Integrand size = 27, antiderivative size = 323 \[ \int (e \cos (c+d x))^{3/2} (a+a \sin (c+d x))^{5/2} \, dx=-\frac {77 a^3 (e \cos (c+d x))^{5/2}}{96 d e \sqrt {a+a \sin (c+d x)}}+\frac {77 a^2 e \sqrt {e \cos (c+d x)} \sqrt {a+a \sin (c+d x)}}{64 d}-\frac {11 a^2 (e \cos (c+d x))^{5/2} \sqrt {a+a \sin (c+d x)}}{24 d e}-\frac {77 a^2 e^{3/2} \text {arcsinh}\left (\frac {\sqrt {e \cos (c+d x)}}{\sqrt {e}}\right ) \sqrt {1+\cos (c+d x)} \sqrt {a+a \sin (c+d x)}}{64 d (1+\cos (c+d x)+\sin (c+d x))}+\frac {77 a^2 e^{3/2} \arctan \left (\frac {\sqrt {e} \sin (c+d x)}{\sqrt {e \cos (c+d x)} \sqrt {1+\cos (c+d x)}}\right ) \sqrt {1+\cos (c+d x)} \sqrt {a+a \sin (c+d x)}}{64 d (1+\cos (c+d x)+\sin (c+d x))}-\frac {a (e \cos (c+d x))^{5/2} (a+a \sin (c+d x))^{3/2}}{4 d e} \]
-1/4*a*(e*cos(d*x+c))^(5/2)*(a+a*sin(d*x+c))^(3/2)/d/e-77/96*a^3*(e*cos(d* x+c))^(5/2)/d/e/(a+a*sin(d*x+c))^(1/2)-11/24*a^2*(e*cos(d*x+c))^(5/2)*(a+a *sin(d*x+c))^(1/2)/d/e+77/64*a^2*e*(e*cos(d*x+c))^(1/2)*(a+a*sin(d*x+c))^( 1/2)/d-77/64*a^2*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))+77/64*a^2*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 = 0.18 (sec) , antiderivative size = 77, normalized size of antiderivative = 0.24 \[ \int (e \cos (c+d x))^{3/2} (a+a \sin (c+d x))^{5/2} \, dx=-\frac {16\ 2^{3/4} (e \cos (c+d x))^{5/2} \operatorname {Hypergeometric2F1}\left (-\frac {11}{4},\frac {5}{4},\frac {9}{4},\frac {1}{2} (1-\sin (c+d x))\right ) (a (1+\sin (c+d x)))^{5/2}}{5 d e (1+\sin (c+d x))^{15/4}} \]
(-16*2^(3/4)*(e*Cos[c + d*x])^(5/2)*Hypergeometric2F1[-11/4, 5/4, 9/4, (1 - Sin[c + d*x])/2]*(a*(1 + Sin[c + d*x]))^(5/2))/(5*d*e*(1 + Sin[c + d*x]) ^(15/4))
Time = 1.38 (sec) , antiderivative size = 335, normalized size of antiderivative = 1.04, number of steps used = 18, number of rules used = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.630, Rules used = {3042, 3157, 3042, 3157, 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.
