\(\int \frac {\sec ^{\frac {7}{2}}(c+d x) (A+B \sec (c+d x))}{a+b \sec (c+d x)} \, dx\) [415]

Optimal result

Integrand size = 33, antiderivative size = 277 \[ \int \frac {\sec ^{\frac {7}{2}}(c+d x) (A+B \sec (c+d x))}{a+b \sec (c+d x)} \, dx=\frac {2 \left (5 a A b-5 a^2 B-3 b^2 B\right ) \sqrt {\cos (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \sqrt {\sec (c+d x)}}{5 b^3 d}+\frac {2 (A b-a B) \sqrt {\cos (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right ) \sqrt {\sec (c+d x)}}{3 b^2 d}+\frac {2 a^2 (A b-a B) \sqrt {\cos (c+d x)} \operatorname {EllipticPi}\left (\frac {2 a}{a+b},\frac {1}{2} (c+d x),2\right ) \sqrt {\sec (c+d x)}}{b^3 (a+b) d}-\frac {2 \left (5 a A b-5 a^2 B-3 b^2 B\right ) \sqrt {\sec (c+d x)} \sin (c+d x)}{5 b^3 d}+\frac {2 (A b-a B) \sec ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{3 b^2 d}+\frac {2 B \sec ^{\frac {5}{2}}(c+d x) \sin (c+d x)}{5 b d} \] Output:

2/5*(5*A*a*b-5*B*a^2-3*B*b^2)*cos(d*x+c)^(1/2)*EllipticE(sin(1/2*d*x+1/2*c 
),2^(1/2))*sec(d*x+c)^(1/2)/b^3/d+2/3*(A*b-B*a)*cos(d*x+c)^(1/2)*InverseJa 
cobiAM(1/2*d*x+1/2*c,2^(1/2))*sec(d*x+c)^(1/2)/b^2/d+2*a^2*(A*b-B*a)*cos(d 
*x+c)^(1/2)*EllipticPi(sin(1/2*d*x+1/2*c),2*a/(a+b),2^(1/2))*sec(d*x+c)^(1 
/2)/b^3/(a+b)/d-2/5*(5*A*a*b-5*B*a^2-3*B*b^2)*sec(d*x+c)^(1/2)*sin(d*x+c)/ 
b^3/d+2/3*(A*b-B*a)*sec(d*x+c)^(3/2)*sin(d*x+c)/b^2/d+2/5*B*sec(d*x+c)^(5/ 
2)*sin(d*x+c)/b/d
 

Mathematica [B] (warning: unable to verify)

Leaf count is larger than twice the leaf count of optimal. \(664\) vs. \(2(277)=554\).

