\(\int \frac {(g \cos (e+f x))^{5/2} \sin ^3(e+f x)}{a+b \sin (e+f x)} \, dx\) [1383]

Optimal result

Integrand size = 33, antiderivative size = 610 \[ \int \frac {(g \cos (e+f x))^{5/2} \sin ^3(e+f x)}{a+b \sin (e+f x)} \, dx=-\frac {a^3 \left (-a^2+b^2\right )^{3/4} g^{5/2} \arctan \left (\frac {\sqrt {b} \sqrt {g \cos (e+f x)}}{\sqrt [4]{-a^2+b^2} \sqrt {g}}\right )}{b^{11/2} f}+\frac {a^3 \left (-a^2+b^2\right )^{3/4} g^{5/2} \text {arctanh}\left (\frac {\sqrt {b} \sqrt {g \cos (e+f x)}}{\sqrt [4]{-a^2+b^2} \sqrt {g}}\right )}{b^{11/2} f}-\frac {2 a^3 g (g \cos (e+f x))^{3/2}}{3 b^4 f}+\frac {2 a (g \cos (e+f x))^{7/2}}{7 b^2 f g}-\frac {2 a^4 g^2 \sqrt {g \cos (e+f x)} E\left (\left .\frac {1}{2} (e+f x)\right |2\right )}{b^5 f \sqrt {\cos (e+f x)}}+\frac {6 a^2 g^2 \sqrt {g \cos (e+f x)} E\left (\left .\frac {1}{2} (e+f x)\right |2\right )}{5 b^3 f \sqrt {\cos (e+f x)}}+\frac {4 g^2 \sqrt {g \cos (e+f x)} E\left (\left .\frac {1}{2} (e+f x)\right |2\right )}{15 b f \sqrt {\cos (e+f x)}}+\frac {a^4 \left (a^2-b^2\right ) g^3 \sqrt {\cos (e+f x)} \operatorname {EllipticPi}\left (\frac {2 b}{b-\sqrt {-a^2+b^2}},\frac {1}{2} (e+f x),2\right )}{b^6 \left (b-\sqrt {-a^2+b^2}\right ) f \sqrt {g \cos (e+f x)}}+\frac {a^4 \left (a^2-b^2\right ) g^3 \sqrt {\cos (e+f x)} \operatorname {EllipticPi}\left (\frac {2 b}{b+\sqrt {-a^2+b^2}},\frac {1}{2} (e+f x),2\right )}{b^6 \left (b+\sqrt {-a^2+b^2}\right ) f \sqrt {g \cos (e+f x)}}+\frac {2 a^2 g (g \cos (e+f x))^{3/2} \sin (e+f x)}{5 b^3 f}+\frac {4 g (g \cos (e+f x))^{3/2} \sin (e+f x)}{45 b f}-\frac {2 (g \cos (e+f x))^{7/2} \sin (e+f x)}{9 b f g} \]

[Out]

-a^3*(-a^2+b^2)^(3/4)*g^(5/2)*arctan(b^(1/2)*(g*cos(f*x+e))^(1/2)/(-a^2+b^2)^(1/4)/g^(1/2))/b^(11/2)/f+a^3*(-a
^2+b^2)^(3/4)*g^(5/2)*arctanh(b^(1/2)*(g*cos(f*x+e))^(1/2)/(-a^2+b^2)^(1/4)/g^(1/2))/b^(11/2)/f-2/3*a^3*g*(g*c
os(f*x+e))^(3/2)/b^4/f+2/7*a*(g*cos(f*x+e))^(7/2)/b^2/f/g+2/5*a^2*g*(g*cos(f*x+e))^(3/2)*sin(f*x+e)/b^3/f+4/45
*g*(g*cos(f*x+e))^(3/2)*sin(f*x+e)/b/f-2/9*(g*cos(f*x+e))^(7/2)*sin(f*x+e)/b/f/g+a^4*(a^2-b^2)*g^3*(cos(1/2*f*
x+1/2*e)^2)^(1/2)/cos(1/2*f*x+1/2*e)*EllipticPi(sin(1/2*f*x+1/2*e),2*b/(b-(-a^2+b^2)^(1/2)),2^(1/2))*cos(f*x+e
)^(1/2)/b^6/f/(b-(-a^2+b^2)^(1/2))/(g*cos(f*x+e))^(1/2)+a^4*(a^2-b^2)*g^3*(cos(1/2*f*x+1/2*e)^2)^(1/2)/cos(1/2
*f*x+1/2*e)*EllipticPi(sin(1/2*f*x+1/2*e),2*b/(b+(-a^2+b^2)^(1/2)),2^(1/2))*cos(f*x+e)^(1/2)/b^6/f/(b+(-a^2+b^
2)^(1/2))/(g*cos(f*x+e))^(1/2)-2*a^4*g^2*(cos(1/2*f*x+1/2*e)^2)^(1/2)/cos(1/2*f*x+1/2*e)*EllipticE(sin(1/2*f*x
+1/2*e),2^(1/2))*(g*cos(f*x+e))^(1/2)/b^5/f/cos(f*x+e)^(1/2)+6/5*a^2*g^2*(cos(1/2*f*x+1/2*e)^2)^(1/2)/cos(1/2*
f*x+1/2*e)*EllipticE(sin(1/2*f*x+1/2*e),2^(1/2))*(g*cos(f*x+e))^(1/2)/b^3/f/cos(f*x+e)^(1/2)+4/15*g^2*(cos(1/2
*f*x+1/2*e)^2)^(1/2)/cos(1/2*f*x+1/2*e)*EllipticE(sin(1/2*f*x+1/2*e),2^(1/2))*(g*cos(f*x+e))^(1/2)/b/f/cos(f*x
+e)^(1/2)

Rubi [A] (verified)

