\(\int \frac {(e \sin (c+d x))^{9/2}}{(a+b \sec (c+d x))^2} \, dx\) [241]

Optimal result

Integrand size = 25, antiderivative size = 1070 \[ \int \frac {(e \sin (c+d x))^{9/2}}{(a+b \sec (c+d x))^2} \, dx=-\frac {7 b^3 \left (a^2-b^2\right )^{3/4} e^{9/2} \arctan \left (\frac {\sqrt {a} \sqrt {e \sin (c+d x)}}{\sqrt [4]{a^2-b^2} \sqrt {e}}\right )}{2 a^{13/2} d}+\frac {2 b \left (a^2-b^2\right )^{7/4} e^{9/2} \arctan \left (\frac {\sqrt {a} \sqrt {e \sin (c+d x)}}{\sqrt [4]{a^2-b^2} \sqrt {e}}\right )}{a^{13/2} d}+\frac {7 b^3 \left (a^2-b^2\right )^{3/4} e^{9/2} \text {arctanh}\left (\frac {\sqrt {a} \sqrt {e \sin (c+d x)}}{\sqrt [4]{a^2-b^2} \sqrt {e}}\right )}{2 a^{13/2} d}-\frac {2 b \left (a^2-b^2\right )^{7/4} e^{9/2} \text {arctanh}\left (\frac {\sqrt {a} \sqrt {e \sin (c+d x)}}{\sqrt [4]{a^2-b^2} \sqrt {e}}\right )}{a^{13/2} d}+\frac {7 b^4 \left (a^2-b^2\right ) e^5 \operatorname {EllipticPi}\left (\frac {2 a}{a-\sqrt {a^2-b^2}},\frac {1}{2} \left (c-\frac {\pi }{2}+d x\right ),2\right ) \sqrt {\sin (c+d x)}}{2 a^7 \left (a-\sqrt {a^2-b^2}\right ) d \sqrt {e \sin (c+d x)}}-\frac {2 b^2 \left (a^2-b^2\right )^2 e^5 \operatorname {EllipticPi}\left (\frac {2 a}{a-\sqrt {a^2-b^2}},\frac {1}{2} \left (c-\frac {\pi }{2}+d x\right ),2\right ) \sqrt {\sin (c+d x)}}{a^7 \left (a-\sqrt {a^2-b^2}\right ) d \sqrt {e \sin (c+d x)}}+\frac {7 b^4 \left (a^2-b^2\right ) e^5 \operatorname {EllipticPi}\left (\frac {2 a}{a+\sqrt {a^2-b^2}},\frac {1}{2} \left (c-\frac {\pi }{2}+d x\right ),2\right ) \sqrt {\sin (c+d x)}}{2 a^7 \left (a+\sqrt {a^2-b^2}\right ) d \sqrt {e \sin (c+d x)}}-\frac {2 b^2 \left (a^2-b^2\right )^2 e^5 \operatorname {EllipticPi}\left (\frac {2 a}{a+\sqrt {a^2-b^2}},\frac {1}{2} \left (c-\frac {\pi }{2}+d x\right ),2\right ) \sqrt {\sin (c+d x)}}{a^7 \left (a+\sqrt {a^2-b^2}\right ) d \sqrt {e \sin (c+d x)}}+\frac {14 e^4 E\left (\left .\frac {1}{2} \left (c-\frac {\pi }{2}+d x\right )\right |2\right ) \sqrt {e \sin (c+d x)}}{15 a^2 d \sqrt {\sin (c+d x)}}-\frac {7 b^2 \left (3 a^2-5 b^2\right ) e^4 E\left (\left .\frac {1}{2} \left (c-\frac {\pi }{2}+d x\right )\right |2\right ) \sqrt {e \sin (c+d x)}}{5 a^6 d \sqrt {\sin (c+d x)}}-\frac {4 b^2 \left (8 a^2-5 b^2\right ) e^4 E\left (\left .\frac {1}{2} \left (c-\frac {\pi }{2}+d x\right )\right |2\right ) \sqrt {e \sin (c+d x)}}{5 a^6 d \sqrt {\sin (c+d x)}}-\frac {14 e^3 \cos (c+d x) (e \sin (c+d x))^{3/2}}{45 a^2 d}-\frac {7 b^2 e^3 (5 b-3 a \cos (c+d x)) (e \sin (c+d x))^{3/2}}{15 a^5 d}+\frac {4 b e^3 \left (5 \left (a^2-b^2\right )+3 a b \cos (c+d x)\right ) (e \sin (c+d x))^{3/2}}{15 a^5 d}+\frac {4 b e (e \sin (c+d x))^{7/2}}{7 a^3 d}-\frac {2 e \cos (c+d x) (e \sin (c+d x))^{7/2}}{9 a^2 d}+\frac {b^2 e (e \sin (c+d x))^{7/2}}{a^3 d (b+a \cos (c+d x))} \]

[Out]

