Integrand size = 37, antiderivative size = 359 \[ \int \frac {(g \cos (e+f x))^{5/2}}{(d \sin (e+f x))^{5/2} (a+b \sin (e+f x))} \, dx=-\frac {2 g (g \cos (e+f x))^{3/2}}{3 a d f (d \sin (e+f x))^{3/2}}+\frac {2 b g (g \cos (e+f x))^{3/2}}{a^2 d^2 f \sqrt {d \sin (e+f x)}}+\frac {2 \sqrt {2} \sqrt {-a+b} \sqrt {a+b} g^{5/2} \operatorname {EllipticPi}\left (-\frac {\sqrt {-a+b}}{\sqrt {a+b}},\arcsin \left (\frac {\sqrt {g \cos (e+f x)}}{\sqrt {g} \sqrt {1+\sin (e+f x)}}\right ),-1\right ) \sqrt {\sin (e+f x)}}{a^2 d^2 f \sqrt {d \sin (e+f x)}}-\frac {2 \sqrt {2} \sqrt {-a+b} \sqrt {a+b} g^{5/2} \operatorname {EllipticPi}\left (\frac {\sqrt {-a+b}}{\sqrt {a+b}},\arcsin \left (\frac {\sqrt {g \cos (e+f x)}}{\sqrt {g} \sqrt {1+\sin (e+f x)}}\right ),-1\right ) \sqrt {\sin (e+f x)}}{a^2 d^2 f \sqrt {d \sin (e+f x)}}+\frac {2 b g^2 \sqrt {g \cos (e+f x)} E\left (\left .e-\frac {\pi }{4}+f x\right |2\right ) \sqrt {d \sin (e+f x)}}{a^2 d^3 f \sqrt {\sin (2 e+2 f x)}} \]
-2/3*g*(g*cos(f*x+e))^(3/2)/a/d/f/(d*sin(f*x+e))^(3/2)+2*b*g*(g*cos(f*x+e) )^(3/2)/a^2/d^2/f/(d*sin(f*x+e))^(1/2)+2*g^(5/2)*EllipticPi((g*cos(f*x+e)) ^(1/2)/g^(1/2)/(1+sin(f*x+e))^(1/2),-(-a+b)^(1/2)/(a+b)^(1/2),I)*2^(1/2)*( -a+b)^(1/2)*(a+b)^(1/2)*sin(f*x+e)^(1/2)/a^2/d^2/f/(d*sin(f*x+e))^(1/2)-2* g^(5/2)*EllipticPi((g*cos(f*x+e))^(1/2)/g^(1/2)/(1+sin(f*x+e))^(1/2),(-a+b )^(1/2)/(a+b)^(1/2),I)*2^(1/2)*(-a+b)^(1/2)*(a+b)^(1/2)*sin(f*x+e)^(1/2)/a ^2/d^2/f/(d*sin(f*x+e))^(1/2)-2*b*g^2*(sin(e+1/4*Pi+f*x)^2)^(1/2)/sin(e+1/ 4*Pi+f*x)*EllipticE(cos(e+1/4*Pi+f*x),2^(1/2))*(g*cos(f*x+e))^(1/2)*(d*sin (f*x+e))^(1/2)/a^2/d^3/f/sin(2*f*x+2*e)^(1/2)
Result contains higher order function than in optimal. Order 6 vs. order 4 in optimal.
