\(\int \frac {(d \sin (e+f x))^{5/2}}{(g \cos (e+f x))^{3/2} (a+b \sin (e+f x))} \, dx\) [1436]

Optimal result

Integrand size = 37, antiderivative size = 1064 \[ \int \frac {(d \sin (e+f x))^{5/2}}{(g \cos (e+f x))^{3/2} (a+b \sin (e+f x))} \, dx=-\frac {a^2 d^{5/2} \arctan \left (1-\frac {\sqrt {2} \sqrt {d} \sqrt {g \cos (e+f x)}}{\sqrt {g} \sqrt {d \sin (e+f x)}}\right )}{\sqrt {2} b \left (a^2-b^2\right ) f g^{3/2}}+\frac {b d^{5/2} \arctan \left (1-\frac {\sqrt {2} \sqrt {d} \sqrt {g \cos (e+f x)}}{\sqrt {g} \sqrt {d \sin (e+f x)}}\right )}{\sqrt {2} \left (a^2-b^2\right ) f g^{3/2}}+\frac {a^2 d^{5/2} \arctan \left (1+\frac {\sqrt {2} \sqrt {d} \sqrt {g \cos (e+f x)}}{\sqrt {g} \sqrt {d \sin (e+f x)}}\right )}{\sqrt {2} b \left (a^2-b^2\right ) f g^{3/2}}-\frac {b d^{5/2} \arctan \left (1+\frac {\sqrt {2} \sqrt {d} \sqrt {g \cos (e+f x)}}{\sqrt {g} \sqrt {d \sin (e+f x)}}\right )}{\sqrt {2} \left (a^2-b^2\right ) f g^{3/2}}+\frac {a^2 d^{5/2} \log \left (\sqrt {g}+\sqrt {g} \cot (e+f x)-\frac {\sqrt {2} \sqrt {d} \sqrt {g \cos (e+f x)}}{\sqrt {d \sin (e+f x)}}\right )}{2 \sqrt {2} b \left (a^2-b^2\right ) f g^{3/2}}-\frac {b d^{5/2} \log \left (\sqrt {g}+\sqrt {g} \cot (e+f x)-\frac {\sqrt {2} \sqrt {d} \sqrt {g \cos (e+f x)}}{\sqrt {d \sin (e+f x)}}\right )}{2 \sqrt {2} \left (a^2-b^2\right ) f g^{3/2}}-\frac {a^2 d^{5/2} \log \left (\sqrt {g}+\sqrt {g} \cot (e+f x)+\frac {\sqrt {2} \sqrt {d} \sqrt {g \cos (e+f x)}}{\sqrt {d \sin (e+f x)}}\right )}{2 \sqrt {2} b \left (a^2-b^2\right ) f g^{3/2}}+\frac {b d^{5/2} \log \left (\sqrt {g}+\sqrt {g} \cot (e+f x)+\frac {\sqrt {2} \sqrt {d} \sqrt {g \cos (e+f x)}}{\sqrt {d \sin (e+f x)}}\right )}{2 \sqrt {2} \left (a^2-b^2\right ) f g^{3/2}}-\frac {2 \sqrt {2} a^3 d^3 \operatorname {EllipticPi}\left (-\frac {\sqrt {-a+b}}{\sqrt {a+b}},\arcsin \left (\frac {\sqrt {g \cos (e+f x)}}{\sqrt {g} \sqrt {1+\sin (e+f x)}}\right ),-1\right ) \sqrt {\sin (e+f x)}}{b (-a+b)^{3/2} (a+b)^{3/2} f g^{3/2} \sqrt {d \sin (e+f x)}}+\frac {2 \sqrt {2} a^3 d^3 \operatorname {EllipticPi}\left (\frac {\sqrt {-a+b}}{\sqrt {a+b}},\arcsin \left (\frac {\sqrt {g \cos (e+f x)}}{\sqrt {g} \sqrt {1+\sin (e+f x)}}\right ),-1\right ) \sqrt {\sin (e+f x)}}{b (-a+b)^{3/2} (a+b)^{3/2} f g^{3/2} \sqrt {d \sin (e+f x)}}-\frac {2 b d^2 \sqrt {d \sin (e+f x)}}{\left (a^2-b^2\right ) f g \sqrt {g \cos (e+f x)}}+\frac {2 a d (d \sin (e+f x))^{3/2}}{\left (a^2-b^2\right ) f g \sqrt {g \cos (e+f x)}}-\frac {2 a d^2 \sqrt {g \cos (e+f x)} E\left (\left .e-\frac {\pi }{4}+f x\right |2\right ) \sqrt {d \sin (e+f x)}}{\left (a^2-b^2\right ) f g^2 \sqrt {\sin (2 e+2 f x)}} \]

[Out]

1/2*a^2*d^(5/2)*arctan(-1+2^(1/2)*d^(1/2)*(g*cos(f*x+e))^(1/2)/g^(1/2)/(d*sin(f*x+e))^(1/2))/b/(a^2-b^2)/f/g^(
3/2)*2^(1/2)-1/2*b*d^(5/2)*arctan(-1+2^(1/2)*d^(1/2)*(g*cos(f*x+e))^(1/2)/g^(1/2)/(d*sin(f*x+e))^(1/2))/(a^2-b
^2)/f/g^(3/2)*2^(1/2)+1/2*a^2*d^(5/2)*arctan(1+2^(1/2)*d^(1/2)*(g*cos(f*x+e))^(1/2)/g^(1/2)/(d*sin(f*x+e))^(1/
2))/b/(a^2-b^2)/f/g^(3/2)*2^(1/2)-1/2*b*d^(5/2)*arctan(1+2^(1/2)*d^(1/2)*(g*cos(f*x+e))^(1/2)/g^(1/2)/(d*sin(f
*x+e))^(1/2))/(a^2-b^2)/f/g^(3/2)*2^(1/2)+1/4*a^2*d^(5/2)*ln(g^(1/2)+cot(f*x+e)*g^(1/2)-2^(1/2)*d^(1/2)*(g*cos
(f*x+e))^(1/2)/(d*sin(f*x+e))^(1/2))/b/(a^2-b^2)/f/g^(3/2)*2^(1/2)-1/4*b*d^(5/2)*ln(g^(1/2)+cot(f*x+e)*g^(1/2)
-2^(1/2)*d^(1/2)*(g*cos(f*x+e))^(1/2)/(d*sin(f*x+e))^(1/2))/(a^2-b^2)/f/g^(3/2)*2^(1/2)-1/4*a^2*d^(5/2)*ln(g^(
1/2)+cot(f*x+e)*g^(1/2)+2^(1/2)*d^(1/2)*(g*cos(f*x+e))^(1/2)/(d*sin(f*x+e))^(1/2))/b/(a^2-b^2)/f/g^(3/2)*2^(1/
2)+1/4*b*d^(5/2)*ln(g^(1/2)+cot(f*x+e)*g^(1/2)+2^(1/2)*d^(1/2)*(g*cos(f*x+e))^(1/2)/(d*sin(f*x+e))^(1/2))/(a^2
-b^2)/f/g^(3/2)*2^(1/2)+2*a*d*(d*sin(f*x+e))^(3/2)/(a^2-b^2)/f/g/(g*cos(f*x+e))^(1/2)-2*a^3*d^3*EllipticPi((g*
cos(f*x+e))^(1/2)/g^(1/2)/(1+sin(f*x+e))^(1/2),-(-a+b)^(1/2)/(a+b)^(1/2),I)*2^(1/2)*sin(f*x+e)^(1/2)/b/(-a+b)^
(3/2)/(a+b)^(3/2)/f/g^(3/2)/(d*sin(f*x+e))^(1/2)+2*a^3*d^3*EllipticPi((g*cos(f*x+e))^(1/2)/g^(1/2)/(1+sin(f*x+
e))^(1/2),(-a+b)^(1/2)/(a+b)^(1/2),I)*2^(1/2)*sin(f*x+e)^(1/2)/b/(-a+b)^(3/2)/(a+b)^(3/2)/f/g^(3/2)/(d*sin(f*x
+e))^(1/2)-2*b*d^2*(d*sin(f*x+e))^(1/2)/(a^2-b^2)/f/g/(g*cos(f*x+e))^(1/2)+2*a*d^2*(sin(e+1/4*Pi+f*x)^2)^(1/2)
/sin(e+1/4*Pi+f*x)*EllipticE(cos(e+1/4*Pi+f*x),2^(1/2))*(g*cos(f*x+e))^(1/2)*(d*sin(f*x+e))^(1/2)/(a^2-b^2)/f/
g^2/sin(2*f*x+2*e)^(1/2)

