Integrand size = 23, antiderivative size = 530 \[ \int \cos ^4(c+d x) (a+b \sec (c+d x))^{5/2} \, dx=\frac {(a-b) \sqrt {a+b} \left (284 a^2+15 b^2\right ) \cot (c+d x) E\left (\arcsin \left (\frac {\sqrt {a+b \sec (c+d x)}}{\sqrt {a+b}}\right )|\frac {a+b}{a-b}\right ) \sqrt {\frac {b (1-\sec (c+d x))}{a+b}} \sqrt {-\frac {b (1+\sec (c+d x))}{a-b}}}{192 a d}+\frac {\sqrt {a+b} \left (72 a^3+284 a^2 b+118 a b^2+15 b^3\right ) \cot (c+d x) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {a+b \sec (c+d x)}}{\sqrt {a+b}}\right ),\frac {a+b}{a-b}\right ) \sqrt {\frac {b (1-\sec (c+d x))}{a+b}} \sqrt {-\frac {b (1+\sec (c+d x))}{a-b}}}{192 a d}-\frac {\sqrt {a+b} \left (48 a^4+120 a^2 b^2-5 b^4\right ) \cot (c+d x) \operatorname {EllipticPi}\left (\frac {a+b}{a},\arcsin \left (\frac {\sqrt {a+b \sec (c+d x)}}{\sqrt {a+b}}\right ),\frac {a+b}{a-b}\right ) \sqrt {\frac {b (1-\sec (c+d x))}{a+b}} \sqrt {-\frac {b (1+\sec (c+d x))}{a-b}}}{64 a^2 d}+\frac {b \left (284 a^2+15 b^2\right ) \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{192 a d}+\frac {\left (36 a^2+59 b^2\right ) \cos (c+d x) \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{96 d}+\frac {17 a b \cos ^2(c+d x) \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{24 d}+\frac {a^2 \cos ^3(c+d x) \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{4 d} \]
1/192*(a-b)*(284*a^2+15*b^2)*cot(d*x+c)*EllipticE((a+b*sec(d*x+c))^(1/2)/( a+b)^(1/2),((a+b)/(a-b))^(1/2))*(a+b)^(1/2)*(b*(1-sec(d*x+c))/(a+b))^(1/2) *(-b*(1+sec(d*x+c))/(a-b))^(1/2)/a/d+1/192*(72*a^3+284*a^2*b+118*a*b^2+15* b^3)*cot(d*x+c)*EllipticF((a+b*sec(d*x+c))^(1/2)/(a+b)^(1/2),((a+b)/(a-b)) ^(1/2))*(a+b)^(1/2)*(b*(1-sec(d*x+c))/(a+b))^(1/2)*(-b*(1+sec(d*x+c))/(a-b ))^(1/2)/a/d-1/64*(48*a^4+120*a^2*b^2-5*b^4)*cot(d*x+c)*EllipticPi((a+b*se c(d*x+c))^(1/2)/(a+b)^(1/2),(a+b)/a,((a+b)/(a-b))^(1/2))*(a+b)^(1/2)*(b*(1 -sec(d*x+c))/(a+b))^(1/2)*(-b*(1+sec(d*x+c))/(a-b))^(1/2)/a^2/d+1/192*b*(2 84*a^2+15*b^2)*sin(d*x+c)*(a+b*sec(d*x+c))^(1/2)/a/d+1/96*(36*a^2+59*b^2)* cos(d*x+c)*sin(d*x+c)*(a+b*sec(d*x+c))^(1/2)/d+17/24*a*b*cos(d*x+c)^2*sin( d*x+c)*(a+b*sec(d*x+c))^(1/2)/d+1/4*a^2*cos(d*x+c)^3*sin(d*x+c)*(a+b*sec(d *x+c))^(1/2)/d
Leaf count is larger than twice the leaf count of optimal. \(1274\) vs. \(2(530)=1060\).