Time = 41.42 (sec) , antiderivative size = 664, normalized size of antiderivative = 2.40 \[ \int \frac {\sec ^{\frac {7}{2}}(c+d x) (A+B \sec (c+d x))}{a+b \sec (c+d x)} \, dx=-\frac {\frac {2 \left (-45 a^2 A b-10 A b^3+45 a^3 B+19 a b^2 B\right ) \cos ^2(c+d x) \left (\operatorname {EllipticF}\left (\arcsin \left (\sqrt {\sec (c+d x)}\right ),-1\right )-\operatorname {EllipticPi}\left (-\frac {b}{a},\arcsin \left (\sqrt {\sec (c+d x)}\right ),-1\right )\right ) (a+b \sec (c+d x)) \sqrt {1-\sec ^2(c+d x)} \sin (c+d x)}{b (b+a \cos (c+d x)) \left (1-\cos ^2(c+d x)\right )}+\frac {2 \left (-40 a A b^2+40 a^2 b B+18 b^3 B\right ) \cos ^2(c+d x) \operatorname {EllipticPi}\left (-\frac {b}{a},\arcsin \left (\sqrt {\sec (c+d x)}\right ),-1\right ) (a+b \sec (c+d x)) \sqrt {1-\sec ^2(c+d x)} \sin (c+d x)}{a (b+a \cos (c+d x)) \left (1-\cos ^2(c+d x)\right )}+\frac {\left (-15 a^2 A b+15 a^3 B+9 a b^2 B\right ) \cos (2 (c+d x)) (a+b \sec (c+d x)) \left (-4 a b+4 a b \sec ^2(c+d x)-4 a b E\left (\left .\arcsin \left (\sqrt {\sec (c+d x)}\right )\right |-1\right ) \sqrt {\sec (c+d x)} \sqrt {1-\sec ^2(c+d x)}-2 a (a-2 b) \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\sec (c+d x)}\right ),-1\right ) \sqrt {\sec (c+d x)} \sqrt {1-\sec ^2(c+d x)}+2 a^2 \operatorname {EllipticPi}\left (-\frac {b}{a},\arcsin \left (\sqrt {\sec (c+d x)}\right ),-1\right ) \sqrt {\sec (c+d x)} \sqrt {1-\sec ^2(c+d x)}-4 b^2 \operatorname {EllipticPi}\left (-\frac {b}{a},\arcsin \left (\sqrt {\sec (c+d x)}\right ),-1\right ) \sqrt {\sec (c+d x)} \sqrt {1-\sec ^2(c+d x)}\right ) \sin (c+d x)}{a^2 b (b+a \cos (c+d x)) \left (1-\cos ^2(c+d x)\right ) \sqrt {\sec (c+d x)} \left (2-\sec ^2(c+d x)\right )}}{30 b^3 d}+\frac {\sqrt {\sec (c+d x)} \left (\frac {2 \left (-5 a A b+5 a^2 B+3 b^2 B\right ) \sin (c+d x)}{5 b^3}+\frac {2 \sec (c+d x) (A b \sin (c+d x)-a B \sin (c+d x))}{3 b^2}+\frac {2 B \sec (c+d x) \tan (c+d x)}{5 b}\right )}{d} \] Input:

Integrate[(Sec[c + d*x]^(7/2)*(A + B*Sec[c + d*x]))/(a + b*Sec[c + d*x]),x 
]
 

Output:

-1/30*((2*(-45*a^2*A*b - 10*A*b^3 + 45*a^3*B + 19*a*b^2*B)*Cos[c + d*x]^2* 
(EllipticF[ArcSin[Sqrt[Sec[c + d*x]]], -1] - EllipticPi[-(b/a), ArcSin[Sqr 
t[Sec[c + d*x]]], -1])*(a + b*Sec[c + d*x])*Sqrt[1 - Sec[c + d*x]^2]*Sin[c 
 + d*x])/(b*(b + a*Cos[c + d*x])*(1 - Cos[c + d*x]^2)) + (2*(-40*a*A*b^2 + 
 40*a^2*b*B + 18*b^3*B)*Cos[c + d*x]^2*EllipticPi[-(b/a), ArcSin[Sqrt[Sec[ 
c + d*x]]], -1]*(a + b*Sec[c + d*x])*Sqrt[1 - Sec[c + d*x]^2]*Sin[c + d*x] 
)/(a*(b + a*Cos[c + d*x])*(1 - Cos[c + d*x]^2)) + ((-15*a^2*A*b + 15*a^3*B 
 + 9*a*b^2*B)*Cos[2*(c + d*x)]*(a + b*Sec[c + d*x])*(-4*a*b + 4*a*b*Sec[c 
+ d*x]^2 - 4*a*b*EllipticE[ArcSin[Sqrt[Sec[c + d*x]]], -1]*Sqrt[Sec[c + d* 
x]]*Sqrt[1 - Sec[c + d*x]^2] - 2*a*(a - 2*b)*EllipticF[ArcSin[Sqrt[Sec[c + 
 d*x]]], -1]*Sqrt[Sec[c + d*x]]*Sqrt[1 - Sec[c + d*x]^2] + 2*a^2*EllipticP 
i[-(b/a), ArcSin[Sqrt[Sec[c + d*x]]], -1]*Sqrt[Sec[c + d*x]]*Sqrt[1 - Sec[ 
c + d*x]^2] - 4*b^2*EllipticPi[-(b/a), ArcSin[Sqrt[Sec[c + d*x]]], -1]*Sqr 
t[Sec[c + d*x]]*Sqrt[1 - Sec[c + d*x]^2])*Sin[c + d*x])/(a^2*b*(b + a*Cos[ 
c + d*x])*(1 - Cos[c + d*x]^2)*Sqrt[Sec[c + d*x]]*(2 - Sec[c + d*x]^2)))/( 
b^3*d) + (Sqrt[Sec[c + d*x]]*((2*(-5*a*A*b + 5*a^2*B + 3*b^2*B)*Sin[c + d* 
x])/(5*b^3) + (2*Sec[c + d*x]*(A*b*Sin[c + d*x] - a*B*Sin[c + d*x]))/(3*b^ 
2) + (2*B*Sec[c + d*x]*Tan[c + d*x])/(5*b)))/d
 

