\(\int \frac {1}{(e \cos (c+d x))^{3/2} (a+b \sin (c+d x))^4} \, dx\) [614]

Optimal result

Integrand size = 25, antiderivative size = 674 \[ \int \frac {1}{(e \cos (c+d x))^{3/2} (a+b \sin (c+d x))^4} \, dx=-\frac {15 a b^{3/2} \left (7 a^2+6 b^2\right ) \arctan \left (\frac {\sqrt {b} \sqrt {e \cos (c+d x)}}{\sqrt [4]{-a^2+b^2} \sqrt {e}}\right )}{16 \left (-a^2+b^2\right )^{17/4} d e^{3/2}}+\frac {15 a b^{3/2} \left (7 a^2+6 b^2\right ) \text {arctanh}\left (\frac {\sqrt {b} \sqrt {e \cos (c+d x)}}{\sqrt [4]{-a^2+b^2} \sqrt {e}}\right )}{16 \left (-a^2+b^2\right )^{17/4} d e^{3/2}}-\frac {\left (16 a^4+151 a^2 b^2+28 b^4\right ) \sqrt {e \cos (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{8 \left (a^2-b^2\right )^4 d e^2 \sqrt {\cos (c+d x)}}-\frac {15 a^2 b \left (7 a^2+6 b^2\right ) \sqrt {\cos (c+d x)} \operatorname {EllipticPi}\left (\frac {2 b}{b-\sqrt {-a^2+b^2}},\frac {1}{2} (c+d x),2\right )}{16 \left (a^2-b^2\right )^4 \left (b-\sqrt {-a^2+b^2}\right ) d e \sqrt {e \cos (c+d x)}}-\frac {15 a^2 b \left (7 a^2+6 b^2\right ) \sqrt {\cos (c+d x)} \operatorname {EllipticPi}\left (\frac {2 b}{b+\sqrt {-a^2+b^2}},\frac {1}{2} (c+d x),2\right )}{16 \left (a^2-b^2\right )^4 \left (b+\sqrt {-a^2+b^2}\right ) d e \sqrt {e \cos (c+d x)}}+\frac {b}{3 \left (a^2-b^2\right ) d e \sqrt {e \cos (c+d x)} (a+b \sin (c+d x))^3}+\frac {13 a b}{12 \left (a^2-b^2\right )^2 d e \sqrt {e \cos (c+d x)} (a+b \sin (c+d x))^2}+\frac {b \left (89 a^2+28 b^2\right )}{24 \left (a^2-b^2\right )^3 d e \sqrt {e \cos (c+d x)} (a+b \sin (c+d x))}-\frac {15 a b \left (7 a^2+6 b^2\right )-\left (16 a^4+151 a^2 b^2+28 b^4\right ) \sin (c+d x)}{8 \left (a^2-b^2\right )^4 d e \sqrt {e \cos (c+d x)}} \]

[Out]

-15/16*a*b^(3/2)*(7*a^2+6*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)^(17/4)
/d/e^(3/2)+15/16*a*b^(3/2)*(7*a^2+6*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)^(17/4)/d/e^(3/2)+1/3*b/(a^2-b^2)/d/e/(a+b*sin(d*x+c))^3/(e*cos(d*x+c))^(1/2)+13/12*a*b/(a^2-b^2)^2/d/e/(a
+b*sin(d*x+c))^2/(e*cos(d*x+c))^(1/2)+1/24*b*(89*a^2+28*b^2)/(a^2-b^2)^3/d/e/(a+b*sin(d*x+c))/(e*cos(d*x+c))^(
1/2)+1/8*(-15*a*b*(7*a^2+6*b^2)+(16*a^4+151*a^2*b^2+28*b^4)*sin(d*x+c))/(a^2-b^2)^4/d/e/(e*cos(d*x+c))^(1/2)-1
5/16*a^2*b*(7*a^2+6*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)^4/d/e/(b-(-a^2+b^2)^(1/2))/(e*cos(d*x+c))^(1/2)-15/16*a^
2*b*(7*a^2+6*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)^4/d/e/(b+(-a^2+b^2)^(1/2))/(e*cos(d*x+c))^(1/2)-1/8*(16*a^4+151
*a^2*b^2+28*b^4)*(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)^4/d/e^2/cos(d*x+c)^(1/2)

