3.76 \(\int \frac{1}{(a+b \cos (c+d x))^2 (e \sin (c+d x))^{5/2}} \, dx\)

Optimal. Leaf size=530 \[ -\frac{7 a b^{5/2} \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{e \sin (c+d x)}}{\sqrt{e} \sqrt [4]{b^2-a^2}}\right )}{2 d e^{5/2} \left (b^2-a^2\right )^{11/4}}-\frac{7 a b^{5/2} \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{e \sin (c+d x)}}{\sqrt{e} \sqrt [4]{b^2-a^2}}\right )}{2 d e^{5/2} \left (b^2-a^2\right )^{11/4}}+\frac{\left (2 a^2+5 b^2\right ) \sqrt{\sin (c+d x)} F\left (\left .\frac{1}{2} \left (c+d x-\frac{\pi }{2}\right )\right |2\right )}{3 d e^2 \left (a^2-b^2\right )^2 \sqrt{e \sin (c+d x)}}-\frac{7 a^2 b^2 \sqrt{\sin (c+d x)} \Pi \left (\frac{2 b}{b-\sqrt{b^2-a^2}};\left .\frac{1}{2} \left (c+d x-\frac{\pi }{2}\right )\right |2\right )}{2 d e^2 \left (a^2-b^2\right )^2 \left (a^2-b \left (b-\sqrt{b^2-a^2}\right )\right ) \sqrt{e \sin (c+d x)}}-\frac{7 a^2 b^2 \sqrt{\sin (c+d x)} \Pi \left (\frac{2 b}{b+\sqrt{b^2-a^2}};\left .\frac{1}{2} \left (c+d x-\frac{\pi }{2}\right )\right |2\right )}{2 d e^2 \left (a^2-b^2\right )^2 \left (a^2-b \left (\sqrt{b^2-a^2}+b\right )\right ) \sqrt{e \sin (c+d x)}}-\frac{b}{d e \left (a^2-b^2\right ) (e \sin (c+d x))^{3/2} (a+b \cos (c+d x))}+\frac{7 a b-\left (2 a^2+5 b^2\right ) \cos (c+d x)}{3 d e \left (a^2-b^2\right )^2 (e \sin (c+d x))^{3/2}} \]

[Out]

(-7*a*b^(5/2)*ArcTan[(Sqrt[b]*Sqrt[e*Sin[c + d*x]])/((-a^2 + b^2)^(1/4)*Sqrt[e])])/(2*(-a^2 + b^2)^(11/4)*d*e^
(5/2)) - (7*a*b^(5/2)*ArcTanh[(Sqrt[b]*Sqrt[e*Sin[c + d*x]])/((-a^2 + b^2)^(1/4)*Sqrt[e])])/(2*(-a^2 + b^2)^(1
1/4)*d*e^(5/2)) - b/((a^2 - b^2)*d*e*(a + b*Cos[c + d*x])*(e*Sin[c + d*x])^(3/2)) + (7*a*b - (2*a^2 + 5*b^2)*C
os[c + d*x])/(3*(a^2 - b^2)^2*d*e*(e*Sin[c + d*x])^(3/2)) + ((2*a^2 + 5*b^2)*EllipticF[(c - Pi/2 + d*x)/2, 2]*
Sqrt[Sin[c + d*x]])/(3*(a^2 - b^2)^2*d*e^2*Sqrt[e*Sin[c + d*x]]) - (7*a^2*b^2*EllipticPi[(2*b)/(b - Sqrt[-a^2
+ b^2]), (c - Pi/2 + d*x)/2, 2]*Sqrt[Sin[c + d*x]])/(2*(a^2 - b^2)^2*(a^2 - b*(b - Sqrt[-a^2 + b^2]))*d*e^2*Sq
rt[e*Sin[c + d*x]]) - (7*a^2*b^2*EllipticPi[(2*b)/(b + Sqrt[-a^2 + b^2]), (c - Pi/2 + d*x)/2, 2]*Sqrt[Sin[c +
d*x]])/(2*(a^2 - b^2)^2*(a^2 - b*(b + Sqrt[-a^2 + b^2]))*d*e^2*Sqrt[e*Sin[c + d*x]])

