Integrand size = 27, antiderivative size = 286 \[ \int \sqrt {e \cos (c+d x)} (a+a \sin (c+d x))^{5/2} \, dx=-\frac {15 a^3 (e \cos (c+d x))^{3/2}}{8 d e \sqrt {a+a \sin (c+d x)}}-\frac {3 a^2 (e \cos (c+d x))^{3/2} \sqrt {a+a \sin (c+d x)}}{4 d e}+\frac {15 a^2 \sqrt {e} \text {arcsinh}\left (\frac {\sqrt {e \cos (c+d x)}}{\sqrt {e}}\right ) \sqrt {1+\cos (c+d x)} \sqrt {a+a \sin (c+d x)}}{8 d (1+\cos (c+d x)+\sin (c+d x))}+\frac {15 a^2 \sqrt {e} \arctan \left (\frac {\sqrt {e} \sin (c+d x)}{\sqrt {e \cos (c+d x)} \sqrt {1+\cos (c+d x)}}\right ) \sqrt {1+\cos (c+d x)} \sqrt {a+a \sin (c+d x)}}{8 d (1+\cos (c+d x)+\sin (c+d x))}-\frac {a (e \cos (c+d x))^{3/2} (a+a \sin (c+d x))^{3/2}}{3 d e} \] Output:
-15/8*a^3*(e*cos(d*x+c))^(3/2)/d/e/(a+a*sin(d*x+c))^(1/2)-3/4*a^2*(e*cos(d *x+c))^(3/2)*(a+a*sin(d*x+c))^(1/2)/d/e+15/8*a^2*e^(1/2)*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))+15/8*a^2*e^(1/2)*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) /d/(1+cos(d*x+c)+sin(d*x+c))-1/3*a*(e*cos(d*x+c))^(3/2)*(a+a*sin(d*x+c))^( 3/2)/d/e
Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.
Time = 0.13 (sec) , antiderivative size = 78, normalized size of antiderivative = 0.27 \[ \int \sqrt {e \cos (c+d x)} (a+a \sin (c+d x))^{5/2} \, dx=-\frac {16 \sqrt [4]{2} a (e \cos (c+d x))^{3/2} \operatorname {Hypergeometric2F1}\left (-\frac {9}{4},\frac {3}{4},\frac {7}{4},\frac {1}{2} (1-\sin (c+d x))\right ) (a (1+\sin (c+d x)))^{3/2}}{3 d e (1+\sin (c+d x))^{9/4}} \] Input:
Integrate[Sqrt[e*Cos[c + d*x]]*(a + a*Sin[c + d*x])^(5/2),x]
Output:
(-16*2^(1/4)*a*(e*Cos[c + d*x])^(3/2)*Hypergeometric2F1[-9/4, 3/4, 7/4, (1 - Sin[c + d*x])/2]*(a*(1 + Sin[c + d*x]))^(3/2))/(3*d*e*(1 + Sin[c + d*x] )^(9/4))
Time = 1.22 (sec) , antiderivative size = 296, normalized size of antiderivative = 1.03, number of steps used = 16, number of rules used = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.556, Rules used = {3042, 3157, 3042, 3157, 3042, 3157, 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.
