Integrand size = 23, antiderivative size = 126 \[ \int \frac {\sec ^{\frac {7}{2}}(c+d x)}{\sqrt {1+\sec (c+d x)}} \, dx=-\frac {\sqrt {2} \text {arcsinh}\left (\frac {\tan (c+d x)}{1+\sec (c+d x)}\right )}{d}+\frac {7 \text {arcsinh}\left (\frac {\tan (c+d x)}{\sqrt {1+\sec (c+d x)}}\right )}{4 d}-\frac {\sec ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{4 d \sqrt {1+\sec (c+d x)}}+\frac {\sec ^{\frac {5}{2}}(c+d x) \sin (c+d x)}{2 d \sqrt {1+\sec (c+d x)}} \]
7/4*arcsinh(tan(d*x+c)/(1+sec(d*x+c))^(1/2))/d-arcsinh(tan(d*x+c)/(1+sec(d *x+c)))*2^(1/2)/d-1/4*sec(d*x+c)^(3/2)*sin(d*x+c)/d/(1+sec(d*x+c))^(1/2)+1 /2*sec(d*x+c)^(5/2)*sin(d*x+c)/d/(1+sec(d*x+c))^(1/2)
Time = 0.44 (sec) , antiderivative size = 140, normalized size of antiderivative = 1.11 \[ \int \frac {\sec ^{\frac {7}{2}}(c+d x)}{\sqrt {1+\sec (c+d x)}} \, dx=\frac {\cot (c+d x) \left (\arcsin \left (\sqrt {1-\sec (c+d x)}\right )+8 \arcsin \left (\sqrt {\sec (c+d x)}\right )-4 \sqrt {2} \arctan \left (\frac {\sqrt {2} \sqrt {\sec (c+d x)}}{\sqrt {1-\sec (c+d x)}}\right )-2 \sqrt {1-\sec (c+d x)} \sec ^{\frac {3}{2}}(c+d x)+\sqrt {-((-1+\sec (c+d x)) \sec (c+d x))}\right ) \sqrt {-\tan ^2(c+d x)}}{4 d} \]
(Cot[c + d*x]*(ArcSin[Sqrt[1 - Sec[c + d*x]]] + 8*ArcSin[Sqrt[Sec[c + d*x] ]] - 4*Sqrt[2]*ArcTan[(Sqrt[2]*Sqrt[Sec[c + d*x]])/Sqrt[1 - Sec[c + d*x]]] - 2*Sqrt[1 - Sec[c + d*x]]*Sec[c + d*x]^(3/2) + Sqrt[-((-1 + Sec[c + d*x] )*Sec[c + d*x])])*Sqrt[-Tan[c + d*x]^2])/(4*d)
Time = 0.82 (sec) , antiderivative size = 132, normalized size of antiderivative = 1.05, number of steps used = 13, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.522, Rules used = {3042, 4309, 3042, 4509, 27, 3042, 4511, 3042, 4288, 222, 4294, 222}
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.
| \(\displaystyle \int \frac {\sec ^{\frac {7}{2}}(c+d x)}{\sqrt {\sec (c+d x)+1}} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\csc \left (c+d x+\frac {\pi }{2}\right )^{7/2}}{\sqrt {\csc \left (c+d x+\frac {\pi }{2}\right )+1}}dx\) |
| \(\Big \downarrow \) 4309 |
| \(\displaystyle \frac {1}{4} \int \frac {(3-\sec (c+d x)) \sec ^{\frac {3}{2}}(c+d x)}{\sqrt {\sec (c+d x)+1}}dx+\frac {\sin (c+d x) \sec ^{\frac {5}{2}}(c+d x)}{2 d \sqrt {\sec (c+d x)+1}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{4} \int \frac {\left (3-\csc \left (c+d x+\frac {\pi }{2}\right )\right ) \csc \left (c+d x+\frac {\pi }{2}\right )^{3/2}}{\sqrt {\csc \left (c+d x+\frac {\pi }{2}\right )+1}}dx+\frac {\sin (c+d x) \sec ^{\frac {5}{2}}(c+d x)}{2 d \sqrt {\sec (c+d x)+1}}\) |
| \(\Big \downarrow \) 4509 |
