Integrand size = 43, antiderivative size = 334 \[ \int \frac {(a+b \sec (c+d x))^3 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right )}{\sqrt {\sec (c+d x)}} \, dx=-\frac {2 \left (15 a^2 b B+3 b^3 B-5 a^3 (A-C)+3 a b^2 (5 A+3 C)\right ) \sqrt {\cos (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \sqrt {\sec (c+d x)}}{5 d}+\frac {2 \left (21 a^3 B+21 a b^2 B+21 a^2 b (3 A+C)+b^3 (7 A+5 C)\right ) \sqrt {\cos (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right ) \sqrt {\sec (c+d x)}}{21 d}+\frac {2 \left (98 a^2 b B+21 b^3 B+24 a^3 C+21 a b^2 (5 A+3 C)\right ) \sqrt {\sec (c+d x)} \sin (c+d x)}{35 d}+\frac {2 b \left (35 A b^2+63 a b B+24 a^2 C+25 b^2 C\right ) \sec ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{105 d}+\frac {2 (7 b B+6 a C) \sqrt {\sec (c+d x)} (a+b \sec (c+d x))^2 \sin (c+d x)}{35 d}+\frac {2 C \sqrt {\sec (c+d x)} (a+b \sec (c+d x))^3 \sin (c+d x)}{7 d} \]
2/105*b*(35*A*b^2+63*B*a*b+24*C*a^2+25*C*b^2)*sec(d*x+c)^(3/2)*sin(d*x+c)/ d+2/35*(98*B*a^2*b+21*B*b^3+24*a^3*C+21*a*b^2*(5*A+3*C))*sin(d*x+c)*sec(d* x+c)^(1/2)/d+2/35*(7*B*b+6*C*a)*(a+b*sec(d*x+c))^2*sin(d*x+c)*sec(d*x+c)^( 1/2)/d+2/7*C*(a+b*sec(d*x+c))^3*sin(d*x+c)*sec(d*x+c)^(1/2)/d-2/5*(15*B*a^ 2*b+3*B*b^3-5*a^3*(A-C)+3*a*b^2*(5*A+3*C))*(cos(1/2*d*x+1/2*c)^2)^(1/2)/co s(1/2*d*x+1/2*c)*EllipticE(sin(1/2*d*x+1/2*c),2^(1/2))*cos(d*x+c)^(1/2)*se c(d*x+c)^(1/2)/d+2/21*(21*B*a^3+21*B*a*b^2+21*a^2*b*(3*A+C)+b^3*(7*A+5*C)) *(cos(1/2*d*x+1/2*c)^2)^(1/2)/cos(1/2*d*x+1/2*c)*EllipticF(sin(1/2*d*x+1/2 *c),2^(1/2))*cos(d*x+c)^(1/2)*sec(d*x+c)^(1/2)/d
Time = 7.98 (sec) , antiderivative size = 377, normalized size of antiderivative = 1.13 \[ \int \frac {(a+b \sec (c+d x))^3 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right )}{\sqrt {\sec (c+d x)}} \, dx=\frac {4 (a+b \sec (c+d x))^3 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \left (21 \left (-15 a^2 b B-3 b^3 B+5 a^3 (A-C)-3 a b^2 (5 A+3 C)\right ) \sqrt {\cos (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right )+5 \left (21 a^3 B+21 a b^2 B+21 a^2 b (3 A+C)+b^3 (7 A+5 C)\right ) \sqrt {\cos (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )+315 a A b^2 \sin (c+d x)+315 a^2 b B \sin (c+d x)+63 b^3 B \sin (c+d x)+105 a^3 C \sin (c+d x)+189 a b^2 C \sin (c+d x)+35 A b^3 \tan (c+d x)+105 a b^2 B \tan (c+d x)+105 a^2 b C \tan (c+d x)+25 b^3 C \tan (c+d x)+21 b^3 B \sec (c+d x) \tan (c+d x)+63 a b^2 C \sec (c+d x) \tan (c+d x)+15 b^3 C \sec ^2(c+d x) \tan (c+d x)\right )}{105 d (b+a \cos (c+d x))^3 (A+2 C+2 B \cos (c+d x)+A \cos (2 (c+d x))) \sec ^{\frac {9}{2}}(c+d x)} \]
Integrate[((a + b*Sec[c + d*x])^3*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)) /Sqrt[Sec[c + d*x]],x]
(4*(a + b*Sec[c + d*x])^3*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*(21*(-15 *a^2*b*B - 3*b^3*B + 5*a^3*(A - C) - 3*a*b^2*(5*A + 3*C))*Sqrt[Cos[c + d*x ]]*EllipticE[(c + d*x)/2, 2] + 5*(21*a^3*B + 21*a*b^2*B + 21*a^2*b*(3*A + C) + b^3*(7*A + 5*C))*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2] + 315*a *A*b^2*Sin[c + d*x] + 315*a^2*b*B*Sin[c + d*x] + 63*b^3*B*Sin[c + d*x] + 1 05*a^3*C*Sin[c + d*x] + 189*a*b^2*C*Sin[c + d*x] + 35*A*b^3*Tan[c + d*x] + 105*a*b^2*B*Tan[c + d*x] + 105*a^2*b*C*Tan[c + d*x] + 25*b^3*C*Tan[c + d* x] + 21*b^3*B*Sec[c + d*x]*Tan[c + d*x] + 63*a*b^2*C*Sec[c + d*x]*Tan[c + d*x] + 15*b^3*C*Sec[c + d*x]^2*Tan[c + d*x]))/(105*d*(b + a*Cos[c + d*x])^ 3*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*(c + d*x)])*Sec[c + d*x]^(9/2))
Time = 2.22 (sec) , antiderivative size = 343, normalized size of antiderivative = 1.03, number of steps used = 20, number of rules used = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.465, Rules used = {3042, 4584, 27, 3042, 4584, 27, 3042, 4564, 27, 3042, 4535, 3042, 4258, 3042, 3120, 4534, 3042, 4258, 3042, 3119}
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.
