\(\int \frac {(e \cos (c+d x))^{9/2}}{(a+b \sin (c+d x))^4} \, dx\) [608]

Optimal result

Integrand size = 25, antiderivative size = 591 \[ \int \frac {(e \cos (c+d x))^{9/2}}{(a+b \sin (c+d x))^4} \, dx=\frac {7 a \left (5 a^2-6 b^2\right ) e^{9/2} \arctan \left (\frac {\sqrt {b} \sqrt {e \cos (c+d x)}}{\sqrt [4]{-a^2+b^2} \sqrt {e}}\right )}{16 b^{9/2} \left (-a^2+b^2\right )^{5/4} d}-\frac {7 a \left (5 a^2-6 b^2\right ) e^{9/2} \text {arctanh}\left (\frac {\sqrt {b} \sqrt {e \cos (c+d x)}}{\sqrt [4]{-a^2+b^2} \sqrt {e}}\right )}{16 b^{9/2} \left (-a^2+b^2\right )^{5/4} d}+\frac {7 \left (5 a^2-4 b^2\right ) e^4 \sqrt {e \cos (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{8 b^4 \left (a^2-b^2\right ) d \sqrt {\cos (c+d x)}}-\frac {7 a^2 \left (5 a^2-6 b^2\right ) e^5 \sqrt {\cos (c+d x)} \operatorname {EllipticPi}\left (\frac {2 b}{b-\sqrt {-a^2+b^2}},\frac {1}{2} (c+d x),2\right )}{16 b^5 \left (a^2-b^2\right ) \left (b-\sqrt {-a^2+b^2}\right ) d \sqrt {e \cos (c+d x)}}-\frac {7 a^2 \left (5 a^2-6 b^2\right ) e^5 \sqrt {\cos (c+d x)} \operatorname {EllipticPi}\left (\frac {2 b}{b+\sqrt {-a^2+b^2}},\frac {1}{2} (c+d x),2\right )}{16 b^5 \left (a^2-b^2\right ) \left (b+\sqrt {-a^2+b^2}\right ) d \sqrt {e \cos (c+d x)}}-\frac {e (e \cos (c+d x))^{7/2}}{3 b d (a+b \sin (c+d x))^3}+\frac {7 \left (5 a^2-4 b^2\right ) e^3 (e \cos (c+d x))^{3/2}}{8 b^3 \left (a^2-b^2\right ) d (a+b \sin (c+d x))}-\frac {7 e^3 (e \cos (c+d x))^{3/2} (5 a+4 b \sin (c+d x))}{12 b^3 d (a+b \sin (c+d x))^2} \]

[Out]

7/16*a*(5*a^2-6*b^2)*e^(9/2)*arctan(b^(1/2)*(e*cos(d*x+c))^(1/2)/(-a^2+b^2)^(1/4)/e^(1/2))/b^(9/2)/(-a^2+b^2)^
(5/4)/d-7/16*a*(5*a^2-6*b^2)*e^(9/2)*arctanh(b^(1/2)*(e*cos(d*x+c))^(1/2)/(-a^2+b^2)^(1/4)/e^(1/2))/b^(9/2)/(-
a^2+b^2)^(5/4)/d-1/3*e*(e*cos(d*x+c))^(7/2)/b/d/(a+b*sin(d*x+c))^3+7/8*(5*a^2-4*b^2)*e^3*(e*cos(d*x+c))^(3/2)/
b^3/(a^2-b^2)/d/(a+b*sin(d*x+c))-7/12*e^3*(e*cos(d*x+c))^(3/2)*(5*a+4*b*sin(d*x+c))/b^3/d/(a+b*sin(d*x+c))^2-7
/16*a^2*(5*a^2-6*b^2)*e^5*(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^5/(a^2-b^2)/d/(b-(-a^2+b^2)^(1/2))/(e*cos(d*x+c))^(1/2)-7/16*a^
2*(5*a^2-6*b^2)*e^5*(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^5/(a^2-b^2)/d/(b+(-a^2+b^2)^(1/2))/(e*cos(d*x+c))^(1/2)+7/8*(5*a^2-4*
b^2)*e^4*(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)/b^4/(a^2-b^2)/d/cos(d*x+c)^(1/2)

