\(\int \frac {1}{(a+b \sec (c+d x))^2 (e \sin (c+d x))^{3/2}} \, dx\) [247]

Optimal result

Integrand size = 25, antiderivative size = 1054 \[ \int \frac {1}{(a+b \sec (c+d x))^2 (e \sin (c+d x))^{3/2}} \, dx=\frac {5 b^3 \arctan \left (\frac {\sqrt {a} \sqrt {e \sin (c+d x)}}{\sqrt [4]{a^2-b^2} \sqrt {e}}\right )}{2 \sqrt {a} \left (a^2-b^2\right )^{9/4} d e^{3/2}}+\frac {2 b \arctan \left (\frac {\sqrt {a} \sqrt {e \sin (c+d x)}}{\sqrt [4]{a^2-b^2} \sqrt {e}}\right )}{\sqrt {a} \left (a^2-b^2\right )^{5/4} d e^{3/2}}-\frac {5 b^3 \text {arctanh}\left (\frac {\sqrt {a} \sqrt {e \sin (c+d x)}}{\sqrt [4]{a^2-b^2} \sqrt {e}}\right )}{2 \sqrt {a} \left (a^2-b^2\right )^{9/4} d e^{3/2}}-\frac {2 b \text {arctanh}\left (\frac {\sqrt {a} \sqrt {e \sin (c+d x)}}{\sqrt [4]{a^2-b^2} \sqrt {e}}\right )}{\sqrt {a} \left (a^2-b^2\right )^{5/4} d e^{3/2}}-\frac {2 \cos (c+d x)}{a^2 d e \sqrt {e \sin (c+d x)}}+\frac {b^2}{a \left (a^2-b^2\right ) d e (b+a \cos (c+d x)) \sqrt {e \sin (c+d x)}}+\frac {4 b (a-b \cos (c+d x))}{a^2 \left (a^2-b^2\right ) d e \sqrt {e \sin (c+d x)}}+\frac {b^2 \left (5 a b-\left (3 a^2+2 b^2\right ) \cos (c+d x)\right )}{a^2 \left (a^2-b^2\right )^2 d e \sqrt {e \sin (c+d x)}}-\frac {5 b^4 \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 \left (a^2-b^2\right )^2 \left (a-\sqrt {a^2-b^2}\right ) d e \sqrt {e \sin (c+d x)}}-\frac {2 b^2 \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 \left (a^2-b^2\right ) \left (a-\sqrt {a^2-b^2}\right ) d e \sqrt {e \sin (c+d x)}}-\frac {5 b^4 \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 \left (a^2-b^2\right )^2 \left (a+\sqrt {a^2-b^2}\right ) d e \sqrt {e \sin (c+d x)}}-\frac {2 b^2 \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 \left (a^2-b^2\right ) \left (a+\sqrt {a^2-b^2}\right ) d e \sqrt {e \sin (c+d x)}}-\frac {2 E\left (\left .\frac {1}{2} \left (c-\frac {\pi }{2}+d x\right )\right |2\right ) \sqrt {e \sin (c+d x)}}{a^2 d e^2 \sqrt {\sin (c+d x)}}-\frac {4 b^2 E\left (\left .\frac {1}{2} \left (c-\frac {\pi }{2}+d x\right )\right |2\right ) \sqrt {e \sin (c+d x)}}{a^2 \left (a^2-b^2\right ) d e^2 \sqrt {\sin (c+d x)}}-\frac {b^2 \left (3 a^2+2 b^2\right ) E\left (\left .\frac {1}{2} \left (c-\frac {\pi }{2}+d x\right )\right |2\right ) \sqrt {e \sin (c+d x)}}{a^2 \left (a^2-b^2\right )^2 d e^2 \sqrt {\sin (c+d x)}} \]

[Out]

