Integrand size = 27, antiderivative size = 223 \[ \int \frac {(3+b \sin (e+f x))^2}{(c+d \sin (e+f x))^{3/2}} \, dx=\frac {2 (b c-3 d)^2 \cos (e+f x)}{d \left (c^2-d^2\right ) f \sqrt {c+d \sin (e+f x)}}+\frac {2 \left (2 b^2 c^2-6 b c d+\left (9-b^2\right ) d^2\right ) E\left (\frac {1}{2} \left (e-\frac {\pi }{2}+f x\right )|\frac {2 d}{c+d}\right ) \sqrt {c+d \sin (e+f x)}}{d^2 \left (c^2-d^2\right ) f \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}-\frac {4 b (b c-3 d) \operatorname {EllipticF}\left (\frac {1}{2} \left (e-\frac {\pi }{2}+f x\right ),\frac {2 d}{c+d}\right ) \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}{d^2 f \sqrt {c+d \sin (e+f x)}} \]
2*(-a*d+b*c)^2*cos(f*x+e)/d/(c^2-d^2)/f/(c+d*sin(f*x+e))^(1/2)-2*(2*b^2*c^ 2-2*a*b*c*d+(a^2-b^2)*d^2)*(sin(1/2*e+1/4*Pi+1/2*f*x)^2)^(1/2)/sin(1/2*e+1 /4*Pi+1/2*f*x)*EllipticE(cos(1/2*e+1/4*Pi+1/2*f*x),2^(1/2)*(d/(c+d))^(1/2) )*(c+d*sin(f*x+e))^(1/2)/d^2/(c^2-d^2)/f/((c+d*sin(f*x+e))/(c+d))^(1/2)+4* b*(-a*d+b*c)*(sin(1/2*e+1/4*Pi+1/2*f*x)^2)^(1/2)/sin(1/2*e+1/4*Pi+1/2*f*x) *EllipticF(cos(1/2*e+1/4*Pi+1/2*f*x),2^(1/2)*(d/(c+d))^(1/2))*((c+d*sin(f* x+e))/(c+d))^(1/2)/d^2/f/(c+d*sin(f*x+e))^(1/2)
Time = 1.01 (sec) , antiderivative size = 166, normalized size of antiderivative = 0.74 \[ \int \frac {(3+b \sin (e+f x))^2}{(c+d \sin (e+f x))^{3/2}} \, dx=\frac {2 \left (\frac {(b c-3 d)^2 \cos (e+f x)}{c^2-d^2}+\frac {\left (\left (6 b c d-9 d^2+b^2 \left (-2 c^2+d^2\right )\right ) E\left (\frac {1}{4} (-2 e+\pi -2 f x)|\frac {2 d}{c+d}\right )+2 b (b c-3 d) (c-d) \operatorname {EllipticF}\left (\frac {1}{4} (-2 e+\pi -2 f x),\frac {2 d}{c+d}\right )\right ) \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}{(c-d) d}\right )}{d f \sqrt {c+d \sin (e+f x)}} \]
(2*(((b*c - 3*d)^2*Cos[e + f*x])/(c^2 - d^2) + (((6*b*c*d - 9*d^2 + b^2*(- 2*c^2 + d^2))*EllipticE[(-2*e + Pi - 2*f*x)/4, (2*d)/(c + d)] + 2*b*(b*c - 3*d)*(c - d)*EllipticF[(-2*e + Pi - 2*f*x)/4, (2*d)/(c + d)])*Sqrt[(c + d *Sin[e + f*x])/(c + d)])/((c - d)*d)))/(d*f*Sqrt[c + d*Sin[e + f*x]])
Time = 1.03 (sec) , antiderivative size = 242, normalized size of antiderivative = 1.09, number of steps used = 12, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.444, Rules used = {3042, 3269, 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.