Rubi [A] (verified)

Time = 0.96 (sec) , antiderivative size = 591, normalized size of antiderivative = 1.00, number of steps used = 15, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.520, Rules used = {2772, 2942, 2943, 2946, 2721, 2719, 2780, 2886, 2884, 335, 304, 211, 214} \[ \int \frac {(e \cos (c+d x))^{9/2}}{(a+b \sin (c+d x))^4} \, dx=\frac {7 a e^{9/2} \left (5 a^2-6 b^2\right ) \arctan \left (\frac {\sqrt {b} \sqrt {e \cos (c+d x)}}{\sqrt {e} \sqrt [4]{b^2-a^2}}\right )}{16 b^{9/2} d \left (b^2-a^2\right )^{5/4}}-\frac {7 a e^{9/2} \left (5 a^2-6 b^2\right ) \text {arctanh}\left (\frac {\sqrt {b} \sqrt {e \cos (c+d x)}}{\sqrt {e} \sqrt [4]{b^2-a^2}}\right )}{16 b^{9/2} d \left (b^2-a^2\right )^{5/4}}-\frac {7 a^2 e^5 \left (5 a^2-6 b^2\right ) \sqrt {\cos (c+d x)} \operatorname {EllipticPi}\left (\frac {2 b}{b-\sqrt {b^2-a^2}},\frac {1}{2} (c+d x),2\right )}{16 b^5 d \left (a^2-b^2\right ) \left (b-\sqrt {b^2-a^2}\right ) \sqrt {e \cos (c+d x)}}-\frac {7 a^2 e^5 \left (5 a^2-6 b^2\right ) \sqrt {\cos (c+d x)} \operatorname {EllipticPi}\left (\frac {2 b}{b+\sqrt {b^2-a^2}},\frac {1}{2} (c+d x),2\right )}{16 b^5 d \left (a^2-b^2\right ) \left (\sqrt {b^2-a^2}+b\right ) \sqrt {e \cos (c+d x)}}+\frac {7 e^4 \left (5 a^2-4 b^2\right ) E\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \sqrt {e \cos (c+d x)}}{8 b^4 d \left (a^2-b^2\right ) \sqrt {\cos (c+d x)}}+\frac {7 e^3 \left (5 a^2-4 b^2\right ) (e \cos (c+d x))^{3/2}}{8 b^3 d \left (a^2-b^2\right ) (a+b \sin (c+d x))}-\frac {7 e^3 (e \cos (c+d x))^{3/2} (5 a+4 b \sin (c+d x))}{12 b^3 d (a+b \sin (c+d x))^2}-\frac {e (e \cos (c+d x))^{7/2}}{3 b d (a+b \sin (c+d x))^3} \]

[In]

Int[(e*Cos[c + d*x])^(9/2)/(a + b*Sin[c + d*x])^4,x]

[Out]

