\(\int \frac {A+B \sec (c+d x)}{\cos ^{\frac {7}{2}}(c+d x) (a+b \sec (c+d x))^2} \, dx\) [586]

Optimal result

Integrand size = 33, antiderivative size = 346 \[ \int \frac {A+B \sec (c+d x)}{\cos ^{\frac {7}{2}}(c+d x) (a+b \sec (c+d x))^2} \, dx=-\frac {\left (3 a^2 A b-2 A b^3-5 a^3 B+4 a b^2 B\right ) E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{b^3 \left (a^2-b^2\right ) d}-\frac {\left (3 a A b-5 a^2 B+2 b^2 B\right ) \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )}{3 b^2 \left (a^2-b^2\right ) d}-\frac {a \left (3 a^2 A b-5 A b^3-5 a^3 B+7 a b^2 B\right ) \operatorname {EllipticPi}\left (\frac {2 a}{a+b},\frac {1}{2} (c+d x),2\right )}{(a-b) b^3 (a+b)^2 d}-\frac {\left (3 a A b-5 a^2 B+2 b^2 B\right ) \sin (c+d x)}{3 b^2 \left (a^2-b^2\right ) d \cos ^{\frac {3}{2}}(c+d x)}+\frac {\left (3 a^2 A b-2 A b^3-5 a^3 B+4 a b^2 B\right ) \sin (c+d x)}{b^3 \left (a^2-b^2\right ) d \sqrt {\cos (c+d x)}}+\frac {a (A b-a B) \sin (c+d x)}{b \left (a^2-b^2\right ) d \cos ^{\frac {3}{2}}(c+d x) (b+a \cos (c+d x))} \] Output:

-(3*A*a^2*b-2*A*b^3-5*B*a^3+4*B*a*b^2)*EllipticE(sin(1/2*d*x+1/2*c),2^(1/2 
))/b^3/(a^2-b^2)/d-1/3*(3*A*a*b-5*B*a^2+2*B*b^2)*InverseJacobiAM(1/2*d*x+1 
/2*c,2^(1/2))/b^2/(a^2-b^2)/d-a*(3*A*a^2*b-5*A*b^3-5*B*a^3+7*B*a*b^2)*Elli 
pticPi(sin(1/2*d*x+1/2*c),2*a/(a+b),2^(1/2))/(a-b)/b^3/(a+b)^2/d-1/3*(3*A* 
a*b-5*B*a^2+2*B*b^2)*sin(d*x+c)/b^2/(a^2-b^2)/d/cos(d*x+c)^(3/2)+(3*A*a^2* 
b-2*A*b^3-5*B*a^3+4*B*a*b^2)*sin(d*x+c)/b^3/(a^2-b^2)/d/cos(d*x+c)^(1/2)+a 
*(A*b-B*a)*sin(d*x+c)/b/(a^2-b^2)/d/cos(d*x+c)^(3/2)/(b+a*cos(d*x+c))
                                                                                    
                                                                                    
 

Mathematica [A] (warning: unable to verify)

Time = 4.76 (sec) , antiderivative size = 363, normalized size of antiderivative = 1.05 \[ \int \frac {A+B \sec (c+d x)}{\cos ^{\frac {7}{2}}(c+d x) (a+b \sec (c+d x))^2} \, dx=\frac {\frac {\frac {2 \left (-27 a^3 A b+30 a A b^3+45 a^4 B-44 a^2 b^2 B-4 b^4 B\right ) \operatorname {EllipticPi}\left (\frac {2 a}{a+b},\frac {1}{2} (c+d x),2\right )}{a+b}+\frac {8 b \left (-6 a^2 A b+3 A b^3+10 a^3 B-7 a b^2 B\right ) \left ((a+b) \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )-b \operatorname {EllipticPi}\left (\frac {2 a}{a+b},\frac {1}{2} (c+d x),2\right )\right )}{a (a+b)}+\frac {6 \left (-3 a^2 A b+2 A b^3+5 a^3 B-4 a b^2 B\right ) \left (-2 a b E\left (\left .\arcsin \left (\sqrt {\cos (c+d x)}\right )\right |-1\right )+2 b (a+b) \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\cos (c+d x)}\right ),-1\right )+\left (a^2-2 b^2\right ) \operatorname {EllipticPi}\left (-\frac {a}{b},\arcsin \left (\sqrt {\cos (c+d x)}\right ),-1\right )\right ) \sin (c+d x)}{a b \sqrt {\sin ^2(c+d x)}}}{(a-b) (a+b)}+4 \sqrt {\cos (c+d x)} \left (-\frac {3 a^3 (-A b+a B) \sin (c+d x)}{\left (a^2-b^2\right ) (b+a \cos (c+d x))}+2 (3 A b-6 a B+b B \sec (c+d x)) \tan (c+d x)\right )}{12 b^3 d} \] Input:

