Integrand size = 33, antiderivative size = 342 \[ \int (a+b \cos (c+d x))^{5/2} \left (A+C \cos ^2(c+d x)\right ) \sec (c+d x) \, dx=\frac {2 a \left (49 A b^2+3 a^2 C+29 b^2 C\right ) \sqrt {a+b \cos (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )}{21 b d \sqrt {\frac {a+b \cos (c+d x)}{a+b}}}+\frac {2 \left (2 a^2 b^2 (7 A-C)-3 a^4 C+b^4 (7 A+5 C)\right ) \sqrt {\frac {a+b \cos (c+d x)}{a+b}} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),\frac {2 b}{a+b}\right )}{21 b d \sqrt {a+b \cos (c+d x)}}+\frac {2 a^3 A \sqrt {\frac {a+b \cos (c+d x)}{a+b}} \operatorname {EllipticPi}\left (2,\frac {1}{2} (c+d x),\frac {2 b}{a+b}\right )}{d \sqrt {a+b \cos (c+d x)}}+\frac {2 \left (3 a^2 C+b^2 (7 A+5 C)\right ) \sqrt {a+b \cos (c+d x)} \sin (c+d x)}{21 d}+\frac {2 a C (a+b \cos (c+d x))^{3/2} \sin (c+d x)}{7 d}+\frac {2 C (a+b \cos (c+d x))^{5/2} \sin (c+d x)}{7 d} \]
2/7*a*C*(a+b*cos(d*x+c))^(3/2)*sin(d*x+c)/d+2/7*C*(a+b*cos(d*x+c))^(5/2)*s in(d*x+c)/d+2/21*(3*a^2*C+b^2*(7*A+5*C))*sin(d*x+c)*(a+b*cos(d*x+c))^(1/2) /d+2/21*a*(49*A*b^2+3*C*a^2+29*C*b^2)*(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)*(b/(a+b))^(1/2))*(a+b*cos (d*x+c))^(1/2)/b/d/((a+b*cos(d*x+c))/(a+b))^(1/2)+2/21*(2*a^2*b^2*(7*A-C)- 3*a^4*C+b^4*(7*A+5*C))*(cos(1/2*d*x+1/2*c)^2)^(1/2)/cos(1/2*d*x+1/2*c)*Ell ipticF(sin(1/2*d*x+1/2*c),2^(1/2)*(b/(a+b))^(1/2))*((a+b*cos(d*x+c))/(a+b) )^(1/2)/b/d/(a+b*cos(d*x+c))^(1/2)+2*a^3*A*(cos(1/2*d*x+1/2*c)^2)^(1/2)/co s(1/2*d*x+1/2*c)*EllipticPi(sin(1/2*d*x+1/2*c),2,2^(1/2)*(b/(a+b))^(1/2))* ((a+b*cos(d*x+c))/(a+b))^(1/2)/d/(a+b*cos(d*x+c))^(1/2)
Result contains complex when optimal does not.
Time = 8.75 (sec) , antiderivative size = 468, normalized size of antiderivative = 1.37 \[ \int (a+b \cos (c+d x))^{5/2} \left (A+C \cos ^2(c+d x)\right ) \sec (c+d x) \, dx=\frac {\frac {4 b \left (9 a^2 (7 A+3 C)+b^2 (7 A+5 C)\right ) \sqrt {\frac {a+b \cos (c+d x)}{a+b}} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),\frac {2 b}{a+b}\right )}{\sqrt {a+b \cos (c+d x)}}+\frac {2 a \left (3 a^2 (14 A+C)+b^2 (49 A+29 C)\right ) \sqrt {\frac {a+b \cos (c+d x)}{a+b}} \operatorname {EllipticPi}\left (2,\frac {1}{2} (c+d x),\frac {2 b}{a+b}\right )}{\sqrt {a+b \cos (c+d x)}}+\frac {2 i \left (49 A b^2+3 a^2 C+29 b^2 C\right ) \sqrt {-\frac {b (-1+\cos (c+d x))}{a+b}} \sqrt {\frac {b (1+\cos (c+d x))}{-a+b}} \csc (c+d x) \left (-2 a (a-b) E\left (i \text {arcsinh}\left (\sqrt {-\frac {1}{a+b}} \sqrt {a+b \cos (c+d x)}\right )|\frac {a+b}{a-b}\right )+b \left (-2 a \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {-\frac {1}{a+b}} \sqrt {a+b \cos (c+d x)}\right ),\frac {a+b}{a-b}\right )+b \operatorname {EllipticPi}\left (\frac {a+b}{a},i \text {arcsinh}\left (\sqrt {-\frac {1}{a+b}} \sqrt {a+b \cos (c+d x)}\right ),\frac {a+b}{a-b}\right )\right )\right )}{b^2 \sqrt {-\frac {1}{a+b}}}+2 \sqrt {a+b \cos (c+d x)} \left (14 A b^2+18 a^2 C+13 b^2 C+18 a b C \cos (c+d x)+3 b^2 C \cos (2 (c+d x))\right ) \sin (c+d x)}{42 d} \]