________________________________________________________________________________________

Rubi [A]  time = 1.36662, antiderivative size = 530, normalized size of antiderivative = 1., number of steps used = 14, number of rules used = 12, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.48, Rules used = {2694, 2866, 2867, 2642, 2641, 2702, 2807, 2805, 329, 212, 208, 205} \[ -\frac{7 a b^{5/2} \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{e \sin (c+d x)}}{\sqrt{e} \sqrt [4]{b^2-a^2}}\right )}{2 d e^{5/2} \left (b^2-a^2\right )^{11/4}}-\frac{7 a b^{5/2} \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{e \sin (c+d x)}}{\sqrt{e} \sqrt [4]{b^2-a^2}}\right )}{2 d e^{5/2} \left (b^2-a^2\right )^{11/4}}+\frac{\left (2 a^2+5 b^2\right ) \sqrt{\sin (c+d x)} F\left (\left .\frac{1}{2} \left (c+d x-\frac{\pi }{2}\right )\right |2\right )}{3 d e^2 \left (a^2-b^2\right )^2 \sqrt{e \sin (c+d x)}}-\frac{7 a^2 b^2 \sqrt{\sin (c+d x)} \Pi \left (\frac{2 b}{b-\sqrt{b^2-a^2}};\left .\frac{1}{2} \left (c+d x-\frac{\pi }{2}\right )\right |2\right )}{2 d e^2 \left (a^2-b^2\right )^2 \left (a^2-b \left (b-\sqrt{b^2-a^2}\right )\right ) \sqrt{e \sin (c+d x)}}-\frac{7 a^2 b^2 \sqrt{\sin (c+d x)} \Pi \left (\frac{2 b}{b+\sqrt{b^2-a^2}};\left .\frac{1}{2} \left (c+d x-\frac{\pi }{2}\right )\right |2\right )}{2 d e^2 \left (a^2-b^2\right )^2 \left (a^2-b \left (\sqrt{b^2-a^2}+b\right )\right ) \sqrt{e \sin (c+d x)}}-\frac{b}{d e \left (a^2-b^2\right ) (e \sin (c+d x))^{3/2} (a+b \cos (c+d x))}+\frac{7 a b-\left (2 a^2+5 b^2\right ) \cos (c+d x)}{3 d e \left (a^2-b^2\right )^2 (e \sin (c+d x))^{3/2}} \]

Antiderivative was successfully verified.

[In]

Int[1/((a + b*Cos[c + d*x])^2*(e*Sin[c + d*x])^(5/2)),x]

[Out]

(-7*a*b^(5/2)*ArcTan[(Sqrt[b]*Sqrt[e*Sin[c + d*x]])/((-a^2 + b^2)^(1/4)*Sqrt[e])])/(2*(-a^2 + b^2)^(11/4)*d*e^
(5/2)) - (7*a*b^(5/2)*ArcTanh[(Sqrt[b]*Sqrt[e*Sin[c + d*x]])/((-a^2 + b^2)^(1/4)*Sqrt[e])])/(2*(-a^2 + b^2)^(1
1/4)*d*e^(5/2)) - b/((a^2 - b^2)*d*e*(a + b*Cos[c + d*x])*(e*Sin[c + d*x])^(3/2)) + (7*a*b - (2*a^2 + 5*b^2)*C
os[c + d*x])/(3*(a^2 - b^2)^2*d*e*(e*Sin[c + d*x])^(3/2)) + ((2*a^2 + 5*b^2)*EllipticF[(c - Pi/2 + d*x)/2, 2]*
Sqrt[Sin[c + d*x]])/(3*(a^2 - b^2)^2*d*e^2*Sqrt[e*Sin[c + d*x]]) - (7*a^2*b^2*EllipticPi[(2*b)/(b - Sqrt[-a^2
+ b^2]), (c - Pi/2 + d*x)/2, 2]*Sqrt[Sin[c + d*x]])/(2*(a^2 - b^2)^2*(a^2 - b*(b - Sqrt[-a^2 + b^2]))*d*e^2*Sq
rt[e*Sin[c + d*x]]) - (7*a^2*b^2*EllipticPi[(2*b)/(b + Sqrt[-a^2 + b^2]), (c - Pi/2 + d*x)/2, 2]*Sqrt[Sin[c +
d*x]])/(2*(a^2 - b^2)^2*(a^2 - b*(b + Sqrt[-a^2 + b^2]))*d*e^2*Sqrt[e*Sin[c + d*x]])

