\(\int \frac {3+3 \sin (e+f x)}{(c+d \sin (e+f x))^{5/2}} \, dx\) [487]

Optimal result

Integrand size = 25, antiderivative size = 225 \[ \int \frac {3+3 \sin (e+f x)}{(c+d \sin (e+f x))^{5/2}} \, dx=-\frac {2 \cos (e+f x)}{(c+d) f (c+d \sin (e+f x))^{3/2}}-\frac {2 (c-3 d) \cos (e+f x)}{(c-d) (c+d)^2 f \sqrt {c+d \sin (e+f x)}}-\frac {2 (c-3 d) E\left (\frac {1}{2} \left (e-\frac {\pi }{2}+f x\right )|\frac {2 d}{c+d}\right ) \sqrt {c+d \sin (e+f x)}}{(c-d) d (c+d)^2 f \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}+\frac {2 \operatorname {EllipticF}\left (\frac {1}{2} \left (e-\frac {\pi }{2}+f x\right ),\frac {2 d}{c+d}\right ) \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}{d (c+d) f \sqrt {c+d \sin (e+f x)}} \]

[Out]

-2/3*a*cos(f*x+e)/(c+d)/f/(c+d*sin(f*x+e))^(3/2)-2/3*a*(c-3*d)*cos(f*x+e)/(c-d)/(c+d)^2/f/(c+d*sin(f*x+e))^(1/
2)+2/3*a*(c-3*d)*(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)/(c-d)/d/(c+d)^2/f/((c+d*sin(f*x+e))/(c+d))^(1/2)-2/3*a*(s
in(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/(c+d)/f/(c+d*sin(f*x+e))^(1/2)

Rubi [A] (verified)

Time = 0.25 (sec) , antiderivative size = 237, normalized size of antiderivative = 1.05, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.240, Rules used = {2833, 2831, 2742, 2740, 2734, 2732} \[ \int \frac {3+3 \sin (e+f x)}{(c+d \sin (e+f x))^{5/2}} \, dx=-\frac {2 a (c-3 d) \cos (e+f x)}{3 f (c-d) (c+d)^2 \sqrt {c+d \sin (e+f x)}}-\frac {2 a \cos (e+f x)}{3 f (c+d) (c+d \sin (e+f x))^{3/2}}+\frac {2 a \sqrt {\frac {c+d \sin (e+f x)}{c+d}} \operatorname {EllipticF}\left (\frac {1}{2} \left (e+f x-\frac {\pi }{2}\right ),\frac {2 d}{c+d}\right )}{3 d f (c+d) \sqrt {c+d \sin (e+f x)}}-\frac {2 a (c-3 d) \sqrt {c+d \sin (e+f x)} E\left (\frac {1}{2} \left (e+f x-\frac {\pi }{2}\right )|\frac {2 d}{c+d}\right )}{3 d f (c-d) (c+d)^2 \sqrt {\frac {c+d \sin (e+f x)}{c+d}}} \]

[In]

Int[(a + a*Sin[e + f*x])/(c + d*Sin[e + f*x])^(5/2),x]

[Out]

(-2*a*Cos[e + f*x])/(3*(c + d)*f*(c + d*Sin[e + f*x])^(3/2)) - (2*a*(c - 3*d)*Cos[e + f*x])/(3*(c - d)*(c + d)
^2*f*Sqrt[c + d*Sin[e + f*x]]) - (2*a*(c - 3*d)*EllipticE[(e - Pi/2 + f*x)/2, (2*d)/(c + d)]*Sqrt[c + d*Sin[e
+ f*x]])/(3*(c - d)*d*(c + d)^2*f*Sqrt[(c + d*Sin[e + f*x])/(c + d)]) + (2*a*EllipticF[(e - Pi/2 + f*x)/2, (2*
d)/(c + d)]*Sqrt[(c + d*Sin[e + f*x])/(c + d)])/(3*d*(c + d)*f*Sqrt[c + d*Sin[e + f*x]])

Rule 2732

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]

