Integrand size = 45, antiderivative size = 453 \[ \int \frac {(a+b \sec (c+d x))^{5/2} \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right )}{\sqrt {\sec (c+d x)}} \, dx=\frac {\left (48 a^3 B+66 a b^2 B+8 b^3 (3 A+2 C)+a^2 b (96 A+59 C)\right ) \sqrt {\frac {b+a \cos (c+d x)}{a+b}} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),\frac {2 a}{a+b}\right ) \sqrt {\sec (c+d x)}}{24 d \sqrt {a+b \sec (c+d x)}}+\frac {\left (30 a^2 b B+8 b^3 B+5 a^3 C+20 a b^2 (2 A+C)\right ) \sqrt {\frac {b+a \cos (c+d x)}{a+b}} \operatorname {EllipticPi}\left (2,\frac {1}{2} (c+d x),\frac {2 a}{a+b}\right ) \sqrt {\sec (c+d x)}}{8 d \sqrt {a+b \sec (c+d x)}}-\frac {\left (54 a b B-a^2 (48 A-33 C)+8 b^2 (3 A+2 C)\right ) E\left (\frac {1}{2} (c+d x)|\frac {2 a}{a+b}\right ) \sqrt {a+b \sec (c+d x)}}{24 d \sqrt {\frac {b+a \cos (c+d x)}{a+b}} \sqrt {\sec (c+d x)}}+\frac {\left (24 A b^2+42 a b B+15 a^2 C+16 b^2 C\right ) \sqrt {\sec (c+d x)} \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{24 d}+\frac {(6 b B+5 a C) \sqrt {\sec (c+d x)} (a+b \sec (c+d x))^{3/2} \sin (c+d x)}{12 d}+\frac {C \sqrt {\sec (c+d x)} (a+b \sec (c+d x))^{5/2} \sin (c+d x)}{3 d} \]
1/12*(6*B*b+5*C*a)*(a+b*sec(d*x+c))^(3/2)*sin(d*x+c)*sec(d*x+c)^(1/2)/d+1/ 3*C*(a+b*sec(d*x+c))^(5/2)*sin(d*x+c)*sec(d*x+c)^(1/2)/d+1/24*(48*B*a^3+66 *B*a*b^2+8*b^3*(3*A+2*C)+a^2*b*(96*A+59*C))*(cos(1/2*d*x+1/2*c)^2)^(1/2)/c os(1/2*d*x+1/2*c)*EllipticF(sin(1/2*d*x+1/2*c),2^(1/2)*(a/(a+b))^(1/2))*(( b+a*cos(d*x+c))/(a+b))^(1/2)*sec(d*x+c)^(1/2)/d/(a+b*sec(d*x+c))^(1/2)+1/8 *(30*B*a^2*b+8*B*b^3+5*a^3*C+20*a*b^2*(2*A+C))*(cos(1/2*d*x+1/2*c)^2)^(1/2 )/cos(1/2*d*x+1/2*c)*EllipticPi(sin(1/2*d*x+1/2*c),2,2^(1/2)*(a/(a+b))^(1/ 2))*((b+a*cos(d*x+c))/(a+b))^(1/2)*sec(d*x+c)^(1/2)/d/(a+b*sec(d*x+c))^(1/ 2)-1/24*(54*B*a*b-a^2*(48*A-33*C)+8*b^2*(3*A+2*C))*(cos(1/2*d*x+1/2*c)^2)^ (1/2)/cos(1/2*d*x+1/2*c)*EllipticE(sin(1/2*d*x+1/2*c),2^(1/2)*(a/(a+b))^(1 /2))*(a+b*sec(d*x+c))^(1/2)/d/((b+a*cos(d*x+c))/(a+b))^(1/2)/sec(d*x+c)^(1 /2)+1/24*(24*A*b^2+42*B*a*b+15*C*a^2+16*C*b^2)*sin(d*x+c)*sec(d*x+c)^(1/2) *(a+b*sec(d*x+c))^(1/2)/d
Result contains complex when optimal does not.
Time = 19.60 (sec) , antiderivative size = 817, normalized size of antiderivative = 1.80 \[ \int \frac {(a+b \sec (c+d x))^{5/2} \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right )}{\sqrt {\sec (c+d x)}} \, dx=\frac {(a+b \sec (c+d x))^{5/2} \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \left (\frac {2 \left (288 a^2 A b+96 a^3 B+24 a b^2 B+52 a^2 b C\right ) \sqrt {\frac {b+a \cos (c+d x)}{a+b}} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),\frac {2 a}{a+b}\right )}{\sqrt {b+a \cos (c+d x)}}+\frac {2 \left (48 a^3 A+216 a A b^2+126 a^2 b B+48 b^3 B-3 a^3 C+104 a b^2 C\right ) \sqrt {\frac {b+a \cos (c+d x)}{a+b}} \operatorname {EllipticPi}\left (2,\frac {1}{2} (c+d x),\frac {2 a}{a+b}\right )}{\sqrt {b+a \cos (c+d x)}}+\frac {2 i \left (48 a^3 A-24 a A b^2-54 a^2 b B-33 a^3 C-16 a b^2 C\right ) \sqrt {\frac {a-a \cos (c+d x)}{a+b}} \sqrt {\frac {a+a \cos (c+d x)}{a-b}} \cos (2 (c+d x)) \left (-2 b (a+b) E\left (i \text {arcsinh}\left (\sqrt {\frac {1}{a-b}} \sqrt {b+a \cos (c+d x)}\right )|\frac {-a+b}{a+b}\right )+a \left (2 b \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {\frac {1}{a-b}} \sqrt {b+a \cos (c+d x)}\right ),\frac {-a+b}{a+b}\right )+a \operatorname {EllipticPi}\left (1-\frac {a}{b},i \text {arcsinh}\left (\sqrt {\frac {1}{a-b}} \sqrt {b+a \cos (c+d x)}\right ),\frac {-a+b}{a+b}\right )\right )\right ) \sin (c+d x)}{\sqrt {\frac {1}{a-b}} b \sqrt {1-\cos ^2(c+d x)} \sqrt {\frac {a^2-a^2 \cos ^2(c+d x)}{a^2}} \left (-a^2+2 b^2-4 b (b+a \cos (c+d x))+2 (b+a \cos (c+d x))^2\right )}\right )}{48 d (b+a \cos (c+d x))^{5/2} (A+2 C+2 B \cos (c+d x)+A \cos (2 c+2 d x)) \sec ^{\frac {9}{2}}(c+d x)}+\frac {(a+b \sec (c+d x))^{5/2} \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \left (\frac {1}{6} \sec ^2(c+d x) \left (6 b^2 B \sin (c+d x)+13 a b C \sin (c+d x)\right )+\frac {1}{12} \sec (c+d x) \left (24 A b^2 \sin (c+d x)+54 a b B \sin (c+d x)+33 a^2 C \sin (c+d x)+16 b^2 C \sin (c+d x)\right )+\frac {2}{3} b^2 C \sec ^2(c+d x) \tan (c+d x)\right )}{d (b+a \cos (c+d x))^2 (A+2 C+2 B \cos (c+d x)+A \cos (2 c+2 d x)) \sec ^{\frac {9}{2}}(c+d x)} \]
Integrate[((a + b*Sec[c + d*x])^(5/2)*(A + B*Sec[c + d*x] + C*Sec[c + d*x] ^2))/Sqrt[Sec[c + d*x]],x]
