Integrand size = 35, antiderivative size = 502 \[ \int \frac {\sec ^6(e+f x) (a+b \sin (e+f x))^{9/2}}{\sqrt {d \sin (e+f x)}} \, dx=-\frac {3 a b \left (-2 a^2+b^2\right ) \cos (e+f x) \sqrt {a+b \sin (e+f x)}}{5 f \sqrt {d \sin (e+f x)}}+\frac {\sec ^5(e+f x) \sqrt {d \sin (e+f x)} (a+b \sin (e+f x))^{9/2}}{5 d f}-\frac {3 a \sec ^3(e+f x) \sqrt {d \sin (e+f x)} \sqrt {a+b \sin (e+f x)} \left (-a \left (7 a^2+b^2\right )+2 b \left (-7 a^2+b^2\right ) \sin (e+f x)+5 a \left (a^2-b^2\right ) \sin ^2(e+f x)+\left (8 a^2 b-4 b^3\right ) \sin ^3(e+f x)\right )}{20 d f}-\frac {3 a (a+b)^{3/2} \left (5 a^2+3 a b-4 b^2\right ) \sqrt {-\frac {a (-1+\csc (e+f x))}{a+b}} \sqrt {\frac {a (1+\csc (e+f x))}{a-b}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {d} \sqrt {a+b \sin (e+f x)}}{\sqrt {a+b} \sqrt {d \sin (e+f x)}}\right ),-\frac {a+b}{a-b}\right ) \tan (e+f x)}{20 \sqrt {d} f}-\frac {3 b \left (2 a^4-3 a^2 b^2+b^4\right ) \sqrt {-\frac {a (-1+\csc (e+f x))}{a+b}} E\left (\arcsin \left (\sqrt {-\frac {b+a \csc (e+f x)}{a-b}}\right )|1-\frac {2 a}{a+b}\right ) \sqrt {d \sin (e+f x)} \sqrt {-\frac {a \csc ^2(e+f x) (1+\sin (e+f x)) (a+b \sin (e+f x))}{(a-b)^2}} \tan (e+f x)}{5 d f \sqrt {a+b \sin (e+f x)}} \]
1/5*sec(f*x+e)^5*(a+b*sin(f*x+e))^(9/2)*(d*sin(f*x+e))^(1/2)/d/f-3/5*a*b*( -2*a^2+b^2)*cos(f*x+e)*(a+b*sin(f*x+e))^(1/2)/f/(d*sin(f*x+e))^(1/2)-3/20* a*sec(f*x+e)^3*(-a*(7*a^2+b^2)+2*b*(-7*a^2+b^2)*sin(f*x+e)+5*a*(a^2-b^2)*s in(f*x+e)^2+(8*a^2*b-4*b^3)*sin(f*x+e)^3)*(d*sin(f*x+e))^(1/2)*(a+b*sin(f* x+e))^(1/2)/d/f-3/20*a*(a+b)^(3/2)*(5*a^2+3*a*b-4*b^2)*EllipticF(d^(1/2)*( a+b*sin(f*x+e))^(1/2)/(a+b)^(1/2)/(d*sin(f*x+e))^(1/2),((-a-b)/(a-b))^(1/2 ))*(-a*(-1+csc(f*x+e))/(a+b))^(1/2)*(a*(1+csc(f*x+e))/(a-b))^(1/2)*tan(f*x +e)/f/d^(1/2)-3/5*b*(2*a^4-3*a^2*b^2+b^4)*EllipticE(((-b-a*csc(f*x+e))/(a- b))^(1/2),(1-2*a/(a+b))^(1/2))*(-a*(-1+csc(f*x+e))/(a+b))^(1/2)*(d*sin(f*x +e))^(1/2)*(-a*csc(f*x+e)^2*(1+sin(f*x+e))*(a+b*sin(f*x+e))/(a-b)^2)^(1/2) *tan(f*x+e)/d/f/(a+b*sin(f*x+e))^(1/2)
Result contains complex when optimal does not.
