Integrand size = 23, antiderivative size = 388 \[ \int \frac {\sec ^{\frac {3}{2}}(c+d x)}{(a+b \cos (c+d x))^3} \, dx=-\frac {\left (8 a^4-29 a^2 b^2+15 b^4\right ) \sqrt {\cos (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \sqrt {\sec (c+d x)}}{4 a^3 \left (a^2-b^2\right )^2 d}+\frac {b \left (11 a^2-5 b^2\right ) \sqrt {\cos (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right ) \sqrt {\sec (c+d x)}}{4 a^2 \left (a^2-b^2\right )^2 d}-\frac {b \left (35 a^4-38 a^2 b^2+15 b^4\right ) \sqrt {\cos (c+d x)} \operatorname {EllipticPi}\left (\frac {2 b}{a+b},\frac {1}{2} (c+d x),2\right ) \sqrt {\sec (c+d x)}}{4 a^3 (a-b)^2 (a+b)^3 d}+\frac {\left (8 a^4-29 a^2 b^2+15 b^4\right ) \sqrt {\sec (c+d x)} \sin (c+d x)}{4 a^3 \left (a^2-b^2\right )^2 d}+\frac {b^2 \sec ^{\frac {5}{2}}(c+d x) \sin (c+d x)}{2 a \left (a^2-b^2\right ) d (b+a \sec (c+d x))^2}+\frac {b^2 \left (11 a^2-5 b^2\right ) \sec ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{4 a^2 \left (a^2-b^2\right )^2 d (b+a \sec (c+d x))} \] Output:
-1/4*(8*a^4-29*a^2*b^2+15*b^4)*cos(d*x+c)^(1/2)*EllipticE(sin(1/2*d*x+1/2* c),2^(1/2))*sec(d*x+c)^(1/2)/a^3/(a^2-b^2)^2/d+1/4*b*(11*a^2-5*b^2)*cos(d* x+c)^(1/2)*InverseJacobiAM(1/2*d*x+1/2*c,2^(1/2))*sec(d*x+c)^(1/2)/a^2/(a^ 2-b^2)^2/d-1/4*b*(35*a^4-38*a^2*b^2+15*b^4)*cos(d*x+c)^(1/2)*EllipticPi(si n(1/2*d*x+1/2*c),2*b/(a+b),2^(1/2))*sec(d*x+c)^(1/2)/a^3/(a-b)^2/(a+b)^3/d +1/4*(8*a^4-29*a^2*b^2+15*b^4)*sec(d*x+c)^(1/2)*sin(d*x+c)/a^3/(a^2-b^2)^2 /d+1/2*b^2*sec(d*x+c)^(5/2)*sin(d*x+c)/a/(a^2-b^2)/d/(b+a*sec(d*x+c))^2+1/ 4*b^2*(11*a^2-5*b^2)*sec(d*x+c)^(3/2)*sin(d*x+c)/a^2/(a^2-b^2)^2/d/(b+a*se c(d*x+c))
Time = 6.48 (sec) , antiderivative size = 532, normalized size of antiderivative = 1.37 \[ \int \frac {\sec ^{\frac {3}{2}}(c+d x)}{(a+b \cos (c+d x))^3} \, dx=\frac {\frac {2 a \left (16 a^6-24 a^4 b^2-13 a^2 b^4+15 b^6+\left (32 a^5 b-94 a^3 b^3+50 a b^5\right ) \cos (c+d x)+\left (8 a^4 b^2-29 a^2 b^4+15 b^6\right ) \cos (2 (c+d x))\right ) \tan (c+d x)}{\left (a^2-b^2\right )^2}-\frac {4 \cos (c+d x) (a+b \cos (c+d x)) \cot (c+d x) (b+a \sec (c+d x)) \left (-8 a^5+29 a^3 b^2-15 a b^4+8 a^5 \sec ^2(c+d x)-29 a^3 b^2 \sec ^2(c+d x)+15 a b^4 \sec ^2(c+d x)-a \left (8 a^4-29 a^2 b^2+15 b^4\right ) E\left (\left .