Rubi [A] (verified)

Time = 2.38 (sec) , antiderivative size = 295, normalized size of antiderivative = 1.06, number of steps used = 21, number of rules used = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.636, Rules used = {3042, 4521, 27, 3042, 4590, 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.

\(\displaystyle \int \frac {\sec ^{\frac {7}{2}}(c+d x) (A+B \sec (c+d x))}{a+b \sec (c+d x)} \, dx\)

\(\Big \downarrow \) 3042

\(\displaystyle \int \frac {\csc \left (c+d x+\frac {\pi }{2}\right )^{7/2} \left (A+B \csc \left (c+d x+\frac {\pi }{2}\right )\right )}{a+b \csc \left (c+d x+\frac {\pi }{2}\right )}dx\)

\(\Big \downarrow \) 4521

\(\displaystyle \frac {2 \int \frac {\sec ^{\frac {3}{2}}(c+d x) \left (5 (A b-a B) \sec ^2(c+d x)+3 b B \sec (c+d x)+3 a B\right )}{2 (a+b \sec (c+d x))}dx}{5 b}+\frac {2 B \sin (c+d x) \sec ^{\frac {5}{2}}(c+d x)}{5 b d}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\int \frac {\sec ^{\frac {3}{2}}(c+d x) \left (5 (A b-a B) \sec ^2(c+d x)+3 b B \sec (c+d x)+3 a B\right )}{a+b \sec (c+d x)}dx}{5 b}+\frac {2 B \sin (c+d x) \sec ^{\frac {5}{2}}(c+d x)}{5 b d}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\int \frac {\csc \left (c+d x+\frac {\pi }{2}\right )^{3/2} \left (5 (A b-a B) \csc \left (c+d x+\frac {\pi }{2}\right )^2+3 b B \csc \left (c+d x+\frac {\pi }{2}\right )+3 a B\right )}{a+b \csc \left (c+d x+\frac {\pi }{2}\right )}dx}{5 b}+\frac {2 B \sin (c+d x) \sec ^{\frac {5}{2}}(c+d x)}{5 b d}\)

\(\Big \downarrow \) 4590

\(\displaystyle \frac {\frac {2 \int \frac {\sqrt {\sec (c+d x)} \left (-3 \left (-5 B a^2+5 A b a-3 b^2 B\right ) \sec ^2(c+d x)+b (5 A b+4 a B) \sec (c+d x)+5 a (A b-a B)\right )}{2 (a+b \sec (c+d x))}dx}{3 b}+\frac {10 (A b-a B) \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x)}{3 b d}}{5 b}+\frac {2 B \sin (c+d x) \sec ^{\frac {5}{2}}(c+d x)}{5 b d}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\frac {\int \frac {\sqrt {\sec (c+d x)} \left (-3 \left (-5 B a^2+5 A b a-3 b^2 B\right ) \sec ^2(c+d x)+b (5 A b+4 a B) \sec (c+d x)+5 a (A b-a B)\right )}{a+b \sec (c+d x)}dx}{3 b}+\frac {10 (A b-a B) \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x)}{3 b d}}{5 b}+\frac {2 B \sin (c+d x) \sec ^{\frac {5}{2}}(c+d x)}{5 b d}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\frac {\int \frac {\sqrt {\csc \left (c+d x+\frac {\pi }{2}\right )} \left (-3 \left (-5 B a^2+5 A b a-3 b^2 B\right ) \csc \left (c+d x+\frac {\pi }{2}\right )^2+b (5 A b+4 a B) \csc \left (c+d x+\frac {\pi }{2}\right )+5 a (A b-a B)\right )}{a+b \csc \left (c+d x+\frac {\pi }{2}\right )}dx}{3 b}+\frac {10 (A b-a B) \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x)}{3 b d}}{5 b}+\frac {2 B \sin (c+d x) \sec ^{\frac {5}{2}}(c+d x)}{5 b d}\)