Time = 23.52 (sec) , antiderivative size = 1600, normalized size of antiderivative = 3.19 \[ \int \frac {\sec ^6(e+f x) (a+b \sin (e+f x))^{9/2}}{\sqrt {d \sin (e+f x)}} \, dx =\text {Too large to display} \]
(Sin[e + f*x]*Sqrt[a + b*Sin[e + f*x]]*((Sec[e + f*x]*(15*a^4 - 15*a^2*b^2 + 4*b^4 + 24*a^3*b*Sin[e + f*x] - 12*a*b^3*Sin[e + f*x]))/20 + (Sec[e + f *x]^3*(3*a^4 - 3*a^2*b^2 - 4*b^4 + 9*a^3*b*Sin[e + f*x] - 5*a*b^3*Sin[e + f*x]))/10 + (Sec[e + f*x]^5*(a^4 + 6*a^2*b^2 + b^4 + 4*a^3*b*Sin[e + f*x] + 4*a*b^3*Sin[e + f*x]))/5))/(f*Sqrt[d*Sin[e + f*x]]) + (3*a*Sqrt[Sin[e + f*x]]*((4*a*(5*a^4 - 9*a^2*b^2 + 4*b^4)*Sqrt[((a + b)*Cot[(-e + Pi/2 - f*x )/2]^2)/(-a + b)]*EllipticF[ArcSin[Sqrt[(Csc[(-e + Pi/2 - f*x)/2]^2*(a + b *Sin[e + f*x]))/a]/Sqrt[2]], (-2*a)/(-a + b)]*Sec[e + f*x]*Sin[(-e + Pi/2 - f*x)/2]^4*Sqrt[-(((a + b)*Csc[(-e + Pi/2 - f*x)/2]^2*Sin[e + f*x])/a)]*S qrt[(Csc[(-e + Pi/2 - f*x)/2]^2*(a + b*Sin[e + f*x]))/a])/((a + b)*Sqrt[Si n[e + f*x]]*Sqrt[a + b*Sin[e + f*x]]) + 4*a*(-8*a^3*b + 4*a*b^3)*((Sqrt[(( a + b)*Cot[(-e + Pi/2 - f*x)/2]^2)/(-a + b)]*EllipticF[ArcSin[Sqrt[(Csc[(- e + Pi/2 - f*x)/2]^2*(a + b*Sin[e + f*x]))/a]/Sqrt[2]], (-2*a)/(-a + b)]*S ec[e + f*x]*Sin[(-e + Pi/2 - f*x)/2]^4*Sqrt[-(((a + b)*Csc[(-e + Pi/2 - f* x)/2]^2*Sin[e + f*x])/a)]*Sqrt[(Csc[(-e + Pi/2 - f*x)/2]^2*(a + b*Sin[e + f*x]))/a])/((a + b)*Sqrt[Sin[e + f*x]]*Sqrt[a + b*Sin[e + f*x]]) - (Sqrt[( (a + b)*Cot[(-e + Pi/2 - f*x)/2]^2)/(-a + b)]*EllipticPi[-(a/b), ArcSin[Sq rt[(Csc[(-e + Pi/2 - f*x)/2]^2*(a + b*Sin[e + f*x]))/a]/Sqrt[2]], (-2*a)/( -a + b)]*Sec[e + f*x]*Sin[(-e + Pi/2 - f*x)/2]^4*Sqrt[-(((a + b)*Csc[(-e + Pi/2 - f*x)/2]^2*Sin[e + f*x])/a)]*Sqrt[(Csc[(-e + Pi/2 - f*x)/2]^2*(a...