\arcsin \left (\sqrt {\sec (c+d x)}\right )\right |-1\right ) \sqrt {\sec (c+d x)} \sqrt {-\tan ^2(c+d x)}+\left (8 a^5+24 a^4 b-29 a^3 b^2-33 a^2 b^3+15 a b^4+15 b^5\right ) \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\sec (c+d x)}\right ),-1\right ) \sqrt {\sec (c+d x)} \sqrt {-\tan ^2(c+d x)}-35 a^4 b \operatorname {EllipticPi}\left (-\frac {a}{b},\arcsin \left (\sqrt {\sec (c+d x)}\right ),-1\right ) \sqrt {\sec (c+d x)} \sqrt {-\tan ^2(c+d x)}+38 a^2 b^3 \operatorname {EllipticPi}\left (-\frac {a}{b},\arcsin \left (\sqrt {\sec (c+d x)}\right ),-1\right ) \sqrt {\sec (c+d x)} \sqrt {-\tan ^2(c+d x)}-15 b^5 \operatorname {EllipticPi}\left (-\frac {a}{b},\arcsin \left (\sqrt {\sec (c+d x)}\right ),-1\right ) \sqrt {\sec (c+d x)} \sqrt {-\tan ^2(c+d x)}\right )}{(a-b)^2 (a+b)^2}}{16 a^4 d (a+b \cos (c+d x))^2 \sqrt {\sec (c+d x)}} \] Input:
Integrate[Sec[c + d*x]^(3/2)/(a + b*Cos[c + d*x])^3,x]
Output:
((2*a*(16*a^6 - 24*a^4*b^2 - 13*a^2*b^4 + 15*b^6 + (32*a^5*b - 94*a^3*b^3 + 50*a*b^5)*Cos[c + d*x] + (8*a^4*b^2 - 29*a^2*b^4 + 15*b^6)*Cos[2*(c + d* x)])*Tan[c + d*x])/(a^2 - b^2)^2 - (4*Cos[c + d*x]*(a + b*Cos[c + d*x])*Co t[c + d*x]*(b + a*Sec[c + d*x])*(-8*a^5 + 29*a^3*b^2 - 15*a*b^4 + 8*a^5*Se c[c + d*x]^2 - 29*a^3*b^2*Sec[c + d*x]^2 + 15*a*b^4*Sec[c + d*x]^2 - a*(8* a^4 - 29*a^2*b^2 + 15*b^4)*EllipticE[ArcSin[Sqrt[Sec[c + d*x]]], -1]*Sqrt[ Sec[c + d*x]]*Sqrt[-Tan[c + d*x]^2] + (8*a^5 + 24*a^4*b - 29*a^3*b^2 - 33* a^2*b^3 + 15*a*b^4 + 15*b^5)*EllipticF[ArcSin[Sqrt[Sec[c + d*x]]], -1]*Sqr t[Sec[c + d*x]]*Sqrt[-Tan[c + d*x]^2] - 35*a^4*b*EllipticPi[-(a/b), ArcSin [Sqrt[Sec[c + d*x]]], -1]*Sqrt[Sec[c + d*x]]*Sqrt[-Tan[c + d*x]^2] + 38*a^ 2*b^3*EllipticPi[-(a/b), ArcSin[Sqrt[Sec[c + d*x]]], -1]*Sqrt[Sec[c + d*x] ]*Sqrt[-Tan[c + d*x]^2] - 15*b^5*EllipticPi[-(a/b), ArcSin[Sqrt[Sec[c + d* x]]], -1]*Sqrt[Sec[c + d*x]]*Sqrt[-Tan[c + d*x]^2]))/((a - b)^2*(a + b)^2) )/(16*a^4*d*(a + b*Cos[c + d*x])^2*Sqrt[Sec[c + d*x]])
Time = 2.91 (sec) , antiderivative size = 383, normalized size of antiderivative = 0.99, number of steps used = 23, number of rules used = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.000, Rules used = {3042, 3717, 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.
