Integrand size = 23, antiderivative size = 323 \[ \int (a+b \cos (c+d x))^{5/2} \sec ^4(c+d x) \, dx=-\frac {\left (16 a^2+33 b^2\right ) \sqrt {a+b \cos (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )}{24 d \sqrt {\frac {a+b \cos (c+d x)}{a+b}}}+\frac {a \left (16 a^2+59 b^2\right ) \sqrt {\frac {a+b \cos (c+d x)}{a+b}} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),\frac {2 b}{a+b}\right )}{24 d \sqrt {a+b \cos (c+d x)}}+\frac {5 b \left (4 a^2+b^2\right ) \sqrt {\frac {a+b \cos (c+d x)}{a+b}} \operatorname {EllipticPi}\left (2,\frac {1}{2} (c+d x),\frac {2 b}{a+b}\right )}{8 d \sqrt {a+b \cos (c+d x)}}+\frac {\left (16 a^2+33 b^2\right ) \sqrt {a+b \cos (c+d x)} \tan (c+d x)}{24 d}+\frac {13 a b \sqrt {a+b \cos (c+d x)} \sec (c+d x) \tan (c+d x)}{12 d}+\frac {a^2 \sqrt {a+b \cos (c+d x)} \sec ^2(c+d x) \tan (c+d x)}{3 d} \]
-1/24*(16*a^2+33*b^2)*(cos(1/2*d*x+1/2*c)^2)^(1/2)/cos(1/2*d*x+1/2*c)*Elli pticE(sin(1/2*d*x+1/2*c),2^(1/2)*(b/(a+b))^(1/2))*(a+b*cos(d*x+c))^(1/2)/d /((a+b*cos(d*x+c))/(a+b))^(1/2)+1/24*a*(16*a^2+59*b^2)*(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)*(b/(a+b) )^(1/2))*((a+b*cos(d*x+c))/(a+b))^(1/2)/d/(a+b*cos(d*x+c))^(1/2)+5/8*b*(4* a^2+b^2)*(cos(1/2*d*x+1/2*c)^2)^(1/2)/cos(1/2*d*x+1/2*c)*EllipticPi(sin(1/ 2*d*x+1/2*c),2,2^(1/2)*(b/(a+b))^(1/2))*((a+b*cos(d*x+c))/(a+b))^(1/2)/d/( a+b*cos(d*x+c))^(1/2)+1/24*(16*a^2+33*b^2)*(a+b*cos(d*x+c))^(1/2)*tan(d*x+ c)/d+13/12*a*b*sec(d*x+c)*(a+b*cos(d*x+c))^(1/2)*tan(d*x+c)/d+1/3*a^2*sec( d*x+c)^2*(a+b*cos(d*x+c))^(1/2)*tan(d*x+c)/d
Result contains complex when optimal does not.
Time = 2.67 (sec) , antiderivative size = 434, normalized size of antiderivative = 1.34 \[ \int (a+b \cos (c+d x))^{5/2} \sec ^4(c+d x) \, dx=\frac {\frac {104 a b^2 \sqrt {\frac {a+b \cos (c+d x)}{a+b}} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),\frac {2 b}{a+b}\right )}{\sqrt {a+b \cos (c+d x)}}+\frac {2 b \left (104 a^2-3 b^2\right ) \sqrt {\frac {a+b \cos (c+d x)}{a+b}} \operatorname {EllipticPi}\left (2,\frac {1}{2} (c+d x),\frac {2 b}{a+b}\right )}{\sqrt {a+b \cos (c+d x)}}-\frac {2 i \left (16 a^2+33 b^2\right ) \sqrt {-\frac {b (-1+\cos (c+d x))}{a+b}} \sqrt {-\frac {b (1+\cos (c+d x))}{a-b}} \csc (c+d x) \left (-2 a (a-b) E\left (i \text {arcsinh}\left (\sqrt {-\frac {1}{a+b}} \sqrt {a+b \cos (c+d x)}\right )|\frac {a+b}{a-b}\right )+b \left (-2 a \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {-\frac {1}{a+b}} \sqrt {a+b \cos (c+d x)}\right ),\frac {a+b}{a-b}\right )+b \operatorname {EllipticPi}\left (\frac {a+b}{a},i \text {arcsinh}\left (\sqrt {-\frac {1}{a+b}} \sqrt {a+b \cos (c+d x)}\right ),\frac {a+b}{a-b}\right )\right )\right )}{a b \sqrt {-\frac {1}{a+b}}}+4 \sqrt {a+b \cos (c+d x)} \sec ^2(c+d x) \left (26 a b \sin (c+d x)+\left (8 a^2+\frac {33 b^2}{2}\right ) \sin (2 (c+d x))+8 a^2 \tan (c+d x)\right )}{96 d} \]
((104*a*b^2*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*b*(104*a^2 - 3*b^2)*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)*(16*a^2 + 33*b^2)*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)*EllipticE[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)])))/(a*b*Sqrt[- (a + b)^(-1)]) + 4*Sqrt[a + b*Cos[c + d*x]]*Sec[c + d*x]^2*(26*a*b*Sin[c + d*x] + (8*a^2 + (33*b^2)/2)*Sin[2*(c + d*x)] + 8*a^2*Tan[c + d*x]))/(96*d )
Time = 3.02 (sec) , antiderivative size = 352, normalized size of antiderivative = 1.09, number of steps used = 24, number of rules used = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.043, Rules used = {3042, 3271, 27, 3042, 3534, 27, 3042, 3534, 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.
