3.584 \(\int \frac{(e \cos (c+d x))^{11/2}}{(a+b \sin (c+d x))^2} \, dx\)

Optimal. Leaf size=543 \[ -\frac{3 e^5 \sqrt{e \cos (c+d x)} \left (21 a \left (a^2-b^2\right )-b \left (7 a^2-5 b^2\right ) \sin (c+d x)\right )}{7 b^5 d}-\frac{9 a e^{11/2} \left (b^2-a^2\right )^{5/4} \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{e \cos (c+d x)}}{\sqrt{e} \sqrt [4]{b^2-a^2}}\right )}{2 b^{11/2} d}-\frac{9 a e^{11/2} \left (b^2-a^2\right )^{5/4} \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{e \cos (c+d x)}}{\sqrt{e} \sqrt [4]{b^2-a^2}}\right )}{2 b^{11/2} d}-\frac{3 e^6 \left (-28 a^2 b^2+21 a^4+5 b^4\right ) \sqrt{\cos (c+d x)} F\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{7 b^6 d \sqrt{e \cos (c+d x)}}+\frac{9 a^2 e^6 \left (a^2-b^2\right )^2 \sqrt{\cos (c+d x)} \Pi \left (\frac{2 b}{b-\sqrt{b^2-a^2}};\left .\frac{1}{2} (c+d x)\right |2\right )}{2 b^6 d \left (a^2-b \left (b-\sqrt{b^2-a^2}\right )\right ) \sqrt{e \cos (c+d x)}}+\frac{9 a^2 e^6 \left (a^2-b^2\right )^2 \sqrt{\cos (c+d x)} \Pi \left (\frac{2 b}{b+\sqrt{b^2-a^2}};\left .\frac{1}{2} (c+d x)\right |2\right )}{2 b^6 d \left (a^2-b \left (\sqrt{b^2-a^2}+b\right )\right ) \sqrt{e \cos (c+d x)}}+\frac{9 e^3 (e \cos (c+d x))^{5/2} (7 a-5 b \sin (c+d x))}{35 b^3 d}-\frac{e (e \cos (c+d x))^{9/2}}{b d (a+b \sin (c+d x))} \]

[Out]

(-9*a*(-a^2 + b^2)^(5/4)*e^(11/2)*ArcTan[(Sqrt[b]*Sqrt[e*Cos[c + d*x]])/((-a^2 + b^2)^(1/4)*Sqrt[e])])/(2*b^(1
1/2)*d) - (9*a*(-a^2 + b^2)^(5/4)*e^(11/2)*ArcTanh[(Sqrt[b]*Sqrt[e*Cos[c + d*x]])/((-a^2 + b^2)^(1/4)*Sqrt[e])
])/(2*b^(11/2)*d) - (3*(21*a^4 - 28*a^2*b^2 + 5*b^4)*e^6*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2])/(7*b^6*
d*Sqrt[e*Cos[c + d*x]]) + (9*a^2*(a^2 - b^2)^2*e^6*Sqrt[Cos[c + d*x]]*EllipticPi[(2*b)/(b - Sqrt[-a^2 + b^2]),
(c + d*x)/2, 2])/(2*b^6*(a^2 - b*(b - Sqrt[-a^2 + b^2]))*d*Sqrt[e*Cos[c + d*x]]) + (9*a^2*(a^2 - b^2)^2*e^6*S
qrt[Cos[c + d*x]]*EllipticPi[(2*b)/(b + Sqrt[-a^2 + b^2]), (c + d*x)/2, 2])/(2*b^6*(a^2 - b*(b + Sqrt[-a^2 + b
^2]))*d*Sqrt[e*Cos[c + d*x]]) + (9*e^3*(e*Cos[c + d*x])^(5/2)*(7*a - 5*b*Sin[c + d*x]))/(35*b^3*d) - (e*(e*Cos
[c + d*x])^(9/2))/(b*d*(a + b*Sin[c + d*x])) - (3*e^5*Sqrt[e*Cos[c + d*x]]*(21*a*(a^2 - b^2) - b*(7*a^2 - 5*b^
2)*Sin[c + d*x]))/(7*b^5*d)