-7/2*b^3*(a^2-b^2)^(3/4)*e^(9/2)*arctan(a^(1/2)*(e*sin(d*x+c))^(1/2)/(a^2-b^2)^(1/4)/e^(1/2))/a^(13/2)/d+2*b*(
a^2-b^2)^(7/4)*e^(9/2)*arctan(a^(1/2)*(e*sin(d*x+c))^(1/2)/(a^2-b^2)^(1/4)/e^(1/2))/a^(13/2)/d+7/2*b^3*(a^2-b^
2)^(3/4)*e^(9/2)*arctanh(a^(1/2)*(e*sin(d*x+c))^(1/2)/(a^2-b^2)^(1/4)/e^(1/2))/a^(13/2)/d-2*b*(a^2-b^2)^(7/4)*
e^(9/2)*arctanh(a^(1/2)*(e*sin(d*x+c))^(1/2)/(a^2-b^2)^(1/4)/e^(1/2))/a^(13/2)/d-14/45*e^3*cos(d*x+c)*(e*sin(d
*x+c))^(3/2)/a^2/d-7/15*b^2*e^3*(5*b-3*a*cos(d*x+c))*(e*sin(d*x+c))^(3/2)/a^5/d+4/15*b*e^3*(5*a^2-5*b^2+3*a*b*
cos(d*x+c))*(e*sin(d*x+c))^(3/2)/a^5/d+4/7*b*e*(e*sin(d*x+c))^(7/2)/a^3/d-2/9*e*cos(d*x+c)*(e*sin(d*x+c))^(7/2
)/a^2/d+b^2*e*(e*sin(d*x+c))^(7/2)/a^3/d/(b+a*cos(d*x+c))-7/2*b^4*(a^2-b^2)*e^5*(sin(1/2*c+1/4*Pi+1/2*d*x)^2)^
(1/2)/sin(1/2*c+1/4*Pi+1/2*d*x)*EllipticPi(cos(1/2*c+1/4*Pi+1/2*d*x),2*a/(a-(a^2-b^2)^(1/2)),2^(1/2))*sin(d*x+
c)^(1/2)/a^7/d/(a-(a^2-b^2)^(1/2))/(e*sin(d*x+c))^(1/2)+2*b^2*(a^2-b^2)^2*e^5*(sin(1/2*c+1/4*Pi+1/2*d*x)^2)^(1
/2)/sin(1/2*c+1/4*Pi+1/2*d*x)*EllipticPi(cos(1/2*c+1/4*Pi+1/2*d*x),2*a/(a-(a^2-b^2)^(1/2)),2^(1/2))*sin(d*x+c)
^(1/2)/a^7/d/(a-(a^2-b^2)^(1/2))/(e*sin(d*x+c))^(1/2)-7/2*b^4*(a^2-b^2)*e^5*(sin(1/2*c+1/4*Pi+1/2*d*x)^2)^(1/2
)/sin(1/2*c+1/4*Pi+1/2*d*x)*EllipticPi(cos(1/2*c+1/4*Pi+1/2*d*x),2*a/(a+(a^2-b^2)^(1/2)),2^(1/2))*sin(d*x+c)^(
1/2)/a^7/d/(a+(a^2-b^2)^(1/2))/(e*sin(d*x+c))^(1/2)+2*b^2*(a^2-b^2)^2*e^5*(sin(1/2*c+1/4*Pi+1/2*d*x)^2)^(1/2)/
sin(1/2*c+1/4*Pi+1/2*d*x)*EllipticPi(cos(1/2*c+1/4*Pi+1/2*d*x),2*a/(a+(a^2-b^2)^(1/2)),2^(1/2))*sin(d*x+c)^(1/
2)/a^7/d/(a+(a^2-b^2)^(1/2))/(e*sin(d*x+c))^(1/2)-14/15*e^4*(sin(1/2*c+1/4*Pi+1/2*d*x)^2)^(1/2)/sin(1/2*c+1/4*
Pi+1/2*d*x)*EllipticE(cos(1/2*c+1/4*Pi+1/2*d*x),2^(1/2))*(e*sin(d*x+c))^(1/2)/a^2/d/sin(d*x+c)^(1/2)+7/5*b^2*(
3*a^2-5*b^2)*e^4*(sin(1/2*c+1/4*Pi+1/2*d*x)^2)^(1/2)/sin(1/2*c+1/4*Pi+1/2*d*x)*EllipticE(cos(1/2*c+1/4*Pi+1/2*
d*x),2^(1/2))*(e*sin(d*x+c))^(1/2)/a^6/d/sin(d*x+c)^(1/2)+4/5*b^2*(8*a^2-5*b^2)*e^4*(sin(1/2*c+1/4*Pi+1/2*d*x)
^2)^(1/2)/sin(1/2*c+1/4*Pi+1/2*d*x)*EllipticE(cos(1/2*c+1/4*Pi+1/2*d*x),2^(1/2))*(e*sin(d*x+c))^(1/2)/a^6/d/si
n(d*x+c)^(1/2)

Rubi [A] (verified)

Time = 3.47 (sec) , antiderivative size = 1070, normalized size of antiderivative = 1.00, number of steps used = 35, number of rules used = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.640, Rules used = {3957, 2991, 2715, 2721, 2719, 2772, 2944, 2946, 2780, 2886, 2884, 335, 304, 211, 214, 2774} \[ \int \frac {(e \sin (c+d x))^{9/2}}{(a+b \sec (c+d x))^2} \, dx=-\frac {2 b^2 \left (a^2-b^2\right )^2 \operatorname {EllipticPi}\left (\frac {2 a}{a-\sqrt {a^2-b^2}},\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right ),2\right ) \sqrt {\sin (c+d x)} e^5}{a^7 \left (a-\sqrt {a^2-b^2}\right ) d \sqrt {e \sin (c+d x)}}+\frac {7 b^4 \left (a^2-b^2\right ) \operatorname {EllipticPi}\left (\frac {2 a}{a-\sqrt {a^2-b^2}},\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right ),2\right ) \sqrt {\sin (c+d x)} e^5}{2 a^7 \left (a-\sqrt {a^2-b^2}\right ) d \sqrt {e \sin (c+d x)}}-\frac {2 b^2 \left (a^2-b^2\right )^2 \operatorname {EllipticPi}\left (\frac {2 a}{a+\sqrt {a^2-b^2}},\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right ),2\right ) \sqrt {\sin (c+d x)} e^5}{a^7 \left (a+\sqrt {a^2-b^2}\right ) d \sqrt {e \sin (c+d x)}}+\frac {7 b^4 \left (a^2-b^2\right ) \operatorname {EllipticPi}\left (\frac {2 a}{a+\sqrt {a^2-b^2}},\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right ),2\right ) \sqrt {\sin (c+d x)} e^5}{2 a^7 \left (a+\sqrt {a^2-b^2}\right ) d \sqrt {e \sin (c+d x)}}+\frac {2 b \left (a^2-b^2\right )^{7/4} \arctan \left (\frac {\sqrt {a} \sqrt {e \sin (c+d x)}}{\sqrt [4]{a^2-b^2} \sqrt {e}}\right ) e^{9/2}}{a^{13/2} d}-\frac {7 b^3 \left (a^2-b^2\right )^{3/4} \arctan \left (\frac {\sqrt {a} \sqrt {e \sin (c+d x)}}{\sqrt [4]{a^2-b^2} \sqrt {e}}\right ) e^{9/2}}{2 a^{13/2} d}-\frac {2 b \left (a^2-b^2\right )^{7/4} \text {arctanh}\left (\frac {\sqrt {a} \sqrt {e \sin (c+d x)}}{\sqrt [4]{a^2-b^2} \sqrt {e}}\right ) e^{9/2}}{a^{13/2} d}+\frac {7 b^3 \left (a^2-b^2\right )^{3/4} \text {arctanh}\left (\frac {\sqrt {a} \sqrt {e \sin (c+d x)}}{\sqrt [4]{a^2-b^2} \sqrt {e}}\right ) e^{9/2}}{2 a^{13/2} d}-\frac {7 b^2 \left (3 a^2-5 b^2\right ) E\left (\left .\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right )\right |2\right ) \sqrt {e \sin (c+d x)} e^4}{5 a^6 d \sqrt {\sin (c+d x)}}-\frac {4 b^2 \left (8 a^2-5 b^2\right ) E\left (\left .\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right )\right |2\right ) \sqrt {e \sin (c+d x)} e^4}{5 a^6 d \sqrt {\sin (c+d x)}}+\frac {14 E\left (\left .\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right )\right |2\right ) \sqrt {e \sin (c+d x)} e^4}{15 a^2 d \sqrt {\sin (c+d x)}}-\frac {14 \cos (c+d x) (e \sin (c+d x))^{3/2} e^3}{45 a^2 d}-\frac {7 b^2 (5 b-3 a \cos (c+d x)) (e \sin (c+d x))^{3/2} e^3}{15 a^5 d}+\frac {4 b \left (5 \left (a^2-b^2\right )+3 a b \cos (c+d x)\right ) (e \sin (c+d x))^{3/2} e^3}{15 a^5 d}-\frac {2 \cos (c+d x) (e \sin (c+d x))^{7/2} e}{9 a^2 d}+\frac {4 b (e \sin (c+d x))^{7/2} e}{7 a^3 d}+\frac {b^2 (e \sin (c+d x))^{7/2} e}{a^3 d (b+a \cos (c+d x))} \]

[In]

Int[(e*Sin[c + d*x])^(9/2)/(a + b*Sec[c + d*x])^2,x]

[Out]