((4*b*(9*a^2*(7*A + 3*C) + b^2*(7*A + 5*C))*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*b)/(a + b)])/Sqrt[a + b*Cos[c + d*x]] + (2* a*(3*a^2*(14*A + C) + b^2*(49*A + 29*C))*Sqrt[(a + b*Cos[c + d*x])/(a + b) ]*EllipticPi[2, (c + d*x)/2, (2*b)/(a + b)])/Sqrt[a + b*Cos[c + d*x]] + (( 2*I)*(49*A*b^2 + 3*a^2*C + 29*b^2*C)*Sqrt[-((b*(-1 + Cos[c + d*x]))/(a + b ))]*Sqrt[(b*(1 + Cos[c + d*x]))/(-a + b)]*Csc[c + d*x]*(-2*a*(a - b)*Ellip ticE[I*ArcSinh[Sqrt[-(a + b)^(-1)]*Sqrt[a + b*Cos[c + d*x]]], (a + b)/(a - b)] + b*(-2*a*EllipticF[I*ArcSinh[Sqrt[-(a + b)^(-1)]*Sqrt[a + b*Cos[c + d*x]]], (a + b)/(a - b)] + b*EllipticPi[(a + b)/a, I*ArcSinh[Sqrt[-(a + b) ^(-1)]*Sqrt[a + b*Cos[c + d*x]]], (a + b)/(a - b)])))/(b^2*Sqrt[-(a + b)^( -1)]) + 2*Sqrt[a + b*Cos[c + d*x]]*(14*A*b^2 + 18*a^2*C + 13*b^2*C + 18*a* b*C*Cos[c + d*x] + 3*b^2*C*Cos[2*(c + d*x)])*Sin[c + d*x])/(42*d)
Time = 2.95 (sec) , antiderivative size = 350, normalized size of antiderivative = 1.02, number of steps used = 24, number of rules used = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.727, Rules used = {3042, 3529, 27, 3042, 3528, 27, 3042, 3528, 27, 3042, 3538, 25, 3042, 3134, 3042, 3132, 3481, 3042, 3142, 3042, 3140, 3286, 3042, 3284}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
\(\displaystyle \int \sec (c+d x) (a+b \cos (c+d x))^{5/2} \left (A+C \cos ^2(c+d x)\right ) \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {\left (a+b \sin \left (c+d x+\frac {\pi }{2}\right )\right )^{5/2} \left (A+C \sin \left (c+d x+\frac {\pi }{2}\right )^2\right )}{\sin \left (c+d x+\frac {\pi }{2}\right )}dx\) |
\(\Big \downarrow \) 3529 |
\(\displaystyle \frac {2}{7} \int \frac {1}{2} (a+b \cos (c+d x))^{3/2} \left (5 a C \cos ^2(c+d x)+b (7 A+5 C) \cos (c+d x)+7 a A\right ) \sec (c+d x)dx+\frac {2 C \sin (c+d x) (a+b \cos (c+d x))^{5/2}}{7 d}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {1}{7} \int (a+b \cos (c+d x))^{3/2} \left (5 a C \cos ^2(c+d x)+b (7 A+5 C) \cos (c+d x)+7 a A\right ) \sec (c+d x)dx+\frac {2 C \sin (c+d x) (a+b \cos (c+d x))^{5/2}}{7 d}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {1}{7} \int \frac {\left (a+b \sin \left (c+d x+\frac {\pi }{2}\right )\right )^{3/2} \left (5 a C \sin \left (c+d x+\frac {\pi }{2}\right )^2+b (7 A+5 C) \sin \left (c+d x+\frac {\pi }{2}\right )+7 a A\right )}{\sin \left (c+d x+\frac {\pi }{2}\right )}dx+\frac {2 C \sin (c+d x) (a+b \cos (c+d x))^{5/2}}{7 d}\) |
\(\Big \downarrow \) 3528 |
\(\displaystyle \frac {1}{7} \left (\frac {2}{5} \int \frac {5}{2} \sqrt {a+b \cos (c+d x)} \left (7 A a^2+2 b (7 A+4 C) \cos (c+d x) a+\left (3 C a^2+b^2 (7 A+5 C)\right ) \cos ^2(c+d x)\right ) \sec (c+d x)dx+\frac {2 a C \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{d}\right )+\frac {2 C \sin (c+d x) (a+b \cos (c+d x))^{5/2}}{7 d}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {1}{7} \left (\int \sqrt {a+b \cos (c+d x)} \left (7 A a^2+2 b (7 A+4 C) \cos (c+d x) a+\left (3 C a^2+b^2 (7 A+5 C)\right ) \cos ^2(c+d x)\right ) \sec (c+d x)dx+\frac {2 a C \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{d}\right )+\frac {2 C \sin (c+d x) (a+b \cos (c+d x))^{5/2}}{7 d}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {1}{7} \left (\int \frac {\sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )} \left (7 A a^2+2 b (7 A+4 C) \sin \left (c+d x+\frac {\pi }{2}\right ) a+\left (3 C a^2+b^2 (7 A+5 C)\right ) \sin \left (c+d x+\frac {\pi }{2}\right )^2\right )}{\sin \left (c+d x+\frac {\pi }{2}\right )}dx+\frac {2 a C \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{d}\right )+\frac {2 C \sin (c+d x) (a+b \cos (c+d x))^{5/2}}{7 d}\) |