| \(\displaystyle \int (a \sin (c+d x)+a)^{5/2} (e \cos (c+d x))^{3/2} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int (a \sin (c+d x)+a)^{5/2} (e \cos (c+d x))^{3/2}dx\) |
| \(\Big \downarrow \) 3157 |
| \(\displaystyle \frac {11}{8} a \int (e \cos (c+d x))^{3/2} (\sin (c+d x) a+a)^{3/2}dx-\frac {a (a \sin (c+d x)+a)^{3/2} (e \cos (c+d x))^{5/2}}{4 d e}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {11}{8} a \int (e \cos (c+d x))^{3/2} (\sin (c+d x) a+a)^{3/2}dx-\frac {a (a \sin (c+d x)+a)^{3/2} (e \cos (c+d x))^{5/2}}{4 d e}\) |
| \(\Big \downarrow \) 3157 |
| \(\displaystyle \frac {11}{8} a \left (\frac {7}{6} a \int (e \cos (c+d x))^{3/2} \sqrt {\sin (c+d x) a+a}dx-\frac {a \sqrt {a \sin (c+d x)+a} (e \cos (c+d x))^{5/2}}{3 d e}\right )-\frac {a (a \sin (c+d x)+a)^{3/2} (e \cos (c+d x))^{5/2}}{4 d e}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {11}{8} a \left (\frac {7}{6} a \int (e \cos (c+d x))^{3/2} \sqrt {\sin (c+d x) a+a}dx-\frac {a \sqrt {a \sin (c+d x)+a} (e \cos (c+d x))^{5/2}}{3 d e}\right )-\frac {a (a \sin (c+d x)+a)^{3/2} (e \cos (c+d x))^{5/2}}{4 d e}\) |
| \(\Big \downarrow \) 3157 |
| \(\displaystyle \frac {11}{8} a \left (\frac {7}{6} a \left (\frac {3}{4} a \int \frac {(e \cos (c+d x))^{3/2}}{\sqrt {\sin (c+d x) a+a}}dx-\frac {a (e \cos (c+d x))^{5/2}}{2 d e \sqrt {a \sin (c+d x)+a}}\right )-\frac {a \sqrt {a \sin (c+d x)+a} (e \cos (c+d x))^{5/2}}{3 d e}\right )-\frac {a (a \sin (c+d x)+a)^{3/2} (e \cos (c+d x))^{5/2}}{4 d e}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {11}{8} a \left (\frac {7}{6} a \left (\frac {3}{4} a \int \frac {(e \cos (c+d x))^{3/2}}{\sqrt {\sin (c+d x) a+a}}dx-\frac {a (e \cos (c+d x))^{5/2}}{2 d e \sqrt {a \sin (c+d x)+a}}\right )-\frac {a \sqrt {a \sin (c+d x)+a} (e \cos (c+d x))^{5/2}}{3 d e}\right )-\frac {a (a \sin (c+d x)+a)^{3/2} (e \cos (c+d x))^{5/2}}{4 d e}\) |
| \(\Big \downarrow \) 3164 |
| \(\displaystyle \frac {11}{8} a \left (\frac {7}{6} a \left (\frac {3}{4} a \left (\frac {e^2 \int \frac {\sqrt {\sin (c+d x) a+a}}{\sqrt {e \cos (c+d x)}}dx}{2 a}+\frac {e \sqrt {a \sin (c+d x)+a} \sqrt {e \cos (c+d x)}}{a d}\right )-\frac {a (e \cos (c+d x))^{5/2}}{2 d e \sqrt {a \sin (c+d x)+a}}\right )-\frac {a \sqrt {a \sin (c+d x)+a} (e \cos (c+d x))^{5/2}}{3 d e}\right )-\frac {a (a \sin (c+d x)+a)^{3/2} (e \cos (c+d x))^{5/2}}{4 d e}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {11}{8} a \left (\frac {7}{6} a \left (\frac {3}{4} a \left (\frac {e^2 \int \frac {\sqrt {\sin (c+d x) a+a}}{\sqrt {e \cos (c+d x)}}dx}{2 a}+\frac {e \sqrt {a \sin (c+d x)+a} \sqrt {e \cos (c+d x)}}{a d}\right )-\frac {a (e \cos (c+d x))^{5/2}}{2 d e \sqrt {a \sin (c+d x)+a}}\right )-\frac {a \sqrt {a \sin (c+d x)+a} (e \cos (c+d x))^{5/2}}{3 d e}\right )-\frac {a (a \sin (c+d x)+a)^{3/2} (e \cos (c+d x))^{5/2}}{4 d e}\) |
| \(\Big \downarrow \) 3156 |