________________________________________________________________________________________

Rubi [A]  time = 1.5189, antiderivative size = 543, normalized size of antiderivative = 1., number of steps used = 15, number of rules used = 12, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.48, Rules used = {2693, 2865, 2867, 2642, 2641, 2702, 2807, 2805, 329, 212, 208, 205} \[ -\frac{3 e^5 \sqrt{e \cos (c+d x)} \left (21 a \left (a^2-b^2\right )-b \left (7 a^2-5 b^2\right ) \sin (c+d x)\right )}{7 b^5 d}-\frac{9 a e^{11/2} \left (b^2-a^2\right )^{5/4} \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{e \cos (c+d x)}}{\sqrt{e} \sqrt [4]{b^2-a^2}}\right )}{2 b^{11/2} d}-\frac{9 a e^{11/2} \left (b^2-a^2\right )^{5/4} \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{e \cos (c+d x)}}{\sqrt{e} \sqrt [4]{b^2-a^2}}\right )}{2 b^{11/2} d}-\frac{3 e^6 \left (-28 a^2 b^2+21 a^4+5 b^4\right ) \sqrt{\cos (c+d x)} F\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{7 b^6 d \sqrt{e \cos (c+d x)}}+\frac{9 a^2 e^6 \left (a^2-b^2\right )^2 \sqrt{\cos (c+d x)} \Pi \left (\frac{2 b}{b-\sqrt{b^2-a^2}};\left .\frac{1}{2} (c+d x)\right |2\right )}{2 b^6 d \left (a^2-b \left (b-\sqrt{b^2-a^2}\right )\right ) \sqrt{e \cos (c+d x)}}+\frac{9 a^2 e^6 \left (a^2-b^2\right )^2 \sqrt{\cos (c+d x)} \Pi \left (\frac{2 b}{b+\sqrt{b^2-a^2}};\left .\frac{1}{2} (c+d x)\right |2\right )}{2 b^6 d \left (a^2-b \left (\sqrt{b^2-a^2}+b\right )\right ) \sqrt{e \cos (c+d x)}}+\frac{9 e^3 (e \cos (c+d x))^{5/2} (7 a-5 b \sin (c+d x))}{35 b^3 d}-\frac{e (e \cos (c+d x))^{9/2}}{b d (a+b \sin (c+d x))} \]

Antiderivative was successfully verified.

[In]

Int[(e*Cos[c + d*x])^(11/2)/(a + b*Sin[c + d*x])^2,x]

[Out]

