Integrand size = 45, antiderivative size = 427 \[ \int \frac {(a+b \sec (c+d x))^{5/2} \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right )}{\sec ^{\frac {3}{2}}(c+d x)} \, dx=\frac {\left (48 a^2 b B+12 b^3 B+8 a^3 (A+3 C)+a b^2 (16 A+33 C)\right ) \sqrt {\frac {b+a \cos (c+d x)}{a+b}} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),\frac {2 a}{a+b}\right ) \sqrt {\sec (c+d x)}}{12 d \sqrt {a+b \sec (c+d x)}}+\frac {b \left (8 A b^2+20 a b B+15 a^2 C+4 b^2 C\right ) \sqrt {\frac {b+a \cos (c+d x)}{a+b}} \operatorname {EllipticPi}\left (2,\frac {1}{2} (c+d x),\frac {2 a}{a+b}\right ) \sqrt {\sec (c+d x)}}{4 d \sqrt {a+b \sec (c+d x)}}+\frac {\left (24 a^2 B-12 b^2 B+a b (56 A-27 C)\right ) E\left (\frac {1}{2} (c+d x)|\frac {2 a}{a+b}\right ) \sqrt {a+b \sec (c+d x)}}{12 d \sqrt {\frac {b+a \cos (c+d x)}{a+b}} \sqrt {\sec (c+d x)}}-\frac {b (8 a A-12 b B-21 a C) \sqrt {\sec (c+d x)} \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{12 d}-\frac {b (4 A-3 C) \sqrt {\sec (c+d x)} (a+b \sec (c+d x))^{3/2} \sin (c+d x)}{6 d}+\frac {2 A (a+b \sec (c+d x))^{5/2} \sin (c+d x)}{3 d \sqrt {\sec (c+d x)}} \] Output:
1/12*(48*B*a^2*b+12*B*b^3+8*a^3*(A+3*C)+a*b^2*(16*A+33*C))*((b+a*cos(d*x+c ))/(a+b))^(1/2)*InverseJacobiAM(1/2*d*x+1/2*c,2^(1/2)*(a/(a+b))^(1/2))*sec (d*x+c)^(1/2)/d/(a+b*sec(d*x+c))^(1/2)+1/4*b*(8*A*b^2+20*B*a*b+15*C*a^2+4* C*b^2)*((b+a*cos(d*x+c))/(a+b))^(1/2)*EllipticPi(sin(1/2*d*x+1/2*c),2,2^(1 /2)*(a/(a+b))^(1/2))*sec(d*x+c)^(1/2)/d/(a+b*sec(d*x+c))^(1/2)+1/12*(24*B* a^2-12*B*b^2+a*b*(56*A-27*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)-1/12*b*(8*A*a-12*B*b-21*C*a)*sec(d*x+c)^(1/2)*(a+b*sec(d*x+c))^(1 /2)*sin(d*x+c)/d-1/6*b*(4*A-3*C)*sec(d*x+c)^(1/2)*(a+b*sec(d*x+c))^(3/2)*s in(d*x+c)/d+2/3*A*(a+b*sec(d*x+c))^(5/2)*sin(d*x+c)/d/sec(d*x+c)^(1/2)
Result contains complex when optimal does not.
Time = 11.72 (sec) , antiderivative size = 766, normalized size of antiderivative = 1.79 \[ \int \frac {(a+b \sec (c+d x))^{5/2} \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right )}{\sec ^{\frac {3}{2}}(c+d x)} \, dx=\frac {(a+b \sec (c+d x))^{5/2} \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \left (\frac {2 \left (16 a^3 A+144 a A b^2+144 a^2 b B+48 a^3 C+12 a b^2 C\right ) \sqrt {\frac {b+a \cos (c+d x)}{a+b}} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),\frac {2 a}{a+b}\right )}{\sqrt {b+a \cos (c+d x)}}+\frac {2 \left (56 a^2 A b+48 A b^3+24 a^3 B+108 a b^2 B+63 a^2 b C+24 b^3 C\right ) \sqrt {\frac {b+a \cos (c+d x)}{a+b}} \operatorname {EllipticPi}\left (2,\frac {1}{2} (c+d x),\frac {2 a}{a+b}\right )}{\sqrt {b+a \cos (c+d x)}}+\frac {2 i \left (56 a^2 A b+24 a^3 B-12 a b^2 B-27 a^2 b C\right ) \sqrt {\frac {a-a \cos (c+d x)}{a+b}} \sqrt {\frac {a+a \cos (c+d x)}{a-b}} \cos (2 (c+d x)) \left (-2 b (a+b) E\left (i \text {arcsinh}\left (\sqrt {\frac {1}{a-b}} \sqrt {b+a \cos (c+d x)}\right )|\frac {-a+b}{a+b}\right )+a \left (2 b \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {\frac {1}{a-b}} \sqrt {b+a \cos (c+d x)}\right ),\frac {-a+b}{a+b}\right )+a \operatorname {EllipticPi}\left (1-\frac {a}{b},i \text {arcsinh}\left (\sqrt {\frac {1}{a-b}} \sqrt {b+a \cos (c+d x)}\right ),\frac {-a+b}{a+b}\right )\right )\right ) \sin (c+d x)}{\sqrt {\frac {1}{a-b}} b \sqrt {1-\cos ^2(c+d x)} \sqrt {\frac {a^2-a^2 \cos ^2(c+d x)}{a^2}} \left (-a^2+2 b^2-4 b (b+a \cos (c+d x))+2 (b+a \cos (c+d x))^2\right )}\right )}{24 d (b+a \cos (c+d x))^{5/2} (A+2 C+2 B \cos (c+d x)+A \cos (2 c+2 d x)) \sec ^{\frac {9}{2}}(c+d x)}+\frac {(a+b \sec (c+d x))^{5/2} \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \left (\frac {4}{3} a^2 A \sin (c+d x)+\frac {1}{2} \sec (c+d x) \left (4 b^2 B \sin (c+d x)+9 a b C \sin (c+d x)\right )+b^2 C \sec (c+d x) \tan (c+d x)\right )}{d (b+a \cos (c+d x))^2 (A+2 C+2 B \cos (c+d x)+A \cos (2 c+2 d x)) \sec ^{\frac {9}{2}}(c+d x)} \] Input:
Integrate[((a + b*Sec[c + d*x])^(5/2)*(A + B*Sec[c + d*x] + C*Sec[c + d*x] ^2))/Sec[c + d*x]^(3/2),x]
Output:
((a + b*Sec[c + d*x])^(5/2)*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*((2*(1 6*a^3*A + 144*a*A*b^2 + 144*a^2*b*B + 48*a^3*C + 12*a*b^2*C)*Sqrt[(b + a*C os[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*a)/(a + b)])/Sqrt[b + a*Co s[c + d*x]] + (2*(56*a^2*A*b + 48*A*b^3 + 24*a^3*B + 108*a*b^2*B + 63*a^2* b*C + 24*b^3*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)*(56*a^2*A*b + 24*a^3 *B - 12*a*b^2*B - 27*a^2*b*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))))/(24*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)*((4*a^2*A*Sin[c + d*x])/3 + (Sec[c + d*x]*(4*b^2*B*Sin[c + d*x] + 9*a*b*C*Sin[c + d*x]))/2 + b^2*C*Sec[c + d*x]*Tan[c + d*x]))/(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[c + d*x]^(9/2))
Time = 4.49 (sec) , antiderivative size = 434, normalized size of antiderivative = 1.02, number of steps used = 29, number of rules used = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.644, Rules used = {3042, 4582, 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.
