\(\int \frac {\sqrt {e \cos (c+d x)}}{(a+b \sin (c+d x))^2} \, dx\) [589]

Optimal result

Integrand size = 25, antiderivative size = 422 \[ \int \frac {\sqrt {e \cos (c+d x)}}{(a+b \sin (c+d x))^2} \, dx=-\frac {a \sqrt {e} \arctan \left (\frac {\sqrt {b} \sqrt {e \cos (c+d x)}}{\sqrt [4]{-a^2+b^2} \sqrt {e}}\right )}{2 \sqrt {b} \left (-a^2+b^2\right )^{5/4} d}+\frac {a \sqrt {e} \text {arctanh}\left (\frac {\sqrt {b} \sqrt {e \cos (c+d x)}}{\sqrt [4]{-a^2+b^2} \sqrt {e}}\right )}{2 \sqrt {b} \left (-a^2+b^2\right )^{5/4} d}+\frac {\sqrt {e \cos (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{\left (a^2-b^2\right ) d \sqrt {\cos (c+d x)}}+\frac {a^2 e \sqrt {\cos (c+d x)} \operatorname {EllipticPi}\left (\frac {2 b}{b-\sqrt {-a^2+b^2}},\frac {1}{2} (c+d x),2\right )}{2 b \left (a^2-b^2\right ) \left (b-\sqrt {-a^2+b^2}\right ) d \sqrt {e \cos (c+d x)}}+\frac {a^2 e \sqrt {\cos (c+d x)} \operatorname {EllipticPi}\left (\frac {2 b}{b+\sqrt {-a^2+b^2}},\frac {1}{2} (c+d x),2\right )}{2 b \left (a^2-b^2\right ) \left (b+\sqrt {-a^2+b^2}\right ) d \sqrt {e \cos (c+d x)}}+\frac {b (e \cos (c+d x))^{3/2}}{\left (a^2-b^2\right ) d e (a+b \sin (c+d x))} \]

[Out]

b*(e*cos(d*x+c))^(3/2)/(a^2-b^2)/d/e/(a+b*sin(d*x+c))-1/2*a*arctan(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)^(5/4)/d/b^(1/2)+1/2*a*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)^(5/4)/d/b^(1/2)+1/2*a^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)/d/(b-(-a^2+b^2)^(1/2))/(e*co
s(d*x+c))^(1/2)+1/2*a^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)/d/(b+(-a^2+b^2)^(1/2))/(e*cos(d*x+c))^(1/2)+(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)/d/
cos(d*x+c)^(1/2)

Rubi [A] (verified)

Time = 0.57 (sec) , antiderivative size = 422, normalized size of antiderivative = 1.00, number of steps used = 13, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.440, Rules used = {2773, 2946, 2721, 2719, 2780, 2886, 2884, 335, 304, 211, 214} \[ \int \frac {\sqrt {e \cos (c+d x)}}{(a+b \sin (c+d x))^2} \, dx=-\frac {a \sqrt {e} \arctan \left (\frac {\sqrt {b} \sqrt {e \cos (c+d x)}}{\sqrt {e} \sqrt [4]{b^2-a^2}}\right )}{2 \sqrt {b} d \left (b^2-a^2\right )^{5/4}}+\frac {a \sqrt {e} \text {arctanh}\left (\frac {\sqrt {b} \sqrt {e \cos (c+d x)}}{\sqrt {e} \sqrt [4]{b^2-a^2}}\right )}{2 \sqrt {b} d \left (b^2-a^2\right )^{5/4}}+\frac {b (e \cos (c+d x))^{3/2}}{d e \left (a^2-b^2\right ) (a+b \sin (c+d x))}+\frac {E\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \sqrt {e \cos (c+d x)}}{d \left (a^2-b^2\right ) \sqrt {\cos (c+d x)}}+\frac {a^2 e \sqrt {\cos (c+d x)} \operatorname {EllipticPi}\left (\frac {2 b}{b-\sqrt {b^2-a^2}},\frac {1}{2} (c+d x),2\right )}{2 b d \left (a^2-b^2\right ) \left (b-\sqrt {b^2-a^2}\right ) \sqrt {e \cos (c+d x)}}+\frac {a^2 e \sqrt {\cos (c+d x)} \operatorname {EllipticPi}\left (\frac {2 b}{b+\sqrt {b^2-a^2}},\frac {1}{2} (c+d x),2\right )}{2 b d \left (a^2-b^2\right ) \left (\sqrt {b^2-a^2}+b\right ) \sqrt {e \cos (c+d x)}} \]