\(\displaystyle \int \frac {(a+b \sec (c+d x))^3 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right )}{\sqrt {\sec (c+d x)}} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {\left (a+b \csc \left (c+d x+\frac {\pi }{2}\right )\right )^3 \left (A+B \csc \left (c+d x+\frac {\pi }{2}\right )+C \csc \left (c+d x+\frac {\pi }{2}\right )^2\right )}{\sqrt {\csc \left (c+d x+\frac {\pi }{2}\right )}}dx\) |
\(\Big \downarrow \) 4584 |
\(\displaystyle \frac {2}{7} \int \frac {(a+b \sec (c+d x))^2 \left ((7 b B+6 a C) \sec ^2(c+d x)+(7 A b+5 C b+7 a B) \sec (c+d x)+a (7 A-C)\right )}{2 \sqrt {\sec (c+d x)}}dx+\frac {2 C \sin (c+d x) \sqrt {\sec (c+d x)} (a+b \sec (c+d x))^3}{7 d}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {1}{7} \int \frac {(a+b \sec (c+d x))^2 \left ((7 b B+6 a C) \sec ^2(c+d x)+(7 A b+5 C b+7 a B) \sec (c+d x)+a (7 A-C)\right )}{\sqrt {\sec (c+d x)}}dx+\frac {2 C \sin (c+d x) \sqrt {\sec (c+d x)} (a+b \sec (c+d x))^3}{7 d}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {1}{7} \int \frac {\left (a+b \csc \left (c+d x+\frac {\pi }{2}\right )\right )^2 \left ((7 b B+6 a C) \csc \left (c+d x+\frac {\pi }{2}\right )^2+(7 A b+5 C b+7 a B) \csc \left (c+d x+\frac {\pi }{2}\right )+a (7 A-C)\right )}{\sqrt {\csc \left (c+d x+\frac {\pi }{2}\right )}}dx+\frac {2 C \sin (c+d x) \sqrt {\sec (c+d x)} (a+b \sec (c+d x))^3}{7 d}\) |
\(\Big \downarrow \) 4584 |
\(\displaystyle \frac {1}{7} \left (\frac {2}{5} \int \frac {(a+b \sec (c+d x)) \left (\left (24 C a^2+63 b B a+35 A b^2+25 b^2 C\right ) \sec ^2(c+d x)+\left (35 B a^2+70 A b a+38 b C a+21 b^2 B\right ) \sec (c+d x)+a (35 a A-7 b B-11 a C)\right )}{2 \sqrt {\sec (c+d x)}}dx+\frac {2 (6 a C+7 b B) \sin (c+d x) \sqrt {\sec (c+d x)} (a+b \sec (c+d x))^2}{5 d}\right )+\frac {2 C \sin (c+d x) \sqrt {\sec (c+d x)} (a+b \sec (c+d x))^3}{7 d}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {1}{7} \left (\frac {1}{5} \int \frac {(a+b \sec (c+d x)) \left (\left (24 C a^2+63 b B a+35 A b^2+25 b^2 C\right ) \sec ^2(c+d x)+\left (35 B a^2+70 A b a+38 b C a+21 b^2 B\right ) \sec (c+d x)+a (35 a A-7 b B-11 a C)\right )}{\sqrt {\sec (c+d x)}}dx+\frac {2 (6 a C+7 b B) \sin (c+d x) \sqrt {\sec (c+d x)} (a+b \sec (c+d x))^2}{5 d}\right )+\frac {2 C \sin (c+d x) \sqrt {\sec (c+d x)} (a+b \sec (c+d x))^3}{7 d}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {1}{7} \left (\frac {1}{5} \int \frac {\left (a+b \csc \left (c+d x+\frac {\pi }{2}\right )\right ) \left (\left (24 C a^2+63 b B a+35 A b^2+25 b^2 C\right ) \csc \left (c+d x+\frac {\pi }{2}\right )^2+\left (35 B a^2+70 A b a+38 b C a+21 b^2 B\right ) \csc \left (c+d x+\frac {\pi }{2}\right )+a (35 a A-7 b B-11 a C)\right )}{\sqrt {\csc \left (c+d x+\frac {\pi }{2}\right )}}dx+\frac {2 (6 a C+7 b B) \sin (c+d x) \sqrt {\sec (c+d x)} (a+b \sec (c+d x))^2}{5 d}\right )+\frac {2 C \sin (c+d x) \sqrt {\sec (c+d x)} (a+b \sec (c+d x))^3}{7 d}\) |
\(\Big \downarrow \) 4564 |
