Integrand size = 25, antiderivative size = 96 \[ \int \frac {(b \tan (e+f x))^{3/2}}{(d \sec (e+f x))^{3/2}} \, dx=\frac {2 b^2 \operatorname {EllipticF}\left (\frac {1}{2} \left (e-\frac {\pi }{2}+f x\right ),2\right ) \sqrt {d \sec (e+f x)} \sqrt {\sin (e+f x)}}{3 d^2 f \sqrt {b \tan (e+f x)}}-\frac {2 b \sqrt {b \tan (e+f x)}}{3 f (d \sec (e+f x))^{3/2}} \]
-2/3*b^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)*Ell ipticF(cos(1/2*e+1/4*Pi+1/2*f*x),2^(1/2))*(d*sec(f*x+e))^(1/2)*sin(f*x+e)^ (1/2)/d^2/f/(b*tan(f*x+e))^(1/2)-2/3*b*(b*tan(f*x+e))^(1/2)/f/(d*sec(f*x+e ))^(3/2)
Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
Time = 0.64 (sec) , antiderivative size = 67, normalized size of antiderivative = 0.70 \[ \int \frac {(b \tan (e+f x))^{3/2}}{(d \sec (e+f x))^{3/2}} \, dx=\frac {2 b \left (-1+\operatorname {Hypergeometric2F1}\left (\frac {1}{4},\frac {3}{4},\frac {5}{4},-\tan ^2(e+f x)\right ) \sec ^2(e+f x)^{3/4}\right ) \sqrt {b \tan (e+f x)}}{3 f (d \sec (e+f x))^{3/2}} \]
(2*b*(-1 + Hypergeometric2F1[1/4, 3/4, 5/4, -Tan[e + f*x]^2]*(Sec[e + f*x] ^2)^(3/4))*Sqrt[b*Tan[e + f*x]])/(3*f*(d*Sec[e + f*x])^(3/2))
Time = 0.53 (sec) , antiderivative size = 96, 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, 3090, 3042, 3096, 3042, 3121, 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.
\(\displaystyle \int \frac {(b \tan (e+f x))^{3/2}}{(d \sec (e+f x))^{3/2}} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {(b \tan (e+f x))^{3/2}}{(d \sec (e+f x))^{3/2}}dx\) |
\(\Big \downarrow \) 3090 |
\(\displaystyle \frac {b^2 \int \frac {\sqrt {d \sec (e+f x)}}{\sqrt {b \tan (e+f x)}}dx}{3 d^2}-\frac {2 b \sqrt {b \tan (e+f x)}}{3 f (d \sec (e+f x))^{3/2}}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {b^2 \int \frac {\sqrt {d \sec (e+f x)}}{\sqrt {b \tan (e+f x)}}dx}{3 d^2}-\frac {2 b \sqrt {b \tan (e+f x)}}{3 f (d \sec (e+f x))^{3/2}}\) |
\(\Big \downarrow \) 3096 |
\(\displaystyle \frac {b^2 \sqrt {b \sin (e+f x)} \sqrt {d \sec (e+f x)} \int \frac {1}{\sqrt {b \sin (e+f x)}}dx}{3 d^2 \sqrt {b \tan (e+f x)}}-\frac {2 b \sqrt {b \tan (e+f x)}}{3 f (d \sec (e+f x))^{3/2}}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {b^2 \sqrt {b \sin (e+f x)} \sqrt {d \sec (e+f x)} \int \frac {1}{\sqrt {b \sin (e+f x)}}dx}{3 d^2 \sqrt {b \tan (e+f x)}}-\frac {2 b \sqrt {b \tan (e+f x)}}{3 f (d \sec (e+f x))^{3/2}}\) |
\(\Big \downarrow \) 3121 |
\(\displaystyle \frac {b^2 \sqrt {\sin (e+f x)} \sqrt {d \sec (e+f x)} \int \frac {1}{\sqrt {\sin (e+f x)}}dx}{3 d^2 \sqrt {b \tan (e+f x)}}-\frac {2 b \sqrt {b \tan (e+f x)}}{3 f (d \sec (e+f x))^{3/2}}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {b^2 \sqrt {\sin (e+f x)} \sqrt {d \sec (e+f x)} \int \frac {1}{\sqrt {\sin (e+f x)}}dx}{3 d^2 \sqrt {b \tan (e+f x)}}-\frac {2 b \sqrt {b \tan (e+f x)}}{3 f (d \sec (e+f x))^{3/2}}\) |
\(\Big \downarrow \) 3120 |
\(\displaystyle \frac {2 b^2 \sqrt {\sin (e+f x)} \operatorname {EllipticF}\left (\frac {1}{2} \left (e+f x-\frac {\pi }{2}\right ),2\right ) \sqrt {d \sec (e+f x)}}{3 d^2 f \sqrt {b \tan (e+f x)}}-\frac {2 b \sqrt {b \tan (e+f x)}}{3 f (d \sec (e+f x))^{3/2}}\) |
(2*b^2*EllipticF[(e - Pi/2 + f*x)/2, 2]*Sqrt[d*Sec[e + f*x]]*Sqrt[Sin[e + f*x]])/(3*d^2*f*Sqrt[b*Tan[e + f*x]]) - (2*b*Sqrt[b*Tan[e + f*x]])/(3*f*(d *Sec[e + f*x])^(3/2))
3.4.3.3.1 Defintions of rubi rules used
Int[((a_.)*sec[(e_.) + (f_.)*(x_)])^(m_)*((b_.)*tan[(e_.) + (f_.)*(x_)])^(n _), x_Symbol] :> Simp[b*(a*Sec[e + f*x])^m*((b*Tan[e + f*x])^(n - 1)/(f*m)) , x] - Simp[b^2*((n - 1)/(a^2*m)) Int[(a*Sec[e + f*x])^(m + 2)*(b*Tan[e + f*x])^(n - 2), x], x] /; FreeQ[{a, b, e, f}, x] && GtQ[n, 1] && (LtQ[m, -1 ] || (EqQ[m, -1] && EqQ[n, 3/2])) && 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[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]