(-9*a*(-a^2 + b^2)^(5/4)*e^(11/2)*ArcTan[(Sqrt[b]*Sqrt[e*Cos[c + d*x]])/((-a^2 + b^2)^(1/4)*Sqrt[e])])/(2*b^(1
1/2)*d) - (9*a*(-a^2 + b^2)^(5/4)*e^(11/2)*ArcTanh[(Sqrt[b]*Sqrt[e*Cos[c + d*x]])/((-a^2 + b^2)^(1/4)*Sqrt[e])
])/(2*b^(11/2)*d) - (3*(21*a^4 - 28*a^2*b^2 + 5*b^4)*e^6*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2])/(7*b^6*
d*Sqrt[e*Cos[c + d*x]]) + (9*a^2*(a^2 - b^2)^2*e^6*Sqrt[Cos[c + d*x]]*EllipticPi[(2*b)/(b - Sqrt[-a^2 + b^2]),
(c + d*x)/2, 2])/(2*b^6*(a^2 - b*(b - Sqrt[-a^2 + b^2]))*d*Sqrt[e*Cos[c + d*x]]) + (9*a^2*(a^2 - b^2)^2*e^6*S
qrt[Cos[c + d*x]]*EllipticPi[(2*b)/(b + Sqrt[-a^2 + b^2]), (c + d*x)/2, 2])/(2*b^6*(a^2 - b*(b + Sqrt[-a^2 + b
^2]))*d*Sqrt[e*Cos[c + d*x]]) + (9*e^3*(e*Cos[c + d*x])^(5/2)*(7*a - 5*b*Sin[c + d*x]))/(35*b^3*d) - (e*(e*Cos
[c + d*x])^(9/2))/(b*d*(a + b*Sin[c + d*x])) - (3*e^5*Sqrt[e*Cos[c + d*x]]*(21*a*(a^2 - b^2) - b*(7*a^2 - 5*b^
2)*Sin[c + d*x]))/(7*b^5*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 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))^{11/2}}{(a+b \sin (c+d x))^2} \, dx &=-\frac{e (e \cos (c+d x))^{9/2}}{b d (a+b \sin (c+d x))}-\frac{\left (9 e^2\right ) \int \frac{(e \cos (c+d x))^{7/2} \sin (c+d x)}{a+b \sin (c+d x)} \, dx}{2 b}\\ &=\frac{9 e^3 (e \cos (c+d x))^{5/2} (7 a-5 b \sin (c+d x))}{35 b^3 d}-\frac{e (e \cos (c+d x))^{9/2}}{b d (a+b \sin (c+d x))}-\frac{\left (9 e^4\right ) \int \frac{(e \cos (c+d x))^{3/2} \left (-a b-\frac{1}{2} \left (7 a^2-5 b^2\right ) \sin (c+d x)\right )}{a+b \sin (c+d x)} \, dx}{7 b^3}\\ &=\frac{9 e^3 (e \cos (c+d x))^{5/2} (7 a-5 b \sin (c+d x))}{35 b^3 d}-\frac{e (e \cos (c+d x))^{9/2}}{b d (a+b \sin (c+d x))}-\frac{3 e^5 \sqrt{e \cos (c+d x)} \left (21 a \left (a^2-b^2\right )-b \left (7 a^2-5 b^2\right ) \sin (c+d x)\right )}{7 b^5 d}-\frac{\left (6 e^6\right ) \int \frac{\frac{1}{2} a b \left (7 a^2-8 b^2\right )+\frac{1}{4} \left (21 a^4-28 a^2 b^2+5 b^4\right ) \sin (c+d x)}{\sqrt{e \cos (c+d x)} (a+b \sin (c+d x))} \, dx}{7 b^5}\\ &=\frac{9 e^3 (e \cos (c+d x))^{5/2} (7 a-5 b \sin (c+d x))}{35 b^3 d}-\frac{e (e \cos (c+d x))^{9/2}}{b d (a+b \sin (c+d x))}-\frac{3 e^5 \sqrt{e \cos (c+d x)} \left (21 a \left (a^2-b^2\right )-b \left (7 a^2-5 b^2\right ) \sin (c+d x)\right )}{7 b^5 d}+\frac{\left (9 a \left (a^2-b^2\right )^2 e^6\right ) \int \frac{1}{\sqrt{e \cos (c+d x)} (a+b \sin (c+d x))} \, dx}{2 b^6}-\frac{\left (3 \left (21 a^4-28 a^2 b^2+5 b^4\right ) e^6\right ) \int \frac{1}{\sqrt{e \cos (c+d x)}} \, dx}{14 b^6}\\ &=\frac{9 e^3 (e \cos (c+d x))^{5/2} (7 a-5 b \sin (c+d x))}{35 b^3 d}-\frac{e (e \cos (c+d x))^{9/2}}{b d (a+b \sin (c+d x))}-\frac{3 e^5 \sqrt{e \cos (c+d x)} \left (21 a \left (a^2-b^2\right )-b \left (7 a^2-5 b^2\right ) \sin (c+d x)\right )}{7 b^5 d}-\frac{\left (9 a^2 \left (-a^2+b^2\right )^{3/2} e^6\right ) \int \frac{1}{\sqrt{e \cos (c+d x)} \left (\sqrt{-a^2+b^2}-b \cos (c+d x)\right )} \, dx}{4 b^6}-\frac{\left (9 a^2 \left (-a^2+b^2\right )^{3/2} e^6\right ) \int \frac{1}{\sqrt{e \cos (c+d x)} \left (\sqrt{-a^2+b^2}+b \cos (c+d x)\right )} \, dx}{4 b^6}+\frac{\left (9 a \left (a^2-b^2\right )^2 e^7\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 )}{2 b^5 d}-\frac{\left (3 \left (21 a^4-28 a^2 b^2+5 b^4\right ) e^6 \sqrt{\cos (c+d x)}\right ) \int \frac{1}{\sqrt{\cos (c+d x)}} \, dx}{14 b^6 \sqrt{e \cos (c+d x)}}\\ &=-\frac{3 \left (21 a^4-28 a^2 b^2+5 b^4\right ) e^6 \sqrt{\cos (c+d x)} F\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{7 b^6 d \sqrt{e \cos (c+d x)}}+\frac{9 e^3 (e \cos (c+d x))^{5/2} (7 a-5 b \sin (c+d x))}{35 b^3 d}-\frac{e (e \cos (c+d x))^{9/2}}{b d (a+b \sin (c+d x))}-\frac{3 e^5 \sqrt{e \cos (c+d x)} \left (21 a \left (a^2-b^2\right )-b \left (7 a^2-5 b^2\right ) \sin (c+d x)\right )}{7 b^5 d}+\frac{\left (9 a \left (a^2-b^2\right )^2 e^7\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 )}{b^5 d}-\frac{\left (9 a^2 \left (-a^2+b^2\right )^{3/2} e^6 \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}{4 b^6 \sqrt{e \cos (c+d x)}}-\frac{\left (9 a^2 \left (-a^2+b^2\right )^{3/2} e^6 \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}{4 b^6 \sqrt{e \cos (c+d x)}}\\ &=-\frac{3 \left (21 a^4-28 a^2 b^2+5 b^4\right ) e^6 \sqrt{\cos (c+d x)} F\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{7 b^6 d \sqrt{e \cos (c+d x)}}+\frac{9 a^2 \left (-a^2+b^2\right )^{3/2} e^6 \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 )}{2 b^6 \left (b-\sqrt{-a^2+b^2}\right ) d \sqrt{e \cos (c+d x)}}-\frac{9 a^2 \left (-a^2+b^2\right )^{3/2} e^6 \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 )}{2 b^6 \left (b+\sqrt{-a^2+b^2}\right ) d \sqrt{e \cos (c+d x)}}+\frac{9 e^3 (e \cos (c+d x))^{5/2} (7 a-5 b \sin (c+d x))}{35 b^3 d}-\frac{e (e \cos (c+d x))^{9/2}}{b d (a+b \sin (c+d x))}-\frac{3 e^5 \sqrt{e \cos (c+d x)} \left (21 a \left (a^2-b^2\right )-b \left (7 a^2-5 b^2\right ) \sin (c+d x)\right )}{7 b^5 d}-\frac{\left (9 a \left (-a^2+b^2\right )^{3/2} e^6\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 )}{2 b^5 d}-\frac{\left (9 a \left (-a^2+b^2\right )^{3/2} e^6\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 )}{2 b^5 d}\\ &=-\frac{9 a \left (-a^2+b^2\right )^{5/4} e^{11/2} \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{e \cos (c+d x)}}{\sqrt [4]{-a^2+b^2} \sqrt{e}}\right )}{2 b^{11/2} d}-\frac{9 a \left (-a^2+b^2\right )^{5/4} e^{11/2} \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{e \cos (c+d x)}}{\sqrt [4]{-a^2+b^2} \sqrt{e}}\right )}{2 b^{11/2} d}-\frac{3 \left (21 a^4-28 a^2 b^2+5 b^4\right ) e^6 \sqrt{\cos (c+d x)} F\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{7 b^6 d \sqrt{e \cos (c+d x)}}+\frac{9 a^2 \left (-a^2+b^2\right )^{3/2} e^6 \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 )}{2 b^6 \left (b-\sqrt{-a^2+b^2}\right ) d \sqrt{e \cos (c+d x)}}-\frac{9 a^2 \left (-a^2+b^2\right )^{3/2} e^6 \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 )}{2 b^6 \left (b+\sqrt{-a^2+b^2}\right ) d \sqrt{e \cos (c+d x)}}+\frac{9 e^3 (e \cos (c+d x))^{5/2} (7 a-5 b \sin (c+d x))}{35 b^3 d}-\frac{e (e \cos (c+d x))^{9/2}}{b d (a+b \sin (c+d x))}-\frac{3 e^5 \sqrt{e \cos (c+d x)} \left (21 a \left (a^2-b^2\right )-b \left (7 a^2-5 b^2\right ) \sin (c+d x)\right )}{7 b^5 d}\\ \end{align*}

