Integrand size = 33, antiderivative size = 492 \[ \int \frac {1}{\cos ^{\frac {5}{2}}(d+e x) (a+b \sec (d+e x)+c \tan (d+e x))^{5/2}} \, dx=-\frac {2 (c \cos (d+e x)-a \sin (d+e x)) (b+a \cos (d+e x)+c \sin (d+e x))}{3 \left (a^2-b^2+c^2\right ) e \cos ^{\frac {5}{2}}(d+e x) (a+b \sec (d+e x)+c \tan (d+e x))^{5/2}}+\frac {8 (b c \cos (d+e x)-a b \sin (d+e x)) (b+a \cos (d+e x)+c \sin (d+e x))^2}{3 \left (a^2-b^2+c^2\right )^2 e \cos ^{\frac {5}{2}}(d+e x) (a+b \sec (d+e x)+c \tan (d+e x))^{5/2}}+\frac {8 b E\left (\frac {1}{2} \left (d+e x-\tan ^{-1}(a,c)\right )|\frac {2 \sqrt {a^2+c^2}}{b+\sqrt {a^2+c^2}}\right ) (b+a \cos (d+e x)+c \sin (d+e x))^3}{3 \left (a^2-b^2+c^2\right )^2 e \cos ^{\frac {5}{2}}(d+e x) \sqrt {\frac {b+a \cos (d+e x)+c \sin (d+e x)}{b+\sqrt {a^2+c^2}}} (a+b \sec (d+e x)+c \tan (d+e x))^{5/2}}+\frac {2 \operatorname {EllipticF}\left (\frac {1}{2} \left (d+e x-\tan ^{-1}(a,c)\right ),\frac {2 \sqrt {a^2+c^2}}{b+\sqrt {a^2+c^2}}\right ) (b+a \cos (d+e x)+c \sin (d+e x))^2 \sqrt {\frac {b+a \cos (d+e x)+c \sin (d+e x)}{b+\sqrt {a^2+c^2}}}}{3 \left (a^2-b^2+c^2\right ) e \cos ^{\frac {5}{2}}(d+e x) (a+b \sec (d+e x)+c \tan (d+e x))^{5/2}} \]
-2/3*(c*cos(e*x+d)-a*sin(e*x+d))*(b+a*cos(e*x+d)+c*sin(e*x+d))/(a^2-b^2+c^ 2)/e/cos(e*x+d)^(5/2)/(a+b*sec(e*x+d)+c*tan(e*x+d))^(5/2)+8/3*(b*c*cos(e*x +d)-a*b*sin(e*x+d))*(b+a*cos(e*x+d)+c*sin(e*x+d))^2/(a^2-b^2+c^2)^2/e/cos( e*x+d)^(5/2)/(a+b*sec(e*x+d)+c*tan(e*x+d))^(5/2)+8/3*b*(cos(1/2*d+1/2*e*x- 1/2*arctan(a,c))^2)^(1/2)/cos(1/2*d+1/2*e*x-1/2*arctan(a,c))*EllipticE(sin (1/2*d+1/2*e*x-1/2*arctan(a,c)),2^(1/2)*((a^2+c^2)^(1/2)/(b+(a^2+c^2)^(1/2 )))^(1/2))*(b+a*cos(e*x+d)+c*sin(e*x+d))^3/(a^2-b^2+c^2)^2/e/cos(e*x+d)^(5 /2)/((b+a*cos(e*x+d)+c*sin(e*x+d))/(b+(a^2+c^2)^(1/2)))^(1/2)/(a+b*sec(e*x +d)+c*tan(e*x+d))^(5/2)+2/3*(cos(1/2*d+1/2*e*x-1/2*arctan(a,c))^2)^(1/2)/c os(1/2*d+1/2*e*x-1/2*arctan(a,c))*EllipticF(sin(1/2*d+1/2*e*x-1/2*arctan(a ,c)),2^(1/2)*((a^2+c^2)^(1/2)/(b+(a^2+c^2)^(1/2)))^(1/2))*(b+a*cos(e*x+d)+ c*sin(e*x+d))^2*((b+a*cos(e*x+d)+c*sin(e*x+d))/(b+(a^2+c^2)^(1/2)))^(1/2)/ (a^2-b^2+c^2)/e/cos(e*x+d)^(5/2)/(a+b*sec(e*x+d)+c*tan(e*x+d))^(5/2)
Result contains complex when optimal does not.