Time = 14.49 (sec) , antiderivative size = 1274, normalized size of antiderivative = 2.40 \[ \int \cos ^4(c+d x) (a+b \sec (c+d x))^{5/2} \, dx =\text {Too large to display} \]
(Cos[c + d*x]^2*(a + b*Sec[c + d*x])^(5/2)*((17*a*b*Sin[c + d*x])/96 + ((4 8*a^2 + 59*b^2)*Sin[2*(c + d*x)])/192 + (17*a*b*Sin[3*(c + d*x)])/96 + (a^ 2*Sin[4*(c + d*x)])/32))/(d*(b + a*Cos[c + d*x])^2) + ((a + b*Sec[c + d*x] )^(5/2)*(-284*a^3*b*Tan[(c + d*x)/2] - 284*a^2*b^2*Tan[(c + d*x)/2] - 15*a *b^3*Tan[(c + d*x)/2] - 15*b^4*Tan[(c + d*x)/2] + 568*a^3*b*Tan[(c + d*x)/ 2]^3 + 30*a*b^3*Tan[(c + d*x)/2]^3 - 284*a^3*b*Tan[(c + d*x)/2]^5 + 284*a^ 2*b^2*Tan[(c + d*x)/2]^5 - 15*a*b^3*Tan[(c + d*x)/2]^5 + 15*b^4*Tan[(c + d *x)/2]^5 - 288*a^4*EllipticPi[-1, ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b )]*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan [(c + d*x)/2]^2)/(a + b)] - 720*a^2*b^2*EllipticPi[-1, ArcSin[Tan[(c + d*x )/2]], (a - b)/(a + b)]*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b - a*Tan[( c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)] + 30*b^4*EllipticPi[-1, Arc Sin[Tan[(c + d*x)/2]], (a - b)/(a + b)]*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[ (a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)] - 288*a^4*E llipticPi[-1, ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)]*Tan[(c + d*x)/2]^ 2*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[ (c + d*x)/2]^2)/(a + b)] - 720*a^2*b^2*EllipticPi[-1, ArcSin[Tan[(c + d*x) /2]], (a - b)/(a + b)]*Tan[(c + d*x)/2]^2*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqr t[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)] + 30*b^4* EllipticPi[-1, ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)]*Tan[(c + d*x)...
Time = 2.88 (sec) , antiderivative size = 547, normalized size of antiderivative = 1.03, number of steps used = 20, number of rules used = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.870, Rules used = {3042, 4328, 27, 3042, 4592, 27, 3042, 4592, 27, 3042, 4592, 27, 3042, 4546, 3042, 4409, 3042, 4271, 4319, 4492}
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.
| \(\displaystyle \int \cos ^4(c+d x) (a+b \sec (c+d x))^{5/2} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\left (a+b \csc \left (c+d x+\frac {\pi }{2}\right )\right )^{5/2}}{\csc \left (c+d x+\frac {\pi }{2}\right )^4}dx\) |
| \(\Big \downarrow \) 4328 |
| \(\displaystyle \frac {1}{4} \int \frac {\cos ^3(c+d x) \left (17 b a^2+6 \left (a^2+4 b^2\right ) \sec (c+d x) a+b \left (5 a^2+8 b^2\right ) \sec ^2(c+d x)\right )}{2 \sqrt {a+b \sec (c+d x)}}dx+\frac {a^2 \sin (c+d x) \cos ^3(c+d x) \sqrt {a+b \sec (c+d x)}}{4 d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{8} \int \frac {\cos ^3(c+d x) \left (17 b a^2+6 \left (a^2+4 b^2\right ) \sec (c+d x) a+b \left (5 a^2+8 b^2\right ) \sec ^2(c+d x)\right )}{\sqrt {a+b \sec (c+d x)}}dx+\frac {a^2 \sin (c+d x) \cos ^3(c+d x) \sqrt {a+b \sec (c+d x)}}{4 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{8} \int \frac {17 b a^2+6 \left (a^2+4 b^2\right ) \csc \left (c+d x+\frac {\pi }{2}\right ) a+b \left (5 a^2+8 b^2\right ) \csc \left (c+d x+\frac {\pi }{2}\right )^2}{\csc \left (c+d x+\frac {\pi }{2}\right )^3 \sqrt {a+b \csc \left (c+d x+\frac {\pi }{2}\right )}}dx+\frac {a^2 \sin (c+d x) \cos ^3(c+d x) \sqrt {a+b \sec (c+d x)}}{4 d}\) |