\(\Big \downarrow \) 4590

\(\displaystyle \frac {\frac {\frac {2 \int \frac {5 \left (3 a^2+b^2\right ) (A b-a B) \sec ^2(c+d x)+b \left (-20 B a^2+20 A b a-9 b^2 B\right ) \sec (c+d x)+3 a \left (-5 B a^2+5 A b a-3 b^2 B\right )}{2 \sqrt {\sec (c+d x)} (a+b \sec (c+d x))}dx}{b}-\frac {6 \left (-5 a^2 B+5 a A b-3 b^2 B\right ) \sin (c+d x) \sqrt {\sec (c+d x)}}{b d}}{3 b}+\frac {10 (A b-a B) \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x)}{3 b d}}{5 b}+\frac {2 B \sin (c+d x) \sec ^{\frac {5}{2}}(c+d x)}{5 b d}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\frac {\frac {\int \frac {5 \left (3 a^2+b^2\right ) (A b-a B) \sec ^2(c+d x)+b \left (-20 B a^2+20 A b a-9 b^2 B\right ) \sec (c+d x)+3 a \left (-5 B a^2+5 A b a-3 b^2 B\right )}{\sqrt {\sec (c+d x)} (a+b \sec (c+d x))}dx}{b}-\frac {6 \left (-5 a^2 B+5 a A b-3 b^2 B\right ) \sin (c+d x) \sqrt {\sec (c+d x)}}{b d}}{3 b}+\frac {10 (A b-a B) \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x)}{3 b d}}{5 b}+\frac {2 B \sin (c+d x) \sec ^{\frac {5}{2}}(c+d x)}{5 b d}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\frac {\frac {\int \frac {5 \left (3 a^2+b^2\right ) (A b-a B) \csc \left (c+d x+\frac {\pi }{2}\right )^2+b \left (-20 B a^2+20 A b a-9 b^2 B\right ) \csc \left (c+d x+\frac {\pi }{2}\right )+3 a \left (-5 B a^2+5 A b a-3 b^2 B\right )}{\sqrt {\csc \left (c+d x+\frac {\pi }{2}\right )} \left (a+b \csc \left (c+d x+\frac {\pi }{2}\right )\right )}dx}{b}-\frac {6 \left (-5 a^2 B+5 a A b-3 b^2 B\right ) \sin (c+d x) \sqrt {\sec (c+d x)}}{b d}}{3 b}+\frac {10 (A b-a B) \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x)}{3 b d}}{5 b}+\frac {2 B \sin (c+d x) \sec ^{\frac {5}{2}}(c+d x)}{5 b d}\)

\(\Big \downarrow \) 4594

\(\displaystyle \frac {\frac {\frac {\frac {\int \frac {3 \left (-5 B a^2+5 A b a-3 b^2 B\right ) a^2+5 b (A b-a B) \sec (c+d x) a^2}{\sqrt {\sec (c+d x)}}dx}{a^2}+15 a^2 (A b-a B) \int \frac {\sec ^{\frac {3}{2}}(c+d x)}{a+b \sec (c+d x)}dx}{b}-\frac {6 \left (-5 a^2 B+5 a A b-3 b^2 B\right ) \sin (c+d x) \sqrt {\sec (c+d x)}}{b d}}{3 b}+\frac {10 (A b-a B) \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x)}{3 b d}}{5 b}+\frac {2 B \sin (c+d x) \sec ^{\frac {5}{2}}(c+d x)}{5 b d}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\frac {\frac {\frac {\int \frac {3 \left (-5 B a^2+5 A b a-3 b^2 B\right ) a^2+5 b (A b-a B) \csc \left (c+d x+\frac {\pi }{2}\right ) a^2}{\sqrt {\csc \left (c+d x+\frac {\pi }{2}\right )}}dx}{a^2}+15 a^2 (A b-a B) \int \frac {\csc \left (c+d x+\frac {\pi }{2}\right )^{3/2}}{a+b \csc \left (c+d x+\frac {\pi }{2}\right )}dx}{b}-\frac {6 \left (-5 a^2 B+5 a A b-3 b^2 B\right ) \sin (c+d x) \sqrt {\sec (c+d x)}}{b d}}{3 b}+\frac {10 (A b-a B) \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x)}{3 b d}}{5 b}+\frac {2 B \sin (c+d x) \sec ^{\frac {5}{2}}(c+d x)}{5 b d}\)

