3.1441 \(\int \frac {1}{(g \cos (e+f x))^{3/2} (d \sin (e+f x))^{5/2} (a+b \sin (e+f x))} \, dx\)

Optimal. Leaf size=673 \[ \frac {2 \sqrt {2} b^4 \sqrt {\sin (e+f x)} \Pi \left (-\frac {\sqrt {b-a}}{\sqrt {a+b}};\left .\sin ^{-1}\left (\frac {\sqrt {g \cos (e+f x)}}{\sqrt {g} \sqrt {\sin (e+f x)+1}}\right )\right |-1\right )}{a^2 d^2 f g^{3/2} (b-a)^{3/2} (a+b)^{3/2} \sqrt {d \sin (e+f x)}}-\frac {2 \sqrt {2} b^4 \sqrt {\sin (e+f x)} \Pi \left (\frac {\sqrt {b-a}}{\sqrt {a+b}};\left .\sin ^{-1}\left (\frac {\sqrt {g \cos (e+f x)}}{\sqrt {g} \sqrt {\sin (e+f x)+1}}\right )\right |-1\right )}{a^2 d^2 f g^{3/2} (b-a)^{3/2} (a+b)^{3/2} \sqrt {d \sin (e+f x)}}-\frac {4 b (d \sin (e+f x))^{3/2}}{d^4 f g \left (a^2-b^2\right ) \sqrt {g \cos (e+f x)}}+\frac {4 b E\left (\left .e+f x-\frac {\pi }{4}\right |2\right ) \sqrt {d \sin (e+f x)} \sqrt {g \cos (e+f x)}}{d^3 f g^2 \left (a^2-b^2\right ) \sqrt {\sin (2 e+2 f x)}}+\frac {8 a \sqrt {d \sin (e+f x)}}{3 d^3 f g \left (a^2-b^2\right ) \sqrt {g \cos (e+f x)}}+\frac {2 b}{d^2 f g \left (a^2-b^2\right ) \sqrt {d \sin (e+f x)} \sqrt {g \cos (e+f x)}}+\frac {2 b^2 (g \cos (e+f x))^{3/2}}{3 a d f g^3 \left (a^2-b^2\right ) (d \sin (e+f x))^{3/2}}-\frac {2 a}{3 d f g \left (a^2-b^2\right ) (d \sin (e+f x))^{3/2} \sqrt {g \cos (e+f x)}}-\frac {2 b^3 E\left (\left .e+f x-\frac {\pi }{4}\right |2\right ) \sqrt {d \sin (e+f x)} \sqrt {g \cos (e+f x)}}{a^2 d^3 f g^2 \left (a^2-b^2\right ) \sqrt {\sin (2 e+2 f x)}}-\frac {2 b^3 (g \cos (e+f x))^{3/2}}{a^2 d^2 f g^3 \left (a^2-b^2\right ) \sqrt {d \sin (e+f x)}} \]

[Out]

2/3*b^2*(g*cos(f*x+e))^(3/2)/a/(a^2-b^2)/d/f/g^3/(d*sin(f*x+e))^(3/2)-2/3*a/(a^2-b^2)/d/f/g/(d*sin(f*x+e))^(3/
2)/(g*cos(f*x+e))^(1/2)-4*b*(d*sin(f*x+e))^(3/2)/(a^2-b^2)/d^4/f/g/(g*cos(f*x+e))^(1/2)-2*b^3*(g*cos(f*x+e))^(
3/2)/a^2/(a^2-b^2)/d^2/f/g^3/(d*sin(f*x+e))^(1/2)+2*b/(a^2-b^2)/d^2/f/g/(g*cos(f*x+e))^(1/2)/(d*sin(f*x+e))^(1
/2)+2*b^4*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)*si
n(f*x+e)^(1/2)/a^2/(-a+b)^(3/2)/(a+b)^(3/2)/d^2/f/g^(3/2)/(d*sin(f*x+e))^(1/2)-2*b^4*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)/a^2/(-a+b)^(3/2)/(a+b
)^(3/2)/d^2/f/g^(3/2)/(d*sin(f*x+e))^(1/2)+8/3*a*(d*sin(f*x+e))^(1/2)/(a^2-b^2)/d^3/f/g/(g*cos(f*x+e))^(1/2)-4
*b*(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)/d^3/f/g^2/sin(2*f*x+2*e)^(1/2)+2*b^3*(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/(a^2-b^2)/d^3/f/g^2/sin(2*
f*x+2*e)^(1/2)