| \(\displaystyle \int \sec ^4(c+d x) (a+b \cos (c+d x))^{5/2} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\left (a+b \sin \left (c+d x+\frac {\pi }{2}\right )\right )^{5/2}}{\sin \left (c+d x+\frac {\pi }{2}\right )^4}dx\) |
| \(\Big \downarrow \) 3271 |
| \(\displaystyle \frac {1}{3} \int \frac {\left (13 b a^2+2 \left (2 a^2+9 b^2\right ) \cos (c+d x) a+3 b \left (a^2+2 b^2\right ) \cos ^2(c+d x)\right ) \sec ^3(c+d x)}{2 \sqrt {a+b \cos (c+d x)}}dx+\frac {a^2 \tan (c+d x) \sec ^2(c+d x) \sqrt {a+b \cos (c+d x)}}{3 d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{6} \int \frac {\left (13 b a^2+2 \left (2 a^2+9 b^2\right ) \cos (c+d x) a+3 b \left (a^2+2 b^2\right ) \cos ^2(c+d x)\right ) \sec ^3(c+d x)}{\sqrt {a+b \cos (c+d x)}}dx+\frac {a^2 \tan (c+d x) \sec ^2(c+d x) \sqrt {a+b \cos (c+d x)}}{3 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{6} \int \frac {13 b a^2+2 \left (2 a^2+9 b^2\right ) \sin \left (c+d x+\frac {\pi }{2}\right ) a+3 b \left (a^2+2 b^2\right ) \sin \left (c+d x+\frac {\pi }{2}\right )^2}{\sin \left (c+d x+\frac {\pi }{2}\right )^3 \sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}}dx+\frac {a^2 \tan (c+d x) \sec ^2(c+d x) \sqrt {a+b \cos (c+d x)}}{3 d}\) |
| \(\Big \downarrow \) 3534 |
| \(\displaystyle \frac {1}{6} \left (\frac {\int \frac {\left (13 b^2 \cos ^2(c+d x) a^2+\left (16 a^2+33 b^2\right ) a^2+2 b \left (19 a^2+12 b^2\right ) \cos (c+d x) a\right ) \sec ^2(c+d x)}{2 \sqrt {a+b \cos (c+d x)}}dx}{2 a}+\frac {13 a b \tan (c+d x) \sec (c+d x) \sqrt {a+b \cos (c+d x)}}{2 d}\right )+\frac {a^2 \tan (c+d x) \sec ^2(c+d x) \sqrt {a+b \cos (c+d x)}}{3 d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{6} \left (\frac {\int \frac {\left (13 b^2 \cos ^2(c+d x) a^2+\left (16 a^2+33 b^2\right ) a^2+2 b \left (19 a^2+12 b^2\right ) \cos (c+d x) a\right ) \sec ^2(c+d x)}{\sqrt {a+b \cos (c+d x)}}dx}{4 a}+\frac {13 a b \tan (c+d x) \sec (c+d x) \sqrt {a+b \cos (c+d x)}}{2 d}\right )+\frac {a^2 \tan (c+d x) \sec ^2(c+d x) \sqrt {a+b \cos (c+d x)}}{3 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{6} \left (\frac {\int \frac {13 b^2 \sin \left (c+d x+\frac {\pi }{2}\right )^2 a^2+\left (16 a^2+33 b^2\right ) a^2+2 b \left (19 a^2+12 b^2\right ) \sin \left (c+d x+\frac {\pi }{2}\right ) a}{\sin \left (c+d x+\frac {\pi }{2}\right )^2 \sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}}dx}{4 a}+\frac {13 a b \tan (c+d x) \sec (c+d x) \sqrt {a+b \cos (c+d x)}}{2 d}\right )+\frac {a^2 \tan (c+d x) \sec ^2(c+d x) \sqrt {a+b \cos (c+d x)}}{3 d}\) |
| \(\Big \downarrow \) 3534 |