| \(\displaystyle \int \frac {(a+b \sec (c+d x))^{5/2} \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right )}{\sec ^{\frac {3}{2}}(c+d x)} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\left (a+b \csc \left (c+d x+\frac {\pi }{2}\right )\right )^{5/2} \left (A+B \csc \left (c+d x+\frac {\pi }{2}\right )+C \csc \left (c+d x+\frac {\pi }{2}\right )^2\right )}{\csc \left (c+d x+\frac {\pi }{2}\right )^{3/2}}dx\) |
| \(\Big \downarrow \) 4582 |
| \(\displaystyle \frac {2}{3} \int \frac {(a+b \sec (c+d x))^{3/2} \left (-b (4 A-3 C) \sec ^2(c+d x)+(3 b B+a (A+3 C)) \sec (c+d x)+5 A b+3 a B\right )}{2 \sqrt {\sec (c+d x)}}dx+\frac {2 A \sin (c+d x) (a+b \sec (c+d x))^{5/2}}{3 d \sqrt {\sec (c+d x)}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{3} \int \frac {(a+b \sec (c+d x))^{3/2} \left (-b (4 A-3 C) \sec ^2(c+d x)+(3 b B+a (A+3 C)) \sec (c+d x)+5 A b+3 a B\right )}{\sqrt {\sec (c+d x)}}dx+\frac {2 A \sin (c+d x) (a+b \sec (c+d x))^{5/2}}{3 d \sqrt {\sec (c+d x)}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{3} \int \frac {\left (a+b \csc \left (c+d x+\frac {\pi }{2}\right )\right )^{3/2} \left (-b (4 A-3 C) \csc \left (c+d x+\frac {\pi }{2}\right )^2+(3 b B+a (A+3 C)) \csc \left (c+d x+\frac {\pi }{2}\right )+5 A b+3 a B\right )}{\sqrt {\csc \left (c+d x+\frac {\pi }{2}\right )}}dx+\frac {2 A \sin (c+d x) (a+b \sec (c+d x))^{5/2}}{3 d \sqrt {\sec (c+d x)}}\) |
| \(\Big \downarrow \) 4584 |
| \(\displaystyle \frac {1}{3} \left (\frac {1}{2} \int \frac {\sqrt {a+b \sec (c+d x)} \left (-b (8 a A-12 b B-21 a C) \sec ^2(c+d x)+2 \left (2 (A+3 C) a^2+12 b B a+3 b^2 (2 A+C)\right ) \sec (c+d x)+3 a (8 A b-C b+4 a B)\right )}{2 \sqrt {\sec (c+d x)}}dx-\frac {b (4 A-3 C) \sin (c+d x) \sqrt {\sec (c+d x)} (a+b \sec (c+d x))^{3/2}}{2 d}\right )+\frac {2 A \sin (c+d x) (a+b \sec (c+d x))^{5/2}}{3 d \sqrt {\sec (c+d x)}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{3} \left (\frac {1}{4} \int \frac {\sqrt {a+b \sec (c+d x)} \left (-b (8 a A-12 b B-21 a C) \sec ^2(c+d x)+2 \left (2 (A+3 C) a^2+12 b B a+3 b^2 (2 A+C)\right ) \sec (c+d x)+3 a (8 A b-C b+4 a B)\right )}{\sqrt {\sec (c+d x)}}dx-\frac {b (4 A-3 C) \sin (c+d x) \sqrt {\sec (c+d x)} (a+b \sec (c+d x))^{3/2}}{2 d}\right )+\frac {2 A \sin (c+d x) (a+b \sec (c+d x))^{5/2}}{3 d \sqrt {\sec (c+d x)}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{3} \left (\frac {1}{4} \int \frac {\sqrt {a+b \csc \left (c+d x+\frac {\pi }{2}\right )} \left (-b (8 a A-12 b B-21 a C) \csc \left (c+d x+\frac {\pi }{2}\right )^2+2 \left (2 (A+3 C) a^2+12 b B a+3 b^2 (2 A+C)\right ) \csc \left (c+d x+\frac {\pi }{2}\right )+3 a (8 A b-C b+4 a B)\right )}{\sqrt {\csc \left (c+d x+\frac {\pi }{2}\right )}}dx-\frac {b (4 A-3 C) \sin (c+d x) \sqrt {\sec (c+d x)} (a+b \sec (c+d x))^{3/2}}{2 d}\right )+\frac {2 A \sin (c+d x) (a+b \sec (c+d x))^{5/2}}{3 d \sqrt {\sec (c+d x)}}\) |
| \(\Big \downarrow \) 4584 |
| \(\displaystyle \frac {1}{3} \left (\frac {1}{4} \left (\int \frac {3 b \left (15 C a^2+20 b B a+8 A b^2+4 b^2 C\right ) \sec ^2(c+d x)+2 a \left (4 (A+3 C) a^2+36 b B a+3 b^2 (12 A+C)\right ) \sec (c+d x)+a \left (24 B a^2+b (56 A-27 C) a-12 b^2 B\right )}{2 \sqrt {\sec (c+d x)} \sqrt {a+b \sec (c+d x)}}dx-\frac {b \sin (c+d x) \sqrt {\sec (c+d x)} (8 a A-21 a C-12 b B) \sqrt {a+b \sec (c+d x)}}{d}\right )-\frac {b (4 A-3 C) \sin (c+d x) \sqrt {\sec (c+d x)} (a+b \sec (c+d x))^{3/2}}{2 d}\right )+\frac {2 A \sin (c+d x) (a+b \sec (c+d x))^{5/2}}{3 d \sqrt {\sec (c+d x)}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{3} \left (\frac {1}{4} \left (\frac {1}{2} \int \frac {3 b \left (15 C a^2+20 b B a+8 A b^2+4 b^2 C\right ) \sec ^2(c+d x)+2 a \left (4 (A+3 C) a^2+36 b B a+3 b^2 (12 A+C)\right ) \sec (c+d x)+a \left (24 B a^2+b (56 A-27 C) a-12 b^2 B\right )}{\sqrt {\sec (c+d x)} \sqrt {a+b \sec (c+d x)}}dx-\frac {b \sin (c+d x) \sqrt {\sec (c+d x)} (8 a A-21 a C-12 b B) \sqrt {a+b \sec (c+d x)}}{d}\right )-\frac {b (4 A-3 C) \sin (c+d x) \sqrt {\sec (c+d x)} (a+b \sec (c+d x))^{3/2}}{2 d}\right )+\frac {2 A \sin (c+d x) (a+b \sec (c+d x))^{5/2}}{3 d \sqrt {\sec (c+d x)}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{3} \left (\frac {1}{4} \left (\frac {1}{2} \int \frac {3 b \left (15 C a^2+20 b B a+8 A b^2+4 b^2 C\right ) \csc \left (c+d x+\frac {\pi }{2}\right )^2+2 a \left (4 (A+3 C) a^2+36 b B a+3 b^2 (12 A+C)\right ) \csc \left (c+d x+\frac {\pi }{2}\right )+a \left (24 B a^2+b (56 A-27 C) a-12 b^2 B\right )}{\sqrt {\csc \left (c+d x+\frac {\pi }{2}\right )} \sqrt {a+b \csc \left (c+d x+\frac {\pi }{2}\right )}}dx-\frac {b \sin (c+d x) \sqrt {\sec (c+d x)} (8 a A-21 a C-12 b B) \sqrt {a+b \sec (c+d x)}}{d}\right )-\frac {b (4 A-3 C) \sin (c+d x) \sqrt {\sec (c+d x)} (a+b \sec (c+d x))^{3/2}}{2 d}\right )+\frac {2 A \sin (c+d x) (a+b \sec (c+d x))^{5/2}}{3 d \sqrt {\sec (c+d x)}}\) |