Rule 2694

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

Rule 2866

Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*((c_.) + (d_.)*sin[(e_.)
+ (f_.)*(x_)]), x_Symbol] :> Simp[((g*Cos[e + f*x])^(p + 1)*(a + b*Sin[e + f*x])^(m + 1)*(b*c - a*d - (a*c -
b*d)*Sin[e + f*x]))/(f*g*(a^2 - b^2)*(p + 1)), x] + Dist[1/(g^2*(a^2 - b^2)*(p + 1)), Int[(g*Cos[e + f*x])^(p
+ 2)*(a + b*Sin[e + f*x])^m*Simp[c*(a^2*(p + 2) - b^2*(m + p + 2)) + a*b*d*m + b*(a*c - b*d)*(m + p + 3)*Sin[e
+ f*x], x], x], x] /; FreeQ[{a, b, c, d, e, f, g, m}, x] && NeQ[a^2 - b^2, 0] && LtQ[p, -1] && 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{1}{(a+b \cos (c+d x))^2 (e \sin (c+d x))^{5/2}} \, dx &=-\frac{b}{\left (a^2-b^2\right ) d e (a+b \cos (c+d x)) (e \sin (c+d x))^{3/2}}+\frac{\int \frac{-a+\frac{5}{2} b \cos (c+d x)}{(a+b \cos (c+d x)) (e \sin (c+d x))^{5/2}} \, dx}{-a^2+b^2}\\ &=-\frac{b}{\left (a^2-b^2\right ) d e (a+b \cos (c+d x)) (e \sin (c+d x))^{3/2}}+\frac{7 a b-\left (2 a^2+5 b^2\right ) \cos (c+d x)}{3 \left (a^2-b^2\right )^2 d e (e \sin (c+d x))^{3/2}}+\frac{2 \int \frac{\frac{1}{2} a \left (a^2-8 b^2\right )+\frac{1}{4} b \left (2 a^2+5 b^2\right ) \cos (c+d x)}{(a+b \cos (c+d x)) \sqrt{e \sin (c+d x)}} \, dx}{3 \left (a^2-b^2\right )^2 e^2}\\ &=-\frac{b}{\left (a^2-b^2\right ) d e (a+b \cos (c+d x)) (e \sin (c+d x))^{3/2}}+\frac{7 a b-\left (2 a^2+5 b^2\right ) \cos (c+d x)}{3 \left (a^2-b^2\right )^2 d e (e \sin (c+d x))^{3/2}}-\frac{\left (7 a b^2\right ) \int \frac{1}{(a+b \cos (c+d x)) \sqrt{e \sin (c+d x)}} \, dx}{2 \left (a^2-b^2\right )^2 e^2}+\frac{\left (2 a^2+5 b^2\right ) \int \frac{1}{\sqrt{e \sin (c+d x)}} \, dx}{6 \left (a^2-b^2\right )^2 e^2}\\ &=-\frac{b}{\left (a^2-b^2\right ) d e (a+b \cos (c+d x)) (e \sin (c+d x))^{3/2}}+\frac{7 a b-\left (2 a^2+5 b^2\right ) \cos (c+d x)}{3 \left (a^2-b^2\right )^2 d e (e \sin (c+d x))^{3/2}}+\frac{\left (7 a^2 b^2\right ) \int \frac{1}{\sqrt{e \sin (c+d x)} \left (\sqrt{-a^2+b^2}-b \sin (c+d x)\right )} \, dx}{4 \left (-a^2+b^2\right )^{5/2} e^2}+\frac{\left (7 a^2 b^2\right ) \int \frac{1}{\sqrt{e \sin (c+d x)} \left (\sqrt{-a^2+b^2}+b \sin (c+d x)\right )} \, dx}{4 \left (-a^2+b^2\right )^{5/2} e^2}+\frac{\left (7 a b^3\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 \sin (c+d x)\right )}{2 \left (a^2-b^2\right )^2 d e}+\frac{\left (\left (2 a^2+5 b^2\right ) \sqrt{\sin (c+d x)}\right ) \int \frac{1}{\sqrt{\sin (c+d x)}} \, dx}{6 \left (a^2-b^2\right )^2 e^2 \sqrt{e \sin (c+d x)}}\\ &=-\frac{b}{\left (a^2-b^2\right ) d e (a+b \cos (c+d x)) (e \sin (c+d x))^{3/2}}+\frac{7 a b-\left (2 a^2+5 b^2\right ) \cos (c+d x)}{3 \left (a^2-b^2\right )^2 d e (e \sin (c+d x))^{3/2}}+\frac{\left (2 a^2+5 b^2\right ) F\left (\left .