3.1.78 \(\int \frac {(e \sin (c+d x))^{13/2}}{(a+b \cos (c+d x))^3} \, dx\) [78]

Optimal. Leaf size=590 \[ \frac {11 \left (9 a^4-11 a^2 b^2+2 b^4\right ) e^{13/2} \text {ArcTan}\left (\frac {\sqrt {b} \sqrt {e \sin (c+d x)}}{\sqrt [4]{-a^2+b^2} \sqrt {e}}\right )}{8 b^{13/2} \sqrt [4]{-a^2+b^2} d}-\frac {11 \left (9 a^4-11 a^2 b^2+2 b^4\right ) e^{13/2} \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {e \sin (c+d x)}}{\sqrt [4]{-a^2+b^2} \sqrt {e}}\right )}{8 b^{13/2} \sqrt [4]{-a^2+b^2} d}-\frac {11 a \left (9 a^4-11 a^2 b^2+2 b^4\right ) e^7 \Pi \left (\frac {2 b}{b-\sqrt {-a^2+b^2}};\left .\frac {1}{2} \left (c-\frac {\pi }{2}+d x\right )\right |2\right ) \sqrt {\sin (c+d x)}}{8 b^7 \left (b-\sqrt {-a^2+b^2}\right ) d \sqrt {e \sin (c+d x)}}-\frac {11 a \left (9 a^4-11 a^2 b^2+2 b^4\right ) e^7 \Pi \left (\frac {2 b}{b+\sqrt {-a^2+b^2}};\left .\frac {1}{2} \left (c-\frac {\pi }{2}+d x\right )\right |2\right ) \sqrt {\sin (c+d x)}}{8 b^7 \left (b+\sqrt {-a^2+b^2}\right ) d \sqrt {e \sin (c+d x)}}+\frac {11 a \left (45 a^2-37 b^2\right ) e^6 E\left (\left .\frac {1}{2} \left (c-\frac {\pi }{2}+d x\right )\right |2\right ) \sqrt {e \sin (c+d x)}}{20 b^6 d \sqrt {\sin (c+d x)}}-\frac {11 e^5 \left (5 \left (9 a^2-2 b^2\right )-27 a b \cos (c+d x)\right ) (e \sin (c+d x))^{3/2}}{60 b^5 d}+\frac {11 e^3 (9 a+2 b \cos (c+d x)) (e \sin (c+d x))^{7/2}}{28 b^3 d (a+b \cos (c+d x))}+\frac {e (e \sin (c+d x))^{11/2}}{2 b d (a+b \cos (c+d x))^2} \]

[Out]

11/8*(9*a^4-11*a^2*b^2+2*b^4)*e^(13/2)*arctan(b^(1/2)*(e*sin(d*x+c))^(1/2)/(-a^2+b^2)^(1/4)/e^(1/2))/b^(13/2)/
(-a^2+b^2)^(1/4)/d-11/8*(9*a^4-11*a^2*b^2+2*b^4)*e^(13/2)*arctanh(b^(1/2)*(e*sin(d*x+c))^(1/2)/(-a^2+b^2)^(1/4
)/e^(1/2))/b^(13/2)/(-a^2+b^2)^(1/4)/d-11/60*e^5*(45*a^2-10*b^2-27*a*b*cos(d*x+c))*(e*sin(d*x+c))^(3/2)/b^5/d+
11/28*e^3*(9*a+2*b*cos(d*x+c))*(e*sin(d*x+c))^(7/2)/b^3/d/(a+b*cos(d*x+c))+1/2*e*(e*sin(d*x+c))^(11/2)/b/d/(a+
b*cos(d*x+c))^2+11/8*a*(9*a^4-11*a^2*b^2+2*b^4)*e^7*(sin(1/2*c+1/4*Pi+1/2*d*x)^2)^(1/2)/sin(1/2*c+1/4*Pi+1/2*d
*x)*EllipticPi(cos(1/2*c+1/4*Pi+1/2*d*x),2*b/(b-(-a^2+b^2)^(1/2)),2^(1/2))*sin(d*x+c)^(1/2)/b^7/d/(b-(-a^2+b^2
)^(1/2))/(e*sin(d*x+c))^(1/2)+11/8*a*(9*a^4-11*a^2*b^2+2*b^4)*e^7*(sin(1/2*c+1/4*Pi+1/2*d*x)^2)^(1/2)/sin(1/2*
c+1/4*Pi+1/2*d*x)*EllipticPi(cos(1/2*c+1/4*Pi+1/2*d*x),2*b/(b+(-a^2+b^2)^(1/2)),2^(1/2))*sin(d*x+c)^(1/2)/b^7/
d/(b+(-a^2+b^2)^(1/2))/(e*sin(d*x+c))^(1/2)-11/20*a*(45*a^2-37*b^2)*e^6*(sin(1/2*c+1/4*Pi+1/2*d*x)^2)^(1/2)/si
n(1/2*c+1/4*Pi+1/2*d*x)*EllipticE(cos(1/2*c+1/4*Pi+1/2*d*x),2^(1/2))*(e*sin(d*x+c))^(1/2)/b^6/d/sin(d*x+c)^(1/
2)