| \(\Big \downarrow \) 4592 |
| \(\displaystyle \frac {1}{8} \left (\frac {17 a b \sin (c+d x) \cos ^2(c+d x) \sqrt {a+b \sec (c+d x)}}{3 d}-\frac {\int -\frac {\cos ^2(c+d x) \left (51 b^2 \sec ^2(c+d x) a^2+\left (36 a^2+59 b^2\right ) a^2+2 b \left (49 a^2+24 b^2\right ) \sec (c+d x) a\right )}{2 \sqrt {a+b \sec (c+d x)}}dx}{3 a}\right )+\frac {a^2 \sin (c+d x) \cos ^3(c+d x) \sqrt {a+b \sec (c+d x)}}{4 d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{8} \left (\frac {\int \frac {\cos ^2(c+d x) \left (51 b^2 \sec ^2(c+d x) a^2+\left (36 a^2+59 b^2\right ) a^2+2 b \left (49 a^2+24 b^2\right ) \sec (c+d x) a\right )}{\sqrt {a+b \sec (c+d x)}}dx}{6 a}+\frac {17 a b \sin (c+d x) \cos ^2(c+d x) \sqrt {a+b \sec (c+d x)}}{3 d}\right )+\frac {a^2 \sin (c+d x) \cos ^3(c+d x) \sqrt {a+b \sec (c+d x)}}{4 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{8} \left (\frac {\int \frac {51 b^2 \csc \left (c+d x+\frac {\pi }{2}\right )^2 a^2+\left (36 a^2+59 b^2\right ) a^2+2 b \left (49 a^2+24 b^2\right ) \csc \left (c+d x+\frac {\pi }{2}\right ) a}{\csc \left (c+d x+\frac {\pi }{2}\right )^2 \sqrt {a+b \csc \left (c+d x+\frac {\pi }{2}\right )}}dx}{6 a}+\frac {17 a b \sin (c+d x) \cos ^2(c+d x) \sqrt {a+b \sec (c+d x)}}{3 d}\right )+\frac {a^2 \sin (c+d x) \cos ^3(c+d x) \sqrt {a+b \sec (c+d x)}}{4 d}\) |
| \(\Big \downarrow \) 4592 |
| \(\displaystyle \frac {1}{8} \left (\frac {\frac {a \left (36 a^2+59 b^2\right ) \sin (c+d x) \cos (c+d x) \sqrt {a+b \sec (c+d x)}}{2 d}-\frac {\int -\frac {\cos (c+d x) \left (2 \left (36 a^2+161 b^2\right ) \sec (c+d x) a^3+b \left (36 a^2+59 b^2\right ) \sec ^2(c+d x) a^2+b \left (284 a^2+15 b^2\right ) a^2\right )}{2 \sqrt {a+b \sec (c+d x)}}dx}{2 a}}{6 a}+\frac {17 a b \sin (c+d x) \cos ^2(c+d x) \sqrt {a+b \sec (c+d x)}}{3 d}\right )+\frac {a^2 \sin (c+d x) \cos ^3(c+d x) \sqrt {a+b \sec (c+d x)}}{4 d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{8} \left (\frac {\frac {\int \frac {\cos (c+d x) \left (2 \left (36 a^2+161 b^2\right ) \sec (c+d x) a^3+b \left (36 a^2+59 b^2\right ) \sec ^2(c+d x) a^2+b \left (284 a^2+15 b^2\right ) a^2\right )}{\sqrt {a+b \sec (c+d x)}}dx}{4 a}+\frac {a \left (36 a^2+59 b^2\right ) \sin (c+d x) \cos (c+d x) \sqrt {a+b \sec (c+d x)}}{2 d}}{6 a}+\frac {17 a b \sin (c+d x) \cos ^2(c+d x) \sqrt {a+b \sec (c+d x)}}{3 d}\right )+\frac {a^2 \sin (c+d x) \cos ^3(c+d x) \sqrt {a+b \sec (c+d x)}}{4 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{8} \left (\frac {\frac {\int \frac {2 \left (36 a^2+161 b^2\right ) \csc \left (c+d x+\frac {\pi }{2}\right ) a^3+b \left (36 a^2+59 b^2\right ) \csc \left (c+d x+\frac {\pi }{2}\right )^2 a^2+b \left (284 a^2+15 b^2\right ) a^2}{\csc \left (c+d x+\frac {\pi }{2}\right ) \sqrt {a+b \csc \left (c+d x+\frac {\pi }{2}\right )}}dx}{4 a}+\frac {a \left (36 a^2+59 b^2\right ) \sin (c+d x) \cos (c+d x) \sqrt {a+b \sec (c+d x)}}{2 d}}{6 a}+\frac {17 a b \sin (c+d x) \cos ^2(c+d x) \sqrt {a+b \sec (c+d x)}}{3 d}\right )+\frac {a^2 \sin (c+d x) \cos ^3(c+d x) \sqrt {a+b \sec (c+d x)}}{4 d}\) |
