Integrand size = 23, antiderivative size = 218 \[ \int \sec ^4(c+d x) (a+b \sin (c+d x))^{3/2} \, dx=-\frac {\sec (c+d x) (b-4 a \sin (c+d x)) \sqrt {a+b \sin (c+d x)}}{6 d}+\frac {\sec ^3(c+d x) (b+a \sin (c+d x)) \sqrt {a+b \sin (c+d x)}}{3 d}-\frac {2 a E\left (\frac {1}{2} \left (c-\frac {\pi }{2}+d x\right )|\frac {2 b}{a+b}\right ) \sqrt {a+b \sin (c+d x)}}{3 d \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}+\frac {\left (4 a^2-b^2\right ) \operatorname {EllipticF}\left (\frac {1}{2} \left (c-\frac {\pi }{2}+d x\right ),\frac {2 b}{a+b}\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}{6 d \sqrt {a+b \sin (c+d x)}} \]
-1/6*sec(d*x+c)*(b-4*a*sin(d*x+c))*(a+b*sin(d*x+c))^(1/2)/d+1/3*sec(d*x+c) ^3*(b+a*sin(d*x+c))*(a+b*sin(d*x+c))^(1/2)/d+2/3*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)*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)/d/((a+b*sin(d*x+c))/(a+b) )^(1/2)-1/6*(4*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 = 1.65 (sec) , antiderivative size = 211, normalized size of antiderivative = 0.97 \[ \int \sec ^4(c+d x) (a+b \sin (c+d x))^{3/2} \, dx=\frac {16 a (a+b) E\left (\frac {1}{4} (-2 c+\pi -2 d x)|\frac {2 b}{a+b}\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}}-4 \left (4 a^2-b^2\right ) \operatorname {EllipticF}\left (\frac {1}{4} (-2 c+\pi -2 d x),\frac {2 b}{a+b}\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}}+\sec ^3(c+d x) \left (12 a b-6 a b \cos (2 (c+d x))-2 a b \cos (4 (c+d x))+12 a^2 \sin (c+d x)+7 b^2 \sin (c+d x)+4 a^2 \sin (3 (c+d x))-b^2 \sin (3 (c+d x))\right )}{24 d \sqrt {a+b \sin (c+d x)}} \]
(16*a*(a + b)*EllipticE[(-2*c + Pi - 2*d*x)/4, (2*b)/(a + b)]*Sqrt[(a + b* Sin[c + d*x])/(a + b)] - 4*(4*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*(12*a*b - 6*a*b*Cos[2*(c + d*x)] - 2*a*b*Cos[4*(c + d*x)] + 12*a^2*Sin[c + d*x] + 7*b^2*Sin[c + d*x] + 4*a^2*Sin[3*(c + d*x)] - b^2*Sin[3*(c + d*x)]))/(24* d*Sqrt[a + b*Sin[c + d*x]])
Time = 1.37 (sec) , antiderivative size = 281, normalized size of antiderivative = 1.29, 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, 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.
| \(\displaystyle \int \sec ^4(c+d x) (a+b \sin (c+d x))^{3/2} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {(a+b \sin (c+d x))^{3/2}}{\cos (c+d x)^4}dx\) |
| \(\Big \downarrow \) 3170 |
| \(\displaystyle \frac {\sec ^3(c+d x) (a \sin (c+d x)+b) \sqrt {a+b \sin (c+d x)}}{3 d}-\frac {1}{3} \int -\frac {\sec ^2(c+d x) \left (4 a^2+3 b \sin (c+d x) a-b^2\right )}{2 \sqrt {a+b \sin (c+d x)}}dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{6} \int \frac {\sec ^2(c+d x) \left (4 a^2+3 b \sin (c+d x) a-b^2\right )}{\sqrt {a+b \sin (c+d x)}}dx+\frac {\sec ^3(c+d x) (a \sin (c+d x)+b) \sqrt {a+b \sin (c+d x)}}{3 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{6} \int \frac {4 a^2+3 b \sin (c+d x) a-b^2}{\cos (c+d x)^2 \sqrt {a+b \sin (c+d x)}}dx+\frac {\sec ^3(c+d x) (a \sin (c+d x)+b) \sqrt {a+b \sin (c+d x)}}{3 d}\) |
| \(\Big \downarrow \) 3345 |