\(\displaystyle \frac {1}{7} \left (\frac {1}{5} \left (\frac {2}{3} \int \frac {3 (35 a A-7 b B-11 a C) a^2+3 \left (24 C a^3+98 b B a^2+21 b^2 (5 A+3 C) a+21 b^3 B\right ) \sec ^2(c+d x)+5 \left (21 B a^3+21 b (3 A+C) a^2+21 b^2 B a+b^3 (7 A+5 C)\right ) \sec (c+d x)}{2 \sqrt {\sec (c+d x)}}dx+\frac {2 b \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x) \left (24 a^2 C+63 a b B+35 A b^2+25 b^2 C\right )}{3 d}\right )+\frac {2 (6 a C+7 b B) \sin (c+d x) \sqrt {\sec (c+d x)} (a+b \sec (c+d x))^2}{5 d}\right )+\frac {2 C \sin (c+d x) \sqrt {\sec (c+d x)} (a+b \sec (c+d x))^3}{7 d}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {1}{7} \left (\frac {1}{5} \left (\frac {1}{3} \int \frac {3 (35 a A-7 b B-11 a C) a^2+3 \left (24 C a^3+98 b B a^2+21 b^2 (5 A+3 C) a+21 b^3 B\right ) \sec ^2(c+d x)+5 \left (21 B a^3+21 b (3 A+C) a^2+21 b^2 B a+b^3 (7 A+5 C)\right ) \sec (c+d x)}{\sqrt {\sec (c+d x)}}dx+\frac {2 b \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x) \left (24 a^2 C+63 a b B+35 A b^2+25 b^2 C\right )}{3 d}\right )+\frac {2 (6 a C+7 b B) \sin (c+d x) \sqrt {\sec (c+d x)} (a+b \sec (c+d x))^2}{5 d}\right )+\frac {2 C \sin (c+d x) \sqrt {\sec (c+d x)} (a+b \sec (c+d x))^3}{7 d}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {1}{7} \left (\frac {1}{5} \left (\frac {1}{3} \int \frac {3 (35 a A-7 b B-11 a C) a^2+3 \left (24 C a^3+98 b B a^2+21 b^2 (5 A+3 C) a+21 b^3 B\right ) \csc \left (c+d x+\frac {\pi }{2}\right )^2+5 \left (21 B a^3+21 b (3 A+C) a^2+21 b^2 B a+b^3 (7 A+5 C)\right ) \csc \left (c+d x+\frac {\pi }{2}\right )}{\sqrt {\csc \left (c+d x+\frac {\pi }{2}\right )}}dx+\frac {2 b \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x) \left (24 a^2 C+63 a b B+35 A b^2+25 b^2 C\right )}{3 d}\right )+\frac {2 (6 a C+7 b B) \sin (c+d x) \sqrt {\sec (c+d x)} (a+b \sec (c+d x))^2}{5 d}\right )+\frac {2 C \sin (c+d x) \sqrt {\sec (c+d x)} (a+b \sec (c+d x))^3}{7 d}\) |
\(\Big \downarrow \) 4535 |
\(\displaystyle \frac {1}{7} \left (\frac {1}{5} \left (\frac {1}{3} \left (\int \frac {3 (35 a A-7 b B-11 a C) a^2+3 \left (24 C a^3+98 b B a^2+21 b^2 (5 A+3 C) a+21 b^3 B\right ) \sec ^2(c+d x)}{\sqrt {\sec (c+d x)}}dx+5 \left (21 a^3 B+21 a^2 b (3 A+C)+21 a b^2 B+b^3 (7 A+5 C)\right ) \int \sqrt {\sec (c+d x)}dx\right )+\frac {2 b \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x) \left (24 a^2 C+63 a b B+35 A b^2+25 b^2 C\right )}{3 d}\right )+\frac {2 (6 a C+7 b B) \sin (c+d x) \sqrt {\sec (c+d x)} (a+b \sec (c+d x))^2}{5 d}\right )+\frac {2 C \sin (c+d x) \sqrt {\sec (c+d x)} (a+b \sec (c+d x))^3}{7 d}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {1}{7} \left (\frac {1}{5} \left (\frac {1}{3} \left (5 \left (21 a^3 B+21 a^2 b (3 A+C)+21 a b^2 B+b^3 (7 A+5 C)\right ) \int \sqrt {\csc \left (c+d x+\frac {\pi }{2}\right )}dx+\int \frac {3 (35 a A-7 b B-11 a C) a^2+3 \left (24 C a^3+98 b B a^2+21 b^2 (5 A+3 C) a+21 b^3 B\right ) \csc \left (c+d x+\frac {\pi }{2}\right )^2}{\sqrt {\csc \left (c+d x+\frac {\pi }{2}\right )}}dx\right )+\frac {2 b \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x) \left (24 a^2 C+63 a b B+35 A b^2+25 b^2 C\right )}{3 d}\right )+\frac {2 (6 a C+7 b B) \sin (c+d x) \sqrt {\sec (c+d x)} (a+b \sec (c+d x))^2}{5 d}\right )+\frac {2 C \sin (c+d x) \sqrt {\sec (c+d x)} (a+b \sec (c+d x))^3}{7 d}\) |