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.
| \(\displaystyle \int \frac {\sec ^6(e+f x) (a+b \sin (e+f x))^{9/2}}{\sqrt {d \sin (e+f x)}} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {(a+b \sin (e+f x))^{9/2}}{\cos (e+f x)^6 \sqrt {d \sin (e+f x)}}dx\) |
| \(\Big \downarrow \) 3367 |
| \(\displaystyle \frac {9}{10} a \int \frac {\sec ^4(e+f x) (a+b \sin (e+f x))^{7/2}}{\sqrt {d \sin (e+f x)}}dx+\frac {\sec ^5(e+f x) \sqrt {d \sin (e+f x)} (a+b \sin (e+f x))^{9/2}}{5 d f}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {9}{10} a \int \frac {(a+b \sin (e+f x))^{7/2}}{\cos (e+f x)^4 \sqrt {d \sin (e+f x)}}dx+\frac {\sec ^5(e+f x) \sqrt {d \sin (e+f x)} (a+b \sin (e+f x))^{9/2}}{5 d f}\) |
| \(\Big \downarrow \) 3404 |
| \(\displaystyle \frac {9}{10} a \int \frac {\sec ^4(e+f x) (a+b \sin (e+f x))^{7/2}}{\sqrt {d \sin (e+f x)}}dx+\frac {\sec ^5(e+f x) \sqrt {d \sin (e+f x)} (a+b \sin (e+f x))^{9/2}}{5 d f}\) |
3.16.15.3.1 Defintions of rubi rules used
Int[((cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((a_) + (b_.)*sin[(e_.) + (f_.)*( x_)])^(m_))/Sqrt[(d_.)*sin[(e_.) + (f_.)*(x_)]], x_Symbol] :> Simp[2*(g*Cos [e + f*x])^(p + 1)*Sqrt[d*Sin[e + f*x]]*((a + b*Sin[e + f*x])^m/(d*f*g*(2*m + 1))), x] + Simp[2*a*(m/(g^2*(2*m + 1))) Int[(g*Cos[e + f*x])^(p + 2)*( (a + b*Sin[e + f*x])^(m - 1)/Sqrt[d*Sin[e + f*x]]), x], x] /; FreeQ[{a, b, e, f, g}, x] && NeQ[a^2 - b^2, 0] && GtQ[m, 0] && EqQ[m + p + 3/2, 0]
Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x _)])^(m_.)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_.), x_Symbol] :> Unin tegrable[(g*Cos[e + f*x])^p*(a + b*Sin[e + f*x])^m*(c + d*Sin[e + f*x])^n, x] /; FreeQ[{a, b, c, d, e, f, g, m, n, p}, x] && NeQ[a^2 - b^2, 0]
Leaf count of result is larger than twice the leaf count of optimal. \(4485\) vs. \(2(461)=922\).
Time = 5.27 (sec) , antiderivative size = 4486, normalized size of antiderivative = 8.94
1/40/f/(a+b*sin(f*x+e))^(1/2)/(d*sin(f*x+e))^(1/2)*(-48*cos(f*x+e)*(-a^2+b ^2)^(1/2)*(1/(b+(-a^2+b^2)^(1/2))*(a*csc(f*x+e)-a*cot(f*x+e)+(-a^2+b^2)^(1 /2)+b))^(1/2)*(-(a*csc(f*x+e)-a*cot(f*x+e)-(-a^2+b^2)^(1/2)+b)/(-a^2+b^2)^ (1/2))^(1/2)*(-a/(b+(-a^2+b^2)^(1/2))*(csc(f*x+e)-cot(f*x+e)))^(1/2)*Ellip ticE((1/(b+(-a^2+b^2)^(1/2))*(a*csc(f*x+e)-a*cot(f*x+e)+(-a^2+b^2)^(1/2)+b ))^(1/2),1/2*2^(1/2)*((b+(-a^2+b^2)^(1/2))/(-a^2+b^2)^(1/2))^(1/2))*a^2*b^ 2-3*cos(f*x+e)*(-a^2+b^2)^(1/2)*(1/(b+(-a^2+b^2)^(1/2))*(a*csc(f*x+e)-a*co t(f*x+e)+(-a^2+b^2)^(1/2)+b))^(1/2)*(-(a*csc(f*x+e)-a*cot(f*x+e)-(-a^2+b^2 )^(1/2)+b)/(-a^2+b^2)^(1/2))^(1/2)*(-a/(b+(-a^2+b^2)^(1/2))*(csc(f*x+e)-co t(f*x+e)))^(1/2)*EllipticF((1/(b+(-a^2+b^2)^(1/2))*(a*csc(f*x+e)-a*cot(f*x +e)+(-a^2+b^2)^(1/2)+b))^(1/2),1/2*2^(1/2)*((b+(-a^2+b^2)^(1/2))/(-a^2+b^2 )^(1/2))^(1/2))*a^2*b^2+24*cos(f*x+e)*(1/(b+(-a^2+b^2)^(1/2))*(a*csc(f*x+e )-a*cot(f*x+e)+(-a^2+b^2)^(1/2)+b))^(1/2)*(-(a*csc(f*x+e)-a*cot(f*x+e)-(-a ^2+b^2)^(1/2)+b)/(-a^2+b^2)^(1/2))^(1/2)*(-a/(b+(-a^2+b^2)^(1/2))*(csc(f*x +e)-cot(f*x+e)))^(1/2)*EllipticE((1/(b+(-a^2+b^2)^(1/2))*(a*csc(f*x+e)-a*c ot(f*x+e)+(-a^2+b^2)^(1/2)+b))^(1/2),1/2*2^(1/2)*((b+(-a^2+b^2)^(1/2))/(-a ^2+b^2)^(1/2))^(1/2))*b^5+24*(-a^2+b^2)^(1/2)*(1/(b+(-a^2+b^2)^(1/2))*(a*c sc(f*x+e)-a*cot(f*x+e)+(-a^2+b^2)^(1/2)+b))^(1/2)*(-(a*csc(f*x+e)-a*cot(f* x+e)-(-a^2+b^2)^(1/2)+b)/(-a^2+b^2)^(1/2))^(1/2)*(-a/(b+(-a^2+b^2)^(1/2))* (csc(f*x+e)-cot(f*x+e)))^(1/2)*EllipticE((1/(b+(-a^2+b^2)^(1/2))*(a*csc...
\[ \int \frac {\sec ^6(e+f x) (a+b \sin (e+f x))^{9/2}}{\sqrt {d \sin (e+f x)}} \, dx=\int { \frac {{\left (b \sin \left (f x + e\right ) + a\right )}^{\frac {9}{2}} \sec \left (f x + e\right )^{6}}{\sqrt {d \sin \left (f x + e\right )}} \,d x } \]
integral(-(4*(a*b^3*cos(f*x + e)^2 - a^3*b - a*b^3)*sec(f*x + e)^6*sin(f*x + e) - (b^4*cos(f*x + e)^4 + a^4 + 6*a^2*b^2 + b^4 - 2*(3*a^2*b^2 + b^4)* cos(f*x + e)^2)*sec(f*x + e)^6)*sqrt(b*sin(f*x + e) + a)*sqrt(d*sin(f*x + e))/(d*sin(f*x + e)), x)
Timed out. \[ \int \frac {\sec ^6(e+f x) (a+b \sin (e+f x))^{9/2}}{\sqrt {d \sin (e+f x)}} \, dx=\text {Timed out} \]
\[ \int \frac {\sec ^6(e+f x) (a+b \sin (e+f x))^{9/2}}{\sqrt {d \sin (e+f x)}} \, dx=\int { \frac {{\left (b \sin \left (f x + e\right ) + a\right )}^{\frac {9}{2}} \sec \left (f x + e\right )^{6}}{\sqrt {d \sin \left (f x + e\right )}} \,d x } \]
Timed out. \[ \int \frac {\sec ^6(e+f x) (a+b \sin (e+f x))^{9/2}}{\sqrt {d \sin (e+f x)}} \, dx=\text {Timed out} \]
Timed out. \[ \int \frac {\sec ^6(e+f x) (a+b \sin (e+f x))^{9/2}}{\sqrt {d \sin (e+f x)}} \, dx=\int \frac {{\left (a+b\,\sin \left (e+f\,x\right )\right )}^{9/2}}{{\cos \left (e+f\,x\right )}^6\,\sqrt {d\,\sin \left (e+f\,x\right )}} \,d x \]