| \(\displaystyle \frac {1}{6} \left (-\frac {\int \frac {\left (a^2-b^2\right ) b^2+4 a \left (a^2-b^2\right ) \sin (c+d x) b}{2 \sqrt {a+b \sin (c+d x)}}dx}{a^2-b^2}-\frac {\sec (c+d x) \sqrt {a+b \sin (c+d x)} \left (b \left (a^2-b^2\right )-4 a \left (a^2-b^2\right ) \sin (c+d x)\right )}{d \left (a^2-b^2\right )}\right )+\frac {\sec ^3(c+d x) (a \sin (c+d x)+b) \sqrt {a+b \sin (c+d x)}}{3 d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{6} \left (-\frac {\int \frac {\left (a^2-b^2\right ) b^2+4 a \left (a^2-b^2\right ) \sin (c+d x) b}{\sqrt {a+b \sin (c+d x)}}dx}{2 \left (a^2-b^2\right )}-\frac {\sec (c+d x) \sqrt {a+b \sin (c+d x)} \left (b \left (a^2-b^2\right )-4 a \left (a^2-b^2\right ) \sin (c+d x)\right )}{d \left (a^2-b^2\right )}\right )+\frac {\sec ^3(c+d x) (a \sin (c+d x)+b) \sqrt {a+b \sin (c+d x)}}{3 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{6} \left (-\frac {\int \frac {\left (a^2-b^2\right ) b^2+4 a \left (a^2-b^2\right ) \sin (c+d x) b}{\sqrt {a+b \sin (c+d x)}}dx}{2 \left (a^2-b^2\right )}-\frac {\sec (c+d x) \sqrt {a+b \sin (c+d x)} \left (b \left (a^2-b^2\right )-4 a \left (a^2-b^2\right ) \sin (c+d x)\right )}{d \left (a^2-b^2\right )}\right )+\frac {\sec ^3(c+d x) (a \sin (c+d x)+b) \sqrt {a+b \sin (c+d x)}}{3 d}\) |
| \(\Big \downarrow \) 3231 |
| \(\displaystyle \frac {1}{6} \left (-\frac {4 a \left (a^2-b^2\right ) \int \sqrt {a+b \sin (c+d x)}dx-\left (a^2-b^2\right ) \left (4 a^2-b^2\right ) \int \frac {1}{\sqrt {a+b \sin (c+d x)}}dx}{2 \left (a^2-b^2\right )}-\frac {\sec (c+d x) \sqrt {a+b \sin (c+d x)} \left (b \left (a^2-b^2\right )-4 a \left (a^2-b^2\right ) \sin (c+d x)\right )}{d \left (a^2-b^2\right )}\right )+\frac {\sec ^3(c+d x) (a \sin (c+d x)+b) \sqrt {a+b \sin (c+d x)}}{3 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{6} \left (-\frac {4 a \left (a^2-b^2\right ) \int \sqrt {a+b \sin (c+d x)}dx-\left (a^2-b^2\right ) \left (4 a^2-b^2\right ) \int \frac {1}{\sqrt {a+b \sin (c+d x)}}dx}{2 \left (a^2-b^2\right )}-\frac {\sec (c+d x) \sqrt {a+b \sin (c+d x)} \left (b \left (a^2-b^2\right )-4 a \left (a^2-b^2\right ) \sin (c+d x)\right )}{d \left (a^2-b^2\right )}\right )+\frac {\sec ^3(c+d x) (a \sin (c+d x)+b) \sqrt {a+b \sin (c+d x)}}{3 d}\) |
| \(\Big \downarrow \) 3134 |
| \(\displaystyle \frac {1}{6} \left (-\frac {\frac {4 a \left (a^2-b^2\right ) \sqrt {a+b \sin (c+d x)} \int \sqrt {\frac {a}{a+b}+\frac {b \sin (c+d x)}{a+b}}dx}{\sqrt {\frac {a+b \sin (c+d x)}{a+b}}}-\left (a^2-b^2\right ) \left (4 a^2-b^2\right ) \int \frac {1}{\sqrt {a+b \sin (c+d x)}}dx}{2 \left (a^2-b^2\right )}-\frac {\sec (c+d x) \sqrt {a+b \sin (c+d x)} \left (b \left (a^2-b^2\right )-4 a \left (a^2-b^2\right ) \sin (c+d x)\right )}{d \left (a^2-b^2\right )}\right )+\frac {\sec ^3(c+d x) (a \sin (c+d x)+b) \sqrt {a+b \sin (c+d x)}}{3 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{6} \left (-\frac {\frac {4 a \left (a^2-b^2\right ) \sqrt {a+b \sin (c+d x)} \int \sqrt {\frac {a}{a+b}+\frac {b \sin (c+d x)}{a+b}}dx}{\sqrt {\frac {a+b \sin (c+d x)}{a+b}}}-\left (a^2-b^2\right ) \left (4 a^2-b^2\right ) \int \frac {1}{\sqrt {a+b \sin (c+d x)}}dx}{2 \left (a^2-b^2\right )}-\frac {\sec (c+d x) \sqrt {a+b \sin (c+d x)} \left (b \left (a^2-b^2\right )-4 a \left (a^2-b^2\right ) \sin (c+d x)\right )}{d \left (a^2-b^2\right )}\right )+\frac {\sec ^3(c+d x) (a \sin (c+d x)+b) \sqrt {a+b \sin (c+d x)}}{3 d}\) |
| \(\Big \downarrow \) 3132 |
