Integrand size = 25, antiderivative size = 184 \[ \int \frac {(a+b \tan (e+f x))^2}{(d \sec (e+f x))^{9/2}} \, dx=-\frac {10 a b}{63 f (d \sec (e+f x))^{9/2}}+\frac {2 \left (7 a^2+2 b^2\right ) E\left (\left .\frac {1}{2} (e+f x)\right |2\right )}{15 d^4 f \sqrt {\cos (e+f x)} \sqrt {d \sec (e+f x)}}+\frac {2 \left (7 a^2+2 b^2\right ) \sin (e+f x)}{63 d f (d \sec (e+f x))^{7/2}}+\frac {2 \left (7 a^2+2 b^2\right ) \sin (e+f x)}{45 d^3 f (d \sec (e+f x))^{3/2}}-\frac {2 b (a+b \tan (e+f x))}{7 f (d \sec (e+f x))^{9/2}} \]
-10/63*a*b/f/(d*sec(f*x+e))^(9/2)+2/63*(7*a^2+2*b^2)*sin(f*x+e)/d/f/(d*sec (f*x+e))^(7/2)+2/45*(7*a^2+2*b^2)*sin(f*x+e)/d^3/f/(d*sec(f*x+e))^(3/2)+2/ 15*(7*a^2+2*b^2)*(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))/d^4/f/cos(f*x+e)^(1/2)/(d*sec(f*x+e))^(1/2)-2 /7*b*(a+b*tan(f*x+e))/f/(d*sec(f*x+e))^(9/2)
Time = 6.01 (sec) , antiderivative size = 126, normalized size of antiderivative = 0.68 \[ \int \frac {(a+b \tan (e+f x))^2}{(d \sec (e+f x))^{9/2}} \, dx=\frac {\frac {48 \left (7 a^2+2 b^2\right ) E\left (\left .\frac {1}{2} (e+f x)\right |2\right )}{\sqrt {\cos (e+f x)}}+4 \cos (e+f x) \left (-30 a b \cos (e+f x)-10 a b \cos (3 (e+f x))+2 \left (19 a^2-b^2+5 \left (a^2-b^2\right ) \cos (2 (e+f x))\right ) \sin (e+f x)\right )}{360 d^4 f \sqrt {d \sec (e+f x)}} \]
((48*(7*a^2 + 2*b^2)*EllipticE[(e + f*x)/2, 2])/Sqrt[Cos[e + f*x]] + 4*Cos [e + f*x]*(-30*a*b*Cos[e + f*x] - 10*a*b*Cos[3*(e + f*x)] + 2*(19*a^2 - b^ 2 + 5*(a^2 - b^2)*Cos[2*(e + f*x)])*Sin[e + f*x]))/(360*d^4*f*Sqrt[d*Sec[e + f*x]])
Time = 0.85 (sec) , antiderivative size = 177, normalized size of antiderivative = 0.96, number of steps used = 13, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.520, Rules used = {3042, 3993, 27, 3042, 3967, 3042, 4256, 3042, 4256, 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.