Time = 24.75 (sec) , antiderivative size = 1656, normalized size of antiderivative = 4.61 \[ \int \frac {(g \cos (e+f x))^{5/2}}{(d \sin (e+f x))^{5/2} (a+b \sin (e+f x))} \, dx =\text {Too large to display} \]
((g*Cos[e + f*x])^(5/2)*((2*b*Cot[e + f*x])/a^2 - (2*Cot[e + f*x]*Csc[e + f*x])/(3*a))*Sin[e + f*x]*Tan[e + f*x]^2)/(f*(d*Sin[e + f*x])^(5/2)) - ((g *Cos[e + f*x])^(5/2)*Sin[e + f*x]^(5/2)*((4*a*b*(-(b*AppellF1[3/4, -1/4, 1 , 7/4, Cos[e + f*x]^2, (b^2*Cos[e + f*x]^2)/(-a^2 + b^2)]) + a*AppellF1[3/ 4, 1/4, 1, 7/4, Cos[e + f*x]^2, (b^2*Cos[e + f*x]^2)/(-a^2 + b^2)])*Cos[e + f*x]^(3/2)*(a + b*Sqrt[1 - Cos[e + f*x]^2])*Sin[e + f*x]^(3/2))/(3*(a^2 - b^2)*(1 - Cos[e + f*x]^2)^(3/4)*(a + b*Sin[e + f*x])) + ((a^2 - 2*b^2)*S qrt[Tan[e + f*x]]*((3*Sqrt[2]*a^(3/2)*(-2*ArcTan[1 - (Sqrt[2]*(a^2 - b^2)^ (1/4)*Sqrt[Tan[e + f*x]])/Sqrt[a]] + 2*ArcTan[1 + (Sqrt[2]*(a^2 - b^2)^(1/ 4)*Sqrt[Tan[e + f*x]])/Sqrt[a]] - Log[-a + Sqrt[2]*Sqrt[a]*(a^2 - b^2)^(1/ 4)*Sqrt[Tan[e + f*x]] - Sqrt[a^2 - b^2]*Tan[e + f*x]] + Log[a + Sqrt[2]*Sq rt[a]*(a^2 - b^2)^(1/4)*Sqrt[Tan[e + f*x]] + Sqrt[a^2 - b^2]*Tan[e + f*x]] ))/(a^2 - b^2)^(1/4) - 8*b*AppellF1[3/4, 1/2, 1, 7/4, -Tan[e + f*x]^2, ((- a^2 + b^2)*Tan[e + f*x]^2)/a^2]*Tan[e + f*x]^(3/2))*(b*Tan[e + f*x] + a*Sq rt[1 + Tan[e + f*x]^2]))/(12*a^2*Cos[e + f*x]^(3/2)*Sqrt[Sin[e + f*x]]*(a + b*Sin[e + f*x])*(1 + Tan[e + f*x]^2)^(3/2)) + (Cos[2*(e + f*x)]*Sqrt[Tan [e + f*x]]*(b*Tan[e + f*x] + a*Sqrt[1 + Tan[e + f*x]^2])*(56*b*(-3*a^2 + b ^2)*AppellF1[3/4, 1/2, 1, 7/4, -Tan[e + f*x]^2, (-1 + b^2/a^2)*Tan[e + f*x ]^2]*Tan[e + f*x]^(3/2) + 24*b*(-a^2 + b^2)*AppellF1[7/4, 1/2, 1, 11/4, -T an[e + f*x]^2, (-1 + b^2/a^2)*Tan[e + f*x]^2]*Tan[e + f*x]^(7/2) + 21*a...
Time = 1.88 (sec) , antiderivative size = 346, normalized size of antiderivative = 0.96, number of steps used = 15, number of rules used = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.378, Rules used = {3042, 3378, 3042, 3043, 3050, 3042, 3052, 3042, 3119, 3385, 3042, 3384, 993, 1542}
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.
