3.7.2 \(\int \frac {1}{(e \cos (c+d x))^{3/2} (a+b \sin (c+d x))^3} \, dx\) [602]

Optimal. Leaf size=596 \[ \frac {5 b^{3/2} \left (7 a^2+2 b^2\right ) \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {e \cos (c+d x)}}{\sqrt [4]{-a^2+b^2} \sqrt {e}}\right )}{8 \left (-a^2+b^2\right )^{13/4} d e^{3/2}}-\frac {5 b^{3/2} \left (7 a^2+2 b^2\right ) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {e \cos (c+d x)}}{\sqrt [4]{-a^2+b^2} \sqrt {e}}\right )}{8 \left (-a^2+b^2\right )^{13/4} d e^{3/2}}-\frac {a \left (8 a^2+37 b^2\right ) \sqrt {e \cos (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{4 \left (a^2-b^2\right )^3 d e^2 \sqrt {\cos (c+d x)}}-\frac {5 a b \left (7 a^2+2 b^2\right ) \sqrt {\cos (c+d x)} \Pi \left (\frac {2 b}{b-\sqrt {-a^2+b^2}};\left .\frac {1}{2} (c+d x)\right |2\right )}{8 \left (a^2-b^2\right )^3 \left (b-\sqrt {-a^2+b^2}\right ) d e \sqrt {e \cos (c+d x)}}-\frac {5 a b \left (7 a^2+2 b^2\right ) \sqrt {\cos (c+d x)} \Pi \left (\frac {2 b}{b+\sqrt {-a^2+b^2}};\left .\frac {1}{2} (c+d x)\right |2\right )}{8 \left (a^2-b^2\right )^3 \left (b+\sqrt {-a^2+b^2}\right ) d e \sqrt {e \cos (c+d x)}}+\frac {b}{2 \left (a^2-b^2\right ) d e \sqrt {e \cos (c+d x)} (a+b \sin (c+d x))^2}+\frac {9 a b}{4 \left (a^2-b^2\right )^2 d e \sqrt {e \cos (c+d x)} (a+b \sin (c+d x))}-\frac {5 b \left (7 a^2+2 b^2\right )-a \left (8 a^2+37 b^2\right ) \sin (c+d x)}{4 \left (a^2-b^2\right )^3 d e \sqrt {e \cos (c+d x)}} \]

[Out]

5/8*b^(3/2)*(7*a^2+2*b^2)*arctan(b^(1/2)*(e*cos(d*x+c))^(1/2)/(-a^2+b^2)^(1/4)/e^(1/2))/(-a^2+b^2)^(13/4)/d/e^
(3/2)-5/8*b^(3/2)*(7*a^2+2*b^2)*arctanh(b^(1/2)*(e*cos(d*x+c))^(1/2)/(-a^2+b^2)^(1/4)/e^(1/2))/(-a^2+b^2)^(13/
4)/d/e^(3/2)+1/2*b/(a^2-b^2)/d/e/(a+b*sin(d*x+c))^2/(e*cos(d*x+c))^(1/2)+9/4*a*b/(a^2-b^2)^2/d/e/(a+b*sin(d*x+
c))/(e*cos(d*x+c))^(1/2)+1/4*(-5*b*(7*a^2+2*b^2)+a*(8*a^2+37*b^2)*sin(d*x+c))/(a^2-b^2)^3/d/e/(e*cos(d*x+c))^(
1/2)-5/8*a*b*(7*a^2+2*b^2)*(cos(1/2*d*x+1/2*c)^2)^(1/2)/cos(1/2*d*x+1/2*c)*EllipticPi(sin(1/2*d*x+1/2*c),2*b/(
b-(-a^2+b^2)^(1/2)),2^(1/2))*cos(d*x+c)^(1/2)/(a^2-b^2)^3/d/e/(b-(-a^2+b^2)^(1/2))/(e*cos(d*x+c))^(1/2)-5/8*a*
b*(7*a^2+2*b^2)*(cos(1/2*d*x+1/2*c)^2)^(1/2)/cos(1/2*d*x+1/2*c)*EllipticPi(sin(1/2*d*x+1/2*c),2*b/(b+(-a^2+b^2
)^(1/2)),2^(1/2))*cos(d*x+c)^(1/2)/(a^2-b^2)^3/d/e/(b+(-a^2+b^2)^(1/2))/(e*cos(d*x+c))^(1/2)-1/4*a*(8*a^2+37*b
^2)*(cos(1/2*d*x+1/2*c)^2)^(1/2)/cos(1/2*d*x+1/2*c)*EllipticE(sin(1/2*d*x+1/2*c),2^(1/2))*(e*cos(d*x+c))^(1/2)
/(a^2-b^2)^3/d/e^2/cos(d*x+c)^(1/2)

