Integrand size = 23, antiderivative size = 188 \[ \int \frac {\sec ^{\frac {5}{2}}(c+d x)}{a+b \cos (c+d x)} \, dx=\frac {2 b \sqrt {\cos (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \sqrt {\sec (c+d x)}}{a^2 d}+\frac {2 \sqrt {\cos (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right ) \sqrt {\sec (c+d x)}}{3 a d}+\frac {2 b^2 \sqrt {\cos (c+d x)} \operatorname {EllipticPi}\left (\frac {2 b}{a+b},\frac {1}{2} (c+d x),2\right ) \sqrt {\sec (c+d x)}}{a^2 (a+b) d}-\frac {2 b \sqrt {\sec (c+d x)} \sin (c+d x)}{a^2 d}+\frac {2 \sec ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{3 a d} \]
2/3*sec(d*x+c)^(3/2)*sin(d*x+c)/a/d-2*b*sin(d*x+c)*sec(d*x+c)^(1/2)/a^2/d+ 2*b*(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)/a^2/d+2/3*(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))*co s(d*x+c)^(1/2)*sec(d*x+c)^(1/2)/a/d+2*b^2*(cos(1/2*d*x+1/2*c)^2)^(1/2)/cos (1/2*d*x+1/2*c)*EllipticPi(sin(1/2*d*x+1/2*c),2*b/(a+b),2^(1/2))*cos(d*x+c )^(1/2)*sec(d*x+c)^(1/2)/a^2/(a+b)/d
Time = 35.18 (sec) , antiderivative size = 165, normalized size of antiderivative = 0.88 \[ \int \frac {\sec ^{\frac {5}{2}}(c+d x)}{a+b \cos (c+d x)} \, dx=-\frac {\cot (c+d x) \left (-a^2 \sec ^{\frac {5}{2}}(c+d x)+a^2 \cos (2 (c+d x)) \sec ^{\frac {5}{2}}(c+d x)+6 a b E\left (\left .\arcsin \left (\sqrt {\sec (c+d x)}\right )\right |-1\right ) \sqrt {-\tan ^2(c+d x)}-2 \left (a^2+3 a b+3 b^2\right ) \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\sec (c+d x)}\right ),-1\right ) \sqrt {-\tan ^2(c+d x)}+6 b^2 \operatorname {EllipticPi}\left (-\frac {a}{b},\arcsin \left (\sqrt {\sec (c+d x)}\right ),-1\right ) \sqrt {-\tan ^2(c+d x)}\right )}{3 a^3 d} \]
-1/3*(Cot[c + d*x]*(-(a^2*Sec[c + d*x]^(5/2)) + a^2*Cos[2*(c + d*x)]*Sec[c + d*x]^(5/2) + 6*a*b*EllipticE[ArcSin[Sqrt[Sec[c + d*x]]], -1]*Sqrt[-Tan[ c + d*x]^2] - 2*(a^2 + 3*a*b + 3*b^2)*EllipticF[ArcSin[Sqrt[Sec[c + d*x]]] , -1]*Sqrt[-Tan[c + d*x]^2] + 6*b^2*EllipticPi[-(a/b), ArcSin[Sqrt[Sec[c + d*x]]], -1]*Sqrt[-Tan[c + d*x]^2]))/(a^3*d)
Time = 1.66 (sec) , antiderivative size = 201, normalized size of antiderivative = 1.07, number of steps used = 20, number of rules used = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.870, Rules used = {3042, 3717, 3042, 4338, 27, 3042, 4590, 27, 3042, 4594, 3042, 4274, 3042, 4258, 3042, 3119, 3120, 4336, 3042, 3284}
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.
