\(\int \frac {1}{(e \cos (c+d x))^{7/2} (a+b \sin (c+d x))^3} \, dx\) [604]

Optimal result

Integrand size = 25, antiderivative size = 685 \[ \int \frac {1}{(e \cos (c+d x))^{7/2} (a+b \sin (c+d x))^3} \, dx=\frac {9 b^{7/2} \left (11 a^2+2 b^2\right ) \arctan \left (\frac {\sqrt {b} \sqrt {e \cos (c+d x)}}{\sqrt [4]{-a^2+b^2} \sqrt {e}}\right )}{8 \left (-a^2+b^2\right )^{17/4} d e^{7/2}}-\frac {9 b^{7/2} \left (11 a^2+2 b^2\right ) \text {arctanh}\left (\frac {\sqrt {b} \sqrt {e \cos (c+d x)}}{\sqrt [4]{-a^2+b^2} \sqrt {e}}\right )}{8 \left (-a^2+b^2\right )^{17/4} d e^{7/2}}-\frac {3 a \left (8 a^4-64 a^2 b^2-139 b^4\right ) \sqrt {e \cos (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{20 \left (a^2-b^2\right )^4 d e^4 \sqrt {\cos (c+d x)}}+\frac {9 a b^3 \left (11 a^2+2 b^2\right ) \sqrt {\cos (c+d x)} \operatorname {EllipticPi}\left (\frac {2 b}{b-\sqrt {-a^2+b^2}},\frac {1}{2} (c+d x),2\right )}{8 \left (a^2-b^2\right )^4 \left (b-\sqrt {-a^2+b^2}\right ) d e^3 \sqrt {e \cos (c+d x)}}+\frac {9 a b^3 \left (11 a^2+2 b^2\right ) \sqrt {\cos (c+d x)} \operatorname {EllipticPi}\left (\frac {2 b}{b+\sqrt {-a^2+b^2}},\frac {1}{2} (c+d x),2\right )}{8 \left (a^2-b^2\right )^4 \left (b+\sqrt {-a^2+b^2}\right ) d e^3 \sqrt {e \cos (c+d x)}}+\frac {b}{2 \left (a^2-b^2\right ) d e (e \cos (c+d x))^{5/2} (a+b \sin (c+d x))^2}+\frac {13 a b}{4 \left (a^2-b^2\right )^2 d e (e \cos (c+d x))^{5/2} (a+b \sin (c+d x))}-\frac {9 b \left (11 a^2+2 b^2\right )-a \left (8 a^2+109 b^2\right ) \sin (c+d x)}{20 \left (a^2-b^2\right )^3 d e (e \cos (c+d x))^{5/2}}+\frac {3 \left (15 b^3 \left (11 a^2+2 b^2\right )+a \left (8 a^4-64 a^2 b^2-139 b^4\right ) \sin (c+d x)\right )}{20 \left (a^2-b^2\right )^4 d e^3 \sqrt {e \cos (c+d x)}} \]

[Out]

9/8*b^(7/2)*(11*a^2+2*b^2)*arctan(b^(1/2)*(e*cos(d*x+c))^(1/2)/(-a^2+b^2)^(1/4)/e^(1/2))/(-a^2+b^2)^(17/4)/d/e
^(7/2)-9/8*b^(7/2)*(11*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))/(-a^2+b^2)^(1
7/4)/d/e^(7/2)+1/2*b/(a^2-b^2)/d/e/(e*cos(d*x+c))^(5/2)/(a+b*sin(d*x+c))^2+13/4*a*b/(a^2-b^2)^2/d/e/(e*cos(d*x
+c))^(5/2)/(a+b*sin(d*x+c))+1/20*(-9*b*(11*a^2+2*b^2)+a*(8*a^2+109*b^2)*sin(d*x+c))/(a^2-b^2)^3/d/e/(e*cos(d*x
+c))^(5/2)+3/20*(15*b^3*(11*a^2+2*b^2)+a*(8*a^4-64*a^2*b^2-139*b^4)*sin(d*x+c))/(a^2-b^2)^4/d/e^3/(e*cos(d*x+c
))^(1/2)+9/8*a*b^3*(11*a^2+2*b^2)*(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)/(a^2-b^2)^4/d/e^3/(b-(-a^2+b^2)^(1/2))/(e*cos(d*x+c))^(1/
2)+9/8*a*b^3*(11*a^2+2*b^2)*(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)/(a^2-b^2)^4/d/e^3/(b+(-a^2+b^2)^(1/2))/(e*cos(d*x+c))^(1/2)-3/2
0*a*(8*a^4-64*a^2*b^2-139*b^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)/(a^2-b^2)^4/d/e^4/cos(d*x+c)^(1/2)