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

Output:

(((2*(-27*a^3*A*b + 30*a*A*b^3 + 45*a^4*B - 44*a^2*b^2*B - 4*b^4*B)*Ellipt 
icPi[(2*a)/(a + b), (c + d*x)/2, 2])/(a + b) + (8*b*(-6*a^2*A*b + 3*A*b^3 
+ 10*a^3*B - 7*a*b^2*B)*((a + b)*EllipticF[(c + d*x)/2, 2] - b*EllipticPi[ 
(2*a)/(a + b), (c + d*x)/2, 2]))/(a*(a + b)) + (6*(-3*a^2*A*b + 2*A*b^3 + 
5*a^3*B - 4*a*b^2*B)*(-2*a*b*EllipticE[ArcSin[Sqrt[Cos[c + d*x]]], -1] + 2 
*b*(a + b)*EllipticF[ArcSin[Sqrt[Cos[c + d*x]]], -1] + (a^2 - 2*b^2)*Ellip 
ticPi[-(a/b), ArcSin[Sqrt[Cos[c + d*x]]], -1])*Sin[c + d*x])/(a*b*Sqrt[Sin 
[c + d*x]^2]))/((a - b)*(a + b)) + 4*Sqrt[Cos[c + d*x]]*((-3*a^3*(-(A*b) + 
 a*B)*Sin[c + d*x])/((a^2 - b^2)*(b + a*Cos[c + d*x])) + 2*(3*A*b - 6*a*B 
+ b*B*Sec[c + d*x])*Tan[c + d*x]))/(12*b^3*d)
 

Rubi [A] (verified)

Time = 2.64 (sec) , antiderivative size = 327, normalized size of antiderivative = 0.95, number of steps used = 20, number of rules used = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.606, Rules used = {3042, 3433, 3042, 3479, 27, 3042, 3534, 27, 3042, 3534, 27, 3042, 3538, 25, 3042, 3119, 3481, 3042, 3120, 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.

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3433

\(\displaystyle \int \frac {A \cos (c+d x)+B}{\cos ^{\frac {5}{2}}(c+d x) (a \cos (c+d x)+b)^2}dx\)