Time = 33.65 (sec) , antiderivative size = 81703, normalized size of antiderivative = 166.06 \[ \int \frac {1}{\cos ^{\frac {5}{2}}(d+e x) (a+b \sec (d+e x)+c \tan (d+e x))^{5/2}} \, dx=\text {Result too large to show} \]
Time = 1.79 (sec) , antiderivative size = 443, normalized size of antiderivative = 0.90, number of steps used = 17, number of rules used = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.515, Rules used = {3042, 3642, 3042, 3608, 27, 3042, 3635, 27, 3042, 3628, 3042, 3598, 3042, 3132, 3606, 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.
| \(\displaystyle \int \frac {1}{\cos ^{\frac {5}{2}}(d+e x) (a+b \sec (d+e x)+c \tan (d+e x))^{5/2}} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {1}{\cos (d+e x)^{5/2} (a+b \sec (d+e x)+c \tan (d+e x))^{5/2}}dx\) |
| \(\Big \downarrow \) 3642 |
| \(\displaystyle \frac {(a \cos (d+e x)+b+c \sin (d+e x))^{5/2} \int \frac {1}{(b+a \cos (d+e x)+c \sin (d+e x))^{5/2}}dx}{\cos ^{\frac {5}{2}}(d+e x) (a+b \sec (d+e x)+c \tan (d+e x))^{5/2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {(a \cos (d+e x)+b+c \sin (d+e x))^{5/2} \int \frac {1}{(b+a \cos (d+e x)+c \sin (d+e x))^{5/2}}dx}{\cos ^{\frac {5}{2}}(d+e x) (a+b \sec (d+e x)+c \tan (d+e x))^{5/2}}\) |
| \(\Big \downarrow \) 3608 |
| \(\displaystyle \frac {(a \cos (d+e x)+b+c \sin (d+e x))^{5/2} \left (\frac {2 \int -\frac {3 b-a \cos (d+e x)-c \sin (d+e x)}{2 (b+a \cos (d+e x)+c \sin (d+e x))^{3/2}}dx}{3 \left (a^2-b^2+c^2\right )}-\frac {2 (c \cos (d+e x)-a \sin (d+e x))}{3 e \left (a^2-b^2+c^2\right ) (a \cos (d+e x)+b+c \sin (d+e x))^{3/2}}\right )}{\cos ^{\frac {5}{2}}(d+e x) (a+b \sec (d+e x)+c \tan (d+e x))^{5/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {(a \cos (d+e x)+b+c \sin (d+e x))^{5/2} \left (-\frac {\int \frac {3 b-a \cos (d+e x)-c \sin (d+e x)}{(b+a \cos (d+e x)+c \sin (d+e x))^{3/2}}dx}{3 \left (a^2-b^2+c^2\right )}-\frac {2 (c \cos (d+e x)-a \sin (d+e x))}{3 e \left (a^2-b^2+c^2\right ) (a \cos (d+e x)+b+c \sin (d+e x))^{3/2}}\right )}{\cos ^{\frac {5}{2}}(d+e x) (a+b \sec (d+e x)+c \tan (d+e x))^{5/2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {(a \cos (d+e x)+b+c \sin (d+e x))^{5/2} \left (-\frac {\int \frac {3 b-a \cos (d+e x)-c \sin (d+e x)}{(b+a \cos (d+e x)+c \sin (d+e x))^{3/2}}dx}{3 \left (a^2-b^2+c^2\right )}-\frac {2 (c \cos (d+e x)-a \sin (d+e x))}{3 e \left (a^2-b^2+c^2\right ) (a \cos (d+e x)+b+c \sin (d+e x))^{3/2}}\right )}{\cos ^{\frac {5}{2}}(d+e x) (a+b \sec (d+e x)+c \tan (d+e x))^{5/2}}\) |
