Integrand size = 23, antiderivative size = 251 \[ \int \frac {\sec ^2(c+d x)}{(a+b \sin (c+d x))^{3/2}} \, dx=\frac {2 b \sec (c+d x)}{\left (a^2-b^2\right ) d \sqrt {a+b \sin (c+d x)}}-\frac {\left (a^2+3 b^2\right ) E\left (\frac {1}{2} \left (c-\frac {\pi }{2}+d x\right )|\frac {2 b}{a+b}\right ) \sqrt {a+b \sin (c+d x)}}{\left (a^2-b^2\right )^2 d \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}+\frac {a \operatorname {EllipticF}\left (\frac {1}{2} \left (c-\frac {\pi }{2}+d x\right ),\frac {2 b}{a+b}\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}{\left (a^2-b^2\right ) d \sqrt {a+b \sin (c+d x)}}-\frac {\sec (c+d x) \sqrt {a+b \sin (c+d x)} \left (4 a b-\left (a^2+3 b^2\right ) \sin (c+d x)\right )}{\left (a^2-b^2\right )^2 d} \]
2*b*sec(d*x+c)/(a^2-b^2)/d/(a+b*sin(d*x+c))^(1/2)-sec(d*x+c)*(4*a*b-(a^2+3 *b^2)*sin(d*x+c))*(a+b*sin(d*x+c))^(1/2)/(a^2-b^2)^2/d+(a^2+3*b^2)*(sin(1/ 2*c+1/4*Pi+1/2*d*x)^2)^(1/2)/sin(1/2*c+1/4*Pi+1/2*d*x)*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) ^2/d/((a+b*sin(d*x+c))/(a+b))^(1/2)-a*(sin(1/2*c+1/4*Pi+1/2*d*x)^2)^(1/2)/ sin(1/2*c+1/4*Pi+1/2*d*x)*EllipticF(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))/(a+b))^(1/2)/(a^2-b^2)/d/(a+b*sin(d*x+c))^( 1/2)
Time = 1.22 (sec) , antiderivative size = 205, normalized size of antiderivative = 0.82 \[ \int \frac {\sec ^2(c+d x)}{(a+b \sin (c+d x))^{3/2}} \, dx=\frac {\left (a^3+a^2 b+3 a b^2+3 b^3\right ) E\left (\frac {1}{4} (-2 c+\pi -2 d x)|\frac {2 b}{a+b}\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}}-a \left (a^2-b^2\right ) \operatorname {EllipticF}\left (\frac {1}{4} (-2 c+\pi -2 d x),\frac {2 b}{a+b}\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}}-\frac {1}{2} \sec (c+d x) \left (3 a^2 b+b^3+b \left (a^2+3 b^2\right ) \cos (2 (c+d x))-2 a \left (a^2-b^2\right ) \sin (c+d x)\right )}{(a-b)^2 (a+b)^2 d \sqrt {a+b \sin (c+d x)}} \]
((a^3 + a^2*b + 3*a*b^2 + 3*b^3)*EllipticE[(-2*c + Pi - 2*d*x)/4, (2*b)/(a + b)]*Sqrt[(a + b*Sin[c + d*x])/(a + b)] - a*(a^2 - b^2)*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*a^2*b + b^3 + b*(a^2 + 3*b^2)*Cos[2*(c + d*x)] - 2*a*(a^2 - b^ 2)*Sin[c + d*x]))/2)/((a - b)^2*(a + b)^2*d*Sqrt[a + b*Sin[c + d*x]])
Time = 1.26 (sec) , antiderivative size = 268, normalized size of antiderivative = 1.07, number of steps used = 15, number of rules used = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.652, Rules used = {3042, 3173, 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.