| \(\displaystyle \int \frac {(a+b \sin (e+f x))^2}{(c+d \sin (e+f x))^{3/2}} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {(a+b \sin (e+f x))^2}{(c+d \sin (e+f x))^{3/2}}dx\) |
| \(\Big \downarrow \) 3269 |
| \(\displaystyle \frac {2 \int \frac {d \left (c a^2-2 b d a+b^2 c\right )+\left (2 b^2 c^2-2 a b d c+\left (a^2-b^2\right ) d^2\right ) \sin (e+f x)}{2 \sqrt {c+d \sin (e+f x)}}dx}{d \left (c^2-d^2\right )}+\frac {2 (b c-a d)^2 \cos (e+f x)}{d f \left (c^2-d^2\right ) \sqrt {c+d \sin (e+f x)}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int \frac {d \left (c a^2-2 b d a+b^2 c\right )+\left (2 b^2 c^2-2 a b d c+\left (a^2-b^2\right ) d^2\right ) \sin (e+f x)}{\sqrt {c+d \sin (e+f x)}}dx}{d \left (c^2-d^2\right )}+\frac {2 (b c-a d)^2 \cos (e+f x)}{d f \left (c^2-d^2\right ) \sqrt {c+d \sin (e+f x)}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\int \frac {d \left (c a^2-2 b d a+b^2 c\right )+\left (2 b^2 c^2-2 a b d c+\left (a^2-b^2\right ) d^2\right ) \sin (e+f x)}{\sqrt {c+d \sin (e+f x)}}dx}{d \left (c^2-d^2\right )}+\frac {2 (b c-a d)^2 \cos (e+f x)}{d f \left (c^2-d^2\right ) \sqrt {c+d \sin (e+f x)}}\) |
| \(\Big \downarrow \) 3231 |
| \(\displaystyle \frac {\frac {\left (d^2 \left (a^2-b^2\right )-2 a b c d+2 b^2 c^2\right ) \int \sqrt {c+d \sin (e+f x)}dx}{d}-\frac {2 b \left (c^2-d^2\right ) (b c-a d) \int \frac {1}{\sqrt {c+d \sin (e+f x)}}dx}{d}}{d \left (c^2-d^2\right )}+\frac {2 (b c-a d)^2 \cos (e+f x)}{d f \left (c^2-d^2\right ) \sqrt {c+d \sin (e+f x)}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {\left (d^2 \left (a^2-b^2\right )-2 a b c d+2 b^2 c^2\right ) \int \sqrt {c+d \sin (e+f x)}dx}{d}-\frac {2 b \left (c^2-d^2\right ) (b c-a d) \int \frac {1}{\sqrt {c+d \sin (e+f x)}}dx}{d}}{d \left (c^2-d^2\right )}+\frac {2 (b c-a d)^2 \cos (e+f x)}{d f \left (c^2-d^2\right ) \sqrt {c+d \sin (e+f x)}}\) |
| \(\Big \downarrow \) 3134 |
| \(\displaystyle \frac {\frac {\left (d^2 \left (a^2-b^2\right )-2 a b c d+2 b^2 c^2\right ) \sqrt {c+d \sin (e+f x)} \int \sqrt {\frac {c}{c+d}+\frac {d \sin (e+f x)}{c+d}}dx}{d \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}-\frac {2 b \left (c^2-d^2\right ) (b c-a d) \int \frac {1}{\sqrt {c+d \sin (e+f x)}}dx}{d}}{d \left (c^2-d^2\right )}+\frac {2 (b c-a d)^2 \cos (e+f x)}{d f \left (c^2-d^2\right ) \sqrt {c+d \sin (e+f