Integrand size = 23, antiderivative size = 388 \[ \int \frac {\sec ^{\frac {9}{2}}(c+d x)}{(a+b \sec (c+d x))^3} \, dx=-\frac {\left (15 a^4-29 a^2 b^2+8 b^4\right ) \sqrt {\cos (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \sqrt {\sec (c+d x)}}{4 b^3 \left (a^2-b^2\right )^2 d}-\frac {a \left (5 a^2-11 b^2\right ) \sqrt {\cos (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right ) \sqrt {\sec (c+d x)}}{4 b^2 \left (a^2-b^2\right )^2 d}-\frac {a \left (15 a^4-38 a^2 b^2+35 b^4\right ) \sqrt {\cos (c+d x)} \operatorname {EllipticPi}\left (\frac {2 a}{a+b},\frac {1}{2} (c+d x),2\right ) \sqrt {\sec (c+d x)}}{4 (a-b)^2 b^3 (a+b)^3 d}+\frac {\left (15 a^4-29 a^2 b^2+8 b^4\right ) \sqrt {\sec (c+d x)} \sin (c+d x)}{4 b^3 \left (a^2-b^2\right )^2 d}-\frac {a^2 \sec ^{\frac {5}{2}}(c+d x) \sin (c+d x)}{2 b \left (a^2-b^2\right ) d (a+b \sec (c+d x))^2}-\frac {a^2 \left (5 a^2-11 b^2\right ) \sec ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{4 b^2 \left (a^2-b^2\right )^2 d (a+b \sec (c+d x))} \]
-1/2*a^2*sec(d*x+c)^(5/2)*sin(d*x+c)/b/(a^2-b^2)/d/(a+b*sec(d*x+c))^2-1/4* a^2*(5*a^2-11*b^2)*sec(d*x+c)^(3/2)*sin(d*x+c)/b^2/(a^2-b^2)^2/d/(a+b*sec( d*x+c))+1/4*(15*a^4-29*a^2*b^2+8*b^4)*sin(d*x+c)*sec(d*x+c)^(1/2)/b^3/(a^2 -b^2)^2/d-1/4*(15*a^4-29*a^2*b^2+8*b^4)*(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))*cos(d*x+c)^(1/2)*sec(d *x+c)^(1/2)/b^3/(a^2-b^2)^2/d-1/4*a*(5*a^2-11*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))*cos(d*x+c)^ (1/2)*sec(d*x+c)^(1/2)/b^2/(a^2-b^2)^2/d-1/4*a*(15*a^4-38*a^2*b^2+35*b^4)* (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*a/(a+b),2^(1/2))*cos(d*x+c)^(1/2)*sec(d*x+c)^(1/2)/(a-b)^2/b^3/(a+b) ^3/d
Time = 6.64 (sec) , antiderivative size = 721, normalized size of antiderivative = 1.86 \[ \int \frac {\sec ^{\frac {9}{2}}(c+d x)}{(a+b \sec (c+d x))^3} \, dx=-\frac {\frac {2 \left (45 a^5-95 a^3 b^2+56 a b^4\right ) \cos ^2(c+d x) \left (\operatorname {EllipticF}\left (\arcsin \left (\sqrt {\sec (c+d x)}\right ),-1\right )-\operatorname {EllipticPi}\left (-\frac {b}{a},\arcsin \left (\sqrt {\sec (c+d x)}\right ),-1\right )\right ) (a+b \sec (c+d x)) \sqrt {1-\sec ^2(c+d x)} \sin (c+d x)}{b (b+a \cos (c+d x)) \left (1-\cos ^2(c+d x)\right )}+\frac {2 \left (40 a^4 b-80 a^2 b^3+16 b^5\right ) \cos ^2(c+d x) \operatorname {EllipticPi}\left (-\frac {b}{a},\arcsin \left (\sqrt {\sec (c+d x)}\right ),-1\right ) (a+b \sec (c+d x)) \sqrt {1-\sec ^2(c+d x)} \sin (c+d x)}{a (b+a \cos (c+d x)) \left (1-\cos ^2(c+d x)\right )}+\frac {\left (15 a^5-29 a^3 b^2+8 a b^4\right ) \cos (2 (c+d x)) (a+b \sec (c+d x)) \left (-4 a b+4 a b \sec ^2(c+d x)-4 a b E\left (\left .