\(\Big \downarrow \) 4274

\(\displaystyle \frac {\frac {\frac {\frac {3 a^2 \left (-5 a^2 B+5 a A b-3 b^2 B\right ) \int \frac {1}{\sqrt {\sec (c+d x)}}dx+5 a^2 b (A b-a B) \int \sqrt {\sec (c+d x)}dx}{a^2}+15 a^2 (A b-a B) \int \frac {\csc \left (c+d x+\frac {\pi }{2}\right )^{3/2}}{a+b \csc \left (c+d x+\frac {\pi }{2}\right )}dx}{b}-\frac {6 \left (-5 a^2 B+5 a A b-3 b^2 B\right ) \sin (c+d x) \sqrt {\sec (c+d x)}}{b d}}{3 b}+\frac {10 (A b-a B) \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x)}{3 b d}}{5 b}+\frac {2 B \sin (c+d x) \sec ^{\frac {5}{2}}(c+d x)}{5 b d}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\frac {\frac {\frac {3 a^2 \left (-5 a^2 B+5 a A b-3 b^2 B\right ) \int \frac {1}{\sqrt {\csc \left (c+d x+\frac {\pi }{2}\right )}}dx+5 a^2 b (A b-a B) \int \sqrt {\csc \left (c+d x+\frac {\pi }{2}\right )}dx}{a^2}+15 a^2 (A b-a B) \int \frac {\csc \left (c+d x+\frac {\pi }{2}\right )^{3/2}}{a+b \csc \left (c+d x+\frac {\pi }{2}\right )}dx}{b}-\frac {6 \left (-5 a^2 B+5 a A b-3 b^2 B\right ) \sin (c+d x) \sqrt {\sec (c+d x)}}{b d}}{3 b}+\frac {10 (A b-a B) \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x)}{3 b d}}{5 b}+\frac {2 B \sin (c+d x) \sec ^{\frac {5}{2}}(c+d x)}{5 b d}\)

\(\Big \downarrow \) 4258

\(\displaystyle \frac {\frac {\frac {\frac {3 a^2 \left (-5 a^2 B+5 a A b-3 b^2 B\right ) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \int \sqrt {\cos (c+d x)}dx+5 a^2 b (A b-a B) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \int \frac {1}{\sqrt {\cos (c+d x)}}dx}{a^2}+15 a^2 (A b-a B) \int \frac {\csc \left (c+d x+\frac {\pi }{2}\right )^{3/2}}{a+b \csc \left (c+d x+\frac {\pi }{2}\right )}dx}{b}-\frac {6 \left (-5 a^2 B+5 a A b-3 b^2 B\right ) \sin (c+d x) \sqrt {\sec (c+d x)}}{b d}}{3 b}+\frac {10 (A b-a B) \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x)}{3 b d}}{5 b}+\frac {2 B \sin (c+d x) \sec ^{\frac {5}{2}}(c+d x)}{5 b d}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\frac {\frac {\frac {3 a^2 \left (-5 a^2 B+5 a A b-3 b^2 B\right ) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \int \sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}dx+5 a^2 b (A b-a B) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \int \frac {1}{\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}}dx}{a^2}+15 a^2 (A b-a B) \int \frac {\csc \left (c+d x+\frac {\pi }{2}\right )^{3/2}}{a+b \csc \left (c+d x+\frac {\pi }{2}\right )}dx}{b}-\frac {6 \left (-5 a^2 B+5 a A b-3 b^2 B\right ) \sin (c+d x) \sqrt {\sec (c+d x)}}{b d}}{3 b}+\frac {10 (A b-a B) \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x)}{3 b d}}{5 b}+\frac {2 B \sin (c+d x) \sec ^{\frac {5}{2}}(c+d x)}{5 b d}\)

