3.248 \(\int \frac {1}{(a+b \sec (c+d x))^2 (e \sin (c+d x))^{5/2}} \, dx\)

Optimal. Leaf size=1089 \[ \frac {7 \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 \left (a^2-b^2\right )^2 \left (a^2-\sqrt {a^2-b^2} a-b^2\right ) d e^2 \sqrt {e \sin (c+d x)}}+\frac {7 \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 \left (a^2-b^2\right )^2 \left (a^2+\sqrt {a^2-b^2} a-b^2\right ) d e^2 \sqrt {e \sin (c+d x)}}-\frac {7 \sqrt {a} \tan ^{-1}\left (\frac {\sqrt {a} \sqrt {e \sin (c+d x)}}{\sqrt [4]{a^2-b^2} \sqrt {e}}\right ) b^3}{2 \left (a^2-b^2\right )^{11/4} d e^{5/2}}-\frac {7 \sqrt {a} \tanh ^{-1}\left (\frac {\sqrt {a} \sqrt {e \sin (c+d x)}}{\sqrt [4]{a^2-b^2} \sqrt {e}}\right ) b^3}{2 \left (a^2-b^2\right )^{11/4} d e^{5/2}}+\frac {\left (5 a^2+2 b^2\right ) F\left (\left .\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right )\right |2\right ) \sqrt {\sin (c+d x)} b^2}{3 a^2 \left (a^2-b^2\right )^2 d e^2 \sqrt {e \sin (c+d x)}}+\frac {4 F\left (\left .\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right )\right |2\right ) \sqrt {\sin (c+d x)} b^2}{3 a^2 \left (a^2-b^2\right ) d e^2 \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}{\left (a^2-b^2\right ) \left (a^2-\sqrt {a^2-b^2} a-b^2\right ) d e^2 \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}{\left (a^2-b^2\right ) \left (a^2+\sqrt {a^2-b^2} a-b^2\right ) d e^2 \sqrt {e \sin (c+d x)}}+\frac {\left (7 a b-\left (5 a^2+2 b^2\right ) \cos (c+d x)\right ) b^2}{3 a^2 \left (a^2-b^2\right )^2 d e (e \sin (c+d x))^{3/2}}+\frac {b^2}{a \left (a^2-b^2\right ) d e (b+a \cos (c+d x)) (e \sin (c+d x))^{3/2}}-\frac {2 \sqrt {a} \tan ^{-1}\left (\frac {\sqrt {a} \sqrt {e \sin (c+d x)}}{\sqrt [4]{a^2-b^2} \sqrt {e}}\right ) b}{\left (a^2-b^2\right )^{7/4} d e^{5/2}}-\frac {2 \sqrt {a} \tanh ^{-1}\left (\frac {\sqrt {a} \sqrt {e \sin (c+d x)}}{\sqrt [4]{a^2-b^2} \sqrt {e}}\right ) b}{\left (a^2-b^2\right )^{7/4} d e^{5/2}}+\frac {4 (a-b \cos (c+d x)) b}{3 a^2 \left (a^2-b^2\right ) d e (e \sin (c+d x))^{3/2}}+\frac {2 F\left (\left .\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right )\right |2\right ) \sqrt {\sin (c+d x)}}{3 a^2 d e^2 \sqrt {e \sin (c+d x)}}-\frac {2 \cos (c+d x)}{3 a^2 d e (e \sin (c+d x))^{3/2}} \]

[Out]