\frac{1}{2} \left (c-\frac{\pi }{2}+d x\right )\right |2\right ) \sqrt{\sin (c+d x)}}{3 \left (a^2-b^2\right )^2 d e^2 \sqrt{e \sin (c+d x)}}+\frac{\left (7 a b^3\right ) \operatorname{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 )}{\left (a^2-b^2\right )^2 d e}+\frac{\left (7 a^2 b^2 \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}{4 \left (-a^2+b^2\right )^{5/2} e^2 \sqrt{e \sin (c+d x)}}+\frac{\left (7 a^2 b^2 \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}{4 \left (-a^2+b^2\right )^{5/2} e^2 \sqrt{e \sin (c+d x)}}\\ &=-\frac{b}{\left (a^2-b^2\right ) d e (a+b \cos (c+d x)) (e \sin (c+d x))^{3/2}}+\frac{7 a b-\left (2 a^2+5 b^2\right ) \cos (c+d x)}{3 \left (a^2-b^2\right )^2 d e (e \sin (c+d x))^{3/2}}+\frac{\left (2 a^2+5 b^2\right ) F\left (\left .\frac{1}{2} \left (c-\frac{\pi }{2}+d x\right )\right |2\right ) \sqrt{\sin (c+d x)}}{3 \left (a^2-b^2\right )^2 d e^2 \sqrt{e \sin (c+d x)}}-\frac{7 a^2 b^2 \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)}}{2 \left (-a^2+b^2\right )^{5/2} \left (b-\sqrt{-a^2+b^2}\right ) d e^2 \sqrt{e \sin (c+d x)}}+\frac{7 a^2 b^2 \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)}}{2 \left (-a^2+b^2\right )^{5/2} \left (b+\sqrt{-a^2+b^2}\right ) d e^2 \sqrt{e \sin (c+d x)}}-\frac{\left (7 a b^3\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{-a^2+b^2} e-b x^2} \, dx,x,\sqrt{e \sin (c+d x)}\right )}{2 \left (-a^2+b^2\right )^{5/2} d e^2}-\frac{\left (7 a b^3\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{-a^2+b^2} e+b x^2} \, dx,x,\sqrt{e \sin (c+d x)}\right )}{2 \left (-a^2+b^2\right )^{5/2} d e^2}\\ &=-\frac{7 a b^{5/2} \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{e \sin (c+d x)}}{\sqrt [4]{-a^2+b^2} \sqrt{e}}\right )}{2 \left (-a^2+b^2\right )^{11/4} d e^{5/2}}-\frac{7 a b^{5/2} \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{e \sin (c+d x)}}{\sqrt [4]{-a^2+b^2} \sqrt{e}}\right )}{2 \left (-a^2+b^2\right )^{11/4} d e^{5/2}}-\frac{b}{\left (a^2-b^2\right ) d e (a+b \cos (c+d x)) (e \sin (c+d x))^{3/2}}+\frac{7 a b-\left (2 a^2+5 b^2\right ) \cos (c+d x)}{3 \left (a^2-b^2\right )^2 d e (e \sin (c+d x))^{3/2}}+\frac{\left (2 a^2+5 b^2\right ) F\left (\left .\frac{1}{2} \left (c-\frac{\pi }{2}+d x\right )\right |2\right ) \sqrt{\sin (c+d x)}}{3 \left (a^2-b^2\right )^2 d e^2 \sqrt{e \sin (c+d x)}}-\frac{7 a^2 b^2 \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)}}{2 \left (-a^2+b^2\right )^{5/2} \left (b-\sqrt{-a^2+b^2}\right ) d e^2 \sqrt{e \sin (c+d x)}}+\frac{7 a^2 b^2 \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)}}{2 \left (-a^2+b^2\right )^{5/2} \left (b+\sqrt{-a^2+b^2}\right ) d e^2 \sqrt{e \sin (c+d x)}}\\ \end{align*}

