\(\int \frac {\sqrt {e \cos (c+d x)}}{(a+b \sin (c+d x))^4} \, dx\) [612]

Optimal result

Integrand size = 25, antiderivative size = 579 \[ \int \frac {\sqrt {e \cos (c+d x)}}{(a+b \sin (c+d x))^4} \, dx=-\frac {5 a \left (a^2+2 b^2\right ) \sqrt {e} \arctan \left (\frac {\sqrt {b} \sqrt {e \cos (c+d x)}}{\sqrt [4]{-a^2+b^2} \sqrt {e}}\right )}{16 \sqrt {b} \left (-a^2+b^2\right )^{13/4} d}+\frac {5 a \left (a^2+2 b^2\right ) \sqrt {e} \text {arctanh}\left (\frac {\sqrt {b} \sqrt {e \cos (c+d x)}}{\sqrt [4]{-a^2+b^2} \sqrt {e}}\right )}{16 \sqrt {b} \left (-a^2+b^2\right )^{13/4} d}+\frac {\left (11 a^2+4 b^2\right ) \sqrt {e \cos (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{8 \left (a^2-b^2\right )^3 d \sqrt {\cos (c+d x)}}+\frac {5 a^2 \left (a^2+2 b^2\right ) e \sqrt {\cos (c+d x)} \operatorname {EllipticPi}\left (\frac {2 b}{b-\sqrt {-a^2+b^2}},\frac {1}{2} (c+d x),2\right )}{16 b \left (a^2-b^2\right )^3 \left (b-\sqrt {-a^2+b^2}\right ) d \sqrt {e \cos (c+d x)}}+\frac {5 a^2 \left (a^2+2 b^2\right ) e \sqrt {\cos (c+d x)} \operatorname {EllipticPi}\left (\frac {2 b}{b+\sqrt {-a^2+b^2}},\frac {1}{2} (c+d x),2\right )}{16 b \left (a^2-b^2\right )^3 \left (b+\sqrt {-a^2+b^2}\right ) d \sqrt {e \cos (c+d x)}}+\frac {b (e \cos (c+d x))^{3/2}}{3 \left (a^2-b^2\right ) d e (a+b \sin (c+d x))^3}+\frac {3 a b (e \cos (c+d x))^{3/2}}{4 \left (a^2-b^2\right )^2 d e (a+b \sin (c+d x))^2}+\frac {b \left (11 a^2+4 b^2\right ) (e \cos (c+d x))^{3/2}}{8 \left (a^2-b^2\right )^3 d e (a+b \sin (c+d x))} \]

[Out]

1/3*b*(e*cos(d*x+c))^(3/2)/(a^2-b^2)/d/e/(a+b*sin(d*x+c))^3+3/4*a*b*(e*cos(d*x+c))^(3/2)/(a^2-b^2)^2/d/e/(a+b*
sin(d*x+c))^2+1/8*b*(11*a^2+4*b^2)*(e*cos(d*x+c))^(3/2)/(a^2-b^2)^3/d/e/(a+b*sin(d*x+c))-5/16*a*(a^2+2*b^2)*ar
ctan(b^(1/2)*(e*cos(d*x+c))^(1/2)/(-a^2+b^2)^(1/4)/e^(1/2))*e^(1/2)/(-a^2+b^2)^(13/4)/d/b^(1/2)+5/16*a*(a^2+2*
b^2)*arctanh(b^(1/2)*(e*cos(d*x+c))^(1/2)/(-a^2+b^2)^(1/4)/e^(1/2))*e^(1/2)/(-a^2+b^2)^(13/4)/d/b^(1/2)+5/16*a
^2*(a^2+2*b^2)*e*(cos(1/2*d*x+1/2*c)^2)^(1/2)/cos(1/2*d*x+1/2*c)*EllipticPi(sin(1/2*d*x+1/2*c),2*b/(b-(-a^2+b^
2)^(1/2)),2^(1/2))*cos(d*x+c)^(1/2)/b/(a^2-b^2)^3/d/(b-(-a^2+b^2)^(1/2))/(e*cos(d*x+c))^(1/2)+5/16*a^2*(a^2+2*
b^2)*e*(cos(1/2*d*x+1/2*c)^2)^(1/2)/cos(1/2*d*x+1/2*c)*EllipticPi(sin(1/2*d*x+1/2*c),2*b/(b+(-a^2+b^2)^(1/2)),
2^(1/2))*cos(d*x+c)^(1/2)/b/(a^2-b^2)^3/d/(b+(-a^2+b^2)^(1/2))/(e*cos(d*x+c))^(1/2)+1/8*(11*a^2+4*b^2)*(cos(1/
2*d*x+1/2*c)^2)^(1/2)/cos(1/2*d*x+1/2*c)*EllipticE(sin(1/2*d*x+1/2*c),2^(1/2))*(e*cos(d*x+c))^(1/2)/(a^2-b^2)^
3/d/cos(d*x+c)^(1/2)

Rubi [A] (verified)