________________________________________________________________________________________

Rubi [A]
time = 1.05, antiderivative size = 596, 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 = {2773, 2943, 2945, 2946, 2721, 2719, 2780, 2886, 2884, 335, 304, 211, 214} \begin {gather*} \frac {5 b^{3/2} \left (7 a^2+2 b^2\right ) \text {ArcTan}\left (\frac {\sqrt {b} \sqrt {e \cos (c+d x)}}{\sqrt {e} \sqrt [4]{b^2-a^2}}\right )}{8 d e^{3/2} \left (b^2-a^2\right )^{13/4}}-\frac {a \left (8 a^2+37 b^2\right ) E\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \sqrt {e \cos (c+d x)}}{4 d e^2 \left (a^2-b^2\right )^3 \sqrt {\cos (c+d x)}}+\frac {9 a b}{4 d e \left (a^2-b^2\right )^2 \sqrt {e \cos (c+d x)} (a+b \sin (c+d x))}+\frac {b}{2 d e \left (a^2-b^2\right ) \sqrt {e \cos (c+d x)} (a+b \sin (c+d x))^2}-\frac {5 b \left (7 a^2+2 b^2\right )-a \left (8 a^2+37 b^2\right ) \sin (c+d x)}{4 d e \left (a^2-b^2\right )^3 \sqrt {e \cos (c+d x)}}-\frac {5 a b \left (7 a^2+2 b^2\right ) \sqrt {\cos (c+d x)} \Pi \left (\frac {2 b}{b-\sqrt {b^2-a^2}};\left .\frac {1}{2} (c+d x)\right |2\right )}{8 d e \left (a^2-b^2\right )^3 \left (b-\sqrt {b^2-a^2}\right ) \sqrt {e \cos (c+d x)}}-\frac {5 a b \left (7 a^2+2 b^2\right ) \sqrt {\cos (c+d x)} \Pi \left (\frac {2 b}{b+\sqrt {b^2-a^2}};\left .\frac {1}{2} (c+d x)\right |2\right )}{8 d e \left (a^2-b^2\right )^3 \left (\sqrt {b^2-a^2}+b\right ) \sqrt {e \cos (c+d x)}}-\frac {5 b^{3/2} \left (7 a^2+2 b^2\right ) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {e \cos (c+d x)}}{\sqrt {e} \sqrt [4]{b^2-a^2}}\right )}{8 d e^{3/2} \left (b^2-a^2\right )^{13/4}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[1/((e*Cos[c + d*x])^(3/2)*(a + b*Sin[c + d*x])^3),x]

[Out]