| \(\displaystyle \frac {11}{8} a \left (\frac {7}{6} a \left (\frac {3}{4} a \left (\frac {e^2 \left (\frac {\sqrt {\cos (c+d x)+1} \sqrt {a \sin (c+d x)+a} \int \frac {\sqrt {\cos (c+d x)+1}}{\sqrt {e \cos (c+d x)}}dx}{\sin (c+d x)+\cos (c+d x)+1}+\frac {\sqrt {\cos (c+d x)+1} \sqrt {a \sin (c+d x)+a} \int \frac {\sin (c+d x)}{\sqrt {e \cos (c+d x)} \sqrt {\cos (c+d x)+1}}dx}{\sin (c+d x)+\cos (c+d x)+1}\right )}{2 a}+\frac {e \sqrt {a \sin (c+d x)+a} \sqrt {e \cos (c+d x)}}{a d}\right )-\frac {a (e \cos (c+d x))^{5/2}}{2 d e \sqrt {a \sin (c+d x)+a}}\right )-\frac {a \sqrt {a \sin (c+d x)+a} (e \cos (c+d x))^{5/2}}{3 d e}\right )-\frac {a (a \sin (c+d x)+a)^{3/2} (e \cos (c+d x))^{5/2}}{4 d e}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {11}{8} a \left (\frac {7}{6} a \left (\frac {3}{4} a \left (\frac {e^2 \left (\frac {\sqrt {\cos (c+d x)+1} \sqrt {a \sin (c+d x)+a} \int -\frac {\cos \left (c+d x+\frac {\pi }{2}\right )}{\sqrt {e \sin \left (c+d x+\frac {\pi }{2}\right )} \sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )+1}}dx}{\sin (c+d x)+\cos (c+d x)+1}+\frac {\sqrt {\cos (c+d x)+1} \sqrt {a \sin (c+d x)+a} \int \frac {\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )+1}}{\sqrt {e \sin \left (c+d x+\frac {\pi }{2}\right )}}dx}{\sin (c+d x)+\cos (c+d x)+1}\right )}{2 a}+\frac {e \sqrt {a \sin (c+d x)+a} \sqrt {e \cos (c+d x)}}{a d}\right )-\frac {a (e \cos (c+d x))^{5/2}}{2 d e \sqrt {a \sin (c+d x)+a}}\right )-\frac {a \sqrt {a \sin (c+d x)+a} (e \cos (c+d x))^{5/2}}{3 d e}\right )-\frac {a (a \sin (c+d x)+a)^{3/2} (e \cos (c+d x))^{5/2}}{4 d e}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {11}{8} a \left (\frac {7}{6} a \left (\frac {3}{4} a \left (\frac {e^2 \left (\frac {\sqrt {\cos (c+d x)+1} \sqrt {a \sin (c+d x)+a} \int \frac {\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )+1}}{\sqrt {e \sin \left (c+d x+\frac {\pi }{2}\right )}}dx}{\sin (c+d x)+\cos (c+d x)+1}-\frac {\sqrt {\cos (c+d x)+1} \sqrt {a \sin (c+d x)+a} \int \frac {\cos \left (\frac {1}{2} (2 c+\pi )+d x\right )}{\sqrt {e \sin \left (\frac {1}{2} (2 c+\pi )+d x\right )} \sqrt {\sin \left (\frac {1}{2} (2 c+\pi )+d x\right )+1}}dx}{\sin (c+d x)+\cos (c+d x)+1}\right )}{2 a}+\frac {e \sqrt {a \sin (c+d x)+a} \sqrt {e \cos (c+d x)}}{a d}\right )-\frac {a (e \cos (c+d x))^{5/2}}{2 d e \sqrt {a \sin (c+d x)+a}}\right )-\frac {a \sqrt {a \sin (c+d x)+a} (e \cos (c+d x))^{5/2}}{3 d e}\right )-\frac {a (a \sin (c+d x)+a)^{3/2} (e \cos (c+d x))^{5/2}}{4 d e}\) |
| \(\Big \downarrow \) 3254 |
| \(\displaystyle \frac {11}{8} a \left (\frac {7}{6} a \left (\frac {3}{4} a \left (\frac {e^2 \left (-\frac {\sqrt {\cos (c+d x)+1} \sqrt {a \sin (c+d x)+a} \int \frac {\cos \left (\frac {1}{2} (2 c+\pi )+d x\right )}{\sqrt {e \sin \left (\frac {1}{2} (2 c+\pi )+d x\right )} \sqrt {\sin \left (\frac {1}{2} (2 c+\pi )+d x\right )+1}}dx}{\sin (c+d x)+\cos (c+d x)+1}-\frac {2 \sqrt {\cos (c+d x)+1} \sqrt {a \sin (c+d x)+a} \int \frac {1}{\frac {\sin (c+d x) \tan (c+d x)}{\cos (c+d x)+1}+1}d\left (-\frac {\sin (c+d x)}{\sqrt {e \cos (c+d x)} \sqrt {\cos (c+d x)+1}}\right )}{d (\sin (c+d x)+\cos (c+d x)+1)}\right )}{2 a}+\frac {e \sqrt {a \sin (c+d x)+a} \sqrt {e \cos (c+d x)}}{a d}\right )-\frac {a (e \cos (c+d x))^{5/2}}{2 d e \sqrt {a \sin (c+d x)+a}}\right )-\frac {a \sqrt {a \sin (c+d x)+a} (e \cos (c+d x))^{5/2}}{3 d e}\right )-\frac {a (a \sin (c+d x)+a)^{3/2} (e \cos (c+d x))^{5/2}}{4 d e}\) |
