Integrand size = 27, antiderivative size = 203 \[ \int \frac {(a+b \sin (e+f x))^2}{\sqrt {c+d \sin (e+f x)}} \, dx=-\frac {2 b^2 \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{3 d f}-\frac {4 b (b c-3 a d) E\left (\frac {1}{2} \left (e-\frac {\pi }{2}+f x\right )|\frac {2 d}{c+d}\right ) \sqrt {c+d \sin (e+f x)}}{3 d^2 f \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}+\frac {2 \left (\left (3 a^2+b^2\right ) d^2+2 b c (b c-3 a d)\right ) \operatorname {EllipticF}\left (\frac {1}{2} \left (e-\frac {\pi }{2}+f x\right ),\frac {2 d}{c+d}\right ) \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}{3 d^2 f \sqrt {c+d \sin (e+f x)}} \] Output:
-2/3*b^2*cos(f*x+e)*(c+d*sin(f*x+e))^(1/2)/d/f+4/3*b*(-3*a*d+b*c)*Elliptic E(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/f/((c+d*sin(f*x+e))/(c+d))^(1/2)+2/3*((3*a^2+b^2)*d^2+2*b*c*(-3*a*d+ b*c))*InverseJacobiAM(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 = 1.84 (sec) , antiderivative size = 173, normalized size of antiderivative = 0.85 \[ \int \frac {(a+b \sin (e+f x))^2}{\sqrt {c+d \sin (e+f x)}} \, dx=-\frac {2 \left (b^2 d \cos (e+f x) (c+d \sin (e+f x))-2 b (c+d) (b c-3 a d) E\left (\frac {1}{4} (-2 e+\pi -2 f x)|\frac {2 d}{c+d}\right ) \sqrt {\frac {c+d \sin (e+f x)}{c+d}}+\left (-6 a b c d+3 a^2 d^2+b^2 \left (2 c^2+d^2\right )\right ) \operatorname {EllipticF}\left (\frac {1}{4} (-2 e+\pi -2 f x),\frac {2 d}{c+d}\right ) \sqrt {\frac {c+d \sin (e+f x)}{c+d}}\right )}{3 d^2 f \sqrt {c+d \sin (e+f x)}} \] Input:
Integrate[(a + b*Sin[e + f*x])^2/Sqrt[c + d*Sin[e + f*x]],x]
Output:
(-2*(b^2*d*Cos[e + f*x]*(c + d*Sin[e + f*x]) - 2*b*(c + d)*(b*c - 3*a*d)*E llipticE[(-2*e + Pi - 2*f*x)/4, (2*d)/(c + d)]*Sqrt[(c + d*Sin[e + f*x])/( c + d)] + (-6*a*b*c*d + 3*a^2*d^2 + b^2*(2*c^2 + d^2))*EllipticF[(-2*e + P i - 2*f*x)/4, (2*d)/(c + d)]*Sqrt[(c + d*Sin[e + f*x])/(c + d)]))/(3*d^2*f *Sqrt[c + d*Sin[e + f*x]])
Time = 0.92 (sec) , antiderivative size = 207, normalized size of antiderivative = 1.02, number of steps used = 12, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.444, Rules used = {3042, 3270, 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.