Rubi [A] (verified)

Time = 1.08 (sec) , antiderivative size = 1064, normalized size of antiderivative = 1.00, number of steps used = 31, number of rules used = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.459, Rules used = {2981, 2651, 2652, 2719, 2646, 2655, 303, 1176, 631, 210, 1179, 642, 2988, 2985, 2984, 504, 1232} \[ \int \frac {(d \sin (e+f x))^{5/2}}{(g \cos (e+f x))^{3/2} (a+b \sin (e+f x))} \, dx=-\frac {2 \sqrt {2} a^3 \operatorname {EllipticPi}\left (-\frac {\sqrt {b-a}}{\sqrt {a+b}},\arcsin \left (\frac {\sqrt {g \cos (e+f x)}}{\sqrt {g} \sqrt {\sin (e+f x)+1}}\right ),-1\right ) \sqrt {\sin (e+f x)} d^3}{b (b-a)^{3/2} (a+b)^{3/2} f g^{3/2} \sqrt {d \sin (e+f x)}}+\frac {2 \sqrt {2} a^3 \operatorname {EllipticPi}\left (\frac {\sqrt {b-a}}{\sqrt {a+b}},\arcsin \left (\frac {\sqrt {g \cos (e+f x)}}{\sqrt {g} \sqrt {\sin (e+f x)+1}}\right ),-1\right ) \sqrt {\sin (e+f x)} d^3}{b (b-a)^{3/2} (a+b)^{3/2} f g^{3/2} \sqrt {d \sin (e+f x)}}+\frac {b \arctan \left (1-\frac {\sqrt {2} \sqrt {d} \sqrt {g \cos (e+f x)}}{\sqrt {g} \sqrt {d \sin (e+f x)}}\right ) d^{5/2}}{\sqrt {2} \left (a^2-b^2\right ) f g^{3/2}}-\frac {a^2 \arctan \left (1-\frac {\sqrt {2} \sqrt {d} \sqrt {g \cos (e+f x)}}{\sqrt {g} \sqrt {d \sin (e+f x)}}\right ) d^{5/2}}{\sqrt {2} b \left (a^2-b^2\right ) f g^{3/2}}-\frac {b \arctan \left (\frac {\sqrt {2} \sqrt {d} \sqrt {g \cos (e+f x)}}{\sqrt {g} \sqrt {d \sin (e+f x)}}+1\right ) d^{5/2}}{\sqrt {2} \left (a^2-b^2\right ) f g^{3/2}}+\frac {a^2 \arctan \left (\frac {\sqrt {2} \sqrt {d} \sqrt {g \cos (e+f x)}}{\sqrt {g} \sqrt {d \sin (e+f x)}}+1\right ) d^{5/2}}{\sqrt {2} b \left (a^2-b^2\right ) f g^{3/2}}-\frac {b \log \left (\sqrt {g} \cot (e+f x)+\sqrt {g}-\frac {\sqrt {2} \sqrt {d} \sqrt {g \cos (e+f x)}}{\sqrt {d \sin (e+f x)}}\right ) d^{5/2}}{2 \sqrt {2} \left (a^2-b^2\right ) f g^{3/2}}+\frac {a^2 \log \left (\sqrt {g} \cot (e+f x)+\sqrt {g}-\frac {\sqrt {2} \sqrt {d} \sqrt {g \cos (e+f x)}}{\sqrt {d \sin (e+f x)}}\right ) d^{5/2}}{2 \sqrt {2} b \left (a^2-b^2\right ) f g^{3/2}}+\frac {b \log \left (\sqrt {g} \cot (e+f x)+\sqrt {g}+\frac {\sqrt {2} \sqrt {d} \sqrt {g \cos (e+f x)}}{\sqrt {d \sin (e+f x)}}\right ) d^{5/2}}{2 \sqrt {2} \left (a^2-b^2\right ) f g^{3/2}}-\frac {a^2 \log \left (\sqrt {g} \cot (e+f x)+\sqrt {g}+\frac {\sqrt {2} \sqrt {d} \sqrt {g \cos (e+f x)}}{\sqrt {d \sin (e+f x)}}\right ) d^{5/2}}{2 \sqrt {2} b \left (a^2-b^2\right ) f g^{3/2}}-\frac {2 b \sqrt {d \sin (e+f x)} d^2}{\left (a^2-b^2\right ) f g \sqrt {g \cos (e+f x)}}-\frac {2 a \sqrt {g \cos (e+f x)} E\left (\left .e+f x-\frac {\pi }{4}\right |2\right ) \sqrt {d \sin (e+f x)} d^2}{\left (a^2-b^2\right ) f g^2 \sqrt {\sin (2 e+2 f x)}}+\frac {2 a (d \sin (e+f x))^{3/2} d}{\left (a^2-b^2\right ) f g \sqrt {g \cos (e+f x)}} \]

[In]

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

[Out]

-((a^2*d^(5/2)*ArcTan[1 - (Sqrt[2]*Sqrt[d]*Sqrt[g*Cos[e + f*x]])/(Sqrt[g]*Sqrt[d*Sin[e + f*x]])])/(Sqrt[2]*b*(
a^2 - b^2)*f*g^(3/2))) + (b*d^(5/2)*ArcTan[1 - (Sqrt[2]*Sqrt[d]*Sqrt[g*Cos[e + f*x]])/(Sqrt[g]*Sqrt[d*Sin[e +
f*x]])])/(Sqrt[2]*(a^2 - b^2)*f*g^(3/2)) + (a^2*d^(5/2)*ArcTan[1 + (Sqrt[2]*Sqrt[d]*Sqrt[g*Cos[e + f*x]])/(Sqr
t[g]*Sqrt[d*Sin[e + f*x]])])/(Sqrt[2]*b*(a^2 - b^2)*f*g^(3/2)) - (b*d^(5/2)*ArcTan[1 + (Sqrt[2]*Sqrt[d]*Sqrt[g
*Cos[e + f*x]])/(Sqrt[g]*Sqrt[d*Sin[e + f*x]])])/(Sqrt[2]*(a^2 - b^2)*f*g^(3/2)) + (a^2*d^(5/2)*Log[Sqrt[g] +
Sqrt[g]*Cot[e + f*x] - (Sqrt[2]*Sqrt[d]*Sqrt[g*Cos[e + f*x]])/Sqrt[d*Sin[e + f*x]]])/(2*Sqrt[2]*b*(a^2 - b^2)*
f*g^(3/2)) - (b*d^(5/2)*Log[Sqrt[g] + Sqrt[g]*Cot[e + f*x] - (Sqrt[2]*Sqrt[d]*Sqrt[g*Cos[e + f*x]])/Sqrt[d*Sin
[e + f*x]]])/(2*Sqrt[2]*(a^2 - b^2)*f*g^(3/2)) - (a^2*d^(5/2)*Log[Sqrt[g] + Sqrt[g]*Cot[e + f*x] + (Sqrt[2]*Sq
rt[d]*Sqrt[g*Cos[e + f*x]])/Sqrt[d*Sin[e + f*x]]])/(2*Sqrt[2]*b*(a^2 - b^2)*f*g^(3/2)) + (b*d^(5/2)*Log[Sqrt[g
] + Sqrt[g]*Cot[e + f*x] + (Sqrt[2]*Sqrt[d]*Sqrt[g*Cos[e + f*x]])/Sqrt[d*Sin[e + f*x]]])/(2*Sqrt[2]*(a^2 - b^2
)*f*g^(3/2)) - (2*Sqrt[2]*a^3*d^3*EllipticPi[-(Sqrt[-a + b]/Sqrt[a + b]), ArcSin[Sqrt[g*Cos[e + f*x]]/(Sqrt[g]
*Sqrt[1 + Sin[e + f*x]])], -1]*Sqrt[Sin[e + f*x]])/(b*(-a + b)^(3/2)*(a + b)^(3/2)*f*g^(3/2)*Sqrt[d*Sin[e + f*
x]]) + (2*Sqrt[2]*a^3*d^3*EllipticPi[Sqrt[-a + b]/Sqrt[a + b], ArcSin[Sqrt[g*Cos[e + f*x]]/(Sqrt[g]*Sqrt[1 + S
in[e + f*x]])], -1]*Sqrt[Sin[e + f*x]])/(b*(-a + b)^(3/2)*(a + b)^(3/2)*f*g^(3/2)*Sqrt[d*Sin[e + f*x]]) - (2*b
*d^2*Sqrt[d*Sin[e + f*x]])/((a^2 - b^2)*f*g*Sqrt[g*Cos[e + f*x]]) + (2*a*d*(d*Sin[e + f*x])^(3/2))/((a^2 - b^2
)*f*g*Sqrt[g*Cos[e + f*x]]) - (2*a*d^2*Sqrt[g*Cos[e + f*x]]*EllipticE[e - Pi/4 + f*x, 2]*Sqrt[d*Sin[e + f*x]])
/((a^2 - b^2)*f*g^2*Sqrt[Sin[2*e + 2*f*x]])