| \(\Big \downarrow \) 216 |
| \(\displaystyle \frac {11}{8} a \left (\frac {7}{6} a \left (\frac {3}{4} a \left (\frac {e^2 \left (\frac {2 \sqrt {\cos (c+d x)+1} \sqrt {a \sin (c+d x)+a} \arctan \left (\frac {\sqrt {e} \sin (c+d x)}{\sqrt {\cos (c+d x)+1} \sqrt {e \cos (c+d x)}}\right )}{d \sqrt {e} (\sin (c+d x)+\cos (c+d x)+1)}-\frac {\sqrt {\cos (c+d x)+1} \sqrt {a \sin (c+d x)+a} \int \frac {\cos \left (\frac {1}{2} (2 c+\pi )+d x\right )}{\sqrt {e \sin \left (\frac {1}{2} (2 c+\pi )+d x\right )} \sqrt {\sin \left (\frac {1}{2} (2 c+\pi )+d x\right )+1}}dx}{\sin (c+d x)+\cos (c+d x)+1}\right )}{2 a}+\frac {e \sqrt {a \sin (c+d x)+a} \sqrt {e \cos (c+d x)}}{a d}\right )-\frac {a (e \cos (c+d x))^{5/2}}{2 d e \sqrt {a \sin (c+d x)+a}}\right )-\frac {a \sqrt {a \sin (c+d x)+a} (e \cos (c+d x))^{5/2}}{3 d e}\right )-\frac {a (a \sin (c+d x)+a)^{3/2} (e \cos (c+d x))^{5/2}}{4 d e}\) |
| \(\Big \downarrow \) 3312 |
| \(\displaystyle \frac {11}{8} a \left (\frac {7}{6} a \left (\frac {3}{4} a \left (\frac {e^2 \left (\frac {2 \sqrt {\cos (c+d x)+1} \sqrt {a \sin (c+d x)+a} \arctan \left (\frac {\sqrt {e} \sin (c+d x)}{\sqrt {\cos (c+d x)+1} \sqrt {e \cos (c+d x)}}\right )}{d \sqrt {e} (\sin (c+d x)+\cos (c+d x)+1)}-\frac {\sqrt {\cos (c+d x)+1} \sqrt {a \sin (c+d x)+a} \int \frac {1}{\sqrt {e \cos (c+d x)} \sqrt {\cos (c+d x)+1}}d\cos (c+d x)}{d (\sin (c+d x)+\cos (c+d x)+1)}\right )}{2 a}+\frac {e \sqrt {a \sin (c+d x)+a} \sqrt {e \cos (c+d x)}}{a d}\right )-\frac {a (e \cos (c+d x))^{5/2}}{2 d e \sqrt {a \sin (c+d x)+a}}\right )-\frac {a \sqrt {a \sin (c+d x)+a} (e \cos (c+d x))^{5/2}}{3 d e}\right )-\frac {a (a \sin (c+d x)+a)^{3/2} (e \cos (c+d x))^{5/2}}{4 d e}\) |
| \(\Big \downarrow \) 63 |
| \(\displaystyle \frac {11}{8} a \left (\frac {7}{6} a \left (\frac {3}{4} a \left (\frac {e^2 \left (\frac {2 \sqrt {\cos (c+d x)+1} \sqrt {a \sin (c+d x)+a} \arctan \left (\frac {\sqrt {e} \sin (c+d x)}{\sqrt {\cos (c+d x)+1} \sqrt {e \cos (c+d x)}}\right )}{d \sqrt {e} (\sin (c+d x)+\cos (c+d x)+1)}-\frac {2 \sqrt {\cos (c+d x)+1} \sqrt {a \sin (c+d x)+a} \int \frac {1}{\sqrt {\cos (c+d x)+1}}d\sqrt {e \cos (c+d x)}}{d e (\sin (c+d x)+\cos (c+d x)+1)}\right )}{2 a}+\frac {e \sqrt {a \sin (c+d x)+a} \sqrt {e \cos (c+d x)}}{a d}\right )-\frac {a (e \cos (c+d x))^{5/2}}{2 d e \sqrt {a \sin (c+d x)+a}}\right )-\frac {a \sqrt {a \sin (c+d x)+a} (e \cos (c+d x))^{5/2}}{3 d e}\right )-\frac {a (a \sin (c+d x)+a)^{3/2} (e \cos (c+d x))^{5/2}}{4 d e}\) |
| \(\Big \downarrow \) 222 |
| \(\displaystyle \frac {11}{8} a \left (\frac {7}{6} a \left (\frac {3}{4} a \left (\frac {e^2 \left (\frac {2 \sqrt {\cos (c+d x)+1} \sqrt {a \sin (c+d x)+a} \arctan \left (\frac {\sqrt {e} \sin (c+d x)}{\sqrt {\cos (c+d x)+1} \sqrt {e \cos (c+d x)}}\right )}{d \sqrt {e} (\sin (c+d x)+\cos (c+d x)+1)}-\frac {2 \sqrt {\cos (c+d x)+1} \sqrt {a \sin (c+d x)+a} \text {arcsinh}\left (\frac {\sqrt {e \cos (c+d x)}}{\sqrt {e}}\right )}{d \sqrt {e} (\sin (c+d x)+\cos (c+d x)+1)}\right )}{2 a}+\frac {e \sqrt {a \sin (c+d x)+a} \sqrt {e \cos (c+d x)}}{a d}\right )-\frac {a (e \cos (c+d x))^{5/2}}{2 d e \sqrt {a \sin (c+d x)+a}}\right )-\frac {a \sqrt {a \sin (c+d x)+a} (e \cos (c+d x))^{5/2}}{3 d e}\right )-\frac {a (a \sin (c+d x)+a)^{3/2} (e \cos (c+d x))^{5/2}}{4 d e}\) |
