\(\int \frac {(e \sin (c+d x))^{11/2}}{(a+b \cos (c+d x))^3} \, dx\) [79]
Optimal result
Integrand size = 25, antiderivative size = 604 \[
\int \frac {(e \sin (c+d x))^{11/2}}{(a+b \cos (c+d x))^3} \, dx=-\frac {9 \left (7 a^4-9 a^2 b^2+2 b^4\right ) e^{11/2} \arctan \left (\frac {\sqrt {b} \sqrt {e \sin (c+d x)}}{\sqrt [4]{-a^2+b^2} \sqrt {e}}\right )}{8 b^{11/2} \left (-a^2+b^2\right )^{3/4} d}-\frac {9 \left (7 a^4-9 a^2 b^2+2 b^4\right ) e^{11/2} \text {arctanh}\left (\frac {\sqrt {b} \sqrt {e \sin (c+d x)}}{\sqrt [4]{-a^2+b^2} \sqrt {e}}\right )}{8 b^{11/2} \left (-a^2+b^2\right )^{3/4} d}+\frac {3 a \left (21 a^2-13 b^2\right ) e^6 \operatorname {EllipticF}\left (\frac {1}{2} \left (c-\frac {\pi }{2}+d x\right ),2\right ) \sqrt {\sin (c+d x)}}{4 b^6 d \sqrt {e \sin (c+d x)}}-\frac {9 a \left (7 a^4-9 a^2 b^2+2 b^4\right ) e^6 \operatorname {EllipticPi}\left (\frac {2 b}{b-\sqrt {-a^2+b^2}},\frac {1}{2} \left (c-\frac {\pi }{2}+d x\right ),2\right ) \sqrt {\sin (c+d x)}}{8 b^6 \left (a^2-b \left (b-\sqrt {-a^2+b^2}\right )\right ) d \sqrt {e \sin (c+d x)}}-\frac {9 a \left (7 a^4-9 a^2 b^2+2 b^4\right ) e^6 \operatorname {EllipticPi}\left (\frac {2 b}{b+\sqrt {-a^2+b^2}},\frac {1}{2} \left (c-\frac {\pi }{2}+d x\right ),2\right ) \sqrt {\sin (c+d x)}}{8 b^6 \left (a^2-b \left (b+\sqrt {-a^2+b^2}\right )\right ) d \sqrt {e \sin (c+d x)}}-\frac {3 e^5 \left (3 \left (7 a^2-2 b^2\right )-7 a b \cos (c+d x)\right ) \sqrt {e \sin (c+d x)}}{4 b^5 d}+\frac {9 e^3 (7 a+2 b \cos (c+d x)) (e \sin (c+d x))^{5/2}}{20 b^3 d (a+b \cos (c+d x))}+\frac {e (e \sin (c+d x))^{9/2}}{2 b d (a+b \cos (c+d x))^2}
\]
[Out]
-9/8*(7*a^4-9*a^2*b^2+2*b^4)*e^(11/2)*arctan(b^(1/2)*(e*sin(d*x+c))^(1/2)/(-a^2+b^2)^(1/4)/e^(1/2))/b^(11/2)/(
-a^2+b^2)^(3/4)/d-9/8*(7*a^4-9*a^2*b^2+2*b^4)*e^(11/2)*arctanh(b^(1/2)*(e*sin(d*x+c))^(1/2)/(-a^2+b^2)^(1/4)/e
^(1/2))/b^(11/2)/(-a^2+b^2)^(3/4)/d+9/20*e^3*(7*a+2*b*cos(d*x+c))*(e*sin(d*x+c))^(5/2)/b^3/d/(a+b*cos(d*x+c))+
1/2*e*(e*sin(d*x+c))^(9/2)/b/d/(a+b*cos(d*x+c))^2-3/4*a*(21*a^2-13*b^2)*e^6*(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)*EllipticF(cos(1/2*c+1/4*Pi+1/2*d*x),2^(1/2))*sin(d*x+c)^(1/2)/b^6/d/(e*sin(d*x+c))
^(1/2)+9/8*a*(7*a^4-9*a^2*b^2+2*b^4)*e^6*(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)*Ellipti
cPi(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^6/d/(a^2-b*(b-(-a^2+b^2)^(1
/2)))/(e*sin(d*x+c))^(1/2)+9/8*a*(7*a^4-9*a^2*b^2+2*b^4)*e^6*(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^6/d/(a^
2-b*(b+(-a^2+b^2)^(1/2)))/(e*sin(d*x+c))^(1/2)-3/4*e^5*(21*a^2-6*b^2-7*a*b*cos(d*x+c))*(e*sin(d*x+c))^(1/2)/b^