Rule 210

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(-(Rt[-a, 2]*Rt[-b, 2])^(-1))*ArcTan[Rt[-b, 2]*(x/Rt[-a, 2])
], x] /; FreeQ[{a, b}, x] && PosQ[a/b] && (LtQ[a, 0] || LtQ[b, 0])

Rule 303

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

Rule 504

Int[(x_)^2/(((a_) + (b_.)*(x_)^4)*Sqrt[(c_) + (d_.)*(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)*Sqrt[c + d*x^4]), x], x] - Dist[s/(2*b), Int[1/
((r - s*x^2)*Sqrt[c + d*x^4]), x], x]] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0]

Rule 631

Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> With[{q = 1 - 4*Simplify[a*(c/b^2)]}, Dist[-2/b, Sub
st[Int[1/(q - x^2), x], x, 1 + 2*c*(x/b)], x] /; RationalQ[q] && (EqQ[q^2, 1] ||  !RationalQ[b^2 - 4*a*c])] /;
FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]

Rule 642

Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Simp[d*(Log[RemoveContent[a + b*x +
c*x^2, x]]/b), x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[2*c*d - b*e, 0]

Rule 1176

Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[2*(d/e), 2]}, Dist[e/(2*c), Int[1/S
imp[d/e + q*x + x^2, x], x], x] + Dist[e/(2*c), Int[1/Simp[d/e - q*x + x^2, x], x], x]] /; FreeQ[{a, c, d, e},
x] && EqQ[c*d^2 - a*e^2, 0] && PosQ[d*e]

Rule 1179

Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[-2*(d/e), 2]}, Dist[e/(2*c*q), Int[
(q - 2*x)/Simp[d/e + q*x - x^2, x], x], x] + Dist[e/(2*c*q), Int[(q + 2*x)/Simp[d/e - q*x - x^2, x], x], x]] /
; FreeQ[{a, c, d, e}, x] && EqQ[c*d^2 - a*e^2, 0] && NegQ[d*e]

Rule 1232

Int[1/(((d_) + (e_.)*(x_)^2)*Sqrt[(a_) + (c_.)*(x_)^4]), x_Symbol] :> With[{q = Rt[-c/a, 4]}, Simp[(1/(d*Sqrt[
a]*q))*EllipticPi[-e/(d*q^2), ArcSin[q*x], -1], x]] /; FreeQ[{a, c, d, e}, x] && NegQ[c/a] && GtQ[a, 0]

Rule 2646

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

Rule 2651

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

Rule 2652

Int[Sqrt[cos[(e_.) + (f_.)*(x_)]*(b_.)]*Sqrt[(a_.)*sin[(e_.) + (f_.)*(x_)]], x_Symbol] :> Dist[Sqrt[a*Sin[e +
f*x]]*(Sqrt[b*Cos[e + f*x]]/Sqrt[Sin[2*e + 2*f*x]]), Int[Sqrt[Sin[2*e + 2*f*x]], x], x] /; FreeQ[{a, b, e, f},
x]

Rule 2655

Int[(cos[(e_.) + (f_.)*(x_)]*(a_.))^(m_)*((b_.)*sin[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> With[{k = Denomina
tor[m]}, Dist[(-k)*a*(b/f), Subst[Int[x^(k*(m + 1) - 1)/(a^2 + b^2*x^(2*k)), x], x, (a*Cos[e + f*x])^(1/k)/(b*
Sin[e + f*x])^(1/k)], x]] /; FreeQ[{a, b, e, f}, x] && EqQ[m + n, 0] && GtQ[m, 0] && LtQ[m, 1]

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 2981

Int[((cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((d_.)*sin[(e_.) + (f_.)*(x_)])^(n_))/((a_) + (b_.)*sin[(e_.) + (f_.
)*(x_)]), x_Symbol] :> Dist[a*(d^2/(a^2 - b^2)), Int[(g*Cos[e + f*x])^p*(d*Sin[e + f*x])^(n - 2), x], x] + (-D
ist[b*(d/(a^2 - b^2)), Int[(g*Cos[e + f*x])^p*(d*Sin[e + f*x])^(n - 1), x], x] - Dist[a^2*(d^2/(g^2*(a^2 - b^2
))), Int[(g*Cos[e + f*x])^(p + 2)*((d*Sin[e + f*x])^(n - 2)/(a + b*Sin[e + f*x])), x], x]) /; FreeQ[{a, b, d,
e, f, g}, x] && NeQ[a^2 - b^2, 0] && IntegersQ[2*n, 2*p] && LtQ[p, -1] && GtQ[n, 1]

Rule 2984

Int[Sqrt[cos[(e_.) + (f_.)*(x_)]*(g_.)]/(Sqrt[sin[(e_.) + (f_.)*(x_)]]*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]))
, x_Symbol] :> Dist[-4*Sqrt[2]*(g/f), Subst[Int[x^2/(((a + b)*g^2 + (a - b)*x^4)*Sqrt[1 - x^4/g^2]), x], x, Sq
rt[g*Cos[e + f*x]]/Sqrt[1 + Sin[e + f*x]]], x] /; FreeQ[{a, b, e, f, g}, x] && NeQ[a^2 - b^2, 0]

Rule 2985

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

Rule 2988

Int[((cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((d_.)*sin[(e_.) + (f_.)*(x_)])^(n_))/((a_) + (b_.)*sin[(e_.) + (f_.
)*(x_)]), x_Symbol] :> Dist[d/b, Int[(g*Cos[e + f*x])^p*(d*Sin[e + f*x])^(n - 1), x], x] - Dist[a*(d/b), Int[(
g*Cos[e + f*x])^p*((d*Sin[e + f*x])^(n - 1)/(a + b*Sin[e + f*x])), x], x] /; FreeQ[{a, b, d, e, f, g}, x] && N
eQ[a^2 - b^2, 0] && IntegersQ[2*n, 2*p] && LtQ[-1, p, 1] && GtQ[n, 0]

