Integrand size = 25, antiderivative size = 128 \[ \int \sqrt {b \sec (e+f x)} (a \sin (e+f x))^{7/2} \, dx=-\frac {5 a^3 b \sqrt {a \sin (e+f x)}}{6 f \sqrt {b \sec (e+f x)}}-\frac {a b (a \sin (e+f x))^{5/2}}{3 f \sqrt {b \sec (e+f x)}}+\frac {5 a^4 \operatorname {EllipticF}\left (e-\frac {\pi }{4}+f x,2\right ) \sqrt {b \sec (e+f x)} \sqrt {\sin (2 e+2 f x)}}{12 f \sqrt {a \sin (e+f x)}} \]
-1/3*a*b*(a*sin(f*x+e))^(5/2)/f/(b*sec(f*x+e))^(1/2)-5/6*a^3*b*(a*sin(f*x+ e))^(1/2)/f/(b*sec(f*x+e))^(1/2)-5/12*a^4*(sin(e+1/4*Pi+f*x)^2)^(1/2)/sin( e+1/4*Pi+f*x)*EllipticF(cos(e+1/4*Pi+f*x),2^(1/2))*(b*sec(f*x+e))^(1/2)*si n(2*f*x+2*e)^(1/2)/f/(a*sin(f*x+e))^(1/2)
Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
Time = 15.84 (sec) , antiderivative size = 90, normalized size of antiderivative = 0.70 \[ \int \sqrt {b \sec (e+f x)} (a \sin (e+f x))^{7/2} \, dx=\frac {a^3 b \sqrt {a \sin (e+f x)} \left (2 (-6+\cos (2 (e+f x)))+5 \csc ^2(e+f x) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {3}{4},\frac {3}{2},\sec ^2(e+f x)\right ) \left (-\tan ^2(e+f x)\right )^{3/4}\right )}{12 f \sqrt {b \sec (e+f x)}} \]
(a^3*b*Sqrt[a*Sin[e + f*x]]*(2*(-6 + Cos[2*(e + f*x)]) + 5*Csc[e + f*x]^2* Hypergeometric2F1[1/2, 3/4, 3/2, Sec[e + f*x]^2]*(-Tan[e + f*x]^2)^(3/4))) /(12*f*Sqrt[b*Sec[e + f*x]])
Time = 0.66 (sec) , antiderivative size = 132, normalized size of antiderivative = 1.03, number of steps used = 10, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {3042, 3063, 3042, 3063, 3042, 3065, 3042, 3053, 3042, 3120}
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.
\(\displaystyle \int (a \sin (e+f x))^{7/2} \sqrt {b \sec (e+f x)} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int (a \sin (e+f x))^{7/2} \sqrt {b \sec (e+f x)}dx\) |
\(\Big \downarrow \) 3063 |
\(\displaystyle \frac {5}{6} a^2 \int \sqrt {b \sec (e+f x)} (a \sin (e+f x))^{3/2}dx-\frac {a b (a \sin (e+f x))^{5/2}}{3 f \sqrt {b \sec (e+f x)}}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {5}{6} a^2 \int \sqrt {b \sec (e+f x)} (a \sin (e+f x))^{3/2}dx-\frac {a b (a \sin (e+f x))^{5/2}}{3 f \sqrt {b \sec (e+f x)}}\) |
\(\Big \downarrow \) 3063 |
\(\displaystyle \frac {5}{6} a^2 \left (\frac {1}{2} a^2 \int \frac {\sqrt {b \sec (e+f x)}}{\sqrt {a \sin (e+f x)}}dx-\frac {a b \sqrt {a \sin (e+f x)}}{f \sqrt {b \sec (e+f x)}}\right )-\frac {a b (a \sin (e+f x))^{5/2}}{3 f \sqrt {b \sec (e+f x)}}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {5}{6} a^2 \left (\frac {1}{2} a^2 \int \frac {\sqrt {b \sec (e+f x)}}{\sqrt {a \sin (e+f x)}}dx-\frac {a b \sqrt {a \sin (e+f x)}}{f \sqrt {b \sec (e+f x)}}\right )-\frac {a b (a \sin (e+f x))^{5/2}}{3 f \sqrt {b \sec (e+f x)}}\) |
\(\Big \downarrow \) 3065 |
