Integrand size = 25, antiderivative size = 93 \[ \int (d \sec (e+f x))^{3/2} \sqrt {b \tan (e+f x)} \, dx=-\frac {d^2 E\left (\left .\frac {1}{2} \left (e-\frac {\pi }{2}+f x\right )\right |2\right ) \sqrt {b \tan (e+f x)}}{f \sqrt {d \sec (e+f x)} \sqrt {\sin (e+f x)}}+\frac {d^2 (b \tan (e+f x))^{3/2}}{b f \sqrt {d \sec (e+f x)}} \]
d^2*(sin(1/2*e+1/4*Pi+1/2*f*x)^2)^(1/2)/sin(1/2*e+1/4*Pi+1/2*f*x)*Elliptic E(cos(1/2*e+1/4*Pi+1/2*f*x),2^(1/2))*(b*tan(f*x+e))^(1/2)/f/(d*sec(f*x+e)) ^(1/2)/sin(f*x+e)^(1/2)+d^2*(b*tan(f*x+e))^(3/2)/b/f/(d*sec(f*x+e))^(1/2)
Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
Time = 10.55 (sec) , antiderivative size = 73, normalized size of antiderivative = 0.78 \[ \int (d \sec (e+f x))^{3/2} \sqrt {b \tan (e+f x)} \, dx=-\frac {d \sqrt {d \sec (e+f x)} \left (-3+\operatorname {Hypergeometric2F1}\left (\frac {3}{4},\frac {5}{4},\frac {7}{4},-\tan ^2(e+f x)\right ) \sqrt [4]{\sec ^2(e+f x)}\right ) \sin (e+f x) \sqrt {b \tan (e+f x)}}{3 f} \]
-1/3*(d*Sqrt[d*Sec[e + f*x]]*(-3 + Hypergeometric2F1[3/4, 5/4, 7/4, -Tan[e + f*x]^2]*(Sec[e + f*x]^2)^(1/4))*Sin[e + f*x]*Sqrt[b*Tan[e + f*x]])/f
Time = 0.53 (sec) , antiderivative size = 93, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.320, Rules used = {3042, 3093, 3042, 3096, 3042, 3121, 3042, 3119}
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.
| \(\displaystyle \int \sqrt {b \tan (e+f x)} (d \sec (e+f x))^{3/2} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \sqrt {b \tan (e+f x)} (d \sec (e+f x))^{3/2}dx\) |
| \(\Big \downarrow \) 3093 |
| \(\displaystyle \frac {d^2 (b \tan (e+f x))^{3/2}}{b f \sqrt {d \sec (e+f x)}}-\frac {1}{2} d^2 \int \frac {\sqrt {b \tan (e+f x)}}{\sqrt {d \sec (e+f x)}}dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {d^2 (b \tan (e+f x))^{3/2}}{b f \sqrt {d \sec (e+f x)}}-\frac {1}{2} d^2 \int \frac {\sqrt {b \tan (e+f x)}}{\sqrt {d \sec (e+f x)}}dx\) |
| \(\Big \downarrow \) 3096 |
| \(\displaystyle \frac {d^2 (b \tan (e+f x))^{3/2}}{b f \sqrt {d \sec (e+f x)}}-\frac {d^2 \sqrt {b \tan (e+f x)} \int \sqrt {b \sin (e+f x)}dx}{2 \sqrt {b \sin (e+f x)} \sqrt {d \sec (e+f x)}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {d^2 (b \tan (e+f x))^{3/2}}{b f \sqrt {d \sec (e+f x)}}-\frac {d^2 \sqrt {b \tan (e+f x)} \int \sqrt {b \sin (e+f x)}dx}{2 \sqrt {b \sin (e+f x)} \sqrt {d \sec (e+f x)}}\) |
| \(\Big \downarrow \) 3121 |
| \(\displaystyle \frac {d^2 (b \tan (e+f x))^{3/2}}{b f \sqrt {d \sec (e+f x)}}-\frac {d^2 \sqrt {b \tan (e+f x)} \int \sqrt {\sin (e+f x)}dx}{2 \sqrt {\sin (e+f x)} \sqrt {d \sec (e+f x)}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {d^2 (b \tan (e+f x))^{3/2}}{b f \sqrt {d \sec (e+f x)}}-\frac {d^2 \sqrt {b \tan (e+f x)} \int \sqrt {\sin (e+f x)}dx}{2 \sqrt {\sin (e+f x)} \sqrt {d \sec (e+f x)}}\) |
| \(\Big \downarrow \) 3119 |