Rubi [A] (verified)

Time = 1.60 (sec) , antiderivative size = 674, normalized size of antiderivative = 1.00, number of steps used = 16, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.520, Rules used = {2773, 2943, 2945, 2946, 2721, 2719, 2780, 2886, 2884, 335, 304, 211, 214} \[ \int \frac {1}{(e \cos (c+d x))^{3/2} (a+b \sin (c+d x))^4} \, dx=-\frac {15 a b^{3/2} \left (7 a^2+6 b^2\right ) \arctan \left (\frac {\sqrt {b} \sqrt {e \cos (c+d x)}}{\sqrt {e} \sqrt [4]{b^2-a^2}}\right )}{16 d e^{3/2} \left (b^2-a^2\right )^{17/4}}+\frac {15 a b^{3/2} \left (7 a^2+6 b^2\right ) \text {arctanh}\left (\frac {\sqrt {b} \sqrt {e \cos (c+d x)}}{\sqrt {e} \sqrt [4]{b^2-a^2}}\right )}{16 d e^{3/2} \left (b^2-a^2\right )^{17/4}}+\frac {13 a b}{12 d e \left (a^2-b^2\right )^2 \sqrt {e \cos (c+d x)} (a+b \sin (c+d x))^2}+\frac {b \left (89 a^2+28 b^2\right )}{24 d e \left (a^2-b^2\right )^3 \sqrt {e \cos (c+d x)} (a+b \sin (c+d x))}+\frac {b}{3 d e \left (a^2-b^2\right ) \sqrt {e \cos (c+d x)} (a+b \sin (c+d x))^3}-\frac {15 a^2 b \left (7 a^2+6 b^2\right ) \sqrt {\cos (c+d x)} \operatorname {EllipticPi}\left (\frac {2 b}{b-\sqrt {b^2-a^2}},\frac {1}{2} (c+d x),2\right )}{16 d e \left (a^2-b^2\right )^4 \left (b-\sqrt {b^2-a^2}\right ) \sqrt {e \cos (c+d x)}}-\frac {15 a^2 b \left (7 a^2+6 b^2\right ) \sqrt {\cos (c+d x)} \operatorname {EllipticPi}\left (\frac {2 b}{b+\sqrt {b^2-a^2}},\frac {1}{2} (c+d x),2\right )}{16 d e \left (a^2-b^2\right )^4 \left (\sqrt {b^2-a^2}+b\right ) \sqrt {e \cos (c+d x)}}-\frac {\left (16 a^4+151 a^2 b^2+28 b^4\right ) E\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \sqrt {e \cos (c+d x)}}{8 d e^2 \left (a^2-b^2\right )^4 \sqrt {\cos (c+d x)}}-\frac {15 a b \left (7 a^2+6 b^2\right )-\left (16 a^4+151 a^2 b^2+28 b^4\right ) \sin (c+d x)}{8 d e \left (a^2-b^2\right )^4 \sqrt {e \cos (c+d x)}} \]

[In]

Int[1/((e*Cos[c + d*x])^(3/2)*(a + b*Sin[c + d*x])^4),x]

[Out]