-2/3*cos(d*x+c)/a^2/d/e/(e*sin(d*x+c))^(3/2)+b^2/a/(a^2-b^2)/d/e/(b+a*cos(d*x+c))/(e*sin(d*x+c))^(3/2)+4/3*b*(
a-b*cos(d*x+c))/a^2/(a^2-b^2)/d/e/(e*sin(d*x+c))^(3/2)+1/3*b^2*(7*a*b-(5*a^2+2*b^2)*cos(d*x+c))/a^2/(a^2-b^2)^
2/d/e/(e*sin(d*x+c))^(3/2)-7/2*b^3*arctan(a^(1/2)*(e*sin(d*x+c))^(1/2)/(a^2-b^2)^(1/4)/e^(1/2))*a^(1/2)/(a^2-b
^2)^(11/4)/d/e^(5/2)-2*b*arctan(a^(1/2)*(e*sin(d*x+c))^(1/2)/(a^2-b^2)^(1/4)/e^(1/2))*a^(1/2)/(a^2-b^2)^(7/4)/
d/e^(5/2)-7/2*b^3*arctanh(a^(1/2)*(e*sin(d*x+c))^(1/2)/(a^2-b^2)^(1/4)/e^(1/2))*a^(1/2)/(a^2-b^2)^(11/4)/d/e^(
5/2)-2*b*arctanh(a^(1/2)*(e*sin(d*x+c))^(1/2)/(a^2-b^2)^(1/4)/e^(1/2))*a^(1/2)/(a^2-b^2)^(7/4)/d/e^(5/2)-2/3*(
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)*EllipticF(cos(1/2*c+1/4*Pi+1/2*d*x),2^(1/2))*sin(
d*x+c)^(1/2)/a^2/d/e^2/(e*sin(d*x+c))^(1/2)-4/3*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)*EllipticF(cos(1/2*c+1/4*Pi+1/2*d*x),2^(1/2))*sin(d*x+c)^(1/2)/a^2/(a^2-b^2)/d/e^2/(e*sin(d*x+c))^(1/2)-1/3
*b^2*(5*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)*EllipticF(cos(1/2*c+1/4*Pi+1/
2*d*x),2^(1/2))*sin(d*x+c)^(1/2)/a^2/(a^2-b^2)^2/d/e^2/(e*sin(d*x+c))^(1/2)-7/2*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^2-b^2)^2/d/e^2/(a^2-b^2-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))*si
n(d*x+c)^(1/2)/(a^2-b^2)/d/e^2/(a^2-b^2-a*(a^2-b^2)^(1/2))/(e*sin(d*x+c))^(1/2)-7/2*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^2-b^2)^2/d/e^2/(a^2-b^2+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^2-b^2)/d/e^2/(a^2-b^2+a*(a^2-b^2)^(1/2))/(e*sin(d*x+c))^(1/2)

________________________________________________________________________________________

Rubi [A]  time = 2.78, antiderivative size = 1089, 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, 2642, 2641, 2694, 2866, 2867, 2702, 2807, 2805, 329, 212, 208, 205, 2696} \[ \frac {7 \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 \left (a^2-b^2\right )^2 \left (a^2-\sqrt {a^2-b^2} a-b^2\right ) d e^2 \sqrt {e \sin (c+d x)}}+\frac {7 \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 \left (a^2-b^2\right )^2 \left (a^2+\sqrt {a^2-b^2} a-b^2\right ) d e^2 \sqrt {e \sin (c+d x)}}-\frac {7 \sqrt {a} \tan ^{-1}\left (\frac {\sqrt {a} \sqrt {e \sin (c+d x)}}{\sqrt [4]{a^2-b^2} \sqrt {e}}\right ) b^3}{2 \left (a^2-b^2\right )^{11/4} d e^{5/2}}-\frac {7 \sqrt {a} \tanh ^{-1}\left (\frac {\sqrt {a} \sqrt {e \sin (c+d x)}}{\sqrt [4]{a^2-b^2} \sqrt {e}}\right ) b^3}{2 \left (a^2-b^2\right )^{11/4} d e^{5/2}}+\frac {\left (5 a^2+2 b^2\right ) F\left (\left .\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right )\right |2\right ) \sqrt {\sin (c+d x)} b^2}{3 a^2 \left (a^2-b^2\right )^2 d e^2 \sqrt {e \sin (c+d x)}}+\frac {4 F\left (\left .\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right )\right |2\right ) \sqrt {\sin (c+d x)} b^2}{3 a^2 \left (a^2-b^2\right ) d e^2 \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}{\left (a^2-b^2\right ) \left (a^2-\sqrt {a^2-b^2} a-b^2\right ) d e^2 \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}{\left (a^2-b^2\right ) \left (a^2+\sqrt {a^2-b^2} a-b^2\right ) d e^2 \sqrt {e \sin (c+d x)}}+\frac {\left (7 a b-\left (5 a^2+2 b^2\right ) \cos (c+d x)\right ) b^2}{3 a^2 \left (a^2-b^2\right )^2 d e (e \sin (c+d x))^{3/2}}+\frac {b^2}{a \left (a^2-b^2\right ) d e (b+a \cos (c+d x)) (e \sin (c+d x))^{3/2}}-\frac {2 \sqrt {a} \tan ^{-1}\left (\frac {\sqrt {a} \sqrt {e \sin (c+d x)}}{\sqrt [4]{a^2-b^2} \sqrt {e}}\right ) b}{\left (a^2-b^2\right )^{7/4} d e^{5/2}}-\frac {2 \sqrt {a} \tanh ^{-1}\left (\frac {\sqrt {a} \sqrt {e \sin (c+d x)}}{\sqrt [4]{a^2-b^2} \sqrt {e}}\right ) b}{\left (a^2-b^2\right )^{7/4} d e^{5/2}}+\frac {4 (a-b \cos (c+d x)) b}{3 a^2 \left (a^2-b^2\right ) d e (e \sin (c+d x))^{3/2}}+\frac {2 F\left (\left .\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right )\right |2\right ) \sqrt {\sin (c+d x)}}{3 a^2 d e^2 \sqrt {e \sin (c+d x)}}-\frac {2 \cos (c+d x)}{3 a^2 d e (e \sin (c+d x))^{3/2}} \]

