Integrand size = 23, antiderivative size = 247 \[ \int \sqrt {\sec (c+d x)} (a+b \sec (c+d x))^4 \, dx=-\frac {8 a b \left (5 a^2+3 b^2\right ) \sqrt {\cos (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \sqrt {\sec (c+d x)}}{5 d}+\frac {2 \left (21 a^4+42 a^2 b^2+5 b^4\right ) \sqrt {\cos (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right ) \sqrt {\sec (c+d x)}}{21 d}+\frac {8 a b \left (5 a^2+3 b^2\right ) \sqrt {\sec (c+d x)} \sin (c+d x)}{5 d}+\frac {2 b^2 \left (39 a^2+5 b^2\right ) \sec ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{21 d}+\frac {36 a b^3 \sec ^{\frac {5}{2}}(c+d x) \sin (c+d x)}{35 d}+\frac {2 b^2 \sec ^{\frac {3}{2}}(c+d x) (a+b \sec (c+d x))^2 \sin (c+d x)}{7 d} \]
2/21*b^2*(39*a^2+5*b^2)*sec(d*x+c)^(3/2)*sin(d*x+c)/d+36/35*a*b^3*sec(d*x+ c)^(5/2)*sin(d*x+c)/d+2/7*b^2*sec(d*x+c)^(3/2)*(a+b*sec(d*x+c))^2*sin(d*x+ c)/d+8/5*a*b*(5*a^2+3*b^2)*sin(d*x+c)*sec(d*x+c)^(1/2)/d-8/5*a*b*(5*a^2+3* b^2)*(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))*cos(d*x+c)^(1/2)*sec(d*x+c)^(1/2)/d+2/21*(21*a^4+42*a^2*b ^2+5*b^4)*(cos(1/2*d*x+1/2*c)^2)^(1/2)/cos(1/2*d*x+1/2*c)*EllipticF(sin(1/ 2*d*x+1/2*c),2^(1/2))*cos(d*x+c)^(1/2)*sec(d*x+c)^(1/2)/d
Time = 1.97 (sec) , antiderivative size = 168, normalized size of antiderivative = 0.68 \[ \int \sqrt {\sec (c+d x)} (a+b \sec (c+d x))^4 \, dx=\frac {2 \sec ^{\frac {7}{2}}(c+d x) \left (-84 a b \left (5 a^2+3 b^2\right ) \cos ^{\frac {7}{2}}(c+d x) E\left (\left .\frac {1}{2} (c+d x)\right |2\right )+5 \left (21 a^4+42 a^2 b^2+5 b^4\right ) \cos ^{\frac {7}{2}}(c+d x) \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )+b \left (15 b^3+5 b \left (42 a^2+5 b^2\right ) \cos ^2(c+d x)+84 a \left (5 a^2+3 b^2\right ) \cos ^3(c+d x)\right ) \sin (c+d x)+42 a b^3 \sin (2 (c+d x))\right )}{105 d} \]
(2*Sec[c + d*x]^(7/2)*(-84*a*b*(5*a^2 + 3*b^2)*Cos[c + d*x]^(7/2)*Elliptic E[(c + d*x)/2, 2] + 5*(21*a^4 + 42*a^2*b^2 + 5*b^4)*Cos[c + d*x]^(7/2)*Ell ipticF[(c + d*x)/2, 2] + b*(15*b^3 + 5*b*(42*a^2 + 5*b^2)*Cos[c + d*x]^2 + 84*a*(5*a^2 + 3*b^2)*Cos[c + d*x]^3)*Sin[c + d*x] + 42*a*b^3*Sin[2*(c + d *x)]))/(105*d)
Time = 1.57 (sec) , antiderivative size = 243, normalized size of antiderivative = 0.98, number of steps used = 19, number of rules used = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.826, Rules used = {3042, 4329, 27, 3042, 4564, 27, 3042, 4535, 3042, 4255, 3042, 4258, 3042, 3119, 4534, 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.