(5*b^(3/2)*(7*a^2 + 2*b^2)*ArcTan[(Sqrt[b]*Sqrt[e*Cos[c + d*x]])/((-a^2 + b^2)^(1/4)*Sqrt[e])])/(8*(-a^2 + b^2
)^(13/4)*d*e^(3/2)) - (5*b^(3/2)*(7*a^2 + 2*b^2)*ArcTanh[(Sqrt[b]*Sqrt[e*Cos[c + d*x]])/((-a^2 + b^2)^(1/4)*Sq
rt[e])])/(8*(-a^2 + b^2)^(13/4)*d*e^(3/2)) - (a*(8*a^2 + 37*b^2)*Sqrt[e*Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2
])/(4*(a^2 - b^2)^3*d*e^2*Sqrt[Cos[c + d*x]]) - (5*a*b*(7*a^2 + 2*b^2)*Sqrt[Cos[c + d*x]]*EllipticPi[(2*b)/(b
- Sqrt[-a^2 + b^2]), (c + d*x)/2, 2])/(8*(a^2 - b^2)^3*(b - Sqrt[-a^2 + b^2])*d*e*Sqrt[e*Cos[c + d*x]]) - (5*a
*b*(7*a^2 + 2*b^2)*Sqrt[Cos[c + d*x]]*EllipticPi[(2*b)/(b + Sqrt[-a^2 + b^2]), (c + d*x)/2, 2])/(8*(a^2 - b^2)
^3*(b + Sqrt[-a^2 + b^2])*d*e*Sqrt[e*Cos[c + d*x]]) + b/(2*(a^2 - b^2)*d*e*Sqrt[e*Cos[c + d*x]]*(a + b*Sin[c +
d*x])^2) + (9*a*b)/(4*(a^2 - b^2)^2*d*e*Sqrt[e*Cos[c + d*x]]*(a + b*Sin[c + d*x])) - (5*b*(7*a^2 + 2*b^2) - a
*(8*a^2 + 37*b^2)*Sin[c + d*x])/(4*(a^2 - b^2)^3*d*e*Sqrt[e*Cos[c + d*x]])

Rule 211

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

Rule 214

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

Rule 304

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

Rule 335

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

Rule 2719

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

Rule 2721

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

Rule 2773

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] /;
FreeQ[{a, b, e, f, g, p}, x] && NeQ[a^2 - b^2, 0] && LtQ[m, -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 2943

