\(\int \frac {a b B-a^2 C+b^2 B \sec (c+d x)+b^2 C \sec ^2(c+d x)}{(a+b \sec (c+d x))^{7/2}} \, dx\) [980]

Optimal result

Integrand size = 50, antiderivative size = 519 \[ \int \frac {a b B-a^2 C+b^2 B \sec (c+d x)+b^2 C \sec ^2(c+d x)}{(a+b \sec (c+d x))^{7/2}} \, dx=\frac {2 \left (7 a^2 b B-3 b^3 B-11 a^3 C+3 a b^2 C\right ) \cot (c+d x) E\left (\arcsin \left (\frac {\sqrt {a+b \sec (c+d x)}}{\sqrt {a+b}}\right )|\frac {a+b}{a-b}\right ) \sqrt {\frac {b (1-\sec (c+d x))}{a+b}} \sqrt {-\frac {b (1+\sec (c+d x))}{a-b}}}{3 a^2 (a-b) (a+b)^{3/2} d}+\frac {2 \left (3 b^3 B+a b^2 (B-3 C)+9 a^3 C-2 a^2 b (3 B+C)\right ) \cot (c+d x) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {a+b \sec (c+d x)}}{\sqrt {a+b}}\right ),\frac {a+b}{a-b}\right ) \sqrt {\frac {b (1-\sec (c+d x))}{a+b}} \sqrt {-\frac {b (1+\sec (c+d x))}{a-b}}}{3 a^2 \sqrt {a+b} \left (a^2-b^2\right ) d}-\frac {2 \sqrt {a+b} (b B-a C) \cot (c+d x) \operatorname {EllipticPi}\left (\frac {a+b}{a},\arcsin \left (\frac {\sqrt {a+b \sec (c+d x)}}{\sqrt {a+b}}\right ),\frac {a+b}{a-b}\right ) \sqrt {\frac {b (1-\sec (c+d x))}{a+b}} \sqrt {-\frac {b (1+\sec (c+d x))}{a-b}}}{a^3 d}+\frac {2 b^2 (b B-2 a C) \tan (c+d x)}{3 a \left (a^2-b^2\right ) d (a+b \sec (c+d x))^{3/2}}+\frac {2 b^2 \left (7 a^2 b B-3 b^3 B-11 a^3 C+3 a b^2 C\right ) \tan (c+d x)}{3 a^2 \left (a^2-b^2\right )^2 d \sqrt {a+b \sec (c+d x)}} \] Output:

2/3*(7*B*a^2*b-3*B*b^3-11*C*a^3+3*C*a*b^2)*cot(d*x+c)*EllipticE((a+b*sec(d 
*x+c))^(1/2)/(a+b)^(1/2),((a+b)/(a-b))^(1/2))*(b*(1-sec(d*x+c))/(a+b))^(1/ 
2)*(-b*(1+sec(d*x+c))/(a-b))^(1/2)/a^2/(a-b)/(a+b)^(3/2)/d+2/3*(3*B*b^3+a* 
b^2*(B-3*C)+9*a^3*C-2*a^2*b*(3*B+C))*cot(d*x+c)*EllipticF((a+b*sec(d*x+c)) 
^(1/2)/(a+b)^(1/2),((a+b)/(a-b))^(1/2))*(b*(1-sec(d*x+c))/(a+b))^(1/2)*(-b 
*(1+sec(d*x+c))/(a-b))^(1/2)/a^2/(a+b)^(1/2)/(a^2-b^2)/d-2*(a+b)^(1/2)*(B* 
b-C*a)*cot(d*x+c)*EllipticPi((a+b*sec(d*x+c))^(1/2)/(a+b)^(1/2),(a+b)/a,(( 
a+b)/(a-b))^(1/2))*(b*(1-sec(d*x+c))/(a+b))^(1/2)*(-b*(1+sec(d*x+c))/(a-b) 
)^(1/2)/a^3/d+2/3*b^2*(B*b-2*C*a)*tan(d*x+c)/a/(a^2-b^2)/d/(a+b*sec(d*x+c) 
)^(3/2)+2/3*b^2*(7*B*a^2*b-3*B*b^3-11*C*a^3+3*C*a*b^2)*tan(d*x+c)/a^2/(a^2 
-b^2)^2/d/(a+b*sec(d*x+c))^(1/2)
 

Mathematica [A] (verified)

