3.1436 \(\int \frac{(d \sin (e+f x))^{5/2}}{(g \cos (e+f x))^{3/2} (a+b \sin (e+f x))} \, dx\)

Optimal. Leaf size=1064 \[ \text{result too large to display} \]

[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]])

________________________________________________________________________________________

Rubi [A]  time = 1.62418, antiderivative size = 1064, normalized size of antiderivative = 1., number of steps used = 31, number of rules used = 17, integrand size = 37, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.46, Rules used = {2902, 2571, 2572, 2639, 2566, 2575, 297, 1162, 617, 204, 1165, 628, 2909, 2906, 2905, 490, 1218} \[ -\frac{2 \sqrt{2} a^3 \Pi \left (-\frac{\sqrt{b-a}}{\sqrt{a+b}};\left .\sin ^{-1}\left (\frac{\sqrt{g \cos (e+f x)}}{\sqrt{g} \sqrt{\sin (e+f x)+1}}\right )\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 \Pi \left (\frac{\sqrt{b-a}}{\sqrt{a+b}};\left .\sin ^{-1}\left (\frac{\sqrt{g \cos (e+f x)}}{\sqrt{g} \sqrt{\sin (e+f x)+1}}\right )\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 \tan ^{-1}\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 \tan ^{-1}\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 \tan ^{-1}\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 \tan ^{-1}\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)}} \]

Antiderivative was successfully verified.

[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 2902

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 2571

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 2572

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 2639

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

Rule 2566

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] && (Integ
ersQ[2*m, 2*n] || EqQ[m + n, 0])

Rule 2575

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*Si
n[e + f*x])^(1/k)], x]] /; FreeQ[{a, b, e, f}, x] && EqQ[m + n, 0] && GtQ[m, 0] && LtQ[m, 1]

Rule 297

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 1162

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 617

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 204

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

Rule 1165

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 628

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 2909

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]

Rule 2906

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 2905

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 490

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), In
t[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 1218

