Integrand size = 25, antiderivative size = 369 \[ \int \sec ^{\frac {3}{2}}(c+d x) (a+b \sec (c+d x))^{5/2} \, dx=\frac {b \left (59 a^2+16 b^2\right ) \sqrt {\frac {b+a \cos (c+d x)}{a+b}} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),\frac {2 a}{a+b}\right ) \sqrt {\sec (c+d x)}}{24 d \sqrt {a+b \sec (c+d x)}}+\frac {5 a \left (a^2+4 b^2\right ) \sqrt {\frac {b+a \cos (c+d x)}{a+b}} \operatorname {EllipticPi}\left (2,\frac {1}{2} (c+d x),\frac {2 a}{a+b}\right ) \sqrt {\sec (c+d x)}}{8 d \sqrt {a+b \sec (c+d x)}}-\frac {\left (33 a^2+16 b^2\right ) E\left (\frac {1}{2} (c+d x)|\frac {2 a}{a+b}\right ) \sqrt {a+b \sec (c+d x)}}{24 d \sqrt {\frac {b+a \cos (c+d x)}{a+b}} \sqrt {\sec (c+d x)}}+\frac {\left (33 a^2+16 b^2\right ) \sqrt {\sec (c+d x)} \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{24 d}+\frac {13 a b \sec ^{\frac {3}{2}}(c+d x) \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{12 d}+\frac {b^2 \sec ^{\frac {5}{2}}(c+d x) \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{3 d} \]
1/24*b*(59*a^2+16*b^2)*(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)*(a/(a+b))^(1/2))*((b+a*cos(d*x+c))/(a+b) )^(1/2)*sec(d*x+c)^(1/2)/d/(a+b*sec(d*x+c))^(1/2)+5/8*a*(a^2+4*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)*(a/(a+b))^(1/2))*((b+a*cos(d*x+c))/(a+b))^(1/2)*sec(d*x+c)^(1/2)/d /(a+b*sec(d*x+c))^(1/2)+13/12*a*b*sec(d*x+c)^(3/2)*sin(d*x+c)*(a+b*sec(d*x +c))^(1/2)/d+1/3*b^2*sec(d*x+c)^(5/2)*sin(d*x+c)*(a+b*sec(d*x+c))^(1/2)/d- 1/24*(33*a^2+16*b^2)*(cos(1/2*d*x+1/2*c)^2)^(1/2)/cos(1/2*d*x+1/2*c)*Ellip ticE(sin(1/2*d*x+1/2*c),2^(1/2)*(a/(a+b))^(1/2))*(a+b*sec(d*x+c))^(1/2)/d/ ((b+a*cos(d*x+c))/(a+b))^(1/2)/sec(d*x+c)^(1/2)+1/24*(33*a^2+16*b^2)*sin(d *x+c)*sec(d*x+c)^(1/2)*(a+b*sec(d*x+c))^(1/2)/d
Result contains complex when optimal does not.
Time = 6.59 (sec) , antiderivative size = 602, normalized size of antiderivative = 1.63 \[ \int \sec ^{\frac {3}{2}}(c+d x) (a+b \sec (c+d x))^{5/2} \, dx=-\frac {a (a+b \sec (c+d x))^{5/2} \left (-\frac {104 a b \sqrt {\frac {b+a \cos (c+d x)}{a+b}} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),\frac {2 a}{a+b}\right )}{\sqrt {b+a \cos (c+d x)}}+\frac {2 \left (3 a^2-104 b^2\right ) \sqrt {\frac {b+a \cos (c+d x)}{a+b}} \operatorname {EllipticPi}\left (2,\frac {1}{2} (c+d x),\frac {2 a}{a+b}\right )}{\sqrt {b+a \cos (c+d x)}}+\frac {2 i \left (33 a^2+16 b^2\right ) \sqrt {\frac {a-a \cos (c+d x)}{a+b}} \sqrt {\frac {a+a \cos (c+d x)}{a-b}} \cos (2 (c+d x)) \left (-2 b (a+b) E\left (i \text {arcsinh}\left (\sqrt {\frac {1}{a-b}} \sqrt {b+a \cos (c+d x)}\right )|\frac {-a+b}{a+b}\right )+a \left (2 b \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {\frac {1}{a-b}} \sqrt {b+a \cos (c+d x)}\right ),\frac {-a+b}{a+b}\right )+a \operatorname {EllipticPi}\left (1-\frac {a}{b},i \text {arcsinh}\left (\sqrt {\frac {1}{a-b}} \sqrt {b+a \cos (c+d x)}\right ),\frac {-a+b}{a+b}\right )\right )\right ) \sin (c+d x)}{\sqrt {\frac {1}{a-b}} b \sqrt {1-\cos ^2(c+d x)} \sqrt {\frac {a^2-a^2 \cos ^2(c+d x)}{a^2}} \left (-a^2+2 b^2-4 b (b+a \cos (c+d x))+2 (b+a \cos (c+d x))^2\right )}\right )}{96 d (b+a \cos (c+d x))^{5/2} \sec ^{\frac {5}{2}}(c+d x)}+\frac {(a+b \sec (c+d x))^{5/2} \left (\frac {1}{24} \sec (c+d x) \left (33 a^2 \sin (c+d x)+16 b^2 \sin (c+d x)\right )+\frac {13}{12} a b \sec (c+d x) \tan (c+d x)+\frac {1}{3} b^2 \sec ^2(c+d x) \tan (c+d x)\right )}{d (b+a \cos (c+d x))^2 \sec ^{\frac {5}{2}}(c+d x)} \]
-1/96*(a*(a + b*Sec[c + d*x])^(5/2)*((-104*a*b*Sqrt[(b + a*Cos[c + d*x])/( a + b)]*EllipticF[(c + d*x)/2, (2*a)/(a + b)])/Sqrt[b + a*Cos[c + d*x]] + (2*(3*a^2 - 104*b^2)*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)*(33*a^2 + 16*b^ 2)*Sqrt[(a - a*Cos[c + d*x])/(a + b)]*Sqrt[(a + a*Cos[c + d*x])/(a - b)]*C os[2*(c + d*x)]*(-2*b*(a + b)*EllipticE[I*ArcSinh[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]*Sq rt[(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))))/(d*(b + a*Cos[c + d*x])^(5/2)*Sec[c + d*x ]^(5/2)) + ((a + b*Sec[c + d*x])^(5/2)*((Sec[c + d*x]*(33*a^2*Sin[c + d*x] + 16*b^2*Sin[c + d*x]))/24 + (13*a*b*Sec[c + d*x]*Tan[c + d*x])/12 + (b^2 *Sec[c + d*x]^2*Tan[c + d*x])/3))/(d*(b + a*Cos[c + d*x])^2*Sec[c + d*x]^( 5/2))
Time = 4.07 (sec) , antiderivative size = 390, normalized size of antiderivative = 1.06, number of steps used = 29, number of rules used = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.160, Rules used = {3042, 4329, 27, 3042, 4590, 27, 3042, 4590, 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.