| \(\displaystyle \int \frac {\sec ^2(c+d x)}{(a+b \sin (c+d x))^{3/2}} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {1}{\cos (c+d x)^2 (a+b \sin (c+d x))^{3/2}}dx\) |
| \(\Big \downarrow \) 3173 |
| \(\displaystyle \frac {2 b \sec (c+d x)}{d \left (a^2-b^2\right ) \sqrt {a+b \sin (c+d x)}}-\frac {2 \int -\frac {\sec ^2(c+d x) (a-3 b \sin (c+d x))}{2 \sqrt {a+b \sin (c+d x)}}dx}{a^2-b^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int \frac {\sec ^2(c+d x) (a-3 b \sin (c+d x))}{\sqrt {a+b \sin (c+d x)}}dx}{a^2-b^2}+\frac {2 b \sec (c+d x)}{d \left (a^2-b^2\right ) \sqrt {a+b \sin (c+d x)}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\int \frac {a-3 b \sin (c+d x)}{\cos (c+d x)^2 \sqrt {a+b \sin (c+d x)}}dx}{a^2-b^2}+\frac {2 b \sec (c+d x)}{d \left (a^2-b^2\right ) \sqrt {a+b \sin (c+d x)}}\) |
| \(\Big \downarrow \) 3345 |
| \(\displaystyle \frac {-\frac {\int \frac {4 a b^2+\left (a^2+3 b^2\right ) \sin (c+d x) b}{2 \sqrt {a+b \sin (c+d x)}}dx}{a^2-b^2}-\frac {\sec (c+d x) \sqrt {a+b \sin (c+d x)} \left (4 a b-\left (a^2+3 b^2\right ) \sin (c+d x)\right )}{d \left (a^2-b^2\right )}}{a^2-b^2}+\frac {2 b \sec (c+d x)}{d \left (a^2-b^2\right ) \sqrt {a+b \sin (c+d x)}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {-\frac {\int \frac {4 a b^2+\left (a^2+3 b^2\right ) \sin (c+d x) b}{\sqrt {a+b \sin (c+d x)}}dx}{2 \left (a^2-b^2\right )}-\frac {\sec (c+d x) \sqrt {a+b \sin (c+d x)} \left (4 a b-\left (a^2+3 b^2\right ) \sin (c+d x)\right )}{d \left (a^2-b^2\right )}}{a^2-b^2}+\frac {2 b \sec (c+d x)}{d \left (a^2-b^2\right ) \sqrt {a+b \sin (c+d x)}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {-\frac {\int \frac {4 a b^2+\left (a^2+3 b^2\right ) \sin (c+d x) b}{\sqrt {a+b \sin (c+d x)}}dx}{2 \left (a^2-b^2\right )}-\frac {\sec (c+d x) \sqrt {a+b \sin (c+d x)} \left (4 a b-\left (a^2+3 b^2\right ) \sin (c+d x)\right )}{d \left (a^2-b^2\right )}}{a^2-b^2}+\frac {2 b \sec (c+d x)}{d \left (a^2-b^2\right ) \sqrt {a+b \sin (c+d x)}}\) |
| \(\Big \downarrow \) 3231 |
| \(\displaystyle \frac {-\frac {\left (a^2+3 b^2\right ) \int \sqrt {a+b \sin (c+d x)}dx-a \left (a^2-b^2\right ) \int \frac {1}{\sqrt {a+b \sin (c+d x)}}dx}{2 \left (a^2-b^2\right )}-\frac {\sec (c+d x) \sqrt {a+b \sin (c+d x)} \left (4 a b-\left (a^2+3 b^2\right ) \sin (c+d x)\right )}{d \left (a^2-b^2\right )}}{a^2-b^2}+\frac {2 b \sec (c+d x)}{d \left (a^2-b^2\right ) \sqrt {a+b \sin (c+d x)}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {-\frac {\left (a^2+3 b^2\right ) \int \sqrt {a+b \sin (c+d x)}dx-a \left (a^2-b^2\right ) \int \frac {1}{\sqrt {a+b \sin (c+d x)}}dx}{2 \left (a^2-b^2\right )}-\frac {\sec (c+d x) \sqrt {a+b \sin (c+d x)} \left (4 a b-\left (a^2+3 b^2\right ) \sin (c+d x)\right )}{d \left (a^2-b^2\right )}}{a^2-b^2}+\frac {2 b \sec (c+d x)}{d \left (a^2-b^2\right ) \sqrt {a+b \sin (c+d x)}}\) |
| \(\Big \downarrow \) 3134 |