Rule 2734

Int[Sqrt[(a_) + (b_.)*sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Dist[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]

Rule 2740

Int[1/Sqrt[(a_) + (b_.)*sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[(2/(d*Sqrt[a + b]))*EllipticF[(1/2)*(c - P
i/2 + d*x), 2*(b/(a + b))], x] /; FreeQ[{a, b, c, d}, x] && NeQ[a^2 - b^2, 0] && GtQ[a + b, 0]

Rule 2742

Int[1/Sqrt[(a_) + (b_.)*sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Dist[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]

Rule 2831

Int[((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])/Sqrt[(a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]], x_Symbol] :> Dist[(b*c
- a*d)/b, Int[1/Sqrt[a + b*Sin[e + f*x]], x], x] + Dist[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]

Rule 2833

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] + Dist[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]

Rubi steps \begin{align*} \text {integral}& = -\frac {2 a \cos (e+f x)}{3 (c+d) f (c+d \sin (e+f x))^{3/2}}-\frac {2 \int \frac {-\frac {3}{2} a (c-d)-\frac {1}{2} a (c-d) \sin (e+f x)}{(c+d \sin (e+f x))^{3/2}} \, dx}{3 \left (c^2-d^2\right )} \\ & = -\frac {2 a \cos (e+f x)}{3 (c+d) f (c+d \sin (e+f x))^{3/2}}-\frac {2 a (c-3 d) \cos (e+f x)}{3 (c-d) (c+d)^2 f \sqrt {c+d \sin (e+f x)}}+\frac {4 \int \frac {\frac {1}{4} a (c-d) (3 c-d)-\frac {1}{4} a (c-3 d) (c-d) \sin (e+f x)}{\sqrt {c+d \sin (e+f x)}} \, dx}{3 \left (c^2-d^2\right )^2} \\ & = -\frac {2 a \cos (e+f x)}{3 (c+d) f (c+d \sin (e+f x))^{3/2}}-\frac {2 a (c-3 d) \cos (e+f x)}{3 (c-d) (c+d)^2 f \sqrt {c+d \sin (e+f x)}}-\frac {(a (c-3 d)) \int \sqrt {c+d \sin (e+f x)} \, dx}{3 (c-d) d (c+d)^2}+\frac {a \int \frac {1}{\sqrt {c+d \sin (e+f x)}} \, dx}{3 d (c+d)} \\ & = -\frac {2 a \cos (e+f x)}{3 (c+d) f (c+d \sin (e+f x))^{3/2}}-\frac {2 a (c-3 d) \cos (e+f x)}{3 (c-d) (c+d)^2 f \sqrt {c+d \sin (e+f x)}}-\frac {\left (a (c-3 d) \sqrt {c+d \sin (e+f x)}\right ) \int \sqrt {\frac {c}{c+d}+\frac {d \sin (e+f x)}{c+d}} \, dx}{3 (c-d) d (c+d)^2 \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}+\frac {\left (a \sqrt {\frac {c+d \sin (e+f x)}{c+d}}\right ) \int \frac {1}{\sqrt {\frac {c}{c+d}+\frac {d \sin (e+f x)}{c+d}}} \, dx}{3 d (c+d) \sqrt {c+d \sin (e+f x)}} \\ & = -\frac {2 a \cos (e+f x)}{3 (c+d) f (c+d \sin (e+f x))^{3/2}}-\frac {2 a (c-3 d) \cos (e+f x)}{3 (c-d) (c+d)^2 f \sqrt {c+d \sin (e+f x)}}-\frac {2 a (c-3 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 (c-d) d (c+d)^2 f \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}+\frac {2 a \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 (c+d) f \sqrt {c+d \sin (e+f x)}} \\ \end{align*}

Mathematica [C] (warning: unable to verify)

Result contains higher order function than in optimal. Order 6 vs. order 4 in optimal.