\(\Big \downarrow \) 4258 |
\(\displaystyle \frac {1}{7} \left (\frac {1}{5} \left (\frac {1}{3} \left (\int \frac {3 (35 a A-7 b B-11 a C) a^2+3 \left (24 C a^3+98 b B a^2+21 b^2 (5 A+3 C) a+21 b^3 B\right ) \csc \left (c+d x+\frac {\pi }{2}\right )^2}{\sqrt {\csc \left (c+d x+\frac {\pi }{2}\right )}}dx+5 \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (21 a^3 B+21 a^2 b (3 A+C)+21 a b^2 B+b^3 (7 A+5 C)\right ) \int \frac {1}{\sqrt {\cos (c+d x)}}dx\right )+\frac {2 b \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x) \left (24 a^2 C+63 a b B+35 A b^2+25 b^2 C\right )}{3 d}\right )+\frac {2 (6 a C+7 b B) \sin (c+d x) \sqrt {\sec (c+d x)} (a+b \sec (c+d x))^2}{5 d}\right )+\frac {2 C \sin (c+d x) \sqrt {\sec (c+d x)} (a+b \sec (c+d x))^3}{7 d}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {1}{7} \left (\frac {1}{5} \left (\frac {1}{3} \left (\int \frac {3 (35 a A-7 b B-11 a C) a^2+3 \left (24 C a^3+98 b B a^2+21 b^2 (5 A+3 C) a+21 b^3 B\right ) \csc \left (c+d x+\frac {\pi }{2}\right )^2}{\sqrt {\csc \left (c+d x+\frac {\pi }{2}\right )}}dx+5 \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (21 a^3 B+21 a^2 b (3 A+C)+21 a b^2 B+b^3 (7 A+5 C)\right ) \int \frac {1}{\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}}dx\right )+\frac {2 b \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x) \left (24 a^2 C+63 a b B+35 A b^2+25 b^2 C\right )}{3 d}\right )+\frac {2 (6 a C+7 b B) \sin (c+d x) \sqrt {\sec (c+d x)} (a+b \sec (c+d x))^2}{5 d}\right )+\frac {2 C \sin (c+d x) \sqrt {\sec (c+d x)} (a+b \sec (c+d x))^3}{7 d}\) |
\(\Big \downarrow \) 3120 |
\(\displaystyle \frac {1}{7} \left (\frac {1}{5} \left (\frac {1}{3} \left (\int \frac {3 (35 a A-7 b B-11 a C) a^2+3 \left (24 C a^3+98 b B a^2+21 b^2 (5 A+3 C) a+21 b^3 B\right ) \csc \left (c+d x+\frac {\pi }{2}\right )^2}{\sqrt {\csc \left (c+d x+\frac {\pi }{2}\right )}}dx+\frac {10 \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right ) \left (21 a^3 B+21 a^2 b (3 A+C)+21 a b^2 B+b^3 (7 A+5 C)\right )}{d}\right )+\frac {2 b \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x) \left (24 a^2 C+63 a b B+35 A b^2+25 b^2 C\right )}{3 d}\right )+\frac {2 (6 a C+7 b B) \sin (c+d x) \sqrt {\sec (c+d x)} (a+b \sec (c+d x))^2}{5 d}\right )+\frac {2 C \sin (c+d x) \sqrt {\sec (c+d x)} (a+b \sec (c+d x))^3}{7 d}\) |
\(\Big \downarrow \) 4534 |
