3.605 \(\int \frac{(e \cos (c+d x))^{15/2}}{(a+b \sin (c+d x))^4} \, dx\)

Optimal. Leaf size=671 \[ \frac{13 e^7 \sqrt{e \cos (c+d x)} \left (21 a \left (11 a^2-6 b^2\right )-b \left (77 a^2-20 b^2\right ) \sin (c+d x)\right )}{56 b^7 d}-\frac{39 e^5 (e \cos (c+d x))^{5/2} \left (77 a^2+22 a b \sin (c+d x)-20 b^2\right )}{280 b^5 d (a+b \sin (c+d x))}+\frac{39 a e^{15/2} \left (-17 a^2 b^2+11 a^4+6 b^4\right ) \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{e \cos (c+d x)}}{\sqrt{e} \sqrt [4]{b^2-a^2}}\right )}{16 b^{15/2} d \left (b^2-a^2\right )^{3/4}}+\frac{39 a e^{15/2} \left (-17 a^2 b^2+11 a^4+6 b^4\right ) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{e \cos (c+d x)}}{\sqrt{e} \sqrt [4]{b^2-a^2}}\right )}{16 b^{15/2} d \left (b^2-a^2\right )^{3/4}}+\frac{13 e^8 \left (-203 a^2 b^2+231 a^4+20 b^4\right ) \sqrt{\cos (c+d x)} F\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{56 b^8 d \sqrt{e \cos (c+d x)}}-\frac{39 a^2 e^8 \left (-17 a^2 b^2+11 a^4+6 b^4\right ) \sqrt{\cos (c+d x)} \Pi \left (\frac{2 b}{b-\sqrt{b^2-a^2}};\left .\frac{1}{2} (c+d x)\right |2\right )}{16 b^8 d \left (a^2-b \left (b-\sqrt{b^2-a^2}\right )\right ) \sqrt{e \cos (c+d x)}}-\frac{39 a^2 e^8 \left (-17 a^2 b^2+11 a^4+6 b^4\right ) \sqrt{\cos (c+d x)} \Pi \left (\frac{2 b}{b+\sqrt{b^2-a^2}};\left .\frac{1}{2} (c+d x)\right |2\right )}{16 b^8 d \left (a^2-b \left (\sqrt{b^2-a^2}+b\right )\right ) \sqrt{e \cos (c+d x)}}-\frac{13 e^3 (e \cos (c+d x))^{9/2} (11 a+4 b \sin (c+d x))}{84 b^3 d (a+b \sin (c+d x))^2}-\frac{e (e \cos (c+d x))^{13/2}}{3 b d (a+b \sin (c+d x))^3} \]

[Out]

(39*a*(11*a^4 - 17*a^2*b^2 + 6*b^4)*e^(15/2)*ArcTan[(Sqrt[b]*Sqrt[e*Cos[c + d*x]])/((-a^2 + b^2)^(1/4)*Sqrt[e]
)])/(16*b^(15/2)*(-a^2 + b^2)^(3/4)*d) + (39*a*(11*a^4 - 17*a^2*b^2 + 6*b^4)*e^(15/2)*ArcTanh[(Sqrt[b]*Sqrt[e*
Cos[c + d*x]])/((-a^2 + b^2)^(1/4)*Sqrt[e])])/(16*b^(15/2)*(-a^2 + b^2)^(3/4)*d) + (13*(231*a^4 - 203*a^2*b^2
+ 20*b^4)*e^8*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2])/(56*b^8*d*Sqrt[e*Cos[c + d*x]]) - (39*a^2*(11*a^4
- 17*a^2*b^2 + 6*b^4)*e^8*Sqrt[Cos[c + d*x]]*EllipticPi[(2*b)/(b - Sqrt[-a^2 + b^2]), (c + d*x)/2, 2])/(16*b^8
*(a^2 - b*(b - Sqrt[-a^2 + b^2]))*d*Sqrt[e*Cos[c + d*x]]) - (39*a^2*(11*a^4 - 17*a^2*b^2 + 6*b^4)*e^8*Sqrt[Cos
[c + d*x]]*EllipticPi[(2*b)/(b + Sqrt[-a^2 + b^2]), (c + d*x)/2, 2])/(16*b^8*(a^2 - b*(b + Sqrt[-a^2 + b^2]))*
d*Sqrt[e*Cos[c + d*x]]) - (e*(e*Cos[c + d*x])^(13/2))/(3*b*d*(a + b*Sin[c + d*x])^3) - (13*e^3*(e*Cos[c + d*x]
)^(9/2)*(11*a + 4*b*Sin[c + d*x]))/(84*b^3*d*(a + b*Sin[c + d*x])^2) - (39*e^5*(e*Cos[c + d*x])^(5/2)*(77*a^2
- 20*b^2 + 22*a*b*Sin[c + d*x]))/(280*b^5*d*(a + b*Sin[c + d*x])) + (13*e^7*Sqrt[e*Cos[c + d*x]]*(21*a*(11*a^2
- 6*b^2) - b*(77*a^2 - 20*b^2)*Sin[c + d*x]))/(56*b^7*d)