| \(\displaystyle \int (a \sin (c+d x)+a)^{5/2} \sqrt {e \cos (c+d x)} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int (a \sin (c+d x)+a)^{5/2} \sqrt {e \cos (c+d x)}dx\) |
| \(\Big \downarrow \) 3157 |
| \(\displaystyle \frac {3}{2} a \int \sqrt {e \cos (c+d x)} (\sin (c+d x) a+a)^{3/2}dx-\frac {a (a \sin (c+d x)+a)^{3/2} (e \cos (c+d x))^{3/2}}{3 d e}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {3}{2} a \int \sqrt {e \cos (c+d x)} (\sin (c+d x) a+a)^{3/2}dx-\frac {a (a \sin (c+d x)+a)^{3/2} (e \cos (c+d x))^{3/2}}{3 d e}\) |
| \(\Big \downarrow \) 3157 |
| \(\displaystyle \frac {3}{2} a \left (\frac {5}{4} a \int \sqrt {e \cos (c+d x)} \sqrt {\sin (c+d x) a+a}dx-\frac {a \sqrt {a \sin (c+d x)+a} (e \cos (c+d x))^{3/2}}{2 d e}\right )-\frac {a (a \sin (c+d x)+a)^{3/2} (e \cos (c+d x))^{3/2}}{3 d e}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {3}{2} a \left (\frac {5}{4} a \int \sqrt {e \cos (c+d x)} \sqrt {\sin (c+d x) a+a}dx-\frac {a \sqrt {a \sin (c+d x)+a} (e \cos (c+d x))^{3/2}}{2 d e}\right )-\frac {a (a \sin (c+d x)+a)^{3/2} (e \cos (c+d x))^{3/2}}{3 d e}\) |
| \(\Big \downarrow \) 3157 |
| \(\displaystyle \frac {3}{2} a \left (\frac {5}{4} a \left (\frac {1}{2} a \int \frac {\sqrt {e \cos (c+d x)}}{\sqrt {\sin (c+d x) a+a}}dx-\frac {a (e \cos (c+d x))^{3/2}}{d e \sqrt {a \sin (c+d x)+a}}\right )-\frac {a \sqrt {a \sin (c+d x)+a} (e \cos (c+d x))^{3/2}}{2 d e}\right )-\frac {a (a \sin (c+d x)+a)^{3/2} (e \cos (c+d x))^{3/2}}{3 d e}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {3}{2} a \left (\frac {5}{4} a \left (\frac {1}{2} a \int \frac {\sqrt {e \cos (c+d x)}}{\sqrt {\sin (c+d x) a+a}}dx-\frac {a (e \cos (c+d x))^{3/2}}{d e \sqrt {a \sin (c+d x)+a}}\right )-\frac {a \sqrt {a \sin (c+d x)+a} (e \cos (c+d x))^{3/2}}{2 d e}\right )-\frac {a (a \sin (c+d x)+a)^{3/2} (e \cos (c+d x))^{3/2}}{3 d e}\) |
| \(\Big \downarrow \) 3163 |
| \(\displaystyle \frac {3}{2} a \left (\frac {5}{4} a \left (\frac {1}{2} a \left (\frac {e \sqrt {\cos (c+d x)+1} \sqrt {a \sin (c+d x)+a} \int \frac {\sqrt {\cos (c+d x)+1}}{\sqrt {e \cos (c+d x)}}dx}{a \sin (c+d x)+a \cos (c+d x)+a}-\frac {e \sqrt {\cos (c+d x)+1} \sqrt {a \sin (c+d x)+a} \int \frac {\sin (c+d x)}{\sqrt {e \cos (c+d x)} \sqrt {\cos (c+d x)+1}}dx}{a \sin (c+d x)+a \cos (c+d x)+a}\right )-\frac {a (e \cos (c+d x))^{3/2}}{d e \sqrt {a \sin (c+d x)+a}}\right )-\frac {a \sqrt {a \sin (c+d x)+a} (e \cos (c+d x))^{3/2}}{2 d e}\right )-\frac {a (a \sin (c+d x)+a)^{3/2} (e \cos (c+d x))^{3/2}}{3 