\(\int \frac {A+C \sec ^2(c+d x)}{(b \sec (c+d x))^{9/2}} \, dx\) [23]

Optimal result

Integrand size = 25, antiderivative size = 112 \[ \int \frac {A+C \sec ^2(c+d x)}{(b \sec (c+d x))^{9/2}} \, dx=\frac {2 (7 A+9 C) E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{15 b^4 d \sqrt {\cos (c+d x)} \sqrt {b \sec (c+d x)}}+\frac {2 (7 A+9 C) \sin (c+d x)}{45 b^3 d (b \sec (c+d x))^{3/2}}+\frac {2 A \tan (c+d x)}{9 d (b \sec (c+d x))^{9/2}} \] Output:

2/15*(7*A+9*C)*EllipticE(sin(1/2*d*x+1/2*c),2^(1/2))/b^4/d/cos(d*x+c)^(1/2 
)/(b*sec(d*x+c))^(1/2)+2/45*(7*A+9*C)*sin(d*x+c)/b^3/d/(b*sec(d*x+c))^(3/2 
)+2/9*A*tan(d*x+c)/d/(b*sec(d*x+c))^(9/2)
 

Mathematica [C] (verified)

Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.

Time = 1.18 (sec) , antiderivative size = 143, normalized size of antiderivative = 1.28 \[ \int \frac {A+C \sec ^2(c+d x)}{(b \sec (c+d x))^{9/2}} \, dx=\frac {e^{-i d x} (\cos (d x)+i \sin (d x)) \left (336 i A+432 i C-\frac {32 i (7 A+9 C) e^{2 i (c+d x)} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {3}{4},\frac {7}{4},-e^{2 i (c+d x)}\right )}{\sqrt {1+e^{2 i (c+d x)}}}+(76 A+72 C) \sin (2 (c+d x))+10 A \sin (4 (c+d x))\right )}{360 b^4 d \sqrt {b \sec (c+d x)}} \] Input:

Integrate[(A + C*Sec[c + d*x]^2)/(b*Sec[c + d*x])^(9/2),x]
 

Output:

((Cos[d*x] + I*Sin[d*x])*((336*I)*A + (432*I)*C - ((32*I)*(7*A + 9*C)*E^(( 
2*I)*(c + d*x))*Hypergeometric2F1[1/2, 3/4, 7/4, -E^((2*I)*(c + d*x))])/Sq 
rt[1 + E^((2*I)*(c + d*x))] + (76*A + 72*C)*Sin[2*(c + d*x)] + 10*A*Sin[4* 
(c + d*x)]))/(360*b^4*d*E^(I*d*x)*Sqrt[b*Sec[c + d*x]])
 

Rubi [A] (verified)

Time = 0.49 (sec) , antiderivative size = 113, normalized size of antiderivative = 1.01, number of steps used = 8, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.320, Rules used = {3042, 4533, 3042, 4256, 3042, 4258, 3042, 3119}

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.

\(\displaystyle \int \frac {A+C \sec ^2(c+d x)}{(b \sec (c+d x))^{9/2}} \, dx\)

\(\Big \downarrow \) 3042

\(\displaystyle \int \frac {A+C \csc \left (c+d x+\frac {\pi }{2}\right )^2}{\left (b \csc \left (c+d x+\frac {\pi }{2}\right )\right )^{9/2}}dx\)

\(\Big \downarrow \) 4533

\(\displaystyle \frac {(7 A+9 C) \int \frac {1}{(b \sec (c+d x))^{5/2}}dx}{9 b^2}+\frac {2 A \tan (c+d x)}{9 d (b \sec (c+d x))^{9/2}}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {(7 A+9 C) \int \frac {1}{\left (b \csc \left (c+d x+\frac {\pi }{2}\right )\right )^{5/2}}dx}{9 b^2}+\frac {2 A \tan (c+d x)}{9 d (b \sec (c+d x))^{9/2}}\)

\(\Big \downarrow \) 4256

\(\displaystyle \frac {(7 A+9 C) \left (\frac {3 \int \frac {1}{\sqrt {b \sec (c+d x)}}dx}{5 b^2}+\frac {2 \sin (c+d x)}{5 b d (b \sec (c+d x))^{3/2}}\right )}{9 b^2}+\frac {2 A \tan (c+d x)}{9 d (b \sec (c+d x))^{9/2}}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {(7 A+9 C) \left (\frac {3 \int \frac {1}{\sqrt {b \csc \left (c+d x+\frac {\pi }{2}\right )}}dx}{5 b^2}+\frac {2 \sin (c+d x)}{5 b d (b \sec (c+d x))^{3/2}}\right )}{9 b^2}+\frac {2 A \tan (c+d x)}{9 d (b \sec (c+d x))^{9/2}}\)