| \(\displaystyle \frac {1}{4} \left (\int -\frac {(1-7 \sec (c+d x)) \sqrt {\sec (c+d x)}}{2 \sqrt {\sec (c+d x)+1}}dx-\frac {\sin (c+d x) \sec ^{\frac {3}{2}}(c+d x)}{d \sqrt {\sec (c+d x)+1}}\right )+\frac {\sin (c+d x) \sec ^{\frac {5}{2}}(c+d x)}{2 d \sqrt {\sec (c+d x)+1}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{4} \left (-\frac {1}{2} \int \frac {(1-7 \sec (c+d x)) \sqrt {\sec (c+d x)}}{\sqrt {\sec (c+d x)+1}}dx-\frac {\sin (c+d x) \sec ^{\frac {3}{2}}(c+d x)}{d \sqrt {\sec (c+d x)+1}}\right )+\frac {\sin (c+d x) \sec ^{\frac {5}{2}}(c+d x)}{2 d \sqrt {\sec (c+d x)+1}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{4} \left (-\frac {1}{2} \int \frac {\left (1-7 \csc \left (c+d x+\frac {\pi }{2}\right )\right ) \sqrt {\csc \left (c+d x+\frac {\pi }{2}\right )}}{\sqrt {\csc \left (c+d x+\frac {\pi }{2}\right )+1}}dx-\frac {\sin (c+d x) \sec ^{\frac {3}{2}}(c+d x)}{d \sqrt {\sec (c+d x)+1}}\right )+\frac {\sin (c+d x) \sec ^{\frac {5}{2}}(c+d x)}{2 d \sqrt {\sec (c+d x)+1}}\) |
| \(\Big \downarrow \) 4511 |
| \(\displaystyle \frac {1}{4} \left (\frac {1}{2} \left (7 \int \sqrt {\sec (c+d x)} \sqrt {\sec (c+d x)+1}dx-8 \int \frac {\sqrt {\sec (c+d x)}}{\sqrt {\sec (c+d x)+1}}dx\right )-\frac {\sin (c+d x) \sec ^{\frac {3}{2}}(c+d x)}{d \sqrt {\sec (c+d x)+1}}\right )+\frac {\sin (c+d x) \sec ^{\frac {5}{2}}(c+d x)}{2 d \sqrt {\sec (c+d x)+1}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{4} \left (\frac {1}{2} \left (7 \int \sqrt {\csc \left (c+d x+\frac {\pi }{2}\right )} \sqrt {\csc \left (c+d x+\frac {\pi }{2}\right )+1}dx-8 \int \frac {\sqrt {\csc \left (c+d x+\frac {\pi }{2}\right )}}{\sqrt {\csc \left (c+d x+\frac {\pi }{2}\right )+1}}dx\right )-\frac {\sin (c+d x) \sec ^{\frac {3}{2}}(c+d x)}{d \sqrt {\sec (c+d x)+1}}\right )+\frac {\sin (c+d x) \sec ^{\frac {5}{2}}(c+d x)}{2 d \sqrt {\sec (c+d x)+1}}\) |
| \(\Big \downarrow \) 4288 |
| \(\displaystyle \frac {1}{4} \left (\frac {1}{2} \left (-8 \int \frac {\sqrt {\csc \left (c+d x+\frac {\pi }{2}\right )}}{\sqrt {\csc \left (c+d x+\frac {\pi }{2}\right )+1}}dx-\frac {14 \int \frac {1}{\sqrt {\frac {\tan ^2(c+d x)}{\sec (c+d x)+1}+1}}d\left (-\frac {\tan (c+d x)}{\sqrt {\sec (c+d x)+1}}\right )}{d}\right )-\frac {\sin (c+d x) \sec ^{\frac {3}{2}}(c+d x)}{d \sqrt {\sec (c+d x)+1}}\right )+\frac {\sin (c+d x) \sec ^{\frac {5}{2}}(c+d x)}{2 d \sqrt {\sec (c+d x)+1}}\) |
| \(\Big \downarrow \) 222 |
| \(\displaystyle \frac {1}{4} \left (\frac {1}{2} \left (\frac {14 \text {arcsinh}\left (\frac {\tan (c+d x)}{\sqrt {\sec (c+d x)+1}}\right )}{d}-8 \int \frac {\sqrt {\csc \left (c+d x+\frac {\pi }{2}\right )}}{\sqrt {\csc \left (c+d x+\frac {\pi }{2}\right )+1}}dx\right )-\frac {\sin (c+d x) \sec ^{\frac {3}{2}}(c+d x)}{d \sqrt {\sec (c+d x)+1}}\right )+\frac {\sin (c+d x) \sec ^{\frac {5}{2}}(c+d x)}{2 d \sqrt {\sec (c+d x)+1}}\) |