| \(\displaystyle \frac {1}{6} \left (\frac {\frac {\int \frac {\left (26 b^2 \cos (c+d x) a^3-b \left (16 a^2+33 b^2\right ) \cos ^2(c+d x) a^2+15 b \left (4 a^2+b^2\right ) a^2\right ) \sec (c+d x)}{2 \sqrt {a+b \cos (c+d x)}}dx}{a}+\frac {a \left (16 a^2+33 b^2\right ) \tan (c+d x) \sqrt {a+b \cos (c+d x)}}{d}}{4 a}+\frac {13 a b \tan (c+d x) \sec (c+d x) \sqrt {a+b \cos (c+d x)}}{2 d}\right )+\frac {a^2 \tan (c+d x) \sec ^2(c+d x) \sqrt {a+b \cos (c+d x)}}{3 d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{6} \left (\frac {\frac {\int \frac {\left (26 b^2 \cos (c+d x) a^3-b \left (16 a^2+33 b^2\right ) \cos ^2(c+d x) a^2+15 b \left (4 a^2+b^2\right ) a^2\right ) \sec (c+d x)}{\sqrt {a+b \cos (c+d x)}}dx}{2 a}+\frac {a \left (16 a^2+33 b^2\right ) \tan (c+d x) \sqrt {a+b \cos (c+d x)}}{d}}{4 a}+\frac {13 a b \tan (c+d x) \sec (c+d x) \sqrt {a+b \cos (c+d x)}}{2 d}\right )+\frac {a^2 \tan (c+d x) \sec ^2(c+d x) \sqrt {a+b \cos (c+d x)}}{3 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{6} \left (\frac {\frac {\int \frac {26 b^2 \sin \left (c+d x+\frac {\pi }{2}\right ) a^3-b \left (16 a^2+33 b^2\right ) \sin \left (c+d x+\frac {\pi }{2}\right )^2 a^2+15 b \left (4 a^2+b^2\right ) a^2}{\sin \left (c+d x+\frac {\pi }{2}\right ) \sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}}dx}{2 a}+\frac {a \left (16 a^2+33 b^2\right ) \tan (c+d x) \sqrt {a+b \cos (c+d x)}}{d}}{4 a}+\frac {13 a b \tan (c+d x) \sec (c+d x) \sqrt {a+b \cos (c+d x)}}{2 d}\right )+\frac {a^2 \tan (c+d x) \sec ^2(c+d x) \sqrt {a+b \cos (c+d x)}}{3 d}\) |
| \(\Big \downarrow \) 3538 |
| \(\displaystyle \frac {1}{6} \left (\frac {\frac {-\left (a^2 \left (16 a^2+33 b^2\right ) \int \sqrt {a+b \cos (c+d x)}dx\right )-\frac {\int -\frac {\left (b \left (16 a^2+59 b^2\right ) \cos (c+d x) a^3+15 b^2 \left (4 a^2+b^2\right ) a^2\right ) \sec (c+d x)}{\sqrt {a+b \cos (c+d x)}}dx}{b}}{2 a}+\frac {a \left (16 a^2+33 b^2\right ) \tan (c+d x) \sqrt {a+b \cos (c+d x)}}{d}}{4 a}+\frac {13 a b \tan (c+d x) \sec (c+d x) \sqrt {a+b \cos (c+d x)}}{2 d}\right )+\frac {a^2 \tan (c+d x) \sec ^2(c+d x) \sqrt {a+b \cos (c+d x)}}{3 d}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {1}{6} \left (\frac {\frac {\frac {\int \frac {\left (b \left (16 a^2+59 b^2\right ) \cos (c+d x) a^3+15 b^2 \left (4 a^2+b^2\right ) a^2\right ) \sec (c+d x)}{\sqrt {a+b \cos (c+d x)}}dx}{b}-a^2 \left (16 a^2+33 b^2\right ) \int \sqrt {a+b \cos (c+d x)}dx}{2 a}+\frac {a \left (16 a^2+33 b^2\right ) \tan (c+d x) \sqrt {a+b \cos (c+d x)}}{d}}{4 a}+\frac {13 a b \tan (c+d x) \sec (c+d x) \sqrt {a+b \cos (c+d x)}}{2 d}\right )+\frac {a^2 \tan (c+d x) \sec ^2(c+d x) \sqrt {a+b \cos (c+d x)}}{3 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{6} \left (\frac {\frac {\frac {\int \frac {b \left (16 a^2+59 b^2\right ) \sin \left (c+d x+\frac {\pi }{2}\right ) a^3+15 b^2 \left (4 a^2+b^2\right ) a^2}{\sin \left (c+d x+\frac {\pi }{2}\right ) \sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}}dx}{b}-a^2 \left (16 a^2+33 b^2\right ) \int \sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}dx}{2 a}+\frac {a \left (16 a^2+33 b^2\right ) \tan (c+d x) \sqrt {a+b \cos (c+d x)}}{d}}{4 a}+\frac {13 a b \tan (c+d x) \sec (c+d x) \sqrt {a+b \cos (c+d x)}}{2 d}\right )+\frac {a^2 \tan (c+d x) \sec ^2(c+d x) \sqrt {a+b \cos (c+d x)}}{3 d}\) |
