Integrand size = 23, antiderivative size = 211 \[ \int \frac {(a+b \sec (c+d x))^4}{\sec ^{\frac {7}{2}}(c+d x)} \, dx=\frac {8 a b \left (3 a^2+5 b^2\right ) \sqrt {\cos (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \sqrt {\sec (c+d x)}}{5 d}+\frac {2 \left (5 a^4+42 a^2 b^2+21 b^4\right ) \sqrt {\cos (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right ) \sqrt {\sec (c+d x)}}{21 d}+\frac {36 a^3 b \sin (c+d x)}{35 d \sec ^{\frac {3}{2}}(c+d x)}+\frac {2 a^2 \left (5 a^2+39 b^2\right ) \sin (c+d x)}{21 d \sqrt {\sec (c+d x)}}+\frac {2 a^2 (a+b \sec (c+d x))^2 \sin (c+d x)}{7 d \sec ^{\frac {5}{2}}(c+d x)} \]
36/35*a^3*b*sin(d*x+c)/d/sec(d*x+c)^(3/2)+2/7*a^2*(a+b*sec(d*x+c))^2*sin(d *x+c)/d/sec(d*x+c)^(5/2)+2/21*a^2*(5*a^2+39*b^2)*sin(d*x+c)/d/sec(d*x+c)^( 1/2)+8/5*a*b*(3*a^2+5*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*(5*a^4+42*a^2*b^2+21*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 = 1.50 (sec) , antiderivative size = 142, normalized size of antiderivative = 0.67 \[ \int \frac {(a+b \sec (c+d x))^4}{\sec ^{\frac {7}{2}}(c+d x)} \, dx=\frac {\sqrt {\sec (c+d x)} \left (336 a b \left (3 a^2+5 b^2\right ) \sqrt {\cos (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right )+20 \left (5 a^4+42 a^2 b^2+21 b^4\right ) \sqrt {\cos (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )+a^2 \left (65 a^2+420 b^2+168 a b \cos (c+d x)+15 a^2 \cos (2 (c+d x))\right ) \sin (2 (c+d x))\right )}{210 d} \]
(Sqrt[Sec[c + d*x]]*(336*a*b*(3*a^2 + 5*b^2)*Sqrt[Cos[c + d*x]]*EllipticE[ (c + d*x)/2, 2] + 20*(5*a^4 + 42*a^2*b^2 + 21*b^4)*Sqrt[Cos[c + d*x]]*Elli pticF[(c + d*x)/2, 2] + a^2*(65*a^2 + 420*b^2 + 168*a*b*Cos[c + d*x] + 15* a^2*Cos[2*(c + d*x)])*Sin[2*(c + d*x)]))/(210*d)
Time = 1.34 (sec) , antiderivative size = 219, normalized size of antiderivative = 1.04, number of steps used = 17, number of rules used = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.739, Rules used = {3042, 4328, 27, 3042, 4562, 27, 3042, 4535, 3042, 4258, 3042, 3119, 4533, 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.