Time = 1.00 (sec) , antiderivative size = 579, normalized size of antiderivative = 1.00, number of steps used = 15, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.480, Rules used = {2773, 2943, 2946, 2721, 2719, 2780, 2886, 2884, 335, 304, 211, 214} \[ \int \frac {\sqrt {e \cos (c+d x)}}{(a+b \sin (c+d x))^4} \, dx=-\frac {5 a \sqrt {e} \left (a^2+2 b^2\right ) \arctan \left (\frac {\sqrt {b} \sqrt {e \cos (c+d x)}}{\sqrt {e} \sqrt [4]{b^2-a^2}}\right )}{16 \sqrt {b} d \left (b^2-a^2\right )^{13/4}}+\frac {5 a \sqrt {e} \left (a^2+2 b^2\right ) \text {arctanh}\left (\frac {\sqrt {b} \sqrt {e \cos (c+d x)}}{\sqrt {e} \sqrt [4]{b^2-a^2}}\right )}{16 \sqrt {b} d \left (b^2-a^2\right )^{13/4}}+\frac {3 a b (e \cos (c+d x))^{3/2}}{4 d e \left (a^2-b^2\right )^2 (a+b \sin (c+d x))^2}+\frac {b \left (11 a^2+4 b^2\right ) (e \cos (c+d x))^{3/2}}{8 d e \left (a^2-b^2\right )^3 (a+b \sin (c+d x))}+\frac {b (e \cos (c+d x))^{3/2}}{3 d e \left (a^2-b^2\right ) (a+b \sin (c+d x))^3}+\frac {\left (11 a^2+4 b^2\right ) E\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \sqrt {e \cos (c+d x)}}{8 d \left (a^2-b^2\right )^3 \sqrt {\cos (c+d x)}}+\frac {5 a^2 e \left (a^2+2 b^2\right ) \sqrt {\cos (c+d x)} \operatorname {EllipticPi}\left (\frac {2 b}{b-\sqrt {b^2-a^2}},\frac {1}{2} (c+d x),2\right )}{16 b d \left (a^2-b^2\right )^3 \left (b-\sqrt {b^2-a^2}\right ) \sqrt {e \cos (c+d x)}}+\frac {5 a^2 e \left (a^2+2 b^2\right ) \sqrt {\cos (c+d x)} \operatorname {EllipticPi}\left (\frac {2 b}{b+\sqrt {b^2-a^2}},\frac {1}{2} (c+d x),2\right )}{16 b d \left (a^2-b^2\right )^3 \left (\sqrt {b^2-a^2}+b\right ) \sqrt {e \cos (c+d x)}} \]

[In]

Int[Sqrt[e*Cos[c + d*x]]/(a + b*Sin[c + d*x])^4,x]

[Out]

(-5*a*(a^2 + 2*b^2)*Sqrt[e]*ArcTan[(Sqrt[b]*Sqrt[e*Cos[c + d*x]])/((-a^2 + b^2)^(1/4)*Sqrt[e])])/(16*Sqrt[b]*(
-a^2 + b^2)^(13/4)*d) + (5*a*(a^2 + 2*b^2)*Sqrt[e]*ArcTanh[(Sqrt[b]*Sqrt[e*Cos[c + d*x]])/((-a^2 + b^2)^(1/4)*
Sqrt[e])])/(16*Sqrt[b]*(-a^2 + b^2)^(13/4)*d) + ((11*a^2 + 4*b^2)*Sqrt[e*Cos[c + d*x]]*EllipticE[(c + d*x)/2,
2])/(8*(a^2 - b^2)^3*d*Sqrt[Cos[c + d*x]]) + (5*a^2*(a^2 + 2*b^2)*e*Sqrt[Cos[c + d*x]]*EllipticPi[(2*b)/(b - S
qrt[-a^2 + b^2]), (c + d*x)/2, 2])/(16*b*(a^2 - b^2)^3*(b - Sqrt[-a^2 + b^2])*d*Sqrt[e*Cos[c + d*x]]) + (5*a^2
*(a^2 + 2*b^2)*e*Sqrt[Cos[c + d*x]]*EllipticPi[(2*b)/(b + Sqrt[-a^2 + b^2]), (c + d*x)/2, 2])/(16*b*(a^2 - b^2
)^3*(b + Sqrt[-a^2 + b^2])*d*Sqrt[e*Cos[c + d*x]]) + (b*(e*Cos[c + d*x])^(3/2))/(3*(a^2 - b^2)*d*e*(a + b*Sin[
c + d*x])^3) + (3*a*b*(e*Cos[c + d*x])^(3/2))/(4*(a^2 - b^2)^2*d*e*(a + b*Sin[c + d*x])^2) + (b*(11*a^2 + 4*b^
2)*(e*Cos[c + d*x])^(3/2))/(8*(a^2 - b^2)^3*d*e*(a + b*Sin[c + d*x]))

Rule 211

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

Rule 214

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

Rule 304

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

Rule 335

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

Rule 2719

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

Rule 2721

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

Rule 2773

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

Rule 2780

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

Rule 2884

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

Rule 2886

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

Rule 2943

Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*sin[(e_.)
+ (f_.)*(x_)]), x_Symbol] :> Simp[(-(b*c - a*d))*(g*Cos[e + f*x])^(p + 1)*((a + b*Sin[e + f*x])^(m + 1)/(f*g*(
a^2 - b^2)*(m + 1))), x] + Dist[1/((a^2 - b^2)*(m + 1)), Int[(g*Cos[e + f*x])^p*(a + b*Sin[e + f*x])^(m + 1)*S
imp[(a*c - b*d)*(m + 1) - (b*c - a*d)*(m + p + 2)*Sin[e + f*x], x], x], x] /; FreeQ[{a, b, c, d, e, f, g, p},
x] && NeQ[a^2 - b^2, 0] && LtQ[m, -1] && IntegerQ[2*m]

