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

Optimal. Leaf size=1054 \[ -\frac {5 \Pi \left (\frac {2 a}{a-\sqrt {a^2-b^2}};\left .\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right )\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 \Pi \left (\frac {2 a}{a+\sqrt {a^2-b^2}};\left .\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right )\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 \tan ^{-1}\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 \tanh ^{-1}\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 \Pi \left (\frac {2 a}{a-\sqrt {a^2-b^2}};\left .\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right )\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 \Pi \left (\frac {2 a}{a+\sqrt {a^2-b^2}};\left .\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right )\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 \tan ^{-1}\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 \tanh ^{-1}\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)}} \]

[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]  time = 2.69, antiderivative size = 1054, normalized size of antiderivative = 1.00, number of steps used = 33, number of rules used = 16, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.640, Rules used = {3872, 2912, 2636, 2640, 2639, 2694, 2866, 2867, 2701, 2807, 2805, 329, 298, 205, 208, 2696} \[ -\frac {5 \Pi \left (\frac {2 a}{a-\sqrt {a^2-b^2}};\left .\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right )\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 \Pi \left (\frac {2 a}{a+\sqrt {a^2-b^2}};\left .\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right )\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 \tan ^{-1}\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 \tanh ^{-1}\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 \Pi \left (\frac {2 a}{a-\sqrt {a^2-b^2}};\left .\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right )\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 \Pi \left (\frac {2 a}{a+\sqrt {a^2-b^2}};\left .\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right )\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 \tan ^{-1}\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 \tanh ^{-1}\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)}} \]

Antiderivative was successfully verified.

[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 205

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

Rule 208

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

Rule 298

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 329

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 2636

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 2639

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

Rule 2640

Int[Sqrt[(b_)*sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Dist[Sqrt[b*Sin[c + d*x]]/Sqrt[Sin[c + d*x]], Int[Sqrt[Si
n[c + d*x]], x], x] /; FreeQ[{b, c, d}, x]

Rule 2694

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] /; F
reeQ[{a, b, e, f, g, p}, x] && NeQ[a^2 - b^2, 0] && LtQ[m, -1] && IntegersQ[2*m, 2*p]

Rule 2696

Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_), x_Symbol] :> Simp[((g*Co
s[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 2701

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 2805

Int[1/(((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])*Sqrt[(c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)]]), x_Symbol] :> Simp
[(2*EllipticPi[(2*b)/(a + b), (1*(e - Pi/2 + f*x))/2, (2*d)/(c + d)])/(f*(a + b)*Sqrt[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 2807

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*
Sin[e + f*x])/(c + d)]), 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 2866

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 2867

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 2912

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 3872

Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_.)*(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_))^(m_.), x_Symbol] :> Int[((g*C
os[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*} \int \frac {1}{(a+b \sec (c+d x))^2 (e \sin (c+d x))^{3/2}} \, dx &=\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) \operatorname {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 ) \operatorname {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) \operatorname {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 \Pi \left (\frac {2 a}{a-\sqrt {a^2-b^2}};\left .\frac {1}{2} \left (c-\frac {\pi }{2}+d x\right )\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 \Pi \left (\frac {2 a}{a+\sqrt {a^2-b^2}};\left .\frac {1}{2} \left (c-\frac {\pi }{2}+d x\right )\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 ) \operatorname {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) \operatorname {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) \operatorname {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 \tan ^{-1}\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 \tanh ^{-1}\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 \Pi \left (\frac {2 a}{a-\sqrt {a^2-b^2}};\left .\frac {1}{2} \left (c-\frac {\pi }{2}+d x\right )\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 \Pi \left (\frac {2 a}{a-\sqrt {a^2-b^2}};\left .\frac {1}{2} \left (c-\frac {\pi }{2}+d x\right )\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 \Pi \left (\frac {2 a}{a+\sqrt {a^2-b^2}};\left .\frac {1}{2} \left (c-\frac {\pi }{2}+d x\right )\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 \Pi \left (\frac {2 a}{a+\sqrt {a^2-b^2}};\left .\frac {1}{2} \left (c-\frac {\pi }{2}+d x\right )\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 ) \operatorname {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 ) \operatorname {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 \tan ^{-1}\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 \tan ^{-1}\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 \tanh ^{-1}\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 \tanh ^{-1}\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 \Pi \left (\frac {2 a}{a-\sqrt {a^2-b^2}};\left .\frac {1}{2} \left (c-\frac {\pi }{2}+d x\right )\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 \Pi \left (\frac {2 a}{a-\sqrt {a^2-b^2}};\left .\frac {1}{2} \left (c-\frac {\pi }{2}+d x\right )\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 \Pi \left (\frac {2 a}{a+\sqrt {a^2-b^2}};\left .\frac {1}{2} \left (c-\frac {\pi }{2}+d x\right )\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 \Pi \left (\frac {2 a}{a+\sqrt {a^2-b^2}};\left .\frac {1}{2} \left (c-\frac {\pi }{2}+d x\right )\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*}