| \(\displaystyle \int \frac {(a+b \sec (c+d x))^4}{\sec ^{\frac {7}{2}}(c+d x)} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\left (a+b \csc \left (c+d x+\frac {\pi }{2}\right )\right )^4}{\csc \left (c+d x+\frac {\pi }{2}\right )^{7/2}}dx\) |
| \(\Big \downarrow \) 4328 |
| \(\displaystyle \frac {2}{7} \int \frac {(a+b \sec (c+d x)) \left (18 b a^2+\left (5 a^2+21 b^2\right ) \sec (c+d x) a+b \left (a^2+7 b^2\right ) \sec ^2(c+d x)\right )}{2 \sec ^{\frac {5}{2}}(c+d x)}dx+\frac {2 a^2 \sin (c+d x) (a+b \sec (c+d x))^2}{7 d \sec ^{\frac {5}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{7} \int \frac {(a+b \sec (c+d x)) \left (18 b a^2+\left (5 a^2+21 b^2\right ) \sec (c+d x) a+b \left (a^2+7 b^2\right ) \sec ^2(c+d x)\right )}{\sec ^{\frac {5}{2}}(c+d x)}dx+\frac {2 a^2 \sin (c+d x) (a+b \sec (c+d x))^2}{7 d \sec ^{\frac {5}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{7} \int \frac {\left (a+b \csc \left (c+d x+\frac {\pi }{2}\right )\right ) \left (18 b a^2+\left (5 a^2+21 b^2\right ) \csc \left (c+d x+\frac {\pi }{2}\right ) a+b \left (a^2+7 b^2\right ) \csc \left (c+d x+\frac {\pi }{2}\right )^2\right )}{\csc \left (c+d x+\frac {\pi }{2}\right )^{5/2}}dx+\frac {2 a^2 \sin (c+d x) (a+b \sec (c+d x))^2}{7 d \sec ^{\frac {5}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 4562 |
| \(\displaystyle \frac {1}{7} \left (\frac {36 a^3 b \sin (c+d x)}{5 d \sec ^{\frac {3}{2}}(c+d x)}-\frac {2}{5} \int -\frac {5 \left (5 a^2+39 b^2\right ) a^2+28 b \left (3 a^2+5 b^2\right ) \sec (c+d x) a+5 b^2 \left (a^2+7 b^2\right ) \sec ^2(c+d x)}{2 \sec ^{\frac {3}{2}}(c+d x)}dx\right )+\frac {2 a^2 \sin (c+d x) (a+b \sec (c+d x))^2}{7 d \sec ^{\frac {5}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{7} \left (\frac {1}{5} \int \frac {5 \left (5 a^2+39 b^2\right ) a^2+28 b \left (3 a^2+5 b^2\right ) \sec (c+d x) a+5 b^2 \left (a^2+7 b^2\right ) \sec ^2(c+d x)}{\sec ^{\frac {3}{2}}(c+d x)}dx+\frac {36 a^3 b \sin (c+d x)}{5 d \sec ^{\frac {3}{2}}(c+d x)}\right )+\frac {2 a^2 \sin (c+d x) (a+b \sec (c+d x))^2}{7 d \sec ^{\frac {5}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{7} \left (\frac {1}{5} \int \frac {5 \left (5 a^2+39 b^2\right ) a^2+28 b \left (3 a^2+5 b^2\right ) \csc \left (c+d x+\frac {\pi }{2}\right ) a+5 b^2 \left (a^2+7 b^2\right ) \csc \left (c+d x+\frac {\pi }{2}\right )^2}{\csc \left (c+d x+\frac {\pi }{2}\right )^{3/2}}dx+\frac {36 a^3 b \sin (c+d x)}{5 d \sec ^{\frac {3}{2}}(c+d x)}\right )+\frac {2 a^2 \sin (c+d x) (a+b \sec (c+d x))^2}{7 d \sec ^{\frac {5}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 4535 |