| \(\Big \downarrow \) 4294 |
| \(\displaystyle \frac {1}{4} \left (\frac {1}{2} \left (\frac {8 \sqrt {2} \int \frac {1}{\sqrt {\frac {\tan ^2(c+d x)}{(\sec (c+d x)+1)^2}+1}}d\left (-\frac {\tan (c+d x)}{\sec (c+d x)+1}\right )}{d}+\frac {14 \text {arcsinh}\left (\frac {\tan (c+d x)}{\sqrt {\sec (c+d x)+1}}\right )}{d}\right )-\frac {\sin (c+d x) \sec ^{\frac {3}{2}}(c+d x)}{d \sqrt {\sec (c+d x)+1}}\right )+\frac {\sin (c+d x) \sec ^{\frac {5}{2}}(c+d x)}{2 d \sqrt {\sec (c+d x)+1}}\) |
| \(\Big \downarrow \) 222 |
| \(\displaystyle \frac {1}{4} \left (\frac {1}{2} \left (\frac {14 \text {arcsinh}\left (\frac {\tan (c+d x)}{\sqrt {\sec (c+d x)+1}}\right )}{d}-\frac {8 \sqrt {2} \text {arcsinh}\left (\frac {\tan (c+d x)}{\sec (c+d x)+1}\right )}{d}\right )-\frac {\sin (c+d x) \sec ^{\frac {3}{2}}(c+d x)}{d \sqrt {\sec (c+d x)+1}}\right )+\frac {\sin (c+d x) \sec ^{\frac {5}{2}}(c+d x)}{2 d \sqrt {\sec (c+d x)+1}}\) |
(Sec[c + d*x]^(5/2)*Sin[c + d*x])/(2*d*Sqrt[1 + Sec[c + d*x]]) + (((-8*Sqr t[2]*ArcSinh[Tan[c + d*x]/(1 + Sec[c + d*x])])/d + (14*ArcSinh[Tan[c + d*x ]/Sqrt[1 + Sec[c + d*x]]])/d)/2 - (Sec[c + d*x]^(3/2)*Sin[c + d*x])/(d*Sqr t[1 + Sec[c + d*x]]))/4
3.3.65.3.1 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[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Simp[ArcSinh[Rt[b, 2]*(x/Sqrt [a])]/Rt[b, 2], x] /; FreeQ[{a, b}, x] && GtQ[a, 0] && PosQ[b]
Int[Sqrt[csc[(e_.) + (f_.)*(x_)]*(d_.)]*Sqrt[csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_)], x_Symbol] :> Simp[-2*(a/(b*f))*Sqrt[a*(d/b)] Subst[Int[1/Sqrt[1 + x^2/a], x], x, b*(Cot[e + f*x]/Sqrt[a + b*Csc[e + f*x]])], x] /; FreeQ[{a , b, d, e, f}, x] && EqQ[a^2 - b^2, 0] && GtQ[a*(d/b), 0]
Int[Sqrt[csc[(e_.) + (f_.)*(x_)]*(d_.)]/Sqrt[csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_)], x_Symbol] :> Simp[(-Sqrt[2])*(Sqrt[a]/(b*f)) Subst[Int[1/Sqrt[1 + x^2], x], x, b*(Cot[e + f*x]/(a + b*Csc[e + f*x]))], x] /; FreeQ[{a, b, d , e, f}, x] && EqQ[a^2 - b^2, 0] && EqQ[d - a/b, 0] && GtQ[a, 0]
Int[(csc[(e_.) + (f_.)*(x_)]*(d_.))^(n_)/Sqrt[csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_)], x_Symbol] :> Simp[-2*d^2*Cot[e + f*x]*((d*Csc[e + f*x])^(n - 2)/( f*(2*n - 3)*Sqrt[a + b*Csc[e + f*x]])), x] + Simp[d^2/(b*(2*n - 3)) Int[( d*Csc[e + f*x])^(n - 2)*((2*b*(n - 2) - a*Csc[e + f*x])/Sqrt[a + b*Csc[e + f*x]]), x], x] /; FreeQ[{a, b, d, e, f}, x] && EqQ[a^2 - b^2, 0] && GtQ[n, 2] && IntegerQ[2*n]