\(\Big \downarrow \) 4258

\(\displaystyle \frac {(7 A+9 C) \left (\frac {3 \int \sqrt {\cos (c+d x)}dx}{5 b^2 \sqrt {\cos (c+d x)} \sqrt {b \sec (c+d x)}}+\frac {2 \sin (c+d x)}{5 b d (b \sec (c+d x))^{3/2}}\right )}{9 b^2}+\frac {2 A \tan (c+d x)}{9 d (b \sec (c+d x))^{9/2}}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {(7 A+9 C) \left (\frac {3 \int \sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}dx}{5 b^2 \sqrt {\cos (c+d x)} \sqrt {b \sec (c+d x)}}+\frac {2 \sin (c+d x)}{5 b d (b \sec (c+d x))^{3/2}}\right )}{9 b^2}+\frac {2 A \tan (c+d x)}{9 d (b \sec (c+d x))^{9/2}}\)

\(\Big \downarrow \) 3119

\(\displaystyle \frac {(7 A+9 C) \left (\frac {6 E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{5 b^2 d \sqrt {\cos (c+d x)} \sqrt {b \sec (c+d x)}}+\frac {2 \sin (c+d x)}{5 b d (b \sec (c+d x))^{3/2}}\right )}{9 b^2}+\frac {2 A \tan (c+d x)}{9 d (b \sec (c+d x))^{9/2}}\)

Input:

Int[(A + C*Sec[c + d*x]^2)/(b*Sec[c + d*x])^(9/2),x]
 

Output:

((7*A + 9*C)*((6*EllipticE[(c + d*x)/2, 2])/(5*b^2*d*Sqrt[Cos[c + d*x]]*Sq 
rt[b*Sec[c + d*x]]) + (2*Sin[c + d*x])/(5*b*d*(b*Sec[c + d*x])^(3/2))))/(9 
*b^2) + (2*A*Tan[c + d*x])/(9*d*(b*Sec[c + d*x])^(9/2))
 

Defintions of rubi rules used

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 4256

Int[(csc[(c_.) + (d_.)*(x_)]*(b_.))^(n_), x_Symbol] :> Simp[Cos[c + d*x]*(( 
b*Csc[c + d*x])^(n + 1)/(b*d*n)), x] + Simp[(n + 1)/(b^2*n)   Int[(b*Csc[c 
+ d*x])^(n + 2), x], x] /; FreeQ[{b, c, d}, x] && LtQ[n, -1] && IntegerQ[2* 
n]
 

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 4533

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]
 

Maple [C] (verified)

Result contains complex when optimal does not.

Time = 11.66 (sec) , antiderivative size = 383, normalized size of antiderivative = 3.42