| \(\displaystyle \frac {-\frac {\frac {\left (a^2+3 b^2\right ) \sqrt {a+b \sin (c+d x)} \int \sqrt {\frac {a}{a+b}+\frac {b \sin (c+d x)}{a+b}}dx}{\sqrt {\frac {a+b \sin (c+d x)}{a+b}}}-a \left (a^2-b^2\right ) \int \frac {1}{\sqrt {a+b \sin (c+d x)}}dx}{2 \left (a^2-b^2\right )}-\frac {\sec (c+d x) \sqrt {a+b \sin (c+d x)} \left (4 a b-\left (a^2+3 b^2\right ) \sin (c+d x)\right )}{d \left (a^2-b^2\right )}}{a^2-b^2}+\frac {2 b \sec (c+d x)}{d \left (a^2-b^2\right ) \sqrt {a+b \sin (c+d x)}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {-\frac {\frac {\left (a^2+3 b^2\right ) \sqrt {a+b \sin (c+d x)} \int \sqrt {\frac {a}{a+b}+\frac {b \sin (c+d x)}{a+b}}dx}{\sqrt {\frac {a+b \sin (c+d x)}{a+b}}}-a \left (a^2-b^2\right ) \int \frac {1}{\sqrt {a+b \sin (c+d x)}}dx}{2 \left (a^2-b^2\right )}-\frac {\sec (c+d x) \sqrt {a+b \sin (c+d x)} \left (4 a b-\left (a^2+3 b^2\right ) \sin (c+d x)\right )}{d \left (a^2-b^2\right )}}{a^2-b^2}+\frac {2 b \sec (c+d x)}{d \left (a^2-b^2\right ) \sqrt {a+b \sin (c+d x)}}\) |
| \(\Big \downarrow \) 3132 |
| \(\displaystyle \frac {-\frac {\frac {2 \left (a^2+3 b^2\right ) \sqrt {a+b \sin (c+d x)} E\left (\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right )|\frac {2 b}{a+b}\right )}{d \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}-a \left (a^2-b^2\right ) \int \frac {1}{\sqrt {a+b \sin (c+d x)}}dx}{2 \left (a^2-b^2\right )}-\frac {\sec (c+d x) \sqrt {a+b \sin (c+d x)} \left (4 a b-\left (a^2+3 b^2\right ) \sin (c+d x)\right )}{d \left (a^2-b^2\right )}}{a^2-b^2}+\frac {2 b \sec (c+d x)}{d \left (a^2-b^2\right ) \sqrt {a+b \sin (c+d x)}}\) |
| \(\Big \downarrow \) 3142 |
| \(\displaystyle \frac {-\frac {\frac {2 \left (a^2+3 b^2\right ) \sqrt {a+b \sin (c+d x)} E\left (\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right )|\frac {2 b}{a+b}\right )}{d \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}-\frac {a \left (a^2-b^2\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}} \int \frac {1}{\sqrt {\frac {a}{a+b}+\frac {b \sin (c+d x)}{a+b}}}dx}{\sqrt {a+b \sin (c+d x)}}}{2 \left (a^2-b^2\right )}-\frac {\sec (c+d x) \sqrt {a+b \sin (c+d x)} \left (4 a b-\left (a^2+3 b^2\right ) \sin (c+d x)\right )}{d \left (a^2-b^2\right )}}{a^2-b^2}+\frac {2 b \sec (c+d x)}{d \left (a^2-b^2\right ) \sqrt {a+b \sin (c+d x)}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {-\frac {\frac {2 \left (a^2+3 b^2\right ) \sqrt {a+b \sin (c+d x)} E\left (\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right )|\frac {2 b}{a+b}\right )}{d \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}-\frac {a \left (a^2-b^2\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}} \int \frac {1}{\sqrt {\frac {a}{a+b}+\frac {b \sin (c+d x)}{a+b}}}dx}{\sqrt {a+b \sin (c+d x)}}}{2 \left (a^2-b^2\right )}-\frac {\sec (c+d x) \sqrt {a+b \sin (c+d x)} \left (4 a b-\left (a^2+3 b^2\right ) \sin (c+d x)\right )}{d \left (a^2-b^2\right )}}{a^2-b^2}+\frac {2 b \sec (c+d x)}{d \left (a^2-b^2\right ) \sqrt {a+b \sin (c+d x)}}\) |
