3.6.93 \(\int \frac {1}{(e \cos (c+d x))^{7/2} (a+b \sin (c+d x))^2} \, dx\) [593]

Optimal. Leaf size=574 \[ -\frac {9 a b^{7/2} \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {e \cos (c+d x)}}{\sqrt [4]{-a^2+b^2} \sqrt {e}}\right )}{2 \left (-a^2+b^2\right )^{13/4} d e^{7/2}}+\frac {9 a b^{7/2} \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {e \cos (c+d x)}}{\sqrt [4]{-a^2+b^2} \sqrt {e}}\right )}{2 \left (-a^2+b^2\right )^{13/4} d e^{7/2}}-\frac {3 \left (2 a^4-10 a^2 b^2-7 b^4\right ) \sqrt {e \cos (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{5 \left (a^2-b^2\right )^3 d e^4 \sqrt {\cos (c+d x)}}+\frac {9 a^2 b^3 \sqrt {\cos (c+d x)} \Pi \left (\frac {2 b}{b-\sqrt {-a^2+b^2}};\left .\frac {1}{2} (c+d x)\right |2\right )}{2 \left (a^2-b^2\right )^3 \left (b-\sqrt {-a^2+b^2}\right ) d e^3 \sqrt {e \cos (c+d x)}}+\frac {9 a^2 b^3 \sqrt {\cos (c+d x)} \Pi \left (\frac {2 b}{b+\sqrt {-a^2+b^2}};\left .\frac {1}{2} (c+d x)\right |2\right )}{2 \left (a^2-b^2\right )^3 \left (b+\sqrt {-a^2+b^2}\right ) d e^3 \sqrt {e \cos (c+d x)}}+\frac {b}{\left (a^2-b^2\right ) d e (e \cos (c+d x))^{5/2} (a+b \sin (c+d x))}-\frac {9 a b-\left (2 a^2+7 b^2\right ) \sin (c+d x)}{5 \left (a^2-b^2\right )^2 d e (e \cos (c+d x))^{5/2}}+\frac {3 \left (15 a b^3+\left (2 a^4-10 a^2 b^2-7 b^4\right ) \sin (c+d x)\right )}{5 \left (a^2-b^2\right )^3 d e^3 \sqrt {e \cos (c+d x)}} \]

[Out]

-9/2*a*b^(7/2)*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^(7/2)+9/2*a
*b^(7/2)*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^(7/2)+b/(a^2-b^2
)/d/e/(e*cos(d*x+c))^(5/2)/(a+b*sin(d*x+c))+1/5*(-9*a*b+(2*a^2+7*b^2)*sin(d*x+c))/(a^2-b^2)^2/d/e/(e*cos(d*x+c
))^(5/2)+3/5*(15*a*b^3+(2*a^4-10*a^2*b^2-7*b^4)*sin(d*x+c))/(a^2-b^2)^3/d/e^3/(e*cos(d*x+c))^(1/2)+9/2*a^2*b^3
*(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^3/(b-(-a^2+b^2)^(1/2))/(e*cos(d*x+c))^(1/2)+9/2*a^2*b^3*(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^3/(b+(-a^2+b^2)^(1/2))/(e*cos(d*x+c))^(1/2)-3/5*(2*a^4-10*a^2*b^2-7*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)^3/d/e^4/cos(d
*x+c)^(1/2)