\(\displaystyle \int \sec ^{\frac {3}{2}}(c+d x) (a+b \sec (c+d x))^{5/2} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \csc \left (c+d x+\frac {\pi }{2}\right )^{3/2} \left (a+b \csc \left (c+d x+\frac {\pi }{2}\right )\right )^{5/2}dx\) |
\(\Big \downarrow \) 4329 |
\(\displaystyle \frac {1}{3} \int \frac {\sec ^{\frac {3}{2}}(c+d x) \left (13 a b^2 \sec ^2(c+d x)+2 b \left (9 a^2+2 b^2\right ) \sec (c+d x)+3 a \left (2 a^2+b^2\right )\right )}{2 \sqrt {a+b \sec (c+d x)}}dx+\frac {b^2 \sin (c+d x) \sec ^{\frac {5}{2}}(c+d x) \sqrt {a+b \sec (c+d x)}}{3 d}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {1}{6} \int \frac {\sec ^{\frac {3}{2}}(c+d x) \left (13 a b^2 \sec ^2(c+d x)+2 b \left (9 a^2+2 b^2\right ) \sec (c+d x)+3 a \left (2 a^2+b^2\right )\right )}{\sqrt {a+b \sec (c+d x)}}dx+\frac {b^2 \sin (c+d x) \sec ^{\frac {5}{2}}(c+d x) \sqrt {a+b \sec (c+d x)}}{3 d}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {1}{6} \int \frac {\csc \left (c+d x+\frac {\pi }{2}\right )^{3/2} \left (13 a b^2 \csc \left (c+d x+\frac {\pi }{2}\right )^2+2 b \left (9 a^2+2 b^2\right ) \csc \left (c+d x+\frac {\pi }{2}\right )+3 a \left (2 a^2+b^2\right )\right )}{\sqrt {a+b \csc \left (c+d x+\frac {\pi }{2}\right )}}dx+\frac {b^2 \sin (c+d x) \sec ^{\frac {5}{2}}(c+d x) \sqrt {a+b \sec (c+d x)}}{3 d}\) |
\(\Big \downarrow \) 4590 |
\(\displaystyle \frac {1}{6} \left (\frac {\int \frac {\sqrt {\sec (c+d x)} \left (13 a^2 b^2+\left (33 a^2+16 b^2\right ) \sec ^2(c+d x) b^2+2 a \left (12 a^2+19 b^2\right ) \sec (c+d x) b\right )}{2 \sqrt {a+b \sec (c+d x)}}dx}{2 b}+\frac {13 a b \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x) \sqrt {a+b \sec (c+d x)}}{2 d}\right )+\frac {b^2 \sin (c+d x) \sec ^{\frac {5}{2}}(c+d x) \sqrt {a+b \sec (c+d x)}}{3 d}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {1}{6} \left (\frac {\int \frac {\sqrt {\sec (c+d x)} \left (13 a^2 b^2+\left (33 a^2+16 b^2\right ) \sec ^2(c+d x) b^2+2 a \left (12 a^2+19 b^2\right ) \sec (c+d x) b\right )}{\sqrt {a+b \sec (c+d x)}}dx}{4 b}+\frac {13 a b \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x) \sqrt {a+b \sec (c+d x)}}{2 d}\right )+\frac {b^2 \sin (c+d x) \sec ^{\frac {5}{2}}(c+d x) \sqrt {a+b \sec (c+d x)}}{3 d}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {1}{6} \left (\frac {\int \frac {\sqrt {\csc \left (c+d x+\frac {\pi }{2}\right )} \left (13 a^2 b^2+\left (33 a^2+16 b^2\right ) \csc \left (c+d x+\frac {\pi }{2}\right )^2 b^2+2 a \left (12 a^2+19 b^2\right ) \csc \left (c+d x+\frac {\pi }{2}\right ) b\right )}{\sqrt {a+b \csc \left (c+d x+\frac {\pi }{2}\right )}}dx}{4 b}+\frac {13 a b \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x) \sqrt {a+b \sec (c+d x)}}{2 d}\right )+\frac {b^2 \sin (c+d x) \sec ^{\frac {5}{2}}(c+d x) \sqrt {a+b \sec (c+d x)}}{3 d}\) |
\(\Big \downarrow \) 4590 |
\(\displaystyle \frac {1}{6} \left (\frac {\frac {\int -\frac {-26 a^2 \sec (c+d x) b^3-15 a \left (a^2+4 b^2\right ) \sec ^2(c+d x) b^2+a \left (33 a^2+16 b^2\right ) b^2}{2 \sqrt {\sec (c+d x)} \sqrt {a+b \sec (c+d x)}}dx}{b}+\frac {b \left (33 a^2+16 b^2\right ) \sin (c+d x) \sqrt {\sec (c+d x)} \sqrt {a+b \sec (c+d x)}}{d}}{4 b}+\frac {13 a b \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x) \sqrt {a+b \sec (c+d x)}}{2 d}\right )+\frac {b^2 \sin (c+d x) \sec ^{\frac {5}{2}}(c+d x) \sqrt {a+b \sec (c+d x)}}{3 d}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {1}{6} \left (\frac {\frac {b \left (33 a^2+16 b^2\right ) \sin (c+d x) \sqrt {\sec (c+d x)} \sqrt {a+b \sec (c+d x)}}{d}-\frac {\int \frac {-26 a^2 \sec (c+d x) b^3-15 a \left (a^2+4 b^2\right ) \sec ^2(c+d x) b^2+a \left (33 a^2+16 b^2\right ) b^2}{\sqrt {\sec (c+d x)} \sqrt {a+b \sec (c+d x)}}dx}{2 b}}{4 b}+\frac {13 a b \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x) \sqrt {a+b \sec (c+d x)}}{2 