________________________________________________________________________________________

Rubi [A]
time = 0.93, antiderivative size = 590, normalized size of antiderivative = 1.00, number of steps used = 15, number of rules used = 13, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.520, Rules used = {2772, 2942, 2944, 2946, 2721, 2719, 2780, 2886, 2884, 335, 304, 211, 214} \begin {gather*} \frac {11 a e^6 \left (45 a^2-37 b^2\right ) E\left (\left .\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right )\right |2\right ) \sqrt {e \sin (c+d x)}}{20 b^6 d \sqrt {\sin (c+d x)}}-\frac {11 e^5 (e \sin (c+d x))^{3/2} \left (5 \left (9 a^2-2 b^2\right )-27 a b \cos (c+d x)\right )}{60 b^5 d}+\frac {11 e^{13/2} \left (9 a^4-11 a^2 b^2+2 b^4\right ) \text {ArcTan}\left (\frac {\sqrt {b} \sqrt {e \sin (c+d x)}}{\sqrt {e} \sqrt [4]{b^2-a^2}}\right )}{8 b^{13/2} d \sqrt [4]{b^2-a^2}}-\frac {11 e^{13/2} \left (9 a^4-11 a^2 b^2+2 b^4\right ) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {e \sin (c+d x)}}{\sqrt {e} \sqrt [4]{b^2-a^2}}\right )}{8 b^{13/2} d \sqrt [4]{b^2-a^2}}-\frac {11 a e^7 \left (9 a^4-11 a^2 b^2+2 b^4\right ) \sqrt {\sin (c+d x)} \Pi \left (\frac {2 b}{b-\sqrt {b^2-a^2}};\left .\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right )\right |2\right )}{8 b^7 d \left (b-\sqrt {b^2-a^2}\right ) \sqrt {e \sin (c+d x)}}-\frac {11 a e^7 \left (9 a^4-11 a^2 b^2+2 b^4\right ) \sqrt {\sin (c+d x)} \Pi \left (\frac {2 b}{b+\sqrt {b^2-a^2}};\left .\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right )\right |2\right )}{8 b^7 d \left (\sqrt {b^2-a^2}+b\right ) \sqrt {e \sin (c+d x)}}+\frac {11 e^3 (e \sin (c+d x))^{7/2} (9 a+2 b \cos (c+d x))}{28 b^3 d (a+b \cos (c+d x))}+\frac {e (e \sin (c+d x))^{11/2}}{2 b d (a+b \cos (c+d x))^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(e*Sin[c + d*x])^(13/2)/(a + b*Cos[c + d*x])^3,x]

[Out]

(11*(9*a^4 - 11*a^2*b^2 + 2*b^4)*e^(13/2)*ArcTan[(Sqrt[b]*Sqrt[e*Sin[c + d*x]])/((-a^2 + b^2)^(1/4)*Sqrt[e])])
/(8*b^(13/2)*(-a^2 + b^2)^(1/4)*d) - (11*(9*a^4 - 11*a^2*b^2 + 2*b^4)*e^(13/2)*ArcTanh[(Sqrt[b]*Sqrt[e*Sin[c +
d*x]])/((-a^2 + b^2)^(1/4)*Sqrt[e])])/(8*b^(13/2)*(-a^2 + b^2)^(1/4)*d) - (11*a*(9*a^4 - 11*a^2*b^2 + 2*b^4)*
e^7*EllipticPi[(2*b)/(b - Sqrt[-a^2 + b^2]), (c - Pi/2 + d*x)/2, 2]*Sqrt[Sin[c + d*x]])/(8*b^7*(b - Sqrt[-a^2
+ b^2])*d*Sqrt[e*Sin[c + d*x]]) - (11*a*(9*a^4 - 11*a^2*b^2 + 2*b^4)*e^7*EllipticPi[(2*b)/(b + Sqrt[-a^2 + b^2
]), (c - Pi/2 + d*x)/2, 2]*Sqrt[Sin[c + d*x]])/(8*b^7*(b + Sqrt[-a^2 + b^2])*d*Sqrt[e*Sin[c + d*x]]) + (11*a*(
45*a^2 - 37*b^2)*e^6*EllipticE[(c - Pi/2 + d*x)/2, 2]*Sqrt[e*Sin[c + d*x]])/(20*b^6*d*Sqrt[Sin[c + d*x]]) - (1
1*e^5*(5*(9*a^2 - 2*b^2) - 27*a*b*Cos[c + d*x])*(e*Sin[c + d*x])^(3/2))/(60*b^5*d) + (11*e^3*(9*a + 2*b*Cos[c
+ d*x])*(e*Sin[c + d*x])^(7/2))/(28*b^3*d*(a + b*Cos[c + d*x])) + (e*(e*Sin[c + d*x])^(11/2))/(2*b*d*(a + b*Co
s[c + d*x])^2)

