Integrand size = 23, antiderivative size = 238 \[ \int \sec ^4(c+d x) (a+b \sin (c+d x))^{5/2} \, dx=\frac {\sec ^3(c+d x) (b+a \sin (c+d x)) (a+b \sin (c+d x))^{3/2}}{3 d}-\frac {\left (4 a^2-3 b^2\right ) E\left (\frac {1}{2} \left (c-\frac {\pi }{2}+d x\right )|\frac {2 b}{a+b}\right ) \sqrt {a+b \sin (c+d x)}}{6 d \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}+\frac {2 a \left (a^2-b^2\right ) \operatorname {EllipticF}\left (\frac {1}{2} \left (c-\frac {\pi }{2}+d x\right ),\frac {2 b}{a+b}\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}{3 d \sqrt {a+b \sin (c+d x)}}+\frac {\sec (c+d x) \sqrt {a+b \sin (c+d x)} \left (a b+\left (4 a^2-3 b^2\right ) \sin (c+d x)\right )}{6 d} \]
1/3*sec(d*x+c)^3*(b+a*sin(d*x+c))*(a+b*sin(d*x+c))^(3/2)/d+1/6*sec(d*x+c)* (a*b+(4*a^2-3*b^2)*sin(d*x+c))*(a+b*sin(d*x+c))^(1/2)/d+1/6*(4*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(co s(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)/d/ ((a+b*sin(d*x+c))/(a+b))^(1/2)-2/3*a*(a^2-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)*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)/d/(a+b*sin(d*x+c))^(1 /2)
Time = 2.40 (sec) , antiderivative size = 259, normalized size of antiderivative = 1.09 \[ \int \sec ^4(c+d x) (a+b \sin (c+d x))^{5/2} \, dx=\frac {\left (4 a^3+4 a^2 b-3 a b^2-3 b^3\right ) E\left (\frac {1}{4} (-2 c+\pi -2 d x)|\frac {2 b}{a+b}\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}}-4 a \left (a^2-b^2\right ) \operatorname {EllipticF}\left (\frac {1}{4} (-2 c+\pi -2 d x),\frac {2 b}{a+b}\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}}+\frac {1}{8} \sec ^3(c+d x) \left (40 a^2 b+5 b^3-4 \left (3 a^2 b+2 b^3\right ) \cos (2 (c+d x))+\left (-4 a^2 b+3 b^3\right ) \cos (4 (c+d x))+24 a^3 \sin (c+d x)+40 a b^2 \sin (c+d x)+8 a^3 \sin (3 (c+d x))-8 a b^2 \sin (3 (c+d x))\right )}{6 d \sqrt {a+b \sin (c+d x)}} \]
((4*a^3 + 4*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)] - 4*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*(40*a^2*b + 5*b^3 - 4*(3*a^2*b + 2*b^3)*Cos[2*(c + d*x)] + (-4*a^2*b + 3*b^3)*Cos[4*(c + d*x)] + 24*a^3*Sin[c + d*x] + 40*a*b^2*Sin [c + d*x] + 8*a^3*Sin[3*(c + d*x)] - 8*a*b^2*Sin[3*(c + d*x)]))/8)/(6*d*Sq rt[a + b*Sin[c + d*x]])
Time = 1.15 (sec) , antiderivative size = 241, normalized size of antiderivative = 1.01, number of steps used = 15, number of rules used = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.652, Rules used = {3042, 3170, 27, 3042, 3340, 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.