| \(\displaystyle \frac {1}{7} \left (\frac {1}{5} \left (\int \frac {5 \left (5 a^2+39 b^2\right ) a^2+5 b^2 \left (a^2+7 b^2\right ) \sec ^2(c+d x)}{\sec ^{\frac {3}{2}}(c+d x)}dx+28 a b \left (3 a^2+5 b^2\right ) \int \frac {1}{\sqrt {\sec (c+d x)}}dx\right )+\frac {36 a^3 b \sin (c+d x)}{5 d \sec ^{\frac {3}{2}}(c+d x)}\right )+\frac {2 a^2 \sin (c+d x) (a+b \sec (c+d x))^2}{7 d \sec ^{\frac {5}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{7} \left (\frac {1}{5} \left (28 a b \left (3 a^2+5 b^2\right ) \int \frac {1}{\sqrt {\csc \left (c+d x+\frac {\pi }{2}\right )}}dx+\int \frac {5 \left (5 a^2+39 b^2\right ) a^2+5 b^2 \left (a^2+7 b^2\right ) \csc \left (c+d x+\frac {\pi }{2}\right )^2}{\csc \left (c+d x+\frac {\pi }{2}\right )^{3/2}}dx\right )+\frac {36 a^3 b \sin (c+d x)}{5 d \sec ^{\frac {3}{2}}(c+d x)}\right )+\frac {2 a^2 \sin (c+d x) (a+b \sec (c+d x))^2}{7 d \sec ^{\frac {5}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 4258 |
| \(\displaystyle \frac {1}{7} \left (\frac {1}{5} \left (\int \frac {5 \left (5 a^2+39 b^2\right ) a^2+5 b^2 \left (a^2+7 b^2\right ) \csc \left (c+d x+\frac {\pi }{2}\right )^2}{\csc \left (c+d x+\frac {\pi }{2}\right )^{3/2}}dx+28 a b \left (3 a^2+5 b^2\right ) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \int \sqrt {\cos (c+d x)}dx\right )+\frac {36 a^3 b \sin (c+d x)}{5 d \sec ^{\frac {3}{2}}(c+d x)}\right )+\frac {2 a^2 \sin (c+d x) (a+b \sec (c+d x))^2}{7 d \sec ^{\frac {5}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{7} \left (\frac {1}{5} \left (\int \frac {5 \left (5 a^2+39 b^2\right ) a^2+5 b^2 \left (a^2+7 b^2\right ) \csc \left (c+d x+\frac {\pi }{2}\right )^2}{\csc \left (c+d x+\frac {\pi }{2}\right )^{3/2}}dx+28 a b \left (3 a^2+5 b^2\right ) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \int \sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}dx\right )+\frac {36 a^3 b \sin (c+d x)}{5 d \sec ^{\frac {3}{2}}(c+d x)}\right )+\frac {2 a^2 \sin (c+d x) (a+b \sec (c+d x))^2}{7 d \sec ^{\frac {5}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 3119 |
| \(\displaystyle \frac {1}{7} \left (\frac {1}{5} \left (\int \frac {5 \left (5 a^2+39 b^2\right ) a^2+5 b^2 \left (a^2+7 b^2\right ) \csc \left (c+d x+\frac {\pi }{2}\right )^2}{\csc \left (c+d x+\frac {\pi }{2}\right )^{3/2}}dx+\frac {56 a b \left (3 a^2+5 b^2\right ) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{d}\right )+\frac {36 a^3 b \sin (c+d x)}{5 d \sec ^{\frac {3}{2}}(c+d x)}\right )+\frac {2 a^2 \sin (c+d x) (a+b \sec (c+d x))^2}{7 d \sec ^{\frac {5}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 4533 |
| \(\displaystyle \frac {1}{7} \left (\frac {1}{5} \left (\frac {5}{3} \left (5 a^4+42 a^2 b^2+21 b^4\right ) \int \sqrt {\sec (c+d x)}dx+\frac {10 a^2 \left (5 a^2+39 b^2\right ) \sin (c+d x)}{3 d \sqrt {\sec (c+d x)}}+\frac {56 a b \left (3 a^2+5 b^2\right ) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{d}\right )+\frac {36 a^3 b \sin (c+d x)}{5 d \sec ^{\frac {3}{2}}(c+d x)}\right )+\frac {2 a^2 \sin (c+d x) (a+b \sec (c+d x))^2}{7 d \sec ^{\frac {5}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{7} \left (\frac {1}{5} \left (\frac {5}{3} \left (5 a^4+42 a^2 b^2+21 b^4\right ) \int \sqrt {\csc \left (c+d x+\frac {\pi }{2}\right )}dx+\frac {10 a^2 \left (5 a^2+39 b^2\right ) \sin (c+d x)}{3 d \sqrt {\sec (c+d x)}}+\frac {56 a b \left (3 a^2+5 b^2\right ) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{d}\right )+\frac {36 a^3 b \sin (c+d x)}{5 d \sec ^{\frac {3}{2}}(c+d x)}\right )+\frac {2 a^2 \sin (c+d x) (a+b \sec (c+d x))^2}{7 d \sec ^{\frac {5}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 4258 |