\(\Big \downarrow \) 3528 |
\(\displaystyle \frac {1}{7} \left (\frac {2}{3} \int \frac {\left (21 A a^3+\left (3 C a^2+49 A b^2+29 b^2 C\right ) \cos ^2(c+d x) a+b \left (9 (7 A+3 C) a^2+b^2 (7 A+5 C)\right ) \cos (c+d x)\right ) \sec (c+d x)}{2 \sqrt {a+b \cos (c+d x)}}dx+\frac {2 \left (3 a^2 C+7 A b^2+5 b^2 C\right ) \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{3 d}+\frac {2 a C \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{d}\right )+\frac {2 C \sin (c+d x) (a+b \cos (c+d x))^{5/2}}{7 d}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {1}{7} \left (\frac {1}{3} \int \frac {\left (21 A a^3+\left (3 C a^2+49 A b^2+29 b^2 C\right ) \cos ^2(c+d x) a+b \left (9 (7 A+3 C) a^2+b^2 (7 A+5 C)\right ) \cos (c+d x)\right ) \sec (c+d x)}{\sqrt {a+b \cos (c+d x)}}dx+\frac {2 \left (3 a^2 C+7 A b^2+5 b^2 C\right ) \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{3 d}+\frac {2 a C \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{d}\right )+\frac {2 C \sin (c+d x) (a+b \cos (c+d x))^{5/2}}{7 d}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {1}{7} \left (\frac {1}{3} \int \frac {21 A a^3+\left (3 C a^2+49 A b^2+29 b^2 C\right ) \sin \left (c+d x+\frac {\pi }{2}\right )^2 a+b \left (9 (7 A+3 C) a^2+b^2 (7 A+5 C)\right ) \sin \left (c+d x+\frac {\pi }{2}\right )}{\sin \left (c+d x+\frac {\pi }{2}\right ) \sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}}dx+\frac {2 \left (3 a^2 C+7 A b^2+5 b^2 C\right ) \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{3 d}+\frac {2 a C \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{d}\right )+\frac {2 C \sin (c+d x) (a+b \cos (c+d x))^{5/2}}{7 d}\) |
\(\Big \downarrow \) 3538 |
\(\displaystyle \frac {1}{7} \left (\frac {1}{3} \left (\frac {a \left (3 a^2 C+49 A b^2+29 b^2 C\right ) \int \sqrt {a+b \cos (c+d x)}dx}{b}-\frac {\int -\frac {\left (21 A b a^3+\left (-3 C a^4+2 b^2 (7 A-C) a^2+b^4 (7 A+5 C)\right ) \cos (c+d x)\right ) \sec (c+d x)}{\sqrt {a+b \cos (c+d x)}}dx}{b}\right )+\frac {2 \left (3 a^2 C+7 A b^2+5 b^2 C\right ) \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{3 d}+\frac {2 a C \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{d}\right )+\frac {2 C \sin (c+d x) (a+b \cos (c+d x))^{5/2}}{7 d}\) |
\(\Big \downarrow \) 25 |
\(\displaystyle \frac {1}{7} \left (\frac {1}{3} \left (\frac {a \left (3 a^2 C+49 A b^2+29 b^2 C\right ) \int \sqrt {a+b \cos (c+d x)}dx}{b}+\frac {\int \frac {\left (21 A b a^3+\left (-3 C a^4+2 b^2 (7 A-C) a^2+b^4 (7 A+5 C)\right ) \cos (c+d x)\right ) \sec (c+d x)}{\sqrt {a+b \cos (c+d x)}}dx}{b}\right )+\frac {2 \left (3 a^2 C+7 A b^2+5 b^2 C\right ) \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{3 d}+\frac {2 a C \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{d}\right )+\frac {2 C \sin (c+d x) (a+b \cos (c+d x))^{5/2}}{7 d}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {1}{7} \left (\frac {1}{3} \left (\frac {a \left (3 a^2 C+49 A b^2+29 b^2 C\right ) \int \sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}dx}{b}+\frac {\int \frac {21 A b a^3+\left (-3 C a^4+2 b^2 (7 A-C) a^2+b^4 (7 A+5 C)\right ) \sin \left (c+d x+\frac {\pi }{2}\right )}{\sin \left (c+d x+\frac {\pi }{2}\right ) \sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}}dx}{b}\right )+\frac {2 \left (3 a^2 C+7 A b^2+5 b^2 C\right ) \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{3 d}+\frac {2 a C \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{d}\right )+\frac {2 C \sin (c+d x) (a+b \cos (c+d x))^{5/2}}{7 d}\) |
\(\Big \downarrow \) 3134 |
\(\displaystyle \frac {1}{7} \left (\frac {1}{3} \left (\frac {a \left (3 a^2 C+49 A b^2+29 b^2 C\right ) \sqrt {a+b \cos (c+d x)} \int \sqrt {\frac {a}{a+b}+\frac {b \cos (c+d x)}{a+b}}dx}{b \sqrt {\frac {a+b \cos (c+d x)}{a+b}}}+\frac {\int \frac {21 A b a^3+\left (-3 C a^4+2 b^2 (7 A-C) a^2+b^4 (7 A+5 C)\right ) \sin \left (c+d x+\frac {\pi }{2}\right )}{\sin \left (c+d x+\frac {\pi }{2}\right ) \sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}}dx}{b}\right )+\frac {2 \left (3 a^2 C+7 A b^2+5 b^2 C\right ) \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{3 d}+\frac {2 a C \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{d}\right )+\frac {2 C \sin (c+d x) (a+b \cos (c+d x))^{5/2}}{7 d}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {1}{7} \left (\frac {1}{3} \left (\frac {a \left (3 a^2 C+49 A b^2+29 b^2 C\right ) \sqrt {a+b \cos (c+d x)} \int \sqrt {\frac {a}{a+b}+\frac {b \sin \left (c+d x+\frac {\pi }{2}\right )}{a+b}}dx}{b \sqrt {\frac {a+b \cos (c+d x)}{a+b}}}+\frac {\int \frac {21 A b a^3+\left (-3 C a^4+2 b^2 (7 A-C) a^2+b^4 (7 A+5 C)\right ) \sin \left (c+d x+\frac {\pi }{2}\right )}{\sin \left (c+d x+\frac {\pi }{2}\right ) \sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}}dx}{b}\right )+\frac {2 \left (3 a^2 C+7 A b^2+5 b^2 C\right ) \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{3 d}+\frac {2 a C \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{d}\right )+\frac {2 C \sin (c+d x) (a+b \cos (c+d x))^{5/2}}{7 d}\) |
\(\Big \downarrow \) 3132 |
\(\displaystyle \frac {1}{7} \left (\frac {1}{3} \left (\frac {\int \frac {21 A b a^3+\left (-3 C a^4+2 b^2 (7 A-C) a^2+b^4 (7 A+5 C)\right ) \sin \left (c+d x+\frac {\pi }{2}\right )}{\sin \left (c+d x+\frac {\pi }{2}\right ) \sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}}dx}{b}+\frac {2 a \left (3 a^2 C+49 A b^2+29 b^2 C\right ) \sqrt {a+b \cos (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )}{b d \sqrt {\frac {a+b \cos (c+d x)}{a+b}}}\right )+\frac {2 \left (3 a^2 C+7 A b^2+5 b^2 C\right ) \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{3 d}+\frac {2 a C \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{d}\right )+\frac {2 C \sin (c+d x) (a+b \cos (c+d x))^{5/2}}{7 d}\) |
\(\Big \downarrow \) 3481 |