(-15*a*b^(3/2)*(7*a^2 + 6*b^2)*ArcTan[(Sqrt[b]*Sqrt[e*Cos[c + d*x]])/((-a^2 + b^2)^(1/4)*Sqrt[e])])/(16*(-a^2
+ b^2)^(17/4)*d*e^(3/2)) + (15*a*b^(3/2)*(7*a^2 + 6*b^2)*ArcTanh[(Sqrt[b]*Sqrt[e*Cos[c + d*x]])/((-a^2 + b^2)^
(1/4)*Sqrt[e])])/(16*(-a^2 + b^2)^(17/4)*d*e^(3/2)) - ((16*a^4 + 151*a^2*b^2 + 28*b^4)*Sqrt[e*Cos[c + d*x]]*El
lipticE[(c + d*x)/2, 2])/(8*(a^2 - b^2)^4*d*e^2*Sqrt[Cos[c + d*x]]) - (15*a^2*b*(7*a^2 + 6*b^2)*Sqrt[Cos[c + d
*x]]*EllipticPi[(2*b)/(b - Sqrt[-a^2 + b^2]), (c + d*x)/2, 2])/(16*(a^2 - b^2)^4*(b - Sqrt[-a^2 + b^2])*d*e*Sq
rt[e*Cos[c + d*x]]) - (15*a^2*b*(7*a^2 + 6*b^2)*Sqrt[Cos[c + d*x]]*EllipticPi[(2*b)/(b + Sqrt[-a^2 + b^2]), (c
+ d*x)/2, 2])/(16*(a^2 - b^2)^4*(b + Sqrt[-a^2 + b^2])*d*e*Sqrt[e*Cos[c + d*x]]) + b/(3*(a^2 - b^2)*d*e*Sqrt[
e*Cos[c + d*x]]*(a + b*Sin[c + d*x])^3) + (13*a*b)/(12*(a^2 - b^2)^2*d*e*Sqrt[e*Cos[c + d*x]]*(a + b*Sin[c + d
*x])^2) + (b*(89*a^2 + 28*b^2))/(24*(a^2 - b^2)^3*d*e*Sqrt[e*Cos[c + d*x]]*(a + b*Sin[c + d*x])) - (15*a*b*(7*
a^2 + 6*b^2) - (16*a^4 + 151*a^2*b^2 + 28*b^4)*Sin[c + d*x])/(8*(a^2 - b^2)^4*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*} \text {integral}& = \frac {b}{3 \left (a^2-b^2\right ) d e \sqrt {e \cos (c+d x)} (a+b \sin (c+d x))^3}-\frac {\int \frac {-3 a+\frac {7}{2} b \sin (c+d x)}{(e \cos (c+d x))^{3/2} (a+b \sin (c+d x))^3} \, dx}{3 \left (a^2-b^2\right )} \\ & = \frac {b}{3 \left (a^2-b^2\right ) d e \sqrt {e \cos (c+d x)} (a+b \sin (c+d x))^3}+\frac {13 a b}{12 \left (a^2-b^2\right )^2 d e \sqrt {e \cos (c+d x)} (a+b \sin (c+d x))^2}+\frac {\int \frac {6 a^2+7 b^2-\frac {65}{4} a b \sin (c+d x)}{(e \cos (c+d x))^{3/2} (a+b \sin (c+d x))^2} \, dx}{6 \left (a^2-b^2\right )^2} \\ & = \frac {b}{3 \left (a^2-b^2\right ) d e \sqrt {e \cos (c+d x)} (a+b \sin (c+d x))^3}+\frac {13 a b}{12 \left (a^2-b^2\right )^2 d e \sqrt {e \cos (c+d x)} (a+b \sin (c+d x))^2}+\frac {b \left (89 a^2+28 b^2\right )}{24 \left (a^2-b^2\right )^3 d e \sqrt {e \cos (c+d x)} (a+b \sin (c+d x))}-\frac {\int \frac {-\frac {3}{4} a \left (8 a^2+31 b^2\right )+\frac {3}{8} b \left (89 a^2+28 b^2\right ) \sin (c+d x)}{(e \cos (c+d x))^{3/2} (a+b \sin (c+d x))} \, dx}{6 \left (a^2-b^2\right )^3} \\ & = \frac {b}{3 \left (a^2-b^2\right ) d e \sqrt {e \cos (c+d x)} (a+b \sin (c+d x))^3}+\frac {13 a b}{12 \left (a^2-b^2\right )^2 d e \sqrt {e \cos (c+d x)} (a+b \sin (c+d x))^2}+\frac {b \left (89 a^2+28 b^2\right )}{24 \left (a^2-b^2\right )^3 d e \sqrt {e \cos (c+d x)} (a+b \sin (c+d x))}-\frac {15 a b \left (7 a^2+6 b^2\right )-\left (16 a^4+151 a^2 b^2+28 b^4\right ) \sin (c+d x)}{8 \left (a^2-b^2\right )^4 d e \sqrt {e \cos (c+d x)}}+\frac {\int \frac {\sqrt {e \cos (c+d x)} \left (-\frac {3}{8} a \left (8 a^4+128 a^2 b^2+59 b^4\right )-\frac {3}{16} b \left (16 a^4+151 a^2 b^2+28 b^4\right ) \sin (c+d x)\right )}{a+b \sin (c+d x)} \, dx}{3 \left (a^2-b^2\right )^4 e^2} \\ & = \frac {b}{3 \left (a^2-b^2\right ) d e \sqrt {e \cos (c+d x)} (a+b \sin (c+d x))^3}+\frac {13 a b}{12 \left (a^2-b^2\right )^2 d e \sqrt {e \cos (c+d x)} (a+b \sin (c+d x))^2}+\frac {b \left (89 a^2+28 b^2\right )}{24 \left (a^2-b^2\right )^3 d e \sqrt {e \cos (c+d x)} (a+b \sin (c+d x))}-\frac {15 a b \left (7 a^2+6 b^2\right )-\left (16 a^4+151 a^2 b^2+28 b^4\right ) \sin (c+d x)}{8 \left (a^2-b^2\right )^4 d e \sqrt {e \cos (c+d x)}}-\frac {\left (15 a b^2 \left (7 a^2+6 b^2\right )\right ) \int \frac {\sqrt {e \cos (c+d x)}}{a+b \sin (c+d x)} \, dx}{16 \left (a^2-b^2\right )^4 e^2}-\frac {\left (16 a^4+151 a^2 b^2+28 b^4\right ) \int \sqrt {e \cos (c+d x)} \, dx}{16 \left (a^2-b^2\right )^4 e^2} \\ & = \frac {b}{3 \left (a^2-b^2\right ) d e \sqrt {e \cos (c+d x)} (a+b \sin (c+d x))^3}+\frac {13 a b}{12 \left (a^2-b^2\right )^2 d e \sqrt {e \cos (c+d x)} (a+b \sin (c+d x))^2}+\frac {b \left (89 a^2+28 b^2\right )}{24 \left (a^2-b^2\right )^3 d e \sqrt {e \cos (c+d x)} (a+b \sin (c+d x))}-\frac {15 a b \left (7 a^2+6 b^2\right )-\left (16 a^4+151 a^2 b^2+28 b^4\right ) \sin (c+d x)}{8 \left (a^2-b^2\right )^4 d e \sqrt {e \cos (c+d x)}}+\frac {\left (15 a^2 b \left (7 a^2+6 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}{32 \left (a^2-b^2\right )^4 e}-\frac {\left (15 a^2 b \left (7 a^2+6 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}{32 \left (a^2-b^2\right )^4 e}-\frac {\left (15 a b^3 \left (7 a^2+6 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 )}{16 \left (a^2-b^2\right )^4 d e}-\frac {\left (\left (16 a^4+151 a^2 b^2+28 b^4\right ) \sqrt {e \cos (c+d x)}\right ) \int \sqrt {\cos (c+d x)} \, dx}{16 \left (a^2-b^2\right )^4 e^2 \sqrt {\cos (c+d x)}} \\ & = -\frac {\left (16 a^4+151 a^2 b^2+28 b^4\right ) \sqrt {e \cos (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{8 \left (a^2-b^2\right )^4 d e^2 \sqrt {\cos (c+d x)}}+\frac {b}{3 \left (a^2-b^2\right ) d e \sqrt {e \cos (c+d x)} (a+b \sin (c+d x))^3}+\frac {13 a b}{12 \left (a^2-b^2\right )^2 d e \sqrt {e \cos (c+d x)} (a+b \sin (c+d x))^2}+\frac {b \left (89 a^2+28 b^2\right )}{24 \left (a^2-b^2\right )^3 d e \sqrt {e \cos (c+d x)} (a+b \sin (c+d x))}-\frac {15 a b \left (7 a^2+6 b^2\right )-\left (16 a^4+151 a^2 b^2+28 b^4\right ) \sin (c+d x)}{8 \left (a^2-b^2\right )^4 d e \sqrt {e \cos (c+d x)}}-\frac {\left (15 a b^3 \left (7 a^2+6 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 )}{8 \left (a^2-b^2\right )^4 d e}+\frac {\left (15 a^2 b \left (7 a^2+6 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}{32 \left (a^2-b^2\right )^4 e \sqrt {e \cos (c+d x)}}-\frac {\left (15 a^2 b \left (7 a^2+6 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}{32 \left (a^2-b^2\right )^4 e \sqrt {e \cos (c+d x)}} \\ & = -\frac {\left (16 a^4+151 a^2 b^2+28 b^4\right ) \sqrt {e \cos (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{8 \left (a^2-b^2\right )^4 d e^2 \sqrt {\cos (c+d x)}}-\frac {15 a^2 b \left (7 a^2+6 b^2\right ) \sqrt {\cos (c+d x)} \operatorname {EllipticPi}\left (\frac {2 b}{b-\sqrt {-a^2+b^2}},\frac {1}{2} (c+d x),2\right )}{16 \left (a^2-b^2\right )^4 \left (b-\sqrt {-a^2+b^2}\right ) d e \sqrt {e \cos (c+d x)}}-\frac {15 a^2 b \left (7 a^2+6 b^2\right ) \sqrt {\cos (c+d x)} \operatorname {EllipticPi}\left (\frac {2 b}{b+\sqrt {-a^2+b^2}},\frac {1}{2} (c+d x),2\right )}{16 \left (a^2-b^2\right )^4 \left (b+\sqrt {-a^2+b^2}\right ) d e \sqrt {e \cos (c+d x)}}+\frac {b}{3 \left (a^2-b^2\right ) d e \sqrt {e \cos (c+d x)} (a+b \sin (c+d x))^3}+\frac {13 a b}{12 \left (a^2-b^2\right )^2 d e \sqrt {e \cos (c+d x)} (a+b \sin (c+d x))^2}+\frac {b \left (89 a^2+28 b^2\right )}{24 \left (a^2-b^2\right )^3 d e \sqrt {e \cos (c+d x)} (a+b \sin (c+d x))}-\frac {15 a b \left (7 a^2+6 b^2\right )-\left (16 a^4+151 a^2 b^2+28 b^4\right ) \sin (c+d x)}{8 \left (a^2-b^2\right )^4 d e \sqrt {e \cos (c+d x)}}+\frac {\left (15 a b^2 \left (7 a^2+6 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 )}{16 \left (a^2-b^2\right )^4 d e}-\frac {\left (15 a b^2 \left (7 a^2+6 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 )}{16 \left (a^2-b^2\right )^4 d e} \\ & = -\frac {15 a b^{3/2} \left (7 a^2+6 b^2\right ) \arctan \left (\frac {\sqrt {b} \sqrt {e \cos (c+d x)}}{\sqrt [4]{-a^2+b^2} \sqrt {e}}\right )}{16 \left (-a^2+b^2\right )^{17/4} d e^{3/2}}+\frac {15 a b^{3/2} \left (7 a^2+6 b^2\right ) \text {arctanh}\left (\frac {\sqrt {b} \sqrt {e \cos (c+d x)}}{\sqrt [4]{-a^2+b^2} \sqrt {e}}\right )}{16 \left (-a^2+b^2\right )^{17/4} d e^{3/2}}-\frac {\left (16 a^4+151 a^2 b^2+28 b^4\right ) \sqrt {e \cos (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{8 \left (a^2-b^2\right )^4 d e^2 \sqrt {\cos (c+d x)}}-\frac {15 a^2 b \left (7 a^2+6 b^2\right ) \sqrt {\cos (c+d x)} \operatorname {EllipticPi}\left (\frac {2 b}{b-\sqrt {-a^2+b^2}},\frac {1}{2} (c+d x),2\right )}{16 \left (a^2-b^2\right )^4 \left (b-\sqrt {-a^2+b^2}\right ) d e \sqrt {e \cos (c+d x)}}-\frac {15 a^2 b \left (7 a^2+6 b^2\right ) \sqrt {\cos (c+d x)} \operatorname {EllipticPi}\left (\frac {2 b}{b+\sqrt {-a^2+b^2}},\frac {1}{2} (c+d x),2\right )}{16 \left (a^2-b^2\right )^4 \left (b+\sqrt {-a^2+b^2}\right ) d e \sqrt {e \cos (c+d x)}}+\frac {b}{3 \left (a^2-b^2\right ) d e \sqrt {e \cos (c+d x)} (a+b \sin (c+d x))^3}+\frac {13 a b}{12 \left (a^2-b^2\right )^2 d e \sqrt {e \cos (c+d x)} (a+b \sin (c+d x))^2}+\frac {b \left (89 a^2+28 b^2\right )}{24 \left (a^2-b^2\right )^3 d e \sqrt {e \cos (c+d x)} (a+b \sin (c+d x))}-\frac {15 a b \left (7 a^2+6 b^2\right )-\left (16 a^4+151 a^2 b^2+28 b^4\right ) \sin (c+d x)}{8 \left (a^2-b^2\right )^4 d e \sqrt {e \cos (c+d x)}} \\ \end{align*}