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3479

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3534

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3534

\(\displaystyle \frac {a (A b-a B) \sin (c+d x)}{b d \left (a^2-b^2\right ) \cos ^{\frac {3}{2}}(c+d x) (a \cos (c+d x)+b)}-\frac {\frac {2 \left (-5 a^2 B+3 a A b+2 b^2 B\right ) \sin (c+d x)}{3 b d \cos ^{\frac {3}{2}}(c+d x)}-\frac {\frac {2 \int -\frac {-15 B a^4+9 A b a^3+16 b^2 B a^2-12 A b^3 a+3 \left (-5 B a^3+3 A b a^2+4 b^2 B a-2 A b^3\right ) \cos ^2(c+d x) a+2 b^4 B+2 b \left (-10 B a^3+6 A b a^2+7 b^2 B a-3 A b^3\right ) \cos (c+d x)}{2 \sqrt {\cos (c+d x)} (b+a \cos (c+d x))}dx}{b}+\frac {6 \left (-5 a^3 B+3 a^2 A b+4 a b^2 B-2 A b^3\right ) \sin (c+d x)}{b d \sqrt {\cos (c+d x)}}}{3 b}}{2 b \left (a^2-b^2\right )}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {a (A b-a B) \sin (c+d x)}{b d \left (a^2-b^2\right ) \cos ^{\frac {3}{2}}(c+d x) (a \cos (c+d x)+b)}-\frac {\frac {2 \left (-5 a^2 B+3 a A b+2 b^2 B\right ) \sin (c+d x)}{3 b d \cos ^{\frac {3}{2}}(c+d x)}-\frac {\frac {6 \left (-5 a^3 B+3 a^2 A b+4 a b^2 B-2 A b^3\right ) \sin (c+d x)}{b d \sqrt {\cos (c+d x)}}-\frac {\int \frac {-15 B a^4+9 A b a^3+16 b^2 B a^2-12 A b^3 a+3 \left (-5 B a^3+3 A b a^2+4 b^2 B a-2 A b^3\right ) \cos ^2(c+d x) a+2 b^4 B+2 b \left (-10 B a^3+6 A b a^2+7 b^2 B a-3 A b^3\right ) \cos (c+d x)}{\sqrt {\cos (c+d x)} (b+a \cos (c+d x))}dx}{b}}{3 b}}{2 b \left (a^2-b^2\right )}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {a (A b-a B) \sin (c+d x)}{b d \left (a^2-b^2\right ) \cos ^{\frac {3}{2}}(c+d x) (a \cos (c+d x)+b)}-\frac {\frac {2 \left (-5 a^2 B+3 a A b+2 b^2 B\right ) \sin (c+d x)}{3 b d \cos ^{\frac {3}{2}}(c+d x)}-\frac {\frac {6 \left (-5 a^3 B+3 a^2 A b+4 a b^2 B-2 A b^3\right ) \sin (c+d x)}{b d \sqrt {\cos (c+d x)}}-\frac {\int \frac {-15 B a^4+9 A b a^3+16 b^2 B a^2-12 A b^3 a+3 \left (-5 B a^3+3 A b a^2+4 b^2 B a-2 A b^3\right ) \sin \left (c+d x+\frac {\pi }{2}\right )^2 a+2 b^4 B+2 b \left (-10 B a^3+6 A b a^2+7 b^2 B a-3 A b^3\right ) \sin \left (c+d x+\frac {\pi }{2}\right )}{\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )} \left (b+a \sin \left (c+d x+\frac {\pi }{2}\right )\right )}dx}{b}}{3 b}}{2 b \left (a^2-b^2\right )}\)