\(\displaystyle \frac {1}{7} \left (\frac {1}{3} \left (\frac {21 a^3 A b \int \frac {\sec (c+d x)}{\sqrt {a+b \cos (c+d x)}}dx+\left (-3 a^4 C+2 a^2 b^2 (7 A-C)+b^4 (7 A+5 C)\right ) \int \frac {1}{\sqrt {a+b \cos (c+d x)}}dx}{b}+\frac {2 a \left (3 a^2 C+49 A b^2+29 b^2 C\right ) \sqrt {a+b \cos (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )}{b d \sqrt {\frac {a+b \cos (c+d x)}{a+b}}}\right )+\frac {2 \left (3 a^2 C+7 A b^2+5 b^2 C\right ) \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{3 d}+\frac {2 a C \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{d}\right )+\frac {2 C \sin (c+d x) (a+b \cos (c+d x))^{5/2}}{7 d}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {1}{7} \left (\frac {1}{3} \left (\frac {21 a^3 A b \int \frac {1}{\sin \left (c+d x+\frac {\pi }{2}\right ) \sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}}dx+\left (-3 a^4 C+2 a^2 b^2 (7 A-C)+b^4 (7 A+5 C)\right ) \int \frac {1}{\sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}}dx}{b}+\frac {2 a \left (3 a^2 C+49 A b^2+29 b^2 C\right ) \sqrt {a+b \cos (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )}{b d \sqrt {\frac {a+b \cos (c+d x)}{a+b}}}\right )+\frac {2 \left (3 a^2 C+7 A b^2+5 b^2 C\right ) \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{3 d}+\frac {2 a C \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{d}\right )+\frac {2 C \sin (c+d x) (a+b \cos (c+d x))^{5/2}}{7 d}\) |
\(\Big \downarrow \) 3142 |
\(\displaystyle \frac {1}{7} \left (\frac {1}{3} \left (\frac {21 a^3 A b \int \frac {1}{\sin \left (c+d x+\frac {\pi }{2}\right ) \sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}}dx+\frac {\left (-3 a^4 C+2 a^2 b^2 (7 A-C)+b^4 (7 A+5 C)\right ) \sqrt {\frac {a+b \cos (c+d x)}{a+b}} \int \frac {1}{\sqrt {\frac {a}{a+b}+\frac {b \cos (c+d x)}{a+b}}}dx}{\sqrt {a+b \cos (c+d x)}}}{b}+\frac {2 a \left (3 a^2 C+49 A b^2+29 b^2 C\right ) \sqrt {a+b \cos (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )}{b d \sqrt {\frac {a+b \cos (c+d x)}{a+b}}}\right )+\frac {2 \left (3 a^2 C+7 A b^2+5 b^2 C\right ) \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{3 d}+\frac {2 a C \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{d}\right )+\frac {2 C \sin (c+d x) (a+b \cos (c+d x))^{5/2}}{7 d}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {1}{7} \left (\frac {1}{3} \left (\frac {21 a^3 A b \int \frac {1}{\sin \left (c+d x+\frac {\pi }{2}\right ) \sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}}dx+\frac {\left (-3 a^4 C+2 a^2 b^2 (7 A-C)+b^4 (7 A+5 C)\right ) \sqrt {\frac {a+b \cos (c+d x)}{a+b}} \int \frac {1}{\sqrt {\frac {a}{a+b}+\frac {b \sin \left (c+d x+\frac {\pi }{2}\right )}{a+b}}}dx}{\sqrt {a+b \cos (c+d x)}}}{b}+\frac {2 a \left (3 a^2 C+49 A b^2+29 b^2 C\right ) \sqrt {a+b \cos (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )}{b d \sqrt {\frac {a+b \cos (c+d x)}{a+b}}}\right )+\frac {2 \left (3 a^2 C+7 A b^2+5 b^2 C\right ) \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{3 d}+\frac {2 a C \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{d}\right )+\frac {2 C \sin (c+d x) (a+b \cos (c+d x))^{5/2}}{7 d}\) |
\(\Big \downarrow \) 3140 |