________________________________________________________________________________________

Rubi [A]  time = 1.83161, antiderivative size = 671, normalized size of antiderivative = 1., number of steps used = 16, number of rules used = 13, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.52, Rules used = {2693, 2863, 2865, 2867, 2642, 2641, 2702, 2807, 2805, 329, 212, 208, 205} \[ \frac{13 e^7 \sqrt{e \cos (c+d x)} \left (21 a \left (11 a^2-6 b^2\right )-b \left (77 a^2-20 b^2\right ) \sin (c+d x)\right )}{56 b^7 d}-\frac{39 e^5 (e \cos (c+d x))^{5/2} \left (77 a^2+22 a b \sin (c+d x)-20 b^2\right )}{280 b^5 d (a+b \sin (c+d x))}+\frac{39 a e^{15/2} \left (-17 a^2 b^2+11 a^4+6 b^4\right ) \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{e \cos (c+d x)}}{\sqrt{e} \sqrt [4]{b^2-a^2}}\right )}{16 b^{15/2} d \left (b^2-a^2\right )^{3/4}}+\frac{39 a e^{15/2} \left (-17 a^2 b^2+11 a^4+6 b^4\right ) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{e \cos (c+d x)}}{\sqrt{e} \sqrt [4]{b^2-a^2}}\right )}{16 b^{15/2} d \left (b^2-a^2\right )^{3/4}}+\frac{13 e^8 \left (-203 a^2 b^2+231 a^4+20 b^4\right ) \sqrt{\cos (c+d x)} F\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{56 b^8 d \sqrt{e \cos (c+d x)}}-\frac{39 a^2 e^8 \left (-17 a^2 b^2+11 a^4+6 b^4\right ) \sqrt{\cos (c+d x)} \Pi \left (\frac{2 b}{b-\sqrt{b^2-a^2}};\left .\frac{1}{2} (c+d x)\right |2\right )}{16 b^8 d \left (a^2-b \left (b-\sqrt{b^2-a^2}\right )\right ) \sqrt{e \cos (c+d x)}}-\frac{39 a^2 e^8 \left (-17 a^2 b^2+11 a^4+6 b^4\right ) \sqrt{\cos (c+d x)} \Pi \left (\frac{2 b}{b+\sqrt{b^2-a^2}};\left .\frac{1}{2} (c+d x)\right |2\right )}{16 b^8 d \left (a^2-b \left (\sqrt{b^2-a^2}+b\right )\right ) \sqrt{e \cos (c+d x)}}-\frac{13 e^3 (e \cos (c+d x))^{9/2} (11 a+4 b \sin (c+d x))}{84 b^3 d (a+b \sin (c+d x))^2}-\frac{e (e \cos (c+d x))^{13/2}}{3 b d (a+b \sin (c+d x))^3} \]

Antiderivative was successfully verified.