5/2*b^3*arctan(a^(1/2)*(e*sin(d*x+c))^(1/2)/(a^2-b^2)^(1/4)/e^(1/2))/(a^2-b^2)^(9/4)/d/e^(3/2)/a^(1/2)+2*b*arc
tan(a^(1/2)*(e*sin(d*x+c))^(1/2)/(a^2-b^2)^(1/4)/e^(1/2))/(a^2-b^2)^(5/4)/d/e^(3/2)/a^(1/2)-5/2*b^3*arctanh(a^
(1/2)*(e*sin(d*x+c))^(1/2)/(a^2-b^2)^(1/4)/e^(1/2))/(a^2-b^2)^(9/4)/d/e^(3/2)/a^(1/2)-2*b*arctanh(a^(1/2)*(e*s
in(d*x+c))^(1/2)/(a^2-b^2)^(1/4)/e^(1/2))/(a^2-b^2)^(5/4)/d/e^(3/2)/a^(1/2)-2*cos(d*x+c)/a^2/d/e/(e*sin(d*x+c)
)^(1/2)+b^2/a/(a^2-b^2)/d/e/(b+a*cos(d*x+c))/(e*sin(d*x+c))^(1/2)+4*b*(a-b*cos(d*x+c))/a^2/(a^2-b^2)/d/e/(e*si
n(d*x+c))^(1/2)+b^2*(5*a*b-(3*a^2+2*b^2)*cos(d*x+c))/a^2/(a^2-b^2)^2/d/e/(e*sin(d*x+c))^(1/2)+5/2*b^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)*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/(a^2-b^2)^2/d/e/(a-(a^2-b^2)^(1/2))/(e*sin(d*x+c))^(1/2)+2*b^2*(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/(a^2-b^2)/d/e/(a-(a^2-b^2)^(1/2))/(e*sin(d*x+c))^(1/2)+5/2*b^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)*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/(a^2-b^2)^2/d/e/(a+(a^2-b^2)^(1/2))/(e*sin(d*x+c))^(1/2)+2*b^2*(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/(a^2-b^2)/d/e/(a+(a^2-b^2)^(1/2))/(e*sin(d*x+c))^(1/2)+2*(sin(1/2*c+1/4*Pi+1/2*d*x)^2)^(1/2)/si
n(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/e^2/sin(d*x+c)
^(1/2)+4*b^2*(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/(a^2-b^2)/d/e^2/sin(d*x+c)^(1/2)+b^2*(3*a^2+2*b^2)*(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/(a
^2-b^2)^2/d/e^2/sin(d*x+c)^(1/2)

Rubi [A] (verified)

Time = 3.29 (sec) , antiderivative size = 1054, normalized size of antiderivative = 1.00, number of steps used = 33, number of rules used = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.640, Rules used = {3957, 2991, 2716, 2721, 2719, 2773, 2945, 2946, 2780, 2886, 2884, 335, 304, 211, 214, 2775} \[ \int \frac {1}{(a+b \sec (c+d x))^2 (e \sin (c+d x))^{3/2}} \, dx=-\frac {5 \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)} b^4}{2 a \left (a^2-b^2\right )^2 \left (a-\sqrt {a^2-b^2}\right ) d e \sqrt {e \sin (c+d x)}}-\frac {5 \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)} b^4}{2 a \left (a^2-b^2\right )^2 \left (a+\sqrt {a^2-b^2}\right ) d e \sqrt {e \sin (c+d x)}}+\frac {5 \arctan \left (\frac {\sqrt {a} \sqrt {e \sin (c+d x)}}{\sqrt [4]{a^2-b^2} \sqrt {e}}\right ) b^3}{2 \sqrt {a} \left (a^2-b^2\right )^{9/4} d e^{3/2}}-\frac {5 \text {arctanh}\left (\frac {\sqrt {a} \sqrt {e \sin (c+d x)}}{\sqrt [4]{a^2-b^2} \sqrt {e}}\right ) b^3}{2 \sqrt {a} \left (a^2-b^2\right )^{9/4} d e^{3/2}}-\frac {\left (3 a^2+2 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)} b^2}{a^2 \left (a^2-b^2\right )^2 d e^2 \sqrt {\sin (c+d x)}}-\frac {4 E\left (\left .\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right )\right |2\right ) \sqrt {e \sin (c+d x)} b^2}{a^2 \left (a^2-b^2\right ) d e^2 \sqrt {\sin (c+d x)}}+\frac {\left (5 a b-\left (3 a^2+2 b^2\right ) \cos (c+d x)\right ) b^2}{a^2 \left (a^2-b^2\right )^2 d e \sqrt {e \sin (c+d x)}}-\frac {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)} b^2}{a \left (a^2-b^2\right ) \left (a-\sqrt {a^2-b^2}\right ) d e \sqrt {e \sin (c+d x)}}-\frac {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)} b^2}{a \left (a^2-b^2\right ) \left (a+\sqrt {a^2-b^2}\right ) d e \sqrt {e \sin (c+d x)}}+\frac {b^2}{a \left (a^2-b^2\right ) d e (b+a \cos (c+d x)) \sqrt {e \sin (c+d x)}}+\frac {2 \arctan \left (\frac {\sqrt {a} \sqrt {e \sin (c+d x)}}{\sqrt [4]{a^2-b^2} \sqrt {e}}\right ) b}{\sqrt {a} \left (a^2-b^2\right )^{5/4} d e^{3/2}}-\frac {2 \text {arctanh}\left (\frac {\sqrt {a} \sqrt {e \sin (c+d x)}}{\sqrt [4]{a^2-b^2} \sqrt {e}}\right ) b}{\sqrt {a} \left (a^2-b^2\right )^{5/4} d e^{3/2}}+\frac {4 (a-b \cos (c+d x)) b}{a^2 \left (a^2-b^2\right ) d e \sqrt {e \sin (c+d x)}}-\frac {2 E\left (\left .\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right )\right |2\right ) \sqrt {e \sin (c+d x)}}{a^2 d e^2 \sqrt {\sin (c+d x)}}-\frac {2 \cos (c+d x)}{a^2 d e \sqrt {e \sin (c+d x)}} \]