| \(\displaystyle \int \frac {\sec ^{\frac {3}{2}}(c+d x)}{(a+b \cos (c+d x))^3} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\csc \left (c+d x+\frac {\pi }{2}\right )^{3/2}}{\left (a+b \sin \left (c+d x+\frac {\pi }{2}\right )\right )^3}dx\) |
| \(\Big \downarrow \) 3717 |
| \(\displaystyle \int \frac {\sec ^{\frac {9}{2}}(c+d x)}{(a \sec (c+d x)+b)^3}dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\csc \left (c+d x+\frac {\pi }{2}\right )^{9/2}}{\left (a \csc \left (c+d x+\frac {\pi }{2}\right )+b\right )^3}dx\) |
| \(\Big \downarrow \) 4332 |
| \(\displaystyle \frac {\int \frac {\sec ^{\frac {3}{2}}(c+d x) \left (3 b^2-4 a \sec (c+d x) b+\left (4 a^2-5 b^2\right ) \sec ^2(c+d x)\right )}{2 (b+a \sec (c+d x))^2}dx}{2 a \left (a^2-b^2\right )}+\frac {b^2 \sin (c+d x) \sec ^{\frac {5}{2}}(c+d x)}{2 a d \left (a^2-b^2\right ) (a \sec (c+d x)+b)^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int \frac {\sec ^{\frac {3}{2}}(c+d x) \left (3 b^2-4 a \sec (c+d x) b+\left (4 a^2-5 b^2\right ) \sec ^2(c+d x)\right )}{(b+a \sec (c+d x))^2}dx}{4 a \left (a^2-b^2\right )}+\frac {b^2 \sin (c+d x) \sec ^{\frac {5}{2}}(c+d x)}{2 a d \left (a^2-b^2\right ) (a \sec (c+d x)+b)^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\int \frac {\csc \left (c+d x+\frac {\pi }{2}\right )^{3/2} \left (3 b^2-4 a \csc \left (c+d x+\frac {\pi }{2}\right ) b+\left (4 a^2-5 b^2\right ) \csc \left (c+d x+\frac {\pi }{2}\right )^2\right )}{\left (b+a \csc \left (c+d x+\frac {\pi }{2}\right )\right )^2}dx}{4 a \left (a^2-b^2\right )}+\frac {b^2 \sin (c+d x) \sec ^{\frac {5}{2}}(c+d x)}{2 a d \left (a^2-b^2\right ) (a \sec (c+d x)+b)^2}\) |
| \(\Big \downarrow \) 4586 |
| \(\displaystyle \frac {\frac {\int \frac {\sqrt {\sec (c+d x)} \left (\left (11 a^2-5 b^2\right ) b^2-4 a \left (4 a^2-b^2\right ) \sec (c+d x) b+\left (8 a^4-29 b^2 a^2+15 b^4\right ) \sec ^2(c+d x)\right )}{2 (b+a \sec (c+d x))}dx}{a \left (a^2-b^2\right )}+\frac {b^2 \left (11 a^2-5 b^2\right ) \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x)}{a d \left (a^2-b^2\right ) (a \sec (c+d x)+b)}}{4 a \left (a^2-b^2\right )}+\frac {b^2 \sin (c+d x) \sec ^{\frac {5}{2}}(c+d x)}{2 a d \left (a^2-b^2\right ) (a \sec (c+d x)+b)^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {\int \frac {\sqrt {\sec (c+d x)} \left (\left (11 a^2-5 b^2\right ) b^2-4 a \left (4 a^2-b^2\right ) \sec (c+d x) b+\left (8 a^4-29 b^2 a^2+15 b^4\right ) \sec ^2(c+d x)\right )}{b+a \sec (c+d x)}dx}{2 a \left (a^2-b^2\right )}+\frac {b^2 \left (11 a^2-5 b^2\right ) \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x)}{a d \left (a^2-b^2\right ) (a \sec (c+d x)+b)}}{4 a \left (a^2-b^2\right )}+\frac {b^2 \sin (c+d x) \sec ^{\frac {5}{2}}(c+d x)}{2 a d \left (a^2-b^2\right ) (a \sec (c+d x)+b)^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {\int \frac {\sqrt {\csc \left (c+d x+\frac {\pi }{2}\right )} \left (\left (11 a^2-5 b^2\right ) b^2-4 a \left (4 a^2-b^2\right ) \csc \left (c+d x+\frac {\pi }{2}\right ) b+\left (8 a^4-29 b^2 a^2+15 b^4\right ) \csc \left (c+d x+\frac {\pi }{2}\right )^2\right )}{b+a \csc \left (c+d x+\frac {\pi }{2}\right )}dx}{2 a \left (a^2-b^2\right )}+\frac {b^2 \left (11 a^2-5 b^2\right ) \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x)}{a d \left (a^2-b^2\right ) (a \sec (c+d x)+b)}}{4 a \left (a^2-b^2\right )}+\frac {b^2 \sin (c+d x) \sec ^{\frac {5}{2}}(c+d x)}{2 a d \left (a^2-b^2\right ) (a \sec (c+d x)+b)^2}\) |