Rule 2946

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

Rubi steps \begin{align*} \text {integral}& = \frac {b (e \cos (c+d x))^{3/2}}{3 \left (a^2-b^2\right ) d e (a+b \sin (c+d x))^3}-\frac {\int \frac {\sqrt {e \cos (c+d x)} \left (-3 a+\frac {3}{2} b \sin (c+d x)\right )}{(a+b \sin (c+d x))^3} \, dx}{3 \left (a^2-b^2\right )} \\ & = \frac {b (e \cos (c+d x))^{3/2}}{3 \left (a^2-b^2\right ) d e (a+b \sin (c+d x))^3}+\frac {3 a b (e \cos (c+d x))^{3/2}}{4 \left (a^2-b^2\right )^2 d e (a+b \sin (c+d x))^2}+\frac {\int \frac {\sqrt {e \cos (c+d x)} \left (3 \left (2 a^2+b^2\right )-\frac {9}{4} a b \sin (c+d x)\right )}{(a+b \sin (c+d x))^2} \, dx}{6 \left (a^2-b^2\right )^2} \\ & = \frac {b (e \cos (c+d x))^{3/2}}{3 \left (a^2-b^2\right ) d e (a+b \sin (c+d x))^3}+\frac {3 a b (e \cos (c+d x))^{3/2}}{4 \left (a^2-b^2\right )^2 d e (a+b \sin (c+d x))^2}+\frac {b \left (11 a^2+4 b^2\right ) (e \cos (c+d x))^{3/2}}{8 \left (a^2-b^2\right )^3 d e (a+b \sin (c+d x))}-\frac {\int \frac {\sqrt {e \cos (c+d x)} \left (-\frac {3}{4} a \left (8 a^2+7 b^2\right )-\frac {3}{8} b \left (11 a^2+4 b^2\right ) \sin (c+d x)\right )}{a+b \sin (c+d x)} \, dx}{6 \left (a^2-b^2\right )^3} \\ & = \frac {b (e \cos (c+d x))^{3/2}}{3 \left (a^2-b^2\right ) d e (a+b \sin (c+d x))^3}+\frac {3 a b (e \cos (c+d x))^{3/2}}{4 \left (a^2-b^2\right )^2 d e (a+b \sin (c+d x))^2}+\frac {b \left (11 a^2+4 b^2\right ) (e \cos (c+d x))^{3/2}}{8 \left (a^2-b^2\right )^3 d e (a+b \sin (c+d x))}+\frac {\left (5 a \left (a^2+2 b^2\right )\right ) \int \frac {\sqrt {e \cos (c+d x)}}{a+b \sin (c+d x)} \, dx}{16 \left (a^2-b^2\right )^3}+\frac {\left (11 a^2+4 b^2\right ) \int \sqrt {e \cos (c+d x)} \, dx}{16 \left (a^2-b^2\right )^3} \\ & = \frac {b (e \cos (c+d x))^{3/2}}{3 \left (a^2-b^2\right ) d e (a+b \sin (c+d x))^3}+\frac {3 a b (e \cos (c+d x))^{3/2}}{4 \left (a^2-b^2\right )^2 d e (a+b \sin (c+d x))^2}+\frac {b \left (11 a^2+4 b^2\right ) (e \cos (c+d x))^{3/2}}{8 \left (a^2-b^2\right )^3 d e (a+b \sin (c+d x))}-\frac {\left (5 a^2 \left (a^2+2 b^2\right ) e\right ) \int \frac {1}{\sqrt {e \cos (c+d x)} \left (\sqrt {-a^2+b^2}-b \cos (c+d x)\right )} \, dx}{32 b \left (a^2-b^2\right )^3}+\frac {\left (5 a^2 \left (a^2+2 b^2\right ) e\right ) \int \frac {1}{\sqrt {e \cos (c+d x)} \left (\sqrt {-a^2+b^2}+b \cos (c+d x)\right )} \, dx}{32 b \left (a^2-b^2\right )^3}+\frac {\left (5 a b \left (a^2+2 b^2\right ) e\right ) \text {Subst}\left (\int \frac {\sqrt {x}}{\left (a^2-b^2\right ) e^2+b^2 x^2} \, dx,x,e \cos (c+d x)\right )}{16 \left (a^2-b^2\right )^3 d}+\frac {\left (\left (11 a^2+4 b^2\right ) \sqrt {e \cos (c+d x)}\right ) \int \sqrt {\cos (c+d x)} \, dx}{16 \left (a^2-b^2\right )^3 \sqrt {\cos (c+d x)}} \\ & = \frac {\left (11 a^2+4 b^2\right ) \sqrt {e \cos (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{8 \left (a^2-b^2\right )^3 d \sqrt {\cos (c+d x)}}+\frac {b (e \cos (c+d x))^{3/2}}{3 \left (a^2-b^2\right ) d e (a+b \sin (c+d x))^3}+\frac {3 a b (e \cos (c+d x))^{3/2}}{4 \left (a^2-b^2\right )^2 d e (a+b \sin (c+d x))^2}+\frac {b \left (11 a^2+4 b^2\right ) (e \cos (c+d x))^{3/2}}{8 \left (a^2-b^2\right )^3 d e (a+b \sin (c+d x))}+\frac {\left (5 a b \left (a^2+2 b^2\right ) e\right ) \text {Subst}\left (\int \frac {x^2}{\left (a^2-b^2\right ) e^2+b^2 x^4} \, dx,x,\sqrt {e \cos (c+d x)}\right )}{8 \left (a^2-b^2\right )^3 d}-\frac {\left (5 a^2 \left (a^2+2 b^2\right ) e \sqrt {\cos (c+d x)}\right ) \int \frac {1}{\sqrt {\cos (c+d x)} \left (\sqrt {-a^2+b^2}-b \cos (c+d x)\right )} \, dx}{32 b \left (a^2-b^2\right )^3 \sqrt {e \cos (c+d x)}}+\frac {\left (5 a^2 \left (a^2+2 b^2\right ) e \sqrt {\cos (c+d x)}\right ) \int \frac {1}{\sqrt {\cos (c+d x)} \left (\sqrt {-a^2+b^2}+b \cos (c+d x)\right )} \, dx}{32 b \left (a^2-b^2\right )^3 \sqrt {e \cos (c+d x)}} \\ & = \frac {\left (11 a^2+4 b^2\right ) \sqrt {e \cos (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{8 \left (a^2-b^2\right )^3 d \sqrt {\cos (c+d x)}}+\frac {5 a^2 \left (a^2+2 b^2\right ) e \sqrt {\cos (c+d x)} \operatorname {EllipticPi}\left (\frac {2 b}{b-\sqrt {-a^2+b^2}},\frac {1}{2} (c+d x),2\right )}{16 b \left (a^2-b^2\right )^3 \left (b-\sqrt {-a^2+b^2}\right ) d \sqrt {e \cos (c+d x)}}+\frac {5 a^2 \left (a^2+2 b^2\right ) e \sqrt {\cos (c+d x)} \operatorname {EllipticPi}\left (\frac {2 b}{b+\sqrt {-a^2+b^2}},\frac {1}{2} (c+d