\(\Big \downarrow \) 3538

\(\displaystyle \frac {a (A b-a B) \sin (c+d x)}{b d \left (a^2-b^2\right ) \cos ^{\frac {3}{2}}(c+d x) (a \cos (c+d x)+b)}-\frac {\frac {2 \left (-5 a^2 B+3 a A b+2 b^2 B\right ) \sin (c+d x)}{3 b d \cos ^{\frac {3}{2}}(c+d x)}-\frac {\frac {6 \left (-5 a^3 B+3 a^2 A b+4 a b^2 B-2 A b^3\right ) \sin (c+d x)}{b d \sqrt {\cos (c+d x)}}-\frac {3 \left (-5 a^3 B+3 a^2 A b+4 a b^2 B-2 A b^3\right ) \int \sqrt {\cos (c+d x)}dx-\frac {\int -\frac {b \left (-5 B a^2+3 A b a+2 b^2 B\right ) \cos (c+d x) a^2+\left (-15 B a^4+9 A b a^3+16 b^2 B a^2-12 A b^3 a+2 b^4 B\right ) a}{\sqrt {\cos (c+d x)} (b+a \cos (c+d x))}dx}{a}}{b}}{3 b}}{2 b \left (a^2-b^2\right )}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {a (A b-a B) \sin (c+d x)}{b d \left (a^2-b^2\right ) \cos ^{\frac {3}{2}}(c+d x) (a \cos (c+d x)+b)}-\frac {\frac {2 \left (-5 a^2 B+3 a A b+2 b^2 B\right ) \sin (c+d x)}{3 b d \cos ^{\frac {3}{2}}(c+d x)}-\frac {\frac {6 \left (-5 a^3 B+3 a^2 A b+4 a b^2 B-2 A b^3\right ) \sin (c+d x)}{b d \sqrt {\cos (c+d x)}}-\frac {3 \left (-5 a^3 B+3 a^2 A b+4 a b^2 B-2 A b^3\right ) \int \sqrt {\cos (c+d x)}dx+\frac {\int \frac {b \left (-5 B a^2+3 A b a+2 b^2 B\right ) \cos (c+d x) a^2+\left (-15 B a^4+9 A b a^3+16 b^2 B a^2-12 A b^3 a+2 b^4 B\right ) a}{\sqrt {\cos (c+d x)} (b+a \cos (c+d x))}dx}{a}}{b}}{3 b}}{2 b \left (a^2-b^2\right )}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {a (A b-a B) \sin (c+d x)}{b d \left (a^2-b^2\right ) \cos ^{\frac {3}{2}}(c+d x) (a \cos (c+d x)+b)}-\frac {\frac {2 \left (-5 a^2 B+3 a A b+2 b^2 B\right ) \sin (c+d x)}{3 b d \cos ^{\frac {3}{2}}(c+d x)}-\frac {\frac {6 \left (-5 a^3 B+3 a^2 A b+4 a b^2 B-2 A b^3\right ) \sin (c+d x)}{b d \sqrt {\cos (c+d x)}}-\frac {3 \left (-5 a^3 B+3 a^2 A b+4 a b^2 B-2 A b^3\right ) \int \sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}dx+\frac {\int \frac {b \left (-5 B a^2+3 A b a+2 b^2 B\right ) \sin \left (c+d x+\frac {\pi }{2}\right ) a^2+\left (-15 B a^4+9 A b a^3+16 b^2 B a^2-12 A b^3 a+2 b^4 B\right ) a}{\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )} \left (b+a \sin \left (c+d x+\frac {\pi }{2}\right )\right )}dx}{a}}{b}}{3 b}}{2 b \left (a^2-b^2\right )}\)

\(\Big \downarrow \) 3119

\(\displaystyle \frac {a (A b-a B) \sin (c+d x)}{b d \left (a^2-b^2\right ) \cos ^{\frac {3}{2}}(c+d x) (a \cos (c+d x)+b)}-\frac {\frac {2 \left (-5 a^2 B+3 a A b+2 b^2 B\right ) \sin (c+d x)}{3 b d \cos ^{\frac {3}{2}}(c+d x)}-\frac {\frac {6 \left (-5 a^3 B+3 a^2 A b+4 a b^2 B-2 A b^3\right ) \sin (c+d x)}{b d \sqrt {\cos (c+d x)}}-\frac {\frac {\int \frac {b \left (-5 B a^2+3 A b a+2 b^2 B\right ) \sin \left (c+d x+\frac {\pi }{2}\right ) a^2+\left (-15 B a^4+9 A b a^3+16 b^2 B a^2-12 A b^3 a+2 b^4 B\right ) a}{\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )} \left (b+a \sin \left (c+d x+\frac {\pi }{2}\right )\right )}dx}{a}+\frac {6 \left (-5 a^3 B+3 a^2 A b+4 a b^2 B-2 A b^3\right ) E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{d}}{b}}{3 b}}{2 b \left (a^2-b^2\right )}\)

