Integrand size = 25, antiderivative size = 172 \[ \int \frac {(a \sin (e+f x))^{7/2}}{(b \sec (e+f x))^{3/2}} \, dx=-\frac {a^3 \sqrt {a \sin (e+f x)}}{12 b f \sqrt {b \sec (e+f x)}}-\frac {a (a \sin (e+f x))^{5/2}}{30 b f \sqrt {b \sec (e+f x)}}+\frac {(a \sin (e+f x))^{9/2}}{5 a b f \sqrt {b \sec (e+f x)}}+\frac {a^4 \operatorname {EllipticF}\left (e-\frac {\pi }{4}+f x,2\right ) \sqrt {b \sec (e+f x)} \sqrt {\sin (2 e+2 f x)}}{24 b^2 f \sqrt {a \sin (e+f x)}} \]
-1/30*a*(a*sin(f*x+e))^(5/2)/b/f/(b*sec(f*x+e))^(1/2)+1/5*(a*sin(f*x+e))^( 9/2)/a/b/f/(b*sec(f*x+e))^(1/2)-1/12*a^3*(a*sin(f*x+e))^(1/2)/b/f/(b*sec(f *x+e))^(1/2)-1/24*a^4*(sin(e+1/4*Pi+f*x)^2)^(1/2)/sin(e+1/4*Pi+f*x)*Ellipt icF(cos(e+1/4*Pi+f*x),2^(1/2))*(b*sec(f*x+e))^(1/2)*sin(2*f*x+2*e)^(1/2)/b ^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 = 1.04 (sec) , antiderivative size = 103, normalized size of antiderivative = 0.60 \[ \int \frac {(a \sin (e+f x))^{7/2}}{(b \sec (e+f x))^{3/2}} \, dx=-\frac {a^5 \left (-4+17 \cos (2 (e+f x))-16 \cos (4 (e+f x))+3 \cos (6 (e+f x))-20 \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {3}{4},\frac {3}{2},\sec ^2(e+f x)\right ) \left (-\tan ^2(e+f x)\right )^{3/4}\right )}{480 b f \sqrt {b \sec (e+f x)} (a \sin (e+f x))^{3/2}} \]
-1/480*(a^5*(-4 + 17*Cos[2*(e + f*x)] - 16*Cos[4*(e + f*x)] + 3*Cos[6*(e + f*x)] - 20*Hypergeometric2F1[1/2, 3/4, 3/2, Sec[e + f*x]^2]*(-Tan[e + f*x ]^2)^(3/4)))/(b*f*Sqrt[b*Sec[e + f*x]]*(a*Sin[e + f*x])^(3/2))
Time = 0.94 (sec) , antiderivative size = 177, normalized size of antiderivative = 1.03, number of steps used = 12, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.480, Rules used = {3042, 3062, 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.
| \(\displaystyle \int \frac {(a \sin (e+f x))^{7/2}}{(b \sec (e+f x))^{3/2}} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {(a \sin (e+f x))^{7/2}}{(b \sec (e+f x))^{3/2}}dx\) |
| \(\Big \downarrow \) 3062 |
| \(\displaystyle \frac {\int \sqrt {b \sec (e+f x)} (a \sin (e+f x))^{7/2}dx}{10 b^2}+\frac {(a \sin (e+f x))^{9/2}}{5 a b f \sqrt {b \sec (e+f x)}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\int \sqrt {b \sec (e+f x)} (a \sin (e+f x))^{7/2}dx}{10 b^2}+\frac {(a \sin (e+f x))^{9/2}}{5 a b f \sqrt {b \sec (e+f x)}}\) |
| \(\Big \downarrow \) 3063 |
| \(\displaystyle \frac {\frac {5}{6} a^2 \int \sqrt {b \sec (e+f x)} (a \sin (e+f x))^{3/2}dx-\frac {a b (a \sin (e+f x))^{5/2}}{3 f \sqrt {b \sec (e+f x)}}}{10 b^2}+\frac {(a \sin (e+f x))^{9/2}}{5 a b f \sqrt {b \sec (e+f x)}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {5}{6} a^2 \int \sqrt {b \sec (e+f x)} (a \sin (e+f x))^{3/2}dx-\frac {a b (a \sin (e+f x))^{5/2}}{3 f \sqrt {b \sec (e+f x)}}}{10 b^2}+\frac {(a \sin (e+f x))^{9/2}}{5 a b f \sqrt {b \sec (e+f x)}}\) |
| \(\Big \downarrow \) 3063 |
