Integrand size = 33, antiderivative size = 147 \[ \int \frac {\left (A+C \cos ^2(c+d x)\right ) \sec ^5(c+d x)}{\sqrt {b \cos (c+d x)}} \, dx=-\frac {2 (7 A+9 C) \sqrt {b \cos (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{15 b d \sqrt {\cos (c+d x)}}+\frac {2 A b^4 \sin (c+d x)}{9 d (b \cos (c+d x))^{9/2}}+\frac {2 b^2 (7 A+9 C) \sin (c+d x)}{45 d (b \cos (c+d x))^{5/2}}+\frac {2 (7 A+9 C) \sin (c+d x)}{15 d \sqrt {b \cos (c+d x)}} \]
2/9*A*b^4*sin(d*x+c)/d/(b*cos(d*x+c))^(9/2)+2/45*b^2*(7*A+9*C)*sin(d*x+c)/ d/(b*cos(d*x+c))^(5/2)+2/15*(7*A+9*C)*sin(d*x+c)/d/(b*cos(d*x+c))^(1/2)-2/ 15*(7*A+9*C)*(cos(1/2*d*x+1/2*c)^2)^(1/2)/cos(1/2*d*x+1/2*c)*EllipticE(sin (1/2*d*x+1/2*c),2^(1/2))*(b*cos(d*x+c))^(1/2)/b/d/cos(d*x+c)^(1/2)
Time = 1.01 (sec) , antiderivative size = 97, normalized size of antiderivative = 0.66 \[ \int \frac {\left (A+C \cos ^2(c+d x)\right ) \sec ^5(c+d x)}{\sqrt {b \cos (c+d x)}} \, dx=\frac {-6 (7 A+9 C) \sqrt {\cos (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right )+6 (7 A+9 C) \sin (c+d x)+2 \sec (c+d x) \left (7 A+9 C+5 A \sec ^2(c+d x)\right ) \tan (c+d x)}{45 d \sqrt {b \cos (c+d x)}} \]
(-6*(7*A + 9*C)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2] + 6*(7*A + 9* C)*Sin[c + d*x] + 2*Sec[c + d*x]*(7*A + 9*C + 5*A*Sec[c + d*x]^2)*Tan[c + d*x])/(45*d*Sqrt[b*Cos[c + d*x]])
Time = 0.66 (sec) , antiderivative size = 152, normalized size of antiderivative = 1.03, number of steps used = 11, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {3042, 2030, 3491, 3042, 3116, 3042, 3116, 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.
| \(\displaystyle \int \frac {\sec ^5(c+d x) \left (A+C \cos ^2(c+d x)\right )}{\sqrt {b \cos (c+d x)}} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {A+C \sin \left (c+d x+\frac {\pi }{2}\right )^2}{\sin \left (c+d x+\frac {\pi }{2}\right )^5 \sqrt {b \sin \left (c+d x+\frac {\pi }{2}\right )}}dx\) |
| \(\Big \downarrow \) 2030 |
| \(\displaystyle b^5 \int \frac {C \sin \left (\frac {1}{2} (2 c+\pi )+d x\right )^2+A}{\left (b \sin \left (\frac {1}{2} (2 c+\pi )+d x\right )\right )^{11/2}}dx\) |
| \(\Big \downarrow \) 3491 |
| \(\displaystyle b^5 \left (\frac {(7 A+9 C) \int \frac {1}{(b \cos (c+d x))^{7/2}}dx}{9 b^2}+\frac {2 A \sin (c+d x)}{9 b d (b \cos (c+d x))^{9/2}}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle b^5 \left (\frac {(7 A+9 C) \int \frac {1}{\left (b \sin \left (c+d x+\frac {\pi }{2}\right )\right )^{7/2}}dx}{9 b^2}+\frac {2 A \sin (c+d x)}{9 b d (b \cos (c+d x))^{9/2}}\right )\) |