| \(\Big \downarrow \) 3140 |
| \(\displaystyle \frac {2 b \sec (c+d x)}{d \left (a^2-b^2\right ) \sqrt {a+b \sin (c+d x)}}+\frac {-\frac {\sec (c+d x) \sqrt {a+b \sin (c+d x)} \left (4 a b-\left (a^2+3 b^2\right ) \sin (c+d x)\right )}{d \left (a^2-b^2\right )}-\frac {\frac {2 \left (a^2+3 b^2\right ) \sqrt {a+b \sin (c+d x)} E\left (\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right )|\frac {2 b}{a+b}\right )}{d \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}-\frac {2 a \left (a^2-b^2\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}} \operatorname {EllipticF}\left (\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right ),\frac {2 b}{a+b}\right )}{d \sqrt {a+b \sin (c+d x)}}}{2 \left (a^2-b^2\right )}}{a^2-b^2}\) |
(2*b*Sec[c + d*x])/((a^2 - b^2)*d*Sqrt[a + b*Sin[c + d*x]]) + (-((Sec[c + d*x]*Sqrt[a + b*Sin[c + d*x]]*(4*a*b - (a^2 + 3*b^2)*Sin[c + d*x]))/((a^2 - b^2)*d)) - ((2*(a^2 + 3*b^2)*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)]) - (2*a* (a^2 - b^2)*EllipticF[(c - Pi/2 + d*x)/2, (2*b)/(a + b)]*Sqrt[(a + b*Sin[c + d*x])/(a + b)])/(d*Sqrt[a + b*Sin[c + d*x]]))/(2*(a^2 - b^2)))/(a^2 - b ^2)
3.6.25.3.1 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[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]
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[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]
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]
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]
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]
Leaf count of result is larger than twice the leaf count of optimal. \(1063\) vs. \(2(305)=610\).
Time = 1.56 (sec) , antiderivative size = 1064, normalized size of antiderivative = 4.24
-1/b*(b*cos(d*x+c)^2*sin(d*x+c)+a*cos(d*x+c)^2)^(1/2)*((b/(a-b)*sin(d*x+c) +a/(a-b))^(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)+a/(a-b))^(1/2),((a-b)/(a+b))^( 1/2))*a^3*b+3*(b/(a-b)*sin(d*x+c)+a/(a-b))^(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)+a/(a-b))^(1/2),((a-b)/(a+b))^(1/2))*a^2*b^2-(b/(a-b)*sin(d*x+c)+a/(a-b) )^(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)+a/(a-b))^(1/2),((a-b)/(a+b))^(1/2))*a* b^3-3*(b/(a-b)*sin(d*x+c)+a/(a-b))^(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)+a/(a- b))^(1/2),((a-b)/(a+b))^(1/2))*b^4-(b/(a-b)*sin(d*x+c)+a/(a-b))^(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)*Ellipt icE((b/(a-b)*sin(d*x+c)+a/(a-b))^(1/2),((a-b)/(a+b))^(1/2))*a^4-2*(b/(a-b) *sin(d*x+c)+a/(a-b))^(1/2)*(-b/(a+b)*sin(d*x+c)+b/(a+b))^(1/2)*(-b/(a-b)*s in(d*x+c)-b/(a-b))^(1/2)*EllipticE((b/(a-b)*sin(d*x+c)+a/(a-b))^(1/2),((a- b)/(a+b))^(1/2))*a^2*b^2+3*(b/(a-b)*sin(d*x+c)+a/(a-b))^(1/2)*(-b/(a+b)*si n(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)+a/(a-b))^(1/2),((a-b)/(a+b))^(1/2))*b^4+a^2*b^2*cos(d*x+c) ^2+3*b^4*cos(d*x+c)^2-a^3*b*sin(d*x+c)+a*b^3*sin(d*x+c)+a^2*b^2-b^4)/(a^2- b^2)/(a-b)/(a+b)/(-(a+b*sin(d*x+c))*(sin(d*x+c)-1)*(1+sin(d*x+c)))^(1/2...