d e}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {3}{2} a \left (\frac {5}{4} a \left (\frac {1}{2} a \left (\frac {e \sqrt {\cos (c+d x)+1} \sqrt {a \sin (c+d x)+a} \int \frac {\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )+1}}{\sqrt {e \sin \left (c+d x+\frac {\pi }{2}\right )}}dx}{a \sin (c+d x)+a \cos (c+d x)+a}-\frac {e \sqrt {\cos (c+d x)+1} \sqrt {a \sin (c+d x)+a} \int -\frac {\cos \left (c+d x+\frac {\pi }{2}\right )}{\sqrt {e \sin \left (c+d x+\frac {\pi }{2}\right )} \sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )+1}}dx}{a \sin (c+d x)+a \cos (c+d x)+a}\right )-\frac {a (e \cos (c+d x))^{3/2}}{d e \sqrt {a \sin (c+d x)+a}}\right )-\frac {a \sqrt {a \sin (c+d x)+a} (e \cos (c+d x))^{3/2}}{2 d e}\right )-\frac {a (a \sin (c+d x)+a)^{3/2} (e \cos (c+d x))^{3/2}}{3 d e}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {3}{2} a \left (\frac {5}{4} a \left (\frac {1}{2} a \left (\frac {e \sqrt {\cos (c+d x)+1} \sqrt {a \sin (c+d x)+a} \int \frac {\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )+1}}{\sqrt {e \sin \left (c+d x+\frac {\pi }{2}\right )}}dx}{a \sin (c+d x)+a \cos (c+d x)+a}+\frac {e \sqrt {\cos (c+d x)+1} \sqrt {a \sin (c+d x)+a} \int \frac {\cos \left (\frac {1}{2} (2 c+\pi )+d x\right )}{\sqrt {e \sin \left (\frac {1}{2} (2 c+\pi )+d x\right )} \sqrt {\sin \left (\frac {1}{2} (2 c+\pi )+d x\right )+1}}dx}{a \sin (c+d x)+a \cos (c+d x)+a}\right )-\frac {a (e \cos (c+d x))^{3/2}}{d e \sqrt {a \sin (c+d x)+a}}\right )-\frac {a \sqrt {a \sin (c+d x)+a} (e \cos (c+d x))^{3/2}}{2 d e}\right )-\frac {a (a \sin (c+d x)+a)^{3/2} (e \cos (c+d x))^{3/2}}{3 d e}\) |
| \(\Big \downarrow \) 3254 |
| \(\displaystyle \frac {3}{2} a \left (\frac {5}{4} a \left (\frac {1}{2} a \left (\frac {e \sqrt {\cos (c+d x)+1} \sqrt {a \sin (c+d x)+a} \int \frac {\cos \left (\frac {1}{2} (2 c+\pi )+d x\right )}{\sqrt {e \sin \left (\frac {1}{2} (2 c+\pi )+d x\right )} \sqrt {\sin \left (\frac {1}{2} (2 c+\pi )+d x\right )+1}}dx}{a \sin (c+d x)+a \cos (c+d x)+a}-\frac {2 e \sqrt {\cos (c+d x)+1} \sqrt {a \sin (c+d x)+a} \int \frac {1}{\frac {\sin (c+d x) \tan (c+d x)}{\cos (c+d x)+1}+1}d\left (-\frac {\sin (c+d x)}{\sqrt {e \cos (c+d x)} \sqrt {\cos (c+d x)+1}}\right )}{d (a \sin (c+d x)+a \cos (c+d x)+a)}\right )-\frac {a (e \cos (c+d x))^{3/2}}{d e \sqrt {a \sin (c+d x)+a}}\right )-\frac {a \sqrt {a \sin (c+d x)+a} (e \cos (c+d x))^{3/2}}{2 d e}\right )-\frac {a (a \sin (c+d x)+a)^{3/2} (e \cos (c+d x))^{3/2}}{3 d e}\) |