x),2\right )}{16 b \left (a^2-b^2\right )^3 \left (b+\sqrt {-a^2+b^2}\right ) d \sqrt {e \cos (c+d x)}}+\frac {b (e \cos (c+d x))^{3/2}}{3 \left (a^2-b^2\right ) d e (a+b \sin (c+d x))^3}+\frac {3 a b (e \cos (c+d x))^{3/2}}{4 \left (a^2-b^2\right )^2 d e (a+b \sin (c+d x))^2}+\frac {b \left (11 a^2+4 b^2\right ) (e \cos (c+d x))^{3/2}}{8 \left (a^2-b^2\right )^3 d e (a+b \sin (c+d x))}-\frac {\left (5 a \left (a^2+2 b^2\right ) e\right ) \text {Subst}\left (\int \frac {1}{\sqrt {-a^2+b^2} e-b x^2} \, dx,x,\sqrt {e \cos (c+d x)}\right )}{16 \left (a^2-b^2\right )^3 d}+\frac {\left (5 a \left (a^2+2 b^2\right ) e\right ) \text {Subst}\left (\int \frac {1}{\sqrt {-a^2+b^2} e+b x^2} \, dx,x,\sqrt {e \cos (c+d x)}\right )}{16 \left (a^2-b^2\right )^3 d} \\ & = -\frac {5 a \left (a^2+2 b^2\right ) \sqrt {e} \arctan \left (\frac {\sqrt {b} \sqrt {e \cos (c+d x)}}{\sqrt [4]{-a^2+b^2} \sqrt {e}}\right )}{16 \sqrt {b} \left (-a^2+b^2\right )^{13/4} d}+\frac {5 a \left (a^2+2 b^2\right ) \sqrt {e} \text {arctanh}\left (\frac {\sqrt {b} \sqrt {e \cos (c+d x)}}{\sqrt [4]{-a^2+b^2} \sqrt {e}}\right )}{16 \sqrt {b} \left (-a^2+b^2\right )^{13/4} d}+\frac {\left (11 a^2+4 b^2\right ) \sqrt {e \cos (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{8 \left (a^2-b^2\right )^3 d \sqrt {\cos (c+d x)}}+\frac {5 a^2 \left (a^2+2 b^2\right ) e \sqrt {\cos (c+d x)} \operatorname {EllipticPi}\left (\frac {2 b}{b-\sqrt {-a^2+b^2}},\frac {1}{2} (c+d x),2\right )}{16 b \left (a^2-b^2\right )^3 \left (b-\sqrt {-a^2+b^2}\right ) d \sqrt {e \cos (c+d x)}}+\frac {5 a^2 \left (a^2+2 b^2\right ) e \sqrt {\cos (c+d x)} \operatorname {EllipticPi}\left (\frac {2 b}{b+\sqrt {-a^2+b^2}},\frac {1}{2} (c+d x),2\right )}{16 b \left (a^2-b^2\right )^3 \left (b+\sqrt {-a^2+b^2}\right ) d \sqrt {e \cos (c+d x)}}+\frac {b (e \cos (c+d x))^{3/2}}{3 \left (a^2-b^2\right ) d e (a+b \sin (c+d x))^3}+\frac {3 a b (e \cos (c+d x))^{3/2}}{4 \left (a^2-b^2\right )^2 d e (a+b \sin (c+d x))^2}+\frac {b \left (11 a^2+4 b^2\right ) (e \cos (c+d x))^{3/2}}{8 \left (a^2-b^2\right )^3 d e (a+b \sin (c+d 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 = 6.56 (sec) , antiderivative size = 900, normalized size of antiderivative = 1.55 \[ \int \frac {\sqrt {e \cos (c+d x)}}{(a+b \sin (c+d x))^4} \, dx=\frac {\sqrt {e \cos (c+d x)} \left (\frac {b \cos (c+d x)}{3 \left (a^2-b^2\right ) (a+b \sin (c+d x))^3}+\frac {3 a b \cos (c+d x)}{4 \left (a^2-b^2\right )^2 (a+b \sin (c+d x))^2}-\frac {-11 a^2 b \cos (c+d x)-4 b^3 \cos (c+d x)}{8 \left (a^2-b^2\right )^3 (a+b \sin (c+d x))}\right )}{d}+\frac {\sqrt {e \cos (c+d x)} \left (-\frac {2 \left (16 a^3+14 a b^2\right ) \left (a+b \sqrt {1-\cos ^2(c+d x)}\right ) \left (\frac {a \operatorname {AppellF1}\left (\frac {3}{4},\frac {1}{2},1,\frac {7}{4},\cos ^2(c+d x),\frac {b^2 \cos ^2(c+d x)}{-a^2+b^2}\right ) \cos ^{\frac {3}{2}}(c+d x)}{3 \left (a^2-b^2\right )}+\frac {\left (\frac {1}{8}+\frac {i}{8}\right ) \left (2 \arctan \left (1-\frac {(1+i) \sqrt {b} \sqrt {\cos (c+d x)}}{\sqrt [4]{-a^2+b^2}}\right )-2 \arctan \left (1+\frac {(1+i) \sqrt {b} \sqrt {\cos (c+d x)}}{\sqrt [4]{-a^2+b^2}}\right )-\log \left (\sqrt {-a^2+b^2}-(1+i) \sqrt {b} \sqrt [4]{-a^2+b^2} \sqrt {\cos (c+d x)}+i b \cos (c+d x)\right )+\log \left (\sqrt {-a^2+b^2}+(1+i) \sqrt {b} \sqrt [4]{-a^2+b^2} \sqrt {\cos (c+d x)}+i b \cos (c+d x)\right )\right )}{\sqrt {b} \sqrt [4]{-a^2+b^2}}\right ) \sin (c+d x)}{\sqrt {1-\cos ^2(c+d x)} (a+b \sin (c+d x))}-\frac {\left (11 a^2 b+4 b^3\right ) \left (a+b \sqrt {1-\cos ^2(c+d x)}\right ) \left (8 b^{5/2} \operatorname {AppellF1}\left (\frac {3}{4},-\frac {1}{2},1,\frac {7}{4},\cos ^2(c+d x),\frac {b^2 \cos ^2(c+d x)}{-a^2+b^2}\right ) \cos ^{\frac {3}{2}}(c+d x)+3 \sqrt {2} a \left (a^2-b^2\right )^{3/4} \left (2 \arctan \left (1-\frac {\sqrt {2} \sqrt {b} \sqrt {\cos (c+d x)}}{\sqrt [4]{a^2-b^2}}\right )-2 \arctan \left (1+\frac {\sqrt {2} \sqrt {b} \sqrt {\cos (c+d x)}}{\sqrt [4]{a^2-b^2}}\right )-\log \left (\sqrt {a^2-b^2}-\sqrt {2} \sqrt {b} \sqrt [4]{a^2-b^2} \sqrt {\cos (c+d x)}+b \cos (c+d x)\right )+\log \left (\sqrt {a^2-b^2}+\sqrt {2} \sqrt {b} \sqrt [4]{a^2-b^2} \sqrt {\cos (c+d x)}+b \cos (c+d x)\right )\right )\right ) \sin ^2(c+d x)}{12 b^{3/2} \left (-a^2+b^2\right ) \left (1-\cos ^2(c+d x)\right ) (a+b \sin (c+d x))}\right )}{16 (a-b)^3 (a+b)^3 d \sqrt {\cos (c+d x)}} \]

