3.606 \(\int \frac{(e \cos (c+d x))^{13/2}}{(a+b \sin (c+d x))^4} \, dx\)

Optimal. Leaf size=557 \[ -\frac{77 e^5 (e \cos (c+d x))^{3/2} \left (15 a^2+6 a b \sin (c+d x)-4 b^2\right )}{120 b^5 d (a+b \sin (c+d x))}+\frac{77 a e^{13/2} \left (3 a^2-2 b^2\right ) \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{e \cos (c+d x)}}{\sqrt{e} \sqrt [4]{b^2-a^2}}\right )}{16 b^{13/2} d \sqrt [4]{b^2-a^2}}-\frac{77 a e^{13/2} \left (3 a^2-2 b^2\right ) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{e \cos (c+d x)}}{\sqrt{e} \sqrt [4]{b^2-a^2}}\right )}{16 b^{13/2} d \sqrt [4]{b^2-a^2}}-\frac{77 e^6 \left (15 a^2-4 b^2\right ) E\left (\left .\frac{1}{2} (c+d x)\right |2\right ) \sqrt{e \cos (c+d x)}}{40 b^6 d \sqrt{\cos (c+d x)}}+\frac{77 a^2 e^7 \left (3 a^2-2 b^2\right ) \sqrt{\cos (c+d x)} \Pi \left (\frac{2 b}{b-\sqrt{b^2-a^2}};\left .\frac{1}{2} (c+d x)\right |2\right )}{16 b^7 d \left (b-\sqrt{b^2-a^2}\right ) \sqrt{e \cos (c+d x)}}+\frac{77 a^2 e^7 \left (3 a^2-2 b^2\right ) \sqrt{\cos (c+d x)} \Pi \left (\frac{2 b}{b+\sqrt{b^2-a^2}};\left .\frac{1}{2} (c+d x)\right |2\right )}{16 b^7 d \left (\sqrt{b^2-a^2}+b\right ) \sqrt{e \cos (c+d x)}}-\frac{11 e^3 (e \cos (c+d x))^{7/2} (9 a+4 b \sin (c+d x))}{60 b^3 d (a+b \sin (c+d x))^2}-\frac{e (e \cos (c+d x))^{11/2}}{3 b d (a+b \sin (c+d x))^3} \]

[Out]

(77*a*(3*a^2 - 2*b^2)*e^(13/2)*ArcTan[(Sqrt[b]*Sqrt[e*Cos[c + d*x]])/((-a^2 + b^2)^(1/4)*Sqrt[e])])/(16*b^(13/
2)*(-a^2 + b^2)^(1/4)*d) - (77*a*(3*a^2 - 2*b^2)*e^(13/2)*ArcTanh[(Sqrt[b]*Sqrt[e*Cos[c + d*x]])/((-a^2 + b^2)
^(1/4)*Sqrt[e])])/(16*b^(13/2)*(-a^2 + b^2)^(1/4)*d) - (77*(15*a^2 - 4*b^2)*e^6*Sqrt[e*Cos[c + d*x]]*EllipticE
[(c + d*x)/2, 2])/(40*b^6*d*Sqrt[Cos[c + d*x]]) + (77*a^2*(3*a^2 - 2*b^2)*e^7*Sqrt[Cos[c + d*x]]*EllipticPi[(2
*b)/(b - Sqrt[-a^2 + b^2]), (c + d*x)/2, 2])/(16*b^7*(b - Sqrt[-a^2 + b^2])*d*Sqrt[e*Cos[c + d*x]]) + (77*a^2*
(3*a^2 - 2*b^2)*e^7*Sqrt[Cos[c + d*x]]*EllipticPi[(2*b)/(b + Sqrt[-a^2 + b^2]), (c + d*x)/2, 2])/(16*b^7*(b +
Sqrt[-a^2 + b^2])*d*Sqrt[e*Cos[c + d*x]]) - (e*(e*Cos[c + d*x])^(11/2))/(3*b*d*(a + b*Sin[c + d*x])^3) - (11*e
^3*(e*Cos[c + d*x])^(7/2)*(9*a + 4*b*Sin[c + d*x]))/(60*b^3*d*(a + b*Sin[c + d*x])^2) - (77*e^5*(e*Cos[c + d*x
])^(3/2)*(15*a^2 - 4*b^2 + 6*a*b*Sin[c + d*x]))/(120*b^5*d*(a + b*Sin[c + d*x]))