\(\Big \downarrow \) 3481

\(\displaystyle \frac {a (A b-a B) \sin (c+d x)}{b d \left (a^2-b^2\right ) \cos ^{\frac {3}{2}}(c+d x) (a \cos (c+d x)+b)}-\frac {\frac {2 \left (-5 a^2 B+3 a A b+2 b^2 B\right ) \sin (c+d x)}{3 b d \cos ^{\frac {3}{2}}(c+d x)}-\frac {\frac {6 \left (-5 a^3 B+3 a^2 A b+4 a b^2 B-2 A b^3\right ) \sin (c+d x)}{b d \sqrt {\cos (c+d x)}}-\frac {\frac {a b \left (-5 a^2 B+3 a A b+2 b^2 B\right ) \int \frac {1}{\sqrt {\cos (c+d x)}}dx+3 a^2 \left (-5 a^3 B+3 a^2 A b+7 a b^2 B-5 A b^3\right ) \int \frac {1}{\sqrt {\cos (c+d x)} (b+a \cos (c+d x))}dx}{a}+\frac {6 \left (-5 a^3 B+3 a^2 A b+4 a b^2 B-2 A b^3\right ) E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{d}}{b}}{3 b}}{2 b \left (a^2-b^2\right )}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {a (A b-a B) \sin (c+d x)}{b d \left (a^2-b^2\right ) \cos ^{\frac {3}{2}}(c+d x) (a \cos (c+d x)+b)}-\frac {\frac {2 \left (-5 a^2 B+3 a A b+2 b^2 B\right ) \sin (c+d x)}{3 b d \cos ^{\frac {3}{2}}(c+d x)}-\frac {\frac {6 \left (-5 a^3 B+3 a^2 A b+4 a b^2 B-2 A b^3\right ) \sin (c+d x)}{b d \sqrt {\cos (c+d x)}}-\frac {\frac {a b \left (-5 a^2 B+3 a A b+2 b^2 B\right ) \int \frac {1}{\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}}dx+3 a^2 \left (-5 a^3 B+3 a^2 A b+7 a b^2 B-5 A b^3\right ) \int \frac {1}{\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )} \left (b+a \sin \left (c+d x+\frac {\pi }{2}\right )\right )}dx}{a}+\frac {6 \left (-5 a^3 B+3 a^2 A b+4 a b^2 B-2 A b^3\right ) E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{d}}{b}}{3 b}}{2 b \left (a^2-b^2\right )}\)

\(\Big \downarrow \) 3120

\(\displaystyle \frac {a (A b-a B) \sin (c+d x)}{b d \left (a^2-b^2\right ) \cos ^{\frac {3}{2}}(c+d x) (a \cos (c+d x)+b)}-\frac {\frac {2 \left (-5 a^2 B+3 a A b+2 b^2 B\right ) \sin (c+d x)}{3 b d \cos ^{\frac {3}{2}}(c+d x)}-\frac {\frac {6 \left (-5 a^3 B+3 a^2 A b+4 a b^2 B-2 A b^3\right ) \sin (c+d x)}{b d \sqrt {\cos (c+d x)}}-\frac {\frac {3 a^2 \left (-5 a^3 B+3 a^2 A b+7 a b^2 B-5 A b^3\right ) \int \frac {1}{\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )} \left (b+a \sin \left (c+d x+\frac {\pi }{2}\right )\right )}dx+\frac {2 a b \left (-5 a^2 B+3 a A b+2 b^2 B\right ) \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )}{d}}{a}+\frac {6 \left (-5 a^3 B+3 a^2 A b+4 a b^2 B-2 A b^3\right ) E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{d}}{b}}{3 b}}{2 b \left (a^2-b^2\right )}\)

\(\Big \downarrow \) 3284

\(\displaystyle \frac {a (A b-a B) \sin (c+d x)}{b d \left (a^2-b^2\right ) \cos ^{\frac {3}{2}}(c+d x) (a \cos (c+d x)+b)}-\frac {\frac {2 \left (-5 a^2 B+3 a A b+2 b^2 B\right ) \sin (c+d x)}{3 b d \cos ^{\frac {3}{2}}(c+d x)}-\frac {\frac {6 \left (-5 a^3 B+3 a^2 A b+4 a b^2 B-2 A b^3\right ) \sin (c+d x)}{b d \sqrt {\cos (c+d x)}}-\frac {\frac {6 \left (-5 a^3 B+3 a^2 A b+4 a b^2 B-2 A b^3\right ) E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{d}+\frac {\frac {2 a b \left (-5 a^2 B+3 a A b+2 b^2 B\right ) \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )}{d}+\frac {6 a^2 \left (-5 a^3 B+3 a^2 A b+7 a b^2 B-5 A b^3\right ) \operatorname {EllipticPi}\left (\frac {2 a}{a+b},\frac {1}{2} (c+d x),2\right )}{d (a+b)}}{a}}{b}}{3 b}}{2 b \left (a^2-b^2\right )}\)