| \(\displaystyle \int \sqrt {\sec (c+d x)} (a+b \sec (c+d x))^4 \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \sqrt {\csc \left (c+d x+\frac {\pi }{2}\right )} \left (a+b \csc \left (c+d x+\frac {\pi }{2}\right )\right )^4dx\) |
| \(\Big \downarrow \) 4329 |
| \(\displaystyle \frac {2}{7} \int \frac {1}{2} \sqrt {\sec (c+d x)} (a+b \sec (c+d x)) \left (18 a b^2 \sec ^2(c+d x)+b \left (21 a^2+5 b^2\right ) \sec (c+d x)+a \left (7 a^2+b^2\right )\right )dx+\frac {2 b^2 \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x) (a+b \sec (c+d x))^2}{7 d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{7} \int \sqrt {\sec (c+d x)} (a+b \sec (c+d x)) \left (18 a b^2 \sec ^2(c+d x)+b \left (21 a^2+5 b^2\right ) \sec (c+d x)+a \left (7 a^2+b^2\right )\right )dx+\frac {2 b^2 \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x) (a+b \sec (c+d x))^2}{7 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{7} \int \sqrt {\csc \left (c+d x+\frac {\pi }{2}\right )} \left (a+b \csc \left (c+d x+\frac {\pi }{2}\right )\right ) \left (18 a b^2 \csc \left (c+d x+\frac {\pi }{2}\right )^2+b \left (21 a^2+5 b^2\right ) \csc \left (c+d x+\frac {\pi }{2}\right )+a \left (7 a^2+b^2\right )\right )dx+\frac {2 b^2 \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x) (a+b \sec (c+d x))^2}{7 d}\) |
| \(\Big \downarrow \) 4564 |
| \(\displaystyle \frac {1}{7} \left (\frac {2}{5} \int \frac {1}{2} \sqrt {\sec (c+d x)} \left (5 \left (7 a^2+b^2\right ) a^2+28 b \left (5 a^2+3 b^2\right ) \sec (c+d x) a+5 b^2 \left (39 a^2+5 b^2\right ) \sec ^2(c+d x)\right )dx+\frac {36 a b^3 \sin (c+d x) \sec ^{\frac {5}{2}}(c+d x)}{5 d}\right )+\frac {2 b^2 \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x) (a+b \sec (c+d x))^2}{7 d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{7} \left (\frac {1}{5} \int \sqrt {\sec (c+d x)} \left (5 \left (7 a^2+b^2\right ) a^2+28 b \left (5 a^2+3 b^2\right ) \sec (c+d x) a+5 b^2 \left (39 a^2+5 b^2\right ) \sec ^2(c+d x)\right )dx+\frac {36 a b^3 \sin (c+d x) \sec ^{\frac {5}{2}}(c+d x)}{5 d}\right )+\frac {2 b^2 \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x) (a+b \sec (c+d x))^2}{7 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{7} \left (\frac {1}{5} \int \sqrt {\csc \left (c+d x+\frac {\pi }{2}\right )} \left (5 \left (7 a^2+b^2\right ) a^2+28 b \left (5 a^2+3 b^2\right ) \csc \left (c+d x+\frac {\pi }{2}\right ) a+5 b^2 \left (39 a^2+5 b^2\right ) \csc \left (c+d x+\frac {\pi }{2}\right )^2\right )dx+\frac {36 a b^3 \sin (c+d x) \sec ^{\frac {5}{2}}(c+d x)}{5 d}\right )+\frac {2 b^2 \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x) (a+b \sec (c+d x))^2}{7 d}\) |
| \(\Big \downarrow \) 4535 |
| \(\displaystyle \frac {1}{7} \left (\frac {1}{5} \left (28 a b \left (5 a^2+3 b^2\right ) \int \sec ^{\frac {3}{2}}(c+d x)dx+\int \sqrt {\sec (c+d x)} \left (5 \left (7 a^2+b^2\right ) a^2+5 b^2 \left (39 a^2+5 b^2\right ) \sec ^2(c+d x)\right )dx\right )+\frac {36 a b^3 \sin (c+d x) \sec ^{\frac {5}{2}}(c+d x)}{5 d}\right )+\frac {2 b^2 \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x) (a+b \sec (c+d x))^2}{7 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{7} \left (\frac {1}{5} \left (28 a b \left (5 a^2+3 b^2\right ) \int \csc \left (c+d x+\frac {\pi }{2}\right )^{3/2}dx+\int \sqrt {\csc \left (c+d x+\frac {\pi }{2}\right )} \left (5 \left (7 a^2+b^2\right ) a^2+5 b^2 \left (39 a^2+5 b^2\right ) \csc \left (c+d x+\frac {\pi }{2}\right )^2\right )dx\right )+\frac {36 a b^3 \sin (c+d x) \sec ^{\frac {5}{2}}(c+d x)}{5 d}\right )+\frac {2 b^2 \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x) (a+b \sec (c+d x))^2}{7 d}\) |