| \(\displaystyle \int \sec ^4(c+d x) (a+b \sin (c+d x))^{5/2} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {(a+b \sin (c+d x))^{5/2}}{\cos (c+d x)^4}dx\) |
| \(\Big \downarrow \) 3170 |
| \(\displaystyle \frac {\sec ^3(c+d x) (a \sin (c+d x)+b) (a+b \sin (c+d x))^{3/2}}{3 d}-\frac {1}{3} \int -\frac {1}{2} \sec ^2(c+d x) \sqrt {a+b \sin (c+d x)} \left (4 a^2+b \sin (c+d x) a-3 b^2\right )dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{6} \int \sec ^2(c+d x) \sqrt {a+b \sin (c+d x)} \left (4 a^2+b \sin (c+d x) a-3 b^2\right )dx+\frac {\sec ^3(c+d x) (a \sin (c+d x)+b) (a+b \sin (c+d x))^{3/2}}{3 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{6} \int \frac {\sqrt {a+b \sin (c+d x)} \left (4 a^2+b \sin (c+d x) a-3 b^2\right )}{\cos (c+d x)^2}dx+\frac {\sec ^3(c+d x) (a \sin (c+d x)+b) (a+b \sin (c+d x))^{3/2}}{3 d}\) |
| \(\Big \downarrow \) 3340 |
| \(\displaystyle \frac {1}{6} \left (\frac {\sec (c+d x) \sqrt {a+b \sin (c+d x)} \left (\left (4 a^2-3 b^2\right ) \sin (c+d x)+a b\right )}{d}-\int \frac {a b^2+\left (4 a^2-3 b^2\right ) \sin (c+d x) b}{2 \sqrt {a+b \sin (c+d x)}}dx\right )+\frac {\sec ^3(c+d x) (a \sin (c+d x)+b) (a+b \sin (c+d x))^{3/2}}{3 d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{6} \left (\frac {\sec (c+d x) \sqrt {a+b \sin (c+d x)} \left (\left (4 a^2-3 b^2\right ) \sin (c+d x)+a b\right )}{d}-\frac {1}{2} \int \frac {a b^2+\left (4 a^2-3 b^2\right ) \sin (c+d x) b}{\sqrt {a+b \sin (c+d x)}}dx\right )+\frac {\sec ^3(c+d x) (a \sin (c+d x)+b) (a+b \sin (c+d x))^{3/2}}{3 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{6} \left (\frac {\sec (c+d x) \sqrt {a+b \sin (c+d x)} \left (\left (4 a^2-3 b^2\right ) \sin (c+d x)+a b\right )}{d}-\frac {1}{2} \int \frac {a b^2+\left (4 a^2-3 b^2\right ) \sin (c+d x) b}{\sqrt {a+b \sin (c+d x)}}dx\right )+\frac {\sec ^3(c+d x) (a \sin (c+d x)+b) (a+b \sin (c+d x))^{3/2}}{3 d}\) |
| \(\Big \downarrow \) 3231 |
| \(\displaystyle \frac {1}{6} \left (\frac {1}{2} \left (4 a \left (a^2-b^2\right ) \int \frac {1}{\sqrt {a+b \sin (c+d x)}}dx-\left (4 a^2-3 b^2\right ) \int \sqrt {a+b \sin (c+d x)}dx\right )+\frac {\sec (c+d x) \sqrt {a+b \sin (c+d x)} \left (\left (4 a^2-3 b^2\right ) \sin (c+d x)+a b\right )}{d}\right )+\frac {\sec ^3(c+d x) (a \sin (c+d x)+b) (a+b \sin (c+d x))^{3/2}}{3 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{6} \left (\frac {1}{2} \left (4 a \left (a^2-b^2\right ) \int \frac {1}{\sqrt {a+b \sin (c+d x)}}dx-\left (4 a^2-3 b^2\right ) \int \sqrt {a+b \sin (c+d x)}dx\right )+\frac {\sec (c+d x) \sqrt {a+b \sin (c+d x)} \left (\left (4 a^2-3 b^2\right ) \sin (c+d x)+a b\right )}{d}\right )+\frac {\sec ^3(c+d x) (a \sin (c+d x)+b) (a+b \sin (c+d x))^{3/2}}{3 d}\) |
| \(\Big \downarrow \) 3134 |
| \(\displaystyle \frac {1}{6} \left (\frac {1}{2} \left (4 a \left (a^2-b^2\right ) \int \frac {1}{\sqrt {a+b \sin (c+d x)}}dx-\frac {\left (4 a^2-3 b^2\right ) \sqrt {a+b \sin (c+d x)} \int \sqrt {\frac {a}{a+b}+\frac {b \sin (c+d x)}{a+b}}dx}{\sqrt {\frac {a+b \sin (c+d x)}{a+b}}}\right )+\frac {\sec (c+d x) \sqrt {a+b \sin (c+d x)} \left (\left (4 a^2-3 b^2\right ) \sin (c+d x)+a b\right )}{d}\right )+\frac {\sec ^3(c+d x) (a \sin (c+d x)+b) (a+b \sin (c+d x))^{3/2}}{3 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{6} \left (\frac {1}{2} \left (4 a \left (a^2-b^2\right ) \int \frac {1}{\sqrt {a+b \sin (c+d x)}}dx-\frac {\left (4 a^2-3 b^2\right ) \sqrt {a+b \sin (c+d x)} \int \sqrt {\frac {a}{a+b}+\frac {b \sin (c+d x)}{a+b}}dx}{\sqrt {\frac {a+b \sin (c+d x)}{a+b}}}\right )+\frac {\sec (c+d x) \sqrt {a+b \sin (c+d x)} \left (\left (4 a^2-3 b^2\right ) \sin (c+d x)+a b\right )}{d}\right )+\frac {\sec ^3(c+d x) (a \sin (c+d x)+b) (a+b \sin (c+d x))^{3/2}}{3 d}\) |
| \(\Big \downarrow \) 3132 |