| \(\displaystyle \int \frac {(a+b \tan (e+f x))^2}{(d \sec (e+f x))^{9/2}} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {(a+b \tan (e+f x))^2}{(d \sec (e+f x))^{9/2}}dx\) |
| \(\Big \downarrow \) 3993 |
| \(\displaystyle -\frac {2}{7} \int -\frac {7 a^2+5 b \tan (e+f x) a+2 b^2}{2 (d \sec (e+f x))^{9/2}}dx-\frac {2 b (a+b \tan (e+f x))}{7 f (d \sec (e+f x))^{9/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{7} \int \frac {7 a^2+5 b \tan (e+f x) a+2 b^2}{(d \sec (e+f x))^{9/2}}dx-\frac {2 b (a+b \tan (e+f x))}{7 f (d \sec (e+f x))^{9/2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{7} \int \frac {7 a^2+5 b \tan (e+f x) a+2 b^2}{(d \sec (e+f x))^{9/2}}dx-\frac {2 b (a+b \tan (e+f x))}{7 f (d \sec (e+f x))^{9/2}}\) |
| \(\Big \downarrow \) 3967 |
| \(\displaystyle \frac {1}{7} \left (\left (7 a^2+2 b^2\right ) \int \frac {1}{(d \sec (e+f x))^{9/2}}dx-\frac {10 a b}{9 f (d \sec (e+f x))^{9/2}}\right )-\frac {2 b (a+b \tan (e+f x))}{7 f (d \sec (e+f x))^{9/2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{7} \left (\left (7 a^2+2 b^2\right ) \int \frac {1}{\left (d \csc \left (e+f x+\frac {\pi }{2}\right )\right )^{9/2}}dx-\frac {10 a b}{9 f (d \sec (e+f x))^{9/2}}\right )-\frac {2 b (a+b \tan (e+f x))}{7 f (d \sec (e+f x))^{9/2}}\) |
| \(\Big \downarrow \) 4256 |
| \(\displaystyle \frac {1}{7} \left (\left (7 a^2+2 b^2\right ) \left (\frac {7 \int \frac {1}{(d \sec (e+f x))^{5/2}}dx}{9 d^2}+\frac {2 \sin (e+f x)}{9 d f (d \sec (e+f x))^{7/2}}\right )-\frac {10 a b}{9 f (d \sec (e+f x))^{9/2}}\right )-\frac {2 b (a+b \tan (e+f x))}{7 f (d \sec (e+f x))^{9/2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{7} \left (\left (7 a^2+2 b^2\right ) \left (\frac {7 \int \frac {1}{\left (d \csc \left (e+f x+\frac {\pi }{2}\right )\right )^{5/2}}dx}{9 d^2}+\frac {2 \sin (e+f x)}{9 d f (d \sec (e+f x))^{7/2}}\right )-\frac {10 a b}{9 f (d \sec (e+f x))^{9/2}}\right )-\frac {2 b (a+b \tan (e+f x))}{7 f (d \sec (e+f x))^{9/2}}\) |
| \(\Big \downarrow \) 4256 |
| \(\displaystyle \frac {1}{7} \left (\left (7 a^2+2 b^2\right ) \left (\frac {7 \left (\frac {3 \int \frac {1}{\sqrt {d \sec (e+f x)}}dx}{5 d^2}+\frac {2 \sin (e+f x)}{5 d f (d \sec (e+f x))^{3/2}}\right )}{9 d^2}+\frac {2 \sin (e+f x)}{9 d f (d \sec (e+f x))^{7/2}}\right )-\frac {10 a b}{9 f (d \sec (e+f x))^{9/2}}\right )-\frac {2 b (a+b \tan (e+f x))}{7 f (d \sec (e+f x))^{9/2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{7} \left (\left (7 a^2+2 b^2\right ) \left (\frac {7 \left (\frac {3 \int \frac {1}{\sqrt {d \csc \left (e+f x+\frac {\pi }{2}\right )}}dx}{5 d^2}+\frac {2 \sin (e+f x)}{5 d f (d \sec (e+f x))^{3/2}}\right )}{9 d^2}+\frac {2 \sin (e+f x)}{9 d f (d \sec (e+f x))^{7/2}}\right )-\frac {10 a b}{9 f (d \sec (e+f x))^{9/2}}\right )-\frac {2 b (a+b \tan (e+f x))}{7 f (d \sec (e+f x))^{9/2}}\) |
| \(\Big \downarrow \) 4258 |
