Integrand size = 27, antiderivative size = 247 \[ \int \frac {(e \cos (c+d x))^{7/2}}{(a+a \sin (c+d x))^{3/2}} \, dx=\frac {e (e \cos (c+d x))^{5/2}}{2 a d \sqrt {a+a \sin (c+d x)}}+\frac {5 e^3 \sqrt {e \cos (c+d x)} \sqrt {a+a \sin (c+d x)}}{4 a^2 d}-\frac {5 e^{7/2} \text {arcsinh}\left (\frac {\sqrt {e \cos (c+d x)}}{\sqrt {e}}\right ) \sqrt {1+\cos (c+d x)} \sqrt {a+a \sin (c+d x)}}{4 a^2 d (1+\cos (c+d x)+\sin (c+d x))}+\frac {5 e^{7/2} \arctan \left (\frac {\sqrt {e} \sin (c+d x)}{\sqrt {e \cos (c+d x)} \sqrt {1+\cos (c+d x)}}\right ) \sqrt {1+\cos (c+d x)} \sqrt {a+a \sin (c+d x)}}{4 a^2 d (1+\cos (c+d x)+\sin (c+d x))} \]
1/2*e*(e*cos(d*x+c))^(5/2)/a/d/(a+a*sin(d*x+c))^(1/2)+5/4*e^3*(e*cos(d*x+c ))^(1/2)*(a+a*sin(d*x+c))^(1/2)/a^2/d-5/4*e^(7/2)*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)/a^2/d/(1+cos(d*x +c)+sin(d*x+c))+5/4*e^(7/2)*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)/a^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 = 0.09 (sec) , antiderivative size = 80, normalized size of antiderivative = 0.32 \[ \int \frac {(e \cos (c+d x))^{7/2}}{(a+a \sin (c+d x))^{3/2}} \, dx=-\frac {2\ 2^{3/4} (e \cos (c+d x))^{9/2} \operatorname {Hypergeometric2F1}\left (\frac {1}{4},\frac {9}{4},\frac {13}{4},\frac {1}{2} (1-\sin (c+d x))\right ) \sqrt {a (1+\sin (c+d x))}}{9 a^2 d e (1+\sin (c+d x))^{11/4}} \]
(-2*2^(3/4)*(e*Cos[c + d*x])^(9/2)*Hypergeometric2F1[1/4, 9/4, 13/4, (1 - Sin[c + d*x])/2]*Sqrt[a*(1 + Sin[c + d*x])])/(9*a^2*d*e*(1 + Sin[c + d*x]) ^(11/4))
Time = 0.98 (sec) , antiderivative size = 254, normalized size of antiderivative = 1.03, number of steps used = 14, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.481, Rules used = {3042, 3158, 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.
| \(\displaystyle \int \frac {(e \cos (c+d x))^{7/2}}{(a \sin (c+d x)+a)^{3/2}} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {(e \cos (c+d x))^{7/2}}{(a \sin (c+d x)+a)^{3/2}}dx\) |
| \(\Big \downarrow \) 3158 |
| \(\displaystyle \frac {5 e^2 \int \frac {(e \cos (c+d x))^{3/2}}{\sqrt {\sin (c+d x) a+a}}dx}{4 a}+\frac {e (e \cos (c+d x))^{5/2}}{2 a d \sqrt {a \sin (c+d x)+a}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {5 e^2 \int \frac {(e \cos (c+d x))^{3/2}}{\sqrt {\sin (c+d x) a+a}}dx}{4 a}+\frac {e (e \cos (c+d x))^{5/2}}{2 a d \sqrt {a \sin (c+d x)+a}}\) |
| \(\Big \downarrow \) 3164 |
| \(\displaystyle \frac {5 e^2 \left (\frac {e^2 \int \frac {\sqrt {\sin (c+d x) a+a}}{\sqrt {e \cos (c+d x)}}dx}{2 a}+\frac {e \sqrt {a \sin (c+d x)+a} \sqrt {e \cos (c+d x)}}{a d}\right )}{4 a}+\frac {e (e \cos (c+d x))^{5/2}}{2 a d \sqrt {a \sin (c+d x)+a}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {5 e^2 \left (\frac {e^2 \int \frac {\sqrt {\sin (c+d x) a+a}}{\sqrt {e \cos (c+d x)}}dx}{2 a}+\frac {e \sqrt {a \sin (c+d x)+a} \sqrt {e \cos (c+d x)}}{a d}\right )}{4 a}+\frac {e (e \cos (c+d x))^{5/2}}{2 a d \sqrt {a \sin (c+d x)+a}}\) |