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

Rubi steps

\begin{align*} \int \frac{(d \sin (e+f x))^{5/2}}{(g \cos (e+f x))^{3/2} (a+b \sin (e+f x))} \, dx &=-\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 ) \operatorname{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 ) \operatorname{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 ) \operatorname{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 ) \operatorname{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 ) \operatorname{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 ) \operatorname{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 ) \operatorname{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 ) \operatorname{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 ) \operatorname{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 ) \operatorname{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 ) \operatorname{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 ) \operatorname{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 ) \operatorname{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 ) \operatorname{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 ) \operatorname{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 ) \operatorname{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 ) \operatorname{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 \Pi \left (-\frac{\sqrt{-a+b}}{\sqrt{a+b}};\left .\sin ^{-1}\left (\frac{\sqrt{g \cos (e+f x)}}{\sqrt{g} \sqrt{1+\sin (e+f x)}}\right )\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 \Pi \left (\frac{\sqrt{-a+b}}{\sqrt{a+b}};\left .\sin ^{-1}\left (\frac{\sqrt{g \cos (e+f x)}}{\sqrt{g} \sqrt{1+\sin (e+f x)}}\right )\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 ) \operatorname{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 ) \operatorname{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 ) \operatorname{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 ) \operatorname{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} \tan ^{-1}\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} \tan ^{-1}\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} \tan ^{-1}\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} \tan ^{-1}\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 \Pi \left (-\frac{\sqrt{-a+b}}{\sqrt{a+b}};\left .\sin ^{-1}\left (\frac{\sqrt{g \cos (e+f x)}}{\sqrt{g} \sqrt{1+\sin (e+f x)}}\right )\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 \Pi \left (\frac{\sqrt{-a+b}}{\sqrt{a+b}};\left .\sin ^{-1}\left (\frac{\sqrt{g \cos (e+f x)}}{\sqrt{g} \sqrt{1+\sin (e+f x)}}\right )\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]  time = 80.9139, size = 1296, normalized size = 1.22 \[ \frac{2 \cot (e+f x) \csc (e+f x) (d \sin (e+f x))^{5/2} (a \sin (e+f x)-b)}{\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 (a F_1\left (\frac{3}{4};\frac{1}{4},1;\frac{7}{4};\cos ^2(e+f x),\frac{b^2 \cos ^2(e+f x)}{b^2-a^2}\right )-b F_1\left (\frac{3}{4};-\frac{1}{4},1;\frac{7}{4};\cos ^2(e+f x),\frac{b^2 \cos ^2(e+f x)}{b^2-a^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 (\sqrt{\tan ^2(e+f x)+1} a+b \tan (e+f x)\right ) \left (24 b \left (b^2-a^2\right ) F_1\left (\frac{7}{4};\frac{1}{2},1;\frac{11}{4};-\tan ^2(e+f x),\frac{\left (b^2-a^2\right ) \tan ^2(e+f x)}{a^2}\right ) \tan ^{\frac{7}{2}}(e+f x)+56 b \left (b^2-3 a^2\right ) F_1\left (\frac{3}{4};\frac{1}{2},1;\frac{7}{4};-\tan ^2(e+f x),\frac{\left (b^2-a^2\right ) \tan ^2(e+f x)}{a^2}\right ) \tan ^{\frac{3}{2}}(e+f x)+21 a^{3/2} \left (-\frac{4 \sqrt{2} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{a^2-b^2} \sqrt{\tan (e+f x)}}{\sqrt{a}}\right ) a^2}{\sqrt [4]{a^2-b^2}}+\frac{4 \sqrt{2} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{a^2-b^2} \sqrt{\tan (e+f x)}}{\sqrt{a}}+1\right ) a^2}{\sqrt [4]{a^2-b^2}}-\frac{2 \sqrt{2} \log \left (-a+\sqrt{2} \sqrt [4]{a^2-b^2} \sqrt{\tan (e+f x)} \sqrt{a}-\sqrt{a^2-b^2} \tan (e+f x)\right ) a^2}{\sqrt [4]{a^2-b^2}}+\frac{2 \sqrt{2} \log \left (a+\sqrt{2} \sqrt [4]{a^2-b^2} \sqrt{\tan (e+f x)} \sqrt{a}+\sqrt{a^2-b^2} \tan (e+f x)\right ) a^2}{\sqrt [4]{a^2-b^2}}+4 \sqrt{2} \tan ^{-1}\left (1-\sqrt{2} \sqrt{\tan (e+f x)}\right ) a^{3/2}-4 \sqrt{2} \tan ^{-1}\left (\sqrt{2} \sqrt{\tan (e+f x)}+1\right ) a^{3/2}+2 \sqrt{2} \log \left (\tan (e+f x)-\sqrt{2} \sqrt{\tan (e+f x)}+1\right ) a^{3/2}-2 \sqrt{2} \log \left (\tan (e+f x)+\sqrt{2} \sqrt{\tan (e+f x)}+1\right ) a^{3/2}+\frac{8 b \tan ^{\frac{3}{2}}(e+f x) \sqrt{a}}{\sqrt{\tan ^2(e+f x)+1}}+\frac{2 \sqrt{2} b^2 \tan ^{-1}\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 \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{a^2-b^2} \sqrt{\tan (e+f x)}}{\sqrt{a}}+1\right )}{\sqrt [4]{a^2-b^2}}+\frac{\sqrt{2} b^2 \log \left (-a+\sqrt{2} \sqrt [4]{a^2-b^2} \sqrt{\tan (e+f x)} \sqrt{a}-\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 [4]{a^2-b^2} \sqrt{\tan (e+f x)} \sqrt{a}+\sqrt{a^2-b^2} \tan (e+f x)\right )}{\sqrt [4]{a^2-b^2}}\right )\right )}{84 a b \cos ^{\frac{3}{2}}(e+f x) (a+b \sin (e+f x)) \left (\tan ^2(e+f x)-1\right ) \sqrt{\tan ^2(e+f x)+1} \sqrt{\sin (e+f x)}}\right )}{(a-b) (a+b) f (g \cos (e+f x))^{3/2} \sin ^{\frac{5}{2}}(e+f x)} \]

Warning: Unable to verify antiderivative.