(7*a*(5*a^2 - 6*b^2)*e^(9/2)*ArcTan[(Sqrt[b]*Sqrt[e*Cos[c + d*x]])/((-a^2 + b^2)^(1/4)*Sqrt[e])])/(16*b^(9/2)*
(-a^2 + b^2)^(5/4)*d) - (7*a*(5*a^2 - 6*b^2)*e^(9/2)*ArcTanh[(Sqrt[b]*Sqrt[e*Cos[c + d*x]])/((-a^2 + b^2)^(1/4
)*Sqrt[e])])/(16*b^(9/2)*(-a^2 + b^2)^(5/4)*d) + (7*(5*a^2 - 4*b^2)*e^4*Sqrt[e*Cos[c + d*x]]*EllipticE[(c + d*
x)/2, 2])/(8*b^4*(a^2 - b^2)*d*Sqrt[Cos[c + d*x]]) - (7*a^2*(5*a^2 - 6*b^2)*e^5*Sqrt[Cos[c + d*x]]*EllipticPi[
(2*b)/(b - Sqrt[-a^2 + b^2]), (c + d*x)/2, 2])/(16*b^5*(a^2 - b^2)*(b - Sqrt[-a^2 + b^2])*d*Sqrt[e*Cos[c + d*x
]]) - (7*a^2*(5*a^2 - 6*b^2)*e^5*Sqrt[Cos[c + d*x]]*EllipticPi[(2*b)/(b + Sqrt[-a^2 + b^2]), (c + d*x)/2, 2])/
(16*b^5*(a^2 - b^2)*(b + Sqrt[-a^2 + b^2])*d*Sqrt[e*Cos[c + d*x]]) - (e*(e*Cos[c + d*x])^(7/2))/(3*b*d*(a + b*
Sin[c + d*x])^3) + (7*(5*a^2 - 4*b^2)*e^3*(e*Cos[c + d*x])^(3/2))/(8*b^3*(a^2 - b^2)*d*(a + b*Sin[c + d*x])) -
(7*e^3*(e*Cos[c + d*x])^(3/2)*(5*a + 4*b*Sin[c + d*x]))/(12*b^3*d*(a + b*Sin[c + d*x])^2)

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 2772

Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_), x_Symbol] :> Simp[g*(g*C
os[e + f*x])^(p - 1)*((a + b*Sin[e + f*x])^(m + 1)/(b*f*(m + 1))), x] + Dist[g^2*((p - 1)/(b*(m + 1))), Int[(g
*Cos[e + f*x])^(p - 2)*(a + b*Sin[e + f*x])^(m + 1)*Sin[e + f*x], x], x] /; FreeQ[{a, b, e, f, g}, x] && NeQ[a
^2 - b^2, 0] && LtQ[m, -1] && GtQ[p, 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 2942