Rubi [A] (verified)

Time = 1.37 (sec) , antiderivative size = 685, normalized size of antiderivative = 1.00, number of steps used = 16, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.520, Rules used = {2773, 2943, 2945, 2946, 2721, 2719, 2780, 2886, 2884, 335, 304, 211, 214} \[ \int \frac {1}{(e \cos (c+d x))^{7/2} (a+b \sin (c+d x))^3} \, dx=\frac {9 b^{7/2} \left (11 a^2+2 b^2\right ) \arctan \left (\frac {\sqrt {b} \sqrt {e \cos (c+d x)}}{\sqrt {e} \sqrt [4]{b^2-a^2}}\right )}{8 d e^{7/2} \left (b^2-a^2\right )^{17/4}}-\frac {9 b^{7/2} \left (11 a^2+2 b^2\right ) \text {arctanh}\left (\frac {\sqrt {b} \sqrt {e \cos (c+d x)}}{\sqrt {e} \sqrt [4]{b^2-a^2}}\right )}{8 d e^{7/2} \left (b^2-a^2\right )^{17/4}}+\frac {13 a b}{4 d e \left (a^2-b^2\right )^2 (e \cos (c+d x))^{5/2} (a+b \sin (c+d x))}+\frac {b}{2 d e \left (a^2-b^2\right ) (e \cos (c+d x))^{5/2} (a+b \sin (c+d x))^2}-\frac {9 b \left (11 a^2+2 b^2\right )-a \left (8 a^2+109 b^2\right ) \sin (c+d x)}{20 d e \left (a^2-b^2\right )^3 (e \cos (c+d x))^{5/2}}+\frac {9 a b^3 \left (11 a^2+2 b^2\right ) \sqrt {\cos (c+d x)} \operatorname {EllipticPi}\left (\frac {2 b}{b-\sqrt {b^2-a^2}},\frac {1}{2} (c+d x),2\right )}{8 d e^3 \left (a^2-b^2\right )^4 \left (b-\sqrt {b^2-a^2}\right ) \sqrt {e \cos (c+d x)}}+\frac {9 a b^3 \left (11 a^2+2 b^2\right ) \sqrt {\cos (c+d x)} \operatorname {EllipticPi}\left (\frac {2 b}{b+\sqrt {b^2-a^2}},\frac {1}{2} (c+d x),2\right )}{8 d e^3 \left (a^2-b^2\right )^4 \left (\sqrt {b^2-a^2}+b\right ) \sqrt {e \cos (c+d x)}}-\frac {3 a \left (8 a^4-64 a^2 b^2-139 b^4\right ) E\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \sqrt {e \cos (c+d x)}}{20 d e^4 \left (a^2-b^2\right )^4 \sqrt {\cos (c+d x)}}+\frac {3 \left (15 b^3 \left (11 a^2+2 b^2\right )+a \left (8 a^4-64 a^2 b^2-139 b^4\right ) \sin (c+d x)\right )}{20 d e^3 \left (a^2-b^2\right )^4 \sqrt {e \cos (c+d x)}} \]

[In]

Int[1/((e*Cos[c + d*x])^(7/2)*(a + b*Sin[c + d*x])^3),x]

[Out]