[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, ((-a^2 + b^2)*Tan[e + f*x]^2
)/a^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, ((-a^2 + b^2)*Tan[e
+ f*x]^2)/a^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[Ta
n[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]])/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]])/S
qrt[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]  time = 0.329, size = 4622, normalized size = 4.3 \begin{align*} \text{output too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/f*2^(1/2)*a/(a+b)/b/(-a^2+b^2)^(1/2)/(a-b+(-a^2+b^2)^(1/2))/(b+(-a^2+b^2)^(1/2)-a)*(I*cos(f*x+e)*(-(-1+cos(f
*x+e)-sin(f*x+e))/sin(f*x+e))^(1/2)*((-1+cos(f*x+e)+sin(f*x+e))/sin(f*x+e))^(1/2)*((-1+cos(f*x+e))/sin(f*x+e))
^(1/2)*EllipticPi((-(-1+cos(f*x+e)-sin(f*x+e))/sin(f*x+e))^(1/2),1/2+1/2*I,1/2*2^(1/2))*(-a^2+b^2)^(1/2)*b^2+2
*2^(1/2)*sin(f*x+e)*(-a^2+b^2)^(1/2)*b^2+EllipticPi((-(-1+cos(f*x+e)-sin(f*x+e))/sin(f*x+e))^(1/2),a/(a-b+(-a^
2+b^2)^(1/2)),1/2*2^(1/2))*cos(f*x+e)*(-(-1+cos(f*x+e)-sin(f*x+e))/sin(f*x+e))^(1/2)*((-1+cos(f*x+e)+sin(f*x+e
))/sin(f*x+e))^(1/2)*((-1+cos(f*x+e))/sin(f*x+e))^(1/2)*a^3+EllipticPi((-(-1+cos(f*x+e)-sin(f*x+e))/sin(f*x+e)
)^(1/2),a/(a-b+(-a^2+b^2)^(1/2)),1/2*2^(1/2))*(-(-1+cos(f*x+e)-sin(f*x+e))/sin(f*x+e))^(1/2)*((-1+cos(f*x+e)+s
in(f*x+e))/sin(f*x+e))^(1/2)*((-1+cos(f*x+e))/sin(f*x+e))^(1/2)*a^3-EllipticPi((-(-1+cos(f*x+e)-sin(f*x+e))/si
n(f*x+e))^(1/2),-a/(b+(-a^2+b^2)^(1/2)-a),1/2*2^(1/2))*(-(-1+cos(f*x+e)-sin(f*x+e))/sin(f*x+e))^(1/2)*((-1+cos
(f*x+e)+sin(f*x+e))/sin(f*x+e))^(1/2)*((-1+cos(f*x+e))/sin(f*x+e))^(1/2)*a^3+2*cos(f*x+e)*2^(1/2)*(-a^2+b^2)^(
1/2)*a*b-2*2^(1/2)*(-a^2+b^2)^(1/2)*a*b-cos(f*x+e)*(-(-1+cos(f*x+e)-sin(f*x+e))/sin(f*x+e))^(1/2)*((-1+cos(f*x
+e)+sin(f*x+e))/sin(f*x+e))^(1/2)*((-1+cos(f*x+e))/sin(f*x+e))^(1/2)*EllipticPi((-(-1+cos(f*x+e)-sin(f*x+e))/s