((a + b*Sec[c + d*x])^(5/2)*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*((2*(2 88*a^2*A*b + 96*a^3*B + 24*a*b^2*B + 52*a^2*b*C)*Sqrt[(b + a*Cos[c + d*x]) /(a + b)]*EllipticF[(c + d*x)/2, (2*a)/(a + b)])/Sqrt[b + a*Cos[c + d*x]] + (2*(48*a^3*A + 216*a*A*b^2 + 126*a^2*b*B + 48*b^3*B - 3*a^3*C + 104*a*b^ 2*C)*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticPi[2, (c + d*x)/2, (2*a)/( a + b)])/Sqrt[b + a*Cos[c + d*x]] + ((2*I)*(48*a^3*A - 24*a*A*b^2 - 54*a^2 *b*B - 33*a^3*C - 16*a*b^2*C)*Sqrt[(a - a*Cos[c + d*x])/(a + b)]*Sqrt[(a + a*Cos[c + d*x])/(a - b)]*Cos[2*(c + d*x)]*(-2*b*(a + b)*EllipticE[I*ArcSi nh[Sqrt[(a - b)^(-1)]*Sqrt[b + a*Cos[c + d*x]]], (-a + b)/(a + b)] + a*(2* b*EllipticF[I*ArcSinh[Sqrt[(a - b)^(-1)]*Sqrt[b + a*Cos[c + d*x]]], (-a + b)/(a + b)] + a*EllipticPi[1 - a/b, I*ArcSinh[Sqrt[(a - b)^(-1)]*Sqrt[b + a*Cos[c + d*x]]], (-a + b)/(a + b)]))*Sin[c + d*x])/(Sqrt[(a - b)^(-1)]*b* Sqrt[1 - Cos[c + d*x]^2]*Sqrt[(a^2 - a^2*Cos[c + d*x]^2)/a^2]*(-a^2 + 2*b^ 2 - 4*b*(b + a*Cos[c + d*x]) + 2*(b + a*Cos[c + d*x])^2))))/(48*d*(b + a*C os[c + d*x])^(5/2)*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*Sec[c + d*x]^(9/2)) + ((a + b*Sec[c + d*x])^(5/2)*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*((Sec[c + d*x]^2*(6*b^2*B*Sin[c + d*x] + 13*a*b*C*Sin[c + d*x]) )/6 + (Sec[c + d*x]*(24*A*b^2*Sin[c + d*x] + 54*a*b*B*Sin[c + d*x] + 33*a^ 2*C*Sin[c + d*x] + 16*b^2*C*Sin[c + d*x]))/12 + (2*b^2*C*Sec[c + d*x]^2*Ta n[c + d*x])/3))/(d*(b + a*Cos[c + d*x])^2*(A + 2*C + 2*B*Cos[c + d*x] +...
Time = 4.60 (sec) , antiderivative size = 459, normalized size of antiderivative = 1.01, number of steps used = 29, number of rules used = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.644, Rules used = {3042, 4584, 27, 3042, 4584, 27, 3042, 4584, 27, 3042, 4596, 3042, 4346, 3042, 3286, 3042, 3284, 4523, 3042, 4343, 3042, 3134, 3042, 3132, 4345, 3042, 3142, 3042, 3140}
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.
| \(\displaystyle \int \frac {(a+b \sec (c+d x))^{5/2} \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right )}{\sqrt {\sec (c+d x)}} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\left (a+b \csc \left (c+d x+\frac {\pi }{2}\right )\right )^{5/2} \left (A+B \csc \left (c+d x+\frac {\pi }{2}\right )+C \csc \left (c+d x+\frac {\pi }{2}\right )^2\right )}{\sqrt {\csc \left (c+d x+\frac {\pi }{2}\right )}}dx\) |
| \(\Big \downarrow \) 4584 |
| \(\displaystyle \frac {1}{3} \int \frac {(a+b \sec (c+d x))^{3/2} \left ((6 b B+5 a C) \sec ^2(c+d x)+2 (3 A b+2 C b+3 a B) \sec (c+d x)+a (6 A-C)\right )}{2 \sqrt {\sec (c+d x)}}dx+\frac {C \sin (c+d x) \sqrt {\sec (c+d x)} (a+b \sec (c+d x))^{5/2}}{3 d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{6} \int \frac {(a+b \sec (c+d x))^{3/2} \left ((6 b B+5 a C) \sec ^2(c+d x)+2 (3 A b+2 C b+3 a B) \sec (c+d x)+a (6 A-C)\right )}{\sqrt {\sec (c+d x)}}dx+\frac {C \sin (c+d x) \sqrt {\sec (c+d x)} (a+b \sec (c+d x))^{5/2}}{3 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{6} \int \frac {\left (a+b \csc \left (c+d x+\frac {\pi }{2}\right )\right )^{3/2} \left ((6 b B+5 a C) \csc \left (c+d x+\frac {\pi }{2}\right )^2+2 (3 A b+2 C b+3 a B) \csc \left (c+d x+\frac {\pi }{2}\right )+a (6 A-C)\right )}{\sqrt {\csc \left (c+d x+\frac {\pi }{2}\right )}}dx+\frac {C \sin (c+d x) \sqrt {\sec (c+d x)} (a+b \sec (c+d x))^{5/2}}{3 d}\) |
| \(\Big \downarrow \) 4584 |
| \(\displaystyle \frac {1}{6} \left (\frac {1}{2} \int \frac {\sqrt {a+b \sec (c+d x)} \left (\left (15 C a^2+42 b B a+24 A b^2+16 b^2 C\right ) \sec ^2(c+d x)+2 \left (12 B a^2+b (24 A+11 C) a+6 b^2 B\right ) \sec (c+d x)+3 a (8 a A-2 b B-3 a C)\right )}{2 \sqrt {\sec (c+d x)}}dx+\frac {(5 a C+6 b B) \sin (c+d x) \sqrt {\sec (c+d x)} (a+b \sec (c+d x))^{3/2}}{2 d}\right )+\frac {C \sin (c+d x) \sqrt {\sec (c+d x)} (a+b \sec (c+d x))^{5/2}}{3 d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{6} \left (\frac {1}{4} \int \frac {\sqrt {a+b \sec (c+d x)} \left (\left (15 C a^2+42 b B a+24 A b^2+16 b^2 C\right ) \sec ^2(c+d x)+2 \left (12 B a^2+b (24 A+11 C) a+6 b^2 B\right ) \sec (c+d x)+3 a (8 a A-2 b B-3 a C)\right )}{\sqrt {\sec (c+d x)}}dx+\frac {(5 a C+6 b B) \sin (c+d x) \sqrt {\sec (c+d x)} (a+b \sec (c+d x))^{3/2}}{2 d}\right )+\frac {C \sin (c+d x) \sqrt {\sec (c+d x)} (a+b \sec (c+d x))^{5/2}}{3 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{6} \left (\frac {1}{4} \int \frac {\sqrt {a+b \csc \left (c+d x+\frac {\pi }{2}\right )} \left (\left (15 C a^2+42 b B a+24 A b^2+16 b^2 C\right ) \csc \left (c+d x+\frac {\pi }{2}\right )^2+2 \left (12 B a^2+b (24 A+11 C) a+6 b^2 B\right ) \csc \left (c+d x+\frac {\pi }{2}\right )+3 a (8 a A-2 b B-3 a C)\right )}{\sqrt {\csc \left (c+d x+\frac {\pi }{2}\right )}}dx+\frac {(5 a C+6 b B) \sin (c+d x) \sqrt {\sec (c+d x)} (a+b \sec (c+d x))^{3/2}}{2 d}\right )+\frac {C \sin (c+d x) \sqrt {\sec (c+d x)} (a+b \sec (c+d x))^{5/2}}{3 d}\) |
| \(\Big \downarrow \) 4584 |