Time = 7.06 (sec) , antiderivative size = 1870, normalized size of antiderivative = 8.31 \[ \int \frac {3+3 \sin (e+f x)}{(c+d \sin (e+f x))^{5/2}} \, dx=3 \left (\frac {(1+\sin (e+f x)) \sqrt {c+d \sin (e+f x)} \left (-\frac {2 (c-3 d) \csc (e) \sec (e)}{3 (c-d) d (c+d)^2 f}+\frac {2 \csc (e) (c \cos (e)+d \sin (f x))}{3 d (c+d) f (c+d \sin (e+f x))^2}-\frac {2 \csc (e) (3 c \cos (e)-d \cos (e)-c \sin (f x)+3 d \sin (f x))}{3 (c-d) (c+d)^2 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 {c \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 )}{3 (c-d) (c+d)^2 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 {d \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) (c+d)^2 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)}}}{3 (c-d) (c+d)^2 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)}}+\frac {2 c \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)}}}{(c-d) d (c+d)^2 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 ) \]

[In]

Integrate[(3 + 3*Sin[e + f*x])/(c + d*Sin[e + f*x])^(5/2),x]

[Out]

3*(((1 + Sin[e + f*x])*Sqrt[c + d*Sin[e + f*x]]*((-2*(c - 3*d)*Csc[e]*Sec[e])/(3*(c - d)*d*(c + d)^2*f) + (2*C
sc[e]*(c*Cos[e] + d*Sin[f*x]))/(3*d*(c + d)*f*(c + d*Sin[e + f*x])^2) - (2*Csc[e]*(3*c*Cos[e] - d*Cos[e] - c*S
in[f*x] + 3*d*Sin[f*x]))/(3*(c - d)*(c + d)^2*f*(c + d*Sin[e + f*x]))))/(Cos[e/2 + (f*x)/2] + Sin[e/2 + (f*x)/
2])^2 - (c*Sec[e]*(1 + Sin[e + f*x])*(-((AppellF1[-1/2, -1/2, -1/2, 1/2, -((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])))), -((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 + Cot[e]^2])/Sqrt[c + d*Cos[f*x - ArcTan
[Cot[e]]]*Sqrt[1 + Cot[e]^2]*Sin[e]]))/(3*(c - d)*(c + d)^2*f*(Cos[e/2 + (f*x)/2] + Sin[e/2 + (f*x)/2])^2) + (
d*Sec[e]*(1 + Sin[e + f*x])*(-((AppellF1[-1/2, -1/2, -1/2, 1/2, -((Csc[e]*(c + d*Cos[f*x - ArcTan[Cot[e]]]*Sqr
t[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[C
ot[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 + Cot[e]^2])/Sqrt[c + d*Cos[f*x - ArcTan[Cot[e]]]
*Sqrt[1 + Cot[e]^2]*Sin[e]]))/((c - d)*(c + d)^2*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[e]*Sec[f*x + ArcTan[Tan[e]]]*(1 +
Sin[e + f*x])*Sqrt[(d*Sqrt[1 + Tan[e]^2] - d*Sin[f*x + ArcTan[Tan[e]]]*Sqrt[1 + Tan[e]^2])/(c*Sec[e] + d*Sqrt
[1 + Tan[e]^2])]*Sqrt[(d*Sqrt[1 + Tan[e]^2] + d*Sin[f*x + ArcTan[Tan[e]]]*Sqrt[1 + Tan[e]^2])/(-(c*Sec[e]) + d
*Sqrt[1 + Tan[e]^2])]*Sqrt[c + d*Cos[e]*Sin[f*x + ArcTan[Tan[e]]]*Sqrt[1 + Tan[e]^2]])/(3*(c - d)*(c + d)^2*f*
(Cos[e/2 + (f*x)/2] + Sin[e/2 + (f*x)/2])^2*Sqrt[1 + Tan[e]^2]) + (2*c*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[e]*Sec[f*x + ArcTan[Tan[e]]]*(1 + Sin[e + f*x])*Sqrt[(d*Sqrt[1
+ Tan[e]^2] - d*Sin[f*x + ArcTan[Tan[e]]]*Sqrt[1 + Tan[e]^2])/(c*Sec[e] + d*Sqrt[1 + Tan[e]^2])]*Sqrt[(d*Sqrt[
1 + Tan[e]^2] + d*Sin[f*x + ArcTan[Tan[e]]]*Sqrt[1 + Tan[e]^2])/(-(c*Sec[e]) + d*Sqrt[1 + Tan[e]^2])]*Sqrt[c +
d*Cos[e]*Sin[f*x + ArcTan[Tan[e]]]*Sqrt[1 + Tan[e]^2]])/((c - d)*d*(c + d)^2*f*(Cos[e/2 + (f*x)/2] + Sin[e/2
+ (f*x)/2])^2*Sqrt[1 + Tan[e]^2]))