5/d
Rubi [A] (verified)
Time = 1.73 (sec) , antiderivative size = 604, normalized size of antiderivative = 1.00, number of
steps used = 15, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.520, Rules used = {2772, 2942, 2944, 2946,
2721, 2720, 2781, 2886, 2884, 335, 218, 214, 211} \[
\int \frac {(e \sin (c+d x))^{11/2}}{(a+b \cos (c+d x))^3} \, dx=\frac {3 a e^6 \left (21 a^2-13 b^2\right ) \sqrt {\sin (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right ),2\right )}{4 b^6 d \sqrt {e \sin (c+d x)}}-\frac {3 e^5 \sqrt {e \sin (c+d x)} \left (3 \left (7 a^2-2 b^2\right )-7 a b \cos (c+d x)\right )}{4 b^5 d}-\frac {9 e^{11/2} \left (7 a^4-9 a^2 b^2+2 b^4\right ) \arctan \left (\frac {\sqrt {b} \sqrt {e \sin (c+d x)}}{\sqrt {e} \sqrt [4]{b^2-a^2}}\right )}{8 b^{11/2} d \left (b^2-a^2\right )^{3/4}}-\frac {9 e^{11/2} \left (7 a^4-9 a^2 b^2+2 b^4\right ) \text {arctanh}\left (\frac {\sqrt {b} \sqrt {e \sin (c+d x)}}{\sqrt {e} \sqrt [4]{b^2-a^2}}\right )}{8 b^{11/2} d \left (b^2-a^2\right )^{3/4}}-\frac {9 a e^6 \left (7 a^4-9 a^2 b^2+2 b^4\right ) \sqrt {\sin (c+d x)} \operatorname {EllipticPi}\left (\frac {2 b}{b-\sqrt {b^2-a^2}},\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right ),2\right )}{8 b^6 d \left (a^2-b \left (b-\sqrt {b^2-a^2}\right )\right ) \sqrt {e \sin (c+d x)}}-\frac {9 a e^6 \left (7 a^4-9 a^2 b^2+2 b^4\right ) \sqrt {\sin (c+d x)} \operatorname {EllipticPi}\left (\frac {2 b}{b+\sqrt {b^2-a^2}},\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right ),2\right )}{8 b^6 d \left (a^2-b \left (\sqrt {b^2-a^2}+b\right )\right ) \sqrt {e \sin (c+d x)}}+\frac {9 e^3 (e \sin (c+d x))^{5/2} (7 a+2 b \cos (c+d x))}{20 b^3 d (a+b \cos (c+d x))}+\frac {e (e \sin (c+d x))^{9/2}}{2 b d (a+b \cos (c+d x))^2}
\]
[In]
Int[(e*Sin[c + d*x])^(11/2)/(a + b*Cos[c + d*x])^3,x]
[Out]
(-9*(7*a^4 - 9*a^2*b^2 + 2*b^4)*e^(11/2)*ArcTan[(Sqrt[b]*Sqrt[e*Sin[c + d*x]])/((-a^2 + b^2)^(1/4)*Sqrt[e])])/
(8*b^(11/2)*(-a^2 + b^2)^(3/4)*d) - (9*(7*a^4 - 9*a^2*b^2 + 2*b^4)*e^(11/2)*ArcTanh[(Sqrt[b]*Sqrt[e*Sin[c + d*
x]])/((-a^2 + b^2)^(1/4)*Sqrt[e])])/(8*b^(11/2)*(-a^2 + b^2)^(3/4)*d) + (3*a*(21*a^2 - 13*b^2)*e^6*EllipticF[(
c - Pi/2 + d*x)/2, 2]*Sqrt[Sin[c + d*x]])/(4*b^6*d*Sqrt[e*Sin[c + d*x]]) - (9*a*(7*a^4 - 9*a^2*b^2 + 2*b^4)*e^
6*EllipticPi[(2*b)/(b - Sqrt[-a^2 + b^2]), (c - Pi/2 + d*x)/2, 2]*Sqrt[Sin[c + d*x]])/(8*b^6*(a^2 - b*(b - Sqr
t[-a^2 + b^2]))*d*Sqrt[e*Sin[c + d*x]]) - (9*a*(7*a^4 - 9*a^2*b^2 + 2*b^4)*e^6*EllipticPi[(2*b)/(b + Sqrt[-a^2
