Integrand size = 23, antiderivative size = 406 \[ \int \frac {1}{\sec ^{\frac {3}{2}}(c+d x) (a+b \sec (c+d x))^3} \, dx=-\frac {b \left (24 a^4-65 a^2 b^2+35 b^4\right ) \sqrt {\cos (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \sqrt {\sec (c+d x)}}{4 a^4 \left (a^2-b^2\right )^2 d}+\frac {\left (8 a^6+128 a^4 b^2-223 a^2 b^4+105 b^6\right ) \sqrt {\cos (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right ) \sqrt {\sec (c+d x)}}{12 a^5 \left (a^2-b^2\right )^2 d}-\frac {b^3 \left (63 a^4-86 a^2 b^2+35 b^4\right ) \sqrt {\cos (c+d x)} \operatorname {EllipticPi}\left (\frac {2 a}{a+b},\frac {1}{2} (c+d x),2\right ) \sqrt {\sec (c+d x)}}{4 a^5 (a-b)^2 (a+b)^3 d}+\frac {\left (8 a^4-61 a^2 b^2+35 b^4\right ) \sin (c+d x)}{12 a^3 \left (a^2-b^2\right )^2 d \sqrt {\sec (c+d x)}}+\frac {b^2 \sin (c+d x)}{2 a \left (a^2-b^2\right ) d \sqrt {\sec (c+d x)} (a+b \sec (c+d x))^2}+\frac {b^2 \left (13 a^2-7 b^2\right ) \sin (c+d x)}{4 a^2 \left (a^2-b^2\right )^2 d \sqrt {\sec (c+d x)} (a+b \sec (c+d x))} \]
1/12*(8*a^4-61*a^2*b^2+35*b^4)*sin(d*x+c)/a^3/(a^2-b^2)^2/d/sec(d*x+c)^(1/ 2)+1/2*b^2*sin(d*x+c)/a/(a^2-b^2)/d/(a+b*sec(d*x+c))^2/sec(d*x+c)^(1/2)+1/ 4*b^2*(13*a^2-7*b^2)*sin(d*x+c)/a^2/(a^2-b^2)^2/d/(a+b*sec(d*x+c))/sec(d*x +c)^(1/2)-1/4*b*(24*a^4-65*a^2*b^2+35*b^4)*(cos(1/2*d*x+1/2*c)^2)^(1/2)/co s(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)*se c(d*x+c)^(1/2)/a^4/(a^2-b^2)^2/d+1/12*(8*a^6+128*a^4*b^2-223*a^2*b^4+105*b ^6)*(cos(1/2*d*x+1/2*c)^2)^(1/2)/cos(1/2*d*x+1/2*c)*EllipticF(sin(1/2*d*x+ 1/2*c),2^(1/2))*cos(d*x+c)^(1/2)*sec(d*x+c)^(1/2)/a^5/(a^2-b^2)^2/d-1/4*b^ 3*(63*a^4-86*a^2*b^2+35*b^4)*(cos(1/2*d*x+1/2*c)^2)^(1/2)/cos(1/2*d*x+1/2* c)*EllipticPi(sin(1/2*d*x+1/2*c),2*a/(a+b),2^(1/2))*cos(d*x+c)^(1/2)*sec(d *x+c)^(1/2)/a^5/(a-b)^2/(a+b)^3/d
Time = 6.86 (sec) , antiderivative size = 731, normalized size of antiderivative = 1.80 \[ \int \frac {1}{\sec ^{\frac {3}{2}}(c+d x) (a+b \sec (c+d x))^3} \, dx=\frac {\frac {2 \left (-56 a^4 b+73 a^2 b^3-35 b^5\right ) \cos ^2(c+d x) \left (\operatorname {EllipticF}\left (\arcsin \left (\sqrt {\sec (c+d x)}\right ),-1\right )-\operatorname {EllipticPi}\left (-\frac {b}{a},\arcsin \left (\sqrt {\sec (c+d x)}\right ),-1\right )\right ) (a+b \sec (c+d x)) \sqrt {1-\sec ^2(c+d x)} \sin (c+d x)}{b (b+a \cos (c+d x)) \left (1-\cos ^2(c+d x)\right )}+\frac {2 \left (16 a^5+112 a^3 b^2-56 a b^4\right ) \cos ^2(c+d x) \operatorname {EllipticPi}\left (-\frac {b}{a},\arcsin \left (\sqrt {\sec (c+d x)}\right ),-1\right ) (a+b \sec (c+d x)) \sqrt {1-\sec ^2(c+d x)} \sin (c+d x)}{a (b+a \cos (c+d x)) \left (1-\cos ^2(c+d x)\right )}+\frac {\left (-72 a^4 b+195 a^2 b^3-105 b^5\right ) \cos (2 (c+d x)) (a+b \sec (c+d x)) \left (-4 a b+4 a b \sec ^2(c+d x)-4 a b E\left (\left .