Time = 0.97 (sec) , antiderivative size = 610, normalized size of antiderivative = 1.00, number of steps used = 24, number of rules used = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.485, Rules used = {2977, 2715, 2721, 2719, 2645, 30, 2648, 2774, 2946, 2780, 2886, 2884, 335, 304, 211, 214} \[ \int \frac {(g \cos (e+f x))^{5/2} \sin ^3(e+f x)}{a+b \sin (e+f x)} \, dx=-\frac {2 a^4 g^2 E\left (\left .\frac {1}{2} (e+f x)\right |2\right ) \sqrt {g \cos (e+f x)}}{b^5 f \sqrt {\cos (e+f x)}}-\frac {2 a^3 g (g \cos (e+f x))^{3/2}}{3 b^4 f}+\frac {6 a^2 g^2 E\left (\left .\frac {1}{2} (e+f x)\right |2\right ) \sqrt {g \cos (e+f x)}}{5 b^3 f \sqrt {\cos (e+f x)}}+\frac {2 a^2 g \sin (e+f x) (g \cos (e+f x))^{3/2}}{5 b^3 f}+\frac {a^4 g^3 \left (a^2-b^2\right ) \sqrt {\cos (e+f x)} \operatorname {EllipticPi}\left (\frac {2 b}{b-\sqrt {b^2-a^2}},\frac {1}{2} (e+f x),2\right )}{b^6 f \left (b-\sqrt {b^2-a^2}\right ) \sqrt {g \cos (e+f x)}}+\frac {a^4 g^3 \left (a^2-b^2\right ) \sqrt {\cos (e+f x)} \operatorname {EllipticPi}\left (\frac {2 b}{b+\sqrt {b^2-a^2}},\frac {1}{2} (e+f x),2\right )}{b^6 f \left (\sqrt {b^2-a^2}+b\right ) \sqrt {g \cos (e+f x)}}-\frac {a^3 g^{5/2} \left (b^2-a^2\right )^{3/4} \arctan \left (\frac {\sqrt {b} \sqrt {g \cos (e+f x)}}{\sqrt {g} \sqrt [4]{b^2-a^2}}\right )}{b^{11/2} f}+\frac {a^3 g^{5/2} \left (b^2-a^2\right )^{3/4} \text {arctanh}\left (\frac {\sqrt {b} \sqrt {g \cos (e+f x)}}{\sqrt {g} \sqrt [4]{b^2-a^2}}\right )}{b^{11/2} f}+\frac {2 a (g \cos (e+f x))^{7/2}}{7 b^2 f g}+\frac {4 g^2 E\left (\left .\frac {1}{2} (e+f x)\right |2\right ) \sqrt {g \cos (e+f x)}}{15 b f \sqrt {\cos (e+f x)}}-\frac {2 \sin (e+f x) (g \cos (e+f x))^{7/2}}{9 b f g}+\frac {4 g \sin (e+f x) (g \cos (e+f x))^{3/2}}{45 b f} \]

[In]

Int[((g*Cos[e + f*x])^(5/2)*Sin[e + f*x]^3)/(a + b*Sin[e + f*x]),x]

[Out]

-((a^3*(-a^2 + b^2)^(3/4)*g^(5/2)*ArcTan[(Sqrt[b]*Sqrt[g*Cos[e + f*x]])/((-a^2 + b^2)^(1/4)*Sqrt[g])])/(b^(11/
2)*f)) + (a^3*(-a^2 + b^2)^(3/4)*g^(5/2)*ArcTanh[(Sqrt[b]*Sqrt[g*Cos[e + f*x]])/((-a^2 + b^2)^(1/4)*Sqrt[g])])
/(b^(11/2)*f) - (2*a^3*g*(g*Cos[e + f*x])^(3/2))/(3*b^4*f) + (2*a*(g*Cos[e + f*x])^(7/2))/(7*b^2*f*g) - (2*a^4
*g^2*Sqrt[g*Cos[e + f*x]]*EllipticE[(e + f*x)/2, 2])/(b^5*f*Sqrt[Cos[e + f*x]]) + (6*a^2*g^2*Sqrt[g*Cos[e + f*
x]]*EllipticE[(e + f*x)/2, 2])/(5*b^3*f*Sqrt[Cos[e + f*x]]) + (4*g^2*Sqrt[g*Cos[e + f*x]]*EllipticE[(e + f*x)/
2, 2])/(15*b*f*Sqrt[Cos[e + f*x]]) + (a^4*(a^2 - b^2)*g^3*Sqrt[Cos[e + f*x]]*EllipticPi[(2*b)/(b - Sqrt[-a^2 +
b^2]), (e + f*x)/2, 2])/(b^6*(b - Sqrt[-a^2 + b^2])*f*Sqrt[g*Cos[e + f*x]]) + (a^4*(a^2 - b^2)*g^3*Sqrt[Cos[e
+ f*x]]*EllipticPi[(2*b)/(b + Sqrt[-a^2 + b^2]), (e + f*x)/2, 2])/(b^6*(b + Sqrt[-a^2 + b^2])*f*Sqrt[g*Cos[e
+ f*x]]) + (2*a^2*g*(g*Cos[e + f*x])^(3/2)*Sin[e + f*x])/(5*b^3*f) + (4*g*(g*Cos[e + f*x])^(3/2)*Sin[e + f*x])
/(45*b*f) - (2*(g*Cos[e + f*x])^(7/2)*Sin[e + f*x])/(9*b*f*g)

Rule 30

Int[(x_)^(m_.), x_Symbol] :> Simp[x^(m + 1)/(m + 1), x] /; FreeQ[m, x] && NeQ[m, -1]

Rule 211

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[a/b, 2]/a)*ArcTan[x/Rt[a/b, 2]], x] /; FreeQ[{a, b}, x]
&& PosQ[a/b]

Rule 214

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-a/b, 2]/a)*ArcTanh[x/Rt[-a/b, 2]], x] /; FreeQ[{a, b},
x] && NegQ[a/b]