| \(\displaystyle \int \frac {(g \cos (e+f x))^{5/2}}{(d \sin (e+f x))^{5/2} (a+b \sin (e+f x))} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {(g \cos (e+f x))^{5/2}}{(d \sin (e+f x))^{5/2} (a+b \sin (e+f x))}dx\) |
| \(\Big \downarrow \) 3378 |
| \(\displaystyle -\frac {g^2 \left (a^2-b^2\right ) \int \frac {\sqrt {g \cos (e+f x)}}{\sqrt {d \sin (e+f x)} (a+b \sin (e+f x))}dx}{a^2 d^2}-\frac {b g^2 \int \frac {\sqrt {g \cos (e+f x)}}{(d \sin (e+f x))^{3/2}}dx}{a^2 d}+\frac {g^2 \int \frac {\sqrt {g \cos (e+f x)}}{(d \sin (e+f x))^{5/2}}dx}{a}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {g^2 \left (a^2-b^2\right ) \int \frac {\sqrt {g \cos (e+f x)}}{\sqrt {d \sin (e+f x)} (a+b \sin (e+f x))}dx}{a^2 d^2}-\frac {b g^2 \int \frac {\sqrt {g \cos (e+f x)}}{(d \sin (e+f x))^{3/2}}dx}{a^2 d}+\frac {g^2 \int \frac {\sqrt {g \cos (e+f x)}}{(d \sin (e+f x))^{5/2}}dx}{a}\) |
| \(\Big \downarrow \) 3043 |
| \(\displaystyle -\frac {g^2 \left (a^2-b^2\right ) \int \frac {\sqrt {g \cos (e+f x)}}{\sqrt {d \sin (e+f x)} (a+b \sin (e+f x))}dx}{a^2 d^2}-\frac {b g^2 \int \frac {\sqrt {g \cos (e+f x)}}{(d \sin (e+f x))^{3/2}}dx}{a^2 d}-\frac {2 g (g \cos (e+f x))^{3/2}}{3 a d f (d \sin (e+f x))^{3/2}}\) |
| \(\Big \downarrow \) 3050 |
| \(\displaystyle -\frac {g^2 \left (a^2-b^2\right ) \int \frac {\sqrt {g \cos (e+f x)}}{\sqrt {d \sin (e+f x)} (a+b \sin (e+f x))}dx}{a^2 d^2}-\frac {b g^2 \left (-\frac {2 \int \sqrt {g \cos (e+f x)} \sqrt {d \sin (e+f x)}dx}{d^2}-\frac {2 (g \cos (e+f x))^{3/2}}{d f g \sqrt {d \sin (e+f x)}}\right )}{a^2 d}-\frac {2 g (g \cos (e+f x))^{3/2}}{3 a d f (d \sin (e+f x))^{3/2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {g^2 \left (a^2-b^2\right ) \int \frac {\sqrt {g \cos (e+f x)}}{\sqrt {d \sin (e+f x)} (a+b \sin (e+f x))}dx}{a^2 d^2}-\frac {b g^2 \left (-\frac {2 \int \sqrt {g \cos (e+f x)} \sqrt {d \sin (e+f x)}dx}{d^2}-\frac {2 (g \cos (e+f x))^{3/2}}{d f g \sqrt {d \sin (e+f x)}}\right )}{a^2 d}-\frac {2 g (g \cos (e+f x))^{3/2}}{3 a d f (d \sin (e+f x))^{3/2}}\) |
| \(\Big \downarrow \) 3052 |