| \(\Big \downarrow \) 3635 |
| \(\displaystyle \frac {(a \cos (d+e x)+b+c \sin (d+e x))^{5/2} \left (-\frac {\frac {2 \int -\frac {a^2+4 b \cos (d+e x) a+3 b^2+c^2+4 b c \sin (d+e x)}{2 \sqrt {b+a \cos (d+e x)+c \sin (d+e x)}}dx}{a^2-b^2+c^2}-\frac {8 (b c \cos (d+e x)-a b \sin (d+e x))}{e \left (a^2-b^2+c^2\right ) \sqrt {a \cos (d+e x)+b+c \sin (d+e x)}}}{3 \left (a^2-b^2+c^2\right )}-\frac {2 (c \cos (d+e x)-a \sin (d+e x))}{3 e \left (a^2-b^2+c^2\right ) (a \cos (d+e x)+b+c \sin (d+e x))^{3/2}}\right )}{\cos ^{\frac {5}{2}}(d+e x) (a+b \sec (d+e x)+c \tan (d+e x))^{5/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {(a \cos (d+e x)+b+c \sin (d+e x))^{5/2} \left (-\frac {-\frac {\int \frac {a^2+4 b \cos (d+e x) a+3 b^2+c^2+4 b c \sin (d+e x)}{\sqrt {b+a \cos (d+e x)+c \sin (d+e x)}}dx}{a^2-b^2+c^2}-\frac {8 (b c \cos (d+e x)-a b \sin (d+e x))}{e \left (a^2-b^2+c^2\right ) \sqrt {a \cos (d+e x)+b+c \sin (d+e x)}}}{3 \left (a^2-b^2+c^2\right )}-\frac {2 (c \cos (d+e x)-a \sin (d+e x))}{3 e \left (a^2-b^2+c^2\right ) (a \cos (d+e x)+b+c \sin (d+e x))^{3/2}}\right )}{\cos ^{\frac {5}{2}}(d+e x) (a+b \sec (d+e x)+c \tan (d+e x))^{5/2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {(a \cos (d+e x)+b+c \sin (d+e x))^{5/2} \left (-\frac {-\frac {\int \frac {a^2+4 b \cos (d+e x) a+3 b^2+c^2+4 b c \sin (d+e x)}{\sqrt {b+a \cos (d+e x)+c \sin (d+e x)}}dx}{a^2-b^2+c^2}-\frac {8 (b c \cos (d+e x)-a b \sin (d+e x))}{e \left (a^2-b^2+c^2\right ) \sqrt {a \cos (d+e x)+b+c \sin (d+e x)}}}{3 \left (a^2-b^2+c^2\right )}-\frac {2 (c \cos (d+e x)-a \sin (d+e x))}{3 e \left (a^2-b^2+c^2\right ) (a \cos (d+e x)+b+c \sin (d+e x))^{3/2}}\right )}{\cos ^{\frac {5}{2}}(d+e x) (a+b \sec (d+e x)+c \tan (d+e x))^{5/2}}\) |
| \(\Big \downarrow \) 3628 |
| \(\displaystyle \frac {(a \cos (d+e x)+b+c \sin (d+e x))^{5/2} \left (-\frac {-\frac {\left (a^2-b^2+c^2\right ) \int \frac {1}{\sqrt {b+a \cos (d+e x)+c \sin (d+e x)}}dx+4 b \int \sqrt {b+a \cos (d+e x)+c \sin (d+e x)}dx}{a^2-b^2+c^2}-\frac {8 (b c \cos (d+e x)-a b \sin (d+e x))}{e \left (a^2-b^2+c^2\right ) \sqrt {a \cos (d+e x)+b+c \sin (d+e x)}}}{3 \left (a^2-b^2+c^2\right )}-\frac {2 (c \cos (d+e x)-a \sin (d+e x))}{3 e \left (a^2-b^2+c^2\right ) (a \cos (d+e x)+b+c \sin (d+e x))^{3/2}}\right )}{\cos ^{\frac {5}{2}}(d+e x) (a+b \sec (d+e