[In]

Int[(e*Cos[c + d*x])^(15/2)/(a + b*Sin[c + d*x])^4,x]

[Out]

(39*a*(11*a^4 - 17*a^2*b^2 + 6*b^4)*e^(15/2)*ArcTan[(Sqrt[b]*Sqrt[e*Cos[c + d*x]])/((-a^2 + b^2)^(1/4)*Sqrt[e]
)])/(16*b^(15/2)*(-a^2 + b^2)^(3/4)*d) + (39*a*(11*a^4 - 17*a^2*b^2 + 6*b^4)*e^(15/2)*ArcTanh[(Sqrt[b]*Sqrt[e*
Cos[c + d*x]])/((-a^2 + b^2)^(1/4)*Sqrt[e])])/(16*b^(15/2)*(-a^2 + b^2)^(3/4)*d) + (13*(231*a^4 - 203*a^2*b^2
+ 20*b^4)*e^8*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2])/(56*b^8*d*Sqrt[e*Cos[c + d*x]]) - (39*a^2*(11*a^4
- 17*a^2*b^2 + 6*b^4)*e^8*Sqrt[Cos[c + d*x]]*EllipticPi[(2*b)/(b - Sqrt[-a^2 + b^2]), (c + d*x)/2, 2])/(16*b^8
*(a^2 - b*(b - Sqrt[-a^2 + b^2]))*d*Sqrt[e*Cos[c + d*x]]) - (39*a^2*(11*a^4 - 17*a^2*b^2 + 6*b^4)*e^8*Sqrt[Cos
[c + d*x]]*EllipticPi[(2*b)/(b + Sqrt[-a^2 + b^2]), (c + d*x)/2, 2])/(16*b^8*(a^2 - b*(b + Sqrt[-a^2 + b^2]))*
d*Sqrt[e*Cos[c + d*x]]) - (e*(e*Cos[c + d*x])^(13/2))/(3*b*d*(a + b*Sin[c + d*x])^3) - (13*e^3*(e*Cos[c + d*x]
)^(9/2)*(11*a + 4*b*Sin[c + d*x]))/(84*b^3*d*(a + b*Sin[c + d*x])^2) - (39*e^5*(e*Cos[c + d*x])^(5/2)*(77*a^2
- 20*b^2 + 22*a*b*Sin[c + d*x]))/(280*b^5*d*(a + b*Sin[c + d*x])) + (13*e^7*Sqrt[e*Cos[c + d*x]]*(21*a*(11*a^2
- 6*b^2) - b*(77*a^2 - 20*b^2)*Sin[c + d*x]))/(56*b^7*d)

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 2865

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 + p)*Sin[e + f*x]))/(b^2*f*(m + p)*(m + p + 1)), x] + Dist[(g^2*(p - 1))/(b^2*(m + p)*(m + p +
1)), Int[(g*Cos[e + f*x])^(p - 2)*(a + b*Sin[e + f*x])^m*Simp[b*(a*d*m + b*c*(m + p + 1)) + (a*b*c*(m + p + 1
) - d*(a^2*p - b^2*(m + p)))*Sin[e + f*x], x], x], x] /; FreeQ[{a, b, c, d, e, f, g, m}, x] && NeQ[a^2 - b^2,
0] && GtQ[p, 1] && NeQ[m + p, 0] && NeQ[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 2642

Int[1/Sqrt[(b_)*sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Dist[Sqrt[Sin[c + d*x]]/Sqrt[b*Sin[c + d*x]], Int[1/Sqr
t[Sin[c + d*x]], x], x] /; FreeQ[{b, c, d}, x]