\arcsin \left (\sqrt {\sec (c+d x)}\right )\right |-1\right ) \sqrt {\sec (c+d x)} \sqrt {1-\sec ^2(c+d x)}-2 a (a-2 b) \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\sec (c+d x)}\right ),-1\right ) \sqrt {\sec (c+d x)} \sqrt {1-\sec ^2(c+d x)}+2 a^2 \operatorname {EllipticPi}\left (-\frac {b}{a},\arcsin \left (\sqrt {\sec (c+d x)}\right ),-1\right ) \sqrt {\sec (c+d x)} \sqrt {1-\sec ^2(c+d x)}-4 b^2 \operatorname {EllipticPi}\left (-\frac {b}{a},\arcsin \left (\sqrt {\sec (c+d x)}\right ),-1\right ) \sqrt {\sec (c+d x)} \sqrt {1-\sec ^2(c+d x)}\right ) \sin (c+d x)}{a^2 b (b+a \cos (c+d x)) \left (1-\cos ^2(c+d x)\right ) \sqrt {\sec (c+d x)} \left (2-\sec ^2(c+d x)\right )}}{48 a^3 (a-b)^2 (a+b)^2 d}+\frac {\sqrt {\sec (c+d x)} \left (\frac {b^3 \left (-17 a^2+11 b^2\right ) \sin (c+d x)}{4 a^4 \left (-a^2+b^2\right )^2}-\frac {b^5 \sin (c+d x)}{2 a^4 \left (a^2-b^2\right ) (b+a \cos (c+d x))^2}+\frac {19 a^2 b^4 \sin (c+d x)-13 b^6 \sin (c+d x)}{4 a^4 \left (a^2-b^2\right )^2 (b+a \cos (c+d x))}+\frac {\sin (2 (c+d x))}{3 a^3}\right )}{d} \]
((2*(-56*a^4*b + 73*a^2*b^3 - 35*b^5)*Cos[c + d*x]^2*(EllipticF[ArcSin[Sqr t[Sec[c + d*x]]], -1] - EllipticPi[-(b/a), ArcSin[Sqrt[Sec[c + d*x]]], -1] )*(a + b*Sec[c + d*x])*Sqrt[1 - Sec[c + d*x]^2]*Sin[c + d*x])/(b*(b + a*Co s[c + d*x])*(1 - Cos[c + d*x]^2)) + (2*(16*a^5 + 112*a^3*b^2 - 56*a*b^4)*C os[c + d*x]^2*EllipticPi[-(b/a), ArcSin[Sqrt[Sec[c + d*x]]], -1]*(a + b*Se c[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)) + ((-72*a^4*b + 195*a^2*b^3 - 105*b^5)*Cos[2*(c + d *x)]*(a + b*Sec[c + d*x])*(-4*a*b + 4*a*b*Sec[c + d*x]^2 - 4*a*b*EllipticE [ArcSin[Sqrt[Sec[c + d*x]]], -1]*Sqrt[Sec[c + d*x]]*Sqrt[1 - Sec[c + d*x]^ 2] - 2*a*(a - 2*b)*EllipticF[ArcSin[Sqrt[Sec[c + d*x]]], -1]*Sqrt[Sec[c + d*x]]*Sqrt[1 - Sec[c + d*x]^2] + 2*a^2*EllipticPi[-(b/a), ArcSin[Sqrt[Sec[ c + d*x]]], -1]*Sqrt[Sec[c + d*x]]*Sqrt[1 - Sec[c + d*x]^2] - 4*b^2*Ellipt icPi[-(b/a), ArcSin[Sqrt[Sec[c + d*x]]], -1]*Sqrt[Sec[c + d*x]]*Sqrt[1 - S ec[c + d*x]^2])*Sin[c + d*x])/(a^2*b*(b + a*Cos[c + d*x])*(1 - Cos[c + d*x ]^2)*Sqrt[Sec[c + d*x]]*(2 - Sec[c + d*x]^2)))/(48*a^3*(a - b)^2*(a + b)^2 *d) + (Sqrt[Sec[c + d*x]]*((b^3*(-17*a^2 + 11*b^2)*Sin[c + d*x])/(4*a^4*(- a^2 + b^2)^2) - (b^5*Sin[c + d*x])/(2*a^4*(a^2 - b^2)*(b + a*Cos[c + d*x]) ^2) + (19*a^2*b^4*Sin[c + d*x] - 13*b^6*Sin[c + d*x])/(4*a^4*(a^2 - b^2)^2 *(b + a*Cos[c + d*x])) + Sin[2*(c + d*x)]/(3*a^3)))/d
Time = 2.91 (sec) , antiderivative size = 403, normalized size of antiderivative = 0.99, number of steps used = 21, number of rules used = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.913, Rules used = {3042, 4334, 27, 3042, 4588, 27, 3042, 4592, 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.