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

Rule 2945

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 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 {1}{(e \cos (c+d x))^{3/2} (a+b \sin (c+d x))^3} \, dx &=\frac {b}{2 \left (a^2-b^2\right ) d e \sqrt {e \cos (c+d x)} (a+b \sin (c+d x))^2}-\frac {\int \frac {-2 a+\frac {5}{2} b \sin (c+d x)}{(e \cos (c+d x))^{3/2} (a+b \sin (c+d x))^2} \, dx}{2 \left (a^2-b^2\right )}\\ &=\frac {b}{2 \left (a^2-b^2\right ) d e \sqrt {e \cos (c+d x)} (a+b \sin (c+d x))^2}+\frac {9 a b}{4 \left (a^2-b^2\right )^2 d e \sqrt {e \cos (c+d x)} (a+b \sin (c+d x))}+\frac {\int \frac {\frac {1}{2} \left (4 a^2+5 b^2\right )-\frac {27}{4} a b \sin (c+d x)}{(e \cos (c+d x))^{3/2} (a+b \sin (c+d x))} \, dx}{2 \left (a^2-b^2\right )^2}\\ &=\frac {b}{2 \left (a^2-b^2\right ) d e \sqrt {e \cos (c+d x)} (a+b \sin (c+d x))^2}+\frac {9 a b}{4 \left (a^2-b^2\right )^2 d e \sqrt {e \cos (c+d x)} (a+b \sin (c+d x))}-\frac {5 b \left (7 a^2+2 b^2\right )-a \left (8 a^2+37 b^2\right ) \sin (c+d x)}{4 \left (a^2-b^2\right )^3 d e \sqrt {e \cos (c+d x)}}-\frac {\int \frac {\sqrt {e \cos (c+d x)} \left (\frac {1}{4} \left (4 a^4+36 a^2 b^2+5 b^4\right )+\frac {1}{8} a b \left (8 a^2+37 b^2\right ) \sin (c+d x)\right )}{a+b \sin (c+d x)} \, dx}{\left (a^2-b^2\right )^3 e^2}\\ &=\frac {b}{2 \left (a^2-b^2\right ) d e \sqrt {e \cos (c+d x)} (a+b \sin (c+d x))^2}+\frac {9 a b}{4 \left (a^2-b^2\right )^2 d e \sqrt {e \cos (c+d x)} (a+b \sin (c+d x))}-\frac {5 b \left (7 a^2+2 b^2\right )-a \left (8 a^2+37 b^2\right ) \sin (c+d x)}{4 \left (a^2-b^2\right )^3 d e \sqrt {e \cos (c+d x)}}-\frac {\left (5 b^2 \left (7 a^2+2 b^2\right )\right ) \int \frac {\sqrt {e \cos (c+d x)}}{a+b \sin (c+d x)} \, dx}{8 \left (a^2-b^2\right )^3 e^2}-\frac {\left (a \left (8 a^2+37 b^2\right )\right ) \int \sqrt {e \cos (c+d x)} \, dx}{8 \left (a^2-b^2\right )^3 e^2}\\ &=\frac {b}{2 \left (a^2-b^2\right ) d e \sqrt {e \cos (c+d x)} (a+b \sin (c+d x))^2}+\frac {9 a b}{4 \left (a^2-b^2\right )^2 d e \sqrt {e \cos (c+d x)} (a+b \sin (c+d x))}-\frac {5 b \left (7 a^2+2 b^2\right )-a \left (8 a^2+37 b^2\right ) \sin (c+d x)}{4 \left (a^2-b^2\right )^3 d e \sqrt {e \cos (c+d x)}}+\frac {\left (5 a b \left (7 a^2+2 b^2\right )\right ) \int \frac {1}{\sqrt {e \cos (c+d x)} \left (\sqrt {-a^2+b^2}-b \cos (c+d x)\right )} \, dx}{16 \left (a^2-b^2\right )^3 e}-\frac {\left (5 a b \left (7 a^2+2 b^2\right )\right ) \int \frac {1}{\sqrt {e \cos (c+d x)} \left (\sqrt {-a^2+b^2}+b \cos (c+d x)\right )} \, dx}{16 \left (a^2-b^2\right )^3 e}-\frac {\left (5 b^3 \left (7 a^2+2 b^2\right )\right ) \text {Subst}\left (\int \frac {\sqrt {x}}{\left (a^2-b^2\right ) e^2+b^2 x^2} \, dx,x,e \cos (c+d x)\right )}{8 \left (a^2-b^2\right )^3 d e}-\frac {\left (a \left (8 a^2+37 b^2\right ) \sqrt {e \cos (c+d x)}\right ) \int \sqrt {\cos (c+d x)} \, dx}{8 \left (a^2-b^2\right )^3 e^2 \sqrt {\cos (c+d x)}}\\ &=-\frac {a \left (8 a^2+37 b^2\right ) \sqrt {e \cos (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{4 \left (a^2-b^2\right )^3 d e^2 \sqrt {\cos (c+d x)}}+\frac {b}{2 \left (a^2-b^2\right ) d e \sqrt {e \cos (c+d x)} (a+b \sin (c+d x))^2}+\frac {9 a b}{4 \left (a^2-b^2\right )^2 d e \sqrt {e \cos (c+d x)} (a+b \sin (c+d x))}-\frac {5 b \left (7 a^2+2 b^2\right )-a \left (8 a^2+37 b^2\right ) \sin (c+d x)}{4 \left (a^2-b^2\right )^3 d e \sqrt {e \cos (c+d x)}}-\frac {\left (5 b^3 \left (7 a^2+2 b^2\right )\right ) \text {Subst}\left (\int \frac {x^2}{\left (a^2-b^2\right ) e^2+b^2 x^4} \, dx,x,\sqrt {e \cos (c+d x)}\right )}{4 \left (a^2-b^2\right )^3 d e}+\frac {\left (5 a b \left (7 a^2+2 b^2\right ) \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}{16 \left (a^2-b^2\right )^3 e \sqrt {e \cos (c+d x)}}-\frac {\left (5 a b \left (7 a^2+2 b^2\right ) \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}{16 \left (a^2-b^2\right )^3 e \sqrt {e \cos (c+d x)}}\\ &=-\frac {a \left (8 a^2+37 b^2\right ) \sqrt {e \cos (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{4 \left (a^2-b^2\right )^3 d e^2 \sqrt {\cos (c+d x)}}-\frac {5 a b \left (7 a^2+2 b^2\right ) \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 )}{8 \left (a^2-b^2\right )^3 \left (b-\sqrt {-a^2+b^2}\right ) d e \sqrt {e \cos (c+d x)}}-\frac {5 a b \left (7 a^2+2 b^2\right ) \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 )}{8 \left (a^2-b^2\right )^3 \left (b+\sqrt {-a^2+b^2}\right ) d e \sqrt {e \cos (c+d x)}}+\frac {b}{2 \left (a^2-b^2\right ) d e \sqrt {e \cos (c+d x)} (a+b \sin (c+d x))^2}+\frac {9 a b}{4 \left (a^2-b^2\right )^2 d e \sqrt {e \cos (c+d x)} (a+b \sin (c+d x))}-\frac {5 b \left (7 a^2+2 b^2\right )-a \left (8 a^2+37 b^2\right ) \sin (c+d x)}{4 \left (a^2-b^2\right )^3 d e \sqrt {e \cos (c+d x)}}+\frac {\left (5 b^2 \left (7 a^2+2 b^2\right )\right ) \text {Subst}\left (\int \frac {1}{\sqrt {-a^2+b^2} e-b x^2} \, dx,x,\sqrt {e \cos (c+d x)}\right )}{8 \left (a^2-b^2\right )^3 d e}-\frac {\left (5 b^2 \left (7 a^2+2 b^2\right )\right ) \text {Subst}\left (\int \frac {1}{\sqrt {-a^2+b^2} e+b x^2} \, dx,x,\sqrt {e \cos (c+d x)}\right )}{8 \left (a^2-b^2\right )^3 d e}\\ &=\frac {5 b^{3/2} \left (7 a^2+2 b^2\right ) \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {e \cos (c+d x)}}{\sqrt [4]{-a^2+b^2} \sqrt {e}}\right )}{8 \left (-a^2+b^2\right )^{13/4} d e^{3/2}}-\frac {5 b^{3/2} \left (7 a^2+2 b^2\right ) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {e \cos (c+d x)}}{\sqrt [4]{-a^2+b^2} \sqrt {e}}\right )}{8 \left (-a^2+b^2\right )^{13/4} d e^{3/2}}-\frac {a \left (8 a^2+37 b^2\right ) \sqrt {e \cos (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{4 \left (a^2-b^2\right )^3 d e^2 \sqrt {\cos (c+d x)}}-\frac {5 a b \left (7 a^2+2 b^2\right ) \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 )}{8 \left (a^2-b^2\right )^3 \left (b-\sqrt {-a^2+b^2}\right ) d e \sqrt {e \cos (c+d x)}}-\frac {5 a b \left (7 a^2+2 b^2\right ) \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 )}{8 \left (a^2-b^2\right )^3 \left (b+\sqrt {-a^2+b^2}\right ) d e \sqrt {e \cos (c+d x)}}+\frac {b}{2 \left (a^2-b^2\right ) d e \sqrt {e \cos (c+d x)} (a+b \sin (c+d x))^2}+\frac {9 a b}{4 \left (a^2-b^2\right )^2 d e \sqrt {e \cos (c+d x)} (a+b \sin (c+d x))}-\frac {5 b \left (7 a^2+2 b^2\right )-a \left (8 a^2+37 b^2\right ) \sin (c+d x)}{4 \left (a^2-b^2\right )^3 d e \sqrt {e \cos (c+d x)}}\\ \end {align*}