\(\displaystyle \frac {1}{7} \left (\frac {1}{3} \left (\frac {21 a^3 A b \int \frac {1}{\sin \left (c+d x+\frac {\pi }{2}\right ) \sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}}dx+\frac {2 \left (-3 a^4 C+2 a^2 b^2 (7 A-C)+b^4 (7 A+5 C)\right ) \sqrt {\frac {a+b \cos (c+d x)}{a+b}} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),\frac {2 b}{a+b}\right )}{d \sqrt {a+b \cos (c+d x)}}}{b}+\frac {2 a \left (3 a^2 C+49 A b^2+29 b^2 C\right ) \sqrt {a+b \cos (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )}{b d \sqrt {\frac {a+b \cos (c+d x)}{a+b}}}\right )+\frac {2 \left (3 a^2 C+7 A b^2+5 b^2 C\right ) \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{3 d}+\frac {2 a C \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{d}\right )+\frac {2 C \sin (c+d x) (a+b \cos (c+d x))^{5/2}}{7 d}\) |
\(\Big \downarrow \) 3286 |
\(\displaystyle \frac {1}{7} \left (\frac {1}{3} \left (\frac {\frac {21 a^3 A b \sqrt {\frac {a+b \cos (c+d x)}{a+b}} \int \frac {\sec (c+d x)}{\sqrt {\frac {a}{a+b}+\frac {b \cos (c+d x)}{a+b}}}dx}{\sqrt {a+b \cos (c+d x)}}+\frac {2 \left (-3 a^4 C+2 a^2 b^2 (7 A-C)+b^4 (7 A+5 C)\right ) \sqrt {\frac {a+b \cos (c+d x)}{a+b}} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),\frac {2 b}{a+b}\right )}{d \sqrt {a+b \cos (c+d x)}}}{b}+\frac {2 a \left (3 a^2 C+49 A b^2+29 b^2 C\right ) \sqrt {a+b \cos (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )}{b d \sqrt {\frac {a+b \cos (c+d x)}{a+b}}}\right )+\frac {2 \left (3 a^2 C+7 A b^2+5 b^2 C\right ) \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{3 d}+\frac {2 a C \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{d}\right )+\frac {2 C \sin (c+d x) (a+b \cos (c+d x))^{5/2}}{7 d}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {1}{7} \left (\frac {1}{3} \left (\frac {\frac {21 a^3 A b \sqrt {\frac {a+b \cos (c+d x)}{a+b}} \int \frac {1}{\sin \left (c+d x+\frac {\pi }{2}\right ) \sqrt {\frac {a}{a+b}+\frac {b \sin \left (c+d x+\frac {\pi }{2}\right )}{a+b}}}dx}{\sqrt {a+b \cos (c+d x)}}+\frac {2 \left (-3 a^4 C+2 a^2 b^2 (7 A-C)+b^4 (7 A+5 C)\right ) \sqrt {\frac {a+b \cos (c+d x)}{a+b}} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),\frac {2 b}{a+b}\right )}{d \sqrt {a+b \cos (c+d x)}}}{b}+\frac {2 a \left (3 a^2 C+49 A b^2+29 b^2 C\right ) \sqrt {a+b \cos (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )}{b d \sqrt {\frac {a+b \cos (c+d x)}{a+b}}}\right )+\frac {2 \left (3 a^2 C+7 A b^2+5 b^2 C\right ) \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{3 d}+\frac {2 a C \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{d}\right )+\frac {2 C \sin (c+d x) (a+b \cos (c+d x))^{5/2}}{7 d}\) |
\(\Big \downarrow \) 3284 |
\(\displaystyle \frac {1}{7} \left (\frac {2 \left (3 a^2 C+7 A b^2+5 b^2 C\right ) \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{3 d}+\frac {1}{3} \left (\frac {2 a \left (3 a^2 C+49 A b^2+29 b^2 C\right ) \sqrt {a+b \cos (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )}{b d \sqrt {\frac {a+b \cos (c+d x)}{a+b}}}+\frac {\frac {42 a^3 A b \sqrt {\frac {a+b \cos (c+d x)}{a+b}} \operatorname {EllipticPi}\left (2,\frac {1}{2} (c+d x),\frac {2 b}{a+b}\right )}{d \sqrt {a+b \cos (c+d x)}}+\frac {2 \left (-3 a^4 C+2 a^2 b^2 (7 A-C)+b^4 (7 A+5 C)\right ) \sqrt {\frac {a+b \cos (c+d x)}{a+b}} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),\frac {2 b}{a+b}\right )}{d \sqrt {a+b \cos (c+d x)}}}{b}\right )+\frac {2 a C \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{d}\right )+\frac {2 C \sin (c+d x) (a+b \cos (c+d x))^{5/2}}{7 d}\) |