(9*b^(7/2)*(11*a^2 + 2*b^2)*ArcTan[(Sqrt[b]*Sqrt[e*Cos[c + d*x]])/((-a^2 + b^2)^(1/4)*Sqrt[e])])/(8*(-a^2 + b^
2)^(17/4)*d*e^(7/2)) - (9*b^(7/2)*(11*a^2 + 2*b^2)*ArcTanh[(Sqrt[b]*Sqrt[e*Cos[c + d*x]])/((-a^2 + b^2)^(1/4)*
Sqrt[e])])/(8*(-a^2 + b^2)^(17/4)*d*e^(7/2)) - (3*a*(8*a^4 - 64*a^2*b^2 - 139*b^4)*Sqrt[e*Cos[c + d*x]]*Ellipt
icE[(c + d*x)/2, 2])/(20*(a^2 - b^2)^4*d*e^4*Sqrt[Cos[c + d*x]]) + (9*a*b^3*(11*a^2 + 2*b^2)*Sqrt[Cos[c + d*x]
]*EllipticPi[(2*b)/(b - Sqrt[-a^2 + b^2]), (c + d*x)/2, 2])/(8*(a^2 - b^2)^4*(b - Sqrt[-a^2 + b^2])*d*e^3*Sqrt
[e*Cos[c + d*x]]) + (9*a*b^3*(11*a^2 + 2*b^2)*Sqrt[Cos[c + d*x]]*EllipticPi[(2*b)/(b + Sqrt[-a^2 + b^2]), (c +
d*x)/2, 2])/(8*(a^2 - b^2)^4*(b + Sqrt[-a^2 + b^2])*d*e^3*Sqrt[e*Cos[c + d*x]]) + b/(2*(a^2 - b^2)*d*e*(e*Cos
[c + d*x])^(5/2)*(a + b*Sin[c + d*x])^2) + (13*a*b)/(4*(a^2 - b^2)^2*d*e*(e*Cos[c + d*x])^(5/2)*(a + b*Sin[c +
d*x])) - (9*b*(11*a^2 + 2*b^2) - a*(8*a^2 + 109*b^2)*Sin[c + d*x])/(20*(a^2 - b^2)^3*d*e*(e*Cos[c + d*x])^(5/
2)) + (3*(15*b^3*(11*a^2 + 2*b^2) + a*(8*a^4 - 64*a^2*b^2 - 139*b^4)*Sin[c + d*x]))/(20*(a^2 - b^2)^4*d*e^3*Sq
rt[e*Cos[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 2945

Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*((c_.) + (d_.)*sin[(e_.)
+ (f_.)*(x_)]), x_Symbol] :> Simp[(g*Cos[e + f*x])^(p + 1)*(a + b*Sin[e + f*x])^(m + 1)*((b*c - a*d - (a*c -
b*d)*Sin[e + f*x])/(f*g*(a^2 - b^2)*(p + 1))), x] + Dist[1/(g^2*(a^2 - b^2)*(p + 1)), Int[(g*Cos[e + f*x])^(p
+ 2)*(a + b*Sin[e + f*x])^m*Simp[c*(a^2*(p + 2) - b^2*(m + p + 2)) + a*b*d*m + b*(a*c - b*d)*(m + p + 3)*Sin[e
+ f*x], x], x], x] /; FreeQ[{a, b, c, d, e, f, g, m}, x] && NeQ[a^2 - b^2, 0] && LtQ[p, -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}{2 \left (a^2-b^2\right ) d e (e \cos (c+d x))^{5/2} (a+b \sin (c+d x))^2}-\frac {\int \frac {-2 a+\frac {9}{2} b \sin (c+d x)}{(e \cos (c+d x))^{7/2} (a+b \sin (c+d x))^2} \, dx}{2 \left (a^2-b^2\right )} \\ & = \frac {b}{2 \left (a^2-b^2\right ) d e (e \cos (c+d x))^{5/2} (a+b \sin (c+d x))^2}+\frac {13 a b}{4 \left (a^2-b^2\right )^2 d e (e \cos (c+d x))^{5/2} (a+b \sin (c+d x))}+\frac {\int \frac {\frac {1}{2} \left (4 a^2+9 b^2\right )-\frac {91}{4} a b \sin (c+d x)}{(e \cos (c+d x))^{7/2} (a+b \sin (c+d x))} \, dx}{2 \left (a^2-b^2\right )^2} \\ & = \frac {b}{2 \left (a^2-b^2\right ) d e (e \cos (c+d x))^{5/2} (a+b \sin (c+d x))^2}+\frac {13 a b}{4 \left (a^2-b^2\right )^2 d e (e \cos (c+d x))^{5/2} (a+b \sin (c+d x))}-\frac {9 b \left (11 a^2+2 b^2\right )-a \left (8 a^2+109 b^2\right ) \sin (c+d x)}{20 \left (a^2-b^2\right )^3 d e (e \cos (c+d x))^{5/2}}-\frac {\int \frac {-\frac {3}{4} \left (4 a^4-28 a^2 b^2-15 b^4\right )-\frac {3}{8} a b \left (8 a^2+109 b^2\right ) \sin (c+d x)}{(e \cos (c+d x))^{3/2} (a+b \sin (c+d x))} \, dx}{5 \left (a^2-b^2\right )^3 e^2} \\ & = \frac {b}{2 \left (a^2-b^2\right ) d e (e \cos (c+d x))^{5/2} (a+b \sin (c+d x))^2}+\frac {13 a b}{4 \left (a^2-b^2\right )^2 d e (e \cos (c+d x))^{5/2} (a+b \sin (c+d x))}-\frac {9 b \left (11 a^2+2 b^2\right )-a \left (8 a^2+109 b^2\right ) \sin (c+d x)}{20 \left (a^2-b^2\right )^3 d e (e \cos (c+d x))^{5/2}}+\frac {3 \left (15 b^3 \left (11 a^2+2 b^2\right )+a \left (8 a^4-64 a^2 b^2-139 b^4\right ) \sin (c+d x)\right )}{20 \left (a^2-b^2\right )^4 d e^3 \sqrt {e \cos (c+d x)}}+\frac {2 \int \frac {\sqrt {e \cos (c+d x)} \left (-\frac {3}{8} \left (4 a^6-32 a^4 b^2-152 a^2 b^4-15 b^6\right )-\frac {3}{16} a b \left (8 a^4-64 a^2 b^2-139 b^4\right ) \sin (c+d x)\right )}{a+b \sin (c+d x)} \, dx}{5 \left (a^2-b^2\right )^4 e^4} \\ & = \frac {b}{2 \left (a^2-b^2\right ) d e (e \cos (c+d x))^{5/2} (a+b \sin (c+d x))^2}+\frac {13 a b}{4 \left (a^2-b^2\right )^2 d e (e \cos (c+d x))^{5/2} (a+b \sin (c+d x))}-\frac {9 b \left (11 a^2+2 b^2\right )-a \left (8 a^2+109 b^2\right ) \sin (c+d x)}{20 \left (a^2-b^2\right )^3 d e (e \cos (c+d x))^{5/2}}+\frac {3 \left (15 b^3 \left (11 a^2+2 b^2\right )+a \left (8 a^4-64 a^2 b^2-139 b^4\right ) \sin (c+d x)\right )}{20 \left (a^2-b^2\right )^4 d e^3 \sqrt {e \cos (c+d x)}}+\frac {\left (9 b^4 \left (11 a^2+2 b^2\right )\right ) \int \frac {\sqrt {e \cos (c+d x)}}{a+b \sin (c+d x)} \, dx}{8 \left (a^2-b^2\right )^4 e^4}-\frac {\left (3 a \left (8 a^4-64 a^2 b^2-139 b^4\right )\right ) \int \sqrt {e \cos (c+d x)} \, dx}{40 \left (a^2-b^2\right )^4 e^4} \\ & = \frac {b}{2 \left (a^2-b^2\right ) d e (e \cos (c+d x))^{5/2} (a+b \sin (c+d x))^2}+\frac {13 a b}{4 \left (a^2-b^2\right )^2 d e (e \cos (c+d x))^{5/2} (a+b \sin (c+d x))}-\frac {9 b \left (11 a^2+2 b^2\right )-a \left (8 a^2+109 b^2\right ) \sin (c+d x)}{20 \left (a^2-b^2\right )^3 d e (e \cos (c+d x))^{5/2}}+\frac {3 \left (15 b^3 \left (11 a^2+2 b^2\right )+a \left (8 a^4-64 a^2 b^2-139 b^4\right ) \sin (c+d x)\right )}{20 \left (a^2-b^2\right )^4 d e^3 \sqrt {e \cos (c+d x)}}-\frac {\left (9 a b^3 \left (11 a^2+2 b^2\right )\right ) \int \frac {1}{\sqrt {e \cos (c+d x)} \left (\sqrt {-a^2+b^2}-b \cos (c+d x)\right )} \, dx}{16 \left (a^2-b^2\right )^4 e^3}+\frac {\left (9 a b^3 \left (11 a^2+2 b^2\right )\right ) \int \frac {1}{\sqrt {e \cos (c+d x)} \left (\sqrt {-a^2+b^2}+b \cos (c+d x)\right )} \, dx}{16 \left (a^2-b^2\right )^4 e^3}+\frac {\left (9 b^5 \left (11 a^2+2 b^2\right )\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 )}{8 \left (a^2-b^2\right )^4 d e^3}-\frac {\left (3 a \left (8 a^4-64 a^2 b^2-139 b^4\right ) \sqrt {e \cos (c+d x)}\right ) \int \sqrt {\cos (c+d x)} \, dx}{40 \left (a^2-b^2\right )^4 e^4 \sqrt {\cos (c+d x)}} \\ & = -\frac {3 a \left (8 a^4-64 a^2 b^2-139 b^4\right ) \sqrt {e \cos (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{20 \left (a^2-b^2\right )^4 d e^4 \sqrt {\cos (c+d x)}}+\frac {b}{2 \left (a^2-b^2\right ) d e (e \cos (c+d x))^{5/2} (a+b \sin (c+d