-1/4*(a*(e*Cos[c + d*x])^(5/2)*(a + a*Sin[c + d*x])^(3/2))/(d*e) + (11*a*( -1/3*(a*(e*Cos[c + d*x])^(5/2)*Sqrt[a + a*Sin[c + d*x]])/(d*e) + (7*a*(-1/ 2*(a*(e*Cos[c + d*x])^(5/2))/(d*e*Sqrt[a + a*Sin[c + d*x]]) + (3*a*((e*Sqr t[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]*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))/6))/8
3.3.89.3.1 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 = 5.84 (sec) , antiderivative size = 360, normalized size of antiderivative = 1.11
| method | result | size |
| default | \(-\frac {\sqrt {e \cos \left (d x +c \right )}\, \sqrt {a \left (1+\sin \left (d x +c \right )\right )}\, a^{2} e \left (48 \sin \left (d x +c \right ) \left (\cos ^{3}\left (d x +c \right )\right )-48 \left (\cos ^{4}\left (d x +c \right )\right )+184 \left (\cos ^{2}\left (d x +c \right )\right ) \sin \left (d x +c \right )+136 \left (\cos ^{3}\left (d x +c \right )\right )-154 \cos \left (d x +c \right ) \sin \left (d x +c \right )+338 \left (\cos ^{2}\left (d x +c \right )\right )-231 \sqrt {-\frac {\cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\, \arctan \left (\sqrt {-\frac {\cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\right )-231 \sqrt {-\frac {\cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\, \operatorname {arctanh}\left (\frac {\sin \left (d x +c \right )}{\left (1+\cos \left (d x +c \right )\right ) \sqrt {-\frac {\cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}}\right )-231 \sin \left (d x +c \right )-77 \cos \left (d x +c \right )-231 \sec \left (d x +c \right ) \sqrt {-\frac {\cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\, \arctan \left (\sqrt {-\frac {\cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\right )-231 \sec \left (d x +c \right ) \sqrt {-\frac {\cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\, \operatorname {arctanh}\left (\frac {\sin \left (d x +c \right )}{\left (1+\cos \left (d x +c \right )\right ) \sqrt {-\frac {\cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}}\right )-231\right )}{192 d \left (1+\cos \left (d x +c \right )+\sin \left (d x +c \right )\right )}\) | \(360\) |
-1/192/d*(e*cos(d*x+c))^(1/2)*(a*(1+sin(d*x+c)))^(1/2)*a^2*e/(1+cos(d*x+c) +sin(d*x+c))*(48*cos(d*x+c)^3*sin(d*x+c)-48*cos(d*x+c)^4+184*cos(d*x+c)^2* sin(d*x+c)+136*cos(d*x+c)^3-154*cos(d*x+c)*sin(d*x+c)+338*cos(d*x+c)^2-231 *(-cos(d*x+c)/(1+cos(d*x+c)))^(1/2)*arctan((-cos(d*x+c)/(1+cos(d*x+c)))^(1 /2))-231*(-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))-231*sin(d*x+c)-77*cos(d*x+c)-231*s ec(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))-231*sec(d*x+c)*(-cos(d*x+c)/(1+cos(d*x+c)))^(1/2)*arctanh(si n(d*x+c)/(1+cos(d*x+c))/(-cos(d*x+c)/(1+cos(d*x+c)))^(1/2))-231)
Result contains complex when optimal does not.