________________________________________________________________________________________

Rubi [A]  time = 1.36407, antiderivative size = 557, normalized size of antiderivative = 1., number of steps used = 15, number of rules used = 12, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.48, Rules used = {2693, 2863, 2867, 2640, 2639, 2701, 2807, 2805, 329, 298, 205, 208} \[ -\frac{77 e^5 (e \cos (c+d x))^{3/2} \left (15 a^2+6 a b \sin (c+d x)-4 b^2\right )}{120 b^5 d (a+b \sin (c+d x))}+\frac{77 a e^{13/2} \left (3 a^2-2 b^2\right ) \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{e \cos (c+d x)}}{\sqrt{e} \sqrt [4]{b^2-a^2}}\right )}{16 b^{13/2} d \sqrt [4]{b^2-a^2}}-\frac{77 a e^{13/2} \left (3 a^2-2 b^2\right ) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{e \cos (c+d x)}}{\sqrt{e} \sqrt [4]{b^2-a^2}}\right )}{16 b^{13/2} d \sqrt [4]{b^2-a^2}}-\frac{77 e^6 \left (15 a^2-4 b^2\right ) E\left (\left .\frac{1}{2} (c+d x)\right |2\right ) \sqrt{e \cos (c+d x)}}{40 b^6 d \sqrt{\cos (c+d x)}}+\frac{77 a^2 e^7 \left (3 a^2-2 b^2\right ) \sqrt{\cos (c+d x)} \Pi \left (\frac{2 b}{b-\sqrt{b^2-a^2}};\left .\frac{1}{2} (c+d x)\right |2\right )}{16 b^7 d \left (b-\sqrt{b^2-a^2}\right ) \sqrt{e \cos (c+d x)}}+\frac{77 a^2 e^7 \left (3 a^2-2 b^2\right ) \sqrt{\cos (c+d x)} \Pi \left (\frac{2 b}{b+\sqrt{b^2-a^2}};\left .\frac{1}{2} (c+d x)\right |2\right )}{16 b^7 d \left (\sqrt{b^2-a^2}+b\right ) \sqrt{e \cos (c+d x)}}-\frac{11 e^3 (e \cos (c+d x))^{7/2} (9 a+4 b \sin (c+d x))}{60 b^3 d (a+b \sin (c+d x))^2}-\frac{e (e \cos (c+d x))^{11/2}}{3 b d (a+b \sin (c+d x))^3} \]

Antiderivative was successfully verified.

[In]

Int[(e*Cos[c + d*x])^(13/2)/(a + b*Sin[c + d*x])^4,x]

[Out]

(77*a*(3*a^2 - 2*b^2)*e^(13/2)*ArcTan[(Sqrt[b]*Sqrt[e*Cos[c + d*x]])/((-a^2 + b^2)^(1/4)*Sqrt[e])])/(16*b^(13/
2)*(-a^2 + b^2)^(1/4)*d) - (77*a*(3*a^2 - 2*b^2)*e^(13/2)*ArcTanh[(Sqrt[b]*Sqrt[e*Cos[c + d*x]])/((-a^2 + b^2)
^(1/4)*Sqrt[e])])/(16*b^(13/2)*(-a^2 + b^2)^(1/4)*d) - (77*(15*a^2 - 4*b^2)*e^6*Sqrt[e*Cos[c + d*x]]*EllipticE
[(c + d*x)/2, 2])/(40*b^6*d*Sqrt[Cos[c + d*x]]) + (77*a^2*(3*a^2 - 2*b^2)*e^7*Sqrt[Cos[c + d*x]]*EllipticPi[(2
*b)/(b - Sqrt[-a^2 + b^2]), (c + d*x)/2, 2])/(16*b^7*(b - Sqrt[-a^2 + b^2])*d*Sqrt[e*Cos[c + d*x]]) + (77*a^2*
(3*a^2 - 2*b^2)*e^7*Sqrt[Cos[c + d*x]]*EllipticPi[(2*b)/(b + Sqrt[-a^2 + b^2]), (c + d*x)/2, 2])/(16*b^7*(b +
Sqrt[-a^2 + b^2])*d*Sqrt[e*Cos[c + d*x]]) - (e*(e*Cos[c + d*x])^(11/2))/(3*b*d*(a + b*Sin[c + d*x])^3) - (11*e
^3*(e*Cos[c + d*x])^(7/2)*(9*a + 4*b*Sin[c + d*x]))/(60*b^3*d*(a + b*Sin[c + d*x])^2) - (77*e^5*(e*Cos[c + d*x
])^(3/2)*(15*a^2 - 4*b^2 + 6*a*b*Sin[c + d*x]))/(120*b^5*d*(a + b*Sin[c + d*x]))