\(\displaystyle \int \frac {\sec ^{\frac {5}{2}}(c+d x)}{a+b \cos (c+d x)} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {\csc \left (c+d x+\frac {\pi }{2}\right )^{5/2}}{a+b \sin \left (c+d x+\frac {\pi }{2}\right )}dx\) |
\(\Big \downarrow \) 3717 |
\(\displaystyle \int \frac {\sec ^{\frac {7}{2}}(c+d x)}{a \sec (c+d x)+b}dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {\csc \left (c+d x+\frac {\pi }{2}\right )^{7/2}}{a \csc \left (c+d x+\frac {\pi }{2}\right )+b}dx\) |
\(\Big \downarrow \) 4338 |
\(\displaystyle \frac {2 \int \frac {\sqrt {\sec (c+d x)} \left (-3 b \sec ^2(c+d x)+a \sec (c+d x)+b\right )}{2 (b+a \sec (c+d x))}dx}{3 a}+\frac {2 \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x)}{3 a d}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {\int \frac {\sqrt {\sec (c+d x)} \left (-3 b \sec ^2(c+d x)+a \sec (c+d x)+b\right )}{b+a \sec (c+d x)}dx}{3 a}+\frac {2 \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x)}{3 a d}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {\int \frac {\sqrt {\csc \left (c+d x+\frac {\pi }{2}\right )} \left (-3 b \csc \left (c+d x+\frac {\pi }{2}\right )^2+a \csc \left (c+d x+\frac {\pi }{2}\right )+b\right )}{b+a \csc \left (c+d x+\frac {\pi }{2}\right )}dx}{3 a}+\frac {2 \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x)}{3 a d}\) |
\(\Big \downarrow \) 4590 |
\(\displaystyle \frac {\frac {2 \int \frac {3 b^2+4 a \sec (c+d x) b+\left (a^2+3 b^2\right ) \sec ^2(c+d x)}{2 \sqrt {\sec (c+d x)} (b+a \sec (c+d x))}dx}{a}-\frac {6 b \sin (c+d x) \sqrt {\sec (c+d x)}}{a d}}{3 a}+\frac {2 \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x)}{3 a d}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {\frac {\int \frac {3 b^2+4 a \sec (c+d x) b+\left (a^2+3 b^2\right ) \sec ^2(c+d x)}{\sqrt {\sec (c+d x)} (b+a \sec (c+d x))}dx}{a}-\frac {6 b \sin (c+d x) \sqrt {\sec (c+d x)}}{a d}}{3 a}+\frac {2 \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x)}{3 a d}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {\frac {\int \frac {3 b^2+4 a \csc \left (c+d x+\frac {\pi }{2}\right ) b+\left (a^2+3 b^2\right ) \csc \left (c+d x+\frac {\pi }{2}\right )^2}{\sqrt {\csc \left (c+d x+\frac {\pi }{2}\right )} \left (b+a \csc \left (c+d x+\frac {\pi }{2}\right )\right )}dx}{a}-\frac {6 b \sin (c+d x) \sqrt {\sec (c+d x)}}{a d}}{3 a}+\frac {2 \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x)}{3 a d}\) |
\(\Big \downarrow \) 4594 |
\(\displaystyle \frac {\frac {3 b^2 \int \frac {\sec ^{\frac {3}{2}}(c+d x)}{b+a \sec (c+d x)}dx+\frac {\int \frac {3 b^3+a \sec (c+d x) b^2}{\sqrt {\sec (c+d x)}}dx}{b^2}}{a}-\frac {6 b \sin (c+d x) \sqrt {\sec (c+d x)}}{a d}}{3 a}+\frac {2 \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x)}{3 a d}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {\frac {3 b^2 \int \frac {\csc \left (c+d x+\frac {\pi }{2}\right )^{3/2}}{b+a \csc \left (c+d x+\frac {\pi }{2}\right )}dx+\frac {\int \frac {3 b^3+a \csc \left (c+d x+\frac {\pi }{2}\right ) b^2}{\sqrt {\csc \left (c+d x+\frac {\pi }{2}\right )}}dx}{b^2}}{a}-\frac {6 b \sin (c+d x) \sqrt {\sec (c+d x)}}{a d}}{3 a}+\frac {2 \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x)}{3 a d}\) |
\(\Big \downarrow \) 4274 |
