Integrand size = 23, antiderivative size = 123 \[ \int \frac {a+b \tan (e+f x)}{(d \sec (e+f x))^{7/2}} \, dx=-\frac {2 b}{7 f (d \sec (e+f x))^{7/2}}+\frac {10 a \sqrt {\cos (e+f x)} \operatorname {EllipticF}\left (\frac {1}{2} (e+f x),2\right ) \sqrt {d \sec (e+f x)}}{21 d^4 f}+\frac {2 a \sin (e+f x)}{7 d f (d \sec (e+f x))^{5/2}}+\frac {10 a \sin (e+f x)}{21 d^3 f \sqrt {d \sec (e+f x)}} \]
-2/7*b/f/(d*sec(f*x+e))^(7/2)+2/7*a*sin(f*x+e)/d/f/(d*sec(f*x+e))^(5/2)+10 /21*a*sin(f*x+e)/d^3/f/(d*sec(f*x+e))^(1/2)+10/21*a*(cos(1/2*f*x+1/2*e)^2) ^(1/2)/cos(1/2*f*x+1/2*e)*EllipticF(sin(1/2*f*x+1/2*e),2^(1/2))*cos(f*x+e) ^(1/2)*(d*sec(f*x+e))^(1/2)/d^4/f
Time = 1.11 (sec) , antiderivative size = 94, normalized size of antiderivative = 0.76 \[ \int \frac {a+b \tan (e+f x)}{(d \sec (e+f x))^{7/2}} \, dx=\frac {\sqrt {d \sec (e+f x)} \left (-9 b-12 b \cos (2 (e+f x))-3 b \cos (4 (e+f x))+40 a \sqrt {\cos (e+f x)} \operatorname {EllipticF}\left (\frac {1}{2} (e+f x),2\right )+26 a \sin (2 (e+f x))+3 a \sin (4 (e+f x))\right )}{84 d^4 f} \]
(Sqrt[d*Sec[e + f*x]]*(-9*b - 12*b*Cos[2*(e + f*x)] - 3*b*Cos[4*(e + f*x)] + 40*a*Sqrt[Cos[e + f*x]]*EllipticF[(e + f*x)/2, 2] + 26*a*Sin[2*(e + f*x )] + 3*a*Sin[4*(e + f*x)]))/(84*d^4*f)
Time = 0.56 (sec) , antiderivative size = 131, normalized size of antiderivative = 1.07, 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, 4256, 3042, 4256, 3042, 4258, 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.
| \(\displaystyle \int \frac {a+b \tan (e+f x)}{(d \sec (e+f x))^{7/2}} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {a+b \tan (e+f x)}{(d \sec (e+f x))^{7/2}}dx\) |
| \(\Big \downarrow \) 3967 |
| \(\displaystyle a \int \frac {1}{(d \sec (e+f x))^{7/2}}dx-\frac {2 b}{7 f (d \sec (e+f x))^{7/2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle a \int \frac {1}{\left (d \csc \left (e+f x+\frac {\pi }{2}\right )\right )^{7/2}}dx-\frac {2 b}{7 f (d \sec (e+f x))^{7/2}}\) |
| \(\Big \downarrow \) 4256 |
| \(\displaystyle a \left (\frac {5 \int \frac {1}{(d \sec (e+f x))^{3/2}}dx}{7 d^2}+\frac {2 \sin (e+f x)}{7 d f (d \sec (e+f x))^{5/2}}\right )-\frac {2 b}{7 f (d \sec (e+f x))^{7/2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle a \left (\frac {5 \int \frac {1}{\left (d \csc \left (e+f x+\frac {\pi }{2}\right )\right )^{3/2}}dx}{7 d^2}+\frac {2 \sin (e+f x)}{7 d f (d \sec (e+f x))^{5/2}}\right )-\frac {2 b}{7 f (d \sec (e+f x))^{7/2}}\) |
| \(\Big \downarrow \) 4256 |
| \(\displaystyle a \left (\frac {5 \left (\frac {\int \sqrt {d \sec (e+f x)}dx}{3 d^2}+\frac {2 \sin (e+f x)}{3 d f \sqrt {d \sec (e+f x)}}\right )}{7 d^2}+\frac {2 \sin (e+f x)}{7 d f (d \sec (e+f x))^{5/2}}\right )-\frac {2 b}{7 f (d \sec (e+f x))^{7/2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle a \left (\frac {5 \left (\frac {\int \sqrt {d \csc \left (e+f x+\frac {\pi }{2}\right )}dx}{3 d^2}+\frac {2 \sin (e+f x)}{3 d f \sqrt {d \sec (e+f x)}}\right )}{7 d^2}+\frac {2 \sin (e+f x)}{7 d f (d \sec (e+f x))^{5/2}}\right )-\frac {2 b}{7 f (d \sec (e+f x))^{7/2}}\) |