| \(\Big \downarrow \) 3134 |
| \(\displaystyle \frac {1}{6} \left (\frac {\frac {\frac {\int \frac {b \left (16 a^2+59 b^2\right ) \sin \left (c+d x+\frac {\pi }{2}\right ) a^3+15 b^2 \left (4 a^2+b^2\right ) a^2}{\sin \left (c+d x+\frac {\pi }{2}\right ) \sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}}dx}{b}-\frac {a^2 \left (16 a^2+33 b^2\right ) \sqrt {a+b \cos (c+d x)} \int \sqrt {\frac {a}{a+b}+\frac {b \cos (c+d x)}{a+b}}dx}{\sqrt {\frac {a+b \cos (c+d x)}{a+b}}}}{2 a}+\frac {a \left (16 a^2+33 b^2\right ) \tan (c+d x) \sqrt {a+b \cos (c+d x)}}{d}}{4 a}+\frac {13 a b \tan (c+d x) \sec (c+d x) \sqrt {a+b \cos (c+d x)}}{2 d}\right )+\frac {a^2 \tan (c+d x) \sec ^2(c+d x) \sqrt {a+b \cos (c+d x)}}{3 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{6} \left (\frac {\frac {\frac {\int \frac {b \left (16 a^2+59 b^2\right ) \sin \left (c+d x+\frac {\pi }{2}\right ) a^3+15 b^2 \left (4 a^2+b^2\right ) a^2}{\sin \left (c+d x+\frac {\pi }{2}\right ) \sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}}dx}{b}-\frac {a^2 \left (16 a^2+33 b^2\right ) \sqrt {a+b \cos (c+d x)} \int \sqrt {\frac {a}{a+b}+\frac {b \sin \left (c+d x+\frac {\pi }{2}\right )}{a+b}}dx}{\sqrt {\frac {a+b \cos (c+d x)}{a+b}}}}{2 a}+\frac {a \left (16 a^2+33 b^2\right ) \tan (c+d x) \sqrt {a+b \cos (c+d x)}}{d}}{4 a}+\frac {13 a b \tan (c+d x) \sec (c+d x) \sqrt {a+b \cos (c+d x)}}{2 d}\right )+\frac {a^2 \tan (c+d x) \sec ^2(c+d x) \sqrt {a+b \cos (c+d x)}}{3 d}\) |
| \(\Big \downarrow \) 3132 |
| \(\displaystyle \frac {1}{6} \left (\frac {\frac {\frac {\int \frac {b \left (16 a^2+59 b^2\right ) \sin \left (c+d x+\frac {\pi }{2}\right ) a^3+15 b^2 \left (4 a^2+b^2\right ) a^2}{\sin \left (c+d x+\frac {\pi }{2}\right ) \sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}}dx}{b}-\frac {2 a^2 \left (16 a^2+33 b^2\right ) \sqrt {a+b \cos (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )}{d \sqrt {\frac {a+b \cos (c+d x)}{a+b}}}}{2 a}+\frac {a \left (16 a^2+33 b^2\right ) \tan (c+d x) \sqrt {a+b \cos (c+d x)}}{d}}{4 a}+\frac {13 a b \tan (c+d x) \sec (c+d x) \sqrt {a+b \cos (c+d x)}}{2 d}\right )+\frac {a^2 \tan (c+d x) \sec ^2(c+d x) \sqrt {a+b \cos (c+d x)}}{3 d}\) |
| \(\Big \downarrow \) 3481 |