x)}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {\left (d^2 \left (a^2-b^2\right )-2 a b c d+2 b^2 c^2\right ) \sqrt {c+d \sin (e+f x)} \int \sqrt {\frac {c}{c+d}+\frac {d \sin (e+f x)}{c+d}}dx}{d \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}-\frac {2 b \left (c^2-d^2\right ) (b c-a d) \int \frac {1}{\sqrt {c+d \sin (e+f x)}}dx}{d}}{d \left (c^2-d^2\right )}+\frac {2 (b c-a d)^2 \cos (e+f x)}{d f \left (c^2-d^2\right ) \sqrt {c+d \sin (e+f x)}}\) |
| \(\Big \downarrow \) 3132 |
| \(\displaystyle \frac {\frac {2 \left (d^2 \left (a^2-b^2\right )-2 a b c d+2 b^2 c^2\right ) \sqrt {c+d \sin (e+f x)} E\left (\frac {1}{2} \left (e+f x-\frac {\pi }{2}\right )|\frac {2 d}{c+d}\right )}{d f \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}-\frac {2 b \left (c^2-d^2\right ) (b c-a d) \int \frac {1}{\sqrt {c+d \sin (e+f x)}}dx}{d}}{d \left (c^2-d^2\right )}+\frac {2 (b c-a d)^2 \cos (e+f x)}{d f \left (c^2-d^2\right ) \sqrt {c+d \sin (e+f x)}}\) |
| \(\Big \downarrow \) 3142 |
| \(\displaystyle \frac {\frac {2 \left (d^2 \left (a^2-b^2\right )-2 a b c d+2 b^2 c^2\right ) \sqrt {c+d \sin (e+f x)} E\left (\frac {1}{2} \left (e+f x-\frac {\pi }{2}\right )|\frac {2 d}{c+d}\right )}{d f \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}-\frac {2 b \left (c^2-d^2\right ) (b c-a d) \sqrt {\frac {c+d \sin (e+f x)}{c+d}} \int \frac {1}{\sqrt {\frac {c}{c+d}+\frac {d \sin (e+f x)}{c+d}}}dx}{d \sqrt {c+d \sin (e+f x)}}}{d \left (c^2-d^2\right )}+\frac {2 (b c-a d)^2 \cos (e+f x)}{d f \left (c^2-d^2\right ) \sqrt {c+d \sin (e+f x)}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {2 \left (d^2 \left (a^2-b^2\right )-2 a b c d+2 b^2 c^2\right ) \sqrt {c+d \sin (e+f x)} E\left (\frac {1}{2} \left (e+f x-\frac {\pi }{2}\right )|\frac {2 d}{c+d}\right )}{d f \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}-\frac {2 b \left (c^2-d^2\right ) (b c-a d) \sqrt {\frac {c+d \sin (e+f x)}{c+d}} \int \frac {1}{\sqrt {\frac {c}{c+d}+\frac {d \sin (e+f x)}{c+d}}}dx}{d \sqrt {c+d \sin (e+f x)}}}{d \left (c^2-d^2\right )}+\frac {2 (b c-a d)^2 \cos (e+f x)}{d f \left (c^2-d^2\right ) \sqrt {c+d \sin (e+f x)}}\) |
| \(\Big \downarrow \) 3140 |