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 2944

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 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 \sin (c+d x))^{13/2}}{(a+b \cos (c+d x))^3} \, dx &=\frac {e (e \sin (c+d x))^{11/2}}{2 b d (a+b \cos (c+d x))^2}-\frac {\left (11 e^2\right ) \int \frac {\cos (c+d x) (e \sin (c+d x))^{9/2}}{(a+b \cos (c+d x))^2} \, dx}{4 b}\\ &=\frac {11 e^3 (9 a+2 b \cos (c+d x)) (e \sin (c+d x))^{7/2}}{28 b^3 d (a+b \cos (c+d x))}+\frac {e (e \sin (c+d x))^{11/2}}{2 b d (a+b \cos (c+d x))^2}+\frac {\left (11 e^4\right ) \int \frac {\left (-b-\frac {9}{2} a \cos (c+d x)\right ) (e \sin (c+d x))^{5/2}}{a+b \cos (c+d x)} \, dx}{4 b^3}\\ &=-\frac {11 e^5 \left (5 \left (9 a^2-2 b^2\right )-27 a b \cos (c+d x)\right ) (e \sin (c+d x))^{3/2}}{60 b^5 d}+\frac {11 e^3 (9 a+2 b \cos (c+d x)) (e \sin (c+d x))^{7/2}}{28 b^3 d (a+b \cos (c+d x))}+\frac {e (e \sin (c+d x))^{11/2}}{2 b d (a+b \cos (c+d x))^2}+\frac {\left (11 e^6\right ) \int \frac {\left (\frac {1}{2} b \left (9 a^2-5 b^2\right )+\frac {1}{4} a \left (45 a^2-37 b^2\right ) \cos (c+d x)\right ) \sqrt {e \sin (c+d x)}}{a+b \cos (c+d x)} \, dx}{10 b^5}\\ &=-\frac {11 e^5 \left (5 \left (9 a^2-2 b^2\right )-27 a b \cos (c+d x)\right ) (e \sin (c+d x))^{3/2}}{60 b^5 d}+\frac {11 e^3 (9 a+2 b \cos (c+d x)) (e \sin (c+d x))^{7/2}}{28 b^3 d (a+b \cos (c+d x))}+\frac {e (e \sin (c+d x))^{11/2}}{2 b d (a+b \cos (c+d x))^2}+\frac {\left (11 a \left (45 a^2-37 b^2\right ) e^6\right ) \int \sqrt {e \sin (c+d x)} \, dx}{40 b^6}-\frac {\left (11 \left (9 a^4-11 a^2 b^2+2 b^4\right ) e^6\right ) \int \frac {\sqrt {e \sin (c+d x)}}{a+b \cos (c+d x)} \, dx}{8 b^6}\\ &=-\frac {11 e^5 \left (5 \left (9 a^2-2 b^2\right )-27 a b \cos (c+d x)\right ) (e \sin (c+d x))^{3/2}}{60 b^5 d}+\frac {11 e^3 (9 a+2 b \cos (c+d x)) (e \sin (c+d x))^{7/2}}{28 b^3 d (a+b \cos (c+d x))}+\frac {e (e \sin (c+d x))^{11/2}}{2 b d (a+b \cos (c+d x))^2}+\frac {\left (11 a \left (9 a^4-11 a^2 b^2+2 b^4\right ) e^7\right ) \int \frac {1}{\sqrt {e \sin (c+d x)} \left (\sqrt {-a^2+b^2}-b \sin (c+d x)\right )} \, dx}{16 b^7}-\frac {\left (11 a \left (9 a^4-11 a^2 b^2+2 b^4\right ) e^7\right ) \int \frac {1}{\sqrt {e \sin (c+d x)} \left (\sqrt {-a^2+b^2}+b \sin (c+d x)\right )} \, dx}{16 b^7}+\frac {\left (11 \left (9 a^4-11 a^2 b^2+2 b^4\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 \sin (c+d x)\right )}{8 b^5 d}+\frac {\left (11 a \left (45 a^2-37 b^2\right ) e^6 \sqrt {e \sin (c+d x)}\right ) \int \sqrt {\sin (c+d x)} \, dx}{40 b^6 \sqrt {\sin (c+d x)}}\\ &=\frac {11 a \left (45 a^2-37 b^2\right ) e^6 E\left (\left .\frac {1}{2} \left (c-\frac {\pi }{2}+d x\right )\right |2\right ) \sqrt {e \sin (c+d x)}}{20 b^6 d \sqrt {\sin (c+d x)}}-\frac {11 e^5 \left (5 \left (9 a^2-2 b^2\right )-27 a b \cos (c+d x)\right ) (e \sin (c+d x))^{3/2}}{60 b^5 d}+\frac {11 e^3 (9 a+2 b \cos (c+d x)) (e \sin (c+d x))^{7/2}}{28 b^3 d (a+b \cos (c+d x))}+\frac {e (e \sin (c+d x))^{11/2}}{2 b d (a+b \cos (c+d x))^2}+\frac {\left (11 \left (9 a^4-11 a^2 b^2+2 b^4\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 \sin (c+d x)}\right )}{4 