| \(\displaystyle \frac {1}{6} \left (\frac {\frac {\frac {15 a^2 b^2 \left (4 a^2+b^2\right ) \int \frac {\sec (c+d x)}{\sqrt {a+b \cos (c+d x)}}dx+a^3 b \left (16 a^2+59 b^2\right ) \int \frac {1}{\sqrt {a+b \cos (c+d x)}}dx}{b}-\frac {2 a^2 \left (16 a^2+33 b^2\right ) \sqrt {a+b \cos (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )}{d \sqrt {\frac {a+b \cos (c+d x)}{a+b}}}}{2 a}+\frac {a \left (16 a^2+33 b^2\right ) \tan (c+d x) \sqrt {a+b \cos (c+d x)}}{d}}{4 a}+\frac {13 a b \tan (c+d x) \sec (c+d x) \sqrt {a+b \cos (c+d x)}}{2 d}\right )+\frac {a^2 \tan (c+d x) \sec ^2(c+d x) \sqrt {a+b \cos (c+d x)}}{3 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{6} \left (\frac {\frac {\frac {15 a^2 b^2 \left (4 a^2+b^2\right ) \int \frac {1}{\sin \left (c+d x+\frac {\pi }{2}\right ) \sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}}dx+a^3 b \left (16 a^2+59 b^2\right ) \int \frac {1}{\sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}}dx}{b}-\frac {2 a^2 \left (16 a^2+33 b^2\right ) \sqrt {a+b \cos (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )}{d \sqrt {\frac {a+b \cos (c+d x)}{a+b}}}}{2 a}+\frac {a \left (16 a^2+33 b^2\right ) \tan (c+d x) \sqrt {a+b \cos (c+d x)}}{d}}{4 a}+\frac {13 a b \tan (c+d x) \sec (c+d x) \sqrt {a+b \cos (c+d x)}}{2 d}\right )+\frac {a^2 \tan (c+d x) \sec ^2(c+d x) \sqrt {a+b \cos (c+d x)}}{3 d}\) |
| \(\Big \downarrow \) 3142 |
| \(\displaystyle \frac {1}{6} \left (\frac {\frac {\frac {15 a^2 b^2 \left (4 a^2+b^2\right ) \int \frac {1}{\sin \left (c+d x+\frac {\pi }{2}\right ) \sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}}dx+\frac {a^3 b \left (16 a^2+59 b^2\right ) \sqrt {\frac {a+b \cos (c+d x)}{a+b}} \int \frac {1}{\sqrt {\frac {a}{a+b}+\frac {b \cos (c+d x)}{a+b}}}dx}{\sqrt {a+b \cos (c+d x)}}}{b}-\frac {2 a^2 \left (16 a^2+33 b^2\right ) \sqrt {a+b \cos (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )}{d \sqrt {\frac {a+b \cos (c+d x)}{a+b}}}}{2 a}+\frac {a \left (16 a^2+33 b^2\right ) \tan (c+d x) \sqrt {a+b \cos (c+d x)}}{d}}{4 a}+\frac {13 a b \tan (c+d x) \sec (c+d x) \sqrt {a+b \cos (c+d x)}}{2 d}\right )+\frac {a^2 \tan (c+d x) \sec ^2(c+d x) \sqrt {a+b \cos (c+d x)}}{3 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{6} \left (\frac {\frac {\frac {15 a^2 b^2 \left (4 a^2+b^2\right ) \int \frac {1}{\sin \left (c+d x+\frac {\pi }{2}\right ) \sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}}dx+\frac {a^3 b \left (16 a^2+59 b^2\right ) \sqrt {\frac {a+b \cos (c+d x)}{a+b}} \int \frac {1}{\sqrt {\frac {a}{a+b}+\frac {b \sin \left (c+d x+\frac {\pi }{2}\right )}{a+b}}}dx}{\sqrt {a+b \cos (c+d x)}}}{b}-\frac {2 a^2 \left (16 a^2+33 b^2\right ) \sqrt {a+b \cos (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )}{d \sqrt {\frac {a+b \cos (c+d x)}{a+b}}}}{2 a}+\frac {a \left (16 a^2+33 b^2\right ) \tan (c+d x) \sqrt {a+b \cos (c+d x)}}{d}}{4 a}+\frac {13 a b \tan (c+d x) \sec (c+d x) \sqrt {a+b \cos (c+d x)}}{2 d}\right )+\frac {a^2 \tan (c+d x) \sec ^2(c+d x) \sqrt {a+b \cos (c+d x)}}{3 d}\) |