Mathematica [C]  time = 14.7895, size = 1257, normalized size = 2.37 \[ \frac{\left (\frac{b^3}{\left (a^2-b^2\right )^2 (a+b \cos (c+d x))}-\frac{2 \left (\cos (c+d x) a^2-2 b a+b^2 \cos (c+d x)\right ) \csc ^2(c+d x)}{3 \left (a^2-b^2\right )^2}\right ) \sin ^3(c+d x)}{d (e \sin (c+d x))^{5/2}}+\frac{\left (\frac{2 \left (5 b^3+2 a^2 b\right ) \left (a+b \sqrt{1-\sin ^2(c+d x)}\right ) \left (\frac{5 b \left (a^2-b^2\right ) \sqrt{\sin (c+d x)} \sqrt{1-\sin ^2(c+d x)} F_1\left (\frac{1}{4};-\frac{1}{2},1;\frac{5}{4};\sin ^2(c+d x),\frac{b^2 \sin ^2(c+d x)}{b^2-a^2}\right )}{\left (2 \left (2 F_1\left (\frac{5}{4};-\frac{1}{2},2;\frac{9}{4};\sin ^2(c+d x),\frac{b^2 \sin ^2(c+d x)}{b^2-a^2}\right ) b^2+\left (a^2-b^2\right ) F_1\left (\frac{5}{4};\frac{1}{2},1;\frac{9}{4};\sin ^2(c+d x),\frac{b^2 \sin ^2(c+d x)}{b^2-a^2}\right )\right ) \sin ^2(c+d x)-5 \left (a^2-b^2\right ) F_1\left (\frac{1}{4};-\frac{1}{2},1;\frac{5}{4};\sin ^2(c+d x),\frac{b^2 \sin ^2(c+d x)}{b^2-a^2}\right )\right ) \left (a^2+b^2 \left (\sin ^2(c+d x)-1\right )\right )}+\frac{a \left (-2 \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt{b} \sqrt{\sin (c+d x)}}{\sqrt [4]{a^2-b^2}}\right )+2 \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{b} \sqrt{\sin (c+d x)}}{\sqrt [4]{a^2-b^2}}+1\right )-\log \left (b \sin (c+d x)-\sqrt{2} \sqrt{b} \sqrt [4]{a^2-b^2} \sqrt{\sin (c+d x)}+\sqrt{a^2-b^2}\right )+\log \left (b \sin (c+d x)+\sqrt{2} \sqrt{b} \sqrt [4]{a^2-b^2} \sqrt{\sin (c+d x)}+\sqrt{a^2-b^2}\right )\right )}{4 \sqrt{2} \sqrt{b} \left (a^2-b^2\right )^{3/4}}\right ) \cos ^2(c+d x)}{(a+b \cos (c+d x)) \left (1-\sin ^2(c+d x)\right )}+\frac{2 \left (2 a^3-16 a b^2\right ) \left (a+b \sqrt{1-\sin ^2(c+d x)}\right ) \left (\frac{5 a \left (a^2-b^2\right ) F_1\left (\frac{1}{4};\frac{1}{2},1;\frac{5}{4};\sin ^2(c+d x),\frac{b^2 \sin ^2(c+d x)}{b^2-a^2}\right ) \sqrt{\sin (c+d x)}}{\sqrt{1-\sin ^2(c+d x)} \left (5 \left (a^2-b^2\right ) F_1\left (\frac{1}{4};\frac{1}{2},1;\frac{5}{4};\sin ^2(c+d x),\frac{b^2 \sin ^2(c+d x)}{b^2-a^2}\right )-2 \left (2 F_1\left (\frac{5}{4};\frac{1}{2},2;\frac{9}{4};\sin ^2(c+d x),\frac{b^2 \sin ^2(c+d x)}{b^2-a^2}\right ) b^2+\left (b^2-a^2\right ) F_1\left (\frac{5}{4};\frac{3}{2},1;\frac{9}{4};\sin ^2(c+d x),\frac{b^2 \sin ^2(c+d x)}{b^2-a^2}\right )\right ) \sin ^2(c+d x)\right ) \left (a^2+b^2 \left (\sin ^2(c+d x)-1\right )\right )}-\frac{\left (\frac{1}{8}-\frac{i}{8}\right ) \sqrt{b} \left (2 \tan ^{-1}\left (1-\frac{(1+i) \sqrt{b} \sqrt{\sin (c+d x)}}{\sqrt [4]{b^2-a^2}}\right )-2 \tan ^{-1}\left (\frac{(1+i) \sqrt{b} \sqrt{\sin (c+d x)}}{\sqrt [4]{b^2-a^2}}+1\right )+\log \left (i b \sin (c+d x)-(1+i) \sqrt{b} \sqrt [4]{b^2-a^2} \sqrt{\sin (c+d x)}+\sqrt{b^2-a^2}\right )-\log \left (i b \sin (c+d x)+(1+i) \sqrt{b} \sqrt [4]{b^2-a^2} \sqrt{\sin (c+d x)}+\sqrt{b^2-a^2}\right )\right )}{\left (b^2-a^2\right )^{3/4}}\right ) \cos (c+d x)}{(a+b \cos (c+d x)) \sqrt{1-\sin ^2(c+d x)}}\right ) \sin ^{\frac{5}{2}}(c+d x)}{6 (a-b)^2 (a+b)^2 d (e \sin (c+d x))^{5/2}} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[1/((a + b*Cos[c + d*x])^2*(e*Sin[c + d*x])^(5/2)),x]

