Integrand size = 33, antiderivative size = 277 \[ \int \frac {\sec ^{\frac {7}{2}}(c+d x) (A+B \sec (c+d x))}{a+b \sec (c+d x)} \, dx=\frac {2 \left (5 a A b-5 a^2 B-3 b^2 B\right ) \sqrt {\cos (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \sqrt {\sec (c+d x)}}{5 b^3 d}+\frac {2 (A b-a B) \sqrt {\cos (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right ) \sqrt {\sec (c+d x)}}{3 b^2 d}+\frac {2 a^2 (A b-a B) \sqrt {\cos (c+d x)} \operatorname {EllipticPi}\left (\frac {2 a}{a+b},\frac {1}{2} (c+d x),2\right ) \sqrt {\sec (c+d x)}}{b^3 (a+b) d}-\frac {2 \left (5 a A b-5 a^2 B-3 b^2 B\right ) \sqrt {\sec (c+d x)} \sin (c+d x)}{5 b^3 d}+\frac {2 (A b-a B) \sec ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{3 b^2 d}+\frac {2 B \sec ^{\frac {5}{2}}(c+d x) \sin (c+d x)}{5 b d} \]
2/3*(A*b-B*a)*sec(d*x+c)^(3/2)*sin(d*x+c)/b^2/d+2/5*B*sec(d*x+c)^(5/2)*sin (d*x+c)/b/d-2/5*(5*A*a*b-5*B*a^2-3*B*b^2)*sin(d*x+c)*sec(d*x+c)^(1/2)/b^3/ d+2/5*(5*A*a*b-5*B*a^2-3*B*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))*cos(d*x+c)^(1/2)*sec(d*x+c)^(1 /2)/b^3/d+2/3*(A*b-B*a)*(cos(1/2*d*x+1/2*c)^2)^(1/2)/cos(1/2*d*x+1/2*c)*El lipticF(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/ d+2*a^2*(A*b-B*a)*(cos(1/2*d*x+1/2*c)^2)^(1/2)/cos(1/2*d*x+1/2*c)*Elliptic Pi(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) /b^3/(a+b)/d
Leaf count is larger than twice the leaf count of optimal. \(664\) vs. \(2(277)=554\).
Time = 10.73 (sec) , antiderivative size = 664, normalized size of antiderivative = 2.40 \[ \int \frac {\sec ^{\frac {7}{2}}(c+d x) (A+B \sec (c+d x))}{a+b \sec (c+d x)} \, dx=-\frac {\frac {2 \left (-45 a^2 A b-10 A b^3+45 a^3 B+19 a b^2 B\right ) \cos ^2(c+d x) \left (\operatorname {EllipticF}\left (\arcsin \left (\sqrt {\sec (c+d x)}\right ),-1\right )-\operatorname {EllipticPi}\left (-\frac {b}{a},\arcsin \left (\sqrt {\sec (c+d x)}\right ),-1\right )\right ) (a+b \sec (c+d x)) \sqrt {1-\sec ^2(c+d x)} \sin (c+d x)}{b (b+a \cos (c+d x)) \left (1-\cos ^2(c+d x)\right )}+\frac {2 \left (-40 a A b^2+40 a^2 b B+18 b^3 B\right ) \cos ^2(c+d x) \operatorname {EllipticPi}\left (-\frac {b}{a},\arcsin \left (\sqrt {\sec (c+d x)}\right ),-1\right ) (a+b \sec (c+d x)) \sqrt {1-\sec ^2(c+d x)} \sin (c+d x)}{a (b+a \cos (c+d x)) \left (1-\cos ^2(c+d x)\right )}+\frac {\left (-15 a^2 A b+15 a^3 B+9 a b^2 B\right ) \cos (2 (c+d x)) (a+b \sec (c+d x)) \left (-4 a b+4 a b \sec ^2(c+d x)-4 a b E\left (\left .\arcsin \left (\sqrt {\sec (c+d x)}\right )\right |-1\right ) \sqrt {\sec (c+d x)} \sqrt {1-\sec ^2(c+d x)}-2 a (a-2 b) \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\sec (c+d x)}\right ),-1\right ) \sqrt {\sec (c+d x)} \sqrt {1-\sec ^2(c+d x)}+2 a^2 \operatorname {EllipticPi}\left (-\frac {b}{a},\arcsin \left (\sqrt {\sec (c+d x)}\right ),-1\right ) \sqrt {\sec (c+d x)} \sqrt {1-\sec ^2(c+d x)}-4 b^2 \operatorname {EllipticPi}\left (-\frac {b}{a},\arcsin \left (\sqrt {\sec (c+d x)}\right ),-1\right ) \sqrt {\sec (c+d x)} \sqrt {1-\sec ^2(c+d x)}\right ) \sin (c+d x)}{a^2 b (b+a \cos (c+d x)) \left (1-\cos ^2(c+d x)\right ) \sqrt {\sec (c+d x)} \left (2-\sec ^2(c+d x)\right )}}{30 b^3 d}+\frac {\sqrt {\sec (c+d x)} \left (\frac {2 \left (-5 a A b+5 a^2 B+3 b^2 B\right ) \sin (c+d x)}{5 b^3}+\frac {2 \sec (c+d x) (A b \sin (c+d x)-a B \sin (c+d x))}{3 b^2}+\frac {2 B \sec (c+d x) \tan (c+d x)}{5 b}\right )}{d} \]
-1/30*((2*(-45*a^2*A*b - 10*A*b^3 + 45*a^3*B + 19*a*b^2*B)*Cos[c + d*x]^2* (EllipticF[ArcSin[Sqrt[Sec[c + d*x]]], -1] - EllipticPi[-(b/a), ArcSin[Sqr t[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*A*b^2 + 40*a^2*b*B + 18*b^3*B)*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^2*A*b + 15*a^3*B + 9*a*b^2*B)*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*EllipticP i[-(b/a), ArcSin[Sqrt[Sec[c + d*x]]], -1]*Sqrt[Sec[c + d*x]]*Sqrt[1 - Sec[ c + d*x]^2] - 4*b^2*EllipticPi[-(b/a), ArcSin[Sqrt[Sec[c + d*x]]], -1]*Sqr t[Sec[c + d*x]]*Sqrt[1 - Sec[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)))/( b^3*d) + (Sqrt[Sec[c + d*x]]*((2*(-5*a*A*b + 5*a^2*B + 3*b^2*B)*Sin[c + d* x])/(5*b^3) + (2*Sec[c + d*x]*(A*b*Sin[c + d*x] - a*B*Sin[c + d*x]))/(3*b^ 2) + (2*B*Sec[c + d*x]*Tan[c + d*x])/(5*b)))/d
Time = 2.33 (sec) , antiderivative size = 295, normalized size of antiderivative = 1.06, number of steps used = 21, number of rules used = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.636, Rules used = {3042, 4521, 27, 3042, 4590, 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.
\(\displaystyle \int \frac {\sec ^{\frac {7}{2}}(c+d x) (A+B \sec (c+d x))}{a+b \sec (c+d x)} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {\csc \left (c+d x+\frac {\pi }{2}\right )^{7/2} \left (A+B \csc \left (c+d x+\frac {\pi }{2}\right )\right )}{a+b \csc \left (c+d x+\frac {\pi }{2}\right )}dx\) |
\(\Big \downarrow \) 4521 |
\(\displaystyle \frac {2 \int \frac {\sec ^{\frac {3}{2}}(c+d x) \left (5 (A b-a B) \sec ^2(c+d x)+3 b B \sec (c+d x)+3 a B\right )}{2 (a+b \sec (c+d x))}dx}{5 b}+\frac {2 B \sin (c+d x) \sec ^{\frac {5}{2}}(c+d x)}{5 b d}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {\int \frac {\sec ^{\frac {3}{2}}(c+d x) \left (5 (A b-a B) \sec ^2(c+d x)+3 b B \sec (c+d x)+3 a B\right )}{a+b \sec (c+d x)}dx}{5 b}+\frac {2 B \sin (c+d x) \sec ^{\frac {5}{2}}(c+d x)}{5 b d}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {\int \frac {\csc \left (c+d x+\frac {\pi }{2}\right )^{3/2} \left (5 (A b-a B) \csc \left (c+d x+\frac {\pi }{2}\right )^2+3 b B \csc \left (c+d x+\frac {\pi }{2}\right )+3 a B\right )}{a+b \csc \left (c+d x+\frac {\pi }{2}\right )}dx}{5 b}+\frac {2 B \sin (c+d x) \sec ^{\frac {5}{2}}(c+d x)}{5 b d}\) |
\(\Big \downarrow \) 4590 |