(-7*b^3*(a^2 - b^2)^(3/4)*e^(9/2)*ArcTan[(Sqrt[a]*Sqrt[e*Sin[c + d*x]])/((a^2 - b^2)^(1/4)*Sqrt[e])])/(2*a^(13
/2)*d) + (2*b*(a^2 - b^2)^(7/4)*e^(9/2)*ArcTan[(Sqrt[a]*Sqrt[e*Sin[c + d*x]])/((a^2 - b^2)^(1/4)*Sqrt[e])])/(a
^(13/2)*d) + (7*b^3*(a^2 - b^2)^(3/4)*e^(9/2)*ArcTanh[(Sqrt[a]*Sqrt[e*Sin[c + d*x]])/((a^2 - b^2)^(1/4)*Sqrt[e
])])/(2*a^(13/2)*d) - (2*b*(a^2 - b^2)^(7/4)*e^(9/2)*ArcTanh[(Sqrt[a]*Sqrt[e*Sin[c + d*x]])/((a^2 - b^2)^(1/4)
*Sqrt[e])])/(a^(13/2)*d) + (7*b^4*(a^2 - b^2)*e^5*EllipticPi[(2*a)/(a - Sqrt[a^2 - b^2]), (c - Pi/2 + d*x)/2,
2]*Sqrt[Sin[c + d*x]])/(2*a^7*(a - Sqrt[a^2 - b^2])*d*Sqrt[e*Sin[c + d*x]]) - (2*b^2*(a^2 - b^2)^2*e^5*Ellipti
cPi[(2*a)/(a - Sqrt[a^2 - b^2]), (c - Pi/2 + d*x)/2, 2]*Sqrt[Sin[c + d*x]])/(a^7*(a - Sqrt[a^2 - b^2])*d*Sqrt[
e*Sin[c + d*x]]) + (7*b^4*(a^2 - b^2)*e^5*EllipticPi[(2*a)/(a + Sqrt[a^2 - b^2]), (c - Pi/2 + d*x)/2, 2]*Sqrt[
Sin[c + d*x]])/(2*a^7*(a + Sqrt[a^2 - b^2])*d*Sqrt[e*Sin[c + d*x]]) - (2*b^2*(a^2 - b^2)^2*e^5*EllipticPi[(2*a
)/(a + Sqrt[a^2 - b^2]), (c - Pi/2 + d*x)/2, 2]*Sqrt[Sin[c + d*x]])/(a^7*(a + Sqrt[a^2 - b^2])*d*Sqrt[e*Sin[c
+ d*x]]) + (14*e^4*EllipticE[(c - Pi/2 + d*x)/2, 2]*Sqrt[e*Sin[c + d*x]])/(15*a^2*d*Sqrt[Sin[c + d*x]]) - (7*b
^2*(3*a^2 - 5*b^2)*e^4*EllipticE[(c - Pi/2 + d*x)/2, 2]*Sqrt[e*Sin[c + d*x]])/(5*a^6*d*Sqrt[Sin[c + d*x]]) - (
4*b^2*(8*a^2 - 5*b^2)*e^4*EllipticE[(c - Pi/2 + d*x)/2, 2]*Sqrt[e*Sin[c + d*x]])/(5*a^6*d*Sqrt[Sin[c + d*x]])
- (14*e^3*Cos[c + d*x]*(e*Sin[c + d*x])^(3/2))/(45*a^2*d) - (7*b^2*e^3*(5*b - 3*a*Cos[c + d*x])*(e*Sin[c + d*x
])^(3/2))/(15*a^5*d) + (4*b*e^3*(5*(a^2 - b^2) + 3*a*b*Cos[c + d*x])*(e*Sin[c + d*x])^(3/2))/(15*a^5*d) + (4*b
*e*(e*Sin[c + d*x])^(7/2))/(7*a^3*d) - (2*e*Cos[c + d*x]*(e*Sin[c + d*x])^(7/2))/(9*a^2*d) + (b^2*e*(e*Sin[c +
d*x])^(7/2))/(a^3*d*(b + a*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 2715

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

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 2772

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

Rule 2774

Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_), x_Symbol] :> Simp[g*(g*C
os[e + f*x])^(p - 1)*((a + b*Sin[e + f*x])^(m + 1)/(b*f*(m + p))), x] + Dist[g^2*((p - 1)/(b*(m + p))), Int[(g
*Cos[e + f*x])^(p - 2)*(a + b*Sin[e + f*x])^m*(b + a*Sin[e + f*x]), x], x] /; FreeQ[{a, b, e, f, g, m}, x] &&
NeQ[a^2 - b^2, 0] && GtQ[p, 1] && NeQ[m + p, 0] && 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 2944

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

Rule 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]

Rule 2991

Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((d_.)*sin[(e_.) + (f_.)*(x_)])^(n_)*((a_) + (b_.)*sin[(e_.) + (f_.)*
(x_)])^(m_), x_Symbol] :> Int[ExpandTrig[(g*cos[e + f*x])^p, (d*sin[e + f*x])^n*(a + b*sin[e + f*x])^m, x], x]
/; FreeQ[{a, b, d, e, f, g, n, p}, x] && NeQ[a^2 - b^2, 0] && IntegerQ[m] && (GtQ[m, 0] || IntegerQ[n])

Rule 3957

Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_.)*(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_))^(m_.), x_Symbol] :> Int[(g*Co
s[e + f*x])^p*((b + a*Sin[e + f*x])^m/Sin[e + f*x]^m), x] /; FreeQ[{a, b, e, f, g, p}, x] && IntegerQ[m]

