Integrand size = 25, antiderivative size = 166 \[ \int \frac {3+3 \sin (e+f x)}{(c+d \sin (e+f x))^{3/2}} \, dx=-\frac {6 \cos (e+f x)}{(c+d) f \sqrt {c+d \sin (e+f x)}}-\frac {6 E\left (\frac {1}{2} \left (e-\frac {\pi }{2}+f x\right )|\frac {2 d}{c+d}\right ) \sqrt {c+d \sin (e+f x)}}{d (c+d) f \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}+\frac {6 \operatorname {EllipticF}\left (\frac {1}{2} \left (e-\frac {\pi }{2}+f x\right ),\frac {2 d}{c+d}\right ) \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}{d f \sqrt {c+d \sin (e+f x)}} \]
-2*a*cos(f*x+e)/(c+d)/f/(c+d*sin(f*x+e))^(1/2)+2*a*(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/(c+d)/f/((c+d*sin(f*x+e ))/(c+d))^(1/2)-2*a*(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/f/(c+d*sin(f*x+e))^(1/2)
Result contains higher order function than in optimal. Order 6 vs. order 4 in optimal.
Time = 6.66 (sec) , antiderivative size = 938, normalized size of antiderivative = 5.65 \[ \int \frac {3+3 \sin (e+f x)}{(c+d \sin (e+f x))^{3/2}} \, dx=3 \left (\frac {(1+\sin (e+f x)) \sqrt {c+d \sin (e+f x)} \left (-\frac {2 \csc (e) \sec (e)}{d (c+d) f}+\frac {2 \csc (e) (c \cos (e)+d \sin (f x))}{d (c+d) f (c+d \sin (e+f x))}\right )}{\left (\cos \left (\frac {e}{2}+\frac {f x}{2}\right )+\sin \left (\frac {e}{2}+\frac {f x}{2}\right )\right )^2}-\frac {\sec (e) (1+\sin (e+f x)) \left (-\frac {\operatorname {AppellF1}\left (-\frac {1}{2},-\frac {1}{2},-\frac {1}{2},\frac {1}{2},-\frac {\csc (e) \left (c+d \cos (f x-\arctan (\cot (e))) \sqrt {1+\cot ^2(e)} \sin (e)\right )}{d \sqrt {1+\cot ^2(e)} \left (1-\frac {c \csc (e)}{d \sqrt {1+\cot ^2(e)}}\right )},-\frac {\csc (e) \left (c+d \cos (f x-\arctan (\cot (e))) \sqrt {1+\cot ^2(e)} \sin (e)\right )}{d \sqrt {1+\cot ^2(e)} \left (-1-\frac {c \csc (e)}{d \sqrt {1+\cot ^2(e)}}\right )}\right ) \cot (e) \sin (f x-\arctan (\cot (e)))}{\sqrt {1+\cot ^2(e)} \sqrt {\frac {d \sqrt {1+\cot ^2(e)}+d \cos (f x-\arctan (\cot (e))) \sqrt {1+\cot ^2(e)}}{d \sqrt {1+\cot ^2(e)}-c \csc (e)}} \sqrt {\frac {d \sqrt {1+\cot ^2(e)}-d \cos (f x-\arctan (\cot (e))) \sqrt {1+\cot ^2(e)}}{d \sqrt {1+\cot ^2(e)}+c \csc (e)}} \sqrt {c+d \cos (f x-\arctan (\cot (e))) \sqrt {1+\cot ^2(e)} \sin (e)}}-\frac {\frac {2 d \sin (e) \left (c+d \cos (f x-\arctan (\cot (e))) \sqrt {1+\cot ^2(e)} \sin (e)\right )}{d^2 \cos ^2(e)+d^2 \sin ^2(e)}-\frac {\cot (e) \sin (f x-\arctan (\cot (e)))}{\sqrt {1+\cot ^2(e)}}}{\sqrt {c+d \cos (f x-\arctan (\cot (e))) \sqrt {1+\cot ^2(e)} \sin (e)}}\right )}{(c+d) f \left (\cos \left (\frac {e}{2}+\frac {f x}{2}\right )+\sin \left (\frac {e}{2}+\frac {f x}{2}\right )\right )^2}+\frac {2 \operatorname {AppellF1}\left (\frac {1}{2},\frac {1}{2},\frac {1}{2},\frac {3}{2},-\frac {\sec (e) \left (c+d \cos (e) \sin (f x+\arctan (\tan (e))) \sqrt {1+\tan ^2(e)}\right )}{d \sqrt {1+\tan ^2(e)} \left (1-\frac {c \sec (e)}{d \sqrt {1+\tan ^2(e)}}\right )},-\frac {\sec (e) \left (c+d \cos (e) \sin (f x+\arctan (\tan (e))) \sqrt {1+\tan ^2(e)}\right )}{d \sqrt {1+\tan ^2(e)} \left (-1-\frac {c \sec (e)}{d \sqrt {1+\tan ^2(e)}}\right )}\right ) \sec (e) \sec (f x+\arctan (\tan (e))) (1+\sin (e+f x)) \sqrt {\frac {d \sqrt {1+\tan ^2(e)}-d \sin (f x+\arctan (\tan (e))) \sqrt {1+\tan ^2(e)}}{c \sec (e)+d \sqrt {1+\tan ^2(e)}}} \sqrt {\frac {d \sqrt {1+\tan ^2(e)}+d \sin (f x+\arctan (\tan (e))) \sqrt {1+\tan ^2(e)}}{-c \sec (e)+d \sqrt {1+\tan ^2(e)}}} \sqrt {c+d \cos (e) \sin (f x+\arctan (\tan (e))) \sqrt {1+\tan ^2(e)}}}{d (c+d) f \left (\cos \left (\frac {e}{2}+\frac {f x}{2}\right )+\sin \left (\frac {e}{2}+\frac {f x}{2}\right )\right )^2 \sqrt {1+\tan ^2(e)}}\right ) \]
3*(((1 + Sin[e + f*x])*Sqrt[c + d*Sin[e + f*x]]*((-2*Csc[e]*Sec[e])/(d*(c + d)*f) + (2*Csc[e]*(c*Cos[e] + d*Sin[f*x]))/(d*(c + d)*f*(c + d*Sin[e + f *x]))))/(Cos[e/2 + (f*x)/2] + Sin[e/2 + (f*x)/2])^2 - (Sec[e]*(1 + Sin[e + f*x])*(-((AppellF1[-1/2, -1/2, -1/2, 1/2, -((Csc[e]*(c + d*Cos[f*x - ArcT an[Cot[e]]]*Sqrt[1 + Cot[e]^2]*Sin[e]))/(d*Sqrt[1 + Cot[e]^2]*(1 - (c*Csc[ e])/(d*Sqrt[1 + Cot[e]^2])))), -((Csc[e]*(c + d*Cos[f*x - ArcTan[Cot[e]]]* Sqrt[1 + Cot[e]^2]*Sin[e]))/(d*Sqrt[1 + Cot[e]^2]*(-1 - (c*Csc[e])/(d*Sqrt [1 + Cot[e]^2]))))]*Cot[e]*Sin[f*x - ArcTan[Cot[e]]])/(Sqrt[1 + Cot[e]^2]* Sqrt[(d*Sqrt[1 + Cot[e]^2] + d*Cos[f*x - ArcTan[Cot[e]]]*Sqrt[1 + Cot[e]^2 ])/(d*Sqrt[1 + Cot[e]^2] - c*Csc[e])]*Sqrt[(d*Sqrt[1 + Cot[e]^2] - d*Cos[f *x - ArcTan[Cot[e]]]*Sqrt[1 + Cot[e]^2])/(d*Sqrt[1 + Cot[e]^2] + c*Csc[e]) ]*Sqrt[c + d*Cos[f*x - ArcTan[Cot[e]]]*Sqrt[1 + Cot[e]^2]*Sin[e]])) - ((2* d*Sin[e]*(c + d*Cos[f*x - ArcTan[Cot[e]]]*Sqrt[1 + Cot[e]^2]*Sin[e]))/(d^2 *Cos[e]^2 + d^2*Sin[e]^2) - (Cot[e]*Sin[f*x - ArcTan[Cot[e]]])/Sqrt[1 + Co t[e]^2])/Sqrt[c + d*Cos[f*x - ArcTan[Cot[e]]]*Sqrt[1 + Cot[e]^2]*Sin[e]])) /((c + d)*f*(Cos[e/2 + (f*x)/2] + Sin[e/2 + (f*x)/2])^2) + (2*AppellF1[1/2 , 1/2, 1/2, 3/2, -((Sec[e]*(c + d*Cos[e]*Sin[f*x + ArcTan[Tan[e]]]*Sqrt[1 + Tan[e]^2]))/(d*Sqrt[1 + Tan[e]^2]*(1 - (c*Sec[e])/(d*Sqrt[1 + Tan[e]^2]) ))), -((Sec[e]*(c + d*Cos[e]*Sin[f*x + ArcTan[Tan[e]]]*Sqrt[1 + Tan[e]^2]) )/(d*Sqrt[1 + Tan[e]^2]*(-1 - (c*Sec[e])/(d*Sqrt[1 + Tan[e]^2]))))]*Sec...