| \(\Big \downarrow \) 4590 |
| \(\displaystyle \frac {\frac {\frac {2 \int -\frac {3 b \left (8 a^4-11 b^2 a^2+5 b^4\right ) \sec ^2(c+d x)+4 a \left (2 a^4-10 b^2 a^2+5 b^4\right ) \sec (c+d x)+b \left (8 a^4-29 b^2 a^2+15 b^4\right )}{2 \sqrt {\sec (c+d x)} (b+a \sec (c+d x))}dx}{a}+\frac {2 \left (8 a^4-29 a^2 b^2+15 b^4\right ) \sin (c+d x) \sqrt {\sec (c+d x)}}{a d}}{2 a \left (a^2-b^2\right )}+\frac {b^2 \left (11 a^2-5 b^2\right ) \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x)}{a d \left (a^2-b^2\right ) (a \sec (c+d x)+b)}}{4 a \left (a^2-b^2\right )}+\frac {b^2 \sin (c+d x) \sec ^{\frac {5}{2}}(c+d x)}{2 a d \left (a^2-b^2\right ) (a \sec (c+d x)+b)^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {\frac {2 \left (8 a^4-29 a^2 b^2+15 b^4\right ) \sin (c+d x) \sqrt {\sec (c+d x)}}{a d}-\frac {\int \frac {3 b \left (8 a^4-11 b^2 a^2+5 b^4\right ) \sec ^2(c+d x)+4 a \left (2 a^4-10 b^2 a^2+5 b^4\right ) \sec (c+d x)+b \left (8 a^4-29 b^2 a^2+15 b^4\right )}{\sqrt {\sec (c+d x)} (b+a \sec (c+d x))}dx}{a}}{2 a \left (a^2-b^2\right )}+\frac {b^2 \left (11 a^2-5 b^2\right ) \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x)}{a d \left (a^2-b^2\right ) (a \sec (c+d x)+b)}}{4 a \left (a^2-b^2\right )}+\frac {b^2 \sin (c+d x) \sec ^{\frac {5}{2}}(c+d x)}{2 a d \left (a^2-b^2\right ) (a \sec (c+d x)+b)^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {\frac {2 \left (8 a^4-29 a^2 b^2+15 b^4\right ) \sin (c+d x) \sqrt {\sec (c+d x)}}{a d}-\frac {\int \frac {3 b \left (8 a^4-11 b^2 a^2+5 b^4\right ) \csc \left (c+d x+\frac {\pi }{2}\right )^2+4 a \left (2 a^4-10 b^2 a^2+5 b^4\right ) \csc \left (c+d x+\frac {\pi }{2}\right )+b \left (8 a^4-29 b^2 a^2+15 b^4\right )}{\sqrt {\csc \left (c+d x+\frac {\pi }{2}\right )} \left (b+a \csc \left (c+d x+\frac {\pi }{2}\right )\right )}dx}{a}}{2 a \left (a^2-b^2\right )}+\frac {b^2 \left (11 a^2-5 b^2\right ) \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x)}{a d \left (a^2-b^2\right ) (a \sec (c+d x)+b)}}{4 a \left (a^2-b^2\right )}+\frac {b^2 \sin (c+d x) \sec ^{\frac {5}{2}}(c+d x)}{2 a d \left (a^2-b^2\right ) (a \sec (c+d x)+b)^2}\) |
| \(\Big \downarrow \) 4594 |
| \(\displaystyle \frac {\frac {\frac {2 \left (8 a^4-29 a^2 b^2+15 b^4\right ) \sin (c+d x) \sqrt {\sec (c+d x)}}{a d}-\frac {b \left (35 a^4-38 a^2 b^2+15 b^4\right ) \int \frac {\sec ^{\frac {3}{2}}(c+d x)}{b+a \sec (c+d x)}dx+\frac {\int \frac {b^2 \left (8 a^4-29 b^2 a^2+15 b^4\right )-a b^3 \left (11 a^2-5 b^2\right ) \sec (c+d x)}{\sqrt {\sec (c+d x)}}dx}{b^2}}{a}}{2 a \left (a^2-b^2\right )}+\frac {b^2 \left (11 a^2-5 b^2\right ) \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x)}{a d \left (a^2-b^2\right ) (a \sec (c+d x)+b)}}{4 a \left (a^2-b^2\right )}+\frac {b^2 \sin (c+d x) \sec ^{\frac {5}{2}}(c+d x)}{2 a d \left (a^2-b^2\right ) (a \sec (c+d x)+b)^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {\frac {2 \left (8 a^4-29 a^2 b^2+15 b^4\right ) \sin (c+d x) \sqrt {\sec (c+d x)}}{a