| \(\Big \downarrow \) 4592 |
| \(\displaystyle \frac {1}{8} \left (\frac {\frac {\frac {a b \left (284 a^2+15 b^2\right ) \sin (c+d x) \sqrt {a+b \sec (c+d x)}}{d}-\frac {\int -\frac {2 b \left (36 a^2+59 b^2\right ) \sec (c+d x) a^3-b^2 \left (284 a^2+15 b^2\right ) \sec ^2(c+d x) a^2+3 \left (48 a^4+120 b^2 a^2-5 b^4\right ) a^2}{2 \sqrt {a+b \sec (c+d x)}}dx}{a}}{4 a}+\frac {a \left (36 a^2+59 b^2\right ) \sin (c+d x) \cos (c+d x) \sqrt {a+b \sec (c+d x)}}{2 d}}{6 a}+\frac {17 a b \sin (c+d x) \cos ^2(c+d x) \sqrt {a+b \sec (c+d x)}}{3 d}\right )+\frac {a^2 \sin (c+d x) \cos ^3(c+d x) \sqrt {a+b \sec (c+d x)}}{4 d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{8} \left (\frac {\frac {\frac {\int \frac {2 b \left (36 a^2+59 b^2\right ) \sec (c+d x) a^3-b^2 \left (284 a^2+15 b^2\right ) \sec ^2(c+d x) a^2+3 \left (48 a^4+120 b^2 a^2-5 b^4\right ) a^2}{\sqrt {a+b \sec (c+d x)}}dx}{2 a}+\frac {a b \left (284 a^2+15 b^2\right ) \sin (c+d x) \sqrt {a+b \sec (c+d x)}}{d}}{4 a}+\frac {a \left (36 a^2+59 b^2\right ) \sin (c+d x) \cos (c+d x) \sqrt {a+b \sec (c+d x)}}{2 d}}{6 a}+\frac {17 a b \sin (c+d x) \cos ^2(c+d x) \sqrt {a+b \sec (c+d x)}}{3 d}\right )+\frac {a^2 \sin (c+d x) \cos ^3(c+d x) \sqrt {a+b \sec (c+d x)}}{4 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{8} \left (\frac {\frac {\frac {\int \frac {2 b \left (36 a^2+59 b^2\right ) \csc \left (c+d x+\frac {\pi }{2}\right ) a^3-b^2 \left (284 a^2+15 b^2\right ) \csc \left (c+d x+\frac {\pi }{2}\right )^2 a^2+3 \left (48 a^4+120 b^2 a^2-5 b^4\right ) a^2}{\sqrt {a+b \csc \left (c+d x+\frac {\pi }{2}\right )}}dx}{2 a}+\frac {a b \left (284 a^2+15 b^2\right ) \sin (c+d x) \sqrt {a+b \sec (c+d x)}}{d}}{4 a}+\frac {a \left (36 a^2+59 b^2\right ) \sin (c+d x) \cos (c+d x) \sqrt {a+b \sec (c+d x)}}{2 d}}{6 a}+\frac {17 a b \sin (c+d x) \cos ^2(c+d x) \sqrt {a+b \sec (c+d x)}}{3 d}\right )+\frac {a^2 \sin (c+d x) \cos ^3(c+d x) \sqrt {a+b \sec (c+d x)}}{4 d}\) |
| \(\Big \downarrow \) 4546 |
| \(\displaystyle \frac {1}{8} \left (\frac {\frac {\frac {\int \frac {3 \left (48 a^4+120 b^2 a^2-5 b^4\right ) a^2+\left (2 b \left (36 a^2+59 b^2\right ) a^3+b^2 \left (284 a^2+15 b^2\right ) a^2\right ) \sec (c+d x)}{\sqrt {a+b \sec (c+d x)}}dx-a^2 b^2 \left (284 a^2+15 b^2\right ) \int \frac {\sec (c+d x) (\sec (c+d x)+1)}{\sqrt {a+b \sec (c+d x)}}dx}{2 a}+\frac {a b \left (284 a^2+15 b^2\right ) \sin (c+d x) \sqrt {a+b \sec (c+d x)}}{d}}{4 a}+\frac {a \left (36 a^2+59 b^2\right ) \sin (c+d x) \cos (c+d x) \sqrt {a+b \sec (c+d x)}}{2 d}}{6 a}+\frac {17 a b \sin (c+d x) \cos ^2(c+d x) \sqrt {a+b \sec (c+d x)}}{3 d}\right )+\frac {a^2 \sin (c+d x) \cos ^3(c+d x) \sqrt {a+b \sec (c+d x)}}{4 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{8} \left (\frac {\frac {\frac {\int \frac {3 \left (48 a^4+120 b^2 a^2-5 b^4\right ) a^2+\left (2 b \left (36 a^2+59 b^2\right ) a^3+b^2 \left (284 a^2+15 b^2\right ) a^2\right ) \csc \left (c+d x+\frac {\pi }{2}\right )}{\sqrt {a+b \csc \left (c+d x+\frac {\pi }{2}\right )}}dx-a^2 b^2 \left (284 a^2+15 b^2\right ) \int \frac {\csc \left (c+d x+\frac {\pi }{2}\right ) \left (\csc \left (c+d x+\frac {\pi }{2}\right )+1\right )}{\sqrt {a+b \csc \left (c+d x+\frac {\pi }{2}\right )}}dx}{2 a}+\frac {a b \left (284 a^2+15 b^2\right ) \sin (c+d x) \sqrt {a+b \sec (c+d x)}}{d}}{4 a}+\frac {a \left (36 a^2+59 b^2\right ) \sin (c+d x) \cos (c+d x) \sqrt {a+b \sec (c+d x)}}{2 d}}{6 a}+\frac {17 a b \sin (c+d x) \cos ^2(c+d x) \sqrt {a+b \sec (c+d x)}}{3 d}\right )+\frac {a^2 \sin (c+d x) \cos ^3(c+d x) \sqrt {a+b \sec (c+d x)}}{4 d}\) |