Int[(csc[(e_.) + (f_.)*(x_)]*(d_.))^(n_)*(csc[(e_.) + (f_.)*(x_)]*(b_.) + ( a_))^(m_)*(csc[(e_.) + (f_.)*(x_)]*(B_.) + (A_)), x_Symbol] :> Simp[(-B)*d* Cot[e + f*x]*(a + b*Csc[e + f*x])^m*((d*Csc[e + f*x])^(n - 1)/(f*(m + n))), x] + Simp[d/(b*(m + n)) Int[(a + b*Csc[e + f*x])^m*(d*Csc[e + f*x])^(n - 1)*Simp[b*B*(n - 1) + (A*b*(m + n) + a*B*m)*Csc[e + f*x], x], x], x] /; Fr eeQ[{a, b, d, e, f, A, B, m}, x] && NeQ[A*b - a*B, 0] && EqQ[a^2 - b^2, 0] && GtQ[n, 1]
Int[(csc[(e_.) + (f_.)*(x_)]*(d_.))^(n_)*(csc[(e_.) + (f_.)*(x_)]*(b_.) + ( a_))^(m_)*(csc[(e_.) + (f_.)*(x_)]*(B_.) + (A_)), x_Symbol] :> Simp[(A*b - a*B)/b Int[(a + b*Csc[e + f*x])^m*(d*Csc[e + f*x])^n, x], x] + Simp[B/b Int[(a + b*Csc[e + f*x])^(m + 1)*(d*Csc[e + f*x])^n, x], x] /; FreeQ[{a, b , d, e, f, A, B, m}, x] && NeQ[A*b - a*B, 0] && EqQ[a^2 - b^2, 0]
Leaf count of result is larger than twice the leaf count of optimal. \(261\) vs. \(2(108)=216\).
Time = 1.63 (sec) , antiderivative size = 262, normalized size of antiderivative = 2.08
| method | result | size |
| default | \(\frac {\sqrt {1+\sec \left (d x +c \right )}\, \sec \left (d x +c \right )^{\frac {7}{2}} \left (8 \arctan \left (\frac {\sin \left (d x +c \right ) \sqrt {2}}{2 \left (\cos \left (d x +c \right )+1\right ) \sqrt {-\frac {1}{\cos \left (d x +c \right )+1}}}\right ) \sqrt {2}\, \cos \left (d x +c \right )^{4}+7 \arctan \left (\frac {\cos \left (d x +c \right )-\sin \left (d x +c \right )+1}{2 \left (\cos \left (d x +c \right )+1\right ) \sqrt {-\frac {1}{\cos \left (d x +c \right )+1}}}\right ) \cos \left (d x +c \right )^{4}-7 \arctan \left (\frac {\cos \left (d x +c \right )+\sin \left (d x +c \right )+1}{2 \left (\cos \left (d x +c \right )+1\right ) \sqrt {-\frac {1}{\cos \left (d x +c \right )+1}}}\right ) \cos \left (d x +c \right )^{4}-2 \sin \left (d x +c \right ) \sqrt {-\frac {1}{\cos \left (d x +c \right )+1}}\, \cos \left (d x +c \right )^{3}+4 \sqrt {-\frac {1}{\cos \left (d x +c \right )+1}}\, \sin \left (d x +c \right ) \cos \left (d x +c \right )^{2}\right )}{8 d \left (\cos \left (d x +c \right )+1\right ) \sqrt {-\frac {1}{\cos \left (d x +c \right )+1}}}\) | \(262\) |
1/8/d*(1+sec(d*x+c))^(1/2)*sec(d*x+c)^(7/2)/(cos(d*x+c)+1)/(-1/(cos(d*x+c) +1))^(1/2)*(8*arctan(1/2*sin(d*x+c)*2^(1/2)/(cos(d*x+c)+1)/(-1/(cos(d*x+c) +1))^(1/2))*2^(1/2)*cos(d*x+c)^4+7*arctan(1/2*(cos(d*x+c)-sin(d*x+c)+1)/(c os(d*x+c)+1)/(-1/(cos(d*x+c)+1))^(1/2))*cos(d*x+c)^4-7*arctan(1/2*(cos(d*x +c)+sin(d*x+c)+1)/(cos(d*x+c)+1)/(-1/(cos(d*x+c)+1))^(1/2))*cos(d*x+c)^4-2 *sin(d*x+c)*(-1/(cos(d*x+c)+1))^(1/2)*cos(d*x+c)^3+4*(-1/(cos(d*x+c)+1))^( 1/2)*sin(d*x+c)*cos(d*x+c)^2)
Leaf count of result is larger than twice the leaf count of optimal. 337 vs. \(2 (108) = 216\).