Rubi steps \begin{align*} \text {integral}& = \int \frac {\cos ^2(c+d x) (e \sin (c+d x))^{9/2}}{(-b-a \cos (c+d x))^2} \, dx \\ & = \int \left (\frac {(e \sin (c+d x))^{9/2}}{a^2}+\frac {b^2 (e \sin (c+d x))^{9/2}}{a^2 (b+a \cos (c+d x))^2}-\frac {2 b (e \sin (c+d x))^{9/2}}{a^2 (b+a \cos (c+d x))}\right ) \, dx \\ & = \frac {\int (e \sin (c+d x))^{9/2} \, dx}{a^2}-\frac {(2 b) \int \frac {(e \sin (c+d x))^{9/2}}{b+a \cos (c+d x)} \, dx}{a^2}+\frac {b^2 \int \frac {(e \sin (c+d x))^{9/2}}{(b+a \cos (c+d x))^2} \, dx}{a^2} \\ & = \frac {4 b e (e \sin (c+d x))^{7/2}}{7 a^3 d}-\frac {2 e \cos (c+d x) (e \sin (c+d x))^{7/2}}{9 a^2 d}+\frac {b^2 e (e \sin (c+d x))^{7/2}}{a^3 d (b+a \cos (c+d x))}+\frac {\left (7 e^2\right ) \int (e \sin (c+d x))^{5/2} \, dx}{9 a^2}+\frac {\left (2 b e^2\right ) \int \frac {(-a-b \cos (c+d x)) (e \sin (c+d x))^{5/2}}{b+a \cos (c+d x)} \, dx}{a^3}-\frac {\left (7 b^2 e^2\right ) \int \frac {\cos (c+d x) (e \sin (c+d x))^{5/2}}{b+a \cos (c+d x)} \, dx}{2 a^3} \\ & = -\frac {14 e^3 \cos (c+d x) (e \sin (c+d x))^{3/2}}{45 a^2 d}-\frac {7 b^2 e^3 (5 b-3 a \cos (c+d x)) (e \sin (c+d x))^{3/2}}{15 a^5 d}+\frac {4 b e^3 \left (5 \left (a^2-b^2\right )+3 a b \cos (c+d x)\right ) (e \sin (c+d x))^{3/2}}{15 a^5 d}+\frac {4 b e (e \sin (c+d x))^{7/2}}{7 a^3 d}-\frac {2 e \cos (c+d x) (e \sin (c+d x))^{7/2}}{9 a^2 d}+\frac {b^2 e (e \sin (c+d x))^{7/2}}{a^3 d (b+a \cos (c+d x))}+\frac {\left (7 e^4\right ) \int \sqrt {e \sin (c+d x)} \, dx}{15 a^2}+\frac {\left (4 b e^4\right ) \int \frac {\left (-\frac {1}{2} a \left (5 a^2-2 b^2\right )-\frac {1}{2} b \left (8 a^2-5 b^2\right ) \cos (c+d x)\right ) \sqrt {e \sin (c+d x)}}{b+a \cos (c+d x)} \, dx}{5 a^5}-\frac {\left (7 b^2 e^4\right ) \int \frac {\left (-a b+\frac {1}{2} \left (3 a^2-5 b^2\right ) \cos (c+d x)\right ) \sqrt {e \sin (c+d x)}}{b+a \cos (c+d x)} \, dx}{5 a^5} \\ & = -\frac {14 e^3 \cos (c+d x) (e \sin (c+d x))^{3/2}}{45 a^2 d}-\frac {7 b^2 e^3 (5 b-3 a \cos (c+d x)) (e \sin (c+d x))^{3/2}}{15 a^5 d}+\frac {4 b e^3 \left (5 \left (a^2-b^2\right )+3 a b \cos (c+d x)\right ) (e \sin (c+d x))^{3/2}}{15 a^5 d}+\frac {4 b e (e \sin (c+d x))^{7/2}}{7 a^3 d}-\frac {2 e \cos (c+d x) (e \sin (c+d x))^{7/2}}{9 a^2 d}+\frac {b^2 e (e \sin (c+d x))^{7/2}}{a^3 d (b+a \cos (c+d x))}-\frac {\left (7 b^2 \left (3 a^2-5 b^2\right ) e^4\right ) \int \sqrt {e \sin (c+d x)} \, dx}{10 a^6}-\frac {\left (2 b^2 \left (8 a^2-5 b^2\right ) e^4\right ) \int \sqrt {e \sin (c+d x)} \, dx}{5 a^6}+\frac {\left (7 b^3 \left (a^2-b^2\right ) e^4\right ) \int \frac {\sqrt {e \sin (c+d x)}}{b+a \cos (c+d x)} \, dx}{2 a^6}-\frac {\left (2 b \left (a^2-b^2\right )^2 e^4\right ) \int \frac {\sqrt {e \sin (c+d x)}}{b+a \cos (c+d x)} \, dx}{a^6}+\frac {\left (7 e^4 \sqrt {e \sin (c+d x)}\right ) \int \sqrt {\sin (c+d x)} \, dx}{15 a^2 \sqrt {\sin (c+d x)}} \\ & = \frac {14 e^4 E\left (\left .\frac {1}{2} \left (c-\frac {\pi }{2}+d x\right )\right |2\right ) \sqrt {e \sin (c+d x)}}{15 a^2 d \sqrt {\sin (c+d x)}}-\frac {14 e^3 \cos (c+d x) (e \sin (c+d x))^{3/2}}{45 a^2 d}-\frac {7 b^2 e^3 (5 b-3 a \cos (c+d x)) (e \sin (c+d x))^{3/2}}{15 a^5 d}+\frac {4 b e^3 \left (5 \left (a^2-b^2\right )+3 a b \cos (c+d x)\right ) (e \sin (c+d x))^{3/2}}{15 a^5 d}+\frac {4 b e (e \sin (c+d x))^{7/2}}{7 a^3 d}-\frac {2 e \cos (c+d x) (e \sin (c+d x))^{7/2}}{9 a^2 d}+\frac {b^2 e (e \sin (c+d x))^{7/2}}{a^3 d (b+a \cos (c+d x))}-\frac {\left (7 b^4 \left (a^2-b^2\right ) e^5\right ) \int \frac {1}{\sqrt {e \sin (c+d x)} \left (\sqrt {a^2-b^2}-a \sin (c+d x)\right )} \, dx}{4 a^7}+\frac {\left (7 b^4 \left (a^2-b^2\right ) e^5\right ) \int \frac {1}{\sqrt {e \sin (c+d x)} \left (\sqrt {a^2-b^2}+a \sin (c+d x)\right )} \, dx}{4 a^7}+\frac {\left (b^2 \left (a^2-b^2\right )^2 e^5\right ) \int \frac {1}{\sqrt {e \sin (c+d x)} \left (\sqrt {a^2-b^2}-a \sin (c+d x)\right )} \, dx}{a^7}-\frac {\left (b^2 \left (a^2-b^2\right )^2 e^5\right ) \int \frac {1}{\sqrt {e \sin (c+d x)} \left (\sqrt {a^2-b^2}+a \sin (c+d x)\right )} \, dx}{a^7}-\frac {\left (7 b^3 \left (a^2-b^2\right ) e^5\right ) \text {Subst}\left (\int \frac {\sqrt {x}}{\left (-a^2+b^2\right ) e^2+a^2 x^2} \, dx,x,e \sin (c+d x)\right )}{2 a^5 d}+\frac {\left (2 b \left (a^2-b^2\right )^2 e^5\right ) \text {Subst}\left (\int \frac {\sqrt {x}}{\left (-a^2+b^2\right ) e^2+a^2 x^2} \, dx,x,e \sin (c+d x)\right )}{a^5 d}-\frac {\left (7 b^2 \left (3 a^2-5 b^2\right ) e^4 \sqrt {e \sin (c+d x)}\right ) \int \sqrt {\sin (c+d x)} \, dx}{10 a^6 \sqrt {\sin (c+d x)}}-\frac {\left (2 b^2 \left (8 a^2-5 b^2\right ) e^4 \sqrt {e \sin (c+d x)}\right ) \int \sqrt {\sin (c+d x)} \, dx}{5 a^6 \sqrt {\sin (c+d x)}} \\ & = \frac {14 e^4 E\left (\left .