(2*C*(a + b*Cos[c + d*x])^(5/2)*Sin[c + d*x])/(7*d) + (((2*a*(49*A*b^2 + 3 *a^2*C + 29*b^2*C)*Sqrt[a + b*Cos[c + d*x]]*EllipticE[(c + d*x)/2, (2*b)/( a + b)])/(b*d*Sqrt[(a + b*Cos[c + d*x])/(a + b)]) + ((2*(2*a^2*b^2*(7*A - C) - 3*a^4*C + b^4*(7*A + 5*C))*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*Ellipti cF[(c + d*x)/2, (2*b)/(a + b)])/(d*Sqrt[a + b*Cos[c + d*x]]) + (42*a^3*A*b *Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticPi[2, (c + d*x)/2, (2*b)/(a + b)])/(d*Sqrt[a + b*Cos[c + d*x]]))/b)/3 + (2*(7*A*b^2 + 3*a^2*C + 5*b^2*C) *Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(3*d) + (2*a*C*(a + b*Cos[c + d*x] )^(3/2)*Sin[c + d*x])/d)/7
3.7.41.3.1 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[(((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)]))/((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[ B/d Int[(a + b*Sin[e + f*x])^m, x], x] - Simp[(B*c - A*d)/d Int[(a + b* Sin[e + f*x])^m/(c + d*Sin[e + f*x]), x], x] /; FreeQ[{a, b, c, d, e, f, A, B, m}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0]
Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_.)*((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)] + (C_.)*sin[(e_ .) + (f_.)*(x_)]^2), x_Symbol] :> Simp[(-C)*Cos[e + f*x]*(a + b*Sin[e + f*x ])^m*((c + d*Sin[e + f*x])^(n + 1)/(d*f*(m + n + 2))), x] + Simp[1/(d*(m + n + 2)) Int[(a + b*Sin[e + f*x])^(m - 1)*(c + d*Sin[e + f*x])^n*Simp[a*A* d*(m + n + 2) + C*(b*c*m + a*d*(n + 1)) + (d*(A*b + a*B)*(m + n + 2) - C*(a *c - b*d*(m + n + 1)))*Sin[e + f*x] + (C*(a*d*m - b*c*(m + 1)) + b*B*d*(m + n + 2))*Sin[e + f*x]^2, x], x], x] /; FreeQ[{a, b, c, d, e, f, A, B, C, n} , x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && GtQ[ m, 0] && !(IGtQ[n, 0] && ( !IntegerQ[m] || (EqQ[a, 0] && NeQ[c, 0])))
Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_.)*((A_.) + (C_.)*sin[(e_.) + (f_.)*(x_)]^2), x_Symbol] : > Simp[(-C)*Cos[e + f*x]*(a + b*Sin[e + f*x])^m*((c + d*Sin[e + f*x])^(n + 1)/(d*f*(m + n + 2))), x] + Simp[1/(d*(m + n + 2)) Int[(a + b*Sin[e + f*x ])^(m - 1)*(c + d*Sin[e + f*x])^n*Simp[a*A*d*(m + n + 2) + C*(b*c*m + a*d*( n + 1)) + (A*b*d*(m + n + 2) - C*(a*c - b*d*(m + n + 1)))*Sin[e + f*x] + C* (a*d*m - b*c*(m + 1))*Sin[e + f*x]^2, x], x], x] /; FreeQ[{a, b, c, d, e, f , A, C, n}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && GtQ[m, 0] && !(IGtQ[n, 0] && ( !IntegerQ[m] || (EqQ[a, 0] && NeQ[c, 0])))
Int[((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)] + (C_.)*sin[(e_.) + (f_.)*(x_)]^ 2)/(Sqrt[(a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)]]*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])), x_Symbol] :> Simp[C/(b*d) Int[Sqrt[a + b*Sin[e + f*x]], x] , x] - Simp[1/(b*d) Int[Simp[a*c*C - A*b*d + (b*c*C - b*B*d + a*C*d)*Sin[ e + f*x], x]/(Sqrt[a + b*Sin[e + f*x]]*(c + d*Sin[e + f*x])), x], x] /; Fre eQ[{a, b, c, d, e, f, A, B, C}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0 ] && NeQ[c^2 - d^2, 0]
Leaf count of result is larger than twice the leaf count of optimal. \(1208\) vs. \(2(401)=802\).