Antiderivative was successfully verified.

[In]

Int[1/((a + b*Sec[c + d*x])^2*(e*Sin[c + d*x])^(5/2)),x]

[Out]

(-7*Sqrt[a]*b^3*ArcTan[(Sqrt[a]*Sqrt[e*Sin[c + d*x]])/((a^2 - b^2)^(1/4)*Sqrt[e])])/(2*(a^2 - b^2)^(11/4)*d*e^
(5/2)) - (2*Sqrt[a]*b*ArcTan[(Sqrt[a]*Sqrt[e*Sin[c + d*x]])/((a^2 - b^2)^(1/4)*Sqrt[e])])/((a^2 - b^2)^(7/4)*d
*e^(5/2)) - (7*Sqrt[a]*b^3*ArcTanh[(Sqrt[a]*Sqrt[e*Sin[c + d*x]])/((a^2 - b^2)^(1/4)*Sqrt[e])])/(2*(a^2 - b^2)
^(11/4)*d*e^(5/2)) - (2*Sqrt[a]*b*ArcTanh[(Sqrt[a]*Sqrt[e*Sin[c + d*x]])/((a^2 - b^2)^(1/4)*Sqrt[e])])/((a^2 -
b^2)^(7/4)*d*e^(5/2)) - (2*Cos[c + d*x])/(3*a^2*d*e*(e*Sin[c + d*x])^(3/2)) + b^2/(a*(a^2 - b^2)*d*e*(b + a*C
os[c + d*x])*(e*Sin[c + d*x])^(3/2)) + (4*b*(a - b*Cos[c + d*x]))/(3*a^2*(a^2 - b^2)*d*e*(e*Sin[c + d*x])^(3/2
)) + (b^2*(7*a*b - (5*a^2 + 2*b^2)*Cos[c + d*x]))/(3*a^2*(a^2 - b^2)^2*d*e*(e*Sin[c + d*x])^(3/2)) + (2*Ellipt
icF[(c - Pi/2 + d*x)/2, 2]*Sqrt[Sin[c + d*x]])/(3*a^2*d*e^2*Sqrt[e*Sin[c + d*x]]) + (4*b^2*EllipticF[(c - Pi/2
+ d*x)/2, 2]*Sqrt[Sin[c + d*x]])/(3*a^2*(a^2 - b^2)*d*e^2*Sqrt[e*Sin[c + d*x]]) + (b^2*(5*a^2 + 2*b^2)*Ellipt
icF[(c - Pi/2 + d*x)/2, 2]*Sqrt[Sin[c + d*x]])/(3*a^2*(a^2 - b^2)^2*d*e^2*Sqrt[e*Sin[c + d*x]]) + (7*b^4*Ellip
ticPi[(2*a)/(a - Sqrt[a^2 - b^2]), (c - Pi/2 + d*x)/2, 2]*Sqrt[Sin[c + d*x]])/(2*(a^2 - b^2)^2*(a^2 - b^2 - a*
Sqrt[a^2 - b^2])*d*e^2*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^2 - b^2)*(a^2 - b^2 - a*Sqrt[a^2 - b^2])*d*e^2*Sqrt[e*Sin[c + d*x]]) + (7*b^4*E
llipticPi[(2*a)/(a + Sqrt[a^2 - b^2]), (c - Pi/2 + d*x)/2, 2]*Sqrt[Sin[c + d*x]])/(2*(a^2 - b^2)^2*(a^2 - b^2
+ a*Sqrt[a^2 - b^2])*d*e^2*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^2 - b^2)*(a^2 - b^2 + a*Sqrt[a^2 - b^2])*d*e^2*Sqrt[e*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 212