| \(\displaystyle \int \frac {(a+b \sin (e+f x))^2}{\sqrt {c+d \sin (e+f x)}} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {(a+b \sin (e+f x))^2}{\sqrt {c+d \sin (e+f x)}}dx\) |
| \(\Big \downarrow \) 3270 |
| \(\displaystyle \frac {2 \int \frac {\left (3 a^2+b^2\right ) d-2 b (b c-3 a d) \sin (e+f x)}{2 \sqrt {c+d \sin (e+f x)}}dx}{3 d}-\frac {2 b^2 \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{3 d f}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int \frac {\left (3 a^2+b^2\right ) d-2 b (b c-3 a d) \sin (e+f x)}{\sqrt {c+d \sin (e+f x)}}dx}{3 d}-\frac {2 b^2 \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{3 d f}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\int \frac {\left (3 a^2+b^2\right ) d-2 b (b c-3 a d) \sin (e+f x)}{\sqrt {c+d \sin (e+f x)}}dx}{3 d}-\frac {2 b^2 \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{3 d f}\) |
| \(\Big \downarrow \) 3231 |
| \(\displaystyle \frac {\frac {\left (d^2 \left (3 a^2+b^2\right )+2 b c (b c-3 a d)\right ) \int \frac {1}{\sqrt {c+d \sin (e+f x)}}dx}{d}-\frac {2 b (b c-3 a d) \int \sqrt {c+d \sin (e+f x)}dx}{d}}{3 d}-\frac {2 b^2 \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{3 d f}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {\left (d^2 \left (3 a^2+b^2\right )+2 b c (b c-3 a d)\right ) \int \frac {1}{\sqrt {c+d \sin (e+f x)}}dx}{d}-\frac {2 b (b c-3 a d) \int \sqrt {c+d \sin (e+f x)}dx}{d}}{3 d}-\frac {2 b^2 \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{3 d f}\) |
| \(\Big \downarrow \) 3134 |
| \(\displaystyle \frac {\frac {\left (d^2 \left (3 a^2+b^2\right )+2 b c (b c-3 a d)\right ) \int \frac {1}{\sqrt {c+d \sin (e+f x)}}dx}{d}-\frac {2 b (b c-3 a d) \sqrt {c+d \sin (e+f x)} \int \sqrt {\frac {c}{c+d}+\frac {d \sin (e+f x)}{c+d}}dx}{d \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}}{3 d}-\frac {2 b^2 \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{3 d f}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {\left (d^2 \left (3 a^2+b^2\right )+2 b c (b c-3 a d)\right ) \int \frac {1}{\sqrt {c+d \sin (e+f x)}}dx}{d}-\frac {2 b (b c-3 a d) \sqrt {c+d \sin (e+f x)} \int \sqrt {\frac {c}{c+d}+\frac {d \sin (e+f x)}{c+d}}dx}{d \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}}{3 d}-\frac {2 b^2 \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{3 d f}\) |
| \(\Big \downarrow \) 3132 |
| \(\displaystyle \frac {\frac {\left (d^2 \left (3 a^2+b^2\right )+2 b c (b c-3 a d)\right ) \int \frac {1}{\sqrt {c+d \sin (e+f x)}}dx}{d}-\frac {4 b (b c-3 a d) \sqrt {c+d \sin (e+f x)} E\left (\frac {1}{2} \left (e+f x-\frac {\pi }{2}\right )|\frac {2 d}{c+d}\right )}{d f \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}}{3 d}-\frac {2 b^2 \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{3 d f}\) |
| \(\Big \downarrow \) 3142 |
