\(\int \frac {\sec ^4(c+d x)}{(a+b \sin (c+d x))^{3/2}} \, dx\) [523]

Optimal result

Integrand size = 23, antiderivative size = 359 \[ \int \frac {\sec ^4(c+d x)}{(a+b \sin (c+d x))^{3/2}} \, dx=\frac {2 b \sec ^3(c+d x)}{\left (a^2-b^2\right ) d \sqrt {a+b \sin (c+d x)}}-\frac {\left (4 a^4-15 a^2 b^2-21 b^4\right ) E\left (\frac {1}{2} \left (c-\frac {\pi }{2}+d x\right )|\frac {2 b}{a+b}\right ) \sqrt {a+b \sin (c+d x)}}{6 \left (a^2-b^2\right )^3 d \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}+\frac {2 a \left (a^2-3 b^2\right ) \operatorname {EllipticF}\left (\frac {1}{2} \left (c-\frac {\pi }{2}+d x\right ),\frac {2 b}{a+b}\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}{3 \left (a^2-b^2\right )^2 d \sqrt {a+b \sin (c+d x)}}-\frac {\sec ^3(c+d x) \sqrt {a+b \sin (c+d x)} \left (8 a b-\left (a^2+7 b^2\right ) \sin (c+d x)\right )}{3 \left (a^2-b^2\right )^2 d}-\frac {\sec (c+d x) \sqrt {a+b \sin (c+d x)} \left (a b \left (a^2-33 b^2\right )-\left (4 a^4-15 a^2 b^2-21 b^4\right ) \sin (c+d x)\right )}{6 \left (a^2-b^2\right )^3 d} \] Output:

2*b*sec(d*x+c)^3/(a^2-b^2)/d/(a+b*sin(d*x+c))^(1/2)+1/6*(4*a^4-15*a^2*b^2- 
21*b^4)*EllipticE(cos(1/2*c+1/4*Pi+1/2*d*x),2^(1/2)*(b/(a+b))^(1/2))*(a+b* 
sin(d*x+c))^(1/2)/(a^2-b^2)^3/d/((a+b*sin(d*x+c))/(a+b))^(1/2)+2/3*a*(a^2- 
3*b^2)*InverseJacobiAM(1/2*c-1/4*Pi+1/2*d*x,2^(1/2)*(b/(a+b))^(1/2))*((a+b 
*sin(d*x+c))/(a+b))^(1/2)/(a^2-b^2)^2/d/(a+b*sin(d*x+c))^(1/2)-1/3*sec(d*x 
+c)^3*(a+b*sin(d*x+c))^(1/2)*(8*a*b-(a^2+7*b^2)*sin(d*x+c))/(a^2-b^2)^2/d- 
1/6*sec(d*x+c)*(a+b*sin(d*x+c))^(1/2)*(a*b*(a^2-33*b^2)-(4*a^4-15*a^2*b^2- 
21*b^4)*sin(d*x+c))/(a^2-b^2)^3/d
 

Mathematica [A] (verified)

Time = 3.47 (sec) , antiderivative size = 348, normalized size of antiderivative = 0.97 \[ \int \frac {\sec ^4(c+d x)}{(a+b \sin (c+d x))^{3/2}} \, dx=\frac {\left (4 a^5+4 a^4 b-15 a^3 b^2-15 a^2 b^3-21 a b^4-21 b^5\right ) E\left (\frac {1}{4} (-2 c+\pi -2 d x)|\frac {2 b}{a+b}\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}}-4 a \left (a^4-4 a^2 b^2+3 b^4\right ) \operatorname {EllipticF}\left (\frac {1}{4} (-2 c+\pi -2 d x),\frac {2 b}{a+b}\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}}+\frac {1}{8} \sec ^3(c+d x) \left (-24 a^4 b+101 a^2 b^3+19 b^5+\left (-12 a^4 b+84 a^2 b^3+56 b^5\right ) \cos (2 (c+d x))+\left (-4 a^4 b+15 a^2 b^3+21 b^5\right ) \cos (4 (c+d x))+24 a^5 \sin (c+d x)-64 a^3 b^2 \sin (c+d x)+40 a b^4 \sin (c+d x)+8 a^5 \sin (3 (c+d x))-32 a^3 b^2 \sin (3 (c+d x))+24 a b^4 \sin (3 (c+d x))\right )}{6 (a-b)^3 (a+b)^3 d \sqrt {a+b \sin (c+d x)}} \] Input:

Integrate[Sec[c + d*x]^4/(a + b*Sin[c + d*x])^(3/2),x]
 

Output:

((4*a^5 + 4*a^4*b - 15*a^3*b^2 - 15*a^2*b^3 - 21*a*b^4 - 21*b^5)*EllipticE 
[(-2*c + Pi - 2*d*x)/4, (2*b)/(a + b)]*Sqrt[(a + b*Sin[c + d*x])/(a + b)] 
- 4*a*(a^4 - 4*a^2*b^2 + 3*b^4)*EllipticF[(-2*c + Pi - 2*d*x)/4, (2*b)/(a 
+ b)]*Sqrt[(a + b*Sin[c + d*x])/(a + b)] + (Sec[c + d*x]^3*(-24*a^4*b + 10 
1*a^2*b^3 + 19*b^5 + (-12*a^4*b + 84*a^2*b^3 + 56*b^5)*Cos[2*(c + d*x)] + 
(-4*a^4*b + 15*a^2*b^3 + 21*b^5)*Cos[4*(c + d*x)] + 24*a^5*Sin[c + d*x] - 
64*a^3*b^2*Sin[c + d*x] + 40*a*b^4*Sin[c + d*x] + 8*a^5*Sin[3*(c + d*x)] - 
 32*a^3*b^2*Sin[3*(c + d*x)] + 24*a*b^4*Sin[3*(c + d*x)]))/8)/(6*(a - b)^3 
*(a + b)^3*d*Sqrt[a + b*Sin[c + d*x]])
 

Rubi [A] (verified)

Time = 1.89 (sec) , antiderivative size = 384, normalized size of antiderivative = 1.07, number of steps used = 18, number of rules used = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.783, Rules used = {3042, 3173, 27, 3042, 3345, 27, 3042, 3345, 27, 3042, 3231, 3042, 3134, 3042, 3132, 3142, 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.

\(\displaystyle \int \frac {\sec ^4(c+d x)}{(a+b \sin (c+d x))^{3/2}} \, dx\)

\(\Big \downarrow \) 3042

\(\displaystyle \int \frac {1}{\cos (c+d x)^4 (a+b \sin (c+d x))^{3/2}}dx\)

\(\Big \downarrow \) 3173

\(\displaystyle \frac {2 b \sec ^3(c+d x)}{d \left (a^2-b^2\right ) \sqrt {a+b \sin (c+d x)}}-\frac {2 \int -\frac {\sec ^4(c+d x) (a-7 b \sin (c+d x))}{2 \sqrt {a+b \sin (c+d x)}}dx}{a^2-b^2}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\int \frac {\sec ^4(c+d x) (a-7 b \sin (c+d x))}{\sqrt {a+b \sin (c+d x)}}dx}{a^2-b^2}+\frac {2 b \sec ^3(c+d x)}{d \left (a^2-b^2\right ) \sqrt {a+b \sin (c+d x)}}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\int \frac {a-7 b \sin (c+d x)}{\cos (c+d x)^4 \sqrt {a+b \sin (c+d x)}}dx}{a^2-b^2}+\frac {2 b \sec ^3(c+d x)}{d \left (a^2-b^2\right ) \sqrt {a+b \sin (c+d x)}}\)

\(\Big \downarrow \) 3345

\(\displaystyle \frac {-\frac {\int -\frac {\sec ^2(c+d x) \left (4 a \left (a^2-3 b^2\right )+3 b \left (a^2+7 b^2\right ) \sin (c+d x)\right )}{2 \sqrt {a+b \sin (c+d x)}}dx}{3 \left (a^2-b^2\right )}-\frac {\sec ^3(c+d x) \sqrt {a+b \sin (c+d x)} \left (8 a b-\left (a^2+7 b^2\right ) \sin (c+d x)\right )}{3 d \left (a^2-b^2\right )}}{a^2-b^2}+\frac {2 b \sec ^3(c+d x)}{d \left (a^2-b^2\right ) \sqrt {a+b \sin (c+d x)}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\frac {\int \frac {\sec ^2(c+d x) \left (4 a \left (a^2-3 b^2\right )+3 b \left (a^2+7 b^2\right ) \sin (c+d x)\right )}{\sqrt {a+b \sin (c+d x)}}dx}{6 \left (a^2-b^2\right )}-\frac {\sec ^3(c+d x) \sqrt {a+b \sin (c+d x)} \left (8 a b-\left (a^2+7 b^2\right ) \sin (c+d x)\right )}{3 d \left (a^2-b^2\right )}}{a^2-b^2}+\frac {2 b \sec ^3(c+d x)}{d \left (a^2-b^2\right ) \sqrt {a+b \sin (c+d x)}}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\frac {\int \frac {4 a \left (a^2-3 b^2\right )+3 b \left (a^2+7 b^2\right ) \sin (c+d x)}{\cos (c+d x)^2 \sqrt {a+b \sin (c+d x)}}dx}{6 \left (a^2-b^2\right )}-\frac {\sec ^3(c+d x) \sqrt {a+b \sin (c+d x)} \left (8 a b-\left (a^2+7 b^2\right ) \sin (c+d x)\right )}{3 d \left (a^2-b^2\right )}}{a^2-b^2}+\frac {2 b \sec ^3(c+d x)}{d \left (a^2-b^2\right ) \sqrt {a+b \sin (c+d x)}}\)