________________________________________________________________________________________

Rubi [A]  time = 1.81, antiderivative size = 673, normalized size of antiderivative = 1.00, number of steps used = 19, number of rules used = 12, integrand size = 37, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.324, Rules used = {2904, 2838, 2570, 2563, 2571, 2572, 2639, 2910, 2906, 2905, 490, 1218} \[ -\frac {2 b^3 (g \cos (e+f x))^{3/2}}{a^2 d^2 f g^3 \left (a^2-b^2\right ) \sqrt {d \sin (e+f x)}}-\frac {2 b^3 E\left (\left .e+f x-\frac {\pi }{4}\right |2\right ) \sqrt {d \sin (e+f x)} \sqrt {g \cos (e+f x)}}{a^2 d^3 f g^2 \left (a^2-b^2\right ) \sqrt {\sin (2 e+2 f x)}}+\frac {4 b E\left (\left .e+f x-\frac {\pi }{4}\right |2\right ) \sqrt {d \sin (e+f x)} \sqrt {g \cos (e+f x)}}{d^3 f g^2 \left (a^2-b^2\right ) \sqrt {\sin (2 e+2 f x)}}+\frac {2 \sqrt {2} b^4 \sqrt {\sin (e+f x)} \Pi \left (-\frac {\sqrt {b-a}}{\sqrt {a+b}};\left .\sin ^{-1}\left (\frac {\sqrt {g \cos (e+f x)}}{\sqrt {g} \sqrt {\sin (e+f x)+1}}\right )\right |-1\right )}{a^2 d^2 f g^{3/2} (b-a)^{3/2} (a+b)^{3/2} \sqrt {d \sin (e+f x)}}-\frac {2 \sqrt {2} b^4 \sqrt {\sin (e+f x)} \Pi \left (\frac {\sqrt {b-a}}{\sqrt {a+b}};\left .\sin ^{-1}\left (\frac {\sqrt {g \cos (e+f x)}}{\sqrt {g} \sqrt {\sin (e+f x)+1}}\right )\right |-1\right )}{a^2 d^2 f g^{3/2} (b-a)^{3/2} (a+b)^{3/2} \sqrt {d \sin (e+f x)}}-\frac {4 b (d \sin (e+f x))^{3/2}}{d^4 f g \left (a^2-b^2\right ) \sqrt {g \cos (e+f x)}}+\frac {2 b}{d^2 f g \left (a^2-b^2\right ) \sqrt {d \sin (e+f x)} \sqrt {g \cos (e+f x)}}+\frac {8 a \sqrt {d \sin (e+f x)}}{3 d^3 f g \left (a^2-b^2\right ) \sqrt {g \cos (e+f x)}}+\frac {2 b^2 (g \cos (e+f x))^{3/2}}{3 a d f g^3 \left (a^2-b^2\right ) (d \sin (e+f x))^{3/2}}-\frac {2 a}{3 d f g \left (a^2-b^2\right ) (d \sin (e+f x))^{3/2} \sqrt {g \cos (e+f x)}} \]

Antiderivative was successfully verified.

[In]

Int[1/((g*Cos[e + f*x])^(3/2)*(d*Sin[e + f*x])^(5/2)*(a + b*Sin[e + f*x])),x]

[Out]