| \(\displaystyle \frac {1}{7} \left (\frac {1}{5} \left (\frac {5}{3} \left (5 a^4+42 a^2 b^2+21 b^4\right ) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \int \frac {1}{\sqrt {\cos (c+d x)}}dx+\frac {10 a^2 \left (5 a^2+39 b^2\right ) \sin (c+d x)}{3 d \sqrt {\sec (c+d x)}}+\frac {56 a b \left (3 a^2+5 b^2\right ) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{d}\right )+\frac {36 a^3 b \sin (c+d x)}{5 d \sec ^{\frac {3}{2}}(c+d x)}\right )+\frac {2 a^2 \sin (c+d x) (a+b \sec (c+d x))^2}{7 d \sec ^{\frac {5}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{7} \left (\frac {1}{5} \left (\frac {5}{3} \left (5 a^4+42 a^2 b^2+21 b^4\right ) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \int \frac {1}{\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}}dx+\frac {10 a^2 \left (5 a^2+39 b^2\right ) \sin (c+d x)}{3 d \sqrt {\sec (c+d x)}}+\frac {56 a b \left (3 a^2+5 b^2\right ) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{d}\right )+\frac {36 a^3 b \sin (c+d x)}{5 d \sec ^{\frac {3}{2}}(c+d x)}\right )+\frac {2 a^2 \sin (c+d x) (a+b \sec (c+d x))^2}{7 d \sec ^{\frac {5}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 3120 |
| \(\displaystyle \frac {2 a^2 \sin (c+d x) (a+b \sec (c+d x))^2}{7 d \sec ^{\frac {5}{2}}(c+d x)}+\frac {1}{7} \left (\frac {36 a^3 b \sin (c+d x)}{5 d \sec ^{\frac {3}{2}}(c+d x)}+\frac {1}{5} \left (\frac {10 a^2 \left (5 a^2+39 b^2\right ) \sin (c+d x)}{3 d \sqrt {\sec (c+d x)}}+\frac {56 a b \left (3 a^2+5 b^2\right ) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{d}+\frac {10 \left (5 a^4+42 a^2 b^2+21 b^4\right ) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )}{3 d}\right )\right )\) |
(2*a^2*(a + b*Sec[c + d*x])^2*Sin[c + d*x])/(7*d*Sec[c + d*x]^(5/2)) + ((3 6*a^3*b*Sin[c + d*x])/(5*d*Sec[c + d*x]^(3/2)) + ((56*a*b*(3*a^2 + 5*b^2)* Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/d + (10*( 5*a^4 + 42*a^2*b^2 + 21*b^4)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]* Sqrt[Sec[c + d*x]])/(3*d) + (10*a^2*(5*a^2 + 39*b^2)*Sin[c + d*x])/(3*d*Sq rt[Sec[c + d*x]]))/5)/7
3.7.5.3.1 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*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[a^2*Cot[e + f*x]*(a + b*Csc[e + f*x])^(m - 2)* ((d*Csc[e + f*x])^n/(f*n)), x] - Simp[1/(d*n) Int[(a + b*Csc[e + f*x])^(m - 3)*(d*Csc[e + f*x])^(n + 1)*Simp[a^2*b*(m - 2*n - 2) - a*(3*b^2*n + a^2* (n + 1))*Csc[e + f*x] - b*(b^2*n + a^2*(m + n - 1))*Csc[e + f*x]^2, x], x], x] /; FreeQ[{a, b, d, e, f}, x] && NeQ[a^2 - b^2, 0] && GtQ[m, 2] && ((Int egerQ[m] && LtQ[n, -1]) || (IntegersQ[m + 1/2, 2*n] && LeQ[n, -1]))