d}-\frac {b \left (35 a^4-38 a^2 b^2+15 b^4\right ) \int \frac {\csc \left (c+d x+\frac {\pi }{2}\right )^{3/2}}{b+a \csc \left (c+d x+\frac {\pi }{2}\right )}dx+\frac {\int \frac {b^2 \left (8 a^4-29 b^2 a^2+15 b^4\right )-a b^3 \left (11 a^2-5 b^2\right ) \csc \left (c+d x+\frac {\pi }{2}\right )}{\sqrt {\csc \left (c+d x+\frac {\pi }{2}\right )}}dx}{b^2}}{a}}{2 a \left (a^2-b^2\right )}+\frac {b^2 \left (11 a^2-5 b^2\right ) \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x)}{a d \left (a^2-b^2\right ) (a \sec (c+d x)+b)}}{4 a \left (a^2-b^2\right )}+\frac {b^2 \sin (c+d x) \sec ^{\frac {5}{2}}(c+d x)}{2 a d \left (a^2-b^2\right ) (a \sec (c+d x)+b)^2}\) |
| \(\Big \downarrow \) 4274 |
| \(\displaystyle \frac {\frac {\frac {2 \left (8 a^4-29 a^2 b^2+15 b^4\right ) \sin (c+d x) \sqrt {\sec (c+d x)}}{a d}-\frac {b \left (35 a^4-38 a^2 b^2+15 b^4\right ) \int \frac {\csc \left (c+d x+\frac {\pi }{2}\right )^{3/2}}{b+a \csc \left (c+d x+\frac {\pi }{2}\right )}dx+\frac {b^2 \left (8 a^4-29 a^2 b^2+15 b^4\right ) \int \frac {1}{\sqrt {\sec (c+d x)}}dx-a b^3 \left (11 a^2-5 b^2\right ) \int \sqrt {\sec (c+d x)}dx}{b^2}}{a}}{2 a \left (a^2-b^2\right )}+\frac {b^2 \left (11 a^2-5 b^2\right ) \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x)}{a d \left (a^2-b^2\right ) (a \sec (c+d x)+b)}}{4 a \left (a^2-b^2\right )}+\frac {b^2 \sin (c+d x) \sec ^{\frac {5}{2}}(c+d x)}{2 a d \left (a^2-b^2\right ) (a \sec (c+d x)+b)^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {\frac {2 \left (8 a^4-29 a^2 b^2+15 b^4\right ) \sin (c+d x) \sqrt {\sec (c+d x)}}{a d}-\frac {b \left (35 a^4-38 a^2 b^2+15 b^4\right ) \int \frac {\csc \left (c+d x+\frac {\pi }{2}\right )^{3/2}}{b+a \csc \left (c+d x+\frac {\pi }{2}\right )}dx+\frac {b^2 \left (8 a^4-29 a^2 b^2+15 b^4\right ) \int \frac {1}{\sqrt {\csc \left (c+d x+\frac {\pi }{2}\right )}}dx-a b^3 \left (11 a^2-5 b^2\right ) \int \sqrt {\csc \left (c+d x+\frac {\pi }{2}\right )}dx}{b^2}}{a}}{2 a \left (a^2-b^2\right )}+\frac {b^2 \left (11 a^2-5 b^2\right ) \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x)}{a d \left (a^2-b^2\right ) (a \sec (c+d x)+b)}}{4 a \left (a^2-b^2\right )}+\frac {b^2 \sin (c+d x) \sec ^{\frac {5}{2}}(c+d x)}{2 a d \left (a^2-b^2\right ) (a \sec (c+d x)+b)^2}\) |
| \(\Big \downarrow \) 4258 |
| \(\displaystyle \frac {\frac {\frac {2 \left (8 a^4-29 a^2 b^2+15 b^4\right ) \sin (c+d x) \sqrt {\sec (c+d x)}}{a d}-\frac {b \left (35 a^4-38 a^2 b^2+15 b^4\right ) \int \frac {\csc \left (c+d x+\frac {\pi }{2}\right )^{3/2}}{b+a \csc \left (c+d x+\frac {\pi }{2}\right )}dx+\frac {b^2 \left (8 a^4-29 a^2 b^2+15 b^4\right ) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \int \sqrt {\cos (c+d x)}dx-a b^3 \left (11 a^2-5 b^2\right ) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \int \frac {1}{\sqrt {\cos (c+d x)}}dx}{b^2}}{a}}{2 a \left (a^2-b^2\right )}+\frac {b^2 \left (11 a^2-5 b^2\right ) \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x)}{a d \left (a^2-b^2\right ) (a \sec (c+d x)+b)}}{4 a \left (a^2-b^2\right )}+\frac {b^2 \sin (c+d x) \sec ^{\frac {5}{2}}(c+d x)}{2 a d \left (a^2-b^2\right ) (a \sec (c+d x)+b)^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {\frac {2 \left (8 a^4-29 a^2 b^2+15 b^4\right ) \sin (c+d x) \sqrt {\sec (c+d x)}}{a d}-\frac {b \left (35 a^4-38 a^2 b^2+15 b^4\right ) \int \frac {\csc \left (c+d x+\frac {\pi }{2}\right )^{3/2}}{b+a \csc \left (c+d x+\frac {\pi }{2}\right )}dx+\frac {b^2 \left (8 a^4-29 a^2 b^2+15 b^4\right ) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \int \sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}dx-a b^3 \left (11 a^2-5 b^2\right ) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \int \frac {1}{\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}}dx}{b^2}}{a}}{2 a \left (a^2-b^2\right )}+\frac {b^2 \left (11 a^2-5 b^2\right ) \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x)}{a d \left (a^2-b^2\right ) (a \sec (c+d x)+b)}}{4 a \left (a^2-b^2\right )}+\frac {b^2 \sin (c+d x) \sec ^{\frac {5}{2}}(c+d x)}{2 a d \left (a^2-b^2\right ) (a \sec (c+d x)+b)^2}\) |
| \(\Big \downarrow \) 3119 |
| \(\displaystyle \frac {\frac {\frac {2 \left (8 a^4-29 a^2 b^2+15 b^4\right ) \sin (c+d x) \sqrt {\sec (c+d x)}}{a d}-\frac {b \left (35 a^4-38 a^2 b^2+15 b^4\right ) \int \frac {\csc \left (c+d x+\frac {\pi }{2}\right )^{3/2}}{b+a \csc \left (c+d x+\frac {\pi }{2}\right )}dx+\frac {\frac {2 b^2 \left (8 a^4-29 a^2 b^2+15 b^4\right ) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{d}-a b^3 \left (11 a^2-5 b^2\right ) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \int \frac {1}{\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}}dx}{b^2}}{a}}{2 a \left (a^2-b^2\right )}+\frac {b^2 \left (11 a^2-5 b^2\right ) \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x)}{a d \left (a^2-b^2\right ) (a \sec (c+d x)+b)}}{4 a \left (a^2-b^2\right )}+\frac {b^2 \sin (c+d x) \sec ^{\frac {5}{2}}(c+d x)}{2 a d \left (a^2-b^2\right ) (a \sec (c+d x)+b)^2}\) |
| \(\Big \downarrow \) 3120 |
| \(\displaystyle \frac {\frac {\frac {2 \left (8 a^4-29 a^2 b^2+15 b^4\right ) \sin (c+d x) \sqrt {\sec (c+d x)}}{a d}-\frac {b \left (35 a^4-38 a^2 b^2+15 b^4\right ) \int \frac {\csc \left (c+d x+\frac {\pi }{2}\right )^{3/2}}{b+a \csc \left (c+d x+\frac {\pi }{2}\right )}dx+\frac {\frac {2 b^2 \left (8 a^4-29 a^2 b^2+15 b^4\right ) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{d}-\frac {2 a b^3 \left (11 a^2-5 b^2\right ) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )}{d}}{b^2}}{a}}{2 a \left (a^2-b^2\right )}+\frac {b^2 \left (11 a^2-5 b^2\right ) \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x)}{a d \left (a^2-b^2\right ) (a \sec (c+d x)+b)}}{4 a \left (a^2-b^2\right )}+\frac {b^2 \sin (c+d x) \sec ^{\frac {5}{2}}(c+d x)}{2 a d \left (a^2-b^2\right ) (a \sec (c+d x)+b)^2}\) |
| \(\Big \downarrow \) 4336 |
| \(\displaystyle \frac {\frac {\frac {2 \left (8 a^4-29 a^2 b^2+15 b^4\right ) \sin (c+d x) \sqrt {\sec (c+d x)}}{a d}-\frac {b \left (35 a^4-38 a^2 b^2+15 b^4\right ) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \int \frac {1}{\sqrt {\cos (c+d x)} (a+b \cos (c+d x))}dx+\frac {\frac {2 b^2 \left (8 a^4-29 a^2 b^2+15 b^4\right ) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{d}-\frac {2 a b^3 \left (11 a^2-5 b^2\right ) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )}{d}}{b^2}}{a}}{2 a \left (a^2-b^2\right )}+\frac {b^2 \left (11 a^2-5 b^2\right ) \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x)}{a d \left (a^2-b^2\right ) (a \sec (c+d x)+b)}}{4 a \left (a^2-b^2\right )}+\frac {b^2 \sin (c+d x) \sec ^{\frac {5}{2}}(c+d x)}{2 a d \left (a^2-b^2\right ) (a \sec (c+d x)+b)^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {\frac {2 \left (8 a^4-29 a^2 b^2+15 b^4\right ) \sin (c+d x) \sqrt {\sec (c+d x)}}{a d}-\frac {b \left (35 a^4-38 a^2 b^2+15 b^4\right ) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \int \frac {1}{\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )} \left (a+b \sin \left (c+d x+\frac {\pi }{2}\right )\right )}dx+\frac {\frac {2 b^2 \left (8 a^4-29 a^2 b^2+15 b^4\right ) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{d}-\frac {2 a b^3 \left (11 a^2-5 b^2\right ) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )}{d}}{b^2}}{a}}{2 a \left (a^2-b^2\right )}+\frac {b^2 \left (11 a^2-5 b^2\right ) \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x)}{a d \left (a^2-b^2\right ) (a \sec (c+d x)+b)}}{4 a \left (a^2-b^2\right )}+\frac {b^2 \sin (c+d x) \sec ^{\frac {5}{2}}(c+d x)}{2 a d \left (a^2-b^2\right ) (a \sec (c+d x)+b)^2}\) |
| \(\Big \downarrow \) 3284 |
| \(\displaystyle \frac {b^2 \sin (c+d x) \sec ^{\frac {5}{2}}(c+d x)}{2 a d \left (a^2-b^2\right ) (a \sec (c+d x)+b)^2}+\frac {\frac {b^2 \left (11 a^2-5 b^2\right ) \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x)}{a d \left (a^2-b^2\right ) (a \sec (c+d x)+b)}+\frac {\frac {2 \left (8 a^4-29 a^2 b^2+15 b^4\right ) \sin (c+d x) \sqrt {\sec (c+d x)}}{a d}-\frac {\frac {2 b \left (35 a^4-38 a^2 b^2+15 b^4\right ) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \operatorname {EllipticPi}\left (\frac {2 b}{a+b},\frac {1}{2} (c+d x),2\right )}{d (a+b)}+\frac {\frac {2 b^2 \left (8 a^4-29 a^2 b^2+15 b^4\right ) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{d}-\frac {2 a b^3 \left (11 a^2-5 b^2\right ) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )}{d}}{b^2}}{a}}{2 a \left (a^2-b^2\right )}}{4 a \left (a^2-b^2\right )}\) |
Input:
Int[Sec[c + d*x]^(3/2)/(a + b*Cos[c + d*x])^3,x]
Output:
(b^2*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(2*a*(a^2 - b^2)*d*(b + a*Sec[c + d* x])^2) + ((b^2*(11*a^2 - 5*b^2)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(a*(a^2 - b^2)*d*(b + a*Sec[c + d*x])) + (-((((2*b^2*(8*a^4 - 29*a^2*b^2 + 15*b^4)* Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/d - (2*a* b^3*(11*a^2 - 5*b^2)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec [c + d*x]])/d)/b^2 + (2*b*(35*a^4 - 38*a^2*b^2 + 15*b^4)*Sqrt[Cos[c + d*x] ]*EllipticPi[(2*b)/(a + b), (c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/((a + b)*d ))/a) + (2*(8*a^4 - 29*a^2*b^2 + 15*b^4)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/ (a*d))/(2*a*(a^2 - b^2)))/(4*a*(a^2 - b^2))
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[(e_.) + (f_.)*(x_)]*(d_.))^(m_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x _)]^(n_.))^(p_.), x_Symbol] :> Simp[d^(n*p) Int[(d*Csc[e + f*x])^(m - n*p )*(b + a*Csc[e + f*x]^n)^p, x], x] /; FreeQ[{a, b, d, e, f, m, n, p}, x] && !IntegerQ[m] && IntegersQ[n, p]
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. \(1964\) vs. \(2(363)=726\).