(-2*a)/(3*(a^2 - b^2)*d*f*g*Sqrt[g*Cos[e + f*x]]*(d*Sin[e + f*x])^(3/2)) + (2*b^2*(g*Cos[e + f*x])^(3/2))/(3*a
*(a^2 - b^2)*d*f*g^3*(d*Sin[e + f*x])^(3/2)) + (2*b)/((a^2 - b^2)*d^2*f*g*Sqrt[g*Cos[e + f*x]]*Sqrt[d*Sin[e +
f*x]]) - (2*b^3*(g*Cos[e + f*x])^(3/2))/(a^2*(a^2 - b^2)*d^2*f*g^3*Sqrt[d*Sin[e + f*x]]) + (2*Sqrt[2]*b^4*Elli
pticPi[-(Sqrt[-a + b]/Sqrt[a + b]), ArcSin[Sqrt[g*Cos[e + f*x]]/(Sqrt[g]*Sqrt[1 + Sin[e + f*x]])], -1]*Sqrt[Si
n[e + f*x]])/(a^2*(-a + b)^(3/2)*(a + b)^(3/2)*d^2*f*g^(3/2)*Sqrt[d*Sin[e + f*x]]) - (2*Sqrt[2]*b^4*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
]])/(a^2*(-a + b)^(3/2)*(a + b)^(3/2)*d^2*f*g^(3/2)*Sqrt[d*Sin[e + f*x]]) + (8*a*Sqrt[d*Sin[e + f*x]])/(3*(a^2
- b^2)*d^3*f*g*Sqrt[g*Cos[e + f*x]]) - (4*b*(d*Sin[e + f*x])^(3/2))/((a^2 - b^2)*d^4*f*g*Sqrt[g*Cos[e + f*x]]
) + (4*b*Sqrt[g*Cos[e + f*x]]*EllipticE[e - Pi/4 + f*x, 2]*Sqrt[d*Sin[e + f*x]])/((a^2 - b^2)*d^3*f*g^2*Sqrt[S
in[2*e + 2*f*x]]) - (2*b^3*Sqrt[g*Cos[e + f*x]]*EllipticE[e - Pi/4 + f*x, 2]*Sqrt[d*Sin[e + f*x]])/(a^2*(a^2 -
b^2)*d^3*f*g^2*Sqrt[Sin[2*e + 2*f*x]])

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]

Rule 2563

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

Rule 2570

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

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 2838

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

Rule 2904

Int[((cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((d_.)*sin[(e_.) + (f_.)*(x_)])^(n_))/((a_) + (b_.)*sin[(e_.) + (f_.
)*(x_)]), x_Symbol] :> Dist[1/(a^2 - b^2), Int[(g*Cos[e + f*x])^p*(d*Sin[e + f*x])^n*(a - b*Sin[e + f*x]), x],
x] - Dist[b^2/(g^2*(a^2 - b^2)), Int[((g*Cos[e + f*x])^(p + 2)*(d*Sin[e + f*x])^n)/(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]

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 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 2910