Rubi steps \begin{align*} \text {integral}& = -\frac {(b d) \int \frac {(d \sin (e+f x))^{3/2}}{(g \cos (e+f x))^{3/2}} \, dx}{a^2-b^2}+\frac {\left (a d^2\right ) \int \frac {\sqrt {d \sin (e+f x)}}{(g \cos (e+f x))^{3/2}} \, dx}{a^2-b^2}-\frac {\left (a^2 d^2\right ) \int \frac {\sqrt {g \cos (e+f x)} \sqrt {d \sin (e+f x)}}{a+b \sin (e+f x)} \, dx}{\left (a^2-b^2\right ) g^2} \\ & = -\frac {2 b d^2 \sqrt {d \sin (e+f x)}}{\left (a^2-b^2\right ) f g \sqrt {g \cos (e+f x)}}+\frac {2 a d (d \sin (e+f x))^{3/2}}{\left (a^2-b^2\right ) f g \sqrt {g \cos (e+f x)}}-\frac {\left (2 a d^2\right ) \int \sqrt {g \cos (e+f x)} \sqrt {d \sin (e+f x)} \, dx}{\left (a^2-b^2\right ) g^2}-\frac {\left (a^2 d^3\right ) \int \frac {\sqrt {g \cos (e+f x)}}{\sqrt {d \sin (e+f x)}} \, dx}{b \left (a^2-b^2\right ) g^2}+\frac {\left (a^3 d^3\right ) \int \frac {\sqrt {g \cos (e+f x)}}{\sqrt {d \sin (e+f x)} (a+b \sin (e+f x))} \, dx}{b \left (a^2-b^2\right ) g^2}+\frac {\left (b d^3\right ) \int \frac {\sqrt {g \cos (e+f x)}}{\sqrt {d \sin (e+f x)}} \, dx}{\left (a^2-b^2\right ) g^2} \\ & = -\frac {2 b d^2 \sqrt {d \sin (e+f x)}}{\left (a^2-b^2\right ) f g \sqrt {g \cos (e+f x)}}+\frac {2 a d (d \sin (e+f x))^{3/2}}{\left (a^2-b^2\right ) f g \sqrt {g \cos (e+f x)}}+\frac {\left (2 a^2 d^4\right ) \text {Subst}\left (\int \frac {x^2}{g^2+d^2 x^4} \, dx,x,\frac {\sqrt {g \cos (e+f x)}}{\sqrt {d \sin (e+f x)}}\right )}{b \left (a^2-b^2\right ) f g}-\frac {\left (2 b d^4\right ) \text {Subst}\left (\int \frac {x^2}{g^2+d^2 x^4} \, dx,x,\frac {\sqrt {g \cos (e+f x)}}{\sqrt {d \sin (e+f x)}}\right )}{\left (a^2-b^2\right ) f g}+\frac {\left (a^3 d^3 \sqrt {\sin (e+f x)}\right ) \int \frac {\sqrt {g \cos (e+f x)}}{\sqrt {\sin (e+f x)} (a+b \sin (e+f x))} \, dx}{b \left (a^2-b^2\right ) g^2 \sqrt {d \sin (e+f x)}}-\frac {\left (2 a d^2 \sqrt {g \cos (e+f x)} \sqrt {d \sin (e+f x)}\right ) \int \sqrt {\sin (2 e+2 f x)} \, dx}{\left (a^2-b^2\right ) g^2 \sqrt {\sin (2 e+2 f x)}} \\ & = -\frac {2 b d^2 \sqrt {d \sin (e+f x)}}{\left (a^2-b^2\right ) f g \sqrt {g \cos (e+f x)}}+\frac {2 a d (d \sin (e+f x))^{3/2}}{\left (a^2-b^2\right ) f g \sqrt {g \cos (e+f x)}}-\frac {2 a d^2 \sqrt {g \cos (e+f x)} E\left (\left .e-\frac {\pi }{4}+f x\right |2\right ) \sqrt {d \sin (e+f x)}}{\left (a^2-b^2\right ) f g^2 \sqrt {\sin (2 e+2 f x)}}-\frac {\left (a^2 d^3\right ) \text {Subst}\left (\int \frac {g-d x^2}{g^2+d^2 x^4} \, dx,x,\frac {\sqrt {g \cos (e+f x)}}{\sqrt {d \sin (e+f x)}}\right )}{b \left (a^2-b^2\right ) f g}+\frac {\left (a^2 d^3\right ) \text {Subst}\left (\int \frac {g+d x^2}{g^2+d^2 x^4} \, dx,x,\frac {\sqrt {g \cos (e+f x)}}{\sqrt {d \sin (e+f x)}}\right )}{b \left (a^2-b^2\right ) f g}+\frac {\left (b d^3\right ) \text {Subst}\left (\int \frac {g-d x^2}{g^2+d^2 x^4} \, dx,x,\frac {\sqrt {g \cos (e+f x)}}{\sqrt {d \sin (e+f x)}}\right )}{\left (a^2-b^2\right ) f g}-\frac {\left (b d^3\right ) \text {Subst}\left (\int \frac {g+d x^2}{g^2+d^2 x^4} \, dx,x,\frac {\sqrt {g \cos (e+f x)}}{\sqrt {d \sin (e+f x)}}\right )}{\left (a^2-b^2\right ) f g}-\frac {\left (4 \sqrt {2} a^3 d^3 \sqrt {\sin (e+f x)}\right ) \text {Subst}\left (\int \frac {x^2}{\left ((a+b) g^2+(a-b) x^4\right ) \sqrt {1-\frac {x^4}{g^2}}} \, dx,x,\frac {\sqrt {g \cos (e+f x)}}{\sqrt {1+\sin (e+f x)}}\right )}{b \left (a^2-b^2\right ) f g \sqrt {d \sin (e+f x)}} \\ & = -\frac {2 b d^2 \sqrt {d \sin (e+f x)}}{\left (a^2-b^2\right ) f g \sqrt {g \cos (e+f x)}}+\frac {2 a d (d \sin (e+f x))^{3/2}}{\left (a^2-b^2\right ) f g \sqrt {g \cos (e+f x)}}-\frac {2 a d^2 \sqrt {g \cos (e+f x)} E\left (\left .e-\frac {\pi }{4}+f x\right |2\right ) \sqrt {d \sin (e+f x)}}{\left (a^2-b^2\right ) f g^2 \sqrt {\sin (2 e+2 f x)}}+\frac {\left (a^2 d^{5/2}\right ) \text {Subst}\left (\int \frac {\frac {\sqrt {2} \sqrt {g}}{\sqrt {d}}+2 x}{-\frac {g}{d}-\frac {\sqrt {2} \sqrt {g} x}{\sqrt {d}}-x^2} \, dx,x,\frac {\sqrt {g \cos (e+f x)}}{\sqrt {d \sin (e+f x)}}\right )}{2 \sqrt {2} b \left (a^2-b^2\right ) f g^{3/2}}+\frac {\left (a^2 d^{5/2}\right ) \text {Subst}\left (\int \frac {\frac {\sqrt {2} \sqrt {g}}{\sqrt {d}}-2 x}{-\frac {g}{d}+\frac {\sqrt {2} \sqrt {g} x}{\sqrt {d}}-x^2} \, dx,x,\frac {\sqrt {g \cos (e+f x)}}{\sqrt {d \sin (e+f x)}}\right )}{2 \sqrt {2} b \left (a^2-b^2\right ) f g^{3/2}}-\frac {\left (b d^{5/2}\right ) \text {Subst}\left (\int \frac {\frac {\sqrt {2} \sqrt {g}}{\sqrt {d}}+2 x}{-\frac {g}{d}-\frac {\sqrt {2} \sqrt {g} x}{\sqrt {d}}-x^2} \, dx,x,\frac {\sqrt {g \cos (e+f x)}}{\sqrt {d \sin (e+f x)}}\right )}{2 \sqrt {2} \left (a^2-b^2\right ) f g^{3/2}}-\frac {\left (b d^{5/2}\right ) \text {Subst}\left (\int \frac {\frac {\sqrt {2} \sqrt {g}}{\sqrt {d}}-2 x}{-\frac {g}{d}+\frac {\sqrt {2} \sqrt {g} x}{\sqrt {d}}-x^2} \, dx,x,\frac {\sqrt {g \cos (e+f x)}}{\sqrt {d \sin (e+f x)}}\right )}{2 \sqrt {2} \left (a^2-b^2\right ) f g^{3/2}}+\frac {\left (a^2 d^2\right ) \text {Subst}\left (\int \frac {1}{\frac {g}{d}-\frac {\sqrt {2} \sqrt {g} x}{\sqrt {d}}+x^2} \, dx,x,\frac {\sqrt {g \cos (e+f x)}}{\sqrt {d \sin (e+f x)}}\right )}{2 b \left (a^2-b^2\right ) f g}+\frac {\left (a^2 d^2\right ) \text {Subst}\left (\int \frac {1}{\frac {g}{d}+\frac {\sqrt {2} \sqrt {g} x}{\sqrt {d}}+x^2} \, dx,x,\frac {\sqrt {g \cos (e+f x)}}{\sqrt {d \sin (e+f x)}}\right )}{2 b \left (a^2-b^2\right ) f g}-\frac {\left (b d^2\right ) \text {Subst}\left (\int \frac {1}{\frac {g}{d}-\frac {\sqrt {2} \sqrt {g} x}{\sqrt {d}}+x^2} \, dx,x,\frac {\sqrt {g \cos (e+f x)}}{\sqrt {d \sin (e+f x)}}\right )}{2 \left (a^2-b^2\right ) f g}-\frac {\left (b d^2\right ) \text {Subst}\left (\int \frac {1}{\frac {g}{d}+\frac {\sqrt {2} \sqrt {g} x}{\sqrt {d}}+x^2} \, dx,x,\frac {\sqrt {g \cos (e+f x)}}{\sqrt {d \sin (e+f x)}}\right )}{2 \left (a^2-b^2\right ) f g}-\frac {\left (2 \sqrt {2} a^3 d^3 \sqrt {\sin (e+f x)}\right ) \text {Subst}\left (\int \frac {1}{\left (\sqrt {a+b} g-\sqrt {-a+b} x^2\right ) \sqrt {1-\frac {x^4}{g^2}}} \, dx,x,\frac {\sqrt {g \cos (e+f x)}}{\sqrt {1+\sin (e+f x)}}\right )}{b \sqrt {-a+b} \left (a^2-b^2\right ) f g \sqrt {d \sin (e+f x)}}+\frac {\left (2 \sqrt {2} a^3 d^3 \sqrt {\sin (e+f x)}\right ) \text {Subst}\left (\int \frac {1}{\left (\sqrt {a+b} g+\sqrt {-a+b} x^2\right ) \sqrt {1-\frac {x^4}{g^2}}} \, dx,x,\frac {\sqrt {g \cos (e+f x)}}{\sqrt {1+\sin (e+f x)}}\right )}{b \sqrt {-a+b} \left (a^2-b^2\right ) f g \sqrt {d \sin (e+f x)}} \\ & = \frac {a^2 d^{5/2} \log \left (\sqrt {g}+\sqrt {g} \cot (e+f x)-\frac {\sqrt {2} \sqrt {d} \sqrt {g \cos (e+f x)}}{\sqrt {d \sin (e+f x)}}\right )}{2 \sqrt {2} b \left (a^2-b^2\right ) f g^{3/2}}-\frac {b d^{5/2} \log \left (\sqrt {g}+\sqrt {g} \cot (e+f x)-\frac {\sqrt {2} \sqrt {d} \sqrt {g \cos (e+f x)}}{\sqrt {d \sin (e+f x)}}\right )}{2 \sqrt {2} \left (a^2-b^2\right ) f g^{3/2}}-\frac {a^2 d^{5/2} \log \left (\sqrt {g}+\sqrt {g} \cot (e+f x)+\frac {\sqrt {2} \sqrt {d} \sqrt {g \cos (e+f x)}}{\sqrt {d \sin (e+f x)}}\right )}{2 \sqrt {2} b \left (a^2-b^2\right ) f g^{3/2}}+\frac {b d^{5/2} \log \left (\sqrt {g}+\sqrt {g} \cot (e+f x)+\frac {\sqrt {2} \sqrt {d} \sqrt {g \cos (e+f x)}}{\sqrt {d \sin (e+f x)}}\right )}{2 \sqrt {2} \left (a^2-b^2\right ) f g^{3/2}}-\frac {2 \sqrt {2} a^3 d^3 \operatorname {EllipticPi}\left (-\frac {\sqrt {-a+b}}{\sqrt {a+b}},\arcsin \left (\frac {\sqrt {g \cos (e+f x)}}{\sqrt {g} \sqrt {1+\sin (e+f x)}}\right ),-1\right ) \sqrt {\sin (e+f x)}}{b (-a+b)^{3/2} (a+b)^{3/2} f g^{3/2} \sqrt {d \sin (e+f x)}}+\frac {2 \sqrt {2} a^3 d^3 \operatorname {EllipticPi}\left (\frac {\sqrt {-a+b}}{\sqrt {a+b}},\arcsin \left (\frac {\sqrt {g \cos (e+f x)}}{\sqrt {g} \sqrt {1+\sin (e+f x)}}\right ),-1\right ) \sqrt {\sin (e+f x)}}{b (-a+b)^{3/2} (a+b)^{3/2} f g^{3/2} \sqrt {d \sin (e+f x)}}-\frac {2 b d^2 \sqrt {d \sin (e+f x)}}{\left (a^2-b^2\right ) f g \sqrt {g \cos (e+f x)}}+\frac {2 a d (d \sin (e+f x))^{3/2}}{\left (a^2-b^2\right ) f g \sqrt {g \cos (e+f x)}}-\frac {2 a d^2 \sqrt {g \cos (e+f x)} E\left (\left .e-\frac {\pi }{4}+f x\right |2\right ) \sqrt {d \sin (e+f x)}}{\left (a^2-b^2\right ) f g^2 \sqrt {\sin (2 e+2 f x)}}+\frac {\left (a^2 d^{5/2}\right ) \text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1-\frac {\sqrt {2} \sqrt {d} \sqrt {g \cos (e+f x)}}{\sqrt {g} \sqrt {d \sin (e+f x)}}\right )}{\sqrt {2} b \left (a^2-b^2\right ) f g^{3/2}}-\frac {\left (a^2 d^{5/2}\right ) \text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1+\frac {\sqrt {2} \sqrt {d} \sqrt {g \cos (e+f x)}}{\sqrt {g} \sqrt {d \sin (e+f x)}}\right )}{\sqrt {2} b \left (a^2-b^2\right ) f g^{3/2}}-\frac {\left (b d^{5/2}\right ) \text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1-\frac {\sqrt {2} \sqrt {d} \sqrt {g \cos (e+f x)}}{\sqrt {g} \sqrt {d \sin (e+f x)}}\right )}{\sqrt {2} \left (a^2-b^2\right ) f g^{3/2}}+\frac {\left (b d^{5/2}\right ) \text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1+\frac {\sqrt {2} \sqrt {d} \sqrt {g \cos (e+f x)}}{\sqrt {g} \sqrt {d \sin (e+f x)}}\right )}{\sqrt {2} \left (a^2-b^2\right ) f g^{3/2}} \\ & = -\frac {a^2 d^{5/2} \arctan \left (1-\frac {\sqrt {2} \sqrt {d} \sqrt {g \cos (e+f x)}}{\sqrt {g} \sqrt {d \sin (e+f x)}}\right )}{\sqrt {2} b \left (a^2-b^2\right ) f g^{3/2}}+\frac {b d^{5/2} \arctan \left (1-\frac {\sqrt {2} \sqrt {d} \sqrt {g \cos (e+f x)}}{\sqrt {g} \sqrt {d \sin (e+f x)}}\right )}{\sqrt {2} \left (a^2-b^2\right ) f g^{3/2}}+\frac {a^2 d^{5/2} \arctan \left (1+\frac {\sqrt {2} \sqrt {d} \sqrt {g \cos (e+f x)}}{\sqrt {g} \sqrt {d \sin (e+f x)}}\right )}{\sqrt {2} b \left (a^2-b^2\right ) f g^{3/2}}-\frac {b d^{5/2} \arctan \left (1+\frac {\sqrt {2} \sqrt {d} \sqrt {g \cos (e+f x)}}{\sqrt {g} \sqrt {d \sin (e+f x)}}\right )}{\sqrt {2} \left (a^2-b^2\right ) f g^{3/2}}+\frac {a^2 d^{5/2} \log \left (\sqrt {g}+\sqrt {g} \cot (e+f