\(\displaystyle \frac {5}{6} a^2 \left (\frac {1}{2} a^2 \sqrt {b \cos (e+f x)} \sqrt {b \sec (e+f x)} \int \frac {1}{\sqrt {b \cos (e+f x)} \sqrt {a \sin (e+f x)}}dx-\frac {a b \sqrt {a \sin (e+f x)}}{f \sqrt {b \sec (e+f x)}}\right )-\frac {a b (a \sin (e+f x))^{5/2}}{3 f \sqrt {b \sec (e+f x)}}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {5}{6} a^2 \left (\frac {1}{2} a^2 \sqrt {b \cos (e+f x)} \sqrt {b \sec (e+f x)} \int \frac {1}{\sqrt {b \cos (e+f x)} \sqrt {a \sin (e+f x)}}dx-\frac {a b \sqrt {a \sin (e+f x)}}{f \sqrt {b \sec (e+f x)}}\right )-\frac {a b (a \sin (e+f x))^{5/2}}{3 f \sqrt {b \sec (e+f x)}}\) |
\(\Big \downarrow \) 3053 |
\(\displaystyle \frac {5}{6} a^2 \left (\frac {a^2 \sqrt {\sin (2 e+2 f x)} \sqrt {b \sec (e+f x)} \int \frac {1}{\sqrt {\sin (2 e+2 f x)}}dx}{2 \sqrt {a \sin (e+f x)}}-\frac {a b \sqrt {a \sin (e+f x)}}{f \sqrt {b \sec (e+f x)}}\right )-\frac {a b (a \sin (e+f x))^{5/2}}{3 f \sqrt {b \sec (e+f x)}}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {5}{6} a^2 \left (\frac {a^2 \sqrt {\sin (2 e+2 f x)} \sqrt {b \sec (e+f x)} \int \frac {1}{\sqrt {\sin (2 e+2 f x)}}dx}{2 \sqrt {a \sin (e+f x)}}-\frac {a b \sqrt {a \sin (e+f x)}}{f \sqrt {b \sec (e+f x)}}\right )-\frac {a b (a \sin (e+f x))^{5/2}}{3 f \sqrt {b \sec (e+f x)}}\) |
\(\Big \downarrow \) 3120 |
\(\displaystyle \frac {5}{6} a^2 \left (\frac {a^2 \sqrt {\sin (2 e+2 f x)} \operatorname {EllipticF}\left (e+f x-\frac {\pi }{4},2\right ) \sqrt {b \sec (e+f x)}}{2 f \sqrt {a \sin (e+f x)}}-\frac {a b \sqrt {a \sin (e+f x)}}{f \sqrt {b \sec (e+f x)}}\right )-\frac {a b (a \sin (e+f x))^{5/2}}{3 f \sqrt {b \sec (e+f x)}}\) |
-1/3*(a*b*(a*Sin[e + f*x])^(5/2))/(f*Sqrt[b*Sec[e + f*x]]) + (5*a^2*(-((a* b*Sqrt[a*Sin[e + f*x]])/(f*Sqrt[b*Sec[e + f*x]])) + (a^2*EllipticF[e - Pi/ 4 + f*x, 2]*Sqrt[b*Sec[e + f*x]]*Sqrt[Sin[2*e + 2*f*x]])/(2*f*Sqrt[a*Sin[e + f*x]])))/6
3.5.56.3.1 Defintions of rubi rules used
Int[1/(Sqrt[cos[(e_.) + (f_.)*(x_)]*(b_.)]*Sqrt[(a_.)*sin[(e_.) + (f_.)*(x_ )]]), x_Symbol] :> Simp[Sqrt[Sin[2*e + 2*f*x]]/(Sqrt[a*Sin[e + f*x]]*Sqrt[b *Cos[e + f*x]]) Int[1/Sqrt[Sin[2*e + 2*f*x]], x], x] /; FreeQ[{a, b, e, f }, x]
Int[((b_.)*sec[(e_.) + (f_.)*(x_)])^(n_)*((a_.)*sin[(e_.) + (f_.)*(x_)])^(m _), x_Symbol] :> Simp[(-a)*b*(a*Sin[e + f*x])^(m - 1)*((b*Sec[e + f*x])^(n - 1)/(f*(m - n))), x] + Simp[a^2*((m - 1)/(m - n)) Int[(a*Sin[e + f*x])^( m - 2)*(b*Sec[e + f*x])^n, x], x] /; FreeQ[{a, b, e, f, n}, x] && GtQ[m, 1] && NeQ[m - n, 0] && IntegersQ[2*m, 2*n]
Int[((b_.)*sec[(e_.) + (f_.)*(x_)])^(n_)*((a_.)*sin[(e_.) + (f_.)*(x_)])^(m _), x_Symbol] :> Simp[(b*Cos[e + f*x])^n*(b*Sec[e + f*x])^n Int[(a*Sin[e + f*x])^m/(b*Cos[e + f*x])^n, x], x] /; FreeQ[{a, b, e, f, m, n}, x] && Int egerQ[m - 1/2] && IntegerQ[n - 1/2]
Int[1/Sqrt[sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[(2/d)*EllipticF[(1/2 )*(c - Pi/2 + d*x), 2], x] /; FreeQ[{c, d}, x]
Result contains complex when optimal does not.