| \(\displaystyle \frac {d^2 (b \tan (e+f x))^{3/2}}{b f \sqrt {d \sec (e+f x)}}-\frac {d^2 E\left (\left .\frac {1}{2} \left (e+f x-\frac {\pi }{2}\right )\right |2\right ) \sqrt {b \tan (e+f x)}}{f \sqrt {\sin (e+f x)} \sqrt {d \sec (e+f x)}}\) |
-((d^2*EllipticE[(e - Pi/2 + f*x)/2, 2]*Sqrt[b*Tan[e + f*x]])/(f*Sqrt[d*Se c[e + f*x]]*Sqrt[Sin[e + f*x]])) + (d^2*(b*Tan[e + f*x])^(3/2))/(b*f*Sqrt[ d*Sec[e + f*x]])
3.3.92.3.1 Defintions of rubi rules used
Int[((a_.)*sec[(e_.) + (f_.)*(x_)])^(m_.)*((b_.)*tan[(e_.) + (f_.)*(x_)])^( n_), x_Symbol] :> Simp[a^2*(a*Sec[e + f*x])^(m - 2)*((b*Tan[e + f*x])^(n + 1)/(b*f*(m + n - 1))), x] + Simp[a^2*((m - 2)/(m + n - 1)) Int[(a*Sec[e + f*x])^(m - 2)*(b*Tan[e + f*x])^n, x], x] /; FreeQ[{a, b, e, f, n}, x] && ( GtQ[m, 1] || (EqQ[m, 1] && EqQ[n, 1/2])) && NeQ[m + n - 1, 0] && IntegersQ[ 2*m, 2*n]
Int[((a_.)*sec[(e_.) + (f_.)*(x_)])^(m_)*((b_.)*tan[(e_.) + (f_.)*(x_)])^(n _), x_Symbol] :> Simp[a^(m + n)*((b*Tan[e + f*x])^n/((a*Sec[e + f*x])^n*(b* Sin[e + f*x])^n)) Int[(b*Sin[e + f*x])^n/Cos[e + f*x]^(m + n), x], x] /; FreeQ[{a, b, e, f, m, n}, x] && IntegerQ[n + 1/2] && IntegerQ[m + 1/2]
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[((b_)*sin[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[(b*Sin[c + d*x]) ^n/Sin[c + d*x]^n Int[Sin[c + d*x]^n, x], x] /; FreeQ[{b, c, d}, x] && Lt Q[-1, n, 1] && IntegerQ[2*n]
Result contains complex when optimal does not.
Time = 2.48 (sec) , antiderivative size = 454, normalized size of antiderivative = 4.88
| method | result | size |
| default | \(-\frac {\csc \left (f x +e \right ) \left (\sqrt {-i \left (i-\cot \left (f x +e \right )+\csc \left (f x +e \right )\right )}\, \sqrt {i \left (-i-\cot \left (f x +e \right )+\csc \left (f x +e \right )\right )}\, \sqrt {-i \left (\cot \left (f x +e \right )-\csc \left (f x +e \right )\right )}\, F\left (\sqrt {-i \left (i-\cot \left (f x +e \right )+\csc \left (f x +e \right )\right )}, \frac {\sqrt {2}}{2}\right ) \left (\cos ^{2}\left (f x +e \right )\right )-2 \sqrt {-i \left (i-\cot \left (f x +e \right )+\csc \left (f x +e \right )\right )}\, \sqrt {i \left (-i-\cot \left (f x +e \right )+\csc \left (f x +e \right )\right )}\, \sqrt {-i \left (\cot \left (f x +e \right )-\csc \left (f x +e \right )\right )}\, E\left (\sqrt {-i \left (i-\cot \left (f x +e \right )+\csc \left (f x +e \right )\right )}, \frac {\sqrt {2}}{2}\right ) \left (\cos ^{2}\left (f x +e \right )\right )+\sqrt {-i \left (i-\cot \left (f x +e \right )+\csc \left (f x +e \right )\right )}\, \sqrt {i \left (-i-\cot \left (f x +e \right )+\csc \left (f x +e \right )\right )}\, \sqrt {-i \left (\cot \left (f x +e \right )-\csc \left (f x +e \right )\right )}\, F\left (\sqrt {-i \left (i-\cot \left (f x +e \right )+\csc \left (f x +e \right )\right )}, \frac {\sqrt {2}}{2}\right ) \cos \left (f x +e \right )-2 \sqrt {-i \left (i-\cot \left (f x +e \right )+\csc \left (f x +e \right )\right )}\, \sqrt {i \left (-i-\cot \left (f x +e \right )+\csc \left (f x +e \right )\right )}\, \sqrt {-i \left (\cot \left (f x +e \right )-\csc \left (f x +e \right )\right )}\, E\left (\sqrt {-i \left (i-\cot \left (f x +e \right )+\csc \left (f x +e \right )\right )}, \frac {\sqrt {2}}{2}\right ) \cos \left (f x +e \right )+\sqrt {2}\, \cos \left (f x +e \right )-\sqrt {2}\right ) \sqrt {b \tan \left (f x +e \right )}\, \sqrt {d \sec \left (f x +e \right )}\, d \sqrt {2}}{2 f}\) | \(454\) |