| \(\displaystyle \frac {1}{7} \left (\left (7 a^2+2 b^2\right ) \left (\frac {7 \left (\frac {3 \int \sqrt {\cos (e+f x)}dx}{5 d^2 \sqrt {\cos (e+f x)} \sqrt {d \sec (e+f x)}}+\frac {2 \sin (e+f x)}{5 d f (d \sec (e+f x))^{3/2}}\right )}{9 d^2}+\frac {2 \sin (e+f x)}{9 d f (d \sec (e+f x))^{7/2}}\right )-\frac {10 a b}{9 f (d \sec (e+f x))^{9/2}}\right )-\frac {2 b (a+b \tan (e+f x))}{7 f (d \sec (e+f x))^{9/2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{7} \left (\left (7 a^2+2 b^2\right ) \left (\frac {7 \left (\frac {3 \int \sqrt {\sin \left (e+f x+\frac {\pi }{2}\right )}dx}{5 d^2 \sqrt {\cos (e+f x)} \sqrt {d \sec (e+f x)}}+\frac {2 \sin (e+f x)}{5 d f (d \sec (e+f x))^{3/2}}\right )}{9 d^2}+\frac {2 \sin (e+f x)}{9 d f (d \sec (e+f x))^{7/2}}\right )-\frac {10 a b}{9 f (d \sec (e+f x))^{9/2}}\right )-\frac {2 b (a+b \tan (e+f x))}{7 f (d \sec (e+f x))^{9/2}}\) |
| \(\Big \downarrow \) 3119 |
| \(\displaystyle \frac {1}{7} \left (\left (7 a^2+2 b^2\right ) \left (\frac {7 \left (\frac {6 E\left (\left .\frac {1}{2} (e+f x)\right |2\right )}{5 d^2 f \sqrt {\cos (e+f x)} \sqrt {d \sec (e+f x)}}+\frac {2 \sin (e+f x)}{5 d f (d \sec (e+f x))^{3/2}}\right )}{9 d^2}+\frac {2 \sin (e+f x)}{9 d f (d \sec (e+f x))^{7/2}}\right )-\frac {10 a b}{9 f (d \sec (e+f x))^{9/2}}\right )-\frac {2 b (a+b \tan (e+f x))}{7 f (d \sec (e+f x))^{9/2}}\) |
((-10*a*b)/(9*f*(d*Sec[e + f*x])^(9/2)) + (7*a^2 + 2*b^2)*((2*Sin[e + f*x] )/(9*d*f*(d*Sec[e + f*x])^(7/2)) + (7*((6*EllipticE[(e + f*x)/2, 2])/(5*d^ 2*f*Sqrt[Cos[e + f*x]]*Sqrt[d*Sec[e + f*x]]) + (2*Sin[e + f*x])/(5*d*f*(d* Sec[e + f*x])^(3/2))))/(9*d^2)))/7 - (2*b*(a + b*Tan[e + f*x]))/(7*f*(d*Se c[e + f*x])^(9/2))
3.6.93.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
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[((d_.)*sec[(e_.) + (f_.)*(x_)])^(m_.)*((a_) + (b_.)*tan[(e_.) + (f_.)*( x_)])^2, x_Symbol] :> Simp[b*(d*Sec[e + f*x])^m*((a + b*Tan[e + f*x])/(f*(m + 1))), x] + Simp[1/(m + 1) Int[(d*Sec[e + f*x])^m*(a^2*(m + 1) - b^2 + a*b*(m + 2)*Tan[e + f*x]), x], x] /; FreeQ[{a, b, d, e, f, m}, x] && NeQ[a^ 2 + b^2, 0] && !IntegerQ[m]
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 = 33.43 (sec) , antiderivative size = 931, normalized size of antiderivative = 5.06
| method | result | size |
| parts | \(\text {Expression too large to display}\) | \(931\) |
| default | \(\text {Expression too large to display}\) | \(968\) |
2/45*a^2/f/(cos(f*x+e)+1)/(d*sec(f*x+e))^(1/2)/d^4*(21*I*cos(f*x+e)*Ellipt icE(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)-21*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)+5*cos(f*x+e)^4*s in(f*x+e)+42*I*(1/(cos(f*x+e)+1))^(1/2)*(cos(f*x+e)/(cos(f*x+e)+1))^(1/2)* EllipticE(I*(csc(f*x+e)-cot(f*x+e)),I)-42*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*cos( f*x+e)^3*sin(f*x+e)+21*I*sec(f*x+e)*(1/(cos(f*x+e)+1))^(1/2)*(cos(f*x+e)/( cos(f*x+e)+1))^(1/2)*EllipticE(I*(csc(f*x+e)-cot(f*x+e)),I)-21*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)+7*sin(f*x+e)*cos(f*x+e)^2+7*sin(f*x+e)*cos(f*x+e )+21*sin(f*x+e))+2/45*b^2/f/(cos(f*x+e)+1)/(d*sec(f*x+e))^(1/2)/d^4*(6*I*c os(f*x+e)*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)-6*I*cos(f*x+e)*EllipticF(I*(csc(f*x+e)-c ot(f*x+e)),I)*(1/(cos(f*x+e)+1))^(1/2)*(cos(f*x+e)/(cos(f*x+e)+1))^(1/2)-5 *cos(f*x+e)^4*sin(f*x+e)+12*I*(1/(cos(f*x+e)+1))^(1/2)*(cos(f*x+e)/(cos(f* x+e)+1))^(1/2)*EllipticE(I*(csc(f*x+e)-cot(f*x+e)),I)-12*I*(1/(cos(f*x+e)+ 1))^(1/2)*(cos(f*x+e)/(cos(f*x+e)+1))^(1/2)*EllipticF(I*(csc(f*x+e)-cot(f* x+e)),I)-5*cos(f*x+e)^3*sin(f*x+e)+6*I*sec(f*x+e)*(1/(cos(f*x+e)+1))^(1/2) *(cos(f*x+e)/(cos(f*x+e)+1))^(1/2)*EllipticE(I*(csc(f*x+e)-cot(f*x+e)),...