\frac {1}{2} \left (c-\frac {\pi }{2}+d x\right )\right |2\right ) \sqrt {e \sin (c+d x)}}{15 a^2 d \sqrt {\sin (c+d x)}}-\frac {7 b^2 \left (3 a^2-5 b^2\right ) e^4 E\left (\left .\frac {1}{2} \left (c-\frac {\pi }{2}+d x\right )\right |2\right ) \sqrt {e \sin (c+d x)}}{5 a^6 d \sqrt {\sin (c+d x)}}-\frac {4 b^2 \left (8 a^2-5 b^2\right ) e^4 E\left (\left .\frac {1}{2} \left (c-\frac {\pi }{2}+d x\right )\right |2\right ) \sqrt {e \sin (c+d x)}}{5 a^6 d \sqrt {\sin (c+d x)}}-\frac {14 e^3 \cos (c+d x) (e \sin (c+d x))^{3/2}}{45 a^2 d}-\frac {7 b^2 e^3 (5 b-3 a \cos (c+d x)) (e \sin (c+d x))^{3/2}}{15 a^5 d}+\frac {4 b e^3 \left (5 \left (a^2-b^2\right )+3 a b \cos (c+d x)\right ) (e \sin (c+d x))^{3/2}}{15 a^5 d}+\frac {4 b e (e \sin (c+d x))^{7/2}}{7 a^3 d}-\frac {2 e \cos (c+d x) (e \sin (c+d x))^{7/2}}{9 a^2 d}+\frac {b^2 e (e \sin (c+d x))^{7/2}}{a^3 d (b+a \cos (c+d x))}-\frac {\left (7 b^3 \left (a^2-b^2\right ) e^5\right ) \text {Subst}\left (\int \frac {x^2}{\left (-a^2+b^2\right ) e^2+a^2 x^4} \, dx,x,\sqrt {e \sin (c+d x)}\right )}{a^5 d}+\frac {\left (4 b \left (a^2-b^2\right )^2 e^5\right ) \text {Subst}\left (\int \frac {x^2}{\left (-a^2+b^2\right ) e^2+a^2 x^4} \, dx,x,\sqrt {e \sin (c+d x)}\right )}{a^5 d}-\frac {\left (7 b^4 \left (a^2-b^2\right ) e^5 \sqrt {\sin (c+d x)}\right ) \int \frac {1}{\sqrt {\sin (c+d x)} \left (\sqrt {a^2-b^2}-a \sin (c+d x)\right )} \, dx}{4 a^7 \sqrt {e \sin (c+d x)}}+\frac {\left (7 b^4 \left (a^2-b^2\right ) e^5 \sqrt {\sin (c+d x)}\right ) \int \frac {1}{\sqrt {\sin (c+d x)} \left (\sqrt {a^2-b^2}+a \sin (c+d x)\right )} \, dx}{4 a^7 \sqrt {e \sin (c+d x)}}+\frac {\left (b^2 \left (a^2-b^2\right )^2 e^5 \sqrt {\sin (c+d x)}\right ) \int \frac {1}{\sqrt {\sin (c+d x)} \left (\sqrt {a^2-b^2}-a \sin (c+d x)\right )} \, dx}{a^7 \sqrt {e \sin (c+d x)}}-\frac {\left (b^2 \left (a^2-b^2\right )^2 e^5 \sqrt {\sin (c+d x)}\right ) \int \frac {1}{\sqrt {\sin (c+d x)} \left (\sqrt {a^2-b^2}+a \sin (c+d x)\right )} \, dx}{a^7 \sqrt {e \sin (c+d x)}} \\ & = \frac {7 b^4 \left (a^2-b^2\right ) e^5 \operatorname {EllipticPi}\left (\frac {2 a}{a-\sqrt {a^2-b^2}},\frac {1}{2} \left (c-\frac {\pi }{2}+d x\right ),2\right ) \sqrt {\sin (c+d x)}}{2 a^7 \left (a-\sqrt {a^2-b^2}\right ) d \sqrt {e \sin (c+d x)}}-\frac {2 b^2 \left (a^2-b^2\right )^2 e^5 \operatorname {EllipticPi}\left (\frac {2 a}{a-\sqrt {a^2-b^2}},\frac {1}{2} \left (c-\frac {\pi }{2}+d x\right ),2\right ) \sqrt {\sin (c+d x)}}{a^7 \left (a-\sqrt {a^2-b^2}\right ) d \sqrt {e \sin (c+d x)}}+\frac {7 b^4 \left (a^2-b^2\right ) e^5 \operatorname {EllipticPi}\left (\frac {2 a}{a+\sqrt {a^2-b^2}},\frac {1}{2} \left (c-\frac {\pi }{2}+d x\right ),2\right ) \sqrt {\sin (c+d x)}}{2 a^7 \left (a+\sqrt {a^2-b^2}\right ) d \sqrt {e \sin (c+d x)}}-\frac {2 b^2 \left (a^2-b^2\right )^2 e^5 \operatorname {EllipticPi}\left (\frac {2 a}{a+\sqrt {a^2-b^2}},\frac {1}{2} \left (c-\frac {\pi }{2}+d x\right ),2\right ) \sqrt {\sin (c+d x)}}{a^7 \left (a+\sqrt {a^2-b^2}\right ) d \sqrt {e \sin (c+d x)}}+\frac {14 e^4 E\left (\left .\frac {1}{2} \left (c-\frac {\pi }{2}+d x\right )\right |2\right ) \sqrt {e \sin (c+d x)}}{15 a^2 d \sqrt {\sin (c+d x)}}-\frac {7 b^2 \left (3 a^2-5 b^2\right ) e^4 E\left (\left .\frac {1}{2} \left (c-\frac {\pi }{2}+d x\right )\right |2\right ) \sqrt {e \sin (c+d x)}}{5 a^6 d \sqrt {\sin (c+d x)}}-\frac {4 b^2 \left (8 a^2-5 b^2\right ) e^4 E\left (\left .\frac {1}{2} \left (c-\frac {\pi }{2}+d x\right )\right |2\right ) \sqrt {e \sin (c+d x)}}{5 a^6 d \sqrt {\sin (c+d x)}}-\frac {14 e^3 \cos (c+d x) (e \sin (c+d x))^{3/2}}{45 a^2 d}-\frac {7 b^2 e^3 (5 b-3 a \cos (c+d x)) (e \sin (c+d x))^{3/2}}{15 a^5 d}+\frac {4 b e^3 \left (5 \left (a^2-b^2\right )+3 a b \cos (c+d x)\right ) (e \sin (c+d x))^{3/2}}{15 a^5 d}+\frac {4 b e (e \sin (c+d x))^{7/2}}{7 a^3 d}-\frac {2 e \cos (c+d x) (e \sin (c+d x))^{7/2}}{9 a^2 d}+\frac {b^2 e (e \sin (c+d x))^{7/2}}{a^3 d (b+a \cos (c+d x))}+\frac {\left (7 b^3 \left (a^2-b^2\right ) e^5\right ) \text {Subst}\left (\int \frac {1}{\sqrt {a^2-b^2} e-a x^2} \, dx,x,\sqrt {e \sin (c+d x)}\right )}{2 a^6 d}-\frac {\left (7 b^3 \left (a^2-b^2\right ) e^5\right ) \text {Subst}\left (\int \frac {1}{\sqrt {a^2-b^2} e+a x^2} \, dx,x,\sqrt {e \sin (c+d x)}\right )}{2 a^6 d}-\frac {\left (2 b \left (a^2-b^2\right )^2 e^5\right ) \text {Subst}\left (\int \frac {1}{\sqrt {a^2-b^2} e-a x^2} \, dx,x,\sqrt {e \sin (c+d x)}\right )}{a^6 d}+\frac {\left (2 b \left (a^2-b^2\right )^2 e^5\right ) \text {Subst}\left (\int \frac {1}{\sqrt {a^2-b^2} e+a x^2} \, dx,x,\sqrt {e \sin (c+d x)}\right )}{a^6 d} \\ & = -\frac {7 b^3 \left (a^2-b^2\right )^{3/4} e^{9/2} \arctan \left (\frac {\sqrt {a} \sqrt {e \sin (c+d x)}}{\sqrt [4]{a^2-b^2} \sqrt {e}}\right )}{2 a^{13/2} d}+\frac {2 b \left (a^2-b^2\right )^{7/4} e^{9/2} \arctan \left (\frac {\sqrt {a} \sqrt {e \sin (c+d x)}}{\sqrt [4]{a^2-b^2} \sqrt {e}}\right )}{a^{13/2} d}+\frac {7 b^3 \left (a^2-b^2\right )^{3/4} e^{9/2} \text {arctanh}\left (\frac {\sqrt {a} \sqrt {e \sin (c+d x)}}{\sqrt [4]{a^2-b^2} \sqrt {e}}\right )}{2 a^{13/2} d}-\frac {2 b \left (a^2-b^2\right )^{7/4} e^{9/2} \text {arctanh}\left (\frac {\sqrt {a} \sqrt {e \sin (c+d x)}}{\sqrt [4]{a^2-b^2} \sqrt {e}}\right )}{a^{13/2} d}+\frac {7 b^4 \left (a^2-b^2\right ) e^5 \operatorname {EllipticPi}\left (\frac {2 a}{a-\sqrt {a^2-b^2}},\frac {1}{2} \left (c-\frac {\pi }{2}+d x\right ),2\right ) \sqrt {\sin (c+d x)}}{2 a^7 \left (a-\sqrt {a^2-b^2}\right ) d \sqrt {e \sin (c+d x)}}-\frac {2 b^2 \left (a^2-b^2\right )^2 e^5 \operatorname {EllipticPi}\left (\frac {2 a}{a-\sqrt {a^2-b^2}},\frac {1}{2} \left (c-\frac {\pi }{2}+d x\right ),2\right ) \sqrt {\sin (c+d x)}}{a^7 \left (a-\sqrt {a^2-b^2}\right ) d \sqrt {e \sin (c+d x)}}+\frac {7 b^4 \left (a^2-b^2\right ) e^5 \operatorname {EllipticPi}\left (\frac {2 a}{a+\sqrt {a^2-b^2}},\frac {1}{2} \left (c-\frac {\pi }{2}+d x\right ),2\right ) \sqrt {\sin (c+d x)}}{2 a^7 \left (a+\sqrt {a^2-b^2}\right ) d \sqrt {e \sin (c+d x)}}-\frac {2 b^2 \left (a^2-b^2\right )^2 e^5 \operatorname {EllipticPi}\left (\frac {2 a}{a+\sqrt {a^2-b^2}},\frac {1}{2} \left (c-\frac {\pi }{2}+d x\right ),2\right ) \sqrt {\sin (c+d x)}}{a^7 \left (a+\sqrt {a^2-b^2}\right ) d \sqrt {e \sin (c+d x)}}+\frac {14 e^4 E\left (\left .\frac {1}{2} \left (c-\frac {\pi }{2}+d x\right )\right |2\right ) \sqrt {e \sin (c+d x)}}{15 a^2 d \sqrt {\sin (c+d x)}}-\frac {7 b^2 \left (3 a^2-5 b^2\right ) e^4 E\left (\left .\frac {1}{2} \left (c-\frac {\pi }{2}+d x\right )\right |2\right ) \sqrt {e \sin (c+d x)}}{5 a^6 d \sqrt {\sin (c+d x)}}-\frac {4 b^2 \left (8 a^2-5 b^2\right ) e^4 E\left (\left .\frac {1}{2} \left (c-\frac {\pi }{2}+d x\right )\right |2\right ) \sqrt {e \sin (c+d x)}}{5 a^6 d \sqrt {\sin (c+d x)}}-\frac {14 e^3 \cos (c+d x) (e \sin (c+d x))^{3/2}}{45 a^2 d}-\frac {7 b^2 e^3 (5 b-3 a \cos (c+d x)) (e \sin (c+d x))^{3/2}}{15 a^5 d}+\frac {4 b e^3 \left (5 \left (a^2-b^2\right )+3 a b \cos (c+d x)\right ) (e \sin (c+d x))^{3/2}}{15 a^5 d}+\frac {4 b e (e \sin (c+d x))^{7/2}}{7 a^3 d}-\frac {2 e \cos (c+d x) (e \sin (c+d x))^{7/2}}{9 a^2 d}+\frac {b^2 e (e \sin (c+d x))^{7/2}}{a^3 d (b+a \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 = 15.32 (sec) , antiderivative size = 974, normalized size of antiderivative = 0.91 \[ \int \frac {(e \sin (c+d x))^{9/2}}{(a+b \sec (c+d x))^2} \, dx=\frac {(b+a \cos (c+d x))^2 \sec ^2(c+d x) (e \sin (c+d x))^{9/2} \left (\frac {\left (14 a^4-159 a^2 b^2+165 b^4\right ) \cos ^2(c+d x) \left (3 \sqrt {2} b \left (-a^2+b^2\right )^{3/4} \left (2 \arctan \left (1-\frac {\sqrt {2} \sqrt {a} \sqrt {\sin (c+d x)}}{\sqrt [4]{-a^2+b^2}}\right )-2 \arctan \left (1+\frac {\sqrt {2} \sqrt {a} \sqrt {\sin (c+d x)}}{\sqrt [4]{-a^2+b^2}}\right )-\log \left (\sqrt {-a^2+b^2}-\sqrt {2} \sqrt {a} \sqrt [4]{-a^2+b^2} \sqrt {\sin (c+d x)}+a \sin (c+d x)\right )+\log \left (\sqrt {-a^2+b^2}+\sqrt {2} \sqrt {a} \sqrt [4]{-a^2+b^2} \sqrt {\sin (c+d x)}+a \sin (c+d x)\right )\right )+8 a^{5/2} \operatorname {AppellF1}\left (\frac {3}{4},-\frac {1}{2},1,\frac {7}{4},\sin ^2(c+d x),\frac {a^2 \sin ^2(c+d x)}{a^2-b^2}\right ) \sin ^{\frac {3}{2}}(c+d x)\right ) \left (b+a \sqrt {1-\sin ^2(c+d x)}\right )}{12 a^{3/2} \left (a^2-b^2\right ) (b+a \cos (c+d x)) \left (1-\sin ^2(c+d x)\right )}+\frac {2 \left (-46 a^3 b+66 a b^3\right ) \cos (c+d x) \left (\frac {\left (\frac {1}{8}+\frac {i}{8}\right ) \left (2 \arctan \left (1-\frac {(1+i) \sqrt {a} \sqrt {\sin (c+d x)}}{\sqrt [4]{a^2-b^2}}\right )-2 \arctan \left (1+\frac {(1+i) \sqrt {a} \sqrt {\sin (c+d x)}}{\sqrt [4]{a^2-b^2}}\right )-\log \left (\sqrt {a^2-b^2}-(1+i) \sqrt {a} \sqrt [4]{a^2-b^2} \sqrt {\sin (c+d x)}+i a \sin (c+d x)\right )+\log \left (\sqrt {a^2-b^2}+(1+i) \sqrt {a} \sqrt [4]{a^2-b^2} \sqrt {\sin (c+d x)}+i a \sin (c+d x)\right )\right )}{\sqrt {a} \sqrt [4]{a^2-b^2}}+\frac {b \operatorname {AppellF1}\left (\frac {3}{4},\frac {1}{2},1,\frac {7}{4},\sin ^2(c+d x),\frac {a^2 \sin ^2(c+d x)}{a^2-b^2}\right ) \sin ^{\frac {3}{2}}(c+d x)}{3 \left (-a^2+b^2\right )}\right ) \left (b+a \sqrt {1-\sin ^2(c+d x)}\right )}{(b+a \cos (c+d x)) \sqrt {1-\sin ^2(c+d x)}}\right )}{30 a^5 d (a+b \sec (c+d x))^2 \sin ^{\frac {9}{2}}(c+d x)}+\frac {(b+a \cos (c+d x))^2 \csc ^4(c+d x) \sec ^2(c+d x) (e \sin (c+d x))^{9/2} \left (-\frac {b \left (-37 a^2+56 b^2\right ) \sin (c+d x)}{21 a^5}+\frac {a^2 b^2 \sin (c+d x)-b^4 \sin (c+d x)}{a^5 (b+a \cos (c+d x))}-\frac {\left (19 a^2-54 b^2\right ) \sin (2 (c+d x))}{90 a^4}-\frac {b \sin (3 (c+d x))}{7 a^3}+\frac {\sin (4 (c+d x))}{36 a^2}\right )}{d (a+b \sec (c+d x))^2} \]