| \(\Big \downarrow \) 4409 |
| \(\displaystyle \frac {1}{8} \left (\frac {\frac {\frac {a^2 \left (-b^2\right ) \left (284 a^2+15 b^2\right ) \int \frac {\csc \left (c+d x+\frac {\pi }{2}\right ) \left (\csc \left (c+d x+\frac {\pi }{2}\right )+1\right )}{\sqrt {a+b \csc \left (c+d x+\frac {\pi }{2}\right )}}dx+3 a^2 \left (48 a^4+120 a^2 b^2-5 b^4\right ) \int \frac {1}{\sqrt {a+b \sec (c+d x)}}dx+a^2 b \left (72 a^3+284 a^2 b+118 a b^2+15 b^3\right ) \int \frac {\sec (c+d x)}{\sqrt {a+b \sec (c+d x)}}dx}{2 a}+\frac {a b \left (284 a^2+15 b^2\right ) \sin (c+d x) \sqrt {a+b \sec (c+d x)}}{d}}{4 a}+\frac {a \left (36 a^2+59 b^2\right ) \sin (c+d x) \cos (c+d x) \sqrt {a+b \sec (c+d x)}}{2 d}}{6 a}+\frac {17 a b \sin (c+d x) \cos ^2(c+d x) \sqrt {a+b \sec (c+d x)}}{3 d}\right )+\frac {a^2 \sin (c+d x) \cos ^3(c+d x) \sqrt {a+b \sec (c+d x)}}{4 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{8} \left (\frac {\frac {\frac {-a^2 b^2 \left (284 a^2+15 b^2\right ) \int \frac {\csc \left (c+d x+\frac {\pi }{2}\right ) \left (\csc \left (c+d x+\frac {\pi }{2}\right )+1\right )}{\sqrt {a+b \csc \left (c+d x+\frac {\pi }{2}\right )}}dx+3 a^2 \left (48 a^4+120 a^2 b^2-5 b^4\right ) \int \frac {1}{\sqrt {a+b \csc \left (c+d x+\frac {\pi }{2}\right )}}dx+a^2 b \left (72 a^3+284 a^2 b+118 a b^2+15 b^3\right ) \int \frac {\csc \left (c+d x+\frac {\pi }{2}\right )}{\sqrt {a+b \csc \left (c+d x+\frac {\pi }{2}\right )}}dx}{2 a}+\frac {a b \left (284 a^2+15 b^2\right ) \sin (c+d x) \sqrt {a+b \sec (c+d x)}}{d}}{4 a}+\frac {a \left (36 a^2+59 b^2\right ) \sin (c+d x) \cos (c+d x) \sqrt {a+b \sec (c+d x)}}{2 d}}{6 a}+\frac {17 a b \sin (c+d x) \cos ^2(c+d x) \sqrt {a+b \sec (c+d x)}}{3 d}\right )+\frac {a^2 \sin (c+d x) \cos ^3(c+d x) \sqrt {a+b \sec (c+d x)}}{4 d}\) |
| \(\Big \downarrow \) 4271 |
| \(\displaystyle \frac {1}{8} \left (\frac {\frac {\frac {-a^2 b^2 \left (284 a^2+15 b^2\right ) \int \frac {\csc \left (c+d x+\frac {\pi }{2}\right ) \left (\csc \left (c+d x+\frac {\pi }{2}\right )+1\right )}{\sqrt {a+b \csc \left (c+d x+\frac {\pi }{2}\right )}}dx+a^2 b \left (72 a^3+284 a^2 b+118 a b^2+15 b^3\right ) \int \frac {\csc \left (c+d x+\frac {\pi }{2}\right )}{\sqrt {a+b \csc \left (c+d x+\frac {\pi }{2}\right )}}dx-\frac {6 a \sqrt {a+b} \left (48 a^4+120 a^2 b^2-5 b^4\right ) \cot (c+d x) \sqrt {\frac {b (1-\sec (c+d x))}{a+b}} \sqrt {-\frac {b (\sec (c+d x)+1)}{a-b}} \operatorname {EllipticPi}\left (\frac {a+b}{a},\arcsin \left (\frac {\sqrt {a+b \sec (c+d x)}}{\sqrt {a+b}}\right ),\frac {a+b}{a-b}\right )}{d}}{2 a}+\frac {a b \left (284 a^2+15 b^2\right ) \sin (c+d x) \sqrt {a+b \sec (c+d x)}}{d}}{4 a}+\frac {a \left (36 a^2+59 b^2\right ) \sin (c+d x) \cos (c+d x) \sqrt {a+b \sec (c+d x)}}{2 d}}{6 a}+\frac {17 a b \sin (c+d x) \cos ^2(c+d x) \sqrt {a+b \sec (c+d x)}}{3 d}\right )+\frac {a^2 \sin (c+d x) \cos ^3(c+d x) \sqrt {a+b \sec (c+d x)}}{4 d}\) |