\(\displaystyle \frac {\frac {2 \int \frac {\sqrt {\sec (c+d x)} \left (-3 \left (-5 B a^2+5 A b a-3 b^2 B\right ) \sec ^2(c+d x)+b (5 A b+4 a B) \sec (c+d x)+5 a (A b-a B)\right )}{2 (a+b \sec (c+d x))}dx}{3 b}+\frac {10 (A b-a B) \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x)}{3 b d}}{5 b}+\frac {2 B \sin (c+d x) \sec ^{\frac {5}{2}}(c+d x)}{5 b d}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {\frac {\int \frac {\sqrt {\sec (c+d x)} \left (-3 \left (-5 B a^2+5 A b a-3 b^2 B\right ) \sec ^2(c+d x)+b (5 A b+4 a B) \sec (c+d x)+5 a (A b-a B)\right )}{a+b \sec (c+d x)}dx}{3 b}+\frac {10 (A b-a B) \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x)}{3 b d}}{5 b}+\frac {2 B \sin (c+d x) \sec ^{\frac {5}{2}}(c+d x)}{5 b d}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {\frac {\int \frac {\sqrt {\csc \left (c+d x+\frac {\pi }{2}\right )} \left (-3 \left (-5 B a^2+5 A b a-3 b^2 B\right ) \csc \left (c+d x+\frac {\pi }{2}\right )^2+b (5 A b+4 a B) \csc \left (c+d x+\frac {\pi }{2}\right )+5 a (A b-a B)\right )}{a+b \csc \left (c+d x+\frac {\pi }{2}\right )}dx}{3 b}+\frac {10 (A b-a B) \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x)}{3 b d}}{5 b}+\frac {2 B \sin (c+d x) \sec ^{\frac {5}{2}}(c+d x)}{5 b d}\) |
\(\Big \downarrow \) 4590 |
\(\displaystyle \frac {\frac {\frac {2 \int \frac {5 \left (3 a^2+b^2\right ) (A b-a B) \sec ^2(c+d x)+b \left (-20 B a^2+20 A b a-9 b^2 B\right ) \sec (c+d x)+3 a \left (-5 B a^2+5 A b a-3 b^2 B\right )}{2 \sqrt {\sec (c+d x)} (a+b \sec (c+d x))}dx}{b}-\frac {6 \left (-5 a^2 B+5 a A b-3 b^2 B\right ) \sin (c+d x) \sqrt {\sec (c+d x)}}{b d}}{3 b}+\frac {10 (A b-a B) \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x)}{3 b d}}{5 b}+\frac {2 B \sin (c+d x) \sec ^{\frac {5}{2}}(c+d x)}{5 b d}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {\frac {\frac {\int \frac {5 \left (3 a^2+b^2\right ) (A b-a B) \sec ^2(c+d x)+b \left (-20 B a^2+20 A b a-9 b^2 B\right ) \sec (c+d x)+3 a \left (-5 B a^2+5 A b a-3 b^2 B\right )}{\sqrt {\sec (c+d x)} (a+b \sec (c+d x))}dx}{b}-\frac {6 \left (-5 a^2 B+5 a A b-3 b^2 B\right ) \sin (c+d x) \sqrt {\sec (c+d x)}}{b d}}{3 b}+\frac {10 (A b-a B) \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x)}{3 b d}}{5 b}+\frac {2 B \sin (c+d x) \sec ^{\frac {5}{2}}(c+d x)}{5 b d}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {\frac {\frac {\int \frac {5 \left (3 a^2+b^2\right ) (A b-a B) \csc \left (c+d x+\frac {\pi }{2}\right )^2+b \left (-20 B a^2+20 A b a-9 b^2 B\right ) \csc \left (c+d x+\frac {\pi }{2}\right )+3 a \left (-5 B a^2+5 A b a-3 b^2 B\right )}{\sqrt {\csc \left (c+d x+\frac {\pi }{2}\right )} \left (a+b \csc \left (c+d x+\frac {\pi }{2}\right )\right )}dx}{b}-\frac {6 \left (-5 a^2 B+5 a A b-3 b^2 B\right ) \sin (c+d x) \sqrt {\sec (c+d x)}}{b d}}{3 b}+\frac {10 (A b-a B) \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x)}{3 b d}}{5 b}+\frac {2 B \sin (c+d x) \sec ^{\frac {5}{2}}(c+d x)}{5 b d}\) |
\(\Big \downarrow \) 4594 |