| \(\displaystyle -\frac {g^2 \left (a^2-b^2\right ) \int \frac {\sqrt {g \cos (e+f x)}}{\sqrt {d \sin (e+f x)} (a+b \sin (e+f x))}dx}{a^2 d^2}-\frac {b g^2 \left (-\frac {2 \sqrt {d \sin (e+f x)} \sqrt {g \cos (e+f x)} \int \sqrt {\sin (2 e+2 f x)}dx}{d^2 \sqrt {\sin (2 e+2 f x)}}-\frac {2 (g \cos (e+f x))^{3/2}}{d f g \sqrt {d \sin (e+f x)}}\right )}{a^2 d}-\frac {2 g (g \cos (e+f x))^{3/2}}{3 a d f (d \sin (e+f x))^{3/2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {g^2 \left (a^2-b^2\right ) \int \frac {\sqrt {g \cos (e+f x)}}{\sqrt {d \sin (e+f x)} (a+b \sin (e+f x))}dx}{a^2 d^2}-\frac {b g^2 \left (-\frac {2 \sqrt {d \sin (e+f x)} \sqrt {g \cos (e+f x)} \int \sqrt {\sin (2 e+2 f x)}dx}{d^2 \sqrt {\sin (2 e+2 f x)}}-\frac {2 (g \cos (e+f x))^{3/2}}{d f g \sqrt {d \sin (e+f x)}}\right )}{a^2 d}-\frac {2 g (g \cos (e+f x))^{3/2}}{3 a d f (d \sin (e+f x))^{3/2}}\) |
| \(\Big \downarrow \) 3119 |
| \(\displaystyle -\frac {g^2 \left (a^2-b^2\right ) \int \frac {\sqrt {g \cos (e+f x)}}{\sqrt {d \sin (e+f x)} (a+b \sin (e+f x))}dx}{a^2 d^2}-\frac {b g^2 \left (-\frac {2 E\left (\left .e+f x-\frac {\pi }{4}\right |2\right ) \sqrt {d \sin (e+f x)} \sqrt {g \cos (e+f x)}}{d^2 f \sqrt {\sin (2 e+2 f x)}}-\frac {2 (g \cos (e+f x))^{3/2}}{d f g \sqrt {d \sin (e+f x)}}\right )}{a^2 d}-\frac {2 g (g \cos (e+f x))^{3/2}}{3 a d f (d \sin (e+f x))^{3/2}}\) |
| \(\Big \downarrow \) 3385 |
| \(\displaystyle -\frac {g^2 \left (a^2-b^2\right ) \sqrt {\sin (e+f x)} \int \frac {\sqrt {g \cos (e+f x)}}{\sqrt {\sin (e+f x)} (a+b \sin (e+f x))}dx}{a^2 d^2 \sqrt {d \sin (e+f x)}}-\frac {b g^2 \left (-\frac {2 E\left (\left .e+f x-\frac {\pi }{4}\right |2\right ) \sqrt {d \sin (e+f x)} \sqrt {g \cos (e+f x)}}{d^2 f \sqrt {\sin (2 e+2 f x)}}-\frac {2 (g \cos (e+f x))^{3/2}}{d f g \sqrt {d \sin (e+f x)}}\right )}{a^2 d}-\frac {2 g (g \cos (e+f x))^{3/2}}{3 a d f (d \sin (e+f x))^{3/2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {g^2 \left (a^2-b^2\right ) \sqrt {\sin (e+f x)} \int \frac {\sqrt {g \cos (e+f x)}}{\sqrt {\sin (e+f x)} (a+b \sin (e+f x))}dx}{a^2 d^2 \sqrt {d \sin (e+f x)}}-\frac {b g^2 \left (-\frac {2 E\left (\left .e+f x-\frac {\pi }{4}\right |2\right ) \sqrt {d \sin (e+f x)} \sqrt {g \cos (e+f x)}}{d^2 f \sqrt {\sin (2 e+2 f x)}}-\frac {2 (g \cos (e+f x))^{3/2}}{d f g \sqrt {d \sin (e+f x)}}\right )}{a^2 d}-\frac {2 g (g \cos (e+f x))^{3/2}}{3 a d f (d \sin (e+f x))^{3/2}}\) |