Time = 12.57 (sec) , antiderivative size = 615, normalized size of antiderivative = 1.18 \[ \int \frac {a b B-a^2 C+b^2 B \sec (c+d x)+b^2 C \sec ^2(c+d x)}{(a+b \sec (c+d x))^{7/2}} \, dx=\frac {2 (b+a \cos (c+d x))^2 (b B-a C+b C \sec (c+d x)) \left (-\frac {a b (a+b) \left (-7 a^2 b B+3 b^3 B+11 a^3 C-3 a b^2 C\right ) E\left (\arcsin \left (\tan \left (\frac {1}{2} (c+d x)\right )\right )|\frac {a-b}{a+b}\right ) \sec ^2\left (\frac {1}{2} (c+d x)\right ) \sqrt {\frac {(b+a \cos (c+d x)) \sec ^2\left (\frac {1}{2} (c+d x)\right )}{a+b}}+b (a+b) \left (3 b^4 B-2 a^2 b^2 (B-3 C)-12 a^4 C-3 a b^3 (2 B+C)+a^3 b (9 B+C)\right ) \operatorname {EllipticF}\left (\arcsin \left (\tan \left (\frac {1}{2} (c+d x)\right )\right ),\frac {a-b}{a+b}\right ) \sec ^2\left (\frac {1}{2} (c+d x)\right ) \sqrt {\frac {(b+a \cos (c+d x)) \sec ^2\left (\frac {1}{2} (c+d x)\right )}{a+b}}+3 (a-b)^2 (a+b)^2 (b B-a C) \left ((a-b) \operatorname {EllipticF}\left (\arcsin \left (\tan \left (\frac {1}{2} (c+d x)\right )\right ),\frac {a-b}{a+b}\right )-2 a \operatorname {EllipticPi}\left (-1,\arcsin \left (\tan \left (\frac {1}{2} (c+d x)\right )\right ),\frac {a-b}{a+b}\right )\right ) \sec ^2\left (\frac {1}{2} (c+d x)\right ) \sqrt {\frac {(b+a \cos (c+d x)) \sec ^2\left (\frac {1}{2} (c+d x)\right )}{a+b}}+a b \left (-7 a^2 b B+3 b^3 B+11 a^3 C-3 a b^2 C\right ) \left (\cos (c+d x) \sec ^2\left (\frac {1}{2} (c+d x)\right )\right )^{3/2} (a+b \sec (c+d x)) \tan \left (\frac {1}{2} (c+d x)\right )}{\left (\cos (c+d x) \sec ^2\left (\frac {1}{2} (c+d x)\right )\right )^{3/2}}+\frac {a^2 b \left (b \left (-6 a^2 b B+2 b^3 B+9 a^3 C-a b^2 C\right )+a \left (-7 a^2 b B+3 b^3 B+11 a^3 C-3 a b^2 C\right ) \cos (c+d x)\right ) \tan (c+d x)}{b+a \cos (c+d x)}\right )}{3 a^3 \left (a^2-b^2\right )^2 d (b C+(b B-a C) \cos (c+d x)) (a+b \sec (c+d x))^{5/2}} \] Input:

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

Output:

(2*(b + a*Cos[c + d*x])^2*(b*B - a*C + b*C*Sec[c + d*x])*(-((a*b*(a + b)*( 
-7*a^2*b*B + 3*b^3*B + 11*a^3*C - 3*a*b^2*C)*EllipticE[ArcSin[Tan[(c + d*x 
)/2]], (a - b)/(a + b)]*Sec[(c + d*x)/2]^2*Sqrt[((b + a*Cos[c + d*x])*Sec[ 
(c + d*x)/2]^2)/(a + b)] + b*(a + b)*(3*b^4*B - 2*a^2*b^2*(B - 3*C) - 12*a 
^4*C - 3*a*b^3*(2*B + C) + a^3*b*(9*B + C))*EllipticF[ArcSin[Tan[(c + d*x) 
/2]], (a - b)/(a + b)]*Sec[(c + d*x)/2]^2*Sqrt[((b + a*Cos[c + d*x])*Sec[( 
c + d*x)/2]^2)/(a + b)] + 3*(a - b)^2*(a + b)^2*(b*B - a*C)*((a - b)*Ellip 
ticF[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)] - 2*a*EllipticPi[-1, ArcSi 
n[Tan[(c + d*x)/2]], (a - b)/(a + b)])*Sec[(c + d*x)/2]^2*Sqrt[((b + a*Cos 
[c + d*x])*Sec[(c + d*x)/2]^2)/(a + b)] + a*b*(-7*a^2*b*B + 3*b^3*B + 11*a 
^3*C - 3*a*b^2*C)*(Cos[c + d*x]*Sec[(c + d*x)/2]^2)^(3/2)*(a + b*Sec[c + d 
*x])*Tan[(c + d*x)/2])/(Cos[c + d*x]*Sec[(c + d*x)/2]^2)^(3/2)) + (a^2*b*( 
b*(-6*a^2*b*B + 2*b^3*B + 9*a^3*C - a*b^2*C) + a*(-7*a^2*b*B + 3*b^3*B + 1 
1*a^3*C - 3*a*b^2*C)*Cos[c + d*x])*Tan[c + d*x])/(b + a*Cos[c + d*x])))/(3 
*a^3*(a^2 - b^2)^2*d*(b*C + (b*B - a*C)*Cos[c + d*x])*(a + b*Sec[c + d*x]) 
^(5/2))
 

Rubi [A] (verified)

Time = 2.25 (sec) , antiderivative size = 558, normalized size of antiderivative = 1.08, number of steps used = 15, number of rules used = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {2014, 3042, 4411, 27, 3042, 4548, 27, 3042, 4546, 3042, 4409, 3042, 4271, 4319, 4492}

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.

