Integrand size = 27, antiderivative size = 239 \[ \int \frac {(e \cos (c+d x))^{7/2}}{(a+a \sin (c+d x))^{5/2}} \, dx=-\frac {4 e (e \cos (c+d x))^{5/2}}{a d (a+a \sin (c+d x))^{3/2}}-\frac {5 e^3 \sqrt {e \cos (c+d x)} \sqrt {a+a \sin (c+d x)}}{a^3 d}+\frac {5 e^{7/2} \text {arcsinh}\left (\frac {\sqrt {e \cos (c+d x)}}{\sqrt {e}}\right ) \sqrt {1+\cos (c+d x)} \sqrt {a+a \sin (c+d x)}}{a^3 d (1+\cos (c+d x)+\sin (c+d x))}-\frac {5 e^{7/2} \arctan \left (\frac {\sqrt {e} \sin (c+d x)}{\sqrt {e \cos (c+d x)} \sqrt {1+\cos (c+d x)}}\right ) \sqrt {1+\cos (c+d x)} \sqrt {a+a \sin (c+d x)}}{a^3 d (1+\cos (c+d x)+\sin (c+d x))} \] Output:
-4*e*(e*cos(d*x+c))^(5/2)/a/d/(a+a*sin(d*x+c))^(3/2)-5*e^3*(e*cos(d*x+c))^ (1/2)*(a+a*sin(d*x+c))^(1/2)/a^3/d+5*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^3/d/(1+cos(d*x+c)+s in(d*x+c))-5*e^(7/2)*arctan(e^(1/2)*sin(d*x+c)/(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^3/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 = 0.13 (sec) , antiderivative size = 80, normalized size of antiderivative = 0.33 \[ \int \frac {(e \cos (c+d x))^{7/2}}{(a+a \sin (c+d x))^{5/2}} \, dx=-\frac {2^{3/4} (e \cos (c+d x))^{9/2} \operatorname {Hypergeometric2F1}\left (\frac {5}{4},\frac {9}{4},\frac {13}{4},\frac {1}{2} (1-\sin (c+d x))\right ) \sqrt {a (1+\sin (c+d x))}}{9 a^3 d e (1+\sin (c+d x))^{11/4}} \] Input:
Integrate[(e*Cos[c + d*x])^(7/2)/(a + a*Sin[c + d*x])^(5/2),x]
Output:
-1/9*(2^(3/4)*(e*Cos[c + d*x])^(9/2)*Hypergeometric2F1[5/4, 9/4, 13/4, (1 - Sin[c + d*x])/2]*Sqrt[a*(1 + Sin[c + d*x])])/(a^3*d*e*(1 + Sin[c + d*x]) ^(11/4))
Time = 1.04 (sec) , antiderivative size = 250, 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, 3159, 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.
| \(\displaystyle \int \frac {(e \cos (c+d x))^{7/2}}{(a \sin (c+d x)+a)^{5/2}} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {(e \cos (c+d x))^{7/2}}{(a \sin (c+d x)+a)^{5/2}}dx\) |
| \(\Big \downarrow \) 3159 |
| \(\displaystyle -\frac {5 e^2 \int \frac {(e \cos (c+d x))^{3/2}}{\sqrt {\sin (c+d x) a+a}}dx}{a^2}-\frac {4 e (e \cos (c+d x))^{5/2}}{a d (a \sin (c+d x)+a)^{3/2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {5 e^2 \int \frac {(e \cos (c+d x))^{3/2}}{\sqrt {\sin (c+d x) a+a}}dx}{a^2}-\frac {4 e (e \cos (c+d x))^{5/2}}{a d (a \sin (c+d x)+a)^{3/2}}\) |
| \(\Big \downarrow \) 3164 |