x)-\frac {\sqrt {2} \sqrt {d} \sqrt {g \cos (e+f x)}}{\sqrt {d \sin (e+f x)}}\right )}{2 \sqrt {2} b \left (a^2-b^2\right ) f g^{3/2}}-\frac {b d^{5/2} \log \left (\sqrt {g}+\sqrt {g} \cot (e+f x)-\frac {\sqrt {2} \sqrt {d} \sqrt {g \cos (e+f x)}}{\sqrt {d \sin (e+f x)}}\right )}{2 \sqrt {2} \left (a^2-b^2\right ) f g^{3/2}}-\frac {a^2 d^{5/2} \log \left (\sqrt {g}+\sqrt {g} \cot (e+f x)+\frac {\sqrt {2} \sqrt {d} \sqrt {g \cos (e+f x)}}{\sqrt {d \sin (e+f x)}}\right )}{2 \sqrt {2} b \left (a^2-b^2\right ) f g^{3/2}}+\frac {b d^{5/2} \log \left (\sqrt {g}+\sqrt {g} \cot (e+f x)+\frac {\sqrt {2} \sqrt {d} \sqrt {g \cos (e+f x)}}{\sqrt {d \sin (e+f x)}}\right )}{2 \sqrt {2} \left (a^2-b^2\right ) f g^{3/2}}-\frac {2 \sqrt {2} a^3 d^3 \operatorname {EllipticPi}\left (-\frac {\sqrt {-a+b}}{\sqrt {a+b}},\arcsin \left (\frac {\sqrt {g \cos (e+f x)}}{\sqrt {g} \sqrt {1+\sin (e+f x)}}\right ),-1\right ) \sqrt {\sin (e+f x)}}{b (-a+b)^{3/2} (a+b)^{3/2} f g^{3/2} \sqrt {d \sin (e+f x)}}+\frac {2 \sqrt {2} a^3 d^3 \operatorname {EllipticPi}\left (\frac {\sqrt {-a+b}}{\sqrt {a+b}},\arcsin \left (\frac {\sqrt {g \cos (e+f x)}}{\sqrt {g} \sqrt {1+\sin (e+f x)}}\right ),-1\right ) \sqrt {\sin (e+f x)}}{b (-a+b)^{3/2} (a+b)^{3/2} f g^{3/2} \sqrt {d \sin (e+f x)}}-\frac {2 b d^2 \sqrt {d \sin (e+f x)}}{\left (a^2-b^2\right ) f g \sqrt {g \cos (e+f x)}}+\frac {2 a d (d \sin (e+f x))^{3/2}}{\left (a^2-b^2\right ) f g \sqrt {g \cos (e+f x)}}-\frac {2 a d^2 \sqrt {g \cos (e+f x)} E\left (\left .e-\frac {\pi }{4}+f x\right |2\right ) \sqrt {d \sin (e+f x)}}{\left (a^2-b^2\right ) f g^2 \sqrt {\sin (2 e+2 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 = 25.90 (sec) , antiderivative size = 1290, normalized size of antiderivative = 1.21 \[ \int \frac {(d \sin (e+f x))^{5/2}}{(g \cos (e+f x))^{3/2} (a+b \sin (e+f x))} \, dx=\frac {2 \cot (e+f x) \csc (e+f x) (d \sin (e+f x))^{5/2} (-b+a \sin (e+f x))}{\left (a^2-b^2\right ) f (g \cos (e+f x))^{3/2}}-\frac {\cos ^{\frac {3}{2}}(e+f x) (d \sin (e+f x))^{5/2} \left (-\frac {2 \left (3 a^2-b^2\right ) \left (-b \operatorname {AppellF1}\left (\frac {3}{4},-\frac {1}{4},1,\frac {7}{4},\cos ^2(e+f x),\frac {b^2 \cos ^2(e+f x)}{-a^2+b^2}\right )+a \operatorname {AppellF1}\left (\frac {3}{4},\frac {1}{4},1,\frac {7}{4},\cos ^2(e+f x),\frac {b^2 \cos ^2(e+f x)}{-a^2+b^2}\right )\right ) \cos ^{\frac {3}{2}}(e+f x) \left (a+b \sqrt {1-\cos ^2(e+f x)}\right ) \sin ^{\frac {3}{2}}(e+f x)}{3 \left (a^2-b^2\right ) \left (1-\cos ^2(e+f x)\right )^{3/4} (a+b \sin (e+f x))}-\frac {\cos (2 (e+f x)) \sqrt {\tan (e+f x)} \left (b \tan (e+f x)+a \sqrt {1+\tan ^2(e+f x)}\right ) \left (56 b \left (-3 a^2+b^2\right ) \operatorname {AppellF1}\left (\frac {3}{4},\frac {1}{2},1,\frac {7}{4},-\tan ^2(e+f x),\left (-1+\frac {b^2}{a^2}\right ) \tan ^2(e+f x)\right ) \tan ^{\frac {3}{2}}(e+f x)+24 b \left (-a^2+b^2\right ) \operatorname {AppellF1}\left (\frac {7}{4},\frac {1}{2},1,\frac {11}{4},-\tan ^2(e+f x),\left (-1+\frac {b^2}{a^2}\right ) \tan ^2(e+f x)\right ) \tan ^{\frac {7}{2}}(e+f x)+21 a^{3/2} \left (4 \sqrt {2} a^{3/2} \arctan \left (1-\sqrt {2} \sqrt {\tan (e+f x)}\right )-4 \sqrt {2} a^{3/2} \arctan \left (1+\sqrt {2} \sqrt {\tan (e+f x)}\right )-\frac {4 \sqrt {2} a^2 \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{a^2-b^2} \sqrt {\tan (e+f x)}}{\sqrt {a}}\right )}{\sqrt [4]{a^2-b^2}}+\frac {2 \sqrt {2} b^2 \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{a^2-b^2} \sqrt {\tan (e+f x)}}{\sqrt {a}}\right )}{\sqrt [4]{a^2-b^2}}+\frac {4 \sqrt {2} a^2 \arctan \left (1+\frac {\sqrt {2} \sqrt [4]{a^2-b^2} \sqrt {\tan (e+f x)}}{\sqrt {a}}\right )}{\sqrt [4]{a^2-b^2}}-\frac {2 \sqrt {2} b^2 \arctan \left (1+\frac {\sqrt {2} \sqrt [4]{a^2-b^2} \sqrt {\tan (e+f x)}}{\sqrt {a}}\right )}{\sqrt [4]{a^2-b^2}}+2 \sqrt {2} a^{3/2} \log \left (1-\sqrt {2} \sqrt {\tan (e+f x)}+\tan (e+f x)\right )-2 \sqrt {2} a^{3/2} \log \left (1+\sqrt {2} \sqrt {\tan (e+f x)}+\tan (e+f x)\right )-\frac {2 \sqrt {2} a^2 \log \left (-a+\sqrt {2} \sqrt {a} \sqrt [4]{a^2-b^2} \sqrt {\tan (e+f x)}-\sqrt {a^2-b^2} \tan (e+f x)\right )}{\sqrt [4]{a^2-b^2}}+\frac {\sqrt {2} b^2 \log \left (-a+\sqrt {2} \sqrt {a} \sqrt [4]{a^2-b^2} \sqrt {\tan (e+f x)}-\sqrt {a^2-b^2} \tan (e+f x)\right )}{\sqrt [4]{a^2-b^2}}+\frac {2 \sqrt {2} a^2 \log \left (a+\sqrt {2} \sqrt {a} \sqrt [4]{a^2-b^2} \sqrt {\tan (e+f x)}+\sqrt {a^2-b^2} \tan (e+f x)\right )}{\sqrt [4]{a^2-b^2}}-\frac {\sqrt {2} b^2 \log \left (a+\sqrt {2} \sqrt {a} \sqrt [4]{a^2-b^2} \sqrt {\tan (e+f x)}+\sqrt {a^2-b^2} \tan (e+f x)\right )}{\sqrt [4]{a^2-b^2}}+\frac {8 \sqrt {a} b \tan ^{\frac {3}{2}}(e+f x)}{\sqrt {1+\tan ^2(e+f x)}}\right )\right )}{84 a b \cos ^{\frac {3}{2}}(e+f x) \sqrt {\sin (e+f x)} (a+b \sin (e+f x)) \left (-1+\tan ^2(e+f x)\right ) \sqrt {1+\tan ^2(e+f x)}}\right )}{(a-b) (a+b) f (g \cos (e+f x))^{3/2} \sin ^{\frac {5}{2}}(e+f x)} \]