| \(\Big \downarrow \) 4255 |
| \(\displaystyle \frac {1}{7} \left (\frac {1}{5} \left (\int \sqrt {\csc \left (c+d x+\frac {\pi }{2}\right )} \left (5 \left (7 a^2+b^2\right ) a^2+5 b^2 \left (39 a^2+5 b^2\right ) \csc \left (c+d x+\frac {\pi }{2}\right )^2\right )dx+28 a b \left (5 a^2+3 b^2\right ) \left (\frac {2 \sin (c+d x) \sqrt {\sec (c+d x)}}{d}-\int \frac {1}{\sqrt {\sec (c+d x)}}dx\right )\right )+\frac {36 a b^3 \sin (c+d x) \sec ^{\frac {5}{2}}(c+d x)}{5 d}\right )+\frac {2 b^2 \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x) (a+b \sec (c+d x))^2}{7 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{7} \left (\frac {1}{5} \left (\int \sqrt {\csc \left (c+d x+\frac {\pi }{2}\right )} \left (5 \left (7 a^2+b^2\right ) a^2+5 b^2 \left (39 a^2+5 b^2\right ) \csc \left (c+d x+\frac {\pi }{2}\right )^2\right )dx+28 a b \left (5 a^2+3 b^2\right ) \left (\frac {2 \sin (c+d x) \sqrt {\sec (c+d x)}}{d}-\int \frac {1}{\sqrt {\csc \left (c+d x+\frac {\pi }{2}\right )}}dx\right )\right )+\frac {36 a b^3 \sin (c+d x) \sec ^{\frac {5}{2}}(c+d x)}{5 d}\right )+\frac {2 b^2 \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x) (a+b \sec (c+d x))^2}{7 d}\) |
| \(\Big \downarrow \) 4258 |
| \(\displaystyle \frac {1}{7} \left (\frac {1}{5} \left (\int \sqrt {\csc \left (c+d x+\frac {\pi }{2}\right )} \left (5 \left (7 a^2+b^2\right ) a^2+5 b^2 \left (39 a^2+5 b^2\right ) \csc \left (c+d x+\frac {\pi }{2}\right )^2\right )dx+28 a b \left (5 a^2+3 b^2\right ) \left (\frac {2 \sin (c+d x) \sqrt {\sec (c+d x)}}{d}-\sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \int \sqrt {\cos (c+d x)}dx\right )\right )+\frac {36 a b^3 \sin (c+d x) \sec ^{\frac {5}{2}}(c+d x)}{5 d}\right )+\frac {2 b^2 \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x) (a+b \sec (c+d x))^2}{7 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{7} \left (\frac {1}{5} \left (\int \sqrt {\csc \left (c+d x+\frac {\pi }{2}\right )} \left (5 \left (7 a^2+b^2\right ) a^2+5 b^2 \left (39 a^2+5 b^2\right ) \csc \left (c+d x+\frac {\pi }{2}\right )^2\right )dx+28 a b \left (5 a^2+3 b^2\right ) \left (\frac {2 \sin (c+d x) \sqrt {\sec (c+d x)}}{d}-\sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \int \sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}dx\right )\right )+\frac {36 a b^3 \sin (c+d x) \sec ^{\frac {5}{2}}(c+d x)}{5 d}\right )+\frac {2 b^2 \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x) (a+b \sec (c+d x))^2}{7 d}\) |
| \(\Big \downarrow \) 3119 |
| \(\displaystyle \frac {1}{7} \left (\frac {1}{5} \left (\int \sqrt {\csc \left (c+d x+\frac {\pi }{2}\right )} \left (5 \left (7 a^2+b^2\right ) a^2+5 b^2 \left (39 a^2+5 b^2\right ) \csc \left (c+d x+\frac {\pi }{2}\right )^2\right )dx+28 a b \left (5 a^2+3 b^2\right ) \left (\frac {2 \sin (c+d x) \sqrt {\sec (c+d x)}}{d}-\frac {2 \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{d}\right )\right )+\frac {36 a b^3 \sin (c+d x) \sec ^{\frac {5}{2}}(c+d x)}{5 d}\right )+\frac {2 b^2 \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x) (a+b \sec (c+d x))^2}{7 d}\) |
| \(\Big \downarrow \) 4534 |