+ b^2]), (c - Pi/2 + d*x)/2, 2]*Sqrt[Sin[c + d*x]])/(8*b^6*(a^2 - b*(b + Sqrt[-a^2 + b^2]))*d*Sqrt[e*Sin[c +
d*x]]) - (3*e^5*(3*(7*a^2 - 2*b^2) - 7*a*b*Cos[c + d*x])*Sqrt[e*Sin[c + d*x]])/(4*b^5*d) + (9*e^3*(7*a + 2*b*C
os[c + d*x])*(e*Sin[c + d*x])^(5/2))/(20*b^3*d*(a + b*Cos[c + d*x])) + (e*(e*Sin[c + d*x])^(9/2))/(2*b*d*(a +
b*Cos[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 218
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] && !Gt
Q[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 2720
Int[1/Sqrt[sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[(2/d)*EllipticF[(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 2781
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 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*}
\text {integral}& = \frac {e (e \sin (c+d x))^{9/2}}{2 b d (a+b \cos (c+d x))^2}-\frac {\left (9 e^2\right ) \int \frac {\cos (c+d x) (e \sin (c+d x))^{7/2}}{(a+b \cos (c+d x))^2} \, dx}{4 b} \\ & = \frac {9 e^3 (7 a+2 b \cos (c+d x)) (e \sin (c+d x))^{5/2}}{20 b^3 d (a+b \cos (c+d x))}+\frac {e (e \sin (c+d x))^{9/2}}{2 b d (a+b \cos (c+d x))^2}+\frac {\left (9 e^4\right ) \int \frac {\left (-b-\frac {7}{2} a \cos (c+d x)\right ) (e \sin (c+d x))^{3/2}}{a+b \cos (c+d x)} \, dx}{4 b^3} \\ & = -\frac {3 e^5 \left (3 \left (7 a^2-2 b^2\right )-7 a b \cos (c+d x)\right ) \sqrt {e \sin (c+d x)}}{4 b^5 d}+\frac {9 e^3 (7 a+2 b \cos (c+d x)) (e \sin (c+d x))^{5/2}}{20 b^3 d (a+b \cos (c+d x))}+\frac {e (e \sin (c+d x))^{9/2}}{2 b d (a+b \cos (c+d x))^2}+\frac {\left (3 e^6\right ) \int \frac {\frac {1}{2} b \left (7 a^2-3 b^2\right )+\frac {1}{4} a \left (21 a^2-13 b^2\right ) \cos (c+d x)}{(a+b \cos (c+d x)) \sqrt {e \sin (c+d x)}} \, dx}{2 b^5} \\ & = -\frac {3 e^5 \left (3 \left (7 a^2-2 b^2\right )-7 a b \cos (c+d x)\right ) \sqrt {e \sin (c+d x)}}{4 b^5 d}+\frac {9 e^3 (7 a+2 b \cos (c+d x)) (e \sin (c+d x))^{5/2}}{20 b^3 d (a+b \cos (c+d x))}+\frac {e (e \sin (c+d x))^{9/2}}{2 b d (a+b \cos (c+d x))^2}+\frac {\left (3 a \left (21 a^2-13 b^2\right ) e^6\right ) \int \frac {1}{\sqrt {e \sin (c+d x)}} \, dx}{8 b^6}-\frac {\left (9 \left (7 a^4-9 a^2 b^2+2 b^4\right ) e^6\right ) \int \frac {1}{(a+b \cos (c+d x)) \sqrt {e \sin (c+d x)}} \, dx}{8 b^6} \\ & = -\frac {3 e^5 \left (3 \left (7 a^2-2 b^2\right )-7 a b \cos (c+d x)\right ) \sqrt {e \sin (c+d x)}}{4 b^5 d}+\frac {9 e^3 (7 a+2 b \cos (c+d x)) (e \sin (c+d x))^{5/2}}{20 b^3 d (a+b \cos (c+d x))}+\frac {e (e \sin (c+d x))^{9/2}}{2 b d (a+b \cos (c+d x))^2}+\frac {\left (9 a \left (7 a^4-9 a^2 b^2+2 b^4\right ) e^6\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^6 \sqrt {-a^2+b^2}}+\frac {\left (9 a \left (7 a^4-9 a^2 b^2+2 b^4\right ) e^6\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^6 \sqrt {-a^2+b^2}}+\frac {\left (9 \left (7 a^4-9 a^2 b^2+2 b^4\right ) e^7\right ) \text {Subst}\left (\int \frac {1}{\sqrt {x} \left (\left (a^2-b^2\right ) e^2+b^2 x^2\right )} \, dx,x,e \sin (c+d x)\right )}{8 b^5 d}+\frac {\left (3 a \left (21 a^2-13 b^2\right ) e^6 \sqrt {\sin (c+d x)}\right ) \int \frac {1}{\sqrt {\sin (c+d x)}} \, dx}{8 b^6 \sqrt {e \sin (c+d x)}} \\ & = \frac {3 a \left (21 a^2-13 b^2\right ) e^6 \operatorname {EllipticF}\left (\frac {1}{2} \left (c-\frac {\pi }{2}+d x\right ),2\right ) \sqrt {\sin (c+d x)}}{4 b^6 d \sqrt {e \sin (c+d x)}}-\frac {3 e^5 \left (3 \left (7 a^2-2 b^2\right )-7 a b \cos (c+d x)\right ) \sqrt {e \sin (c+d x)}}{4 b^5 d}+\frac {9 e^3 (7 a+2 b \cos (c+d x)) (e \sin (c+d x))^{5/2}}{20 b^3 d (a+b \cos (c+d x))}+\frac {e (e \sin (c+d x))^{9/2}}{2 b d (a+b \cos (c+d x))^2}+\frac {\left (9 \left (7 a^4-9 a^2 b^2+2 b^4\right ) e^7\right ) \text {Subst}\left (\int \frac {1}{\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 (9 a \left (7 a^4-9 a^2 b^2+2 b^4\right ) e^6 \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^6 \sqrt {-a^2+b^2} \sqrt {e \sin (c+d x)}}+\frac {\left (9 a \left (7 a^4-9 a^2 b^2+2 b^4\right ) e^6 \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^6 \sqrt {-a^2+b^2} \sqrt {e \sin (c+d x)}} \\ & = \frac {3 a \left (21 a^2-13 b^2\right ) e^6 \operatorname {EllipticF}\left (\frac {1}{2} \left (c-\frac {\pi }{2}+d x\right ),2\right ) \sqrt {\sin (c+d x)}}{4 b^6 d \sqrt {e \sin (c+d x)}}-\frac {9 a \left (7 a^4-9 a^2 b^2+2 b^4\right ) e^6 \operatorname {EllipticPi}\left (\frac {2 b}{b-\sqrt {-a^2+b^2}},\frac {1}{2} \left (c-\frac {\pi }{2}+d x\right ),2\right ) \sqrt {\sin (c+d x)}}{8 b^6 \left (a^2-b \left (b-\sqrt {-a^2+b^2}\right )\right ) d \sqrt {e \sin (c+d x)}}+\frac {9 a \left (7 a^4-9 a^2 b^2+2 b^4\right ) e^6 \operatorname {EllipticPi}\left (\frac {2 b}{b+\sqrt {-a^2+b^2}},\frac {1}{2} \left (c-\frac {\pi }{2}+d x\right ),2\right ) \sqrt {\sin (c+d x)}}{8 b^6 \sqrt {-a^2+b^2} \left (b+\sqrt {-a^2+b^2}\right ) d \sqrt {e \sin (c+d x)}}-\frac {3 e^5 \left (3 \left (7 a^2-2 b^2\right )-7 a b \cos (c+d x)\right ) \sqrt {e \sin (c+d x)}}{4 b^5 d}+\frac {9 e^3 (7 a+2 b \cos (c+d x)) (e \sin (c+d x))^{5/2}}{20 b^3 d (a+b \cos (c+d x))}+\frac {e (e \sin (c+d x))^{9/2}}{2 b d (a+b \cos (c+d x))^2}-\frac {\left (9 \left (7 a^4-9 a^2 b^2+2 b^4\right ) e^6\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^5 \sqrt {-a^2+b^2} d}-\frac {\left (9 \left (7 a^4-9 a^2 b^2+2 b^4\right ) e^6\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^5 \sqrt {-a^2+b^2} d} \\ & = -\frac {9 \left (7 a^4-9 a^2 b^2+2 b^4\right ) e^{11/2} \arctan \left (\frac {\sqrt {b} \sqrt {e \sin (c+d x)}}{\sqrt [4]{-a^2+b^2} \sqrt {e}}\right )}{8 b^{11/2} \left (-a^2+b^2\right )^{3/4} d}-\frac {9 \left (7 a^4-9 a^2 b^2+2 b^4\right ) e^{11/2} \text {arctanh}\left (\frac {\sqrt {b} \sqrt {e \sin (c+d x)}}{\sqrt [4]{-a^2+b^2} \sqrt {e}}\right )}{8 b^{11/2} \left (-a^2+b^2\right )^{3/4} d}+\frac {3 a \left (21 a^2-13 b^2\right ) e^6 \operatorname {EllipticF}\left (\frac {1}{2} \left (c-\frac {\pi }{2}+d x\right ),2\right ) \sqrt {\sin (c+d x)}}{4 b^6 d \sqrt {e \sin (c+d x)}}-\frac {9 a \left (7 a^4-9 a^2 b^2+2 b^4\right ) e^6 \operatorname {EllipticPi}\left (\frac {2 b}{b-\sqrt {-a^2+b^2}},\frac {1}{2} \left (c-\frac {\pi }{2}+d x\right ),2\right ) \sqrt {\sin (c+d x)}}{8 b^6 \left (a^2-b \left (b-\sqrt {-a^2+b^2}\right )\right ) d \sqrt {e \sin (c+d x)}}+\frac {9 a \left (7 a^4-9 a^2 b^2+2 b^4\right ) e^6 \operatorname {EllipticPi}\left (\frac {2 b}{b+\sqrt {-a^2+b^2}},\frac {1}{2} \left (c-\frac {\pi }{2}+d x\right ),2\right ) \sqrt {\sin (c+d x)}}{8 b^6 \sqrt {-a^2+b^2} \left (b+\sqrt {-a^2+b^2}\right ) d \sqrt {e \sin (c+d x)}}-\frac {3 e^5 \left (3 \left (7 a^2-2 b^2\right )-7 a b \cos (c+d x)\right ) \sqrt {e \sin (c+d x)}}{4 b^5 d}+\frac {9 e^3 (7 a+2 b \cos (c+d x)) (e \sin (c+d x))^{5/2}}{20 b^3 d (a+b \cos (c+d x))}+\frac {e (e \sin (c+d x))^{9/2}}{2 b d (a+b \cos (c+d x))^2} \\
\end{align*}
Mathematica [C] (warning: unable to verify)
Result contains higher order function than in optimal. Order 6 vs. order 4 in optimal.
Time = 15.20 (sec) , antiderivative size = 2024, normalized size of antiderivative = 3.35
\[
\int \frac {(e \sin (c+d x))^{11/2}}{(a+b \cos (c+d x))^3} \, dx=\text {Result too large to show}
\]
[In]
Integrate[(e*Sin[c + d*x])^(11/2)/(a + b*Cos[c + d*x])^3,x]
[Out]
(((2*a*Cos[c + d*x])/b^4 + (-a^2 + b^2)^2/(2*b^5*(a + b*Cos[c + d*x])^2) - (17*a*(a^2 - b^2))/(4*b^5*(a + b*Co
s[c + d*x])) - Cos[2*(c + d*x)]/(5*b^3))*Csc[c + d*x]^5*(e*Sin[c + d*x])^(11/2))/d + (3*(e*Sin[c + d*x])^(11/2
)*((2*(25*a^3 - 37*a*b^2)*Cos[c + d*x]^2*(a + b*Sqrt[1 - Sin[c + d*x]^2])*((a*(-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]]))/(4*Sqrt[2]*Sqrt[b]*(a^2 - b^
2)^(3/4)) + (5*b*(a^2 - b^2)*AppellF1[1/4, -1/2, 1, 5/4, Sin[c + d*x]^2, (b^2*Sin[c + d*x]^2)/(-a^2 + b^2)]*Sq