| \(\Big \downarrow \) 3116 |
| \(\displaystyle b^5 \left (\frac {(7 A+9 C) \left (\frac {3 \int \frac {1}{(b \cos (c+d x))^{3/2}}dx}{5 b^2}+\frac {2 \sin (c+d x)}{5 b d (b \cos (c+d x))^{5/2}}\right )}{9 b^2}+\frac {2 A \sin (c+d x)}{9 b d (b \cos (c+d x))^{9/2}}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle b^5 \left (\frac {(7 A+9 C) \left (\frac {3 \int \frac {1}{\left (b \sin \left (c+d x+\frac {\pi }{2}\right )\right )^{3/2}}dx}{5 b^2}+\frac {2 \sin (c+d x)}{5 b d (b \cos (c+d x))^{5/2}}\right )}{9 b^2}+\frac {2 A \sin (c+d x)}{9 b d (b \cos (c+d x))^{9/2}}\right )\) |
| \(\Big \downarrow \) 3116 |
| \(\displaystyle b^5 \left (\frac {(7 A+9 C) \left (\frac {3 \left (\frac {2 \sin (c+d x)}{b d \sqrt {b \cos (c+d x)}}-\frac {\int \sqrt {b \cos (c+d x)}dx}{b^2}\right )}{5 b^2}+\frac {2 \sin (c+d x)}{5 b d (b \cos (c+d x))^{5/2}}\right )}{9 b^2}+\frac {2 A \sin (c+d x)}{9 b d (b \cos (c+d x))^{9/2}}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle b^5 \left (\frac {(7 A+9 C) \left (\frac {3 \left (\frac {2 \sin (c+d x)}{b d \sqrt {b \cos (c+d x)}}-\frac {\int \sqrt {b \sin \left (c+d x+\frac {\pi }{2}\right )}dx}{b^2}\right )}{5 b^2}+\frac {2 \sin (c+d x)}{5 b d (b \cos (c+d x))^{5/2}}\right )}{9 b^2}+\frac {2 A \sin (c+d x)}{9 b d (b \cos (c+d x))^{9/2}}\right )\) |
| \(\Big \downarrow \) 3121 |
| \(\displaystyle b^5 \left (\frac {(7 A+9 C) \left (\frac {3 \left (\frac {2 \sin (c+d x)}{b d \sqrt {b \cos (c+d x)}}-\frac {\sqrt {b \cos (c+d x)} \int \sqrt {\cos (c+d x)}dx}{b^2 \sqrt {\cos (c+d x)}}\right )}{5 b^2}+\frac {2 \sin (c+d x)}{5 b d (b \cos (c+d x))^{5/2}}\right )}{9 b^2}+\frac {2 A \sin (c+d x)}{9 b d (b \cos (c+d x))^{9/2}}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle b^5 \left (\frac {(7 A+9 C) \left (\frac {3 \left (\frac {2 \sin (c+d x)}{b d \sqrt {b \cos (c+d x)}}-\frac {\sqrt {b \cos (c+d x)} \int \sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}dx}{b^2 \sqrt {\cos (c+d x)}}\right )}{5 b^2}+\frac {2 \sin (c+d x)}{5 b d (b \cos (c+d x))^{5/2}}\right )}{9 b^2}+\frac {2 A \sin (c+d x)}{9 b d (b \cos (c+d x))^{9/2}}\right )\) |
| \(\Big \downarrow \) 3119 |
| \(\displaystyle b^5 \left (\frac {(7 A+9 C) \left (\frac {3 \left (\frac {2 \sin (c+d x)}{b d \sqrt {b \cos (c+d x)}}-\frac {2 E\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \sqrt {b \cos (c+d x)}}{b^2 d \sqrt {\cos (c+d x)}}\right )}{5 b^2}+\frac {2 \sin (c+d x)}{5 b d (b \cos (c+d x))^{5/2}}\right )}{9 b^2}+\frac {2 A \sin (c+d x)}{9 b d (b \cos (c+d x))^{9/2}}\right )\) |