| \(\Big \downarrow \) 4258 |
| \(\displaystyle a \left (\frac {5 \left (\frac {\sqrt {\cos (e+f x)} \sqrt {d \sec (e+f x)} \int \frac {1}{\sqrt {\cos (e+f x)}}dx}{3 d^2}+\frac {2 \sin (e+f x)}{3 d f \sqrt {d \sec (e+f x)}}\right )}{7 d^2}+\frac {2 \sin (e+f x)}{7 d f (d \sec (e+f x))^{5/2}}\right )-\frac {2 b}{7 f (d \sec (e+f x))^{7/2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle a \left (\frac {5 \left (\frac {\sqrt {\cos (e+f x)} \sqrt {d \sec (e+f x)} \int \frac {1}{\sqrt {\sin \left (e+f x+\frac {\pi }{2}\right )}}dx}{3 d^2}+\frac {2 \sin (e+f x)}{3 d f \sqrt {d \sec (e+f x)}}\right )}{7 d^2}+\frac {2 \sin (e+f x)}{7 d f (d \sec (e+f x))^{5/2}}\right )-\frac {2 b}{7 f (d \sec (e+f x))^{7/2}}\) |
| \(\Big \downarrow \) 3120 |
| \(\displaystyle a \left (\frac {5 \left (\frac {2 \sqrt {\cos (e+f x)} \operatorname {EllipticF}\left (\frac {1}{2} (e+f x),2\right ) \sqrt {d \sec (e+f x)}}{3 d^2 f}+\frac {2 \sin (e+f x)}{3 d f \sqrt {d \sec (e+f x)}}\right )}{7 d^2}+\frac {2 \sin (e+f x)}{7 d f (d \sec (e+f x))^{5/2}}\right )-\frac {2 b}{7 f (d \sec (e+f x))^{7/2}}\) |
(-2*b)/(7*f*(d*Sec[e + f*x])^(7/2)) + a*((2*Sin[e + f*x])/(7*d*f*(d*Sec[e + f*x])^(5/2)) + (5*((2*Sqrt[Cos[e + f*x]]*EllipticF[(e + f*x)/2, 2]*Sqrt[ d*Sec[e + f*x]])/(3*d^2*f) + (2*Sin[e + f*x])/(3*d*f*Sqrt[d*Sec[e + f*x]]) ))/(7*d^2))
3.6.85.3.1 Defintions of rubi rules used
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[((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[Cos[c + d*x]*(( b*Csc[c + d*x])^(n + 1)/(b*d*n)), x] + Simp[(n + 1)/(b^2*n) Int[(b*Csc[c + d*x])^(n + 2), x], x] /; FreeQ[{b, c, d}, x] && LtQ[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 = 6.18 (sec) , antiderivative size = 178, normalized size of antiderivative = 1.45
| method | result | size |
| default | \(-\frac {2 a \left (5 i \sqrt {\frac {\cos \left (f x +e \right )}{\cos \left (f x +e \right )+1}}\, F\left (i \left (\csc \left (f x +e \right )-\cot \left (f x +e \right )\right ), i\right ) \sqrt {\frac {1}{\cos \left (f x +e \right )+1}}+5 i \sec \left (f x +e \right ) \sqrt {\frac {\cos \left (f x +e \right )}{\cos \left (f x +e \right )+1}}\, F\left (i \left (\csc \left (f x +e \right )-\cot \left (f x +e \right )\right ), i\right ) \sqrt {\frac {1}{\cos \left (f x +e \right )+1}}-3 \sin \left (f x +e \right ) \left (\cos ^{2}\left (f x +e \right )\right )-5 \sin \left (f x +e \right )\right )}{21 f \sqrt {d \sec \left (f x +e \right )}\, d^{3}}-\frac {2 b}{7 f \left (d \sec \left (f x +e \right )\right )^{\frac {7}{2}}}\) | \(178\) |