\(\displaystyle \int \frac {1}{\sec ^{\frac {3}{2}}(c+d x) (a+b \sec (c+d x))^3} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {1}{\csc \left (c+d x+\frac {\pi }{2}\right )^{3/2} \left (a+b \csc \left (c+d x+\frac {\pi }{2}\right )\right )^3}dx\) |
\(\Big \downarrow \) 4334 |
\(\displaystyle \frac {b^2 \sin (c+d x)}{2 a d \left (a^2-b^2\right ) \sqrt {\sec (c+d x)} (a+b \sec (c+d x))^2}-\frac {\int -\frac {4 a^2-4 b \sec (c+d x) a-7 b^2+5 b^2 \sec ^2(c+d x)}{2 \sec ^{\frac {3}{2}}(c+d x) (a+b \sec (c+d x))^2}dx}{2 a \left (a^2-b^2\right )}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {\int \frac {4 a^2-4 b \sec (c+d x) a-7 b^2+5 b^2 \sec ^2(c+d x)}{\sec ^{\frac {3}{2}}(c+d x) (a+b \sec (c+d x))^2}dx}{4 a \left (a^2-b^2\right )}+\frac {b^2 \sin (c+d x)}{2 a d \left (a^2-b^2\right ) \sqrt {\sec (c+d x)} (a+b \sec (c+d x))^2}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {\int \frac {4 a^2-4 b \csc \left (c+d x+\frac {\pi }{2}\right ) a-7 b^2+5 b^2 \csc \left (c+d x+\frac {\pi }{2}\right )^2}{\csc \left (c+d x+\frac {\pi }{2}\right )^{3/2} \left (a+b \csc \left (c+d x+\frac {\pi }{2}\right )\right )^2}dx}{4 a \left (a^2-b^2\right )}+\frac {b^2 \sin (c+d x)}{2 a d \left (a^2-b^2\right ) \sqrt {\sec (c+d x)} (a+b \sec (c+d x))^2}\) |
\(\Big \downarrow \) 4588 |
\(\displaystyle \frac {\frac {b^2 \left (13 a^2-7 b^2\right ) \sin (c+d x)}{a d \left (a^2-b^2\right ) \sqrt {\sec (c+d x)} (a+b \sec (c+d x))}-\frac {\int -\frac {8 a^4-61 b^2 a^2-4 b \left (4 a^2-b^2\right ) \sec (c+d x) a+35 b^4+3 b^2 \left (13 a^2-7 b^2\right ) \sec ^2(c+d x)}{2 \sec ^{\frac {3}{2}}(c+d x) (a+b \sec (c+d x))}dx}{a \left (a^2-b^2\right )}}{4 a \left (a^2-b^2\right )}+\frac {b^2 \sin (c+d x)}{2 a d \left (a^2-b^2\right ) \sqrt {\sec (c+d x)} (a+b \sec (c+d x))^2}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {\frac {\int \frac {8 a^4-61 b^2 a^2-4 b \left (4 a^2-b^2\right ) \sec (c+d x) a+35 b^4+3 b^2 \left (13 a^2-7 b^2\right ) \sec ^2(c+d x)}{\sec ^{\frac {3}{2}}(c+d x) (a+b \sec (c+d x))}dx}{2 a \left (a^2-b^2\right )}+\frac {b^2 \left (13 a^2-7 b^2\right ) \sin (c+d x)}{a d \left (a^2-b^2\right ) \sqrt {\sec (c+d x)} (a+b \sec (c+d x))}}{4 a \left (a^2-b^2\right )}+\frac {b^2 \sin (c+d x)}{2 a d \left (a^2-b^2\right ) \sqrt {\sec (c+d x)} (a+b \sec (c+d x))^2}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {\frac {\int \frac {8 a^4-61 b^2 a^2-4 b \left (4 a^2-b^2\right ) \csc \left (c+d x+\frac {\pi }{2}\right ) a+35 b^4+3 b^2 \left (13 a^2-7 b^2\right ) \csc \left (c+d x+\frac {\pi }{2}\right )^2}{\csc \left (c+d x+\frac {\pi }{2}\right )^{3/2} \left (a+b \csc \left (c+d x+\frac {\pi }{2}\right )\right )}dx}{2 a \left (a^2-b^2\right )}+\frac {b^2 \left (13 a^2-7 b^2\right ) \sin (c+d x)}{a d \left (a^2-b^2\right ) \sqrt {\sec (c+d x)} (a+b \sec (c+d x))}}{4 a \left (a^2-b^2\right )}+\frac {b^2 \sin (c+d x)}{2 a d \left (a^2-b^2\right ) \sqrt {\sec (c+d x)} (a+b \sec (c+d x))^2}\) |
\(\Big \downarrow \) 4592 |
