Sin[e + f*x]]/(b + b*Cos[e + f*x] + a*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_)/Sqrt[(a_) + (b_.)*sin[(e_.) + (f_. )*(x_)]], x_Symbol] :> Simp[-2*b*((g*Cos[e + f*x])^(p + 1)/(f*g*(2*p - 1)*( a + b*Sin[e + f*x])^(3/2))), x] + Simp[2*a*((p - 2)/(2*p - 1)) Int[(g*Cos [e + f*x])^p/(a + b*Sin[e + f*x])^(3/2), x], x] /; FreeQ[{a, b, e, f, g}, x ] && EqQ[a^2 - b^2, 0] && GtQ[p, 2] && IntegerQ[2*p]
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 = 6.77 (sec) , antiderivative size = 326, normalized size of antiderivative = 1.34
| method | result | size |
| default | \(\frac {\left (3 \,\operatorname {arctanh}\left (\frac {\sin \left (d x +c \right )}{\left (1+\cos \left (d x +c \right )\right ) \sqrt {-\frac {\cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}}\right ) \sqrt {-\frac {\cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\, \cos \left (d x +c \right )-3 \arctan \left (\sqrt {-\frac {\cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\right ) \sqrt {-\frac {\cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\, \cos \left (d x +c \right )+2 \left (\cos ^{3}\left (d x +c \right )\right )+2 \left (\cos ^{2}\left (d x +c \right )\right ) \sin \left (d x +c \right )+3 \sqrt {-\frac {\cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\, \operatorname {arctanh}\left (\frac {\sin \left (d x +c \right )}{\left (1+\cos \left (d x +c \right )\right ) \sqrt {-\frac {\cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}}\right )-3 \sqrt {-\frac {\cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\, \arctan \left (\sqrt {-\frac {\cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\right )-\left (\cos ^{2}\left (d x +c \right )\right )+3 \cos \left (d x +c \right ) \sin \left (d x +c \right )-3 \cos \left (d x +c \right )\right ) \sqrt {e \cos \left (d x +c \right )}\, e^{2}}{4 d \left (\cos \left (d x +c \right )-\sin \left (d x +c \right )+1\right ) \sqrt {a \left (1+\sin \left (d x +c \right )\right )}}\) | \(326\) |
1/4/d*(3*arctanh(sin(d*x+c)/(1+cos(d*x+c))/(-cos(d*x+c)/(1+cos(d*x+c)))^(1 /2))*(-cos(d*x+c)/(1+cos(d*x+c)))^(1/2)*cos(d*x+c)-3*arctan((-cos(d*x+c)/( 1+cos(d*x+c)))^(1/2))*(-cos(d*x+c)/(1+cos(d*x+c)))^(1/2)*cos(d*x+c)+2*cos( d*x+c)^3+2*cos(d*x+c)^2*sin(d*x+c)+3*(-cos(d*x+c)/(1+cos(d*x+c)))^(1/2)*ar ctanh(sin(d*x+c)/(1+cos(d*x+c))/(-cos(d*x+c)/(1+cos(d*x+c)))^(1/2))-3*(-co s(d*x+c)/(1+cos(d*x+c)))^(1/2)*arctan((-cos(d*x+c)/(1+cos(d*x+c)))^(1/2))- cos(d*x+c)^2+3*cos(d*x+c)*sin(d*x+c)-3*cos(d*x+c))*(e*cos(d*x+c))^(1/2)*e^ 2/(cos(d*x+c)-sin(d*x+c)+1)/(a*(1+sin(d*x+c)))^(1/2)
Result contains complex when optimal does not.