d}\right )+\frac {b^2 \sin (c+d x) \sec ^{\frac {5}{2}}(c+d x) \sqrt {a+b \sec (c+d x)}}{3 d}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {1}{6} \left (\frac {\frac {b \left (33 a^2+16 b^2\right ) \sin (c+d x) \sqrt {\sec (c+d x)} \sqrt {a+b \sec (c+d x)}}{d}-\frac {\int \frac {-26 a^2 \csc \left (c+d x+\frac {\pi }{2}\right ) b^3-15 a \left (a^2+4 b^2\right ) \csc \left (c+d x+\frac {\pi }{2}\right )^2 b^2+a \left (33 a^2+16 b^2\right ) b^2}{\sqrt {\csc \left (c+d x+\frac {\pi }{2}\right )} \sqrt {a+b \csc \left (c+d x+\frac {\pi }{2}\right )}}dx}{2 b}}{4 b}+\frac {13 a b \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x) \sqrt {a+b \sec (c+d x)}}{2 d}\right )+\frac {b^2 \sin (c+d x) \sec ^{\frac {5}{2}}(c+d x) \sqrt {a+b \sec (c+d x)}}{3 d}\) |
\(\Big \downarrow \) 4596 |
\(\displaystyle \frac {1}{6} \left (\frac {\frac {b \left (33 a^2+16 b^2\right ) \sin (c+d x) \sqrt {\sec (c+d x)} \sqrt {a+b \sec (c+d x)}}{d}-\frac {\int \frac {a b^2 \left (33 a^2+16 b^2\right )-26 a^2 b^3 \sec (c+d x)}{\sqrt {\sec (c+d x)} \sqrt {a+b \sec (c+d x)}}dx-15 a b^2 \left (a^2+4 b^2\right ) \int \frac {\sec ^{\frac {3}{2}}(c+d x)}{\sqrt {a+b \sec (c+d x)}}dx}{2 b}}{4 b}+\frac {13 a b \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x) \sqrt {a+b \sec (c+d x)}}{2 d}\right )+\frac {b^2 \sin (c+d x) \sec ^{\frac {5}{2}}(c+d x) \sqrt {a+b \sec (c+d x)}}{3 d}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {1}{6} \left (\frac {\frac {b \left (33 a^2+16 b^2\right ) \sin (c+d x) \sqrt {\sec (c+d x)} \sqrt {a+b \sec (c+d x)}}{d}-\frac {\int \frac {a b^2 \left (33 a^2+16 b^2\right )-26 a^2 b^3 \csc \left (c+d x+\frac {\pi }{2}\right )}{\sqrt {\csc \left (c+d x+\frac {\pi }{2}\right )} \sqrt {a+b \csc \left (c+d x+\frac {\pi }{2}\right )}}dx-15 a b^2 \left (a^2+4 b^2\right ) \int \frac {\csc \left (c+d x+\frac {\pi }{2}\right )^{3/2}}{\sqrt {a+b \csc \left (c+d x+\frac {\pi }{2}\right )}}dx}{2 b}}{4 b}+\frac {13 a b \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x) \sqrt {a+b \sec (c+d x)}}{2 d}\right )+\frac {b^2 \sin (c+d x) \sec ^{\frac {5}{2}}(c+d x) \sqrt {a+b \sec (c+d x)}}{3 d}\) |
\(\Big \downarrow \) 4346 |
\(\displaystyle \frac {1}{6} \left (\frac {\frac {b \left (33 a^2+16 b^2\right ) \sin (c+d x) \sqrt {\sec (c+d x)} \sqrt {a+b \sec (c+d x)}}{d}-\frac {\int \frac {a b^2 \left (33 a^2+16 b^2\right )-26 a^2 b^3 \csc \left (c+d x+\frac {\pi }{2}\right )}{\sqrt {\csc \left (c+d x+\frac {\pi }{2}\right )} \sqrt {a+b \csc \left (c+d x+\frac {\pi }{2}\right )}}dx-\frac {15 a b^2 \left (a^2+4 b^2\right ) \sqrt {\sec (c+d x)} \sqrt {a \cos (c+d x)+b} \int \frac {\sec (c+d x)}{\sqrt {b+a \cos (c+d x)}}dx}{\sqrt {a+b \sec (c+d x)}}}{2 b}}{4 b}+\frac {13 a b \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x) \sqrt {a+b \sec (c+d x)}}{2 d}\right )+\frac {b^2 \sin (c+d x) \sec ^{\frac {5}{2}}(c+d x) \sqrt {a+b \sec (c+d x)}}{3 d}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {1}{6} \left (\frac {\frac {b \left (33 a^2+16 b^2\right ) \sin (c+d x) \sqrt {\sec (c+d x)} \sqrt {a+b \sec (c+d x)}}{d}-\frac {\int \frac {a b^2 \left (33 a^2+16 b^2\right )-26 a^2 b^3 \csc \left (c+d x+\frac {\pi }{2}\right )}{\sqrt {\csc \left (c+d x+\frac {\pi }{2}\right )} \sqrt {a+b \csc \left (c+d x+\frac {\pi }{2}\right )}}dx-\frac {15 a b^2 \left (a^2+4 b^2\right ) \sqrt {\sec (c+d x)} \sqrt {a \cos (c+d x)+b} \int \frac {1}{\sin \left (c+d x+\frac {\pi }{2}\right ) \sqrt {b+a \sin \left (c+d x+\frac {\pi }{2}\right )}}dx}{\sqrt {a+b \sec (c+d x)}}}{2 b}}{4 b}+\frac {13 a b \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x) \sqrt {a+b \sec (c+d x)}}{2 d}\right )+\frac {b^2 \sin (c+d x) \sec ^{\frac {5}{2}}(c+d x) \sqrt {a+b \sec (c+d x)}}{3 d}\) |
\(\Big \downarrow \) 3286 |