\(\Big \downarrow \) 3119

\(\displaystyle \frac {\frac {\frac {\frac {5 a^2 b (A b-a B) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \int \frac {1}{\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}}dx+\frac {6 a^2 \left (-5 a^2 B+5 a A b-3 b^2 B\right ) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{d}}{a^2}+15 a^2 (A b-a B) \int \frac {\csc \left (c+d x+\frac {\pi }{2}\right )^{3/2}}{a+b \csc \left (c+d x+\frac {\pi }{2}\right )}dx}{b}-\frac {6 \left (-5 a^2 B+5 a A b-3 b^2 B\right ) \sin (c+d x) \sqrt {\sec (c+d x)}}{b d}}{3 b}+\frac {10 (A b-a B) \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x)}{3 b d}}{5 b}+\frac {2 B \sin (c+d x) \sec ^{\frac {5}{2}}(c+d x)}{5 b d}\)

\(\Big \downarrow \) 3120

\(\displaystyle \frac {\frac {\frac {15 a^2 (A b-a B) \int \frac {\csc \left (c+d x+\frac {\pi }{2}\right )^{3/2}}{a+b \csc \left (c+d x+\frac {\pi }{2}\right )}dx+\frac {\frac {6 a^2 \left (-5 a^2 B+5 a A b-3 b^2 B\right ) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{d}+\frac {10 a^2 b (A b-a B) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )}{d}}{a^2}}{b}-\frac {6 \left (-5 a^2 B+5 a A b-3 b^2 B\right ) \sin (c+d x) \sqrt {\sec (c+d x)}}{b d}}{3 b}+\frac {10 (A b-a B) \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x)}{3 b d}}{5 b}+\frac {2 B \sin (c+d x) \sec ^{\frac {5}{2}}(c+d x)}{5 b d}\)

\(\Big \downarrow \) 4336

\(\displaystyle \frac {\frac {\frac {15 a^2 (A b-a B) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \int \frac {1}{\sqrt {\cos (c+d x)} (b+a \cos (c+d x))}dx+\frac {\frac {6 a^2 \left (-5 a^2 B+5 a A b-3 b^2 B\right ) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{d}+\frac {10 a^2 b (A b-a B) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )}{d}}{a^2}}{b}-\frac {6 \left (-5 a^2 B+5 a A b-3 b^2 B\right ) \sin (c+d x) \sqrt {\sec (c+d x)}}{b d}}{3 b}+\frac {10 (A b-a B) \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x)}{3 b d}}{5 b}+\frac {2 B \sin (c+d x) \sec ^{\frac {5}{2}}(c+d x)}{5 b d}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\frac {\frac {15 a^2 (A b-a B) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \int \frac {1}{\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )} \left (b+a \sin \left (c+d x+\frac {\pi }{2}\right )\right )}dx+\frac {\frac {6 a^2 \left (-5 a^2 B+5 a A b-3 b^2 B\right ) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{d}+\frac {10 a^2 b (A b-a B) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )}{d}}{a^2}}{b}-\frac {6 \left (-5 a^2 B+5 a A b-3 b^2 B\right ) \sin (c+d x) \sqrt {\sec (c+d x)}}{b d}}{3 b}+\frac {10 (A b-a B) \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x)}{3 b d}}{5 b}+\frac {2 B \sin (c+d x) \sec ^{\frac {5}{2}}(c+d x)}{5 b d}\)

\(\Big \downarrow \) 3284

\(\displaystyle \frac {\frac {\frac {\frac {\frac {6 a^2 \left (-5 a^2 B+5 a A b-3 b^2 B\right ) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{d}+\frac {10 a^2 b (A b-a B) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )}{d}}{a^2}+\frac {30 a^2 (A b-a B) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \operatorname {EllipticPi}\left (\frac {2 a}{a+b},\frac {1}{2} (c+d x),2\right )}{d (a+b)}}{b}-\frac {6 \left (-5 a^2 B+5 a A b-3 b^2 B\right ) \sin (c+d x) \sqrt {\sec (c+d x)}}{b d}}{3 b}+\frac {10 (A b-a B) \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x)}{3 b d}}{5 b}+\frac {2 B \sin (c+d x) \sec ^{\frac {5}{2}}(c+d x)}{5 b d}\)