| \(\Big \downarrow \) 216 |
| \(\displaystyle \frac {3}{2} a \left (\frac {5}{4} a \left (\frac {1}{2} a \left (\frac {e \sqrt {\cos (c+d x)+1} \sqrt {a \sin (c+d x)+a} \int \frac {\cos \left (\frac {1}{2} (2 c+\pi )+d x\right )}{\sqrt {e \sin \left (\frac {1}{2} (2 c+\pi )+d x\right )} \sqrt {\sin \left (\frac {1}{2} (2 c+\pi )+d x\right )+1}}dx}{a \sin (c+d x)+a \cos (c+d x)+a}+\frac {2 \sqrt {e} \sqrt {\cos (c+d x)+1} \sqrt {a \sin (c+d x)+a} \arctan \left (\frac {\sqrt {e} \sin (c+d x)}{\sqrt {\cos (c+d x)+1} \sqrt {e \cos (c+d x)}}\right )}{d (a \sin (c+d x)+a \cos (c+d x)+a)}\right )-\frac {a (e \cos (c+d x))^{3/2}}{d e \sqrt {a \sin (c+d x)+a}}\right )-\frac {a \sqrt {a \sin (c+d x)+a} (e \cos (c+d x))^{3/2}}{2 d e}\right )-\frac {a (a \sin (c+d x)+a)^{3/2} (e \cos (c+d x))^{3/2}}{3 d e}\) |
| \(\Big \downarrow \) 3312 |
| \(\displaystyle \frac {3}{2} a \left (\frac {5}{4} a \left (\frac {1}{2} a \left (\frac {e \sqrt {\cos (c+d x)+1} \sqrt {a \sin (c+d x)+a} \int \frac {1}{\sqrt {e \cos (c+d x)} \sqrt {\cos (c+d x)+1}}d\cos (c+d x)}{d (a \sin (c+d x)+a \cos (c+d x)+a)}+\frac {2 \sqrt {e} \sqrt {\cos (c+d x)+1} \sqrt {a \sin (c+d x)+a} \arctan \left (\frac {\sqrt {e} \sin (c+d x)}{\sqrt {\cos (c+d x)+1} \sqrt {e \cos (c+d x)}}\right )}{d (a \sin (c+d x)+a \cos (c+d x)+a)}\right )-\frac {a (e \cos (c+d x))^{3/2}}{d e \sqrt {a \sin (c+d x)+a}}\right )-\frac {a \sqrt {a \sin (c+d x)+a} (e \cos (c+d x))^{3/2}}{2 d e}\right )-\frac {a (a \sin (c+d x)+a)^{3/2} (e \cos (c+d x))^{3/2}}{3 d e}\) |
| \(\Big \downarrow \) 63 |
| \(\displaystyle \frac {3}{2} a \left (\frac {5}{4} a \left (\frac {1}{2} a \left (\frac {2 \sqrt {\cos (c+d x)+1} \sqrt {a \sin (c+d x)+a} \int \frac {1}{\sqrt {\cos (c+d x)+1}}d\sqrt {e \cos (c+d x)}}{d (a \sin (c+d x)+a \cos (c+d x)+a)}+\frac {2 \sqrt {e} \sqrt {\cos (c+d x)+1} \sqrt {a \sin (c+d x)+a} \arctan \left (\frac {\sqrt {e} \sin (c+d x)}{\sqrt {\cos (c+d x)+1} \sqrt {e \cos (c+d x)}}\right )}{d (a \sin (c+d x)+a \cos (c+d x)+a)}\right )-\frac {a (e \cos (c+d x))^{3/2}}{d e \sqrt {a \sin (c+d x)+a}}\right )-\frac {a \sqrt {a \sin (c+d x)+a} (e \cos (c+d x))^{3/2}}{2 d e}\right )-\frac {a (a \sin (c+d x)+a)^{3/2} (e \cos (c+d x))^{3/2}}{3 d e}\) |
| \(\Big \downarrow \) 222 |