in(f*x+e))^(1/2),1/2-1/2*I,1/2*2^(1/2))*(-a^2+b^2)^(1/2)*b^2+cos(f*x+e)*(-(-1+cos(f*x+e)-sin(f*x+e))/sin(f*x+e
))^(1/2)*((-1+cos(f*x+e)+sin(f*x+e))/sin(f*x+e))^(1/2)*((-1+cos(f*x+e))/sin(f*x+e))^(1/2)*EllipticPi((-(-1+cos
(f*x+e)-sin(f*x+e))/sin(f*x+e))^(1/2),1/2+1/2*I,1/2*2^(1/2))*(-a^2+b^2)^(1/2)*a^2-cos(f*x+e)*(-(-1+cos(f*x+e)-
sin(f*x+e))/sin(f*x+e))^(1/2)*((-1+cos(f*x+e)+sin(f*x+e))/sin(f*x+e))^(1/2)*((-1+cos(f*x+e))/sin(f*x+e))^(1/2)
*EllipticPi((-(-1+cos(f*x+e)-sin(f*x+e))/sin(f*x+e))^(1/2),1/2+1/2*I,1/2*2^(1/2))*(-a^2+b^2)^(1/2)*b^2+2*cos(f
*x+e)*(-(-1+cos(f*x+e)-sin(f*x+e))/sin(f*x+e))^(1/2)*((-1+cos(f*x+e)+sin(f*x+e))/sin(f*x+e))^(1/2)*((-1+cos(f*
x+e))/sin(f*x+e))^(1/2)*EllipticF((-(-1+cos(f*x+e)-sin(f*x+e))/sin(f*x+e))^(1/2),1/2*2^(1/2))*(-a^2+b^2)^(1/2)
*b^2+I*(-(-1+cos(f*x+e)-sin(f*x+e))/sin(f*x+e))^(1/2)*((-1+cos(f*x+e)+sin(f*x+e))/sin(f*x+e))^(1/2)*((-1+cos(f
*x+e))/sin(f*x+e))^(1/2)*EllipticPi((-(-1+cos(f*x+e)-sin(f*x+e))/sin(f*x+e))^(1/2),1/2+1/2*I,1/2*2^(1/2))*(-a^
2+b^2)^(1/2)*b^2-4*(-(-1+cos(f*x+e)-sin(f*x+e))/sin(f*x+e))^(1/2)*((-1+cos(f*x+e)+sin(f*x+e))/sin(f*x+e))^(1/2
)*((-1+cos(f*x+e))/sin(f*x+e))^(1/2)*EllipticE((-(-1+cos(f*x+e)-sin(f*x+e))/sin(f*x+e))^(1/2),1/2*2^(1/2))*(-a
^2+b^2)^(1/2)*a*b+2*(-(-1+cos(f*x+e)-sin(f*x+e))/sin(f*x+e))^(1/2)*((-1+cos(f*x+e)+sin(f*x+e))/sin(f*x+e))^(1/
2)*((-1+cos(f*x+e))/sin(f*x+e))^(1/2)*EllipticF((-(-1+cos(f*x+e)-sin(f*x+e))/sin(f*x+e))^(1/2),1/2*2^(1/2))*(-
a^2+b^2)^(1/2)*a*b-I*(-(-1+cos(f*x+e)-sin(f*x+e))/sin(f*x+e))^(1/2)*((-1+cos(f*x+e)+sin(f*x+e))/sin(f*x+e))^(1
/2)*((-1+cos(f*x+e))/sin(f*x+e))^(1/2)*EllipticPi((-(-1+cos(f*x+e)-sin(f*x+e))/sin(f*x+e))^(1/2),1/2-1/2*I,1/2
*2^(1/2))*(-a^2+b^2)^(1/2)*b^2-EllipticPi((-(-1+cos(f*x+e)-sin(f*x+e))/sin(f*x+e))^(1/2),a/(a-b+(-a^2+b^2)^(1/
2)),1/2*2^(1/2))*cos(f*x+e)*(-(-1+cos(f*x+e)-sin(f*x+e))/sin(f*x+e))^(1/2)*((-1+cos(f*x+e)+sin(f*x+e))/sin(f*x
+e))^(1/2)*((-1+cos(f*x+e))/sin(f*x+e))^(1/2)*(-a^2+b^2)^(1/2)*a^2+EllipticPi((-(-1+cos(f*x+e)-sin(f*x+e))/sin