x)+c \tan (d+e x))^{5/2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {(a \cos (d+e x)+b+c \sin (d+e x))^{5/2} \left (-\frac {-\frac {\left (a^2-b^2+c^2\right ) \int \frac {1}{\sqrt {b+a \cos (d+e x)+c \sin (d+e x)}}dx+4 b \int \sqrt {b+a \cos (d+e x)+c \sin (d+e x)}dx}{a^2-b^2+c^2}-\frac {8 (b c \cos (d+e x)-a b \sin (d+e x))}{e \left (a^2-b^2+c^2\right ) \sqrt {a \cos (d+e x)+b+c \sin (d+e x)}}}{3 \left (a^2-b^2+c^2\right )}-\frac {2 (c \cos (d+e x)-a \sin (d+e x))}{3 e \left (a^2-b^2+c^2\right ) (a \cos (d+e x)+b+c \sin (d+e x))^{3/2}}\right )}{\cos ^{\frac {5}{2}}(d+e x) (a+b \sec (d+e x)+c \tan (d+e x))^{5/2}}\) |
| \(\Big \downarrow \) 3598 |
| \(\displaystyle \frac {(a \cos (d+e x)+b+c \sin (d+e x))^{5/2} \left (-\frac {-\frac {\left (a^2-b^2+c^2\right ) \int \frac {1}{\sqrt {b+a \cos (d+e x)+c \sin (d+e x)}}dx+\frac {4 b \sqrt {a \cos (d+e x)+b+c \sin (d+e x)} \int \sqrt {\frac {b}{b+\sqrt {a^2+c^2}}+\frac {\sqrt {a^2+c^2} \cos \left (d+e x-\tan ^{-1}(a,c)\right )}{b+\sqrt {a^2+c^2}}}dx}{\sqrt {\frac {a \cos (d+e x)+b+c \sin (d+e x)}{\sqrt {a^2+c^2}+b}}}}{a^2-b^2+c^2}-\frac {8 (b c \cos (d+e x)-a b \sin (d+e x))}{e \left (a^2-b^2+c^2\right ) \sqrt {a \cos (d+e x)+b+c \sin (d+e x)}}}{3 \left (a^2-b^2+c^2\right )}-\frac {2 (c \cos (d+e x)-a \sin (d+e x))}{3 e \left (a^2-b^2+c^2\right ) (a \cos (d+e x)+b+c \sin (d+e x))^{3/2}}\right )}{\cos ^{\frac {5}{2}}(d+e x) (a+b \sec (d+e x)+c \tan (d+e x))^{5/2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {(a \cos (d+e x)+b+c \sin (d+e x))^{5/2} \left (-\frac {-\frac {\left (a^2-b^2+c^2\right ) \int \frac {1}{\sqrt {b+a \cos (d+e x)+c \sin (d+e x)}}dx+\frac {4 b \sqrt {a \cos (d+e x)+b+c \sin (d+e x)} \int \sqrt {\frac {b}{b+\sqrt {a^2+c^2}}+\frac {\sqrt {a^2+c^2} \sin \left (d+e x-\tan ^{-1}(a,c)+\frac {\pi }{2}\right )}{b+\sqrt {a^2+c^2}}}dx}{\sqrt {\frac {a \cos (d+e x)+b+c \sin (d+e x)}{\sqrt {a^2+c^2}+b}}}}{a^2-b^2+c^2}-\frac {8 (b c \cos (d+e x)-a b \sin (d+e x))}{e \left (a^2-b^2+c^2\right ) \sqrt {a \cos (d+e x)+b+c \sin (d+e x)}}}{3 \left (a^2-b^2+c^2\right )}-\frac {2 (c \cos (d+e x)-a \sin (d+e x))}{3 e \left (a^2-b^2+c^2\right ) (a \cos (d+e x)+b+c \sin (d+e x))^{3/2}}\right )}{\cos ^{\frac {5}{2}}(d+e x) (a+b \sec (d+e x)+c \tan (d+e x))^{5/2}}\) |
| \(\Big \downarrow \) 3132 |