| \(\displaystyle \frac {3}{2} a \left (\frac {5}{4} a \left (\frac {1}{2} a \left (\frac {2 \sqrt {e} \sqrt {\cos (c+d x)+1} \sqrt {a \sin (c+d x)+a} \text {arcsinh}\left (\frac {\sqrt {e \cos (c+d x)}}{\sqrt {e}}\right )}{d (a \sin (c+d x)+a \cos (c+d x)+a)}+\frac {2 \sqrt {e} \sqrt {\cos (c+d x)+1} \sqrt {a \sin (c+d x)+a} \arctan \left (\frac {\sqrt {e} \sin (c+d x)}{\sqrt {\cos (c+d x)+1} \sqrt {e \cos (c+d x)}}\right )}{d (a \sin (c+d x)+a \cos (c+d x)+a)}\right )-\frac {a (e \cos (c+d x))^{3/2}}{d e \sqrt {a \sin (c+d x)+a}}\right )-\frac {a \sqrt {a \sin (c+d x)+a} (e \cos (c+d x))^{3/2}}{2 d e}\right )-\frac {a (a \sin (c+d x)+a)^{3/2} (e \cos (c+d x))^{3/2}}{3 d e}\) |
Input:
Int[Sqrt[e*Cos[c + d*x]]*(a + a*Sin[c + d*x])^(5/2),x]
Output:
-1/3*(a*(e*Cos[c + d*x])^(3/2)*(a + a*Sin[c + d*x])^(3/2))/(d*e) + (3*a*(- 1/2*(a*(e*Cos[c + d*x])^(3/2)*Sqrt[a + a*Sin[c + d*x]])/(d*e) + (5*a*(-((a *(e*Cos[c + d*x])^(3/2))/(d*e*Sqrt[a + a*Sin[c + d*x]])) + (a*((2*Sqrt[e]* ArcSinh[Sqrt[e*Cos[c + d*x]]/Sqrt[e]]*Sqrt[1 + Cos[c + d*x]]*Sqrt[a + a*Si n[c + d*x]])/(d*(a + a*Cos[c + d*x] + a*Sin[c + d*x])) + (2*Sqrt[e]*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*(a + a*Cos[c + d*x] + a*S in[c + d*x]))))/2))/4))/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[(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[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[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 = 11.04 (sec) , antiderivative size = 793, normalized size of antiderivative = 2.77
Input:
int((e*cos(d*x+c))^(1/2)*(a+a*sin(d*x+c))^(5/2),x,method=_RETURNVERBOSE)
Output:
1/48/d/(2*2^(1/2)+3)^(1/2)/(1+2^(1/2))*a^2/e*(180*(2+(cos(1/2*d*x+1/2*c)+1 )*2^(1/2)+2*cos(1/2*d*x+1/2*c))*(-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^(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*Ell ipticPi((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))+45*(4+3*(cos(1/2*d*x+ 1/2*c)+1)*2^(1/2)+4*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^(1/2)*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))+90*(-2+(-cos(1/2*d*x+1/2*c)-1)*2^(1/2)-2*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*(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*cos(1/2*d*x+1/2*c)^2-1)* (2*(-(32*cos(1/2*d*x+1/2*c)^4+36*cos(1/2*d*x+1/2*c)^2-71)*sin(1/2*d*x+1/2* c)+(32*cos(1/2*d*x+1/2*c)^4-100*cos(1/2*d*x+1/2*c)^2-3)*cos(1/2*d*x+1/2*c) )*2^(1/2)-3*(32*cos(1/2*d*x+1/2*c)^4+36*cos(1/2*d*x+1/2*c)^2-71)*sin(1/2*d *x+1/2*c)+3*(32*cos(1/2*d*x+1/2*c)^4-100*cos(1/2*d*x+1/2*c)^2-3)*cos(1/2*d *x+1/2*c))*e)*(e*(2*cos(1/2*d*x+1/2*c)^2-1))^(1/2)*((1+2*cos(1/2*d*x+1/2*c )*sin(1/2*d*x+1/2*c))*a)^(1/2)/(2*cos(1/2*d*x+1/2*c)^3+2*sin(1/2*d*x+1/...
Leaf count of result is larger than twice the leaf count of optimal. 516 vs. \(2 (242) = 484\).