| \(\Big \downarrow \) 3156 |
| \(\displaystyle \frac {5 e^2 \left (\frac {e^2 \left (\frac {\sqrt {\cos (c+d x)+1} \sqrt {a \sin (c+d x)+a} \int \frac {\sqrt {\cos (c+d x)+1}}{\sqrt {e \cos (c+d x)}}dx}{\sin (c+d x)+\cos (c+d x)+1}+\frac {\sqrt {\cos (c+d x)+1} \sqrt {a \sin (c+d x)+a} \int \frac {\sin (c+d x)}{\sqrt {e \cos (c+d x)} \sqrt {\cos (c+d x)+1}}dx}{\sin (c+d x)+\cos (c+d x)+1}\right )}{2 a}+\frac {e \sqrt {a \sin (c+d x)+a} \sqrt {e \cos (c+d x)}}{a d}\right )}{4 a}+\frac {e (e \cos (c+d x))^{5/2}}{2 a d \sqrt {a \sin (c+d x)+a}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {5 e^2 \left (\frac {e^2 \left (\frac {\sqrt {\cos (c+d x)+1} \sqrt {a \sin (c+d x)+a} \int -\frac {\cos \left (c+d x+\frac {\pi }{2}\right )}{\sqrt {e \sin \left (c+d x+\frac {\pi }{2}\right )} \sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )+1}}dx}{\sin (c+d x)+\cos (c+d x)+1}+\frac {\sqrt {\cos (c+d x)+1} \sqrt {a \sin (c+d x)+a} \int \frac {\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )+1}}{\sqrt {e \sin \left (c+d x+\frac {\pi }{2}\right )}}dx}{\sin (c+d x)+\cos (c+d x)+1}\right )}{2 a}+\frac {e \sqrt {a \sin (c+d x)+a} \sqrt {e \cos (c+d x)}}{a d}\right )}{4 a}+\frac {e (e \cos (c+d x))^{5/2}}{2 a d \sqrt {a \sin (c+d x)+a}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {5 e^2 \left (\frac {e^2 \left (\frac {\sqrt {\cos (c+d x)+1} \sqrt {a \sin (c+d x)+a} \int \frac {\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )+1}}{\sqrt {e \sin \left (c+d x+\frac {\pi }{2}\right )}}dx}{\sin (c+d x)+\cos (c+d x)+1}-\frac {\sqrt {\cos (c+d x)+1} \sqrt {a \sin (c+d x)+a} \int \frac {\cos \left (\frac {1}{2} (2 c+\pi )+d x\right )}{\sqrt {e \sin \left (\frac {1}{2} (2 c+\pi )+d x\right )} \sqrt {\sin \left (\frac {1}{2} (2 c+\pi )+d x\right )+1}}dx}{\sin (c+d x)+\cos (c+d x)+1}\right )}{2 a}+\frac {e \sqrt {a \sin (c+d x)+a} \sqrt {e \cos (c+d x)}}{a d}\right )}{4 a}+\frac {e (e \cos (c+d x))^{5/2}}{2 a d \sqrt {a \sin (c+d x)+a}}\) |
| \(\Big \downarrow \) 3254 |
| \(\displaystyle \frac {5 e^2 \left (\frac {e^2 \left (-\frac {\sqrt {\cos (c+d x)+1} \sqrt {a \sin (c+d x)+a} \int \frac {\cos \left (\frac {1}{2} (2 c+\pi )+d x\right )}{\sqrt {e \sin \left (\frac {1}{2} (2 c+\pi )+d x\right )} \sqrt {\sin \left (\frac {1}{2} (2 c+\pi )+d x\right )+1}}dx}{\sin (c+d x)+\cos (c+d x)+1}-\frac {2 \sqrt {\cos (c+d x)+1} \sqrt {a \sin (c+d x)+a} \int \frac {1}{\frac {\sin (c+d x) \tan (c+d x)}{\cos (c+d x)+1}+1}d\left (-\frac {\sin (c+d x)}{\sqrt {e \cos (c+d x)} \sqrt {\cos (c+d x)+1}}\right )}{d (\sin (c+d x)+\cos (c+d x)+1)}\right )}{2 a}+\frac {e \sqrt {a \sin (c+d x)+a} \sqrt {e \cos (c+d x)}}{a d}\right )}{4 a}+\frac {e (e \cos (c+d x))^{5/2}}{2 a d \sqrt {a \sin (c+d x)+a}}\) |