| \(\displaystyle \frac {1}{7} \left (\frac {1}{5} \left (\frac {5}{3} \left (21 a^4+42 a^2 b^2+5 b^4\right ) \int \sqrt {\sec (c+d x)}dx+\frac {10 b^2 \left (39 a^2+5 b^2\right ) \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x)}{3 d}+28 a b \left (5 a^2+3 b^2\right ) \left (\frac {2 \sin (c+d x) \sqrt {\sec (c+d x)}}{d}-\frac {2 \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{d}\right )\right )+\frac {36 a b^3 \sin (c+d x) \sec ^{\frac {5}{2}}(c+d x)}{5 d}\right )+\frac {2 b^2 \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x) (a+b \sec (c+d x))^2}{7 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{7} \left (\frac {1}{5} \left (\frac {5}{3} \left (21 a^4+42 a^2 b^2+5 b^4\right ) \int \sqrt {\csc \left (c+d x+\frac {\pi }{2}\right )}dx+\frac {10 b^2 \left (39 a^2+5 b^2\right ) \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x)}{3 d}+28 a b \left (5 a^2+3 b^2\right ) \left (\frac {2 \sin (c+d x) \sqrt {\sec (c+d x)}}{d}-\frac {2 \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{d}\right )\right )+\frac {36 a b^3 \sin (c+d x) \sec ^{\frac {5}{2}}(c+d x)}{5 d}\right )+\frac {2 b^2 \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x) (a+b \sec (c+d x))^2}{7 d}\) |
| \(\Big \downarrow \) 4258 |
| \(\displaystyle \frac {1}{7} \left (\frac {1}{5} \left (\frac {5}{3} \left (21 a^4+42 a^2 b^2+5 b^4\right ) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \int \frac {1}{\sqrt {\cos (c+d x)}}dx+\frac {10 b^2 \left (39 a^2+5 b^2\right ) \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x)}{3 d}+28 a b \left (5 a^2+3 b^2\right ) \left (\frac {2 \sin (c+d x) \sqrt {\sec (c+d x)}}{d}-\frac {2 \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{d}\right )\right )+\frac {36 a b^3 \sin (c+d x) \sec ^{\frac {5}{2}}(c+d x)}{5 d}\right )+\frac {2 b^2 \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x) (a+b \sec (c+d x))^2}{7 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{7} \left (\frac {1}{5} \left (\frac {5}{3} \left (21 a^4+42 a^2 b^2+5 b^4\right ) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \int \frac {1}{\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}}dx+\frac {10 b^2 \left (39 a^2+5 b^2\right ) \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x)}{3 d}+28 a b \left (5 a^2+3 b^2\right ) \left (\frac {2 \sin (c+d x) \sqrt {\sec (c+d x)}}{d}-\frac {2 \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{d}\right )\right )+\frac {36 a b^3 \sin (c+d x) \sec ^{\frac {5}{2}}(c+d x)}{5 d}\right )+\frac {2 b^2 \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x) (a+b \sec (c+d x))^2}{7 d}\) |
| \(\Big \downarrow \) 3120 |
| \(\displaystyle \frac {1}{7} \left (\frac {1}{5} \left (\frac {10 b^2 \left (39 a^2+5 b^2\right ) \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x)}{3 d}+28 a b \left (5 a^2+3 b^2\right ) \left (\frac {2 \sin (c+d x) \sqrt {\sec (c+d x)}}{d}-\frac {2 \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{d}\right )+\frac {10 \left (21 a^4+42 a^2 b^2+5 b^4\right ) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )}{3 d}\right )+\frac {36 a b^3 \sin (c+d x) \sec ^{\frac {5}{2}}(c+d x)}{5 d}\right )+\frac {2 b^2 \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x) (a+b \sec (c+d x))^2}{7 d}\) |
(2*b^2*Sec[c + d*x]^(3/2)*(a + b*Sec[c + d*x])^2*Sin[c + d*x])/(7*d) + ((3 6*a*b^3*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(5*d) + ((10*(21*a^4 + 42*a^2*b^2 + 5*b^4)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]]) /(3*d) + (10*b^2*(39*a^2 + 5*b^2)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(3*d) + 28*a*b*(5*a^2 + 3*b^2)*((-2*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]* Sqrt[Sec[c + d*x]])/d + (2*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/d))/5)/7