b^5 d}+\frac {\left (11 a \left (9 a^4-11 a^2 b^2+2 b^4\right ) e^7 \sqrt {\sin (c+d x)}\right ) \int \frac {1}{\sqrt {\sin (c+d x)} \left (\sqrt {-a^2+b^2}-b \sin (c+d x)\right )} \, dx}{16 b^7 \sqrt {e \sin (c+d x)}}-\frac {\left (11 a \left (9 a^4-11 a^2 b^2+2 b^4\right ) e^7 \sqrt {\sin (c+d x)}\right ) \int \frac {1}{\sqrt {\sin (c+d x)} \left (\sqrt {-a^2+b^2}+b \sin (c+d x)\right )} \, dx}{16 b^7 \sqrt {e \sin (c+d x)}}\\ &=-\frac {11 a \left (9 a^4-11 a^2 b^2+2 b^4\right ) e^7 \Pi \left (\frac {2 b}{b-\sqrt {-a^2+b^2}};\left .\frac {1}{2} \left (c-\frac {\pi }{2}+d x\right )\right |2\right ) \sqrt {\sin (c+d x)}}{8 b^7 \left (b-\sqrt {-a^2+b^2}\right ) d \sqrt {e \sin (c+d x)}}-\frac {11 a \left (9 a^4-11 a^2 b^2+2 b^4\right ) e^7 \Pi \left (\frac {2 b}{b+\sqrt {-a^2+b^2}};\left .\frac {1}{2} \left (c-\frac {\pi }{2}+d x\right )\right |2\right ) \sqrt {\sin (c+d x)}}{8 b^7 \left (b+\sqrt {-a^2+b^2}\right ) d \sqrt {e \sin (c+d x)}}+\frac {11 a \left (45 a^2-37 b^2\right ) e^6 E\left (\left .\frac {1}{2} \left (c-\frac {\pi }{2}+d x\right )\right |2\right ) \sqrt {e \sin (c+d x)}}{20 b^6 d \sqrt {\sin (c+d x)}}-\frac {11 e^5 \left (5 \left (9 a^2-2 b^2\right )-27 a b \cos (c+d x)\right ) (e \sin (c+d x))^{3/2}}{60 b^5 d}+\frac {11 e^3 (9 a+2 b \cos (c+d x)) (e \sin (c+d x))^{7/2}}{28 b^3 d (a+b \cos (c+d x))}+\frac {e (e \sin (c+d x))^{11/2}}{2 b d (a+b \cos (c+d x))^2}-\frac {\left (11 \left (9 a^4-11 a^2 b^2+2 b^4\right ) e^7\right ) \text {Subst}\left (\int \frac {1}{\sqrt {-a^2+b^2} e-b x^2} \, dx,x,\sqrt {e \sin (c+d x)}\right )}{8 b^6 d}+\frac {\left (11 \left (9 a^4-11 a^2 b^2+2 b^4\right ) e^7\right ) \text {Subst}\left (\int \frac {1}{\sqrt {-a^2+b^2} e+b x^2} \, dx,x,\sqrt {e \sin (c+d x)}\right )}{8 b^6 d}\\ &=\frac {11 \left (9 a^4-11 a^2 b^2+2 b^4\right ) e^{13/2} \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {e \sin (c+d x)}}{\sqrt [4]{-a^2+b^2} \sqrt {e}}\right )}{8 b^{13/2} \sqrt [4]{-a^2+b^2} d}-\frac {11 \left (9 a^4-11 a^2 b^2+2 b^4\right ) e^{13/2} \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {e \sin (c+d x)}}{\sqrt [4]{-a^2+b^2} \sqrt {e}}\right )}{8 b^{13/2} \sqrt [4]{-a^2+b^2} d}-\frac {11 a \left (9 a^4-11 a^2 b^2+2 b^4\right ) e^7 \Pi \left (\frac {2 b}{b-\sqrt {-a^2+b^2}};\left .\frac {1}{2} \left (c-\frac {\pi }{2}+d x\right )\right |2\right ) \sqrt {\sin (c+d x)}}{8 b^7 \left (b-\sqrt {-a^2+b^2}\right ) d \sqrt {e \sin (c+d x)}}-\frac {11 a \left (9 a^4-11 a^2 b^2+2 b^4\right ) e^7 \Pi \left (\frac {2 b}{b+\sqrt {-a^2+b^2}};\left .\frac {1}{2} \left (c-\frac {\pi }{2}+d x\right )\right |2\right ) \sqrt {\sin (c+d x)}}{8 b^7 \left (b+\sqrt {-a^2+b^2}\right ) d \sqrt {e \sin (c+d x)}}+\frac {11 a \left (45 a^2-37 b^2\right ) e^6 E\left (\left .\frac {1}{2} \left (c-\frac {\pi }{2}+d x\right )\right |2\right ) \sqrt {e \sin (c+d x)}}{20 b^6 d \sqrt {\sin (c+d x)}}-\frac {11 e^5 \left (5 \left (9 a^2-2 b^2\right )-27 a b \cos (c+d x)\right ) (e \sin (c+d x))^{3/2}}{60 b^5 d}+\frac {11 e^3 (9 a+2 b \cos (c+d x)) (e \sin (c+d x))^{7/2}}{28 b^3 d (a+b \cos (c+d x))}+\frac {e (e \sin (c+d x))^{11/2}}{2 b d (a+b \cos (c+d x))^2}\\ \end {align*}