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

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 {e (e \cos (c+d x))^{7/2}}{3 b d (a+b \sin (c+d x))^3}-\frac {\left (7 e^2\right ) \int \frac {(e \cos (c+d x))^{5/2} \sin (c+d x)}{(a+b \sin (c+d x))^3} \, dx}{6 b} \\ & = -\frac {e (e \cos (c+d x))^{7/2}}{3 b d (a+b \sin (c+d x))^3}-\frac {7 e^3 (e \cos (c+d x))^{3/2} (5 a+4 b \sin (c+d x))}{12 b^3 d (a+b \sin (c+d x))^2}+\frac {\left (7 e^4\right ) \int \frac {\sqrt {e \cos (c+d x)} \left (-2 b-\frac {5}{2} a \sin (c+d x)\right )}{(a+b \sin (c+d x))^2} \, dx}{4 b^3} \\ & = -\frac {e (e \cos (c+d x))^{7/2}}{3 b d (a+b \sin (c+d x))^3}+\frac {7 \left (5 a^2-4 b^2\right ) e^3 (e \cos (c+d x))^{3/2}}{8 b^3 \left (a^2-b^2\right ) d (a+b \sin (c+d x))}-\frac {7 e^3 (e \cos (c+d x))^{3/2} (5 a+4 b \sin (c+d x))}{12 b^3 d (a+b \sin (c+d x))^2}-\frac {\left (7 e^4\right ) \int \frac {\sqrt {e \cos (c+d x)} \left (-\frac {a b}{2}-\frac {1}{4} \left (5 a^2-4 b^2\right ) \sin (c+d x)\right )}{a+b \sin (c+d x)} \, dx}{4 b^3 \left (a^2-b^2\right )} \\ & = -\frac {e (e \cos (c+d x))^{7/2}}{3 b d (a+b \sin (c+d x))^3}+\frac {7 \left (5 a^2-4 b^2\right ) e^3 (e \cos (c+d x))^{3/2}}{8 b^3 \left (a^2-b^2\right ) d (a+b \sin (c+d x))}-\frac {7 e^3 (e \cos (c+d x))^{3/2} (5 a+4 b \sin (c+d x))}{12 b^3 d (a+b \sin (c+d x))^2}-\frac {\left (7 a \left (5 a^2-6 b^2\right ) e^4\right ) \int \frac {\sqrt {e \cos (c+d x)}}{a+b \sin (c+d x)} \, dx}{16 b^4 \left (a^2-b^2\right )}+\frac {\left (7 \left (5 a^2-4 b^2\right ) e^4\right ) \int \sqrt {e \cos (c+d x)} \, dx}{16 b^4 \left (a^2-b^2\right )} \\ & = -\frac {e (e \cos (c+d x))^{7/2}}{3 b d (a+b \sin (c+d x))^3}+\frac {7 \left (5 a^2-4 b^2\right ) e^3 (e \cos (c+d x))^{3/2}}{8 b^3 \left (a^2-b^2\right ) d (a+b \sin (c+d x))}-\frac {7 e^3 (e \cos (c+d x))^{3/2} (5 a+4 b \sin (c+d x))}{12 b^3 d (a+b \sin (c+d x))^2}+\frac {\left (7 a^2 \left (5 a^2-6 b^2\right ) e^5\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^5 \left (a^2-b^2\right )}-\frac {\left (7 a^2 \left (5 a^2-6 b^2\right ) e^5\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^5 \left (a^2-b^2\right )}-\frac {\left (7 a \left (5 a^2-6 b^2\right ) e^5\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 b^3 \left (a^2-b^2\right ) d}+\frac {\left (7 \left (5 a^2-4 b^2\right ) e^4 \sqrt {e \cos (c+d x)}\right ) \int \sqrt {\cos (c+d x)} \, dx}{16 b^4 \left (a^2-b^2\right ) \sqrt {\cos (c+d x)}} \\ & = \frac {7 \left (5 a^2-4 b^2\right ) e^4 \sqrt {e \cos (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{8 b^4 \left (a^2-b^2\right ) d \sqrt {\cos (c+d x)}}-\frac {e (e \cos (c+d x))^{7/2}}{3 b d (a+b \sin (c+d x))^3}+\frac {7 \left (5 a^2-4 b^2\right ) e^3 (e \cos (c+d x))^{3/2}}{8 b^3 \left (a^2-b^2\right ) d (a+b \sin (c+d x))}-\frac {7 