| \(\Big \downarrow \) 3384 |
| \(\displaystyle \frac {4 \sqrt {2} g^3 \left (a^2-b^2\right ) \sqrt {\sin (e+f x)} \int \frac {g \cos (e+f x)}{(\sin (e+f x)+1) \sqrt {1-\frac {\cos ^2(e+f x)}{(\sin (e+f x)+1)^2}} \left ((a+b) g^2+\frac {(a-b) \cos ^2(e+f x) g^2}{(\sin (e+f x)+1)^2}\right )}d\frac {\sqrt {g \cos (e+f x)}}{\sqrt {\sin (e+f x)+1}}}{a^2 d^2 f \sqrt {d \sin (e+f x)}}-\frac {b g^2 \left (-\frac {2 E\left (\left .e+f x-\frac {\pi }{4}\right |2\right ) \sqrt {d \sin (e+f x)} \sqrt {g \cos (e+f x)}}{d^2 f \sqrt {\sin (2 e+2 f x)}}-\frac {2 (g \cos (e+f x))^{3/2}}{d f g \sqrt {d \sin (e+f x)}}\right )}{a^2 d}-\frac {2 g (g \cos (e+f x))^{3/2}}{3 a d f (d \sin (e+f x))^{3/2}}\) |
| \(\Big \downarrow \) 993 |
| \(\displaystyle \frac {4 \sqrt {2} g^3 \left (a^2-b^2\right ) \sqrt {\sin (e+f x)} \left (\frac {\int \frac {1}{\sqrt {1-\frac {\cos ^2(e+f x)}{(\sin (e+f x)+1)^2}} \left (\sqrt {a+b} g-\frac {\sqrt {b-a} g \cos (e+f x)}{\sin (e+f x)+1}\right )}d\frac {\sqrt {g \cos (e+f x)}}{\sqrt {\sin (e+f x)+1}}}{2 \sqrt {b-a}}-\frac {\int \frac {1}{\sqrt {1-\frac {\cos ^2(e+f x)}{(\sin (e+f x)+1)^2}} \left (\sqrt {a+b} g+\frac {\sqrt {b-a} \cos (e+f x) g}{\sin (e+f x)+1}\right )}d\frac {\sqrt {g \cos (e+f x)}}{\sqrt {\sin (e+f x)+1}}}{2 \sqrt {b-a}}\right )}{a^2 d^2 f \sqrt {d \sin (e+f x)}}-\frac {b g^2 \left (-\frac {2 E\left (\left .e+f x-\frac {\pi }{4}\right |2\right ) \sqrt {d \sin (e+f x)} \sqrt {g \cos (e+f x)}}{d^2 f \sqrt {\sin (2 e+2 f x)}}-\frac {2 (g \cos (e+f x))^{3/2}}{d f g \sqrt {d \sin (e+f x)}}\right )}{a^2 d}-\frac {2 g (g \cos (e+f x))^{3/2}}{3 a d f (d \sin (e+f x))^{3/2}}\) |
| \(\Big \downarrow \) 1542 |
| \(\displaystyle \frac {4 \sqrt {2} g^3 \left (a^2-b^2\right ) \sqrt {\sin (e+f x)} \left (\frac {\operatorname {EllipticPi}\left (\frac {\sqrt {b-a}}{\sqrt {a+b}},\arcsin \left (\frac {\sqrt {g \cos (e+f x)}}{\sqrt {g} \sqrt {\sin (e+f x)+1}}\right ),-1\right )}{2 \sqrt {g} \sqrt {b-a} \sqrt {a+b}}-\frac {\operatorname {EllipticPi}\left (-\frac {\sqrt {b-a}}{\sqrt {a+b}},\arcsin \left (\frac {\sqrt {g \cos (e+f