(f*x+e))^(1/2),a/(a-b+(-a^2+b^2)^(1/2)),1/2*2^(1/2))*cos(f*x+e)*(-(-1+cos(f*x+e)-sin(f*x+e))/sin(f*x+e))^(1/2)
*((-1+cos(f*x+e)+sin(f*x+e))/sin(f*x+e))^(1/2)*((-1+cos(f*x+e))/sin(f*x+e))^(1/2)*a^2*b-EllipticPi((-(-1+cos(f
*x+e)-sin(f*x+e))/sin(f*x+e))^(1/2),-a/(b+(-a^2+b^2)^(1/2)-a),1/2*2^(1/2))*cos(f*x+e)*(-(-1+cos(f*x+e)-sin(f*x
+e))/sin(f*x+e))^(1/2)*((-1+cos(f*x+e)+sin(f*x+e))/sin(f*x+e))^(1/2)*((-1+cos(f*x+e))/sin(f*x+e))^(1/2)*(-a^2+
b^2)^(1/2)*a^2-EllipticPi((-(-1+cos(f*x+e)-sin(f*x+e))/sin(f*x+e))^(1/2),-a/(b+(-a^2+b^2)^(1/2)-a),1/2*2^(1/2)
)*cos(f*x+e)*(-(-1+cos(f*x+e)-sin(f*x+e))/sin(f*x+e))^(1/2)*((-1+cos(f*x+e)+sin(f*x+e))/sin(f*x+e))^(1/2)*((-1
+cos(f*x+e))/sin(f*x+e))^(1/2)*a^2*b+cos(f*x+e)*(-(-1+cos(f*x+e)-sin(f*x+e))/sin(f*x+e))^(1/2)*((-1+cos(f*x+e)
+sin(f*x+e))/sin(f*x+e))^(1/2)*((-1+cos(f*x+e))/sin(f*x+e))^(1/2)*EllipticPi((-(-1+cos(f*x+e)-sin(f*x+e))/sin(
f*x+e))^(1/2),1/2-1/2*I,1/2*2^(1/2))*(-a^2+b^2)^(1/2)*a^2-4*cos(f*x+e)*(-(-1+cos(f*x+e)-sin(f*x+e))/sin(f*x+e)
)^(1/2)*((-1+cos(f*x+e)+sin(f*x+e))/sin(f*x+e))^(1/2)*((-1+cos(f*x+e))/sin(f*x+e))^(1/2)*EllipticE((-(-1+cos(f
*x+e)-sin(f*x+e))/sin(f*x+e))^(1/2),1/2*2^(1/2))*(-a^2+b^2)^(1/2)*a*b+2*cos(f*x+e)*(-(-1+cos(f*x+e)-sin(f*x+e)
)/sin(f*x+e))^(1/2)*((-1+cos(f*x+e)+sin(f*x+e))/sin(f*x+e))^(1/2)*((-1+cos(f*x+e))/sin(f*x+e))^(1/2)*EllipticF
((-(-1+cos(f*x+e)-sin(f*x+e))/sin(f*x+e))^(1/2),1/2*2^(1/2))*(-a^2+b^2)^(1/2)*a*b-I*cos(f*x+e)*(-(-1+cos(f*x+e
)-sin(f*x+e))/sin(f*x+e))^(1/2)*((-1+cos(f*x+e)+sin(f*x+e))/sin(f*x+e))^(1/2)*((-1+cos(f*x+e))/sin(f*x+e))^(1/
2)*EllipticPi((-(-1+cos(f*x+e)-sin(f*x+e))/sin(f*x+e))^(1/2),1/2-1/2*I,1/2*2^(1/2))*(-a^2+b^2)^(1/2)*b^2-I*cos
(f*x+e)*(-(-1+cos(f*x+e)-sin(f*x+e))/sin(f*x+e))^(1/2)*((-1+cos(f*x+e)+sin(f*x+e))/sin(f*x+e))^(1/2)*((-1+cos(
f*x+e))/sin(f*x+e))^(1/2)*EllipticPi((-(-1+cos(f*x+e)-sin(f*x+e))/sin(f*x+e))^(1/2),1/2+1/2*I,1/2*2^(1/2))*(-a