\(\Big \downarrow \) 3345

\(\displaystyle \frac {\frac {-\frac {\int \frac {a \left (a^2-33 b^2\right ) b^2+\left (4 a^4-15 b^2 a^2-21 b^4\right ) \sin (c+d x) b}{2 \sqrt {a+b \sin (c+d x)}}dx}{a^2-b^2}-\frac {\sec (c+d x) \sqrt {a+b \sin (c+d x)} \left (a b \left (a^2-33 b^2\right )-\left (4 a^4-15 a^2 b^2-21 b^4\right ) \sin (c+d x)\right )}{d \left (a^2-b^2\right )}}{6 \left (a^2-b^2\right )}-\frac {\sec ^3(c+d x) \sqrt {a+b \sin (c+d x)} \left (8 a b-\left (a^2+7 b^2\right ) \sin (c+d x)\right )}{3 d \left (a^2-b^2\right )}}{a^2-b^2}+\frac {2 b \sec ^3(c+d x)}{d \left (a^2-b^2\right ) \sqrt {a+b \sin (c+d x)}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\frac {-\frac {\int \frac {a \left (a^2-33 b^2\right ) b^2+\left (4 a^4-15 b^2 a^2-21 b^4\right ) \sin (c+d x) b}{\sqrt {a+b \sin (c+d x)}}dx}{2 \left (a^2-b^2\right )}-\frac {\sec (c+d x) \sqrt {a+b \sin (c+d x)} \left (a b \left (a^2-33 b^2\right )-\left (4 a^4-15 a^2 b^2-21 b^4\right ) \sin (c+d x)\right )}{d \left (a^2-b^2\right )}}{6 \left (a^2-b^2\right )}-\frac {\sec ^3(c+d x) \sqrt {a+b \sin (c+d x)} \left (8 a b-\left (a^2+7 b^2\right ) \sin (c+d x)\right )}{3 d \left (a^2-b^2\right )}}{a^2-b^2}+\frac {2 b \sec ^3(c+d x)}{d \left (a^2-b^2\right ) \sqrt {a+b \sin (c+d x)}}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\frac {-\frac {\int \frac {a \left (a^2-33 b^2\right ) b^2+\left (4 a^4-15 b^2 a^2-21 b^4\right ) \sin (c+d x) b}{\sqrt {a+b \sin (c+d x)}}dx}{2 \left (a^2-b^2\right )}-\frac {\sec (c+d x) \sqrt {a+b \sin (c+d x)} \left (a b \left (a^2-33 b^2\right )-\left (4 a^4-15 a^2 b^2-21 b^4\right ) \sin (c+d x)\right )}{d \left (a^2-b^2\right )}}{6 \left (a^2-b^2\right )}-\frac {\sec ^3(c+d x) \sqrt {a+b \sin (c+d x)} \left (8 a b-\left (a^2+7 b^2\right ) \sin (c+d x)\right )}{3 d \left (a^2-b^2\right )}}{a^2-b^2}+\frac {2 b \sec ^3(c+d x)}{d \left (a^2-b^2\right ) \sqrt {a+b \sin (c+d x)}}\)

\(\Big \downarrow \) 3231

\(\displaystyle \frac {\frac {-\frac {\left (4 a^4-15 a^2 b^2-21 b^4\right ) \int \sqrt {a+b \sin (c+d x)}dx-4 a \left (a^4-4 a^2 b^2+3 b^4\right ) \int \frac {1}{\sqrt {a+b \sin (c+d x)}}dx}{2 \left (a^2-b^2\right )}-\frac {\sec (c+d x) \sqrt {a+b \sin (c+d x)} \left (a b \left (a^2-33 b^2\right )-\left (4 a^4-15 a^2 b^2-21 b^4\right ) \sin (c+d x)\right )}{d \left (a^2-b^2\right )}}{6 \left (a^2-b^2\right )}-\frac {\sec ^3(c+d x) \sqrt {a+b \sin (c+d x)} \left (8 a b-\left (a^2+7 b^2\right ) \sin (c+d x)\right )}{3 d \left (a^2-b^2\right )}}{a^2-b^2}+\frac {2 b \sec ^3(c+d x)}{d \left (a^2-b^2\right ) \sqrt {a+b \sin (c+d x)}}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\frac {-\frac {\left (4 a^4-15 a^2 b^2-21 b^4\right ) \int \sqrt {a+b \sin (c+d x)}dx-4 a \left (a^4-4 a^2 b^2+3 b^4\right ) \int \frac {1}{\sqrt {a+b \sin (c+d x)}}dx}{2 \left (a^2-b^2\right )}-\frac {\sec (c+d x) \sqrt {a+b \sin (c+d x)} \left (a b \left (a^2-33 b^2\right )-\left (4 a^4-15 a^2 b^2-21 b^4\right ) \sin (c+d x)\right )}{d \left (a^2-b^2\right )}}{6 \left (a^2-b^2\right )}-\frac {\sec ^3(c+d x) \sqrt {a+b \sin (c+d x)} \left (8 a b-\left (a^2+7 b^2\right ) \sin (c+d x)\right )}{3 d \left (a^2-b^2\right )}}{a^2-b^2}+\frac {2 b \sec ^3(c+d x)}{d \left (a^2-b^2\right ) \sqrt {a+b \sin (c+d x)}}\)