[In]

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

[Out]

(2*Cot[e + f*x]*Csc[e + f*x]*(d*Sin[e + f*x])^(5/2)*(-b + a*Sin[e + f*x]))/((a^2 - b^2)*f*(g*Cos[e + f*x])^(3/
2)) - (Cos[e + f*x]^(3/2)*(d*Sin[e + f*x])^(5/2)*((-2*(3*a^2 - b^2)*(-(b*AppellF1[3/4, -1/4, 1, 7/4, Cos[e + f
*x]^2, (b^2*Cos[e + f*x]^2)/(-a^2 + b^2)]) + a*AppellF1[3/4, 1/4, 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)*(a + b*Sqrt[1 - Cos[e + f*x]^2])*Sin[e + f*x]^(3/2))/(3*(a^2 - b^2)*(1 - Co
s[e + f*x]^2)^(3/4)*(a + b*Sin[e + f*x])) - (Cos[2*(e + f*x)]*Sqrt[Tan[e + f*x]]*(b*Tan[e + f*x] + a*Sqrt[1 +
Tan[e + f*x]^2])*(56*b*(-3*a^2 + b^2)*AppellF1[3/4, 1/2, 1, 7/4, -Tan[e + f*x]^2, (-1 + b^2/a^2)*Tan[e + f*x]^
2]*Tan[e + f*x]^(3/2) + 24*b*(-a^2 + b^2)*AppellF1[7/4, 1/2, 1, 11/4, -Tan[e + f*x]^2, (-1 + b^2/a^2)*Tan[e +
f*x]^2]*Tan[e + f*x]^(7/2) + 21*a^(3/2)*(4*Sqrt[2]*a^(3/2)*ArcTan[1 - Sqrt[2]*Sqrt[Tan[e + f*x]]] - 4*Sqrt[2]*
a^(3/2)*ArcTan[1 + Sqrt[2]*Sqrt[Tan[e + f*x]]] - (4*Sqrt[2]*a^2*ArcTan[1 - (Sqrt[2]*(a^2 - b^2)^(1/4)*Sqrt[Tan
[e + f*x]])/Sqrt[a]])/(a^2 - b^2)^(1/4) + (2*Sqrt[2]*b^2*ArcTan[1 - (Sqrt[2]*(a^2 - b^2)^(1/4)*Sqrt[Tan[e + f*
x]])/Sqrt[a]])/(a^2 - b^2)^(1/4) + (4*Sqrt[2]*a^2*ArcTan[1 + (Sqrt[2]*(a^2 - b^2)^(1/4)*Sqrt[Tan[e + f*x]])/Sq
rt[a]])/(a^2 - b^2)^(1/4) - (2*Sqrt[2]*b^2*ArcTan[1 + (Sqrt[2]*(a^2 - b^2)^(1/4)*Sqrt[Tan[e + f*x]])/Sqrt[a]])
/(a^2 - b^2)^(1/4) + 2*Sqrt[2]*a^(3/2)*Log[1 - Sqrt[2]*Sqrt[Tan[e + f*x]] + Tan[e + f*x]] - 2*Sqrt[2]*a^(3/2)*
Log[1 + Sqrt[2]*Sqrt[Tan[e + f*x]] + Tan[e + f*x]] - (2*Sqrt[2]*a^2*Log[-a + Sqrt[2]*Sqrt[a]*(a^2 - b^2)^(1/4)
*Sqrt[Tan[e + f*x]] - Sqrt[a^2 - b^2]*Tan[e + f*x]])/(a^2 - b^2)^(1/4) + (Sqrt[2]*b^2*Log[-a + Sqrt[2]*Sqrt[a]
*(a^2 - b^2)^(1/4)*Sqrt[Tan[e + f*x]] - Sqrt[a^2 - b^2]*Tan[e + f*x]])/(a^2 - b^2)^(1/4) + (2*Sqrt[2]*a^2*Log[
a + Sqrt[2]*Sqrt[a]*(a^2 - b^2)^(1/4)*Sqrt[Tan[e + f*x]] + Sqrt[a^2 - b^2]*Tan[e + f*x]])/(a^2 - b^2)^(1/4) -
(Sqrt[2]*b^2*Log[a + Sqrt[2]*Sqrt[a]*(a^2 - b^2)^(1/4)*Sqrt[Tan[e + f*x]] + Sqrt[a^2 - b^2]*Tan[e + f*x]])/(a^
2 - b^2)^(1/4) + (8*Sqrt[a]*b*Tan[e + f*x]^(3/2))/Sqrt[1 + Tan[e + f*x]^2])))/(84*a*b*Cos[e + f*x]^(3/2)*Sqrt[
Sin[e + f*x]]*(a + b*Sin[e + f*x])*(-1 + Tan[e + f*x]^2)*Sqrt[1 + Tan[e + f*x]^2])))/((a - b)*(a + b)*f*(g*Cos
[e + f*x])^(3/2)*Sin[e + f*x]^(5/2))

Maple [B] (warning: unable to verify)

Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 2012 vs. \(2 (887 ) = 1774\).