Time = 0.30 (sec) , antiderivative size = 337, normalized size of antiderivative = 2.67 \[ \int \frac {\sec ^{\frac {7}{2}}(c+d x)}{\sqrt {1+\sec (c+d x)}} \, dx=\frac {8 \, {\left (\sqrt {2} \cos \left (d x + c\right )^{2} + \sqrt {2} \cos \left (d x + c\right )\right )} \log \left (-\frac {2 \, \sqrt {2} \sqrt {\frac {\cos \left (d x + c\right ) + 1}{\cos \left (d x + c\right )}} \sqrt {\cos \left (d x + c\right )} \sin \left (d x + c\right ) + \cos \left (d x + c\right )^{2} - 2 \, \cos \left (d x + c\right ) - 3}{\cos \left (d x + c\right )^{2} + 2 \, \cos \left (d x + c\right ) + 1}\right ) - 7 \, {\left (\cos \left (d x + c\right )^{2} + \cos \left (d x + c\right )\right )} \log \left (-\frac {\cos \left (d x + c\right )^{2} + 2 \, \sqrt {\frac {\cos \left (d x + c\right ) + 1}{\cos \left (d x + c\right )}} \sqrt {\cos \left (d x + c\right )} \sin \left (d x + c\right ) - \cos \left (d x + c\right ) - 2}{\cos \left (d x + c\right ) + 1}\right ) + 7 \, {\left (\cos \left (d x + c\right )^{2} + \cos \left (d x + c\right )\right )} \log \left (-\frac {\cos \left (d x + c\right )^{2} - 2 \, \sqrt {\frac {\cos \left (d x + c\right ) + 1}{\cos \left (d x + c\right )}} \sqrt {\cos \left (d x + c\right )} \sin \left (d x + c\right ) - \cos \left (d x + c\right ) - 2}{\cos \left (d x + c\right ) + 1}\right ) - \frac {4 \, \sqrt {\frac {\cos \left (d x + c\right ) + 1}{\cos \left (d x + c\right )}} {\left (\cos \left (d x + c\right ) - 2\right )} \sin \left (d x + c\right )}{\sqrt {\cos \left (d x + c\right )}}}{16 \, {\left (d \cos \left (d x + c\right )^{2} + d \cos \left (d x + c\right )\right )}} \]
1/16*(8*(sqrt(2)*cos(d*x + c)^2 + sqrt(2)*cos(d*x + c))*log(-(2*sqrt(2)*sq rt((cos(d*x + c) + 1)/cos(d*x + c))*sqrt(cos(d*x + c))*sin(d*x + c) + cos( d*x + c)^2 - 2*cos(d*x + c) - 3)/(cos(d*x + c)^2 + 2*cos(d*x + c) + 1)) - 7*(cos(d*x + c)^2 + cos(d*x + c))*log(-(cos(d*x + c)^2 + 2*sqrt((cos(d*x + c) + 1)/cos(d*x + c))*sqrt(cos(d*x + c))*sin(d*x + c) - cos(d*x + c) - 2) /(cos(d*x + c) + 1)) + 7*(cos(d*x + c)^2 + cos(d*x + c))*log(-(cos(d*x + c )^2 - 2*sqrt((cos(d*x + c) + 1)/cos(d*x + c))*sqrt(cos(d*x + c))*sin(d*x + c) - cos(d*x + c) - 2)/(cos(d*x + c) + 1)) - 4*sqrt((cos(d*x + c) + 1)/co s(d*x + c))*(cos(d*x + c) - 2)*sin(d*x + c)/sqrt(cos(d*x + c)))/(d*cos(d*x + c)^2 + d*cos(d*x + c))
Timed out. \[ \int \frac {\sec ^{\frac {7}{2}}(c+d x)}{\sqrt {1+\sec (c+d x)}} \, dx=\text {Timed out} \]
Leaf count of result is larger than twice the leaf count of optimal. 1643 vs. \(2 (108) = 216\).