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

Rubi steps

\begin {align*} \int \frac {1}{(g \cos (e+f x))^{3/2} (d \sin (e+f x))^{5/2} (a+b \sin (e+f x))} \, dx &=\frac {\int \frac {a-b \sin (e+f x)}{(g \cos (e+f x))^{3/2} (d \sin (e+f x))^{5/2}} \, dx}{a^2-b^2}-\frac {b^2 \int \frac {\sqrt {g \cos (e+f x)}}{(d \sin (e+f x))^{5/2} (a+b \sin (e+f x))} \, dx}{\left (a^2-b^2\right ) g^2}\\ &=\frac {a \int \frac {1}{(g \cos (e+f x))^{3/2} (d \sin (e+f x))^{5/2}} \, dx}{a^2-b^2}-\frac {b \int \frac {1}{(g \cos (e+f x))^{3/2} (d \sin (e+f x))^{3/2}} \, dx}{\left (a^2-b^2\right ) d}-\frac {b^2 \int \frac {\sqrt {g \cos (e+f x)}}{(d \sin (e+f x))^{5/2}} \, dx}{a \left (a^2-b^2\right ) g^2}+\frac {b^3 \int \frac {\sqrt {g \cos (e+f x)}}{(d \sin (e+f x))^{3/2} (a+b \sin (e+f x))} \, dx}{a \left (a^2-b^2\right ) d g^2}\\ &=-\frac {2 a}{3 \left (a^2-b^2\right ) d f g \sqrt {g \cos (e+f x)} (d \sin (e+f x))^{3/2}}+\frac {2 b^2 (g \cos (e+f x))^{3/2}}{3 a \left (a^2-b^2\right ) d f g^3 (d \sin (e+f x))^{3/2}}+\frac {2 b}{\left (a^2-b^2\right ) d^2 f g \sqrt {g \cos (e+f x)} \sqrt {d \sin (e+f x)}}-\frac {(2 b) \int \frac {\sqrt {d \sin (e+f x)}}{(g \cos (e+f x))^{3/2}} \, dx}{\left (a^2-b^2\right ) d^3}+\frac {(4 a) \int \frac {1}{(g \cos (e+f x))^{3/2} \sqrt {d \sin (e+f x)}} \, dx}{3 \left (a^2-b^2\right ) d^2}-\frac {b^4 \int \frac {\sqrt {g \cos (e+f x)}}{\sqrt {d \sin (e+f x)} (a+b \sin (e+f x))} \, dx}{a^2 \left (a^2-b^2\right ) d^2 g^2}+\frac {b^3 \int \frac {\sqrt {g \cos (e+f x)}}{(d \sin (e+f x))^{3/2}} \, dx}{a^2 \left (a^2-b^2\right ) d g^2}\\ &=-\frac {2 a}{3 \left (a^2-b^2\right ) d f g \sqrt {g \cos (e+f x)} (d \sin (e+f x))^{3/2}}+\frac {2 b^2 (g \cos (e+f x))^{3/2}}{3 a \left (a^2-b^2\right ) d f g^3 (d \sin (e+f x))^{3/2}}+\frac {2 b}{\left (a^2-b^2\right ) d^2 f g \sqrt {g \cos (e+f x)} \sqrt {d \sin (e+f x)}}-\frac {2 b^3 (g \cos (e+f x))^{3/2}}{a^2 \left (a^2-b^2\right ) d^2 f g^3 \sqrt {d \sin (e+f x)}}+\frac {8 a \sqrt {d \sin (e+f x)}}{3 \left (a^2-b^2\right ) d^3 f g \sqrt {g \cos (e+f x)}}-\frac {4 b (d \sin (e+f x))^{3/2}}{\left (a^2-b^2\right ) d^4 f g \sqrt {g \cos (e+f x)}}+\frac {(4 b) \int \sqrt {g \cos (e+f x)} \sqrt {d \sin (e+f x)} \, dx}{\left (a^2-b^2\right ) d^3 g^2}-\frac {\left (2 b^3\right ) \int \sqrt {g \cos (e+f x)} \sqrt {d \sin (e+f x)} \, dx}{a^2 \left (a^2-b^2\right ) d^3 g^2}-\frac {\left (b^4 \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}{a^2 \left (a^2-b^2\right ) d^2 g^2 \sqrt {d \sin (e+f x)}}\\ &=-\frac {2 a}{3 \left (a^2-b^2\right ) d f g \sqrt {g \cos (e+f x)} (d \sin (e+f x))^{3/2}}+\frac {2 b^2 (g \cos (e+f x))^{3/2}}{3 a \left (a^2-b^2\right ) d f g^3 (d \sin (e+f x))^{3/2}}+\frac {2 b}{\left (a^2-b^2\right ) d^2 f g \sqrt {g \cos (e+f x)} \sqrt {d \sin (e+f x)}}-\frac {2 b^3 (g \cos (e+f x))^{3/2}}{a^2 \left (a^2-b^2\right ) d^2 f g^3 \sqrt {d \sin (e+f x)}}+\frac {8 a \sqrt {d \sin (e+f x)}}{3 \left (a^2-b^2\right ) d^3 f g \sqrt {g \cos (e+f x)}}-\frac {4 b (d \sin (e+f x))^{3/2}}{\left (a^2-b^2\right ) d^4 f g \sqrt {g \cos (e+f x)}}+\frac {\left (4 \sqrt {2} b^4 \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 )}{a^2 \left (a^2-b^2\right ) d^2 f g \sqrt {d \sin (e+f x)}}+\frac {\left (4 b \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 ) d^3 g^2 \sqrt {\sin (2 e+2 f x)}}-\frac {\left (2 b^3 \sqrt {g \cos (e+f x)} \sqrt {d \sin (e+f x)}\right ) \int \sqrt {\sin (2 e+2 f x)} \, dx}{a^2 \left (a^2-b^2\right ) d^3 g^2 \sqrt {\sin (2 e+2 f x)}}\\ &=-\frac {2 a}{3 \left (a^2-b^2\right ) d f g \sqrt {g \cos (e+f x)} (d \sin (e+f x))^{3/2}}+\frac {2 b^2 (g \cos (e+f x))^{3/2}}{3 a \left (a^2-b^2\right ) d f g^3 (d \sin (e+f x))^{3/2}}+\frac {2 b}{\left (a^2-b^2\right ) d^2 f g \sqrt {g \cos (e+f x)} \sqrt {d \sin (e+f x)}}-\frac {2 b^3 (g \cos (e+f x))^{3/2}}{a^2 \left (a^2-b^2\right ) d^2 f g^3 \sqrt {d \sin (e+f x)}}+\frac {8 a \sqrt {d \sin (e+f x)}}{3 \left (a^2-b^2\right ) d^3 f g \sqrt {g \cos (e+f x)}}-\frac {4 b (d \sin (e+f x))^{3/2}}{\left (a^2-b^2\right ) d^4 f g \sqrt {g \cos (e+f x)}}+\frac {4 b \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 ) d^3 f g^2 \sqrt {\sin (2 e+2 f x)}}-\frac {2 b^3 \sqrt {g \cos (e+f x)} E\left (\left .e-\frac {\pi }{4}+f x\right |2\right ) \sqrt {d \sin (e+f x)}}{a^2 \left (a^2-b^2\right ) d^3 f g^2 \sqrt {\sin (2 e+2 f x)}}+\frac {\left (2 \sqrt {2} b^4 \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 )}{a^2 \sqrt {-a+b} \left (a^2-b^2\right ) d^2 f g \sqrt {d \sin (e+f x)}}-\frac {\left (2 \sqrt {2} b^4 \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 )}{a^2 \sqrt {-a+b} \left (a^2-b^2\right ) d^2 f g \sqrt {d \sin (e+f x)}}\\ &=-\frac {2 a}{3 \left (a^2-b^2\right ) d f g \sqrt {g \cos (e+f x)} (d \sin (e+f x))^{3/2}}+\frac {2 b^2 (g \cos (e+f x))^{3/2}}{3 a \left (a^2-b^2\right ) d f g^3 (d \sin (e+f x))^{3/2}}+\frac {2 b}{\left (a^2-b^2\right ) d^2 f g \sqrt {g \cos (e+f x)} \sqrt {d \sin (e+f x)}}-\frac {2 b^3 (g \cos (e+f x))^{3/2}}{a^2 \left (a^2-b^2\right ) d^2 f g^3 \sqrt {d \sin (e+f x)}}+\frac {2 \sqrt {2} b^4 \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)}}{a^2 (-a+b)^{3/2} (a+b)^{3/2} d^2 f g^{3/2} \sqrt {d \sin (e+f x)}}-\frac {2 \sqrt {2} b^4 \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)}}{a^2 (-a+b)^{3/2} (a+b)^{3/2} d^2 f g^{3/2} \sqrt {d \sin (e+f x)}}+\frac {8 a \sqrt {d \sin (e+f x)}}{3 \left (a^2-b^2\right ) d^3 f g \sqrt {g \cos (e+f x)}}-\frac {4 b (d \sin (e+f x))^{3/2}}{\left (a^2-b^2\right ) d^4 f g \sqrt {g \cos (e+f x)}}+\frac {4 b \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 ) d^3 f g^2 \sqrt {\sin (2 e+2 f x)}}-\frac {2 b^3 \sqrt {g \cos (e+f x)} E\left (\left .e-\frac {\pi }{4}+f x\right |2\right ) \sqrt {d \sin (e+f x)}}{a^2 \left (a^2-b^2\right ) d^3 f g^2 \sqrt {\sin (2 e+2 f x)}}\\ \end {align*}