\(\displaystyle \frac {1}{6} \left (\frac {\frac {b \left (33 a^2+16 b^2\right ) \sin (c+d x) \sqrt {\sec (c+d x)} \sqrt {a+b \sec (c+d x)}}{d}-\frac {\int \frac {a b^2 \left (33 a^2+16 b^2\right )-26 a^2 b^3 \csc \left (c+d x+\frac {\pi }{2}\right )}{\sqrt {\csc \left (c+d x+\frac {\pi }{2}\right )} \sqrt {a+b \csc \left (c+d x+\frac {\pi }{2}\right )}}dx-\frac {15 a b^2 \left (a^2+4 b^2\right ) \sqrt {\sec (c+d x)} \sqrt {\frac {a \cos (c+d x)+b}{a+b}} \int \frac {\sec (c+d x)}{\sqrt {\frac {b}{a+b}+\frac {a \cos (c+d x)}{a+b}}}dx}{\sqrt {a+b \sec (c+d x)}}}{2 b}}{4 b}+\frac {13 a b \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x) \sqrt {a+b \sec (c+d x)}}{2 d}\right )+\frac {b^2 \sin (c+d x) \sec ^{\frac {5}{2}}(c+d x) \sqrt {a+b \sec (c+d x)}}{3 d}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {1}{6} \left (\frac {\frac {b \left (33 a^2+16 b^2\right ) \sin (c+d x) \sqrt {\sec (c+d x)} \sqrt {a+b \sec (c+d x)}}{d}-\frac {\int \frac {a b^2 \left (33 a^2+16 b^2\right )-26 a^2 b^3 \csc \left (c+d x+\frac {\pi }{2}\right )}{\sqrt {\csc \left (c+d x+\frac {\pi }{2}\right )} \sqrt {a+b \csc \left (c+d x+\frac {\pi }{2}\right )}}dx-\frac {15 a b^2 \left (a^2+4 b^2\right ) \sqrt {\sec (c+d x)} \sqrt {\frac {a \cos (c+d x)+b}{a+b}} \int \frac {1}{\sin \left (c+d x+\frac {\pi }{2}\right ) \sqrt {\frac {b}{a+b}+\frac {a \sin \left (c+d x+\frac {\pi }{2}\right )}{a+b}}}dx}{\sqrt {a+b \sec (c+d x)}}}{2 b}}{4 b}+\frac {13 a b \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x) \sqrt {a+b \sec (c+d x)}}{2 d}\right )+\frac {b^2 \sin (c+d x) \sec ^{\frac {5}{2}}(c+d x) \sqrt {a+b \sec (c+d x)}}{3 d}\) |
\(\Big \downarrow \) 3284 |
\(\displaystyle \frac {1}{6} \left (\frac {\frac {b \left (33 a^2+16 b^2\right ) \sin (c+d x) \sqrt {\sec (c+d x)} \sqrt {a+b \sec (c+d x)}}{d}-\frac {\int \frac {a b^2 \left (33 a^2+16 b^2\right )-26 a^2 b^3 \csc \left (c+d x+\frac {\pi }{2}\right )}{\sqrt {\csc \left (c+d x+\frac {\pi }{2}\right )} \sqrt {a+b \csc \left (c+d x+\frac {\pi }{2}\right )}}dx-\frac {30 a b^2 \left (a^2+4 b^2\right ) \sqrt {\sec (c+d x)} \sqrt {\frac {a \cos (c+d x)+b}{a+b}} \operatorname {EllipticPi}\left (2,\frac {1}{2} (c+d x),\frac {2 a}{a+b}\right )}{d \sqrt {a+b \sec (c+d x)}}}{2 b}}{4 b}+\frac {13 a b \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x) \sqrt {a+b \sec (c+d x)}}{2 d}\right )+\frac {b^2 \sin (c+d x) \sec ^{\frac {5}{2}}(c+d x) \sqrt {a+b \sec (c+d x)}}{3 d}\) |
\(\Big \downarrow \) 4523 |
\(\displaystyle \frac {1}{6} \left (\frac {\frac {b \left (33 a^2+16 b^2\right ) \sin (c+d x) \sqrt {\sec (c+d x)} \sqrt {a+b \sec (c+d x)}}{d}-\frac {b^2 \left (33 a^2+16 b^2\right ) \int \frac {\sqrt {a+b \sec (c+d x)}}{\sqrt {\sec (c+d x)}}dx-\left (b^3 \left (59 a^2+16 b^2\right ) \int \frac {\sqrt {\sec (c+d x)}}{\sqrt {a+b \sec (c+d x)}}dx\right )-\frac {30 a b^2 \left (a^2+4 b^2\right ) \sqrt {\sec (c+d x)} \sqrt {\frac {a \cos (c+d x)+b}{a+b}} \operatorname {EllipticPi}\left (2,\frac {1}{2} (c+d x),\frac {2 a}{a+b}\right )}{d \sqrt {a+b \sec (c+d x)}}}{2 b}}{4 b}+\frac {13 a b \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x) \sqrt {a+b \sec (c+d x)}}{2 d}\right )+\frac {b^2 \sin (c+d x) \sec ^{\frac {5}{2}}(c+d x) \sqrt {a+b \sec (c+d x)}}{3 d}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {1}{6} \left (\frac {\frac {b \left (33 a^2+16 b^2\right ) \sin (c+d x) \sqrt {\sec (c+d x)} \sqrt {a+b \sec (c+d x)}}{d}-\frac {b^2 \left (33 a^2+16 b^2\right ) \int \frac {\sqrt {a+b \csc \left (c+d x+\frac {\pi }{2}\right )}}{\sqrt {\csc \left (c+d x+\frac {\pi }{2}\right )}}dx-\left (b^3 \left (59 a^2+16 b^2\right ) \int \frac {\sqrt {\csc \left (c+d x+\frac {\pi }{2}\right )}}{\sqrt {a+b \csc \left (c+d x+\frac {\pi }{2}\right )}}dx\right )-\frac {30 a b^2 \left (a^2+4 b^2\right ) \sqrt {\sec (c+d x)} \sqrt {\frac {a \cos (c+d x)+b}{a+b}} \operatorname {EllipticPi}\left (2,\frac {1}{2} (c+d x),\frac {2 a}{a+b}\right )}{d \sqrt {a+b \sec (c+d x)}}}{2 b}}{4 b}+\frac {13 a b \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x) \sqrt {a+b \sec (c+d x)}}{2 d}\right )+\frac {b^2 \sin (c+d x) \sec ^{\frac {5}{2}}(c+d x) \sqrt {a+b \sec (c+d x)}}{3 d}\) |