| \(\Big \downarrow \) 3140 |
| \(\displaystyle \frac {1}{6} \left (\frac {\frac {\frac {15 a^2 b^2 \left (4 a^2+b^2\right ) \int \frac {1}{\sin \left (c+d x+\frac {\pi }{2}\right ) \sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}}dx+\frac {2 a^3 b \left (16 a^2+59 b^2\right ) \sqrt {\frac {a+b \cos (c+d x)}{a+b}} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),\frac {2 b}{a+b}\right )}{d \sqrt {a+b \cos (c+d x)}}}{b}-\frac {2 a^2 \left (16 a^2+33 b^2\right ) \sqrt {a+b \cos (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )}{d \sqrt {\frac {a+b \cos (c+d x)}{a+b}}}}{2 a}+\frac {a \left (16 a^2+33 b^2\right ) \tan (c+d x) \sqrt {a+b \cos (c+d x)}}{d}}{4 a}+\frac {13 a b \tan (c+d x) \sec (c+d x) \sqrt {a+b \cos (c+d x)}}{2 d}\right )+\frac {a^2 \tan (c+d x) \sec ^2(c+d x) \sqrt {a+b \cos (c+d x)}}{3 d}\) |
| \(\Big \downarrow \) 3286 |
| \(\displaystyle \frac {1}{6} \left (\frac {\frac {\frac {\frac {15 a^2 b^2 \left (4 a^2+b^2\right ) \sqrt {\frac {a+b \cos (c+d x)}{a+b}} \int \frac {\sec (c+d x)}{\sqrt {\frac {a}{a+b}+\frac {b \cos (c+d x)}{a+b}}}dx}{\sqrt {a+b \cos (c+d x)}}+\frac {2 a^3 b \left (16 a^2+59 b^2\right ) \sqrt {\frac {a+b \cos (c+d x)}{a+b}} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),\frac {2 b}{a+b}\right )}{d \sqrt {a+b \cos (c+d x)}}}{b}-\frac {2 a^2 \left (16 a^2+33 b^2\right ) \sqrt {a+b \cos (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )}{d \sqrt {\frac {a+b \cos (c+d x)}{a+b}}}}{2 a}+\frac {a \left (16 a^2+33 b^2\right ) \tan (c+d x) \sqrt {a+b \cos (c+d x)}}{d}}{4 a}+\frac {13 a b \tan (c+d x) \sec (c+d x) \sqrt {a+b \cos (c+d x)}}{2 d}\right )+\frac {a^2 \tan (c+d x) \sec ^2(c+d x) \sqrt {a+b \cos (c+d x)}}{3 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{6} \left (\frac {\frac {\frac {\frac {15 a^2 b^2 \left (4 a^2+b^2\right ) \sqrt {\frac {a+b \cos (c+d x)}{a+b}} \int \frac {1}{\sin \left (c+d x+\frac {\pi }{2}\right ) \sqrt {\frac {a}{a+b}+\frac {b \sin \left (c+d x+\frac {\pi }{2}\right )}{a+b}}}dx}{\sqrt {a+b \cos (c+d x)}}+\frac {2 a^3 b \left (16 a^2+59 b^2\right ) \sqrt {\frac {a+b \cos (c+d x)}{a+b}} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),\frac {2 b}{a+b}\right )}{d \sqrt {a+b \cos (c+d x)}}}{b}-\frac {2 a^2 \left (16 a^2+33 b^2\right ) \sqrt {a+b \cos (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )}{d \sqrt {\frac {a+b \cos (c+d x)}{a+b}}}}{2 a}+\frac {a \left (16 a^2+33 b^2\right ) \tan (c+d x) \sqrt {a+b \cos (c+d x)}}{d}}{4 a}+\frac {13 a b \tan (c+d x) \sec (c+d x) \sqrt {a+b \cos (c+d x)}}{2 d}\right )+\frac {a^2 \tan (c+d x) \sec ^2(c+d x) \sqrt {a+b \cos (c+d x)}}{3 d}\) |
| \(\Big \downarrow \) 3284 |
