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

Optimal. Leaf size=685 \[ \frac {13 a b}{4 d e \left (a^2-b^2\right )^2 (e \cos (c+d x))^{5/2} (a+b \sin (c+d x))}+\frac {b}{2 d e \left (a^2-b^2\right ) (e \cos (c+d x))^{5/2} (a+b \sin (c+d x))^2}-\frac {9 b \left (11 a^2+2 b^2\right )-a \left (8 a^2+109 b^2\right ) \sin (c+d x)}{20 d e \left (a^2-b^2\right )^3 (e \cos (c+d x))^{5/2}}+\frac {9 b^{7/2} \left (11 a^2+2 b^2\right ) \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {e \cos (c+d x)}}{\sqrt {e} \sqrt [4]{b^2-a^2}}\right )}{8 d e^{7/2} \left (b^2-a^2\right )^{17/4}}-\frac {9 b^{7/2} \left (11 a^2+2 b^2\right ) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {e \cos (c+d x)}}{\sqrt {e} \sqrt [4]{b^2-a^2}}\right )}{8 d e^{7/2} \left (b^2-a^2\right )^{17/4}}+\frac {9 a b^3 \left (11 a^2+2 b^2\right ) \sqrt {\cos (c+d x)} \Pi \left (\frac {2 b}{b-\sqrt {b^2-a^2}};\left .\frac {1}{2} (c+d x)\right |2\right )}{8 d e^3 \left (a^2-b^2\right )^4 \left (b-\sqrt {b^2-a^2}\right ) \sqrt {e \cos (c+d x)}}+\frac {9 a b^3 \left (11 a^2+2 b^2\right ) \sqrt {\cos (c+d x)} \Pi \left (\frac {2 b}{b+\sqrt {b^2-a^2}};\left .\frac {1}{2} (c+d x)\right |2\right )}{8 d e^3 \left (a^2-b^2\right )^4 \left (\sqrt {b^2-a^2}+b\right ) \sqrt {e \cos (c+d x)}}-\frac {3 a \left (8 a^4-64 a^2 b^2-139 b^4\right ) E\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \sqrt {e \cos (c+d x)}}{20 d e^4 \left (a^2-b^2\right )^4 \sqrt {\cos (c+d x)}}+\frac {3 \left (15 b^3 \left (11 a^2+2 b^2\right )+a \left (8 a^4-64 a^2 b^2-139 b^4\right ) \sin (c+d x)\right )}{20 d e^3 \left (a^2-b^2\right )^4 \sqrt {e \cos (c+d x)}} \]