\(\displaystyle \frac {\frac {3 b^2 \int \frac {\csc \left (c+d x+\frac {\pi }{2}\right )^{3/2}}{b+a \csc \left (c+d x+\frac {\pi }{2}\right )}dx+\frac {a b^2 \int \sqrt {\sec (c+d x)}dx+3 b^3 \int \frac {1}{\sqrt {\sec (c+d x)}}dx}{b^2}}{a}-\frac {6 b \sin (c+d x) \sqrt {\sec (c+d x)}}{a d}}{3 a}+\frac {2 \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x)}{3 a d}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {\frac {3 b^2 \int \frac {\csc \left (c+d x+\frac {\pi }{2}\right )^{3/2}}{b+a \csc \left (c+d x+\frac {\pi }{2}\right )}dx+\frac {a b^2 \int \sqrt {\csc \left (c+d x+\frac {\pi }{2}\right )}dx+3 b^3 \int \frac {1}{\sqrt {\csc \left (c+d x+\frac {\pi }{2}\right )}}dx}{b^2}}{a}-\frac {6 b \sin (c+d x) \sqrt {\sec (c+d x)}}{a d}}{3 a}+\frac {2 \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x)}{3 a d}\) |
\(\Big \downarrow \) 4258 |
\(\displaystyle \frac {\frac {3 b^2 \int \frac {\csc \left (c+d x+\frac {\pi }{2}\right )^{3/2}}{b+a \csc \left (c+d x+\frac {\pi }{2}\right )}dx+\frac {a b^2 \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \int \frac {1}{\sqrt {\cos (c+d x)}}dx+3 b^3 \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \int \sqrt {\cos (c+d x)}dx}{b^2}}{a}-\frac {6 b \sin (c+d x) \sqrt {\sec (c+d x)}}{a d}}{3 a}+\frac {2 \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x)}{3 a d}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {\frac {3 b^2 \int \frac {\csc \left (c+d x+\frac {\pi }{2}\right )^{3/2}}{b+a \csc \left (c+d x+\frac {\pi }{2}\right )}dx+\frac {a b^2 \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \int \frac {1}{\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}}dx+3 b^3 \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \int \sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}dx}{b^2}}{a}-\frac {6 b \sin (c+d x) \sqrt {\sec (c+d x)}}{a d}}{3 a}+\frac {2 \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x)}{3 a d}\) |
\(\Big \downarrow \) 3119 |
\(\displaystyle \frac {\frac {3 b^2 \int \frac {\csc \left (c+d x+\frac {\pi }{2}\right )^{3/2}}{b+a \csc \left (c+d x+\frac {\pi }{2}\right )}dx+\frac {a b^2 \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \int \frac {1}{\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}}dx+\frac {6 b^3 \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{d}}{b^2}}{a}-\frac {6 b \sin (c+d x) \sqrt {\sec (c+d x)}}{a d}}{3 a}+\frac {2 \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x)}{3 a d}\) |
\(\Big \downarrow \) 3120 |
\(\displaystyle \frac {\frac {3 b^2 \int \frac {\csc \left (c+d x+\frac {\pi }{2}\right )^{3/2}}{b+a \csc \left (c+d x+\frac {\pi }{2}\right )}dx+\frac {\frac {2 a b^2 \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )}{d}+\frac {6 b^3 \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{d}}{b^2}}{a}-\frac {6 b \sin (c+d x) \sqrt {\sec (c+d x)}}{a d}}{3 a}+\frac {2 \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x)}{3 a d}\) |
\(\Big \downarrow \) 4336 |
\(\displaystyle \frac {\frac {3 b^2 \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \int \frac {1}{\sqrt {\cos (c+d x)} (a+b \cos (c+d x))}dx+\frac {\frac {2 a b^2 \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )}{d}+\frac {6 b^3 \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{d}}{b^2}}{a}-\frac {6 b \sin (c+d x) \sqrt {\sec (c+d x)}}{a d}}{3 a}+\frac {2 \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x)}{3 a d}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {\frac {3 b^2 \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \int \frac {1}{\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )} \left (a+b \sin \left (c+d x+\frac {\pi }{2}\right )\right )}dx+\frac {\frac {2 a b^2 \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )}{d}+\frac {6 b^3 \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{d}}{b^2}}{a}-\frac {6 b \sin (c+d x) \sqrt {\sec (c+d x)}}{a d}}{3 a}+\frac {2 \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x)}{3 a d}\) |