| \(\Big \downarrow \) 216 |
| \(\displaystyle \frac {5 e^2 \left (\frac {e^2 \left (\frac {2 \sqrt {\cos (c+d x)+1} \sqrt {a \sin (c+d x)+a} \arctan \left (\frac {\sqrt {e} \sin (c+d x)}{\sqrt {\cos (c+d x)+1} \sqrt {e \cos (c+d x)}}\right )}{d \sqrt {e} (\sin (c+d x)+\cos (c+d x)+1)}-\frac {\sqrt {\cos (c+d x)+1} \sqrt {a \sin (c+d x)+a} \int \frac {\cos \left (\frac {1}{2} (2 c+\pi )+d x\right )}{\sqrt {e \sin \left (\frac {1}{2} (2 c+\pi )+d x\right )} \sqrt {\sin \left (\frac {1}{2} (2 c+\pi )+d x\right )+1}}dx}{\sin (c+d x)+\cos (c+d x)+1}\right )}{2 a}+\frac {e \sqrt {a \sin (c+d x)+a} \sqrt {e \cos (c+d x)}}{a d}\right )}{4 a}+\frac {e (e \cos (c+d x))^{5/2}}{2 a d \sqrt {a \sin (c+d x)+a}}\) |
| \(\Big \downarrow \) 3312 |
| \(\displaystyle \frac {5 e^2 \left (\frac {e^2 \left (\frac {2 \sqrt {\cos (c+d x)+1} \sqrt {a \sin (c+d x)+a} \arctan \left (\frac {\sqrt {e} \sin (c+d x)}{\sqrt {\cos (c+d x)+1} \sqrt {e \cos (c+d x)}}\right )}{d \sqrt {e} (\sin (c+d x)+\cos (c+d x)+1)}-\frac {\sqrt {\cos (c+d x)+1} \sqrt {a \sin (c+d x)+a} \int \frac {1}{\sqrt {e \cos (c+d x)} \sqrt {\cos (c+d x)+1}}d\cos (c+d x)}{d (\sin (c+d x)+\cos (c+d x)+1)}\right )}{2 a}+\frac {e \sqrt {a \sin (c+d x)+a} \sqrt {e \cos (c+d x)}}{a d}\right )}{4 a}+\frac {e (e \cos (c+d x))^{5/2}}{2 a d \sqrt {a \sin (c+d x)+a}}\) |
| \(\Big \downarrow \) 63 |
| \(\displaystyle \frac {5 e^2 \left (\frac {e^2 \left (\frac {2 \sqrt {\cos (c+d x)+1} \sqrt {a \sin (c+d x)+a} \arctan \left (\frac {\sqrt {e} \sin (c+d x)}{\sqrt {\cos (c+d x)+1} \sqrt {e \cos (c+d x)}}\right )}{d \sqrt {e} (\sin (c+d x)+\cos (c+d x)+1)}-\frac {2 \sqrt {\cos (c+d x)+1} \sqrt {a \sin (c+d x)+a} \int \frac {1}{\sqrt {\cos (c+d x)+1}}d\sqrt {e \cos (c+d x)}}{d e (\sin (c+d x)+\cos (c+d x)+1)}\right )}{2 a}+\frac {e \sqrt {a \sin (c+d x)+a} \sqrt {e \cos (c+d x)}}{a d}\right )}{4 a}+\frac {e (e \cos (c+d x))^{5/2}}{2 a d \sqrt {a \sin (c+d x)+a}}\) |
| \(\Big \downarrow \) 222 |
| \(\displaystyle \frac {5 e^2 \left (\frac {e^2 \left (\frac {2 \sqrt {\cos (c+d x)+1} \sqrt {a \sin (c+d x)+a} \arctan \left (\frac {\sqrt {e} \sin (c+d x)}{\sqrt {\cos (c+d x)+1} \sqrt {e \cos (c+d x)}}\right )}{d \sqrt {e} (\sin (c+d x)+\cos (c+d x)+1)}-\frac {2 \sqrt {\cos (c+d x)+1} \sqrt {a \sin (c+d x)+a} \text {arcsinh}\left (\frac {\sqrt {e \cos (c+d x)}}{\sqrt {e}}\right )}{d \sqrt {e} (\sin (c+d x)+\cos (c+d x)+1)}\right )}{2 a}+\frac {e \sqrt {a \sin (c+d x)+a} \sqrt {e \cos (c+d x)}}{a d}\right )}{4 a}+\frac {e (e \cos (c+d x))^{5/2}}{2 a d \sqrt {a \sin (c+d x)+a}}\) |