[In]

Integrate[(e*Sin[c + d*x])^(9/2)/(a + b*Sec[c + d*x])^2,x]

[Out]

((b + a*Cos[c + d*x])^2*Sec[c + d*x]^2*(e*Sin[c + d*x])^(9/2)*(((14*a^4 - 159*a^2*b^2 + 165*b^4)*Cos[c + d*x]^
2*(3*Sqrt[2]*b*(-a^2 + b^2)^(3/4)*(2*ArcTan[1 - (Sqrt[2]*Sqrt[a]*Sqrt[Sin[c + d*x]])/(-a^2 + b^2)^(1/4)] - 2*A
rcTan[1 + (Sqrt[2]*Sqrt[a]*Sqrt[Sin[c + d*x]])/(-a^2 + b^2)^(1/4)] - Log[Sqrt[-a^2 + b^2] - Sqrt[2]*Sqrt[a]*(-
a^2 + b^2)^(1/4)*Sqrt[Sin[c + d*x]] + a*Sin[c + d*x]] + Log[Sqrt[-a^2 + b^2] + Sqrt[2]*Sqrt[a]*(-a^2 + b^2)^(1
/4)*Sqrt[Sin[c + d*x]] + a*Sin[c + d*x]]) + 8*a^(5/2)*AppellF1[3/4, -1/2, 1, 7/4, Sin[c + d*x]^2, (a^2*Sin[c +
d*x]^2)/(a^2 - b^2)]*Sin[c + d*x]^(3/2))*(b + a*Sqrt[1 - Sin[c + d*x]^2]))/(12*a^(3/2)*(a^2 - b^2)*(b + a*Cos
[c + d*x])*(1 - Sin[c + d*x]^2)) + (2*(-46*a^3*b + 66*a*b^3)*Cos[c + d*x]*(((1/8 + I/8)*(2*ArcTan[1 - ((1 + I)
*Sqrt[a]*Sqrt[Sin[c + d*x]])/(a^2 - b^2)^(1/4)] - 2*ArcTan[1 + ((1 + I)*Sqrt[a]*Sqrt[Sin[c + d*x]])/(a^2 - b^2
)^(1/4)] - Log[Sqrt[a^2 - b^2] - (1 + I)*Sqrt[a]*(a^2 - b^2)^(1/4)*Sqrt[Sin[c + d*x]] + I*a*Sin[c + d*x]] + Lo
g[Sqrt[a^2 - b^2] + (1 + I)*Sqrt[a]*(a^2 - b^2)^(1/4)*Sqrt[Sin[c + d*x]] + I*a*Sin[c + d*x]]))/(Sqrt[a]*(a^2 -
b^2)^(1/4)) + (b*AppellF1[3/4, 1/2, 1, 7/4, Sin[c + d*x]^2, (a^2*Sin[c + d*x]^2)/(a^2 - b^2)]*Sin[c + d*x]^(3
/2))/(3*(-a^2 + b^2)))*(b + a*Sqrt[1 - Sin[c + d*x]^2]))/((b + a*Cos[c + d*x])*Sqrt[1 - Sin[c + d*x]^2])))/(30
*a^5*d*(a + b*Sec[c + d*x])^2*Sin[c + d*x]^(9/2)) + ((b + a*Cos[c + d*x])^2*Csc[c + d*x]^4*Sec[c + d*x]^2*(e*S
in[c + d*x])^(9/2)*(-1/21*(b*(-37*a^2 + 56*b^2)*Sin[c + d*x])/a^5 + (a^2*b^2*Sin[c + d*x] - b^4*Sin[c + d*x])/
(a^5*(b + a*Cos[c + d*x])) - ((19*a^2 - 54*b^2)*Sin[2*(c + d*x)])/(90*a^4) - (b*Sin[3*(c + d*x)])/(7*a^3) + Si
n[4*(c + d*x)]/(36*a^2)))/(d*(a + b*Sec[c + d*x])^2)

Maple [A] (warning: unable to verify)

Time = 44.55 (sec) , antiderivative size = 1725, normalized size of antiderivative = 1.61

method	result	size
default	\(\text {Expression too large to display}\)	\(1725\)

[In]

int((e*sin(d*x+c))^(9/2)/(a+b*sec(d*x+c))^2,x,method=_RETURNVERBOSE)

[Out]