Rule 2641

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

Rule 2702

Int[1/(Sqrt[cos[(e_.) + (f_.)*(x_)]*(g_.)]*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])), x_Symbol] :> With[{q = Rt[
-a^2 + b^2, 2]}, -Dist[a/(2*q), Int[1/(Sqrt[g*Cos[e + f*x]]*(q + b*Cos[e + f*x])), x], x] + (Dist[(b*g)/f, Sub
st[Int[1/(Sqrt[x]*(g^2*(a^2 - b^2) + b^2*x^2)), x], x, g*Cos[e + f*x]], x] - Dist[a/(2*q), Int[1/(Sqrt[g*Cos[e
+ f*x]]*(q - b*Cos[e + f*x])), 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 212

Int[((a_) + (b_.)*(x_)^4)^(-1), x_Symbol] :> With[{r = Numerator[Rt[-(a/b), 2]], s = Denominator[Rt[-(a/b), 2]
]}, Dist[r/(2*a), Int[1/(r - s*x^2), x], x] + Dist[r/(2*a), Int[1/(r + s*x^2), x], x]] /; FreeQ[{a, b}, x] &&
!GtQ[a/b, 0]

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 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]

Rubi steps

\begin{align*} \int \frac{(e \cos (c+d x))^{15/2}}{(a+b \sin (c+d x))^4} \, dx &=-\frac{e (e \cos (c+d x))^{13/2}}{3 b d (a+b \sin (c+d x))^3}-\frac{\left (13 e^2\right ) \int \frac{(e \cos (c+d x))^{11/2} \sin (c+d x)}{(a+b \sin (c+d x))^3} \, dx}{6 b}\\ &=-\frac{e (e \cos (c+d x))^{13/2}}{3 b d (a+b \sin (c+d x))^3}-\frac{13 e^3 (e \cos (c+d x))^{9/2} (11 a+4 b \sin (c+d x))}{84 b^3 d (a+b \sin (c+d x))^2}+\frac{\left (39 e^4\right ) \int \frac{(e \cos (c+d x))^{7/2} \left (-2 b-\frac{11}{2} a \sin (c+d x)\right )}{(a+b \sin (c+d x))^2} \, dx}{28 b^3}\\ &=-\frac{e (e \cos (c+d x))^{13/2}}{3 b d (a+b \sin (c+d x))^3}-\frac{13 e^3 (e \cos (c+d x))^{9/2} (11 a+4 b \sin (c+d x))}{84 b^3 d (a+b \sin (c+d x))^2}-\frac{39 e^5 (e \cos (c+d x))^{5/2} \left (77 a^2-20 b^2+22 a b \sin (c+d x)\right )}{280 b^5 d (a+b \sin (c+d x))}-\frac{\left (39 e^6\right ) \int \frac{(e \cos (c+d x))^{3/2} \left (\frac{11 a b}{2}+\frac{1}{4} \left (77 a^2-20 b^2\right ) \sin (c+d x)\right )}{a+b \sin (c+d x)} \, dx}{28 b^5}\\ &=-\frac{e (e \cos (c+d x))^{13/2}}{3 b d (a+b \sin (c+d x))^3}-\frac{13 e^3 (e \cos (c+d x))^{9/2} (11 a+4 b \sin (c+d x))}{84 b^3 d (a+b \sin (c+d x))^2}-\frac{39 e^5 (e \cos (c+d x))^{5/2} \left (77 a^2-20 b^2+22 a b \sin (c+d x)\right )}{280 b^5 d (a+b \sin (c+d x))}+\frac{13 e^7 \sqrt{e \cos (c+d x)} \left (21 a \left (11 a^2-6 b^2\right )-b \left (77 a^2-20 b^2\right ) \sin (c+d x)\right )}{56 b^7 d}-\frac{\left (13 e^8\right ) \int \frac{-\frac{1}{4} a b \left (77 a^2-53 b^2\right )-\frac{1}{8} \left (231 a^4-203 a^2 b^2+20 b^4\right ) \sin (c+d x)}{\sqrt{e \cos (c+d x)} (a+b \sin (c+d x))} \, dx}{14 b^7}\\ &=-\frac{e (e \cos (c+d x))^{13/2}}{3 b d (a+b \sin (c+d x))^3}-\frac{13 e^3 (e \cos (c+d x))^{9/2} (11 a+4 b \sin (c+d x))}{84 b^3 d (a+b \sin (c+d x))^2}-\frac{39 e^5 (e \cos (c+d x))^{5/2} \left (77 a^2-20 b^2+22 a b \sin (c+d x)\right )}{280 b^5 d (a+b \sin (c+d x))}+\frac{13 e^7 \sqrt{e \cos (c+d x)} \left (21 a \left (11 a^2-6 b^2\right )-b \left (77 a^2-20 b^2\right ) \sin (c+d x)\right )}{56 b^7 d}-\frac{\left (39 a \left (11 a^4-17 a^2 b^2+6 b^4\right ) e^8\right ) \int \frac{1}{\sqrt{e \cos (c+d x)} (a+b \sin (c+d x))} \, dx}{16 b^8}+\frac{\left (13 \left (231 a^4-203 a^2 b^2+20 b^4\right ) e^8\right ) \int \frac{1}{\sqrt{e \cos (c+d x)}} \, dx}{112 b^8}\\ &=-\frac{e (e \cos (c+d x))^{13/2}}{3 b d (a+b \sin (c+d x))^3}-\frac{13 e^3 (e \cos (c+d x))^{9/2} (11 a+4 b \sin (c+d x))}{84 b^3 d (a+b \sin (c+d x))^2}-\frac{39 e^5 (e \cos (c+d x))^{5/2} \left (77 a^2-20 b^2+22 a b \sin (c+d x)\right )}{280 b^5 d (a+b \sin (c+d x))}+\frac{13 e^7 \sqrt{e \cos (c+d x)} \left (21 a \left (11 a^2-6 b^2\right )-b \left (77 a^2-20 b^2\right ) \sin (c+d x)\right )}{56 b^7 d}+\frac{\left (39 a^2 \left (11 a^4-17 a^2 b^2+6 b^4\right ) e^8\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^8 \sqrt{-a^2+b^2}}+\frac{\left (39 a^2 \left (11 a^4-17 a^2 b^2+6 b^4\right ) e^8\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^8 \sqrt{-a^2+b^2}}-\frac{\left (39 a \left (11 a^4-17 a^2 b^2+6 b^4\right ) e^9\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{x} \left (\left (a^2-b^2\right ) e^2+b^2 x^2\right )} \, dx,x,e \cos (c+d x)\right )}{16 b^7 d}+\frac{\left (13 \left (231 a^4-203 a^2 b^2+20 b^4\right ) e^8 \sqrt{\cos (c+d x)}\right ) \int \frac{1}{\sqrt{\cos (c+d x)}} \, dx}{112 b^8 \sqrt{e \cos (c+d x)}}\\ &=\frac{13 \left (231 a^4-203 a^2 b^2+20 b^4\right ) e^8 \sqrt{\cos (c+d x)} F\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{56 b^8 d \sqrt{e \cos (c+d x)}}-\frac{e (e \cos (c+d x))^{13/2}}{3 b d (a+b \sin (c+d x))^3}-\frac{13 e^3 (e \cos (c+d x))^{9/2} (11 a+4 b \sin (c+d x))}{84 b^3 d (a+b \sin (c+d x))^2}-\frac{39 e^5 (e \cos (c+d x))^{5/2} \left (77 a^2-20 b^2+22 a b \sin (c+d x)\right )}{280 b^5 d (a+b \sin (c+d x))}+\frac{13 e^7 \sqrt{e \cos (c+d x)} \left (21 a \left (11 a^2-6 b^2\right )-b \left (77 a^2-20 b^2\right ) \sin (c+d x)\right )}{56 b^7 d}-\frac{\left (39 a \left (11 a^4-17 a^2 b^2+6 b^4\right ) e^9\right ) \operatorname{Subst}\left (\int \frac{1}{\left (a^2-b^2\right ) e^2+b^2 x^4} \, dx,x,\sqrt{e \cos (c+d x)}\right )}{8 b^7 d}+\frac{\left (39 a^2 \left (11 a^4-17 a^2 b^2+6 b^4\right ) e^8 \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^8 \sqrt{-a^2+b^2} \sqrt{e \cos (c+d x)}}+\frac{\left (39 a^2 \left (11 a^4-17 a^2 b^2+6 b^4\right ) e^8 \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^8 \sqrt{-a^2+b^2} \sqrt{e \cos (c+d x)}}\\ &=\frac{13 \left (231 a^4-203 a^2 b^2+20 b^4\right ) e^8 \sqrt{\cos (c+d x)} F\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{56 b^8 d \sqrt{e \cos (c+d x)}}-\frac{39 a^2 \left (11 a^4-17 a^2 b^2+6 b^4\right ) e^8 \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^8 \sqrt{-a^2+b^2} \left (b-\sqrt{-a^2+b^2}\right ) d \sqrt{e \cos (c+d x)}}+\frac{39 a^2 \left (11 a^4-17 a^2 b^2+6 b^4\right ) e^8 \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^8 \sqrt{-a^2+b^2} \left (b+\sqrt{-a^2+b^2}\right ) d \sqrt{e \cos (c+d x)}}-\frac{e (e \cos (c+d x))^{13/2}}{3 b d (a+b \sin (c+d x))^3}-\frac{13 e^3 (e \cos (c+d x))^{9/2} (11 a+4 b \sin (c+d x))}{84 b^3 d (a+b \sin (c+d x))^2}-\frac{39 e^5 (e \cos (c+d x))^{5/2} \left (77 a^2-20 b^2+22 a b \sin (c+d x)\right )}{280 b^5 d (a+b \sin (c+d x))}+\frac{13 e^7 \sqrt{e \cos (c+d x)} \left (21 a \left (11 a^2-6 b^2\right )-b \left (77 a^2-20 b^2\right ) \sin (c+d x)\right )}{56 b^7 d}+\frac{\left (39 a \left (11 a^4-17 a^2 b^2+6 b^4\right ) e^8\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^7 \sqrt{-a^2+b^2} d}+\frac{\left (39 a \left (11 a^4-17 a^2 b^2+6 b^4\right ) e^8\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^7 \sqrt{-a^2+b^2} d}\\ &=\frac{39 a \left (11 a^4-17 a^2 b^2+6 b^4\right ) e^{15/2} \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{e \cos (c+d x)}}{\sqrt [4]{-a^2+b^2} \sqrt{e}}\right )}{16 b^{15/2} \left (-a^2+b^2\right )^{3/4} d}+\frac{39 a \left (11 a^4-17 a^2 b^2+6 b^4\right ) e^{15/2} \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{e \cos (c+d x)}}{\sqrt [4]{-a^2+b^2} \sqrt{e}}\right )}{16 b^{15/2} \left (-a^2+b^2\right )^{3/4} d}+\frac{13 \left (231 a^4-203 a^2 b^2+20 b^4\right ) e^8 \sqrt{\cos (c+d x)} F\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{56 b^8 d \sqrt{e \cos (c+d x)}}-\frac{39 a^2 \left (11 a^4-17 a^2 b^2+6 b^4\right ) e^8 \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^8 \sqrt{-a^2+b^2} \left (b-\sqrt{-a^2+b^2}\right ) d \sqrt{e \cos (c+d x)}}+\frac{39 a^2 \left (11 a^4-17 a^2 b^2+6 b^4\right ) e^8 \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^8 \sqrt{-a^2+b^2} \left (b+\sqrt{-a^2+b^2}\right ) d \sqrt{e \cos (c+d x)}}-\frac{e (e \cos (c+d x))^{13/2}}{3 b d (a+b \sin (c+d x))^3}-\frac{13 e^3 (e \cos (c+d x))^{9/2} (11 a+4 b \sin (c+d x))}{84 b^3 d (a+b \sin (c+d x))^2}-\frac{39 e^5 (e \cos (c+d x))^{5/2} \left (77 a^2-20 b^2+22 a b \sin (c+d x)\right )}{280 b^5 d (a+b \sin (c+d x))}+\frac{13 e^7 \sqrt{e \cos (c+d x)} \left (21 a \left (11 a^2-6 b^2\right )-b \left (77 a^2-20 b^2\right ) \sin (c+d x)\right )}{56 b^7 d}\\ \end{align*}