| \(\displaystyle \frac {1}{6} \left (\frac {1}{2} \left (4 a \left (a^2-b^2\right ) \int \frac {1}{\sqrt {a+b \sin (c+d x)}}dx-\frac {2 \left (4 a^2-3 b^2\right ) \sqrt {a+b \sin (c+d x)} E\left (\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right )|\frac {2 b}{a+b}\right )}{d \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}\right )+\frac {\sec (c+d x) \sqrt {a+b \sin (c+d x)} \left (\left (4 a^2-3 b^2\right ) \sin (c+d x)+a b\right )}{d}\right )+\frac {\sec ^3(c+d x) (a \sin (c+d x)+b) (a+b \sin (c+d x))^{3/2}}{3 d}\) |
| \(\Big \downarrow \) 3142 |
| \(\displaystyle \frac {1}{6} \left (\frac {1}{2} \left (\frac {4 a \left (a^2-b^2\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}} \int \frac {1}{\sqrt {\frac {a}{a+b}+\frac {b \sin (c+d x)}{a+b}}}dx}{\sqrt {a+b \sin (c+d x)}}-\frac {2 \left (4 a^2-3 b^2\right ) \sqrt {a+b \sin (c+d x)} E\left (\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right )|\frac {2 b}{a+b}\right )}{d \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}\right )+\frac {\sec (c+d x) \sqrt {a+b \sin (c+d x)} \left (\left (4 a^2-3 b^2\right ) \sin (c+d x)+a b\right )}{d}\right )+\frac {\sec ^3(c+d x) (a \sin (c+d x)+b) (a+b \sin (c+d x))^{3/2}}{3 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{6} \left (\frac {1}{2} \left (\frac {4 a \left (a^2-b^2\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}} \int \frac {1}{\sqrt {\frac {a}{a+b}+\frac {b \sin (c+d x)}{a+b}}}dx}{\sqrt {a+b \sin (c+d x)}}-\frac {2 \left (4 a^2-3 b^2\right ) \sqrt {a+b \sin (c+d x)} E\left (\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right )|\frac {2 b}{a+b}\right )}{d \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}\right )+\frac {\sec (c+d x) \sqrt {a+b \sin (c+d x)} \left (\left (4 a^2-3 b^2\right ) \sin (c+d x)+a b\right )}{d}\right )+\frac {\sec ^3(c+d x) (a \sin (c+d x)+b) (a+b \sin (c+d x))^{3/2}}{3 d}\) |
| \(\Big \downarrow \) 3140 |
| \(\displaystyle \frac {1}{6} \left (\frac {\sec (c+d x) \sqrt {a+b \sin (c+d x)} \left (\left (4 a^2-3 b^2\right ) \sin (c+d x)+a b\right )}{d}+\frac {1}{2} \left (\frac {8 a \left (a^2-b^2\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}} \operatorname {EllipticF}\left (\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right ),\frac {2 b}{a+b}\right )}{d \sqrt {a+b \sin (c+d x)}}-\frac {2 \left (4 a^2-3 b^2\right ) \sqrt {a+b \sin (c+d x)} E\left (\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right )|\frac {2 b}{a+b}\right )}{d \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}\right )\right )+\frac {\sec ^3(c+d x) (a \sin (c+d x)+b) (a+b \sin (c+d x))^{3/2}}{3 d}\) |
(Sec[c + d*x]^3*(b + a*Sin[c + d*x])*(a + b*Sin[c + d*x])^(3/2))/(3*d) + ( (Sec[c + d*x]*Sqrt[a + b*Sin[c + d*x]]*(a*b + (4*a^2 - 3*b^2)*Sin[c + d*x] ))/d + ((-2*(4*a^2 - 3*b^2)*EllipticE[(c - Pi/2 + d*x)/2, (2*b)/(a + b)]*S qrt[a + b*Sin[c + d*x]])/(d*Sqrt[(a + b*Sin[c + d*x])/(a + b)]) + (8*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)/6
3.6.3.3.1 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[(-(g*Cos[e + f*x])^(p + 1))*(a + b*Sin[e + f*x ])^(m - 1)*((b + a*Sin[e + f*x])/(f*g*(p + 1))), x] + Simp[1/(g^2*(p + 1)) Int[(g*Cos[e + f*x])^(p + 2)*(a + b*Sin[e + f*x])^(m - 2)*(b^2*(m - 1) + a^2*(p + 2) + a*b*(m + p + 1)*Sin[e + f*x]), x], x] /; FreeQ[{a, b, e, f, g }, x] && NeQ[a^2 - b^2, 0] && GtQ[m, 1] && LtQ[p, -1] && (IntegersQ[2*m, 2* p] || IntegerQ[m])
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* Cos[e + f*x])^(p + 1))*(a + b*Sin[e + f*x])^m*((d + c*Sin[e + f*x])/(f*g*(p + 1))), x] + Simp[1/(g^2*(p + 1)) Int[(g*Cos[e + f*x])^(p + 2)*(a + b*Si n[e + f*x])^(m - 1)*Simp[a*c*(p + 2) + b*d*m + b*c*(m + p + 2)*Sin[e + f*x] , x], x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] && NeQ[a^2 - b^2, 0] && GtQ [m, 0] && LtQ[p, -1] && IntegerQ[2*m] && !(EqQ[m, 1] && NeQ[c^2 - d^2, 0] && SimplerQ[c + d*x, a + b*x])
Leaf count of result is larger than twice the leaf count of optimal. \(1248\) vs. \(2(284)=568\).