[In]

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

[Out]

(5*b^3*ArcTan[(Sqrt[a]*Sqrt[e*Sin[c + d*x]])/((a^2 - b^2)^(1/4)*Sqrt[e])])/(2*Sqrt[a]*(a^2 - b^2)^(9/4)*d*e^(3
/2)) + (2*b*ArcTan[(Sqrt[a]*Sqrt[e*Sin[c + d*x]])/((a^2 - b^2)^(1/4)*Sqrt[e])])/(Sqrt[a]*(a^2 - b^2)^(5/4)*d*e
^(3/2)) - (5*b^3*ArcTanh[(Sqrt[a]*Sqrt[e*Sin[c + d*x]])/((a^2 - b^2)^(1/4)*Sqrt[e])])/(2*Sqrt[a]*(a^2 - b^2)^(
9/4)*d*e^(3/2)) - (2*b*ArcTanh[(Sqrt[a]*Sqrt[e*Sin[c + d*x]])/((a^2 - b^2)^(1/4)*Sqrt[e])])/(Sqrt[a]*(a^2 - b^
2)^(5/4)*d*e^(3/2)) - (2*Cos[c + d*x])/(a^2*d*e*Sqrt[e*Sin[c + d*x]]) + b^2/(a*(a^2 - b^2)*d*e*(b + a*Cos[c +
d*x])*Sqrt[e*Sin[c + d*x]]) + (4*b*(a - b*Cos[c + d*x]))/(a^2*(a^2 - b^2)*d*e*Sqrt[e*Sin[c + d*x]]) + (b^2*(5*
a*b - (3*a^2 + 2*b^2)*Cos[c + d*x]))/(a^2*(a^2 - b^2)^2*d*e*Sqrt[e*Sin[c + d*x]]) - (5*b^4*EllipticPi[(2*a)/(a
- Sqrt[a^2 - b^2]), (c - Pi/2 + d*x)/2, 2]*Sqrt[Sin[c + d*x]])/(2*a*(a^2 - b^2)^2*(a - Sqrt[a^2 - b^2])*d*e*S
qrt[e*Sin[c + d*x]]) - (2*b^2*EllipticPi[(2*a)/(a - Sqrt[a^2 - b^2]), (c - Pi/2 + d*x)/2, 2]*Sqrt[Sin[c + d*x]
])/(a*(a^2 - b^2)*(a - Sqrt[a^2 - b^2])*d*e*Sqrt[e*Sin[c + d*x]]) - (5*b^4*EllipticPi[(2*a)/(a + Sqrt[a^2 - b^
2]), (c - Pi/2 + d*x)/2, 2]*Sqrt[Sin[c + d*x]])/(2*a*(a^2 - b^2)^2*(a + Sqrt[a^2 - b^2])*d*e*Sqrt[e*Sin[c + d*
x]]) - (2*b^2*EllipticPi[(2*a)/(a + Sqrt[a^2 - b^2]), (c - Pi/2 + d*x)/2, 2]*Sqrt[Sin[c + d*x]])/(a*(a^2 - b^2
)*(a + Sqrt[a^2 - b^2])*d*e*Sqrt[e*Sin[c + d*x]]) - (2*EllipticE[(c - Pi/2 + d*x)/2, 2]*Sqrt[e*Sin[c + d*x]])/
(a^2*d*e^2*Sqrt[Sin[c + d*x]]) - (4*b^2*EllipticE[(c - Pi/2 + d*x)/2, 2]*Sqrt[e*Sin[c + d*x]])/(a^2*(a^2 - b^2
)*d*e^2*Sqrt[Sin[c + d*x]]) - (b^2*(3*a^2 + 2*b^2)*EllipticE[(c - Pi/2 + d*x)/2, 2]*Sqrt[e*Sin[c + d*x]])/(a^2
*(a^2 - b^2)^2*d*e^2*Sqrt[Sin[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 2716