method	result	size
default	\(\frac {\frac {14 i A \sqrt {\frac {1}{\cos \left (d x +c \right )+1}}\, \sqrt {\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}\, \operatorname {EllipticE}\left (i \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right ), i\right ) \left (\cos \left (d x +c \right )+2+\sec \left (d x +c \right )\right )}{15}+\frac {6 i \sqrt {\frac {1}{\cos \left (d x +c \right )+1}}\, \sqrt {\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}\, \left (\cos \left (d x +c \right )+2+\sec \left (d x +c \right )\right ) C \operatorname {EllipticE}\left (i \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right ), i\right )}{5}-\frac {14 i \sqrt {\frac {1}{\cos \left (d x +c \right )+1}}\, \sqrt {\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}\, \left (\cos \left (d x +c \right )+2+\sec \left (d x +c \right )\right ) A \operatorname {EllipticF}\left (i \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right ), i\right )}{15}-\frac {6 i C \sqrt {\frac {1}{\cos \left (d x +c \right )+1}}\, \sqrt {\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}\, \operatorname {EllipticF}\left (i \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right ), i\right ) \left (\cos \left (d x +c \right )+2+\sec \left (d x +c \right )\right )}{5}+\frac {2 \sin \left (d x +c \right ) \left (5 \cos \left (d x +c \right )^{4}+5 \cos \left (d x +c \right )^{3}+7 \cos \left (d x +c \right )^{2}+7 \cos \left (d x +c \right )+21\right ) A}{45}+\frac {2 \sin \left (d x +c \right ) \left (\cos \left (d x +c \right )^{2}+\cos \left (d x +c \right )+3\right ) C}{5}}{d \,b^{4} \left (\cos \left (d x +c \right )+1\right ) \sqrt {b \sec \left (d x +c \right )}}\)	\(383\)
parts	\(\frac {2 A \left (\sin \left (d x +c \right ) \left (5 \cos \left (d x +c \right )^{4}+5 \cos \left (d x +c \right )^{3}+7 \cos \left (d x +c \right )^{2}+7 \cos \left (d x +c \right )+21\right )-21 i \sqrt {\frac {1}{\cos \left (d x +c \right )+1}}\, \sqrt {\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}\, \left (\cos \left (d x +c \right )+2+\sec \left (d x +c \right )\right ) \operatorname {EllipticF}\left (i \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right ), i\right )+21 i \sqrt {\frac {1}{\cos \left (d x +c \right )+1}}\, \sqrt {\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}\, \operatorname {EllipticE}\left (i \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right ), i\right ) \left (\cos \left (d x +c \right )+2+\sec \left (d x +c \right )\right )\right )}{45 d \left (\cos \left (d x +c \right )+1\right ) \sqrt {b \sec \left (d x +c \right )}\, b^{4}}+\frac {2 C \left (\sin \left (d x +c \right ) \left (\cos \left (d x +c \right )^{2}+\cos \left (d x +c \right )+3\right )-3 i \sqrt {\frac {1}{\cos \left (d x +c \right )+1}}\, \sqrt {\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}\, \left (\cos \left (d x +c \right )+2+\sec \left (d x +c \right )\right ) \operatorname {EllipticF}\left (i \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right ), i\right )+3 i \sqrt {\frac {1}{\cos \left (d x +c \right )+1}}\, \sqrt {\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}\, \operatorname {EllipticE}\left (i \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right ), i\right ) \left (\cos \left (d x +c \right )+2+\sec \left (d x +c \right )\right )\right )}{5 d \left (\cos \left (d x +c \right )+1\right ) \sqrt {b \sec \left (d x +c \right )}\, b^{4}}\)	\(408\)

Input:

int((A+C*sec(d*x+c)^2)/(b*sec(d*x+c))^(9/2),x,method=_RETURNVERBOSE)
 

Output:

2/45/d/b^4*(21*I*A*(1/(cos(d*x+c)+1))^(1/2)*(cos(d*x+c)/(cos(d*x+c)+1))^(1 
/2)*EllipticE(I*(csc(d*x+c)-cot(d*x+c)),I)*(cos(d*x+c)+2+sec(d*x+c))+27*I* 
(1/(cos(d*x+c)+1))^(1/2)*(cos(d*x+c)/(cos(d*x+c)+1))^(1/2)*(cos(d*x+c)+2+s 
ec(d*x+c))*C*EllipticE(I*(csc(d*x+c)-cot(d*x+c)),I)-21*I*(1/(cos(d*x+c)+1) 
)^(1/2)*(cos(d*x+c)/(cos(d*x+c)+1))^(1/2)*(cos(d*x+c)+2+sec(d*x+c))*A*Elli 
pticF(I*(csc(d*x+c)-cot(d*x+c)),I)-27*I*C*(1/(cos(d*x+c)+1))^(1/2)*(cos(d* 
x+c)/(cos(d*x+c)+1))^(1/2)*EllipticF(I*(csc(d*x+c)-cot(d*x+c)),I)*(cos(d*x 
+c)+2+sec(d*x+c))+sin(d*x+c)*(5*cos(d*x+c)^4+5*cos(d*x+c)^3+7*cos(d*x+c)^2 
+7*cos(d*x+c)+21)*A+9*sin(d*x+c)*(cos(d*x+c)^2+cos(d*x+c)+3)*C)/(cos(d*x+c 
)+1)/(b*sec(d*x+c))^(1/2)
 