(4*e*a*b*(-1/21/a^6*(e*sin(d*x+c))^(3/2)*e^2*(3*cos(d*x+c)^2*a^2-10*a^2+14*b^2)+e^4/a^6*((1/4*a^2*b^2-1/4*b^4)
*(e*sin(d*x+c))^(3/2)/(-a^2*e^2*cos(d*x+c)^2+b^2*e^2)+1/4*(a^4-15/4*a^2*b^2+11/4*b^4)/a^2/(e^2*(a^2-b^2)/a^2)^
(1/4)*(2*arctan((e*sin(d*x+c))^(1/2)/(e^2*(a^2-b^2)/a^2)^(1/4))-ln(((e*sin(d*x+c))^(1/2)+(e^2*(a^2-b^2)/a^2)^(
1/4))/((e*sin(d*x+c))^(1/2)-(e^2*(a^2-b^2)/a^2)^(1/4))))))+(cos(d*x+c)^2*e*sin(d*x+c))^(1/2)*e^5*(-1/45/a^6/(c
os(d*x+c)^2*e*sin(d*x+c))^(1/2)*(10*a^4*cos(d*x+c)^6+42*(-sin(d*x+c)+1)^(1/2)*(2*sin(d*x+c)+2)^(1/2)*sin(d*x+c
)^(1/2)*EllipticE((-sin(d*x+c)+1)^(1/2),1/2*2^(1/2))*a^4-432*(-sin(d*x+c)+1)^(1/2)*(2*sin(d*x+c)+2)^(1/2)*sin(
d*x+c)^(1/2)*EllipticE((-sin(d*x+c)+1)^(1/2),1/2*2^(1/2))*a^2*b^2+450*(-sin(d*x+c)+1)^(1/2)*(2*sin(d*x+c)+2)^(
1/2)*sin(d*x+c)^(1/2)*EllipticE((-sin(d*x+c)+1)^(1/2),1/2*2^(1/2))*b^4-21*(-sin(d*x+c)+1)^(1/2)*(2*sin(d*x+c)+
2)^(1/2)*sin(d*x+c)^(1/2)*EllipticF((-sin(d*x+c)+1)^(1/2),1/2*2^(1/2))*a^4+216*a^2*b^2*(-sin(d*x+c)+1)^(1/2)*(
2*sin(d*x+c)+2)^(1/2)*sin(d*x+c)^(1/2)*EllipticF((-sin(d*x+c)+1)^(1/2),1/2*2^(1/2))-225*b^4*(-sin(d*x+c)+1)^(1
/2)*(2*sin(d*x+c)+2)^(1/2)*sin(d*x+c)^(1/2)*EllipticF((-sin(d*x+c)+1)^(1/2),1/2*2^(1/2))-34*cos(d*x+c)^4*a^4+5
4*cos(d*x+c)^4*a^2*b^2+24*cos(d*x+c)^2*a^4-54*cos(d*x+c)^2*a^2*b^2)+2*b^4*(a^4-2*a^2*b^2+b^4)/a^6*(-1/2*a^2/e/
b^2/(a^2-b^2)*sin(d*x+c)*(cos(d*x+c)^2*e*sin(d*x+c))^(1/2)/(-cos(d*x+c)^2*a^2+b^2)+1/2/b^2/(a^2-b^2)*(-sin(d*x
+c)+1)^(1/2)*(2*sin(d*x+c)+2)^(1/2)*sin(d*x+c)^(1/2)/(cos(d*x+c)^2*e*sin(d*x+c))^(1/2)*EllipticE((-sin(d*x+c)+
1)^(1/2),1/2*2^(1/2))-1/4/b^2/(a^2-b^2)*(-sin(d*x+c)+1)^(1/2)*(2*sin(d*x+c)+2)^(1/2)*sin(d*x+c)^(1/2)/(cos(d*x
+c)^2*e*sin(d*x+c))^(1/2)*EllipticF((-sin(d*x+c)+1)^(1/2),1/2*2^(1/2))-1/4/b^2/(a^2-b^2)*(-sin(d*x+c)+1)^(1/2)
*(2*sin(d*x+c)+2)^(1/2)*sin(d*x+c)^(1/2)/(cos(d*x+c)^2*e*sin(d*x+c))^(1/2)/(1-(a^2-b^2)^(1/2)/a)*EllipticPi((-
sin(d*x+c)+1)^(1/2),1/(1-(a^2-b^2)^(1/2)/a),1/2*2^(1/2))+3/8/(a^2-b^2)/a^2*(-sin(d*x+c)+1)^(1/2)*(2*sin(d*x+c)
+2)^(1/2)*sin(d*x+c)^(1/2)/(cos(d*x+c)^2*e*sin(d*x+c))^(1/2)/(1-(a^2-b^2)^(1/2)/a)*EllipticPi((-sin(d*x+c)+1)^
(1/2),1/(1-(a^2-b^2)^(1/2)/a),1/2*2^(1/2))-1/4/b^2/(a^2-b^2)*(-sin(d*x+c)+1)^(1/2)*(2*sin(d*x+c)+2)^(1/2)*sin(
d*x+c)^(1/2)/(cos(d*x+c)^2*e*sin(d*x+c))^(1/2)/(1+(a^2-b^2)^(1/2)/a)*EllipticPi((-sin(d*x+c)+1)^(1/2),1/(1+(a^
2-b^2)^(1/2)/a),1/2*2^(1/2))+3/8/(a^2-b^2)/a^2*(-sin(d*x+c)+1)^(1/2)*(2*sin(d*x+c)+2)^(1/2)*sin(d*x+c)^(1/2)/(
cos(d*x+c)^2*e*sin(d*x+c))^(1/2)/(1+(a^2-b^2)^(1/2)/a)*EllipticPi((-sin(d*x+c)+1)^(1/2),1/(1+(a^2-b^2)^(1/2)/a
),1/2*2^(1/2)))-b^2*(3*a^4-10*a^2*b^2+7*b^4)/a^6*(-1/2/a^2*(-sin(d*x+c)+1)^(1/2)*(2*sin(d*x+c)+2)^(1/2)*sin(d*
x+c)^(1/2)/(cos(d*x+c)^2*e*sin(d*x+c))^(1/2)/(1-(a^2-b^2)^(1/2)/a)*EllipticPi((-sin(d*x+c)+1)^(1/2),1/(1-(a^2-
b^2)^(1/2)/a),1/2*2^(1/2))-1/2/a^2*(-sin(d*x+c)+1)^(1/2)*(2*sin(d*x+c)+2)^(1/2)*sin(d*x+c)^(1/2)/(cos(d*x+c)^2
*e*sin(d*x+c))^(1/2)/(1+(a^2-b^2)^(1/2)/a)*EllipticPi((-sin(d*x+c)+1)^(1/2),1/(1+(a^2-b^2)^(1/2)/a),1/2*2^(1/2
))))/cos(d*x+c)/(e*sin(d*x+c))^(1/2))/d

Fricas [F(-1)]

Timed out. \[ \int \frac {(e \sin (c+d x))^{9/2}}{(a+b \sec (c+d x))^2} \, dx=\text {Timed out} \]

[In]

integrate((e*sin(d*x+c))^(9/2)/(a+b*sec(d*x+c))^2,x, algorithm="fricas")

[Out]

Timed out

Sympy [F(-1)]

Timed out. \[ \int \frac {(e \sin (c+d x))^{9/2}}{(a+b \sec (c+d x))^2} \, dx=\text {Timed out} \]

[In]

integrate((e*sin(d*x+c))**(9/2)/(a+b*sec(d*x+c))**2,x)

[Out]

Timed out

Maxima [F(-1)]

Timed out. \[ \int \frac {(e \sin (c+d x))^{9/2}}{(a+b \sec (c+d x))^2} \, dx=\text {Timed out} \]

[In]

integrate((e*sin(d*x+c))^(9/2)/(a+b*sec(d*x+c))^2,x, algorithm="maxima")

[Out]

Timed out

Giac [F]

\[ \int \frac {(e \sin (c+d x))^{9/2}}{(a+b \sec (c+d x))^2} \, dx=\int { \frac {\left (e \sin \left (d x + c\right )\right )^{\frac {9}{2}}}{{\left (b \sec \left (d x + c\right ) + a\right )}^{2}} \,d x } \]

[In]

integrate((e*sin(d*x+c))^(9/2)/(a+b*sec(d*x+c))^2,x, algorithm="giac")

[Out]

integrate((e*sin(d*x + c))^(9/2)/(b*sec(d*x + c) + a)^2, x)

Mupad [F(-1)]

Timed out. \[ \int \frac {(e \sin (c+d x))^{9/2}}{(a+b \sec (c+d x))^2} \, dx=\int \frac {{\cos \left (c+d\,x\right )}^2\,{\left (e\,\sin \left (c+d\,x\right )\right )}^{9/2}}{{\left (b+a\,\cos \left (c+d\,x\right )\right )}^2} \,d x \]

[In]

int((e*sin(c + d*x))^(9/2)/(a + b/cos(c + d*x))^2,x)

[Out]

int((cos(c + d*x)^2*(e*sin(c + d*x))^(9/2))/(b + a*cos(c + d*x))^2, x)