[In]

Integrate[Sqrt[e*Cos[c + d*x]]/(a + b*Sin[c + d*x])^4,x]

[Out]

(Sqrt[e*Cos[c + d*x]]*((b*Cos[c + d*x])/(3*(a^2 - b^2)*(a + b*Sin[c + d*x])^3) + (3*a*b*Cos[c + d*x])/(4*(a^2
- b^2)^2*(a + b*Sin[c + d*x])^2) - (-11*a^2*b*Cos[c + d*x] - 4*b^3*Cos[c + d*x])/(8*(a^2 - b^2)^3*(a + b*Sin[c
+ d*x]))))/d + (Sqrt[e*Cos[c + d*x]]*((-2*(16*a^3 + 14*a*b^2)*(a + b*Sqrt[1 - Cos[c + d*x]^2])*((a*AppellF1[3
/4, 1/2, 1, 7/4, Cos[c + d*x]^2, (b^2*Cos[c + d*x]^2)/(-a^2 + b^2)]*Cos[c + d*x]^(3/2))/(3*(a^2 - b^2)) + ((1/
8 + I/8)*(2*ArcTan[1 - ((1 + I)*Sqrt[b]*Sqrt[Cos[c + d*x]])/(-a^2 + b^2)^(1/4)] - 2*ArcTan[1 + ((1 + I)*Sqrt[b
]*Sqrt[Cos[c + d*x]])/(-a^2 + b^2)^(1/4)] - Log[Sqrt[-a^2 + b^2] - (1 + I)*Sqrt[b]*(-a^2 + b^2)^(1/4)*Sqrt[Cos
[c + d*x]] + I*b*Cos[c + d*x]] + Log[Sqrt[-a^2 + b^2] + (1 + I)*Sqrt[b]*(-a^2 + b^2)^(1/4)*Sqrt[Cos[c + d*x]]
+ I*b*Cos[c + d*x]]))/(Sqrt[b]*(-a^2 + b^2)^(1/4)))*Sin[c + d*x])/(Sqrt[1 - Cos[c + d*x]^2]*(a + b*Sin[c + d*x
])) - ((11*a^2*b + 4*b^3)*(a + b*Sqrt[1 - Cos[c + d*x]^2])*(8*b^(5/2)*AppellF1[3/4, -1/2, 1, 7/4, Cos[c + d*x]
^2, (b^2*Cos[c + d*x]^2)/(-a^2 + b^2)]*Cos[c + d*x]^(3/2) + 3*Sqrt[2]*a*(a^2 - b^2)^(3/4)*(2*ArcTan[1 - (Sqrt[
2]*Sqrt[b]*Sqrt[Cos[c + d*x]])/(a^2 - b^2)^(1/4)] - 2*ArcTan[1 + (Sqrt[2]*Sqrt[b]*Sqrt[Cos[c + d*x]])/(a^2 - b
^2)^(1/4)] - Log[Sqrt[a^2 - b^2] - Sqrt[2]*Sqrt[b]*(a^2 - b^2)^(1/4)*Sqrt[Cos[c + d*x]] + b*Cos[c + d*x]] + Lo
g[Sqrt[a^2 - b^2] + Sqrt[2]*Sqrt[b]*(a^2 - b^2)^(1/4)*Sqrt[Cos[c + d*x]] + b*Cos[c + d*x]]))*Sin[c + d*x]^2)/(
12*b^(3/2)*(-a^2 + b^2)*(1 - Cos[c + d*x]^2)*(a + b*Sin[c + d*x]))))/(16*(a - b)^3*(a + b)^3*d*Sqrt[Cos[c + d*
x]])

Maple [C] (warning: unable to verify)

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

Time = 15.48 (sec) , antiderivative size = 3711, normalized size of antiderivative = 6.41

method	result	size
default	\(\text {Expression too large to display}\)	\(3711\)

[In]

int((e*cos(d*x+c))^(1/2)/(a+b*sin(d*x+c))^4,x,method=_RETURNVERBOSE)

[Out]