\(\Big \downarrow \) 3134

\(\displaystyle \frac {\frac {-\frac {\frac {\left (4 a^4-15 a^2 b^2-21 b^4\right ) \sqrt {a+b \sin (c+d x)} \int \sqrt {\frac {a}{a+b}+\frac {b \sin (c+d x)}{a+b}}dx}{\sqrt {\frac {a+b \sin (c+d x)}{a+b}}}-4 a \left (a^4-4 a^2 b^2+3 b^4\right ) \int \frac {1}{\sqrt {a+b \sin (c+d x)}}dx}{2 \left (a^2-b^2\right )}-\frac {\sec (c+d x) \sqrt {a+b \sin (c+d x)} \left (a b \left (a^2-33 b^2\right )-\left (4 a^4-15 a^2 b^2-21 b^4\right ) \sin (c+d x)\right )}{d \left (a^2-b^2\right )}}{6 \left (a^2-b^2\right )}-\frac {\sec ^3(c+d x) \sqrt {a+b \sin (c+d x)} \left (8 a b-\left (a^2+7 b^2\right ) \sin (c+d x)\right )}{3 d \left (a^2-b^2\right )}}{a^2-b^2}+\frac {2 b \sec ^3(c+d x)}{d \left (a^2-b^2\right ) \sqrt {a+b \sin (c+d x)}}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\frac {-\frac {\frac {\left (4 a^4-15 a^2 b^2-21 b^4\right ) \sqrt {a+b \sin (c+d x)} \int \sqrt {\frac {a}{a+b}+\frac {b \sin (c+d x)}{a+b}}dx}{\sqrt {\frac {a+b \sin (c+d x)}{a+b}}}-4 a \left (a^4-4 a^2 b^2+3 b^4\right ) \int \frac {1}{\sqrt {a+b \sin (c+d x)}}dx}{2 \left (a^2-b^2\right )}-\frac {\sec (c+d x) \sqrt {a+b \sin (c+d x)} \left (a b \left (a^2-33 b^2\right )-\left (4 a^4-15 a^2 b^2-21 b^4\right ) \sin (c+d x)\right )}{d \left (a^2-b^2\right )}}{6 \left (a^2-b^2\right )}-\frac {\sec ^3(c+d x) \sqrt {a+b \sin (c+d x)} \left (8 a b-\left (a^2+7 b^2\right ) \sin (c+d x)\right )}{3 d \left (a^2-b^2\right )}}{a^2-b^2}+\frac {2 b \sec ^3(c+d x)}{d \left (a^2-b^2\right ) \sqrt {a+b \sin (c+d x)}}\)

\(\Big \downarrow \) 3132

\(\displaystyle \frac {\frac {-\frac {\frac {2 \left (4 a^4-15 a^2 b^2-21 b^4\right ) \sqrt {a+b \sin (c+d x)} E\left (\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right )|\frac {2 b}{a+b}\right )}{d \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}-4 a \left (a^4-4 a^2 b^2+3 b^4\right ) \int \frac {1}{\sqrt {a+b \sin (c+d x)}}dx}{2 \left (a^2-b^2\right )}-\frac {\sec (c+d x) \sqrt {a+b \sin (c+d x)} \left (a b \left (a^2-33 b^2\right )-\left (4 a^4-15 a^2 b^2-21 b^4\right ) \sin (c+d x)\right )}{d \left (a^2-b^2\right )}}{6 \left (a^2-b^2\right )}-\frac {\sec ^3(c+d x) \sqrt {a+b \sin (c+d x)} \left (8 a b-\left (a^2+7 b^2\right ) \sin (c+d x)\right )}{3 d \left (a^2-b^2\right )}}{a^2-b^2}+\frac {2 b \sec ^3(c+d x)}{d \left (a^2-b^2\right ) \sqrt {a+b \sin (c+d x)}}\)