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Mathematica [C]  time = 24.37, size = 1727, normalized size = 2.57 \[ \text {result too large to display} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[1/((g*Cos[e + f*x])^(3/2)*(d*Sin[e + f*x])^(5/2)*(a + b*Sin[e + f*x])),x]

[Out]

(Cos[e + f*x]^2*Sin[e + f*x]^3*((2*b*Cot[e + f*x])/a^2 - (2*Cot[e + f*x]*Csc[e + f*x])/(3*a) + (2*Sec[e + f*x]
*(a - b*Sin[e + f*x]))/(a^2 - b^2)))/(f*(g*Cos[e + f*x])^(3/2)*(d*Sin[e + f*x])^(5/2)) - (b*Cos[e + f*x]^(3/2)
*Sin[e + f*x]^(5/2)*((-2*(4*a^3 - 2*a*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 - Cos[e + f*x]^2)^(3/4)*(a +
b*Sin[e + f*x])) + ((2*a^2*b - 2*b^3)*Sqrt[Tan[e + f*x]]*((3*Sqrt[2]*a^(3/2)*(-2*ArcTan[1 - (Sqrt[2]*(a^2 - b^
2)^(1/4)*Sqrt[Tan[e + f*x]])/Sqrt[a]] + 2*ArcTan[1 + (Sqrt[2]*(a^2 - b^2)^(1/4)*Sqrt[Tan[e + f*x]])/Sqrt[a]] -
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]] + 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*b*Appe
llF1[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))*(b*Tan[e + f*x]
+ a*Sqrt[1 + Tan[e + f*x]^2]))/(12*a^2*Cos[e + f*x]^(3/2)*Sqrt[Sin[e + f*x]]*(a + b*Sin[e + f*x])*(1 + Tan[e
+ f*x]^2)^(3/2)) + ((-2*a^2*b + b^3)*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]])/Sqr
t[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]])/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[Ta
n[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^2*b^2*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^2*(-a + b)*(a + b)*f*(g*Co
s[e + f*x])^(3/2)*(d*Sin[e + f*x])^(5/2))

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fricas [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(g*cos(f*x+e))^(3/2)/(d*sin(f*x+e))^(5/2)/(a+b*sin(f*x+e)),x, algorithm="fricas")

[Out]

Timed out

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\left (g \cos \left (f x + e\right )\right )^{\frac {3}{2}} {\left (b \sin \left (f x + e\right ) + a\right )} \left (d \sin \left (f x + e\right )\right )^{\frac {5}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(g*cos(f*x+e))^(3/2)/(d*sin(f*x+e))^(5/2)/(a+b*sin(f*x+e)),x, algorithm="giac")

[Out]

integrate(1/((g*cos(f*x + e))^(3/2)*(b*sin(f*x + e) + a)*(d*sin(f*x + e))^(5/2)), x)

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maple [B]  time = 0.78, size = 3315, normalized size = 4.93 \[ \text {output too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(g*cos(f*x+e))^(3/2)/(d*sin(f*x+e))^(5/2)/(a+b*sin(f*x+e)),x)

[Out]