________________________________________________________________________________________

Mathematica [C] Result contains higher order function than in optimal. Order 6 vs. order 4 in optimal.
time = 16.37, size = 922, normalized size = 1.55 \begin {gather*} -\frac {\cos ^{\frac {3}{2}}(c+d x) \left (-\frac {2 \left (8 a^4+72 a^2 b^2+10 b^4\right ) \left (a+b \sqrt {1-\cos ^2(c+d x)}\right ) \left (\frac {a F_1\left (\frac {3}{4};\frac {1}{2},1;\frac {7}{4};\cos ^2(c+d x),\frac {b^2 \cos ^2(c+d x)}{-a^2+b^2}\right ) \cos ^{\frac {3}{2}}(c+d x)}{3 \left (a^2-b^2\right )}+\frac {\left (\frac {1}{8}+\frac {i}{8}\right ) \left (2 \tan ^{-1}\left (1-\frac {(1+i) \sqrt {b} \sqrt {\cos (c+d x)}}{\sqrt [4]{-a^2+b^2}}\right )-2 \tan ^{-1}\left (1+\frac {(1+i) \sqrt {b} \sqrt {\cos (c+d x)}}{\sqrt [4]{-a^2+b^2}}\right )-\log \left (\sqrt {-a^2+b^2}-(1+i) \sqrt {b} \sqrt [4]{-a^2+b^2} \sqrt {\cos (c+d x)}+i b \cos (c+d x)\right )+\log \left (\sqrt {-a^2+b^2}+(1+i) \sqrt {b} \sqrt [4]{-a^2+b^2} \sqrt {\cos (c+d x)}+i b \cos (c+d x)\right )\right )}{\sqrt {b} \sqrt [4]{-a^2+b^2}}\right ) \sin (c+d x)}{\sqrt {1-\cos ^2(c+d x)} (a+b \sin (c+d x))}-\frac {\left (8 a^3 b+37 a b^3\right ) \left (a+b \sqrt {1-\cos ^2(c+d x)}\right ) \left (8 b^{5/2} F_1\left (\frac {3}{4};-\frac {1}{2},1;\frac {7}{4};\cos ^2(c+d x),\frac {b^2 \cos ^2(c+d x)}{-a^2+b^2}\right ) \cos ^{\frac {3}{2}}(c+d x)+3 \sqrt {2} a \left (a^2-b^2\right )^{3/4} \left (2 \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt {b} \sqrt {\cos (c+d x)}}{\sqrt [4]{a^2-b^2}}\right )-2 \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt {b} \sqrt {\cos (c+d x)}}{\sqrt [4]{a^2-b^2}}\right )-\log \left (\sqrt {a^2-b^2}-\sqrt {2} \sqrt {b} \sqrt [4]{a^2-b^2} \sqrt {\cos (c+d x)}+b \cos (c+d x)\right )+\log \left (\sqrt {a^2-b^2}+\sqrt {2} \sqrt {b} \sqrt [4]{a^2-b^2} \sqrt {\cos (c+d x)}+b \cos (c+d x)\right )\right )\right ) \sin ^2(c+d x)}{12 b^{3/2} \left (-a^2+b^2\right ) \left (1-\cos ^2(c+d x)\right ) (a+b \sin (c+d x))}\right )}{8 (a-b)^3 (a+b)^3 d (e \cos (c+d x))^{3/2}}+\frac {\cos ^2(c+d x) \left (-\frac {b^3 \cos (c+d x)}{2 \left (a^2-b^2\right )^2 (a+b \sin (c+d x))^2}-\frac {13 a b^3 \cos (c+d x)}{4 \left (a^2-b^2\right )^3 (a+b \sin (c+d x))}+\frac {2 \sec (c+d x) \left (-3 a^2 b-b^3+a^3 \sin (c+d x)+3 a b^2 \sin (c+d x)\right )}{\left (a^2-b^2\right )^3}\right )}{d (e \cos (c+d x))^{3/2}} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

Integrate[1/((e*Cos[c + d*x])^(3/2)*(a + b*Sin[c + d*x])^3),x]

[Out]