Rule 304

Int[(x_)^2/((a_) + (b_.)*(x_)^4), x_Symbol] :> With[{r = Numerator[Rt[-a/b, 2]], s = Denominator[Rt[-a/b, 2]]}
, Dist[s/(2*b), Int[1/(r + s*x^2), x], x] - Dist[s/(2*b), Int[1/(r - s*x^2), x], x]] /; FreeQ[{a, b}, x] &&  !
GtQ[a/b, 0]

Rule 335

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> With[{k = Denominator[m]}, Dist[k/c, Subst[I
nt[x^(k*(m + 1) - 1)*(a + b*(x^(k*n)/c^n))^p, x], x, (c*x)^(1/k)], x]] /; FreeQ[{a, b, c, p}, x] && IGtQ[n, 0]
&& FractionQ[m] && IntBinomialQ[a, b, c, n, m, p, x]

Rule 2645

Int[(cos[(e_.) + (f_.)*(x_)]*(a_.))^(m_.)*sin[(e_.) + (f_.)*(x_)]^(n_.), x_Symbol] :> Dist[-(a*f)^(-1), Subst[
Int[x^m*(1 - x^2/a^2)^((n - 1)/2), x], x, a*Cos[e + f*x]], x] /; FreeQ[{a, e, f, m}, x] && IntegerQ[(n - 1)/2]
&&  !(IntegerQ[(m - 1)/2] && GtQ[m, 0] && LeQ[m, n])

Rule 2648

Int[(cos[(e_.) + (f_.)*(x_)]*(b_.))^(n_)*((a_.)*sin[(e_.) + (f_.)*(x_)])^(m_), x_Symbol] :> Simp[(-a)*(b*Cos[e
+ f*x])^(n + 1)*((a*Sin[e + f*x])^(m - 1)/(b*f*(m + n))), x] + Dist[a^2*((m - 1)/(m + n)), Int[(b*Cos[e + f*x
])^n*(a*Sin[e + f*x])^(m - 2), x], x] /; FreeQ[{a, b, e, f, n}, x] && GtQ[m, 1] && NeQ[m + n, 0] && IntegersQ[
2*m, 2*n]

Rule 2715

Int[((b_.)*sin[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[(-b)*Cos[c + d*x]*((b*Sin[c + d*x])^(n - 1)/(d*n))
, x] + Dist[b^2*((n - 1)/n), Int[(b*Sin[c + d*x])^(n - 2), x], x] /; FreeQ[{b, c, d}, x] && GtQ[n, 1] && Integ
erQ[2*n]

Rule 2719

Int[Sqrt[sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[(2/d)*EllipticE[(1/2)*(c - Pi/2 + d*x), 2], x] /; FreeQ[{
c, d}, x]

Rule 2721

Int[((b_)*sin[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Dist[(b*Sin[c + d*x])^n/Sin[c + d*x]^n, Int[Sin[c + d*x]
^n, x], x] /; FreeQ[{b, c, d}, x] && LtQ[-1, n, 1] && IntegerQ[2*n]

Rule 2774

Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_), x_Symbol] :> Simp[g*(g*C
os[e + f*x])^(p - 1)*((a + b*Sin[e + f*x])^(m + 1)/(b*f*(m + p))), x] + Dist[g^2*((p - 1)/(b*(m + p))), Int[(g
*Cos[e + f*x])^(p - 2)*(a + b*Sin[e + f*x])^m*(b + a*Sin[e + f*x]), x], x] /; FreeQ[{a, b, e, f, g, m}, x] &&
NeQ[a^2 - b^2, 0] && GtQ[p, 1] && NeQ[m + p, 0] && IntegersQ[2*m, 2*p]

Rule 2780

Int[Sqrt[cos[(e_.) + (f_.)*(x_)]*(g_.)]/((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]), x_Symbol] :> With[{q = Rt[-a^2
+ b^2, 2]}, Dist[a*(g/(2*b)), Int[1/(Sqrt[g*Cos[e + f*x]]*(q + b*Cos[e + f*x])), x], x] + (-Dist[a*(g/(2*b)),
Int[1/(Sqrt[g*Cos[e + f*x]]*(q - b*Cos[e + f*x])), x], x] + Dist[b*(g/f), Subst[Int[Sqrt[x]/(g^2*(a^2 - b^2)
+ b^2*x^2), x], x, g*Cos[e + f*x]], x])] /; FreeQ[{a, b, e, f, g}, x] && NeQ[a^2 - b^2, 0]

Rule 2884

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

Rule 2886

Int[1/(((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])*Sqrt[(c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)]]), x_Symbol] :> Dist
[Sqrt[(c + d*Sin[e + f*x])/(c + d)]/Sqrt[c + d*Sin[e + f*x]], Int[1/((a + b*Sin[e + f*x])*Sqrt[c/(c + d) + (d/
(c + d))*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] && N
eQ[c^2 - d^2, 0] &&  !GtQ[c + d, 0]

Rule 2946

Int[((cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)]))/((a_) + (b_.)*sin[(e_.) + (
f_.)*(x_)]), x_Symbol] :> Dist[d/b, Int[(g*Cos[e + f*x])^p, x], x] + Dist[(b*c - a*d)/b, Int[(g*Cos[e + f*x])^
p/(a + b*Sin[e + f*x]), x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] && NeQ[a^2 - b^2, 0]

Rule 2977

Int[((cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*sin[(e_.) + (f_.)*(x_)]^(n_))/((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])
, x_Symbol] :> Int[ExpandTrig[(g*cos[e + f*x])^p, sin[e + f*x]^n/(a + b*sin[e + f*x]), x], x] /; FreeQ[{a, b,
e, f, g, p}, x] && NeQ[a^2 - b^2, 0] && IntegerQ[n] && (LtQ[n, 0] || IGtQ[p + 1/2, 0])

Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {a^2 (g \cos (e+f x))^{5/2}}{b^3}-\frac {a (g \cos (e+f x))^{5/2} \sin (e+f x)}{b^2}+\frac {(g \cos (e+f x))^{5/2} \sin ^2(e+f x)}{b}-\frac {a^3 (g \cos (e+f x))^{5/2}}{b^3 (a+b \sin (e+f x))}\right ) \, dx \\ & = \frac {a^2 \int (g \cos (e+f x))^{5/2} \, dx}{b^3}-\frac {a^3 \int \frac {(g \cos (e+f x))^{5/2}}{a+b \sin (e+f x)} \, dx}{b^3}-\frac {a \int (g \cos (e+f x))^{5/2} \sin (e+f x) \, dx}{b^2}+\frac {\int (g \cos (e+f x))^{5/2} \sin ^2(e+f x) \, dx}{b} \\ & = -\frac {2 a^3 g (g \cos (e+f x))^{3/2}}{3 b^4 f}+\frac {2 a^2 g (g \cos (e+f x))^{3/2} \sin (e+f x)}{5 b^3 f}-\frac {2 (g \cos (e+f x))^{7/2} \sin (e+f x)}{9 b f g}+\frac {2 \int (g \cos (e+f x))^{5/2} \, dx}{9 b}+\frac {a \text {Subst}\left (\int x^{5/2} \, dx,x,g \cos (e+f x)\right )}{b^2 f g}-\frac {\left (a^3 g^2\right ) \int \frac {\sqrt {g \cos (e+f x)} (b+a \sin (e+f x))}{a+b \sin (e+f x)} \, dx}{b^4}+\frac {\left (3 a^2 g^2\right ) \int \sqrt {g \cos (e+f x)} \, dx}{5 b^3} \\ & = -\frac {2 a^3 g (g \cos (e+f x))^{3/2}}{3 b^4 f}+\frac {2 a (g \cos (e+f x))^{7/2}}{7 b^2 f g}+\frac {2 a^2 g (g \cos (e+f x))^{3/2} \sin (e+f x)}{5 b^3 f}+\frac {4 g (g \cos (e+f x))^{3/2} \sin (e+f x)}{45 b f}-\frac {2 (g \cos (e+f x))^{7/2} \sin (e+f x)}{9 b f g}-\frac {\left (a^4 g^2\right ) \int \sqrt {g \cos (e+f x)} \, dx}{b^5}+\frac {\left (2 g^2\right ) \int \sqrt {g \cos (e+f x)} \, dx}{15 b}+\frac {\left (a^3 \left (a^2-b^2\right ) g^2\right ) \int \frac {\sqrt {g \cos (e+f x)}}{a+b \sin (e+f x)} \, dx}{b^5}+\frac {\left (3 a^2 g^2 \sqrt {g \cos (e+f x)}\right ) \int \sqrt {\cos (e+f x)} \, dx}{5 b^3 \sqrt {\cos (e+f x)}} \\ & = -\frac {2 a^3 g (g \cos (e+f x))^{3/2}}{3 b^4 f}+\frac {2 a (g \cos (e+f x))^{7/2}}{7 b^2 f g}+\frac {6 a^2 g^2 \sqrt {g \cos (e+f x)} E\left (\left .\frac {1}{2} (e+f x)\right |2\right )}{5 b^3 f \sqrt {\cos (e+f x)}}+\frac {2 a^2 g (g \cos (e+f x))^{3/2} \sin (e+f x)}{5 b^3 f}+\frac {4 g (g \cos (e+f x))^{3/2} \sin (e+f x)}{45 b f}-\frac {2 (g \cos (e+f x))^{7/2} \sin (e+f x)}{9 b f g}-\frac {\left (a^4 \left (a^2-b^2\right ) g^3\right ) \int \frac {1}{\sqrt {g \cos (e+f x)} \left (\sqrt {-a^2+b^2}-b \cos (e+f x)\right )} \, dx}{2 b^6}+\frac {\left (a^4 \left (a^2-b^2\right ) g^3\right ) \int \frac {1}{\sqrt {g \cos (e+f x)} \left (\sqrt {-a^2+b^2}+b \cos (e+f x)\right )} \, dx}{2 b^6}+\frac {\left (a^3 \left (a^2-b^2\right ) g^3\right ) \text {Subst}\left (\int \frac {\sqrt {x}}{\left (a^2-b^2\right ) g^2+b^2 x^2} \, dx,x,g \cos (e+f x)\right )}{b^4 f}-\frac {\left (a^4 g^2 \sqrt {g \cos (e+f x)}\right ) \int \sqrt {\cos (e+f x)} \, dx}{b^5 \sqrt {\cos (e+f x)}}+\frac {\left (2 g^2 \sqrt {g \cos (e+f x)}\right ) \int \sqrt {\cos (e+f x)} \, dx}{15 b \sqrt {\cos (e+f x)}} \\ & = -\frac {2 a^3 g (g \cos (e+f x))^{3/2}}{3 b^4 f}+\frac {2 a (g \cos (e+f x))^{7/2}}{7 b^2 f g}-\frac {2 a^4 g^2 \sqrt {g \cos (e+f x)} E\left (\left .\frac {1}{2} (e+f x)\right |2\right )}{b^5 f \sqrt {\cos (e+f x)}}+\frac {6 a^2 g^2 \sqrt {g \cos (e+f x)} E\left (\left .\frac {1}{2} (e+f x)\right |2\right )}{5 b^3 f \sqrt {\cos (e+f x)}}+\frac {4 g^2 \sqrt {g \cos (e+f x)} E\left (\left .