[In]

Int[Sqrt[e*Cos[c + d*x]]/(a + b*Sin[c + d*x])^2,x]

[Out]

-1/2*(a*Sqrt[e]*ArcTan[(Sqrt[b]*Sqrt[e*Cos[c + d*x]])/((-a^2 + b^2)^(1/4)*Sqrt[e])])/(Sqrt[b]*(-a^2 + b^2)^(5/
4)*d) + (a*Sqrt[e]*ArcTanh[(Sqrt[b]*Sqrt[e*Cos[c + d*x]])/((-a^2 + b^2)^(1/4)*Sqrt[e])])/(2*Sqrt[b]*(-a^2 + b^
2)^(5/4)*d) + (Sqrt[e*Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2])/((a^2 - b^2)*d*Sqrt[Cos[c + d*x]]) + (a^2*e*Sqr
t[Cos[c + d*x]]*EllipticPi[(2*b)/(b - Sqrt[-a^2 + b^2]), (c + d*x)/2, 2])/(2*b*(a^2 - b^2)*(b - Sqrt[-a^2 + b^
2])*d*Sqrt[e*Cos[c + d*x]]) + (a^2*e*Sqrt[Cos[c + d*x]]*EllipticPi[(2*b)/(b + Sqrt[-a^2 + b^2]), (c + d*x)/2,
2])/(2*b*(a^2 - b^2)*(b + Sqrt[-a^2 + b^2])*d*Sqrt[e*Cos[c + d*x]]) + (b*(e*Cos[c + d*x])^(3/2))/((a^2 - b^2)*
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 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}}{\left (a^2-b^2\right ) d e (a+b \sin (c+d x))}+\frac {\int \frac {\sqrt {e \cos (c+d x)} \left (-a-\frac {1}{2} b \sin (c+d x)\right )}{a+b \sin (c+d x)} \, dx}{-a^2+b^2} \\ & = \frac {b (e \cos (c+d x))^{3/2}}{\left (a^2-b^2\right ) d e (a+b \sin (c+d x))}+\frac {\int \sqrt {e \cos (c+d x)} \, dx}{2 \left (a^2-b^2\right )}+\frac {a \int \frac {\sqrt {e \cos (c+d x)}}{a+b \sin (c+d x)} \, dx}{2 \left (a^2-b^2\right )} \\ & = \frac {b (e \cos (c+d x))^{3/2}}{\left (a^2-b^2\right ) d e (a+b \sin (c+d x))}-\frac {\left (a^2 e\right ) \int \frac {1}{\sqrt {e \cos (c+d x)} \left (\sqrt {-a^2+b^2}-b \cos (c+d x)\right )} \, dx}{4 b \left (a^2-b^2\right )}+\frac {\left (a^2 e\right ) \int \frac {1}{\sqrt {e \cos (c+d x)} \left (\sqrt {-a^2+b^2}+b \cos (c+d x)\right )} \, dx}{4 b \left (a^2-b^2\right )}+\frac {(a b e) \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 )}{2 \left (a^2-b^2\right ) d}+\frac {\sqrt {e \cos (c+d x)} \int \sqrt {\cos (c+d x)} \, dx}{2 \left (a^2-b^2\right ) \sqrt {\cos (c+d x)}} \\ & = \frac {\sqrt {e \cos (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{\left (a^2-b^2\right ) d \sqrt {\cos (c+d x)}}+\frac {b (e \cos (c+d x))^{3/2}}{\left (a^2-b^2\right ) d e (a+b \sin (c+d x))}+\frac {(a b e) \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 )}{\left (a^2-b^2\right ) d}-\frac {\left (a^2 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}{4 b \left (a^2-b^2\right ) \sqrt {e \cos (c+d x)}}+\frac {\left (a^2 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}{4 b \left (a^2-b^2\right ) \sqrt {e \cos (c+d x)}} \\ & = \frac {\sqrt {e \cos (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{\left (a^2-b^2\right ) d \sqrt {\cos (c+d x)}}+\frac {a^2 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 )}{2 b \left (a^2-b^2\right ) \left (b-\sqrt {-a^2+b^2}\right ) d \sqrt {e \cos (c+d x)}}+\frac {a^2 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 )}{2 b \left (a^2-b^2\right ) \left (b+\sqrt {-a^2+b^2}\right ) d \sqrt {e \cos (c+d x)}}+\frac {b (e \cos (c+d x))^{3/2}}{\left (a^2-b^2\right ) d e (a+b \sin (c+d x))}-\frac {(a e) \text {Subst}\left (\int \frac {1}{\sqrt {-a^2+b^2} e-b x^2} \, dx,x,\sqrt {e \cos (c+d x)}\right )}{2 \left (a^2-b^2\right ) d}+\frac {(a e) \text {Subst}\left (\int \frac {1}{\sqrt {-a^2+b^2} e+b x^2} \, dx,x,\sqrt {e \cos (c+d x)}\right )}{2 \left (a^2-b^2\right ) d} \\ & = -\frac {a \sqrt {e} \arctan \left (\frac {\sqrt {b} \sqrt {e \cos (c+d x)}}{\sqrt [4]{-a^2+b^2} \sqrt {e}}\right )}{2 \sqrt {b} \left (-a^2+b^2\right )^{5/4} d}+\frac {a \sqrt {e} \text {arctanh}\left (\frac {\sqrt {b} \sqrt {e \cos (c+d x)}}{\sqrt [4]{-a^2+b^2} \sqrt {e}}\right )}{2 \sqrt {b} \left (-a^2+b^2\right )^{5/4} d}+\frac {\sqrt {e \cos (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{\left (a^2-b^2\right ) d \sqrt {\cos (c+d x)}}+\frac {a^2 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 )}{2 b \left (a^2-b^2\right ) \left (b-\sqrt {-a^2+b^2}\right ) d \sqrt {e \cos (c+d x)}}+\frac {a^2 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 )}{2 b \left (a^2-b^2\right ) \left (b+\sqrt {-a^2+b^2}\right ) d \sqrt {e \cos (c+d x)}}+\frac {b (e \cos (c+d x))^{3/2}}{\left (a^2-b^2\right ) 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 = 13.53 (sec) , antiderivative size = 787, normalized size of antiderivative = 1.86 \[ \int \frac {\sqrt {e \cos (c+d x)}}{(a+b \sin (c+d x))^2} \, dx=-\frac {b \cos (c+d x) \sqrt {e \cos (c+d x)}}{\left (-a^2+b^2\right ) d (a+b \sin (c+d x))}+\frac {\sqrt {e \cos (c+d x)} \left (-\frac {4 a \left (a+b \sqrt {1-\cos ^2(c+d x)}\right ) \left (\frac {a \operatorname {AppellF1}\left (\frac {3}{4},\frac {1}{2},1,\frac {7}{4},\cos ^2(c+d x),\frac {b^2 \cos ^2(c+d x)}{-a^2+b^2}\right ) \cos ^{\frac {3}{2}}(c+d x)}{3 \left (a^2-b^2\right )}+\frac {\left (\frac {1}{8}+\frac {i}{8}\right ) \left (2 \arctan \left (1-\frac {(1+i) \sqrt {b} \sqrt {\cos (c+d x)}}{\sqrt [4]{-a^2+b^2}}\right )-2 \arctan \left (1+\frac {(1+i) \sqrt {b} \sqrt {\cos (c+d x)}}{\sqrt [4]{-a^2+b^2}}\right )-\log \left (\sqrt {-a^2+b^2}-(1+i) \sqrt {b} \sqrt [4]{-a^2+b^2} \sqrt {\cos (c+d x)}+i b \cos (c+d x)\right )+\log \left (\sqrt {-a^2+b^2}+(1+i) \sqrt {b} \sqrt [4]{-a^2+b^2} \sqrt {\cos (c+d x)}+i b \cos (c+d x)\right )\right )}{\sqrt {b} \sqrt [4]{-a^2+b^2}}\right ) \sin (c+d x)}{\sqrt {1-\cos ^2(c+d x)} (a+b \sin (c+d x))}-\frac {\left (a+b \sqrt {1-\cos ^2(c+d x)}\right ) \left (8 b^{5/2} \operatorname {AppellF1}\left (\frac {3}{4},-\frac {1}{2},1,\frac {7}{4},\cos ^2(c+d x),\frac {b^2 \cos ^2(c+d x)}{-a^2+b^2}\right ) \cos ^{\frac {3}{2}}(c+d x)+3 \sqrt {2} a \left (a^2-b^2\right )^{3/4} \left (2 \arctan \left (1-\frac {\sqrt {2} \sqrt {b} \sqrt {\cos (c+d x)}}{\sqrt [4]{a^2-b^2}}\right )-2 \arctan \left (1+\frac {\sqrt {2} \sqrt {b} \sqrt {\cos (c+d x)}}{\sqrt [4]{a^2-b^2}}\right )-\log \left (\sqrt {a^2-b^2}-\sqrt {2} \sqrt {b} \sqrt [4]{a^2-b^2} \sqrt {\cos (c+d x)}+b \cos (c+d x)\right )+\log \left (\sqrt {a^2-b^2}+\sqrt {2} \sqrt {b} \sqrt [4]{a^2-b^2} \sqrt {\cos (c+d x)}+b \cos (c+d x)\right )\right )\right ) \sin ^2(c+d x)}{12 \sqrt {b} \left (-a^2+b^2\right ) \left (1-\cos ^2(c+d x)\right ) (a+b \sin (c+d x))}\right )}{2 (a-b) (a+b) d \sqrt {\cos (c+d x)}} \]