Mathematica [C]  time = 27.7086, size = 2102, normalized size = 3.13 \[ \text{Result too large to show} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[(e*Cos[c + d*x])^(15/2)/(a + b*Sin[c + d*x])^4,x]

[Out]

((e*Cos[c + d*x])^(15/2)*((-2*(4410*a^3*b - 3418*a*b^3)*(a + b*Sqrt[1 - Cos[c + d*x]^2])*((5*a*(a^2 - b^2)*App
ellF1[1/4, 1/2, 1, 5/4, Cos[c + d*x]^2, (b^2*Cos[c + d*x]^2)/(-a^2 + b^2)]*Sqrt[Cos[c + d*x]])/(Sqrt[1 - Cos[c
+ d*x]^2]*(5*(a^2 - b^2)*AppellF1[1/4, 1/2, 1, 5/4, Cos[c + d*x]^2, (b^2*Cos[c + d*x]^2)/(-a^2 + b^2)] - 2*(2
*b^2*AppellF1[5/4, 1/2, 2, 9/4, Cos[c + d*x]^2, (b^2*Cos[c + d*x]^2)/(-a^2 + b^2)] + (-a^2 + b^2)*AppellF1[5/4
, 3/2, 1, 9/4, Cos[c + d*x]^2, (b^2*Cos[c + d*x]^2)/(-a^2 + b^2)])*Cos[c + d*x]^2)*(a^2 + b^2*(-1 + Cos[c + d*
x]^2))) - ((1/8 - I/8)*Sqrt[b]*(2*ArcTan[1 - ((1 + I)*Sqrt[b]*Sqrt[Cos[c + d*x]])/(-a^2 + b^2)^(1/4)] - 2*ArcT
an[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]]))/(-a^2 + b^2)^(3/4))*Sin[c + d*x])/(Sqrt[1 - Cos[c + d*x]^2]*(a + b
*Sin[c + d*x])) + ((5600*a^3*b - 3472*a*b^3)*(a + b*Sqrt[1 - Cos[c + d*x]^2])*Cos[2*(c + d*x)]*(((1/2 - I/2)*(
-2*a^2 + b^2)*ArcTan[1 - ((1 + I)*Sqrt[b]*Sqrt[Cos[c + d*x]])/(-a^2 + b^2)^(1/4)])/(b^(3/2)*(-a^2 + b^2)^(3/4)
) - ((1/2 - I/2)*(-2*a^2 + b^2)*ArcTan[1 + ((1 + I)*Sqrt[b]*Sqrt[Cos[c + d*x]])/(-a^2 + b^2)^(1/4)])/(b^(3/2)*
(-a^2 + b^2)^(3/4)) + (4*Sqrt[Cos[c + d*x]])/b - (4*a*AppellF1[5/4, 1/2, 1, 9/4, Cos[c + d*x]^2, (b^2*Cos[c +
d*x]^2)/(-a^2 + b^2)]*Cos[c + d*x]^(5/2))/(5*(a^2 - b^2)) + (10*a*(a^2 - b^2)*AppellF1[1/4, 1/2, 1, 5/4, Cos[c
+ d*x]^2, (b^2*Cos[c + d*x]^2)/(-a^2 + b^2)]*Sqrt[Cos[c + d*x]])/(Sqrt[1 - Cos[c + d*x]^2]*(5*(a^2 - b^2)*App
ellF1[1/4, 1/2, 1, 5/4, Cos[c + d*x]^2, (b^2*Cos[c + d*x]^2)/(-a^2 + b^2)] - 2*(2*b^2*AppellF1[5/4, 1/2, 2, 9/
4, Cos[c + d*x]^2, (b^2*Cos[c + d*x]^2)/(-a^2 + b^2)] + (-a^2 + b^2)*AppellF1[5/4, 3/2, 1, 9/4, Cos[c + d*x]^2
, (b^2*Cos[c + d*x]^2)/(-a^2 + b^2)])*Cos[c + d*x]^2)*(a^2 + b^2*(-1 + Cos[c + d*x]^2))) + ((1/4 - I/4)*(-2*a^
2 + b^2)*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]])/(b^