| \(\displaystyle \frac {\frac {\left (d^2 \left (3 a^2+b^2\right )+2 b c (b c-3 a d)\right ) \sqrt {\frac {c+d \sin (e+f x)}{c+d}} \int \frac {1}{\sqrt {\frac {c}{c+d}+\frac {d \sin (e+f x)}{c+d}}}dx}{d \sqrt {c+d \sin (e+f x)}}-\frac {4 b (b c-3 a d) \sqrt {c+d \sin (e+f x)} E\left (\frac {1}{2} \left (e+f x-\frac {\pi }{2}\right )|\frac {2 d}{c+d}\right )}{d f \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}}{3 d}-\frac {2 b^2 \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{3 d f}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {\left (d^2 \left (3 a^2+b^2\right )+2 b c (b c-3 a d)\right ) \sqrt {\frac {c+d \sin (e+f x)}{c+d}} \int \frac {1}{\sqrt {\frac {c}{c+d}+\frac {d \sin (e+f x)}{c+d}}}dx}{d \sqrt {c+d \sin (e+f x)}}-\frac {4 b (b c-3 a d) \sqrt {c+d \sin (e+f x)} E\left (\frac {1}{2} \left (e+f x-\frac {\pi }{2}\right )|\frac {2 d}{c+d}\right )}{d f \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}}{3 d}-\frac {2 b^2 \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{3 d f}\) |
| \(\Big \downarrow \) 3140 |
| \(\displaystyle \frac {\frac {2 \left (d^2 \left (3 a^2+b^2\right )+2 b c (b c-3 a d)\right ) \sqrt {\frac {c+d \sin (e+f x)}{c+d}} \operatorname {EllipticF}\left (\frac {1}{2} \left (e+f x-\frac {\pi }{2}\right ),\frac {2 d}{c+d}\right )}{d f \sqrt {c+d \sin (e+f x)}}-\frac {4 b (b c-3 a d) \sqrt {c+d \sin (e+f x)} E\left (\frac {1}{2} \left (e+f x-\frac {\pi }{2}\right )|\frac {2 d}{c+d}\right )}{d f \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}}{3 d}-\frac {2 b^2 \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{3 d f}\) |
Input:
Int[(a + b*Sin[e + f*x])^2/Sqrt[c + d*Sin[e + f*x]],x]
Output:
(-2*b^2*Cos[e + f*x]*Sqrt[c + d*Sin[e + f*x]])/(3*d*f) + ((-4*b*(b*c - 3*a *d)*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)]) + (2*((3*a^2 + b^2)*d^2 + 2*b*c* (b*c - 3*a*d))*EllipticF[(e - Pi/2 + f*x)/2, (2*d)/(c + d)]*Sqrt[(c + d*Si n[e + f*x])/(c + d)])/(d*f*Sqrt[c + d*Sin[e + f*x]]))/(3*d)
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[(-d^2)*Cos[e + f*x]*((a + b*Sin[e + f*x]) ^(m + 1)/(b*f*(m + 2))), x] + Simp[1/(b*(m + 2)) Int[(a + b*Sin[e + f*x]) ^m*Simp[b*(d^2*(m + 1) + c^2*(m + 2)) - d*(a*d - 2*b*c*(m + 2))*Sin[e + f*x ], x], x], x] /; FreeQ[{a, b, c, d, e, f, m}, x] && NeQ[b*c - a*d, 0] && Ne Q[a^2 - b^2, 0] && !LtQ[m, -1]
Leaf count of result is larger than twice the leaf count of optimal. \(690\) vs. \(2(192)=384\).
Time = 2.49 (sec) , antiderivative size = 691, normalized size of antiderivative = 3.40
| method | result | size |
