Integrand size = 23, antiderivative size = 121 \[ \int (d \sec (e+f x))^{7/2} (a+b \tan (e+f x)) \, dx=-\frac {6 a d^4 E\left (\left .\frac {1}{2} (e+f x)\right |2\right )}{5 f \sqrt {\cos (e+f x)} \sqrt {d \sec (e+f x)}}+\frac {2 b (d \sec (e+f x))^{7/2}}{7 f}+\frac {6 a d^3 \sqrt {d \sec (e+f x)} \sin (e+f x)}{5 f}+\frac {2 a d (d \sec (e+f x))^{5/2} \sin (e+f x)}{5 f} \]
2/7*b*(d*sec(f*x+e))^(7/2)/f+2/5*a*d*(d*sec(f*x+e))^(5/2)*sin(f*x+e)/f-6/5 *a*d^4*(cos(1/2*f*x+1/2*e)^2)^(1/2)/cos(1/2*f*x+1/2*e)*EllipticE(sin(1/2*f *x+1/2*e),2^(1/2))/f/cos(f*x+e)^(1/2)/(d*sec(f*x+e))^(1/2)+6/5*a*d^3*sin(f *x+e)*(d*sec(f*x+e))^(1/2)/f
Time = 1.38 (sec) , antiderivative size = 69, normalized size of antiderivative = 0.57 \[ \int (d \sec (e+f x))^{7/2} (a+b \tan (e+f x)) \, dx=\frac {(d \sec (e+f x))^{7/2} \left (40 b-168 a \cos ^{\frac {7}{2}}(e+f x) E\left (\left .\frac {1}{2} (e+f x)\right |2\right )+70 a \sin (2 (e+f x))+21 a \sin (4 (e+f x))\right )}{140 f} \]
((d*Sec[e + f*x])^(7/2)*(40*b - 168*a*Cos[e + f*x]^(7/2)*EllipticE[(e + f* x)/2, 2] + 70*a*Sin[2*(e + f*x)] + 21*a*Sin[4*(e + f*x)]))/(140*f)
Time = 0.61 (sec) , antiderivative size = 123, normalized size of antiderivative = 1.02, number of steps used = 10, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.435, Rules used = {3042, 3967, 3042, 4255, 3042, 4255, 3042, 4258, 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.
| \(\displaystyle \int (d \sec (e+f x))^{7/2} (a+b \tan (e+f x)) \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int (d \sec (e+f x))^{7/2} (a+b \tan (e+f x))dx\) |
| \(\Big \downarrow \) 3967 |
| \(\displaystyle a \int (d \sec (e+f x))^{7/2}dx+\frac {2 b (d \sec (e+f x))^{7/2}}{7 f}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle a \int \left (d \csc \left (e+f x+\frac {\pi }{2}\right )\right )^{7/2}dx+\frac {2 b (d \sec (e+f x))^{7/2}}{7 f}\) |
| \(\Big \downarrow \) 4255 |
| \(\displaystyle a \left (\frac {3}{5} d^2 \int (d \sec (e+f x))^{3/2}dx+\frac {2 d \sin (e+f x) (d \sec (e+f x))^{5/2}}{5 f}\right )+\frac {2 b (d \sec (e+f x))^{7/2}}{7 f}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle a \left (\frac {3}{5} d^2 \int \left (d \csc \left (e+f x+\frac {\pi }{2}\right )\right )^{3/2}dx+\frac {2 d \sin (e+f x) (d \sec (e+f x))^{5/2}}{5 f}\right )+\frac {2 b (d \sec (e+f x))^{7/2}}{7 f}\) |
| \(\Big \downarrow \) 4255 |