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Mathematica [C]  time = 6.93, size = 922, normalized size = 0.87 \[ \frac {(b+a \cos (c+d x))^2 \left (\frac {a b^2 \sin (c+d x)}{\left (b^2-a^2\right )^2 (b+a \cos (c+d x))}-\frac {2 \left (\cos (c+d x) a^2-2 b a+b^2 \cos (c+d x)\right ) \csc (c+d x)}{\left (b^2-a^2\right )^2}\right ) \tan ^2(c+d x)}{d (a+b \sec (c+d x))^2 (e \sin (c+d x))^{3/2}}-\frac {(b+a \cos (c+d x))^2 \sec ^2(c+d x) \sin ^{\frac {3}{2}}(c+d x) \left (\frac {\left (2 a^3+3 b^2 a\right ) \left (8 F_1\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) a^{5/2}+3 \sqrt {2} b \left (b^2-a^2\right )^{3/4} \left (2 \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt {a} \sqrt {\sin (c+d x)}}{\sqrt [4]{b^2-a^2}}\right )-2 \tan ^{-1}\left (\frac {\sqrt {2} \sqrt {a} \sqrt {\sin (c+d x)}}{\sqrt [4]{b^2-a^2}}+1\right )-\log \left (a \sin (c+d x)-\sqrt {2} \sqrt {a} \sqrt [4]{b^2-a^2} \sqrt {\sin (c+d x)}+\sqrt {b^2-a^2}\right )+\log \left (a \sin (c+d x)+\sqrt {2} \sqrt {a} \sqrt [4]{b^2-a^2} \sqrt {\sin (c+d x)}+\sqrt {b^2-a^2}\right )\right )\right ) \left (\sqrt {1-\sin ^2(c+d x)} a+b\right ) \cos ^2(c+d x)}{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 (4 b^3+6 a^2 b\right ) \left (\frac {b F_1\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 (b^2-a^2\right )}+\frac {\left (\frac {1}{8}+\frac {i}{8}\right ) \left (2 \tan ^{-1}\left (1-\frac {(1+i) \sqrt {a} \sqrt {\sin (c+d x)}}{\sqrt [4]{a^2-b^2}}\right )-2 \tan ^{-1}\left (\frac {(1+i) \sqrt {a} \sqrt {\sin (c+d x)}}{\sqrt [4]{a^2-b^2}}+1\right )-\log \left (i a \sin (c+d x)-(1+i) \sqrt {a} \sqrt [4]{a^2-b^2} \sqrt {\sin (c+d x)}+\sqrt {a^2-b^2}\right )+\log \left (i a \sin (c+d x)+(1+i) \sqrt {a} \sqrt [4]{a^2-b^2} \sqrt {\sin (c+d x)}+\sqrt {a^2-b^2}\right )\right )}{\sqrt {a} \sqrt [4]{a^2-b^2}}\right ) \left (\sqrt {1-\sin ^2(c+d x)} a+b\right ) \cos (c+d x)}{(b+a \cos (c+d x)) \sqrt {1-\sin ^2(c+d x)}}\right )}{2 (a-b)^2 (a+b)^2 d (a+b \sec (c+d x))^2 (e \sin (c+d x))^{3/2}} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

-1/2*((b + a*Cos[c + d*x])^2*Sec[c + d*x]^2*Sin[c + d*x]^(3/2)*(((2*a^3 + 3*a*b^2)*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*ArcTan[1 + (Sqr
t[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*(6*a^2*b + 4*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]] + 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]]))/(Sqrt[a]*(a^2 - b^2)^(1/4)) + (b*A
ppellF1[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])))/((a - b)^2*(a + b)^2*d
*(a + b*Sec[c + d*x])^2*(e*Sin[c + d*x])^(3/2)) + ((b + a*Cos[c + d*x])^2*((-2*(-2*a*b + a^2*Cos[c + d*x] + b^
2*Cos[c + d*x])*Csc[c + d*x])/(-a^2 + b^2)^2 + (a*b^2*Sin[c + d*x])/((-a^2 + b^2)^2*(b + a*Cos[c + d*x])))*Tan
[c + d*x]^2)/(d*(a + b*Sec[c + d*x])^2*(e*Sin[c + d*x])^(3/2))