Int[((a_) + (b_.)*(x_)^4)^(-1), x_Symbol] :> With[{r = Numerator[Rt[-(a/b), 2]], s = Denominator[Rt[-(a/b), 2]
]}, Dist[r/(2*a), Int[1/(r - s*x^2), x], x] + Dist[r/(2*a), 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 2641

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

Rule 2642

Int[1/Sqrt[(b_)*sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Dist[Sqrt[Sin[c + d*x]]/Sqrt[b*Sin[c + d*x]], Int[1/Sqr
t[Sin[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 2702

Int[1/(Sqrt[cos[(e_.) + (f_.)*(x_)]*(g_.)]*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])), x_Symbol] :> With[{q = Rt[
-a^2 + b^2, 2]}, -Dist[a/(2*q), Int[1/(Sqrt[g*Cos[e + f*x]]*(q + b*Cos[e + f*x])), x], x] + (Dist[(b*g)/f, Sub
st[Int[1/(Sqrt[x]*(g^2*(a^2 - b^2) + b^2*x^2)), x], x, g*Cos[e + f*x]], x] - Dist[a/(2*q), Int[1/(Sqrt[g*Cos[e
+ f*x]]*(q - b*Cos[e + f*x])), 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))^{5/2}} \, dx &=\int \frac {\cos ^2(c+d x)}{(-b-a \cos (c+d x))^2 (e \sin (c+d x))^{5/2}} \, dx\\ &=\int \left (\frac {1}{a^2 (e \sin (c+d x))^{5/2}}+\frac {b^2}{a^2 (-b-a \cos (c+d x))^2 (e \sin (c+d x))^{5/2}}+\frac {2 b}{a^2 (-b-a \cos (c+d x)) (e \sin (c+d x))^{5/2}}\right ) \, dx\\ &=\frac {\int \frac {1}{(e \sin (c+d x))^{5/2}} \, dx}{a^2}+\frac {(2 b) \int \frac {1}{(-b-a \cos (c+d x)) (e \sin (c+d x))^{5/2}} \, dx}{a^2}+\frac {b^2 \int \frac {1}{(-b-a \cos (c+d x))^2 (e \sin (c+d x))^{5/2}} \, dx}{a^2}\\ &=-\frac {2 \cos (c+d x)}{3 a^2 d e (e \sin (c+d x))^{3/2}}+\frac {b^2}{a \left (a^2-b^2\right ) d e (b+a \cos (c+d x)) (e \sin (c+d x))^{3/2}}+\frac {4 b (a-b \cos (c+d x))}{3 a^2 \left (a^2-b^2\right ) d e (e \sin (c+d x))^{3/2}}+\frac {b^2 \int \frac {b-\frac {5}{2} a \cos (c+d x)}{(-b-a \cos (c+d x)) (e \sin (c+d x))^{5/2}} \, dx}{a^2 \left (a^2-b^2\right )}+\frac {\int \frac {1}{\sqrt {e \sin (c+d x)}} \, dx}{3 a^2 e^2}+\frac {(4 b) \int \frac {\frac {3 a^2}{2}-\frac {b^2}{2}-\frac {1}{2} a b \cos (c+d x)}{(-b-a \cos (c+d x)) \sqrt {e \sin (c+d x)}} \, dx}{3 a^2 \left (a^2-b^2\right ) e^2}\\ &=-\frac {2 \cos (c+d x)}{3 a^2 d e (e \sin (c+d x))^{3/2}}+\frac {b^2}{a \left (a^2-b^2\right ) d e (b+a \cos (c+d x)) (e \sin (c+d x))^{3/2}}+\frac {4 b (a-b \cos (c+d x))}{3 a^2 \left (a^2-b^2\right ) d e (e \sin (c+d x))^{3/2}}+\frac {b^2 \left (7 a b-\left (5 a^2+2 b^2\right ) \cos (c+d x)\right )}{3 a^2 \left (a^2-b^2\right )^2 d e (e \sin (c+d x))^{3/2}}+\frac {\left (2 b^2\right ) \int \frac {\frac {1}{2} b \left (8 a^2-b^2\right )-\frac {1}{4} a \left (5 a^2+2 b^2\right ) \cos (c+d x)}{(-b-a \cos (c+d x)) \sqrt {e \sin (c+d x)}} \, dx}{3 a^2 \left (a^2-b^2\right )^2 e^2}+\frac {(2 b) \int \frac {1}{(-b-a \cos (c+d x)) \sqrt {e \sin (c+d x)}} \, dx}{\left (a^2-b^2\right ) e^2}+\frac {\left (2 b^2\right ) \int \frac {1}{\sqrt {e \sin (c+d