Time = 0.44 (sec) , antiderivative size = 1643, normalized size of antiderivative = 13.04 \[ \int \frac {\sec ^{\frac {7}{2}}(c+d x)}{\sqrt {1+\sec (c+d x)}} \, dx=\text {Too large to display} \]
1/16*(4*(sqrt(2)*sin(4*d*x + 4*c) + 2*sqrt(2)*sin(2*d*x + 2*c))*cos(7/2*ar ctan2(sin(d*x + c), cos(d*x + c))) - 20*(sqrt(2)*sin(4*d*x + 4*c) + 2*sqrt (2)*sin(2*d*x + 2*c))*cos(5/2*arctan2(sin(d*x + c), cos(d*x + c))) + 20*(s qrt(2)*sin(4*d*x + 4*c) + 2*sqrt(2)*sin(2*d*x + 2*c))*cos(3/2*arctan2(sin( d*x + c), cos(d*x + c))) - 4*(sqrt(2)*sin(4*d*x + 4*c) + 2*sqrt(2)*sin(2*d *x + 2*c))*cos(1/2*arctan2(sin(d*x + c), cos(d*x + c))) + 7*(2*(2*cos(2*d* x + 2*c) + 1)*cos(4*d*x + 4*c) + cos(4*d*x + 4*c)^2 + 4*cos(2*d*x + 2*c)^2 + sin(4*d*x + 4*c)^2 + 4*sin(4*d*x + 4*c)*sin(2*d*x + 2*c) + 4*sin(2*d*x + 2*c)^2 + 4*cos(2*d*x + 2*c) + 1)*log(2*cos(1/2*arctan2(sin(d*x + c), cos (d*x + c)))^2 + 2*sin(1/2*arctan2(sin(d*x + c), cos(d*x + c)))^2 + 2*sqrt( 2)*cos(1/2*arctan2(sin(d*x + c), cos(d*x + c))) + 2*sqrt(2)*sin(1/2*arctan 2(sin(d*x + c), cos(d*x + c))) + 2) - 7*(2*(2*cos(2*d*x + 2*c) + 1)*cos(4* d*x + 4*c) + cos(4*d*x + 4*c)^2 + 4*cos(2*d*x + 2*c)^2 + sin(4*d*x + 4*c)^ 2 + 4*sin(4*d*x + 4*c)*sin(2*d*x + 2*c) + 4*sin(2*d*x + 2*c)^2 + 4*cos(2*d *x + 2*c) + 1)*log(2*cos(1/2*arctan2(sin(d*x + c), cos(d*x + c)))^2 + 2*si n(1/2*arctan2(sin(d*x + c), cos(d*x + c)))^2 + 2*sqrt(2)*cos(1/2*arctan2(s in(d*x + c), cos(d*x + c))) - 2*sqrt(2)*sin(1/2*arctan2(sin(d*x + c), cos( d*x + c))) + 2) + 7*(2*(2*cos(2*d*x + 2*c) + 1)*cos(4*d*x + 4*c) + cos(4*d *x + 4*c)^2 + 4*cos(2*d*x + 2*c)^2 + sin(4*d*x + 4*c)^2 + 4*sin(4*d*x + 4* c)*sin(2*d*x + 2*c) + 4*sin(2*d*x + 2*c)^2 + 4*cos(2*d*x + 2*c) + 1)*lo...
\[ \int \frac {\sec ^{\frac {7}{2}}(c+d x)}{\sqrt {1+\sec (c+d x)}} \, dx=\int { \frac {\sec \left (d x + c\right )^{\frac {7}{2}}}{\sqrt {\sec \left (d x + c\right ) + 1}} \,d x } \]
Timed out. \[ \int \frac {\sec ^{\frac {7}{2}}(c+d x)}{\sqrt {1+\sec (c+d x)}} \, dx=\int \frac {{\left (\frac {1}{\cos \left (c+d\,x\right )}\right )}^{7/2}}{\sqrt {\frac {1}{\cos \left (c+d\,x\right )}+1}} \,d x \]