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

\(\Big \downarrow \) 2014

\(\displaystyle \frac {\int \frac {C \sec (c+d x) b^3+(b B-a C) b^2}{(a+b \sec (c+d x))^{5/2}}dx}{b^2}\)

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4411

\(\displaystyle \frac {\frac {2 b^4 (b B-2 a C) \tan (c+d x)}{3 a d \left (a^2-b^2\right ) (a+b \sec (c+d x))^{3/2}}-\frac {2 \int -\frac {(b B-2 a C) \sec ^2(c+d x) b^4-3 a (b B-2 a C) \sec (c+d x) b^3+3 \left (a^2-b^2\right ) (b B-a C) b^2}{2 (a+b \sec (c+d x))^{3/2}}dx}{3 a \left (a^2-b^2\right )}}{b^2}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\frac {\int \frac {(b B-2 a C) \sec ^2(c+d x) b^4-3 a (b B-2 a C) \sec (c+d x) b^3+3 \left (a^2-b^2\right ) (b B-a C) b^2}{(a+b \sec (c+d x))^{3/2}}dx}{3 a \left (a^2-b^2\right )}+\frac {2 b^4 (b B-2 a C) \tan (c+d x)}{3 a d \left (a^2-b^2\right ) (a+b \sec (c+d x))^{3/2}}}{b^2}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\frac {\int \frac {(b B-2 a C) \csc \left (c+d x+\frac {\pi }{2}\right )^2 b^4-3 a (b B-2 a C) \csc \left (c+d x+\frac {\pi }{2}\right ) b^3+3 \left (a^2-b^2\right ) (b B-a C) b^2}{\left (a+b \csc \left (c+d x+\frac {\pi }{2}\right )\right )^{3/2}}dx}{3 a \left (a^2-b^2\right )}+\frac {2 b^4 (b B-2 a C) \tan (c+d x)}{3 a d \left (a^2-b^2\right ) (a+b \sec (c+d x))^{3/2}}}{b^2}\)

\(\Big \downarrow \) 4548

\(\displaystyle \frac {\frac {\frac {2 b^4 \left (-11 a^3 C+7 a^2 b B+3 a b^2 C-3 b^3 B\right ) \tan (c+d x)}{a d \left (a^2-b^2\right ) \sqrt {a+b \sec (c+d x)}}-\frac {2 \int -\frac {-\left (\left (-11 C a^3+7 b B a^2+3 b^2 C a-3 b^3 B\right ) \sec ^2(c+d x) b^4\right )-a \left (-9 C a^3+6 b B a^2+b^2 C a-2 b^3 B\right ) \sec (c+d x) b^3+3 \left (a^2-b^2\right )^2 (b B-a C) b^2}{2 \sqrt {a+b \sec (c+d x)}}dx}{a \left (a^2-b^2\right )}}{3 a \left (a^2-b^2\right )}+\frac {2 b^4 (b B-2 a C) \tan (c+d x)}{3 a d \left (a^2-b^2\right ) (a+b \sec (c+d x))^{3/2}}}{b^2}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\frac {\frac {\int \frac {-\left (\left (-11 C a^3+7 b B a^2+3 b^2 C a-3 b^3 B\right ) \sec ^2(c+d x) b^4\right )-a \left (-9 C a^3+6 b B a^2+b^2 C a-2 b^3 B\right ) \sec (c+d x) b^3+3 \left (a^2-b^2\right )^2 (b B-a C) b^2}{\sqrt {a+b \sec (c+d x)}}dx}{a \left (a^2-b^2\right )}+\frac {2 b^4 \left (-11 a^3 C+7 a^2 b B+3 a b^2 C-3 b^3 B\right ) \tan (c+d x)}{a d \left (a^2-b^2\right ) \sqrt {a+b \sec (c+d x)}}}{3 a \left (a^2-b^2\right )}+\frac {2 b^4 (b B-2 a C) \tan (c+d x)}{3 a d \left (a^2-b^2\right ) (a+b \sec (c+d x))^{3/2}}}{b^2}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\frac {\frac {\int \frac {-\left (\left (-11 C a^3+7 b B a^2+3 b^2 C a-3 b^3 B\right ) \csc \left (c+d x+\frac {\pi }{2}\right )^2 b^4\right )-a \left (-9 C a^3+6 b B a^2+b^2 C a-2 b^3 B\right ) \csc \left (c+d x+\frac {\pi }{2}\right ) b^3+3 \left (a^2-b^2\right )^2 (b B-a C) b^2}{\sqrt {a+b \csc \left (c+d x+\frac {\pi }{2}\right )}}dx}{a \left (a^2-b^2\right )}+\frac {2 b^4 \left (-11 a^3 C+7 a^2 b B+3 a b^2 C-3 b^3 B\right ) \tan (c+d x)}{a d \left (a^2-b^2\right ) \sqrt {a+b \sec (c+d x)}}}{3 a \left (a^2-b^2\right )}+\frac {2 b^4 (b B-2 a C) \tan (c+d x)}{3 a d \left (a^2-b^2\right ) (a+b \sec (c+d x))^{3/2}}}{b^2}\)

