\(\int \frac {\sec ^{\frac {5}{2}}(c+d x)}{a+b \cos (c+d x)} \, dx\) [712]

Optimal result

Integrand size = 23, antiderivative size = 188 \[ \int \frac {\sec ^{\frac {5}{2}}(c+d x)}{a+b \cos (c+d x)} \, dx=\frac {2 b \sqrt {\cos (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \sqrt {\sec (c+d x)}}{a^2 d}+\frac {2 \sqrt {\cos (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right ) \sqrt {\sec (c+d x)}}{3 a d}+\frac {2 b^2 \sqrt {\cos (c+d x)} \operatorname {EllipticPi}\left (\frac {2 b}{a+b},\frac {1}{2} (c+d x),2\right ) \sqrt {\sec (c+d x)}}{a^2 (a+b) d}-\frac {2 b \sqrt {\sec (c+d x)} \sin (c+d x)}{a^2 d}+\frac {2 \sec ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{3 a d} \] Output:

2*b*cos(d*x+c)^(1/2)*EllipticE(sin(1/2*d*x+1/2*c),2^(1/2))*sec(d*x+c)^(1/2 
)/a^2/d+2/3*cos(d*x+c)^(1/2)*InverseJacobiAM(1/2*d*x+1/2*c,2^(1/2))*sec(d* 
x+c)^(1/2)/a/d+2*b^2*cos(d*x+c)^(1/2)*EllipticPi(sin(1/2*d*x+1/2*c),2*b/(a 
+b),2^(1/2))*sec(d*x+c)^(1/2)/a^2/(a+b)/d-2*b*sec(d*x+c)^(1/2)*sin(d*x+c)/ 
a^2/d+2/3*sec(d*x+c)^(3/2)*sin(d*x+c)/a/d
 

Mathematica [A] (warning: unable to verify)

Time = 35.38 (sec) , antiderivative size = 165, normalized size of antiderivative = 0.88 \[ \int \frac {\sec ^{\frac {5}{2}}(c+d x)}{a+b \cos (c+d x)} \, dx=-\frac {\cot (c+d x) \left (-a^2 \sec ^{\frac {5}{2}}(c+d x)+a^2 \cos (2 (c+d x)) \sec ^{\frac {5}{2}}(c+d x)+6 a b E\left (\left .\arcsin \left (\sqrt {\sec (c+d x)}\right )\right |-1\right ) \sqrt {-\tan ^2(c+d x)}-2 \left (a^2+3 a b+3 b^2\right ) \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\sec (c+d x)}\right ),-1\right ) \sqrt {-\tan ^2(c+d x)}+6 b^2 \operatorname {EllipticPi}\left (-\frac {a}{b},\arcsin \left (\sqrt {\sec (c+d x)}\right ),-1\right ) \sqrt {-\tan ^2(c+d x)}\right )}{3 a^3 d} \] Input:

Integrate[Sec[c + d*x]^(5/2)/(a + b*Cos[c + d*x]),x]
 

Output:

-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)
 

Rubi [A] (verified)

Time = 1.62 (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

\(\displaystyle \frac {\frac {\frac {6 b^2 \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \operatorname {EllipticPi}\left (\frac {2 b}{a+b},\frac {1}{2} (c+d x),2\right )}{d (a+b)}+\frac {\frac {2 a b^2 \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )}{d}+\frac {6 b^3 \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{d}}{b^2}}{a}-\frac {6 b \sin (c+d x) \sqrt {\sec (c+d x)}}{a d}}{3 a}+\frac {2 \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x)}{3 a d}\)

Input:

Int[Sec[c + d*x]^(5/2)/(a + b*Cos[c + d*x]),x]
 

Output:

(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)
 

Defintions of rubi rules used

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 3042

Int[u_, x_Symbol] :> Int[DeactivateTrig[u, x], x] /; FunctionOfTrigOfLinear 
Q[u, x]
 

rule 3119

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]
 

rule 3120

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]
 

rule 3284

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]
 

rule 3717

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]
 

rule 4258

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]
 

rule 4274

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]
 

rule 4336

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]
 

rule 4338

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]
 

rule 4590

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]
 

rule 4594

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]
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(424\) vs. \(2(173)=346\).