\(\displaystyle \frac {\frac {\frac {\frac {\int \frac {3 \left (-5 B a^2+5 A b a-3 b^2 B\right ) a^2+5 b (A b-a B) \sec (c+d x) a^2}{\sqrt {\sec (c+d x)}}dx}{a^2}+15 a^2 (A b-a B) \int \frac {\sec ^{\frac {3}{2}}(c+d x)}{a+b \sec (c+d x)}dx}{b}-\frac {6 \left (-5 a^2 B+5 a A b-3 b^2 B\right ) \sin (c+d x) \sqrt {\sec (c+d x)}}{b d}}{3 b}+\frac {10 (A b-a B) \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x)}{3 b d}}{5 b}+\frac {2 B \sin (c+d x) \sec ^{\frac {5}{2}}(c+d x)}{5 b d}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {\frac {\frac {\frac {\int \frac {3 \left (-5 B a^2+5 A b a-3 b^2 B\right ) a^2+5 b (A b-a B) \csc \left (c+d x+\frac {\pi }{2}\right ) a^2}{\sqrt {\csc \left (c+d x+\frac {\pi }{2}\right )}}dx}{a^2}+15 a^2 (A b-a B) \int \frac {\csc \left (c+d x+\frac {\pi }{2}\right )^{3/2}}{a+b \csc \left (c+d x+\frac {\pi }{2}\right )}dx}{b}-\frac {6 \left (-5 a^2 B+5 a A b-3 b^2 B\right ) \sin (c+d x) \sqrt {\sec (c+d x)}}{b d}}{3 b}+\frac {10 (A b-a B) \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x)}{3 b d}}{5 b}+\frac {2 B \sin (c+d x) \sec ^{\frac {5}{2}}(c+d x)}{5 b d}\) |
\(\Big \downarrow \) 4274 |
\(\displaystyle \frac {\frac {\frac {\frac {3 a^2 \left (-5 a^2 B+5 a A b-3 b^2 B\right ) \int \frac {1}{\sqrt {\sec (c+d x)}}dx+5 a^2 b (A b-a B) \int \sqrt {\sec (c+d x)}dx}{a^2}+15 a^2 (A b-a B) \int \frac {\csc \left (c+d x+\frac {\pi }{2}\right )^{3/2}}{a+b \csc \left (c+d x+\frac {\pi }{2}\right )}dx}{b}-\frac {6 \left (-5 a^2 B+5 a A b-3 b^2 B\right ) \sin (c+d x) \sqrt {\sec (c+d x)}}{b d}}{3 b}+\frac {10 (A b-a B) \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x)}{3 b d}}{5 b}+\frac {2 B \sin (c+d x) \sec ^{\frac {5}{2}}(c+d x)}{5 b d}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {\frac {\frac {\frac {3 a^2 \left (-5 a^2 B+5 a A b-3 b^2 B\right ) \int \frac {1}{\sqrt {\csc \left (c+d x+\frac {\pi }{2}\right )}}dx+5 a^2 b (A b-a B) \int \sqrt {\csc \left (c+d x+\frac {\pi }{2}\right )}dx}{a^2}+15 a^2 (A b-a B) \int \frac {\csc \left (c+d x+\frac {\pi }{2}\right )^{3/2}}{a+b \csc \left (c+d x+\frac {\pi }{2}\right )}dx}{b}-\frac {6 \left (-5 a^2 B+5 a A b-3 b^2 B\right ) \sin (c+d x) \sqrt {\sec (c+d x)}}{b d}}{3 b}+\frac {10 (A b-a B) \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x)}{3 b d}}{5 b}+\frac {2 B \sin (c+d x) \sec ^{\frac {5}{2}}(c+d x)}{5 b d}\) |
\(\Big \downarrow \) 4258 |
\(\displaystyle \frac {\frac {\frac {\frac {3 a^2 \left (-5 a^2 B+5 a A b-3 b^2 B\right ) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \int \sqrt {\cos (c+d x)}dx+5 a^2 b (A b-a B) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \int \frac {1}{\sqrt {\cos (c+d x)}}dx}{a^2}+15 a^2 (A b-a B) \int \frac {\csc \left (c+d x+\frac {\pi }{2}\right )^{3/2}}{a+b \csc \left (c+d x+\frac {\pi }{2}\right )}dx}{b}-\frac {6 \left (-5 a^2 B+5 a A b-3 b^2 B\right ) \sin (c+d x) \sqrt {\sec (c+d x)}}{b d}}{3 b}+\frac {10 (A b-a