\(\displaystyle \frac {\frac {\frac {2 \left (8 a^4-61 a^2 b^2+35 b^4\right ) \sin (c+d x)}{3 a d \sqrt {\sec (c+d x)}}-\frac {2 \int \frac {-b \left (8 a^4-61 b^2 a^2+35 b^4\right ) \sec ^2(c+d x)-4 a \left (2 a^4+14 b^2 a^2-7 b^4\right ) \sec (c+d x)+3 b \left (24 a^4-65 b^2 a^2+35 b^4\right )}{2 \sqrt {\sec (c+d x)} (a+b \sec (c+d x))}dx}{3 a}}{2 a \left (a^2-b^2\right )}+\frac {b^2 \left (13 a^2-7 b^2\right ) \sin (c+d x)}{a d \left (a^2-b^2\right ) \sqrt {\sec (c+d x)} (a+b \sec (c+d x))}}{4 a \left (a^2-b^2\right )}+\frac {b^2 \sin (c+d x)}{2 a d \left (a^2-b^2\right ) \sqrt {\sec (c+d x)} (a+b \sec (c+d x))^2}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {\frac {\frac {2 \left (8 a^4-61 a^2 b^2+35 b^4\right ) \sin (c+d x)}{3 a d \sqrt {\sec (c+d x)}}-\frac {\int \frac {-b \left (8 a^4-61 b^2 a^2+35 b^4\right ) \sec ^2(c+d x)-4 a \left (2 a^4+14 b^2 a^2-7 b^4\right ) \sec (c+d x)+3 b \left (24 a^4-65 b^2 a^2+35 b^4\right )}{\sqrt {\sec (c+d x)} (a+b \sec (c+d x))}dx}{3 a}}{2 a \left (a^2-b^2\right )}+\frac {b^2 \left (13 a^2-7 b^2\right ) \sin (c+d x)}{a d \left (a^2-b^2\right ) \sqrt {\sec (c+d x)} (a+b \sec (c+d x))}}{4 a \left (a^2-b^2\right )}+\frac {b^2 \sin (c+d x)}{2 a d \left (a^2-b^2\right ) \sqrt {\sec (c+d x)} (a+b \sec (c+d x))^2}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {\frac {\frac {2 \left (8 a^4-61 a^2 b^2+35 b^4\right ) \sin (c+d x)}{3 a d \sqrt {\sec (c+d x)}}-\frac {\int \frac {-b \left (8 a^4-61 b^2 a^2+35 b^4\right ) \csc \left (c+d x+\frac {\pi }{2}\right )^2-4 a \left (2 a^4+14 b^2 a^2-7 b^4\right ) \csc \left (c+d x+\frac {\pi }{2}\right )+3 b \left (24 a^4-65 b^2 a^2+35 b^4\right )}{\sqrt {\csc \left (c+d x+\frac {\pi }{2}\right )} \left (a+b \csc \left (c+d x+\frac {\pi }{2}\right )\right )}dx}{3 a}}{2 a \left (a^2-b^2\right )}+\frac {b^2 \left (13 a^2-7 b^2\right ) \sin (c+d x)}{a d \left (a^2-b^2\right ) \sqrt {\sec (c+d x)} (a+b \sec (c+d x))}}{4 a \left (a^2-b^2\right )}+\frac {b^2 \sin (c+d x)}{2 a d \left (a^2-b^2\right ) \sqrt {\sec (c+d x)} (a+b \sec (c+d x))^2}\) |
\(\Big \downarrow \) 4594 |
\(\displaystyle \frac {\frac {\frac {2 \left (8 a^4-61 a^2 b^2+35 b^4\right ) \sin (c+d x)}{3 a d \sqrt {\sec (c+d x)}}-\frac {\frac {3 b^3 \left (63 a^4-86 a^2 b^2+35 b^4\right ) \int \frac {\sec ^{\frac {3}{2}}(c+d x)}{a+b \sec (c+d x)}dx}{a^2}+\frac {\int \frac {3 a b \left (24 a^4-65 b^2 a^2+35 b^4\right )-\left (8 a^6+128 b^2 a^4-223 b^4 a^2+105 b^6\right ) \sec (c+d x)}{\sqrt {\sec (c+d x)}}dx}{a^2}}{3 a}}{2 a \left (a^2-b^2\right )}+\frac {b^2 \left (13 a^2-7 b^2\right ) \sin (c+d x)}{a d \left (a^2-b^2\right ) \sqrt {\sec (c+d x)} (a+b \sec (c+d x))}}{4 a \left (a^2-b^2\right )}+\frac {b^2 \sin (c+d x)}{2 a d \left (a^2-b^2\right ) \sqrt {\sec (c+d x)} (a+b \sec (c+d x))^2}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {\frac {\frac {2 \left (8 a^4-61 a^2 b^2+35 b^4\right ) \sin (c+d x)}{3 a d \sqrt {\sec (c+d x)}}-\frac {\frac {3 b^3 \left (63 a^4-86 a^2 b^2+35 b^4\right ) \int \frac {\csc \left (c+d x+\frac {\pi }{2}\right )^{3/2}}{a+b \csc \left (c+d x+\frac {\pi }{2}\right )}dx}{a^2}+\frac {\int \frac {3 a b \left (24 a^4-65 b^2 a^2+35 b^4\right )+\left (-8 a^6-128 b^2 a^4+223 b^4 a^2-105 b^6\right ) \csc \left (c+d x+\frac {\pi }{2}\right )}{\sqrt {\csc \left (c+d x+\frac {\pi }{2}\right )}}dx}{a^2}}{3 a}}{2 a \left (a^2-b^2\right )}+\frac {b^2 \left (13 a^2-7 b^2\right ) \sin (c+d x)}{a d \left (a^2-b^2\right ) \sqrt {\sec (c+d x)} (a+b \sec (c+d x))}}{4 a \left (a^2-b^2\right )}+\frac {b^2 \sin (c+d x)}{2 a d \left (a^2-b^2\right ) \sqrt {\sec (c+d x)} (a+b \sec (c+d x))^2}\) |
\(\Big \downarrow \) 4274 |