| \(\displaystyle \frac {\frac {2 \left (d^2 \left (a^2-b^2\right )-2 a b c d+2 b^2 c^2\right ) \sqrt {c+d \sin (e+f x)} E\left (\frac {1}{2} \left (e+f x-\frac {\pi }{2}\right )|\frac {2 d}{c+d}\right )}{d f \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}-\frac {4 b \left (c^2-d^2\right ) (b c-a d) \sqrt {\frac {c+d \sin (e+f x)}{c+d}} \operatorname {EllipticF}\left (\frac {1}{2} \left (e+f x-\frac {\pi }{2}\right ),\frac {2 d}{c+d}\right )}{d f \sqrt {c+d \sin (e+f x)}}}{d \left (c^2-d^2\right )}+\frac {2 (b c-a d)^2 \cos (e+f x)}{d f \left (c^2-d^2\right ) \sqrt {c+d \sin (e+f x)}}\) |
(2*(b*c - a*d)^2*Cos[e + f*x])/(d*(c^2 - d^2)*f*Sqrt[c + d*Sin[e + f*x]]) + ((2*(2*b^2*c^2 - 2*a*b*c*d + (a^2 - b^2)*d^2)*EllipticE[(e - Pi/2 + f*x) /2, (2*d)/(c + d)]*Sqrt[c + d*Sin[e + f*x]])/(d*f*Sqrt[(c + d*Sin[e + f*x] )/(c + d)]) - (4*b*(b*c - a*d)*(c^2 - d^2)*EllipticF[(e - Pi/2 + f*x)/2, ( 2*d)/(c + d)]*Sqrt[(c + d*Sin[e + f*x])/(c + d)])/(d*f*Sqrt[c + d*Sin[e + f*x]]))/(d*(c^2 - d^2))
3.8.34.3.1 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[((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[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^2, x_Symbol] :> Simp[(-(b^2*c^2 - 2*a*b*c*d + a^2*d^2))*Cos[e + f*x]*((a + b*Sin[e + f*x])^(m + 1)/(b*f*(m + 1)*(a^2 - b^2))), x] - Simp[ 1/(b*(m + 1)*(a^2 - b^2)) Int[(a + b*Sin[e + f*x])^(m + 1)*Simp[b*(m + 1) *(2*b*c*d - a*(c^2 + d^2)) + (a^2*d^2 - 2*a*b*c*d*(m + 2) + b^2*(d^2*(m + 1 ) + c^2*(m + 2)))*Sin[e + f*x], x], x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && LtQ[m, -1]
Leaf count of result is larger than twice the leaf count of optimal. \(887\) vs. \(2(284)=568\).
Time = 10.38 (sec) , antiderivative size = 888, normalized size of antiderivative = 3.98
| method | result | size |
| default | \(\frac {\sqrt {-\left (-d \sin \left (f x +e \right )-c \right ) \left (\cos ^{2}\left (f x +e \right )\right )}\, \left (\frac {b \left (\frac {4 d a \left (\frac {c}{d}-1\right ) \sqrt {\frac {c +d \sin \left (f x +e \right )}{c -d}}\, \sqrt {\frac {d \left (1-\sin \left (f x +e \right )\right )}{c +d}}\, \sqrt {\frac {\left (-\sin \left (f x +e \right )-1\right ) d}{c -d}}\, F\left (\sqrt {\frac {c +d \sin \left (f x +e \right )}{c -d}}, \sqrt {\frac {c -d}{c +d}}\right )}{\sqrt {-\left (-d \sin \left (f x +e \right )-c \right ) \left (\cos ^{2}\left (f x +e \right )\right )}}-\frac {2 c b \left (\frac {c}{d}-1\right ) \sqrt {\frac {c +d \sin \left (f x +e \right )}{c -d}}\, \sqrt {\frac {d \left (1-\sin \left (f x +e \right )\right )}{c +d}}\, \sqrt {\frac {\left (-\sin \left (f x +e \right )-1\right ) d}{c -d}}\, F\left (\sqrt {\frac {c +d \sin \left (f x +e \right )}{c -d}}, \sqrt {\frac {c -d}{c +d}}\right )}{\sqrt {-\left (-d \sin \left (f x +e \right )-c \right ) \left (\cos ^{2}\left (f x +e \right )\right )}}+\frac {2 b d \left (\frac {c}{d}-1\right ) \sqrt {\frac {c +d \sin \left (f x +e \right )}{c -d}}\, \sqrt {\frac {d \left (1-\sin \left (f x +e \right )\right )}{c +d}}\, \sqrt {\frac {\left (-\sin \left (f x +e \right )-1\right ) d}{c -d}}\, \left (\left (-\frac {c}{d}-1\right ) E\left (\sqrt {\frac {c +d \sin \left (f x +e \right )}{c -d}}, \sqrt {\frac {c -d}{c +d}}\right )+F\left (\sqrt {\frac {c +d \sin \left (f x +e \right )}{c -d}}, \sqrt {\frac {c -d}{c +d}}\right )\right )}{\sqrt {-\left (-d \sin \left (f x +e \right )-c \right ) \left (\cos ^{2}\left (f x +e \right )\right )}}\right )}{d^{2}}+\frac {\left (d^{2} a^{2}-2 a b c d +b^{2} c^{2}\right ) \left (\frac {2 d \left (\cos ^{2}\left (f x +e \right )\right )}{\left (c^{2}-d^{2}\right ) \sqrt {-\left (-d \sin \left (f x +e \right )-c \right ) \left (\cos ^{2}\left (f x +e \right )\right )}}+\frac {2 c \left (\frac {c}{d}-1\right ) \sqrt {\frac {c +d \sin \left (f x +e \right )}{c -d}}\, \sqrt {\frac {d \left (1-\sin \left (f x +e \right )\right )}{c +d}}\, \sqrt {\frac {\left (-\sin \left (f x +e \right )-1\right ) d}{c -d}}\, F\left (\sqrt {\frac {c +d \sin \left (f x +e \right )}{c -d}}, \sqrt {\frac {c -d}{c +d}}\right )}{\left (c^{2}-d^{2}\right ) \sqrt {-\left (-d \sin \left (f x +e \right )-c \right ) \left (\cos ^{2}\left (f x +e \right )\right )}}+\frac {2 d \left (\frac {c}{d}-1\right ) \sqrt {\frac {c +d \sin \left (f x +e \right )}{c -d}}\, \sqrt {\frac {d \left (1-\sin \left (f x +e \right )\right )}{c +d}}\, \sqrt {\frac {\left (-\sin \left (f x +e \right )-1\right ) d}{c -d}}\, \left (\left (-\frac {c}{d}-1\right ) E\left (\sqrt {\frac {c +d \sin \left (f x +e \right )}{c -d}}, \sqrt {\frac {c -d}{c +d}}\right )+F\left (\sqrt {\frac {c +d \sin \left (f x +e \right )}{c -d}}, \sqrt {\frac {c -d}{c +d}}\right )\right )}{\left (c^{2}-d^{2}\right ) \sqrt {-\left (-d \sin \left (f x +e \right )-c \right ) \left (\cos ^{2}\left (f x +e \right )\right )}}\right )}{d^{2}}\right )}{\cos \left (f x +e \right ) \sqrt {c +d \sin \left (f x +e \right )}\, f}\) | \(888\) |