\(\Big \downarrow \) 4546

\(\displaystyle \frac {\frac {\frac {\int \frac {3 b^2 (b B-a C) \left (a^2-b^2\right )^2+\left (b^4 \left (-11 C a^3+7 b B a^2+3 b^2 C a-3 b^3 B\right )-a b^3 \left (-9 C a^3+6 b B a^2+b^2 C a-2 b^3 B\right )\right ) \sec (c+d x)}{\sqrt {a+b \sec (c+d x)}}dx-b^4 \left (-11 a^3 C+7 a^2 b B+3 a b^2 C-3 b^3 B\right ) \int \frac {\sec (c+d x) (\sec (c+d x)+1)}{\sqrt {a+b \sec (c+d x)}}dx}{a \left (a^2-b^2\right )}+\frac {2 b^4 \left (-11 a^3 C+7 a^2 b B+3 a b^2 C-3 b^3 B\right ) \tan (c+d x)}{a d \left (a^2-b^2\right ) \sqrt {a+b \sec (c+d x)}}}{3 a \left (a^2-b^2\right )}+\frac {2 b^4 (b B-2 a C) \tan (c+d x)}{3 a d \left (a^2-b^2\right ) (a+b \sec (c+d x))^{3/2}}}{b^2}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\frac {\frac {\int \frac {3 b^2 (b B-a C) \left (a^2-b^2\right )^2+\left (b^4 \left (-11 C a^3+7 b B a^2+3 b^2 C a-3 b^3 B\right )-a b^3 \left (-9 C a^3+6 b B a^2+b^2 C a-2 b^3 B\right )\right ) \csc \left (c+d x+\frac {\pi }{2}\right )}{\sqrt {a+b \csc \left (c+d x+\frac {\pi }{2}\right )}}dx-b^4 \left (-11 a^3 C+7 a^2 b B+3 a b^2 C-3 b^3 B\right ) \int \frac {\csc \left (c+d x+\frac {\pi }{2}\right ) \left (\csc \left (c+d x+\frac {\pi }{2}\right )+1\right )}{\sqrt {a+b \csc \left (c+d x+\frac {\pi }{2}\right )}}dx}{a \left (a^2-b^2\right )}+\frac {2 b^4 \left (-11 a^3 C+7 a^2 b B+3 a b^2 C-3 b^3 B\right ) \tan (c+d x)}{a d \left (a^2-b^2\right ) \sqrt {a+b \sec (c+d x)}}}{3 a \left (a^2-b^2\right )}+\frac {2 b^4 (b B-2 a C) \tan (c+d x)}{3 a d \left (a^2-b^2\right ) (a+b \sec (c+d x))^{3/2}}}{b^2}\)

\(\Big \downarrow \) 4409

\(\displaystyle \frac {\frac {\frac {3 b^2 \left (a^2-b^2\right )^2 (b B-a C) \int \frac {1}{\sqrt {a+b \sec (c+d x)}}dx+b^3 (a-b) \left (9 a^3 C-2 a^2 b (3 B+C)+a b^2 (B-3 C)+3 b^3 B\right ) \int \frac {\sec (c+d x)}{\sqrt {a+b \sec (c+d x)}}dx-\left (b^4 \left (-11 a^3 C+7 a^2 b B+3 a b^2 C-3 b^3 B\right ) \int \frac {\csc \left (c+d x+\frac {\pi }{2}\right ) \left (\csc \left (c+d x+\frac {\pi }{2}\right )+1\right )}{\sqrt {a+b \csc \left (c+d x+\frac {\pi }{2}\right )}}dx\right )}{a \left (a^2-b^2\right )}+\frac {2 b^4 \left (-11 a^3 C+7 a^2 b B+3 a b^2 C-3 b^3 B\right ) \tan (c+d x)}{a d \left (a^2-b^2\right ) \sqrt {a+b \sec (c+d x)}}}{3 a \left (a^2-b^2\right )}+\frac {2 b^4 (b B-2 a C) \tan (c+d x)}{3 a d \left (a^2-b^2\right ) (a+b \sec (c+d x))^{3/2}}}{b^2}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\frac {\frac {3 b^2 \left (a^2-b^2\right )^2 (b B-a C) \int \frac {1}{\sqrt {a+b \csc \left (c+d x+\frac {\pi }{2}\right )}}dx+b^3 (a-b) \left (9 a^3 C-2 a^2 b (3 B+C)+a b^2 (B-3 C)+3 b^3 B\right ) \int \frac {\csc \left (c+d x+\frac {\pi }{2}\right )}{\sqrt {a+b \csc \left (c+d x+\frac {\pi }{2}\right )}}dx-\left (b^4 \left (-11 a^3 C+7 a^2 b B+3 a b^2 C-3 b^3 B\right ) \int \frac {\csc \left (c+d x+\frac {\pi }{2}\right ) \left (\csc \left (c+d x+\frac {\pi }{2}\right )+1\right )}{\sqrt {a+b \csc \left (c+d x+\frac {\pi }{2}\right )}}dx\right )}{a \left (a^2-b^2\right )}+\frac {2 b^4 \left (-11 a^3 C+7 a^2 b B+3 a b^2 C-3 b^3 B\right ) \tan (c+d x)}{a d \left (a^2-b^2\right ) \sqrt {a+b \sec (c+d x)}}}{3 a \left (a^2-b^2\right )}+\frac {2 b^4 (b B-2 a C) \tan (c+d x)}{3 a d \left (a^2-b^2\right ) (a+b \sec (c+d x))^{3/2}}}{b^2}\)