Input:

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

Output:

(a*(A*b - a*B)*Sin[c + d*x])/(b*(a^2 - b^2)*d*Cos[c + d*x]^(3/2)*(b + a*Co 
s[c + d*x])) - ((2*(3*a*A*b - 5*a^2*B + 2*b^2*B)*Sin[c + d*x])/(3*b*d*Cos[ 
c + d*x]^(3/2)) - (-(((6*(3*a^2*A*b - 2*A*b^3 - 5*a^3*B + 4*a*b^2*B)*Ellip 
ticE[(c + d*x)/2, 2])/d + ((2*a*b*(3*a*A*b - 5*a^2*B + 2*b^2*B)*EllipticF[ 
(c + d*x)/2, 2])/d + (6*a^2*(3*a^2*A*b - 5*A*b^3 - 5*a^3*B + 7*a*b^2*B)*El 
lipticPi[(2*a)/(a + b), (c + d*x)/2, 2])/((a + b)*d))/a)/b) + (6*(3*a^2*A* 
b - 2*A*b^3 - 5*a^3*B + 4*a*b^2*B)*Sin[c + d*x])/(b*d*Sqrt[Cos[c + d*x]])) 
/(3*b))/(2*b*(a^2 - b^2))
 

Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

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 3433

Int[((a_.) + csc[(e_.) + (f_.)*(x_)]*(b_.))^(m_.)*(csc[(e_.) + (f_.)*(x_)]* 
(d_.) + (c_))^(n_.)*((g_.)*sin[(e_.) + (f_.)*(x_)])^(p_.), x_Symbol] :> Sim 
p[g^(m + n)   Int[(g*Sin[e + f*x])^(p - m - n)*(b + a*Sin[e + f*x])^m*(d + 
c*Sin[e + f*x])^n, x], x] /; FreeQ[{a, b, c, d, e, f, g, p}, x] && NeQ[b*c 
- a*d, 0] &&  !IntegerQ[p] && IntegerQ[m] && IntegerQ[n]
 

rule 3479

Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((A_.) + (B_.)*sin[(e_.) + 
 (f_.)*(x_)])*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Si 
mp[(-(A*b^2 - a*b*B))*Cos[e + f*x]*(a + b*Sin[e + f*x])^(m + 1)*((c + d*Sin 
[e + f*x])^(1 + n)/(f*(m + 1)*(b*c - a*d)*(a^2 - b^2))), x] + Simp[1/((m + 
1)*(b*c - a*d)*(a^2 - b^2))   Int[(a + b*Sin[e + f*x])^(m + 1)*(c + d*Sin[e 
 + f*x])^n*Simp[(a*A - b*B)*(b*c - a*d)*(m + 1) + b*d*(A*b - a*B)*(m + n + 
2) + (A*b - a*B)*(a*d*(m + 1) - b*c*(m + 2))*Sin[e + f*x] - b*d*(A*b - a*B) 
*(m + n + 3)*Sin[e + f*x]^2, x], x], x] /; FreeQ[{a, b, c, d, e, f, A, B, n 
}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && Rat 
ionalQ[m] && m < -1 && ((EqQ[a, 0] && IntegerQ[m] &&  !IntegerQ[n]) ||  !(I 
ntegerQ[2*n] && LtQ[n, -1] && ((IntegerQ[n] &&  !IntegerQ[m]) || EqQ[a, 0]) 
))
 

rule 3481

Int[(((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((A_.) + (B_.)*sin[(e_.) 
+ (f_.)*(x_)]))/((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[ 
B/d   Int[(a + b*Sin[e + f*x])^m, x], x] - Simp[(B*c - A*d)/d   Int[(a + b* 
Sin[e + f*x])^m/(c + d*Sin[e + f*x]), x], x] /; FreeQ[{a, b, c, d, e, f, A, 
 B, m}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0]
 