\arcsin \left (\sqrt {\sec (c+d x)}\right )\right |-1\right ) \sqrt {\sec (c+d x)} \sqrt {1-\sec ^2(c+d x)}-2 a (a-2 b) \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\sec (c+d x)}\right ),-1\right ) \sqrt {\sec (c+d x)} \sqrt {1-\sec ^2(c+d x)}+2 a^2 \operatorname {EllipticPi}\left (-\frac {b}{a},\arcsin \left (\sqrt {\sec (c+d x)}\right ),-1\right ) \sqrt {\sec (c+d x)} \sqrt {1-\sec ^2(c+d x)}-4 b^2 \operatorname {EllipticPi}\left (-\frac {b}{a},\arcsin \left (\sqrt {\sec (c+d x)}\right ),-1\right ) \sqrt {\sec (c+d x)} \sqrt {1-\sec ^2(c+d x)}\right ) \sin (c+d x)}{a^2 b (b+a \cos (c+d x)) \left (1-\cos ^2(c+d x)\right ) \sqrt {\sec (c+d x)} \left (2-\sec ^2(c+d x)\right )}}{16 (a-b)^2 b^3 (a+b)^2 d}+\frac {\sqrt {\sec (c+d x)} \left (\frac {\left (15 a^4-29 a^2 b^2+8 b^4\right ) \sin (c+d x)}{4 b^3 \left (-a^2+b^2\right )^2}+\frac {a^2 \sin (c+d x)}{2 b \left (-a^2+b^2\right ) (b+a \cos (c+d x))^2}+\frac {-5 a^4 \sin (c+d x)+11 a^2 b^2 \sin (c+d x)}{4 b^2 \left (-a^2+b^2\right )^2 (b+a \cos (c+d x))}\right )}{d} \]
-1/16*((2*(45*a^5 - 95*a^3*b^2 + 56*a*b^4)*Cos[c + d*x]^2*(EllipticF[ArcSi n[Sqrt[Sec[c + d*x]]], -1] - EllipticPi[-(b/a), ArcSin[Sqrt[Sec[c + d*x]]] , -1])*(a + b*Sec[c + d*x])*Sqrt[1 - Sec[c + d*x]^2]*Sin[c + d*x])/(b*(b + a*Cos[c + d*x])*(1 - Cos[c + d*x]^2)) + (2*(40*a^4*b - 80*a^2*b^3 + 16*b^ 5)*Cos[c + d*x]^2*EllipticPi[-(b/a), ArcSin[Sqrt[Sec[c + d*x]]], -1]*(a + b*Sec[c + d*x])*Sqrt[1 - Sec[c + d*x]^2]*Sin[c + d*x])/(a*(b + a*Cos[c + d *x])*(1 - Cos[c + d*x]^2)) + ((15*a^5 - 29*a^3*b^2 + 8*a*b^4)*Cos[2*(c + d *x)]*(a + b*Sec[c + d*x])*(-4*a*b + 4*a*b*Sec[c + d*x]^2 - 4*a*b*EllipticE [ArcSin[Sqrt[Sec[c + d*x]]], -1]*Sqrt[Sec[c + d*x]]*Sqrt[1 - Sec[c + d*x]^ 2] - 2*a*(a - 2*b)*EllipticF[ArcSin[Sqrt[Sec[c + d*x]]], -1]*Sqrt[Sec[c + d*x]]*Sqrt[1 - Sec[c + d*x]^2] + 2*a^2*EllipticPi[-(b/a), ArcSin[Sqrt[Sec[ c + d*x]]], -1]*Sqrt[Sec[c + d*x]]*Sqrt[1 - Sec[c + d*x]^2] - 4*b^2*Ellipt icPi[-(b/a), ArcSin[Sqrt[Sec[c + d*x]]], -1]*Sqrt[Sec[c + d*x]]*Sqrt[1 - S ec[c + d*x]^2])*Sin[c + d*x])/(a^2*b*(b + a*Cos[c + d*x])*(1 - Cos[c + d*x ]^2)*Sqrt[Sec[c + d*x]]*(2 - Sec[c + d*x]^2)))/((a - b)^2*b^3*(a + b)^2*d) + (Sqrt[Sec[c + d*x]]*(((15*a^4 - 29*a^2*b^2 + 8*b^4)*Sin[c + d*x])/(4*b^ 3*(-a^2 + b^2)^2) + (a^2*Sin[c + d*x])/(2*b*(-a^2 + b^2)*(b + a*Cos[c + d* x])^2) + (-5*a^4*Sin[c + d*x] + 11*a^2*b^2*Sin[c + d*x])/(4*b^2*(-a^2 + b^ 2)^2*(b + a*Cos[c + d*x]))))/d
Time = 2.79 (sec) , antiderivative size = 382, normalized size of antiderivative = 0.98, number of steps used = 21, number of rules used = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.913, Rules used = {3042, 4332, 27, 3042, 4586, 27, 3042, 4590, 27, 3042, 4594, 3042, 4274, 3042, 4258, 3042, 3119, 3120, 4336, 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.