\(\Big \downarrow \) 4271

\(\displaystyle \frac {\frac {\frac {b^3 (a-b) \left (9 a^3 C-2 a^2 b (3 B+C)+a b^2 (B-3 C)+3 b^3 B\right ) \int \frac {\csc \left (c+d x+\frac {\pi }{2}\right )}{\sqrt {a+b \csc \left (c+d x+\frac {\pi }{2}\right )}}dx-\left (b^4 \left (-11 a^3 C+7 a^2 b B+3 a b^2 C-3 b^3 B\right ) \int \frac {\csc \left (c+d x+\frac {\pi }{2}\right ) \left (\csc \left (c+d x+\frac {\pi }{2}\right )+1\right )}{\sqrt {a+b \csc \left (c+d x+\frac {\pi }{2}\right )}}dx\right )-\frac {6 b^2 \sqrt {a+b} \left (a^2-b^2\right )^2 (b B-a C) \cot (c+d x) \sqrt {\frac {b (1-\sec (c+d x))}{a+b}} \sqrt {-\frac {b (\sec (c+d x)+1)}{a-b}} \operatorname {EllipticPi}\left (\frac {a+b}{a},\arcsin \left (\frac {\sqrt {a+b \sec (c+d x)}}{\sqrt {a+b}}\right ),\frac {a+b}{a-b}\right )}{a d}}{a \left (a^2-b^2\right )}+\frac {2 b^4 \left (-11 a^3 C+7 a^2 b B+3 a b^2 C-3 b^3 B\right ) \tan (c+d x)}{a d \left (a^2-b^2\right ) \sqrt {a+b \sec (c+d x)}}}{3 a \left (a^2-b^2\right )}+\frac {2 b^4 (b B-2 a C) \tan (c+d x)}{3 a d \left (a^2-b^2\right ) (a+b \sec (c+d x))^{3/2}}}{b^2}\)

\(\Big \downarrow \) 4319

\(\displaystyle \frac {\frac {\frac {-\left (b^4 \left (-11 a^3 C+7 a^2 b B+3 a b^2 C-3 b^3 B\right ) \int \frac {\csc \left (c+d x+\frac {\pi }{2}\right ) \left (\csc \left (c+d x+\frac {\pi }{2}\right )+1\right )}{\sqrt {a+b \csc \left (c+d x+\frac {\pi }{2}\right )}}dx\right )-\frac {6 b^2 \sqrt {a+b} \left (a^2-b^2\right )^2 (b B-a C) \cot (c+d x) \sqrt {\frac {b (1-\sec (c+d x))}{a+b}} \sqrt {-\frac {b (\sec (c+d x)+1)}{a-b}} \operatorname {EllipticPi}\left (\frac {a+b}{a},\arcsin \left (\frac {\sqrt {a+b \sec (c+d x)}}{\sqrt {a+b}}\right ),\frac {a+b}{a-b}\right )}{a d}+\frac {2 b^2 (a-b) \sqrt {a+b} \left (9 a^3 C-2 a^2 b (3 B+C)+a b^2 (B-3 C)+3 b^3 B\right ) \cot (c+d x) \sqrt {\frac {b (1-\sec (c+d x))}{a+b}} \sqrt {-\frac {b (\sec (c+d x)+1)}{a-b}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {a+b \sec (c+d x)}}{\sqrt {a+b}}\right ),\frac {a+b}{a-b}\right )}{d}}{a \left (a^2-b^2\right )}+\frac {2 b^4 \left (-11 a^3 C+7 a^2 b B+3 a b^2 C-3 b^3 B\right ) \tan (c+d x)}{a d \left (a^2-b^2\right ) \sqrt {a+b \sec (c+d x)}}}{3 a \left (a^2-b^2\right )}+\frac {2 b^4 (b B-2 a C) \tan (c+d x)}{3 a d \left (a^2-b^2\right ) (a+b \sec (c+d x))^{3/2}}}{b^2}\)