Input:

Int[(Sec[c + d*x]^(7/2)*(A + B*Sec[c + d*x]))/(a + b*Sec[c + d*x]),x]
 

Output:

(2*B*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(5*b*d) + ((10*(A*b - a*B)*Sec[c + d 
*x]^(3/2)*Sin[c + d*x])/(3*b*d) + ((((6*a^2*(5*a*A*b - 5*a^2*B - 3*b^2*B)* 
Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/d + (10*a 
^2*b*(A*b - a*B)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + 
 d*x]])/d)/a^2 + (30*a^2*(A*b - a*B)*Sqrt[Cos[c + d*x]]*EllipticPi[(2*a)/( 
a + b), (c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/((a + b)*d))/b - (6*(5*a*A*b - 
 5*a^2*B - 3*b^2*B)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(b*d))/(3*b))/(5*b)
 

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 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 4521

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^ 
2*Cot[e + f*x]*(a + b*Csc[e + f*x])^(m + 1)*((d*Csc[e + f*x])^(n - 2)/(b*f* 
(m + n))), x] + Simp[d^2/(b*(m + n))   Int[(a + b*Csc[e + f*x])^m*(d*Csc[e 
+ f*x])^(n - 2)*Simp[a*B*(n - 2) + B*b*(m + n - 1)*Csc[e + f*x] + (A*b*(m + 
 n) - a*B*(n - 1))*Csc[e + f*x]^2, x], x], x] /; FreeQ[{a, b, d, e, f, A, B 
, m}, x] && NeQ[A*b - a*B, 0] && NeQ[a^2 - b^2, 0] && GtQ[n, 1] && NeQ[m + 
n, 0] &&  !IGtQ[m, 1]
 

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. \(757\) vs. \(2(254)=508\).

Time = 32.77 (sec) , antiderivative size = 758, normalized size of antiderivative = 2.74

method	result	size
default	\(\text {Expression too large to display}\)	\(758\)

Input:

int(sec(d*x+c)^(7/2)*(A+B*sec(d*x+c))/(a+b*sec(d*x+c)),x,method=_RETURNVER 
BOSE)
 

Output:

-(-(-2*cos(1/2*d*x+1/2*c)^2+1)*sin(1/2*d*x+1/2*c)^2)^(1/2)*(2/5*B/b/(8*sin 
(1/2*d*x+1/2*c)^6-12*sin(1/2*d*x+1/2*c)^4+6*sin(1/2*d*x+1/2*c)^2-1)/sin(1/ 
2*d*x+1/2*c)^2*(24*sin(1/2*d*x+1/2*c)^6*cos(1/2*d*x+1/2*c)-12*EllipticE(co 
s(1/2*d*x+1/2*c),2^(1/2))*(sin(1/2*d*x+1/2*c)^2)^(1/2)*(2*sin(1/2*d*x+1/2* 
c)^2-1)^(1/2)*sin(1/2*d*x+1/2*c)^4-24*cos(1/2*d*x+1/2*c)*sin(1/2*d*x+1/2*c 
)^4+12*EllipticE(cos(1/2*d*x+1/2*c),2^(1/2))*(sin(1/2*d*x+1/2*c)^2)^(1/2)* 
(2*sin(1/2*d*x+1/2*c)^2-1)^(1/2)*sin(1/2*d*x+1/2*c)^2+8*cos(1/2*d*x+1/2*c) 
*sin(1/2*d*x+1/2*c)^2-3*EllipticE(cos(1/2*d*x+1/2*c),2^(1/2))*(sin(1/2*d*x 
+1/2*c)^2)^(1/2)*(2*sin(1/2*d*x+1/2*c)^2-1)^(1/2))*(-2*sin(1/2*d*x+1/2*c)^ 
4+sin(1/2*d*x+1/2*c)^2)^(1/2)+2*(A*b-B*a)/b^2*(-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)*EllipticF(cos(1/2*d*x+1 
/2*c),2^(1/2)))-2*(A*b-B*a)*a^3/b^3/(a^2-a*b)*(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*a/(a-b),2^(1/2))-2*(A*b-B*a) 
/b^3*a/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*cos(1/2*d*x+1/2*c)*sin(1/2*d*x+1/2*c) 
^2-EllipticE(cos(1/2*d*x+1/2*c),2^(1/2))*(sin(1/2*d*x+1/2*c)^2)^(1/2)*(2*s 
in(1/2*d*x+1/2*c)^2-1)^(1/2)))/sin(1/2*d*x+1/2*c)/(2*cos(1/2*d*x+1/2*c)...
 