[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]  time = 2.03, antiderivative size = 685, normalized size of antiderivative = 1.00, number of steps used = 16, number of rules used = 13, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.520, Rules used = {2694, 2864, 2866, 2867, 2640, 2639, 2701, 2807, 2805, 329, 298, 205, 208} \[ \frac {3 \left (a \left (-64 a^2 b^2+8 a^4-139 b^4\right ) \sin (c+d x)+15 b^3 \left (11 a^2+2 b^2\right )\right )}{20 d e^3 \left (a^2-b^2\right )^4 \sqrt {e \cos (c+d x)}}+\frac {9 b^{7/2} \left (11 a^2+2 b^2\right ) \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {e \cos (c+d x)}}{\sqrt {e} \sqrt [4]{b^2-a^2}}\right )}{8 d e^{7/2} \left (b^2-a^2\right )^{17/4}}-\frac {9 b^{7/2} \left (11 a^2+2 b^2\right ) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {e \cos (c+d x)}}{\sqrt {e} \sqrt [4]{b^2-a^2}}\right )}{8 d e^{7/2} \left (b^2-a^2\right )^{17/4}}-\frac {3 a \left (-64 a^2 b^2+8 a^4-139 b^4\right ) E\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \sqrt {e \cos (c+d x)}}{20 d e^4 \left (a^2-b^2\right )^4 \sqrt {\cos (c+d x)}}+\frac {9 a b^3 \left (11 a^2+2 b^2\right ) \sqrt {\cos (c+d x)} \Pi \left (\frac {2 b}{b-\sqrt {b^2-a^2}};\left .\frac {1}{2} (c+d x)\right |2\right )}{8 d e^3 \left (a^2-b^2\right )^4 \left (b-\sqrt {b^2-a^2}\right ) \sqrt {e \cos (c+d x)}}+\frac {9 a b^3 \left (11 a^2+2 b^2\right ) \sqrt {\cos (c+d x)} \Pi \left (\frac {2 b}{b+\sqrt {b^2-a^2}};\left .\frac {1}{2} (c+d x)\right |2\right )}{8 d e^3 \left (a^2-b^2\right )^4 \left (\sqrt {b^2-a^2}+b\right ) \sqrt {e \cos (c+d x)}}+\frac {13 a b}{4 d e \left (a^2-b^2\right )^2 (e \cos (c+d x))^{5/2} (a+b \sin (c+d x))}+\frac {b}{2 d e \left (a^2-b^2\right ) (e \cos (c+d x))^{5/2} (a+b \sin (c+d x))^2}-\frac {9 b \left (11 a^2+2 b^2\right )-a \left (8 a^2+109 b^2\right ) \sin (c+d x)}{20 d e \left (a^2-b^2\right )^3 (e \cos (c+d x))^{5/2}} \]

Antiderivative was successfully verified.

[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 205

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

Rule 208

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

Rule 298

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 329

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 2639

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

Rule 2640

Int[Sqrt[(b_)*sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Dist[Sqrt[b*Sin[c + d*x]]/Sqrt[Sin[c + d*x]], Int[Sqrt[Si
n[c + d*x]], x], x] /; FreeQ[{b, c, d}, x]

Rule 2694

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] /; F
reeQ[{a, b, e, f, g, p}, x] && NeQ[a^2 - b^2, 0] && LtQ[m, -1] && IntegersQ[2*m, 2*p]

Rule 2701

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 2805

Int[1/(((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])*Sqrt[(c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)]]), x_Symbol] :> Simp
[(2*EllipticPi[(2*b)/(a + b), (1*(e - Pi/2 + f*x))/2, (2*d)/(c + d)])/(f*(a + b)*Sqrt[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 2807

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*
Sin[e + f*x])/(c + d)]), 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 2864

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)*Sim
p[(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 2866

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 2867

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*} \int \frac {1}{(e \cos (c+d x))^{7/2} (a+b \sin (c+d x))^3} \, dx &=\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 ) \operatorname {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 ) \operatorname {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)} \Pi \left (\frac {2 b}{b-\sqrt {-a^2+b^2}};\left .\frac {1}{2} (c+d x)\right |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)} \Pi \left (\frac {2 b}{b+\sqrt {-a^2+b^2}};\left .\frac {1}{2} (c+d x)\right |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 ) \operatorname {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 ) \operatorname {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 ) \tan ^{-1}\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 ) \tanh ^{-1}\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)} \Pi \left (\frac {2 b}{b-\sqrt {-a^2+b^2}};\left .\frac {1}{2} (c+d x)\right |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)} \Pi \left (\frac {2 b}{b+\sqrt {-a^2+b^2}};\left .\frac {1}{2} (c+d x)\right |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*}