| \(\displaystyle \frac {a^2 \tan (c+d x) \sec ^2(c+d x) \sqrt {a+b \cos (c+d x)}}{3 d}+\frac {1}{6} \left (\frac {\frac {a \left (16 a^2+33 b^2\right ) \tan (c+d x) \sqrt {a+b \cos (c+d x)}}{d}+\frac {\frac {\frac {30 a^2 b^2 \left (4 a^2+b^2\right ) \sqrt {\frac {a+b \cos (c+d x)}{a+b}} \operatorname {EllipticPi}\left (2,\frac {1}{2} (c+d x),\frac {2 b}{a+b}\right )}{d \sqrt {a+b \cos (c+d x)}}+\frac {2 a^3 b \left (16 a^2+59 b^2\right ) \sqrt {\frac {a+b \cos (c+d x)}{a+b}} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),\frac {2 b}{a+b}\right )}{d \sqrt {a+b \cos (c+d x)}}}{b}-\frac {2 a^2 \left (16 a^2+33 b^2\right ) \sqrt {a+b \cos (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )}{d \sqrt {\frac {a+b \cos (c+d x)}{a+b}}}}{2 a}}{4 a}+\frac {13 a b \tan (c+d x) \sec (c+d x) \sqrt {a+b \cos (c+d x)}}{2 d}\right )\) |
(a^2*Sqrt[a + b*Cos[c + d*x]]*Sec[c + d*x]^2*Tan[c + d*x])/(3*d) + ((13*a* b*Sqrt[a + b*Cos[c + d*x]]*Sec[c + d*x]*Tan[c + d*x])/(2*d) + (((-2*a^2*(1 6*a^2 + 33*b^2)*Sqrt[a + b*Cos[c + d*x]]*EllipticE[(c + d*x)/2, (2*b)/(a + b)])/(d*Sqrt[(a + b*Cos[c + d*x])/(a + b)]) + ((2*a^3*b*(16*a^2 + 59*b^2) *Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*b)/(a + b)]) /(d*Sqrt[a + b*Cos[c + d*x]]) + (30*a^2*b^2*(4*a^2 + b^2)*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)/(2*a) + (a*(16*a^2 + 33*b^2)*Sqrt[a + b*Cos[c + d*x]] *Tan[c + d*x])/d)/(4*a))/6
3.6.7.3.1 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[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Simp[(-(b^2*c^2 - 2*a*b*c*d + a^2*d^2))*Co s[e + f*x]*(a + b*Sin[e + f*x])^(m - 2)*((c + d*Sin[e + f*x])^(n + 1)/(d*f* (n + 1)*(c^2 - d^2))), x] + Simp[1/(d*(n + 1)*(c^2 - d^2)) Int[(a + b*Sin [e + f*x])^(m - 3)*(c + d*Sin[e + f*x])^(n + 1)*Simp[b*(m - 2)*(b*c - a*d)^ 2 + a*d*(n + 1)*(c*(a^2 + b^2) - 2*a*b*d) + (b*(n + 1)*(a*b*c^2 + c*d*(a^2 + b^2) - 3*a*b*d^2) - a*(n + 2)*(b*c - a*d)^2)*Sin[e + f*x] + b*(b^2*(c^2 - d^2) - m*(b*c - a*d)^2 + d*n*(2*a*b*c - d*(a^2 + b^2)))*Sin[e + f*x]^2, 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[m, 2] && LtQ[n, -1] && (IntegerQ[m] || IntegersQ[2*m, 2*n])
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[(-(A*b^2 - a*b*B + a^2*C))*Cos[e + f*x ]*(a + b*Sin[e + f*x])^(m + 1)*((c + d*Sin[e + f*x])^(n + 1)/(f*(m + 1)*(b* c - a*d)*(a^2 - b^2))), x] + Simp[1/((m + 1)*(b*c - a*d)*(a^2 - b^2)) Int [(a + b*Sin[e + f*x])^(m + 1)*(c + d*Sin[e + f*x])^n*Simp[(m + 1)*(b*c - a* d)*(a*A - b*B + a*C) + d*(A*b^2 - a*b*B + a^2*C)*(m + n + 2) - (c*(A*b^2 - a*b*B + a^2*C) + (m + 1)*(b*c - a*d)*(A*b - a*B + b*C))*Sin[e + f*x] - d*(A *b^2 - a*b*B + a^2*C)*(m + n + 3)*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] && LtQ[m, -1] && ((EqQ[a, 0] && IntegerQ[m] && !IntegerQ [n]) || !(IntegerQ[2*n] && LtQ[n, -1] && ((IntegerQ[n] && !IntegerQ[m]) | | EqQ[a, 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. \(1741\) vs. \(2(380)=760\).