| \(\displaystyle \frac {\frac {5}{6} a^2 \left (\frac {1}{2} a^2 \int \frac {\sqrt {b \sec (e+f x)}}{\sqrt {a \sin (e+f x)}}dx-\frac {a b \sqrt {a \sin (e+f x)}}{f \sqrt {b \sec (e+f x)}}\right )-\frac {a b (a \sin (e+f x))^{5/2}}{3 f \sqrt {b \sec (e+f x)}}}{10 b^2}+\frac {(a \sin (e+f x))^{9/2}}{5 a b f \sqrt {b \sec (e+f x)}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {5}{6} a^2 \left (\frac {1}{2} a^2 \int \frac {\sqrt {b \sec (e+f x)}}{\sqrt {a \sin (e+f x)}}dx-\frac {a b \sqrt {a \sin (e+f x)}}{f \sqrt {b \sec (e+f x)}}\right )-\frac {a b (a \sin (e+f x))^{5/2}}{3 f \sqrt {b \sec (e+f x)}}}{10 b^2}+\frac {(a \sin (e+f x))^{9/2}}{5 a b f \sqrt {b \sec (e+f x)}}\) |
| \(\Big \downarrow \) 3065 |
| \(\displaystyle \frac {\frac {5}{6} a^2 \left (\frac {1}{2} a^2 \sqrt {b \cos (e+f x)} \sqrt {b \sec (e+f x)} \int \frac {1}{\sqrt {b \cos (e+f x)} \sqrt {a \sin (e+f x)}}dx-\frac {a b \sqrt {a \sin (e+f x)}}{f \sqrt {b \sec (e+f x)}}\right )-\frac {a b (a \sin (e+f x))^{5/2}}{3 f \sqrt {b \sec (e+f x)}}}{10 b^2}+\frac {(a \sin (e+f x))^{9/2}}{5 a b f \sqrt {b \sec (e+f x)}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {5}{6} a^2 \left (\frac {1}{2} a^2 \sqrt {b \cos (e+f x)} \sqrt {b \sec (e+f x)} \int \frac {1}{\sqrt {b \cos (e+f x)} \sqrt {a \sin (e+f x)}}dx-\frac {a b \sqrt {a \sin (e+f x)}}{f \sqrt {b \sec (e+f x)}}\right )-\frac {a b (a \sin (e+f x))^{5/2}}{3 f \sqrt {b \sec (e+f x)}}}{10 b^2}+\frac {(a \sin (e+f x))^{9/2}}{5 a b f \sqrt {b \sec (e+f x)}}\) |
| \(\Big \downarrow \) 3053 |
| \(\displaystyle \frac {\frac {5}{6} a^2 \left (\frac {a^2 \sqrt {\sin (2 e+2 f x)} \sqrt {b \sec (e+f x)} \int \frac {1}{\sqrt {\sin (2 e+2 f x)}}dx}{2 \sqrt {a \sin (e+f x)}}-\frac {a b \sqrt {a \sin (e+f x)}}{f \sqrt {b \sec (e+f x)}}\right )-\frac {a b (a \sin (e+f x))^{5/2}}{3 f \sqrt {b \sec (e+f x)}}}{10 b^2}+\frac {(a \sin (e+f x))^{9/2}}{5 a b f \sqrt {b \sec (e+f x)}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {5}{6} a^2 \left (\frac {a^2 \sqrt {\sin (2 e+2 f x)} \sqrt {b \sec (e+f x)} \int \frac {1}{\sqrt {\sin (2 e+2 f x)}}dx}{2 \sqrt {a \sin (e+f x)}}-\frac {a b \sqrt {a \sin (e+f x)}}{f \sqrt {b \sec (e+f x)}}\right )-\frac {a b (a \sin (e+f x))^{5/2}}{3 f \sqrt {b \sec (e+f x)}}}{10 b^2}+\frac {(a \sin (e+f x))^{9/2}}{5 a b f \sqrt {b \sec (e+f x)}}\) |
| \(\Big \downarrow \) 3120 |
| \(\displaystyle \frac {\frac {5}{6} a^2 \left (\frac {a^2 \sqrt {\sin (2 e+2 f x)} \operatorname {EllipticF}\left (e+f x-\frac {\pi }{4},2\right ) \sqrt {b \sec (e+f x)}}{2 f \sqrt {a \sin (e+f x)}}-\frac {a b \sqrt {a \sin (e+f x)}}{f \sqrt {b \sec (e+f x)}}\right )-\frac {a b (a \sin (e+f x))^{5/2}}{3 f \sqrt {b \sec (e+f x)}}}{10 b^2}+\frac {(a \sin (e+f x))^{9/2}}{5 a b f \sqrt {b \sec (e+f x)}}\) |
(a*Sin[e + f*x])^(9/2)/(5*a*b*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)/(1 0*b^2)