| \(\displaystyle -\frac {5 e^2 \left (\frac {e^2 \int \frac {\sqrt {\sin (c+d x) a+a}}{\sqrt {e \cos (c+d x)}}dx}{2 a}+\frac {e \sqrt {a \sin (c+d x)+a} \sqrt {e \cos (c+d x)}}{a d}\right )}{a^2}-\frac {4 e (e \cos (c+d x))^{5/2}}{a d (a \sin (c+d x)+a)^{3/2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {5 e^2 \left (\frac {e^2 \int \frac {\sqrt {\sin (c+d x) a+a}}{\sqrt {e \cos (c+d x)}}dx}{2 a}+\frac {e \sqrt {a \sin (c+d x)+a} \sqrt {e \cos (c+d x)}}{a d}\right )}{a^2}-\frac {4 e (e \cos (c+d x))^{5/2}}{a d (a \sin (c+d x)+a)^{3/2}}\) |
| \(\Big \downarrow \) 3156 |
| \(\displaystyle -\frac {5 e^2 \left (\frac {e^2 \left (\frac {\sqrt {\cos (c+d x)+1} \sqrt {a \sin (c+d x)+a} \int \frac {\sqrt {\cos (c+d x)+1}}{\sqrt {e \cos (c+d x)}}dx}{\sin (c+d x)+\cos (c+d x)+1}+\frac {\sqrt {\cos (c+d x)+1} \sqrt {a \sin (c+d x)+a} \int \frac {\sin (c+d x)}{\sqrt {e \cos (c+d x)} \sqrt {\cos (c+d x)+1}}dx}{\sin (c+d x)+\cos (c+d x)+1}\right )}{2 a}+\frac {e \sqrt {a \sin (c+d x)+a} \sqrt {e \cos (c+d x)}}{a d}\right )}{a^2}-\frac {4 e (e \cos (c+d x))^{5/2}}{a d (a \sin (c+d x)+a)^{3/2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {5 e^2 \left (\frac {e^2 \left (\frac {\sqrt {\cos (c+d x)+1} \sqrt {a \sin (c+d x)+a} \int -\frac {\cos \left (c+d x+\frac {\pi }{2}\right )}{\sqrt {e \sin \left (c+d x+\frac {\pi }{2}\right )} \sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )+1}}dx}{\sin (c+d x)+\cos (c+d x)+1}+\frac {\sqrt {\cos (c+d x)+1} \sqrt {a \sin (c+d x)+a} \int \frac {\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )+1}}{\sqrt {e \sin \left (c+d x+\frac {\pi }{2}\right )}}dx}{\sin (c+d x)+\cos (c+d x)+1}\right )}{2 a}+\frac {e \sqrt {a \sin (c+d x)+a} \sqrt {e \cos (c+d x)}}{a d}\right )}{a^2}-\frac {4 e (e \cos (c+d x))^{5/2}}{a d (a \sin (c+d x)+a)^{3/2}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {5 e^2 \left (\frac {e^2 \left (\frac {\sqrt {\cos (c+d x)+1} \sqrt {a \sin (c+d x)+a} \int \frac {\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )+1}}{\sqrt {e \sin \left (c+d x+\frac {\pi }{2}\right )}}dx}{\sin (c+d x)+\cos (c+d x)+1}-\frac {\sqrt {\cos (c+d x)+1} \sqrt {a \sin (c+d x)+a} \int \frac {\cos \left (\frac {1}{2} (2 c+\pi )+d x\right )}{\sqrt {e \sin \left (\frac {1}{2} (2 c+\pi )+d x\right )} \sqrt {\sin \left (\frac {1}{2} (2 c+\pi )+d x\right )+1}}dx}{\sin (c+d x)+\cos (c+d x)+1}\right )}{2 a}+\frac {e \sqrt {a \sin (c+d x)+a} \sqrt {e \cos (c+d x)}}{a d}\right )}{a^2}-\frac {4 e (e \cos (c+d x))^{5/2}}{a d (a \sin (c+d x)+a)^{3/2}}\) |
| \(\Big \downarrow \) 3254 |