\(\displaystyle \frac {\frac {\frac {2 \left (8 a^4-61 a^2 b^2+35 b^4\right ) \sin (c+d x)}{3 a d \sqrt {\sec (c+d x)}}-\frac {\frac {3 b^3 \left (63 a^4-86 a^2 b^2+35 b^4\right ) \int \frac {\csc \left (c+d x+\frac {\pi }{2}\right )^{3/2}}{a+b \csc \left (c+d x+\frac {\pi }{2}\right )}dx}{a^2}+\frac {3 a b \left (24 a^4-65 a^2 b^2+35 b^4\right ) \int \frac {1}{\sqrt {\sec (c+d x)}}dx-\left (8 a^6+128 a^4 b^2-223 a^2 b^4+105 b^6\right ) \int \sqrt {\sec (c+d x)}dx}{a^2}}{3 a}}{2 a \left (a^2-b^2\right )}+\frac {b^2 \left (13 a^2-7 b^2\right ) \sin (c+d x)}{a d \left (a^2-b^2\right ) \sqrt {\sec (c+d x)} (a+b \sec (c+d x))}}{4 a \left (a^2-b^2\right )}+\frac {b^2 \sin (c+d x)}{2 a d \left (a^2-b^2\right ) \sqrt {\sec (c+d x)} (a+b \sec (c+d x))^2}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {\frac {\frac {2 \left (8 a^4-61 a^2 b^2+35 b^4\right ) \sin (c+d x)}{3 a d \sqrt {\sec (c+d x)}}-\frac {\frac {3 b^3 \left (63 a^4-86 a^2 b^2+35 b^4\right ) \int \frac {\csc \left (c+d x+\frac {\pi }{2}\right )^{3/2}}{a+b \csc \left (c+d x+\frac {\pi }{2}\right )}dx}{a^2}+\frac {3 a b \left (24 a^4-65 a^2 b^2+35 b^4\right ) \int \frac {1}{\sqrt {\csc \left (c+d x+\frac {\pi }{2}\right )}}dx-\left (8 a^6+128 a^4 b^2-223 a^2 b^4+105 b^6\right ) \int \sqrt {\csc \left (c+d x+\frac {\pi }{2}\right )}dx}{a^2}}{3 a}}{2 a \left (a^2-b^2\right )}+\frac {b^2 \left (13 a^2-7 b^2\right ) \sin (c+d x)}{a d \left (a^2-b^2\right ) \sqrt {\sec (c+d x)} (a+b \sec (c+d x))}}{4 a \left (a^2-b^2\right )}+\frac {b^2 \sin (c+d x)}{2 a d \left (a^2-b^2\right ) \sqrt {\sec (c+d x)} (a+b \sec (c+d x))^2}\) |
\(\Big \downarrow \) 4258 |
\(\displaystyle \frac {\frac {\frac {2 \left (8 a^4-61 a^2 b^2+35 b^4\right ) \sin (c+d x)}{3 a d \sqrt {\sec (c+d x)}}-\frac {\frac {3 b^3 \left (63 a^4-86 a^2 b^2+35 b^4\right ) \int \frac {\csc \left (c+d x+\frac {\pi }{2}\right )^{3/2}}{a+b \csc \left (c+d x+\frac {\pi }{2}\right )}dx}{a^2}+\frac {3 a b \left (24 a^4-65 a^2 b^2+35 b^4\right ) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \int \sqrt {\cos (c+d x)}dx-\left (8 a^6+128 a^4 b^2-223 a^2 b^4+105 b^6\right ) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \int \frac {1}{\sqrt {\cos (c+d x)}}dx}{a^2}}{3 a}}{2 a \left (a^2-b^2\right )}+\frac {b^2 \left (13 a^2-7 b^2\right ) \sin (c+d x)}{a d \left (a^2-b^2\right ) \sqrt {\sec (c+d x)} (a+b \sec (c+d x))}}{4 a \left (a^2-b^2\right )}+\frac {b^2 \sin (c+d x)}{2 a d \left (a^2-b^2\right ) \sqrt {\sec (c+d x)} (a+b \sec (c+d x))^2}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {\frac {\frac {2 \left (8 a^4-61 a^2 b^2+35 b^4\right ) \sin (c+d x)}{3 a d \sqrt {\sec (c+d x)}}-\frac {\frac {3 b^3 \left (63 a^4-86 a^2 b^2+35 b^4\right ) \int \frac {\csc \left (c+d x+\frac {\pi }{2}\right )^{3/2}}{a+b \csc \left (c+d x+\frac {\pi }{2}\right )}dx}{a^2}+\frac {3 a b \left (24 a^4-65 a^2 b^2+35 b^4\right ) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \int \sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}dx-\left (8 a^6+128 a^4 b^2-223 a^2 b^4+105 b^6\right ) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \int \frac {1}{\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}}dx}{a^2}}{3 a}}{2 a \left (a^2-b^2\right )}+\frac {b^2 \left (13 a^2-7 b^2\right ) \sin (c+d x)}{a d \left (a^2-b^2\right ) \sqrt {\sec (c+d x)} (a+b \sec (c+d x))}}{4 a \left (a^2-b^2\right )}+\frac {b^2 \sin (c+d x)}{2 a d \left (a^2-b^2\right ) \sqrt {\sec (c+d x)} (a+b \sec (c+d x))^2}\) |
\(\Big \downarrow \) 3119 |