Mathematica [C] (warning: unable to verify)

Result contains higher order function than in optimal. Order 6 vs. order 4 in optimal.

Time = 6.71 (sec) , antiderivative size = 996, normalized size of antiderivative = 1.48 \[ \int \frac {1}{(e \cos (c+d x))^{3/2} (a+b \sin (c+d x))^4} \, dx=-\frac {\cos ^{\frac {3}{2}}(c+d x) \left (-\frac {2 \left (16 a^5+256 a^3 b^2+118 a b^4\right ) \left (a+b \sqrt {1-\cos ^2(c+d x)}\right ) \left (\frac {a \operatorname {AppellF1}\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 \arctan \left (1-\frac {(1+i) \sqrt {b} \sqrt {\cos (c+d x)}}{\sqrt [4]{-a^2+b^2}}\right )-2 \arctan \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 (16 a^4 b+151 a^2 b^3+28 b^5\right ) \left (a+b \sqrt {1-\cos ^2(c+d x)}\right ) \left (8 b^{5/2} \operatorname {AppellF1}\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 \arctan \left (1-\frac {\sqrt {2} \sqrt {b} \sqrt {\cos (c+d x)}}{\sqrt [4]{a^2-b^2}}\right )-2 \arctan \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 )}{16 (a-b)^4 (a+b)^4 d (e \cos (c+d x))^{3/2}}+\frac {\cos ^2(c+d x) \left (-\frac {b^3 \cos (c+d x)}{3 \left (a^2-b^2\right )^2 (a+b \sin (c+d x))^3}-\frac {7 a b^3 \cos (c+d x)}{4 \left (a^2-b^2\right )^3 (a+b \sin (c+d x))^2}+\frac {-55 a^2 b^3 \cos (c+d x)-12 b^5 \cos (c+d x)}{8 \left (a^2-b^2\right )^4 (a+b \sin (c+d x))}+\frac {2 \sec (c+d x) \left (-4 a^3 b-4 a b^3+a^4 \sin (c+d x)+6 a^2 b^2 \sin (c+d x)+b^4 \sin (c+d x)\right )}{\left (a^2-b^2\right )^4}\right )}{d (e \cos (c+d x))^{3/2}} \]

[In]

Integrate[1/((e*Cos[c + d*x])^(3/2)*(a + b*Sin[c + d*x])^4),x]

[Out]