B) \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x)}{3 b d}}{5 b}+\frac {2 B \sin (c+d x) \sec ^{\frac {5}{2}}(c+d x)}{5 b d}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {\frac {\frac {\frac {3 a^2 \left (-5 a^2 B+5 a A b-3 b^2 B\right ) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \int \sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}dx+5 a^2 b (A b-a B) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \int \frac {1}{\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}}dx}{a^2}+15 a^2 (A b-a B) \int \frac {\csc \left (c+d x+\frac {\pi }{2}\right )^{3/2}}{a+b \csc \left (c+d x+\frac {\pi }{2}\right )}dx}{b}-\frac {6 \left (-5 a^2 B+5 a A b-3 b^2 B\right ) \sin (c+d x) \sqrt {\sec (c+d x)}}{b d}}{3 b}+\frac {10 (A b-a B) \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x)}{3 b d}}{5 b}+\frac {2 B \sin (c+d x) \sec ^{\frac {5}{2}}(c+d x)}{5 b d}\) |
\(\Big \downarrow \) 3119 |
\(\displaystyle \frac {\frac {\frac {\frac {5 a^2 b (A b-a B) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \int \frac {1}{\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}}dx+\frac {6 a^2 \left (-5 a^2 B+5 a A b-3 b^2 B\right ) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{d}}{a^2}+15 a^2 (A b-a B) \int \frac {\csc \left (c+d x+\frac {\pi }{2}\right )^{3/2}}{a+b \csc \left (c+d x+\frac {\pi }{2}\right )}dx}{b}-\frac {6 \left (-5 a^2 B+5 a A b-3 b^2 B\right ) \sin (c+d x) \sqrt {\sec (c+d x)}}{b d}}{3 b}+\frac {10 (A b-a B) \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x)}{3 b d}}{5 b}+\frac {2 B \sin (c+d x) \sec ^{\frac {5}{2}}(c+d x)}{5 b d}\) |
\(\Big \downarrow \) 3120 |
\(\displaystyle \frac {\frac {\frac {15 a^2 (A b-a B) \int \frac {\csc \left (c+d x+\frac {\pi }{2}\right )^{3/2}}{a+b \csc \left (c+d x+\frac {\pi }{2}\right )}dx+\frac {\frac {6 a^2 \left (-5 a^2 B+5 a A b-3 b^2 B\right ) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{d}+\frac {10 a^2 b (A b-a B) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )}{d}}{a^2}}{b}-\frac {6 \left (-5 a^2 B+5 a A b-3 b^2 B\right ) \sin (c+d x) \sqrt {\sec (c+d x)}}{b d}}{3 b}+\frac {10 (A b-a B) \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x)}{3 b d}}{5 b}+\frac {2 B \sin (c+d x) \sec ^{\frac {5}{2}}(c+d x)}{5 b d}\) |
\(\Big \downarrow \) 4336 |
\(\displaystyle \frac {\frac {\frac {15 a^2 (A b-a B) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \int \frac {1}{\sqrt {\cos (c+d x)} (b+a \cos (c+d x))}dx+\frac {\frac {6 a^2 \left (-5 a^2 B+5 a A b-3 b^2 B\right ) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{d}+\frac {10 a^2 b (A b-a B) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )}{d}}{a^2}}{b}-\frac {6 \left (-5 a^2 B+5 a A b-3 b^2 B\right ) \sin (c+d x) \sqrt {\sec (c+d x)}}{b d}}{3 b}+\frac {10 (A b-a B) \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x)}{3 b d}}{5 b}+\frac {2 B \sin (c+d x) \sec ^{\frac {5}{2}}(c+d x)}{5 b