(-16*e*a*b*(5/256/(e^2*(a^2-b^2)/b^2)^(1/4)*(72/5*(e^2*(a^2-b^2)/b^2)^(1/4)*(2*e*cos(1/2*d*x+1/2*c)^2-e)^(1/2)
*(4/9*(5*cos(1/2*d*x+1/2*c)^4-5*cos(1/2*d*x+1/2*c)^2-1)*b^2+a^2)*(cos(1/2*d*x+1/2*c)^2-1/2)*b^2+(arctan((2^(1/
2)*(2*e*cos(1/2*d*x+1/2*c)^2-e)^(1/2)+(e^2*(a^2-b^2)/b^2)^(1/4))/(e^2*(a^2-b^2)/b^2)^(1/4))+1/2*ln((2*e*cos(1/
2*d*x+1/2*c)^2-e-(e^2*(a^2-b^2)/b^2)^(1/4)*(2*e*cos(1/2*d*x+1/2*c)^2-e)^(1/2)*2^(1/2)+(e^2*(a^2-b^2)/b^2)^(1/2
))/(2*e*cos(1/2*d*x+1/2*c)^2-e+(e^2*(a^2-b^2)/b^2)^(1/4)*(2*e*cos(1/2*d*x+1/2*c)^2-e)^(1/2)*2^(1/2)+(e^2*(a^2-
b^2)/b^2)^(1/2)))+arctan((2^(1/2)*(2*e*cos(1/2*d*x+1/2*c)^2-e)^(1/2)-(e^2*(a^2-b^2)/b^2)^(1/4))/(e^2*(a^2-b^2)
/b^2)^(1/4)))*e*2^(1/2)*(4*sin(1/2*d*x+1/2*c)^4*b^2-4*sin(1/2*d*x+1/2*c)^2*b^2+a^2)^2)/(a+b)^2/e/(a-b)^2/b^2/(
4*sin(1/2*d*x+1/2*c)^4*b^2-4*sin(1/2*d*x+1/2*c)^2*b^2+a^2)^2-15/1024*a^2/(e^2*(a^2-b^2)/b^2)^(1/4)*(1808/45*(e
^2*(a^2-b^2)/b^2)^(1/4)*(2*e*cos(1/2*d*x+1/2*c)^2-e)^(1/2)*(cos(1/2*d*x+1/2*c)^2-1/2)*(16/113*(45*cos(1/2*d*x+
1/2*c)^8-90*cos(1/2*d*x+1/2*c)^6+36*cos(1/2*d*x+1/2*c)^4+9*cos(1/2*d*x+1/2*c)^2+2)*b^4+504/113*(cos(1/2*d*x+1/
2*c)^4-cos(1/2*d*x+1/2*c)^2-25/126)*a^2*b^2+a^4)*b^2+(4*sin(1/2*d*x+1/2*c)^4*b^2-4*sin(1/2*d*x+1/2*c)^2*b^2+a^
2)^3*e*(ln((2*e*cos(1/2*d*x+1/2*c)^2-e-(e^2*(a^2-b^2)/b^2)^(1/4)*(2*e*cos(1/2*d*x+1/2*c)^2-e)^(1/2)*2^(1/2)+(e
^2*(a^2-b^2)/b^2)^(1/2))/(2*e*cos(1/2*d*x+1/2*c)^2-e+(e^2*(a^2-b^2)/b^2)^(1/4)*(2*e*cos(1/2*d*x+1/2*c)^2-e)^(1
/2)*2^(1/2)+(e^2*(a^2-b^2)/b^2)^(1/2)))+2*arctan((2^(1/2)*(2*e*cos(1/2*d*x+1/2*c)^2-e)^(1/2)+(e^2*(a^2-b^2)/b^
2)^(1/4))/(e^2*(a^2-b^2)/b^2)^(1/4))+2*arctan((2^(1/2)*(2*e*cos(1/2*d*x+1/2*c)^2-e)^(1/2)-(e^2*(a^2-b^2)/b^2)^
(1/4))/(e^2*(a^2-b^2)/b^2)^(1/4)))*2^(1/2))/(a+b)^3/e/(a-b)^3/b^2/(4*sin(1/2*d*x+1/2*c)^4*b^2-4*sin(1/2*d*x+1/
2*c)^2*b^2+a^2)^3)-2*(e*(2*cos(1/2*d*x+1/2*c)^2-1)*sin(1/2*d*x+1/2*c)^2)^(1/2)*e*(-(-b^2/(a^2-b^2)/a^2/e*cos(1
/2*d*x+1/2*c)^3+1/2*b^2/(a^2-b^2)/a^2/e*cos(1/2*d*x+1/2*c))*(-e*(2*sin(1/2*d*x+1/2*c)^4-sin(1/2*d*x+1/2*c)^2))
^(1/2)/(4*cos(1/2*d*x+1/2*c)^4*b^2-4*cos(1/2*d*x+1/2*c)^2*b^2+a^2)-1/4/(a^2-b^2)/a^2*(sin(1/2*d*x+1/2*c)^2)^(1
/2)*(-2*cos(1/2*d*x+1/2*c)^2+1)^(1/2)/(-e*(2*sin(1/2*d*x+1/2*c)^4-sin(1/2*d*x+1/2*c)^2))^(1/2)*EllipticF(cos(1
/2*d*x+1/2*c),2^(1/2))+1/4/(a^2-b^2)/a^2*(sin(1/2*d*x+1/2*c)^2)^(1/2)*(-2*cos(1/2*d*x+1/2*c)^2+1)^(1/2)/(-e*(2
*sin(1/2*d*x+1/2*c)^4-sin(1/2*d*x+1/2*c)^2))^(1/2)*(EllipticF(cos(1/2*d*x+1/2*c),2^(1/2))-EllipticE(cos(1/2*d*
x+1/2*c),2^(1/2)))-1/64/a^2/b^2*sum((-3*a^2+2*b^2)/(a-b)/(a+b)/_alpha*(2^(1/2)/(e*(2*_alpha^2*b^2+a^2-2*b^2)/b
^2)^(1/2)*arctanh(1/2*e*(4*_alpha^2-3)/(4*a^2-3*b^2)*(4*a^2*cos(1/2*d*x+1/2*c)^2-3*cos(1/2*d*x+1/2*c)^2*b^2+b^
2*_alpha^2-3*a^2+2*b^2)*2^(1/2)/(e*(2*_alpha^2*b^2+a^2-2*b^2)/b^2)^(1/2)/(-e*(2*sin(1/2*d*x+1/2*c)^4-sin(1/2*d
*x+1/2*c)^2))^(1/2))+8*b^2/a^2*_alpha*(_alpha^2-1)*(sin(1/2*d*x+1/2*c)^2)^(1/2)*(-2*cos(1/2*d*x+1/2*c)^2+1)^(1
/2)/(-e*sin(1/2*d*x+1/2*c)^2*(2*sin(1/2*d*x+1/2*c)^2-1))^(1/2)*EllipticPi(cos(1/2*d*x+1/2*c),-4*b^2/a^2*(_alph
a^2-1),2^(1/2))),_alpha=RootOf(4*_Z^4*b^2-4*_Z^2*b^2+a^2))-8*a^2*(-1/2*(-b^2/(a^2-b^2)/a^2/e*cos(1/2*d*x+1/2*c
)^3+1/2*b^2/(a^2-b^2)/a^2/e*cos(1/2*d*x+1/2*c))*(-e*(2*sin(1/2*d*x+1/2*c)^4-sin(1/2*d*x+1/2*c)^2))^(1/2)/(4*co