1/3/f*(-6*(-a^2+b^2)^(1/2)*2^(1/2)*a^4+3*cos(f*x+e)*sin(f*x+e)*(-a^2+b^2)^(1/2)*(-(-1+cos(f*x+e)-sin(f*x+e))/s
in(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),-a/(b+(-a^2+b^2)^(1/2)-a),1/2*2^(1/2))*b^4-6*cos(f*x+e)*sin(f*x+
e)*(-a^2+b^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)*EllipticF((-(-1+cos(f*x+e)-sin(f*x+e))/sin(f*x+e))^(1/2),1/2*2^(1/2))*b^
4+3*cos(f*x+e)*sin(f*x+e)*(-a^2+b^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)*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))*b^4+3*cos(f*x+e)*sin(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),-a/(b+(-a^2+b^2)^(1/2)-a),1/2*2^(1/2))*a*b^4-3*cos(f*x+e)*sin(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),a/(a-b+(-a^2+b^2)^(1/2)),1/2*2^(1/
2))*a*b^4+24*sin(f*x+e)*(-a^2+b^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)*EllipticE((-(-1+cos(f*x+e)-sin(f*x+e))/sin(f*x+e))^
(1/2),1/2*2^(1/2))*a^3*b-12*sin(f*x+e)*(-a^2+b^2)^(1/2)*(-(-1+cos(f*x+e)-sin(f*x+e))/sin(f*x+e))^(1/2)*((-1+co
s(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*b^3-12*sin(f*x+e)*(-a^2+b^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)*EllipticF((-(-1+cos(f
*x+e)-sin(f*x+e))/sin(f*x+e))^(1/2),1/2*2^(1/2))*a^3*b+6*sin(f*x+e)*(-a^2+b^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)*Ellipti
cF((-(-1+cos(f*x+e)-sin(f*x+e))/sin(f*x+e))^(1/2),1/2*2^(1/2))*a*b^3+8*(-a^2+b^2)^(1/2)*cos(f*x+e)^2*2^(1/2)*a
^4+6*(-a^2+b^2)^(1/2)*sin(f*x+e)*2^(1/2)*a^3*b-12*cos(f*x+e)*sin(f*x+e)*(-a^2+b^2)^(1/2)*2^(1/2)*a^3*b+6*cos(f
*x+e)*sin(f*x+e)*(-a^2+b^2)^(1/2)*2^(1/2)*a*b^3+3*sin(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),-a/(b+(-a^2+b^2)^(1/2)-a),1/2*2^(1/2))*b^5-3*sin(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)*EllipticP
i((-(-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))*b^5+3*cos(f*x+e)*sin(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),-a/(b+(-a^2+b^2)^(1/2)-a),1/2
*2^(1/2))*b^5-3*cos(f*x+e)*sin(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),a/(a-b+(-a^2+b^2)^(1/2)),1/2*2^(1/2))*b^5+3*sin(f*x+e)*(-a^2+b^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)*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))*b^4-6*sin(f*x+e)*(-a^2+b^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)*EllipticF((-(-1+cos(f*x+e)-sin(f*x+e))/sin(f*x+e))^(1/2),1/2*2^(1/2))*b^4+3*sin(f*x+e)*
(-a^2+b^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)*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))*b^4+3*sin(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),-a/(b+(-a^2+b^2)^(1/2)-a),1/2*2^(1/2))*a*b^4-3*sin(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)-si
n(f*x+e))/sin(f*x+e))^(1/2),a/(a-b+(-a^2+b^2)^(1/2)),1/2*2^(1/2))*a*b^4-2*cos(f*x+e)^2*(-a^2+b^2)^(1/2)*2^(1/2
)*a^2*b^2+24*cos(f*x+e)*sin(f*x+e)*(-a^2+b^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)*EllipticE((-(-1+cos(f*x+e)-sin(f*x+e))/s
in(f*x+e))^(1/2),1/2*2^(1/2))*a^3*b-12*cos(f*x+e)*sin(f*x+e)*(-a^2+b^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)*EllipticE((-(-
1+cos(f*x+e)-sin(f*x+e))/sin(f*x+e))^(1/2),1/2*2^(1/2))*a*b^3-12*cos(f*x+e)*sin(f*x+e)*(-a^2+b^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)*EllipticF((-(-1+cos(f*x+e)-sin(f*x+e))/sin(f*x+e))^(1/2),1/2*2^(1/2))*a^3*b+6*cos(f*x+e)*sin(f*x+e
)*(-a^2+b^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)*EllipticF((-(-1+cos(f*x+e)-sin(f*x+e))/sin(f*x+e))^(1/2),1/2*2^(1/2))*a*b
^3)*cos(f*x+e)*sin(f*x+e)/(g*cos(f*x+e))^(3/2)/(d*sin(f*x+e))^(5/2)*2^(1/2)/(a+b)/(-a^2+b^2)^(1/2)/(a-b+(-a^2+
b^2)^(1/2))/(b+(-a^2+b^2)^(1/2)-a)/a^2

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\left (g \cos \left (f x + e\right )\right )^{\frac {3}{2}} {\left (b \sin \left (f x + e\right ) + a\right )} \left (d \sin \left (f x + e\right )\right )^{\frac {5}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(g*cos(f*x+e))^(3/2)/(d*sin(f*x+e))^(5/2)/(a+b*sin(f*x+e)),x, algorithm="maxima")

[Out]

integrate(1/((g*cos(f*x + e))^(3/2)*(b*sin(f*x + e) + a)*(d*sin(f*x + e))^(5/2)), x)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {1}{{\left (g\,\cos \left (e+f\,x\right )\right )}^{3/2}\,{\left (d\,\sin \left (e+f\,x\right )\right )}^{5/2}\,\left (a+b\,\sin \left (e+f\,x\right )\right )} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/((g*cos(e + f*x))^(3/2)*(d*sin(e + f*x))^(5/2)*(a + b*sin(e + f*x))),x)

[Out]

int(1/((g*cos(e + f*x))^(3/2)*(d*sin(e + f*x))^(5/2)*(a + b*sin(e + f*x))), x)

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(g*cos(f*x+e))**(3/2)/(d*sin(f*x+e))**(5/2)/(a+b*sin(f*x+e)),x)

[Out]

Timed out

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