Rule 2693

Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_), x_Symbol] :> Simp[(g*(g*
Cos[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 2863

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 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]

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 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 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 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 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 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 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 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]

Rubi steps

\begin{align*} \int \frac{(e \cos (c+d x))^{13/2}}{(a+b \sin (c+d x))^4} \, dx &=-\frac{e (e \cos (c+d x))^{11/2}}{3 b d (a+b \sin (c+d x))^3}-\frac{\left (11 e^2\right ) \int \frac{(e \cos (c+d x))^{9/2} \sin (c+d x)}{(a+b \sin (c+d x))^3} \, dx}{6 b}\\ &=-\frac{e (e \cos (c+d x))^{11/2}}{3 b d (a+b \sin (c+d x))^3}-\frac{11 e^3 (e \cos (c+d x))^{7/2} (9 a+4 b \sin (c+d x))}{60 b^3 d (a+b \sin (c+d x))^2}+\frac{\left (77 e^4\right ) \int \frac{(e \cos (c+d x))^{5/2} \left (-2 b-\frac{9}{2} a \sin (c+d x)\right )}{(a+b \sin (c+d x))^2} \, dx}{60 b^3}\\ &=-\frac{e (e \cos (c+d x))^{11/2}}{3 b d (a+b \sin (c+d x))^3}-\frac{11 e^3 (e \cos (c+d x))^{7/2} (9 a+4 b \sin (c+d x))}{60 b^3 d (a+b \sin (c+d x))^2}-\frac{77 e^5 (e \cos (c+d x))^{3/2} \left (15 a^2-4 b^2+6 a b \sin (c+d x)\right )}{120 b^5 d (a+b \sin (c+d x))}-\frac{\left (77 e^6\right ) \int \frac{\sqrt{e \cos (c+d x)} \left (\frac{9 a b}{2}+\frac{3}{4} \left (15 a^2-4 b^2\right ) \sin (c+d x)\right )}{a+b \sin (c+d x)} \, dx}{60 b^5}\\ &=-\frac{e (e \cos (c+d x))^{11/2}}{3 b d (a+b \sin (c+d x))^3}-\frac{11 e^3 (e \cos (c+d x))^{7/2} (9 a+4 b \sin (c+d x))}{60 b^3 d (a+b \sin (c+d x))^2}-\frac{77 e^5 (e \cos (c+d x))^{3/2} \left (15 a^2-4 b^2+6 a b \sin (c+d x)\right )}{120 b^5 d (a+b \sin (c+d x))}-\frac{\left (77 \left (15 a^2-4 b^2\right ) e^6\right ) \int \sqrt{e \cos (c+d x)} \, dx}{80 b^6}+\frac{\left (77 a \left (3 a^2-2 b^2\right ) e^6\right ) \int \frac{\sqrt{e \cos (c+d x)}}{a+b \sin (c+d x)} \, dx}{16 b^6}\\ &=-\frac{e (e \cos (c+d x))^{11/2}}{3 b d (a+b \sin (c+d x))^3}-\frac{11 e^3 (e \cos (c+d x))^{7/2} (9 a+4 b \sin (c+d x))}{60 b^3 d (a+b \sin (c+d x))^2}-\frac{77 e^5 (e \cos (c+d x))^{3/2} \left (15 a^2-4 b^2+6 a b \sin (c+d x)\right )}{120 b^5 d (a+b \sin (c+d x))}-\frac{\left (77 a^2 \left (3 a^2-2 b^2\right ) e^7\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^7}+\frac{\left (77 a^2 \left (3 a^2-2 b^2\right ) e^7\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^7}+\frac{\left (77 a \left (3 a^2-2 b^2\right ) e^7\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 )}{16 b^5 d}-\frac{\left (77 \left (15 a^2-4 b^2\right ) e^6 \sqrt{e \cos (c+d x)}\right ) \int \sqrt{\cos (c+d x)} \, dx}{80 b^6 \sqrt{\cos (c+d x)}}\\ &=-\frac{77 \left (15 a^2-4 b^2\right ) e^6 \sqrt{e \cos (c+d x)} E\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{40 b^6 d \sqrt{\cos (c+d x)}}-\frac{e (e \cos (c+d x))^{11/2}}{3 b d (a+b \sin (c+d x))^3}-\frac{11 e^3 (e \cos (c+d x))^{7/2} (9 a+4 b \sin (c+d x))}{60 b^3 d (a+b \sin (c+d x))^2}-\frac{77 e^5 (e \cos (c+d x))^{3/2} \left (15 a^2-4 b^2+6 a b \sin (c+d x)\right )}{120 b^5 d (a+b \sin (c+d x))}+\frac{\left (77 a \left (3 a^2-2 b^2\right ) e^7\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 )}{8 b^5 d}-\frac{\left (77 a^2 \left (3 a^2-2 b^2\right ) e^7 \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^7 \sqrt{e \cos (c+d x)}}+\frac{\left (77 a^2 \left (3 a^2-2 b^2\right ) e^7 \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^7 \sqrt{e \cos (c+d x)}}\\ &=-\frac{77 \left (15 a^2-4 b^2\right ) e^6 \sqrt{e \cos (c+d x)} E\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{40 b^6 d \sqrt{\cos (c+d x)}}+\frac{77 a^2 \left (3 a^2-2 b^2\right ) e^7 \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 )}{16 b^7 \left (b-\sqrt{-a^2+b^2}\right ) d \sqrt{e \cos (c+d x)}}+\frac{77 a^2 \left (3 a^2-2 b^2\right ) e^7 \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 )}{16 b^7 \left (b+\sqrt{-a^2+b^2}\right ) d \sqrt{e \cos (c+d x)}}-\frac{e (e \cos (c+d x))^{11/2}}{3 b d (a+b \sin (c+d x))^3}-\frac{11 e^3 (e \cos (c+d x))^{7/2} (9 a+4 b \sin (c+d x))}{60 b^3 d (a+b \sin (c+d x))^2}-\frac{77 e^5 (e \cos (c+d x))^{3/2} \left (15 a^2-4 b^2+6 a b \sin (c+d x)\right )}{120 b^5 d (a+b \sin (c+d x))}-\frac{\left (77 a \left (3 a^2-2 b^2\right ) e^7\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 )}{16 b^6 d}+\frac{\left (77 a \left (3 a^2-2 b^2\right ) e^7\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 )}{16 b^6 d}\\ &=\frac{77 a \left (3 a^2-2 b^2\right ) e^{13/2} \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{e \cos (c+d x)}}{\sqrt [4]{-a^2+b^2} \sqrt{e}}\right )}{16 b^{13/2} \sqrt [4]{-a^2+b^2} d}-\frac{77 a \left (3 a^2-2 b^2\right ) e^{13/2} \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{e \cos (c+d x)}}{\sqrt [4]{-a^2+b^2} \sqrt{e}}\right )}{16 b^{13/2} \sqrt [4]{-a^2+b^2} d}-\frac{77 \left (15 a^2-4 b^2\right ) e^6 \sqrt{e \cos (c+d x)} E\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{40 b^6 d \sqrt{\cos (c+d x)}}+\frac{77 a^2 \left (3 a^2-2 b^2\right ) e^7 \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 )}{16 b^7 \left (b-\sqrt{-a^2+b^2}\right ) d \sqrt{e \cos (c+d x)}}+\frac{77 a^2 \left (3 a^2-2 b^2\right ) e^7 \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 )}{16 b^7 \left (b+\sqrt{-a^2+b^2}\right ) d \sqrt{e \cos (c+d x)}}-\frac{e (e \cos (c+d x))^{11/2}}{3 b d (a+b \sin (c+d x))^3}-\frac{11 e^3 (e \cos (c+d x))^{7/2} (9 a+4 b \sin (c+d x))}{60 b^3 d (a+b \sin (c+d x))^2}-\frac{77 e^5 (e \cos (c+d x))^{3/2} \left (15 a^2-4 b^2+6 a b \sin (c+d x)\right )}{120 b^5 d (a+b \sin (c+d x))}\\ \end{align*}