| \(\displaystyle -\frac {5 e^2 \left (\frac {e^2 \left (-\frac {\sqrt {\cos (c+d x)+1} \sqrt {a \sin (c+d x)+a} \int \frac {\cos \left (\frac {1}{2} (2 c+\pi )+d x\right )}{\sqrt {e \sin \left (\frac {1}{2} (2 c+\pi )+d x\right )} \sqrt {\sin \left (\frac {1}{2} (2 c+\pi )+d x\right )+1}}dx}{\sin (c+d x)+\cos (c+d x)+1}-\frac {2 \sqrt {\cos (c+d x)+1} \sqrt {a \sin (c+d x)+a} \int \frac {1}{\frac {\sin (c+d x) \tan (c+d x)}{\cos (c+d x)+1}+1}d\left (-\frac {\sin (c+d x)}{\sqrt {e \cos (c+d x)} \sqrt {\cos (c+d x)+1}}\right )}{d (\sin (c+d x)+\cos (c+d x)+1)}\right )}{2 a}+\frac {e \sqrt {a \sin (c+d x)+a} \sqrt {e \cos (c+d x)}}{a d}\right )}{a^2}-\frac {4 e (e \cos (c+d x))^{5/2}}{a d (a \sin (c+d x)+a)^{3/2}}\) |
| \(\Big \downarrow \) 216 |
| \(\displaystyle -\frac {5 e^2 \left (\frac {e^2 \left (\frac {2 \sqrt {\cos (c+d x)+1} \sqrt {a \sin (c+d x)+a} \arctan \left (\frac {\sqrt {e} \sin (c+d x)}{\sqrt {\cos (c+d x)+1} \sqrt {e \cos (c+d x)}}\right )}{d \sqrt {e} (\sin (c+d x)+\cos (c+d x)+1)}-\frac {\sqrt {\cos (c+d x)+1} \sqrt {a \sin (c+d x)+a} \int \frac {\cos \left (\frac {1}{2} (2 c+\pi )+d x\right )}{\sqrt {e \sin \left (\frac {1}{2} (2 c+\pi )+d x\right )} \sqrt {\sin \left (\frac {1}{2} (2 c+\pi )+d x\right )+1}}dx}{\sin (c+d x)+\cos (c+d x)+1}\right )}{2 a}+\frac {e \sqrt {a \sin (c+d x)+a} \sqrt {e \cos (c+d x)}}{a d}\right )}{a^2}-\frac {4 e (e \cos (c+d x))^{5/2}}{a d (a \sin (c+d x)+a)^{3/2}}\) |
| \(\Big \downarrow \) 3312 |
| \(\displaystyle -\frac {5 e^2 \left (\frac {e^2 \left (\frac {2 \sqrt {\cos (c+d x)+1} \sqrt {a \sin (c+d x)+a} \arctan \left (\frac {\sqrt {e} \sin (c+d x)}{\sqrt {\cos (c+d x)+1} \sqrt {e \cos (c+d x)}}\right )}{d \sqrt {e} (\sin (c+d x)+\cos (c+d x)+1)}-\frac {\sqrt {\cos (c+d x)+1} \sqrt {a \sin (c+d x)+a} \int \frac {1}{\sqrt {e \cos (c+d x)} \sqrt {\cos (c+d x)+1}}d\cos (c+d x)}{d (\sin (c+d x)+\cos (c+d x)+1)}\right )}{2 a}+\frac {e \sqrt {a \sin (c+d x)+a} \sqrt {e \cos (c+d x)}}{a d}\right )}{a^2}-\frac {4 e (e \cos (c+d x))^{5/2}}{a d (a \sin (c+d x)+a)^{3/2}}\) |
| \(\Big \downarrow \) 63 |
| \(\displaystyle -\frac {5 e^2 \left (\frac {e^2 \left (\frac {2 \sqrt {\cos (c+d x)+1} \sqrt {a \sin (c+d x)+a} \arctan \left (\frac {\sqrt {e} \sin (c+d x)}{\sqrt {\cos (c+d x)+1} \sqrt {e \cos (c+d x)}}\right )}{d \sqrt {e} (\sin (c+d x)+\cos (c+d x)+1)}-\frac {2 \sqrt {\cos (c+d x)+1} \sqrt {a \sin (c+d x)+a} \int \frac {1}{\sqrt {\cos (c+d x)+1}}d\sqrt {e \cos (c+d x)}}{d e (\sin (c+d x)+\cos (c+d x)+1)}\right )}{2 a}+\frac {e \sqrt {a \sin (c+d x)+a} \sqrt {e \cos (c+d x)}}{a d}\right )}{a^2}-\frac {4 e (e \cos (c+d x))^{5/2}}{a d (a \sin (c+d x)+a)^{3/2}}\) |