| \(\Big \downarrow \) 4596 |
| \(\displaystyle \frac {1}{3} \left (\frac {1}{4} \left (\frac {1}{2} \left (3 b \left (15 a^2 C+20 a b B+8 A b^2+4 b^2 C\right ) \int \frac {\sec ^{\frac {3}{2}}(c+d x)}{\sqrt {a+b \sec (c+d x)}}dx+\int \frac {a \left (24 B a^2+b (56 A-27 C) a-12 b^2 B\right )+2 a \left (4 (A+3 C) a^2+36 b B a+3 b^2 (12 A+C)\right ) \sec (c+d x)}{\sqrt {\sec (c+d x)} \sqrt {a+b \sec (c+d x)}}dx\right )-\frac {b \sin (c+d x) \sqrt {\sec (c+d x)} (8 a A-21 a C-12 b B) \sqrt {a+b \sec (c+d x)}}{d}\right )-\frac {b (4 A-3 C) \sin (c+d x) \sqrt {\sec (c+d x)} (a+b \sec (c+d x))^{3/2}}{2 d}\right )+\frac {2 A \sin (c+d x) (a+b \sec (c+d x))^{5/2}}{3 d \sqrt {\sec (c+d x)}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{3} \left (\frac {1}{4} \left (\frac {1}{2} \left (3 b \left (15 a^2 C+20 a b B+8 A b^2+4 b^2 C\right ) \int \frac {\csc \left (c+d x+\frac {\pi }{2}\right )^{3/2}}{\sqrt {a+b \csc \left (c+d x+\frac {\pi }{2}\right )}}dx+\int \frac {a \left (24 B a^2+b (56 A-27 C) a-12 b^2 B\right )+2 a \left (4 (A+3 C) a^2+36 b B a+3 b^2 (12 A+C)\right ) \csc \left (c+d x+\frac {\pi }{2}\right )}{\sqrt {\csc \left (c+d x+\frac {\pi }{2}\right )} \sqrt {a+b \csc \left (c+d x+\frac {\pi }{2}\right )}}dx\right )-\frac {b \sin (c+d x) \sqrt {\sec (c+d x)} (8 a A-21 a C-12 b B) \sqrt {a+b \sec (c+d x)}}{d}\right )-\frac {b (4 A-3 C) \sin (c+d x) \sqrt {\sec (c+d x)} (a+b \sec (c+d x))^{3/2}}{2 d}\right )+\frac {2 A \sin (c+d x) (a+b \sec (c+d x))^{5/2}}{3 d \sqrt {\sec (c+d x)}}\) |
| \(\Big \downarrow \) 4346 |
| \(\displaystyle \frac {1}{3} \left (\frac {1}{4} \left (\frac {1}{2} \left (\int \frac {a \left (24 B a^2+b (56 A-27 C) a-12 b^2 B\right )+2 a \left (4 (A+3 C) a^2+36 b B a+3 b^2 (12 A+C)\right ) \csc \left (c+d x+\frac {\pi }{2}\right )}{\sqrt {\csc \left (c+d x+\frac {\pi }{2}\right )} \sqrt {a+b \csc \left (c+d x+\frac {\pi }{2}\right )}}dx+\frac {3 b \sqrt {\sec (c+d x)} \left (15 a^2 C+20 a b B+8 A b^2+4 b^2 C\right ) \sqrt {a \cos (c+d x)+b} \int \frac {\sec (c+d x)}{\sqrt {b+a \cos (c+d x)}}dx}{\sqrt {a+b \sec (c+d x)}}\right )-\frac {b \sin (c+d x) \sqrt {\sec (c+d x)} (8 a A-21 a C-12 b B) \sqrt {a+b \sec (c+d x)}}{d}\right )-\frac {b (4 A-3 C) \sin (c+d x) \sqrt {\sec (c+d x)} (a+b \sec (c+d x))^{3/2}}{2 d}\right )+\frac {2 A \sin (c+d x) (a+b \sec (c+d x))^{5/2}}{3 d \sqrt {\sec (c+d x)}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{3} \left (\frac {1}{4} \left (\frac {1}{2} \left (\int \frac {a \left (24 B a^2+b (56 A-27 C) a-12 b^2 B\right )+2 a \left (4 (A+3 C) a^2+36 b B a+3 b^2 (12 A+C)\right ) \csc \left (c+d x+\frac {\pi }{2}\right )}{\sqrt {\csc \left (c+d x+\frac {\pi }{2}\right )} \sqrt {a+b \csc \left (c+d x+\frac {\pi }{2}\right )}}dx+\frac {3 b \sqrt {\sec (c+d x)} \left (15 a^2 C+20 a b B+8 A b^2+4 b^2 C\right ) \sqrt {a \cos (c+d x)+b} \int \frac {1}{\sin \left (c+d x+\frac {\pi }{2}\right ) \sqrt {b+a \sin \left (c+d x+\frac {\pi }{2}\right )}}dx}{\sqrt {a+b \sec (c+d x)}}\right )-\frac {b \sin (c+d x) \sqrt {\sec (c+d x)} (8 a A-21 a C-12 b B) \sqrt {a+b \sec (c+d x)}}{d}\right )-\frac {b (4 A-3 C) \sin (c+d x) \sqrt {\sec (c+d x)} (a+b \sec (c+d x))^{3/2}}{2 d}\right )+\frac {2 A \sin (c+d x) (a+b \sec (c+d x))^{5/2}}{3 d \sqrt {\sec (c+d x)}}\) |
| \(\Big \downarrow \) 3286 |