________________________________________________________________________________________

Rubi [A]
time = 1.09, antiderivative size = 574, normalized size of antiderivative = 1.00, number of steps used = 15, number of rules used = 12, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.480, Rules used = {2773, 2945, 2946, 2721, 2719, 2780, 2886, 2884, 335, 304, 211, 214} \begin {gather*} -\frac {9 a b^{7/2} \text {ArcTan}\left (\frac {\sqrt {b} \sqrt {e \cos (c+d x)}}{\sqrt {e} \sqrt [4]{b^2-a^2}}\right )}{2 d e^{7/2} \left (b^2-a^2\right )^{13/4}}+\frac {b}{d e \left (a^2-b^2\right ) (e \cos (c+d x))^{5/2} (a+b \sin (c+d x))}-\frac {9 a b-\left (2 a^2+7 b^2\right ) \sin (c+d x)}{5 d e \left (a^2-b^2\right )^2 (e \cos (c+d x))^{5/2}}+\frac {9 a b^{7/2} \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {e \cos (c+d x)}}{\sqrt {e} \sqrt [4]{b^2-a^2}}\right )}{2 d e^{7/2} \left (b^2-a^2\right )^{13/4}}+\frac {9 a^2 b^3 \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 d e^3 \left (a^2-b^2\right )^3 \left (b-\sqrt {b^2-a^2}\right ) \sqrt {e \cos (c+d x)}}+\frac {9 a^2 b^3 \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 d e^3 \left (a^2-b^2\right )^3 \left (\sqrt {b^2-a^2}+b\right ) \sqrt {e \cos (c+d x)}}-\frac {3 \left (2 a^4-10 a^2 b^2-7 b^4\right ) E\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \sqrt {e \cos (c+d x)}}{5 d e^4 \left (a^2-b^2\right )^3 \sqrt {\cos (c+d x)}}+\frac {3 \left (\left (2 a^4-10 a^2 b^2-7 b^4\right ) \sin (c+d x)+15 a b^3\right )}{5 d e^3 \left (a^2-b^2\right )^3 \sqrt {e \cos (c+d x)}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[1/((e*Cos[c + d*x])^(7/2)*(a + b*Sin[c + d*x])^2),x]

[Out]