| parts | \(\text {Expression too large to display}\) | \(1648\) |
(-(-d*sin(f*x+e)-c)*cos(f*x+e)^2)^(1/2)*(b/d^2*(4*d*a*(c/d-1)*((c+d*sin(f* x+e))/(c-d))^(1/2)*(d*(1-sin(f*x+e))/(c+d))^(1/2)*(1/(c-d)*(-sin(f*x+e)-1) *d)^(1/2)/(-(-d*sin(f*x+e)-c)*cos(f*x+e)^2)^(1/2)*EllipticF(((c+d*sin(f*x+ e))/(c-d))^(1/2),((c-d)/(c+d))^(1/2))-2*c*b*(c/d-1)*((c+d*sin(f*x+e))/(c-d ))^(1/2)*(d*(1-sin(f*x+e))/(c+d))^(1/2)*(1/(c-d)*(-sin(f*x+e)-1)*d)^(1/2)/ (-(-d*sin(f*x+e)-c)*cos(f*x+e)^2)^(1/2)*EllipticF(((c+d*sin(f*x+e))/(c-d)) ^(1/2),((c-d)/(c+d))^(1/2))+2*b*d*(c/d-1)*((c+d*sin(f*x+e))/(c-d))^(1/2)*( d*(1-sin(f*x+e))/(c+d))^(1/2)*(1/(c-d)*(-sin(f*x+e)-1)*d)^(1/2)/(-(-d*sin( f*x+e)-c)*cos(f*x+e)^2)^(1/2)*((-c/d-1)*EllipticE(((c+d*sin(f*x+e))/(c-d)) ^(1/2),((c-d)/(c+d))^(1/2))+EllipticF(((c+d*sin(f*x+e))/(c-d))^(1/2),((c-d )/(c+d))^(1/2))))+(a^2*d^2-2*a*b*c*d+b^2*c^2)/d^2*(2*d*cos(f*x+e)^2/(c^2-d ^2)/(-(-d*sin(f*x+e)-c)*cos(f*x+e)^2)^(1/2)+2*c/(c^2-d^2)*(c/d-1)*((c+d*si n(f*x+e))/(c-d))^(1/2)*(d*(1-sin(f*x+e))/(c+d))^(1/2)*(1/(c-d)*(-sin(f*x+e )-1)*d)^(1/2)/(-(-d*sin(f*x+e)-c)*cos(f*x+e)^2)^(1/2)*EllipticF(((c+d*sin( f*x+e))/(c-d))^(1/2),((c-d)/(c+d))^(1/2))+2/(c^2-d^2)*d*(c/d-1)*((c+d*sin( f*x+e))/(c-d))^(1/2)*(d*(1-sin(f*x+e))/(c+d))^(1/2)*(1/(c-d)*(-sin(f*x+e)- 1)*d)^(1/2)/(-(-d*sin(f*x+e)-c)*cos(f*x+e)^2)^(1/2)*((-c/d-1)*EllipticE((( c+d*sin(f*x+e))/(c-d))^(1/2),((c-d)/(c+d))^(1/2))+EllipticF(((c+d*sin(f*x+ e))/(c-d))^(1/2),((c-d)/(c+d))^(1/2)))))/cos(f*x+e)/(c+d*sin(f*x+e))^(1/2) /f
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.13 (sec) , antiderivative size = 795, normalized size of antiderivative = 3.57 \[ \int \frac {(3+b \sin (e+f x))^2}{(c+d \sin (e+f x))^{3/2}} \, dx=\frac {6 \, {\left (b^{2} c^{2} d^{2} - 2 \, a b c d^{3} + a^{2} d^{4}\right )} \sqrt {d \sin \left (f x + e\right ) + c} \cos \left (f x + e\right ) - {\left (\sqrt {2} {\left (4 \, b^{2} c^{3} d - 4 \, a b c^{2} d^{2} + 6 \, a b d^{4} - {\left (a^{2} + 5 \, b^{2}\right )} c d^{3}\right )} \sin \left (f x + e\right ) + \sqrt {2} {\left (4 \, b^{2} c^{4} - 4 \, a b c^{3} d + 6 \, a b c d^{3} - {\left (a^{2} + 5 \, b^{2}\right )} c^{2} d^{2}\right )}\right )} \sqrt {i \, d} {\rm weierstrassPInverse}\left (-\frac {4 \, {\left (4 \, c^{2} - 3 \, d^{2}\right )}}{3 \, d^{2}}, -\frac {8 \, {\left (8 