-1/16*(Cos[c + d*x]^(3/2)*((-2*(16*a^5 + 256*a^3*b^2 + 118*a*b^4)*(a + b*Sqrt[1 - Cos[c + d*x]^2])*((a*AppellF
1[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)*Sqr
t[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]]))/(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])) - ((16*a^4*b + 151*a^2*b^3 + 28*b^5)*(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)/(12*b^(3/2)*(-a^2 + b^2)*(1 - Cos[c + d*x]^2)*(a + b*Sin[c + d*x]))))/((a - b)^4*(a + b)^4*d*
(e*Cos[c + d*x])^(3/2)) + (Cos[c + d*x]^2*(-1/3*(b^3*Cos[c + d*x])/((a^2 - b^2)^2*(a + b*Sin[c + d*x])^3) - (7
*a*b^3*Cos[c + d*x])/(4*(a^2 - b^2)^3*(a + b*Sin[c + d*x])^2) + (-55*a^2*b^3*Cos[c + d*x] - 12*b^5*Cos[c + d*x
])/(8*(a^2 - b^2)^4*(a + b*Sin[c + d*x])) + (2*Sec[c + d*x]*(-4*a^3*b - 4*a*b^3 + a^4*Sin[c + d*x] + 6*a^2*b^2
*Sin[c + d*x] + b^4*Sin[c + d*x]))/(a^2 - b^2)^4))/(d*(e*Cos[c + d*x])^(3/2))

Maple [C] (warning: unable to verify)

Result contains higher order function than in optimal. Order 9 vs. order 4.

Time = 23.01 (sec) , antiderivative size = 5260, normalized size of antiderivative = 7.80

method	result	size
default	\(\text {Expression too large to display}\)	\(5260\)

[In]

int(1/(e*cos(d*x+c))^(3/2)/(a+b*sin(d*x+c))^4,x,method=_RETURNVERBOSE)

[Out]

result too large to display

Fricas [F(-1)]

Timed out. \[ \int \frac {1}{(e \cos (c+d x))^{3/2} (a+b \sin (c+d x))^4} \, dx=\text {Timed out} \]

[In]

integrate(1/(e*cos(d*x+c))^(3/2)/(a+b*sin(d*x+c))^4,x, algorithm="fricas")

[Out]

Timed out

Sympy [F(-1)]

Timed out. \[ \int \frac {1}{(e \cos (c+d x))^{3/2} (a+b \sin (c+d x))^4} \, dx=\text {Timed out} \]

[In]

integrate(1/(e*cos(d*x+c))**(3/2)/(a+b*sin(d*x+c))**4,x)

[Out]

Timed out

Maxima [F(-1)]

Timed out. \[ \int \frac {1}{(e \cos (c+d x))^{3/2} (a+b \sin (c+d x))^4} \, dx=\text {Timed out} \]

[In]

integrate(1/(e*cos(d*x+c))^(3/2)/(a+b*sin(d*x+c))^4,x, algorithm="maxima")

[Out]

Timed out

Giac [F]

\[ \int \frac {1}{(e \cos (c+d x))^{3/2} (a+b \sin (c+d x))^4} \, dx=\int { \frac {1}{\left (e \cos \left (d x + c\right )\right )^{\frac {3}{2}} {\left (b \sin \left (d x + c\right ) + a\right )}^{4}} \,d x } \]

[In]

integrate(1/(e*cos(d*x+c))^(3/2)/(a+b*sin(d*x+c))^4,x, algorithm="giac")

[Out]

integrate(1/((e*cos(d*x + c))^(3/2)*(b*sin(d*x + c) + a)^4), x)

Mupad [F(-1)]

Timed out. \[ \int \frac {1}{(e \cos (c+d x))^{3/2} (a+b \sin (c+d x))^4} \, dx=\int \frac {1}{{\left (e\,\cos \left (c+d\,x\right )\right )}^{3/2}\,{\left (a+b\,\sin \left (c+d\,x\right )\right )}^4} \,d x \]

[In]

int(1/((e*cos(c + d*x))^(3/2)*(a + b*sin(c + d*x))^4),x)

[Out]

int(1/((e*cos(c + d*x))^(3/2)*(a + b*sin(c + d*x))^4), x)