3.5.77.3.1 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*Sin[e + f*x])^(m + 1)*((b*Sec[e + f*x])^(n + 1)/(a *b*f*(m - n))), x] - Simp[(n + 1)/(b^2*(m - n)) Int[(a*Sin[e + f*x])^m*(b *Sec[e + f*x])^(n + 2), x], x] /; FreeQ[{a, b, e, f, m}, x] && LtQ[n, -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[(-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]
Time = 2.28 (sec) , antiderivative size = 243, normalized size of antiderivative = 1.41
| method | result | size |
| default | \(\frac {\sqrt {2}\, \sqrt {a \sin \left (f x +e \right )}\, a^{3} \left (12 \left (\cos ^{4}\left (f x +e \right )\right ) \sqrt {2}-22 \left (\cos ^{2}\left (f x +e \right )\right ) \sqrt {2}+5 \sqrt {-\cot \left (f x +e \right )+\csc \left (f x +e \right )+1}\, \sqrt {\cot \left (f x +e \right )-\csc \left (f x +e \right )+1}\, \sqrt {\cot \left (f x +e \right )-\csc \left (f x +e \right )}\, F\left (\sqrt {-\cot \left (f x +e \right )+\csc \left (f x +e \right )+1}, \frac {\sqrt {2}}{2}\right ) \csc \left (f x +e \right )+5 \sqrt {-\cot \left (f x +e \right )+\csc \left (f x +e \right )+1}\, \sqrt {\cot \left (f x +e \right )-\csc \left (f x +e \right )+1}\, \sqrt {\cot \left (f x +e \right )-\csc \left (f x +e \right )}\, F\left (\sqrt {-\cot \left (f x +e \right )+\csc \left (f x +e \right )+1}, \frac {\sqrt {2}}{2}\right ) \sec \left (f x +e \right ) \csc \left (f x +e \right )+5 \sqrt {2}\right )}{120 f \sqrt {b \sec \left (f x +e \right )}\, b}\) | \(243\) |
1/120/f*2^(1/2)*(a*sin(f*x+e))^(1/2)*a^3/(b*sec(f*x+e))^(1/2)/b*(12*cos(f* x+e)^4*2^(1/2)-22*cos(f*x+e)^2*2^(1/2)+5*(-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))*csc(f*x+e)+5*(-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))*sec(f*x+e)*csc (f*x+e)+5*2^(1/2))
\[ \int \frac {(a \sin (e+f x))^{7/2}}{(b \sec (e+f x))^{3/2}} \, dx=\int { \frac {\left (a \sin \left (f x + e\right )\right )^{\frac {7}{2}}}{\left (b \sec \left (f x + e\right )\right )^{\frac {3}{2}}} \,d x } \]
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)/(b^2*sec(f*x + e)^2), x)
Timed out. \[ \int \frac {(a \sin (e+f x))^{7/2}}{(b \sec (e+f x))^{3/2}} \, dx=\text {Timed out} \]
\[ \int \frac {(a \sin (e+f x))^{7/2}}{(b \sec (e+f x))^{3/2}} \, dx=\int { \frac {\left (a \sin \left (f x + e\right )\right )^{\frac {7}{2}}}{\left (b \sec \left (f x + e\right )\right )^{\frac {3}{2}}} \,d x } \]
\[ \int \frac {(a \sin (e+f x))^{7/2}}{(b \sec (e+f x))^{3/2}} \, dx=\int { \frac {\left (a \sin \left (f x + e\right )\right )^{\frac {7}{2}}}{\left (b \sec \left (f x + e\right )\right )^{\frac {3}{2}}} \,d x } \]
Timed out. \[ \int \frac {(a \sin (e+f x))^{7/2}}{(b \sec (e+f x))^{3/2}} \, dx=\int \frac {{\left (a\,\sin \left (e+f\,x\right )\right )}^{7/2}}{{\left (\frac {b}{\cos \left (e+f\,x\right )}\right )}^{3/2}} \,d x \]