| \(\displaystyle \frac {(a \cos (d+e x)+b+c \sin (d+e x))^{5/2} \left (-\frac {-\frac {\left (a^2-b^2+c^2\right ) \int \frac {1}{\sqrt {b+a \cos (d+e x)+c \sin (d+e x)}}dx+\frac {8 b \sqrt {a \cos (d+e x)+b+c \sin (d+e x)} E\left (\frac {1}{2} \left (d+e x-\tan ^{-1}(a,c)\right )|\frac {2 \sqrt {a^2+c^2}}{b+\sqrt {a^2+c^2}}\right )}{e \sqrt {\frac {a \cos (d+e x)+b+c \sin (d+e x)}{\sqrt {a^2+c^2}+b}}}}{a^2-b^2+c^2}-\frac {8 (b c \cos (d+e x)-a b \sin (d+e x))}{e \left (a^2-b^2+c^2\right ) \sqrt {a \cos (d+e x)+b+c \sin (d+e x)}}}{3 \left (a^2-b^2+c^2\right )}-\frac {2 (c \cos (d+e x)-a \sin (d+e x))}{3 e \left (a^2-b^2+c^2\right ) (a \cos (d+e x)+b+c \sin (d+e x))^{3/2}}\right )}{\cos ^{\frac {5}{2}}(d+e x) (a+b \sec (d+e x)+c \tan (d+e x))^{5/2}}\) |
| \(\Big \downarrow \) 3606 |
| \(\displaystyle \frac {(a \cos (d+e x)+b+c \sin (d+e x))^{5/2} \left (-\frac {-\frac {\frac {\left (a^2-b^2+c^2\right ) \sqrt {\frac {a \cos (d+e x)+b+c \sin (d+e x)}{\sqrt {a^2+c^2}+b}} \int \frac {1}{\sqrt {\frac {b}{b+\sqrt {a^2+c^2}}+\frac {\sqrt {a^2+c^2} \cos \left (d+e x-\tan ^{-1}(a,c)\right )}{b+\sqrt {a^2+c^2}}}}dx}{\sqrt {a \cos (d+e x)+b+c \sin (d+e x)}}+\frac {8 b \sqrt {a \cos (d+e x)+b+c \sin (d+e x)} E\left (\frac {1}{2} \left (d+e x-\tan ^{-1}(a,c)\right )|\frac {2 \sqrt {a^2+c^2}}{b+\sqrt {a^2+c^2}}\right )}{e \sqrt {\frac {a \cos (d+e x)+b+c \sin (d+e x)}{\sqrt {a^2+c^2}+b}}}}{a^2-b^2+c^2}-\frac {8 (b c \cos (d+e x)-a b \sin (d+e x))}{e \left (a^2-b^2+c^2\right ) \sqrt {a \cos (d+e x)+b+c \sin (d+e x)}}}{3 \left (a^2-b^2+c^2\right )}-\frac {2 (c \cos (d+e x)-a \sin (d+e x))}{3 e \left (a^2-b^2+c^2\right ) (a \cos (d+e x)+b+c \sin (d+e x))^{3/2}}\right )}{\cos ^{\frac {5}{2}}(d+e x) (a+b \sec (d+e x)+c \tan (d+e x))^{5/2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {(a \cos (d+e x)+b+c \sin (d+e x))^{5/2} \left (-\frac {-\frac {\frac {\left (a^2-b^2+c^2\right ) \sqrt {\frac {a \cos (d+e x)+b+c \sin (d+e x)}{\sqrt {a^2+c^2}+b}} \int \frac {1}{\sqrt {\frac {b}{b+\sqrt {a^2+c^2}}+\frac {\sqrt {a^2+c^2} \sin \left (d+e x-\tan ^{-1}(a,c)+\frac {\pi }{2}\right )}{b+\sqrt {a^2+c^2}}}}dx}{\sqrt {a \cos (d+e x)+b+c \sin (d+e x)}}+\frac {8 b \sqrt {a \cos (d+e x)+b+c \sin (d+e x)} E\left (\frac {1}{2} \left (d+e x-\tan ^{-1}(a,c)\right )|\frac {2 \sqrt {a^2+c^2}}{b+\sqrt {a^2+c^2}}\right )}{e \sqrt {\frac {a \cos (d+e x)+b+c \sin (d+e x)}{\sqrt {a^2+c^2}+b}}}}{a^2-b^2+c^2}-\frac {8 (b c \cos (d+e x)-a b \sin (d+e x))}{e \left (a^2-b^2+c^2\right ) \sqrt {a \cos (d+e x)+b+c \sin (d+e x)}}}{3 \left (a^2-b^2+c^2\right )}-\frac {2 (c \cos (d+e x)-a \sin (d+e x))}{3 e \left (a^2-b^2+c^2\right ) (a \cos (d+e x)+b+c \sin (d+e x))^{3/2}}\right )}{\cos ^{\frac {5}{2}}(d+e x) (a+b \sec (d+e x)+c \tan (d+e x))^{5/2}}\) |