\(\Big \downarrow \) 3142

\(\displaystyle \frac {\frac {-\frac {\frac {2 \left (4 a^4-15 a^2 b^2-21 b^4\right ) \sqrt {a+b \sin (c+d x)} E\left (\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right )|\frac {2 b}{a+b}\right )}{d \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}-\frac {4 a \left (a^4-4 a^2 b^2+3 b^4\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}} \int \frac {1}{\sqrt {\frac {a}{a+b}+\frac {b \sin (c+d x)}{a+b}}}dx}{\sqrt {a+b \sin (c+d x)}}}{2 \left (a^2-b^2\right )}-\frac {\sec (c+d x) \sqrt {a+b \sin (c+d x)} \left (a b \left (a^2-33 b^2\right )-\left (4 a^4-15 a^2 b^2-21 b^4\right ) \sin (c+d x)\right )}{d \left (a^2-b^2\right )}}{6 \left (a^2-b^2\right )}-\frac {\sec ^3(c+d x) \sqrt {a+b \sin (c+d x)} \left (8 a b-\left (a^2+7 b^2\right ) \sin (c+d x)\right )}{3 d \left (a^2-b^2\right )}}{a^2-b^2}+\frac {2 b \sec ^3(c+d x)}{d \left (a^2-b^2\right ) \sqrt {a+b \sin (c+d x)}}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\frac {-\frac {\frac {2 \left (4 a^4-15 a^2 b^2-21 b^4\right ) \sqrt {a+b \sin (c+d x)} E\left (\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right )|\frac {2 b}{a+b}\right )}{d \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}-\frac {4 a \left (a^4-4 a^2 b^2+3 b^4\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}} \int \frac {1}{\sqrt {\frac {a}{a+b}+\frac {b \sin (c+d x)}{a+b}}}dx}{\sqrt {a+b \sin (c+d x)}}}{2 \left (a^2-b^2\right )}-\frac {\sec (c+d x) \sqrt {a+b \sin (c+d x)} \left (a b \left (a^2-33 b^2\right )-\left (4 a^4-15 a^2 b^2-21 b^4\right ) \sin (c+d x)\right )}{d \left (a^2-b^2\right )}}{6 \left (a^2-b^2\right )}-\frac {\sec ^3(c+d x) \sqrt {a+b \sin (c+d x)} \left (8 a b-\left (a^2+7 b^2\right ) \sin (c+d x)\right )}{3 d \left (a^2-b^2\right )}}{a^2-b^2}+\frac {2 b \sec ^3(c+d x)}{d \left (a^2-b^2\right ) \sqrt {a+b \sin (c+d x)}}\)

\(\Big \downarrow \) 3140

\(\displaystyle \frac {2 b \sec ^3(c+d x)}{d \left (a^2-b^2\right ) \sqrt {a+b \sin (c+d x)}}+\frac {\frac {-\frac {\sec (c+d x) \sqrt {a+b \sin (c+d x)} \left (a b \left (a^2-33 b^2\right )-\left (4 a^4-15 a^2 b^2-21 b^4\right ) \sin (c+d x)\right )}{d \left (a^2-b^2\right )}-\frac {\frac {2 \left (4 a^4-15 a^2 b^2-21 b^4\right ) \sqrt {a+b \sin (c+d x)} E\left (\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right )|\frac {2 b}{a+b}\right )}{d \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}-\frac {8 a \left (a^4-4 a^2 b^2+3 b^4\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}} \operatorname {EllipticF}\left (\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right ),\frac {2 b}{a+b}\right )}{d \sqrt {a+b \sin (c+d x)}}}{2 \left (a^2-b^2\right )}}{6 \left (a^2-b^2\right )}-\frac {\sec ^3(c+d x) \sqrt {a+b \sin (c+d x)} \left (8 a b-\left (a^2+7 b^2\right ) \sin (c+d x)\right )}{3 d \left (a^2-b^2\right )}}{a^2-b^2}\)

Input:

Int[Sec[c + d*x]^4/(a + b*Sin[c + d*x])^(3/2),x]
 

Output:

(2*b*Sec[c + d*x]^3)/((a^2 - b^2)*d*Sqrt[a + b*Sin[c + d*x]]) + (-1/3*(Sec 
[c + d*x]^3*Sqrt[a + b*Sin[c + d*x]]*(8*a*b - (a^2 + 7*b^2)*Sin[c + d*x])) 
/((a^2 - b^2)*d) + (-((Sec[c + d*x]*Sqrt[a + b*Sin[c + d*x]]*(a*b*(a^2 - 3 
3*b^2) - (4*a^4 - 15*a^2*b^2 - 21*b^4)*Sin[c + d*x]))/((a^2 - b^2)*d)) - ( 
(2*(4*a^4 - 15*a^2*b^2 - 21*b^4)*EllipticE[(c - Pi/2 + d*x)/2, (2*b)/(a + 
b)]*Sqrt[a + b*Sin[c + d*x]])/(d*Sqrt[(a + b*Sin[c + d*x])/(a + b)]) - (8* 
a*(a^4 - 4*a^2*b^2 + 3*b^4)*EllipticF[(c - Pi/2 + d*x)/2, (2*b)/(a + b)]*S 
qrt[(a + b*Sin[c + d*x])/(a + b)])/(d*Sqrt[a + b*Sin[c + d*x]]))/(2*(a^2 - 
 b^2)))/(6*(a^2 - b^2)))/(a^2 - b^2)
 