rule 3534

Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*sin[(e_.) + 
 (f_.)*(x_)])^(n_)*((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)] + (C_.)*sin[(e_.) 
 + (f_.)*(x_)]^2), x_Symbol] :> Simp[(-(A*b^2 - a*b*B + a^2*C))*Cos[e + f*x 
]*(a + b*Sin[e + f*x])^(m + 1)*((c + d*Sin[e + f*x])^(n + 1)/(f*(m + 1)*(b* 
c - a*d)*(a^2 - b^2))), x] + Simp[1/((m + 1)*(b*c - a*d)*(a^2 - b^2))   Int 
[(a + b*Sin[e + f*x])^(m + 1)*(c + d*Sin[e + f*x])^n*Simp[(m + 1)*(b*c - a* 
d)*(a*A - b*B + a*C) + d*(A*b^2 - a*b*B + a^2*C)*(m + n + 2) - (c*(A*b^2 - 
a*b*B + a^2*C) + (m + 1)*(b*c - a*d)*(A*b - a*B + b*C))*Sin[e + f*x] - d*(A 
*b^2 - a*b*B + a^2*C)*(m + n + 3)*Sin[e + f*x]^2, x], x], x] /; FreeQ[{a, b 
, c, d, e, f, A, B, C, n}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && 
NeQ[c^2 - d^2, 0] && LtQ[m, -1] && ((EqQ[a, 0] && IntegerQ[m] &&  !IntegerQ 
[n]) ||  !(IntegerQ[2*n] && LtQ[n, -1] && ((IntegerQ[n] &&  !IntegerQ[m]) | 
| EqQ[a, 0])))
 

rule 3538

Int[((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)] + (C_.)*sin[(e_.) + (f_.)*(x_)]^ 
2)/(Sqrt[(a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)]]*((c_.) + (d_.)*sin[(e_.) + 
(f_.)*(x_)])), x_Symbol] :> Simp[C/(b*d)   Int[Sqrt[a + b*Sin[e + f*x]], x] 
, x] - Simp[1/(b*d)   Int[Simp[a*c*C - A*b*d + (b*c*C - b*B*d + a*C*d)*Sin[ 
e + f*x], x]/(Sqrt[a + b*Sin[e + f*x]]*(c + d*Sin[e + f*x])), x], x] /; Fre 
eQ[{a, b, c, d, e, f, A, B, C}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0 
] && NeQ[c^2 - d^2, 0]
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(996\) vs. \(2(341)=682\).

Time = 6.16 (sec) , antiderivative size = 997, normalized size of antiderivative = 2.88

method	result	size
default	\(\text {Expression too large to display}\)	\(997\)

Input:

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

Output:

-(-(-2*cos(1/2*d*x+1/2*c)^2+1)*sin(1/2*d*x+1/2*c)^2)^(1/2)*(2*B/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)/(c 
os(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)*E 
llipticF(cos(1/2*d*x+1/2*c),2^(1/2)))+2*(A*b-2*B*a)/b^3/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*sin(1/2*d*x+1/2*c)^2-1)^(1 
/2))+2*a^2*(A*b-2*B*a)/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)*a/b^2*(a 
^2/b/(a^2-b^2)*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)/(2*a*cos(1/2*d*x+1/2*c)^2-a+b)-1/2/(a+b)/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)*EllipticF(cos(1/2*d*x+1/2*c),2^(1/2))+1/2*a/b/(a^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*s 
in(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))-1/2*a/b/(a^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)*El 
lipticE(cos(1/2*d*x+1/2*c),2^(1/2))-1/2/b/(a^2-b^2)/(a^2-a*b)*a^3*(sin(...
 

Fricas [F(-1)]

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

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

Output:

Timed out
 

Sympy [F(-1)]

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

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

Output:

Timed out
 

Maxima [F(-1)]

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

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

Output:

Timed out
 

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int \frac {A+B \sec (c+d x)}{\cos ^{\frac {7}{2}}(c+d x) (a+b \sec (c+d x))^2} \, dx=\int \frac {\sqrt {\cos \left (d x +c \right )}}{\cos \left (d x +c \right )^{4} \sec \left (d x +c \right ) b +\cos \left (d x +c \right )^{4} a}d x \] Input:

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

Output:

int(sqrt(cos(c + d*x))/(cos(c + d*x)**4*sec(c + d*x)*b + cos(c + d*x)**4*a 
),x)