i \, c^{3} - 9 i \, c d^{2}\right )}}{27 \, d^{3}}, \frac {3 \, d \cos \left (f x + e\right ) - 3 i \, d \sin \left (f x + e\right ) - 2 i \, c}{3 \, d}\right ) - {\left (\sqrt {2} {\left (4 \, b^{2} c^{3} d - 4 \, a b c^{2} d^{2} + 6 \, a b d^{4} - {\left (a^{2} + 5 \, b^{2}\right )} c d^{3}\right )} \sin \left (f x + e\right ) + \sqrt {2} {\left (4 \, b^{2} c^{4} - 4 \, a b c^{3} d + 6 \, a b c d^{3} - {\left (a^{2} + 5 \, b^{2}\right )} c^{2} d^{2}\right )}\right )} \sqrt {-i \, d} {\rm weierstrassPInverse}\left (-\frac {4 \, {\left (4 \, c^{2} - 3 \, d^{2}\right )}}{3 \, d^{2}}, -\frac {8 \, {\left (-8 i \, c^{3} + 9 i \, c d^{2}\right )}}{27 \, d^{3}}, \frac {3 \, d \cos \left (f x + e\right ) + 3 i \, d \sin \left (f x + e\right ) + 2 i \, c}{3 \, d}\right ) + 3 \, {\left (\sqrt {2} {\left (-2 i \, b^{2} c^{2} d^{2} + 2 i \, a b c d^{3} - i \, {\left (a^{2} - b^{2}\right )} d^{4}\right )} \sin \left (f x + e\right ) + \sqrt {2} {\left (-2 i \, b^{2} c^{3} d + 2 i \, a b c^{2} d^{2} - i \, {\left (a^{2} - b^{2}\right )} c d^{3}\right )}\right )} \sqrt {i \, d} {\rm weierstrassZeta}\left (-\frac {4 \, {\left (4 \, c^{2} - 3 \, d^{2}\right )}}{3 \, d^{2}}, -\frac {8 \, {\left (8 i \, c^{3} - 9 i \, c d^{2}\right )}}{27 \, d^{3}}, {\rm weierstrassPInverse}\left (-\frac {4 \, {\left (4 \, c^{2} - 3 \, d^{2}\right )}}{3 \, d^{2}}, -\frac {8 \, {\left (8 i \, c^{3} - 9 i \, c d^{2}\right )}}{27 \, d^{3}}, \frac {3 \, d \cos \left (f x + e\right ) - 3 i \, d \sin \left (f x + e\right ) - 2 i \, c}{3 \, d}\right )\right ) + 3 \, {\left (\sqrt {2} {\left (2 i \, b^{2} c^{2} d^{2} - 2 i \, a b c d^{3} + i \, {\left (a^{2} - b^{2}\right )} d^{4}\right )} \sin \left (f x + e\right ) + \sqrt {2} {\left (2 i \, b^{2} c^{3} d - 2 i \, a b c^{2} d^{2} + i \, {\left (a^{2} - b^{2}\right )} c d^{3}\right )}\right )} \sqrt {-i \, d} {\rm weierstrassZeta}\left (-\frac {4 \, {\left (4 \, c^{2} - 3 \, d^{2}\right )}}{3 \, d^{2}}, -\frac {8 \, {\left (-8 i \, c^{3} + 9 i \, c d^{2}\right )}}{27 \, d^{3}}, {\rm weierstrassPInverse}\left (-\frac {4 \, {\left (4 \, c^{2} - 3 \, d^{2}\right )}}{3 \, d^{2}}, -\frac {8 \, {\left (-8 i \, c^{3} + 9 i \, c d^{2}\right )}}{27 \, d^{3}}, \frac {3 \, d \cos \left (f x + e\right ) + 3 i \, d \sin \left (f x + e\right ) + 2 i \, c}{3 \, d}\right )\right )}{3 \, {\left ({\left (c^{2} d^{4} - d^{6}\right )} f \sin \left (f x + e\right ) + {\left (c^{3} d^{3} - c d^{5}\right )} f\right )}} \]