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Mathematica [C]  time = 6.90, size = 1014, normalized size = 1.48 \[ \frac {\cos ^4(c+d x) \left (\frac {21 a \cos (c+d x) b^5}{4 \left (a^2-b^2\right )^4 (a+b \sin (c+d x))}+\frac {\cos (c+d x) b^5}{2 \left (a^2-b^2\right )^3 (a+b \sin (c+d x))^2}+\frac {2 \sec ^3(c+d x) \left (\sin (c+d x) a^3-3 b a^2+3 b^2 \sin (c+d x) a-b^3\right )}{5 \left (a^2-b^2\right )^3}+\frac {2 \sec (c+d x) \left (3 \sin (c+d x) a^5-24 b^2 \sin (c+d x) a^3+50 b^3 a^2-39 b^4 \sin (c+d x) a+10 b^5\right )}{5 \left (a^2-b^2\right )^4}\right )}{d (e \cos (c+d x))^{7/2}}-\frac {3 \cos ^{\frac {7}{2}}(c+d x) \left (-\frac {\left (8 b a^5-64 b^3 a^3-139 b^5 a\right ) \left (a+b \sqrt {1-\cos ^2(c+d x)}\right ) \left (8 F_1\left (\frac {3}{4};-\frac {1}{2},1;\frac {7}{4};\cos ^2(c+d x),\frac {b^2 \cos ^2(c+d x)}{b^2-a^2}\right ) \cos ^{\frac {3}{2}}(c+d x) b^{5/2}+3 \sqrt {2} a \left (a^2-b^2\right )^{3/4} \left (2 \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt {b} \sqrt {\cos (c+d x)}}{\sqrt [4]{a^2-b^2}}\right )-2 \tan ^{-1}\left (\frac {\sqrt {2} \sqrt {b} \sqrt {\cos (c+d x)}}{\sqrt [4]{a^2-b^2}}+1\right )-\log \left (b \cos (c+d x)-\sqrt {2} \sqrt {b} \sqrt [4]{a^2-b^2} \sqrt {\cos (c+d x)}+\sqrt {a^2-b^2}\right )+\log \left (b \cos (c+d x)+\sqrt {2} \sqrt {b} \sqrt [4]{a^2-b^2} \sqrt {\cos (c+d x)}+\sqrt {a^2-b^2}\right )\right )\right ) \sin ^2(c+d x)}{12 b^{3/2} \left (b^2-a^2\right ) \left (1-\cos ^2(c+d x)\right ) (a+b \sin (c+d x))}-\frac {2 \left (8 a^6-64 b^2 a^4-304 b^4 a^2-30 b^6\right ) \left (a+b \sqrt {1-\cos ^2(c+d x)}\right ) \left (\frac {a F_1\left (\frac {3}{4};\frac {1}{2},1;\frac {7}{4};\cos ^2(c+d x),\frac {b^2 \cos ^2(c+d x)}{b^2-a^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 \tan ^{-1}\left (1-\frac {(1+i) \sqrt {b} \sqrt {\cos (c+d x)}}{\sqrt [4]{b^2-a^2}}\right )-2 \tan ^{-1}\left (\frac {(1+i) \sqrt {b} \sqrt {\cos (c+d x)}}{\sqrt [4]{b^2-a^2}}+1\right )-\log \left (i b \cos (c+d x)-(1+i) \sqrt {b} \sqrt [4]{b^2-a^2} \sqrt {\cos (c+d x)}+\sqrt {b^2-a^2}\right )+\log \left (i b \cos (c+d x)+(1+i) \sqrt {b} \sqrt [4]{b^2-a^2} \sqrt {\cos (c+d x)}+\sqrt {b^2-a^2}\right )\right )}{\sqrt {b} \sqrt [4]{b^2-a^2}}\right ) \sin (c+d x)}{\sqrt {1-\cos ^2(c+d x)} (a+b \sin (c+d x))}\right )}{40 (a-b)^4 (a+b)^4 d (e \cos (c+d x))^{7/2}} \]

Warning: Unable to verify antiderivative.

[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))

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\left (e \cos \left (d x + c\right )\right )^{\frac {7}{2}} {\left (b \sin \left (d x + c\right ) + a\right )}^{3}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[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)

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maple [C]  time = 111.40, size = 49016, normalized size = 71.56 \[ \text {output too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

result too large to display

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {1}{{\left (e\,\cos \left (c+d\,x\right )\right )}^{7/2}\,{\left (a+b\,\sin \left (c+d\,x\right )\right )}^3} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[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)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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