Int[((b_.)*sin[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[Cos[c + d*x]*((b*Sin[c + d*x])^(n + 1)/(b*d*(n + 1
))), x] + Dist[(n + 2)/(b^2*(n + 1)), Int[(b*Sin[c + d*x])^(n + 2), x], x] /; FreeQ[{b, c, d}, x] && LtQ[n, -1
] && IntegerQ[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 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 2775

Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_), x_Symbol] :> Simp[(g*Cos
[e + f*x])^(p + 1)*(a + b*Sin[e + f*x])^(m + 1)*((b - a*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*(a^2*(p + 2) - b^2*(m + p + 2)
+ a*b*(m + p + 3)*Sin[e + f*x]), x], x] /; FreeQ[{a, b, e, f, g, m}, x] && NeQ[a^2 - b^2, 0] && LtQ[p, -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]

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)}{(-b-a \cos (c+d x))^2 (e \sin (c+d x))^{3/2}} \, dx \\ & = \int \left (\frac {1}{a^2 (e \sin (c+d x))^{3/2}}+\frac {b^2}{a^2 (-b-a \cos (c+d x))^2 (e \sin (c+d x))^{3/2}}+\frac {2 b}{a^2 (-b-a \cos (c+d x)) (e \sin (c+d x))^{3/2}}\right ) \, dx \\ & = \frac {\int \frac {1}{(e \sin (c+d x))^{3/2}} \, dx}{a^2}+\frac {(2 b) \int \frac {1}{(-b-a \cos (c+d x)) (e \sin (c+d x))^{3/2}} \, dx}{a^2}+\frac {b^2 \int \frac {1}{(-b-a \cos (c+d x))^2 (e \sin (c+d x))^{3/2}} \, dx}{a^2} \\ & = -\frac {2 \cos (c+d x)}{a^2 d e \sqrt {e \sin (c+d x)}}+\frac {b^2}{a \left (a^2-b^2\right ) d e (b+a \cos (c+d x)) \sqrt {e \sin (c+d x)}}+\frac {4 b (a-b \cos (c+d x))}{a^2 \left (a^2-b^2\right ) d e \sqrt {e \sin (c+d x)}}+\frac {b^2 \int \frac {b-\frac {3}{2} a \cos (c+d x)}{(-b-a \cos (c+d x)) (e \sin (c+d x))^{3/2}} \, dx}{a^2 \left (a^2-b^2\right )}-\frac {\int \sqrt {e \sin (c+d x)} \, dx}{a^2 e^2}+\frac {(4 b) \int \frac {\left (\frac {a^2}{2}+\frac {b^2}{2}+\frac {1}{2} a b \cos (c+d x)\right ) \sqrt {e \sin (c+d x)}}{-b-a \cos (c+d x)} \, dx}{a^2 \left (a^2-b^2\right ) e^2} \\ & = -\frac {2 \cos (c+d x)}{a^2 d e \sqrt {e \sin (c+d x)}}+\frac {b^2}{a \left (a^2-b^2\right ) d e (b+a \cos (c+d x)) \sqrt {e \sin (c+d x)}}+\frac {4 b (a-b \cos (c+d x))}{a^2 \left (a^2-b^2\right ) d e \sqrt {e \sin (c+d x)}}+\frac {b^2 \left (5 a b-\left (3 a^2+2 b^2\right ) \cos (c+d x)\right )}{a^2 \left (a^2-b^2\right )^2 d e \sqrt {e \sin (c+d x)}}+\frac {\left (2 b^2\right ) \int \frac {\left (\frac {1}{2} b \left (4 a^2+b^2\right )+\frac {1}{4} a \left (3 a^2+2 b^2\right ) \cos (c+d x)\right ) \sqrt {e \sin (c+d x)}}{-b-a \cos (c+d x)} \, dx}{a^2 \left (a^2-b^2\right )^2 e^2}+\frac {(2 b) \int \frac {\sqrt {e \sin (c+d x)}}{-b-a \cos (c+d x)} \, dx}{\left (a^2-b^2\right ) e^2}-\frac {\left (2 b^2\right ) \int \sqrt {e \sin (c+d x)} \, dx}{a^2 \left (a^2-b^2\right ) e^2}-\frac {\sqrt {e \sin (c+d x)} \int \sqrt {\sin (c+d x)} \, dx}{a^2 e^2 \sqrt {\sin (c+d x)}} \\ & = -\frac {2 \cos (c+d x)}{a^2 d e \sqrt {e \sin (c+d x)}}+\frac {b^2}{a \left (a^2-b^2\right ) d e (b+a \cos (c+d x)) \sqrt {e \sin (c+d x)}}+\frac {4 b (a-b \cos (c+d x))}{a^2 \left (a^2-b^2\right ) d e \sqrt {e \sin (c+d x)}}+\frac {b^2 \left (5 a b-\left (3 a^2+2 b^2\right ) \cos (c+d x)\right )}{a^2 \left (a^2-b^2\right )^2 d e \sqrt {e \sin (c+d x)}}-\frac {2 E\left (\left .