| \(\displaystyle \int \frac {\sec ^{\frac {9}{2}}(c+d x)}{(a+b \sec (c+d x))^3} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\csc \left (c+d x+\frac {\pi }{2}\right )^{9/2}}{\left (a+b \csc \left (c+d x+\frac {\pi }{2}\right )\right )^3}dx\) |
| \(\Big \downarrow \) 4332 |
| \(\displaystyle -\frac {\int \frac {\sec ^{\frac {3}{2}}(c+d x) \left (3 a^2-4 b \sec (c+d x) a-\left (5 a^2-4 b^2\right ) \sec ^2(c+d x)\right )}{2 (a+b \sec (c+d x))^2}dx}{2 b \left (a^2-b^2\right )}-\frac {a^2 \sin (c+d x) \sec ^{\frac {5}{2}}(c+d x)}{2 b d \left (a^2-b^2\right ) (a+b \sec (c+d x))^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {\int \frac {\sec ^{\frac {3}{2}}(c+d x) \left (3 a^2-4 b \sec (c+d x) a-\left (5 a^2-4 b^2\right ) \sec ^2(c+d x)\right )}{(a+b \sec (c+d x))^2}dx}{4 b \left (a^2-b^2\right )}-\frac {a^2 \sin (c+d x) \sec ^{\frac {5}{2}}(c+d x)}{2 b d \left (a^2-b^2\right ) (a+b \sec (c+d x))^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {\int \frac {\csc \left (c+d x+\frac {\pi }{2}\right )^{3/2} \left (3 a^2-4 b \csc \left (c+d x+\frac {\pi }{2}\right ) a+\left (4 b^2-5 a^2\right ) \csc \left (c+d x+\frac {\pi }{2}\right )^2\right )}{\left (a+b \csc \left (c+d x+\frac {\pi }{2}\right )\right )^2}dx}{4 b \left (a^2-b^2\right )}-\frac {a^2 \sin (c+d x) \sec ^{\frac {5}{2}}(c+d x)}{2 b d \left (a^2-b^2\right ) (a+b \sec (c+d x))^2}\) |
| \(\Big \downarrow \) 4586 |
| \(\displaystyle -\frac {\frac {a^2 \left (5 a^2-11 b^2\right ) \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x)}{b d \left (a^2-b^2\right ) (a+b \sec (c+d x))}-\frac {\int -\frac {\sqrt {\sec (c+d x)} \left (\left (5 a^2-11 b^2\right ) a^2-4 b \left (a^2-4 b^2\right ) \sec (c+d x) a-\left (15 a^4-29 b^2 a^2+8 b^4\right ) \sec ^2(c+d x)\right )}{2 (a+b \sec (c+d x))}dx}{b \left (a^2-b^2\right )}}{4 b \left (a^2-b^2\right )}-\frac {a^2 \sin (c+d x) \sec ^{\frac {5}{2}}(c+d x)}{2 b d \left (a^2-b^2\right ) (a+b \sec (c+d x))^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {\frac {\int \frac {\sqrt {\sec (c+d x)} \left (\left (5 a^2-11 b^2\right ) a^2-4 b \left (a^2-4 b^2\right ) \sec (c+d x) a-\left (15 a^4-29 b^2 a^2+8 b^4\right ) \sec ^2(c+d x)\right )}{a+b \sec (c+d x)}dx}{2 b \left (a^2-b^2\right )}+\frac {a^2 \left (5 a^2-11 b^2\right ) \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x)}{b d \left (a^2-b^2\right ) (a+b \sec (c+d x))}}{4 b \left (a^2-b^2\right )}-\frac {a^2 \sin (c+d x) \sec ^{\frac {5}{2}}(c+d x)}{2 b d \left (a^2-b^2\right ) (a+b \sec (c+d x))^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {\frac {\int \frac {\sqrt {\csc \left (c+d x+\frac {\pi }{2}\right )} \left (\left (5 a^2-11 b^2\right ) a^2-4 b \left (a^2-4 b^2\right ) \csc \left (c+d x+\frac {\pi }{2}\right ) a+\left (-15 a^4+29 b^2 a^2-8 b^4\right ) \csc \left (c+d x+\frac {\pi }{2}\right )^2\right )}{a+b \csc \left (c+d x+\frac {\pi }{2}\right )}dx}{2 b \left (a^2-b^2\right )}+\frac {a^2 \left (5 a^2-11 b^2\right ) \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x)}{b d \left (a^2-b^2\right ) (a+b \sec (c+d x))}}{4 b \left (a^2-b^2\right )}-\frac {a^2 \sin (c+d x) \sec ^{\frac {5}{2}}(c+d x)}{2 b d \left (a^2-b^2\right ) (a+b \sec (c+d x))^2}\) |
| \(\Big \downarrow \) 4590 |