| \(\displaystyle \frac {1}{3} \left (\frac {1}{4} \left (\frac {1}{2} \left (\int \frac {a \left (24 B a^2+b (56 A-27 C) a-12 b^2 B\right )+2 a \left (4 (A+3 C) a^2+36 b B a+3 b^2 (12 A+C)\right ) \csc \left (c+d x+\frac {\pi }{2}\right )}{\sqrt {\csc \left (c+d x+\frac {\pi }{2}\right )} \sqrt {a+b \csc \left (c+d x+\frac {\pi }{2}\right )}}dx+\frac {3 b \sqrt {\sec (c+d x)} \left (15 a^2 C+20 a b B+8 A b^2+4 b^2 C\right ) \sqrt {\frac {a \cos (c+d x)+b}{a+b}} \int \frac {\sec (c+d x)}{\sqrt {\frac {b}{a+b}+\frac {a \cos (c+d x)}{a+b}}}dx}{\sqrt {a+b \sec (c+d x)}}\right )-\frac {b \sin (c+d x) \sqrt {\sec (c+d x)} (8 a A-21 a C-12 b B) \sqrt {a+b \sec (c+d x)}}{d}\right )-\frac {b (4 A-3 C) \sin (c+d x) \sqrt {\sec (c+d x)} (a+b \sec (c+d x))^{3/2}}{2 d}\right )+\frac {2 A \sin (c+d x) (a+b \sec (c+d x))^{5/2}}{3 d \sqrt {\sec (c+d x)}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{3} \left (\frac {1}{4} \left (\frac {1}{2} \left (\int \frac {a \left (24 B a^2+b (56 A-27 C) a-12 b^2 B\right )+2 a \left (4 (A+3 C) a^2+36 b B a+3 b^2 (12 A+C)\right ) \csc \left (c+d x+\frac {\pi }{2}\right )}{\sqrt {\csc \left (c+d x+\frac {\pi }{2}\right )} \sqrt {a+b \csc \left (c+d x+\frac {\pi }{2}\right )}}dx+\frac {3 b \sqrt {\sec (c+d x)} \left (15 a^2 C+20 a b B+8 A b^2+4 b^2 C\right ) \sqrt {\frac {a \cos (c+d x)+b}{a+b}} \int \frac {1}{\sin \left (c+d x+\frac {\pi }{2}\right ) \sqrt {\frac {b}{a+b}+\frac {a \sin \left (c+d x+\frac {\pi }{2}\right )}{a+b}}}dx}{\sqrt {a+b \sec (c+d x)}}\right )-\frac {b \sin (c+d x) \sqrt {\sec (c+d x)} (8 a A-21 a C-12 b B) \sqrt {a+b \sec (c+d x)}}{d}\right )-\frac {b (4 A-3 C) \sin (c+d x) \sqrt {\sec (c+d x)} (a+b \sec (c+d x))^{3/2}}{2 d}\right )+\frac {2 A \sin (c+d x) (a+b \sec (c+d x))^{5/2}}{3 d \sqrt {\sec (c+d x)}}\) |
| \(\Big \downarrow \) 3284 |
| \(\displaystyle \frac {1}{3} \left (\frac {1}{4} \left (\frac {1}{2} \left (\int \frac {a \left (24 B a^2+b (56 A-27 C) a-12 b^2 B\right )+2 a \left (4 (A+3 C) a^2+36 b B a+3 b^2 (12 A+C)\right ) \csc \left (c+d x+\frac {\pi }{2}\right )}{\sqrt {\csc \left (c+d x+\frac {\pi }{2}\right )} \sqrt {a+b \csc \left (c+d x+\frac {\pi }{2}\right )}}dx+\frac {6 b \sqrt {\sec (c+d x)} \left (15 a^2 C+20 a b B+8 A b^2+4 b^2 C\right ) \sqrt {\frac {a \cos (c+d x)+b}{a+b}} \operatorname {EllipticPi}\left (2,\frac {1}{2} (c+d x),\frac {2 a}{a+b}\right )}{d \sqrt {a+b \sec (c+d x)}}\right )-\frac {b \sin (c+d x) \sqrt {\sec (c+d x)} (8 a A-21 a C-12 b B) \sqrt {a+b \sec (c+d x)}}{d}\right )-\frac {b (4 A-3 C) \sin (c+d x) \sqrt {\sec (c+d x)} (a+b \sec (c+d x))^{3/2}}{2 d}\right )+\frac {2 A \sin (c+d x) (a+b \sec (c+d x))^{5/2}}{3 d \sqrt {\sec (c+d x)}}\) |
| \(\Big \downarrow \) 4523 |
| \(\displaystyle \frac {1}{3} \left (\frac {1}{4} \left (\frac {1}{2} \left (\left (24 a^2 B+a b (56 A-27 C)-12 b^2 B\right ) \int \frac {\sqrt {a+b \sec (c+d x)}}{\sqrt {\sec (c+d x)}}dx+\left (8 a^3 (A+3 C)+48 a^2 b B+a b^2 (16 A+33 C)+12 b^3 B\right ) \int \frac {\sqrt {\sec (c+d x)}}{\sqrt {a+b \sec (c+d x)}}dx+\frac {6 b \sqrt {\sec (c+d x)} \left (15 a^2 C+20 a b B+8 A b^2+4 b^2 C\right ) \sqrt {\frac {a \cos (c+d x)+b}{a+b}} \operatorname {EllipticPi}\left (2,\frac {1}{2} (c+d x),\frac {2 a}{a+b}\right )}{d \sqrt {a+b \sec (c+d x)}}\right )-\frac {b \sin (c+d x) \sqrt {\sec (c+d x)} (8 a A-21 a C-12 b B) \sqrt {a+b \sec (c+d x)}}{d}\right )-\frac {b (4 A-3 C) \sin (c+d x) \sqrt {\sec (c+d x)} (a+b \sec (c+d x))^{3/2}}{2 d}\right )+\frac {2 A \sin (c+d x) (a+b \sec (c+d x))^{5/2}}{3 d \sqrt {\sec (c+d x)}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{3} \left (\frac {1}{4} \left (\frac {1}{2} \left (\left (24 a^2 B+a b (56 A-27 C)-12 b^2 B\right ) \int \frac {\sqrt {a+b \csc \left (c+d x+\frac {\pi }{2}\right )}}{\sqrt {\csc \left (c+d x+\frac {\pi }{2}\right )}}dx+\left (8 a^3 (A+3 C)+48 a^2 b B+a b^2 (16 A+33 C)+12 b^3 B\right ) \int \frac {\sqrt {\csc \left (c+d x+\frac {\pi }{2}\right )}}{\sqrt {a+b \csc \left (c+d x+\frac {\pi }{2}\right )}}dx+\frac {6 b \sqrt {\sec (c+d x)} \left (15 a^2 C+20 a b B+8 A b^2+4 b^2 C\right ) \sqrt {\frac {a \cos (c+d x)+b}{a+b}} \operatorname {EllipticPi}\left (2,\frac {1}{2} (c+d x),\frac {2 a}{a+b}\right )}{d \sqrt {a+b \sec (c+d x)}}\right )-\frac {b \sin (c+d x) \sqrt {\sec (c+d x)} (8 a A-21 a C-12 b B) \sqrt {a+b \sec (c+d x)}}{d}\right )-\frac {b (4 A-3 C) \sin (c+d x) \sqrt {\sec (c+d x)} (a+b \sec (c+d x))^{3/2}}{2 d}\right )+\frac {2 A \sin (c+d x) (a+b \sec (c+d x))^{5/2}}{3 d \sqrt {\sec (c+d x)}}\) |
| \(\Big \downarrow \) 4343 |