Time = 10.43 (sec) , antiderivative size = 1965, normalized size of antiderivative = 5.06
Input:
int(sec(d*x+c)^(3/2)/(a+cos(d*x+c)*b)^3,x,method=_RETURNVERBOSE)
Output:
-(-(-2*cos(1/2*d*x+1/2*c)^2+1)*sin(1/2*d*x+1/2*c)^2)^(1/2)*(2/a^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*sin(1/2*d*x+1/2*c)^2*cos(1/2*d*x+1/2*c)-(sin(1/2*d*x +1/2*c)^2)^(1/2)*(2*sin(1/2*d*x+1/2*c)^2-1)^(1/2)*EllipticE(cos(1/2*d*x+1/ 2*c),2^(1/2)))-2*b/a*(-1/2*b^2/a/(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*b*cos(1/2*d*x+1/2*c)^2+a-b)^2- 3/4*b^2*(3*a^2-b^2)/a^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*b*cos(1/2*d*x+1/2*c)^2+a-b)-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))+1/4/(a+b)/(a^2-b^2)/a*(sin(1/2*d*x+1/2*c)^2)^(1/2)*(-2*co s(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))*b+3/8/(a+b)/(a^2-b^2)/a^2*(si n(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 ))*b^2-9/8*b/(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)*Ellipti cF(cos(1/2*d*x+1/2*c),2^(1/2))+3/8*b^3/a^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)*EllipticF(cos(1/2*d*x+1/2*c),2^(1/2))+9/8*b/(a^2...
Timed out. \[ \int \frac {\sec ^{\frac {3}{2}}(c+d x)}{(a+b \cos (c+d x))^3} \, dx=\text {Timed out} \] Input:
integrate(sec(d*x+c)^(3/2)/(a+b*cos(d*x+c))^3,x, algorithm="fricas")
Output:
Timed out
\[ \int \frac {\sec ^{\frac {3}{2}}(c+d x)}{(a+b \cos (c+d x))^3} \, dx=\int \frac {\sec ^{\frac {3}{2}}{\left (c + d x \right )}}{\left (a + b \cos {\left (c + d x \right )}\right )^{3}}\, dx \] Input:
integrate(sec(d*x+c)**(3/2)/(a+b*cos(d*x+c))**3,x)
Output:
Integral(sec(c + d*x)**(3/2)/(a + b*cos(c + d*x))**3, x)
Timed out. \[ \int \frac {\sec ^{\frac {3}{2}}(c+d x)}{(a+b \cos (c+d x))^3} \, dx=\text {Timed out} \] Input:
integrate(sec(d*x+c)^(3/2)/(a+b*cos(d*x+c))^3,x, algorithm="maxima")
Output:
Timed out
\[ \int \frac {\sec ^{\frac {3}{2}}(c+d x)}{(a+b \cos (c+d x))^3} \, dx=\int { \frac {\sec \left (d x + c\right )^{\frac {3}{2}}}{{\left (b \cos \left (d x + c\right ) + a\right )}^{3}} \,d x } \] Input:
integrate(sec(d*x+c)^(3/2)/(a+b*cos(d*x+c))^3,x, algorithm="giac")
Output:
integrate(sec(d*x + c)^(3/2)/(b*cos(d*x + c) + a)^3, x)
Timed out. \[ \int \frac {\sec ^{\frac {3}{2}}(c+d x)}{(a+b \cos (c+d x))^3} \, dx=\int \frac {{\left (\frac {1}{\cos \left (c+d\,x\right )}\right )}^{3/2}}{{\left (a+b\,\cos \left (c+d\,x\right )\right )}^3} \,d x \] Input:
int((1/cos(c + d*x))^(3/2)/(a + b*cos(c + d*x))^3,x)
Output:
int((1/cos(c + d*x))^(3/2)/(a + b*cos(c + d*x))^3, x)
\[ \int \frac {\sec ^{\frac {3}{2}}(c+d x)}{(a+b \cos (c+d x))^3} \, dx=\int \frac {\sqrt {\sec \left (d x +c \right )}\, \sec \left (d x +c \right )}{\cos \left (d x +c \right )^{3} b^{3}+3 \cos \left (d x +c \right )^{2} a \,b^{2}+3 \cos \left (d x +c \right ) a^{2} b +a^{3}}d x \] Input:
int(sec(d*x+c)^(3/2)/(a+b*cos(d*x+c))^3,x)
Output:
int((sqrt(sec(c + d*x))*sec(c + d*x))/(cos(c + d*x)**3*b**3 + 3*cos(c + d* x)**2*a*b**2 + 3*cos(c + d*x)*a**2*b + a**3),x)