\(\Big \downarrow \) 4492

\(\displaystyle \frac {\frac {2 b^4 (b B-2 a C) \tan (c+d x)}{3 a d \left (a^2-b^2\right ) (a+b \sec (c+d x))^{3/2}}+\frac {\frac {-\frac {6 b^2 \sqrt {a+b} \left (a^2-b^2\right )^2 (b B-a C) \cot (c+d x) \sqrt {\frac {b (1-\sec (c+d x))}{a+b}} \sqrt {-\frac {b (\sec (c+d x)+1)}{a-b}} \operatorname {EllipticPi}\left (\frac {a+b}{a},\arcsin \left (\frac {\sqrt {a+b \sec (c+d x)}}{\sqrt {a+b}}\right ),\frac {a+b}{a-b}\right )}{a d}+\frac {2 b^2 (a-b) \sqrt {a+b} \left (9 a^3 C-2 a^2 b (3 B+C)+a b^2 (B-3 C)+3 b^3 B\right ) \cot (c+d x) \sqrt {\frac {b (1-\sec (c+d x))}{a+b}} \sqrt {-\frac {b (\sec (c+d x)+1)}{a-b}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {a+b \sec (c+d x)}}{\sqrt {a+b}}\right ),\frac {a+b}{a-b}\right )}{d}+\frac {2 b^2 (a-b) \sqrt {a+b} \left (-11 a^3 C+7 a^2 b B+3 a b^2 C-3 b^3 B\right ) \cot (c+d x) \sqrt {\frac {b (1-\sec (c+d x))}{a+b}} \sqrt {-\frac {b (\sec (c+d x)+1)}{a-b}} E\left (\arcsin \left (\frac {\sqrt {a+b \sec (c+d x)}}{\sqrt {a+b}}\right )|\frac {a+b}{a-b}\right )}{d}}{a \left (a^2-b^2\right )}+\frac {2 b^4 \left (-11 a^3 C+7 a^2 b B+3 a b^2 C-3 b^3 B\right ) \tan (c+d x)}{a d \left (a^2-b^2\right ) \sqrt {a+b \sec (c+d x)}}}{3 a \left (a^2-b^2\right )}}{b^2}\)

Input:

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

Output:

((2*b^4*(b*B - 2*a*C)*Tan[c + d*x])/(3*a*(a^2 - b^2)*d*(a + b*Sec[c + d*x] 
)^(3/2)) + (((2*(a - b)*b^2*Sqrt[a + b]*(7*a^2*b*B - 3*b^3*B - 11*a^3*C + 
3*a*b^2*C)*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + 
 b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + 
 Sec[c + d*x]))/(a - b))])/d + (2*(a - b)*b^2*Sqrt[a + b]*(3*b^3*B + a*b^2 
*(B - 3*C) + 9*a^3*C - 2*a^2*b*(3*B + C))*Cot[c + d*x]*EllipticF[ArcSin[Sq 
rt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + 
 d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/d - (6*b^2*Sqrt[ 
a + b]*(a^2 - b^2)^2*(b*B - a*C)*Cot[c + d*x]*EllipticPi[(a + b)/a, ArcSin 
[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[ 
c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(a*d))/(a*(a^ 
2 - b^2)) + (2*b^4*(7*a^2*b*B - 3*b^3*B - 11*a^3*C + 3*a*b^2*C)*Tan[c + d* 
x])/(a*(a^2 - b^2)*d*Sqrt[a + b*Sec[c + d*x]]))/(3*a*(a^2 - b^2)))/b^2
 

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 2014

Int[(u_.)*((a_) + (b_.)*(v_))^(m_)*((A_.) + (B_.)*(v_) + (C_.)*(v_)^2), x_S 
ymbol] :> Simp[1/b^2   Int[u*(a + b*v)^(m + 1)*Simp[b*B - a*C + b*C*v, x], 
x], x] /; FreeQ[{a, b, A, B, C}, x] && EqQ[A*b^2 - a*b*B + a^2*C, 0] && LeQ 
[m, -1]
 

rule 3042

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

rule 4271

Int[1/Sqrt[csc[(c_.) + (d_.)*(x_)]*(b_.) + (a_)], x_Symbol] :> Simp[2*(Rt[a 
 + b, 2]/(a*d*Cot[c + d*x]))*Sqrt[b*((1 - Csc[c + d*x])/(a + b))]*Sqrt[(-b) 
*((1 + Csc[c + d*x])/(a - b))]*EllipticPi[(a + b)/a, ArcSin[Sqrt[a + b*Csc[ 
c + d*x]]/Rt[a + b, 2]], (a + b)/(a - b)], x] /; FreeQ[{a, b, c, d}, x] && 
NeQ[a^2 - b^2, 0]
 

rule 4319

Int[csc[(e_.) + (f_.)*(x_)]/Sqrt[csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_)], x_S 
ymbol] :> Simp[-2*(Rt[a + b, 2]/(b*f*Cot[e + f*x]))*Sqrt[(b*(1 - Csc[e + f* 
x]))/(a + b)]*Sqrt[(-b)*((1 + Csc[e + f*x])/(a - b))]*EllipticF[ArcSin[Sqrt 
[a + b*Csc[e + f*x]]/Rt[a + b, 2]], (a + b)/(a - b)], x] /; FreeQ[{a, b, e, 
 f}, x] && NeQ[a^2 - b^2, 0]
 