| \(\displaystyle \frac {1}{3} \left (\frac {1}{4} \left (\frac {1}{2} \left (\frac {\left (24 a^2 B+a b (56 A-27 C)-12 b^2 B\right ) \sqrt {a+b \sec (c+d x)} \int \sqrt {b+a \cos (c+d x)}dx}{\sqrt {\sec (c+d x)} \sqrt {a \cos (c+d x)+b}}+\left (8 a^3 (A+3 C)+48 a^2 b B+a b^2 (16 A+33 C)+12 b^3 B\right ) \int \frac {\sqrt {\csc \left (c+d x+\frac {\pi }{2}\right )}}{\sqrt {a+b \csc \left (c+d x+\frac {\pi }{2}\right )}}dx+\frac {6 b \sqrt {\sec (c+d x)} \left (15 a^2 C+20 a b B+8 A b^2+4 b^2 C\right ) \sqrt {\frac {a \cos (c+d x)+b}{a+b}} \operatorname {EllipticPi}\left (2,\frac {1}{2} (c+d x),\frac {2 a}{a+b}\right )}{d \sqrt {a+b \sec (c+d x)}}\right )-\frac {b \sin (c+d x) \sqrt {\sec (c+d x)} (8 a A-21 a C-12 b B) \sqrt {a+b \sec (c+d x)}}{d}\right )-\frac {b (4 A-3 C) \sin (c+d x) \sqrt {\sec (c+d x)} (a+b \sec (c+d x))^{3/2}}{2 d}\right )+\frac {2 A \sin (c+d x) (a+b \sec (c+d x))^{5/2}}{3 d \sqrt {\sec (c+d x)}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{3} \left (\frac {1}{4} \left (\frac {1}{2} \left (\frac {\left (24 a^2 B+a b (56 A-27 C)-12 b^2 B\right ) \sqrt {a+b \sec (c+d x)} \int \sqrt {b+a \sin \left (c+d x+\frac {\pi }{2}\right )}dx}{\sqrt {\sec (c+d x)} \sqrt {a \cos (c+d x)+b}}+\left (8 a^3 (A+3 C)+48 a^2 b B+a b^2 (16 A+33 C)+12 b^3 B\right ) \int \frac {\sqrt {\csc \left (c+d x+\frac {\pi }{2}\right )}}{\sqrt {a+b \csc \left (c+d x+\frac {\pi }{2}\right )}}dx+\frac {6 b \sqrt {\sec (c+d x)} \left (15 a^2 C+20 a b B+8 A b^2+4 b^2 C\right ) \sqrt {\frac {a \cos (c+d x)+b}{a+b}} \operatorname {EllipticPi}\left (2,\frac {1}{2} (c+d x),\frac {2 a}{a+b}\right )}{d \sqrt {a+b \sec (c+d x)}}\right )-\frac {b \sin (c+d x) \sqrt {\sec (c+d x)} (8 a A-21 a C-12 b B) \sqrt {a+b \sec (c+d x)}}{d}\right )-\frac {b (4 A-3 C) \sin (c+d x) \sqrt {\sec (c+d x)} (a+b \sec (c+d x))^{3/2}}{2 d}\right )+\frac {2 A \sin (c+d x) (a+b \sec (c+d x))^{5/2}}{3 d \sqrt {\sec (c+d x)}}\) |
| \(\Big \downarrow \) 3134 |
| \(\displaystyle \frac {1}{3} \left (\frac {1}{4} \left (\frac {1}{2} \left (\frac {\left (24 a^2 B+a b (56 A-27 C)-12 b^2 B\right ) \sqrt {a+b \sec (c+d x)} \int \sqrt {\frac {b}{a+b}+\frac {a \cos (c+d x)}{a+b}}dx}{\sqrt {\sec (c+d x)} \sqrt {\frac {a \cos (c+d x)+b}{a+b}}}+\left (8 a^3 (A+3 C)+48 a^2 b B+a b^2 (16 A+33 C)+12 b^3 B\right ) \int \frac {\sqrt {\csc \left (c+d x+\frac {\pi }{2}\right )}}{\sqrt {a+b \csc \left (c+d x+\frac {\pi }{2}\right )}}dx+\frac {6 b \sqrt {\sec (c+d x)} \left (15 a^2 C+20 a b B+8 A b^2+4 b^2 C\right ) \sqrt {\frac {a \cos (c+d x)+b}{a+b}} \operatorname {EllipticPi}\left (2,\frac {1}{2} (c+d x),\frac {2 a}{a+b}\right )}{d \sqrt {a+b \sec (c+d x)}}\right )-\frac {b \sin (c+d x) \sqrt {\sec (c+d x)} (8 a A-21 a C-12 b B) \sqrt {a+b \sec (c+d x)}}{d}\right )-\frac {b (4 A-3 C) \sin (c+d x) \sqrt {\sec (c+d x)} (a+b \sec (c+d x))^{3/2}}{2 d}\right )+\frac {2 A \sin (c+d x) (a+b \sec (c+d x))^{5/2}}{3 d \sqrt {\sec (c+d x)}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{3} \left (\frac {1}{4} \left (\frac {1}{2} \left (\frac {\left (24 a^2 B+a b (56 A-27 C)-12 b^2 B\right ) \sqrt {a+b \sec (c+d x)} \int \sqrt {\frac {b}{a+b}+\frac {a \sin \left (c+d x+\frac {\pi }{2}\right )}{a+b}}dx}{\sqrt {\sec (c+d x)} \sqrt {\frac {a \cos (c+d x)+b}{a+b}}}+\left (8 a^3 (A+3 C)+48 a^2 b B+a b^2 (16 A+33 C)+12 b^3 B\right ) \int \frac {\sqrt {\csc \left (c+d x+\frac {\pi }{2}\right )}}{\sqrt {a+b \csc \left (c+d x+\frac {\pi }{2}\right )}}dx+\frac {6 b \sqrt {\sec (c+d x)} \left (15 a^2 C+20 a b B+8 A b^2+4 b^2 C\right ) \sqrt {\frac {a \cos (c+d x)+b}{a+b}} \operatorname {EllipticPi}\left (2,\frac {1}{2} (c+d x),\frac {2 a}{a+b}\right )}{d \sqrt {a+b \sec (c+d x)}}\right )-\frac {b \sin (c+d x) \sqrt {\sec (c+d x)} (8 a A-21 a C-12 b B) \sqrt {a+b \sec (c+d x)}}{d}\right )-\frac {b (4 A-3 C) \sin (c+d x) \sqrt {\sec (c+d x)} (a+b \sec (c+d x))^{3/2}}{2 d}\right )+\frac {2 A \sin (c+d x) (a+b \sec (c+d x))^{5/2}}{3 d \sqrt {\sec (c+d x)}}\) |
| \(\Big \downarrow \) 3132 |
| \(\displaystyle \frac {1}{3} \left (\frac {1}{4} \left (\frac {1}{2} \left (\left (8 a^3 (A+3 C)+48 a^2 b B+a b^2 (16 A+33 C)+12 b^3 B\right ) \int \frac {\sqrt {\csc \left (c+d x+\frac {\pi }{2}\right )}}{\sqrt {a+b \csc \left (c+d x+\frac {\pi }{2}\right )}}dx+\frac {2 \left (24 a^2 B+a b (56 A-27 C)-12 b^2 B\right ) \sqrt {a+b \sec (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 a}{a+b}\right )}{d \sqrt {\sec (c+d x)} \sqrt {\frac {a \cos (c+d x)+b}{a+b}}}+\frac {6 b \sqrt {\sec (c+d x)} \left (15 a^2 C+20 a b B+8 A b^2+4 b^2 C\right ) \sqrt {\frac {a \cos (c+d x)+b}{a+b}} \operatorname {EllipticPi}\left (2,\frac {1}{2} (c+d x),\frac {2 a}{a+b}\right )}{d \sqrt {a+b \sec (c+d x)}}\right )-\frac {b \sin (c+d x) \sqrt {\sec (c+d x)} (8 a A-21 a C-12 b B) \sqrt {a+b \sec (c+d x)}}{d}\right )-\frac {b (4 A-3 C) \sin (c+d x) \sqrt {\sec (c+d x)} (a+b \sec (c+d x))^{3/2}}{2 d}\right )+\frac {2 A \sin (c+d x) (a+b \sec (c+d x))^{5/2}}{3 d \sqrt {\sec (c+d x)}}\) |