-1/2/f*csc(f*x+e)*((-I*(I-cot(f*x+e)+csc(f*x+e)))^(1/2)*(I*(-I-cot(f*x+e)+ csc(f*x+e)))^(1/2)*(-I*(cot(f*x+e)-csc(f*x+e)))^(1/2)*EllipticF((-I*(I-cot (f*x+e)+csc(f*x+e)))^(1/2),1/2*2^(1/2))*cos(f*x+e)^2-2*(-I*(I-cot(f*x+e)+c sc(f*x+e)))^(1/2)*(I*(-I-cot(f*x+e)+csc(f*x+e)))^(1/2)*(-I*(cot(f*x+e)-csc (f*x+e)))^(1/2)*EllipticE((-I*(I-cot(f*x+e)+csc(f*x+e)))^(1/2),1/2*2^(1/2) )*cos(f*x+e)^2+(-I*(I-cot(f*x+e)+csc(f*x+e)))^(1/2)*(I*(-I-cot(f*x+e)+csc( f*x+e)))^(1/2)*(-I*(cot(f*x+e)-csc(f*x+e)))^(1/2)*EllipticF((-I*(I-cot(f*x +e)+csc(f*x+e)))^(1/2),1/2*2^(1/2))*cos(f*x+e)-2*(-I*(I-cot(f*x+e)+csc(f*x +e)))^(1/2)*(I*(-I-cot(f*x+e)+csc(f*x+e)))^(1/2)*(-I*(cot(f*x+e)-csc(f*x+e )))^(1/2)*EllipticE((-I*(I-cot(f*x+e)+csc(f*x+e)))^(1/2),1/2*2^(1/2))*cos( f*x+e)+2^(1/2)*cos(f*x+e)-2^(1/2))*(b*tan(f*x+e))^(1/2)*(d*sec(f*x+e))^(1/ 2)*d*2^(1/2)
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.10 (sec) , antiderivative size = 105, normalized size of antiderivative = 1.13 \[ \int (d \sec (e+f x))^{3/2} \sqrt {b \tan (e+f x)} \, dx=\frac {2 \, d \sqrt {\frac {b \sin \left (f x + e\right )}{\cos \left (f x + e\right )}} \sqrt {\frac {d}{\cos \left (f x + e\right )}} \sin \left (f x + e\right ) - i \, \sqrt {-2 i \, b d} d {\rm weierstrassZeta}\left (4, 0, {\rm weierstrassPInverse}\left (4, 0, \cos \left (f x + e\right ) + i \, \sin \left (f x + e\right )\right )\right ) + i \, \sqrt {2 i \, b d} d {\rm weierstrassZeta}\left (4, 0, {\rm weierstrassPInverse}\left (4, 0, \cos \left (f x + e\right ) - i \, \sin \left (f x + e\right )\right )\right )}{2 \, f} \]
1/2*(2*d*sqrt(b*sin(f*x + e)/cos(f*x + e))*sqrt(d/cos(f*x + e))*sin(f*x + e) - I*sqrt(-2*I*b*d)*d*weierstrassZeta(4, 0, weierstrassPInverse(4, 0, co s(f*x + e) + I*sin(f*x + e))) + I*sqrt(2*I*b*d)*d*weierstrassZeta(4, 0, we ierstrassPInverse(4, 0, cos(f*x + e) - I*sin(f*x + e))))/f
\[ \int (d \sec (e+f x))^{3/2} \sqrt {b \tan (e+f x)} \, dx=\int \sqrt {b \tan {\left (e + f x \right )}} \left (d \sec {\left (e + f x \right )}\right )^{\frac {3}{2}}\, dx \]
\[ \int (d \sec (e+f x))^{3/2} \sqrt {b \tan (e+f x)} \, dx=\int { \left (d \sec \left (f x + e\right )\right )^{\frac {3}{2}} \sqrt {b \tan \left (f x + e\right )} \,d x } \]
\[ \int (d \sec (e+f x))^{3/2} \sqrt {b \tan (e+f x)} \, dx=\int { \left (d \sec \left (f x + e\right )\right )^{\frac {3}{2}} \sqrt {b \tan \left (f x + e\right )} \,d x } \]
Timed out. \[ \int (d \sec (e+f x))^{3/2} \sqrt {b \tan (e+f x)} \, dx=\int \sqrt {b\,\mathrm {tan}\left (e+f\,x\right )}\,{\left (\frac {d}{\cos \left (e+f\,x\right )}\right )}^{3/2} \,d x \]