Time = 0.16 (sec) , antiderivative size = 516, normalized size of antiderivative = 1.80 \[ \int \sqrt {e \cos (c+d x)} (a+a \sin (c+d x))^{5/2} \, dx=\frac {180 \, \sqrt {2} {\left (a^{2} \cos \left (d x + c\right ) + a^{2} \sin \left (d x + c\right ) + a^{2}\right )} \sqrt {a e} \arctan \left (\frac {\sqrt {2} \sqrt {a e} \sqrt {e \cos \left (d x + c\right )} \sqrt {a \sin \left (d x + c\right ) + a} \sin \left (d x + c\right )}{a e \cos \left (d x + c\right )^{2} + a e \cos \left (d x + c\right ) \sin \left (d x + c\right ) + a e \cos \left (d x + c\right )}\right ) + 45 \, \sqrt {2} {\left (a^{2} \cos \left (d x + c\right ) + a^{2} \sin \left (d x + c\right ) + a^{2}\right )} \sqrt {a e} \log \left (\frac {2 \, a e \cos \left (d x + c\right )^{2} + 2 \, \sqrt {2} \sqrt {a e} \sqrt {e \cos \left (d x + c\right )} \sqrt {a \sin \left (d x + c\right ) + a} {\left (\cos \left (d x + c\right ) + 1\right )} + 3 \, a e \cos \left (d x + c\right ) + a e + {\left (2 \, a e \cos \left (d x + c\right ) + a e\right )} \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + \sin \left (d x + c\right ) + 1}\right ) - 45 \, \sqrt {2} {\left (a^{2} \cos \left (d x + c\right ) + a^{2} \sin \left (d x + c\right ) + a^{2}\right )} \sqrt {a e} \log \left (\frac {2 \, a e \cos \left (d x + c\right )^{2} - 2 \, \sqrt {2} \sqrt {a e} \sqrt {e \cos \left (d x + c\right )} \sqrt {a \sin \left (d x + c\right ) + a} {\left (\cos \left (d x + c\right ) + 1\right )} + 3 \, a e \cos \left (d x + c\right ) + a e + {\left (2 \, a e \cos \left (d x + c\right ) + a e\right )} \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + \sin \left (d x + c\right ) + 1}\right ) + 8 \, {\left (8 \, a^{2} \cos \left (d x + c\right )^{3} - 26 \, a^{2} \cos \left (d x + c\right )^{2} - 79 \, a^{2} \cos \left (d x + c\right ) - 45 \, a^{2} - {\left (8 \, a^{2} \cos \left (d x + c\right )^{2} + 34 \, a^{2} \cos \left (d x + c\right ) - 45 \, a^{2}\right )} \sin \left (d x + c\right )\right )} \sqrt {e \cos \left (d x + c\right )} \sqrt {a \sin \left (d x + c\right ) + a}}{192 \, {\left (d \cos \left (d x + c\right ) + d \sin \left (d x + c\right ) + d\right )}} \] Input:
integrate((e*cos(d*x+c))^(1/2)*(a+a*sin(d*x+c))^(5/2),x, algorithm="fricas ")
Output:
1/192*(180*sqrt(2)*(a^2*cos(d*x + c) + a^2*sin(d*x + c) + a^2)*sqrt(a*e)*a rctan(sqrt(2)*sqrt(a*e)*sqrt(e*cos(d*x + c))*sqrt(a*sin(d*x + c) + a)*sin( d*x + c)/(a*e*cos(d*x + c)^2 + a*e*cos(d*x + c)*sin(d*x + c) + a*e*cos(d*x + c))) + 45*sqrt(2)*(a^2*cos(d*x + c) + a^2*sin(d*x + c) + a^2)*sqrt(a*e) *log((2*a*e*cos(d*x + c)^2 + 2*sqrt(2)*sqrt(a*e)*sqrt(e*cos(d*x + c))*sqrt (a*sin(d*x + c) + a)*(cos(d*x + c) + 1) + 3*a*e*cos(d*x + c) + a*e + (2*a* e*cos(d*x + c) + a*e)*sin(d*x + c))/(cos(d*x + c) + sin(d*x + c) + 1)) - 4 5*sqrt(2)*(a^2*cos(d*x + c) + a^2*sin(d*x + c) + a^2)*sqrt(a*e)*log((2*a*e *cos(d*x + c)^2 - 2*sqrt(2)*sqrt(a*e)*sqrt(e*cos(d*x + c))*sqrt(a*sin(d*x + c) + a)*(cos(d*x + c) + 1) + 3*a*e*cos(d*x + c) + a*e + (2*a*e*cos(d*x + c) + a*e)*sin(d*x + c))/(cos(d*x + c) + sin(d*x + c) + 1)) + 8*(8*a^2*cos (d*x + c)^3 - 26*a^2*cos(d*x + c)^2 - 79*a^2*cos(d*x + c) - 45*a^2 - (8*a^ 2*cos(d*x + c)^2 + 34*a^2*cos(d*x + c) - 45*a^2)*sin(d*x + c))*sqrt(e*cos( d*x + c))*sqrt(a*sin(d*x + c) + a))/(d*cos(d*x + c) + d*sin(d*x + c) + d)
Timed out. \[ \int \sqrt {e \cos (c+d x)} (a+a \sin (c+d x))^{5/2} \, dx=\text {Timed out} \] Input:
integrate((e*cos(d*x+c))**(1/2)*(a+a*sin(d*x+c))**(5/2),x)
Output:
Timed out
\[ \int \sqrt {e \cos (c+d x)} (a+a \sin (c+d x))^{5/2} \, dx=\int { \sqrt {e \cos \left (d x + c\right )} {\left (a \sin \left (d x + c\right ) + a\right )}^{\frac {5}{2}} \,d x } \] Input:
integrate((e*cos(d*x+c))^(1/2)*(a+a*sin(d*x+c))^(5/2),x, algorithm="maxima ")
Output:
integrate(sqrt(e*cos(d*x + c))*(a*sin(d*x + c) + a)^(5/2), x)
\[ \int \sqrt {e \cos (c+d x)} (a+a \sin (c+d x))^{5/2} \, dx=\int { \sqrt {e \cos \left (d x + c\right )} {\left (a \sin \left (d x + c\right ) + a\right )}^{\frac {5}{2}} \,d x } \] Input:
integrate((e*cos(d*x+c))^(1/2)*(a+a*sin(d*x+c))^(5/2),x, algorithm="giac")
Output:
integrate(sqrt(e*cos(d*x + c))*(a*sin(d*x + c) + a)^(5/2), x)
Timed out. \[ \int \sqrt {e \cos (c+d x)} (a+a \sin (c+d x))^{5/2} \, dx=\int \sqrt {e\,\cos \left (c+d\,x\right )}\,{\left (a+a\,\sin \left (c+d\,x\right )\right )}^{5/2} \,d x \] Input:
int((e*cos(c + d*x))^(1/2)*(a + a*sin(c + d*x))^(5/2),x)
Output:
int((e*cos(c + d*x))^(1/2)*(a + a*sin(c + d*x))^(5/2), x)
\[ \int \sqrt {e \cos (c+d x)} (a+a \sin (c+d x))^{5/2} \, dx=\sqrt {e}\, \sqrt {a}\, a^{2} \left (\int \sqrt {\sin \left (d x +c \right )+1}\, \sqrt {\cos \left (d x +c \right )}\, \sin \left (d x +c \right )^{2}d x +2 \left (\int \sqrt {\sin \left (d x +c \right )+1}\, \sqrt {\cos \left (d x +c \right )}\, \sin \left (d x +c \right )d x \right )+\int \sqrt {\sin \left (d x +c \right )+1}\, \sqrt {\cos \left (d x +c \right )}d x \right ) \] Input:
int((e*cos(d*x+c))^(1/2)*(a+a*sin(d*x+c))^(5/2),x)
Output:
sqrt(e)*sqrt(a)*a**2*(int(sqrt(sin(c + d*x) + 1)*sqrt(cos(c + d*x))*sin(c + d*x)**2,x) + 2*int(sqrt(sin(c + d*x) + 1)*sqrt(cos(c + d*x))*sin(c + d*x ),x) + int(sqrt(sin(c + d*x) + 1)*sqrt(cos(c + d*x)),x))