| \(\displaystyle \frac {1}{6} \left (-\frac {\frac {8 a \left (a^2-b^2\right ) \sqrt {a+b \sin (c+d x)} E\left (\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right )|\frac {2 b}{a+b}\right )}{d \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}-\left (a^2-b^2\right ) \left (4 a^2-b^2\right ) \int \frac {1}{\sqrt {a+b \sin (c+d x)}}dx}{2 \left (a^2-b^2\right )}-\frac {\sec (c+d x) \sqrt {a+b \sin (c+d x)} \left (b \left (a^2-b^2\right )-4 a \left (a^2-b^2\right ) \sin (c+d x)\right )}{d \left (a^2-b^2\right )}\right )+\frac {\sec ^3(c+d x) (a \sin (c+d x)+b) \sqrt {a+b \sin (c+d x)}}{3 d}\) |
| \(\Big \downarrow \) 3142 |
| \(\displaystyle \frac {1}{6} \left (-\frac {\frac {8 a \left (a^2-b^2\right ) \sqrt {a+b \sin (c+d x)} E\left (\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right )|\frac {2 b}{a+b}\right )}{d \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}-\frac {\left (a^2-b^2\right ) \left (4 a^2-b^2\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}} \int \frac {1}{\sqrt {\frac {a}{a+b}+\frac {b \sin (c+d x)}{a+b}}}dx}{\sqrt {a+b \sin (c+d x)}}}{2 \left (a^2-b^2\right )}-\frac {\sec (c+d x) \sqrt {a+b \sin (c+d x)} \left (b \left (a^2-b^2\right )-4 a \left (a^2-b^2\right ) \sin (c+d x)\right )}{d \left (a^2-b^2\right )}\right )+\frac {\sec ^3(c+d x) (a \sin (c+d x)+b) \sqrt {a+b \sin (c+d x)}}{3 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{6} \left (-\frac {\frac {8 a \left (a^2-b^2\right ) \sqrt {a+b \sin (c+d x)} E\left (\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right )|\frac {2 b}{a+b}\right )}{d \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}-\frac {\left (a^2-b^2\right ) \left (4 a^2-b^2\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}} \int \frac {1}{\sqrt {\frac {a}{a+b}+\frac {b \sin (c+d x)}{a+b}}}dx}{\sqrt {a+b \sin (c+d x)}}}{2 \left (a^2-b^2\right )}-\frac {\sec (c+d x) \sqrt {a+b \sin (c+d x)} \left (b \left (a^2-b^2\right )-4 a \left (a^2-b^2\right ) \sin (c+d x)\right )}{d \left (a^2-b^2\right )}\right )+\frac {\sec ^3(c+d x) (a \sin (c+d x)+b) \sqrt {a+b \sin (c+d x)}}{3 d}\) |
| \(\Big \downarrow \) 3140 |
| \(\displaystyle \frac {1}{6} \left (-\frac {\sec (c+d x) \sqrt {a+b \sin (c+d x)} \left (b \left (a^2-b^2\right )-4 a \left (a^2-b^2\right ) \sin (c+d x)\right )}{d \left (a^2-b^2\right )}-\frac {\frac {8 a \left (a^2-b^2\right ) \sqrt {a+b \sin (c+d x)} E\left (\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right )|\frac {2 b}{a+b}\right )}{d \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}-\frac {2 \left (a^2-b^2\right ) \left (4 a^2-b^2\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}} \operatorname {EllipticF}\left (\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right ),\frac {2 b}{a+b}\right )}{d \sqrt {a+b \sin (c+d x)}}}{2 \left (a^2-b^2\right )}\right )+\frac {\sec ^3(c+d x) (a \sin (c+d x)+b) \sqrt {a+b \sin (c+d x)}}{3 d}\) |
(Sec[c + d*x]^3*(b + a*Sin[c + d*x])*Sqrt[a + b*Sin[c + d*x]])/(3*d) + (-( (Sec[c + d*x]*Sqrt[a + b*Sin[c + d*x]]*(b*(a^2 - b^2) - 4*a*(a^2 - b^2)*Si n[c + d*x]))/((a^2 - b^2)*d)) - ((8*a*(a^2 - 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^2 - b^2)*(4*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)))/6
3.5.92.3.1 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*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. \(876\) vs. \(2(264)=528\).