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 = 2.55 (sec) , antiderivative size = 272, normalized size of antiderivative = 2.83
method | result | size |
default | \(\frac {\sin \left (f x +e \right ) \left (-i \sqrt {-i \left (i-\cot \left (f x +e \right )+\csc \left (f x +e \right )\right )}\, \sqrt {-i \left (\cot \left (f x +e \right )-\csc \left (f x +e \right )+i\right )}\, \sqrt {-i \left (\cot \left (f x +e \right )-\csc \left (f x +e \right )\right )}\, F\left (\sqrt {-i \left (i-\cot \left (f x +e \right )+\csc \left (f x +e \right )\right )}, \frac {\sqrt {2}}{2}\right ) \cos \left (f x +e \right )-i \sqrt {-i \left (i-\cot \left (f x +e \right )+\csc \left (f x +e \right )\right )}\, \sqrt {-i \left (\cot \left (f x +e \right )-\csc \left (f x +e \right )+i\right )}\, \sqrt {-i \left (\cot \left (f x +e \right )-\csc \left (f x +e \right )\right )}\, F\left (\sqrt {-i \left (i-\cot \left (f x +e \right )+\csc \left (f x +e \right )\right )}, \frac {\sqrt {2}}{2}\right )+\sqrt {2}\, \cos \left (f x +e \right ) \sin \left (f x +e \right )\right ) b \sqrt {b \tan \left (f x +e \right )}\, \sqrt {2}}{3 f \left (\cos \left (f x +e \right )-1\right ) \sqrt {d \sec \left (f x +e \right )}\, d \left (\cos \left (f x +e \right )+1\right )}\) | \(272\) |
1/3/f*sin(f*x+e)*(-I*(-I*(I-cot(f*x+e)+csc(f*x+e)))^(1/2)*(-I*(cot(f*x+e)- csc(f*x+e)+I))^(1/2)*(-I*(cot(f*x+e)-csc(f*x+e)))^(1/2)*EllipticF((-I*(I-c ot(f*x+e)+csc(f*x+e)))^(1/2),1/2*2^(1/2))*cos(f*x+e)-I*(-I*(I-cot(f*x+e)+c sc(f*x+e)))^(1/2)*(-I*(cot(f*x+e)-csc(f*x+e)+I))^(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) )+2^(1/2)*cos(f*x+e)*sin(f*x+e))*b*(b*tan(f*x+e))^(1/2)/(cos(f*x+e)-1)/(d* sec(f*x+e))^(1/2)/d/(cos(f*x+e)+1)*2^(1/2)
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.09 (sec) , antiderivative size = 104, normalized size of antiderivative = 1.08 \[ \int \frac {(b \tan (e+f x))^{3/2}}{(d \sec (e+f x))^{3/2}} \, dx=-\frac {2 \, b \sqrt {\frac {b \sin \left (f x + e\right )}{\cos \left (f x + e\right )}} \sqrt {\frac {d}{\cos \left (f x + e\right )}} \cos \left (f x + e\right )^{2} - \sqrt {-2 i \, b d} b {\rm weierstrassPInverse}\left (4, 0, \cos \left (f x + e\right ) + i \, \sin \left (f x + e\right )\right ) - \sqrt {2 i \, b d} b {\rm weierstrassPInverse}\left (4, 0, \cos \left (f x + e\right ) - i \, \sin \left (f x + e\right )\right )}{3 \, d^{2} f} \]
-1/3*(2*b*sqrt(b*sin(f*x + e)/cos(f*x + e))*sqrt(d/cos(f*x + e))*cos(f*x + e)^2 - sqrt(-2*I*b*d)*b*weierstrassPInverse(4, 0, cos(f*x + e) + I*sin(f* x + e)) - sqrt(2*I*b*d)*b*weierstrassPInverse(4, 0, cos(f*x + e) - I*sin(f *x + e)))/(d^2*f)
\[ \int \frac {(b \tan (e+f x))^{3/2}}{(d \sec (e+f x))^{3/2}} \, dx=\int \frac {\left (b \tan {\left (e + f x \right )}\right )^{\frac {3}{2}}}{\left (d \sec {\left (e + f x \right )}\right )^{\frac {3}{2}}}\, dx \]
\[ \int \frac {(b \tan (e+f x))^{3/2}}{(d \sec (e+f x))^{3/2}} \, dx=\int { \frac {\left (b \tan \left (f x + e\right )\right )^{\frac {3}{2}}}{\left (d \sec \left (f x + e\right )\right )^{\frac {3}{2}}} \,d x } \]
\[ \int \frac {(b \tan (e+f x))^{3/2}}{(d \sec (e+f x))^{3/2}} \, dx=\int { \frac {\left (b \tan \left (f x + e\right )\right )^{\frac {3}{2}}}{\left (d \sec \left (f x + e\right )\right )^{\frac {3}{2}}} \,d x } \]
Timed out. \[ \int \frac {(b \tan (e+f x))^{3/2}}{(d \sec (e+f x))^{3/2}} \, dx=\int \frac {{\left (b\,\mathrm {tan}\left (e+f\,x\right )\right )}^{3/2}}{{\left (\frac {d}{\cos \left (e+f\,x\right )}\right )}^{3/2}} \,d x \]