| \(\Big \downarrow \) 3140 |
| \(\displaystyle \frac {(a \cos (d+e x)+b+c \sin (d+e x))^{5/2} \left (-\frac {-\frac {\frac {2 \left (a^2-b^2+c^2\right ) \sqrt {\frac {a \cos (d+e x)+b+c \sin (d+e x)}{\sqrt {a^2+c^2}+b}} \operatorname {EllipticF}\left (\frac {1}{2} \left (d+e x-\tan ^{-1}(a,c)\right ),\frac {2 \sqrt {a^2+c^2}}{b+\sqrt {a^2+c^2}}\right )}{e \sqrt {a \cos (d+e x)+b+c \sin (d+e x)}}+\frac {8 b \sqrt {a \cos (d+e x)+b+c \sin (d+e x)} E\left (\frac {1}{2} \left (d+e x-\tan ^{-1}(a,c)\right )|\frac {2 \sqrt {a^2+c^2}}{b+\sqrt {a^2+c^2}}\right )}{e \sqrt {\frac {a \cos (d+e x)+b+c \sin (d+e x)}{\sqrt {a^2+c^2}+b}}}}{a^2-b^2+c^2}-\frac {8 (b c \cos (d+e x)-a b \sin (d+e x))}{e \left (a^2-b^2+c^2\right ) \sqrt {a \cos (d+e x)+b+c \sin (d+e x)}}}{3 \left (a^2-b^2+c^2\right )}-\frac {2 (c \cos (d+e x)-a \sin (d+e x))}{3 e \left (a^2-b^2+c^2\right ) (a \cos (d+e x)+b+c \sin (d+e x))^{3/2}}\right )}{\cos ^{\frac {5}{2}}(d+e x) (a+b \sec (d+e x)+c \tan (d+e x))^{5/2}}\) |
((b + a*Cos[d + e*x] + c*Sin[d + e*x])^(5/2)*((-2*(c*Cos[d + e*x] - a*Sin[ d + e*x]))/(3*(a^2 - b^2 + c^2)*e*(b + a*Cos[d + e*x] + c*Sin[d + e*x])^(3 /2)) - ((-8*(b*c*Cos[d + e*x] - a*b*Sin[d + e*x]))/((a^2 - b^2 + c^2)*e*Sq rt[b + a*Cos[d + e*x] + c*Sin[d + e*x]]) - ((8*b*EllipticE[(d + e*x - ArcT an[a, c])/2, (2*Sqrt[a^2 + c^2])/(b + Sqrt[a^2 + c^2])]*Sqrt[b + a*Cos[d + e*x] + c*Sin[d + e*x]])/(e*Sqrt[(b + a*Cos[d + e*x] + c*Sin[d + e*x])/(b + Sqrt[a^2 + c^2])]) + (2*(a^2 - b^2 + c^2)*EllipticF[(d + e*x - ArcTan[a, c])/2, (2*Sqrt[a^2 + c^2])/(b + Sqrt[a^2 + c^2])]*Sqrt[(b + a*Cos[d + e*x ] + c*Sin[d + e*x])/(b + Sqrt[a^2 + c^2])])/(e*Sqrt[b + a*Cos[d + e*x] + c *Sin[d + e*x]]))/(a^2 - b^2 + c^2))/(3*(a^2 - b^2 + c^2))))/(Cos[d + e*x]^ (5/2)*(a + b*Sec[d + e*x] + c*Tan[d + e*x])^(5/2))
3.5.57.3.1 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[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[Sqrt[cos[(d_.) + (e_.)*(x_)]*(b_.) + (a_) + (c_.)*sin[(d_.) + (e_.)*(x_ )]], x_Symbol] :> Simp[Sqrt[a + b*Cos[d + e*x] + c*Sin[d + e*x]]/Sqrt[(a + b*Cos[d + e*x] + c*Sin[d + e*x])/(a + Sqrt[b^2 + c^2])] Int[Sqrt[a/(a + S qrt[b^2 + c^2]) + (Sqrt[b^2 + c^2]/(a + Sqrt[b^2 + c^2]))*Cos[d + e*x - Arc Tan[b, c]]], x], x] /; FreeQ[{a, b, c, d, e}, x] && NeQ[a^2 - b^2 - c^2, 0] && NeQ[b^2 + c^2, 0] && !GtQ[a + Sqrt[b^2 + c^2], 0]