Fricas [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.09 (sec) , antiderivative size = 129, normalized size of antiderivative = 1.15 \[ \int \frac {A+C \sec ^2(c+d x)}{(b \sec (c+d x))^{9/2}} \, dx=-\frac {3 \, \sqrt {2} {\left (-7 i \, A - 9 i \, C\right )} \sqrt {b} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right )\right ) + 3 \, \sqrt {2} {\left (7 i \, A + 9 i \, C\right )} \sqrt {b} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right )\right ) - 2 \, {\left (5 \, A \cos \left (d x + c\right )^{4} + {\left (7 \, A + 9 \, C\right )} \cos \left (d x + c\right )^{2}\right )} \sqrt {\frac {b}{\cos \left (d x + c\right )}} \sin \left (d x + c\right )}{45 \, b^{5} d} \] Input:

integrate((A+C*sec(d*x+c)^2)/(b*sec(d*x+c))^(9/2),x, algorithm="fricas")
 

Output:

-1/45*(3*sqrt(2)*(-7*I*A - 9*I*C)*sqrt(b)*weierstrassZeta(-4, 0, weierstra 
ssPInverse(-4, 0, cos(d*x + c) + I*sin(d*x + c))) + 3*sqrt(2)*(7*I*A + 9*I 
*C)*sqrt(b)*weierstrassZeta(-4, 0, weierstrassPInverse(-4, 0, cos(d*x + c) 
 - I*sin(d*x + c))) - 2*(5*A*cos(d*x + c)^4 + (7*A + 9*C)*cos(d*x + c)^2)* 
sqrt(b/cos(d*x + c))*sin(d*x + c))/(b^5*d)
 

Sympy [F(-1)]

Timed out. \[ \int \frac {A+C \sec ^2(c+d x)}{(b \sec (c+d x))^{9/2}} \, dx=\text {Timed out} \] Input:

integrate((A+C*sec(d*x+c)**2)/(b*sec(d*x+c))**(9/2),x)
 

Output:

Timed out
 

Maxima [F]

\[ \int \frac {A+C \sec ^2(c+d x)}{(b \sec (c+d x))^{9/2}} \, dx=\int { \frac {C \sec \left (d x + c\right )^{2} + A}{\left (b \sec \left (d x + c\right )\right )^{\frac {9}{2}}} \,d x } \] Input:

integrate((A+C*sec(d*x+c)^2)/(b*sec(d*x+c))^(9/2),x, algorithm="maxima")
 

Output:

integrate((C*sec(d*x + c)^2 + A)/(b*sec(d*x + c))^(9/2), x)
 

Giac [F]

\[ \int \frac {A+C \sec ^2(c+d x)}{(b \sec (c+d x))^{9/2}} \, dx=\int { \frac {C \sec \left (d x + c\right )^{2} + A}{\left (b \sec \left (d x + c\right )\right )^{\frac {9}{2}}} \,d x } \] Input:

integrate((A+C*sec(d*x+c)^2)/(b*sec(d*x+c))^(9/2),x, algorithm="giac")
 

Output:

integrate((C*sec(d*x + c)^2 + A)/(b*sec(d*x + c))^(9/2), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {A+C \sec ^2(c+d x)}{(b \sec (c+d x))^{9/2}} \, dx=\int \frac {A+\frac {C}{{\cos \left (c+d\,x\right )}^2}}{{\left (\frac {b}{\cos \left (c+d\,x\right )}\right )}^{9/2}} \,d x \] Input:

int((A + C/cos(c + d*x)^2)/(b/cos(c + d*x))^(9/2),x)
 

Output:

int((A + C/cos(c + d*x)^2)/(b/cos(c + d*x))^(9/2), x)
 

Reduce [F]

\[ \int \frac {A+C \sec ^2(c+d x)}{(b \sec (c+d x))^{9/2}} \, dx=\frac {\sqrt {b}\, \left (\left (\int \frac {\sqrt {\sec \left (d x +c \right )}}{\sec \left (d x +c \right )^{5}}d x \right ) a +\left (\int \frac {\sqrt {\sec \left (d x +c \right )}}{\sec \left (d x +c \right )^{3}}d x \right ) c \right )}{b^{5}} \] Input:

int((A+C*sec(d*x+c)^2)/(b*sec(d*x+c))^(9/2),x)
 

Output:

(sqrt(b)*(int(sqrt(sec(c + d*x))/sec(c + d*x)**5,x)*a + int(sqrt(sec(c + d 
*x))/sec(c + d*x)**3,x)*c))/b**5