Int[(csc[(e_.) + (f_.)*(x_)]*(b_.))^(m_.)*(csc[(e_.) + (f_.)*(x_)]^2*(C_.) + (A_)), x_Symbol] :> Simp[A*Cot[e + f*x]*((b*Csc[e + f*x])^m/(f*m)), x] + Simp[(C*m + A*(m + 1))/(b^2*m) Int[(b*Csc[e + f*x])^(m + 2), x], x] /; Fr eeQ[{b, e, f, A, C}, 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[A*a*Cot[e + f*x]*((d*Csc[e + f*x])^n/(f*n)), x] + Si mp[1/(d*n) Int[(d*Csc[e + f*x])^(n + 1)*Simp[n*(B*a + A*b) + (n*(a*C + B* b) + A*a*(n + 1))*Csc[e + f*x] + b*C*n*Csc[e + f*x]^2, x], x], x] /; FreeQ[ {a, b, d, e, f, A, B, C}, x] && LtQ[n, -1]
Time = 30.49 (sec) , antiderivative size = 476, normalized size of antiderivative = 2.26
| method | result | size |
| default | \(-\frac {2 \sqrt {\left (2 \cos \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-1\right ) \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}\, \left (240 a^{4} \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{8}+\left (-360 a^{4}-672 a^{3} b \right ) \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{6} \cos \left (\frac {d x}{2}+\frac {c}{2}\right )+\left (280 a^{4}+672 a^{3} b +840 a^{2} b^{2}\right ) \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{4} \cos \left (\frac {d x}{2}+\frac {c}{2}\right )+\left (-80 a^{4}-168 a^{3} b -420 a^{2} b^{2}\right ) \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} \cos \left (\frac {d x}{2}+\frac {c}{2}\right )+25 a^{4} \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {2 \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-1}\, \operatorname {EllipticF}\left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right )+210 a^{2} b^{2} \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {2 \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-1}\, \operatorname {EllipticF}\left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right )+105 b^{4} \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {2 \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-1}\, \operatorname {EllipticF}\left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right )-252 \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {2 \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-1}\, \operatorname {EllipticE}\left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right ) a^{3} b -420 \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {2 \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-1}\, \operatorname {EllipticE}\left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right ) a \,b^{3}\right )}{105 \sqrt {-2 \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}+\sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}\, \sin \left (\frac {d x}{2}+\frac {c}{2}\right ) \sqrt {2 \cos \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-1}\, d}\) | \(476\) |
| parts | \(\text {Expression too large to display}\) | \(863\) |