| \(\Big \downarrow \) 4345 |
| \(\displaystyle \frac {1}{3} \left (\frac {1}{4} \left (\frac {1}{2} \left (\frac {\sqrt {\sec (c+d x)} \left (8 a^3 (A+3 C)+48 a^2 b B+a b^2 (16 A+33 C)+12 b^3 B\right ) \sqrt {a \cos (c+d x)+b} \int \frac {1}{\sqrt {b+a \cos (c+d x)}}dx}{\sqrt {a+b \sec (c+d x)}}+\frac {2 \left (24 a^2 B+a b (56 A-27 C)-12 b^2 B\right ) \sqrt {a+b \sec (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 a}{a+b}\right )}{d \sqrt {\sec (c+d x)} \sqrt {\frac {a \cos (c+d x)+b}{a+b}}}+\frac {6 b \sqrt {\sec (c+d x)} \left (15 a^2 C+20 a b B+8 A b^2+4 b^2 C\right ) \sqrt {\frac {a \cos (c+d x)+b}{a+b}} \operatorname {EllipticPi}\left (2,\frac {1}{2} (c+d x),\frac {2 a}{a+b}\right )}{d \sqrt {a+b \sec (c+d x)}}\right )-\frac {b \sin (c+d x) \sqrt {\sec (c+d x)} (8 a A-21 a C-12 b B) \sqrt {a+b \sec (c+d x)}}{d}\right )-\frac {b (4 A-3 C) \sin (c+d x) \sqrt {\sec (c+d x)} (a+b \sec (c+d x))^{3/2}}{2 d}\right )+\frac {2 A \sin (c+d x) (a+b \sec (c+d x))^{5/2}}{3 d \sqrt {\sec (c+d x)}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{3} \left (\frac {1}{4} \left (\frac {1}{2} \left (\frac {\sqrt {\sec (c+d x)} \left (8 a^3 (A+3 C)+48 a^2 b B+a b^2 (16 A+33 C)+12 b^3 B\right ) \sqrt {a \cos (c+d x)+b} \int \frac {1}{\sqrt {b+a \sin \left (c+d x+\frac {\pi }{2}\right )}}dx}{\sqrt {a+b \sec (c+d x)}}+\frac {2 \left (24 a^2 B+a b (56 A-27 C)-12 b^2 B\right ) \sqrt {a+b \sec (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 a}{a+b}\right )}{d \sqrt {\sec (c+d x)} \sqrt {\frac {a \cos (c+d x)+b}{a+b}}}+\frac {6 b \sqrt {\sec (c+d x)} \left (15 a^2 C+20 a b B+8 A b^2+4 b^2 C\right ) \sqrt {\frac {a \cos (c+d x)+b}{a+b}} \operatorname {EllipticPi}\left (2,\frac {1}{2} (c+d x),\frac {2 a}{a+b}\right )}{d \sqrt {a+b \sec (c+d x)}}\right )-\frac {b \sin (c+d x) \sqrt {\sec (c+d x)} (8 a A-21 a C-12 b B) \sqrt {a+b \sec (c+d x)}}{d}\right )-\frac {b (4 A-3 C) \sin (c+d x) \sqrt {\sec (c+d x)} (a+b \sec (c+d x))^{3/2}}{2 d}\right )+\frac {2 A \sin (c+d x) (a+b \sec (c+d x))^{5/2}}{3 d \sqrt {\sec (c+d x)}}\) |
| \(\Big \downarrow \) 3142 |
| \(\displaystyle \frac {1}{3} \left (\frac {1}{4} \left (\frac {1}{2} \left (\frac {\sqrt {\sec (c+d x)} \left (8 a^3 (A+3 C)+48 a^2 b B+a b^2 (16 A+33 C)+12 b^3 B\right ) \sqrt {\frac {a \cos (c+d x)+b}{a+b}} \int \frac {1}{\sqrt {\frac {b}{a+b}+\frac {a \cos (c+d x)}{a+b}}}dx}{\sqrt {a+b \sec (c+d x)}}+\frac {2 \left (24 a^2 B+a b (56 A-27 C)-12 b^2 B\right ) \sqrt {a+b \sec (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 a}{a+b}\right )}{d \sqrt {\sec (c+d x)} \sqrt {\frac {a \cos (c+d x)+b}{a+b}}}+\frac {6 b \sqrt {\sec (c+d x)} \left (15 a^2 C+20 a b B+8 A b^2+4 b^2 C\right ) \sqrt {\frac {a \cos (c+d x)+b}{a+b}} \operatorname {EllipticPi}\left (2,\frac {1}{2} (c+d x),\frac {2 a}{a+b}\right )}{d \sqrt {a+b \sec (c+d x)}}\right )-\frac {b \sin (c+d x) \sqrt {\sec (c+d x)} (8 a A-21 a C-12 b B) \sqrt {a+b \sec (c+d x)}}{d}\right )-\frac {b (4 A-3 C) \sin (c+d x) \sqrt {\sec (c+d x)} (a+b \sec (c+d x))^{3/2}}{2 d}\right )+\frac {2 A \sin (c+d x) (a+b \sec (c+d x))^{5/2}}{3 d \sqrt {\sec (c+d x)}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{3} \left (\frac {1}{4} \left (\frac {1}{2} \left (\frac {\sqrt {\sec (c+d x)} \left (8 a^3 (A+3 C)+48 a^2 b B+a b^2 (16 A+33 C)+12 b^3 B\right ) \sqrt {\frac {a \cos (c+d x)+b}{a+b}} \int \frac {1}{\sqrt {\frac {b}{a+b}+\frac {a \sin \left (c+d x+\frac {\pi }{2}\right )}{a+b}}}dx}{\sqrt {a+b \sec (c+d x)}}+\frac {2 \left (24 a^2 B+a b (56 A-27 C)-12 b^2 B\right ) \sqrt {a+b \sec (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 a}{a+b}\right )}{d \sqrt {\sec (c+d x)} \sqrt {\frac {a \cos (c+d x)+b}{a+b}}}+\frac {6 b \sqrt {\sec (c+d x)} \left (15 a^2 C+20 a b B+8 A b^2+4 b^2 C\right ) \sqrt {\frac {a \cos (c+d x)+b}{a+b}} \operatorname {EllipticPi}\left (2,\frac {1}{2} (c+d x),\frac {2 a}{a+b}\right )}{d \sqrt {a+b \sec (c+d x)}}\right )-\frac {b \sin (c+d x) \sqrt {\sec (c+d x)} (8 a A-21 a C-12 b B) \sqrt {a+b \sec (c+d x)}}{d}\right )-\frac {b (4 A-3 C) \sin (c+d x) \sqrt {\sec (c+d x)} (a+b \sec (c+d x))^{3/2}}{2 d}\right )+\frac {2 A \sin (c+d x) (a+b \sec (c+d x))^{5/2}}{3 d \sqrt {\sec (c+d x)}}\) |