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(883\) vs. \(2(283)=566\).

Time = 5.83 (sec) , antiderivative size = 884, normalized size of antiderivative = 3.93

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 )}\, a \left (\frac {\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 )}}}{d}+\frac {\left (-c +d \right ) \left (\frac {2 \sqrt {-\left (-d \sin \left (f x +e \right )-c \right ) \left (\cos ^{2}\left (f x +e \right )\right )}}{3 \left (c^{2}-d^{2}\right ) d \left (\sin \left (f x +e \right )+\frac {c}{d}\right )^{2}}+\frac {8 d \left (\cos ^{2}\left (f x +e \right )\right ) c}{3 \left (c^{2}-d^{2}\right )^{2} \sqrt {-\left (-d \sin \left (f x +e \right )-c \right ) \left (\cos ^{2}\left (f x +e \right )\right )}}+\frac {2 \left (3 c^{2}+d^{2}\right ) \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 (3 c^{4}-6 c^{2} d^{2}+3 d^{4}\right ) \sqrt {-\left (-d \sin \left (f x +e \right )-c \right ) \left (\cos ^{2}\left (f x +e \right )\right )}}+\frac {8 c 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 )}{3 \left (c^{2}-d^{2}\right )^{2} \sqrt {-\left (-d \sin \left (f x +e \right )-c \right ) \left (\cos ^{2}\left (f x +e \right )\right )}}\right )}{d}\right )}{\cos \left (f x +e \right ) \sqrt {c +d \sin \left (f x +e \right )}\, f}\)	\(884\)
parts	\(\text {Expression too large to display}\)	\(1379\)

[In]

int((a+a*sin(f*x+e))/(c+d*sin(f*x+e))^(5/2),x,method=_RETURNVERBOSE)

[Out]

(-(-d*sin(f*x+e)-c)*cos(f*x+e)^2)^(1/2)*a*(1/d*(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*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^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))))+(-c+d)/d*(2/3/(c^2-d^2)/d*(-(-d
*sin(f*x+e)-c)*cos(f*x+e)^2)^(1/2)/(sin(f*x+e)+c/d)^2+8/3*d*cos(f*x+e)^2/(c^2-d^2)^2*c/(-(-d*sin(f*x+e)-c)*cos
(f*x+e)^2)^(1/2)+2*(3*c^2+d^2)/(3*c^4-6*c^2*d^2+3*d^4)*(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))+8/3*c*d/(c^2-d^2)^2*(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)*Ellip
ticE(((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

Fricas [C] (verification not implemented)

Result contains higher order function than in optimal. Order 9 vs. order 4.

Time = 0.14 (sec) , antiderivative size = 920, normalized size of antiderivative = 4.09 \[ \int \frac {3+3 \sin (e+f x)}{(c+d \sin (e+f x))^{5/2}} \, dx=\text {Too large to display} \]

[In]

integrate((a+a*sin(f*x+e))/(c+d*sin(f*x+e))^(5/2),x, algorithm="fricas")

[Out]