\(\displaystyle \frac {\frac {\frac {2 \left (8 a^4-61 a^2 b^2+35 b^4\right ) \sin (c+d x)}{3 a d \sqrt {\sec (c+d x)}}-\frac {\frac {3 b^3 \left (63 a^4-86 a^2 b^2+35 b^4\right ) \int \frac {\csc \left (c+d x+\frac {\pi }{2}\right )^{3/2}}{a+b \csc \left (c+d x+\frac {\pi }{2}\right )}dx}{a^2}+\frac {\frac {6 a b \left (24 a^4-65 a^2 b^2+35 b^4\right ) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{d}-\left (8 a^6+128 a^4 b^2-223 a^2 b^4+105 b^6\right ) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \int \frac {1}{\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}}dx}{a^2}}{3 a}}{2 a \left (a^2-b^2\right )}+\frac {b^2 \left (13 a^2-7 b^2\right ) \sin (c+d x)}{a d \left (a^2-b^2\right ) \sqrt {\sec (c+d x)} (a+b \sec (c+d x))}}{4 a \left (a^2-b^2\right )}+\frac {b^2 \sin (c+d x)}{2 a d \left (a^2-b^2\right ) \sqrt {\sec (c+d x)} (a+b \sec (c+d x))^2}\) |
\(\Big \downarrow \) 3120 |
\(\displaystyle \frac {\frac {\frac {2 \left (8 a^4-61 a^2 b^2+35 b^4\right ) \sin (c+d x)}{3 a d \sqrt {\sec (c+d x)}}-\frac {\frac {3 b^3 \left (63 a^4-86 a^2 b^2+35 b^4\right ) \int \frac {\csc \left (c+d x+\frac {\pi }{2}\right )^{3/2}}{a+b \csc \left (c+d x+\frac {\pi }{2}\right )}dx}{a^2}+\frac {\frac {6 a b \left (24 a^4-65 a^2 b^2+35 b^4\right ) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{d}-\frac {2 \left (8 a^6+128 a^4 b^2-223 a^2 b^4+105 b^6\right ) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )}{d}}{a^2}}{3 a}}{2 a \left (a^2-b^2\right )}+\frac {b^2 \left (13 a^2-7 b^2\right ) \sin (c+d x)}{a d \left (a^2-b^2\right ) \sqrt {\sec (c+d x)} (a+b \sec (c+d x))}}{4 a \left (a^2-b^2\right )}+\frac {b^2 \sin (c+d x)}{2 a d \left (a^2-b^2\right ) \sqrt {\sec (c+d x)} (a+b \sec (c+d x))^2}\) |
\(\Big \downarrow \) 4336 |
\(\displaystyle \frac {\frac {\frac {2 \left (8 a^4-61 a^2 b^2+35 b^4\right ) \sin (c+d x)}{3 a d \sqrt {\sec (c+d x)}}-\frac {\frac {3 b^3 \left (63 a^4-86 a^2 b^2+35 b^4\right ) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \int \frac {1}{\sqrt {\cos (c+d x)} (b+a \cos (c+d x))}dx}{a^2}+\frac {\frac {6 a b \left (24 a^4-65 a^2 b^2+35 b^4\right ) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{d}-\frac {2 \left (8 a^6+128 a^4 b^2-223 a^2 b^4+105 b^6\right ) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )}{d}}{a^2}}{3 a}}{2 a \left (a^2-b^2\right )}+\frac {b^2 \left (13 a^2-7 b^2\right ) \sin (c+d x)}{a d \left (a^2-b^2\right ) \sqrt {\sec (c+d x)} (a+b \sec (c+d x))}}{4 a \left (a^2-b^2\right )}+\frac {b^2 \sin (c+d x)}{2 a d \left (a^2-b^2\right ) \sqrt {\sec (c+d x)} (a+b \sec (c+d x))^2}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {\frac {\frac {2 \left (8 a^4-61 a^2 b^2+35 b^4\right ) \sin (c+d x)}{3 a d \sqrt {\sec (c+d x)}}-\frac {\frac {3 b^3 \left (63 a^4-86 a^2 b^2+35 b^4\right ) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \int \frac {1}{\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )} \left (b+a \sin \left (c+d x+\frac {\pi }{2}\right )\right )}dx}{a^2}+\frac {\frac {6 a b \left (24 a^4-65 a^2 b^2+35 b^4\right ) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{d}-\frac {2 \left (8 a^6+128 a^4 b^2-223 a^2 b^4+105 b^6\right ) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )}{d}}{a^2}}{3 a}}{2 a \left (a^2-b^2\right )}+\frac {b^2 \left (13 a^2-7 b^2\right ) \sin (c+d x)}{a d \left (a^2-b^2\right ) \sqrt {\sec (c+d x)} (a+b \sec (c+d x))}}{4 a \left (a^2-b^2\right )}+\frac {b^2 \sin (c+d x)}{2 a d \left (a^2-b^2\right ) \sqrt {\sec (c+d x)} (a+b \sec (c+d x))^2}\) |
\(\Big \downarrow \) 3284 |