\frac {1}{2} (e+f x)\right |2\right )}{15 b f \sqrt {\cos (e+f x)}}+\frac {2 a^2 g (g \cos (e+f x))^{3/2} \sin (e+f x)}{5 b^3 f}+\frac {4 g (g \cos (e+f x))^{3/2} \sin (e+f x)}{45 b f}-\frac {2 (g \cos (e+f x))^{7/2} \sin (e+f x)}{9 b f g}+\frac {\left (2 a^3 \left (a^2-b^2\right ) g^3\right ) \text {Subst}\left (\int \frac {x^2}{\left (a^2-b^2\right ) g^2+b^2 x^4} \, dx,x,\sqrt {g \cos (e+f x)}\right )}{b^4 f}-\frac {\left (a^4 \left (a^2-b^2\right ) g^3 \sqrt {\cos (e+f x)}\right ) \int \frac {1}{\sqrt {\cos (e+f x)} \left (\sqrt {-a^2+b^2}-b \cos (e+f x)\right )} \, dx}{2 b^6 \sqrt {g \cos (e+f x)}}+\frac {\left (a^4 \left (a^2-b^2\right ) g^3 \sqrt {\cos (e+f x)}\right ) \int \frac {1}{\sqrt {\cos (e+f x)} \left (\sqrt {-a^2+b^2}+b \cos (e+f x)\right )} \, dx}{2 b^6 \sqrt {g \cos (e+f x)}} \\ & = -\frac {2 a^3 g (g \cos (e+f x))^{3/2}}{3 b^4 f}+\frac {2 a (g \cos (e+f x))^{7/2}}{7 b^2 f g}-\frac {2 a^4 g^2 \sqrt {g \cos (e+f x)} E\left (\left .\frac {1}{2} (e+f x)\right |2\right )}{b^5 f \sqrt {\cos (e+f x)}}+\frac {6 a^2 g^2 \sqrt {g \cos (e+f x)} E\left (\left .\frac {1}{2} (e+f x)\right |2\right )}{5 b^3 f \sqrt {\cos (e+f x)}}+\frac {4 g^2 \sqrt {g \cos (e+f x)} E\left (\left .\frac {1}{2} (e+f x)\right |2\right )}{15 b f \sqrt {\cos (e+f x)}}+\frac {a^4 \left (a^2-b^2\right ) g^3 \sqrt {\cos (e+f x)} \operatorname {EllipticPi}\left (\frac {2 b}{b-\sqrt {-a^2+b^2}},\frac {1}{2} (e+f x),2\right )}{b^6 \left (b-\sqrt {-a^2+b^2}\right ) f \sqrt {g \cos (e+f x)}}+\frac {a^4 \left (a^2-b^2\right ) g^3 \sqrt {\cos (e+f x)} \operatorname {EllipticPi}\left (\frac {2 b}{b+\sqrt {-a^2+b^2}},\frac {1}{2} (e+f x),2\right )}{b^6 \left (b+\sqrt {-a^2+b^2}\right ) f \sqrt {g \cos (e+f x)}}+\frac {2 a^2 g (g \cos (e+f x))^{3/2} \sin (e+f x)}{5 b^3 f}+\frac {4 g (g \cos (e+f x))^{3/2} \sin (e+f x)}{45 b f}-\frac {2 (g \cos (e+f x))^{7/2} \sin (e+f x)}{9 b f g}-\frac {\left (a^3 \left (a^2-b^2\right ) g^3\right ) \text {Subst}\left (\int \frac {1}{\sqrt {-a^2+b^2} g-b x^2} \, dx,x,\sqrt {g \cos (e+f x)}\right )}{b^5 f}+\frac {\left (a^3 \left (a^2-b^2\right ) g^3\right ) \text {Subst}\left (\int \frac {1}{\sqrt {-a^2+b^2} g+b x^2} \, dx,x,\sqrt {g \cos (e+f x)}\right )}{b^5 f} \\ & = -\frac {a^3 \left (-a^2+b^2\right )^{3/4} g^{5/2} \arctan \left (\frac {\sqrt {b} \sqrt {g \cos (e+f x)}}{\sqrt [4]{-a^2+b^2} \sqrt {g}}\right )}{b^{11/2} f}+\frac {a^3 \left (-a^2+b^2\right )^{3/4} g^{5/2} \text {arctanh}\left (\frac {\sqrt {b} \sqrt {g \cos (e+f x)}}{\sqrt [4]{-a^2+b^2} \sqrt {g}}\right )}{b^{11/2} f}-\frac {2 a^3 g (g \cos (e+f x))^{3/2}}{3 b^4 f}+\frac {2 a (g \cos (e+f x))^{7/2}}{7 b^2 f g}-\frac {2 a^4 g^2 \sqrt {g \cos (e+f x)} E\left (\left .\frac {1}{2} (e+f x)\right |2\right )}{b^5 f \sqrt {\cos (e+f x)}}+\frac {6 a^2 g^2 \sqrt {g \cos (e+f x)} E\left (\left .\frac {1}{2} (e+f x)\right |2\right )}{5 b^3 f \sqrt {\cos (e+f x)}}+\frac {4 g^2 \sqrt {g \cos (e+f x)} E\left (\left .\frac {1}{2} (e+f x)\right |2\right )}{15 b f \sqrt {\cos (e+f x)}}+\frac {a^4 \left (a^2-b^2\right ) g^3 \sqrt {\cos (e+f x)} \operatorname {EllipticPi}\left (\frac {2 b}{b-\sqrt {-a^2+b^2}},\frac {1}{2} (e+f x),2\right )}{b^6 \left (b-\sqrt {-a^2+b^2}\right ) f \sqrt {g \cos (e+f x)}}+\frac {a^4 \left (a^2-b^2\right ) g^3 \sqrt {\cos (e+f x)} \operatorname {EllipticPi}\left (\frac {2 b}{b+\sqrt {-a^2+b^2}},\frac {1}{2} (e+f x),2\right )}{b^6 \left (b+\sqrt {-a^2+b^2}\right ) f \sqrt {g \cos (e+f x)}}+\frac {2 a^2 g (g \cos (e+f x))^{3/2} \sin (e+f x)}{5 b^3 f}+\frac {4 g (g \cos (e+f x))^{3/2} \sin (e+f x)}{45 b f}-\frac {2 (g \cos (e+f x))^{7/2} \sin (e+f x)}{9 b f g} \\ \end{align*}