| \(\displaystyle \frac {1}{6} \left (\frac {1}{4} \left (\int -\frac {-3 \left (5 C a^3+30 b B a^2+20 b^2 (2 A+C) a+8 b^3 B\right ) \sec ^2(c+d x)-2 a \left (24 B a^2+b (72 A+13 C) a+6 b^2 B\right ) \sec (c+d x)+a \left (-\left ((48 A-33 C) a^2\right )+54 b B a+8 b^2 (3 A+2 C)\right )}{2 \sqrt {\sec (c+d x)} \sqrt {a+b \sec (c+d x)}}dx+\frac {\sin (c+d x) \sqrt {\sec (c+d x)} \left (15 a^2 C+42 a b B+24 A b^2+16 b^2 C\right ) \sqrt {a+b \sec (c+d x)}}{d}\right )+\frac {(5 a C+6 b B) \sin (c+d x) \sqrt {\sec (c+d x)} (a+b \sec (c+d x))^{3/2}}{2 d}\right )+\frac {C \sin (c+d x) \sqrt {\sec (c+d x)} (a+b \sec (c+d x))^{5/2}}{3 d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{6} \left (\frac {1}{4} \left (\frac {\sin (c+d x) \sqrt {\sec (c+d x)} \left (15 a^2 C+42 a b B+24 A b^2+16 b^2 C\right ) \sqrt {a+b \sec (c+d x)}}{d}-\frac {1}{2} \int \frac {-3 \left (5 C a^3+30 b B a^2+20 b^2 (2 A+C) a+8 b^3 B\right ) \sec ^2(c+d x)-2 a \left (24 B a^2+b (72 A+13 C) a+6 b^2 B\right ) \sec (c+d x)+a \left (-\left ((48 A-33 C) a^2\right )+54 b B a+8 b^2 (3 A+2 C)\right )}{\sqrt {\sec (c+d x)} \sqrt {a+b \sec (c+d x)}}dx\right )+\frac {(5 a C+6 b B) \sin (c+d x) \sqrt {\sec (c+d x)} (a+b \sec (c+d x))^{3/2}}{2 d}\right )+\frac {C \sin (c+d x) \sqrt {\sec (c+d x)} (a+b \sec (c+d x))^{5/2}}{3 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{6} \left (\frac {1}{4} \left (\frac {\sin (c+d x) \sqrt {\sec (c+d x)} \left (15 a^2 C+42 a b B+24 A b^2+16 b^2 C\right ) \sqrt {a+b \sec (c+d x)}}{d}-\frac {1}{2} \int \frac {-3 \left (5 C a^3+30 b B a^2+20 b^2 (2 A+C) a+8 b^3 B\right ) \csc \left (c+d x+\frac {\pi }{2}\right )^2-2 a \left (24 B a^2+b (72 A+13 C) a+6 b^2 B\right ) \csc \left (c+d x+\frac {\pi }{2}\right )+a \left (-\left ((48 A-33 C) a^2\right )+54 b B a+8 b^2 (3 A+2 C)\right )}{\sqrt {\csc \left (c+d x+\frac {\pi }{2}\right )} \sqrt {a+b \csc \left (c+d x+\frac {\pi }{2}\right )}}dx\right )+\frac {(5 a C+6 b B) \sin (c+d x) \sqrt {\sec (c+d x)} (a+b \sec (c+d x))^{3/2}}{2 d}\right )+\frac {C \sin (c+d x) \sqrt {\sec (c+d x)} (a+b \sec (c+d x))^{5/2}}{3 d}\) |
| \(\Big \downarrow \) 4596 |
| \(\displaystyle \frac {1}{6} \left (\frac {1}{4} \left (\frac {1}{2} \left (3 \left (5 a^3 C+30 a^2 b B+20 a b^2 (2 A+C)+8 b^3 B\right ) \int \frac {\sec ^{\frac {3}{2}}(c+d x)}{\sqrt {a+b \sec (c+d x)}}dx-\int \frac {a \left (-\left ((48 A-33 C) a^2\right )+54 b B a+8 b^2 (3 A+2 C)\right )-2 a \left (24 B a^2+b (72 A+13 C) a+6 b^2 B\right ) \sec (c+d x)}{\sqrt {\sec (c+d x)} \sqrt {a+b \sec (c+d x)}}dx\right )+\frac {\sin (c+d x) \sqrt {\sec (c+d x)} \left (15 a^2 C+42 a b B+24 A b^2+16 b^2 C\right ) \sqrt {a+b \sec (c+d x)}}{d}\right )+\frac {(5 a C+6 b B) \sin (c+d x) \sqrt {\sec (c+d x)} (a+b \sec (c+d x))^{3/2}}{2 d}\right )+\frac {C \sin (c+d x) \sqrt {\sec (c+d x)} (a+b \sec (c+d x))^{5/2}}{3 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{6} \left (\frac {1}{4} \left (\frac {1}{2} \left (3 \left (5 a^3 C+30 a^2 b B+20 a b^2 (2 A+C)+8 b^3 B\right ) \int \frac {\csc \left (c+d x+\frac {\pi }{2}\right )^{3/2}}{\sqrt {a+b \csc \left (c+d x+\frac {\pi }{2}\right )}}dx-\int \frac {a \left (-\left ((48 A-33 C) a^2\right )+54 b B a+8 b^2 (3 A+2 C)\right )-2 a \left (24 B a^2+b (72 A+13 C) a+6 b^2 B\right ) \csc \left (c+d x+\frac {\pi }{2}\right )}{\sqrt {\csc \left (c+d x+\frac {\pi }{2}\right )} \sqrt {a+b \csc \left (c+d x+\frac {\pi }{2}\right )}}dx\right )+\frac {\sin (c+d x) \sqrt {\sec (c+d x)} \left (15 a^2 C+42 a b B+24 A b^2+16 b^2 C\right ) \sqrt {a+b \sec (c+d x)}}{d}\right )+\frac {(5 a C+6 b B) \sin (c+d x) \sqrt {\sec (c+d x)} (a+b \sec (c+d x))^{3/2}}{2 d}\right )+\frac {C \sin (c+d x) \sqrt {\sec (c+d x)} (a+b \sec (c+d x))^{5/2}}{3 d}\) |
| \(\Big \downarrow \) 4346 |
| \(\displaystyle \frac {1}{6} \left (\frac {1}{4} \left (\frac {1}{2} \left (\frac {3 \sqrt {\sec (c+d x)} \left (5 a^3 C+30 a^2 b B+20 a b^2 (2 A+C)+8 b^3 B\right ) \sqrt {a \cos (c+d x)+b} \int \frac {\sec (c+d x)}{\sqrt {b+a \cos (c+d x)}}dx}{\sqrt {a+b \sec (c+d x)}}-\int \frac {a \left (-\left ((48 A-33 C) a^2\right )+54 b B a+8 b^2 (3 A+2 C)\right )-2 a \left (24 B a^2+b (72 A+13 C) a+6 b^2 B\right ) \csc \left (c+d x+\frac {\pi }{2}\right )}{\sqrt {\csc \left (c+d x+\frac {\pi }{2}\right )} \sqrt {a+b \csc \left (c+d x+\frac {\pi }{2}\right )}}dx\right )+\frac {\sin (c+d x) \sqrt {\sec (c+d x)} \left (15 a^2 C+42 a b B+24 A b^2+16 b^2 C\right ) \sqrt {a+b \sec (c+d x)}}{d}\right )+\frac {(5 a C+6 b B) \sin (c+d x) \sqrt {\sec (c+d x)} (a+b \sec (c+d x))^{3/2}}{2 d}\right )+\frac {C \sin (c+d x) \sqrt {\sec (c+d x)} (a+b \sec (c+d x))^{5/2}}{3 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{6} \left (\frac {1}{4} \left (\frac {1}{2} \left (\frac {3 \sqrt {\sec (c+d x)} \left (5 a^3 C+30 a^2 b B+20 a b^2 (2 A+C)+8 b^3 B\right ) \sqrt {a \cos (c+d x)+b} \int \frac {1}{\sin \left (c+d x+\frac {\pi }{2}\right ) \sqrt {b+a \sin \left (c+d x+\frac {\pi }{2}\right )}}dx}{\sqrt {a+b \sec (c+d x)}}-\int \frac {a \left (-\left ((48 A-33 C) a^2\right )+54 b B a+8 b^2 (3 A+2 C)\right )-2 a \left (24 B a^2+b (72 A+13 C) a+6 b^2 B\right ) \csc \left (c+d x+\frac {\pi }{2}\right )}{\sqrt {\csc \left (c+d x+\frac {\pi }{2}\right )} \sqrt {a+b \csc \left (c+d x+\frac {\pi }{2}\right )}}dx\right )+\frac {\sin (c+d x) \sqrt {\sec (c+d x)} \left (15 a^2 C+42 a b B+24 A b^2+16 b^2 C\right ) \sqrt {a+b \sec (c+d x)}}{d}\right )+\frac {(5 a C+6 b B) \sin (c+d x) \sqrt {\sec (c+d x)} (a+b \sec (c+d x))^{3/2}}{2 d}\right )+\frac {C \sin (c+d x) \sqrt {\sec (c+d x)} (a+b \sec (c+d x))^{5/2}}{3 d}\) |
| \(\Big \downarrow \) 3286 |
| \(\displaystyle \frac {1}{6} \left (\frac {1}{4} \left (\frac {1}{2} \left (\frac {3 \sqrt {\sec (c+d x)} \left (5 a^3 C+30 a^2 b B+20 a b^2 (2 A+C)+8 b^3 B\right ) \sqrt {\frac {a \cos (c+d x)+b}{a+b}} \int \frac {\sec (c+d x)}{\sqrt {\frac {b}{a+b}+\frac {a \cos (c+d x)}{a+b}}}dx}{\sqrt {a+b \sec (c+d x)}}-\int \frac {a \left (-\left ((48 A-33 C) a^2\right )+54 b B a+8 b^2 (3 A+2 C)\right )-2 a \left (24 B a^2+b (72 A+13 C) a+6 b^2 B\right ) \csc \left (c+d x+\frac {\pi }{2}\right )}{\sqrt {\csc \left (c+d x+\frac {\pi }{2}\right )} \sqrt {a+b \csc \left (c+d x+\frac {\pi }{2}\right )}}dx\right )+\frac {\sin (c+d x) \sqrt {\sec (c+d x)} \left (15 a^2 C+42 a b B+24 A b^2+16 b^2 C\right ) \sqrt {a+b \sec (c+d x)}}{d}\right )+\frac {(5 a C+6 b B) \sin (c+d x) \sqrt {\sec (c+d x)} (a+b \sec (c+d x))^{3/2}}{2 d}\right )+\frac {C \sin (c+d x) \sqrt {\sec (c+d x)} (a+b \sec (c+d x))^{5/2}}{3 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{6} \left (\frac {1}{4} \left (\frac {1}{2} \left (\frac {3 \sqrt {\sec (c+d x)} \left (5 a^3 C+30 a^2 b B+20 a b^2 (2 A+C)+8 b^3 B\right ) \sqrt {\frac {a \cos (c+d x)+b}{a+b}} \int \frac {1}{\sin \left (c+d x+\frac {\pi }{2}\right ) \sqrt {\frac {b}{a+b}+\frac {a \sin \left (c+d x+\frac {\pi }{2}\right )}{a+b}}}dx}{\sqrt {a+b \sec (c+d x)}}-\int \frac {a \left (-\left ((48 A-33 C) a^2\right )+54 b B a+8 b^2 (3 A+2 C)\right )-2 a \left (24 B a^2+b (72 A+13 C) a+6 b^2 B\right ) \csc \left (c+d x+\frac {\pi }{2}\right )}{\sqrt {\csc \left (c+d x+\frac {\pi }{2}\right )} \sqrt {a+b \csc \left (c+d x+\frac {\pi }{2}\right )}}dx\right )+\frac {\sin (c+d x) \sqrt {\sec (c+d x)} \left (15 a^2 C+42 a b B+24 A b^2+16 b^2 C\right ) \sqrt {a+b \sec (c+d x)}}{d}\right )+\frac {(5 a C+6 b B) \sin (c+d x) \sqrt {\sec (c+d x)} (a+b \sec (c+d x))^{3/2}}{2 d}\right )+\frac {C \sin (c+d x) \sqrt {\sec (c+d x)} (a+b \sec (c+d x))^{5/2}}{3 d}\) |
| \(\Big \downarrow \) 3284 |
| \(\displaystyle \frac {1}{6} \left (\frac {1}{4} \left (\frac {1}{2} \left (\frac {6 \sqrt {\sec (c+d x)} \left (5 a^3 C+30 a^2 b B+20 a b^2 (2 A+C)+8 b^3 B\right ) \sqrt {\frac {a \cos (c+d x)+b}{a+b}} \operatorname {EllipticPi}\left (2,\frac {1}{2} (c+d x),\frac {2 a}{a+b}\right )}{d \sqrt {a+b \sec (c+d x)}}-\int \frac {a \left (-\left ((48 A-33 C) a^2\right )+54 b B a+8 b^2 (3 A+2 C)\right )-2 a \left (24 B a^2+b (72 A+13 C) a+6 b^2 B\right ) \csc \left (c+d x+\frac {\pi }{2}\right )}{\sqrt {\csc \left (c+d x+\frac {\pi }{2}\right )} \sqrt {a+b \csc \left (c+d x+\frac {\pi }{2}\right )}}dx\right )+\frac {\sin (c+d x) \sqrt {\sec (c+d x)} \left (15 a^2 C+42 a b B+24 A b^2+16 b^2 C\right ) \sqrt {a+b \sec (c+d x)}}{d}\right )+\frac {(5 a C+6 b B) \sin (c+d x) \sqrt {\sec (c+d x)} (a+b \sec (c+d x))^{3/2}}{2 d}\right )+\frac {C \sin (c+d x) \sqrt {\sec (c+d x)} (a+b \sec (c+d x))^{5/2}}{3 d}\) |
| \(\Big \downarrow \) 4523 |
| \(\displaystyle \frac {1}{6} \left (\frac {1}{4} \left (\frac {1}{2} \left (-\left (-\left (a^2 (48 A-33 C)\right )+54 a b B+8 b^2 (3 A+2 C)\right ) \int \frac {\sqrt {a+b \sec (c+d x)}}{\sqrt {\sec (c+d x)}}dx+\left (48 a^3 B+a^2 b (96 A+59 C)+66 a b^2 B+8 b^3 (3 A+2 C)\right ) \int \frac {\sqrt {\sec (c+d x)}}{\sqrt {a+b \sec (c+d x)}}dx+\frac {6 \sqrt {\sec (c+d x)} \left (5 a^3 C+30 a^2 b B+20 a b^2 (2 A+C)+8 b^3 B\right ) \sqrt {\frac {a \cos (c+d x)+b}{a+b}} \operatorname {EllipticPi}\left (2,\frac {1}{2} (c+d x),\frac {2 a}{a+b}\right )}{d \sqrt {a+b \sec (c+d x)}}\right )+\frac {\sin (c+d x) \sqrt {\sec (c+d x)} \left (15 a^2 C+42 a b B+24 A b^2+16 b^2 C\right ) \sqrt {a+b \sec (c+d x)}}{d}\right )+\frac {(5 a C+6 b B) \sin (c+d x) \sqrt {\sec (c+d x)} (a+b \sec (c+d x))^{3/2}}{2 d}\right )+\frac {C \sin (c+d x) \sqrt {\sec (c+d x)} (a+b \sec (c+d x))^{5/2}}{3 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{6} \left (\frac {1}{4} \left (\frac {1}{2} \left (-\left (-\left (a^2 (48 A-33 C)\right )+54 a b B+8 b^2 (3 A+2 C)\right ) \int \frac {\sqrt {a+b \csc \left (c+d x+\frac {\pi }{2}\right )}}{\sqrt {\csc \left (c+d x+\frac {\pi }{2}\right )}}dx+\left (48 a^3 B+a^2 b (96 A+59 C)+66 a b^2 B+8 b^3 (3 A+2 C)\right ) \int \frac {\sqrt {\csc \left (c+d x+\frac {\pi }{2}\right )}}{\sqrt {a+b \csc \left (c+d x+\frac {\pi }{2}\right )}}dx+\frac {6 \sqrt {\sec (c+d x)} \left (5 a^3 C+30 a^2 b B+20 a b^2 (2 A+C)+8 b^3 B\right ) \sqrt {\frac {a \cos (c+d x)+b}{a+b}} \operatorname {EllipticPi}\left (2,\frac {1}{2} (c+d x),\frac {2 a}{a+b}\right )}{d \sqrt {a+b \sec (c+d x)}}\right )+\frac {\sin (c+d x) \sqrt {\sec (c+d x)} \left (15 a^2 C+42 a b B+24 A b^2+16 b^2 C\right ) \sqrt {a+b \sec (c+d x)}}{d}\right )+\frac {(5 a C+6 b B) \sin (c+d x) \sqrt {\sec (c+d x)} (a+b \sec (c+d x))^{3/2}}{2 d}\right )+\frac {C \sin (c+d x) \sqrt {\sec (c+d x)} (a+b \sec (c+d x))^{5/2}}{3 d}\) |
| \(\Big \downarrow \) 4343 |
| \(\displaystyle \frac {1}{6} \left (\frac {1}{4} \left (\frac {1}{2} \left (-\frac {\left (-\left (a^2 (48 A-33 C)\right )+54 a b B+8 b^2 (3 A+2 C)\right ) \sqrt {a+b \sec (c+d x)} \int \sqrt {b+a \cos (c+d x)}dx}{\sqrt {\sec (c+d x)} \sqrt {a \cos (c+d x)+b}}+\left (48 a^3 B+a^2 b (96 A+59 C)+66 a b^2 B+8 b^3 (3 A+2 C)\right ) \int \frac {\sqrt {\csc \left (c+d x+\frac {\pi }{2}\right )}}{\sqrt {a+b \csc \left (c+d x+\frac {\pi }{2}\right )}}dx+\frac {6 \sqrt {\sec (c+d x)} \left (5 a^3 C+30 a^2 b B+20 a b^2 (2 A+C)+8 b^3 B\right ) \sqrt {\frac {a \cos (c+d x)+b}{a+b}} \operatorname {EllipticPi}\left (2,\frac {1}{2} (c+d x),\frac {2 a}{a+b}\right )}{d \sqrt {a+b \sec (c+d x)}}\right )+\frac {\sin (c+d x) \sqrt {\sec (c+d x)} \left (15 a^2 C+42 a b B+24 A b^2+16 b^2 C\right ) \sqrt {a+b \sec (c+d x)}}{d}\right )+\frac {(5 a C+6 b B) \sin (c+d x) \sqrt {\sec (c+d x)} (a+b \sec (c+d x))^{3/2}}{2 d}\right )+\frac {C \sin (c+d x) \sqrt {\sec (c+d x)} (a+b \sec (c+d x))^{5/2}}{3 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{6} \left (\frac {1}{4} \left (\frac {1}{2} \left (-\frac {\left (-\left (a^2 (48 A-33 C)\right )+54 a b B+8 b^2 (3 A+2 C)\right ) \sqrt {a+b \sec (c+d x)} \int \sqrt {b+a \sin \left (c+d x+\frac {\pi }{2}\right )}dx}{\sqrt {\sec (c+d x)} \sqrt {a \cos (c+d x)+b}}+\left (48 a^3 B+a^2 b (96 A+59 C)+66 a b^2 B+8 b^3 (3 A+2 C)\right ) \int \frac {\sqrt {\csc \left (c+d x+\frac {\pi }{2}\right )}}{\sqrt {a+b \csc \left (c+d x+\frac {\pi }{2}\right )}}dx+\frac {6 \sqrt {\sec (c+d x)} \left (5 a^3 C+30 a^2 b B+20 a b^2 (2 A+C)+8 b^3 B\right ) \sqrt {\frac {a \cos (c+d x)+b}{a+b}} \operatorname {EllipticPi}\left (2,\frac {1}{2} (c+d x),\frac {2 a}{a+b}\right )}{d \sqrt {a+b \sec (c+d x)}}\right )+\frac {\sin (c+d x) \sqrt {\sec (c+d x)} \left (15 a^2 C+42 a b B+24 A b^2+16 b^2 C\right ) \sqrt {a+b \sec (c+d x)}}{d}\right )+\frac {(5 a C+6 b B) \sin (c+d x) \sqrt {\sec (c+d x)} (a+b \sec (c+d x))^{3/2}}{2 d}\right )+\frac {C \sin (c+d x) \sqrt {\sec (c+d x)} (a+b \sec (c+d x))^{5/2}}{3 d}\) |
| \(\Big \downarrow \) 3134 |
| \(\displaystyle \frac {1}{6} \left (\frac {1}{4} \left (\frac {1}{2} \left (-\frac {\left (-\left (a^2 (48 A-33 C)\right )+54 a b B+8 b^2 (3 A+2 C)\right ) \sqrt {a+b \sec (c+d x)} \int \sqrt {\frac {b}{a+b}+\frac {a \cos (c+d x)}{a+b}}dx}{\sqrt {\sec (c+d x)} \sqrt {\frac {a \cos (c+d x)+b}{a+b}}}+\left (48 a^3 B+a^2 b (96 A+59 C)+66 a b^2 B+8 b^3 (3 A+2 C)\right ) \int \frac {\sqrt {\csc \left (c+d x+\frac {\pi }{2}\right )}}{\sqrt {a+b \csc \left (c+d x+\frac {\pi }{2}\right )}}dx+\frac {6 \sqrt {\sec (c+d x)} \left (5 a^3 C+30 a^2 b B+20 a b^2 (2 A+C)+8 b^3 B\right ) \sqrt {\frac {a \cos (c+d x)+b}{a+b}} \operatorname {EllipticPi}\left (2,\frac {1}{2} (c+d x),\frac {2 a}{a+b}\right )}{d \sqrt {a+b \sec (c+d x)}}\right )+\frac {\sin (c+d x) \sqrt {\sec (c+d x)} \left (15 a^2 C+42 a b B+24 A b^2+16 b^2 C\right ) \sqrt {a+b \sec (c+d x)}}{d}\right )+\frac {(5 a C+6 b B) \sin (c+d x) \sqrt {\sec (c+d x)} (a+b \sec (c+d x))^{3/2}}{2 d}\right )+\frac {C \sin (c+d x) \sqrt {\sec (c+d x)} (a+b \sec (c+d x))^{5/2}}{3 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{6} \left (\frac {1}{4} \left (\frac {1}{2} \left (-\frac {\left (-\left (a^2 (48 A-33 C)\right )+54 a b B+8 b^2 (3 A+2 C)\right ) \sqrt {a+b \sec (c+d x)} \int \sqrt {\frac {b}{a+b}+\frac {a \sin \left (c+d x+\frac {\pi }{2}\right )}{a+b}}dx}{\sqrt {\sec (c+d x)} \sqrt {\frac {a \cos (c+d x)+b}{a+b}}}+\left (48 a^3 B+a^2 b (96 A+59 C)+66 a b^2 B+8 b^3 (3 A+2 C)\right ) \int \frac {\sqrt {\csc \left (c+d x+\frac {\pi }{2}\right )}}{\sqrt {a+b \csc \left (c+d x+\frac {\pi }{2}\right )}}dx+\frac {6 \sqrt {\sec (c+d x)} \left (5 a^3 C+30 a^2 b B+20 a b^2 (2 A+C)+8 b^3 B\right ) \sqrt {\frac {a \cos (c+d x)+b}{a+b}} \operatorname {EllipticPi}\left (2,\frac {1}{2} (c+d x),\frac {2 a}{a+b}\right )}{d \sqrt {a+b \sec (c+d x)}}\right )+\frac {\sin (c+d x) \sqrt {\sec (c+d x)} \left (15 a^2 C+42 a b B+24 A b^2+16 b^2 C\right ) \sqrt {a+b \sec (c+d x)}}{d}\right )+\frac {(5 a C+6 b B) \sin (c+d x) \sqrt {\sec (c+d x)} (a+b \sec (c+d x))^{3/2}}{2 d}\right )+\frac {C \sin (c+d x) \sqrt {\sec (c+d x)} (a+b \sec (c+d x))^{5/2}}{3 d}\) |