x)}}{\sqrt {g} \sqrt {\sin (e+f x)+1}}\right ),-1\right )}{2 \sqrt {g} \sqrt {b-a} \sqrt {a+b}}\right )}{a^2 d^2 f \sqrt {d \sin (e+f x)}}-\frac {b g^2 \left (-\frac {2 E\left (\left .e+f x-\frac {\pi }{4}\right |2\right ) \sqrt {d \sin (e+f x)} \sqrt {g \cos (e+f x)}}{d^2 f \sqrt {\sin (2 e+2 f x)}}-\frac {2 (g \cos (e+f x))^{3/2}}{d f g \sqrt {d \sin (e+f x)}}\right )}{a^2 d}-\frac {2 g (g \cos (e+f x))^{3/2}}{3 a d f (d \sin (e+f x))^{3/2}}\) |
(-2*g*(g*Cos[e + f*x])^(3/2))/(3*a*d*f*(d*Sin[e + f*x])^(3/2)) + (4*Sqrt[2 ]*(a^2 - b^2)*g^3*(-1/2*EllipticPi[-(Sqrt[-a + b]/Sqrt[a + b]), ArcSin[Sqr t[g*Cos[e + f*x]]/(Sqrt[g]*Sqrt[1 + Sin[e + f*x]])], -1]/(Sqrt[-a + b]*Sqr t[a + b]*Sqrt[g]) + EllipticPi[Sqrt[-a + b]/Sqrt[a + b], ArcSin[Sqrt[g*Cos [e + f*x]]/(Sqrt[g]*Sqrt[1 + Sin[e + f*x]])], -1]/(2*Sqrt[-a + b]*Sqrt[a + b]*Sqrt[g]))*Sqrt[Sin[e + f*x]])/(a^2*d^2*f*Sqrt[d*Sin[e + f*x]]) - (b*g^ 2*((-2*(g*Cos[e + f*x])^(3/2))/(d*f*g*Sqrt[d*Sin[e + f*x]]) - (2*Sqrt[g*Co s[e + f*x]]*EllipticE[e - Pi/4 + f*x, 2]*Sqrt[d*Sin[e + f*x]])/(d^2*f*Sqrt [Sin[2*e + 2*f*x]])))/(a^2*d)
3.15.26.3.1 Defintions of rubi rules used
Int[(x_)^2/(((a_) + (b_.)*(x_)^4)*Sqrt[(c_) + (d_.)*(x_)^4]), x_Symbol] :> With[{r = Numerator[Rt[-a/b, 2]], s = Denominator[Rt[-a/b, 2]]}, Simp[s/(2* b) Int[1/((r + s*x^2)*Sqrt[c + d*x^4]), x], x] - Simp[s/(2*b) Int[1/((r - s*x^2)*Sqrt[c + d*x^4]), x], x]] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0]
Int[1/(((d_) + (e_.)*(x_)^2)*Sqrt[(a_) + (c_.)*(x_)^4]), x_Symbol] :> With[ {q = Rt[-c/a, 4]}, Simp[(1/(d*Sqrt[a]*q))*EllipticPi[-e/(d*q^2), ArcSin[q*x ], -1], x]] /; FreeQ[{a, c, d, e}, x] && NegQ[c/a] && GtQ[a, 0]
Int[(cos[(e_.) + (f_.)*(x_)]*(b_.))^(n_.)*((a_.)*sin[(e_.) + (f_.)*(x_)])^( m_.), x_Symbol] :> Simp[(a*Sin[e + f*x])^(m + 1)*((b*Cos[e + f*x])^(n + 1)/ (a*b*f*(m + 1))), x] /; FreeQ[{a, b, e, f, m, n}, x] && EqQ[m + n + 2, 0] & & NeQ[m, -1]
Int[(cos[(e_.) + (f_.)*(x_)]*(b_.))^(n_)*((a_.)*sin[(e_.) + (f_.)*(x_)])^(m _), x_Symbol] :> Simp[(b*Cos[e + f*x])^(n + 1)*((a*Sin[e + f*x])^(m + 1)/(a *b*f*(m + 1))), x] + Simp[(m + n + 2)/(a^2*(m + 1)) Int[(b*Cos[e + f*x])^ n*(a*Sin[e + f*x])^(m + 2), x], x] /; FreeQ[{a, b, e, f, n}, x] && LtQ[m, - 1] && IntegersQ[2*m, 2*n]