^2+b^2)^(1/2)*a^2+I*cos(f*x+e)*EllipticPi((-(-1+cos(f*x+e)-sin(f*x+e))/sin(f*x+e))^(1/2),1/2-1/2*I,1/2*2^(1/2)
)*(-(-1+cos(f*x+e)-sin(f*x+e))/sin(f*x+e))^(1/2)*((-1+cos(f*x+e)+sin(f*x+e))/sin(f*x+e))^(1/2)*((-1+cos(f*x+e)
)/sin(f*x+e))^(1/2)*(-a^2+b^2)^(1/2)*a^2-EllipticPi((-(-1+cos(f*x+e)-sin(f*x+e))/sin(f*x+e))^(1/2),-a/(b+(-a^2
+b^2)^(1/2)-a),1/2*2^(1/2))*cos(f*x+e)*(-(-1+cos(f*x+e)-sin(f*x+e))/sin(f*x+e))^(1/2)*((-1+cos(f*x+e)+sin(f*x+
e))/sin(f*x+e))^(1/2)*((-1+cos(f*x+e))/sin(f*x+e))^(1/2)*a^3-EllipticPi((-(-1+cos(f*x+e)-sin(f*x+e))/sin(f*x+e
))^(1/2),a/(a-b+(-a^2+b^2)^(1/2)),1/2*2^(1/2))*(-(-1+cos(f*x+e)-sin(f*x+e))/sin(f*x+e))^(1/2)*((-1+cos(f*x+e)+
sin(f*x+e))/sin(f*x+e))^(1/2)*((-1+cos(f*x+e))/sin(f*x+e))^(1/2)*(-a^2+b^2)^(1/2)*a^2+EllipticPi((-(-1+cos(f*x
+e)-sin(f*x+e))/sin(f*x+e))^(1/2),a/(a-b+(-a^2+b^2)^(1/2)),1/2*2^(1/2))*(-(-1+cos(f*x+e)-sin(f*x+e))/sin(f*x+e
))^(1/2)*((-1+cos(f*x+e)+sin(f*x+e))/sin(f*x+e))^(1/2)*((-1+cos(f*x+e))/sin(f*x+e))^(1/2)*a^2*b-EllipticPi((-(
-1+cos(f*x+e)-sin(f*x+e))/sin(f*x+e))^(1/2),-a/(b+(-a^2+b^2)^(1/2)-a),1/2*2^(1/2))*(-(-1+cos(f*x+e)-sin(f*x+e)
)/sin(f*x+e))^(1/2)*((-1+cos(f*x+e)+sin(f*x+e))/sin(f*x+e))^(1/2)*((-1+cos(f*x+e))/sin(f*x+e))^(1/2)*(-a^2+b^2
)^(1/2)*a^2-EllipticPi((-(-1+cos(f*x+e)-sin(f*x+e))/sin(f*x+e))^(1/2),-a/(b+(-a^2+b^2)^(1/2)-a),1/2*2^(1/2))*(
-(-1+cos(f*x+e)-sin(f*x+e))/sin(f*x+e))^(1/2)*((-1+cos(f*x+e)+sin(f*x+e))/sin(f*x+e))^(1/2)*((-1+cos(f*x+e))/s
in(f*x+e))^(1/2)*a^2*b+(-(-1+cos(f*x+e)-sin(f*x+e))/sin(f*x+e))^(1/2)*((-1+cos(f*x+e)+sin(f*x+e))/sin(f*x+e))^
(1/2)*((-1+cos(f*x+e))/sin(f*x+e))^(1/2)*EllipticPi((-(-1+cos(f*x+e)-sin(f*x+e))/sin(f*x+e))^(1/2),1/2-1/2*I,1
/2*2^(1/2))*(-a^2+b^2)^(1/2)*a^2-(-(-1+cos(f*x+e)-sin(f*x+e))/sin(f*x+e))^(1/2)*((-1+cos(f*x+e)+sin(f*x+e))/si
n(f*x+e))^(1/2)*((-1+cos(f*x+e))/sin(f*x+e))^(1/2)*EllipticPi((-(-1+cos(f*x+e)-sin(f*x+e))/sin(f*x+e))^(1/2),1