Time = 99.78 (sec) , antiderivative size = 1249, normalized size of antiderivative = 5.25
1/6*((b*cos(d*x+c)^2*sin(d*x+c)+a*cos(d*x+c)^2)^(1/2)*b^2*(4*a^2-3*b^2)*co s(d*x+c)^4-4*(b*cos(d*x+c)^2*sin(d*x+c)+a*cos(d*x+c)^2)^(1/2)*a*b*(a^2-b^2 )*cos(d*x+c)^2*sin(d*x+c)-(b*cos(d*x+c)^2*sin(d*x+c)+a*cos(d*x+c)^2)^(1/2) *(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)*EllipticE((b/(a-b)*sin(d*x+c)+a/(a-b) )^(1/2),((a-b)/(a+b))^(1/2))*a^4-7*(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^2*b^2+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)*EllipticE((b/(a-b)*sin(d*x+c)+a/(a-b))^(1/2), ((a-b)/(a+b))^(1/2))*b^4-4*(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)*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+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)*EllipticF((b/(a-b)*s in(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))...
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.14 (sec) , antiderivative size = 522, normalized size of antiderivative = 2.19 \[ \int \sec ^4(c+d x) (a+b \sin (c+d x))^{5/2} \, dx=\frac {\sqrt {2} {\left (8 \, a^{3} - 9 \, a b^{2}\right )} \sqrt {i \, b} \cos \left (d x + c\right )^{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 ) + \sqrt {2} {\left (8 \, a^{3} - 9 \, a b^{2}\right )} \sqrt {-i \, b} \cos \left (d x + c\right )^{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 ) - 3 \, \sqrt {2} {\left (-4 i \, a^{2} b + 3 i \, b^{3}\right )} \sqrt {i \, b} \cos \left (d x + c\right )^{3} {\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 \, \sqrt {2} {\left (4 i \, a^{2} b - 3 i \, b^{3}\right )} \sqrt {-i \, b} \cos \left (d x + c\right )^{3} {\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 b^{2} \cos \left (d x + c\right )^{2} - 4 \, a b^{2} - {\left (2 \, a^{2} b + 2 \, b^{3} + {\left (4 \, a^{2} b - 3 \, b^{3}\right )} \cos \left (d x + c\right )^{2}\right )} \sin \left (d x + c\right )\right )} \sqrt {b \sin \left (d x + c\right ) + a}}{36 \, b d \cos \left (d x + c\right )^{3}} \]
1/36*(sqrt(2)*(8*a^3 - 9*a*b^2)*sqrt(I*b)*cos(d*x + c)^3*weierstrassPInver se(-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) + sqrt(2)*(8*a^3 - 9*a*b^2)*sqr t(-I*b)*cos(d*x + c)^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)*(-4*I*a^2*b + 3*I*b^3)*sqrt(I*b)*cos(d*x + c)^3*weie rstrassZeta(-4/3*(4*a^2 - 3*b^2)/b^2, -8/27*(8*I*a^3 - 9*I*a*b^2)/b^3, wei erstrassPInverse(-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)*(4*I *a^2*b - 3*I*b^3)*sqrt(-I*b)*cos(d*x + c)^3*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*b^2*cos(d*x + c)^2 - 4*a*b^2 - (2 *a^2*b + 2*b^3 + (4*a^2*b - 3*b^3)*cos(d*x + c)^2)*sin(d*x + c))*sqrt(b*si n(d*x + c) + a))/(b*d*cos(d*x + c)^3)
Timed out. \[ \int \sec ^4(c+d x) (a+b \sin (c+d x))^{5/2} \, dx=\text {Timed out} \]
\[ \int \sec ^4(c+d x) (a+b \sin (c+d x))^{5/2} \, dx=\int { {\left (b \sin \left (d x + c\right ) + a\right )}^{\frac {5}{2}} \sec \left (d x + c\right )^{4} \,d x } \]
Timed out. \[ \int \sec ^4(c+d x) (a+b \sin (c+d x))^{5/2} \, dx=\text {Timed out} \]
Timed out. \[ \int \sec ^4(c+d x) (a+b \sin (c+d x))^{5/2} \, dx=\int \frac {{\left (a+b\,\sin \left (c+d\,x\right )\right )}^{5/2}}{{\cos \left (c+d\,x\right )}^4} \,d x \]