s(1/2*d*x+1/2*c)^4*b^2-4*cos(1/2*d*x+1/2*c)^2*b^2+a^2)^2-(-1/8*b^2*(11*a^2-6*b^2)/a^4/(a^2-b^2)^2/e*cos(1/2*d*
x+1/2*c)^3+1/16*b^2*(11*a^2-6*b^2)/a^4/(a^2-b^2)^2/e*cos(1/2*d*x+1/2*c))*(-e*(2*sin(1/2*d*x+1/2*c)^4-sin(1/2*d
*x+1/2*c)^2))^(1/2)/(4*cos(1/2*d*x+1/2*c)^4*b^2-4*cos(1/2*d*x+1/2*c)^2*b^2+a^2)-1/32*(11*a^2-6*b^2)/a^4/(a^2-b
^2)^2*(sin(1/2*d*x+1/2*c)^2)^(1/2)*(-2*cos(1/2*d*x+1/2*c)^2+1)^(1/2)/(-e*(2*sin(1/2*d*x+1/2*c)^4-sin(1/2*d*x+1
/2*c)^2))^(1/2)*EllipticF(cos(1/2*d*x+1/2*c),2^(1/2))+1/32*(11*a^2-6*b^2)/a^4/(a^2-b^2)^2*(sin(1/2*d*x+1/2*c)^
2)^(1/2)*(-2*cos(1/2*d*x+1/2*c)^2+1)^(1/2)/(-e*(2*sin(1/2*d*x+1/2*c)^4-sin(1/2*d*x+1/2*c)^2))^(1/2)*(EllipticF
(cos(1/2*d*x+1/2*c),2^(1/2))-EllipticE(cos(1/2*d*x+1/2*c),2^(1/2)))-1/512/a^4/b^2*sum((-21*a^4+28*a^2*b^2-12*b
^4)/(a-b)^2/(a+b)^2/_alpha*(2^(1/2)/(e*(2*_alpha^2*b^2+a^2-2*b^2)/b^2)^(1/2)*arctanh(1/2*e*(4*_alpha^2-3)/(4*a
^2-3*b^2)*(4*a^2*cos(1/2*d*x+1/2*c)^2-3*cos(1/2*d*x+1/2*c)^2*b^2+b^2*_alpha^2-3*a^2+2*b^2)*2^(1/2)/(e*(2*_alph
a^2*b^2+a^2-2*b^2)/b^2)^(1/2)/(-e*(2*sin(1/2*d*x+1/2*c)^4-sin(1/2*d*x+1/2*c)^2))^(1/2))+8*b^2/a^2*_alpha*(_alp
ha^2-1)*(sin(1/2*d*x+1/2*c)^2)^(1/2)*(-2*cos(1/2*d*x+1/2*c)^2+1)^(1/2)/(-e*sin(1/2*d*x+1/2*c)^2*(2*sin(1/2*d*x
+1/2*c)^2-1))^(1/2)*EllipticPi(cos(1/2*d*x+1/2*c),-4*b^2/a^2*(_alpha^2-1),2^(1/2))),_alpha=RootOf(4*_Z^4*b^2-4
*_Z^2*b^2+a^2)))+8*a^4*(-1/3*(-b^2/(a^2-b^2)/a^2/e*cos(1/2*d*x+1/2*c)^3+1/2*b^2/(a^2-b^2)/a^2/e*cos(1/2*d*x+1/
2*c))*(-e*(2*sin(1/2*d*x+1/2*c)^4-sin(1/2*d*x+1/2*c)^2))^(1/2)/(4*cos(1/2*d*x+1/2*c)^4*b^2-4*cos(1/2*d*x+1/2*c
)^2*b^2+a^2)^3-1/2*(-1/12*b^2*(19*a^2-10*b^2)/(a^2-b^2)^2/a^4/e*cos(1/2*d*x+1/2*c)^3+1/24*b^2*(19*a^2-10*b^2)/
(a^2-b^2)^2/a^4/e*cos(1/2*d*x+1/2*c))*(-e*(2*sin(1/2*d*x+1/2*c)^4-sin(1/2*d*x+1/2*c)^2))^(1/2)/(4*cos(1/2*d*x+
1/2*c)^4*b^2-4*cos(1/2*d*x+1/2*c)^2*b^2+a^2)^2-(-1/32*b^2*(51*a^4-56*a^2*b^2+20*b^4)/(a^2-b^2)^3/a^6/e*cos(1/2
*d*x+1/2*c)^3+1/64*b^2*(51*a^4-56*a^2*b^2+20*b^4)/(a^2-b^2)^3/a^6/e*cos(1/2*d*x+1/2*c))*(-e*(2*sin(1/2*d*x+1/2
*c)^4-sin(1/2*d*x+1/2*c)^2))^(1/2)/(4*cos(1/2*d*x+1/2*c)^4*b^2-4*cos(1/2*d*x+1/2*c)^2*b^2+a^2)-1/128*(51*a^4-5
6*a^2*b^2+20*b^4)/(a^2-b^2)^3/a^6*(sin(1/2*d*x+1/2*c)^2)^(1/2)*(-2*cos(1/2*d*x+1/2*c)^2+1)^(1/2)/(-e*(2*sin(1/
2*d*x+1/2*c)^4-sin(1/2*d*x+1/2*c)^2))^(1/2)*EllipticF(cos(1/2*d*x+1/2*c),2^(1/2))+1/128*(51*a^4-56*a^2*b^2+20*
b^4)/(a^2-b^2)^3/a^6*(sin(1/2*d*x+1/2*c)^2)^(1/2)*(-2*cos(1/2*d*x+1/2*c)^2+1)^(1/2)/(-e*(2*sin(1/2*d*x+1/2*c)^
4-sin(1/2*d*x+1/2*c)^2))^(1/2)*(EllipticF(cos(1/2*d*x+1/2*c),2^(1/2))-EllipticE(cos(1/2*d*x+1/2*c),2^(1/2)))-1
/2048/a^6/b^2*sum((-77*a^6+154*a^4*b^2-132*a^2*b^4+40*b^6)/(a-b)^3/(a+b)^3/_alpha*(2^(1/2)/(e*(2*_alpha^2*b^2+
a^2-2*b^2)/b^2)^(1/2)*arctanh(1/2*e*(4*_alpha^2-3)/(4*a^2-3*b^2)*(4*a^2*cos(1/2*d*x+1/2*c)^2-3*cos(1/2*d*x+1/2
*c)^2*b^2+b^2*_alpha^2-3*a^2+2*b^2)*2^(1/2)/(e*(2*_alpha^2*b^2+a^2-2*b^2)/b^2)^(1/2)/(-e*(2*sin(1/2*d*x+1/2*c)
^4-sin(1/2*d*x+1/2*c)^2))^(1/2))+8*b^2/a^2*_alpha*(_alpha^2-1)*(sin(1/2*d*x+1/2*c)^2)^(1/2)*(-2*cos(1/2*d*x+1/
2*c)^2+1)^(1/2)/(-e*sin(1/2*d*x+1/2*c)^2*(2*sin(1/2*d*x+1/2*c)^2-1))^(1/2)*EllipticPi(cos(1/2*d*x+1/2*c),-4*b^
2/a^2*(_alpha^2-1),2^(1/2))),_alpha=RootOf(4*_Z^4*b^2-4*_Z^2*b^2+a^2))))/sin(1/2*d*x+1/2*c)/(e*(2*cos(1/2*d*x+
1/2*c)^2-1))^(1/2))/d