| \(\Big \downarrow \) 3132 |
| \(\displaystyle \frac {1}{6} \left (\frac {1}{4} \left (\frac {1}{2} \left (\left (48 a^3 B+a^2 b (96 A+59 C)+66 a b^2 B+8 b^3 (3 A+2 C)\right ) \int \frac {\sqrt {\csc \left (c+d x+\frac {\pi }{2}\right )}}{\sqrt {a+b \csc \left (c+d x+\frac {\pi }{2}\right )}}dx-\frac {2 \left (-\left (a^2 (48 A-33 C)\right )+54 a b B+8 b^2 (3 A+2 C)\right ) \sqrt {a+b \sec (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 a}{a+b}\right )}{d \sqrt {\sec (c+d x)} \sqrt {\frac {a \cos (c+d x)+b}{a+b}}}+\frac {6 \sqrt {\sec (c+d x)} \left (5 a^3 C+30 a^2 b B+20 a b^2 (2 A+C)+8 b^3 B\right ) \sqrt {\frac {a \cos (c+d x)+b}{a+b}} \operatorname {EllipticPi}\left (2,\frac {1}{2} (c+d x),\frac {2 a}{a+b}\right )}{d \sqrt {a+b \sec (c+d x)}}\right )+\frac {\sin (c+d x) \sqrt {\sec (c+d x)} \left (15 a^2 C+42 a b B+24 A b^2+16 b^2 C\right ) \sqrt {a+b \sec (c+d x)}}{d}\right )+\frac {(5 a C+6 b B) \sin (c+d x) \sqrt {\sec (c+d x)} (a+b \sec (c+d x))^{3/2}}{2 d}\right )+\frac {C \sin (c+d x) \sqrt {\sec (c+d x)} (a+b \sec (c+d x))^{5/2}}{3 d}\) |
| \(\Big \downarrow \) 4345 |
| \(\displaystyle \frac {1}{6} \left (\frac {1}{4} \left (\frac {1}{2} \left (\frac {\sqrt {\sec (c+d x)} \left (48 a^3 B+a^2 b (96 A+59 C)+66 a b^2 B+8 b^3 (3 A+2 C)\right ) \sqrt {a \cos (c+d x)+b} \int \frac {1}{\sqrt {b+a \cos (c+d x)}}dx}{\sqrt {a+b \sec (c+d x)}}-\frac {2 \left (-\left (a^2 (48 A-33 C)\right )+54 a b B+8 b^2 (3 A+2 C)\right ) \sqrt {a+b \sec (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 a}{a+b}\right )}{d \sqrt {\sec (c+d x)} \sqrt {\frac {a \cos (c+d x)+b}{a+b}}}+\frac {6 \sqrt {\sec (c+d x)} \left (5 a^3 C+30 a^2 b B+20 a b^2 (2 A+C)+8 b^3 B\right ) \sqrt {\frac {a \cos (c+d x)+b}{a+b}} \operatorname {EllipticPi}\left (2,\frac {1}{2} (c+d x),\frac {2 a}{a+b}\right )}{d \sqrt {a+b \sec (c+d x)}}\right )+\frac {\sin (c+d x) \sqrt {\sec (c+d x)} \left (15 a^2 C+42 a b B+24 A b^2+16 b^2 C\right ) \sqrt {a+b \sec (c+d x)}}{d}\right )+\frac {(5 a C+6 b B) \sin (c+d x) \sqrt {\sec (c+d x)} (a+b \sec (c+d x))^{3/2}}{2 d}\right )+\frac {C \sin (c+d x) \sqrt {\sec (c+d x)} (a+b \sec (c+d x))^{5/2}}{3 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{6} \left (\frac {1}{4} \left (\frac {1}{2} \left (\frac {\sqrt {\sec (c+d x)} \left (48 a^3 B+a^2 b (96 A+59 C)+66 a b^2 B+8 b^3 (3 A+2 C)\right ) \sqrt {a \cos (c+d x)+b} \int \frac {1}{\sqrt {b+a \sin \left (c+d x+\frac {\pi }{2}\right )}}dx}{\sqrt {a+b \sec (c+d x)}}-\frac {2 \left (-\left (a^2 (48 A-33 C)\right )+54 a b B+8 b^2 (3 A+2 C)\right ) \sqrt {a+b \sec (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 a}{a+b}\right )}{d \sqrt {\sec (c+d x)} \sqrt {\frac {a \cos (c+d x)+b}{a+b}}}+\frac {6 \sqrt {\sec (c+d x)} \left (5 a^3 C+30 a^2 b B+20 a b^2 (2 A+C)+8 b^3 B\right ) \sqrt {\frac {a \cos (c+d x)+b}{a+b}} \operatorname {EllipticPi}\left (2,\frac {1}{2} (c+d x),\frac {2 a}{a+b}\right )}{d \sqrt {a+b \sec (c+d x)}}\right )+\frac {\sin (c+d x) \sqrt {\sec (c+d x)} \left (15 a^2 C+42 a b B+24 A b^2+16 b^2 C\right ) \sqrt {a+b \sec (c+d x)}}{d}\right )+\frac {(5 a C+6 b B) \sin (c+d x) \sqrt {\sec (c+d x)} (a+b \sec (c+d x))^{3/2}}{2 d}\right )+\frac {C \sin (c+d x) \sqrt {\sec (c+d x)} (a+b \sec (c+d x))^{5/2}}{3 d}\) |
| \(\Big \downarrow \) 3142 |
| \(\displaystyle \frac {1}{6} \left (\frac {1}{4} \left (\frac {1}{2} \left (\frac {\sqrt {\sec (c+d x)} \left (48 a^3 B+a^2 b (96 A+59 C)+66 a b^2 B+8 b^3 (3 A+2 C)\right ) \sqrt {\frac {a \cos (c+d x)+b}{a+b}} \int \frac {1}{\sqrt {\frac {b}{a+b}+\frac {a \cos (c+d x)}{a+b}}}dx}{\sqrt {a+b \sec (c+d x)}}-\frac {2 \left (-\left (a^2 (48 A-33 C)\right )+54 a b B+8 b^2 (3 A+2 C)\right ) \sqrt {a+b \sec (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 a}{a+b}\right )}{d \sqrt {\sec (c+d x)} \sqrt {\frac {a \cos (c+d x)+b}{a+b}}}+\frac {6 \sqrt {\sec (c+d x)} \left (5 a^3 C+30 a^2 b B+20 a b^2 (2 A+C)+8 b^3 B\right ) \sqrt {\frac {a \cos (c+d x)+b}{a+b}} \operatorname {EllipticPi}\left (2,\frac {1}{2} (c+d x),\frac {2 a}{a+b}\right )}{d \sqrt {a+b \sec (c+d x)}}\right )+\frac {\sin (c+d x) \sqrt {\sec (c+d x)} \left (15 a^2 C+42 a b B+24 A b^2+16 b^2 C\right ) \sqrt {a+b \sec (c+d x)}}{d}\right )+\frac {(5 a C+6 b B) \sin (c+d x) \sqrt {\sec (c+d x)} (a+b \sec (c+d x))^{3/2}}{2 d}\right )+\frac {C \sin (c+d x) \sqrt {\sec (c+d x)} (a+b \sec (c+d x))^{5/2}}{3 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{6} \left (\frac {1}{4} \left (\frac {1}{2} \left (\frac {\sqrt {\sec (c+d x)} \left (48 a^3 B+a^2 b (96 A+59 C)+66 a b^2 B+8 b^3 (3 A+2 C)\right ) \sqrt {\frac {a \cos (c+d x)+b}{a+b}} \int \frac {1}{\sqrt {\frac {b}{a+b}+\frac {a \sin \left (c+d x+\frac {\pi }{2}\right )}{a+b}}}dx}{\sqrt {a+b \sec (c+d x)}}-\frac {2 \left (-\left (a^2 (48 A-33 C)\right )+54 a b B+8 b^2 (3 A+2 C)\right ) \sqrt {a+b \sec (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 a}{a+b}\right )}{d \sqrt {\sec (c+d x)} \sqrt {\frac {a \cos (c+d x)+b}{a+b}}}+\frac {6 \sqrt {\sec (c+d x)} \left (5 a^3 C+30 a^2 b B+20 a b^2 (2 A+C)+8 b^3 B\right ) \sqrt {\frac {a \cos (c+d x)+b}{a+b}} \operatorname {EllipticPi}\left (2,\frac {1}{2} (c+d x),\frac {2 a}{a+b}\right )}{d \sqrt {a+b \sec (c+d x)}}\right )+\frac {\sin (c+d x) \sqrt {\sec (c+d x)} \left (15 a^2 C+42 a b B+24 A b^2+16 b^2 C\right ) \sqrt {a+b \sec (c+d x)}}{d}\right )+\frac {(5 a C+6 b B) \sin (c+d x) \sqrt {\sec (c+d x)} (a+b \sec (c+d x))^{3/2}}{2 d}\right )+\frac {C \sin (c+d x) \sqrt {\sec (c+d x)} (a+b \sec (c+d x))^{5/2}}{3 d}\) |
| \(\Big \downarrow \) 3140 |
| \(\displaystyle \frac {1}{6} \left (\frac {1}{4} \left (\frac {\sin (c+d x) \sqrt {\sec (c+d x)} \left (15 a^2 C+42 a b B+24 A b^2+16 b^2 C\right ) \sqrt {a+b \sec (c+d x)}}{d}+\frac {1}{2} \left (-\frac {2 \left (-\left (a^2 (48 A-33 C)\right )+54 a b B+8 b^2 (3 A+2 C)\right ) \sqrt {a+b \sec (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 a}{a+b}\right )}{d \sqrt {\sec (c+d x)} \sqrt {\frac {a \cos (c+d x)+b}{a+b}}}+\frac {2 \sqrt {\sec (c+d x)} \left (48 a^3 B+a^2 b (96 A+59 C)+66 a b^2 B+8 b^3 (3 A+2 C)\right ) \sqrt {\frac {a \cos (c+d x)+b}{a+b}} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),\frac {2 a}{a+b}\right )}{d \sqrt {a+b \sec (c+d x)}}+\frac {6 \sqrt {\sec (c+d x)} \left (5 a^3 C+30 a^2 b B+20 a b^2 (2 A+C)+8 b^3 B\right ) \sqrt {\frac {a \cos (c+d x)+b}{a+b}} \operatorname {EllipticPi}\left (2,\frac {1}{2} (c+d x),\frac {2 a}{a+b}\right )}{d \sqrt {a+b \sec (c+d x)}}\right )\right )+\frac {(5 a C+6 b B) \sin (c+d x) \sqrt {\sec (c+d x)} (a+b \sec (c+d x))^{3/2}}{2 d}\right )+\frac {C \sin (c+d x) \sqrt {\sec (c+d x)} (a+b \sec (c+d x))^{5/2}}{3 d}\) |
Int[((a + b*Sec[c + d*x])^(5/2)*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/S qrt[Sec[c + d*x]],x]
(C*Sqrt[Sec[c + d*x]]*(a + b*Sec[c + d*x])^(5/2)*Sin[c + d*x])/(3*d) + ((( 6*b*B + 5*a*C)*Sqrt[Sec[c + d*x]]*(a + b*Sec[c + d*x])^(3/2)*Sin[c + d*x]) /(2*d) + (((2*(48*a^3*B + 66*a*b^2*B + 8*b^3*(3*A + 2*C) + a^2*b*(96*A + 5 9*C))*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*a)/(a + b)]*Sqrt[Sec[c + d*x]])/(d*Sqrt[a + b*Sec[c + d*x]]) + (6*(30*a^2*b*B + 8 *b^3*B + 5*a^3*C + 20*a*b^2*(2*A + C))*Sqrt[(b + a*Cos[c + d*x])/(a + b)]* EllipticPi[2, (c + d*x)/2, (2*a)/(a + b)]*Sqrt[Sec[c + d*x]])/(d*Sqrt[a + b*Sec[c + d*x]]) - (2*(54*a*b*B - a^2*(48*A - 33*C) + 8*b^2*(3*A + 2*C))*E llipticE[(c + d*x)/2, (2*a)/(a + b)]*Sqrt[a + b*Sec[c + d*x]])/(d*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*Sqrt[Sec[c + d*x]]))/2 + ((24*A*b^2 + 42*a*b*B + 15*a^2*C + 16*b^2*C)*Sqrt[Sec[c + d*x]]*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/d)/4)/6
3.11.46.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[Sqrt[(a_) + (b_.)*sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[2*(Sqrt[a + b]/d)*EllipticE[(1/2)*(c - Pi/2 + d*x), 2*(b/(a + b))], x] /; FreeQ[{a, b, c, d}, x] && NeQ[a^2 - b^2, 0] && GtQ[a + b, 0]
Int[Sqrt[(a_) + (b_.)*sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[Sqrt[a + b*Sin[c + d*x]]/Sqrt[(a + b*Sin[c + d*x])/(a + b)] Int[Sqrt[a/(a + b) + ( b/(a + b))*Sin[c + d*x]], x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[a^2 - b^2 , 0] && !GtQ[a + b, 0]
Int[1/Sqrt[(a_) + (b_.)*sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[(2/(d*S qrt[a + b]))*EllipticF[(1/2)*(c - Pi/2 + d*x), 2*(b/(a + b))], x] /; FreeQ[ {a, b, c, d}, x] && NeQ[a^2 - b^2, 0] && GtQ[a + b, 0]
Int[1/Sqrt[(a_) + (b_.)*sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[Sqrt[(a + b*Sin[c + d*x])/(a + b)]/Sqrt[a + b*Sin[c + d*x]] Int[1/Sqrt[a/(a + b) + (b/(a + b))*Sin[c + d*x]], x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[a^2 - b^2, 0] && !GtQ[a + b, 0]
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]