\(\displaystyle \frac {1}{7} \left (\frac {1}{5} \left (\frac {1}{3} \left (-21 \left (-5 a^3 (A-C)+15 a^2 b B+3 a b^2 (5 A+3 C)+3 b^3 B\right ) \int \frac {1}{\sqrt {\sec (c+d x)}}dx+\frac {6 \sin (c+d x) \sqrt {\sec (c+d x)} \left (24 a^3 C+98 a^2 b B+21 a b^2 (5 A+3 C)+21 b^3 B\right )}{d}+\frac {10 \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right ) \left (21 a^3 B+21 a^2 b (3 A+C)+21 a b^2 B+b^3 (7 A+5 C)\right )}{d}\right )+\frac {2 b \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x) \left (24 a^2 C+63 a b B+35 A b^2+25 b^2 C\right )}{3 d}\right )+\frac {2 (6 a C+7 b B) \sin (c+d x) \sqrt {\sec (c+d x)} (a+b \sec (c+d x))^2}{5 d}\right )+\frac {2 C \sin (c+d x) \sqrt {\sec (c+d x)} (a+b \sec (c+d x))^3}{7 d}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {1}{7} \left (\frac {1}{5} \left (\frac {1}{3} \left (-21 \left (-5 a^3 (A-C)+15 a^2 b B+3 a b^2 (5 A+3 C)+3 b^3 B\right ) \int \frac {1}{\sqrt {\csc \left (c+d x+\frac {\pi }{2}\right )}}dx+\frac {6 \sin (c+d x) \sqrt {\sec (c+d x)} \left (24 a^3 C+98 a^2 b B+21 a b^2 (5 A+3 C)+21 b^3 B\right )}{d}+\frac {10 \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right ) \left (21 a^3 B+21 a^2 b (3 A+C)+21 a b^2 B+b^3 (7 A+5 C)\right )}{d}\right )+\frac {2 b \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x) \left (24 a^2 C+63 a b B+35 A b^2+25 b^2 C\right )}{3 d}\right )+\frac {2 (6 a C+7 b B) \sin (c+d x) \sqrt {\sec (c+d x)} (a+b \sec (c+d x))^2}{5 d}\right )+\frac {2 C \sin (c+d x) \sqrt {\sec (c+d x)} (a+b \sec (c+d x))^3}{7 d}\) |
\(\Big \downarrow \) 4258 |
\(\displaystyle \frac {1}{7} \left (\frac {1}{5} \left (\frac {1}{3} \left (-21 \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (-5 a^3 (A-C)+15 a^2 b B+3 a b^2 (5 A+3 C)+3 b^3 B\right ) \int \sqrt {\cos (c+d x)}dx+\frac {6 \sin (c+d x) \sqrt {\sec (c+d x)} \left (24 a^3 C+98 a^2 b B+21 a b^2 (5 A+3 C)+21 b^3 B\right )}{d}+\frac {10 \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right ) \left (21 a^3 B+21 a^2 b (3 A+C)+21 a b^2 B+b^3 (7 A+5 C)\right )}{d}\right )+\frac {2 b \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x) \left (24 a^2 C+63 a b B+35 A b^2+25 b^2 C\right )}{3 d}\right )+\frac {2 (6 a C+7 b B) \sin (c+d x) \sqrt {\sec (c+d x)} (a+b \sec (c+d x))^2}{5 d}\right )+\frac {2 C \sin (c+d x) \sqrt {\sec (c+d x)} (a+b \sec (c+d x))^3}{7 d}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {1}{7} \left (\frac {1}{5} \left (\frac {1}{3} \left (-21 \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (-5 a^3 (A-C)+15 a^2 b B+3 a b^2 (5 A+3 C)+3 b^3 B\right ) \int \sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}dx+\frac {6 \sin (c+d x) \sqrt {\sec (c+d x)} \left (24 a^3 C+98 a^2 b B+21 a b^2 (5 A+3 C)+21 b^3 B\right )}{d}+\frac {10 \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right ) \left (21 a^3 B+21 a^2 b (3 A+C)+21 a b^2 B+b^3 (7 A+5 C)\right )}{d}\right )+\frac {2 b \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x) \left (24 a^2 C+63 a b B+35 A b^2+25 b^2 C\right )}{3 d}\right )+\frac {2 (6 a C+7 b B) \sin (c+d x) \sqrt {\sec (c+d x)} (a+b \sec (c+d x))^2}{5 d}\right )+\frac {2 C \sin (c+d x) \sqrt {\sec (c+d x)} (a+b \sec (c+d x))^3}{7 d}\) |