\(\Big \downarrow \) 3284 |
\(\displaystyle \frac {\frac {\frac {6 b^2 \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \operatorname {EllipticPi}\left (\frac {2 b}{a+b},\frac {1}{2} (c+d x),2\right )}{d (a+b)}+\frac {\frac {2 a b^2 \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )}{d}+\frac {6 b^3 \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{d}}{b^2}}{a}-\frac {6 b \sin (c+d x) \sqrt {\sec (c+d x)}}{a d}}{3 a}+\frac {2 \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x)}{3 a d}\) |
(2*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(3*a*d) + ((((6*b^3*Sqrt[Cos[c + d*x]] *EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/d + (2*a*b^2*Sqrt[Cos[c + d *x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/d)/b^2 + (6*b^2*Sqrt[Co s[c + d*x]]*EllipticPi[(2*b)/(a + b), (c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/ ((a + b)*d))/a - (6*b*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(a*d))/(3*a)
3.8.12.3.1 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[1/(((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])*Sqrt[(c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)]]), x_Symbol] :> Simp[(2/(f*(a + b)*Sqrt[c + d]))*EllipticPi[ 2*(b/(a + b)), (1/2)*(e - Pi/2 + f*x), 2*(d/(c + d))], x] /; FreeQ[{a, b, c , d, e, f}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && GtQ[c + d, 0]
Int[(csc[(e_.) + (f_.)*(x_)]*(d_.))^(m_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x _)]^(n_.))^(p_.), x_Symbol] :> Simp[d^(n*p) Int[(d*Csc[e + f*x])^(m - n*p )*(b + a*Csc[e + f*x]^n)^p, x], x] /; FreeQ[{a, b, d, e, f, m, n, p}, x] && !IntegerQ[m] && IntegersQ[n, p]
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_)), x_Symbol] :> Simp[a Int[(d*Csc[e + f*x])^n, x], x] + Simp[b/d In t[(d*Csc[e + f*x])^(n + 1), x], x] /; FreeQ[{a, b, d, e, f, n}, x]
Int[(csc[(e_.) + (f_.)*(x_)]*(d_.))^(3/2)/(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_)), x_Symbol] :> Simp[d*Sqrt[d*Sin[e + f*x]]*Sqrt[d*Csc[e + f*x]] Int[ 1/(Sqrt[d*Sin[e + f*x]]*(b + a*Sin[e + f*x])), x], x] /; FreeQ[{a, b, d, e, f}, x] && NeQ[a^2 - b^2, 0]
Int[(csc[(e_.) + (f_.)*(x_)]*(d_.))^(n_)/(csc[(e_.) + (f_.)*(x_)]*(b_.) + ( a_)), x_Symbol] :> Simp[(-d^3)*Cot[e + f*x]*((d*Csc[e + f*x])^(n - 3)/(b*f* (n - 2))), x] + Simp[d^3/(b*(n - 2)) Int[(d*Csc[e + f*x])^(n - 3)*(Simp[a *(n - 3) + b*(n - 3)*Csc[e + f*x] - a*(n - 2)*Csc[e + f*x]^2, x]/(a + b*Csc [e + f*x])), x], x] /; FreeQ[{a, b, d, e, f}, x] && NeQ[a^2 - b^2, 0] && Gt Q[n, 3]
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 _))^(m_), x_Symbol] :> Simp[(-C)*d*Cot[e + f*x]*(a + b*Csc[e + f*x])^(m + 1 )*((d*Csc[e + f*x])^(n - 1)/(b*f*(m + n + 1))), x] + Simp[d/(b*(m + n + 1)) Int[(a + b*Csc[e + f*x])^m*(d*Csc[e + f*x])^(n - 1)*Simp[a*C*(n - 1) + ( A*b*(m + n + 1) + b*C*(m + n))*Csc[e + f*x] + (b*B*(m + n + 1) - a*C*n)*Csc [e + f*x]^2, x], x], x] /; FreeQ[{a, b, d, e, f, A, B, C, m}, x] && NeQ[a^2 - b^2, 0] && GtQ[n, 0]