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.10 (sec) , antiderivative size = 164, normalized size of antiderivative = 0.89 \[ \int \frac {(a+b \tan (e+f x))^2}{(d \sec (e+f x))^{9/2}} \, dx=-\frac {3 \, \sqrt {2} {\left (-7 i \, a^{2} - 2 i \, b^{2}\right )} \sqrt {d} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (f x + e\right ) + i \, \sin \left (f x + e\right )\right )\right ) + 3 \, \sqrt {2} {\left (7 i \, a^{2} + 2 i \, b^{2}\right )} \sqrt {d} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (f x + e\right ) - i \, \sin \left (f x + e\right )\right )\right ) + 2 \, {\left (10 \, a b \cos \left (f x + e\right )^{5} - {\left (5 \, {\left (a^{2} - b^{2}\right )} \cos \left (f x + e\right )^{4} + {\left (7 \, a^{2} + 2 \, b^{2}\right )} \cos \left (f x + e\right )^{2}\right )} \sin \left (f x + e\right )\right )} \sqrt {\frac {d}{\cos \left (f x + e\right )}}}{45 \, d^{5} f} \]
-1/45*(3*sqrt(2)*(-7*I*a^2 - 2*I*b^2)*sqrt(d)*weierstrassZeta(-4, 0, weier strassPInverse(-4, 0, cos(f*x + e) + I*sin(f*x + e))) + 3*sqrt(2)*(7*I*a^2 + 2*I*b^2)*sqrt(d)*weierstrassZeta(-4, 0, weierstrassPInverse(-4, 0, cos( f*x + e) - I*sin(f*x + e))) + 2*(10*a*b*cos(f*x + e)^5 - (5*(a^2 - b^2)*co s(f*x + e)^4 + (7*a^2 + 2*b^2)*cos(f*x + e)^2)*sin(f*x + e))*sqrt(d/cos(f* x + e)))/(d^5*f)
Timed out. \[ \int \frac {(a+b \tan (e+f x))^2}{(d \sec (e+f x))^{9/2}} \, dx=\text {Timed out} \]
\[ \int \frac {(a+b \tan (e+f x))^2}{(d \sec (e+f x))^{9/2}} \, dx=\int { \frac {{\left (b \tan \left (f x + e\right ) + a\right )}^{2}}{\left (d \sec \left (f x + e\right )\right )^{\frac {9}{2}}} \,d x } \]
\[ \int \frac {(a+b \tan (e+f x))^2}{(d \sec (e+f x))^{9/2}} \, dx=\int { \frac {{\left (b \tan \left (f x + e\right ) + a\right )}^{2}}{\left (d \sec \left (f x + e\right )\right )^{\frac {9}{2}}} \,d x } \]
Timed out. \[ \int \frac {(a+b \tan (e+f x))^2}{(d \sec (e+f x))^{9/2}} \, dx=\int \frac {{\left (a+b\,\mathrm {tan}\left (e+f\,x\right )\right )}^2}{{\left (\frac {d}{\cos \left (e+f\,x\right )}\right )}^{9/2}} \,d x \]