\(\Big \downarrow \) 3119 |
\(\displaystyle \frac {1}{7} \left (\frac {1}{5} \left (\frac {2 b \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x) \left (24 a^2 C+63 a b B+35 A b^2+25 b^2 C\right )}{3 d}+\frac {1}{3} \left (\frac {6 \sin (c+d x) \sqrt {\sec (c+d x)} \left (24 a^3 C+98 a^2 b B+21 a b^2 (5 A+3 C)+21 b^3 B\right )}{d}+\frac {10 \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right ) \left (21 a^3 B+21 a^2 b (3 A+C)+21 a b^2 B+b^3 (7 A+5 C)\right )}{d}-\frac {42 \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \left (-5 a^3 (A-C)+15 a^2 b B+3 a b^2 (5 A+3 C)+3 b^3 B\right )}{d}\right )\right )+\frac {2 (6 a C+7 b B) \sin (c+d x) \sqrt {\sec (c+d x)} (a+b \sec (c+d x))^2}{5 d}\right )+\frac {2 C \sin (c+d x) \sqrt {\sec (c+d x)} (a+b \sec (c+d x))^3}{7 d}\) |
(2*C*Sqrt[Sec[c + d*x]]*(a + b*Sec[c + d*x])^3*Sin[c + d*x])/(7*d) + ((2*( 7*b*B + 6*a*C)*Sqrt[Sec[c + d*x]]*(a + b*Sec[c + d*x])^2*Sin[c + d*x])/(5* d) + ((2*b*(35*A*b^2 + 63*a*b*B + 24*a^2*C + 25*b^2*C)*Sec[c + d*x]^(3/2)* Sin[c + d*x])/(3*d) + ((-42*(15*a^2*b*B + 3*b^3*B - 5*a^3*(A - C) + 3*a*b^ 2*(5*A + 3*C))*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d *x]])/d + (10*(21*a^3*B + 21*a*b^2*B + 21*a^2*b*(3*A + C) + b^3*(7*A + 5*C ))*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/d + (6 *(98*a^2*b*B + 21*b^3*B + 24*a^3*C + 21*a*b^2*(5*A + 3*C))*Sqrt[Sec[c + d* x]]*Sin[c + d*x])/d)/3)/5)/7
3.10.98.3.1 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[sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[(2/d)*EllipticE[(1/2)* (c - Pi/2 + d*x), 2], x] /; FreeQ[{c, d}, x]
Int[1/Sqrt[sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[(2/d)*EllipticF[(1/2 )*(c - Pi/2 + d*x), 2], x] /; FreeQ[{c, d}, x]
Int[(csc[(c_.) + (d_.)*(x_)]*(b_.))^(n_), x_Symbol] :> Simp[(b*Csc[c + d*x] )^n*Sin[c + d*x]^n Int[1/Sin[c + d*x]^n, x], x] /; FreeQ[{b, c, d}, x] && EqQ[n^2, 1/4]
Int[(csc[(e_.) + (f_.)*(x_)]*(b_.))^(m_.)*(csc[(e_.) + (f_.)*(x_)]^2*(C_.) + (A_)), x_Symbol] :> Simp[(-C)*Cot[e + f*x]*((b*Csc[e + f*x])^m/(f*(m + 1) )), x] + Simp[(C*m + A*(m + 1))/(m + 1) Int[(b*Csc[e + f*x])^m, x], x] /; FreeQ[{b, e, f, A, C, m}, x] && NeQ[C*m + A*(m + 1), 0] && !LeQ[m, -1]
Int[(csc[(e_.) + (f_.)*(x_)]*(b_.))^(m_.)*((A_.) + csc[(e_.) + (f_.)*(x_)]* (B_.) + csc[(e_.) + (f_.)*(x_)]^2*(C_.)), x_Symbol] :> Simp[B/b Int[(b*Cs c[e + f*x])^(m + 1), x], x] + Int[(b*Csc[e + f*x])^m*(A + C*Csc[e + f*x]^2) , x] /; FreeQ[{b, e, f, A, B, C, m}, x]
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_)), x_Symbol] :> Simp[(-b)*C*Csc[e + f*x]*Cot[e + f*x]*((d*Csc[e + f*x])^ n/(f*(n + 2))), x] + Simp[1/(n + 2) Int[(d*Csc[e + f*x])^n*Simp[A*a*(n + 2) + (B*a*(n + 2) + b*(C*(n + 1) + A*(n + 2)))*Csc[e + f*x] + (a*C + B*b)*( n + 2)*Csc[e + f*x]^2, x], x], x] /; FreeQ[{a, b, d, e, f, A, B, C, n}, x] && !LtQ[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]
Leaf count of result is larger than twice the leaf count of optimal. \(1177\) vs. \(2(358)=716\).