(-9*a*b^(7/2)*ArcTan[(Sqrt[b]*Sqrt[e*Cos[c + d*x]])/((-a^2 + b^2)^(1/4)*Sqrt[e])])/(2*(-a^2 + b^2)^(13/4)*d*e^
(7/2)) + (9*a*b^(7/2)*ArcTanh[(Sqrt[b]*Sqrt[e*Cos[c + d*x]])/((-a^2 + b^2)^(1/4)*Sqrt[e])])/(2*(-a^2 + b^2)^(1
3/4)*d*e^(7/2)) - (3*(2*a^4 - 10*a^2*b^2 - 7*b^4)*Sqrt[e*Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2])/(5*(a^2 - b^
2)^3*d*e^4*Sqrt[Cos[c + d*x]]) + (9*a^2*b^3*Sqrt[Cos[c + d*x]]*EllipticPi[(2*b)/(b - Sqrt[-a^2 + b^2]), (c + d
*x)/2, 2])/(2*(a^2 - b^2)^3*(b - Sqrt[-a^2 + b^2])*d*e^3*Sqrt[e*Cos[c + d*x]]) + (9*a^2*b^3*Sqrt[Cos[c + d*x]]
*EllipticPi[(2*b)/(b + Sqrt[-a^2 + b^2]), (c + d*x)/2, 2])/(2*(a^2 - b^2)^3*(b + Sqrt[-a^2 + b^2])*d*e^3*Sqrt[
e*Cos[c + d*x]]) + b/((a^2 - b^2)*d*e*(e*Cos[c + d*x])^(5/2)*(a + b*Sin[c + d*x])) - (9*a*b - (2*a^2 + 7*b^2)*
Sin[c + d*x])/(5*(a^2 - b^2)^2*d*e*(e*Cos[c + d*x])^(5/2)) + (3*(15*a*b^3 + (2*a^4 - 10*a^2*b^2 - 7*b^4)*Sin[c
+ d*x]))/(5*(a^2 - b^2)^3*d*e^3*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 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))^{7/2} (a+b \sin (c+d x))^2} \, dx &=\frac {b}{\left (a^2-b^2\right ) d e (e \cos (c+d x))^{5/2} (a+b \sin (c+d x))}+\frac {\int \frac {-a+\frac {7}{2} b \sin (c+d x)}{(e \cos (c+d x))^{7/2} (a+b \sin (c+d x))} \, dx}{-a^2+b^2}\\ &=\frac {b}{\left (a^2-b^2\right ) d e (e \cos (c+d x))^{5/2} (a+b \sin (c+d x))}-\frac {9 a b-\left (2 a^2+7 b^2\right ) \sin (c+d x)}{5 \left (a^2-b^2\right )^2 d e (e \cos (c+d x))^{5/2}}+\frac {2 \int \frac {\frac {3}{2} a \left (a^2-4 b^2\right )+\frac {3}{4} b \left (2 a^2+7 b^2\right ) \sin (c+d x)}{(e \cos (c+d x))^{3/2} (a+b \sin (c+d x))} \, dx}{5 \left (a^2-b^2\right )^2 e^2}\\ &=\frac {b}{\left (a^2-b^2\right ) d e (e \cos (c+d x))^{5/2} (a+b \sin (c+d x))}-\frac {9 a b-\left (2 a^2+7 b^2\right ) \sin (c+d x)}{5 \left (a^2-b^2\right )^2 d e (e \cos (c+d x))^{5/2}}+\frac {3 \left (15 a b^3+\left (2 a^4-10 a^2 b^2-7 b^4\right ) \sin (c+d x)\right )}{5 \left (a^2-b^2\right )^3 d e^3 \sqrt {e \cos (c+d x)}}-\frac {4 \int \frac {\sqrt {e \cos (c+d x)} \left (\frac {3}{4} a \left (a^4-5 a^2 b^2-11 b^4\right )+\frac {3}{8} b \left (2 a^4-10 a^2 b^2-7 b^4\right ) \sin (c+d x)\right )}{a+b \sin (c+d x)} \, dx}{5 \left (a^2-b^2\right )^3 e^4}\\ &=\frac {b}{\left (a^2-b^2\right ) d e (e \cos (c+d x))^{5/2} (a+b \sin (c+d x))}-\frac {9 a b-\left (2 a^2+7 b^2\right ) \sin (c+d x)}{5 \left (a^2-b^2\right )^2 d e (e \cos (c+d x))^{5/2}}+\frac {3 \left (15 a b^3+\left (2 a^4-10 a^2 b^2-7 b^4\right ) \sin (c+d x)\right )}{5 \left (a^2-b^2\right )^3 d e^3 \sqrt {e \cos (c+d x)}}+\frac {\left (9 a b^4\right ) \int \frac {\sqrt {e \cos (c+d x)}}{a+b \sin (c+d x)} \, dx}{2 \left (a^2-b^2\right )^3 e^4}-\frac {\left (3 \left (2 a^4-10 a^2 b^2-7 b^4\right )\right ) \int \sqrt {e \cos (c+d x)} \, dx}{10 \left (a^2-b^2\right )^3 e^4}\\ &=\frac {b}{\left (a^2-b^2\right ) d e (e \cos (c+d x))^{5/2} (a+b \sin (c+d x))}-\frac {9 a b-\left (2 a^2+7 b^2\right ) \sin (c+d x)}{5 \left (a^2-b^2\right )^2 d e (e \cos (c+d x))^{5/2}}+\frac {3 \left (15 a b^3+\left (2 a^4-10 a^2 b^2-7 b^4\right ) \sin (c+d x)\right )}{5 \left (a^2-b^2\right )^3 d e^3 \sqrt {e \cos (c+d x)}}-\frac {\left (9 a^2 b^3\right ) \int \frac {1}{\sqrt {e \cos (c+d x)} \left (\sqrt {-a^2+b^2}-b \cos (c+d x)\right )} \, dx}{4 \left (a^2-b^2\right )^3 e^3}+\frac {\left (9 a^2 b^3\right ) \int \frac {1}{\sqrt {e \cos (c+d x)} \left (\sqrt {-a^2+b^2}+b \cos (c+d x)\right )} \, dx}{4 \left (a^2-b^2\right )^3 e^3}+\frac {\left (9 a b^5\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 )}{2 \left (a^2-b^2\right )^3 d e^3}-\frac {\left (3 \left (2 a^4-10 a^2 b^2-7 b^4\right ) \sqrt {e \cos (c+d x)}\right ) \int \sqrt {\cos (c+d x)} \, dx}{10 \left (a^2-b^2\right )^3 e^4 \sqrt {\cos (c+d x)}}\\ &=-\frac {3 \left (2 a^4-10 a^2 b^2-7 b^4\right ) \sqrt {e \cos (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{5 \left (a^2-b^2\right )^3 d e^4 \sqrt {\cos (c+d x)}}+\frac {b}{\left (a^2-b^2\right ) d e (e \cos (c+d x))^{5/2} (a+b \sin (c+d x))}-\frac {9 a b-\left (2 a^2+7 b^2\right ) \sin (c+d x)}{5 \left (a^2-b^2\right )^2 d e (e \cos (c+d x))^{5/2}}+\frac {3 \left (15 a b^3+\left (2 a^4-10 a^2 b^2-7 b^4\right ) \sin (c+d x)\right )}{5 \left (a^2-b^2\right )^3 d e^3 \sqrt {e \cos (c+d x)}}+\frac {\left (9 a b^5\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 )}{\left (a^2-b^2\right )^3 d e^3}-\frac {\left (9 a^2 b^3 \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 \left (a^2-b^2\right )^3 e^3 \sqrt {e \cos (c+d x)}}+\frac {\left (9 a^2 b^3 \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 \left (a^2-b^2\right )^3 e^3 \sqrt {e \cos (c+d x)}}\\ &=-\frac {3 \left (2 a^4-10 a^2 b^2-7 b^4\right ) \sqrt {e \cos (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{5 \left (a^2-b^2\right )^3 d e^4 \sqrt {\cos (c+d x)}}+\frac {9 a^2 b^3 \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 \left (a^2-b^2\right )^3 \left (b-\sqrt {-a^2+b^2}\right ) d e^3 \sqrt {e \cos (c+d x)}}+\frac {9 a^2 b^3 \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 \left (a^2-b^2\right )^3 \left (b+\sqrt {-a^2+b^2}\right ) d e^3 \sqrt {e \cos (c+d x)}}+\frac {b}{\left (a^2-b^2\right ) d e (e \cos (c+d x))^{5/2} (a+b \sin (c+d x))}-\frac {9 a b-\left (2 a^2+7 b^2\right ) \sin (c+d x)}{5 \left (a^2-b^2\right )^2 d e (e \cos (c+d x))^{5/2}}+\frac {3 \left (15 a b^3+\left (2 a^4-10 a^2 b^2-7 b^4\right ) \sin (c+d x)\right )}{5 \left (a^2-b^2\right )^3 d e^3 \sqrt {e \cos (c+d x)}}-\frac {\left (9 a b^4\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 )}{2 \left (a^2-b^2\right )^3 d e^3}+\frac {\left (9 a b^4\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 )}{2 \left (a^2-b^2\right )^3 d e^3}\\ &=-\frac {9 a b^{7/2} \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {e \cos (c+d x)}}{\sqrt [4]{-a^2+b^2} \sqrt {e}}\right )}{2 \left (-a^2+b^2\right )^{13/4} d e^{7/2}}+\frac {9 a b^{7/2} \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {e \cos (c+d x)}}{\sqrt [4]{-a^2+b^2} \sqrt {e}}\right )}{2 \left (-a^2+b^2\right )^{13/4} d e^{7/2}}-\frac {3 \left (2 a^4-10 a^2 b^2-7 b^4\right ) \sqrt {e \cos (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{5 \left (a^2-b^2\right )^3 d e^4 \sqrt {\cos (c+d x)}}+\frac {9 a^2 b^3 \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 \left (a^2-b^2\right )^3 \left (b-\sqrt {-a^2+b^2}\right ) d e^3 \sqrt {e \cos (c+d x)}}+\frac {9 a^2 b^3 \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 \left (a^2-b^2\right )^3 \left (b+\sqrt {-a^2+b^2}\right ) d e^3 \sqrt {e \cos (c+d x)}}+\frac {b}{\left (a^2-b^2\right ) d e (e \cos (c+d x))^{5/2} (a+b \sin (c+d x))}-\frac {9 a b-\left (2 a^2+7 b^2\right ) \sin (c+d x)}{5 \left (a^2-b^2\right )^2 d e (e \cos (c+d x))^{5/2}}+\frac {3 \left (15 a b^3+\left (2 a^4-10 a^2 b^2-7 b^4\right ) \sin (c+d x)\right )}{5 \left (a^2-b^2\right )^3 d e^3 \sqrt {e \cos (c+d x)}}\\ \end {align*}