x)}} \, dx}{3 a^2 \left (a^2-b^2\right ) e^2}+\frac {\sqrt {\sin (c+d x)} \int \frac {1}{\sqrt {\sin (c+d x)}} \, dx}{3 a^2 e^2 \sqrt {e \sin (c+d x)}}\\ &=-\frac {2 \cos (c+d x)}{3 a^2 d e (e \sin (c+d x))^{3/2}}+\frac {b^2}{a \left (a^2-b^2\right ) d e (b+a \cos (c+d x)) (e \sin (c+d x))^{3/2}}+\frac {4 b (a-b \cos (c+d x))}{3 a^2 \left (a^2-b^2\right ) d e (e \sin (c+d x))^{3/2}}+\frac {b^2 \left (7 a b-\left (5 a^2+2 b^2\right ) \cos (c+d x)\right )}{3 a^2 \left (a^2-b^2\right )^2 d e (e \sin (c+d x))^{3/2}}+\frac {2 F\left (\left .\frac {1}{2} \left (c-\frac {\pi }{2}+d x\right )\right |2\right ) \sqrt {\sin (c+d x)}}{3 a^2 d e^2 \sqrt {e \sin (c+d x)}}+\frac {\left (7 b^3\right ) \int \frac {1}{(-b-a \cos (c+d x)) \sqrt {e \sin (c+d x)}} \, dx}{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}{\left (a^2-b^2\right )^{3/2} 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}{\left (a^2-b^2\right )^{3/2} e^2}+\frac {\left (b^2 \left (5 a^2+2 b^2\right )\right ) \int \frac {1}{\sqrt {e \sin (c+d x)}} \, dx}{6 a^2 \left (a^2-b^2\right )^2 e^2}+\frac {(2 a b) \operatorname {Subst}\left (\int \frac {1}{\sqrt {x} \left (\left (-a^2+b^2\right ) e^2+a^2 x^2\right )} \, dx,x,e \sin (c+d x)\right )}{\left (a^2-b^2\right ) d e}+\frac {\left (2 b^2 \sqrt {\sin (c+d x)}\right ) \int \frac {1}{\sqrt {\sin (c+d x)}} \, dx}{3 a^2 \left (a^2-b^2\right ) e^2 \sqrt {e \sin (c+d x)}}\\ &=-\frac {2 \cos (c+d x)}{3 a^2 d e (e \sin (c+d x))^{3/2}}+\frac {b^2}{a \left (a^2-b^2\right ) d e (b+a \cos (c+d x)) (e \sin (c+d x))^{3/2}}+\frac {4 b (a-b \cos (c+d x))}{3 a^2 \left (a^2-b^2\right ) d e (e \sin (c+d x))^{3/2}}+\frac {b^2 \left (7 a b-\left (5 a^2+2 b^2\right ) \cos (c+d x)\right )}{3 a^2 \left (a^2-b^2\right )^2 d e (e \sin (c+d x))^{3/2}}+\frac {2 F\left (\left .\frac {1}{2} \left (c-\frac {\pi }{2}+d x\right )\right |2\right ) \sqrt {\sin (c+d x)}}{3 a^2 d e^2 \sqrt {e \sin (c+d x)}}+\frac {4 b^2 F\left (\left .\frac {1}{2} \left (c-\frac {\pi }{2}+d x\right )\right |2\right ) \sqrt {\sin (c+d x)}}{3 a^2 \left (a^2-b^2\right ) d e^2 \sqrt {e \sin (c+d x)}}+\frac {\left (7 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 \left (a^2-b^2\right )^{5/2} e^2}+\frac {\left (7 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 \left (a^2-b^2\right )^{5/2} e^2}+\frac {\left (7 a b^3\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {x} \left (\left (-a^2+b^2\right ) e^2+a^2 x^2\right )} \, 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 {1}{\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}{\left (a^2-b^2\right )^{3/2} e^2 \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}{\left (a^2-b^2\right )^{3/2} e^2 \sqrt {e \sin (c+d x)}}+\frac {\left (b^2 \left (5 a^2+2 b^2\right ) \sqrt {\sin (c+d x)}\right ) \int \frac {1}{\sqrt {\sin (c+d x)}} \, dx}{6 