3.7.1.3.1 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[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[(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]
Int[(csc[(e_.) + (f_.)*(x_)]*(d_.))^(n_)*(csc[(e_.) + (f_.)*(x_)]*(b_.) + ( a_))^(m_), x_Symbol] :> Simp[(-b^2)*Cot[e + f*x]*(a + b*Csc[e + f*x])^(m - 2)*((d*Csc[e + f*x])^n/(f*(m + n - 1))), x] + Simp[1/(d*(m + n - 1)) Int[ (a + b*Csc[e + f*x])^(m - 3)*(d*Csc[e + f*x])^n*Simp[a^3*d*(m + n - 1) + a* b^2*d*n + b*(b^2*d*(m + n - 2) + 3*a^2*d*(m + n - 1))*Csc[e + f*x] + a*b^2* d*(3*m + 2*n - 4)*Csc[e + f*x]^2, x], x], x] /; FreeQ[{a, b, d, e, f, n}, x ] && NeQ[a^2 - b^2, 0] && GtQ[m, 2] && (IntegerQ[m] || IntegersQ[2*m, 2*n]) && !(IGtQ[n, 2] && !IntegerQ[m])
Int[(csc[(e_.) + (f_.)*(x_)]*(b_.))^(m_.)*(csc[(e_.) + (f_.)*(x_)]^2*(C_.) + (A_)), x_Symbol] :> Simp[(-C)*Cot[e + f*x]*((b*Csc[e + f*x])^m/(f*(m + 1) )), x] + Simp[(C*m + A*(m + 1))/(m + 1) Int[(b*Csc[e + f*x])^m, x], x] /; FreeQ[{b, e, f, A, C, m}, x] && NeQ[C*m + A*(m + 1), 0] && !LeQ[m, -1]
Int[(csc[(e_.) + (f_.)*(x_)]*(b_.))^(m_.)*((A_.) + csc[(e_.) + (f_.)*(x_)]* (B_.) + csc[(e_.) + (f_.)*(x_)]^2*(C_.)), x_Symbol] :> Simp[B/b Int[(b*Cs c[e + f*x])^(m + 1), x], x] + Int[(b*Csc[e + f*x])^m*(A + C*Csc[e + f*x]^2) , x] /; FreeQ[{b, e, f, A, B, C, m}, x]
Int[((A_.) + csc[(e_.) + (f_.)*(x_)]*(B_.) + csc[(e_.) + (f_.)*(x_)]^2*(C_. ))*(csc[(e_.) + (f_.)*(x_)]*(d_.))^(n_.)*(csc[(e_.) + (f_.)*(x_)]*(b_.) + ( a_)), x_Symbol] :> Simp[(-b)*C*Csc[e + f*x]*Cot[e + f*x]*((d*Csc[e + f*x])^ n/(f*(n + 2))), x] + Simp[1/(n + 2) Int[(d*Csc[e + f*x])^n*Simp[A*a*(n + 2) + (B*a*(n + 2) + b*(C*(n + 1) + A*(n + 2)))*Csc[e + f*x] + (a*C + B*b)*( n + 2)*Csc[e + f*x]^2, x], x], x] /; FreeQ[{a, b, d, e, f, A, B, C, n}, x] && !LtQ[n, -1]
Leaf count of result is larger than twice the leaf count of optimal. \(897\) vs. \(2(271)=542\).
Time = 50.84 (sec) , antiderivative size = 898, normalized size of antiderivative = 3.64
| method | result | size |
| default | \(\text {Expression too large to display}\) | \(898\) |
| parts | \(\text {Expression too large to display}\) | \(1146\) |
-(-(-2*cos(1/2*d*x+1/2*c)^2+1)*sin(1/2*d*x+1/2*c)^2)^(1/2)*(2*a^4*(sin(1/2 *d*x+1/2*c)^2)^(1/2)*(-2*cos(1/2*d*x+1/2*c)^2+1)^(1/2)/(-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))+2* b^4*(-1/56*cos(1/2*d*x+1/2*c)*(-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)^4-5/42*cos(1/2*d*x+1/2*c)*(-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)^2+5/21 *(sin(1/2*d*x+1/2*c)^2)^(1/2)*(-2*cos(1/2*d*x+1/2*c)^2+1)^(1/2)/(-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)))+12*a^2*b^2*(-1/6*cos(1/2*d*x+1/2*c)*(-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)^2+1/3*(sin(1/2*d*x+1/2*c )^2)^(1/2)*(-2*cos(1/2*d*x+1/2*c)^2+1)^(1/2)/(-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)))+8*a^3*b/sin (1/2*d*x+1/2*c)^2/(2*sin(1/2*d*x+1/2*c)^2-1)*(-2*sin(1/2*d*x+1/2*c)^4+sin( 1/2*d*x+1/2*c)^2)^(1/2)*(2*cos(1/2*d*x+1/2*c)*sin(1/2*d*x+1/2*c)^2-(sin(1/ 2*d*x+1/2*c)^2)^(1/2)*EllipticE(cos(1/2*d*x+1/2*c),2^(1/2))*(2*sin(1/2*d*x +1/2*c)^2-1)^(1/2))+8/5*a*b^3/(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)/sin(1/2*d*x+1/2*c)^2*(24*sin(1/2*d*x+1/2*c)^ 6*cos(1/2*d*x+1/2*c)-12*(sin(1/2*d*x+1/2*c)^2)^(1/2)*EllipticE(cos(1/2*d*x +1/2*c),2^(1/2))*(2*sin(1/2*d*x+1/2*c)^2-1)^(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*(sin(1/2*d*x+1/2*c)^2)^(1/2)...