Int[((A_.) + csc[(e_.) + (f_.)*(x_)]*(B_.) + csc[(e_.) + (f_.)*(x_)]^2*(C_. ))/(Sqrt[csc[(e_.) + (f_.)*(x_)]*(d_.)]*(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a _))), x_Symbol] :> Simp[(A*b^2 - a*b*B + a^2*C)/(a^2*d^2) Int[(d*Csc[e + f*x])^(3/2)/(a + b*Csc[e + f*x]), x], x] + Simp[1/a^2 Int[(a*A - (A*b - a *B)*Csc[e + f*x])/Sqrt[d*Csc[e + f*x]], x], x] /; FreeQ[{a, b, d, e, f, A, B, C}, x] && NeQ[a^2 - b^2, 0]
Time = 11.16 (sec) , antiderivative size = 425, normalized size of antiderivative = 2.26
method | result | size |
default | \(-\frac {\sqrt {-\left (-2 \left (\cos ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+1\right ) \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}\, \left (\frac {-\frac {\cos \left (\frac {d x}{2}+\frac {c}{2}\right ) \sqrt {-2 \left (\sin ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )}}{3 \left (\cos ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )-\frac {1}{2}\right )^{2}}+\frac {2 \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {-2 \left (\cos ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+1}\, F\left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right )}{3 \sqrt {-2 \left (\sin ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )}}}{a}-\frac {2 b \sqrt {-2 \left (\sin ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )}\, \left (2 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \cos \left (\frac {d x}{2}+\frac {c}{2}\right )-\sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {2 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1}\, E\left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right )\right )}{a^{2} \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} \left (2 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1\right )}-\frac {4 b^{3} \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {-2 \left (\cos ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+1}\, \Pi \left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), -\frac {2 b}{a -b}, \sqrt {2}\right )}{a^{2} \left (-2 a b +2 b^{2}\right ) \sqrt {-2 \left (\sin ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )}}\right )}{\sin \left (\frac {d x}{2}+\frac {c}{2}\right ) \sqrt {2 \left (\cos ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1}\, d}\) | \(425\) |
-(-(-2*cos(1/2*d*x+1/2*c)^2+1)*sin(1/2*d*x+1/2*c)^2)^(1/2)*(2/a*(-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)*Ellip ticF(cos(1/2*d*x+1/2*c),2^(1/2)))-2/a^2*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*si n(1/2*d*x+1/2*c)^2*cos(1/2*d*x+1/2*c)-(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)))-4*b^3/a^2 /(-2*a*b+2*b^2)*(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)*EllipticPi(cos(1 /2*d*x+1/2*c),-2*b/(a-b),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
Timed out. \[ \int \frac {\sec ^{\frac {5}{2}}(c+d x)}{a+b \cos (c+d x)} \, dx=\text {Timed out} \]
Timed out. \[ \int \frac {\sec ^{\frac {5}{2}}(c+d x)}{a+b \cos (c+d x)} \, dx=\text {Timed out} \]
\[ \int \frac {\sec ^{\frac {5}{2}}(c+d x)}{a+b \cos (c+d x)} \, dx=\int { \frac {\sec \left (d x + c\right )^{\frac {5}{2}}}{b \cos \left (d x + c\right ) + a} \,d x } \]
\[ \int \frac {\sec ^{\frac {5}{2}}(c+d x)}{a+b \cos (c+d x)} \, dx=\int { \frac {\sec \left (d x + c\right )^{\frac {5}{2}}}{b \cos \left (d x + c\right ) + a} \,d x } \]
Timed out. \[ \int \frac {\sec ^{\frac {5}{2}}(c+d x)}{a+b \cos (c+d x)} \, dx=\int \frac {{\left (\frac {1}{\cos \left (c+d\,x\right )}\right )}^{5/2}}{a+b\,\cos \left (c+d\,x\right )} \,d x \]