Fricas [F]

\[ \int \frac {\sqrt {e \cos (c+d x)}}{(a+b \sin (c+d x))^4} \, dx=\int { \frac {\sqrt {e \cos \left (d x + c\right )}}{{\left (b \sin \left (d x + c\right ) + a\right )}^{4}} \,d x } \]

[In]

integrate((e*cos(d*x+c))^(1/2)/(a+b*sin(d*x+c))^4,x, algorithm="fricas")

[Out]

integral(sqrt(e*cos(d*x + c))/(b^4*cos(d*x + c)^4 + a^4 + 6*a^2*b^2 + b^4 - 2*(3*a^2*b^2 + b^4)*cos(d*x + c)^2
- 4*(a*b^3*cos(d*x + c)^2 - a^3*b - a*b^3)*sin(d*x + c)), x)

Sympy [F(-1)]

Timed out. \[ \int \frac {\sqrt {e \cos (c+d x)}}{(a+b \sin (c+d x))^4} \, dx=\text {Timed out} \]

[In]

integrate((e*cos(d*x+c))**(1/2)/(a+b*sin(d*x+c))**4,x)

[Out]

Timed out

Maxima [F]

\[ \int \frac {\sqrt {e \cos (c+d x)}}{(a+b \sin (c+d x))^4} \, dx=\int { \frac {\sqrt {e \cos \left (d x + c\right )}}{{\left (b \sin \left (d x + c\right ) + a\right )}^{4}} \,d x } \]

[In]

integrate((e*cos(d*x+c))^(1/2)/(a+b*sin(d*x+c))^4,x, algorithm="maxima")

[Out]

integrate(sqrt(e*cos(d*x + c))/(b*sin(d*x + c) + a)^4, x)

Giac [F]

\[ \int \frac {\sqrt {e \cos (c+d x)}}{(a+b \sin (c+d x))^4} \, dx=\int { \frac {\sqrt {e \cos \left (d x + c\right )}}{{\left (b \sin \left (d x + c\right ) + a\right )}^{4}} \,d x } \]

[In]

integrate((e*cos(d*x+c))^(1/2)/(a+b*sin(d*x+c))^4,x, algorithm="giac")

[Out]

integrate(sqrt(e*cos(d*x + c))/(b*sin(d*x + c) + a)^4, x)

Mupad [F(-1)]

Timed out. \[ \int \frac {\sqrt {e \cos (c+d x)}}{(a+b \sin (c+d x))^4} \, dx=\int \frac {\sqrt {e\,\cos \left (c+d\,x\right )}}{{\left (a+b\,\sin \left (c+d\,x\right )\right )}^4} \,d x \]

[In]

int((e*cos(c + d*x))^(1/2)/(a + b*sin(c + d*x))^4,x)

[Out]

int((e*cos(c + d*x))^(1/2)/(a + b*sin(c + d*x))^4, x)