| \(\displaystyle -\frac {\frac {\frac {2 \int \frac {3 a \left (5 a^4-11 b^2 a^2+8 b^4\right ) \sec ^2(c+d x)+4 b \left (5 a^4-10 b^2 a^2+2 b^4\right ) \sec (c+d x)+a \left (15 a^4-29 b^2 a^2+8 b^4\right )}{2 \sqrt {\sec (c+d x)} (a+b \sec (c+d x))}dx}{b}-\frac {2 \left (15 a^4-29 a^2 b^2+8 b^4\right ) \sin (c+d x) \sqrt {\sec (c+d x)}}{b d}}{2 b \left (a^2-b^2\right )}+\frac {a^2 \left (5 a^2-11 b^2\right ) \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x)}{b d \left (a^2-b^2\right ) (a+b \sec (c+d x))}}{4 b \left (a^2-b^2\right )}-\frac {a^2 \sin (c+d x) \sec ^{\frac {5}{2}}(c+d x)}{2 b d \left (a^2-b^2\right ) (a+b \sec (c+d x))^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {\frac {\frac {\int \frac {3 a \left (5 a^4-11 b^2 a^2+8 b^4\right ) \sec ^2(c+d x)+4 b \left (5 a^4-10 b^2 a^2+2 b^4\right ) \sec (c+d x)+a \left (15 a^4-29 b^2 a^2+8 b^4\right )}{\sqrt {\sec (c+d x)} (a+b \sec (c+d x))}dx}{b}-\frac {2 \left (15 a^4-29 a^2 b^2+8 b^4\right ) \sin (c+d x) \sqrt {\sec (c+d x)}}{b d}}{2 b \left (a^2-b^2\right )}+\frac {a^2 \left (5 a^2-11 b^2\right ) \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x)}{b d \left (a^2-b^2\right ) (a+b \sec (c+d x))}}{4 b \left (a^2-b^2\right )}-\frac {a^2 \sin (c+d x) \sec ^{\frac {5}{2}}(c+d x)}{2 b d \left (a^2-b^2\right ) (a+b \sec (c+d x))^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {\frac {\frac {\int \frac {3 a \left (5 a^4-11 b^2 a^2+8 b^4\right ) \csc \left (c+d x+\frac {\pi }{2}\right )^2+4 b \left (5 a^4-10 b^2 a^2+2 b^4\right ) \csc \left (c+d x+\frac {\pi }{2}\right )+a \left (15 a^4-29 b^2 a^2+8 b^4\right )}{\sqrt {\csc \left (c+d x+\frac {\pi }{2}\right )} \left (a+b \csc \left (c+d x+\frac {\pi }{2}\right )\right )}dx}{b}-\frac {2 \left (15 a^4-29 a^2 b^2+8 b^4\right ) \sin (c+d x) \sqrt {\sec (c+d x)}}{b d}}{2 b \left (a^2-b^2\right )}+\frac {a^2 \left (5 a^2-11 b^2\right ) \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x)}{b d \left (a^2-b^2\right ) (a+b \sec (c+d x))}}{4 b \left (a^2-b^2\right )}-\frac {a^2 \sin (c+d x) \sec ^{\frac {5}{2}}(c+d x)}{2 b d \left (a^2-b^2\right ) (a+b \sec (c+d x))^2}\) |
| \(\Big \downarrow \) 4594 |
| \(\displaystyle -\frac {\frac {\frac {a \left (15 a^4-38 a^2 b^2+35 b^4\right ) \int \frac {\sec ^{\frac {3}{2}}(c+d x)}{a+b \sec (c+d x)}dx+\frac {\int \frac {b \left (5 a^2-11 b^2\right ) \sec (c+d x) a^3+\left (15 a^4-29 b^2 a^2+8 b^4\right ) a^2}{\sqrt {\sec (c+d x)}}dx}{a^2}}{b}-\frac {2 \left (15 a^4-29 a^2 b^2+8 b^4\right ) \sin (c+d x) \sqrt {\sec (c+d x)}}{b d}}{2 b \left (a^2-b^2\right )}+\frac {a^2 \left (5 a^2-11 b^2\right ) \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x)}{b d \left (a^2-b^2\right ) (a+b \sec (c+d x))}}{4 b \left (a^2-b^2\right )}-\frac {a^2 \sin (c+d x) \sec ^{\frac {5}{2}}(c+d x)}{2 b d \left (a^2-b^2\right ) (a+b \sec (c+d x))^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {\frac {\frac {a \left (15 a^4-38 a^2 b^2+35 b^4\right ) \int \frac {\csc \left (c+d x+\frac {\pi }{2}\right )^{3/2}}{a+b \csc \left (c+d x+\frac {\pi }{2}\right )}dx+\frac {\int \frac {b \left (5 a^2-11 b^2\right ) \csc \left (c+d x+\frac {\pi }{2}\right ) a^3+\left (15 a^4-29 b^2 a^2+8 b^4\right ) a^2}{\sqrt {\csc \left (c+d x+\frac {\pi }{2}\right )}}dx}{a^2}}{b}-\frac {2 \left (15 a^4-29 a^2 b^2+8 b^4\right ) \sin (c+d x) \sqrt {\sec (c+d x)}}{b d}}{2 b \left (a^2-b^2\right )}+\frac {a^2 \left (5 a^2-11 b^2\right ) \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x)}{b d \left (a^2-b^2\right ) (a+b \sec (c+d x))}}{4 b \left (a^2-b^2\right )}-\frac {a^2 \sin (c+d x) \sec ^{\frac {5}{2}}(c+d x)}{2 b d \left (a^2-b^2\right ) (a+b \sec (c+d x))^2}\) |