\(\Big \downarrow \) 4343 |
\(\displaystyle \frac {1}{6} \left (\frac {\frac {b \left (33 a^2+16 b^2\right ) \sin (c+d x) \sqrt {\sec (c+d x)} \sqrt {a+b \sec (c+d x)}}{d}-\frac {\frac {b^2 \left (33 a^2+16 b^2\right ) \sqrt {a+b \sec (c+d x)} \int \sqrt {b+a \cos (c+d x)}dx}{\sqrt {\sec (c+d x)} \sqrt {a \cos (c+d x)+b}}-\left (b^3 \left (59 a^2+16 b^2\right ) \int \frac {\sqrt {\csc \left (c+d x+\frac {\pi }{2}\right )}}{\sqrt {a+b \csc \left (c+d x+\frac {\pi }{2}\right )}}dx\right )-\frac {30 a b^2 \left (a^2+4 b^2\right ) \sqrt {\sec (c+d x)} \sqrt {\frac {a \cos (c+d x)+b}{a+b}} \operatorname {EllipticPi}\left (2,\frac {1}{2} (c+d x),\frac {2 a}{a+b}\right )}{d \sqrt {a+b \sec (c+d x)}}}{2 b}}{4 b}+\frac {13 a b \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x) \sqrt {a+b \sec (c+d x)}}{2 d}\right )+\frac {b^2 \sin (c+d x) \sec ^{\frac {5}{2}}(c+d x) \sqrt {a+b \sec (c+d x)}}{3 d}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {1}{6} \left (\frac {\frac {b \left (33 a^2+16 b^2\right ) \sin (c+d x) \sqrt {\sec (c+d x)} \sqrt {a+b \sec (c+d x)}}{d}-\frac {\frac {b^2 \left (33 a^2+16 b^2\right ) \sqrt {a+b \sec (c+d x)} \int \sqrt {b+a \sin \left (c+d x+\frac {\pi }{2}\right )}dx}{\sqrt {\sec (c+d x)} \sqrt {a \cos (c+d x)+b}}-\left (b^3 \left (59 a^2+16 b^2\right ) \int \frac {\sqrt {\csc \left (c+d x+\frac {\pi }{2}\right )}}{\sqrt {a+b \csc \left (c+d x+\frac {\pi }{2}\right )}}dx\right )-\frac {30 a b^2 \left (a^2+4 b^2\right ) \sqrt {\sec (c+d x)} \sqrt {\frac {a \cos (c+d x)+b}{a+b}} \operatorname {EllipticPi}\left (2,\frac {1}{2} (c+d x),\frac {2 a}{a+b}\right )}{d \sqrt {a+b \sec (c+d x)}}}{2 b}}{4 b}+\frac {13 a b \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x) \sqrt {a+b \sec (c+d x)}}{2 d}\right )+\frac {b^2 \sin (c+d x) \sec ^{\frac {5}{2}}(c+d x) \sqrt {a+b \sec (c+d x)}}{3 d}\) |
\(\Big \downarrow \) 3134 |
\(\displaystyle \frac {1}{6} \left (\frac {\frac {b \left (33 a^2+16 b^2\right ) \sin (c+d x) \sqrt {\sec (c+d x)} \sqrt {a+b \sec (c+d x)}}{d}-\frac {\frac {b^2 \left (33 a^2+16 b^2\right ) \sqrt {a+b \sec (c+d x)} \int \sqrt {\frac {b}{a+b}+\frac {a \cos (c+d x)}{a+b}}dx}{\sqrt {\sec (c+d x)} \sqrt {\frac {a \cos (c+d x)+b}{a+b}}}-\left (b^3 \left (59 a^2+16 b^2\right ) \int \frac {\sqrt {\csc \left (c+d x+\frac {\pi }{2}\right )}}{\sqrt {a+b \csc \left (c+d x+\frac {\pi }{2}\right )}}dx\right )-\frac {30 a b^2 \left (a^2+4 b^2\right ) \sqrt {\sec (c+d x)} \sqrt {\frac {a \cos (c+d x)+b}{a+b}} \operatorname {EllipticPi}\left (2,\frac {1}{2} (c+d x),\frac {2 a}{a+b}\right )}{d \sqrt {a+b \sec (c+d x)}}}{2 b}}{4 b}+\frac {13 a b \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x) \sqrt {a+b \sec (c+d x)}}{2 d}\right )+\frac {b^2 \sin (c+d x) \sec ^{\frac {5}{2}}(c+d x) \sqrt {a+b \sec (c+d x)}}{3 d}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {1}{6} \left (\frac {\frac {b \left (33 a^2+16 b^2\right ) \sin (c+d x) \sqrt {\sec (c+d x)} \sqrt {a+b \sec (c+d x)}}{d}-\frac {\frac {b^2 \left (33 a^2+16 b^2\right ) \sqrt {a+b \sec (c+d x)} \int \sqrt {\frac {b}{a+b}+\frac {a \sin \left (c+d x+\frac {\pi }{2}\right )}{a+b}}dx}{\sqrt {\sec (c+d x)} \sqrt {\frac {a \cos (c+d x)+b}{a+b}}}-\left (b^3 \left (59 a^2+16 b^2\right ) \int \frac {\sqrt {\csc \left (c+d x+\frac {\pi }{2}\right )}}{\sqrt {a+b \csc \left (c+d x+\frac {\pi }{2}\right )}}dx\right )-\frac {30 a b^2 \left (a^2+4 b^2\right ) \sqrt {\sec (c+d x)} \sqrt {\frac {a \cos (c+d x)+b}{a+b}} \operatorname {EllipticPi}\left (2,\frac {1}{2} (c+d x),\frac {2 a}{a+b}\right )}{d \sqrt {a+b \sec (c+d x)}}}{2 b}}{4 b}+\frac {13 a b \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x) \sqrt {a+b \sec (c+d x)}}{2 d}\right )+\frac {b^2 \sin (c+d x) \sec ^{\frac {5}{2}}(c+d x) \sqrt {a+b \sec (c+d x)}}{3 d}\) |