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fricas [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{{\left (b \sec \left (d x + c\right ) + a\right )}^{2} \left (e \sin \left (d x + c\right )\right )^{\frac {3}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[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)

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maple [A]  time = 17.21, size = 2263, normalized size = 2.15 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

4/d/e*a*b/(a^2-b^2)^2/(e*sin(d*x+c))^(1/2)+1/d*a/e*b^3/(a-b)^2/(a+b)^2*(e*sin(d*x+c))^(3/2)/(-a^2*cos(d*x+c)^2
*e^2+b^2*e^2)+2/d*a/e*b/(a-b)^2/(a+b)^2/(e^2*(a^2-b^2)/a^2)^(1/4)*arctan((e*sin(d*x+c))^(1/2)/(e^2*(a^2-b^2)/a
^2)^(1/4))-1/d*a/e*b/(a-b)^2/(a+b)^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)))+1/2/d/a/e*b^3/(a-b)^2/(a+b)^2/(e^2*(a^2-b^2)/a^2)^(1/4)
*arctan((e*sin(d*x+c))^(1/2)/(e^2*(a^2-b^2)/a^2)^(1/4))-1/4/d/a/e*b^3/(a-b)^2/(a+b)^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)))-1/d/e*
sin(d*x+c)^2*cos(d*x+c)/(e*sin(d*x+c))^(1/2)*b^2/(a-b)/(a+b)*a^2/(a^2-b^2)/(-cos(d*x+c)^2*a^2+b^2)+1/d/e/cos(d
*x+c)/(e*sin(d*x+c))^(1/2)*b^2/(a-b)/(a+b)/(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)*EllipticE((-sin(d*x+c)+1)^(1/2),1/2*2^(1/2))-1/2/d/e/cos(d*x+c)/(e*sin(d*x+c))^(1/2)*b^2/(a-b)/(a+b)/(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
))-1/2/d/e/cos(d*x+c)/(e*sin(d*x+c))^(1/2)*b^2/(a-b)/(a+b)/(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)/(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
))+3/4/d/e/cos(d*x+c)/(e*sin(d*x+c))^(1/2)*b^4/(a-b)/(a+b)/(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)/(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/d/e/cos(d*x+c)/(e*sin(d*x+c))^(1/2)*b^2/(a-b)/(a+b)/(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)/(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))+3/4/d/e/cos(d*x+c)/(e*sin(d*x+c))^(1/2)*b^4/(a-b)/(a+b)/(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)/(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))+3/2/d/e/cos(d*x+c)/(e*sin(d*x+c))^(1/2)*b^2/(a-b)^2/(a+b)^2*(-sin(d*x+c)+1)^(1/2)*(2*sin(d*x+c)+2)^
(1/2)*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/d/e/cos(d*x+c)/(e*sin(d*x+c))^(1/2)*b^4/(a-b)^2/(a+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)/(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
))+3/2/d/e/cos(d*x+c)/(e*sin(d*x+c))^(1/2)*b^2/(a-b)^2/(a+b)^2*(-sin(d*x+c)+1)^(1/2)*(2*sin(d*x+c)+2)^(1/2)*si
n(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
/d/e/cos(d*x+c)/(e*sin(d*x+c))^(1/2)*b^4/(a-b)^2/(a+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)/(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))+2/d/e
/cos(d*x+c)/(e*sin(d*x+c))^(1/2)/(a^2-b^2)^2*(-sin(d*x+c)+1)^(1/2)*(2*sin(d*x+c)+2)^(1/2)*sin(d*x+c)^(1/2)*Ell
ipticE((-sin(d*x+c)+1)^(1/2),1/2*2^(1/2))*a^2+2/d/e/cos(d*x+c)/(e*sin(d*x+c))^(1/2)/(a^2-b^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))*b^2-1/d/e/cos(d*x
+c)/(e*sin(d*x+c))^(1/2)/(a^2-b^2)^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))*a^2-1/d/e/cos(d*x+c)/(e*sin(d*x+c))^(1/2)/(a^2-b^2)^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))*b^2-2/d/e*cos(d*x+c)/(e*s
in(d*x+c))^(1/2)/(a^2-b^2)^2*a^2-2/d/e*cos(d*x+c)/(e*sin(d*x+c))^(1/2)/(a^2-b^2)^2*b^2

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maxima [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

[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)

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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