| \(\Big \downarrow \) 4274 |
| \(\displaystyle -\frac {\frac {\frac {a \left (15 a^4-38 a^2 b^2+35 b^4\right ) \int \frac {\csc \left (c+d x+\frac {\pi }{2}\right )^{3/2}}{a+b \csc \left (c+d x+\frac {\pi }{2}\right )}dx+\frac {a^2 \left (15 a^4-29 a^2 b^2+8 b^4\right ) \int \frac {1}{\sqrt {\sec (c+d x)}}dx+a^3 b \left (5 a^2-11 b^2\right ) \int \sqrt {\sec (c+d x)}dx}{a^2}}{b}-\frac {2 \left (15 a^4-29 a^2 b^2+8 b^4\right ) \sin (c+d x) \sqrt {\sec (c+d x)}}{b d}}{2 b \left (a^2-b^2\right )}+\frac {a^2 \left (5 a^2-11 b^2\right ) \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x)}{b d \left (a^2-b^2\right ) (a+b \sec (c+d x))}}{4 b \left (a^2-b^2\right )}-\frac {a^2 \sin (c+d x) \sec ^{\frac {5}{2}}(c+d x)}{2 b d \left (a^2-b^2\right ) (a+b \sec (c+d x))^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {\frac {\frac {a \left (15 a^4-38 a^2 b^2+35 b^4\right ) \int \frac {\csc \left (c+d x+\frac {\pi }{2}\right )^{3/2}}{a+b \csc \left (c+d x+\frac {\pi }{2}\right )}dx+\frac {a^2 \left (15 a^4-29 a^2 b^2+8 b^4\right ) \int \frac {1}{\sqrt {\csc \left (c+d x+\frac {\pi }{2}\right )}}dx+a^3 b \left (5 a^2-11 b^2\right ) \int \sqrt {\csc \left (c+d x+\frac {\pi }{2}\right )}dx}{a^2}}{b}-\frac {2 \left (15 a^4-29 a^2 b^2+8 b^4\right ) \sin (c+d x) \sqrt {\sec (c+d x)}}{b d}}{2 b \left (a^2-b^2\right )}+\frac {a^2 \left (5 a^2-11 b^2\right ) \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x)}{b d \left (a^2-b^2\right ) (a+b \sec (c+d x))}}{4 b \left (a^2-b^2\right )}-\frac {a^2 \sin (c+d x) \sec ^{\frac {5}{2}}(c+d x)}{2 b d \left (a^2-b^2\right ) (a+b \sec (c+d x))^2}\) |
| \(\Big \downarrow \) 4258 |
| \(\displaystyle -\frac {\frac {\frac {a \left (15 a^4-38 a^2 b^2+35 b^4\right ) \int \frac {\csc \left (c+d x+\frac {\pi }{2}\right )^{3/2}}{a+b \csc \left (c+d x+\frac {\pi }{2}\right )}dx+\frac {a^2 \left (15 a^4-29 a^2 b^2+8 b^4\right ) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \int \sqrt {\cos (c+d x)}dx+a^3 b \left (5 a^2-11 b^2\right ) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \int \frac {1}{\sqrt {\cos (c+d x)}}dx}{a^2}}{b}-\frac {2 \left (15 a^4-29 a^2 b^2+8 b^4\right ) \sin (c+d x) \sqrt {\sec (c+d x)}}{b d}}{2 b \left (a^2-b^2\right )}+\frac {a^2 \left (5 a^2-11 b^2\right ) \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x)}{b d \left (a^2-b^2\right ) (a+b \sec (c+d x))}}{4 b \left (a^2-b^2\right )}-\frac {a^2 \sin (c+d x) \sec ^{\frac {5}{2}}(c+d x)}{2 b d \left (a^2-b^2\right ) (a+b \sec (c+d x))^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {\frac {\frac {a \left (15 a^4-38 a^2 b^2+35 b^4\right ) \int \frac {\csc \left (c+d x+\frac {\pi }{2}\right )^{3/2}}{a+b \csc \left (c+d x+\frac {\pi }{2}\right )}dx+\frac {a^2 \left (15 a^4-29 a^2 b^2+8 b^4\right ) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \int \sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}dx+a^3 b \left (5 a^2-11 b^2\right ) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \int \frac {1}{\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}}dx}{a^2}}{b}-\frac {2 \left (15 a^4-29 a^2 b^2+8 b^4\right ) \sin (c+d x) \sqrt {\sec (c+d x)}}{b d}}{2 b \left (a^2-b^2\right )}+\frac {a^2 \left (5 a^2-11 b^2\right ) \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x)}{b d \left (a^2-b^2\right ) (a+b \sec (c+d x))}}{4 b \left (a^2-b^2\right )}-\frac {a^2 \sin (c+d x) \sec ^{\frac {5}{2}}(c+d x)}{2 b d \left (a^2-b^2\right ) (a+b \sec (c+d x))^2}\) |
| \(\Big \downarrow \) 3119 |
| \(\displaystyle -\frac {\frac {\frac {a \left (15 a^4-38 a^2 b^2+35 b^4\right ) \int \frac {\csc \left (c+d x+\frac {\pi }{2}\right )^{3/2}}{a+b \csc \left (c+d x+\frac {\pi }{2}\right )}dx+\frac {a^3 b \left (5 a^2-11 b^2\right ) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \int \frac {1}{\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}}dx+\frac {2 a^2 \left (15 a^4-29 a^2 b^2+8 b^4\right ) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{d}}{a^2}}{b}-\frac {2 \left (15 a^4-29 a^2 b^2+8 b^4\right ) \sin (c+d x) \sqrt {\sec (c+d x)}}{b d}}{2 b \left (a^2-b^2\right )}+\frac {a^2 \left (5 a^2-11 b^2\right ) \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x)}{b d \left (a^2-b^2\right ) (a+b \sec (c+d x))}}{4 b \left (a^2-b^2\right )}-\frac {a^2 \sin (c+d x) \sec ^{\frac {5}{2}}(c+d x)}{2 b d \left (a^2-b^2\right ) (a+b \sec (c+d x))^2}\) |
| \(\Big \downarrow \) 3120 |
| \(\displaystyle -\frac {\frac {\frac {a \left (15 a^4-38 a^2 b^2+35 b^4\right ) \int \frac {\csc \left (c+d x+\frac {\pi }{2}\right )^{3/2}}{a+b \csc \left (c+d x+\frac {\pi }{2}\right )}dx+\frac {\frac {2 a^2 \left (15 a^4-29 a^2 b^2+8 b^4\right ) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{d}+\frac {2 a^3 b \left (5 a^2-11 b^2\right ) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )}{d}}{a^2}}{b}-\frac {2 \left (15 a^4-29 a^2 b^2+8 b^4\right ) \sin (c+d x) \sqrt {\sec (c+d x)}}{b d}}{2 b \left (a^2-b^2\right )}+\frac {a^2 \left (5 a^2-11 b^2\right ) \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x)}{b d \left (a^2-b^2\right ) (a+b \sec (c+d x))}}{4 b \left (a^2-b^2\right )}-\frac {a^2 \sin (c+d x) \sec ^{\frac {5}{2}}(c+d x)}{2 b d \left (a^2-b^2\right ) (a+b \sec (c+d x))^2}\) |
| \(\Big \downarrow \) 4336 |
| \(\displaystyle -\frac {\frac {\frac {a \left (15 a^4-38 a^2 b^2+35 b^4\right ) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \int \frac {1}{\sqrt {\cos (c+d x)} (b+a \cos (c+d x))}dx+\frac {\frac {2 a^2 \left (15 a^4-29 a^2 b^2+8 b^4\right ) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{d}+\frac {2 a^3 b \left (5 a^2-11 b^2\right ) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )}{d}}{a^2}}{b}-\frac {2 \left (15 a^4-29 a^2 b^2+8 b^4\right ) \sin (c+d x) \sqrt {\sec (c+d x)}}{b d}}{2 b \left (a^2-b^2\right )}+\frac {a^2 \left (5 a^2-11 b^2\right ) \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x)}{b d \left (a^2-b^2\right ) (a+b \sec (c+d x))}}{4 b \left (a^2-b^2\right )}-\frac {a^2 \sin (c+d x) \sec ^{\frac {5}{2}}(c+d x)}{2 b d \left (a^2-b^2\right ) (a+b \sec (c+d x))^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {\frac {\frac {a \left (15 a^4-38 a^2 b^2+35 b^4\right ) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \int \frac {1}{\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )} \left (b+a \sin \left (c+d x+\frac {\pi }{2}\right )\right )}dx+\frac {\frac {2 a^2 \left (15 a^4-29 a^2 b^2+8 b^4\right ) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{d}+\frac {2 a^3 b \left (5 a^2-11 b^2\right ) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )}{d}}{a^2}}{b}-\frac {2 \left (15 a^4-29 a^2 b^2+8 b^4\right ) \sin (c+d x) \sqrt {\sec (c+d x)}}{b d}}{2 b \left (a^2-b^2\right )}+\frac {a^2 \left (5 a^2-11 b^2\right ) \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x)}{b d \left (a^2-b^2\right ) (a+b \sec (c+d x))}}{4 b \left (a^2-b^2\right )}-\frac {a^2 \sin (c+d x) \sec ^{\frac {5}{2}}(c+d x)}{2 b d \left (a^2-b^2\right ) (a+b \sec (c+d x))^2}\) |
| \(\Big \downarrow \) 3284 |
| \(\displaystyle -\frac {a^2 \sin (c+d x) \sec ^{\frac {5}{2}}(c+d x)}{2 b d \left (a^2-b^2\right ) (a+b \sec (c+d x))^2}-\frac {\frac {a^2 \left (5 a^2-11 b^2\right ) \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x)}{b d \left (a^2-b^2\right ) (a+b \sec (c+d x))}+\frac {\frac {\frac {2 a \left (15 a^4-38 a^2 b^2+35 b^4\right ) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \operatorname {EllipticPi}\left (\frac {2 a}{a+b},\frac {1}{2} (c+d x),2\right )}{d (a+b)}+\frac {\frac {2 a^2 \left (15 a^4-29 a^2 b^2+8 b^4\right ) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{d}+\frac {2 a^3 b \left (5 a^2-11 b^2\right ) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )}{d}}{a^2}}{b}-\frac {2 \left (15 a^4-29 a^2 b^2+8 b^4\right ) \sin (c+d x) \sqrt {\sec (c+d x)}}{b d}}{2 b \left (a^2-b^2\right )}}{4 b \left (a^2-b^2\right )}\) |
-1/2*(a^2*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(b*(a^2 - b^2)*d*(a + b*Sec[c + d*x])^2) - ((a^2*(5*a^2 - 11*b^2)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(b*(a^ 2 - b^2)*d*(a + b*Sec[c + d*x])) + ((((2*a^2*(15*a^4 - 29*a^2*b^2 + 8*b^4) *Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/d + (2*a ^3*b*(5*a^2 - 11*b^2)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Se c[c + d*x]])/d)/a^2 + (2*a*(15*a^4 - 38*a^2*b^2 + 35*b^4)*Sqrt[Cos[c + d*x ]]*EllipticPi[(2*a)/(a + b), (c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/((a + b)* d))/b - (2*(15*a^4 - 29*a^2*b^2 + 8*b^4)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/ (b*d))/(2*b*(a^2 - b^2)))/(4*b*(a^2 - b^2))
3.7.21.3.1 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[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[(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_)]*(d_.))^(n_.)*(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_)), x_Symbol] :> Simp[a Int[(d*Csc[e + f*x])^n, x], x] + Simp[b/d In t[(d*Csc[e + f*x])^(n + 1), x], x] /; FreeQ[{a, b, d, e, f, n}, x]
Int[(csc[(e_.) + (f_.)*(x_)]*(d_.))^(n_)*(csc[(e_.) + (f_.)*(x_)]*(b_.) + ( a_))^(m_), x_Symbol] :> Simp[(-a^2)*d^3*Cot[e + f*x]*(a + b*Csc[e + f*x])^( m + 1)*((d*Csc[e + f*x])^(n - 3)/(b*f*(m + 1)*(a^2 - b^2))), x] + Simp[d^3/ (b*(m + 1)*(a^2 - b^2)) Int[(a + b*Csc[e + f*x])^(m + 1)*(d*Csc[e + f*x]) ^(n - 3)*Simp[a^2*(n - 3) + a*b*(m + 1)*Csc[e + f*x] - (a^2*(n - 2) + b^2*( m + 1))*Csc[e + f*x]^2, x], x], x] /; FreeQ[{a, b, d, e, f}, x] && NeQ[a^2 - b^2, 0] && LtQ[m, -1] && (IGtQ[n, 3] || (IntegersQ[n + 1/2, 2*m] && GtQ[n , 2]))