Time = 2.34 (sec) , antiderivative size = 877, normalized size of antiderivative = 4.02
| method | result | size |
| default | \(\frac {\sqrt {b \left (\cos ^{2}\left (d x +c \right )\right ) \sin \left (d x +c \right )+a \left (\cos ^{2}\left (d x +c \right )\right )}\, \left (4 \sqrt {-\frac {b \sin \left (d x +c \right )}{a -b}-\frac {b}{a -b}}\, \sqrt {\frac {b \sin \left (d x +c \right )}{a -b}+\frac {a}{a -b}}\, F\left (\sqrt {\frac {b \sin \left (d x +c \right )}{a -b}+\frac {a}{a -b}}, \sqrt {\frac {a -b}{a +b}}\right ) \sqrt {-\frac {b \sin \left (d x +c \right )}{a +b}+\frac {b}{a +b}}\, a^{2} b \left (\cos ^{2}\left (d x +c \right )\right )-3 \sqrt {-\frac {b \sin \left (d x +c \right )}{a -b}-\frac {b}{a -b}}\, \sqrt {\frac {b \sin \left (d x +c \right )}{a -b}+\frac {a}{a -b}}\, F\left (\sqrt {\frac {b \sin \left (d x +c \right )}{a -b}+\frac {a}{a -b}}, \sqrt {\frac {a -b}{a +b}}\right ) \sqrt {-\frac {b \sin \left (d x +c \right )}{a +b}+\frac {b}{a +b}}\, a \,b^{2} \left (\cos ^{2}\left (d x +c \right )\right )-\sqrt {-\frac {b \sin \left (d x +c \right )}{a -b}-\frac {b}{a -b}}\, \sqrt {\frac {b \sin \left (d x +c \right )}{a -b}+\frac {a}{a -b}}\, F\left (\sqrt {\frac {b \sin \left (d x +c \right )}{a -b}+\frac {a}{a -b}}, \sqrt {\frac {a -b}{a +b}}\right ) \sqrt {-\frac {b \sin \left (d x +c \right )}{a +b}+\frac {b}{a +b}}\, b^{3} \left (\cos ^{2}\left (d x +c \right )\right )-4 \sqrt {-\frac {b \sin \left (d x +c \right )}{a -b}-\frac {b}{a -b}}\, \sqrt {\frac {b \sin \left (d x +c \right )}{a -b}+\frac {a}{a -b}}\, \sqrt {-\frac {b \sin \left (d x +c \right )}{a +b}+\frac {b}{a +b}}\, E\left (\sqrt {\frac {b \sin \left (d x +c \right )}{a -b}+\frac {a}{a -b}}, \sqrt {\frac {a -b}{a +b}}\right ) a^{3} \left (\cos ^{2}\left (d x +c \right )\right )+4 \sqrt {-\frac {b \sin \left (d x +c \right )}{a -b}-\frac {b}{a -b}}\, \sqrt {\frac {b \sin \left (d x +c \right )}{a -b}+\frac {a}{a -b}}\, \sqrt {-\frac {b \sin \left (d x +c \right )}{a +b}+\frac {b}{a +b}}\, E\left (\sqrt {\frac {b \sin \left (d x +c \right )}{a -b}+\frac {a}{a -b}}, \sqrt {\frac {a -b}{a +b}}\right ) a \,b^{2} \left (\cos ^{2}\left (d x +c \right )\right )+4 \left (\cos ^{4}\left (d x +c \right )\right ) a \,b^{2}-4 \left (\cos ^{2}\left (d x +c \right )\right ) \sin \left (d x +c \right ) a^{2} b +\sin \left (d x +c \right ) \left (\cos ^{2}\left (d x +c \right )\right ) b^{3}-\left (\cos ^{2}\left (d x +c \right )\right ) a \,b^{2}-2 \sin \left (d x +c \right ) a^{2} b -2 \sin \left (d x +c \right ) b^{3}-4 a \,b^{2}\right )}{6 \sqrt {-\left (a +b \sin \left (d x +c \right )\right ) \left (\sin \left (d x +c \right )-1\right ) \left (1+\sin \left (d x +c \right )\right )}\, \left (1+\sin \left (d x +c \right )\right ) \left (\sin \left (d x +c \right )-1\right ) b \cos \left (d x +c \right ) \sqrt {a +b \sin \left (d x +c \right )}\, d}\) | \(877\) |