\(\displaystyle \frac {b^2 \sin (c+d x)}{2 a d \left (a^2-b^2\right ) \sqrt {\sec (c+d x)} (a+b \sec (c+d x))^2}+\frac {\frac {b^2 \left (13 a^2-7 b^2\right ) \sin (c+d x)}{a d \left (a^2-b^2\right ) \sqrt {\sec (c+d x)} (a+b \sec (c+d x))}+\frac {\frac {2 \left (8 a^4-61 a^2 b^2+35 b^4\right ) \sin (c+d x)}{3 a d \sqrt {\sec (c+d x)}}-\frac {\frac {6 b^3 \left (63 a^4-86 a^2 b^2+35 b^4\right ) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \operatorname {EllipticPi}\left (\frac {2 a}{a+b},\frac {1}{2} (c+d x),2\right )}{a^2 d (a+b)}+\frac {\frac {6 a b \left (24 a^4-65 a^2 b^2+35 b^4\right ) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{d}-\frac {2 \left (8 a^6+128 a^4 b^2-223 a^2 b^4+105 b^6\right ) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )}{d}}{a^2}}{3 a}}{2 a \left (a^2-b^2\right )}}{4 a \left (a^2-b^2\right )}\) |
(b^2*Sin[c + d*x])/(2*a*(a^2 - b^2)*d*Sqrt[Sec[c + d*x]]*(a + b*Sec[c + d* x])^2) + ((b^2*(13*a^2 - 7*b^2)*Sin[c + d*x])/(a*(a^2 - b^2)*d*Sqrt[Sec[c + d*x]]*(a + b*Sec[c + d*x])) + (-1/3*(((6*a*b*(24*a^4 - 65*a^2*b^2 + 35*b ^4)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/d - ( 2*(8*a^6 + 128*a^4*b^2 - 223*a^2*b^4 + 105*b^6)*Sqrt[Cos[c + d*x]]*Ellipti cF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/d)/a^2 + (6*b^3*(63*a^4 - 86*a^2*b^ 2 + 35*b^4)*Sqrt[Cos[c + d*x]]*EllipticPi[(2*a)/(a + b), (c + d*x)/2, 2]*S qrt[Sec[c + d*x]])/(a^2*(a + b)*d))/a + (2*(8*a^4 - 61*a^2*b^2 + 35*b^4)*S in[c + d*x])/(3*a*d*Sqrt[Sec[c + d*x]]))/(2*a*(a^2 - b^2)))/(4*a*(a^2 - b^ 2))
3.7.27.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[Sqrt[sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[(2/d)*EllipticE[(1/2)* (c - Pi/2 + d*x), 2], x] /; FreeQ[{c, d}, x]
Int[1/Sqrt[sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[(2/d)*EllipticF[(1/2 )*(c - Pi/2 + d*x), 2], x] /; FreeQ[{c, d}, x]
Int[1/(((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])*Sqrt[(c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)]]), x_Symbol] :> Simp[(2/(f*(a + b)*Sqrt[c + d]))*EllipticPi[ 2*(b/(a + b)), (1/2)*(e - Pi/2 + f*x), 2*(d/(c + d))], x] /; FreeQ[{a, b, c , d, e, f}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && GtQ[c + d, 0]
Int[(csc[(c_.) + (d_.)*(x_)]*(b_.))^(n_), x_Symbol] :> Simp[(b*Csc[c + d*x] )^n*Sin[c + d*x]^n Int[1/Sin[c + d*x]^n, x], x] /; FreeQ[{b, c, d}, x] && EqQ[n^2, 1/4]
Int[(csc[(e_.) + (f_.)*(x_)]*(d_.))^(n_.)*(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_)), x_Symbol] :> Simp[a Int[(d*Csc[e + f*x])^n, x], x] + Simp[b/d In t[(d*Csc[e + f*x])^(n + 1), x], x] /; FreeQ[{a, b, d, e, f, n}, x]
Int[(csc[(e_.) + (f_.)*(x_)]*(d_.))^(n_)*(csc[(e_.) + (f_.)*(x_)]*(b_.) + ( a_))^(m_), x_Symbol] :> Simp[b^2*Cot[e + f*x]*(a + b*Csc[e + f*x])^(m + 1)* ((d*Csc[e + f*x])^n/(a*f*(m + 1)*(a^2 - b^2))), x] + Simp[1/(a*(m + 1)*(a^2 - b^2)) Int[(a + b*Csc[e + f*x])^(m + 1)*(d*Csc[e + f*x])^n*(a^2*(m + 1) - b^2*(m + n + 1) - a*b*(m + 1)*Csc[e + f*x] + b^2*(m + n + 2)*Csc[e + f*x ]^2), x], x] /; FreeQ[{a, b, d, e, f, n}, x] && NeQ[a^2 - b^2, 0] && LtQ[m, -1] && IntegersQ[2*m, 2*n]
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[(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/(a*f*(m + 1)*(a^2 - b^2))), x] + Sim p[1/(a*(m + 1)*(a^2 - b^2)) Int[(a + b*Csc[e + f*x])^(m + 1)*(d*Csc[e + f *x])^n*Simp[a*(a*A - b*B + a*C)*(m + 1) - (A*b^2 - a*b*B + a^2*C)*(m + n + 1) - a*(A*b - a*B + b*C)*(m + 1)*Csc[e + f*x] + (A*b^2 - a*b*B + a^2*C)*(m + n + 2)*Csc[e + f*x]^2, x], x], x] /; FreeQ[{a, b, d, e, f, A, B, C, n}, x ] && NeQ[a^2 - b^2, 0] && LtQ[m, -1] && !(ILtQ[m + 1/2, 0] && ILtQ[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[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]
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. \(2215\) vs. \(2(454)=908\).