| default | \(\frac {\sqrt {-\left (-c -d \sin \left (f x +e \right )\right ) \cos \left (f x +e \right )^{2}}\, \left (\frac {2 a^{2} \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 {d \left (1+\sin \left (f x +e \right )\right )}{c -d}}\, \operatorname {EllipticF}\left (\sqrt {\frac {c +d \sin \left (f x +e \right )}{c -d}}, \sqrt {\frac {c -d}{c +d}}\right )}{\sqrt {-\left (-c -d \sin \left (f x +e \right )\right ) \cos \left (f x +e \right )^{2}}}+b^{2} \left (-\frac {2 \sqrt {-\left (-c -d \sin \left (f x +e \right )\right ) \cos \left (f x +e \right )^{2}}}{3 d}+\frac {2 \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 {d \left (1+\sin \left (f x +e \right )\right )}{c -d}}\, \operatorname {EllipticF}\left (\sqrt {\frac {c +d \sin \left (f x +e \right )}{c -d}}, \sqrt {\frac {c -d}{c +d}}\right )}{3 \sqrt {-\left (-c -d \sin \left (f x +e \right )\right ) \cos \left (f x +e \right )^{2}}}-\frac {4 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 {d \left (1+\sin \left (f x +e \right )\right )}{c -d}}\, \left (\left (-\frac {c}{d}-1\right ) \operatorname {EllipticE}\left (\sqrt {\frac {c +d \sin \left (f x +e \right )}{c -d}}, \sqrt {\frac {c -d}{c +d}}\right )+\operatorname {EllipticF}\left (\sqrt {\frac {c +d \sin \left (f x +e \right )}{c -d}}, \sqrt {\frac {c -d}{c +d}}\right )\right )}{3 d \sqrt {-\left (-c -d \sin \left (f x +e \right )\right ) \cos \left (f x +e \right )^{2}}}\right )+\frac {4 a 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 {d \left (1+\sin \left (f x +e \right )\right )}{c -d}}\, \left (\left (-\frac {c}{d}-1\right ) \operatorname {EllipticE}\left (\sqrt {\frac {c +d \sin \left (f x +e \right )}{c -d}}, \sqrt {\frac {c -d}{c +d}}\right )+\operatorname {EllipticF}\left (\sqrt {\frac {c +d \sin \left (f x +e \right )}{c -d}}, \sqrt {\frac {c -d}{c +d}}\right )\right )}{\sqrt {-\left (-c -d \sin \left (f x +e \right )\right ) \cos \left (f x +e \right )^{2}}}\right )}{\cos \left (f x +e \right ) \sqrt {c +d \sin \left (f x +e \right )}\, f}\) | \(691\) |
| parts | \(\frac {2 a^{2} \left (c -d \right ) \sqrt {\frac {c +d \sin \left (f x +e \right )}{c -d}}\, \sqrt {-\frac {\left (\sin \left (f x +e \right )-1\right ) d}{c +d}}\, \sqrt {-\frac {d \left (1+\sin \left (f x +e \right )\right )}{c -d}}\, \operatorname {EllipticF}\left (\sqrt {\frac {c +d \sin \left (f x +e \right )}{c -d}}, \sqrt {\frac {c -d}{c +d}}\right )}{d \cos \left (f x +e \right ) \sqrt {c +d \sin \left (f x +e \right )}\, f}+\frac {b^{2} \sqrt {-\left (-c -d \sin \left (f x +e \right )\right ) \cos \left (f x +e \right )^{2}}\, \left (-\frac {2 \sqrt {-\left (-c -d \sin \left (f x +e \right )\right ) \cos \left (f x +e \right )^{2}}}{3 d}+\frac {2 \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 {d \left (1+\sin \left (f x +e \right )\right )}{c -d}}\, \operatorname {EllipticF}\left (\sqrt {\frac {c +d \sin \left (f x +e \right )}{c -d}}, \sqrt {\frac {c -d}{c +d}}\right )}{3 \sqrt {-\left (-c -d \sin \left (f x +e \right )\right ) \cos \left (f x +e \right )^{2}}}-\frac {4 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 {d \left (1+\sin \left (f x +e \right )\right )}{c -d}}\, \left (\left (-\frac {c}{d}-1\right ) \operatorname {EllipticE}\left (\sqrt {\frac {c +d \sin \left (f x +e \right )}{c -d}}, \sqrt {\frac {c -d}{c +d}}\right )+\operatorname {EllipticF}\left (\sqrt {\frac {c +d \sin \left (f x +e \right )}{c -d}}, \sqrt {\frac {c -d}{c +d}}\right )\right )}{3 d \sqrt {-\left (-c -d \sin \left (f x +e \right )\right ) \cos \left (f x +e \right )^{2}}}\right )}{\cos \left (f x +e \right ) \sqrt {c +d \sin \left (f x +e \right )}\, f}-\frac {4 a b \left (c -d \right ) \sqrt {\frac {c +d \sin \left (f x +e \right )}{c -d}}\, \sqrt {-\frac {\left (\sin \left (f x +e \right )-1\right ) d}{c +d}}\, \sqrt {-\frac {d \left (1+\sin \left (f x +e \right )\right )}{c -d}}\, \left (\operatorname {EllipticE}\left (\sqrt {\frac {c +d \sin \left (f x +e \right )}{c -d}}, \sqrt {\frac {c -d}{c +d}}\right ) c +\operatorname {EllipticE}\left (\sqrt {\frac {c +d \sin \left (f x +e \right )}{c -d}}, \sqrt {\frac {c -d}{c +d}}\right ) d -\operatorname {EllipticF}\left (\sqrt {\frac {c +d \sin \left (f x +e \right )}{c -d}}, \sqrt {\frac {c -d}{c +d}}\right ) d \right )}{d^{2} \cos \left (f x +e \right ) \sqrt {c +d \sin \left (f x +e \right )}\, f}\) | \(718\) |
Input:
int((a+b*sin(f*x+e))^2/(c+d*sin(f*x+e))^(1/2),x,method=_RETURNVERBOSE)
Output:
(-(-c-d*sin(f*x+e))*cos(f*x+e)^2)^(1/2)*(2*a^2*(c/d-1)*((c+d*sin(f*x+e))/( c-d))^(1/2)*(d*(1-sin(f*x+e))/(c+d))^(1/2)*(-d*(1+sin(f*x+e))/(c-d))^(1/2) /(-(-c-d*sin(f*x+e))*cos(f*x+e)^2)^(1/2)*EllipticF(((c+d*sin(f*x+e))/(c-d) )^(1/2),((c-d)/(c+d))^(1/2))+b^2*(-2/3/d*(-(-c-d*sin(f*x+e))*cos(f*x+e)^2) ^(1/2)+2/3*(c/d-1)*((c+d*sin(f*x+e))/(c-d))^(1/2)*(d*(1-sin(f*x+e))/(c+d)) ^(1/2)*(-d*(1+sin(f*x+e))/(c-d))^(1/2)/(-(-c-d*sin(f*x+e))*cos(f*x+e)^2)^( 1/2)*EllipticF(((c+d*sin(f*x+e))/(c-d))^(1/2),((c-d)/(c+d))^(1/2))-4/3*c/d *(c/d-1)*((c+d*sin(f*x+e))/(c-d))^(1/2)*(d*(1-sin(f*x+e))/(c+d))^(1/2)*(-d *(1+sin(f*x+e))/(c-d))^(1/2)/(-(-c-d*sin(f*x+e))*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))+Ellipti cF(((c+d*sin(f*x+e))/(c-d))^(1/2),((c-d)/(c+d))^(1/2))))+4*a*b*(c/d-1)*((c +d*sin(f*x+e))/(c-d))^(1/2)*(d*(1-sin(f*x+e))/(c+d))^(1/2)*(-d*(1+sin(f*x+ e))/(c-d))^(1/2)/(-(-c-d*sin(f*x+e))*cos(f*x+e)^2)^(1/2)*((-c/d-1)*Ellipti cE(((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 complex when optimal does not.
Time = 0.11 (sec) , antiderivative size = 463, normalized size of antiderivative = 2.28 \[ \int \frac {(a+b \sin (e+f x))^2}{\sqrt {c+d \sin (e+f x)}} \, dx=-\frac {2 \, {\left (3 \, \sqrt {d \sin \left (f x + e\right ) + c} b^{2} d^{2} \cos \left (f x + e\right ) - {\left (4 \, b^{2} c^{2} - 12 \, a b c d + 3 \, {\left (3 \, a^{2} + b^{2}\right )} d^{2}\right )} \sqrt {\frac {1}{2} 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 (4 \, b^{2} c^{2} - 12 \, a b c d + 3 \, {\left (3 \, a^{2} + b^{2}\right )} d^{2}\right )} \sqrt {-\frac {1}{2} 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 ) + 6 \, {\left (-i \, b^{2} c d + 3 i \, a b d^{2}\right )} \sqrt {\frac {1}{2} 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 ) + 6 \, {\left (i \, b^{2} c d - 3 i \, a b d^{2}\right )} \sqrt {-\frac {1}{2} 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 )\right )}}{9 \, d^{3} f} \] Input:
integrate((a+b*sin(f*x+e))^2/(c+d*sin(f*x+e))^(1/2),x, algorithm="fricas")
Output:
-2/9*(3*sqrt(d*sin(f*x + e) + c)*b^2*d^2*cos(f*x + e) - (4*b^2*c^2 - 12*a* b*c*d + 3*(3*a^2 + b^2)*d^2)*sqrt(1/2*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) - (4*b^2*c^2 - 12*a*b*c*d + 3*(3*a^2 + b^2)*d ^2)*sqrt(-1/2*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) + 6*(-I*b^2*c*d + 3*I*a*b*d^2)*sqrt(1/2*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)) + 6*(I*b^2*c*d - 3*I*a*b*d^2)*sqrt( -1/2*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))) /(d^3*f)
\[ \int \frac {(a+b \sin (e+f x))^2}{\sqrt {c+d \sin (e+f x)}} \, dx=\int \frac {\left (a + b \sin {\left (e + f x \right )}\right )^{2}}{\sqrt {c + d \sin {\left (e + f x \right )}}}\, dx \] Input:
integrate((a+b*sin(f*x+e))**2/(c+d*sin(f*x+e))**(1/2),x)
Output:
Integral((a + b*sin(e + f*x))**2/sqrt(c + d*sin(e + f*x)), x)
\[ \int \frac {(a+b \sin (e+f x))^2}{\sqrt {c+d \sin (e+f x)}} \, dx=\int { \frac {{\left (b \sin \left (f x + e\right ) + a\right )}^{2}}{\sqrt {d \sin \left (f x + e\right ) + c}} \,d x } \] Input:
integrate((a+b*sin(f*x+e))^2/(c+d*sin(f*x+e))^(1/2),x, algorithm="maxima")
Output:
integrate((b*sin(f*x + e) + a)^2/sqrt(d*sin(f*x + e) + c), x)
\[ \int \frac {(a+b \sin (e+f x))^2}{\sqrt {c+d \sin (e+f x)}} \, dx=\int { \frac {{\left (b \sin \left (f x + e\right ) + a\right )}^{2}}{\sqrt {d \sin \left (f x + e\right ) + c}} \,d x } \] Input:
integrate((a+b*sin(f*x+e))^2/(c+d*sin(f*x+e))^(1/2),x, algorithm="giac")
Output:
integrate((b*sin(f*x + e) + a)^2/sqrt(d*sin(f*x + e) + c), x)
Timed out. \[ \int \frac {(a+b \sin (e+f x))^2}{\sqrt {c+d \sin (e+f x)}} \, dx=\int \frac {{\left (a+b\,\sin \left (e+f\,x\right )\right )}^2}{\sqrt {c+d\,\sin \left (e+f\,x\right )}} \,d x \] Input:
int((a + b*sin(e + f*x))^2/(c + d*sin(e + f*x))^(1/2),x)
Output:
int((a + b*sin(e + f*x))^2/(c + d*sin(e + f*x))^(1/2), x)
\[ \int \frac {(a+b \sin (e+f x))^2}{\sqrt {c+d \sin (e+f x)}} \, dx=\left (\int \frac {\sqrt {\sin \left (f x +e \right ) d +c}}{\sin \left (f x +e \right ) d +c}d x \right ) a^{2}+\left (\int \frac {\sqrt {\sin \left (f x +e \right ) d +c}\, \sin \left (f x +e \right )^{2}}{\sin \left (f x +e \right ) d +c}d x \right ) b^{2}+2 \left (\int \frac {\sqrt {\sin \left (f x +e \right ) d +c}\, \sin \left (f x +e \right )}{\sin \left (f x +e \right ) d +c}d x \right ) a b \] Input:
int((a+b*sin(f*x+e))^2/(c+d*sin(f*x+e))^(1/2),x)
Output:
int(sqrt(sin(e + f*x)*d + c)/(sin(e + f*x)*d + c),x)*a**2 + int((sqrt(sin( e + f*x)*d + c)*sin(e + f*x)**2)/(sin(e + f*x)*d + c),x)*b**2 + 2*int((sqr t(sin(e + f*x)*d + c)*sin(e + f*x))/(sin(e + f*x)*d + c),x)*a*b