rt[Sin[c + d*x]]*Sqrt[1 - Sin[c + d*x]^2])/((-5*(a^2 - b^2)*AppellF1[1/4, -1/2, 1, 5/4, Sin[c + d*x]^2, (b^2*S
in[c + d*x]^2)/(-a^2 + b^2)] + 2*(2*b^2*AppellF1[5/4, -1/2, 2, 9/4, Sin[c + d*x]^2, (b^2*Sin[c + d*x]^2)/(-a^2
+ b^2)] + (a^2 - b^2)*AppellF1[5/4, 1/2, 1, 9/4, Sin[c + d*x]^2, (b^2*Sin[c + d*x]^2)/(-a^2 + b^2)])*Sin[c +
d*x]^2)*(a^2 + b^2*(-1 + Sin[c + d*x]^2)))))/((a + b*Cos[c + d*x])*(1 - Sin[c + d*x]^2)) + (2*(30*a^2*b - 16*b
^3)*Cos[c + d*x]*(a + b*Sqrt[1 - Sin[c + d*x]^2])*(((-1/8 + I/8)*Sqrt[b]*(2*ArcTan[1 - ((1 + I)*Sqrt[b]*Sqrt[S
in[c + d*x]])/(-a^2 + b^2)^(1/4)] - 2*ArcTan[1 + ((1 + I)*Sqrt[b]*Sqrt[Sin[c + d*x]])/(-a^2 + b^2)^(1/4)] + Lo
g[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]]))/(-a^2 + b^2)^(3/4) + (5*
a*(a^2 - b^2)*AppellF1[1/4, 1/2, 1, 5/4, Sin[c + d*x]^2, (b^2*Sin[c + d*x]^2)/(-a^2 + b^2)]*Sqrt[Sin[c + d*x]]
)/(Sqrt[1 - Sin[c + d*x]^2]*(5*(a^2 - b^2)*AppellF1[1/4, 1/2, 1, 5/4, Sin[c + d*x]^2, (b^2*Sin[c + d*x]^2)/(-a
^2 + b^2)] - 2*(2*b^2*AppellF1[5/4, 1/2, 2, 9/4, Sin[c + d*x]^2, (b^2*Sin[c + d*x]^2)/(-a^2 + b^2)] + (-a^2 +
b^2)*AppellF1[5/4, 3/2, 1, 9/4, Sin[c + d*x]^2, (b^2*Sin[c + d*x]^2)/(-a^2 + b^2)])*Sin[c + d*x]^2)*(a^2 + b^2
*(-1 + Sin[c + d*x]^2)))))/((a + b*Cos[c + d*x])*Sqrt[1 - Sin[c + d*x]^2]) + ((-40*a^2*b + 14*b^3)*Cos[c + d*x
]*Cos[2*(c + d*x)]*(a + b*Sqrt[1 - Sin[c + d*x]^2])*(((1/2 - I/2)*(-2*a^2 + b^2)*ArcTan[1 - ((1 + I)*Sqrt[b]*S
qrt[Sin[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[Sin[c + d*x]])/(-a^2 + b^2)^(1/4)])/(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[Sin[c + d*x]] + I*b*Sin[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[Sin[c + d*x]] + I*b*Sin[c + d*x]])/(b^(3/2)*(-a^2 + b^2)^(3/4)) + (4*Sqrt[Sin[c + d*x]])/b - (4*a*Appel
lF1[5/4, 1/2, 1, 9/4, Sin[c + d*x]^2, (b^2*Sin[c + d*x]^2)/(-a^2 + b^2)]*Sin[c + d*x]^(5/2))/(5*(a^2 - b^2)) +
(10*a*(a^2 - b^2)*AppellF1[1/4, 1/2, 1, 5/4, Sin[c + d*x]^2, (b^2*Sin[c + d*x]^2)/(-a^2 + b^2)]*Sqrt[Sin[c +
d*x]])/(Sqrt[1 - Sin[c + d*x]^2]*(5*(a^2 - b^2)*AppellF1[1/4, 1/2, 1, 5/4, Sin[c + d*x]^2, (b^2*Sin[c + d*x]^2
)/(-a^2 + b^2)] - 2*(2*b^2*AppellF1[5/4, 1/2, 2, 9/4, Sin[c + d*x]^2, (b^2*Sin[c + d*x]^2)/(-a^2 + b^2)] + (-a
^2 + b^2)*AppellF1[5/4, 3/2, 1, 9/4, Sin[c + d*x]^2, (b^2*Sin[c + d*x]^2)/(-a^2 + b^2)])*Sin[c + d*x]^2)*(a^2
+ b^2*(-1 + Sin[c + d*x]^2)))))/((a + b*Cos[c + d*x])*(1 - 2*Sin[c + d*x]^2)*Sqrt[1 - Sin[c + d*x]^2])))/(40*b
^5*d*Sin[c + d*x]^(11/2))
Maple [B] (warning: unable to verify)
Leaf count of result is larger than twice the leaf count of optimal. \(2751\) vs. \(2(621)=1242\).