| \(\displaystyle a \left (\frac {3}{5} d^2 \left (\frac {2 d \sin (e+f x) \sqrt {d \sec (e+f x)}}{f}-d^2 \int \frac {1}{\sqrt {d \sec (e+f x)}}dx\right )+\frac {2 d \sin (e+f x) (d \sec (e+f x))^{5/2}}{5 f}\right )+\frac {2 b (d \sec (e+f x))^{7/2}}{7 f}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle a \left (\frac {3}{5} d^2 \left (\frac {2 d \sin (e+f x) \sqrt {d \sec (e+f x)}}{f}-d^2 \int \frac {1}{\sqrt {d \csc \left (e+f x+\frac {\pi }{2}\right )}}dx\right )+\frac {2 d \sin (e+f x) (d \sec (e+f x))^{5/2}}{5 f}\right )+\frac {2 b (d \sec (e+f x))^{7/2}}{7 f}\) |
| \(\Big \downarrow \) 4258 |
| \(\displaystyle a \left (\frac {3}{5} d^2 \left (\frac {2 d \sin (e+f x) \sqrt {d \sec (e+f x)}}{f}-\frac {d^2 \int \sqrt {\cos (e+f x)}dx}{\sqrt {\cos (e+f x)} \sqrt {d \sec (e+f x)}}\right )+\frac {2 d \sin (e+f x) (d \sec (e+f x))^{5/2}}{5 f}\right )+\frac {2 b (d \sec (e+f x))^{7/2}}{7 f}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle a \left (\frac {3}{5} d^2 \left (\frac {2 d \sin (e+f x) \sqrt {d \sec (e+f x)}}{f}-\frac {d^2 \int \sqrt {\sin \left (e+f x+\frac {\pi }{2}\right )}dx}{\sqrt {\cos (e+f x)} \sqrt {d \sec (e+f x)}}\right )+\frac {2 d \sin (e+f x) (d \sec (e+f x))^{5/2}}{5 f}\right )+\frac {2 b (d \sec (e+f x))^{7/2}}{7 f}\) |
| \(\Big \downarrow \) 3119 |
| \(\displaystyle a \left (\frac {3}{5} d^2 \left (\frac {2 d \sin (e+f x) \sqrt {d \sec (e+f x)}}{f}-\frac {2 d^2 E\left (\left .\frac {1}{2} (e+f x)\right |2\right )}{f \sqrt {\cos (e+f x)} \sqrt {d \sec (e+f x)}}\right )+\frac {2 d \sin (e+f x) (d \sec (e+f x))^{5/2}}{5 f}\right )+\frac {2 b (d \sec (e+f x))^{7/2}}{7 f}\) |
(2*b*(d*Sec[e + f*x])^(7/2))/(7*f) + a*((2*d*(d*Sec[e + f*x])^(5/2)*Sin[e + f*x])/(5*f) + (3*d^2*((-2*d^2*EllipticE[(e + f*x)/2, 2])/(f*Sqrt[Cos[e + f*x]]*Sqrt[d*Sec[e + f*x]]) + (2*d*Sqrt[d*Sec[e + f*x]]*Sin[e + f*x])/f)) /5)
3.6.78.3.1 Defintions of rubi rules used
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[((d_.)*sec[(e_.) + (f_.)*(x_)])^(m_.)*((a_) + (b_.)*tan[(e_.) + (f_.)*( x_)]), x_Symbol] :> Simp[b*((d*Sec[e + f*x])^m/(f*m)), x] + Simp[a Int[(d *Sec[e + f*x])^m, x], x] /; FreeQ[{a, b, d, e, f, m}, x] && (IntegerQ[2*m] || NeQ[a^2 + b^2, 0])
Int[(csc[(c_.) + (d_.)*(x_)]*(b_.))^(n_), x_Symbol] :> Simp[(-b)*Cos[c + d* x]*((b*Csc[c + d*x])^(n - 1)/(d*(n - 1))), x] + Simp[b^2*((n - 2)/(n - 1)) Int[(b*Csc[c + d*x])^(n - 2), x], x] /; FreeQ[{b, c, d}, x] && GtQ[n, 1] && IntegerQ[2*n]
Int[(csc[(c_.) + (d_.)*(x_)]*(b_.))^(n_), x_Symbol] :> Simp[(b*Csc[c + d*x] )^n*Sin[c + d*x]^n Int[1/Sin[c + d*x]^n, x], x] /; FreeQ[{b, c, d}, x] && EqQ[n^2, 1/4]
Result contains complex when optimal does not.