Int[Sqrt[cos[(e_.) + (f_.)*(x_)]*(b_.)]*Sqrt[(a_.)*sin[(e_.) + (f_.)*(x_)]] , x_Symbol] :> Simp[Sqrt[a*Sin[e + f*x]]*(Sqrt[b*Cos[e + f*x]]/Sqrt[Sin[2*e + 2*f*x]]) Int[Sqrt[Sin[2*e + 2*f*x]], x], x] /; FreeQ[{a, b, e, f}, x]
Int[Sqrt[sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[(2/d)*EllipticE[(1/2)* (c - Pi/2 + d*x), 2], x] /; FreeQ[{c, d}, x]
Int[((cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((d_.)*sin[(e_.) + (f_.)*(x_)])^( n_))/((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[g^2/a Int[ (g*Cos[e + f*x])^(p - 2)*(d*Sin[e + f*x])^n, x], x] + (-Simp[b*(g^2/(a^2*d) ) Int[(g*Cos[e + f*x])^(p - 2)*(d*Sin[e + f*x])^(n + 1), x], x] - Simp[g^ 2*((a^2 - b^2)/(a^2*d^2)) Int[(g*Cos[e + f*x])^(p - 2)*((d*Sin[e + f*x])^ (n + 2)/(a + b*Sin[e + f*x])), x], x]) /; FreeQ[{a, b, d, e, f, g}, x] && N eQ[a^2 - b^2, 0] && IntegersQ[2*n, 2*p] && GtQ[p, 1] && (LeQ[n, -2] || (EqQ [n, -3/2] && EqQ[p, 3/2]))
Int[Sqrt[cos[(e_.) + (f_.)*(x_)]*(g_.)]/(Sqrt[sin[(e_.) + (f_.)*(x_)]]*((a_ ) + (b_.)*sin[(e_.) + (f_.)*(x_)])), x_Symbol] :> Simp[-4*Sqrt[2]*(g/f) S ubst[Int[x^2/(((a + b)*g^2 + (a - b)*x^4)*Sqrt[1 - x^4/g^2]), x], x, Sqrt[g *Cos[e + f*x]]/Sqrt[1 + Sin[e + f*x]]], x] /; FreeQ[{a, b, e, f, g}, x] && NeQ[a^2 - b^2, 0]
Int[Sqrt[cos[(e_.) + (f_.)*(x_)]*(g_.)]/(Sqrt[(d_)*sin[(e_.) + (f_.)*(x_)]] *((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])), x_Symbol] :> Simp[Sqrt[Sin[e + f* x]]/Sqrt[d*Sin[e + f*x]] Int[Sqrt[g*Cos[e + f*x]]/(Sqrt[Sin[e + f*x]]*(a + b*Sin[e + f*x])), x], x] /; FreeQ[{a, b, d, e, f, g}, x] && NeQ[a^2 - b^2 , 0]
Leaf count of result is larger than twice the leaf count of optimal. \(2836\) vs. \(2(324)=648\).
Time = 1.94 (sec) , antiderivative size = 2837, normalized size of antiderivative = 7.90
1/3/f*csc(f*x+e)/(d/((1-cos(f*x+e))^2*csc(f*x+e)^2+1)*(csc(f*x+e)-cot(f*x+ e)))^(5/2)*(1-cos(f*x+e))*(-g*((1-cos(f*x+e))^2*csc(f*x+e)^2-1)/((1-cos(f* x+e))^2*csc(f*x+e)^2+1))^(5/2)*(3*EllipticPi((-cot(f*x+e)+csc(f*x+e)+1)^(1 /2),a/(-b+(-a^2+b^2)^(1/2)+a),1/2*2^(1/2))*a^2*b*(2+2*cot(f*x+e)-2*csc(f*x +e))^(1/2)*(-csc(f*x+e)+cot(f*x+e))^(1/2)*(-a^2+b^2)^(1/2)*(-cot(f*x+e)+cs