\frac {1}{2} \left (c-\frac {\pi }{2}+d x\right )\right |2\right ) \sqrt {e \sin (c+d x)}}{a^2 d e^2 \sqrt {\sin (c+d x)}}+\frac {\left (5 b^3\right ) \int \frac {\sqrt {e \sin (c+d x)}}{-b-a \cos (c+d x)} \, dx}{2 \left (a^2-b^2\right )^2 e^2}-\frac {\left (b^2 \left (3 a^2+2 b^2\right )\right ) \int \sqrt {e \sin (c+d x)} \, dx}{2 a^2 \left (a^2-b^2\right )^2 e^2}+\frac {b^2 \int \frac {1}{\sqrt {e \sin (c+d x)} \left (\sqrt {a^2-b^2}-a \sin (c+d x)\right )} \, dx}{a \left (a^2-b^2\right ) e}-\frac {b^2 \int \frac {1}{\sqrt {e \sin (c+d x)} \left (\sqrt {a^2-b^2}+a \sin (c+d x)\right )} \, dx}{a \left (a^2-b^2\right ) e}+\frac {(2 a b) \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 )}{\left (a^2-b^2\right ) d e}-\frac {\left (2 b^2 \sqrt {e \sin (c+d x)}\right ) \int \sqrt {\sin (c+d x)} \, dx}{a^2 \left (a^2-b^2\right ) e^2 \sqrt {\sin (c+d x)}} \\ & = -\frac {2 \cos (c+d x)}{a^2 d e \sqrt {e \sin (c+d x)}}+\frac {b^2}{a \left (a^2-b^2\right ) d e (b+a \cos (c+d x)) \sqrt {e \sin (c+d x)}}+\frac {4 b (a-b \cos (c+d x))}{a^2 \left (a^2-b^2\right ) d e \sqrt {e \sin (c+d x)}}+\frac {b^2 \left (5 a b-\left (3 a^2+2 b^2\right ) \cos (c+d x)\right )}{a^2 \left (a^2-b^2\right )^2 d e \sqrt {e \sin (c+d x)}}-\frac {2 E\left (\left .\frac {1}{2} \left (c-\frac {\pi }{2}+d x\right )\right |2\right ) \sqrt {e \sin (c+d x)}}{a^2 d e^2 \sqrt {\sin (c+d x)}}-\frac {4 b^2 E\left (\left .\frac {1}{2} \left (c-\frac {\pi }{2}+d x\right )\right |2\right ) \sqrt {e \sin (c+d x)}}{a^2 \left (a^2-b^2\right ) d e^2 \sqrt {\sin (c+d x)}}+\frac {\left (5 b^4\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 \left (a^2-b^2\right )^2 e}-\frac {\left (5 b^4\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 \left (a^2-b^2\right )^2 e}+\frac {\left (5 a b^3\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 \left (a^2-b^2\right )^2 d e}+\frac {(4 a b) \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 )}{\left (a^2-b^2\right ) d e}+\frac {\left (b^2 \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 \left (a^2-b^2\right ) e \sqrt {e \sin (c+d x)}}-\frac {\left (b^2 \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 \left (a^2-b^2\right ) e \sqrt {e \sin (c+d x)}}-\frac {\left (b^2 \left (3 a^2+2 b^2\right ) \sqrt {e \sin (c+d x)}\right ) \int \sqrt {\sin (c+d x)} \, dx}{2 a^2 \left (a^2-b^2\right )^2 e^2 \sqrt {\sin (c+d x)}} \\ & = -\frac {2 \cos (c+d x)}{a^2 d e \sqrt {e \sin (c+d x)}}+\frac {b^2}{a \left (a^2-b^2\right ) d e (b+a \cos (c+d x)) \sqrt {e \sin (c+d x)}}+\frac {4 b (a-b \cos (c+d x))}{a^2 \left (a^2-b^2\right ) d e \sqrt {e \sin (c+d x)}}+\frac {b^2 \left (5 a b-\left (3 a^2+2 b^2\right ) \cos (c+d x)\right )}{a^2 \left (a^2-b^2\right )^2 d e \sqrt {e \sin (c+d x)}}-\frac {2 b^2 \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 \left (a^2-b^2\right ) \left (a-\sqrt {a^2-b^2}\right ) d e \sqrt {e \sin (c+d x)}}-\frac {2 b^2 \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 \left (a^2-b^2\right ) \left (a+\sqrt {a^2-b^2}\right ) d e \sqrt {e \sin (c+d x)}}-\frac {2 E\left (\left .\frac {1}{2} \left (c-\frac {\pi }{2}+d x\right )\right |2\right ) \sqrt {e \sin (c+d x)}}{a^2 d e^2 \sqrt {\sin (c+d x)}}-\frac {4 b^2 E\left (\left .\frac {1}{2} \left (c-\frac {\pi }{2}+d x\right )\right |2\right ) \sqrt {e \sin (c+d x)}}{a^2 \left (a^2-b^2\right ) d e^2 \sqrt {\sin (c+d x)}}-\frac {b^2 \left (3 a^2+2 b^2\right ) E\left (\left .