e^3 (e \cos (c+d x))^{3/2} (5 a+4 b \sin (c+d x))}{12 b^3 d (a+b \sin (c+d x))^2}-\frac {\left (7 a \left (5 a^2-6 b^2\right ) e^5\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 b^3 \left (a^2-b^2\right ) d}+\frac {\left (7 a^2 \left (5 a^2-6 b^2\right ) e^5 \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^5 \left (a^2-b^2\right ) \sqrt {e \cos (c+d x)}}-\frac {\left (7 a^2 \left (5 a^2-6 b^2\right ) e^5 \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^5 \left (a^2-b^2\right ) \sqrt {e \cos (c+d x)}} \\ & = \frac {7 \left (5 a^2-4 b^2\right ) e^4 \sqrt {e \cos (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{8 b^4 \left (a^2-b^2\right ) d \sqrt {\cos (c+d x)}}-\frac {7 a^2 \left (5 a^2-6 b^2\right ) e^5 \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^5 \left (a^2-b^2\right ) \left (b-\sqrt {-a^2+b^2}\right ) d \sqrt {e \cos (c+d x)}}-\frac {7 a^2 \left (5 a^2-6 b^2\right ) e^5 \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^5 \left (a^2-b^2\right ) \left (b+\sqrt {-a^2+b^2}\right ) d \sqrt {e \cos (c+d x)}}-\frac {e (e \cos (c+d x))^{7/2}}{3 b d (a+b \sin (c+d x))^3}+\frac {7 \left (5 a^2-4 b^2\right ) e^3 (e \cos (c+d x))^{3/2}}{8 b^3 \left (a^2-b^2\right ) d (a+b \sin (c+d x))}-\frac {7 e^3 (e \cos (c+d x))^{3/2} (5 a+4 b \sin (c+d x))}{12 b^3 d (a+b \sin (c+d x))^2}+\frac {\left (7 a \left (5 a^2-6 b^2\right ) e^5\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 b^4 \left (a^2-b^2\right ) d}-\frac {\left (7 a \left (5 a^2-6 b^2\right ) e^5\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 b^4 \left (a^2-b^2\right ) d} \\ & = \frac {7 a \left (5 a^2-6 b^2\right ) e^{9/2} \arctan \left (\frac {\sqrt {b} \sqrt {e \cos (c+d x)}}{\sqrt [4]{-a^2+b^2} \sqrt {e}}\right )}{16 b^{9/2} \left (-a^2+b^2\right )^{5/4} d}-\frac {7 a \left (5 a^2-6 b^2\right ) e^{9/2} \text {arctanh}\left (\frac {\sqrt {b} \sqrt {e \cos (c+d x)}}{\sqrt [4]{-a^2+b^2} \sqrt {e}}\right )}{16 b^{9/2} \left (-a^2+b^2\right )^{5/4} d}+\frac {7 \left (5 a^2-4 b^2\right ) e^4 \sqrt {e \cos (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{8 b^4 \left (a^2-b^2\right ) d \sqrt {\cos (c+d x)}}-\frac {7 a^2 \left (5 a^2-6 b^2\right ) e^5 \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^5 \left (a^2-b^2\right ) \left (b-\sqrt {-a^2+b^2}\right ) d \sqrt {e \cos (c+d x)}}-\frac {7 a^2 \left (5 a^2-6 b^2\right ) e^5 \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^5 \left (a^2-b^2\right ) \left (b+\sqrt {-a^2+b^2}\right ) d \sqrt {e \cos (c+d x)}}-\frac {e (e \cos (c+d x))^{7/2}}{3 b d (a+b \sin (c+d x))^3}+\frac {7 \left (5 a^2-4 b^2\right ) e^3 (e \cos (c+d x))^{3/2}}{8 b^3 \left (a^2-b^2\right ) d (a+b \sin (c+d x))}-\frac {7 e^3 (e \cos (c+d x))^{3/2} (5 a+4 b \sin (c+d x))}{12 b^3 d (a+b \sin (c+d x))^2} \\ \end{align*}

