3.7.6 \(\int \frac {(e \cos (c+d x))^{13/2}}{(a+b \sin (c+d x))^4} \, dx\) [606]

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

[Out]

77/16*a*(3*a^2-2*b^2)*e^(13/2)*arctan(b^(1/2)*(e*cos(d*x+c))^(1/2)/(-a^2+b^2)^(1/4)/e^(1/2))/b^(13/2)/(-a^2+b^
2)^(1/4)/d-77/16*a*(3*a^2-2*b^2)*e^(13/2)*arctanh(b^(1/2)*(e*cos(d*x+c))^(1/2)/(-a^2+b^2)^(1/4)/e^(1/2))/b^(13
/2)/(-a^2+b^2)^(1/4)/d-1/3*e*(e*cos(d*x+c))^(11/2)/b/d/(a+b*sin(d*x+c))^3-11/60*e^3*(e*cos(d*x+c))^(7/2)*(9*a+
4*b*sin(d*x+c))/b^3/d/(a+b*sin(d*x+c))^2-77/120*e^5*(e*cos(d*x+c))^(3/2)*(15*a^2-4*b^2+6*a*b*sin(d*x+c))/b^5/d
/(a+b*sin(d*x+c))+77/16*a^2*(3*a^2-2*b^2)*e^7*(cos(1/2*d*x+1/2*c)^2)^(1/2)/cos(1/2*d*x+1/2*c)*EllipticPi(sin(1
/2*d*x+1/2*c),2*b/(b-(-a^2+b^2)^(1/2)),2^(1/2))*cos(d*x+c)^(1/2)/b^7/d/(b-(-a^2+b^2)^(1/2))/(e*cos(d*x+c))^(1/
2)+77/16*a^2*(3*a^2-2*b^2)*e^7*(cos(1/2*d*x+1/2*c)^2)^(1/2)/cos(1/2*d*x+1/2*c)*EllipticPi(sin(1/2*d*x+1/2*c),2
*b/(b+(-a^2+b^2)^(1/2)),2^(1/2))*cos(d*x+c)^(1/2)/b^7/d/(b+(-a^2+b^2)^(1/2))/(e*cos(d*x+c))^(1/2)-77/40*(15*a^
2-4*b^2)*e^6*(cos(1/2*d*x+1/2*c)^2)^(1/2)/cos(1/2*d*x+1/2*c)*EllipticE(sin(1/2*d*x+1/2*c),2^(1/2))*(e*cos(d*x+
c))^(1/2)/b^6/d/cos(d*x+c)^(1/2)

________________________________________________________________________________________

Rubi [A]
time = 0.87, antiderivative size = 557, normalized size of antiderivative = 1.00, number of steps used = 15, number of rules used = 12, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.480, Rules used = {2772, 2942, 2946, 2721, 2719, 2780, 2886, 2884, 335, 304, 211, 214} \begin {gather*} \frac {77 a e^{13/2} \left (3 a^2-2 b^2\right ) \text {ArcTan}\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 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 {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 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 {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} \end {gather*}

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 211

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

Rule 214

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

Rule 304

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

Rule 335

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> With[{k = Denominator[m]}, Dist[k/c, Subst[I
nt[x^(k*(m + 1) - 1)*(a + b*(x^(k*n)/c^n))^p, x], x, (c*x)^(1/k)], x]] /; FreeQ[{a, b, c, p}, x] && IGtQ[n, 0]
&& FractionQ[m] && IntBinomialQ[a, b, c, n, m, p, x]

Rule 2719

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

Rule 2721

Int[((b_)*sin[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Dist[(b*Sin[c + d*x])^n/Sin[c + d*x]^n, Int[Sin[c + d*x]
^n, x], x] /; FreeQ[{b, c, d}, x] && LtQ[-1, n, 1] && IntegerQ[2*n]