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Mathematica [C] Result contains higher order function than in optimal. Order 6 vs. order 4 in optimal.
time = 35.16, size = 930, normalized size = 1.58 \begin {gather*} \frac {11 (e \sin (c+d x))^{13/2} \left (\frac {\left (45 a^3-37 a b^2\right ) \cos ^2(c+d x) \left (3 \sqrt {2} a \left (a^2-b^2\right )^{3/4} \left (2 \text {ArcTan}\left (1-\frac {\sqrt {2} \sqrt {b} \sqrt {\sin (c+d x)}}{\sqrt [4]{a^2-b^2}}\right )-2 \text {ArcTan}\left (1+\frac {\sqrt {2} \sqrt {b} \sqrt {\sin (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 {\sin (c+d x)}+b \sin (c+d x)\right )+\log \left (\sqrt {a^2-b^2}+\sqrt {2} \sqrt {b} \sqrt [4]{a^2-b^2} \sqrt {\sin (c+d x)}+b \sin (c+d x)\right )\right )+8 b^{5/2} F_1\left (\frac {3}{4};-\frac {1}{2},1;\frac {7}{4};\sin ^2(c+d x),\frac {b^2 \sin ^2(c+d x)}{-a^2+b^2}\right ) \sin ^{\frac {3}{2}}(c+d x)\right ) \left (a+b \sqrt {1-\sin ^2(c+d x)}\right )}{12 b^{3/2} \left (-a^2+b^2\right ) (a+b \cos (c+d x)) \left (1-\sin ^2(c+d x)\right )}+\frac {2 \left (18 a^2 b-10 b^3\right ) \cos (c+d x) \left (\frac {\left (\frac {1}{8}+\frac {i}{8}\right ) \left (2 \text {ArcTan}\left (1-\frac {(1+i) \sqrt {b} \sqrt {\sin (c+d x)}}{\sqrt [4]{-a^2+b^2}}\right )-2 \text {ArcTan}\left (1+\frac {(1+i) \sqrt {b} \sqrt {\sin (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 {\sin (c+d x)}+i b \sin (c+d x)\right )+\log \left (\sqrt {-a^2+b^2}+(1+i) \sqrt {b} \sqrt [4]{-a^2+b^2} \sqrt {\sin (c+d x)}+i b \sin (c+d x)\right )\right )}{\sqrt {b} \sqrt [4]{-a^2+b^2}}+\frac {a F_1\left (\frac {3}{4};\frac {1}{2},1;\frac {7}{4};\sin ^2(c+d x),\frac {b^2 \sin ^2(c+d x)}{-a^2+b^2}\right ) \sin ^{\frac {3}{2}}(c+d x)}{3 \left (a^2-b^2\right )}\right ) \left (a+b \sqrt {1-\sin ^2(c+d x)}\right )}{(a+b \cos (c+d x)) \sqrt {1-\sin ^2(c+d x)}}\right )}{40 b^5 d \sin ^{\frac {13}{2}}(c+d x)}+\frac {\csc ^6(c+d x) (e \sin (c+d x))^{13/2} \left (\frac {\left (-168 a^2+65 b^2\right ) \sin (c+d x)}{42 b^5}-\frac {19 \left (a^3 \sin (c+d x)-a b^2 \sin (c+d x)\right )}{4 b^5 (a+b \cos (c+d x))}+\frac {a^4 \sin (c+d x)-2 a^2 b^2 \sin (c+d x)+b^4 \sin (c+d x)}{2 b^5 (a+b \cos (c+d x))^2}+\frac {3 a \sin (2 (c+d x))}{5 b^4}-\frac {\sin (3 (c+d x))}{14 b^3}\right )}{d} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

Integrate[(e*Sin[c + d*x])^(13/2)/(a + b*Cos[c + d*x])^3,x]

[Out]