[In]

Integrate[Sqrt[e*Cos[c + d*x]]/(a + b*Sin[c + d*x])^2,x]

[Out]

-((b*Cos[c + d*x]*Sqrt[e*Cos[c + d*x]])/((-a^2 + b^2)*d*(a + b*Sin[c + d*x]))) + (Sqrt[e*Cos[c + d*x]]*((-4*a*
(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])) - ((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]] + 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]]))*Sin[c + d*x]^2)/(12*Sqrt[b]*(-a^2 + b^2)*(1 - Cos[c + d*x]^2)*(a + b*Sin[c + d*x]))))/(2*(a - b)*(a
+ b)*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 = 6.73 (sec) , antiderivative size = 1495, normalized size of antiderivative = 3.54

method	result	size
default	\(\text {Expression too large to display}\)	\(1495\)

[In]

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

[Out]

(1/8*a/b/(e^2*(a^2-b^2)/b^2)^(1/4)*(16*(e^2*(a^2-b^2)/b^2)^(1/4)*(cos(1/2*d*x+1/2*c)^2-1/2)*(2*e*cos(1/2*d*x+1
/2*c)^2-e)^(1/2)*b^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)*e*2^(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)))+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))))/(a-b)/(a+b)/(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*(e*(2*cos(1/2*d*x+1/2*c)^2
-1)*sin(1/2*d*x+1/2*c)^2)^(1/2)*e*(-1/16/b^2*sum(1/_alpha*(2^(1/2)/(e*(2*_alpha^2*b^2+a^2-2*b^2)/b^2)^(1/2)*ar
ctanh(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))+2*a^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*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*(_alpha^2-1),2^(1/2))),_alpha=Root
Of(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]

\[ \int \frac {\sqrt {e \cos (c+d x)}}{(a+b \sin (c+d x))^2} \, dx=\int { \frac {\sqrt {e \cos \left (d x + c\right )}}{{\left (b \sin \left (d x + c\right ) + a\right )}^{2}} \,d x } \]

[In]

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

[Out]

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

Sympy [F(-1)]

Timed out. \[ \int \frac {\sqrt {e \cos (c+d x)}}{(a+b \sin (c+d x))^2} \, dx=\text {Timed out} \]

[In]

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

[Out]

Timed out

Maxima [F]

\[ \int \frac {\sqrt {e \cos (c+d x)}}{(a+b \sin (c+d x))^2} \, dx=\int { \frac {\sqrt {e \cos \left (d x + c\right )}}{{\left (b \sin \left (d x + c\right ) + a\right )}^{2}} \,d x } \]

[In]

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

[Out]

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

Giac [F]

\[ \int \frac {\sqrt {e \cos (c+d x)}}{(a+b \sin (c+d x))^2} \, dx=\int { \frac {\sqrt {e \cos \left (d x + c\right )}}{{\left (b \sin \left (d x + c\right ) + a\right )}^{2}} \,d x } \]

[In]

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

[Out]

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

Mupad [F(-1)]

Timed out. \[ \int \frac {\sqrt {e \cos (c+d x)}}{(a+b \sin (c+d x))^2} \, dx=\int \frac {\sqrt {e\,\cos \left (c+d\,x\right )}}{{\left (a+b\,\sin \left (c+d\,x\right )\right )}^2} \,d x \]

[In]

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

[Out]

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