| \(\Big \downarrow \) 222 |
| \(\displaystyle -\frac {5 e^2 \left (\frac {e^2 \left (\frac {2 \sqrt {\cos (c+d x)+1} \sqrt {a \sin (c+d x)+a} \arctan \left (\frac {\sqrt {e} \sin (c+d x)}{\sqrt {\cos (c+d x)+1} \sqrt {e \cos (c+d x)}}\right )}{d \sqrt {e} (\sin (c+d x)+\cos (c+d x)+1)}-\frac {2 \sqrt {\cos (c+d x)+1} \sqrt {a \sin (c+d x)+a} \text {arcsinh}\left (\frac {\sqrt {e \cos (c+d x)}}{\sqrt {e}}\right )}{d \sqrt {e} (\sin (c+d x)+\cos (c+d x)+1)}\right )}{2 a}+\frac {e \sqrt {a \sin (c+d x)+a} \sqrt {e \cos (c+d x)}}{a d}\right )}{a^2}-\frac {4 e (e \cos (c+d x))^{5/2}}{a d (a \sin (c+d x)+a)^{3/2}}\) |
Input:
Int[(e*Cos[c + d*x])^(7/2)/(a + a*Sin[c + d*x])^(5/2),x]
Output:
(-4*e*(e*Cos[c + d*x])^(5/2))/(a*d*(a + a*Sin[c + d*x])^(3/2)) - (5*e^2*(( e*Sqrt[e*Cos[c + d*x]]*Sqrt[a + a*Sin[c + d*x]])/(a*d) + (e^2*((-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]*S in[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)))/a^2
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[2*g*(g*Cos[e + f*x])^(p - 1)*((a + b*Sin[e + f *x])^(m + 1)/(b*f*(2*m + p + 1))), x] + Simp[g^2*((p - 1)/(b^2*(2*m + p + 1 ))) Int[(g*Cos[e + f*x])^(p - 2)*(a + b*Sin[e + f*x])^(m + 2), x], x] /; FreeQ[{a, b, e, f, g}, x] && EqQ[a^2 - b^2, 0] && LeQ[m, -2] && GtQ[p, 1] & & NeQ[2*m + p + 1, 0] && !ILtQ[m + p + 1, 0] && IntegersQ[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]
Result contains higher order function than in optimal. Order 4 vs. order 3.
Time = 14.15 (sec) , antiderivative size = 890, normalized size of antiderivative = 3.72
Input:
int((e*cos(d*x+c))^(7/2)/(a+a*sin(d*x+c))^(5/2),x,method=_RETURNVERBOSE)
Output:
-1/2/d/(2*2^(1/2)+3)^(1/2)/(1+2^(1/2))/e/a^2*(20*(2+(cos(1/2*d*x+1/2*c)^2+ 2*cos(1/2*d*x+1/2*c)+1)*2^(1/2)+2*cos(1/2*d*x+1/2*c)^2+4*cos(1/2*d*x+1/2*c ))*((2^(1/2)*cos(1/2*d*x+1/2*c)-2^(1/2)+2*cos(1/2*d*x+1/2*c)-1)/(cos(1/2*d *x+1/2*c)+1))^(1/2)*(-2*(2^(1/2)*cos(1/2*d*x+1/2*c)-2^(1/2)-2*cos(1/2*d*x+ 1/2*c)+1)/(cos(1/2*d*x+1/2*c)+1))^(1/2)*(e*(2*cos(1/2*d*x+1/2*c)^2-1)/(cos (1/2*d*x+1/2*c)+1)^2)^(1/2)*e^4*EllipticPi((2*2^(1/2)+3)^(1/2)*(csc(1/2*d* x+1/2*c)-cot(1/2*d*x+1/2*c)),-1/(2*2^(1/2)+3),(-2*2^(1/2)+3)^(1/2)/(2*2^(1 /2)+3)^(1/2))+5*(4+3*(-2*cos(1/2*d*x+1/2*c)^2+1)*2^(1/2)-8*cos(1/2*d*x+1/2 *c)^2)*e^(9/2)*arctanh(2^(1/2)*e^(1/2)*cos(1/2*d*x+1/2*c)/(cos(1/2*d*x+1/2 *c)+1)/(e*(2*cos(1/2*d*x+1/2*c)^2-1)/(cos(1/2*d*x+1/2*c)+1)^2)^(1/2))+10*( -2+(-cos(1/2*d*x+1/2*c)^2-2*cos(1/2*d*x+1/2*c)-1)*2^(1/2)-2*cos(1/2*d*x+1/ 2*c)^2-4*cos(1/2*d*x+1/2*c))*EllipticF((1+2^(1/2))*(csc(1/2*d*x+1/2*c)-cot (1/2*d*x+1/2*c)),-2*2^(1/2)+3)*((2^(1/2)*cos(1/2*d*x+1/2*c)-2^(1/2)+2*cos( 1/2*d*x+1/2*c)-1)/(cos(1/2*d*x+1/2*c)+1))^(1/2)*(e*(2*cos(1/2*d*x+1/2*c)^2 -1)/(cos(1/2*d*x+1/2*c)+1)^2)^(1/2)*e^4*(-2*(2^(1/2)*cos(1/2*d*x+1/2*c)-2^ (1/2)-2*cos(1/2*d*x+1/2*c)+1)/(cos(1/2*d*x+1/2*c)+1))^(1/2)+2*(cos(1/2*d*x +1/2*c)+1)*(2*((2*cos(1/2*d*x+1/2*c)^2-9)*sin(1/2*d*x+1/2*c)+(2*cos(1/2*d* x+1/2*c)^2+7)*cos(1/2*d*x+1/2*c))*2^(1/2)+3*(2*cos(1/2*d*x+1/2*c)^2-9)*sin (1/2*d*x+1/2*c)+3*(2*cos(1/2*d*x+1/2*c)^2+7)*cos(1/2*d*x+1/2*c))*(e*(2*cos (1/2*d*x+1/2*c)^2-1)/(cos(1/2*d*x+1/2*c)+1)^2)^(1/2)*e^4)*(e*(2*cos(1/2...
Leaf count of result is larger than twice the leaf count of optimal. 428 vs. \(2 (209) = 418\).
Time = 0.15 (sec) , antiderivative size = 428, normalized size of antiderivative = 1.79 \[ \int \frac {(e \cos (c+d x))^{7/2}}{(a+a \sin (c+d x))^{5/2}} \, dx=-\frac {20 \, \sqrt {2} {\left (a e^{3} \sin \left (d x + c\right ) + a e^{3}\right )} \sqrt {\frac {e}{a}} \arctan \left (\frac {\sqrt {2} \sqrt {e \cos \left (d x + c\right )} \sqrt {a \sin \left (d x + c\right ) + a} \sqrt {\frac {e}{a}} \sin \left (d x + c\right )}{e \cos \left (d x + c\right )^{2} + e \cos \left (d x + c\right ) \sin \left (d x + c\right ) + e \cos \left (d x + c\right )}\right ) - 5 \, \sqrt {2} {\left (a e^{3} \sin \left (d x + c\right ) + a e^{3}\right )} \sqrt {\frac {e}{a}} \log \left (\frac {2 \, \sqrt {2} \sqrt {e \cos \left (d x + c\right )} \sqrt {a \sin \left (d x + c\right ) + a} \sqrt {\frac {e}{a}} {\left (\cos \left (d x + c\right ) + 1\right )} + 2 \, e \cos \left (d x + c\right )^{2} + 3 \, e \cos \left (d x + c\right ) + {\left (2 \, e \cos \left (d x + c\right ) + e\right )} \sin \left (d x + c\right ) + e}{\cos \left (d x + c\right ) + \sin \left (d x + c\right ) + 1}\right ) + 5 \, \sqrt {2} {\left (a e^{3} \sin \left (d x + c\right ) + a e^{3}\right )} \sqrt {\frac {e}{a}} \log \left (-\frac {2 \, \sqrt {2} \sqrt {e \cos \left (d x + c\right )} \sqrt {a \sin \left (d x + c\right ) + a} \sqrt {\frac {e}{a}} {\left (\cos \left (d x + c\right ) + 1\right )} - 2 \, e \cos \left (d x + c\right )^{2} - 3 \, e \cos \left (d x + c\right ) - {\left (2 \, e \cos \left (d x + c\right ) + e\right )} \sin \left (d x + c\right ) - e}{\cos \left (d x + c\right ) + \sin \left (d x + c\right ) + 1}\right ) + 8 \, {\left (e^{3} \sin \left (d x + c\right ) + 9 \, e^{3}\right )} \sqrt {e \cos \left (d x + c\right )} \sqrt {a \sin \left (d x + c\right ) + a}}{8 \, {\left (a^{3} d \sin \left (d x + c\right ) + a^{3} d\right )}} \] Input:
integrate((e*cos(d*x+c))^(7/2)/(a+a*sin(d*x+c))^(5/2),x, algorithm="fricas ")
Output:
-1/8*(20*sqrt(2)*(a*e^3*sin(d*x + c) + a*e^3)*sqrt(e/a)*arctan(sqrt(2)*sqr t(e*cos(d*x + c))*sqrt(a*sin(d*x + c) + a)*sqrt(e/a)*sin(d*x + c)/(e*cos(d *x + c)^2 + e*cos(d*x + c)*sin(d*x + c) + e*cos(d*x + c))) - 5*sqrt(2)*(a* e^3*sin(d*x + c) + a*e^3)*sqrt(e/a)*log((2*sqrt(2)*sqrt(e*cos(d*x + c))*sq rt(a*sin(d*x + c) + a)*sqrt(e/a)*(cos(d*x + c) + 1) + 2*e*cos(d*x + c)^2 + 3*e*cos(d*x + c) + (2*e*cos(d*x + c) + e)*sin(d*x + c) + e)/(cos(d*x + c) + sin(d*x + c) + 1)) + 5*sqrt(2)*(a*e^3*sin(d*x + c) + a*e^3)*sqrt(e/a)*l og(-(2*sqrt(2)*sqrt(e*cos(d*x + c))*sqrt(a*sin(d*x + c) + a)*sqrt(e/a)*(co s(d*x + c) + 1) - 2*e*cos(d*x + c)^2 - 3*e*cos(d*x + c) - (2*e*cos(d*x + c ) + e)*sin(d*x + c) - e)/(cos(d*x + c) + sin(d*x + c) + 1)) + 8*(e^3*sin(d *x + c) + 9*e^3)*sqrt(e*cos(d*x + c))*sqrt(a*sin(d*x + c) + a))/(a^3*d*sin (d*x + c) + a^3*d)
Timed out. \[ \int \frac {(e \cos (c+d x))^{7/2}}{(a+a \sin (c+d x))^{5/2}} \, dx=\text {Timed out} \] Input:
integrate((e*cos(d*x+c))**(7/2)/(a+a*sin(d*x+c))**(5/2),x)
Output:
Timed out
\[ \int \frac {(e \cos (c+d x))^{7/2}}{(a+a \sin (c+d x))^{5/2}} \, dx=\int { \frac {\left (e \cos \left (d x + c\right )\right )^{\frac {7}{2}}}{{\left (a \sin \left (d x + c\right ) + a\right )}^{\frac {5}{2}}} \,d x } \] Input:
integrate((e*cos(d*x+c))^(7/2)/(a+a*sin(d*x+c))^(5/2),x, algorithm="maxima ")
Output:
integrate((e*cos(d*x + c))^(7/2)/(a*sin(d*x + c) + a)^(5/2), x)
Timed out. \[ \int \frac {(e \cos (c+d x))^{7/2}}{(a+a \sin (c+d x))^{5/2}} \, dx=\text {Timed out} \] Input:
integrate((e*cos(d*x+c))^(7/2)/(a+a*sin(d*x+c))^(5/2),x, algorithm="giac")
Output:
Timed out
Timed out. \[ \int \frac {(e \cos (c+d x))^{7/2}}{(a+a \sin (c+d x))^{5/2}} \, dx=\int \frac {{\left (e\,\cos \left (c+d\,x\right )\right )}^{7/2}}{{\left (a+a\,\sin \left (c+d\,x\right )\right )}^{5/2}} \,d x \] Input:
int((e*cos(c + d*x))^(7/2)/(a + a*sin(c + d*x))^(5/2),x)
Output:
int((e*cos(c + d*x))^(7/2)/(a + a*sin(c + d*x))^(5/2), x)
\[ \int \frac {(e \cos (c+d x))^{7/2}}{(a+a \sin (c+d x))^{5/2}} \, dx=\frac {\sqrt {e}\, \sqrt {a}\, e^{3} \left (-2 \sqrt {\sin \left (d x +c \right )+1}\, \sqrt {\cos \left (d x +c \right )}\, \cos \left (d x +c \right )^{2}-10 \sqrt {\sin \left (d x +c \right )+1}\, \sqrt {\cos \left (d x +c \right )}\, \sin \left (d x +c \right )^{2}-30 \sqrt {\sin \left (d x +c \right )+1}\, \sqrt {\cos \left (d x +c \right )}\, \sin \left (d x +c \right )-20 \sqrt {\sin \left (d x +c \right )+1}\, \sqrt {\cos \left (d x +c \right )}-5 \left (\int \frac {\sqrt {\sin \left (d x +c \right )+1}\, \sqrt {\cos \left (d x +c \right )}\, \sin \left (d x +c \right )}{\cos \left (d x +c \right )}d x \right ) \sin \left (d x +c \right )^{2} d -10 \left (\int \frac {\sqrt {\sin \left (d x +c \right )+1}\, \sqrt {\cos \left (d x +c \right )}\, \sin \left (d x +c \right )}{\cos \left (d x +c \right )}d x \right ) \sin \left (d x +c \right ) d -5 \left (\int \frac {\sqrt {\sin \left (d x +c \right )+1}\, \sqrt {\cos \left (d x +c \right )}\, \sin \left (d x +c \right )}{\cos \left (d x +c \right )}d x \right ) d -5 \left (\int \frac {\sqrt {\sin \left (d x +c \right )+1}\, \sqrt {\cos \left (d x +c \right )}\, \sin \left (d x +c \right )}{\cos \left (d x +c \right ) \sin \left (d x +c \right )+\cos \left (d x +c \right )}d x \right ) \sin \left (d x +c \right )^{2} d -10 \left (\int \frac {\sqrt {\sin \left (d x +c \right )+1}\, \sqrt {\cos \left (d x +c \right )}\, \sin \left (d x +c \right )}{\cos \left (d x +c \right ) \sin \left (d x +c \right )+\cos \left (d x +c \right )}d x \right ) \sin \left (d x +c \right ) d -5 \left (\int \frac {\sqrt {\sin \left (d x +c \right )+1}\, \sqrt {\cos \left (d x +c \right )}\, \sin \left (d x +c \right )}{\cos \left (d x +c \right ) \sin \left (d x +c \right )+\cos \left (d x +c \right )}d x \right ) d \right )}{3 a^{3} d \left (\sin \left (d x +c \right )^{2}+2 \sin \left (d x +c \right )+1\right )} \] Input:
int((e*cos(d*x+c))^(7/2)/(a+a*sin(d*x+c))^(5/2),x)
Output:
(sqrt(e)*sqrt(a)*e**3*( - 2*sqrt(sin(c + d*x) + 1)*sqrt(cos(c + d*x))*cos( c + d*x)**2 - 10*sqrt(sin(c + d*x) + 1)*sqrt(cos(c + d*x))*sin(c + d*x)**2 - 30*sqrt(sin(c + d*x) + 1)*sqrt(cos(c + d*x))*sin(c + d*x) - 20*sqrt(sin (c + d*x) + 1)*sqrt(cos(c + d*x)) - 5*int((sqrt(sin(c + d*x) + 1)*sqrt(cos (c + d*x))*sin(c + d*x))/cos(c + d*x),x)*sin(c + d*x)**2*d - 10*int((sqrt( sin(c + d*x) + 1)*sqrt(cos(c + d*x))*sin(c + d*x))/cos(c + d*x),x)*sin(c + d*x)*d - 5*int((sqrt(sin(c + d*x) + 1)*sqrt(cos(c + d*x))*sin(c + d*x))/c os(c + d*x),x)*d - 5*int((sqrt(sin(c + d*x) + 1)*sqrt(cos(c + d*x))*sin(c + d*x))/(cos(c + d*x)*sin(c + d*x) + cos(c + d*x)),x)*sin(c + d*x)**2*d - 10*int((sqrt(sin(c + d*x) + 1)*sqrt(cos(c + d*x))*sin(c + d*x))/(cos(c + d *x)*sin(c + d*x) + cos(c + d*x)),x)*sin(c + d*x)*d - 5*int((sqrt(sin(c + d *x) + 1)*sqrt(cos(c + d*x))*sin(c + d*x))/(cos(c + d*x)*sin(c + d*x) + cos (c + d*x)),x)*d))/(3*a**3*d*(sin(c + d*x)**2 + 2*sin(c + d*x) + 1))