1/3*(6*(b^2*c^2*d^2 - 2*a*b*c*d^3 + a^2*d^4)*sqrt(d*sin(f*x + e) + c)*cos( f*x + e) - (sqrt(2)*(4*b^2*c^3*d - 4*a*b*c^2*d^2 + 6*a*b*d^4 - (a^2 + 5*b^ 2)*c*d^3)*sin(f*x + e) + sqrt(2)*(4*b^2*c^4 - 4*a*b*c^3*d + 6*a*b*c*d^3 - (a^2 + 5*b^2)*c^2*d^2))*sqrt(I*d)*weierstrassPInverse(-4/3*(4*c^2 - 3*d^2) /d^2, -8/27*(8*I*c^3 - 9*I*c*d^2)/d^3, 1/3*(3*d*cos(f*x + e) - 3*I*d*sin(f *x + e) - 2*I*c)/d) - (sqrt(2)*(4*b^2*c^3*d - 4*a*b*c^2*d^2 + 6*a*b*d^4 - (a^2 + 5*b^2)*c*d^3)*sin(f*x + e) + sqrt(2)*(4*b^2*c^4 - 4*a*b*c^3*d + 6*a *b*c*d^3 - (a^2 + 5*b^2)*c^2*d^2))*sqrt(-I*d)*weierstrassPInverse(-4/3*(4* c^2 - 3*d^2)/d^2, -8/27*(-8*I*c^3 + 9*I*c*d^2)/d^3, 1/3*(3*d*cos(f*x + e) + 3*I*d*sin(f*x + e) + 2*I*c)/d) + 3*(sqrt(2)*(-2*I*b^2*c^2*d^2 + 2*I*a*b* c*d^3 - I*(a^2 - b^2)*d^4)*sin(f*x + e) + sqrt(2)*(-2*I*b^2*c^3*d + 2*I*a* b*c^2*d^2 - I*(a^2 - b^2)*c*d^3))*sqrt(I*d)*weierstrassZeta(-4/3*(4*c^2 - 3*d^2)/d^2, -8/27*(8*I*c^3 - 9*I*c*d^2)/d^3, weierstrassPInverse(-4/3*(4*c ^2 - 3*d^2)/d^2, -8/27*(8*I*c^3 - 9*I*c*d^2)/d^3, 1/3*(3*d*cos(f*x + e) - 3*I*d*sin(f*x + e) - 2*I*c)/d)) + 3*(sqrt(2)*(2*I*b^2*c^2*d^2 - 2*I*a*b*c* d^3 + I*(a^2 - b^2)*d^4)*sin(f*x + e) + sqrt(2)*(2*I*b^2*c^3*d - 2*I*a*b*c ^2*d^2 + I*(a^2 - b^2)*c*d^3))*sqrt(-I*d)*weierstrassZeta(-4/3*(4*c^2 - 3* d^2)/d^2, -8/27*(-8*I*c^3 + 9*I*c*d^2)/d^3, weierstrassPInverse(-4/3*(4*c^ 2 - 3*d^2)/d^2, -8/27*(-8*I*c^3 + 9*I*c*d^2)/d^3, 1/3*(3*d*cos(f*x + e) + 3*I*d*sin(f*x + e) + 2*I*c)/d)))/((c^2*d^4 - d^6)*f*sin(f*x + e) + (c^3...
Timed out. \[ \int \frac {(3+b \sin (e+f x))^2}{(c+d \sin (e+f x))^{3/2}} \, dx=\text {Timed out} \]
\[ \int \frac {(3+b \sin (e+f x))^2}{(c+d \sin (e+f x))^{3/2}} \, dx=\int { \frac {{\left (b \sin \left (f x + e\right ) + a\right )}^{2}}{{\left (d \sin \left (f x + e\right ) + c\right )}^{\frac {3}{2}}} \,d x } \]
\[ \int \frac {(3+b \sin (e+f x))^2}{(c+d \sin (e+f x))^{3/2}} \, dx=\int { \frac {{\left (b \sin \left (f x + e\right ) + a\right )}^{2}}{{\left (d \sin \left (f x + e\right ) + c\right )}^{\frac {3}{2}}} \,d x } \]
Timed out. \[ \int \frac {(3+b \sin (e+f x))^2}{(c+d \sin (e+f x))^{3/2}} \, dx=\int \frac {{\left (a+b\,\sin \left (e+f\,x\right )\right )}^2}{{\left (c+d\,\sin \left (e+f\,x\right )\right )}^{3/2}} \,d x \]