c(f*x+e)+1)^(1/2)*(csc(f*x+e)-cot(f*x+e))+3*EllipticPi((-cot(f*x+e)+csc(f* x+e)+1)^(1/2),-a/(b+(-a^2+b^2)^(1/2)-a),1/2*2^(1/2))*a^2*b*(2+2*cot(f*x+e) -2*csc(f*x+e))^(1/2)*(-csc(f*x+e)+cot(f*x+e))^(1/2)*(-a^2+b^2)^(1/2)*(-cot (f*x+e)+csc(f*x+e)+1)^(1/2)*(csc(f*x+e)-cot(f*x+e))-csc(f*x+e)^4*a^3*(-a^2 +b^2)^(1/2)*(1-cos(f*x+e))^4+2*csc(f*x+e)^2*a^3*(-a^2+b^2)^(1/2)*(1-cos(f* x+e))^2+6*a^2*b*(-a^2+b^2)^(1/2)*(csc(f*x+e)-cot(f*x+e))-6*a*b^2*(-a^2+b^2 )^(1/2)*(csc(f*x+e)-cot(f*x+e))-3*EllipticPi((-cot(f*x+e)+csc(f*x+e)+1)^(1 /2),-a/(b+(-a^2+b^2)^(1/2)-a),1/2*2^(1/2))*a^3*(-cot(f*x+e)+csc(f*x+e)+1)^ (1/2)*(2+2*cot(f*x+e)-2*csc(f*x+e))^(1/2)*(-csc(f*x+e)+cot(f*x+e))^(1/2)*( -a^2+b^2)^(1/2)*(csc(f*x+e)-cot(f*x+e))+3*EllipticPi((-cot(f*x+e)+csc(f*x+ e)+1)^(1/2),a/(-b+(-a^2+b^2)^(1/2)+a),1/2*2^(1/2))*a^4*(-cot(f*x+e)+csc(f* x+e)+1)^(1/2)*(2+2*cot(f*x+e)-2*csc(f*x+e))^(1/2)*(-csc(f*x+e)+cot(f*x+e)) ^(1/2)*(csc(f*x+e)-cot(f*x+e))+3*EllipticPi((-cot(f*x+e)+csc(f*x+e)+1)^(1/ 2),a/(-b+(-a^2+b^2)^(1/2)+a),1/2*2^(1/2))*b^4*(-cot(f*x+e)+csc(f*x+e)+1)^( 1/2)*(2+2*cot(f*x+e)-2*csc(f*x+e))^(1/2)*(-csc(f*x+e)+cot(f*x+e))^(1/2)...
Timed out. \[ \int \frac {(g \cos (e+f x))^{5/2}}{(d \sin (e+f x))^{5/2} (a+b \sin (e+f x))} \, dx=\text {Timed out} \]
Timed out. \[ \int \frac {(g \cos (e+f x))^{5/2}}{(d \sin (e+f x))^{5/2} (a+b \sin (e+f x))} \, dx=\text {Timed out} \]
\[ \int \frac {(g \cos (e+f x))^{5/2}}{(d \sin (e+f x))^{5/2} (a+b \sin (e+f x))} \, dx=\int { \frac {\left (g \cos \left (f x + e\right )\right )^{\frac {5}{2}}}{{\left (b \sin \left (f x + e\right ) + a\right )} \left (d \sin \left (f x + e\right )\right )^{\frac {5}{2}}} \,d x } \]
\[ \int \frac {(g \cos (e+f x))^{5/2}}{(d \sin (e+f x))^{5/2} (a+b \sin (e+f x))} \, dx=\int { \frac {\left (g \cos \left (f x + e\right )\right )^{\frac {5}{2}}}{{\left (b \sin \left (f x + e\right ) + a\right )} \left (d \sin \left (f x + e\right )\right )^{\frac {5}{2}}} \,d x } \]
Timed out. \[ \int \frac {(g \cos (e+f x))^{5/2}}{(d \sin (e+f x))^{5/2} (a+b \sin (e+f x))} \, dx=\int \frac {{\left (g\,\cos \left (e+f\,x\right )\right )}^{5/2}}{{\left (d\,\sin \left (e+f\,x\right )\right )}^{5/2}\,\left (a+b\,\sin \left (e+f\,x\right )\right )} \,d x \]