Int[1/Sqrt[cos[(d_.) + (e_.)*(x_)]*(b_.) + (a_) + (c_.)*sin[(d_.) + (e_.)*( x_)]], x_Symbol] :> Simp[Sqrt[(a + b*Cos[d + e*x] + c*Sin[d + e*x])/(a + Sq rt[b^2 + c^2])]/Sqrt[a + b*Cos[d + e*x] + c*Sin[d + e*x]] Int[1/Sqrt[a/(a + Sqrt[b^2 + c^2]) + (Sqrt[b^2 + c^2]/(a + Sqrt[b^2 + c^2]))*Cos[d + e*x - ArcTan[b, c]]], x], x] /; FreeQ[{a, b, c, d, e}, x] && NeQ[a^2 - b^2 - c^2 , 0] && NeQ[b^2 + c^2, 0] && !GtQ[a + Sqrt[b^2 + c^2], 0]
Int[(cos[(d_.) + (e_.)*(x_)]*(b_.) + (a_) + (c_.)*sin[(d_.) + (e_.)*(x_)])^ (n_), x_Symbol] :> Simp[((-c)*Cos[d + e*x] + b*Sin[d + e*x])*((a + b*Cos[d + e*x] + c*Sin[d + e*x])^(n + 1)/(e*(n + 1)*(a^2 - b^2 - c^2))), x] + Simp[ 1/((n + 1)*(a^2 - b^2 - c^2)) Int[(a*(n + 1) - b*(n + 2)*Cos[d + e*x] - c *(n + 2)*Sin[d + e*x])*(a + b*Cos[d + e*x] + c*Sin[d + e*x])^(n + 1), x], x ] /; FreeQ[{a, b, c, d, e}, x] && NeQ[a^2 - b^2 - c^2, 0] && LtQ[n, -1] && NeQ[n, -3/2]
Int[((A_.) + cos[(d_.) + (e_.)*(x_)]*(B_.) + (C_.)*sin[(d_.) + (e_.)*(x_)]) /Sqrt[cos[(d_.) + (e_.)*(x_)]*(b_.) + (a_) + (c_.)*sin[(d_.) + (e_.)*(x_)]] , x_Symbol] :> Simp[B/b Int[Sqrt[a + b*Cos[d + e*x] + c*Sin[d + e*x]], x] , x] + Simp[(A*b - a*B)/b Int[1/Sqrt[a + b*Cos[d + e*x] + c*Sin[d + e*x]] , x], x] /; FreeQ[{a, b, c, d, e, A, B, C}, x] && EqQ[B*c - b*C, 0] && NeQ[ A*b - a*B, 0]
Int[((a_.) + cos[(d_.) + (e_.)*(x_)]*(b_.) + (c_.)*sin[(d_.) + (e_.)*(x_)]) ^(n_)*((A_.) + cos[(d_.) + (e_.)*(x_)]*(B_.) + (C_.)*sin[(d_.) + (e_.)*(x_) ]), x_Symbol] :> Simp[(-(c*B - b*C - (a*C - c*A)*Cos[d + e*x] + (a*B - b*A) *Sin[d + e*x]))*((a + b*Cos[d + e*x] + c*Sin[d + e*x])^(n + 1)/(e*(n + 1)*( a^2 - b^2 - c^2))), x] + Simp[1/((n + 1)*(a^2 - b^2 - c^2)) Int[(a + b*Co s[d + e*x] + c*Sin[d + e*x])^(n + 1)*Simp[(n + 1)*(a*A - b*B - c*C) + (n + 2)*(a*B - b*A)*Cos[d + e*x] + (n + 2)*(a*C - c*A)*Sin[d + e*x], x], x], x] /; FreeQ[{a, b, c, d, e, A, B, C}, x] && LtQ[n, -1] && NeQ[a^2 - b^2 - c^2, 0] && NeQ[n, -2]
Int[cos[(d_.) + (e_.)*(x_)]^(n_)*((a_.) + (b_.)*sec[(d_.) + (e_.)*(x_)] + ( c_.)*tan[(d_.) + (e_.)*(x_)])^(n_), x_Symbol] :> Simp[Cos[d + e*x]^n*((a + b*Sec[d + e*x] + c*Tan[d + e*x])^n/(b + a*Cos[d + e*x] + c*Sin[d + e*x])^n) Int[(b + a*Cos[d + e*x] + c*Sin[d + e*x])^n, x], x] /; FreeQ[{a, b, c, d , e}, x] && !IntegerQ[n]
Result contains complex when optimal does not.