\frac {1}{2} \left (c-\frac {\pi }{2}+d x\right )\right |2\right ) \sqrt {e \sin (c+d x)}}{a^2 \left (a^2-b^2\right )^2 d e^2 \sqrt {\sin (c+d x)}}+\frac {\left (5 a b^3\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 )}{\left (a^2-b^2\right )^2 d e}-\frac {(2 b) \text {Subst}\left (\int \frac {1}{\sqrt {a^2-b^2} e-a x^2} \, dx,x,\sqrt {e \sin (c+d x)}\right )}{\left (a^2-b^2\right ) d e}+\frac {(2 b) \text {Subst}\left (\int \frac {1}{\sqrt {a^2-b^2} e+a x^2} \, dx,x,\sqrt {e \sin (c+d x)}\right )}{\left (a^2-b^2\right ) d e}+\frac {\left (5 b^4 \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 \left (a^2-b^2\right )^2 e \sqrt {e \sin (c+d x)}}-\frac {\left (5 b^4 \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 \left (a^2-b^2\right )^2 e \sqrt {e \sin (c+d x)}} \\ & = \frac {2 b \arctan \left (\frac {\sqrt {a} \sqrt {e \sin (c+d x)}}{\sqrt [4]{a^2-b^2} \sqrt {e}}\right )}{\sqrt {a} \left (a^2-b^2\right )^{5/4} d e^{3/2}}-\frac {2 b \text {arctanh}\left (\frac {\sqrt {a} \sqrt {e \sin (c+d x)}}{\sqrt [4]{a^2-b^2} \sqrt {e}}\right )}{\sqrt {a} \left (a^2-b^2\right )^{5/4} d e^{3/2}}-\frac {2 \cos (c+d x)}{a^2 d e \sqrt {e \sin (c+d x)}}+\frac {b^2}{a \left (a^2-b^2\right ) d e (b+a \cos (c+d x)) \sqrt {e \sin (c+d x)}}+\frac {4 b (a-b \cos (c+d x))}{a^2 \left (a^2-b^2\right ) d e \sqrt {e \sin (c+d x)}}+\frac {b^2 \left (5 a b-\left (3 a^2+2 b^2\right ) \cos (c+d x)\right )}{a^2 \left (a^2-b^2\right )^2 d e \sqrt {e \sin (c+d x)}}-\frac {5 b^4 \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 \left (a^2-b^2\right )^2 \left (a-\sqrt {a^2-b^2}\right ) d e \sqrt {e \sin (c+d x)}}-\frac {2 b^2 \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 \left (a^2-b^2\right ) \left (a-\sqrt {a^2-b^2}\right ) d e \sqrt {e \sin (c+d x)}}-\frac {5 b^4 \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 \left (a^2-b^2\right )^2 \left (a+\sqrt {a^2-b^2}\right ) d e \sqrt {e \sin (c+d x)}}-\frac {2 b^2 \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 \left (a^2-b^2\right ) \left (a+\sqrt {a^2-b^2}\right ) d e \sqrt {e \sin (c+d x)}}-\frac {2 E\left (\left .\frac {1}{2} \left (c-\frac {\pi }{2}+d x\right )\right |2\right ) \sqrt {e \sin (c+d x)}}{a^2 d e^2 \sqrt {\sin (c+d x)}}-\frac {4 b^2 E\left (\left .\frac {1}{2} \left (c-\frac {\pi }{2}+d x\right )\right |2\right ) \sqrt {e \sin (c+d x)}}{a^2 \left (a^2-b^2\right ) d e^2 \sqrt {\sin (c+d x)}}-\frac {b^2 \left (3 a^2+2 b^2\right ) E\left (\left .\frac {1}{2} \left (c-\frac {\pi }{2}+d x\right )\right |2\right ) \sqrt {e \sin (c+d x)}}{a^2 \left (a^2-b^2\right )^2 d e^2 \sqrt {\sin (c+d x)}}-\frac {\left (5 b^3\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 \left (a^2-b^2\right )^2 d e}+\frac {\left (5 b^3\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 \left (a^2-b^2\right )^2 d e} \\ & = \frac {5 b^3 \arctan \left (\frac {\sqrt {a} \sqrt {e \sin (c+d x)}}{\sqrt [4]{a^2-b^2} \sqrt {e}}\right )}{2 \sqrt {a} \left (a^2-b^2\right )^{9/4} d e^{3/2}}+\frac {2 b \arctan \left (\frac {\sqrt {a} \sqrt {e \sin (c+d x)}}{\sqrt [4]{a^2-b^2} \sqrt {e}}\right )}{\sqrt {a} \left (a^2-b^2\right )^{5/4} d e^{3/2}}-\frac {5 b^3 \text {arctanh}\left (\frac {\sqrt {a} \sqrt {e \sin (c+d x)}}{\sqrt [4]{a^2-b^2} \sqrt {e}}\right )}{2 \sqrt {a} \left (a^2-b^2\right )^{9/4} d e^{3/2}}-\frac {2 b \text {arctanh}\left (\frac {\sqrt {a} \sqrt {e \sin (c+d x)}}{\sqrt [4]{a^2-b^2} \sqrt {e}}\right )}{\sqrt {a} \left (a^2-b^2\right )^{5/4} d e^{3/2}}-\frac {2 \cos (c+d x)}{a^2 d e \sqrt {e \sin (c+d x)}}+\frac {b^2}{a \left (a^2-b^2\right ) d e (b+a \cos (c+d x)) \sqrt {e \sin (c+d x)}}+\frac {4 b (a-b \cos (c+d x))}{a^2 \left (a^2-b^2\right ) d e \sqrt {e \sin (c+d x)}}+\frac {b^2 \left (5 a b-\left (3 a^2+2 b^2\right ) \cos (c+d x)\right )}{a^2 \left (a^2-b^2\right )^2 d e \sqrt {e \sin (c+d x)}}-\frac {5 b^4 \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 \left (a^2-b^2\right )^2 \left (a-\sqrt {a^2-b^2}\right ) d e \sqrt {e \sin (c+d x)}}-\frac {2 b^2 \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 \left (a^2-b^2\right ) \left (a-\sqrt {a^2-b^2}\right ) d e \sqrt {e \sin (c+d x)}}-\frac {5 b^4 \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 \left (a^2-b^2\right )^2 \left (a+\sqrt {a^2-b^2}\right ) d e \sqrt {e \sin (c+d x)}}-\frac {2 b^2 \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 \left (a^2-b^2\right ) \left (a+\sqrt {a^2-b^2}\right ) d e \sqrt {e \sin (c+d x)}}-\frac {2 E\left (\left .\frac {1}{2} \left (c-\frac {\pi }{2}+d x\right )\right |2\right ) \sqrt {e \sin (c+d x)}}{a^2 d e^2 \sqrt {\sin (c+d x)}}-\frac {4 b^2 E\left (\left .\frac {1}{2} \left (c-\frac {\pi }{2}+d x\right )\right |2\right ) \sqrt {e \sin (c+d x)}}{a^2 \left (a^2-b^2\right ) d e^2 \sqrt {\sin (c+d x)}}-\frac {b^2 \left (3 a^2+2 b^2\right ) E\left (\left .\frac {1}{2} \left (c-\frac {\pi }{2}+d x\right )\right |2\right ) \sqrt {e \sin (c+d x)}}{a^2 \left (a^2-b^2\right )^2 d e^2 \sqrt {\sin (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 = 7.23 (sec) , antiderivative size = 772, normalized size of antiderivative = 0.73 \[ \int \frac {1}{(a+b \sec (c+d x))^2 (e \sin (c+d x))^{3/2}} \, dx=\frac {(b+a \cos (c+d x)) \left (-\frac {\left (a^2-b^2\right ) \left (b+a \sqrt {\cos ^2(c+d x)}\right ) \sec ^3(c+d x) \sin ^{\frac {3}{2}}(c+d x) \left (\left (2 a^3+3 a b^2\right ) \cos (c+d x) \left (3 \sqrt {2} b \left (-a^2+b^2\right )^{3/4} \left (2 \arctan \left (1-\frac {\sqrt {2} \sqrt {a} \sqrt {\sin (c+d x)}}{\sqrt [4]{-a^2+b^2}}\right )-2 \arctan \left (1+\frac {\sqrt {2} \sqrt {a} \sqrt {\sin (c+d x)}}{\sqrt [4]{-a^2+b^2}}\right )-\log \left (\sqrt {-a^2+b^2}-\sqrt {2} \sqrt {a} \sqrt [4]{-a^2+b^2} \sqrt {\sin (c+d x)}+a \sin (c+d x)\right )+\log \left (\sqrt {-a^2+b^2}+\sqrt {2} \sqrt {a} \sqrt [4]{-a^2+b^2} \sqrt {\sin (c+d x)}+a \sin (c+d x)\right )\right )+8 a^{5/2} \operatorname {AppellF1}\left (\frac {3}{4},-\frac {1}{2},1,\frac {7}{4},\sin ^2(c+d x),\frac {a^2 \sin ^2(c+d x)}{a^2-b^2}\right ) \sin ^{\frac {3}{2}}(c+d x)\right )+(1+i) a \left (6 a^2 b+4 b^3\right ) \sqrt {\cos ^2(c+d x)} \left (3 \left (a^2-b^2\right )^{3/4} \left (2 \arctan \left (1-\frac {(1+i) \sqrt {a} \sqrt {\sin (c+d x)}}{\sqrt [4]{a^2-b^2}}\right )-2 \arctan \left (1+\frac {(1+i) \sqrt {a} \sqrt {\sin (c+d x)}}{\sqrt [4]{a^2-b^2}}\right )-\log \left (\sqrt {a^2-b^2}-(1+i) \sqrt {a} \sqrt [4]{a^2-b^2} \sqrt {\sin (c+d x)}+i a \sin (c+d x)\right )+\log \left (\sqrt {a^2-b^2}+(1+i) \sqrt {a} \sqrt [4]{a^2-b^2} \sqrt {\sin (c+d x)}+i a \sin (c+d x)\right )\right )-(4-4 i) \sqrt {a} b \operatorname {AppellF1}\left (\frac {3}{4},\frac {1}{2},1,\frac {7}{4},\sin ^2(c+d x),\frac {a^2 \sin ^2(c+d x)}{a^2-b^2}\right ) \sin ^{\frac {3}{2}}(c+d x)\right )\right )}{a^{3/2} (a-b)^2 (a+b)^2}+24 \left (-2 (b+a \cos (c+d x)) \left (-2 a b+\left (a^2+b^2\right ) \cos (c+d x)\right ) \csc (c+d x)+a b^2 \sin (c+d x)\right ) \tan ^2(c+d x)\right )}{24 \left (a^2-b^2\right )^2 d (a+b \sec (c+d x))^2 (e \sin (c+d x))^{3/2}} \]