Mathematica [C] (warning: unable to verify)

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

Time = 19.23 (sec) , antiderivative size = 790, normalized size of antiderivative = 1.30 \[ \int \frac {(g \cos (e+f x))^{5/2} \sin ^3(e+f x)}{a+b \sin (e+f x)} \, dx=\frac {(g \cos (e+f x))^{5/2} \left (\frac {\sin (e+f x) \left (-\frac {\left (15 a^4-9 a^2 b^2-2 b^4\right ) \csc (e+f x) \left (8 b^{5/2} \operatorname {AppellF1}\left (\frac {3}{4},-\frac {1}{2},1,\frac {7}{4},\cos ^2(e+f x),\frac {b^2 \cos ^2(e+f x)}{-a^2+b^2}\right ) \cos ^{\frac {3}{2}}(e+f x)+3 \sqrt {2} a \left (a^2-b^2\right )^{3/4} \left (2 \arctan \left (1-\frac {\sqrt {2} \sqrt {b} \sqrt {\cos (e+f x)}}{\sqrt [4]{a^2-b^2}}\right )-2 \arctan \left (1+\frac {\sqrt {2} \sqrt {b} \sqrt {\cos (e+f x)}}{\sqrt [4]{a^2-b^2}}\right )-\log \left (\sqrt {a^2-b^2}-\sqrt {2} \sqrt {b} \sqrt [4]{a^2-b^2} \sqrt {\cos (e+f x)}+b \cos (e+f x)\right )+\log \left (\sqrt {a^2-b^2}+\sqrt {2} \sqrt {b} \sqrt [4]{a^2-b^2} \sqrt {\cos (e+f x)}+b \cos (e+f x)\right )\right )\right )}{a^2-b^2}+\frac {(2+2 i) b^2 \left (-3 a^3+a b^2\right ) \left ((4-4 i) a \sqrt {b} \sqrt [4]{-a^2+b^2} \operatorname {AppellF1}\left (\frac {3}{4},\frac {1}{2},1,\frac {7}{4},\cos ^2(e+f x),\frac {b^2 \cos ^2(e+f x)}{-a^2+b^2}\right ) \cos ^{\frac {3}{2}}(e+f x)+3 \left (a^2-b^2\right ) \left (2 \arctan \left (1-\frac {(1+i) \sqrt {b} \sqrt {\cos (e+f x)}}{\sqrt [4]{-a^2+b^2}}\right )-2 \arctan \left (1+\frac {(1+i) \sqrt {b} \sqrt {\cos (e+f x)}}{\sqrt [4]{-a^2+b^2}}\right )-\log \left (\sqrt {-a^2+b^2}-(1+i) \sqrt {b} \sqrt [4]{-a^2+b^2} \sqrt {\cos (e+f x)}+i b \cos (e+f x)\right )+\log \left (\sqrt {-a^2+b^2}+(1+i) \sqrt {b} \sqrt [4]{-a^2+b^2} \sqrt {\cos (e+f x)}+i b \cos (e+f x)\right )\right )\right )}{\left (-a^2+b^2\right )^{5/4} \sqrt {\sin ^2(e+f x)}}\right ) \left (a+b \sqrt {\sin ^2(e+f x)}\right )}{12 b^{11/2} (a+b \sin (e+f x))}+\frac {\cos ^{\frac {3}{2}}(e+f x) \left (90 a b^2 \cos (2 (e+f x))+21 b \left (12 a^2+b^2\right ) \sin (e+f x)-5 \left (84 a^3-18 a b^2+7 b^3 \sin (3 (e+f x))\right )\right )}{42 b^4}\right )}{15 f \cos ^{\frac {5}{2}}(e+f x)} \]

[In]

Integrate[((g*Cos[e + f*x])^(5/2)*Sin[e + f*x]^3)/(a + b*Sin[e + f*x]),x]

[Out]

((g*Cos[e + f*x])^(5/2)*((Sin[e + f*x]*(-(((15*a^4 - 9*a^2*b^2 - 2*b^4)*Csc[e + f*x]*(8*b^(5/2)*AppellF1[3/4,
-1/2, 1, 7/4, Cos[e + f*x]^2, (b^2*Cos[e + f*x]^2)/(-a^2 + b^2)]*Cos[e + f*x]^(3/2) + 3*Sqrt[2]*a*(a^2 - b^2)^
(3/4)*(2*ArcTan[1 - (Sqrt[2]*Sqrt[b]*Sqrt[Cos[e + f*x]])/(a^2 - b^2)^(1/4)] - 2*ArcTan[1 + (Sqrt[2]*Sqrt[b]*Sq
rt[Cos[e + f*x]])/(a^2 - b^2)^(1/4)] - Log[Sqrt[a^2 - b^2] - Sqrt[2]*Sqrt[b]*(a^2 - b^2)^(1/4)*Sqrt[Cos[e + f*
x]] + b*Cos[e + f*x]] + Log[Sqrt[a^2 - b^2] + Sqrt[2]*Sqrt[b]*(a^2 - b^2)^(1/4)*Sqrt[Cos[e + f*x]] + b*Cos[e +
f*x]])))/(a^2 - b^2)) + ((2 + 2*I)*b^2*(-3*a^3 + a*b^2)*((4 - 4*I)*a*Sqrt[b]*(-a^2 + b^2)^(1/4)*AppellF1[3/4,
1/2, 1, 7/4, Cos[e + f*x]^2, (b^2*Cos[e + f*x]^2)/(-a^2 + b^2)]*Cos[e + f*x]^(3/2) + 3*(a^2 - b^2)*(2*ArcTan[
1 - ((1 + I)*Sqrt[b]*Sqrt[Cos[e + f*x]])/(-a^2 + b^2)^(1/4)] - 2*ArcTan[1 + ((1 + I)*Sqrt[b]*Sqrt[Cos[e + f*x]
])/(-a^2 + b^2)^(1/4)] - Log[Sqrt[-a^2 + b^2] - (1 + I)*Sqrt[b]*(-a^2 + b^2)^(1/4)*Sqrt[Cos[e + f*x]] + I*b*Co
s[e + f*x]] + Log[Sqrt[-a^2 + b^2] + (1 + I)*Sqrt[b]*(-a^2 + b^2)^(1/4)*Sqrt[Cos[e + f*x]] + I*b*Cos[e + f*x]]
)))/((-a^2 + b^2)^(5/4)*Sqrt[Sin[e + f*x]^2]))*(a + b*Sqrt[Sin[e + f*x]^2]))/(12*b^(11/2)*(a + b*Sin[e + f*x])
) + (Cos[e + f*x]^(3/2)*(90*a*b^2*Cos[2*(e + f*x)] + 21*b*(12*a^2 + b^2)*Sin[e + f*x] - 5*(84*a^3 - 18*a*b^2 +
7*b^3*Sin[3*(e + f*x)])))/(42*b^4)))/(15*f*Cos[e + f*x]^(5/2))