/2-1/2*I,1/2*2^(1/2))*(-a^2+b^2)^(1/2)*b^2+(-(-1+cos(f*x+e)-sin(f*x+e))/sin(f*x+e))^(1/2)*((-1+cos(f*x+e)+sin(
f*x+e))/sin(f*x+e))^(1/2)*((-1+cos(f*x+e))/sin(f*x+e))^(1/2)*EllipticPi((-(-1+cos(f*x+e)-sin(f*x+e))/sin(f*x+e
))^(1/2),1/2+1/2*I,1/2*2^(1/2))*(-a^2+b^2)^(1/2)*a^2-(-(-1+cos(f*x+e)-sin(f*x+e))/sin(f*x+e))^(1/2)*((-1+cos(f
*x+e)+sin(f*x+e))/sin(f*x+e))^(1/2)*((-1+cos(f*x+e))/sin(f*x+e))^(1/2)*EllipticPi((-(-1+cos(f*x+e)-sin(f*x+e))
/sin(f*x+e))^(1/2),1/2+1/2*I,1/2*2^(1/2))*(-a^2+b^2)^(1/2)*b^2+2*(-(-1+cos(f*x+e)-sin(f*x+e))/sin(f*x+e))^(1/2
)*((-1+cos(f*x+e)+sin(f*x+e))/sin(f*x+e))^(1/2)*((-1+cos(f*x+e))/sin(f*x+e))^(1/2)*EllipticF((-(-1+cos(f*x+e)-
sin(f*x+e))/sin(f*x+e))^(1/2),1/2*2^(1/2))*(-a^2+b^2)^(1/2)*b^2-I*(-(-1+cos(f*x+e)-sin(f*x+e))/sin(f*x+e))^(1/
2)*((-1+cos(f*x+e)+sin(f*x+e))/sin(f*x+e))^(1/2)*((-1+cos(f*x+e))/sin(f*x+e))^(1/2)*EllipticPi((-(-1+cos(f*x+e
)-sin(f*x+e))/sin(f*x+e))^(1/2),1/2+1/2*I,1/2*2^(1/2))*(-a^2+b^2)^(1/2)*a^2+I*EllipticPi((-(-1+cos(f*x+e)-sin(
f*x+e))/sin(f*x+e))^(1/2),1/2-1/2*I,1/2*2^(1/2))*(-(-1+cos(f*x+e)-sin(f*x+e))/sin(f*x+e))^(1/2)*((-1+cos(f*x+e
)+sin(f*x+e))/sin(f*x+e))^(1/2)*((-1+cos(f*x+e))/sin(f*x+e))^(1/2)*(-a^2+b^2)^(1/2)*a^2)*(d*sin(f*x+e))^(5/2)*
cos(f*x+e)/sin(f*x+e)^3/(g*cos(f*x+e))^(3/2)

________________________________________________________________________________________

Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \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} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

________________________________________________________________________________________

Fricas [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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

________________________________________________________________________________________

Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \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} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)