d}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {\frac {\frac {15 a^2 (A b-a B) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \int \frac {1}{\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )} \left (b+a \sin \left (c+d x+\frac {\pi }{2}\right )\right )}dx+\frac {\frac {6 a^2 \left (-5 a^2 B+5 a A b-3 b^2 B\right ) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{d}+\frac {10 a^2 b (A b-a B) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )}{d}}{a^2}}{b}-\frac {6 \left (-5 a^2 B+5 a A b-3 b^2 B\right ) \sin (c+d x) \sqrt {\sec (c+d x)}}{b d}}{3 b}+\frac {10 (A b-a B) \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x)}{3 b d}}{5 b}+\frac {2 B \sin (c+d x) \sec ^{\frac {5}{2}}(c+d x)}{5 b d}\) |
\(\Big \downarrow \) 3284 |
\(\displaystyle \frac {\frac {\frac {\frac {\frac {6 a^2 \left (-5 a^2 B+5 a A b-3 b^2 B\right ) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{d}+\frac {10 a^2 b (A b-a B) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )}{d}}{a^2}+\frac {30 a^2 (A b-a B) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \operatorname {EllipticPi}\left (\frac {2 a}{a+b},\frac {1}{2} (c+d x),2\right )}{d (a+b)}}{b}-\frac {6 \left (-5 a^2 B+5 a A b-3 b^2 B\right ) \sin (c+d x) \sqrt {\sec (c+d x)}}{b d}}{3 b}+\frac {10 (A b-a B) \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x)}{3 b d}}{5 b}+\frac {2 B \sin (c+d x) \sec ^{\frac {5}{2}}(c+d x)}{5 b d}\) |
(2*B*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(5*b*d) + ((10*(A*b - a*B)*Sec[c + d *x]^(3/2)*Sin[c + d*x])/(3*b*d) + ((((6*a^2*(5*a*A*b - 5*a^2*B - 3*b^2*B)* Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/d + (10*a ^2*b*(A*b - a*B)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/d)/a^2 + (30*a^2*(A*b - a*B)*Sqrt[Cos[c + d*x]]*EllipticPi[(2*a)/( a + b), (c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/((a + b)*d))/b - (6*(5*a*A*b - 5*a^2*B - 3*b^2*B)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(b*d))/(3*b))/(5*b)
3.5.15.3.1 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_.))^(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[(csc[(e_.) + (f_.)*(x_)]*(d_.))^(n_)*(csc[(e_.) + (f_.)*(x_)]*(b_.) + ( a_))^(m_)*(csc[(e_.) + (f_.)*(x_)]*(B_.) + (A_)), x_Symbol] :> Simp[(-B)*d^ 2*Cot[e + f*x]*(a + b*Csc[e + f*x])^(m + 1)*((d*Csc[e + f*x])^(n - 2)/(b*f* (m + n))), x] + Simp[d^2/(b*(m + n)) Int[(a + b*Csc[e + f*x])^m*(d*Csc[e + f*x])^(n - 2)*Simp[a*B*(n - 2) + B*b*(m + n - 1)*Csc[e + f*x] + (A*b*(m + n) - a*B*(n - 1))*Csc[e + f*x]^2, x], x], x] /; FreeQ[{a, b, d, e, f, A, B , m}, x] && NeQ[A*b - a*B, 0] && NeQ[a^2 - b^2, 0] && GtQ[n, 1] && NeQ[m + n, 0] && !IGtQ[m, 1]
Int[((A_.) + csc[(e_.) + (f_.)*(x_)]*(B_.) + csc[(e_.) + (f_.)*(x_)]^2*(C_. ))*(csc[(e_.) + (f_.)*(x_)]*(d_.))^(n_)*(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a _))^(m_), x_Symbol] :> Simp[(-C)*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. \(757\) vs. \(2(327)=654\).