-2/105*((2*cos(1/2*d*x+1/2*c)^2-1)*sin(1/2*d*x+1/2*c)^2)^(1/2)*(240*a^4*co s(1/2*d*x+1/2*c)*sin(1/2*d*x+1/2*c)^8+(-360*a^4-672*a^3*b)*sin(1/2*d*x+1/2 *c)^6*cos(1/2*d*x+1/2*c)+(280*a^4+672*a^3*b+840*a^2*b^2)*sin(1/2*d*x+1/2*c )^4*cos(1/2*d*x+1/2*c)+(-80*a^4-168*a^3*b-420*a^2*b^2)*sin(1/2*d*x+1/2*c)^ 2*cos(1/2*d*x+1/2*c)+25*a^4*(sin(1/2*d*x+1/2*c)^2)^(1/2)*(2*sin(1/2*d*x+1/ 2*c)^2-1)^(1/2)*EllipticF(cos(1/2*d*x+1/2*c),2^(1/2))+210*a^2*b^2*(sin(1/2 *d*x+1/2*c)^2)^(1/2)*(2*sin(1/2*d*x+1/2*c)^2-1)^(1/2)*EllipticF(cos(1/2*d* x+1/2*c),2^(1/2))+105*b^4*(sin(1/2*d*x+1/2*c)^2)^(1/2)*(2*sin(1/2*d*x+1/2* c)^2-1)^(1/2)*EllipticF(cos(1/2*d*x+1/2*c),2^(1/2))-252*(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))*a^3*b-420*(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))*a*b^3)/(-2*sin(1/2*d*x+1/2*c)^ 4+sin(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 )^(1/2)/d
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.11 (sec) , antiderivative size = 234, normalized size of antiderivative = 1.11 \[ \int \frac {(a+b \sec (c+d x))^4}{\sec ^{\frac {7}{2}}(c+d x)} \, dx=-\frac {5 \, \sqrt {2} {\left (5 i \, a^{4} + 42 i \, a^{2} b^{2} + 21 i \, b^{4}\right )} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right ) + 5 \, \sqrt {2} {\left (-5 i \, a^{4} - 42 i \, a^{2} b^{2} - 21 i \, b^{4}\right )} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right ) + 84 \, \sqrt {2} {\left (-3 i \, a^{3} b - 5 i \, a b^{3}\right )} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right )\right ) + 84 \, \sqrt {2} {\left (3 i \, a^{3} b + 5 i \, a b^{3}\right )} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right )\right ) - \frac {2 \, {\left (15 \, a^{4} \cos \left (d x + c\right )^{3} + 84 \, a^{3} b \cos \left (d x + c\right )^{2} + 5 \, {\left (5 \, a^{4} + 42 \, a^{2} b^{2}\right )} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{\sqrt {\cos \left (d x + c\right )}}}{105 \, d} \]
-1/105*(5*sqrt(2)*(5*I*a^4 + 42*I*a^2*b^2 + 21*I*b^4)*weierstrassPInverse( -4, 0, cos(d*x + c) + I*sin(d*x + c)) + 5*sqrt(2)*(-5*I*a^4 - 42*I*a^2*b^2 - 21*I*b^4)*weierstrassPInverse(-4, 0, cos(d*x + c) - I*sin(d*x + c)) + 8 4*sqrt(2)*(-3*I*a^3*b - 5*I*a*b^3)*weierstrassZeta(-4, 0, weierstrassPInve rse(-4, 0, cos(d*x + c) + I*sin(d*x + c))) + 84*sqrt(2)*(3*I*a^3*b + 5*I*a *b^3)*weierstrassZeta(-4, 0, weierstrassPInverse(-4, 0, cos(d*x + c) - I*s in(d*x + c))) - 2*(15*a^4*cos(d*x + c)^3 + 84*a^3*b*cos(d*x + c)^2 + 5*(5* a^4 + 42*a^2*b^2)*cos(d*x + c))*sin(d*x + c)/sqrt(cos(d*x + c)))/d
\[ \int \frac {(a+b \sec (c+d x))^4}{\sec ^{\frac {7}{2}}(c+d x)} \, dx=\int \frac {\left (a + b \sec {\left (c + d x \right )}\right )^{4}}{\sec ^{\frac {7}{2}}{\left (c + d x \right )}}\, dx \]
\[ \int \frac {(a+b \sec (c+d x))^4}{\sec ^{\frac {7}{2}}(c+d x)} \, dx=\int { \frac {{\left (b \sec \left (d x + c\right ) + a\right )}^{4}}{\sec \left (d x + c\right )^{\frac {7}{2}}} \,d x } \]
\[ \int \frac {(a+b \sec (c+d x))^4}{\sec ^{\frac {7}{2}}(c+d x)} \, dx=\int { \frac {{\left (b \sec \left (d x + c\right ) + a\right )}^{4}}{\sec \left (d x + c\right )^{\frac {7}{2}}} \,d x } \]
Timed out. \[ \int \frac {(a+b \sec (c+d x))^4}{\sec ^{\frac {7}{2}}(c+d x)} \, dx=\int \frac {{\left (a+\frac {b}{\cos \left (c+d\,x\right )}\right )}^4}{{\left (\frac {1}{\cos \left (c+d\,x\right )}\right )}^{7/2}} \,d x \]