| \(\Big \downarrow \) 4319 |
| \(\displaystyle \frac {1}{8} \left (\frac {\frac {\frac {a^2 \left (-b^2\right ) \left (284 a^2+15 b^2\right ) \int \frac {\csc \left (c+d x+\frac {\pi }{2}\right ) \left (\csc \left (c+d x+\frac {\pi }{2}\right )+1\right )}{\sqrt {a+b \csc \left (c+d x+\frac {\pi }{2}\right )}}dx-\frac {6 a \sqrt {a+b} \left (48 a^4+120 a^2 b^2-5 b^4\right ) \cot (c+d x) \sqrt {\frac {b (1-\sec (c+d x))}{a+b}} \sqrt {-\frac {b (\sec (c+d x)+1)}{a-b}} \operatorname {EllipticPi}\left (\frac {a+b}{a},\arcsin \left (\frac {\sqrt {a+b \sec (c+d x)}}{\sqrt {a+b}}\right ),\frac {a+b}{a-b}\right )}{d}+\frac {2 a^2 \sqrt {a+b} \left (72 a^3+284 a^2 b+118 a b^2+15 b^3\right ) \cot (c+d x) \sqrt {\frac {b (1-\sec (c+d x))}{a+b}} \sqrt {-\frac {b (\sec (c+d x)+1)}{a-b}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {a+b \sec (c+d x)}}{\sqrt {a+b}}\right ),\frac {a+b}{a-b}\right )}{d}}{2 a}+\frac {a b \left (284 a^2+15 b^2\right ) \sin (c+d x) \sqrt {a+b \sec (c+d x)}}{d}}{4 a}+\frac {a \left (36 a^2+59 b^2\right ) \sin (c+d x) \cos (c+d x) \sqrt {a+b \sec (c+d x)}}{2 d}}{6 a}+\frac {17 a b \sin (c+d x) \cos ^2(c+d x) \sqrt {a+b \sec (c+d x)}}{3 d}\right )+\frac {a^2 \sin (c+d x) \cos ^3(c+d x) \sqrt {a+b \sec (c+d x)}}{4 d}\) |
| \(\Big \downarrow \) 4492 |
| \(\displaystyle \frac {a^2 \sin (c+d x) \cos ^3(c+d x) \sqrt {a+b \sec (c+d x)}}{4 d}+\frac {1}{8} \left (\frac {\frac {a \left (36 a^2+59 b^2\right ) \sin (c+d x) \cos (c+d x) \sqrt {a+b \sec (c+d x)}}{2 d}+\frac {\frac {a b \left (284 a^2+15 b^2\right ) \sin (c+d x) \sqrt {a+b \sec (c+d x)}}{d}+\frac {\frac {2 a^2 (a-b) \sqrt {a+b} \left (284 a^2+15 b^2\right ) \cot (c+d x) \sqrt {\frac {b (1-\sec (c+d x))}{a+b}} \sqrt {-\frac {b (\sec (c+d x)+1)}{a-b}} E\left (\arcsin \left (\frac {\sqrt {a+b \sec (c+d x)}}{\sqrt {a+b}}\right )|\frac {a+b}{a-b}\right )}{d}-\frac {6 a \sqrt {a+b} \left (48 a^4+120 a^2 b^2-5 b^4\right ) \cot (c+d x) \sqrt {\frac {b (1-\sec (c+d x))}{a+b}} \sqrt {-\frac {b (\sec (c+d x)+1)}{a-b}} \operatorname {EllipticPi}\left (\frac {a+b}{a},\arcsin \left (\frac {\sqrt {a+b \sec (c+d x)}}{\sqrt {a+b}}\right ),\frac {a+b}{a-b}\right )}{d}+\frac {2 a^2 \sqrt {a+b} \left (72 a^3+284 a^2 b+118 a b^2+15 b^3\right ) \cot (c+d x) \sqrt {\frac {b (1-\sec (c+d x))}{a+b}} \sqrt {-\frac {b (\sec (c+d x)+1)}{a-b}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {a+b \sec (c+d x)}}{\sqrt {a+b}}\right ),\frac {a+b}{a-b}\right )}{d}}{2 a}}{4 a}}{6 a}+\frac {17 a b \sin (c+d x) \cos ^2(c+d x) \sqrt {a+b \sec (c+d x)}}{3 d}\right )\) |
(a^2*Cos[c + d*x]^3*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(4*d) + ((17*a* b*Cos[c + d*x]^2*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(3*d) + ((a*(36*a^ 2 + 59*b^2)*Cos[c + d*x]*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(2*d) + (( (2*a^2*(a - b)*Sqrt[a + b]*(284*a^2 + 15*b^2)*Cot[c + d*x]*EllipticE[ArcSi n[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec [c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/d + (2*a^2*S qrt[a + b]*(72*a^3 + 284*a^2*b + 118*a*b^2 + 15*b^3)*Cot[c + d*x]*Elliptic F[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*( 1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/d - ( 6*a*Sqrt[a + b]*(48*a^4 + 120*a^2*b^2 - 5*b^4)*Cot[c + d*x]*EllipticPi[(a + b)/a, ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqr t[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))]) /d)/(2*a) + (a*b*(284*a^2 + 15*b^2)*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x]) /d)/(4*a))/(6*a))/8