Mathematica [C]  time = 27.7261, size = 2030, normalized size = 3.74 \[ \text{Result too large to show} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[(e*Cos[c + d*x])^(11/2)/(a + b*Sin[c + d*x])^2,x]

[Out]

-((e*Cos[c + d*x])^(11/2)*((-2*(70*a^3*b - 93*a*b^3)*(a + b*Sqrt[1 - Cos[c + d*x]^2])*((5*a*(a^2 - b^2)*Appell
F1[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*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]]))/(-a^2 + b^2)^(3/4))*Sin[c + d*x])/(Sqrt[1 - Cos[c + d*x]^2]*(a + b*Si
n[c + d*x])) + ((140*a^3*b - 147*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)*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, Co
s[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)*S
qrt[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*(35*a^4 - 126*a^2*b^2 + 75*b^4)*(a + b*Sqrt[1 - Cos[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] - 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]]))/(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]))))/(70*b^5*d*Cos[c + d*x]^(11/2)) + ((e*Cos[c + d*x])^(11/2)*Sec[c + d*x]^5*
((2*a*Cos[2*(c + d*x)])/(5*b^3) - ((-28*a^2 + 17*b^2)*Sin[c + d*x])/(14*b^4) - (-a^2 + b^2)^2/(b^5*(a + b*Sin[
c + d*x])) - Sin[3*(c + d*x)]/(14*b^2)))/d

________________________________________________________________________________________

Maple [C]  time = 10.645, size = 19829, normalized size = 36.5 \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))^(11/2)/(a+b*sin(d*x+c))^2,x)

[Out]

result too large to display

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\left (e \cos \left (d x + c\right )\right )^{\frac{11}{2}}}{{\left (b \sin \left (d x + c\right ) + a\right )}^{2}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*cos(d*x+c))^(11/2)/(a+b*sin(d*x+c))^2,x, algorithm="maxima")

[Out]

integrate((e*cos(d*x + c))^(11/2)/(b*sin(d*x + c) + a)^2, x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*cos(d*x+c))^(11/2)/(a+b*sin(d*x+c))^2,x, algorithm="fricas")

[Out]

Timed out

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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))**(11/2)/(a+b*sin(d*x+c))**2,x)

[Out]

Timed out

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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))^(11/2)/(a+b*sin(d*x+c))^2,x, algorithm="giac")

[Out]

Timed out