Int[(csc[(e_.) + (f_.)*(x_)]*(d_.))^(3/2)/(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_)), x_Symbol] :> Simp[d*Sqrt[d*Sin[e + f*x]]*Sqrt[d*Csc[e + f*x]] Int[ 1/(Sqrt[d*Sin[e + f*x]]*(b + a*Sin[e + f*x])), x], x] /; FreeQ[{a, b, d, e, f}, x] && 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[(-d)*(A*b^2 - a*b*B + a^2*C)*Cot[e + f*x]*(a + b*Csc[e + f*x])^(m + 1)*((d*Csc[e + f*x])^(n - 1)/(b*f*(a^2 - b^2)*(m + 1)) ), x] + Simp[d/(b*(a^2 - b^2)*(m + 1)) Int[(a + b*Csc[e + f*x])^(m + 1)*( d*Csc[e + f*x])^(n - 1)*Simp[A*b^2*(n - 1) - a*(b*B - a*C)*(n - 1) + b*(a*A - b*B + a*C)*(m + 1)*Csc[e + f*x] - (b*(A*b - a*B)*(m + n + 1) + C*(a^2*n + b^2*(m + 1)))*Csc[e + f*x]^2, x], x], x] /; FreeQ[{a, b, d, e, f, A, B, C }, x] && NeQ[a^2 - b^2, 0] && LtQ[m, -1] && GtQ[n, 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_.)]*(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a _))), x_Symbol] :> Simp[(A*b^2 - a*b*B + a^2*C)/(a^2*d^2) Int[(d*Csc[e + f*x])^(3/2)/(a + b*Csc[e + f*x]), x], x] + Simp[1/a^2 Int[(a*A - (A*b - a *B)*Csc[e + f*x])/Sqrt[d*Csc[e + f*x]], x], x] /; FreeQ[{a, b, d, e, f, A, B, C}, x] && NeQ[a^2 - b^2, 0]
Leaf count of result is larger than twice the leaf count of optimal. \(1986\) vs. \(2(436)=872\).
Time = 488.59 (sec) , antiderivative size = 1987, normalized size of antiderivative = 5.12
-(-(-2*cos(1/2*d*x+1/2*c)^2+1)*sin(1/2*d*x+1/2*c)^2)^(1/2)*(2/b^3/sin(1/2* d*x+1/2*c)^2/(2*sin(1/2*d*x+1/2*c)^2-1)*(-2*sin(1/2*d*x+1/2*c)^4+sin(1/2*d *x+1/2*c)^2)^(1/2)*(2*cos(1/2*d*x+1/2*c)*sin(1/2*d*x+1/2*c)^2-(sin(1/2*d*x +1/2*c)^2)^(1/2)*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))-2*a/b*(1/2*a^2/b/(a^2-b^2)*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)/(2*a*cos(1/2*d*x+1/2*c)^2-a+b)^2+3 /4*a^2*(a^2-3*b^2)/b^2/(a^2-b^2)^2*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)/(2*a*cos(1/2*d*x+1/2*c)^2-a+b)-3/8/(a+b)/ (a^2-b^2)/b^2*(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))*a^2-1/4/(a+b)/(a^2-b^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)*EllipticF(cos(1/2*d*x+1/2*c),2^(1/2))*a+7/8/(a+b)/(a^2-b^2)* (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))+3/8*a^3/b^2/(a^2-b^2)^2*(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)*El lipticF(cos(1/2*d*x+1/2*c),2^(1/2))-9/8*a/(a^2-b^2)^2*(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))-3/8*a^3/b^2...
Timed out. \[ \int \frac {\sec ^{\frac {9}{2}}(c+d x)}{(a+b \sec (c+d x))^3} \, dx=\text {Timed out} \]
Timed out. \[ \int \frac {\sec ^{\frac {9}{2}}(c+d x)}{(a+b \sec (c+d x))^3} \, dx=\text {Timed out} \]
Timed out. \[ \int \frac {\sec ^{\frac {9}{2}}(c+d x)}{(a+b \sec (c+d x))^3} \, dx=\text {Timed out} \]
\[ \int \frac {\sec ^{\frac {9}{2}}(c+d x)}{(a+b \sec (c+d x))^3} \, dx=\int { \frac {\sec \left (d x + c\right )^{\frac {9}{2}}}{{\left (b \sec \left (d x + c\right ) + a\right )}^{3}} \,d x } \]
Timed out. \[ \int \frac {\sec ^{\frac {9}{2}}(c+d x)}{(a+b \sec (c+d x))^3} \, dx=\int \frac {{\left (\frac {1}{\cos \left (c+d\,x\right )}\right )}^{9/2}}{{\left (a+\frac {b}{\cos \left (c+d\,x\right )}\right )}^3} \,d x \]