Int[1/(((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])*Sqrt[(c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)]]), x_Symbol] :> Simp[Sqrt[(c + d*Sin[e + f*x])/(c + d)]/Sqrt [c + d*Sin[e + f*x]] Int[1/((a + b*Sin[e + f*x])*Sqrt[c/(c + d) + (d/(c + d))*Sin[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] && NeQ[c^2 - d^2, 0] && !GtQ[c + d, 0]
Int[Sqrt[csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_)]/Sqrt[csc[(e_.) + (f_.)*(x_)] *(d_.)], x_Symbol] :> Simp[Sqrt[a + b*Csc[e + f*x]]/(Sqrt[d*Csc[e + f*x]]*S qrt[b + a*Sin[e + f*x]]) Int[Sqrt[b + a*Sin[e + f*x]], x], x] /; FreeQ[{a , b, d, e, f}, x] && NeQ[a^2 - b^2, 0]
Int[Sqrt[csc[(e_.) + (f_.)*(x_)]*(d_.)]/Sqrt[csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_)], x_Symbol] :> Simp[Sqrt[d*Csc[e + f*x]]*(Sqrt[b + a*Sin[e + f*x]]/S qrt[a + b*Csc[e + f*x]]) Int[1/Sqrt[b + a*Sin[e + f*x]], x], x] /; FreeQ[ {a, b, d, e, f}, x] && NeQ[a^2 - b^2, 0]
Int[(csc[(e_.) + (f_.)*(x_)]*(d_.))^(3/2)/Sqrt[csc[(e_.) + (f_.)*(x_)]*(b_. ) + (a_)], x_Symbol] :> Simp[d*Sqrt[d*Csc[e + f*x]]*(Sqrt[b + a*Sin[e + f*x ]]/Sqrt[a + b*Csc[e + f*x]]) Int[1/(Sin[e + f*x]*Sqrt[b + a*Sin[e + f*x]] ), x], x] /; FreeQ[{a, b, d, e, f}, x] && NeQ[a^2 - b^2, 0]
Int[(csc[(e_.) + (f_.)*(x_)]*(B_.) + (A_))/(Sqrt[csc[(e_.) + (f_.)*(x_)]*(d _.)]*Sqrt[csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_)]), x_Symbol] :> Simp[A/a I nt[Sqrt[a + b*Csc[e + f*x]]/Sqrt[d*Csc[e + f*x]], x], x] - Simp[(A*b - a*B) /(a*d) Int[Sqrt[d*Csc[e + f*x]]/Sqrt[a + b*Csc[e + f*x]], x], x] /; FreeQ [{a, b, d, e, f, A, B}, x] && NeQ[A*b - a*B, 0] && NeQ[a^2 - b^2, 0]
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)*Cot[e + f*x]*(a + b*Csc[e + f*x])^m*((d*Cs c[e + f*x])^n/(f*(m + n + 1))), x] + Simp[1/(m + n + 1) Int[(a + b*Csc[e + f*x])^(m - 1)*(d*Csc[e + f*x])^n*Simp[a*A*(m + n + 1) + a*C*n + ((A*b + a *B)*(m + n + 1) + b*C*(m + n))*Csc[e + f*x] + (b*B*(m + n + 1) + a*C*m)*Csc [e + f*x]^2, x], x], x] /; FreeQ[{a, b, d, e, f, A, B, C, n}, x] && NeQ[a^2 - b^2, 0] && GtQ[m, 0] && !LeQ[n, -1]
Int[((A_.) + csc[(e_.) + (f_.)*(x_)]*(B_.) + csc[(e_.) + (f_.)*(x_)]^2*(C_. ))/(Sqrt[csc[(e_.) + (f_.)*(x_)]*(d_.)]*Sqrt[csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_)]), x_Symbol] :> Simp[C/d^2 Int[(d*Csc[e + f*x])^(3/2)/Sqrt[a + b*C sc[e + f*x]], x], x] + Int[(A + B*Csc[e + f*x])/(Sqrt[d*Csc[e + f*x]]*Sqrt[ a + b*Csc[e + f*x]]), x] /; FreeQ[{a, b, d, e, f, A, B, C}, x] && NeQ[a^2 - b^2, 0]
Result contains complex when optimal does not.
Time = 26.13 (sec) , antiderivative size = 7894, normalized size of antiderivative = 17.43
| method | result | size |
| default | \(\text {Expression too large to display}\) | \(7894\) |
| parts | \(\text {Expression too large to display}\) | \(8104\) |
int((a+b*sec(d*x+c))^(5/2)*(A+B*sec(d*x+c)+C*sec(d*x+c)^2)/sec(d*x+c)^(1/2 ),x,method=_RETURNVERBOSE)
Timed out. \[ \int \frac {(a+b \sec (c+d x))^{5/2} \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right )}{\sqrt {\sec (c+d x)}} \, dx=\text {Timed out} \]
integrate((a+b*sec(d*x+c))^(5/2)*(A+B*sec(d*x+c)+C*sec(d*x+c)^2)/sec(d*x+c )^(1/2),x, algorithm="fricas")
Timed out. \[ \int \frac {(a+b \sec (c+d x))^{5/2} \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right )}{\sqrt {\sec (c+d x)}} \, dx=\text {Timed out} \]
\[ \int \frac {(a+b \sec (c+d x))^{5/2} \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right )}{\sqrt {\sec (c+d x)}} \, dx=\int { \frac {{\left (C \sec \left (d x + c\right )^{2} + B \sec \left (d x + c\right ) + A\right )} {\left (b \sec \left (d x + c\right ) + a\right )}^{\frac {5}{2}}}{\sqrt {\sec \left (d x + c\right )}} \,d x } \]
integrate((a+b*sec(d*x+c))^(5/2)*(A+B*sec(d*x+c)+C*sec(d*x+c)^2)/sec(d*x+c )^(1/2),x, algorithm="maxima")
integrate((C*sec(d*x + c)^2 + B*sec(d*x + c) + A)*(b*sec(d*x + c) + a)^(5/ 2)/sqrt(sec(d*x + c)), x)
\[ \int \frac {(a+b \sec (c+d x))^{5/2} \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right )}{\sqrt {\sec (c+d x)}} \, dx=\int { \frac {{\left (C \sec \left (d x + c\right )^{2} + B \sec \left (d x + c\right ) + A\right )} {\left (b \sec \left (d x + c\right ) + a\right )}^{\frac {5}{2}}}{\sqrt {\sec \left (d x + c\right )}} \,d x } \]
integrate((a+b*sec(d*x+c))^(5/2)*(A+B*sec(d*x+c)+C*sec(d*x+c)^2)/sec(d*x+c )^(1/2),x, algorithm="giac")
integrate((C*sec(d*x + c)^2 + B*sec(d*x + c) + A)*(b*sec(d*x + c) + a)^(5/ 2)/sqrt(sec(d*x + c)), x)
Timed out. \[ \int \frac {(a+b \sec (c+d x))^{5/2} \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right )}{\sqrt {\sec (c+d x)}} \, dx=\int \frac {{\left (a+\frac {b}{\cos \left (c+d\,x\right )}\right )}^{5/2}\,\left (A+\frac {B}{\cos \left (c+d\,x\right )}+\frac {C}{{\cos \left (c+d\,x\right )}^2}\right )}{\sqrt {\frac {1}{\cos \left (c+d\,x\right )}}} \,d x \]
int(((a + b/cos(c + d*x))^(5/2)*(A + B/cos(c + d*x) + C/cos(c + d*x)^2))/( 1/cos(c + d*x))^(1/2),x)