[Out]

((b^3/((a^2 - b^2)^2*(a + b*Cos[c + d*x])) - (2*(-2*a*b + a^2*Cos[c + d*x] + b^2*Cos[c + d*x])*Csc[c + d*x]^2)
/(3*(a^2 - b^2)^2))*Sin[c + d*x]^3)/(d*(e*Sin[c + d*x])^(5/2)) + (Sin[c + d*x]^(5/2)*((2*(2*a^2*b + 5*b^3)*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)]*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)*Appell
F1[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*(2*a^3 - 16*a*b^2)*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[Sin[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)] + Log[Sqrt[-a^2 + b^2] - (1 + I)*S
qrt[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])))/(6*(a - b)^2*(a + b)^2*d*(e*Sin[c + d*x])^(5/2))

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Maple [B]  time = 11.063, size = 2143, normalized size = 4. \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(a+b*cos(d*x+c))^2/(e*sin(d*x+c))^(5/2),x)

[Out]

1/d/e*a*b^3/(a-b)^2/(a+b)^2*(e*sin(d*x+c))^(1/2)/(-b^2*cos(d*x+c)^2*e^2+e^2*a^2)+7/8/d/e*a*b^3/(a-b)^2/(a+b)^2
*(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)))+7/4/d/e*a*b^3/(a-b)^2/(a+b)^2*(e^2*(a^2-b^2)/b^2)^(1/4)/(a^2*e^2-b^2*e^2)*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)+7/4/d/e*a*b^3/(a-b)^2/(a+b)^2*(e^2*(a^2-b^2)
/b^2)^(1/4)/(a^2*e^2-b^2*e^2)*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)+4/3/d/e
*a*b/(a^2-b^2)^2/(e*sin(d*x+c))^(3/2)+1/3/d/e^2/cos(d*x+c)/(e*sin(d*x+c))^(1/2)/(a^2-b^2)^2/(cos(d*x+c)^2-1)*(
1-sin(d*x+c))^(1/2)*(2+2*sin(d*x+c))^(1/2)*sin(d*x+c)^(5/2)*EllipticF((1-sin(d*x+c))^(1/2),1/2*2^(1/2))*a^2+1/
3/d/e^2/cos(d*x+c)/(e*sin(d*x+c))^(1/2)/(a^2-b^2)^2/(cos(d*x+c)^2-1)*(1-sin(d*x+c))^(1/2)*(2+2*sin(d*x+c))^(1/
2)*sin(d*x+c)^(5/2)*EllipticF((1-sin(d*x+c))^(1/2),1/2*2^(1/2))*b^2+2/3/d/e^2*cos(d*x+c)/(e*sin(d*x+c))^(1/2)/
(a^2-b^2)^2/(cos(d*x+c)^2-1)*sin(d*x+c)*a^2+2/3/d/e^2*cos(d*x+c)/(e*sin(d*x+c))^(1/2)/(a^2-b^2)^2/(cos(d*x+c)^
2-1)*sin(d*x+c)*b^2-1/d/e^2*sin(d*x+c)*cos(d*x+c)/(e*sin(d*x+c))^(1/2)*b^4/(a-b)/(a+b)/(a^2-b^2)/(-b^2*cos(d*x
+c)^2+a^2)-1/2/d/e^2/cos(d*x+c)/(e*sin(d*x+c))^(1/2)*b^2/(a-b)/(a+b)/(a^2-b^2)*(1-sin(d*x+c))^(1/2)*(2+2*sin(d
*x+c))^(1/2)*sin(d*x+c)^(1/2)*EllipticF((1-sin(d*x+c))^(1/2),1/2*2^(1/2))+5/4/d/e^2/cos(d*x+c)/(e*sin(d*x+c))^