Mathematica [C]  time = 27.0614, size = 937, normalized size = 1.68 \[ \frac{(e \cos (c+d x))^{13/2} \sec ^6(c+d x) \left (-\frac{8 a \cos (c+d x)}{3 b^5}+\frac{\sin (2 (c+d x))}{5 b^4}+\frac{20 b^2 \cos (c+d x)-71 a^2 \cos (c+d x)}{8 b^5 (a+b \sin (c+d x))}+\frac{9 \left (a^3 \cos (c+d x)-a b^2 \cos (c+d x)\right )}{4 b^5 (a+b \sin (c+d x))^2}+\frac{-\cos (c+d x) a^4+2 b^2 \cos (c+d x) a^2-b^4 \cos (c+d x)}{3 b^5 (a+b \sin (c+d x))^3}\right )}{d}-\frac{77 (e \cos (c+d x))^{13/2} \left (-\frac{\left (15 a^2-4 b^2\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{12 a b \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 )}{80 b^5 d \cos ^{\frac{13}{2}}(c+d x)} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[(e*Cos[c + d*x])^(13/2)/(a + b*Sin[c + d*x])^4,x]

[Out]

(-77*(e*Cos[c + d*x])^(13/2)*((-12*a*b*(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]]))/(S
qrt[b]*(-a^2 + b^2)^(1/4)))*Sin[c + d*x])/(Sqrt[1 - Cos[c + d*x]^2]*(a + b*Sin[c + d*x])) - ((15*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]))))/(80*b^5*d*Cos[c + d*x]^(13/2)) + ((e*Cos[c + d*x])^(13/2)*Sec[c + d
*x]^6*((-8*a*Cos[c + d*x])/(3*b^5) + (-(a^4*Cos[c + d*x]) + 2*a^2*b^2*Cos[c + d*x] - b^4*Cos[c + d*x])/(3*b^5*
(a + b*Sin[c + d*x])^3) + (9*(a^3*Cos[c + d*x] - a*b^2*Cos[c + d*x]))/(4*b^5*(a + b*Sin[c + d*x])^2) + (-71*a^
2*Cos[c + d*x] + 20*b^2*Cos[c + d*x])/(8*b^5*(a + b*Sin[c + d*x])) + Sin[2*(c + d*x)]/(5*b^4)))/d

________________________________________________________________________________________

Maple [C]  time = 78.119, size = 180834, normalized size = 324.7 \begin{align*} \text{output too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((e*cos(d*x+c))^(13/2)/(a+b*sin(d*x+c))^4,x)

[Out]

result too large to display

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Maxima [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*cos(d*x+c))^(13/2)/(a+b*sin(d*x+c))^4,x, algorithm="maxima")

[Out]

Timed out

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Fricas [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*cos(d*x+c))^(13/2)/(a+b*sin(d*x+c))^4,x, algorithm="fricas")

[Out]

Timed out

________________________________________________________________________________________

Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*cos(d*x+c))**(13/2)/(a+b*sin(d*x+c))**4,x)

[Out]

Timed out

________________________________________________________________________________________

Giac [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*cos(d*x+c))^(13/2)/(a+b*sin(d*x+c))^4,x, algorithm="giac")

[Out]

Timed out