-1/8*(Cos[c + d*x]^(3/2)*((-2*(8*a^4 + 72*a^2*b^2 + 10*b^4)*(a + b*Sqrt[1 - Cos[c + d*x]^2])*((a*AppellF1[3/4,
1/2, 1, 7/4, Cos[c + d*x]^2, (b^2*Cos[c + d*x]^2)/(-a^2 + b^2)]*Cos[c + d*x]^(3/2))/(3*(a^2 - b^2)) + ((1/8 +
I/8)*(2*ArcTan[1 - ((1 + I)*Sqrt[b]*Sqrt[Cos[c + d*x]])/(-a^2 + b^2)^(1/4)] - 2*ArcTan[1 + ((1 + I)*Sqrt[b]*S
qrt[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]]))/(Sqrt[b]*(-a^2 + b^2)^(1/4)))*Sin[c + d*x])/(Sqrt[1 - Cos[c + d*x]^2]*(a + b*Sin[c + d*x]))
- ((8*a^3*b + 37*a*b^3)*(a + b*Sqrt[1 - Cos[c + d*x]^2])*(8*b^(5/2)*AppellF1[3/4, -1/2, 1, 7/4, Cos[c + d*x]^
2, (b^2*Cos[c + d*x]^2)/(-a^2 + b^2)]*Cos[c + d*x]^(3/2) + 3*Sqrt[2]*a*(a^2 - b^2)^(3/4)*(2*ArcTan[1 - (Sqrt[2
]*Sqrt[b]*Sqrt[Cos[c + d*x]])/(a^2 - b^2)^(1/4)] - 2*ArcTan[1 + (Sqrt[2]*Sqrt[b]*Sqrt[Cos[c + d*x]])/(a^2 - b^
2)^(1/4)] - Log[Sqrt[a^2 - b^2] - Sqrt[2]*Sqrt[b]*(a^2 - b^2)^(1/4)*Sqrt[Cos[c + d*x]] + b*Cos[c + d*x]] + Log
[Sqrt[a^2 - b^2] + Sqrt[2]*Sqrt[b]*(a^2 - b^2)^(1/4)*Sqrt[Cos[c + d*x]] + b*Cos[c + d*x]]))*Sin[c + d*x]^2)/(1
2*b^(3/2)*(-a^2 + b^2)*(1 - Cos[c + d*x]^2)*(a + b*Sin[c + d*x]))))/((a - b)^3*(a + b)^3*d*(e*Cos[c + d*x])^(3
/2)) + (Cos[c + d*x]^2*(-1/2*(b^3*Cos[c + d*x])/((a^2 - b^2)^2*(a + b*Sin[c + d*x])^2) - (13*a*b^3*Cos[c + d*x
])/(4*(a^2 - b^2)^3*(a + b*Sin[c + d*x])) + (2*Sec[c + d*x]*(-3*a^2*b - b^3 + a^3*Sin[c + d*x] + 3*a*b^2*Sin[c
+ d*x]))/(a^2 - b^2)^3))/(d*(e*Cos[c + d*x])^(3/2))

________________________________________________________________________________________

Maple [C] Result contains higher order function than in optimal. Order 9 vs. order 4.
time = 170.83, size = 43749, normalized size = 73.40

method	result	size
default	\(\text {Expression too large to display}\)	\(43749\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(e*cos(d*x+c))^(3/2)/(a+b*sin(d*x+c))^3,x,method=_RETURNVERBOSE)

[Out]

result too large to display

________________________________________________________________________________________

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(1/(e*cos(d*x+c))^(3/2)/(a+b*sin(d*x+c))^3,x, algorithm="maxima")

[Out]

Timed out

________________________________________________________________________________________

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(1/(e*cos(d*x+c))^(3/2)/(a+b*sin(d*x+c))^3,x, algorithm="fricas")

[Out]

Timed out

________________________________________________________________________________________

Sympy [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: SystemError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(e*cos(d*x+c))**(3/2)/(a+b*sin(d*x+c))**3,x)

[Out]

Exception raised: SystemError >> excessive stack use: stack is 3067 deep

________________________________________________________________________________________

Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(e*cos(d*x+c))^(3/2)/(a+b*sin(d*x+c))^3,x, algorithm="giac")

[Out]

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

________________________________________________________________________________________

Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {1}{{\left (e\,\cos \left (c+d\,x\right )\right )}^{3/2}\,{\left (a+b\,\sin \left (c+d\,x\right )\right )}^3} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/((e*cos(c + d*x))^(3/2)*(a + b*sin(c + d*x))^3),x)

[Out]

int(1/((e*cos(c + d*x))^(3/2)*(a + b*sin(c + d*x))^3), x)

________________________________________________________________________________________