(11*(e*Sin[c + d*x])^(13/2)*(((45*a^3 - 37*a*b^2)*Cos[c + d*x]^2*(3*Sqrt[2]*a*(a^2 - b^2)^(3/4)*(2*ArcTan[1 -
(Sqrt[2]*Sqrt[b]*Sqrt[Sin[c + d*x]])/(a^2 - b^2)^(1/4)] - 2*ArcTan[1 + (Sqrt[2]*Sqrt[b]*Sqrt[Sin[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[Sin[c + d*x]] + b*Sin[c + d*x]
] + Log[Sqrt[a^2 - b^2] + Sqrt[2]*Sqrt[b]*(a^2 - b^2)^(1/4)*Sqrt[Sin[c + d*x]] + b*Sin[c + d*x]]) + 8*b^(5/2)*
AppellF1[3/4, -1/2, 1, 7/4, Sin[c + d*x]^2, (b^2*Sin[c + d*x]^2)/(-a^2 + b^2)]*Sin[c + d*x]^(3/2))*(a + b*Sqrt
[1 - Sin[c + d*x]^2]))/(12*b^(3/2)*(-a^2 + b^2)*(a + b*Cos[c + d*x])*(1 - Sin[c + d*x]^2)) + (2*(18*a^2*b - 10
*b^3)*Cos[c + d*x]*(((1/8 + I/8)*(2*ArcTan[1 - ((1 + I)*Sqrt[b]*Sqrt[Sin[c + d*x]])/(-a^2 + b^2)^(1/4)] - 2*Ar
cTan[1 + ((1 + I)*Sqrt[b]*Sqrt[Sin[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[Sin[c + d*x]] + I*b*Sin[c + d*x]] + Log[Sqrt[-a^2 + b^2] + (1 + I)*Sqrt[b]*(-a^2 + b^2)^(
1/4)*Sqrt[Sin[c + d*x]] + I*b*Sin[c + d*x]]))/(Sqrt[b]*(-a^2 + b^2)^(1/4)) + (a*AppellF1[3/4, 1/2, 1, 7/4, Sin
[c + d*x]^2, (b^2*Sin[c + d*x]^2)/(-a^2 + b^2)]*Sin[c + d*x]^(3/2))/(3*(a^2 - b^2)))*(a + b*Sqrt[1 - Sin[c + d
*x]^2]))/((a + b*Cos[c + d*x])*Sqrt[1 - Sin[c + d*x]^2])))/(40*b^5*d*Sin[c + d*x]^(13/2)) + (Csc[c + d*x]^6*(e
*Sin[c + d*x])^(13/2)*(((-168*a^2 + 65*b^2)*Sin[c + d*x])/(42*b^5) - (19*(a^3*Sin[c + d*x] - a*b^2*Sin[c + d*x
]))/(4*b^5*(a + b*Cos[c + d*x])) + (a^4*Sin[c + d*x] - 2*a^2*b^2*Sin[c + d*x] + b^4*Sin[c + d*x])/(2*b^5*(a +
b*Cos[c + d*x])^2) + (3*a*Sin[2*(c + d*x)])/(5*b^4) - Sin[3*(c + d*x)]/(14*b^3)))/d

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(3820\) vs. \(2(607)=1214\).
time = 0.76, size = 3821, normalized size = 6.48

method	result	size
default	\(\text {Expression too large to display}\)	\(3821\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((e*sin(d*x+c))^(13/2)/(a+b*cos(d*x+c))^3,x,method=_RETURNVERBOSE)

[Out]