Rule 2772

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

Rule 2780

Int[Sqrt[cos[(e_.) + (f_.)*(x_)]*(g_.)]/((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]), x_Symbol] :> With[{q = Rt[-a^2
+ b^2, 2]}, Dist[a*(g/(2*b)), Int[1/(Sqrt[g*Cos[e + f*x]]*(q + b*Cos[e + f*x])), x], x] + (-Dist[a*(g/(2*b)),
Int[1/(Sqrt[g*Cos[e + f*x]]*(q - b*Cos[e + f*x])), x], x] + Dist[b*(g/f), Subst[Int[Sqrt[x]/(g^2*(a^2 - b^2)
+ b^2*x^2), x], x, g*Cos[e + f*x]], x])] /; FreeQ[{a, b, e, f, g}, x] && NeQ[a^2 - b^2, 0]

Rule 2884

Int[1/(((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])*Sqrt[(c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)]]), x_Symbol] :> Simp
[(2/(f*(a + b)*Sqrt[c + d]))*EllipticPi[2*(b/(a + b)), (1/2)*(e - Pi/2 + f*x), 2*(d/(c + d))], x] /; FreeQ[{a,
b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && GtQ[c + d, 0]

Rule 2886

Int[1/(((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])*Sqrt[(c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)]]), x_Symbol] :> Dist
[Sqrt[(c + d*Sin[e + f*x])/(c + d)]/Sqrt[c + d*Sin[e + f*x]], Int[1/((a + b*Sin[e + f*x])*Sqrt[c/(c + d) + (d/
(c + d))*Sin[e + f*x]]), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && N
eQ[c^2 - d^2, 0] &&  !GtQ[c + d, 0]

Rule 2942

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

Rule 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*} \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 ) \text {Subst}\left (\int \frac {\sqrt {x}}{\left (a^2-b^2\right ) e^2+b^2 x^2} \, dx,x,e \cos (c+d x)\right )}{16 b^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 ) \text {Subst}\left (\int \frac {x^2}{\left (a^2-b^2\right ) e^2+b^2 x^4} \, dx,x,\sqrt {e \cos (c+d x)}\right )}{8 b^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 ) \text {Subst}\left (\int \frac {1}{\sqrt {-a^2+b^2} e-b x^2} \, dx,x,\sqrt {e \cos (c+d x)}\right )}{16 b^6 d}+\frac {\left (77 a \left (3 a^2-2 b^2\right ) e^7\right ) \text {Subst}\left (\int \frac {1}{\sqrt {-a^2+b^2} e+b x^2} \, dx,x,\sqrt {e \cos (c+d x)}\right )}{16 b^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*}

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Mathematica [C] Result contains higher order function than in optimal. Order 6 vs. order 4 in optimal.
time = 33.78, size = 937, normalized size = 1.68 \begin {gather*} -\frac {77 (e \cos (c+d x))^{13/2} \left (-\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)}{-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 \tan ^{-1}\left (1-\frac {(1+i) \sqrt {b} \sqrt {\cos (c+d x)}}{\sqrt [4]{-a^2+b^2}}\right )-2 \tan ^{-1}\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 (15 a^2-4 b^2\right ) \left (a+b \sqrt {1-\cos ^2(c+d x)}\right ) \left (8 b^{5/2} 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)}{-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 \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 (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 )}{80 b^5 d \cos ^{\frac {13}{2}}(c+d x)}+\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 {-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}+\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 {-71 a^2 \cos (c+d x)+20 b^2 \cos (c+d x)}{8 b^5 (a+b \sin (c+d x))}+\frac {\sin (2 (c+d x))}{5 b^4}\right )}{d} \end {gather*}

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

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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order 4.
time = 284.54, size = 177735, normalized size = 319.09

method	result	size
default	\(\text {Expression too large to display}\)	\(177735\)

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,method=_RETURNVERBOSE)

[Out]

result too large to display

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Maxima [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

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)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

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

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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

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

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Giac [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

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

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {{\left (e\,\cos \left (c+d\,x\right )\right )}^{13/2}}{{\left (a+b\,\sin \left (c+d\,x\right )\right )}^4} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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