3.6.52.3.1 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[1/Sqrt[csc[(c_.) + (d_.)*(x_)]*(b_.) + (a_)], x_Symbol] :> Simp[2*(Rt[a + b, 2]/(a*d*Cot[c + d*x]))*Sqrt[b*((1 - Csc[c + d*x])/(a + b))]*Sqrt[(-b) *((1 + Csc[c + d*x])/(a - b))]*EllipticPi[(a + b)/a, ArcSin[Sqrt[a + b*Csc[ c + d*x]]/Rt[a + b, 2]], (a + b)/(a - b)], x] /; FreeQ[{a, b, c, d}, x] && NeQ[a^2 - b^2, 0]
Int[csc[(e_.) + (f_.)*(x_)]/Sqrt[csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_)], x_S ymbol] :> Simp[-2*(Rt[a + b, 2]/(b*f*Cot[e + f*x]))*Sqrt[(b*(1 - Csc[e + f* x]))/(a + b)]*Sqrt[(-b)*((1 + Csc[e + f*x])/(a - b))]*EllipticF[ArcSin[Sqrt [a + b*Csc[e + f*x]]/Rt[a + b, 2]], (a + b)/(a - b)], x] /; FreeQ[{a, b, e, f}, x] && NeQ[a^2 - b^2, 0]
Int[(csc[(e_.) + (f_.)*(x_)]*(d_.))^(n_)*(csc[(e_.) + (f_.)*(x_)]*(b_.) + ( a_))^(m_), x_Symbol] :> Simp[a^2*Cot[e + f*x]*(a + b*Csc[e + f*x])^(m - 2)* ((d*Csc[e + f*x])^n/(f*n)), x] - Simp[1/(d*n) Int[(a + b*Csc[e + f*x])^(m - 3)*(d*Csc[e + f*x])^(n + 1)*Simp[a^2*b*(m - 2*n - 2) - a*(3*b^2*n + a^2* (n + 1))*Csc[e + f*x] - b*(b^2*n + a^2*(m + n - 1))*Csc[e + f*x]^2, x], x], x] /; FreeQ[{a, b, d, e, f}, x] && NeQ[a^2 - b^2, 0] && GtQ[m, 2] && ((Int egerQ[m] && LtQ[n, -1]) || (IntegersQ[m + 1/2, 2*n] && LeQ[n, -1]))
Int[(csc[(e_.) + (f_.)*(x_)]*(d_.) + (c_))/Sqrt[csc[(e_.) + (f_.)*(x_)]*(b_ .) + (a_)], x_Symbol] :> Simp[c Int[1/Sqrt[a + b*Csc[e + f*x]], x], x] + Simp[d Int[Csc[e + f*x]/Sqrt[a + b*Csc[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]
Int[(csc[(e_.) + (f_.)*(x_)]*(csc[(e_.) + (f_.)*(x_)]*(B_.) + (A_)))/Sqrt[c sc[(e_.) + (f_.)*(x_)]*(b_.) + (a_)], x_Symbol] :> Simp[-2*(A*b - a*B)*Rt[a + b*(B/A), 2]*Sqrt[b*((1 - Csc[e + f*x])/(a + b))]*(Sqrt[(-b)*((1 + Csc[e + f*x])/(a - b))]/(b^2*f*Cot[e + f*x]))*EllipticE[ArcSin[Sqrt[a + b*Csc[e + f*x]]/Rt[a + b*(B/A), 2]], (a*A + b*B)/(a*A - b*B)], x] /; FreeQ[{a, b, e, f, A, B}, x] && NeQ[a^2 - b^2, 0] && EqQ[A^2 - B^2, 0]
Int[((A_.) + csc[(e_.) + (f_.)*(x_)]*(B_.) + csc[(e_.) + (f_.)*(x_)]^2*(C_. ))/Sqrt[csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_)], x_Symbol] :> Int[(A + (B - C )*Csc[e + f*x])/Sqrt[a + b*Csc[e + f*x]], x] + Simp[C Int[Csc[e + f*x]*(( 1 + Csc[e + f*x])/Sqrt[a + b*Csc[e + f*x]]), x], x] /; FreeQ[{a, b, e, f, A , B, C}, 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[A*Cot[e + f*x]*(a + b*Csc[e + f*x])^(m + 1)*((d *Csc[e + f*x])^n/(a*f*n)), x] + Simp[1/(a*d*n) Int[(a + b*Csc[e + f*x])^m *(d*Csc[e + f*x])^(n + 1)*Simp[a*B*n - A*b*(m + n + 1) + a*(A + A*n + C*n)* Csc[e + f*x] + A*b*(m + n + 2)*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] && LeQ[n, -1]
Leaf count of result is larger than twice the leaf count of optimal. \(3191\) vs. \(2(481)=962\).