Fricas [F(-1)]

Timed out. \[ \int \frac {\sec ^{\frac {7}{2}}(c+d x) (A+B \sec (c+d x))}{a+b \sec (c+d x)} \, dx=\text {Timed out} \] Input:

integrate(sec(d*x+c)^(7/2)*(A+B*sec(d*x+c))/(a+b*sec(d*x+c)),x, algorithm= 
"fricas")
 

Output:

Timed out
 

Sympy [F(-1)]

Timed out. \[ \int \frac {\sec ^{\frac {7}{2}}(c+d x) (A+B \sec (c+d x))}{a+b \sec (c+d x)} \, dx=\text {Timed out} \] Input:

integrate(sec(d*x+c)**(7/2)*(A+B*sec(d*x+c))/(a+b*sec(d*x+c)),x)
                                                                                    
                                                                                    
 

Output:

Timed out
 

Maxima [F]

\[ \int \frac {\sec ^{\frac {7}{2}}(c+d x) (A+B \sec (c+d x))}{a+b \sec (c+d x)} \, dx=\int { \frac {{\left (B \sec \left (d x + c\right ) + A\right )} \sec \left (d x + c\right )^{\frac {7}{2}}}{b \sec \left (d x + c\right ) + a} \,d x } \] Input:

integrate(sec(d*x+c)^(7/2)*(A+B*sec(d*x+c))/(a+b*sec(d*x+c)),x, algorithm= 
"maxima")
 

Output:

integrate((B*sec(d*x + c) + A)*sec(d*x + c)^(7/2)/(b*sec(d*x + c) + a), x)
 

Giac [F]

\[ \int \frac {\sec ^{\frac {7}{2}}(c+d x) (A+B \sec (c+d x))}{a+b \sec (c+d x)} \, dx=\int { \frac {{\left (B \sec \left (d x + c\right ) + A\right )} \sec \left (d x + c\right )^{\frac {7}{2}}}{b \sec \left (d x + c\right ) + a} \,d x } \] Input:

integrate(sec(d*x+c)^(7/2)*(A+B*sec(d*x+c))/(a+b*sec(d*x+c)),x, algorithm= 
"giac")
 

Output:

integrate((B*sec(d*x + c) + A)*sec(d*x + c)^(7/2)/(b*sec(d*x + c) + a), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {\sec ^{\frac {7}{2}}(c+d x) (A+B \sec (c+d x))}{a+b \sec (c+d x)} \, dx=\int \frac {\left (A+\frac {B}{\cos \left (c+d\,x\right )}\right )\,{\left (\frac {1}{\cos \left (c+d\,x\right )}\right )}^{7/2}}{a+\frac {b}{\cos \left (c+d\,x\right )}} \,d x \] Input:

int(((A + B/cos(c + d*x))*(1/cos(c + d*x))^(7/2))/(a + b/cos(c + d*x)),x)
 

Output:

int(((A + B/cos(c + d*x))*(1/cos(c + d*x))^(7/2))/(a + b/cos(c + d*x)), x)
 

Reduce [F]

\[ \int \frac {\sec ^{\frac {7}{2}}(c+d x) (A+B \sec (c+d x))}{a+b \sec (c+d x)} \, dx=\int \sqrt {\sec \left (d x +c \right )}\, \sec \left (d x +c \right )^{3}d x \] Input:

int(sec(d*x+c)^(7/2)*(A+B*sec(d*x+c))/(a+b*sec(d*x+c)),x)
 

Output:

int(sqrt(sec(c + d*x))*sec(c + d*x)**3,x)