b^5*((2*A*Sin[c + d*x])/(9*b*d*(b*Cos[c + d*x])^(9/2)) + ((7*A + 9*C)*((2* Sin[c + d*x])/(5*b*d*(b*Cos[c + d*x])^(5/2)) + (3*((-2*Sqrt[b*Cos[c + d*x] ]*EllipticE[(c + d*x)/2, 2])/(b^2*d*Sqrt[Cos[c + d*x]]) + (2*Sin[c + d*x]) /(b*d*Sqrt[b*Cos[c + d*x]])))/(5*b^2)))/(9*b^2))
3.1.70.3.1 Defintions of rubi rules used
Int[(Fx_.)*(v_)^(m_.)*((b_)*(v_))^(n_), x_Symbol] :> Simp[1/b^m Int[(b*v) ^(m + n)*Fx, x], x] /; FreeQ[{b, n}, x] && IntegerQ[m]
Int[((b_.)*sin[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[Cos[c + d*x]*(( b*Sin[c + d*x])^(n + 1)/(b*d*(n + 1))), x] + Simp[(n + 2)/(b^2*(n + 1)) I nt[(b*Sin[c + d*x])^(n + 2), x], x] /; FreeQ[{b, c, d}, x] && LtQ[n, -1] && IntegerQ[2*n]
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]
Int[((b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((A_) + (C_.)*sin[(e_.) + (f_.)*(x _)]^2), x_Symbol] :> Simp[A*Cos[e + f*x]*((b*Sin[e + f*x])^(m + 1)/(b*f*(m + 1))), x] + Simp[(A*(m + 2) + C*(m + 1))/(b^2*(m + 1)) Int[(b*Sin[e + f* x])^(m + 2), x], x] /; FreeQ[{b, e, f, A, C}, x] && LtQ[m, -1]
Leaf count of result is larger than twice the leaf count of optimal. \(730\) vs. \(2(155)=310\).
Time = 16.57 (sec) , antiderivative size = 731, normalized size of antiderivative = 4.97
| method | result | size |
| default | \(\text {Expression too large to display}\) | \(731\) |
| parts | \(\text {Expression too large to display}\) | \(782\) |
-(-(-2*cos(1/2*d*x+1/2*c)^2+1)*b*sin(1/2*d*x+1/2*c)^2)^(1/2)*(2/5*C/b/sin( 1/2*d*x+1/2*c)^2/(8*sin(1/2*d*x+1/2*c)^6-12*sin(1/2*d*x+1/2*c)^4+6*sin(1/2 *d*x+1/2*c)^2-1)*(24*cos(1/2*d*x+1/2*c)*sin(1/2*d*x+1/2*c)^6-12*(2*sin(1/2 *d*x+1/2*c)^2-1)^(1/2)*(sin(1/2*d*x+1/2*c)^2)^(1/2)*EllipticE(cos(1/2*d*x+ 1/2*c),2^(1/2))*sin(1/2*d*x+1/2*c)^4-24*sin(1/2*d*x+1/2*c)^4*cos(1/2*d*x+1 /2*c)+12*(2*sin(1/2*d*x+1/2*c)^2-1)^(1/2)*(sin(1/2*d*x+1/2*c)^2)^(1/2)*Ell ipticE(cos(1/2*d*x+1/2*c),2^(1/2))*sin(1/2*d*x+1/2*c)^2+8*sin(1/2*d*x+1/2* c)^2*cos(1/2*d*x+1/2*c)-3*(sin(1/2*d*x+1/2*c)^2)^(1/2)*(2*sin(1/2*d*x+1/2* c)^2-1)^(1/2)*EllipticE(cos(1/2*d*x+1/2*c),2^(1/2)))*(-2*sin(1/2*d*x+1/2*c )^4*b+b*sin(1/2*d*x+1/2*c)^2)^(1/2)+2*A*(-1/144*cos(1/2*d*x+1/2*c)/b*(-b*( 2*sin(1/2*d*x+1/2*c)^4-sin(1/2*d*x+1/2*c)^2))^(1/2)/(cos(1/2*d*x+1/2*c)^2- 1/2)^5-7/180*cos(1/2*d*x+1/2*c)/b*(-b*(2*sin(1/2*d*x+1/2*c)^4-sin(1/2*d*x+ 1/2*c)^2))^(1/2)/(cos(1/2*d*x+1/2*c)^2-1/2)^3-14/15*sin(1/2*d*x+1/2*c)^2*c os(1/2*d*x+1/2*c)/(-(-2*cos(1/2*d*x+1/2*c)^2+1)*b*sin(1/2*d*x+1/2*c)^2)^(1 /2)+7/15*(sin(1/2*d*x+1/2*c)^2)^(1/2)*(-2*cos(1/2*d*x+1/2*c)^2+1)^(1/2)/(- b*(2*sin(1/2*d*x+1/2*c)^4-sin(1/2*d*x+1/2*c)^2))^(1/2)*EllipticF(cos(1/2*d *x+1/2*c),2^(1/2))-7/15*(sin(1/2*d*x+1/2*c)^2)^(1/2)*(-2*cos(1/2*d*x+1/2*c )^2+1)^(1/2)/(-b*(2*sin(1/2*d*x+1/2*c)^4-sin(1/2*d*x+1/2*c)^2))^(1/2)*(Ell ipticF(cos(1/2*d*x+1/2*c),2^(1/2))-EllipticE(cos(1/2*d*x+1/2*c),2^(1/2)))) )/sin(1/2*d*x+1/2*c)/((2*cos(1/2*d*x+1/2*c)^2-1)*b)^(1/2)/d
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.10 (sec) , antiderivative size = 160, normalized size of antiderivative = 1.09 \[ \int \frac {\left (A+C \cos ^2(c+d x)\right ) \sec ^5(c+d x)}{\sqrt {b \cos (c+d x)}} \, dx=-\frac {3 \, \sqrt {2} {\left (7 i \, A + 9 i \, C\right )} \sqrt {b} \cos \left (d x + c\right )^{5} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right )\right ) + 3 \, \sqrt {2} {\left (-7 i \, A - 9 i \, C\right )} \sqrt {b} \cos \left (d x + c\right )^{5} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right )\right ) - 2 \, {\left (3 \, {\left (7 \, A + 9 \, C\right )} \cos \left (d x + c\right )^{4} + {\left (7 \, A + 9 \, C\right )} \cos \left (d x + c\right )^{2} + 5 \, A\right )} \sqrt {b \cos \left (d x + c\right )} \sin \left (d x + c\right )}{45 \, b d \cos \left (d x + c\right )^{5}} \]
-1/45*(3*sqrt(2)*(7*I*A + 9*I*C)*sqrt(b)*cos(d*x + c)^5*weierstrassZeta(-4 , 0, weierstrassPInverse(-4, 0, cos(d*x + c) + I*sin(d*x + c))) + 3*sqrt(2 )*(-7*I*A - 9*I*C)*sqrt(b)*cos(d*x + c)^5*weierstrassZeta(-4, 0, weierstra ssPInverse(-4, 0, cos(d*x + c) - I*sin(d*x + c))) - 2*(3*(7*A + 9*C)*cos(d *x + c)^4 + (7*A + 9*C)*cos(d*x + c)^2 + 5*A)*sqrt(b*cos(d*x + c))*sin(d*x + c))/(b*d*cos(d*x + c)^5)
Timed out. \[ \int \frac {\left (A+C \cos ^2(c+d x)\right ) \sec ^5(c+d x)}{\sqrt {b \cos (c+d x)}} \, dx=\text {Timed out} \]
\[ \int \frac {\left (A+C \cos ^2(c+d x)\right ) \sec ^5(c+d x)}{\sqrt {b \cos (c+d x)}} \, dx=\int { \frac {{\left (C \cos \left (d x + c\right )^{2} + A\right )} \sec \left (d x + c\right )^{5}}{\sqrt {b \cos \left (d x + c\right )}} \,d x } \]
\[ \int \frac {\left (A+C \cos ^2(c+d x)\right ) \sec ^5(c+d x)}{\sqrt {b \cos (c+d x)}} \, dx=\int { \frac {{\left (C \cos \left (d x + c\right )^{2} + A\right )} \sec \left (d x + c\right )^{5}}{\sqrt {b \cos \left (d x + c\right )}} \,d x } \]
Timed out. \[ \int \frac {\left (A+C \cos ^2(c+d x)\right ) \sec ^5(c+d x)}{\sqrt {b \cos (c+d x)}} \, dx=\int \frac {C\,{\cos \left (c+d\,x\right )}^2+A}{{\cos \left (c+d\,x\right )}^5\,\sqrt {b\,\cos \left (c+d\,x\right )}} \,d x \]