| \(\Big \downarrow \) 3140 |
| \(\displaystyle \frac {1}{3} \left (\frac {1}{4} \left (\frac {1}{2} \left (\frac {2 \left (24 a^2 B+a b (56 A-27 C)-12 b^2 B\right ) \sqrt {a+b \sec (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 a}{a+b}\right )}{d \sqrt {\sec (c+d x)} \sqrt {\frac {a \cos (c+d x)+b}{a+b}}}+\frac {6 b \sqrt {\sec (c+d x)} \left (15 a^2 C+20 a b B+8 A b^2+4 b^2 C\right ) \sqrt {\frac {a \cos (c+d x)+b}{a+b}} \operatorname {EllipticPi}\left (2,\frac {1}{2} (c+d x),\frac {2 a}{a+b}\right )}{d \sqrt {a+b \sec (c+d x)}}+\frac {2 \sqrt {\sec (c+d x)} \left (8 a^3 (A+3 C)+48 a^2 b B+a b^2 (16 A+33 C)+12 b^3 B\right ) \sqrt {\frac {a \cos (c+d x)+b}{a+b}} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),\frac {2 a}{a+b}\right )}{d \sqrt {a+b \sec (c+d x)}}\right )-\frac {b \sin (c+d x) \sqrt {\sec (c+d x)} (8 a A-21 a C-12 b B) \sqrt {a+b \sec (c+d x)}}{d}\right )-\frac {b (4 A-3 C) \sin (c+d x) \sqrt {\sec (c+d x)} (a+b \sec (c+d x))^{3/2}}{2 d}\right )+\frac {2 A \sin (c+d x) (a+b \sec (c+d x))^{5/2}}{3 d \sqrt {\sec (c+d x)}}\) |
Input:
Int[((a + b*Sec[c + d*x])^(5/2)*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/S ec[c + d*x]^(3/2),x]
Output:
(2*A*(a + b*Sec[c + d*x])^(5/2)*Sin[c + d*x])/(3*d*Sqrt[Sec[c + d*x]]) + ( -1/2*(b*(4*A - 3*C)*Sqrt[Sec[c + d*x]]*(a + b*Sec[c + d*x])^(3/2)*Sin[c + d*x])/d + (((2*(48*a^2*b*B + 12*b^3*B + 8*a^3*(A + 3*C) + a*b^2*(16*A + 33 *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*b*(8*A*b^2 + 20* a*b*B + 15*a^2*C + 4*b^2*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*(24*a^2*B - 12*b^2*B + a*b*(56*A - 27*C))*EllipticE[(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 - (b*(8*a*A - 12*b*B - 21*a*C)*Sqrt[Sec[c + d*x]]*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/d)/4)/3
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[A*Cot[e + f*x]*(a + b*Csc[e + f*x])^m*((d*Csc[e + f*x])^n/(f*n)), x] - Simp[1/(d*n) Int[(a + b*Csc[e + f*x])^(m - 1)*(d* Csc[e + f*x])^(n + 1)*Simp[A*b*m - a*B*n - (b*B*n + a*(C*n + A*(n + 1)))*Cs c[e + f*x] - b*(C*n + A*(m + n + 1))*Csc[e + f*x]^2, x], x], x] /; FreeQ[{a , b, d, e, f, A, B, C}, 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_. ))*(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 = 48.77 (sec) , antiderivative size = 2911, normalized size of antiderivative = 6.82
| method | result | size |
| default | \(\text {Expression too large to display}\) | \(2911\) |
| parts | \(\text {Expression too large to display}\) | \(3103\) |
Input:
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)^(3/2 ),x,method=_RETURNVERBOSE)
Output:
1/12/d/((a-b)/(a+b))^(1/2)*(a+b*sec(d*x+c))^(1/2)/(cos(d*x+c)^2*a+a*cos(d* x+c)+b*cos(d*x+c)+b)/sec(d*x+c)^(3/2)*(A*(1/(a+b)*(b+a*cos(d*x+c))/(cos(d* x+c)+1))^(1/2)*(1/(cos(d*x+c)+1))^(1/2)*a^3*EllipticF(((a-b)/(a+b))^(1/2)* (-csc(d*x+c)+cot(d*x+c)),(-(a+b)/(a-b))^(1/2))*(-8*cos(d*x+c)-16-8*sec(d*x +c))+A*(1/(a+b)*(b+a*cos(d*x+c))/(cos(d*x+c)+1))^(1/2)*(1/(cos(d*x+c)+1))^ (1/2)*b^3*EllipticF(((a-b)/(a+b))^(1/2)*(-csc(d*x+c)+cot(d*x+c)),(-(a+b)/( a-b))^(1/2))*(24*cos(d*x+c)+48+24*sec(d*x+c))+B*(1/(a+b)*(b+a*cos(d*x+c))/ (cos(d*x+c)+1))^(1/2)*(1/(cos(d*x+c)+1))^(1/2)*a^3*EllipticF(((a-b)/(a+b)) ^(1/2)*(-csc(d*x+c)+cot(d*x+c)),(-(a+b)/(a-b))^(1/2))*(24*cos(d*x+c)+48+24 *sec(d*x+c))+C*(1/(a+b)*(b+a*cos(d*x+c))/(cos(d*x+c)+1))^(1/2)*(1/(cos(d*x +c)+1))^(1/2)*a^3*EllipticF(((a-b)/(a+b))^(1/2)*(-csc(d*x+c)+cot(d*x+c)),( -(a+b)/(a-b))^(1/2))*(-24*cos(d*x+c)-48-24*sec(d*x+c))+C*(1/(a+b)*(b+a*cos (d*x+c))/(cos(d*x+c)+1))^(1/2)*(1/(cos(d*x+c)+1))^(1/2)*b^3*EllipticF(((a- b)/(a+b))^(1/2)*(-csc(d*x+c)+cot(d*x+c)),(-(a+b)/(a-b))^(1/2))*(12*cos(d*x +c)+24+12*sec(d*x+c))+A*(1/(a+b)*(b+a*cos(d*x+c))/(cos(d*x+c)+1))^(1/2)*(1 /(cos(d*x+c)+1))^(1/2)*b^3*EllipticPi(((a-b)/(a+b))^(1/2)*(-csc(d*x+c)+cot (d*x+c)),(a+b)/(a-b),I/((a-b)/(a+b))^(1/2))*(-48*cos(d*x+c)-96-48*sec(d*x+ c))+C*(1/(a+b)*(b+a*cos(d*x+c))/(cos(d*x+c)+1))^(1/2)*(1/(cos(d*x+c)+1))^( 1/2)*b^3*EllipticPi(((a-b)/(a+b))^(1/2)*(-csc(d*x+c)+cot(d*x+c)),(a+b)/(a- b),I/((a-b)/(a+b))^(1/2))*(-24*cos(d*x+c)-48-24*sec(d*x+c))+B*(1/(a+b)*...