Time = 25.70 (sec) , antiderivative size = 1209, normalized size of antiderivative = 3.54
method | result | size |
default | \(\text {Expression too large to display}\) | \(1209\) |
parts | \(\text {Expression too large to display}\) | \(1357\) |
-2/21*((2*b*cos(1/2*d*x+1/2*c)^2+a-b)*sin(1/2*d*x+1/2*c)^2)^(1/2)*(48*C*co s(1/2*d*x+1/2*c)*sin(1/2*d*x+1/2*c)^8*b^4+(-96*C*a*b^3-72*C*b^4)*sin(1/2*d *x+1/2*c)^6*cos(1/2*d*x+1/2*c)+(28*A*b^4+72*C*a^2*b^2+96*C*a*b^3+56*C*b^4) *sin(1/2*d*x+1/2*c)^4*cos(1/2*d*x+1/2*c)+(-14*A*a*b^3-14*A*b^4-18*C*a^3*b- 36*C*a^2*b^2-34*C*a*b^3-16*C*b^4)*sin(1/2*d*x+1/2*c)^2*cos(1/2*d*x+1/2*c)+ 14*A*(sin(1/2*d*x+1/2*c)^2)^(1/2)*(-2*b/(a-b)*sin(1/2*d*x+1/2*c)^2+(a+b)/( a-b))^(1/2)*EllipticF(cos(1/2*d*x+1/2*c),(-2*b/(a-b))^(1/2))*a^2*b^2+7*A*b ^4*(sin(1/2*d*x+1/2*c)^2)^(1/2)*(-2*b/(a-b)*sin(1/2*d*x+1/2*c)^2+(a+b)/(a- b))^(1/2)*EllipticF(cos(1/2*d*x+1/2*c),(-2*b/(a-b))^(1/2))+49*A*(sin(1/2*d *x+1/2*c)^2)^(1/2)*(-2*b/(a-b)*sin(1/2*d*x+1/2*c)^2+(a+b)/(a-b))^(1/2)*Ell ipticE(cos(1/2*d*x+1/2*c),(-2*b/(a-b))^(1/2))*a^2*b^2-49*A*(sin(1/2*d*x+1/ 2*c)^2)^(1/2)*(-2*b/(a-b)*sin(1/2*d*x+1/2*c)^2+(a+b)/(a-b))^(1/2)*Elliptic E(cos(1/2*d*x+1/2*c),(-2*b/(a-b))^(1/2))*a*b^3-21*A*a^3*(sin(1/2*d*x+1/2*c )^2)^(1/2)*(-2*b/(a-b)*sin(1/2*d*x+1/2*c)^2+(a+b)/(a-b))^(1/2)*EllipticPi( cos(1/2*d*x+1/2*c),2,(-2*b/(a-b))^(1/2))*b-3*C*(sin(1/2*d*x+1/2*c)^2)^(1/2 )*(-2*b/(a-b)*sin(1/2*d*x+1/2*c)^2+(a+b)/(a-b))^(1/2)*EllipticF(cos(1/2*d* x+1/2*c),(-2*b/(a-b))^(1/2))*a^4-2*C*(sin(1/2*d*x+1/2*c)^2)^(1/2)*(-2*b/(a -b)*sin(1/2*d*x+1/2*c)^2+(a+b)/(a-b))^(1/2)*EllipticF(cos(1/2*d*x+1/2*c),( -2*b/(a-b))^(1/2))*a^2*b^2+5*C*b^4*(sin(1/2*d*x+1/2*c)^2)^(1/2)*(-2*b/(a-b )*sin(1/2*d*x+1/2*c)^2+(a+b)/(a-b))^(1/2)*EllipticF(cos(1/2*d*x+1/2*c),...
\[ \int (a+b \cos (c+d x))^{5/2} \left (A+C \cos ^2(c+d x)\right ) \sec (c+d x) \, dx=\int { {\left (C \cos \left (d x + c\right )^{2} + A\right )} {\left (b \cos \left (d x + c\right ) + a\right )}^{\frac {5}{2}} \sec \left (d x + c\right ) \,d x } \]
integral((C*b^2*cos(d*x + c)^4 + 2*C*a*b*cos(d*x + c)^3 + 2*A*a*b*cos(d*x + c) + A*a^2 + (C*a^2 + A*b^2)*cos(d*x + c)^2)*sqrt(b*cos(d*x + c) + a)*se c(d*x + c), x)
Timed out. \[ \int (a+b \cos (c+d x))^{5/2} \left (A+C \cos ^2(c+d x)\right ) \sec (c+d x) \, dx=\text {Timed out} \]
\[ \int (a+b \cos (c+d x))^{5/2} \left (A+C \cos ^2(c+d x)\right ) \sec (c+d x) \, dx=\int { {\left (C \cos \left (d x + c\right )^{2} + A\right )} {\left (b \cos \left (d x + c\right ) + a\right )}^{\frac {5}{2}} \sec \left (d x + c\right ) \,d x } \]
\[ \int (a+b \cos (c+d x))^{5/2} \left (A+C \cos ^2(c+d x)\right ) \sec (c+d x) \, dx=\int { {\left (C \cos \left (d x + c\right )^{2} + A\right )} {\left (b \cos \left (d x + c\right ) + a\right )}^{\frac {5}{2}} \sec \left (d x + c\right ) \,d x } \]
Timed out. \[ \int (a+b \cos (c+d x))^{5/2} \left (A+C \cos ^2(c+d x)\right ) \sec (c+d x) \, dx=\int \frac {\left (C\,{\cos \left (c+d\,x\right )}^2+A\right )\,{\left (a+b\,\cos \left (c+d\,x\right )\right )}^{5/2}}{\cos \left (c+d\,x\right )} \,d x \]