Mathematica [C] (warning: unable to verify)

Result contains higher order function than in optimal. Order 6 vs. order 4 in optimal.

Time = 20.76 (sec) , antiderivative size = 900, normalized size of antiderivative = 1.52 \[ \int \frac {(e \cos (c+d x))^{9/2}}{(a+b \sin (c+d x))^4} \, dx=\frac {(e \cos (c+d x))^{9/2} \sec ^4(c+d x) \left (\frac {a^2 \cos (c+d x)-b^2 \cos (c+d x)}{3 b^3 (a+b \sin (c+d x))^3}-\frac {5 a \cos (c+d x)}{4 b^3 (a+b \sin (c+d x))^2}+\frac {-19 a^2 \cos (c+d x)+12 b^2 \cos (c+d x)}{8 b^3 \left (-a^2+b^2\right ) (a+b \sin (c+d x))}\right )}{d}+\frac {7 (e \cos (c+d x))^{9/2} \left (-\frac {4 a b \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 (5 a^2-4 b^2\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) b^3 (a+b) d \cos ^{\frac {9}{2}}(c+d x)} \]

[In]

Integrate[(e*Cos[c + d*x])^(9/2)/(a + b*Sin[c + d*x])^4,x]

[Out]

((e*Cos[c + d*x])^(9/2)*Sec[c + d*x]^4*((a^2*Cos[c + d*x] - b^2*Cos[c + d*x])/(3*b^3*(a + b*Sin[c + d*x])^3) -
(5*a*Cos[c + d*x])/(4*b^3*(a + b*Sin[c + d*x])^2) + (-19*a^2*Cos[c + d*x] + 12*b^2*Cos[c + d*x])/(8*b^3*(-a^2
+ b^2)*(a + b*Sin[c + d*x]))))/d + (7*(e*Cos[c + d*x])^(9/2)*((-4*a*b*(a + b*Sqrt[1 - Cos[c + d*x]^2])*((a*Ap
pellF1[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])) - ((5*a^2 - 4*b^2)*(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*b^(3/2)*(-a^2 + b^2)*(1 - Cos[c + d*x]^2)*(a + b*Sin[c + d*x]))))/(16*(a - b)*b^3*(a + b)*d*Cos[c + d*
x]^(9/2))

Maple [C] (warning: unable to verify)

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

Time = 341.36 (sec) , antiderivative size = 5039, normalized size of antiderivative = 8.53

method	result	size
default	\(\text {Expression too large to display}\)	\(5039\)

[In]

int((e*cos(d*x+c))^(9/2)/(a+b*sin(d*x+c))^4,x,method=_RETURNVERBOSE)

[Out]

result too large to display

Fricas [F(-1)]

Timed out. \[ \int \frac {(e \cos (c+d x))^{9/2}}{(a+b \sin (c+d x))^4} \, dx=\text {Timed out} \]

[In]

integrate((e*cos(d*x+c))^(9/2)/(a+b*sin(d*x+c))^4,x, algorithm="fricas")

[Out]

Timed out

Sympy [F(-1)]

Timed out. \[ \int \frac {(e \cos (c+d x))^{9/2}}{(a+b \sin (c+d x))^4} \, dx=\text {Timed out} \]

[In]

integrate((e*cos(d*x+c))**(9/2)/(a+b*sin(d*x+c))**4,x)

[Out]

Timed out

Maxima [F]

\[ \int \frac {(e \cos (c+d x))^{9/2}}{(a+b \sin (c+d x))^4} \, dx=\int { \frac {\left (e \cos \left (d x + c\right )\right )^{\frac {9}{2}}}{{\left (b \sin \left (d x + c\right ) + a\right )}^{4}} \,d x } \]

[In]

integrate((e*cos(d*x+c))^(9/2)/(a+b*sin(d*x+c))^4,x, algorithm="maxima")

[Out]

integrate((e*cos(d*x + c))^(9/2)/(b*sin(d*x + c) + a)^4, x)

Giac [F]

\[ \int \frac {(e \cos (c+d x))^{9/2}}{(a+b \sin (c+d x))^4} \, dx=\int { \frac {\left (e \cos \left (d x + c\right )\right )^{\frac {9}{2}}}{{\left (b \sin \left (d x + c\right ) + a\right )}^{4}} \,d x } \]

[In]

integrate((e*cos(d*x+c))^(9/2)/(a+b*sin(d*x+c))^4,x, algorithm="giac")

[Out]

sage0*x

Mupad [F(-1)]

Timed out. \[ \int \frac {(e \cos (c+d x))^{9/2}}{(a+b \sin (c+d x))^4} \, dx=\int \frac {{\left (e\,\cos \left (c+d\,x\right )\right )}^{9/2}}{{\left (a+b\,\sin \left (c+d\,x\right )\right )}^4} \,d x \]

[In]

int((e*cos(c + d*x))^(9/2)/(a + b*sin(c + d*x))^4,x)

[Out]

int((e*cos(c + d*x))^(9/2)/(a + b*sin(c + d*x))^4, x)