rule 4409

Int[(csc[(e_.) + (f_.)*(x_)]*(d_.) + (c_))/Sqrt[csc[(e_.) + (f_.)*(x_)]*(b_ 
.) + (a_)], x_Symbol] :> Simp[c   Int[1/Sqrt[a + b*Csc[e + f*x]], x], x] + 
Simp[d   Int[Csc[e + f*x]/Sqrt[a + b*Csc[e + f*x]], x], x] /; FreeQ[{a, b, 
c, d, e, f}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0]
 

rule 4411

Int[(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_))^(m_)*(csc[(e_.) + (f_.)*(x_)]*(d 
_.) + (c_)), x_Symbol] :> Simp[b*(b*c - a*d)*Cot[e + f*x]*((a + b*Csc[e + f 
*x])^(m + 1)/(a*f*(m + 1)*(a^2 - b^2))), x] + Simp[1/(a*(m + 1)*(a^2 - b^2) 
)   Int[(a + b*Csc[e + f*x])^(m + 1)*Simp[c*(a^2 - b^2)*(m + 1) - (a*(b*c - 
 a*d)*(m + 1))*Csc[e + f*x] + b*(b*c - a*d)*(m + 2)*Csc[e + f*x]^2, x], x], 
 x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && LtQ[m, -1] && N 
eQ[a^2 - b^2, 0] && IntegerQ[2*m]
 

rule 4492

Int[(csc[(e_.) + (f_.)*(x_)]*(csc[(e_.) + (f_.)*(x_)]*(B_.) + (A_)))/Sqrt[c 
sc[(e_.) + (f_.)*(x_)]*(b_.) + (a_)], x_Symbol] :> Simp[-2*(A*b - a*B)*Rt[a 
 + b*(B/A), 2]*Sqrt[b*((1 - Csc[e + f*x])/(a + b))]*(Sqrt[(-b)*((1 + Csc[e 
+ f*x])/(a - b))]/(b^2*f*Cot[e + f*x]))*EllipticE[ArcSin[Sqrt[a + b*Csc[e + 
 f*x]]/Rt[a + b*(B/A), 2]], (a*A + b*B)/(a*A - b*B)], x] /; FreeQ[{a, b, e, 
 f, A, B}, x] && NeQ[a^2 - b^2, 0] && EqQ[A^2 - B^2, 0]
 

rule 4546

Int[((A_.) + csc[(e_.) + (f_.)*(x_)]*(B_.) + csc[(e_.) + (f_.)*(x_)]^2*(C_. 
))/Sqrt[csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_)], x_Symbol] :> Int[(A + (B - C 
)*Csc[e + f*x])/Sqrt[a + b*Csc[e + f*x]], x] + Simp[C   Int[Csc[e + f*x]*(( 
1 + Csc[e + f*x])/Sqrt[a + b*Csc[e + f*x]]), x], x] /; FreeQ[{a, b, e, f, A 
, B, C}, x] && NeQ[a^2 - b^2, 0]
 

rule 4548

Int[((A_.) + csc[(e_.) + (f_.)*(x_)]*(B_.) + csc[(e_.) + (f_.)*(x_)]^2*(C_. 
))*(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_))^(m_), x_Symbol] :> Simp[(A*b^2 - 
a*b*B + a^2*C)*Cot[e + f*x]*((a + b*Csc[e + f*x])^(m + 1)/(a*f*(m + 1)*(a^2 
 - b^2))), x] + Simp[1/(a*(m + 1)*(a^2 - b^2))   Int[(a + b*Csc[e + f*x])^( 
m + 1)*Simp[A*(a^2 - b^2)*(m + 1) - a*(A*b - a*B + b*C)*(m + 1)*Csc[e + f*x 
] + (A*b^2 - a*b*B + a^2*C)*(m + 2)*Csc[e + f*x]^2, x], x], x] /; FreeQ[{a, 
 b, e, f, A, B, C}, x] && NeQ[a^2 - b^2, 0] && LtQ[m, -1]
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(3854\) vs. \(2(480)=960\).

Time = 50.28 (sec) , antiderivative size = 3855, normalized size of antiderivative = 7.43

method	result	size
default	\(\text {Expression too large to display}\)	\(3855\)
parts	\(\text {Expression too large to display}\)	\(6640\)

Input:

int((B*a*b-C*a^2+b^2*B*sec(d*x+c)+b^2*C*sec(d*x+c)^2)/(a+b*sec(d*x+c))^(7/ 
2),x,method=_RETURNVERBOSE)
 

Output:

-2/3/d/(a-b)^2/(a+b)^2/a^2*(7*B*a^4*b^2*cos(d*x+c)^2*sin(d*x+c)+C*(1/(a+b) 
*(b+a*cos(d*x+c))/(cos(d*x+c)+1))^(1/2)*(cos(d*x+c)/(cos(d*x+c)+1))^(1/2)* 
a^6*EllipticF(-csc(d*x+c)+cot(d*x+c),((a-b)/(a+b))^(1/2))*(3*cos(d*x+c)^3+ 
6*cos(d*x+c)^2+3*cos(d*x+c))+3*B*b^6*cos(d*x+c)*sin(d*x+c)-3*C*a*b^5*cos(d 
*x+c)*sin(d*x+c)+sin(d*x+c)*cos(d*x+c)*(-8*cos(d*x+c)+6)*a^3*b^3*B+sin(d*x 
+c)*cos(d*x+c)*(-3*cos(d*x+c)-7)*B*a^2*b^4+sin(d*x+c)*cos(d*x+c)*(4*cos(d* 
x+c)-2)*a*b^5*B+sin(d*x+c)*cos(d*x+c)*(13*cos(d*x+c)-9)*a^4*b^2*C+sin(d*x+ 
c)*cos(d*x+c)*(3*cos(d*x+c)+11)*C*a^3*b^3+sin(d*x+c)*cos(d*x+c)*(-5*cos(d* 
x+c)+1)*C*a^2*b^4+(-3*cos(d*x+c)^2-6*cos(d*x+c)-3)*B*(1/(a+b)*(b+a*cos(d*x 
+c))/(cos(d*x+c)+1))^(1/2)*(cos(d*x+c)/(cos(d*x+c)+1))^(1/2)*b^6*EllipticE 
(-csc(d*x+c)+cot(d*x+c),((a-b)/(a+b))^(1/2))+(6*cos(d*x+c)^2+12*cos(d*x+c) 
+6)*B*(1/(a+b)*(b+a*cos(d*x+c))/(cos(d*x+c)+1))^(1/2)*(cos(d*x+c)/(cos(d*x 
+c)+1))^(1/2)*b^6*EllipticPi(-csc(d*x+c)+cot(d*x+c),-1,((a-b)/(a+b))^(1/2) 
)+(-6*cos(d*x+c)^2-12*cos(d*x+c)-6)*C*(1/(a+b)*(b+a*cos(d*x+c))/(cos(d*x+c 
)+1))^(1/2)*(cos(d*x+c)/(cos(d*x+c)+1))^(1/2)*a*b^5*EllipticPi(-csc(d*x+c) 
+cot(d*x+c),-1,((a-b)/(a+b))^(1/2))+(7*cos(d*x+c)^3+21*cos(d*x+c)^2+21*cos 
(d*x+c)+7)*B*(1/(a+b)*(b+a*cos(d*x+c))/(cos(d*x+c)+1))^(1/2)*(cos(d*x+c)/( 
cos(d*x+c)+1))^(1/2)*a^3*b^3*EllipticE(-csc(d*x+c)+cot(d*x+c),((a-b)/(a+b) 
)^(1/2))+(-3*cos(d*x+c)^3+cos(d*x+c)^2+11*cos(d*x+c)+7)*B*(1/(a+b)*(b+a*co 
s(d*x+c))/(cos(d*x+c)+1))^(1/2)*(cos(d*x+c)/(cos(d*x+c)+1))^(1/2)*a^2*b...
 

Fricas [F(-1)]

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

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

Output:

Timed out
 

Sympy [F]

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

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

Output:

-Integral(-B*b/(a**2*sqrt(a + b*sec(c + d*x)) + 2*a*b*sqrt(a + b*sec(c + d 
*x))*sec(c + d*x) + b**2*sqrt(a + b*sec(c + d*x))*sec(c + d*x)**2), x) - I 
ntegral(C*a/(a**2*sqrt(a + b*sec(c + d*x)) + 2*a*b*sqrt(a + b*sec(c + d*x) 
)*sec(c + d*x) + b**2*sqrt(a + b*sec(c + d*x))*sec(c + d*x)**2), x) - Inte 
gral(-C*b*sec(c + d*x)/(a**2*sqrt(a + b*sec(c + d*x)) + 2*a*b*sqrt(a + b*s 
ec(c + d*x))*sec(c + d*x) + b**2*sqrt(a + b*sec(c + d*x))*sec(c + d*x)**2) 
, x)
 

Maxima [F(-1)]

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

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

Output:

Timed out
 

Giac [F]

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

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

Output:

integrate((C*b^2*sec(d*x + c)^2 + B*b^2*sec(d*x + c) - C*a^2 + B*a*b)/(b*s 
ec(d*x + c) + a)^(7/2), x)
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

 - int(sqrt(sec(c + d*x)*b + a)/(sec(c + d*x)**3*b**3 + 3*sec(c + d*x)**2* 
a*b**2 + 3*sec(c + d*x)*a**2*b + a**3),x)*a*c + int(sqrt(sec(c + d*x)*b + 
a)/(sec(c + d*x)**3*b**3 + 3*sec(c + d*x)**2*a*b**2 + 3*sec(c + d*x)*a**2* 
b + a**3),x)*b**2 + int((sqrt(sec(c + d*x)*b + a)*sec(c + d*x))/(sec(c + d 
*x)**3*b**3 + 3*sec(c + d*x)**2*a*b**2 + 3*sec(c + d*x)*a**2*b + a**3),x)* 
b*c