(1/2)*a^2*b/(a-b)/(a+b)/(a^2-b^2)/(-a^2+b^2)^(1/2)*(1-sin(d*x+c))^(1/2)*(2+2*sin(d*x+c))^(1/2)*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/d/e^2/cos(d
*x+c)/(e*sin(d*x+c))^(1/2)*b^3/(a-b)/(a+b)/(a^2-b^2)/(-a^2+b^2)^(1/2)*(1-sin(d*x+c))^(1/2)*(2+2*sin(d*x+c))^(1
/2)*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))-5/4/d/e^2/cos(d*x+c)/(e*sin(d*x+c))^(1/2)*a^2*b/(a-b)/(a+b)/(a^2-b^2)/(-a^2+b^2)^(1/2)*(1-sin(d*x+c))^(1/2
)*(2+2*sin(d*x+c))^(1/2)*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/d/e^2/cos(d*x+c)/(e*sin(d*x+c))^(1/2)*b^3/(a-b)/(a+b)/(a^2-b^2)/(-a^2+b^2)^(1/2)*
(1-sin(d*x+c))^(1/2)*(2+2*sin(d*x+c))^(1/2)*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/d/e^2/cos(d*x+c)/(e*sin(d*x+c))^(1/2)*b/(a-b)^2/(a+b)^2/(-a^2+
b^2)^(1/2)*(1-sin(d*x+c))^(1/2)*(2+2*sin(d*x+c))^(1/2)*sin(d*x+c)^(1/2)/(1-(-a^2+b^2)^(1/2)/b)*EllipticPi((1-s
in(d*x+c))^(1/2),1/(1-(-a^2+b^2)^(1/2)/b),1/2*2^(1/2))*a^2+1/2/d/e^2/cos(d*x+c)/(e*sin(d*x+c))^(1/2)*b^3/(a-b)
^2/(a+b)^2/(-a^2+b^2)^(1/2)*(1-sin(d*x+c))^(1/2)*(2+2*sin(d*x+c))^(1/2)*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/d/e^2/cos(d*x+c)/(e*sin(d*x+c))^(1
/2)*b/(a-b)^2/(a+b)^2/(-a^2+b^2)^(1/2)*(1-sin(d*x+c))^(1/2)*(2+2*sin(d*x+c))^(1/2)*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))*a^2-1/2/d/e^2/cos(d*x+c)/(e
*sin(d*x+c))^(1/2)*b^3/(a-b)^2/(a+b)^2/(-a^2+b^2)^(1/2)*(1-sin(d*x+c))^(1/2)*(2+2*sin(d*x+c))^(1/2)*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))

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+b*cos(d*x+c))^2/(e*sin(d*x+c))^(5/2),x, algorithm="maxima")

[Out]

Timed out

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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(1/(a+b*cos(d*x+c))^2/(e*sin(d*x+c))^(5/2),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(1/(a+b*cos(d*x+c))**2/(e*sin(d*x+c))**(5/2),x)

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{{\left (b \cos \left (d x + c\right ) + a\right )}^{2} \left (e \sin \left (d x + c\right )\right )^{\frac{5}{2}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+b*cos(d*x+c))^2/(e*sin(d*x+c))^(5/2),x, algorithm="giac")

[Out]

integrate(1/((b*cos(d*x + c) + a)^2*(e*sin(d*x + c))^(5/2)), x)