Time = 26.10 (sec) , antiderivative size = 436, normalized size of antiderivative = 3.60
| method | result | size |
| default | \(-\frac {2 a \sqrt {d \sec \left (f x +e \right )}\, d^{3} \left (3 i E\left (i \left (\csc \left (f x +e \right )-\cot \left (f x +e \right )\right ), i\right ) \sqrt {\frac {1}{\cos \left (f x +e \right )+1}}\, \sqrt {\frac {\cos \left (f x +e \right )}{\cos \left (f x +e \right )+1}}\, \left (\cos ^{2}\left (f x +e \right )\right )-3 i F\left (i \left (\csc \left (f x +e \right )-\cot \left (f x +e \right )\right ), i\right ) \sqrt {\frac {1}{\cos \left (f x +e \right )+1}}\, \sqrt {\frac {\cos \left (f x +e \right )}{\cos \left (f x +e \right )+1}}\, \left (\cos ^{2}\left (f x +e \right )\right )+6 i \cos \left (f x +e \right ) E\left (i \left (\csc \left (f x +e \right )-\cot \left (f x +e \right )\right ), i\right ) \sqrt {\frac {1}{\cos \left (f x +e \right )+1}}\, \sqrt {\frac {\cos \left (f x +e \right )}{\cos \left (f x +e \right )+1}}-6 i \cos \left (f x +e \right ) F\left (i \left (\csc \left (f x +e \right )-\cot \left (f x +e \right )\right ), i\right ) \sqrt {\frac {1}{\cos \left (f x +e \right )+1}}\, \sqrt {\frac {\cos \left (f x +e \right )}{\cos \left (f x +e \right )+1}}+3 i \sqrt {\frac {1}{\cos \left (f x +e \right )+1}}\, E\left (i \left (\csc \left (f x +e \right )-\cot \left (f x +e \right )\right ), i\right ) \sqrt {\frac {\cos \left (f x +e \right )}{\cos \left (f x +e \right )+1}}-3 i \sqrt {\frac {\cos \left (f x +e \right )}{\cos \left (f x +e \right )+1}}\, F\left (i \left (\csc \left (f x +e \right )-\cot \left (f x +e \right )\right ), i\right ) \sqrt {\frac {1}{\cos \left (f x +e \right )+1}}-3 \sin \left (f x +e \right )-\tan \left (f x +e \right )-\sec \left (f x +e \right ) \tan \left (f x +e \right )\right )}{5 f \left (\cos \left (f x +e \right )+1\right )}+\frac {2 b \left (d \sec \left (f x +e \right )\right )^{\frac {7}{2}}}{7 f}\) | \(436\) |
| parts | \(-\frac {2 a \sqrt {d \sec \left (f x +e \right )}\, d^{3} \left (3 i E\left (i \left (\csc \left (f x +e \right )-\cot \left (f x +e \right )\right ), i\right ) \sqrt {\frac {1}{\cos \left (f x +e \right )+1}}\, \sqrt {\frac {\cos \left (f x +e \right )}{\cos \left (f x +e \right )+1}}\, \left (\cos ^{2}\left (f x +e \right )\right )-3 i F\left (i \left (\csc \left (f x +e \right )-\cot \left (f x +e \right )\right ), i\right ) \sqrt {\frac {1}{\cos \left (f x +e \right )+1}}\, \sqrt {\frac {\cos \left (f x +e \right )}{\cos \left (f x +e \right )+1}}\, \left (\cos ^{2}\left (f x +e \right )\right )+6 i \cos \left (f x +e \right ) E\left (i \left (\csc \left (f x +e \right )-\cot \left (f x +e \right )\right ), i\right ) \sqrt {\frac {1}{\cos \left (f x +e \right )+1}}\, \sqrt {\frac {\cos \left (f x +e \right )}{\cos \left (f x +e \right )+1}}-6 i \cos \left (f x +e \right ) F\left (i \left (\csc \left (f x +e \right )-\cot \left (f x +e \right )\right ), i\right ) \sqrt {\frac {1}{\cos \left (f x +e \right )+1}}\, \sqrt {\frac {\cos \left (f x +e \right )}{\cos \left (f x +e \right )+1}}+3 i \sqrt {\frac {1}{\cos \left (f x +e \right )+1}}\, E\left (i \left (\csc \left (f x +e \right )-\cot \left (f x +e \right )\right ), i\right ) \sqrt {\frac {\cos \left (f x +e \right )}{\cos \left (f x +e \right )+1}}-3 i \sqrt {\frac {\cos \left (f x +e \right )}{\cos \left (f x +e \right )+1}}\, F\left (i \left (\csc \left (f x +e \right )-\cot \left (f x +e \right )\right ), i\right ) \sqrt {\frac {1}{\cos \left (f x +e \right )+1}}-3 \sin \left (f x +e \right )-\tan \left (f x +e \right )-\sec \left (f x +e \right ) \tan \left (f x +e \right )\right )}{5 f \left (\cos \left (f x +e \right )+1\right )}+\frac {2 b \left (d \sec \left (f x +e \right )\right )^{\frac {7}{2}}}{7 f}\) | \(436\) |
-2/5*a/f*(d*sec(f*x+e))^(1/2)*d^3/(cos(f*x+e)+1)*(3*I*EllipticE(I*(csc(f*x +e)-cot(f*x+e)),I)*(1/(cos(f*x+e)+1))^(1/2)*(cos(f*x+e)/(cos(f*x+e)+1))^(1 /2)*cos(f*x+e)^2-3*I*EllipticF(I*(csc(f*x+e)-cot(f*x+e)),I)*(1/(cos(f*x+e) +1))^(1/2)*(cos(f*x+e)/(cos(f*x+e)+1))^(1/2)*cos(f*x+e)^2+6*I*cos(f*x+e)*E llipticE(I*(csc(f*x+e)-cot(f*x+e)),I)*(1/(cos(f*x+e)+1))^(1/2)*(cos(f*x+e) /(cos(f*x+e)+1))^(1/2)-6*I*cos(f*x+e)*EllipticF(I*(csc(f*x+e)-cot(f*x+e)), I)*(1/(cos(f*x+e)+1))^(1/2)*(cos(f*x+e)/(cos(f*x+e)+1))^(1/2)+3*I*(1/(cos( f*x+e)+1))^(1/2)*EllipticE(I*(csc(f*x+e)-cot(f*x+e)),I)*(cos(f*x+e)/(cos(f *x+e)+1))^(1/2)-3*I*(cos(f*x+e)/(cos(f*x+e)+1))^(1/2)*EllipticF(I*(csc(f*x +e)-cot(f*x+e)),I)*(1/(cos(f*x+e)+1))^(1/2)-3*sin(f*x+e)-tan(f*x+e)-sec(f* x+e)*tan(f*x+e))+2/7*b*(d*sec(f*x+e))^(7/2)/f
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.10 (sec) , antiderivative size = 145, normalized size of antiderivative = 1.20 \[ \int (d \sec (e+f x))^{7/2} (a+b \tan (e+f x)) \, dx=\frac {-21 i \, \sqrt {2} a d^{\frac {7}{2}} \cos \left (f x + e\right )^{3} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (f x + e\right ) + i \, \sin \left (f x + e\right )\right )\right ) + 21 i \, \sqrt {2} a d^{\frac {7}{2}} \cos \left (f x + e\right )^{3} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (f x + e\right ) - i \, \sin \left (f x + e\right )\right )\right ) + 2 \, {\left (5 \, b d^{3} + 7 \, {\left (3 \, a d^{3} \cos \left (f x + e\right )^{3} + a d^{3} \cos \left (f x + e\right )\right )} \sin \left (f x + e\right )\right )} \sqrt {\frac {d}{\cos \left (f x + e\right )}}}{35 \, f \cos \left (f x + e\right )^{3}} \]
1/35*(-21*I*sqrt(2)*a*d^(7/2)*cos(f*x + e)^3*weierstrassZeta(-4, 0, weiers trassPInverse(-4, 0, cos(f*x + e) + I*sin(f*x + e))) + 21*I*sqrt(2)*a*d^(7 /2)*cos(f*x + e)^3*weierstrassZeta(-4, 0, weierstrassPInverse(-4, 0, cos(f *x + e) - I*sin(f*x + e))) + 2*(5*b*d^3 + 7*(3*a*d^3*cos(f*x + e)^3 + a*d^ 3*cos(f*x + e))*sin(f*x + e))*sqrt(d/cos(f*x + e)))/(f*cos(f*x + e)^3)
Timed out. \[ \int (d \sec (e+f x))^{7/2} (a+b \tan (e+f x)) \, dx=\text {Timed out} \]
\[ \int (d \sec (e+f x))^{7/2} (a+b \tan (e+f x)) \, dx=\int { \left (d \sec \left (f x + e\right )\right )^{\frac {7}{2}} {\left (b \tan \left (f x + e\right ) + a\right )} \,d x } \]
\[ \int (d \sec (e+f x))^{7/2} (a+b \tan (e+f x)) \, dx=\int { \left (d \sec \left (f x + e\right )\right )^{\frac {7}{2}} {\left (b \tan \left (f x + e\right ) + a\right )} \,d x } \]
Timed out. \[ \int (d \sec (e+f x))^{7/2} (a+b \tan (e+f x)) \, dx=\int {\left (\frac {d}{\cos \left (e+f\,x\right )}\right )}^{7/2}\,\left (a+b\,\mathrm {tan}\left (e+f\,x\right )\right ) \,d x \]