\(\Big \downarrow \) 3132 |
\(\displaystyle \frac {1}{6} \left (\frac {\frac {b \left (33 a^2+16 b^2\right ) \sin (c+d x) \sqrt {\sec (c+d x)} \sqrt {a+b \sec (c+d x)}}{d}-\frac {-\left (b^3 \left (59 a^2+16 b^2\right ) \int \frac {\sqrt {\csc \left (c+d x+\frac {\pi }{2}\right )}}{\sqrt {a+b \csc \left (c+d x+\frac {\pi }{2}\right )}}dx\right )+\frac {2 b^2 \left (33 a^2+16 b^2\right ) \sqrt {a+b \sec (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 a}{a+b}\right )}{d \sqrt {\sec (c+d x)} \sqrt {\frac {a \cos (c+d x)+b}{a+b}}}-\frac {30 a b^2 \left (a^2+4 b^2\right ) \sqrt {\sec (c+d x)} \sqrt {\frac {a \cos (c+d x)+b}{a+b}} \operatorname {EllipticPi}\left (2,\frac {1}{2} (c+d x),\frac {2 a}{a+b}\right )}{d \sqrt {a+b \sec (c+d x)}}}{2 b}}{4 b}+\frac {13 a b \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x) \sqrt {a+b \sec (c+d x)}}{2 d}\right )+\frac {b^2 \sin (c+d x) \sec ^{\frac {5}{2}}(c+d x) \sqrt {a+b \sec (c+d x)}}{3 d}\) |
\(\Big \downarrow \) 4345 |
\(\displaystyle \frac {1}{6} \left (\frac {\frac {b \left (33 a^2+16 b^2\right ) \sin (c+d x) \sqrt {\sec (c+d x)} \sqrt {a+b \sec (c+d x)}}{d}-\frac {-\frac {b^3 \left (59 a^2+16 b^2\right ) \sqrt {\sec (c+d x)} \sqrt {a \cos (c+d x)+b} \int \frac {1}{\sqrt {b+a \cos (c+d x)}}dx}{\sqrt {a+b \sec (c+d x)}}+\frac {2 b^2 \left (33 a^2+16 b^2\right ) \sqrt {a+b \sec (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 a}{a+b}\right )}{d \sqrt {\sec (c+d x)} \sqrt {\frac {a \cos (c+d x)+b}{a+b}}}-\frac {30 a b^2 \left (a^2+4 b^2\right ) \sqrt {\sec (c+d x)} \sqrt {\frac {a \cos (c+d x)+b}{a+b}} \operatorname {EllipticPi}\left (2,\frac {1}{2} (c+d x),\frac {2 a}{a+b}\right )}{d \sqrt {a+b \sec (c+d x)}}}{2 b}}{4 b}+\frac {13 a b \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x) \sqrt {a+b \sec (c+d x)}}{2 d}\right )+\frac {b^2 \sin (c+d x) \sec ^{\frac {5}{2}}(c+d x) \sqrt {a+b \sec (c+d x)}}{3 d}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {1}{6} \left (\frac {\frac {b \left (33 a^2+16 b^2\right ) \sin (c+d x) \sqrt {\sec (c+d x)} \sqrt {a+b \sec (c+d x)}}{d}-\frac {-\frac {b^3 \left (59 a^2+16 b^2\right ) \sqrt {\sec (c+d x)} \sqrt {a \cos (c+d x)+b} \int \frac {1}{\sqrt {b+a \sin \left (c+d x+\frac {\pi }{2}\right )}}dx}{\sqrt {a+b \sec (c+d x)}}+\frac {2 b^2 \left (33 a^2+16 b^2\right ) \sqrt {a+b \sec (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 a}{a+b}\right )}{d \sqrt {\sec (c+d x)} \sqrt {\frac {a \cos (c+d x)+b}{a+b}}}-\frac {30 a b^2 \left (a^2+4 b^2\right ) \sqrt {\sec (c+d x)} \sqrt {\frac {a \cos (c+d x)+b}{a+b}} \operatorname {EllipticPi}\left (2,\frac {1}{2} (c+d x),\frac {2 a}{a+b}\right )}{d \sqrt {a+b \sec (c+d x)}}}{2 b}}{4 b}+\frac {13 a b \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x) \sqrt {a+b \sec (c+d x)}}{2 d}\right )+\frac {b^2 \sin (c+d x) \sec ^{\frac {5}{2}}(c+d x) \sqrt {a+b \sec (c+d x)}}{3 d}\) |
\(\Big \downarrow \) 3142 |
\(\displaystyle \frac {1}{6} \left (\frac {\frac {b \left (33 a^2+16 b^2\right ) \sin (c+d x) \sqrt {\sec (c+d x)} \sqrt {a+b \sec (c+d x)}}{d}-\frac {-\frac {b^3 \left (59 a^2+16 b^2\right ) \sqrt {\sec (c+d x)} \sqrt {\frac {a \cos (c+d x)+b}{a+b}} \int \frac {1}{\sqrt {\frac {b}{a+b}+\frac {a \cos (c+d x)}{a+b}}}dx}{\sqrt {a+b \sec (c+d x)}}+\frac {2 b^2 \left (33 a^2+16 b^2\right ) \sqrt {a+b \sec (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 a}{a+b}\right )}{d \sqrt {\sec (c+d x)} \sqrt {\frac {a \cos (c+d x)+b}{a+b}}}-\frac {30 a b^2 \left (a^2+4 b^2\right ) \sqrt {\sec (c+d x)} \sqrt {\frac {a \cos (c+d x)+b}{a+b}} \operatorname {EllipticPi}\left (2,\frac {1}{2} (c+d x),\frac {2 a}{a+b}\right )}{d \sqrt {a+b \sec (c+d x)}}}{2 b}}{4 b}+\frac {13 a b \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x) \sqrt {a+b \sec (c+d x)}}{2 d}\right )+\frac {b^2 \sin (c+d x) \sec ^{\frac {5}{2}}(c+d x) \sqrt {a+b \sec (c+d x)}}{3 d}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {1}{6} \left (\frac {\frac {b \left (33 a^2+16 b^2\right ) \sin (c+d x) \sqrt {\sec (c+d x)} \sqrt {a+b \sec (c+d x)}}{d}-\frac {-\frac {b^3 \left (59 a^2+16 b^2\right ) \sqrt {\sec (c+d x)} \sqrt {\frac {a \cos (c+d x)+b}{a+b}} \int \frac {1}{\sqrt {\frac {b}{a+b}+\frac {a \sin \left (c+d x+\frac {\pi }{2}\right )}{a+b}}}dx}{\sqrt {a+b \sec (c+d x)}}+\frac {2 b^2 \left (33 a^2+16 b^2\right ) \sqrt {a+b \sec (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 a}{a+b}\right )}{d \sqrt {\sec (c+d x)} \sqrt {\frac {a \cos (c+d x)+b}{a+b}}}-\frac {30 a b^2 \left (a^2+4 b^2\right ) \sqrt {\sec (c+d x)} \sqrt {\frac {a \cos (c+d x)+b}{a+b}} \operatorname {EllipticPi}\left (2,\frac {1}{2} (c+d x),\frac {2 a}{a+b}\right )}{d \sqrt {a+b \sec (c+d x)}}}{2 b}}{4 b}+\frac {13 a b \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x) \sqrt {a+b \sec (c+d x)}}{2 d}\right )+\frac {b^2 \sin (c+d x) \sec ^{\frac {5}{2}}(c+d x) \sqrt {a+b \sec (c+d x)}}{3 d}\) |
\(\Big \downarrow \) 3140 |
\(\displaystyle \frac {1}{6} \left (\frac {\frac {b \left (33 a^2+16 b^2\right ) \sin (c+d x) \sqrt {\sec (c+d x)} \sqrt {a+b \sec (c+d x)}}{d}-\frac {\frac {2 b^2 \left (33 a^2+16 b^2\right ) \sqrt {a+b \sec (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 a}{a+b}\right )}{d \sqrt {\sec (c+d x)} \sqrt {\frac {a \cos (c+d x)+b}{a+b}}}-\frac {30 a b^2 \left (a^2+4 b^2\right ) \sqrt {\sec (c+d x)} \sqrt {\frac {a \cos (c+d x)+b}{a+b}} \operatorname {EllipticPi}\left (2,\frac {1}{2} (c+d x),\frac {2 a}{a+b}\right )}{d \sqrt {a+b \sec (c+d x)}}-\frac {2 b^3 \left (59 a^2+16 b^2\right ) \sqrt {\sec (c+d x)} \sqrt {\frac {a \cos (c+d x)+b}{a+b}} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),\frac {2 a}{a+b}\right )}{d \sqrt {a+b \sec (c+d x)}}}{2 b}}{4 b}+\frac {13 a b \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x) \sqrt {a+b \sec (c+d x)}}{2 d}\right )+\frac {b^2 \sin (c+d x) \sec ^{\frac {5}{2}}(c+d x) \sqrt {a+b \sec (c+d x)}}{3 d}\) |
(b^2*Sec[c + d*x]^(5/2)*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(3*d) + ((1 3*a*b*Sec[c + d*x]^(3/2)*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(2*d) + (- 1/2*((-2*b^3*(59*a^2 + 16*b^2)*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*Elliptic F[(c + d*x)/2, (2*a)/(a + b)]*Sqrt[Sec[c + d*x]])/(d*Sqrt[a + b*Sec[c + d* x]]) - (30*a*b^2*(a^2 + 4*b^2)*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*Elliptic Pi[2, (c + d*x)/2, (2*a)/(a + b)]*Sqrt[Sec[c + d*x]])/(d*Sqrt[a + b*Sec[c + d*x]]) + (2*b^2*(33*a^2 + 16*b^2)*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]]))/b + (b*(33*a^2 + 16*b^2)*Sqrt[Sec[c + d*x]]*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/d)/(4*b))/6
3.7.40.3.1 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[(csc[(e_.) + (f_.)*(x_)]*(d_.))^(n_)*(csc[(e_.) + (f_.)*(x_)]*(b_.) + ( a_))^(m_), x_Symbol] :> Simp[(-b^2)*Cot[e + f*x]*(a + b*Csc[e + f*x])^(m - 2)*((d*Csc[e + f*x])^n/(f*(m + n - 1))), x] + Simp[1/(d*(m + n - 1)) Int[ (a + b*Csc[e + f*x])^(m - 3)*(d*Csc[e + f*x])^n*Simp[a^3*d*(m + n - 1) + a* b^2*d*n + b*(b^2*d*(m + n - 2) + 3*a^2*d*(m + n - 1))*Csc[e + f*x] + a*b^2* d*(3*m + 2*n - 4)*Csc[e + f*x]^2, x], x], x] /; FreeQ[{a, b, d, e, f, n}, x ] && NeQ[a^2 - b^2, 0] && GtQ[m, 2] && (IntegerQ[m] || IntegersQ[2*m, 2*n]) && !(IGtQ[n, 2] && !IntegerQ[m])
Int[Sqrt[csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_)]/Sqrt[csc[(e_.) + (f_.)*(x_)] *(d_.)], x_Symbol] :> Simp[Sqrt[a + b*Csc[e + f*x]]/(Sqrt[d*Csc[e + f*x]]*S qrt[b + a*Sin[e + f*x]]) Int[Sqrt[b + a*Sin[e + f*x]], x], x] /; FreeQ[{a , b, d, e, f}, x] && NeQ[a^2 - b^2, 0]