x))^2}+\frac {13 a b}{4 \left (a^2-b^2\right )^2 d e (e \cos (c+d x))^{5/2} (a+b \sin (c+d x))}-\frac {9 b \left (11 a^2+2 b^2\right )-a \left (8 a^2+109 b^2\right ) \sin (c+d x)}{20 \left (a^2-b^2\right )^3 d e (e \cos (c+d x))^{5/2}}+\frac {3 \left (15 b^3 \left (11 a^2+2 b^2\right )+a \left (8 a^4-64 a^2 b^2-139 b^4\right ) \sin (c+d x)\right )}{20 \left (a^2-b^2\right )^4 d e^3 \sqrt {e \cos (c+d x)}}+\frac {\left (9 b^5 \left (11 a^2+2 b^2\right )\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 )}{4 \left (a^2-b^2\right )^4 d e^3}-\frac {\left (9 a b^3 \left (11 a^2+2 b^2\right ) \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}{16 \left (a^2-b^2\right )^4 e^3 \sqrt {e \cos (c+d x)}}+\frac {\left (9 a b^3 \left (11 a^2+2 b^2\right ) \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}{16 \left (a^2-b^2\right )^4 e^3 \sqrt {e \cos (c+d x)}} \\ & = -\frac {3 a \left (8 a^4-64 a^2 b^2-139 b^4\right ) \sqrt {e \cos (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{20 \left (a^2-b^2\right )^4 d e^4 \sqrt {\cos (c+d x)}}+\frac {9 a b^3 \left (11 a^2+2 b^2\right ) \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 )}{8 \left (a^2-b^2\right )^4 \left (b-\sqrt {-a^2+b^2}\right ) d e^3 \sqrt {e \cos (c+d x)}}+\frac {9 a b^3 \left (11 a^2+2 b^2\right ) \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 )}{8 \left (a^2-b^2\right )^4 \left (b+\sqrt {-a^2+b^2}\right ) d e^3 \sqrt {e \cos (c+d x)}}+\frac {b}{2 \left (a^2-b^2\right ) d e (e \cos (c+d x))^{5/2} (a+b \sin (c+d x))^2}+\frac {13 a b}{4 \left (a^2-b^2\right )^2 d e (e \cos (c+d x))^{5/2} (a+b \sin (c+d x))}-\frac {9 b \left (11 a^2+2 b^2\right )-a \left (8 a^2+109 b^2\right ) \sin (c+d x)}{20 \left (a^2-b^2\right )^3 d e (e \cos (c+d x))^{5/2}}+\frac {3 \left (15 b^3 \left (11 a^2+2 b^2\right )+a \left (8 a^4-64 a^2 b^2-139 b^4\right ) \sin (c+d x)\right )}{20 \left (a^2-b^2\right )^4 d e^3 \sqrt {e \cos (c+d x)}}-\frac {\left (9 b^4 \left (11 a^2+2 b^2\right )\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 )}{8 \left (a^2-b^2\right )^4 d e^3}+\frac {\left (9 b^4 \left (11 a^2+2 b^2\right )\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 )}{8 \left (a^2-b^2\right )^4 d e^3} \\ & = \frac {9 b^{7/2} \left (11 a^2+2 b^2\right ) \arctan \left (\frac {\sqrt {b} \sqrt {e \cos (c+d x)}}{\sqrt [4]{-a^2+b^2} \sqrt {e}}\right )}{8 \left (-a^2+b^2\right )^{17/4} d e^{7/2}}-\frac {9 b^{7/2} \left (11 a^2+2 b^2\right ) \text {arctanh}\left (\frac {\sqrt {b} \sqrt {e \cos (c+d x)}}{\sqrt [4]{-a^2+b^2} \sqrt {e}}\right )}{8 \left (-a^2+b^2\right )^{17/4} d e^{7/2}}-\frac {3 a \left (8 a^4-64 a^2 b^2-139 b^4\right ) \sqrt {e \cos (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{20 \left (a^2-b^2\right )^4 d e^4 \sqrt {\cos (c+d x)}}+\frac {9 a b^3 \left (11 a^2+2 b^2\right ) \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 )}{8 \left (a^2-b^2\right )^4 \left (b-\sqrt {-a^2+b^2}\right ) d e^3 \sqrt {e \cos (c+d x)}}+\frac {9 a b^3 \left (11 a^2+2 b^2\right ) \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 )}{8 \left (a^2-b^2\right )^4 \left (b+\sqrt {-a^2+b^2}\right ) d e^3 \sqrt {e \cos (c+d x)}}+\frac {b}{2 \left (a^2-b^2\right ) d e (e \cos (c+d x))^{5/2} (a+b \sin (c+d x))^2}+\frac {13 a b}{4 \left (a^2-b^2\right )^2 d e (e \cos (c+d x))^{5/2} (a+b \sin (c+d x))}-\frac {9 b \left (11 a^2+2 b^2\right )-a \left (8 a^2+109 b^2\right ) \sin (c+d x)}{20 \left (a^2-b^2\right )^3 d e (e \cos (c+d x))^{5/2}}+\frac {3 \left (15 b^3 \left (11 a^2+2 b^2\right )+a \left (8 a^4-64 a^2 b^2-139 b^4\right ) \sin (c+d x)\right )}{20 \left (a^2-b^2\right )^4 d e^3 \sqrt {e \cos (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 = 6.74 (sec) , antiderivative size = 1014, normalized size of antiderivative = 1.48 \[ \int \frac {1}{(e \cos (c+d x))^{7/2} (a+b \sin (c+d x))^3} \, dx=-\frac {3 \cos ^{\frac {7}{2}}(c+d x) \left (-\frac {2 \left (8 a^6-64 a^4 b^2-304 a^2 b^4-30 b^6\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 (8 a^5 b-64 a^3 b^3-139 a b^5\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 )}{40 (a-b)^4 (a+b)^4 d (e \cos (c+d x))^{7/2}}+\frac {\cos ^4(c+d x) \left (\frac {b^5 \cos (c+d x)}{2 \left (a^2-b^2\right )^3 (a+b \sin (c+d x))^2}+\frac {21 a b^5 \cos (c+d x)}{4 \left (a^2-b^2\right )^4 (a+b \sin (c+d x))}+\frac {2 \sec ^3(c+d x) \left (-3 a^2 b-b^3+a^3 \sin (c+d x)+3 a b^2 \sin (c+d x)\right )}{5 \left (a^2-b^2\right )^3}+\frac {2 \sec (c+d x) \left (50 a^2 b^3+10 b^5+3 a^5 \sin (c+d x)-24 a^3 b^2 \sin (c+d x)-39 a b^4 \sin (c+d x)\right )}{5 \left (a^2-b^2\right )^4}\right )}{d (e \cos (c+d x))^{7/2}} \]