(e*(e*Cos[c + d*x])^(5/2))/(2*a*d*Sqrt[a + a*Sin[c + d*x]]) + (5*e^2*((e*S qrt[e*Cos[c + d*x]]*Sqrt[a + a*Sin[c + d*x]])/(a*d) + (e^2*((-2*ArcSinh[Sq rt[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*a)
3.4.5.3.1 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[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[(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 = 7.54 (sec) , antiderivative size = 329, normalized size of antiderivative = 1.33
| method | result | size |
| default | \(\frac {\left (2 \left (\cos ^{2}\left (d x +c \right )\right ) \sin \left (d x +c \right )-2 \left (\cos ^{3}\left (d x +c \right )\right )-5 \arctan \left (\sqrt {-\frac {\cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\right ) \sqrt {-\frac {\cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\, \cos \left (d x +c \right )-5 \,\operatorname {arctanh}\left (\frac {\sin \left (d x +c \right )}{\left (1+\cos \left (d x +c \right )\right ) \sqrt {-\frac {\cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}}\right ) \sqrt {-\frac {\cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\, \cos \left (d x +c \right )-5 \cos \left (d x +c \right ) \sin \left (d x +c \right )-7 \left (\cos ^{2}\left (d x +c \right )\right )-5 \sqrt {-\frac {\cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\, \arctan \left (\sqrt {-\frac {\cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\right )-5 \sqrt {-\frac {\cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\, \operatorname {arctanh}\left (\frac {\sin \left (d x +c \right )}{\left (1+\cos \left (d x +c \right )\right ) \sqrt {-\frac {\cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}}\right )-5 \cos \left (d x +c \right )\right ) \sqrt {e \cos \left (d x +c \right )}\, e^{3}}{4 d \left (-\cos \left (d x +c \right )+\sin \left (d x +c \right )-1\right ) a \sqrt {a \left (1+\sin \left (d x +c \right )\right )}}\) | \(329\) |
1/4/d*(2*cos(d*x+c)^2*sin(d*x+c)-2*cos(d*x+c)^3-5*arctan((-cos(d*x+c)/(1+c os(d*x+c)))^(1/2))*(-cos(d*x+c)/(1+cos(d*x+c)))^(1/2)*cos(d*x+c)-5*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)-5*cos(d*x+c)*sin(d*x+c)-7*cos(d*x+c)^2- 5*(-cos(d*x+c)/(1+cos(d*x+c)))^(1/2)*arctan((-cos(d*x+c)/(1+cos(d*x+c)))^( 1/2))-5*(-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))-5*cos(d*x+c))*(e*cos(d*x+c))^(1/2)* e^3/(-cos(d*x+c)+sin(d*x+c)-1)/a/(a*(1+sin(d*x+c)))^(1/2)
Result contains complex when optimal does not.