a^2 \left (a^2-b^2\right )^2 e^2 \sqrt {e \sin (c+d x)}}\\ &=-\frac {2 \cos (c+d x)}{3 a^2 d e (e \sin (c+d x))^{3/2}}+\frac {b^2}{a \left (a^2-b^2\right ) d e (b+a \cos (c+d x)) (e \sin (c+d x))^{3/2}}+\frac {4 b (a-b \cos (c+d x))}{3 a^2 \left (a^2-b^2\right ) d e (e \sin (c+d x))^{3/2}}+\frac {b^2 \left (7 a b-\left (5 a^2+2 b^2\right ) \cos (c+d x)\right )}{3 a^2 \left (a^2-b^2\right )^2 d e (e \sin (c+d x))^{3/2}}+\frac {2 F\left (\left .\frac {1}{2} \left (c-\frac {\pi }{2}+d x\right )\right |2\right ) \sqrt {\sin (c+d x)}}{3 a^2 d e^2 \sqrt {e \sin (c+d x)}}+\frac {4 b^2 F\left (\left .\frac {1}{2} \left (c-\frac {\pi }{2}+d x\right )\right |2\right ) \sqrt {\sin (c+d x)}}{3 a^2 \left (a^2-b^2\right ) d e^2 \sqrt {e \sin (c+d x)}}+\frac {b^2 \left (5 a^2+2 b^2\right ) F\left (\left .\frac {1}{2} \left (c-\frac {\pi }{2}+d x\right )\right |2\right ) \sqrt {\sin (c+d x)}}{3 a^2 \left (a^2-b^2\right )^2 d e^2 \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)}}{\left (a^2-b^2\right )^{3/2} \left (a-\sqrt {a^2-b^2}\right ) d e^2 \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)}}{\left (a^2-b^2\right )^{3/2} \left (a+\sqrt {a^2-b^2}\right ) d e^2 \sqrt {e \sin (c+d x)}}-\frac {(2 a 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 )^{3/2} d e^2}-\frac {(2 a 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 )^{3/2} d e^2}+\frac {\left (7 a b^3\right ) \operatorname {Subst}\left (\int \frac {1}{\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 {\left (7 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 \left (a^2-b^2\right )^{5/2} e^2 \sqrt {e \sin (c+d x)}}+\frac {\left (7 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 \left (a^2-b^2\right )^{5/2} e^2 \sqrt {e \sin (c+d x)}}\\ &=-\frac {2 \sqrt {a} b \tan ^{-1}\left (\frac {\sqrt {a} \sqrt {e \sin (c+d x)}}{\sqrt [4]{a^2-b^2} \sqrt {e}}\right )}{\left (a^2-b^2\right )^{7/4} d e^{5/2}}-\frac {2 \sqrt {a} b \tanh ^{-1}\left (\frac {\sqrt {a} \sqrt {e \sin (c+d x)}}{\sqrt [4]{a^2-b^2} \sqrt {e}}\right )}{\left (a^2-b^2\right )^{7/4} d e^{5/2}}-\frac {2 \cos (c+d x)}{3 a^2 d e (e \sin (c+d x))^{3/2}}+\frac {b^2}{a \left (a^2-b^2\right ) d e (b+a \cos (c+d x)) (e \sin (c+d x))^{3/2}}+\frac {4 b (a-b \cos (c+d x))}{3 a^2 \left (a^2-b^2\right ) d e (e \sin (c+d x))^{3/2}}+\frac {b^2 \left (7 a b-\left (5 a^2+2 b^2\right ) \cos (c+d x)\right )}{3 a^2 \left (a^2-b^2\right )^2 d e (e \sin (c+d x))^{3/2}}+\frac {2 F\left (\left .\frac {1}{2} \left (c-\frac {\pi }{2}+d x\right )\right |2\right ) \sqrt {\sin (c+d x)}}{3 a^2 d e^2 \sqrt {e \sin (c+d x)}}+\frac {4 b^2 F\left (\left .\frac {1}{2} \left (c-\frac {\pi }{2}+d x\right )\right |2\right ) \sqrt {\sin (c+d x)}}{3 a^2 \left (a^2-b^2\right ) d e^2 \sqrt {e \sin (c+d x)}}+\frac {b^2 \left (5 a^2+2 b^2\right ) F\left (\left .