(3/2)*(-a^2 + b^2)^(3/4)) - ((1/4 - I/4)*(-2*a^2 + b^2)*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]])/(b^(3/2)*(-a^2 + b^2)^(3/4)))*Sin[c + d*x])/(Sqrt[1 - Cos[c + d*x]
^2]*(-1 + 2*Cos[c + d*x]^2)*(a + b*Sin[c + d*x])) - (2*(3815*a^4 - 6251*a^2*b^2 + 1300*b^4)*(a + b*Sqrt[1 - Co
s[c + d*x]^2])*((5*b*(a^2 - b^2)*AppellF1[1/4, -1/2, 1, 5/4, Cos[c + d*x]^2, (b^2*Cos[c + d*x]^2)/(-a^2 + b^2)
]*Sqrt[Cos[c + d*x]]*Sqrt[1 - Cos[c + d*x]^2])/((-5*(a^2 - b^2)*AppellF1[1/4, -1/2, 1, 5/4, Cos[c + d*x]^2, (b
^2*Cos[c + d*x]^2)/(-a^2 + b^2)] + 2*(2*b^2*AppellF1[5/4, -1/2, 2, 9/4, Cos[c + d*x]^2, (b^2*Cos[c + d*x]^2)/(
-a^2 + b^2)] + (a^2 - b^2)*AppellF1[5/4, 1/2, 1, 9/4, Cos[c + d*x]^2, (b^2*Cos[c + d*x]^2)/(-a^2 + b^2)])*Cos[
c + d*x]^2)*(a^2 + b^2*(-1 + Cos[c + d*x]^2))) + (a*(-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] - Sq
rt[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]]))/(4*Sqrt[2]*Sqrt[b]*(a^2 - b^2)^(3/4)))*Sin[c + d*x]^2)
/((1 - Cos[c + d*x]^2)*(a + b*Sin[c + d*x]))))/(560*b^7*d*Cos[c + d*x]^(15/2)) + ((e*Cos[c + d*x])^(15/2)*Sec[
c + d*x]^7*((-4*a*Cos[2*(c + d*x)])/(5*b^5) + ((-280*a^2 + 79*b^2)*Sin[c + d*x])/(42*b^6) - (-a^2 + b^2)^3/(3*
b^7*(a + b*Sin[c + d*x])^3) - (37*a*(a^2 - b^2)^2)/(12*b^7*(a + b*Sin[c + d*x])^2) + ((-a^2 + b^2)*(-393*a^2 +
76*b^2))/(24*b^7*(a + b*Sin[c + d*x])) + Sin[3*(c + d*x)]/(14*b^4)))/d

________________________________________________________________________________________

Maple [C]  time = 111.302, size = 300244, normalized size = 447.5 \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))^(15/2)/(a+b*sin(d*x+c))^4,x)

[Out]

result too large to display

________________________________________________________________________________________

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))^(15/2)/(a+b*sin(d*x+c))^4,x, algorithm="maxima")

[Out]

Timed out

________________________________________________________________________________________

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))^(15/2)/(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))**(15/2)/(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))^(15/2)/(a+b*sin(d*x+c))^4,x, algorithm="giac")

[Out]

Timed out