Time = 0.84 (sec) , antiderivative size = 191, normalized size of antiderivative = 1.15, number of steps used = 12, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.480, Rules used = {3042, 3233, 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.
| \(\displaystyle \int \frac {a \sin (e+f x)+a}{(c+d \sin (e+f x))^{3/2}} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {a \sin (e+f x)+a}{(c+d \sin (e+f x))^{3/2}}dx\) |
| \(\Big \downarrow \) 3233 |
| \(\displaystyle -\frac {2 \int -\frac {a (c-d)-a (c-d) \sin (e+f x)}{2 \sqrt {c+d \sin (e+f x)}}dx}{c^2-d^2}-\frac {2 a \cos (e+f x)}{f (c+d) \sqrt {c+d \sin (e+f x)}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int \frac {a (c-d)-a (c-d) \sin (e+f x)}{\sqrt {c+d \sin (e+f x)}}dx}{c^2-d^2}-\frac {2 a \cos (e+f x)}{f (c+d) \sqrt {c+d \sin (e+f x)}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\int \frac {a (c-d)-a (c-d) \sin (e+f x)}{\sqrt {c+d \sin (e+f x)}}dx}{c^2-d^2}-\frac {2 a \cos (e+f x)}{f (c+d) \sqrt {c+d \sin (e+f x)}}\) |
| \(\Big \downarrow \) 3231 |
| \(\displaystyle \frac {\frac {a \left (c^2-d^2\right ) \int \frac {1}{\sqrt {c+d \sin (e+f x)}}dx}{d}-\frac {a (c-d) \int \sqrt {c+d \sin (e+f x)}dx}{d}}{c^2-d^2}-\frac {2 a \cos (e+f x)}{f (c+d) \sqrt {c+d \sin (e+f x)}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {a \left (c^2-d^2\right ) \int \frac {1}{\sqrt {c+d \sin (e+f x)}}dx}{d}-\frac {a (c-d) \int \sqrt {c+d \sin (e+f x)}dx}{d}}{c^2-d^2}-\frac {2 a \cos (e+f x)}{f (c+d) \sqrt {c+d \sin (e+f x)}}\) |
| \(\Big \downarrow \) 3134 |
| \(\displaystyle \frac {\frac {a \left (c^2-d^2\right ) \int \frac {1}{\sqrt {c+d \sin (e+f x)}}dx}{d}-\frac {a (c-d) \sqrt {c+d \sin (e+f x)} \int \sqrt {\frac {c}{c+d}+\frac {d \sin (e+f x)}{c+d}}dx}{d \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}}{c^2-d^2}-\frac {2 a \cos (e+f x)}{f (c+d) \sqrt {c+d \sin (e+f x)}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {a \left (c^2-d^2\right ) \int \frac {1}{\sqrt {c+d \sin (e+f x)}}dx}{d}-\frac {a (c-d) \sqrt {c+d \sin (e+f x)} \int \sqrt {\frac {c}{c+d}+\frac {d \sin (e+f x)}{c+d}}dx}{d \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}}{c^2-d^2}-\frac {2 a \cos (e+f x)}{f (c+d) \sqrt {c+d \sin (e+f x)}}\) |
| \(\Big \downarrow \) 3132 |
| \(\displaystyle \frac {\frac {a \left (c^2-d^2\right ) \int \frac {1}{\sqrt {c+d \sin (e+f x)}}dx}{d}-\frac {2 a (c-d) \sqrt {c+d \sin (e+f x)} E\left (\frac {1}{2} \left (e+f x-\frac {\pi }{2}\right )|\frac {2 d}{c+d}\right )}{d f \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}}{c^2-d^2}-\frac {2 a \cos (e+f x)}{f (c+d) \sqrt {c+d \sin (e+f x)}}\) |