Time = 0.47 (sec) , antiderivative size = 1020, normalized size of antiderivative = 3.16 \[ \int (e \cos (c+d x))^{3/2} (a+a \sin (c+d x))^{5/2} \, dx=\text {Too large to display} \]
-1/768*(231*I*(-a^10*e^6/d^4)^(1/4)*d*log(-456533/2*(2*(a^7*e^4*sin(d*x + c) + sqrt(-a^10*e^6/d^4)*(a^2*d^2*e*cos(d*x + c) + a^2*d^2*e))*sqrt(e*cos( d*x + c))*sqrt(a*sin(d*x + c) + a) + (-a^10*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)) + (-a^10*e^6/d ^4)^(1/4)*(-2*I*a^5*d*e^3*cos(d*x + c)^2 - I*a^5*d*e^3*cos(d*x + c) + I*a^ 5*d*e^3*sin(d*x + c) + I*a^5*d*e^3))/(cos(d*x + c) + sin(d*x + c) + 1)) - 231*I*(-a^10*e^6/d^4)^(1/4)*d*log(-456533/2*(2*(a^7*e^4*sin(d*x + c) + sqr t(-a^10*e^6/d^4)*(a^2*d^2*e*cos(d*x + c) + a^2*d^2*e))*sqrt(e*cos(d*x + c) )*sqrt(a*sin(d*x + c) + a) + (-a^10*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)) + (-a^10*e^6/d^4)^(1 /4)*(2*I*a^5*d*e^3*cos(d*x + c)^2 + I*a^5*d*e^3*cos(d*x + c) - I*a^5*d*e^3 *sin(d*x + c) - I*a^5*d*e^3))/(cos(d*x + c) + sin(d*x + c) + 1)) - 231*(-a ^10*e^6/d^4)^(1/4)*d*log(-456533/2*(2*(a^7*e^4*sin(d*x + c) - sqrt(-a^10*e ^6/d^4)*(a^2*d^2*e*cos(d*x + c) + a^2*d^2*e))*sqrt(e*cos(d*x + c))*sqrt(a* sin(d*x + c) + a) + (-a^10*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)) + (-a^10*e^6/d^4)^(1/4)*(2*a^5*d*e^3*co s(d*x + c)^2 + a^5*d*e^3*cos(d*x + c) - a^5*d*e^3*sin(d*x + c) - a^5*d*e^3 ))/(cos(d*x + c) + sin(d*x + c) + 1)) + 231*(-a^10*e^6/d^4)^(1/4)*d*log(-4 56533/2*(2*(a^7*e^4*sin(d*x + c) - sqrt(-a^10*e^6/d^4)*(a^2*d^2*e*cos(d*x + c) + a^2*d^2*e))*sqrt(e*cos(d*x + c))*sqrt(a*sin(d*x + c) + a) - (-a^...
Timed out. \[ \int (e \cos (c+d x))^{3/2} (a+a \sin (c+d x))^{5/2} \, dx=\text {Timed out} \]
\[ \int (e \cos (c+d x))^{3/2} (a+a \sin (c+d x))^{5/2} \, dx=\int { \left (e \cos \left (d x + c\right )\right )^{\frac {3}{2}} {\left (a \sin \left (d x + c\right ) + a\right )}^{\frac {5}{2}} \,d x } \]
\[ \int (e \cos (c+d x))^{3/2} (a+a \sin (c+d x))^{5/2} \, dx=\int { \left (e \cos \left (d x + c\right )\right )^{\frac {3}{2}} {\left (a \sin \left (d x + c\right ) + a\right )}^{\frac {5}{2}} \,d x } \]
Timed out. \[ \int (e \cos (c+d x))^{3/2} (a+a \sin (c+d x))^{5/2} \, dx=\int {\left (e\,\cos \left (c+d\,x\right )\right )}^{3/2}\,{\left (a+a\,\sin \left (c+d\,x\right )\right )}^{5/2} \,d x \]