Time = 7.71 (sec) , antiderivative size = 1739, normalized size of antiderivative = 13.59
1/48/f*2^(1/2)*(-6*I*(-cot(f*x+e)+csc(f*x+e)+1)^(1/2)*(cot(f*x+e)-csc(f*x+ e)+1)^(1/2)*(cot(f*x+e)-csc(f*x+e))^(1/2)*EllipticPi((-cot(f*x+e)+csc(f*x+ e)+1)^(1/2),1/2-1/2*I,1/2*2^(1/2))*cos(f*x+e)-6*I*(-cot(f*x+e)+csc(f*x+e)+ 1)^(1/2)*(cot(f*x+e)-csc(f*x+e)+1)^(1/2)*(cot(f*x+e)-csc(f*x+e))^(1/2)*Ell ipticPi((-cot(f*x+e)+csc(f*x+e)+1)^(1/2),1/2-1/2*I,1/2*2^(1/2))+8*2^(1/2)* cos(f*x+e)^3*sin(f*x+e)+6*I*(-cot(f*x+e)+csc(f*x+e)+1)^(1/2)*(cot(f*x+e)-c sc(f*x+e)+1)^(1/2)*(cot(f*x+e)-csc(f*x+e))^(1/2)*EllipticPi((-cot(f*x+e)+c sc(f*x+e)+1)^(1/2),1/2+1/2*I,1/2*2^(1/2))+6*I*(-cot(f*x+e)+csc(f*x+e)+1)^( 1/2)*(cot(f*x+e)-csc(f*x+e)+1)^(1/2)*(cot(f*x+e)-csc(f*x+e))^(1/2)*Ellipti cPi((-cot(f*x+e)+csc(f*x+e)+1)^(1/2),1/2+1/2*I,1/2*2^(1/2))*cos(f*x+e)-6*( -cot(f*x+e)+csc(f*x+e)+1)^(1/2)*(cot(f*x+e)-csc(f*x+e)+1)^(1/2)*(cot(f*x+e )-csc(f*x+e))^(1/2)*EllipticPi((-cot(f*x+e)+csc(f*x+e)+1)^(1/2),1/2+1/2*I, 1/2*2^(1/2))*cos(f*x+e)-6*(-cot(f*x+e)+csc(f*x+e)+1)^(1/2)*(cot(f*x+e)-csc (f*x+e)+1)^(1/2)*(cot(f*x+e)-csc(f*x+e))^(1/2)*EllipticPi((-cot(f*x+e)+csc (f*x+e)+1)^(1/2),1/2-1/2*I,1/2*2^(1/2))*cos(f*x+e)+32*(-cot(f*x+e)+csc(f*x +e)+1)^(1/2)*(cot(f*x+e)-csc(f*x+e)+1)^(1/2)*(cot(f*x+e)-csc(f*x+e))^(1/2) *EllipticF((-cot(f*x+e)+csc(f*x+e)+1)^(1/2),1/2*2^(1/2))*cos(f*x+e)-6*(-co t(f*x+e)+csc(f*x+e)+1)^(1/2)*(cot(f*x+e)-csc(f*x+e)+1)^(1/2)*(cot(f*x+e)-c sc(f*x+e))^(1/2)*EllipticPi((-cot(f*x+e)+csc(f*x+e)+1)^(1/2),1/2+1/2*I,1/2 *2^(1/2))-6*(-cot(f*x+e)+csc(f*x+e)+1)^(1/2)*(cot(f*x+e)-csc(f*x+e)+1)^...
\[ \int \sqrt {b \sec (e+f x)} (a \sin (e+f x))^{7/2} \, dx=\int { \sqrt {b \sec \left (f x + e\right )} \left (a \sin \left (f x + e\right )\right )^{\frac {7}{2}} \,d x } \]
integral(-(a^3*cos(f*x + e)^2 - a^3)*sqrt(b*sec(f*x + e))*sqrt(a*sin(f*x + e))*sin(f*x + e), x)
Timed out. \[ \int \sqrt {b \sec (e+f x)} (a \sin (e+f x))^{7/2} \, dx=\text {Timed out} \]
\[ \int \sqrt {b \sec (e+f x)} (a \sin (e+f x))^{7/2} \, dx=\int { \sqrt {b \sec \left (f x + e\right )} \left (a \sin \left (f x + e\right )\right )^{\frac {7}{2}} \,d x } \]
\[ \int \sqrt {b \sec (e+f x)} (a \sin (e+f x))^{7/2} \, dx=\int { \sqrt {b \sec \left (f x + e\right )} \left (a \sin \left (f x + e\right )\right )^{\frac {7}{2}} \,d x } \]
Timed out. \[ \int \sqrt {b \sec (e+f x)} (a \sin (e+f x))^{7/2} \, dx=\int {\left (a\,\sin \left (e+f\,x\right )\right )}^{7/2}\,\sqrt {\frac {b}{\cos \left (e+f\,x\right )}} \,d x \]