Time = 7.37 (sec) , antiderivative size = 1178, normalized size of antiderivative = 3.53
method | result | size |
default | \(\text {Expression too large to display}\) | \(1178\) |
parts | \(\text {Expression too large to display}\) | \(1332\) |
int((a+b*sec(d*x+c))^3*(A+B*sec(d*x+c)+C*sec(d*x+c)^2)/sec(d*x+c)^(1/2),x, method=_RETURNVERBOSE)
-(-(-2*cos(1/2*d*x+1/2*c)^2+1)*sin(1/2*d*x+1/2*c)^2)^(1/2)*(2*B*a^3*(sin(1 /2*d*x+1/2*c)^2)^(1/2)*(-2*cos(1/2*d*x+1/2*c)^2+1)^(1/2)/(-2*sin(1/2*d*x+1 /2*c)^4+sin(1/2*d*x+1/2*c)^2)^(1/2)*EllipticF(cos(1/2*d*x+1/2*c),2^(1/2))- 2*A*a^3*(sin(1/2*d*x+1/2*c)^2)^(1/2)*(-2*cos(1/2*d*x+1/2*c)^2+1)^(1/2)/(-2 *sin(1/2*d*x+1/2*c)^4+sin(1/2*d*x+1/2*c)^2)^(1/2)*EllipticF(cos(1/2*d*x+1/ 2*c),2^(1/2))+6*A*a^2*b*(sin(1/2*d*x+1/2*c)^2)^(1/2)*(-2*cos(1/2*d*x+1/2*c )^2+1)^(1/2)/(-2*sin(1/2*d*x+1/2*c)^4+sin(1/2*d*x+1/2*c)^2)^(1/2)*Elliptic F(cos(1/2*d*x+1/2*c),2^(1/2))+2*A*a^3*(sin(1/2*d*x+1/2*c)^2)^(1/2)*(-2*cos (1/2*d*x+1/2*c)^2+1)^(1/2)/(-2*sin(1/2*d*x+1/2*c)^4+sin(1/2*d*x+1/2*c)^2)^ (1/2)*(EllipticF(cos(1/2*d*x+1/2*c),2^(1/2))-EllipticE(cos(1/2*d*x+1/2*c), 2^(1/2)))+2*C*b^3*(-1/56*cos(1/2*d*x+1/2*c)*(-2*sin(1/2*d*x+1/2*c)^4+sin(1 /2*d*x+1/2*c)^2)^(1/2)/(cos(1/2*d*x+1/2*c)^2-1/2)^4-5/42*cos(1/2*d*x+1/2*c )*(-2*sin(1/2*d*x+1/2*c)^4+sin(1/2*d*x+1/2*c)^2)^(1/2)/(cos(1/2*d*x+1/2*c) ^2-1/2)^2+5/21*(sin(1/2*d*x+1/2*c)^2)^(1/2)*(-2*cos(1/2*d*x+1/2*c)^2+1)^(1 /2)/(-2*sin(1/2*d*x+1/2*c)^4+sin(1/2*d*x+1/2*c)^2)^(1/2)*EllipticF(cos(1/2 *d*x+1/2*c),2^(1/2)))+2/5*b^2*(B*b+3*C*a)/(8*sin(1/2*d*x+1/2*c)^6-12*sin(1 /2*d*x+1/2*c)^4+6*sin(1/2*d*x+1/2*c)^2-1)/sin(1/2*d*x+1/2*c)^2*(24*sin(1/2 *d*x+1/2*c)^6*cos(1/2*d*x+1/2*c)-12*EllipticE(cos(1/2*d*x+1/2*c),2^(1/2))* (2*sin(1/2*d*x+1/2*c)^2-1)^(1/2)*(sin(1/2*d*x+1/2*c)^2)^(1/2)*sin(1/2*d*x+ 1/2*c)^4-24*cos(1/2*d*x+1/2*c)*sin(1/2*d*x+1/2*c)^4+12*EllipticE(cos(1/...