Defintions of rubi rules used

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 3042

Int[u_, x_Symbol] :> Int[DeactivateTrig[u, x], x] /; FunctionOfTrigOfLinear 
Q[u, x]
 

rule 3132

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]
 

rule 3134

Int[Sqrt[(a_) + (b_.)*sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[Sqrt[a + 
b*Sin[c + d*x]]/Sqrt[(a + b*Sin[c + d*x])/(a + b)]   Int[Sqrt[a/(a + b) + ( 
b/(a + b))*Sin[c + d*x]], x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[a^2 - b^2 
, 0] &&  !GtQ[a + b, 0]
 

rule 3140

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]
 

rule 3142

Int[1/Sqrt[(a_) + (b_.)*sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[Sqrt[(a 
 + b*Sin[c + d*x])/(a + b)]/Sqrt[a + b*Sin[c + d*x]]   Int[1/Sqrt[a/(a + b) 
 + (b/(a + b))*Sin[c + d*x]], x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[a^2 - 
 b^2, 0] &&  !GtQ[a + b, 0]
 

rule 3173

Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x 
_)])^(m_), x_Symbol] :> Simp[(-b)*(g*Cos[e + f*x])^(p + 1)*((a + b*Sin[e + 
f*x])^(m + 1)/(f*g*(a^2 - b^2)*(m + 1))), x] + Simp[1/((a^2 - b^2)*(m + 1)) 
   Int[(g*Cos[e + f*x])^p*(a + b*Sin[e + f*x])^(m + 1)*(a*(m + 1) - b*(m + 
p + 2)*Sin[e + f*x]), x], x] /; FreeQ[{a, b, e, f, g, p}, x] && NeQ[a^2 - b 
^2, 0] && LtQ[m, -1] && IntegersQ[2*m, 2*p]
 

rule 3231

Int[((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])/Sqrt[(a_) + (b_.)*sin[(e_.) + ( 
f_.)*(x_)]], x_Symbol] :> Simp[(b*c - a*d)/b   Int[1/Sqrt[a + b*Sin[e + f*x 
]], x], x] + Simp[d/b   Int[Sqrt[a + b*Sin[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]
 

rule 3345

Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x 
_)])^(m_.)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[(g*Co 
s[e + f*x])^(p + 1)*(a + b*Sin[e + f*x])^(m + 1)*((b*c - a*d - (a*c - b*d)* 
Sin[e + f*x])/(f*g*(a^2 - b^2)*(p + 1))), x] + Simp[1/(g^2*(a^2 - b^2)*(p + 
 1))   Int[(g*Cos[e + f*x])^(p + 2)*(a + b*Sin[e + f*x])^m*Simp[c*(a^2*(p + 
 2) - b^2*(m + p + 2)) + a*b*d*m + b*(a*c - b*d)*(m + p + 3)*Sin[e + f*x], 
x], x], x] /; FreeQ[{a, b, c, d, e, f, g, m}, x] && NeQ[a^2 - b^2, 0] && Lt 
Q[p, -1] && IntegerQ[2*m]
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(1645\) vs. \(2(342)=684\).

Time = 1.57 (sec) , antiderivative size = 1646, normalized size of antiderivative = 4.58

method	result	size
default	\(\text {Expression too large to display}\)	\(1646\)

Input:

int(sec(d*x+c)^4/(a+b*sin(d*x+c))^(3/2),x,method=_RETURNVERBOSE)
 

Output:

1/6*(-(-a-b*sin(d*x+c))*cos(d*x+c)^2)^(1/2)/cos(d*x+c)^5/b/(a+b*sin(d*x+c) 
)^(3/2)/(a^6-3*a^4*b^2+3*a^2*b^4-b^6)*(-2*(cos(d*x+c)^2*sin(d*x+c)*b+cos(d 
*x+c)^2*a)^(1/2)*b^2*(a^4-2*a^2*b^2+b^4)-cos(d*x+c)^4*(cos(d*x+c)^2*sin(d* 
x+c)*b+cos(d*x+c)^2*a)^(1/2)*b^2*(4*a^4-15*a^2*b^2-21*b^4)+4*cos(d*x+c)^2* 
(cos(d*x+c)^2*sin(d*x+c)*b+cos(d*x+c)^2*a)^(1/2)*a*b*(a^4-4*a^2*b^2+3*b^4) 
*sin(d*x+c)+2*(cos(d*x+c)^2*sin(d*x+c)*b+cos(d*x+c)^2*a)^(1/2)*a*b*(a^4-2* 
a^2*b^2+b^4)*sin(d*x+c)+cos(d*x+c)^2*(cos(d*x+c)^2*sin(d*x+c)*b+cos(d*x+c) 
^2*a)^(1/2)*(4*(b/(a-b)*sin(d*x+c)+1/(a-b)*a)^(1/2)*(-b/(a+b)*sin(d*x+c)+b 
/(a+b))^(1/2)*(-b/(a-b)*sin(d*x+c)-b/(a-b))^(1/2)*EllipticE((b/(a-b)*sin(d 
*x+c)+1/(a-b)*a)^(1/2),((a-b)/(a+b))^(1/2))*a^6-19*(b/(a-b)*sin(d*x+c)+1/( 
a-b)*a)^(1/2)*(-b/(a+b)*sin(d*x+c)+b/(a+b))^(1/2)*(-b/(a-b)*sin(d*x+c)-b/( 
a-b))^(1/2)*EllipticE((b/(a-b)*sin(d*x+c)+1/(a-b)*a)^(1/2),((a-b)/(a+b))^( 
1/2))*a^4*b^2-6*(b/(a-b)*sin(d*x+c)+1/(a-b)*a)^(1/2)*(-b/(a+b)*sin(d*x+c)+ 
b/(a+b))^(1/2)*(-b/(a-b)*sin(d*x+c)-b/(a-b))^(1/2)*EllipticE((b/(a-b)*sin( 
d*x+c)+1/(a-b)*a)^(1/2),((a-b)/(a+b))^(1/2))*a^2*b^4+21*(b/(a-b)*sin(d*x+c 
)+1/(a-b)*a)^(1/2)*(-b/(a+b)*sin(d*x+c)+b/(a+b))^(1/2)*(-b/(a-b)*sin(d*x+c 
)-b/(a-b))^(1/2)*EllipticE((b/(a-b)*sin(d*x+c)+1/(a-b)*a)^(1/2),((a-b)/(a+ 
b))^(1/2))*b^6-4*(b/(a-b)*sin(d*x+c)+1/(a-b)*a)^(1/2)*(-b/(a+b)*sin(d*x+c) 
+b/(a+b))^(1/2)*(-b/(a-b)*sin(d*x+c)-b/(a-b))^(1/2)*EllipticF((b/(a-b)*sin 
(d*x+c)+1/(a-b)*a)^(1/2),((a-b)/(a+b))^(1/2))*a^5*b+3*(b/(a-b)*sin(d*x+...
 

Fricas [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.19 (sec) , antiderivative size = 854, normalized size of antiderivative = 2.38 \[ \int \frac {\sec ^4(c+d x)}{(a+b \sin (c+d x))^{3/2}} \, dx=\text {Too large to display} \] Input:

integrate(sec(d*x+c)^4/(a+b*sin(d*x+c))^(3/2),x, algorithm="fricas")
 

Output:

1/18*(((8*a^5*b - 33*a^3*b^3 + 57*a*b^5)*cos(d*x + c)^3*sin(d*x + c) + (8* 
a^6 - 33*a^4*b^2 + 57*a^2*b^4)*cos(d*x + c)^3)*sqrt(1/2*I*b)*weierstrassPI 
nverse(-4/3*(4*a^2 - 3*b^2)/b^2, -8/27*(8*I*a^3 - 9*I*a*b^2)/b^3, 1/3*(3*b 
*cos(d*x + c) - 3*I*b*sin(d*x + c) - 2*I*a)/b) + ((8*a^5*b - 33*a^3*b^3 + 
57*a*b^5)*cos(d*x + c)^3*sin(d*x + c) + (8*a^6 - 33*a^4*b^2 + 57*a^2*b^4)* 
cos(d*x + c)^3)*sqrt(-1/2*I*b)*weierstrassPInverse(-4/3*(4*a^2 - 3*b^2)/b^ 
2, -8/27*(-8*I*a^3 + 9*I*a*b^2)/b^3, 1/3*(3*b*cos(d*x + c) + 3*I*b*sin(d*x 
 + c) + 2*I*a)/b) + 3*((4*I*a^4*b^2 - 15*I*a^2*b^4 - 21*I*b^6)*cos(d*x + c 
)^3*sin(d*x + c) + (4*I*a^5*b - 15*I*a^3*b^3 - 21*I*a*b^5)*cos(d*x + c)^3) 
*sqrt(1/2*I*b)*weierstrassZeta(-4/3*(4*a^2 - 3*b^2)/b^2, -8/27*(8*I*a^3 - 
9*I*a*b^2)/b^3, weierstrassPInverse(-4/3*(4*a^2 - 3*b^2)/b^2, -8/27*(8*I*a 
^3 - 9*I*a*b^2)/b^3, 1/3*(3*b*cos(d*x + c) - 3*I*b*sin(d*x + c) - 2*I*a)/b 
)) + 3*((-4*I*a^4*b^2 + 15*I*a^2*b^4 + 21*I*b^6)*cos(d*x + c)^3*sin(d*x + 
c) + (-4*I*a^5*b + 15*I*a^3*b^3 + 21*I*a*b^5)*cos(d*x + c)^3)*sqrt(-1/2*I* 
b)*weierstrassZeta(-4/3*(4*a^2 - 3*b^2)/b^2, -8/27*(-8*I*a^3 + 9*I*a*b^2)/ 
b^3, weierstrassPInverse(-4/3*(4*a^2 - 3*b^2)/b^2, -8/27*(-8*I*a^3 + 9*I*a 
*b^2)/b^3, 1/3*(3*b*cos(d*x + c) + 3*I*b*sin(d*x + c) + 2*I*a)/b)) - 3*(2* 
a^4*b^2 - 4*a^2*b^4 + 2*b^6 + (4*a^4*b^2 - 15*a^2*b^4 - 21*b^6)*cos(d*x + 
c)^4 - (a^4*b^2 + 6*a^2*b^4 - 7*b^6)*cos(d*x + c)^2 - 2*(a^5*b - 2*a^3*b^3 
 + a*b^5 + 2*(a^5*b - 4*a^3*b^3 + 3*a*b^5)*cos(d*x + c)^2)*sin(d*x + c)...
 

Sympy [F]

\[ \int \frac {\sec ^4(c+d x)}{(a+b \sin (c+d x))^{3/2}} \, dx=\int \frac {\sec ^{4}{\left (c + d x \right )}}{\left (a + b \sin {\left (c + d x \right )}\right )^{\frac {3}{2}}}\, dx \] Input:

integrate(sec(d*x+c)**4/(a+b*sin(d*x+c))**(3/2),x)
 

Output:

Integral(sec(c + d*x)**4/(a + b*sin(c + d*x))**(3/2), x)
                                                                                    
                                                                                    
 

Maxima [F]

\[ \int \frac {\sec ^4(c+d x)}{(a+b \sin (c+d x))^{3/2}} \, dx=\int { \frac {\sec \left (d x + c\right )^{4}}{{\left (b \sin \left (d x + c\right ) + a\right )}^{\frac {3}{2}}} \,d x } \] Input:

integrate(sec(d*x+c)^4/(a+b*sin(d*x+c))^(3/2),x, algorithm="maxima")
 

Output:

integrate(sec(d*x + c)^4/(b*sin(d*x + c) + a)^(3/2), x)
 

Giac [F]

\[ \int \frac {\sec ^4(c+d x)}{(a+b \sin (c+d x))^{3/2}} \, dx=\int { \frac {\sec \left (d x + c\right )^{4}}{{\left (b \sin \left (d x + c\right ) + a\right )}^{\frac {3}{2}}} \,d x } \] Input:

integrate(sec(d*x+c)^4/(a+b*sin(d*x+c))^(3/2),x, algorithm="giac")
 

Output:

integrate(sec(d*x + c)^4/(b*sin(d*x + c) + a)^(3/2), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {\sec ^4(c+d x)}{(a+b \sin (c+d x))^{3/2}} \, dx=\int \frac {1}{{\cos \left (c+d\,x\right )}^4\,{\left (a+b\,\sin \left (c+d\,x\right )\right )}^{3/2}} \,d x \] Input:

int(1/(cos(c + d*x)^4*(a + b*sin(c + d*x))^(3/2)),x)
 

Output:

int(1/(cos(c + d*x)^4*(a + b*sin(c + d*x))^(3/2)), x)
 

Reduce [F]

\[ \int \frac {\sec ^4(c+d x)}{(a+b \sin (c+d x))^{3/2}} \, dx=\int \frac {\sqrt {\sin \left (d x +c \right ) b +a}\, \sec \left (d x +c \right )^{4}}{\sin \left (d x +c \right )^{2} b^{2}+2 \sin \left (d x +c \right ) a b +a^{2}}d x \] Input:

int(sec(d*x+c)^4/(a+b*sin(d*x+c))^(3/2),x)
 

Output:

int((sqrt(sin(c + d*x)*b + a)*sec(c + d*x)**4)/(sin(c + d*x)**2*b**2 + 2*s 
in(c + d*x)*a*b + a**2),x)