Time = 103.95 (sec) , antiderivative size = 1742, normalized size of antiderivative = 5.39
-1/24*((2*b*cos(1/2*d*x+1/2*c)^2+a-b)*sin(1/2*d*x+1/2*c)^2)^(1/2)*((256*a^ 2*b+528*b^3)*cos(1/2*d*x+1/2*c)*sin(1/2*d*x+1/2*c)^8+(-128*a^3-384*a^2*b-4 72*a*b^2-792*b^3)*sin(1/2*d*x+1/2*c)^6*cos(1/2*d*x+1/2*c)+(128*a^3+328*a^2 *b+472*a*b^2+396*b^3)*sin(1/2*d*x+1/2*c)^4*cos(1/2*d*x+1/2*c)+(-48*a^3-100 *a^2*b-118*a*b^2-66*b^3)*sin(1/2*d*x+1/2*c)^2*cos(1/2*d*x+1/2*c)+8*(-2*b/( a-b)*sin(1/2*d*x+1/2*c)^2+(a+b)/(a-b))^(1/2)*(sin(1/2*d*x+1/2*c)^2)^(1/2)* (60*EllipticPi(cos(1/2*d*x+1/2*c),2,(-2*b/(a-b))^(1/2))*a^2*b+15*b^3*Ellip ticPi(cos(1/2*d*x+1/2*c),2,(-2*b/(a-b))^(1/2))-16*EllipticF(cos(1/2*d*x+1/ 2*c),(-2*b/(a-b))^(1/2))*a^3-59*EllipticF(cos(1/2*d*x+1/2*c),(-2*b/(a-b))^ (1/2))*a*b^2+16*EllipticE(cos(1/2*d*x+1/2*c),(-2*b/(a-b))^(1/2))*a^3-16*El lipticE(cos(1/2*d*x+1/2*c),(-2*b/(a-b))^(1/2))*a^2*b+33*EllipticE(cos(1/2* d*x+1/2*c),(-2*b/(a-b))^(1/2))*a*b^2-33*EllipticE(cos(1/2*d*x+1/2*c),(-2*b /(a-b))^(1/2))*b^3)*sin(1/2*d*x+1/2*c)^6-12*(-2*b/(a-b)*sin(1/2*d*x+1/2*c) ^2+(a+b)/(a-b))^(1/2)*(sin(1/2*d*x+1/2*c)^2)^(1/2)*(60*EllipticPi(cos(1/2* d*x+1/2*c),2,(-2*b/(a-b))^(1/2))*a^2*b+15*b^3*EllipticPi(cos(1/2*d*x+1/2*c ),2,(-2*b/(a-b))^(1/2))-16*EllipticF(cos(1/2*d*x+1/2*c),(-2*b/(a-b))^(1/2) )*a^3-59*EllipticF(cos(1/2*d*x+1/2*c),(-2*b/(a-b))^(1/2))*a*b^2+16*Ellipti cE(cos(1/2*d*x+1/2*c),(-2*b/(a-b))^(1/2))*a^3-16*EllipticE(cos(1/2*d*x+1/2 *c),(-2*b/(a-b))^(1/2))*a^2*b+33*EllipticE(cos(1/2*d*x+1/2*c),(-2*b/(a-b)) ^(1/2))*a*b^2-33*EllipticE(cos(1/2*d*x+1/2*c),(-2*b/(a-b))^(1/2))*b^3)*...
Timed out. \[ \int (a+b \cos (c+d x))^{5/2} \sec ^4(c+d x) \, dx=\text {Timed out} \]
Timed out. \[ \int (a+b \cos (c+d x))^{5/2} \sec ^4(c+d x) \, dx=\text {Timed out} \]
\[ \int (a+b \cos (c+d x))^{5/2} \sec ^4(c+d x) \, dx=\int { {\left (b \cos \left (d x + c\right ) + a\right )}^{\frac {5}{2}} \sec \left (d x + c\right )^{4} \,d x } \]
\[ \int (a+b \cos (c+d x))^{5/2} \sec ^4(c+d x) \, dx=\int { {\left (b \cos \left (d x + c\right ) + a\right )}^{\frac {5}{2}} \sec \left (d x + c\right )^{4} \,d x } \]
Timed out. \[ \int (a+b \cos (c+d x))^{5/2} \sec ^4(c+d x) \, dx=\int \frac {{\left (a+b\,\cos \left (c+d\,x\right )\right )}^{5/2}}{{\cos \left (c+d\,x\right )}^4} \,d x \]