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Mathematica [C] Result contains higher order function than in optimal. Order 6 vs. order 4 in optimal.
time = 26.63, size = 949, normalized size = 1.65 \begin {gather*} -\frac {3 \cos ^{\frac {7}{2}}(c+d x) \left (-\frac {2 \left (2 a^5-10 a^3 b^2-22 a 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 (2 a^4 b-10 a^2 b^3-7 b^5\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 )}{10 (a-b)^3 (a+b)^3 d (e \cos (c+d x))^{7/2}}+\frac {\cos ^4(c+d x) \left (\frac {b^5 \cos (c+d x)}{\left (a^2-b^2\right )^3 (a+b \sin (c+d x))}+\frac {2 \sec ^3(c+d x) \left (-2 a b+a^2 \sin (c+d x)+b^2 \sin (c+d x)\right )}{5 \left (a^2-b^2\right )^2}+\frac {2 \sec (c+d x) \left (20 a b^3+3 a^4 \sin (c+d x)-15 a^2 b^2 \sin (c+d x)-8 b^4 \sin (c+d x)\right )}{5 \left (a^2-b^2\right )^3}\right )}{d (e \cos (c+d x))^{7/2}} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

Integrate[1/((e*Cos[c + d*x])^(7/2)*(a + b*Sin[c + d*x])^2),x]

[Out]

(-3*Cos[c + d*x]^(7/2)*((-2*(2*a^5 - 10*a^3*b^2 - 22*a*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]))
- ((2*a^4*b - 10*a^2*b^3 - 7*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]))))/(10*(a - b)^3*(a + b)^3*d*(e*Cos
[c + d*x])^(7/2)) + (Cos[c + d*x]^4*((b^5*Cos[c + d*x])/((a^2 - b^2)^3*(a + b*Sin[c + d*x])) + (2*Sec[c + d*x]
^3*(-2*a*b + a^2*Sin[c + d*x] + b^2*Sin[c + d*x]))/(5*(a^2 - b^2)^2) + (2*Sec[c + d*x]*(20*a*b^3 + 3*a^4*Sin[c
+ d*x] - 15*a^2*b^2*Sin[c + d*x] - 8*b^4*Sin[c + d*x]))/(5*(a^2 - b^2)^3)))/(d*(e*Cos[c + d*x])^(7/2))

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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order 4.
time = 82.80, size = 8202, normalized size = 14.29

method	result	size
default	\(\text {Expression too large to display}\)	\(8202\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(e*cos(d*x+c))^(7/2)/(a+b*sin(d*x+c))^2,x,method=_RETURNVERBOSE)

[Out]

result too large to display

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(e*cos(d*x+c))^(7/2)/(a+b*sin(d*x+c))^2,x, algorithm="maxima")

[Out]

e^(-7/2)*integrate(1/((b*sin(d*x + c) + a)^2*cos(d*x + c)^(7/2)), x)

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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))^(7/2)/(a+b*sin(d*x+c))^2,x, algorithm="fricas")

[Out]

Timed out

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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))**(7/2)/(a+b*sin(d*x+c))**2,x)

[Out]

Exception raised: SystemError >> excessive stack use: stack is 3063 deep

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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))^(7/2)/(a+b*sin(d*x+c))^2,x, algorithm="giac")

[Out]

integrate(e^(-7/2)/((b*sin(d*x + c) + a)^2*cos(d*x + c)^(7/2)), x)

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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 )}^{7/2}\,{\left (a+b\,\sin \left (c+d\,x\right )\right )}^2} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/((e*cos(c + d*x))^(7/2)*(a + b*sin(c + d*x))^2),x)

[Out]

int(1/((e*cos(c + d*x))^(7/2)*(a + b*sin(c + d*x))^2), x)

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