1/9*((sqrt(2)*(2*a*c^2*d^2 + 3*a*c*d^3 - 3*a*d^4)*cos(f*x + e)^2 - 2*sqrt(2)*(2*a*c^3*d + 3*a*c^2*d^2 - 3*a*c*
d^3)*sin(f*x + e) - sqrt(2)*(2*a*c^4 + 3*a*c^3*d - a*c^2*d^2 + 3*a*c*d^3 - 3*a*d^4))*sqrt(I*d)*weierstrassPInv
erse(-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^2*d^2 + 3*a*c*d^3 - 3*a*d^4)*cos(f*x + e)^2 - 2*sqrt(2)*(2*a*c^3*d + 3*a*c^2*d^2 -
3*a*c*d^3)*sin(f*x + e) - sqrt(2)*(2*a*c^4 + 3*a*c^3*d - a*c^2*d^2 + 3*a*c*d^3 - 3*a*d^4))*sqrt(-I*d)*weierstr
assPInverse(-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)*(-I*a*c*d^3 + 3*I*a*d^4)*cos(f*x + e)^2 + 2*sqrt(2)*(I*a*c^2*d^2 - 3*I*a*c*d^3)*
sin(f*x + e) + sqrt(2)*(I*a*c^3*d - 3*I*a*c^2*d^2 + I*a*c*d^3 - 3*I*a*d^4))*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)*(I*a*c*d^3 - 3*I*a*d^
4)*cos(f*x + e)^2 + 2*sqrt(2)*(-I*a*c^2*d^2 + 3*I*a*c*d^3)*sin(f*x + e) + sqrt(2)*(-I*a*c^3*d + 3*I*a*c^2*d^2
- I*a*c*d^3 + 3*I*a*d^4))*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)) + 6*((a*c*d^3 - 3*a*d^4)*cos(f*x + e)*sin(f*x + e) + (2*a*c^2*d^2 - 3*a*c*d^3 - a
*d^4)*cos(f*x + e))*sqrt(d*sin(f*x + e) + c))/((c^3*d^4 + c^2*d^5 - c*d^6 - d^7)*f*cos(f*x + e)^2 - 2*(c^4*d^3
+ c^3*d^4 - c^2*d^5 - c*d^6)*f*sin(f*x + e) - (c^5*d^2 + c^4*d^3 - c*d^6 - d^7)*f)

Sympy [F(-1)]

Timed out. \[ \int \frac {3+3 \sin (e+f x)}{(c+d \sin (e+f x))^{5/2}} \, dx=\text {Timed out} \]

[In]

integrate((a+a*sin(f*x+e))/(c+d*sin(f*x+e))**(5/2),x)

[Out]

Timed out

Maxima [F]

\[ \int \frac {3+3 \sin (e+f x)}{(c+d \sin (e+f x))^{5/2}} \, dx=\int { \frac {a \sin \left (f x + e\right ) + a}{{\left (d \sin \left (f x + e\right ) + c\right )}^{\frac {5}{2}}} \,d x } \]

[In]

integrate((a+a*sin(f*x+e))/(c+d*sin(f*x+e))^(5/2),x, algorithm="maxima")

[Out]

integrate((a*sin(f*x + e) + a)/(d*sin(f*x + e) + c)^(5/2), x)

Giac [F]

\[ \int \frac {3+3 \sin (e+f x)}{(c+d \sin (e+f x))^{5/2}} \, dx=\int { \frac {a \sin \left (f x + e\right ) + a}{{\left (d \sin \left (f x + e\right ) + c\right )}^{\frac {5}{2}}} \,d x } \]

[In]

integrate((a+a*sin(f*x+e))/(c+d*sin(f*x+e))^(5/2),x, algorithm="giac")

[Out]

integrate((a*sin(f*x + e) + a)/(d*sin(f*x + e) + c)^(5/2), x)

Mupad [F(-1)]

Timed out. \[ \int \frac {3+3 \sin (e+f x)}{(c+d \sin (e+f x))^{5/2}} \, dx=\int \frac {a+a\,\sin \left (e+f\,x\right )}{{\left (c+d\,\sin \left (e+f\,x\right )\right )}^{5/2}} \,d x \]

[In]

int((a + a*sin(e + f*x))/(c + d*sin(e + f*x))^(5/2),x)

[Out]

int((a + a*sin(e + f*x))/(c + d*sin(e + f*x))^(5/2), x)