Time = 37.93 (sec) , antiderivative size = 758, normalized size of antiderivative = 2.74
-(-(-2*cos(1/2*d*x+1/2*c)^2+1)*sin(1/2*d*x+1/2*c)^2)^(1/2)*(2/5*B/b/(8*sin (1/2*d*x+1/2*c)^6-12*sin(1/2*d*x+1/2*c)^4+6*sin(1/2*d*x+1/2*c)^2-1)/sin(1/ 2*d*x+1/2*c)^2*(24*sin(1/2*d*x+1/2*c)^6*cos(1/2*d*x+1/2*c)-12*(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)*sin(1/2*d*x+1/2*c)^4-24*sin(1/2*d*x+1/2*c)^4*cos(1/2*d*x+1/2 *c)+12*(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)*sin(1/2*d*x+1/2*c)^2+8*cos(1/2*d*x+1/2*c) *sin(1/2*d*x+1/2*c)^2-3*(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*sin(1/2*d*x+1/2*c)^ 4+sin(1/2*d*x+1/2*c)^2)^(1/2)+2*(A*b-B*a)/b^2*(-1/6*cos(1/2*d*x+1/2*c)*(-2 *sin(1/2*d*x+1/2*c)^4+sin(1/2*d*x+1/2*c)^2)^(1/2)/(cos(1/2*d*x+1/2*c)^2-1/ 2)^2+1/3*(sin(1/2*d*x+1/2*c)^2)^(1/2)*(-2*cos(1/2*d*x+1/2*c)^2+1)^(1/2)/(- 2*sin(1/2*d*x+1/2*c)^4+sin(1/2*d*x+1/2*c)^2)^(1/2)*EllipticF(cos(1/2*d*x+1 /2*c),2^(1/2)))-2*(A*b-B*a)/b^3*a/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-B*a)*a^3/b ^3/(a^2-a*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)*EllipticPi(cos(1/2* d*x+1/2*c),2*a/(a-b),2^(1/2)))/sin(1/2*d*x+1/2*c)/(2*cos(1/2*d*x+1/2*c)...
Timed out. \[ \int \frac {\sec ^{\frac {7}{2}}(c+d x) (A+B \sec (c+d x))}{a+b \sec (c+d x)} \, dx=\text {Timed out} \]
Timed out. \[ \int \frac {\sec ^{\frac {7}{2}}(c+d x) (A+B \sec (c+d x))}{a+b \sec (c+d x)} \, dx=\text {Timed out} \]
\[ \int \frac {\sec ^{\frac {7}{2}}(c+d x) (A+B \sec (c+d x))}{a+b \sec (c+d x)} \, dx=\int { \frac {{\left (B \sec \left (d x + c\right ) + A\right )} \sec \left (d x + c\right )^{\frac {7}{2}}}{b \sec \left (d x + c\right ) + a} \,d x } \]
\[ \int \frac {\sec ^{\frac {7}{2}}(c+d x) (A+B \sec (c+d x))}{a+b \sec (c+d x)} \, dx=\int { \frac {{\left (B \sec \left (d x + c\right ) + A\right )} \sec \left (d x + c\right )^{\frac {7}{2}}}{b \sec \left (d x + c\right ) + a} \,d x } \]
Timed out. \[ \int \frac {\sec ^{\frac {7}{2}}(c+d x) (A+B \sec (c+d x))}{a+b \sec (c+d x)} \, dx=\int \frac {\left (A+\frac {B}{\cos \left (c+d\,x\right )}\right )\,{\left (\frac {1}{\cos \left (c+d\,x\right )}\right )}^{7/2}}{a+\frac {b}{\cos \left (c+d\,x\right )}} \,d x \]