| \(\Big \downarrow \) 3142 |
| \(\displaystyle \frac {\frac {a \left (c^2-d^2\right ) \sqrt {\frac {c+d \sin (e+f x)}{c+d}} \int \frac {1}{\sqrt {\frac {c}{c+d}+\frac {d \sin (e+f x)}{c+d}}}dx}{d \sqrt {c+d \sin (e+f x)}}-\frac {2 a (c-d) \sqrt {c+d \sin (e+f x)} E\left (\frac {1}{2} \left (e+f x-\frac {\pi }{2}\right )|\frac {2 d}{c+d}\right )}{d f \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}}{c^2-d^2}-\frac {2 a \cos (e+f x)}{f (c+d) \sqrt {c+d \sin (e+f x)}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {a \left (c^2-d^2\right ) \sqrt {\frac {c+d \sin (e+f x)}{c+d}} \int \frac {1}{\sqrt {\frac {c}{c+d}+\frac {d \sin (e+f x)}{c+d}}}dx}{d \sqrt {c+d \sin (e+f x)}}-\frac {2 a (c-d) \sqrt {c+d \sin (e+f x)} E\left (\frac {1}{2} \left (e+f x-\frac {\pi }{2}\right )|\frac {2 d}{c+d}\right )}{d f \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}}{c^2-d^2}-\frac {2 a \cos (e+f x)}{f (c+d) \sqrt {c+d \sin (e+f x)}}\) |
| \(\Big \downarrow \) 3140 |
| \(\displaystyle \frac {\frac {2 a \left (c^2-d^2\right ) \sqrt {\frac {c+d \sin (e+f x)}{c+d}} \operatorname {EllipticF}\left (\frac {1}{2} \left (e+f x-\frac {\pi }{2}\right ),\frac {2 d}{c+d}\right )}{d f \sqrt {c+d \sin (e+f x)}}-\frac {2 a (c-d) \sqrt {c+d \sin (e+f x)} E\left (\frac {1}{2} \left (e+f x-\frac {\pi }{2}\right )|\frac {2 d}{c+d}\right )}{d f \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}}{c^2-d^2}-\frac {2 a \cos (e+f x)}{f (c+d) \sqrt {c+d \sin (e+f x)}}\) |
(-2*a*Cos[e + f*x])/((c + d)*f*Sqrt[c + d*Sin[e + f*x]]) + ((-2*a*(c - 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*a*(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]]))/(c^2 - d^2)
3.5.86.3.1 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_)]), x_Symbol] :> Simp[(-(b*c - a*d))*Cos[e + f*x]*((a + b*Sin[e + f*x])^(m + 1)/(f*(m + 1)*(a^2 - b^2))), x] + Simp[1/((m + 1)*(a^2 - b^2)) Int[(a + b*Sin[e + f*x])^(m + 1)*Simp[(a*c - b*d)*(m + 1) - (b*c - a*d)*( 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] && IntegerQ[2*m]
Time = 2.57 (sec) , antiderivative size = 246, normalized size of antiderivative = 1.48
| method | result | size |
| default | \(\frac {2 \left (\sqrt {\frac {c +d \sin \left (f x +e \right )}{c -d}}\, \sqrt {-\frac {\left (\sin \left (f x +e \right )-1\right ) d}{c +d}}\, \sqrt {-\frac {d \left (\sin \left (f x +e \right )+1\right )}{c -d}}\, E\left (\sqrt {\frac {c +d \sin \left (f x +e \right )}{c -d}}, \sqrt {\frac {c -d}{c +d}}\right ) c^{2}-\sqrt {\frac {c +d \sin \left (f x +e \right )}{c -d}}\, \sqrt {-\frac {\left (\sin \left (f x +e \right )-1\right ) d}{c +d}}\, \sqrt {-\frac {d \left (\sin \left (f x +e \right )+1\right )}{c -d}}\, E\left (\sqrt {\frac {c +d \sin \left (f x +e \right )}{c -d}}, \sqrt {\frac {c -d}{c +d}}\right ) d^{2}+\left (\sin ^{2}\left (f x +e \right )\right ) d^{2}-d^{2}\right ) a}{d^{2} \left (c +d \right ) \cos \left (f x +e \right ) \sqrt {c +d \sin \left (f x +e \right )}\, f}\) | \(246\) |
| parts | \(\frac {2 a \left (c^{2} \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 (\sin \left (f x +e \right )+1\right )}{c -d}}\, F\left (\sqrt {\frac {c +d \sin \left (f x +e \right )}{c -d}}, \sqrt {\frac {c -d}{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 (\sin \left (f x +e \right )+1\right )}{c -d}}\, F\left (\sqrt {\frac {c +d \sin \left (f x +e \right )}{c -d}}, \sqrt {\frac {c -d}{c +d}}\right ) d^{2}-\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 (\sin \left (f x +e \right )+1\right )}{c -d}}\, E\left (\sqrt {\frac {c +d \sin \left (f x +e \right )}{c -d}}, \sqrt {\frac {c -d}{c +d}}\right ) c^{2}+\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 (\sin \left (f x +e \right )+1\right )}{c -d}}\, E\left (\sqrt {\frac {c +d \sin \left (f x +e \right )}{c -d}}, \sqrt {\frac {c -d}{c +d}}\right ) d^{2}-\left (\sin ^{2}\left (f x +e \right )\right ) d^{2}+d^{2}\right )}{d \left (c^{2}-d^{2}\right ) \cos \left (f x +e \right ) \sqrt {c +d \sin \left (f x +e \right )}\, f}-\frac {2 a \left (\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 (\sin \left (f x +e \right )+1\right )}{c -d}}\, F\left (\sqrt {\frac {c +d \sin \left (f x +e \right )}{c -d}}, \sqrt {\frac {c -d}{c +d}}\right ) c^{2} d -\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 (\sin \left (f x +e \right )+1\right )}{c -d}}\, F\left (\sqrt {\frac {c +d \sin \left (f x +e \right )}{c -d}}, \sqrt {\frac {c -d}{c +d}}\right ) d^{3}-\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 (\sin \left (f x +e \right )+1\right )}{c -d}}\, E\left (\sqrt {\frac {c +d \sin \left (f x +e \right )}{c -d}}, \sqrt {\frac {c -d}{c +d}}\right ) c^{3}+\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 (\sin \left (f x +e \right )+1\right )}{c -d}}\, E\left (\sqrt {\frac {c +d \sin \left (f x +e \right )}{c -d}}, \sqrt {\frac {c -d}{c +d}}\right ) c \,d^{2}-\left (\sin ^{2}\left (f x +e \right )\right ) c \,d^{2}+c \,d^{2}\right )}{d^{2} \left (c^{2}-d^{2}\right ) \cos \left (f x +e \right ) \sqrt {c +d \sin \left (f x +e \right )}\, f}\) | \(893\) |