[In]

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

[Out]

((b + a*Cos[c + d*x])*(-(((a^2 - b^2)*(b + a*Sqrt[Cos[c + d*x]^2])*Sec[c + d*x]^3*Sin[c + d*x]^(3/2)*((2*a^3 +
3*a*b^2)*Cos[c + d*x]*(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*ArcTan[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]*Sq
rt[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)) + (1 + I)*a*(6*a^2*b + 4*b^3)*Sqrt[Cos[c + d*x]
^2]*(3*(a^2 - b^2)^(3/4)*(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]] + 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]]) - (4 - 4*I)*Sqrt[a]*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))))/(a^(3/2)*(a - b)^2*(a + b)^2)) + 24*(-2*(b + a*Cos[c + d*x])*(-2*a*b
+ (a^2 + b^2)*Cos[c + d*x])*Csc[c + d*x] + a*b^2*Sin[c + d*x])*Tan[c + d*x]^2))/(24*(a^2 - b^2)^2*d*(a + b*Se
c[c + d*x])^2*(e*Sin[c + d*x])^(3/2))

Maple [A] (warning: unable to verify)

Time = 22.48 (sec) , antiderivative size = 1417, normalized size of antiderivative = 1.34

method	result	size
default	\(\text {Expression too large to display}\)	\(1417\)

[In]

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

[Out]

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

[In]

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

[Out]

Timed out

Sympy [F(-1)]

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

[In]

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

[Out]

Timed out

Maxima [F(-1)]

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

[In]

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

[Out]

Timed out

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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