[In]

Integrate[1/((e*Cos[c + d*x])^(7/2)*(a + b*Sin[c + d*x])^3),x]

[Out]

(-3*Cos[c + d*x]^(7/2)*((-2*(8*a^6 - 64*a^4*b^2 - 304*a^2*b^4 - 30*b^6)*(a + b*Sqrt[1 - Cos[c + d*x]^2])*((a*A
ppellF1[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*Si
n[c + d*x])) - ((8*a^5*b - 64*a^3*b^3 - 139*a*b^5)*(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]*Sqr
t[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]))))/(40*(a - b)^4*(a
+ b)^4*d*(e*Cos[c + d*x])^(7/2)) + (Cos[c + d*x]^4*((b^5*Cos[c + d*x])/(2*(a^2 - b^2)^3*(a + b*Sin[c + d*x])^
2) + (21*a*b^5*Cos[c + d*x])/(4*(a^2 - b^2)^4*(a + b*Sin[c + d*x])) + (2*Sec[c + d*x]^3*(-3*a^2*b - b^3 + a^3*
Sin[c + d*x] + 3*a*b^2*Sin[c + d*x]))/(5*(a^2 - b^2)^3) + (2*Sec[c + d*x]*(50*a^2*b^3 + 10*b^5 + 3*a^5*Sin[c +
d*x] - 24*a^3*b^2*Sin[c + d*x] - 39*a*b^4*Sin[c + d*x]))/(5*(a^2 - b^2)^4)))/(d*(e*Cos[c + d*x])^(7/2))