2*(((c+d*sin(f*x+e))/(c-d))^(1/2)*(-(sin(f*x+e)-1)*d/(c+d))^(1/2)*(-d*(sin (f*x+e)+1)/(c-d))^(1/2)*EllipticE(((c+d*sin(f*x+e))/(c-d))^(1/2),((c-d)/(c +d))^(1/2))*c^2-((c+d*sin(f*x+e))/(c-d))^(1/2)*(-(sin(f*x+e)-1)*d/(c+d))^( 1/2)*(-d*(sin(f*x+e)+1)/(c-d))^(1/2)*EllipticE(((c+d*sin(f*x+e))/(c-d))^(1 /2),((c-d)/(c+d))^(1/2))*d^2+sin(f*x+e)^2*d^2-d^2)/d^2*a/(c+d)/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 = 0.11 (sec) , antiderivative size = 528, normalized size of antiderivative = 3.18 \[ \int \frac {3+3 \sin (e+f x)}{(c+d \sin (e+f x))^{3/2}} \, dx=-\frac {6 \, \sqrt {d \sin \left (f x + e\right ) + c} a d^{2} \cos \left (f x + e\right ) - {\left (\sqrt {2} {\left (2 \, a c d + 3 \, a d^{2}\right )} \sin \left (f x + e\right ) + \sqrt {2} {\left (2 \, a c^{2} + 3 \, a c d\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 (2 \, a c d + 3 \, a d^{2}\right )} \sin \left (f x + e\right ) + \sqrt {2} {\left (2 \, a c^{2} + 3 \, a c d\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 (-i \, \sqrt {2} a d^{2} \sin \left (f x + e\right ) - i \, \sqrt {2} a c d\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 (i \, \sqrt {2} a d^{2} \sin \left (f x + e\right ) + i \, \sqrt {2} a c d\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 d^{3} + d^{4}\right )} f \sin \left (f x + e\right ) + {\left (c^{2} d^{2} + c d^{3}\right )} f\right )}} \]
-1/3*(6*sqrt(d*sin(f*x + e) + c)*a*d^2*cos(f*x + e) - (sqrt(2)*(2*a*c*d + 3*a*d^2)*sin(f*x + e) + sqrt(2)*(2*a*c^2 + 3*a*c*d))*sqrt(I*d)*weierstrass PInverse(-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)*(2*a*c*d + 3*a *d^2)*sin(f*x + e) + sqrt(2)*(2*a*c^2 + 3*a*c*d))*sqrt(-I*d)*weierstrassPI nverse(-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*(-I*sqrt(2)*a*d^2*sin( f*x + e) - I*sqrt(2)*a*c*d)*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*(I*sqrt(2)*a*d^2*sin(f*x + e) + I*sqrt(2)*a* c*d)*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*d^3 + d^4)*f*sin(f*x + e) + (c^2*d^2 + c*d^3)*f)
Timed out. \[ \int \frac {3+3 \sin (e+f x)}{(c+d \sin (e+f x))^{3/2}} \, dx=\text {Timed out} \]
\[ \int \frac {3+3 \sin (e+f x)}{(c+d \sin (e+f x))^{3/2}} \, dx=\int { \frac {a \sin \left (f x + e\right ) + a}{{\left (d \sin \left (f x + e\right ) + c\right )}^{\frac {3}{2}}} \,d x } \]
\[ \int \frac {3+3 \sin (e+f x)}{(c+d \sin (e+f x))^{3/2}} \, dx=\int { \frac {a \sin \left (f x + e\right ) + a}{{\left (d \sin \left (f x + e\right ) + c\right )}^{\frac {3}{2}}} \,d x } \]
Timed out. \[ \int \frac {3+3 \sin (e+f x)}{(c+d \sin (e+f x))^{3/2}} \, dx=\int \frac {a+a\,\sin \left (e+f\,x\right )}{{\left (c+d\,\sin \left (e+f\,x\right )\right )}^{3/2}} \,d x \]