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.12 (sec) , antiderivative size = 685, normalized size of antiderivative = 2.73 \[ \int \frac {\sec ^2(c+d x)}{(a+b \sin (c+d x))^{3/2}} \, dx=\frac {2 \, {\left (\sqrt {2} {\left (a^{3} b - 3 \, a b^{3}\right )} \cos \left (d x + c\right ) \sin \left (d x + c\right ) + \sqrt {2} {\left (a^{4} - 3 \, a^{2} b^{2}\right )} \cos \left (d x + c\right )\right )} \sqrt {i \, b} {\rm weierstrassPInverse}\left (-\frac {4 \, {\left (4 \, a^{2} - 3 \, b^{2}\right )}}{3 \, b^{2}}, -\frac {8 \, {\left (8 i \, a^{3} - 9 i \, a b^{2}\right )}}{27 \, b^{3}}, \frac {3 \, b \cos \left (d x + c\right ) - 3 i \, b \sin \left (d x + c\right ) - 2 i \, a}{3 \, b}\right ) + 2 \, {\left (\sqrt {2} {\left (a^{3} b - 3 \, a b^{3}\right )} \cos \left (d x + c\right ) \sin \left (d x + c\right ) + \sqrt {2} {\left (a^{4} - 3 \, a^{2} b^{2}\right )} \cos \left (d x + c\right )\right )} \sqrt {-i \, b} {\rm weierstrassPInverse}\left (-\frac {4 \, {\left (4 \, a^{2} - 3 \, b^{2}\right )}}{3 \, b^{2}}, -\frac {8 \, {\left (-8 i \, a^{3} + 9 i \, a b^{2}\right )}}{27 \, b^{3}}, \frac {3 \, b \cos \left (d x + c\right ) + 3 i \, b \sin \left (d x + c\right ) + 2 i \, a}{3 \, b}\right ) - 3 \, {\left (\sqrt {2} {\left (-i \, a^{2} b^{2} - 3 i \, b^{4}\right )} \cos \left (d x + c\right ) \sin \left (d x + c\right ) + \sqrt {2} {\left (-i \, a^{3} b - 3 i \, a b^{3}\right )} \cos \left (d x + c\right )\right )} \sqrt {i \, b} {\rm weierstrassZeta}\left (-\frac {4 \, {\left (4 \, a^{2} - 3 \, b^{2}\right )}}{3 \, b^{2}}, -\frac {8 \, {\left (8 i \, a^{3} - 9 i \, a b^{2}\right )}}{27 \, b^{3}}, {\rm weierstrassPInverse}\left (-\frac {4 \, {\left (4 \, a^{2} - 3 \, b^{2}\right )}}{3 \, b^{2}}, -\frac {8 \, {\left (8 i \, a^{3} - 9 i \, a b^{2}\right )}}{27 \, b^{3}}, \frac {3 \, b \cos \left (d x + c\right ) - 3 i \, b \sin \left (d x + c\right ) - 2 i \, a}{3 \, b}\right )\right ) - 3 \, {\left (\sqrt {2} {\left (i \, a^{2} b^{2} + 3 i \, b^{4}\right )} \cos \left (d x + c\right ) \sin \left (d x + c\right ) + \sqrt {2} {\left (i \, a^{3} b + 3 i \, a b^{3}\right )} \cos \left (d x + c\right )\right )} \sqrt {-i \, b} {\rm weierstrassZeta}\left (-\frac {4 \, {\left (4 \, a^{2} - 3 \, b^{2}\right )}}{3 \, b^{2}}, -\frac {8 \, {\left (-8 i \, a^{3} + 9 i \, a b^{2}\right )}}{27 \, b^{3}}, {\rm weierstrassPInverse}\left (-\frac {4 \, {\left (4 \, a^{2} - 3 \, b^{2}\right )}}{3 \, b^{2}}, -\frac {8 \, {\left (-8 i \, a^{3} + 9 i \, a b^{2}\right )}}{27 \, b^{3}}, \frac {3 \, b \cos \left (d x + c\right ) + 3 i \, b \sin \left (d x + c\right ) + 2 i \, a}{3 \, b}\right )\right ) - 6 \, {\left (a^{2} b^{2} - b^{4} + {\left (a^{2} b^{2} + 3 \, b^{4}\right )} \cos \left (d x + c\right )^{2} - {\left (a^{3} b - a b^{3}\right )} \sin \left (d x + c\right )\right )} \sqrt {b \sin \left (d x + c\right ) + a}}{6 \, {\left ({\left (a^{4} b^{2} - 2 \, a^{2} b^{4} + b^{6}\right )} d \cos \left (d x + c\right ) \sin \left (d x + c\right ) + {\left (a^{5} b - 2 \, a^{3} b^{3} + a b^{5}\right )} d \cos \left (d x + c\right )\right )}} \]
1/6*(2*(sqrt(2)*(a^3*b - 3*a*b^3)*cos(d*x + c)*sin(d*x + c) + sqrt(2)*(a^4 - 3*a^2*b^2)*cos(d*x + c))*sqrt(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*s in(d*x + c) - 2*I*a)/b) + 2*(sqrt(2)*(a^3*b - 3*a*b^3)*cos(d*x + c)*sin(d* x + c) + sqrt(2)*(a^4 - 3*a^2*b^2)*cos(d*x + c))*sqrt(-I*b)*weierstrassPIn verse(-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*(sqrt(2)*(-I*a^2*b^2 - 3*I*b^4)*cos(d*x + c)*sin(d*x + c) + sqrt(2)*(-I*a^3*b - 3*I*a*b^3)*cos(d* x + c))*sqrt(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*(sqrt(2)*(I*a^2*b^2 + 3*I*b^4)*cos(d*x + c)*sin(d*x + c) + sqrt( 2)*(I*a^3*b + 3*I*a*b^3)*cos(d*x + c))*sqrt(-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)) - 6*(a^2*b^2 - b^4 + (a^2*b^2 + 3* b^4)*cos(d*x + c)^2 - (a^3*b - a*b^3)*sin(d*x + c))*sqrt(b*sin(d*x + c) + a))/((a^4*b^2 - 2*a^2*b^4 + b^6)*d*cos(d*x + c)*sin(d*x + c) + (a^5*b - 2* a^3*b^3 + a*b^5)*d*cos(d*x + c))
\[ \int \frac {\sec ^2(c+d x)}{(a+b \sin (c+d x))^{3/2}} \, dx=\int \frac {\sec ^{2}{\left (c + d x \right )}}{\left (a + b \sin {\left (c + d x \right )}\right )^{\frac {3}{2}}}\, dx \]
\[ \int \frac {\sec ^2(c+d x)}{(a+b \sin (c+d x))^{3/2}} \, dx=\int { \frac {\sec \left (d x + c\right )^{2}}{{\left (b \sin \left (d x + c\right ) + a\right )}^{\frac {3}{2}}} \,d x } \]
\[ \int \frac {\sec ^2(c+d x)}{(a+b \sin (c+d x))^{3/2}} \, dx=\int { \frac {\sec \left (d x + c\right )^{2}}{{\left (b \sin \left (d x + c\right ) + a\right )}^{\frac {3}{2}}} \,d x } \]
Timed out. \[ \int \frac {\sec ^2(c+d x)}{(a+b \sin (c+d x))^{3/2}} \, dx=\int \frac {1}{{\cos \left (c+d\,x\right )}^2\,{\left (a+b\,\sin \left (c+d\,x\right )\right )}^{3/2}} \,d x \]