Time = 0.49 (sec) , antiderivative size = 1058, normalized size of antiderivative = 4.28 \[ \int \frac {(e \cos (c+d x))^{7/2}}{(a+a \sin (c+d x))^{3/2}} \, dx=\text {Too large to display} \]
-1/16*(5*I*(-e^14/(a^6*d^4))^(1/4)*a^2*d*log(-125/2*(2*(e^10*sin(d*x + c) + (a^3*d^2*e^3*cos(d*x + c) + a^3*d^2*e^3)*sqrt(-e^14/(a^6*d^4)))*sqrt(e*c os(d*x + c))*sqrt(a*sin(d*x + c) + a) + (I*a^5*d^3*cos(d*x + c) + I*a^5*d^ 3 + (2*I*a^5*d^3*cos(d*x + c) + I*a^5*d^3)*sin(d*x + c))*(-e^14/(a^6*d^4)) ^(3/4) + (-2*I*a^2*d*e^7*cos(d*x + c)^2 - I*a^2*d*e^7*cos(d*x + c) + I*a^2 *d*e^7*sin(d*x + c) + I*a^2*d*e^7)*(-e^14/(a^6*d^4))^(1/4))/(cos(d*x + c) + sin(d*x + c) + 1)) - 5*I*(-e^14/(a^6*d^4))^(1/4)*a^2*d*log(-125/2*(2*(e^ 10*sin(d*x + c) + (a^3*d^2*e^3*cos(d*x + c) + a^3*d^2*e^3)*sqrt(-e^14/(a^6 *d^4)))*sqrt(e*cos(d*x + c))*sqrt(a*sin(d*x + c) + a) + (-I*a^5*d^3*cos(d* x + c) - I*a^5*d^3 + (-2*I*a^5*d^3*cos(d*x + c) - I*a^5*d^3)*sin(d*x + c)) *(-e^14/(a^6*d^4))^(3/4) + (2*I*a^2*d*e^7*cos(d*x + c)^2 + I*a^2*d*e^7*cos (d*x + c) - I*a^2*d*e^7*sin(d*x + c) - I*a^2*d*e^7)*(-e^14/(a^6*d^4))^(1/4 ))/(cos(d*x + c) + sin(d*x + c) + 1)) - 5*(-e^14/(a^6*d^4))^(1/4)*a^2*d*lo g(-125/2*(2*(e^10*sin(d*x + c) - (a^3*d^2*e^3*cos(d*x + c) + a^3*d^2*e^3)* sqrt(-e^14/(a^6*d^4)))*sqrt(e*cos(d*x + c))*sqrt(a*sin(d*x + c) + a) + (a^ 5*d^3*cos(d*x + c) + a^5*d^3 + (2*a^5*d^3*cos(d*x + c) + a^5*d^3)*sin(d*x + c))*(-e^14/(a^6*d^4))^(3/4) + (2*a^2*d*e^7*cos(d*x + c)^2 + a^2*d*e^7*co s(d*x + c) - a^2*d*e^7*sin(d*x + c) - a^2*d*e^7)*(-e^14/(a^6*d^4))^(1/4))/ (cos(d*x + c) + sin(d*x + c) + 1)) + 5*(-e^14/(a^6*d^4))^(1/4)*a^2*d*log(- 125/2*(2*(e^10*sin(d*x + c) - (a^3*d^2*e^3*cos(d*x + c) + a^3*d^2*e^3)*...
Timed out. \[ \int \frac {(e \cos (c+d x))^{7/2}}{(a+a \sin (c+d x))^{3/2}} \, dx=\text {Timed out} \]
\[ \int \frac {(e \cos (c+d x))^{7/2}}{(a+a \sin (c+d x))^{3/2}} \, dx=\int { \frac {\left (e \cos \left (d x + c\right )\right )^{\frac {7}{2}}}{{\left (a \sin \left (d x + c\right ) + a\right )}^{\frac {3}{2}}} \,d x } \]
Timed out. \[ \int \frac {(e \cos (c+d x))^{7/2}}{(a+a \sin (c+d x))^{3/2}} \, dx=\text {Timed out} \]
Timed out. \[ \int \frac {(e \cos (c+d x))^{7/2}}{(a+a \sin (c+d x))^{3/2}} \, dx=\int \frac {{\left (e\,\cos \left (c+d\,x\right )\right )}^{7/2}}{{\left (a+a\,\sin \left (c+d\,x\right )\right )}^{3/2}} \,d x \]