Time = 410.77 (sec) , antiderivative size = 3192, normalized size of antiderivative = 6.02
1/192/d/a*(-1440*(cos(d*x+c)/(cos(d*x+c)+1))^(1/2)*(1/(a+b)*(b+a*cos(d*x+c ))/(cos(d*x+c)+1))^(1/2)*EllipticPi(cot(d*x+c)-csc(d*x+c),-1,((a-b)/(a+b)) ^(1/2))*a^2*b^2*cos(d*x+c)-720*(cos(d*x+c)/(cos(d*x+c)+1))^(1/2)*(1/(a+b)* (b+a*cos(d*x+c))/(cos(d*x+c)+1))^(1/2)*EllipticPi(cot(d*x+c)-csc(d*x+c),-1 ,((a-b)/(a+b))^(1/2))*a^2*b^2*cos(d*x+c)^2-720*EllipticPi(cot(d*x+c)-csc(d *x+c),-1,((a-b)/(a+b))^(1/2))*(cos(d*x+c)/(cos(d*x+c)+1))^(1/2)*(1/(a+b)*( b+a*cos(d*x+c))/(cos(d*x+c)+1))^(1/2)*a^2*b^2+60*(cos(d*x+c)/(cos(d*x+c)+1 ))^(1/2)*(1/(a+b)*(b+a*cos(d*x+c))/(cos(d*x+c)+1))^(1/2)*EllipticPi(cot(d* x+c)-csc(d*x+c),-1,((a-b)/(a+b))^(1/2))*b^4*cos(d*x+c)-30*(cos(d*x+c)/(cos (d*x+c)+1))^(1/2)*(1/(a+b)*(b+a*cos(d*x+c))/(cos(d*x+c)+1))^(1/2)*Elliptic E(cot(d*x+c)-csc(d*x+c),((a-b)/(a+b))^(1/2))*b^4*cos(d*x+c)+144*(cos(d*x+c )/(cos(d*x+c)+1))^(1/2)*(1/(a+b)*(b+a*cos(d*x+c))/(cos(d*x+c)+1))^(1/2)*El lipticF(cot(d*x+c)-csc(d*x+c),((a-b)/(a+b))^(1/2))*a^4*cos(d*x+c)^2-288*(c os(d*x+c)/(cos(d*x+c)+1))^(1/2)*(1/(a+b)*(b+a*cos(d*x+c))/(cos(d*x+c)+1))^ (1/2)*EllipticPi(cot(d*x+c)-csc(d*x+c),-1,((a-b)/(a+b))^(1/2))*a^4*cos(d*x +c)^2+30*(cos(d*x+c)/(cos(d*x+c)+1))^(1/2)*(1/(a+b)*(b+a*cos(d*x+c))/(cos( d*x+c)+1))^(1/2)*EllipticPi(cot(d*x+c)-csc(d*x+c),-1,((a-b)/(a+b))^(1/2))* b^4*cos(d*x+c)^2-15*(cos(d*x+c)/(cos(d*x+c)+1))^(1/2)*(1/(a+b)*(b+a*cos(d* x+c))/(cos(d*x+c)+1))^(1/2)*EllipticE(cot(d*x+c)-csc(d*x+c),((a-b)/(a+b))^ (1/2))*b^4*cos(d*x+c)^2+288*(cos(d*x+c)/(cos(d*x+c)+1))^(1/2)*(1/(a+b)*...
\[ \int \cos ^4(c+d x) (a+b \sec (c+d x))^{5/2} \, dx=\int { {\left (b \sec \left (d x + c\right ) + a\right )}^{\frac {5}{2}} \cos \left (d x + c\right )^{4} \,d x } \]
integral((b^2*cos(d*x + c)^4*sec(d*x + c)^2 + 2*a*b*cos(d*x + c)^4*sec(d*x + c) + a^2*cos(d*x + c)^4)*sqrt(b*sec(d*x + c) + a), x)
Timed out. \[ \int \cos ^4(c+d x) (a+b \sec (c+d x))^{5/2} \, dx=\text {Timed out} \]
\[ \int \cos ^4(c+d x) (a+b \sec (c+d x))^{5/2} \, dx=\int { {\left (b \sec \left (d x + c\right ) + a\right )}^{\frac {5}{2}} \cos \left (d x + c\right )^{4} \,d x } \]
\[ \int \cos ^4(c+d x) (a+b \sec (c+d x))^{5/2} \, dx=\int { {\left (b \sec \left (d x + c\right ) + a\right )}^{\frac {5}{2}} \cos \left (d x + c\right )^{4} \,d x } \]
Timed out. \[ \int \cos ^4(c+d x) (a+b \sec (c+d x))^{5/2} \, dx=\int {\cos \left (c+d\,x\right )}^4\,{\left (a+\frac {b}{\cos \left (c+d\,x\right )}\right )}^{5/2} \,d x \]