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.11 (sec) , antiderivative size = 289, normalized size of antiderivative = 1.17 \[ \int \sqrt {\sec (c+d x)} (a+b \sec (c+d x))^4 \, dx=-\frac {5 \, \sqrt {2} {\left (21 i \, a^{4} + 42 i \, a^{2} b^{2} + 5 i \, b^{4}\right )} \cos \left (d x + c\right )^{3} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right ) + 5 \, \sqrt {2} {\left (-21 i \, a^{4} - 42 i \, a^{2} b^{2} - 5 i \, b^{4}\right )} \cos \left (d x + c\right )^{3} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right ) + 84 \, \sqrt {2} {\left (5 i \, a^{3} b + 3 i \, a b^{3}\right )} \cos \left (d x + c\right )^{3} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right )\right ) + 84 \, \sqrt {2} {\left (-5 i \, a^{3} b - 3 i \, a b^{3}\right )} \cos \left (d x + c\right )^{3} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right )\right ) - \frac {2 \, {\left (84 \, a b^{3} \cos \left (d x + c\right ) + 15 \, b^{4} + 84 \, {\left (5 \, a^{3} b + 3 \, a b^{3}\right )} \cos \left (d x + c\right )^{3} + 5 \, {\left (42 \, a^{2} b^{2} + 5 \, b^{4}\right )} \cos \left (d x + c\right )^{2}\right )} \sin \left (d x + c\right )}{\sqrt {\cos \left (d x + c\right )}}}{105 \, d \cos \left (d x + c\right )^{3}} \]
-1/105*(5*sqrt(2)*(21*I*a^4 + 42*I*a^2*b^2 + 5*I*b^4)*cos(d*x + c)^3*weier strassPInverse(-4, 0, cos(d*x + c) + I*sin(d*x + c)) + 5*sqrt(2)*(-21*I*a^ 4 - 42*I*a^2*b^2 - 5*I*b^4)*cos(d*x + c)^3*weierstrassPInverse(-4, 0, cos( d*x + c) - I*sin(d*x + c)) + 84*sqrt(2)*(5*I*a^3*b + 3*I*a*b^3)*cos(d*x + c)^3*weierstrassZeta(-4, 0, weierstrassPInverse(-4, 0, cos(d*x + c) + I*si n(d*x + c))) + 84*sqrt(2)*(-5*I*a^3*b - 3*I*a*b^3)*cos(d*x + c)^3*weierstr assZeta(-4, 0, weierstrassPInverse(-4, 0, cos(d*x + c) - I*sin(d*x + c))) - 2*(84*a*b^3*cos(d*x + c) + 15*b^4 + 84*(5*a^3*b + 3*a*b^3)*cos(d*x + c)^ 3 + 5*(42*a^2*b^2 + 5*b^4)*cos(d*x + c)^2)*sin(d*x + c)/sqrt(cos(d*x + c)) )/(d*cos(d*x + c)^3)
Timed out. \[ \int \sqrt {\sec (c+d x)} (a+b \sec (c+d x))^4 \, dx=\text {Timed out} \]
\[ \int \sqrt {\sec (c+d x)} (a+b \sec (c+d x))^4 \, dx=\int { {\left (b \sec \left (d x + c\right ) + a\right )}^{4} \sqrt {\sec \left (d x + c\right )} \,d x } \]
\[ \int \sqrt {\sec (c+d x)} (a+b \sec (c+d x))^4 \, dx=\int { {\left (b \sec \left (d x + c\right ) + a\right )}^{4} \sqrt {\sec \left (d x + c\right )} \,d x } \]
Timed out. \[ \int \sqrt {\sec (c+d x)} (a+b \sec (c+d x))^4 \, dx=\int {\left (a+\frac {b}{\cos \left (c+d\,x\right )}\right )}^4\,\sqrt {\frac {1}{\cos \left (c+d\,x\right )}} \,d x \]