| parts | \(-\frac {2 a \left (5 i \sqrt {\frac {\cos \left (f x +e \right )}{\cos \left (f x +e \right )+1}}\, F\left (i \left (\csc \left (f x +e \right )-\cot \left (f x +e \right )\right ), i\right ) \sqrt {\frac {1}{\cos \left (f x +e \right )+1}}+5 i \sec \left (f x +e \right ) \sqrt {\frac {\cos \left (f x +e \right )}{\cos \left (f x +e \right )+1}}\, F\left (i \left (\csc \left (f x +e \right )-\cot \left (f x +e \right )\right ), i\right ) \sqrt {\frac {1}{\cos \left (f x +e \right )+1}}-3 \sin \left (f x +e \right ) \left (\cos ^{2}\left (f x +e \right )\right )-5 \sin \left (f x +e \right )\right )}{21 f \sqrt {d \sec \left (f x +e \right )}\, d^{3}}-\frac {2 b}{7 f \left (d \sec \left (f x +e \right )\right )^{\frac {7}{2}}}\) | \(178\) |
-2/21*a/f/(d*sec(f*x+e))^(1/2)/d^3*(5*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)+5*I*sec(f* x+e)*(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)*cos(f*x+e)^2-5*sin(f*x+e))-2/7*b /f/(d*sec(f*x+e))^(7/2)
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.09 (sec) , antiderivative size = 118, normalized size of antiderivative = 0.96 \[ \int \frac {a+b \tan (e+f x)}{(d \sec (e+f x))^{7/2}} \, dx=\frac {-5 i \, \sqrt {2} a \sqrt {d} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (f x + e\right ) + i \, \sin \left (f x + e\right )\right ) + 5 i \, \sqrt {2} a \sqrt {d} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (f x + e\right ) - i \, \sin \left (f x + e\right )\right ) - 2 \, {\left (3 \, b \cos \left (f x + e\right )^{4} - {\left (3 \, a \cos \left (f x + e\right )^{3} + 5 \, a \cos \left (f x + e\right )\right )} \sin \left (f x + e\right )\right )} \sqrt {\frac {d}{\cos \left (f x + e\right )}}}{21 \, d^{4} f} \]
1/21*(-5*I*sqrt(2)*a*sqrt(d)*weierstrassPInverse(-4, 0, cos(f*x + e) + I*s in(f*x + e)) + 5*I*sqrt(2)*a*sqrt(d)*weierstrassPInverse(-4, 0, cos(f*x + e) - I*sin(f*x + e)) - 2*(3*b*cos(f*x + e)^4 - (3*a*cos(f*x + e)^3 + 5*a*c os(f*x + e))*sin(f*x + e))*sqrt(d/cos(f*x + e)))/(d^4*f)
\[ \int \frac {a+b \tan (e+f x)}{(d \sec (e+f x))^{7/2}} \, dx=\int \frac {a + b \tan {\left (e + f x \right )}}{\left (d \sec {\left (e + f x \right )}\right )^{\frac {7}{2}}}\, dx \]
\[ \int \frac {a+b \tan (e+f x)}{(d \sec (e+f x))^{7/2}} \, dx=\int { \frac {b \tan \left (f x + e\right ) + a}{\left (d \sec \left (f x + e\right )\right )^{\frac {7}{2}}} \,d x } \]
\[ \int \frac {a+b \tan (e+f x)}{(d \sec (e+f x))^{7/2}} \, dx=\int { \frac {b \tan \left (f x + e\right ) + a}{\left (d \sec \left (f x + e\right )\right )^{\frac {7}{2}}} \,d x } \]
Timed out. \[ \int \frac {a+b \tan (e+f x)}{(d \sec (e+f x))^{7/2}} \, dx=\int \frac {a+b\,\mathrm {tan}\left (e+f\,x\right )}{{\left (\frac {d}{\cos \left (e+f\,x\right )}\right )}^{7/2}} \,d x \]