(2/7*e^3/b^3*(e*sin(d*x+c))^(7/2)-4*e^5/b^5*(e*sin(d*x+c))^(3/2)*a^2+4/3*e^5/b^3*(e*sin(d*x+c))^(3/2)-21/4*e^7
/b^3/(-b^2*cos(d*x+c)^2*e^2+a^2*e^2)^2*(e*sin(d*x+c))^(7/2)*a^4+23/4*e^7/b/(-b^2*cos(d*x+c)^2*e^2+a^2*e^2)^2*(
e*sin(d*x+c))^(7/2)*a^2-1/2*e^7*b/(-b^2*cos(d*x+c)^2*e^2+a^2*e^2)^2*(e*sin(d*x+c))^(7/2)-17/4*e^9/b^5/(-b^2*co
s(d*x+c)^2*e^2+a^2*e^2)^2*(e*sin(d*x+c))^(3/2)*a^6+9*e^9/b^3/(-b^2*cos(d*x+c)^2*e^2+a^2*e^2)^2*(e*sin(d*x+c))^
(3/2)*a^4-21/4*e^9/b/(-b^2*cos(d*x+c)^2*e^2+a^2*e^2)^2*(e*sin(d*x+c))^(3/2)*a^2+1/2*e^9*b/(-b^2*cos(d*x+c)^2*e
^2+a^2*e^2)^2*(e*sin(d*x+c))^(3/2)+99/32*e^7/b^7/(e^2*(a^2-b^2)/b^2)^(1/4)*2^(1/2)*a^4*ln((e*sin(d*x+c)-(e^2*(
a^2-b^2)/b^2)^(1/4)*(e*sin(d*x+c))^(1/2)*2^(1/2)+(e^2*(a^2-b^2)/b^2)^(1/2))/(e*sin(d*x+c)+(e^2*(a^2-b^2)/b^2)^
(1/4)*(e*sin(d*x+c))^(1/2)*2^(1/2)+(e^2*(a^2-b^2)/b^2)^(1/2)))+99/16*e^7/b^7/(e^2*(a^2-b^2)/b^2)^(1/4)*2^(1/2)
*a^4*arctan(2^(1/2)/(e^2*(a^2-b^2)/b^2)^(1/4)*(e*sin(d*x+c))^(1/2)+1)+99/16*e^7/b^7/(e^2*(a^2-b^2)/b^2)^(1/4)*
2^(1/2)*a^4*arctan(2^(1/2)/(e^2*(a^2-b^2)/b^2)^(1/4)*(e*sin(d*x+c))^(1/2)-1)-121/32*e^7/b^5/(e^2*(a^2-b^2)/b^2
)^(1/4)*2^(1/2)*a^2*ln((e*sin(d*x+c)-(e^2*(a^2-b^2)/b^2)^(1/4)*(e*sin(d*x+c))^(1/2)*2^(1/2)+(e^2*(a^2-b^2)/b^2
)^(1/2))/(e*sin(d*x+c)+(e^2*(a^2-b^2)/b^2)^(1/4)*(e*sin(d*x+c))^(1/2)*2^(1/2)+(e^2*(a^2-b^2)/b^2)^(1/2)))-121/
16*e^7/b^5/(e^2*(a^2-b^2)/b^2)^(1/4)*2^(1/2)*a^2*arctan(2^(1/2)/(e^2*(a^2-b^2)/b^2)^(1/4)*(e*sin(d*x+c))^(1/2)
+1)-121/16*e^7/b^5/(e^2*(a^2-b^2)/b^2)^(1/4)*2^(1/2)*a^2*arctan(2^(1/2)/(e^2*(a^2-b^2)/b^2)^(1/4)*(e*sin(d*x+c
))^(1/2)-1)+11/16*e^7/b^3/(e^2*(a^2-b^2)/b^2)^(1/4)*2^(1/2)*ln((e*sin(d*x+c)-(e^2*(a^2-b^2)/b^2)^(1/4)*(e*sin(
d*x+c))^(1/2)*2^(1/2)+(e^2*(a^2-b^2)/b^2)^(1/2))/(e*sin(d*x+c)+(e^2*(a^2-b^2)/b^2)^(1/4)*(e*sin(d*x+c))^(1/2)*
2^(1/2)+(e^2*(a^2-b^2)/b^2)^(1/2)))+11/8*e^7/b^3/(e^2*(a^2-b^2)/b^2)^(1/4)*2^(1/2)*arctan(2^(1/2)/(e^2*(a^2-b^
2)/b^2)^(1/4)*(e*sin(d*x+c))^(1/2)+1)+11/8*e^7/b^3/(e^2*(a^2-b^2)/b^2)^(1/4)*2^(1/2)*arctan(2^(1/2)/(e^2*(a^2-
b^2)/b^2)^(1/4)*(e*sin(d*x+c))^(1/2)-1)-(cos(d*x+c)^2*e*sin(d*x+c))^(1/2)*e^7*a*(1/5/b^6/(cos(d*x+c)^2*e*sin(d
*x+c))^(1/2)*(100*(-sin(d*x+c)+1)^(1/2)*(2*sin(d*x+c)+2)^(1/2)*sin(d*x+c)^(1/2)*EllipticE((-sin(d*x+c)+1)^(1/2
),1/2*2^(1/2))*a^2-78*(-sin(d*x+c)+1)^(1/2)*(2*sin(d*x+c)+2)^(1/2)*sin(d*x+c)^(1/2)*EllipticE((-sin(d*x+c)+1)^
(1/2),1/2*2^(1/2))*b^2-50*(-sin(d*x+c)+1)^(1/2)*(2*sin(d*x+c)+2)^(1/2)*sin(d*x+c)^(1/2)*EllipticF((-sin(d*x+c)
+1)^(1/2),1/2*2^(1/2))*a^2+39*(-sin(d*x+c)+1)^(1/2)*(2*sin(d*x+c)+2)^(1/2)*sin(d*x+c)^(1/2)*EllipticF((-sin(d*
x+c)+1)^(1/2),1/2*2^(1/2))*b^2+6*b^2*cos(d*x+c)^4-6*cos(d*x+c)^2*b^2)+3*(7*a^4-10*a^2*b^2+3*b^4)/b^6*(-1/2/b^2
*(-sin(d*x+c)+1)^(1/2)*(2*sin(d*x+c)+2)^(1/2)*sin(d*x+c)^(1/2)/(cos(d*x+c)^2*e*sin(d*x+c))^(1/2)/(1-(-a^2+b^2)
^(1/2)/b)*EllipticPi((-sin(d*x+c)+1)^(1/2),1/(1-(-a^2+b^2)^(1/2)/b),1/2*2^(1/2))-1/2/b^2*(-sin(d*x+c)+1)^(1/2)