Timed out. \[ \int \frac {(a+b \sec (c+d x))^{5/2} \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right )}{\sec ^{\frac {3}{2}}(c+d x)} \, dx=\text {Timed out} \] Input:
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 )^(3/2),x, algorithm="fricas")
Output:
Timed out
Timed out. \[ \int \frac {(a+b \sec (c+d x))^{5/2} \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right )}{\sec ^{\frac {3}{2}}(c+d x)} \, dx=\text {Timed out} \] Input:
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)**(3/2),x)
Output:
Timed out
\[ \int \frac {(a+b \sec (c+d x))^{5/2} \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right )}{\sec ^{\frac {3}{2}}(c+d x)} \, dx=\int { \frac {{\left (C \sec \left (d x + c\right )^{2} + B \sec \left (d x + c\right ) + A\right )} {\left (b \sec \left (d x + c\right ) + a\right )}^{\frac {5}{2}}}{\sec \left (d x + c\right )^{\frac {3}{2}}} \,d x } \] Input:
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 )^(3/2),x, algorithm="maxima")
Output:
integrate((C*sec(d*x + c)^2 + B*sec(d*x + c) + A)*(b*sec(d*x + c) + a)^(5/ 2)/sec(d*x + c)^(3/2), x)
\[ \int \frac {(a+b \sec (c+d x))^{5/2} \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right )}{\sec ^{\frac {3}{2}}(c+d x)} \, dx=\int { \frac {{\left (C \sec \left (d x + c\right )^{2} + B \sec \left (d x + c\right ) + A\right )} {\left (b \sec \left (d x + c\right ) + a\right )}^{\frac {5}{2}}}{\sec \left (d x + c\right )^{\frac {3}{2}}} \,d x } \] Input:
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 )^(3/2),x, algorithm="giac")
Output:
integrate((C*sec(d*x + c)^2 + B*sec(d*x + c) + A)*(b*sec(d*x + c) + a)^(5/ 2)/sec(d*x + c)^(3/2), x)
Timed out. \[ \int \frac {(a+b \sec (c+d x))^{5/2} \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right )}{\sec ^{\frac {3}{2}}(c+d x)} \, dx=\int \frac {{\left (a+\frac {b}{\cos \left (c+d\,x\right )}\right )}^{5/2}\,\left (A+\frac {B}{\cos \left (c+d\,x\right )}+\frac {C}{{\cos \left (c+d\,x\right )}^2}\right )}{{\left (\frac {1}{\cos \left (c+d\,x\right )}\right )}^{3/2}} \,d x \] Input:
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))^(3/2),x)
Output:
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))^(3/2), x)
\[ \int \frac {(a+b \sec (c+d x))^{5/2} \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right )}{\sec ^{\frac {3}{2}}(c+d x)} \, dx=\left (\int \frac {\sqrt {\sec \left (d x +c \right )}\, \sqrt {\sec \left (d x +c \right ) b +a}}{\sec \left (d x +c \right )^{2}}d x \right ) a^{3}+3 \left (\int \frac {\sqrt {\sec \left (d x +c \right )}\, \sqrt {\sec \left (d x +c \right ) b +a}}{\sec \left (d x +c \right )}d x \right ) a^{2} b +\left (\int \sqrt {\sec \left (d x +c \right )}\, \sqrt {\sec \left (d x +c \right ) b +a}\, \sec \left (d x +c \right )^{2}d x \right ) b^{2} c +2 \left (\int \sqrt {\sec \left (d x +c \right )}\, \sqrt {\sec \left (d x +c \right ) b +a}\, \sec \left (d x +c \right )d x \right ) a b c +\left (\int \sqrt {\sec \left (d x +c \right )}\, \sqrt {\sec \left (d x +c \right ) b +a}\, \sec \left (d x +c \right )d x \right ) b^{3}+\left (\int \sqrt {\sec \left (d x +c \right )}\, \sqrt {\sec \left (d x +c \right ) b +a}d x \right ) a^{2} c +3 \left (\int \sqrt {\sec \left (d x +c \right )}\, \sqrt {\sec \left (d x +c \right ) b +a}d x \right ) a \,b^{2} \] Input:
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)^(3/2 ),x)
Output:
int((sqrt(sec(c + d*x))*sqrt(sec(c + d*x)*b + a))/sec(c + d*x)**2,x)*a**3 + 3*int((sqrt(sec(c + d*x))*sqrt(sec(c + d*x)*b + a))/sec(c + d*x),x)*a**2 *b + int(sqrt(sec(c + d*x))*sqrt(sec(c + d*x)*b + a)*sec(c + d*x)**2,x)*b* *2*c + 2*int(sqrt(sec(c + d*x))*sqrt(sec(c + d*x)*b + a)*sec(c + d*x),x)*a *b*c + int(sqrt(sec(c + d*x))*sqrt(sec(c + d*x)*b + a)*sec(c + d*x),x)*b** 3 + int(sqrt(sec(c + d*x))*sqrt(sec(c + d*x)*b + a),x)*a**2*c + 3*int(sqrt (sec(c + d*x))*sqrt(sec(c + d*x)*b + a),x)*a*b**2