Time = 25.23 (sec) , antiderivative size = 155452, normalized size of antiderivative = 315.96
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.25 (sec) , antiderivative size = 2816, normalized size of antiderivative = 5.72 \[ \int \frac {1}{\cos ^{\frac {5}{2}}(d+e x) (a+b \sec (d+e x)+c \tan (d+e x))^{5/2}} \, dx=\text {Too large to display} \]
1/9*((sqrt(2)*(-3*I*a^5 - I*a^3*b^2 + I*a*b^2*c^2 - b^2*c^3 + 3*I*a*c^4 - 3*c^5 + (3*a^4 + a^2*b^2)*c)*cos(e*x + d)^2 - 2*sqrt(2)*(3*I*a^4*b + I*a^2 *b^3 + 3*I*a^2*b*c^2 - 3*a*b*c^3 - (3*a^3*b + a*b^3)*c)*cos(e*x + d) - 2*( sqrt(2)*(3*I*a^2*c^3 - 3*a*c^4 - (3*a^3 + a*b^2)*c^2 + I*(3*a^4 + a^2*b^2) *c)*cos(e*x + d) + sqrt(2)*(3*I*a*b*c^3 - 3*b*c^4 - (3*a^2*b + b^3)*c^2 + I*(3*a^3*b + a*b^3)*c))*sin(e*x + d) + sqrt(2)*(-3*I*a^3*b^2 - I*a*b^4 - 3 *I*a*c^4 + 3*c^5 + (3*a^2 + 4*b^2)*c^3 - I*(3*a^3 + 4*a*b^2)*c^2 + (3*a^2* b^2 + b^4)*c))*sqrt(a - I*c)*weierstrassPInverse(-4/3*(3*a^4 - 4*a^2*b^2 + 4*b^2*c^2 + 6*I*a*c^3 - 3*c^4 + 2*I*(3*a^3 - 4*a*b^2)*c)/(a^4 + 2*a^2*c^2 + c^4), 8/27*(9*a^5*b - 8*a^3*b^3 - 27*a*b*c^4 - 9*I*b*c^5 + 2*I*(9*a^2*b + 4*b^3)*c^3 - 6*(3*a^3*b - 4*a*b^3)*c^2 + 3*I*(9*a^4*b - 8*a^2*b^3)*c)/( a^6 + 3*a^4*c^2 + 3*a^2*c^4 + c^6), 1/3*(2*a*b + 2*I*b*c + 3*(a^2 + c^2)*c os(e*x + d) - 3*(-I*a^2 - I*c^2)*sin(e*x + d))/(a^2 + c^2)) + (sqrt(2)*(3* I*a^5 + I*a^3*b^2 - I*a*b^2*c^2 - b^2*c^3 - 3*I*a*c^4 - 3*c^5 + (3*a^4 + a ^2*b^2)*c)*cos(e*x + d)^2 - 2*sqrt(2)*(-3*I*a^4*b - I*a^2*b^3 - 3*I*a^2*b* c^2 - 3*a*b*c^3 - (3*a^3*b + a*b^3)*c)*cos(e*x + d) - 2*(sqrt(2)*(-3*I*a^2 *c^3 - 3*a*c^4 - (3*a^3 + a*b^2)*c^2 - I*(3*a^4 + a^2*b^2)*c)*cos(e*x + d) + sqrt(2)*(-3*I*a*b*c^3 - 3*b*c^4 - (3*a^2*b + b^3)*c^2 - I*(3*a^3*b + a* b^3)*c))*sin(e*x + d) + sqrt(2)*(3*I*a^3*b^2 + I*a*b^4 + 3*I*a*c^4 + 3*c^5 + (3*a^2 + 4*b^2)*c^3 + I*(3*a^3 + 4*a*b^2)*c^2 + (3*a^2*b^2 + b^4)*c)...
Timed out. \[ \int \frac {1}{\cos ^{\frac {5}{2}}(d+e x) (a+b \sec (d+e x)+c \tan (d+e x))^{5/2}} \, dx=\text {Timed out} \]
\[ \int \frac {1}{\cos ^{\frac {5}{2}}(d+e x) (a+b \sec (d+e x)+c \tan (d+e x))^{5/2}} \, dx=\int { \frac {1}{{\left (b \sec \left (e x + d\right ) + c \tan \left (e x + d\right ) + a\right )}^{\frac {5}{2}} \cos \left (e x + d\right )^{\frac {5}{2}}} \,d x } \]
Timed out. \[ \int \frac {1}{\cos ^{\frac {5}{2}}(d+e x) (a+b \sec (d+e x)+c \tan (d+e x))^{5/2}} \, dx=\text {Timed out} \]
Timed out. \[ \int \frac {1}{\cos ^{\frac {5}{2}}(d+e x) (a+b \sec (d+e x)+c \tan (d+e x))^{5/2}} \, dx=\int \frac {1}{{\cos \left (d+e\,x\right )}^{5/2}\,{\left (a+c\,\mathrm {tan}\left (d+e\,x\right )+\frac {b}{\cos \left (d+e\,x\right )}\right )}^{5/2}} \,d x \]