Maple [C] (warning: unable to verify)

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

Time = 25.10 (sec) , antiderivative size = 4390, normalized size of antiderivative = 6.41

method	result	size
default	\(\text {Expression too large to display}\)	\(4390\)

[In]

int(1/(e*cos(d*x+c))^(7/2)/(a+b*sin(d*x+c))^3,x,method=_RETURNVERBOSE)

[Out]

(-4/e^3*b*(-1/4*b^2*(5*a^2+b^2)/(a^2-b^2)^4/e*2^(1/2)/(cos(1/2*d*x+1/2*c)-1/2*2^(1/2))*(-2*sin(1/2*d*x+1/2*c)^
2*e+e)^(1/2)+1/4*b^2*(5*a^2+b^2)/(a^2-b^2)^4/e*2^(1/2)/(cos(1/2*d*x+1/2*c)+1/2*2^(1/2))*(-2*sin(1/2*d*x+1/2*c)
^2*e+e)^(1/2)-1/240*(-3*a^2-b^2)/(a^2-b^2)^3/e/(4*sin(1/2*d*x+1/2*c)^2*cos(1/2*d*x+1/2*c)+6*2^(1/2)*sin(1/2*d*
x+1/2*c)^2-10*cos(1/2*d*x+1/2*c)-7*2^(1/2))*(-2*sin(1/2*d*x+1/2*c)^2*e+e)^(1/2)*(-4*2^(1/2)*sin(1/2*d*x+1/2*c)
^2+12*cos(1/2*d*x+1/2*c)+11*2^(1/2))-1/240*(-3*a^2-b^2)/(a^2-b^2)^3/e/(4*sin(1/2*d*x+1/2*c)^2*cos(1/2*d*x+1/2*
c)-6*2^(1/2)*sin(1/2*d*x+1/2*c)^2-10*cos(1/2*d*x+1/2*c)+7*2^(1/2))*(-2*sin(1/2*d*x+1/2*c)^2*e+e)^(1/2)*(4*2^(1
/2)*sin(1/2*d*x+1/2*c)^2+12*cos(1/2*d*x+1/2*c)-11*2^(1/2))-1/96*2^(1/2)*(3*a^2+b^2)/(a^2-b^2)^3*(-2*sin(1/2*d*
x+1/2*c)^2*e+e)^(1/2)*(-2^(1/2)+cos(1/2*d*x+1/2*c))/e/(2*cos(1/2*d*x+1/2*c)*2^(1/2)+2*sin(1/2*d*x+1/2*c)^2-3)-
1/96*2^(1/2)*(3*a^2+b^2)/(a^2-b^2)^3*(-2*sin(1/2*d*x+1/2*c)^2*e+e)^(1/2)*(2^(1/2)+cos(1/2*d*x+1/2*c))/e/(2*cos
(1/2*d*x+1/2*c)*2^(1/2)-2*sin(1/2*d*x+1/2*c)^2+3)-5/128*b^2*a^2/(a+b)^4/(a-b)^4*(144/5*(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+(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)))*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)/(e^2*(a^2-b^2)/b^2)^(1/4)/e/(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)^2-7/64*
b^2/(a+b)^4/(a-b)^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*ar
ctan((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*
(a^2-b^2)/b^2)^(1/4)*(a^2+1/7*b^2)/e/(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)-1/8*b^2/(a-b)
^4/(a+b)^4*(5*a^2+b^2)/(e^2*(a^2-b^2)/b^2)^(1/4)*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))))+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^3*a*(1/5*(-a^2-3*b^2)/(a^2-b^2)^3/e/sin(1/2*d*x+1/2*c)^2/(8*sin(1/2*d*x+
1/2*c)^6-12*sin(1/2*d*x+1/2*c)^4+6*sin(1/2*d*x+1/2*c)^2-1)*(24*sin(1/2*d*x+1/2*c)^6*cos(1/2*d*x+1/2*c)-12*(2*s
in(1/2*d*x+1/2*c)^2-1)^(1/2)*EllipticE(cos(1/2*d*x+1/2*c),2^(1/2))*(sin(1/2*d*x+1/2*c)^2)^(1/2)*sin(1/2*d*x+1/
2*c)^4-24*sin(1/2*d*x+1/2*c)^4*cos(1/2*d*x+1/2*c)+12*(2*sin(1/2*d*x+1/2*c)^2-1)^(1/2)*EllipticE(cos(1/2*d*x+1/
2*c),2^(1/2))*(sin(1/2*d*x+1/2*c)^2)^(1/2)*sin(1/2*d*x+1/2*c)^2+8*sin(1/2*d*x+1/2*c)^2*cos(1/2*d*x+1/2*c)-3*(s
in(1/2*d*x+1/2*c)^2)^(1/2)*(2*sin(1/2*d*x+1/2*c)^2-1)^(1/2)*EllipticE(cos(1/2*d*x+1/2*c),2^(1/2)))*(-2*sin(1/2
*d*x+1/2*c)^4*e+sin(1/2*d*x+1/2*c)^2*e)^(1/2)+6*b^2*(a^2+b^2)/(a^2-b^2)^4/sin(1/2*d*x+1/2*c)^2/e/(2*sin(1/2*d*
x+1/2*c)^2-1)*(-2*sin(1/2*d*x+1/2*c)^4*e+sin(1/2*d*x+1/2*c)^2*e)^(1/2)*(2*sin(1/2*d*x+1/2*c)^2*cos(1/2*d*x+1/2
*c)-(sin(1/2*d*x+1/2*c)^2)^(1/2)*(2*sin(1/2*d*x+1/2*c)^2-1)^(1/2)*EllipticE(cos(1/2*d*x+1/2*c),2^(1/2)))-4*b^4
*a^2/(a+b)^2/(a-b)^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*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-(-1/8*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*_a
lpha^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)))+3/8*b^2/(a-b)^4/(a+b)^4*sum
((-a^2-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*_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))-b^4/(a+b)^3/(a-b)^3*(-(-b^2*(5*a^2+3*b^2)/(a^2-b^2)/a^2/e*cos(1/2*d*x+1/2*c)^3+1/2*b^2*(5*a^2+3*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*(5*a^2+3*b^2)/a^2/(a^2-b^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*(5*a^2+3*b^2)/a^2/(a^2-b^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((-15*a^4+a^2*b^2+6*b^4)/(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=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(-1)]