Maple [C] (warning: unable to verify)

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

Time = 24.72 (sec) , antiderivative size = 2001, normalized size of antiderivative = 3.28

method	result	size
default	\(\text {Expression too large to display}\)	\(2001\)

[In]

int((g*cos(f*x+e))^(5/2)*sin(f*x+e)^3/(a+b*sin(f*x+e)),x,method=_RETURNVERBOSE)

[Out]

(-16*g^3*a*(1/168/b^4/g*(-2*g*sin(1/2*f*x+1/2*e)^2+g)^(1/2)*(24*sin(1/2*f*x+1/2*e)^6*b^2-36*sin(1/2*f*x+1/2*e)
^4*b^2-14*sin(1/2*f*x+1/2*e)^2*a^2+18*sin(1/2*f*x+1/2*e)^2*b^2+7*a^2-3*b^2)-1/64/b^6*a^2*(a^2-b^2)/(g^2*(a^2-b
^2)/b^2)^(1/4)*2^(1/2)*(ln((2*g*cos(1/2*f*x+1/2*e)^2-g-(g^2*(a^2-b^2)/b^2)^(1/4)*(2*g*cos(1/2*f*x+1/2*e)^2-g)^
(1/2)*2^(1/2)+(g^2*(a^2-b^2)/b^2)^(1/2))/(2*g*cos(1/2*f*x+1/2*e)^2-g+(g^2*(a^2-b^2)/b^2)^(1/4)*(2*g*cos(1/2*f*
x+1/2*e)^2-g)^(1/2)*2^(1/2)+(g^2*(a^2-b^2)/b^2)^(1/2)))+2*arctan((2^(1/2)*(2*g*cos(1/2*f*x+1/2*e)^2-g)^(1/2)+(
g^2*(a^2-b^2)/b^2)^(1/4))/(g^2*(a^2-b^2)/b^2)^(1/4))+2*arctan((2^(1/2)*(2*g*cos(1/2*f*x+1/2*e)^2-g)^(1/2)-(g^2
*(a^2-b^2)/b^2)^(1/4))/(g^2*(a^2-b^2)/b^2)^(1/4))))+1/420*(g*(2*cos(1/2*f*x+1/2*e)^2-1)*sin(1/2*f*x+1/2*e)^2)^
(1/2)*g^3*(1680*EllipticF(cos(1/2*f*x+1/2*e),2^(1/2))*(sin(1/2*f*x+1/2*e)^2)^(1/2)*(1-2*cos(1/2*f*x+1/2*e)^2)^
(1/2)*sin(1/2*f*x+1/2*e)^2*a^4*b^2-2240*EllipticF(cos(1/2*f*x+1/2*e),2^(1/2))*(sin(1/2*f*x+1/2*e)^2)^(1/2)*(1-
2*cos(1/2*f*x+1/2*e)^2)^(1/2)*sin(1/2*f*x+1/2*e)^2*a^2*b^4+1680*EllipticE(cos(1/2*f*x+1/2*e),2^(1/2))*(sin(1/2
*f*x+1/2*e)^2)^(1/2)*(1-2*cos(1/2*f*x+1/2*e)^2)^(1/2)*sin(1/2*f*x+1/2*e)^2*a^2*b^4-4864*cos(1/2*f*x+1/2*e)^5*b
^6+2240*cos(1/2*f*x+1/2*e)^5*a^2*b^4-3360*cos(1/2*f*x+1/2*e)^3*a^2*b^4+1008*EllipticE(cos(1/2*f*x+1/2*e),2^(1/
2))*(sin(1/2*f*x+1/2*e)^2)^(1/2)*(1-2*cos(1/2*f*x+1/2*e)^2)^(1/2)*b^6-1120*cos(1/2*f*x+1/2*e)*sin(1/2*f*x+1/2*
e)^2*a^2*b^4-400*EllipticF(cos(1/2*f*x+1/2*e),2^(1/2))*(sin(1/2*f*x+1/2*e)^2)^(1/2)*(1-2*cos(1/2*f*x+1/2*e)^2)
^(1/2)*b^6-2240*cos(1/2*f*x+1/2*e)^5*sin(1/2*f*x+1/2*e)^2*a^2*b^4+3360*cos(1/2*f*x+1/2*e)^3*sin(1/2*f*x+1/2*e)
^2*a^2*b^4+400*EllipticF(cos(1/2*f*x+1/2*e),2^(1/2))*(sin(1/2*f*x+1/2*e)^2)^(1/2)*(1-2*cos(1/2*f*x+1/2*e)^2)^(
1/2)*sin(1/2*f*x+1/2*e)^2*b^6-1008*EllipticE(cos(1/2*f*x+1/2*e),2^(1/2))*(sin(1/2*f*x+1/2*e)^2)^(1/2)*(1-2*cos
(1/2*f*x+1/2*e)^2)^(1/2)*sin(1/2*f*x+1/2*e)^2*b^6-608*b^6*cos(1/2*f*x+1/2*e)-1680*EllipticF(cos(1/2*f*x+1/2*e)
,2^(1/2))*(sin(1/2*f*x+1/2*e)^2)^(1/2)*(1-2*cos(1/2*f*x+1/2*e)^2)^(1/2)*a^4*b^2+2240*EllipticF(cos(1/2*f*x+1/2
*e),2^(1/2))*(sin(1/2*f*x+1/2*e)^2)^(1/2)*(1-2*cos(1/2*f*x+1/2*e)^2)^(1/2)*a^2*b^4-1680*EllipticE(cos(1/2*f*x+