Time = 31.01 (sec) , antiderivative size = 2216, normalized size of antiderivative = 5.46
-(-(-2*cos(1/2*d*x+1/2*c)^2+1)*sin(1/2*d*x+1/2*c)^2)^(1/2)*(4/3/a^3*(2*sin (1/2*d*x+1/2*c)^4*cos(1/2*d*x+1/2*c)-cos(1/2*d*x+1/2*c)*sin(1/2*d*x+1/2*c) ^2+2*(sin(1/2*d*x+1/2*c)^2)^(1/2)*(2*sin(1/2*d*x+1/2*c)^2-1)^(1/2)*Ellipti cF(cos(1/2*d*x+1/2*c),2^(1/2))-3*(sin(1/2*d*x+1/2*c)^2)^(1/2)*EllipticE(co s(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^4*(2*a+3*b)*(sin(1/2*d*x+1/2*c) ^2)^(1/2)*(-2*cos(1/2*d*x+1/2*c)^2+1)^(1/2)/(-2*sin(1/2*d*x+1/2*c)^4+sin(1 /2*d*x+1/2*c)^2)^(1/2)*(EllipticF(cos(1/2*d*x+1/2*c),2^(1/2))-EllipticE(co s(1/2*d*x+1/2*c),2^(1/2)))+2*(a^2+3*a*b+6*b^2)/a^5*(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^5*b^5*(1/2*a^ 2/b/(a^2-b^2)*cos(1/2*d*x+1/2*c)*(-2*sin(1/2*d*x+1/2*c)^4+sin(1/2*d*x+1/2* c)^2)^(1/2)/(2*a*cos(1/2*d*x+1/2*c)^2-a+b)^2+3/4*a^2*(a^2-3*b^2)/b^2/(a^2- b^2)^2*cos(1/2*d*x+1/2*c)*(-2*sin(1/2*d*x+1/2*c)^4+sin(1/2*d*x+1/2*c)^2)^( 1/2)/(2*a*cos(1/2*d*x+1/2*c)^2-a+b)-3/8/(a+b)/(a^2-b^2)/b^2*(sin(1/2*d*x+1 /2*c)^2)^(1/2)*(-2*cos(1/2*d*x+1/2*c)^2+1)^(1/2)/(-2*sin(1/2*d*x+1/2*c)^4+ sin(1/2*d*x+1/2*c)^2)^(1/2)*EllipticF(cos(1/2*d*x+1/2*c),2^(1/2))*a^2-1/4/ (a+b)/(a^2-b^2)/b*(sin(1/2*d*x+1/2*c)^2)^(1/2)*(-2*cos(1/2*d*x+1/2*c)^2+1) ^(1/2)/(-2*sin(1/2*d*x+1/2*c)^4+sin(1/2*d*x+1/2*c)^2)^(1/2)*EllipticF(cos( 1/2*d*x+1/2*c),2^(1/2))*a+7/8/(a+b)/(a^2-b^2)*(sin(1/2*d*x+1/2*c)^2)^(1...
\[ \int \frac {1}{\sec ^{\frac {3}{2}}(c+d x) (a+b \sec (c+d x))^3} \, dx=\int { \frac {1}{{\left (b \sec \left (d x + c\right ) + a\right )}^{3} \sec \left (d x + c\right )^{\frac {3}{2}}} \,d x } \]
integral(sqrt(sec(d*x + c))/(b^3*sec(d*x + c)^5 + 3*a*b^2*sec(d*x + c)^4 + 3*a^2*b*sec(d*x + c)^3 + a^3*sec(d*x + c)^2), x)
\[ \int \frac {1}{\sec ^{\frac {3}{2}}(c+d x) (a+b \sec (c+d x))^3} \, dx=\int \frac {1}{\left (a + b \sec {\left (c + d x \right )}\right )^{3} \sec ^{\frac {3}{2}}{\left (c + d x \right )}}\, dx \]
Exception generated. \[ \int \frac {1}{\sec ^{\frac {3}{2}}(c+d x) (a+b \sec (c+d x))^3} \, dx=\text {Exception raised: RuntimeError} \]
\[ \int \frac {1}{\sec ^{\frac {3}{2}}(c+d x) (a+b \sec (c+d x))^3} \, dx=\int { \frac {1}{{\left (b \sec \left (d x + c\right ) + a\right )}^{3} \sec \left (d x + c\right )^{\frac {3}{2}}} \,d x } \]
Timed out. \[ \int \frac {1}{\sec ^{\frac {3}{2}}(c+d x) (a+b \sec (c+d x))^3} \, dx=\int \frac {1}{{\left (a+\frac {b}{\cos \left (c+d\,x\right )}\right )}^3\,{\left (\frac {1}{\cos \left (c+d\,x\right )}\right )}^{3/2}} \,d x \]