Time = 3.75 (sec) , antiderivative size = 2013, normalized size of antiderivative = 1.89

method	result	size
default	\(\text {Expression too large to display}\)	\(2013\)

[In]

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

[Out]

-1/f*(d/((1-cos(f*x+e))^2*csc(f*x+e)^2+1)*(csc(f*x+e)-cot(f*x+e)))^(5/2)/(1-cos(f*x+e))^3*sin(f*x+e)^3*((1-cos
(f*x+e))^2*csc(f*x+e)^2+1)*(I*(-a^2+b^2)^(1/2)*(-cot(f*x+e)+csc(f*x+e)+1)^(1/2)*(2+2*cot(f*x+e)-2*csc(f*x+e))^
(1/2)*(-csc(f*x+e)+cot(f*x+e))^(1/2)*EllipticPi((-cot(f*x+e)+csc(f*x+e)+1)^(1/2),1/2+1/2*I,1/2*2^(1/2))*b^2+I*
(-a^2+b^2)^(1/2)*(-cot(f*x+e)+csc(f*x+e)+1)^(1/2)*(2+2*cot(f*x+e)-2*csc(f*x+e))^(1/2)*(-csc(f*x+e)+cot(f*x+e))
^(1/2)*EllipticPi((-cot(f*x+e)+csc(f*x+e)+1)^(1/2),1/2-1/2*I,1/2*2^(1/2))*a^2-I*(-a^2+b^2)^(1/2)*(-cot(f*x+e)+
csc(f*x+e)+1)^(1/2)*(2+2*cot(f*x+e)-2*csc(f*x+e))^(1/2)*(-csc(f*x+e)+cot(f*x+e))^(1/2)*EllipticPi((-cot(f*x+e)
+csc(f*x+e)+1)^(1/2),1/2-1/2*I,1/2*2^(1/2))*b^2-I*(-a^2+b^2)^(1/2)*(-cot(f*x+e)+csc(f*x+e)+1)^(1/2)*(2+2*cot(f
*x+e)-2*csc(f*x+e))^(1/2)*(-csc(f*x+e)+cot(f*x+e))^(1/2)*EllipticPi((-cot(f*x+e)+csc(f*x+e)+1)^(1/2),1/2+1/2*I
,1/2*2^(1/2))*a^2+2*EllipticF((-cot(f*x+e)+csc(f*x+e)+1)^(1/2),1/2*2^(1/2))*a*b*(-a^2+b^2)^(1/2)*(-cot(f*x+e)+
csc(f*x+e)+1)^(1/2)*(2+2*cot(f*x+e)-2*csc(f*x+e))^(1/2)*(-csc(f*x+e)+cot(f*x+e))^(1/2)+2*EllipticF((-cot(f*x+e
)+csc(f*x+e)+1)^(1/2),1/2*2^(1/2))*b^2*(-a^2+b^2)^(1/2)*(-cot(f*x+e)+csc(f*x+e)+1)^(1/2)*(2+2*cot(f*x+e)-2*csc
(f*x+e))^(1/2)*(-csc(f*x+e)+cot(f*x+e))^(1/2)+(-a^2+b^2)^(1/2)*(-cot(f*x+e)+csc(f*x+e)+1)^(1/2)*(2+2*cot(f*x+e
)-2*csc(f*x+e))^(1/2)*(-csc(f*x+e)+cot(f*x+e))^(1/2)*EllipticPi((-cot(f*x+e)+csc(f*x+e)+1)^(1/2),1/2-1/2*I,1/2
*2^(1/2))*a^2-(-a^2+b^2)^(1/2)*(-cot(f*x+e)+csc(f*x+e)+1)^(1/2)*(2+2*cot(f*x+e)-2*csc(f*x+e))^(1/2)*(-csc(f*x+
e)+cot(f*x+e))^(1/2)*EllipticPi((-cot(f*x+e)+csc(f*x+e)+1)^(1/2),1/2-1/2*I,1/2*2^(1/2))*b^2+(-a^2+b^2)^(1/2)*(
-cot(f*x+e)+csc(f*x+e)+1)^(1/2)*(2+2*cot(f*x+e)-2*csc(f*x+e))^(1/2)*(-csc(f*x+e)+cot(f*x+e))^(1/2)*EllipticPi(
(-cot(f*x+e)+csc(f*x+e)+1)^(1/2),1/2+1/2*I,1/2*2^(1/2))*a^2-(-a^2+b^2)^(1/2)*(-cot(f*x+e)+csc(f*x+e)+1)^(1/2)*
(2+2*cot(f*x+e)-2*csc(f*x+e))^(1/2)*(-csc(f*x+e)+cot(f*x+e))^(1/2)*EllipticPi((-cot(f*x+e)+csc(f*x+e)+1)^(1/2)
,1/2+1/2*I,1/2*2^(1/2))*b^2+EllipticPi((-cot(f*x+e)+csc(f*x+e)+1)^(1/2),a/(-b+(-a^2+b^2)^(1/2)+a),1/2*2^(1/2))
*a^3*(-cot(f*x+e)+csc(f*x+e)+1)^(1/2)*(2+2*cot(f*x+e)-2*csc(f*x+e))^(1/2)*(-csc(f*x+e)+cot(f*x+e))^(1/2)+Ellip
ticPi((-cot(f*x+e)+csc(f*x+e)+1)^(1/2),a/(-b+(-a^2+b^2)^(1/2)+a),1/2*2^(1/2))*a^2*b*(-cot(f*x+e)+csc(f*x+e)+1)
^(1/2)*(2+2*cot(f*x+e)-2*csc(f*x+e))^(1/2)*(-csc(f*x+e)+cot(f*x+e))^(1/2)-EllipticPi((-cot(f*x+e)+csc(f*x+e)+1
)^(1/2),a/(-b+(-a^2+b^2)^(1/2)+a),1/2*2^(1/2))*a^2*(-a^2+b^2)^(1/2)*(-cot(f*x+e)+csc(f*x+e)+1)^(1/2)*(2+2*cot(
f*x+e)-2*csc(f*x+e))^(1/2)*(-csc(f*x+e)+cot(f*x+e))^(1/2)-4*EllipticE((-cot(f*x+e)+csc(f*x+e)+1)^(1/2),1/2*2^(
1/2))*a*b*(-a^2+b^2)^(1/2)*(-cot(f*x+e)+csc(f*x+e)+1)^(1/2)*(2+2*cot(f*x+e)-2*csc(f*x+e))^(1/2)*(-csc(f*x+e)+c
ot(f*x+e))^(1/2)-EllipticPi((-cot(f*x+e)+csc(f*x+e)+1)^(1/2),-a/(b+(-a^2+b^2)^(1/2)-a),1/2*2^(1/2))*a^3*(-cot(
f*x+e)+csc(f*x+e)+1)^(1/2)*(2+2*cot(f*x+e)-2*csc(f*x+e))^(1/2)*(-csc(f*x+e)+cot(f*x+e))^(1/2)-EllipticPi((-cot
(f*x+e)+csc(f*x+e)+1)^(1/2),-a/(b+(-a^2+b^2)^(1/2)-a),1/2*2^(1/2))*a^2*b*(-cot(f*x+e)+csc(f*x+e)+1)^(1/2)*(2+2
*cot(f*x+e)-2*csc(f*x+e))^(1/2)*(-csc(f*x+e)+cot(f*x+e))^(1/2)-EllipticPi((-cot(f*x+e)+csc(f*x+e)+1)^(1/2),-a/
(b+(-a^2+b^2)^(1/2)-a),1/2*2^(1/2))*a^2*(-a^2+b^2)^(1/2)*(-cot(f*x+e)+csc(f*x+e)+1)^(1/2)*(2+2*cot(f*x+e)-2*cs
c(f*x+e))^(1/2)*(-csc(f*x+e)+cot(f*x+e))^(1/2)-4*csc(f*x+e)^2*a*b*(-a^2+b^2)^(1/2)*(1-cos(f*x+e))^2+4*(-a^2+b^
2)^(1/2)*b^2*(csc(f*x+e)-cot(f*x+e)))*((1-cos(f*x+e))^2*csc(f*x+e)^2-1)/(-g*((1-cos(f*x+e))^2*csc(f*x+e)^2-1)/
((1-cos(f*x+e))^2*csc(f*x+e)^2+1))^(3/2)*2^(1/2)*a/(a+b)/b/(-a^2+b^2)^(1/2)/(-b+(-a^2+b^2)^(1/2)+a)/(b+(-a^2+b
^2)^(1/2)-a)

Fricas [F(-1)]

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

[In]

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

[Out]

Timed out

Sympy [F(-1)]

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

[In]

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

[Out]

Timed out

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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