1/2*e),2^(1/2))*(sin(1/2*f*x+1/2*e)^2)^(1/2)*(1-2*cos(1/2*f*x+1/2*e)^2)^(1/2)*a^2*b^4+1120*a^2*b^4*cos(1/2*f*x
+1/2*e)+3840*cos(1/2*f*x+1/2*e)^9*sin(1/2*f*x+1/2*e)^2*b^6-6912*cos(1/2*f*x+1/2*e)^7*sin(1/2*f*x+1/2*e)^2*b^6+
4864*cos(1/2*f*x+1/2*e)^5*sin(1/2*f*x+1/2*e)^2*b^6-2400*cos(1/2*f*x+1/2*e)^3*sin(1/2*f*x+1/2*e)^2*b^6+608*cos(
1/2*f*x+1/2*e)*sin(1/2*f*x+1/2*e)^2*b^6+105*sum((sin(1/2*f*x+1/2*e)^2*(2*_alpha^2*a^2*b^2-2*_alpha^2*b^4-a^4+a
^2*b^2)-2*_alpha^2*a^2*b^2+2*_alpha^2*b^4+a^4-a^2*b^2)/_alpha/(2*_alpha^2-1)*(8*(g*(2*_alpha^2*b^2+a^2-2*b^2)/
b^2)^(1/2)*(sin(1/2*f*x+1/2*e)^2)^(1/2)*(1-2*cos(1/2*f*x+1/2*e)^2)^(1/2)*EllipticPi(cos(1/2*f*x+1/2*e),-4*b^2/
a^2*(_alpha^2-1),2^(1/2))*_alpha^3*b^2-8*b^2*_alpha*(sin(1/2*f*x+1/2*e)^2)^(1/2)*(1-2*cos(1/2*f*x+1/2*e)^2)^(1
/2)*EllipticPi(cos(1/2*f*x+1/2*e),-4*b^2/a^2*(_alpha^2-1),2^(1/2))*(g*(2*_alpha^2*b^2+a^2-2*b^2)/b^2)^(1/2)+a^
2*2^(1/2)*arctanh(1/2*g*(4*_alpha^2-3)/(4*a^2-3*b^2)*(b^2*_alpha^2+4*a^2*cos(1/2*f*x+1/2*e)^2-3*b^2*cos(1/2*f*
x+1/2*e)^2-3*a^2+2*b^2)*2^(1/2)/(g*(2*_alpha^2*b^2+a^2-2*b^2)/b^2)^(1/2)/(-g*(2*sin(1/2*f*x+1/2*e)^4-sin(1/2*f
*x+1/2*e)^2))^(1/2))*(-g*sin(1/2*f*x+1/2*e)^2*(2*sin(1/2*f*x+1/2*e)^2-1))^(1/2))/(g*(2*_alpha^2*b^2+a^2-2*b^2)
/b^2)^(1/2)/(-g*sin(1/2*f*x+1/2*e)^2*(2*sin(1/2*f*x+1/2*e)^2-1))^(1/2),_alpha=RootOf(4*_Z^4*b^2-4*_Z^2*b^2+a^2
))*(-g*(2*sin(1/2*f*x+1/2*e)^4-sin(1/2*f*x+1/2*e)^2))^(1/2)-3840*cos(1/2*f*x+1/2*e)^9*b^6+6912*cos(1/2*f*x+1/2
*e)^7*b^6+2400*cos(1/2*f*x+1/2*e)^3*b^6)/b^7/(-g*(2*sin(1/2*f*x+1/2*e)^4-sin(1/2*f*x+1/2*e)^2))^(1/2)/sin(1/2*
f*x+1/2*e)/(g*(2*cos(1/2*f*x+1/2*e)^2-1))^(1/2))/f

Fricas [F(-1)]

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

[In]

integrate((g*cos(f*x+e))^(5/2)*sin(f*x+e)^3/(a+b*sin(f*x+e)),x, algorithm="fricas")

[Out]

Timed out

Sympy [F(-1)]

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

[In]

integrate((g*cos(f*x+e))**(5/2)*sin(f*x+e)**3/(a+b*sin(f*x+e)),x)

[Out]

Timed out

Maxima [F]

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

[In]

integrate((g*cos(f*x+e))^(5/2)*sin(f*x+e)^3/(a+b*sin(f*x+e)),x, algorithm="maxima")

[Out]

integrate((g*cos(f*x + e))^(5/2)*sin(f*x + e)^3/(b*sin(f*x + e) + a), x)

Giac [F]

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

[In]

integrate((g*cos(f*x+e))^(5/2)*sin(f*x+e)^3/(a+b*sin(f*x+e)),x, algorithm="giac")

[Out]

integrate((g*cos(f*x + e))^(5/2)*sin(f*x + e)^3/(b*sin(f*x + e) + a), x)

Mupad [F(-1)]

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

[In]

int((sin(e + f*x)^3*(g*cos(e + f*x))^(5/2))/(a + b*sin(e + f*x)),x)

[Out]

int((sin(e + f*x)^3*(g*cos(e + f*x))^(5/2))/(a + b*sin(e + f*x)), x)