Time = 100.88 (sec) , antiderivative size = 2752, normalized size of antiderivative =
4.56
| | |
method | result | size |
| | |
default |
\(\text {Expression too large to display}\) |
\(2752\) |
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|
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[In]
int((e*sin(d*x+c))^(11/2)/(a+cos(d*x+c)*b)^3,x,method=_RETURNVERBOSE)
[Out]
(2*e^3*b*(-1/5/b^6*(e*sin(d*x+c))^(1/2)*e^2*(b^2*cos(d*x+c)^2+30*a^2-11*b^2)+e^4/b^6*(-1/8*(e*sin(d*x+c))^(1/2
)*e^2*(-19*a^4*b^2*cos(d*x+c)^2+21*a^2*b^4*cos(d*x+c)^2-2*b^6*cos(d*x+c)^2+15*a^6-13*a^4*b^2-2*a^2*b^4)/(-b^2*
cos(d*x+c)^2*e^2+a^2*e^2)^2+9/64*(7*a^4-9*a^2*b^2+2*b^4)*(e^2*(a^2-b^2)/b^2)^(1/4)/(a^2*e^2-b^2*e^2)*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)))+2*arctan(2^(1/2)/(e^2*(a
^2-b^2)/b^2)^(1/4)*(e*sin(d*x+c))^(1/2)+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^6*a*(1/b^6/(cos(d*x+c)^2*e*sin(d*x+c))^(1/2)*(10*(1-sin(d*x+c))^(1/2)*(
2*sin(d*x+c)+2)^(1/2)*sin(d*x+c)^(1/2)*EllipticF((1-sin(d*x+c))^(1/2),1/2*2^(1/2))*a^2-7*(1-sin(d*x+c))^(1/2)*
(2*sin(d*x+c)+2)^(1/2)*sin(d*x+c)^(1/2)*EllipticF((1-sin(d*x+c))^(1/2),1/2*2^(1/2))*b^2-2*b^2*cos(d*x+c)^2*sin
(d*x+c))+(21*a^4-30*a^2*b^2+9*b^4)/b^6*(-1/2/(-a^2+b^2)^(1/2)/b*(1-sin(d*x+c))^(1/2)*(2*sin(d*x+c)+2)^(1/2)*si
n(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((1-sin(d*x+c))^(1/2),1/(1-(
-a^2+b^2)^(1/2)/b),1/2*2^(1/2))+1/2/(-a^2+b^2)^(1/2)/b*(1-sin(d*x+c))^(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((1-sin(d*x+c))^(1/2),1/(1+(-a^2+b^2)
^(1/2)/b),1/2*2^(1/2)))+(-15*a^6+33*a^4*b^2-21*a^2*b^4+3*b^6)/b^6*(1/2*b^2/e/a^2/(a^2-b^2)*(cos(d*x+c)^2*e*sin
(d*x+c))^(1/2)/(-b^2*cos(d*x+c)^2+a^2)+1/4/a^2/(a^2-b^2)*(1-sin(d*x+c))^(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((1-sin(d*x+c))^(1/2),1/2*2^(1/2))-5/8/(a^2-b^2)/(-a^2+b^2)
^(1/2)/b*(1-sin(d*x+c))^(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((1-sin(d*x+c))^(1/2),1/(1-(-a^2+b^2)^(1/2)/b),1/2*2^(1/2))+1/4/a^2/(a^2-b^2)/(-a^2
+b^2)^(1/2)*b*(1-sin(d*x+c))^(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((1-sin(d*x+c))^(1/2),1/(1-(-a^2+b^2)^(1/2)/b),1/2*2^(1/2))+5/8/(a^2-b^2)/(-a^
2+b^2)^(1/2)/b*(1-sin(d*x+c))^(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((1-sin(d*x+c))^(1/2),1/(1+(-a^2+b^2)^(1/2)/b),1/2*2^(1/2))-1/4/a^2/(a^2-b^2)
/(-a^2+b^2)^(1/2)*b*(1-sin(d*x+c))^(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((1-sin(d*x+c))^(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)*(cos(d*x+c)^2*e*sin(d*x+c))^(1/2)/(-b^2*cos(d*x+c)^2+a^2
)^2+1/16*b^2*(13*a^2-6*b^2)/a^4/(a^2-b^2)^2/e*(cos(d*x+c)^2*e*sin(d*x+c))^(1/2)/(-b^2*cos(d*x+c)^2+a^2)+13/32/