*(2*sin(d*x+c)+2)^(1/2)*sin(d*x+c)^(1/2)/(cos(d*x+c)^2*e*sin(d*x+c))^(1/2)/(1+(-a^2+b^2)^(1/2)/b)*EllipticPi((
-sin(d*x+c)+1)^(1/2),1/(1+(-a^2+b^2)^(1/2)/b),1/2*2^(1/2)))-3*(5*a^6-11*a^4*b^2+7*a^2*b^4-b^6)/b^6*(1/2*b^2/e/
a^2/(a^2-b^2)*sin(d*x+c)*(cos(d*x+c)^2*e*sin(d*x+c))^(1/2)/(-cos(d*x+c)^2*b^2+a^2)-1/2/a^2/(a^2-b^2)*(-sin(d*x
+c)+1)^(1/2)*(2*sin(d*x+c)+2)^(1/2)*sin(d*x+c)^(1/2)/(cos(d*x+c)^2*e*sin(d*x+c))^(1/2)*EllipticE((-sin(d*x+c)+
1)^(1/2),1/2*2^(1/2))+1/4/a^2/(a^2-b^2)*(-sin(d*x+c)+1)^(1/2)*(2*sin(d*x+c)+2)^(1/2)*sin(d*x+c)^(1/2)/(cos(d*x
+c)^2*e*sin(d*x+c))^(1/2)*EllipticF((-sin(d*x+c)+1)^(1/2),1/2*2^(1/2))-3/8/(a^2-b^2)/b^2*(-sin(d*x+c)+1)^(1/2)
*(2*sin(d*x+c)+2)^(1/2)*sin(d*x+c)^(1/2)/(cos(d*x+c)^2*e*sin(d*x+c))^(1/2)/(1-(-a^2+b^2)^(1/2)/b)*EllipticPi((
-sin(d*x+c)+1)^(1/2),1/(1-(-a^2+b^2)^(1/2)/b),1/2*2^(1/2))+1/4/a^2/(a^2-b^2)*(-sin(d*x+c)+1)^(1/2)*(2*sin(d*x+
c)+2)^(1/2)*sin(d*x+c)^(1/2)/(cos(d*x+c)^2*e*sin(d*x+c))^(1/2)/(1-(-a^2+b^2)^(1/2)/b)*EllipticPi((-sin(d*x+c)+
1)^(1/2),1/(1-(-a^2+b^2)^(1/2)/b),1/2*2^(1/2))-3/8/(a^2-b^2)/b^2*(-sin(d*x+c)+1)^(1/2)*(2*sin(d*x+c)+2)^(1/2)*
sin(d*x+c)^(1/2)/(cos(d*x+c)^2*e*sin(d*x+c))^(1/2)/(1+(-a^2+b^2)^(1/2)/b)*EllipticPi((-sin(d*x+c)+1)^(1/2),1/(
1+(-a^2+b^2)^(1/2)/b),1/2*2^(1/2))+1/4/a^2/(a^2-b^2)*(-sin(d*x+c)+1)^(1/2)*(2*sin(d*x+c)+2)^(1/2)*sin(d*x+c)^(
1/2)/(cos(d*x+c)^2*e*sin(d*x+c))^(1/2)/(1+(-a^2+b^2)^(1/2)/b)*EllipticPi((-sin(d*x+c)+1)^(1/2),1/(1+(-a^2+b^2)
^(1/2)/b),1/2*2^(1/2)))+4*a^2*(a^6-3*a^4*b^2+3*a^2*b^4-b^6)/b^6*(1/4*b^2/e/a^2/(a^2-b^2)*sin(d*x+c)*(cos(d*x+c
)^2*e*sin(d*x+c))^(1/2)/(-cos(d*x+c)^2*b^2+a^2)^2+1/16*b^2*(11*a^2-6*b^2)/a^4/(a^2-b^2)^2/e*sin(d*x+c)*(cos(d*
x+c)^2*e*sin(d*x+c))^(1/2)/(-cos(d*x+c)^2*b^2+a^2)-11/16/a^2/(a^2-b^2)^2*(-sin(d*x+c)+1)^(1/2)*(2*sin(d*x+c)+2
)^(1/2)*sin(d*x+c)^(1/2)/(cos(d*x+c)^2*e*sin(d*x+c))^(1/2)*EllipticE((-sin(d*x+c)+1)^(1/2),1/2*2^(1/2))+3/8/a^
4/(a^2-b^2)^2*(-sin(d*x+c)+1)^(1/2)*(2*sin(d*x+c)+2)^(1/2)*sin(d*x+c)^(1/2)/(cos(d*x+c)^2*e*sin(d*x+c))^(1/2)*
EllipticE((-sin(d*x+c)+1)^(1/2),1/2*2^(1/2))*b^2+11/32/a^2/(a^2-b^2)^2*(-sin(d*x+c)+1)^(1/2)*(2*sin(d*x+c)+2)^
(1/2)*sin(d*x+c)^(1/2)/(cos(d*x+c)^2*e*sin(d*x+...

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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*sin(d*x+c))^(13/2)/(a+b*cos(d*x+c))^3,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*sin(d*x+c))^(13/2)/(a+b*cos(d*x+c))^3,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*sin(d*x+c))**(13/2)/(a+b*cos(d*x+c))**3,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*sin(d*x+c))^(13/2)/(a+b*cos(d*x+c))^3,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\,\sin \left (c+d\,x\right )\right )}^{13/2}}{{\left (a+b\,\cos \left (c+d\,x\right )\right )}^3} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((e*sin(c + d*x))^(13/2)/(a + b*cos(c + d*x))^3,x)

[Out]

int((e*sin(c + d*x))^(13/2)/(a + b*cos(c + d*x))^3, x)

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