Int[Sqrt[csc[(e_.) + (f_.)*(x_)]*(d_.)]/Sqrt[csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_)], x_Symbol] :> Simp[Sqrt[d*Csc[e + f*x]]*(Sqrt[b + a*Sin[e + f*x]]/S qrt[a + b*Csc[e + f*x]]) Int[1/Sqrt[b + a*Sin[e + f*x]], x], x] /; FreeQ[ {a, b, d, e, f}, x] && NeQ[a^2 - b^2, 0]
Int[(csc[(e_.) + (f_.)*(x_)]*(d_.))^(3/2)/Sqrt[csc[(e_.) + (f_.)*(x_)]*(b_. ) + (a_)], x_Symbol] :> Simp[d*Sqrt[d*Csc[e + f*x]]*(Sqrt[b + a*Sin[e + f*x ]]/Sqrt[a + b*Csc[e + f*x]]) Int[1/(Sin[e + f*x]*Sqrt[b + a*Sin[e + f*x]] ), x], x] /; FreeQ[{a, b, d, e, f}, x] && NeQ[a^2 - b^2, 0]
Int[(csc[(e_.) + (f_.)*(x_)]*(B_.) + (A_))/(Sqrt[csc[(e_.) + (f_.)*(x_)]*(d _.)]*Sqrt[csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_)]), x_Symbol] :> Simp[A/a I nt[Sqrt[a + b*Csc[e + f*x]]/Sqrt[d*Csc[e + f*x]], x], x] - Simp[(A*b - a*B) /(a*d) Int[Sqrt[d*Csc[e + f*x]]/Sqrt[a + b*Csc[e + f*x]], x], x] /; FreeQ [{a, b, d, e, f, A, B}, x] && NeQ[A*b - a*B, 0] && NeQ[a^2 - b^2, 0]
Int[((A_.) + csc[(e_.) + (f_.)*(x_)]*(B_.) + csc[(e_.) + (f_.)*(x_)]^2*(C_. ))*(csc[(e_.) + (f_.)*(x_)]*(d_.))^(n_)*(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a _))^(m_), x_Symbol] :> Simp[(-C)*d*Cot[e + f*x]*(a + b*Csc[e + f*x])^(m + 1 )*((d*Csc[e + f*x])^(n - 1)/(b*f*(m + n + 1))), x] + Simp[d/(b*(m + n + 1)) Int[(a + b*Csc[e + f*x])^m*(d*Csc[e + f*x])^(n - 1)*Simp[a*C*(n - 1) + ( A*b*(m + n + 1) + b*C*(m + n))*Csc[e + f*x] + (b*B*(m + n + 1) - a*C*n)*Csc [e + f*x]^2, x], x], x] /; FreeQ[{a, b, d, e, f, A, B, C, m}, x] && NeQ[a^2 - b^2, 0] && GtQ[n, 0]
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 = 12.22 (sec) , antiderivative size = 3037, normalized size of antiderivative = 8.23
1/24/d/((a-b)/(a+b))^(1/2)*(a+b*sec(d*x+c))^(1/2)*sec(d*x+c)^(3/2)/(b+a*co s(d*x+c))/(cos(d*x+c)+1)*(-33*EllipticE(((a-b)/(a+b))^(1/2)*(cot(d*x+c)-cs c(d*x+c)),(-(a+b)/(a-b))^(1/2))*(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^2*b*cos(d*x+c)^4+16*EllipticE(((a-b)/(a+b ))^(1/2)*(cot(d*x+c)-csc(d*x+c)),(-(a+b)/(a-b))^(1/2))*(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*b^2*cos(d*x+c)^4-2 6*EllipticF(((a-b)/(a+b))^(1/2)*(cot(d*x+c)-csc(d*x+c)),(-(a+b)/(a-b))^(1/ 2))*(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^2*b*cos(d*x+c)^4+44*EllipticF(((a-b)/(a+b))^(1/2)*(cot(d*x+c)-csc(d*x +c)),(-(a+b)/(a-b))^(1/2))*(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*b^2*cos(d*x+c)^4-120*EllipticPi(((a-b)/(a+b))^ (1/2)*(cot(d*x+c)-csc(d*x+c)),(a+b)/(a-b),I/((a-b)/(a+b))^(1/2))*(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*b^2*cos( d*x+c)^4-66*EllipticE(((a-b)/(a+b))^(1/2)*(cot(d*x+c)-csc(d*x+c)),(-(a+b)/ (a-b))^(1/2))*(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^2*b*cos(d*x+c)^3+32*EllipticE(((a-b)/(a+b))^(1/2)*(cot(d*x+ c)-csc(d*x+c)),(-(a+b)/(a-b))^(1/2))*(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*b^2*cos(d*x+c)^3-52*EllipticF(((a-b) /(a+b))^(1/2)*(cot(d*x+c)-csc(d*x+c)),(-(a+b)/(a-b))^(1/2))*(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^2*b*cos(d*...
Timed out. \[ \int \sec ^{\frac {3}{2}}(c+d x) (a+b \sec (c+d x))^{5/2} \, dx=\text {Timed out} \]
Timed out. \[ \int \sec ^{\frac {3}{2}}(c+d x) (a+b \sec (c+d x))^{5/2} \, dx=\text {Timed out} \]
\[ \int \sec ^{\frac {3}{2}}(c+d x) (a+b \sec (c+d x))^{5/2} \, dx=\int { {\left (b \sec \left (d x + c\right ) + a\right )}^{\frac {5}{2}} \sec \left (d x + c\right )^{\frac {3}{2}} \,d x } \]
\[ \int \sec ^{\frac {3}{2}}(c+d x) (a+b \sec (c+d x))^{5/2} \, dx=\int { {\left (b \sec \left (d x + c\right ) + a\right )}^{\frac {5}{2}} \sec \left (d x + c\right )^{\frac {3}{2}} \,d x } \]
Timed out. \[ \int \sec ^{\frac {3}{2}}(c+d x) (a+b \sec (c+d x))^{5/2} \, dx=\int {\left (a+\frac {b}{\cos \left (c+d\,x\right )}\right )}^{5/2}\,{\left (\frac {1}{\cos \left (c+d\,x\right )}\right )}^{3/2} \,d x \]