a^2/(a^2-b^2)^2*(1-sin(d*x+c))^(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((1-sin(d*x+c))^(1/2),1/2*2^(1/2))-3/16/a^4/(a^2-b^2)^2*(1-sin(d*x+c))^(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((1-sin(d*x+c))^(1/2),1/2*2^(1/2))*b^2-45/64/(a^2-
b^2)^2/(-a^2+b^2)^(1/2)/b*(1-sin(d*x+c))^(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((1-sin(d*x+c))^(1/2),1/(1-(-a^2+b^2)^(1/2)/b),1/2*2^(1/2))+9/16/a
^2/(a^2-b^2)^2/(-a^2+b^2)^(1/2)*b*(1-sin(d*x+c))^(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((1-sin(d*x+c))^(1/2),1/(1-(-a^2+b^2)^(1/2)/b),1/2*2^(1/2)
)-3/16/a^4/(a^2-b^2)^2/(-a^2+b^2)^(1/2)*b^3*(1-sin(d*x+c))^(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((1-sin(d*x+c))^(1/2),1/(1-(-a^2+b^2)^(1/2)/b),1
/2*2^(1/2))+45/64/(a^2-b^2)^2/(-a^2+b^2)^(1/2)/b*(1-sin(d*x+c))^(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((1-sin(d*x+c))^(1/2),1/(1+(-a^2+b^2)^(1/2)
/b),1/2*2^(1/2))-9/16/a^2/(a^2-b^2)^2/(-a^2+b^2)^(1/2)*b*(1-sin(d*x+c))^(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((1-sin(d*x+c))^(1/2),1/(1+(-a^2+b^
2)^(1/2)/b),1/2*2^(1/2))+3/16/a^4/(a^2-b^2)^2/(-a^2+b^2)^(1/2)*b^3*(1-sin(d*x+c))^(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((1-sin(d*x+c))^(1/2),1/(
1+(-a^2+b^2)^(1/2)/b),1/2*2^(1/2))))/cos(d*x+c)/(e*sin(d*x+c))^(1/2))/d
Fricas [F(-1)]
Timed out. \[
\int \frac {(e \sin (c+d x))^{11/2}}{(a+b \cos (c+d x))^3} \, dx=\text {Timed out}
\]
[In]
integrate((e*sin(d*x+c))^(11/2)/(a+b*cos(d*x+c))^3,x, algorithm="fricas")
[Out]
Timed out
Sympy [F(-1)]
Timed out. \[
\int \frac {(e \sin (c+d x))^{11/2}}{(a+b \cos (c+d x))^3} \, dx=\text {Timed out}
\]
[In]
integrate((e*sin(d*x+c))**(11/2)/(a+b*cos(d*x+c))**3,x)
[Out]
Timed out
Maxima [F(-1)]
Timed out. \[
\int \frac {(e \sin (c+d x))^{11/2}}{(a+b \cos (c+d x))^3} \, dx=\text {Timed out}
\]
[In]
integrate((e*sin(d*x+c))^(11/2)/(a+b*cos(d*x+c))^3,x, algorithm="maxima")
[Out]
Timed out
Giac [F]
\[
\int \frac {(e \sin (c+d x))^{11/2}}{(a+b \cos (c+d x))^3} \, dx=\int { \frac {\left (e \sin \left (d x + c\right )\right )^{\frac {11}{2}}}{{\left (b \cos \left (d x + c\right ) + a\right )}^{3}} \,d x }
\]
[In]
integrate((e*sin(d*x+c))^(11/2)/(a+b*cos(d*x+c))^3,x, algorithm="giac")
[Out]
integrate((e*sin(d*x + c))^(11/2)/(b*cos(d*x + c) + a)^3, x)
Mupad [F(-1)]
Timed out. \[
\int \frac {(e \sin (c+d x))^{11/2}}{(a+b \cos (c+d x))^3} \, dx=\int \frac {{\left (e\,\sin \left (c+d\,x\right )\right )}^{11/2}}{{\left (a+b\,\cos \left (c+d\,x\right )\right )}^3} \,d x
\]
[In]
int((e*sin(c + d*x))^(11/2)/(a + b*cos(c + d*x))^3,x)
[Out]
int((e*sin(c + d*x))^(11/2)/(a + b*cos(c + d*x))^3, x)