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.15 (sec) , antiderivative size = 415, normalized size of antiderivative = 1.24 \[ \int \frac {(a+b \sec (c+d x))^3 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right )}{\sqrt {\sec (c+d x)}} \, dx=-\frac {5 \, \sqrt {2} {\left (21 i \, B a^{3} + 21 i \, {\left (3 \, A + C\right )} a^{2} b + 21 i \, B a b^{2} + i \, {\left (7 \, A + 5 \, C\right )} b^{3}\right )} \cos \left (d x + c\right )^{3} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right ) + 5 \, \sqrt {2} {\left (-21 i \, B a^{3} - 21 i \, {\left (3 \, A + C\right )} a^{2} b - 21 i \, B a b^{2} - i \, {\left (7 \, A + 5 \, C\right )} b^{3}\right )} \cos \left (d x + c\right )^{3} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right ) + 21 \, \sqrt {2} {\left (-5 i \, {\left (A - C\right )} a^{3} + 15 i \, B a^{2} b + 3 i \, {\left (5 \, A + 3 \, C\right )} a b^{2} + 3 i \, B b^{3}\right )} \cos \left (d x + c\right )^{3} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right )\right ) + 21 \, \sqrt {2} {\left (5 i \, {\left (A - C\right )} a^{3} - 15 i \, B a^{2} b - 3 i \, {\left (5 \, A + 3 \, C\right )} a b^{2} - 3 i \, B b^{3}\right )} \cos \left (d x + c\right )^{3} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right )\right ) - \frac {2 \, {\left (15 \, C b^{3} + 21 \, {\left (5 \, C a^{3} + 15 \, B a^{2} b + 3 \, {\left (5 \, A + 3 \, C\right )} a b^{2} + 3 \, B b^{3}\right )} \cos \left (d x + c\right )^{3} + 5 \, {\left (21 \, C a^{2} b + 21 \, B a b^{2} + {\left (7 \, A + 5 \, C\right )} b^{3}\right )} \cos \left (d x + c\right )^{2} + 21 \, {\left (3 \, C a b^{2} + B b^{3}\right )} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{\sqrt {\cos \left (d x + c\right )}}}{105 \, d \cos \left (d x + c\right )^{3}} \]
integrate((a+b*sec(d*x+c))^3*(A+B*sec(d*x+c)+C*sec(d*x+c)^2)/sec(d*x+c)^(1 /2),x, algorithm="fricas")
-1/105*(5*sqrt(2)*(21*I*B*a^3 + 21*I*(3*A + C)*a^2*b + 21*I*B*a*b^2 + I*(7 *A + 5*C)*b^3)*cos(d*x + c)^3*weierstrassPInverse(-4, 0, cos(d*x + c) + I* sin(d*x + c)) + 5*sqrt(2)*(-21*I*B*a^3 - 21*I*(3*A + C)*a^2*b - 21*I*B*a*b ^2 - I*(7*A + 5*C)*b^3)*cos(d*x + c)^3*weierstrassPInverse(-4, 0, cos(d*x + c) - I*sin(d*x + c)) + 21*sqrt(2)*(-5*I*(A - C)*a^3 + 15*I*B*a^2*b + 3*I *(5*A + 3*C)*a*b^2 + 3*I*B*b^3)*cos(d*x + c)^3*weierstrassZeta(-4, 0, weie rstrassPInverse(-4, 0, cos(d*x + c) + I*sin(d*x + c))) + 21*sqrt(2)*(5*I*( A - C)*a^3 - 15*I*B*a^2*b - 3*I*(5*A + 3*C)*a*b^2 - 3*I*B*b^3)*cos(d*x + c )^3*weierstrassZeta(-4, 0, weierstrassPInverse(-4, 0, cos(d*x + c) - I*sin (d*x + c))) - 2*(15*C*b^3 + 21*(5*C*a^3 + 15*B*a^2*b + 3*(5*A + 3*C)*a*b^2 + 3*B*b^3)*cos(d*x + c)^3 + 5*(21*C*a^2*b + 21*B*a*b^2 + (7*A + 5*C)*b^3) *cos(d*x + c)^2 + 21*(3*C*a*b^2 + B*b^3)*cos(d*x + c))*sin(d*x + c)/sqrt(c os(d*x + c)))/(d*cos(d*x + c)^3)
Timed out. \[ \int \frac {(a+b \sec (c+d x))^3 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right )}{\sqrt {\sec (c+d x)}} \, dx=\text {Timed out} \]
\[ \int \frac {(a+b \sec (c+d x))^3 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right )}{\sqrt {\sec (c+d x)}} \, dx=\int { \frac {{\left (C \sec \left (d x + c\right )^{2} + B \sec \left (d x + c\right ) + A\right )} {\left (b \sec \left (d x + c\right ) + a\right )}^{3}}{\sqrt {\sec \left (d x + c\right )}} \,d x } \]
integrate((a+b*sec(d*x+c))^3*(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)^3/s qrt(sec(d*x + c)), x)
\[ \int \frac {(a+b \sec (c+d x))^3 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right )}{\sqrt {\sec (c+d x)}} \, dx=\int { \frac {{\left (C \sec \left (d x + c\right )^{2} + B \sec \left (d x + c\right ) + A\right )} {\left (b \sec \left (d x + c\right ) + a\right )}^{3}}{\sqrt {\sec \left (d x + c\right )}} \,d x } \]
integrate((a+b*sec(d*x+c))^3*(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)^3/s qrt(sec(d*x + c)), x)
Timed out. \[ \int \frac {(a+b \sec (c+d x))^3 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right )}{\sqrt {\sec (c+d x)}} \, dx=\int \frac {{\left (a+\frac {b}{\cos \left (c+d\,x\right )}\right )}^3\,\left (A+\frac {B}{\cos \left (c+d\,x\right )}+\frac {C}{{\cos \left (c+d\,x\right )}^2}\right )}{\sqrt {\frac {1}{\cos \left (c+d\,x\right )}}} \,d x \]
int(((a + b/cos(c + d*x))^3*(A + B/cos(c + d*x) + C/cos(c + d*x)^2))/(1/co s(c + d*x))^(1/2),x)