Timed out. \[ \int \frac {1}{(e \cos (c+d x))^{7/2} (a+b \sin (c+d x))^3} \, dx=\text {Timed out} \]

[In]

integrate(1/(e*cos(d*x+c))^(7/2)/(a+b*sin(d*x+c))^3,x, algorithm="fricas")

[Out]

Timed out

Sympy [F(-1)]

Timed out. \[ \int \frac {1}{(e \cos (c+d x))^{7/2} (a+b \sin (c+d x))^3} \, dx=\text {Timed out} \]

[In]

integrate(1/(e*cos(d*x+c))**(7/2)/(a+b*sin(d*x+c))**3,x)

[Out]

Timed out

Maxima [F(-1)]

Timed out. \[ \int \frac {1}{(e \cos (c+d x))^{7/2} (a+b \sin (c+d x))^3} \, dx=\text {Timed out} \]

[In]

integrate(1/(e*cos(d*x+c))^(7/2)/(a+b*sin(d*x+c))^3,x, algorithm="maxima")

[Out]

Timed out

Giac [F]

\[ \int \frac {1}{(e \cos (c+d x))^{7/2} (a+b \sin (c+d x))^3} \, dx=\int { \frac {1}{\left (e \cos \left (d x + c\right )\right )^{\frac {7}{2}} {\left (b \sin \left (d x + c\right ) + a\right )}^{3}} \,d x } \]

[In]

integrate(1/(e*cos(d*x+c))^(7/2)/(a+b*sin(d*x+c))^3,x, algorithm="giac")

[Out]

integrate(1/((e*cos(d*x + c))^(7/2)*(b*sin(d*x + c) + a)^3), x)

Mupad [F(-1)]

Timed out. \[ \int \frac {1}{(e \cos (c+d x))^{7/2} (a+b \sin (c+d x))^3} \, dx=\int \frac {1}{{\left (e\,\cos \left (c+d\,x\right )\right )}^{7/2}\,{\left (a+b\,\sin \left (c+d\,x\right )\right )}^3} \,d x \]

[In]

int(1/((e*cos(c + d*x))^(7/2)*(a + b*sin(c + d*x))^3),x)

[Out]

int(1/((e*cos(c + d*x))^(7/2)*(a + b*sin(c + d*x))^3), x)