Time = 0.48 (sec) , antiderivative size = 1147, normalized size of antiderivative = 4.70 \[ \int \frac {(e \cos (c+d x))^{5/2}}{\sqrt {a+a \sin (c+d x)}} \, dx=\text {Too large to display} \]
1/16*(3*(-e^10/(a^2*d^4))^(1/4)*(a*d*cos(d*x + c) + a*d*sin(d*x + c) + a*d )*log(27/2*(2*(e^7*sin(d*x + c) + (a*d^2*e^2*cos(d*x + c) + a*d^2*e^2)*sqr t(-e^10/(a^2*d^4)))*sqrt(e*cos(d*x + c))*sqrt(a*sin(d*x + c) + a) + (2*a^2 *d^3*cos(d*x + c)^2 + a^2*d^3*cos(d*x + c) - a^2*d^3*sin(d*x + c) - a^2*d^ 3)*(-e^10/(a^2*d^4))^(3/4) + (a*d*e^5*cos(d*x + c) + a*d*e^5 + (2*a*d*e^5* cos(d*x + c) + a*d*e^5)*sin(d*x + c))*(-e^10/(a^2*d^4))^(1/4))/(cos(d*x + c) + sin(d*x + c) + 1)) - 3*(-e^10/(a^2*d^4))^(1/4)*(a*d*cos(d*x + c) + a* d*sin(d*x + c) + a*d)*log(27/2*(2*(e^7*sin(d*x + c) + (a*d^2*e^2*cos(d*x + c) + a*d^2*e^2)*sqrt(-e^10/(a^2*d^4)))*sqrt(e*cos(d*x + c))*sqrt(a*sin(d* x + c) + a) - (2*a^2*d^3*cos(d*x + c)^2 + a^2*d^3*cos(d*x + c) - a^2*d^3*s in(d*x + c) - a^2*d^3)*(-e^10/(a^2*d^4))^(3/4) - (a*d*e^5*cos(d*x + c) + a *d*e^5 + (2*a*d*e^5*cos(d*x + c) + a*d*e^5)*sin(d*x + c))*(-e^10/(a^2*d^4) )^(1/4))/(cos(d*x + c) + sin(d*x + c) + 1)) - 3*(-e^10/(a^2*d^4))^(1/4)*(- I*a*d*cos(d*x + c) - I*a*d*sin(d*x + c) - I*a*d)*log(27/2*(2*(e^7*sin(d*x + c) - (a*d^2*e^2*cos(d*x + c) + a*d^2*e^2)*sqrt(-e^10/(a^2*d^4)))*sqrt(e* cos(d*x + c))*sqrt(a*sin(d*x + c) + a) - (2*I*a^2*d^3*cos(d*x + c)^2 + I*a ^2*d^3*cos(d*x + c) - I*a^2*d^3*sin(d*x + c) - I*a^2*d^3)*(-e^10/(a^2*d^4) )^(3/4) - (-I*a*d*e^5*cos(d*x + c) - I*a*d*e^5 + (-2*I*a*d*e^5*cos(d*x + c ) - I*a*d*e^5)*sin(d*x + c))*(-e^10/(a^2*d^4))^(1/4))/(cos(d*x + c) + sin( d*x + c) + 1)) - 3*(-e^10/(a^2*d^4))^(1/4)*(I*a*d*cos(d*x + c) + I*a*d*...
Timed out. \[ \int \frac {(e \cos (c+d x))^{5/2}}{\sqrt {a+a \sin (c+d x)}} \, dx=\text {Timed out} \]
\[ \int \frac {(e \cos (c+d x))^{5/2}}{\sqrt {a+a \sin (c+d x)}} \, dx=\int { \frac {\left (e \cos \left (d x + c\right )\right )^{\frac {5}{2}}}{\sqrt {a \sin \left (d x + c\right ) + a}} \,d x } \]
Timed out. \[ \int \frac {(e \cos (c+d x))^{5/2}}{\sqrt {a+a \sin (c+d x)}} \, dx=\text {Timed out} \]
Timed out. \[ \int \frac {(e \cos (c+d x))^{5/2}}{\sqrt {a+a \sin (c+d x)}} \, dx=\int \frac {{\left (e\,\cos \left (c+d\,x\right )\right )}^{5/2}}{\sqrt {a+a\,\sin \left (c+d\,x\right )}} \,d x \]