1/6*(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)-b/(a-b))^(1/2)*(b/(a-b)*sin(d*x+c)+a/(a-b))^(1/2)*EllipticF((b/(a-b)*si n(d*x+c)+a/(a-b))^(1/2),((a-b)/(a+b))^(1/2))*(-b/(a+b)*sin(d*x+c)+b/(a+b)) ^(1/2)*a^2*b*cos(d*x+c)^2-3*(-b/(a-b)*sin(d*x+c)-b/(a-b))^(1/2)*(b/(a-b)*s in(d*x+c)+a/(a-b))^(1/2)*EllipticF((b/(a-b)*sin(d*x+c)+a/(a-b))^(1/2),((a- b)/(a+b))^(1/2))*(-b/(a+b)*sin(d*x+c)+b/(a+b))^(1/2)*a*b^2*cos(d*x+c)^2-(- b/(a-b)*sin(d*x+c)-b/(a-b))^(1/2)*(b/(a-b)*sin(d*x+c)+a/(a-b))^(1/2)*Ellip ticF((b/(a-b)*sin(d*x+c)+a/(a-b))^(1/2),((a-b)/(a+b))^(1/2))*(-b/(a+b)*sin (d*x+c)+b/(a+b))^(1/2)*b^3*cos(d*x+c)^2-4*(-b/(a-b)*sin(d*x+c)-b/(a-b))^(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) *EllipticE((b/(a-b)*sin(d*x+c)+a/(a-b))^(1/2),((a-b)/(a+b))^(1/2))*a^3*cos (d*x+c)^2+4*(-b/(a-b)*sin(d*x+c)-b/(a-b))^(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)*EllipticE((b/(a-b)*sin(d*x+c) +a/(a-b))^(1/2),((a-b)/(a+b))^(1/2))*a*b^2*cos(d*x+c)^2+4*cos(d*x+c)^4*a*b ^2-4*cos(d*x+c)^2*sin(d*x+c)*a^2*b+sin(d*x+c)*cos(d*x+c)^2*b^3-cos(d*x+c)^ 2*a*b^2-2*sin(d*x+c)*a^2*b-2*sin(d*x+c)*b^3-4*a*b^2)/(-(a+b*sin(d*x+c))*(s in(d*x+c)-1)*(1+sin(d*x+c)))^(1/2)/(1+sin(d*x+c))/(sin(d*x+c)-1)/b/cos(d*x +c)/(a+b*sin(d*x+c))^(1/2)/d
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.13 (sec) , antiderivative size = 481, normalized size of antiderivative = 2.21 \[ \int \sec ^4(c+d x) (a+b \sin (c+d x))^{3/2} \, dx=\frac {12 i \, \sqrt {2} a \sqrt {i \, b} 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 ) - 12 i \, \sqrt {2} a \sqrt {-i \, b} 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 ) + \sqrt {2} {\left (8 \, a^{2} - 3 \, 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^{2} - 3 \, 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 ) - 6 \, {\left (b^{2} \cos \left (d x + c\right )^{2} - 2 \, b^{2} - 2 \, {\left (2 \, a b \cos \left (d x + c\right )^{2} + a b\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*(12*I*sqrt(2)*a*sqrt(I*b)*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)) - 12*I*sqrt(2)*a*sqrt(-I*b)*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)) + s qrt(2)*(8*a^2 - 3*b^2)*sqrt(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) + sqrt(2)*(8*a^2 - 3*b^2)*sqrt(-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) - 6*(b^2*cos(d*x + c)^2 - 2*b^2 - 2*(2*a*b*cos(d*x + c)^2 + a*b)*sin(d*x + c))*sqrt(b*sin(d*x + c) + a))/(b*d*cos(d*x + c)^3)
Timed out. \[ \int \sec ^4(c+d x) (a+b \sin (c+d x))^{3/2} \, dx=\text {Timed out} \]
\[ \int \sec ^4(c+d x) (a+b \sin (c+d x))^{3/2} \, dx=\int { {\left (b \sin \left (d x + c\right ) + a\right )}^{\frac {3}{2}} \sec \left (d x + c\right )^{4} \,d x } \]
Timed out. \[ \int \sec ^4(c+d x) (a+b \sin (c+d x))^{3/2} \, dx=\text {Timed out} \]
Timed out. \[ \int \sec ^4(c+d x) (a+b \sin (c+d x))^{3/2} \, dx=\int \frac {{\left (a+b\,\sin \left (c+d\,x\right )\right )}^{3/2}}{{\cos \left (c+d\,x\right )}^4} \,d x \]