Time = 19.20 (sec) , antiderivative size = 425, normalized size of antiderivative = 2.26

method	result	size
default	\(-\frac {\sqrt {-\left (-2 \cos \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}+1\right ) \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}\, \left (\frac {-\frac {\cos \left (\frac {d x}{2}+\frac {c}{2}\right ) \sqrt {-2 \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}+\sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}}{3 \left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-\frac {1}{2}\right )^{2}}+\frac {2 \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {-2 \cos \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}+1}\, \operatorname {EllipticF}\left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right )}{3 \sqrt {-2 \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}+\sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}}}{a}-\frac {4 b^{3} \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {-2 \cos \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}+1}\, \operatorname {EllipticPi}\left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), -\frac {2 b}{a -b}, \sqrt {2}\right )}{a^{2} \left (-2 a b +2 b^{2}\right ) \sqrt {-2 \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}+\sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}}-\frac {2 b \sqrt {-2 \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}+\sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}\, \left (2 \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} \cos \left (\frac {d x}{2}+\frac {c}{2}\right )-\sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {2 \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-1}\, \operatorname {EllipticE}\left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right )\right )}{a^{2} \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} \left (2 \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-1\right )}\right )}{\sin \left (\frac {d x}{2}+\frac {c}{2}\right ) \sqrt {2 \cos \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-1}\, d}\)	\(425\)

Input:

int(sec(d*x+c)^(5/2)/(a+cos(d*x+c)*b),x,method=_RETURNVERBOSE)
 

Output:

-(-(-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)))-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+s 
in(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 
))-2*b/a^2/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*sin(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)*Ellip 
ticE(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)^(1/2)/d
 

Fricas [F(-1)]

Timed out. \[ \int \frac {\sec ^{\frac {5}{2}}(c+d x)}{a+b \cos (c+d x)} \, dx=\text {Timed out} \] Input:

integrate(sec(d*x+c)^(5/2)/(a+b*cos(d*x+c)),x, algorithm="fricas")
 

Output:

Timed out
 

Sympy [F(-1)]

Timed out. \[ \int \frac {\sec ^{\frac {5}{2}}(c+d x)}{a+b \cos (c+d x)} \, dx=\text {Timed out} \] Input:

integrate(sec(d*x+c)**(5/2)/(a+b*cos(d*x+c)),x)
 

Output:

Timed out
 

Maxima [F]

\[ \int \frac {\sec ^{\frac {5}{2}}(c+d x)}{a+b \cos (c+d x)} \, dx=\int { \frac {\sec \left (d x + c\right )^{\frac {5}{2}}}{b \cos \left (d x + c\right ) + a} \,d x } \] Input:

integrate(sec(d*x+c)^(5/2)/(a+b*cos(d*x+c)),x, algorithm="maxima")
 

Output:

integrate(sec(d*x + c)^(5/2)/(b*cos(d*x + c) + a), x)
 

Giac [F]

\[ \int \frac {\sec ^{\frac {5}{2}}(c+d x)}{a+b \cos (c+d x)} \, dx=\int { \frac {\sec \left (d x + c\right )^{\frac {5}{2}}}{b \cos \left (d x + c\right ) + a} \,d x } \] Input:

integrate(sec(d*x+c)^(5/2)/(a+b*cos(d*x+c)),x, algorithm="giac")
 

Output:

integrate(sec(d*x + c)^(5/2)/(b*cos(d*x + c) + a), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {\sec ^{\frac {5}{2}}(c+d x)}{a+b \cos (c+d x)} \, dx=\int \frac {{\left (\frac {1}{\cos \left (c+d\,x\right )}\right )}^{5/2}}{a+b\,\cos \left (c+d\,x\right )} \,d x \] Input:

int((1/cos(c + d*x))^(5/2)/(a + b*cos(c + d*x)),x)
 

Output:

int((1/cos(c + d*x))^(5/2)/(a + b*cos(c + d*x)), x)
 

Reduce [F]

\[ \int \frac {\sec ^{\frac {5}{2}}(c+d x)}{a+b \cos (c+d x)} \, dx=\int \frac {\sqrt {\sec \left (d x +c \right )}\, \sec \left (d x +c \right )^{2}}{\cos \left (d x +c \right ) b +a}d x \] Input:

int(sec(d*x+c)^(5/2)/(a+b*cos(d*x+c)),x)
 

Output:

int((sqrt(sec(c + d*x))*sec(c + d*x)**2)/(cos(c + d*x)*b + a),x)