\frac {1}{2} \left (c-\frac {\pi }{2}+d x\right )\right |2\right ) \sqrt {\sin (c+d x)}}{3 a^2 \left (a^2-b^2\right )^2 d e^2 \sqrt {e \sin (c+d x)}}-\frac {7 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 \left (a^2-b^2\right )^{5/2} \left (a-\sqrt {a^2-b^2}\right ) d e^2 \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)}}{\left (a^2-b^2\right )^{3/2} \left (a-\sqrt {a^2-b^2}\right ) d e^2 \sqrt {e \sin (c+d x)}}+\frac {7 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 \left (a^2-b^2\right )^{5/2} \left (a+\sqrt {a^2-b^2}\right ) d e^2 \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)}}{\left (a^2-b^2\right )^{3/2} \left (a+\sqrt {a^2-b^2}\right ) d e^2 \sqrt {e \sin (c+d x)}}-\frac {\left (7 a 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 )^{5/2} d e^2}-\frac {\left (7 a 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 )^{5/2} d e^2}\\ &=-\frac {7 \sqrt {a} b^3 \tan ^{-1}\left (\frac {\sqrt {a} \sqrt {e \sin (c+d x)}}{\sqrt [4]{a^2-b^2} \sqrt {e}}\right )}{2 \left (a^2-b^2\right )^{11/4} d e^{5/2}}-\frac {2 \sqrt {a} b \tan ^{-1}\left (\frac {\sqrt {a} \sqrt {e \sin (c+d x)}}{\sqrt [4]{a^2-b^2} \sqrt {e}}\right )}{\left (a^2-b^2\right )^{7/4} d e^{5/2}}-\frac {7 \sqrt {a} b^3 \tanh ^{-1}\left (\frac {\sqrt {a} \sqrt {e \sin (c+d x)}}{\sqrt [4]{a^2-b^2} \sqrt {e}}\right )}{2 \left (a^2-b^2\right )^{11/4} d e^{5/2}}-\frac {2 \sqrt {a} b \tanh ^{-1}\left (\frac {\sqrt {a} \sqrt {e \sin (c+d x)}}{\sqrt [4]{a^2-b^2} \sqrt {e}}\right )}{\left (a^2-b^2\right )^{7/4} d e^{5/2}}-\frac {2 \cos (c+d x)}{3 a^2 d e (e \sin (c+d x))^{3/2}}+\frac {b^2}{a \left (a^2-b^2\right ) d e (b+a \cos (c+d x)) (e \sin (c+d x))^{3/2}}+\frac {4 b (a-b \cos (c+d x))}{3 a^2 \left (a^2-b^2\right ) d e (e \sin (c+d x))^{3/2}}+\frac {b^2 \left (7 a b-\left (5 a^2+2 b^2\right ) \cos (c+d x)\right )}{3 a^2 \left (a^2-b^2\right )^2 d e (e \sin (c+d x))^{3/2}}+\frac {2 F\left (\left .\frac {1}{2} \left (c-\frac {\pi }{2}+d x\right )\right |2\right ) \sqrt {\sin (c+d x)}}{3 a^2 d e^2 \sqrt {e \sin (c+d x)}}+\frac {4 b^2 F\left (\left .\frac {1}{2} \left (c-\frac {\pi }{2}+d x\right )\right |2\right ) \sqrt {\sin (c+d x)}}{3 a^2 \left (a^2-b^2\right ) d e^2 \sqrt {e \sin (c+d x)}}+\frac {b^2 \left (5 a^2+2 b^2\right ) F\left (\left .\frac {1}{2} \left (c-\frac {\pi }{2}+d x\right )\right |2\right ) \sqrt {\sin (c+d x)}}{3 a^2 \left (a^2-b^2\right )^2 d e^2 \sqrt {e \sin (c+d x)}}-\frac {7 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 \left (a^2-b^2\right )^{5/2} \left (a-\sqrt {a^2-b^2}\right ) d e^2 \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)}}{\left (a^2-b^2\right )^{3/2} \left (a-\sqrt {a^2-b^2}\right ) d e^2 \sqrt {e \sin (c+d x)}}+\frac {7 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 \left (a^2-b^2\right )^{5/2} \left (a+\sqrt {a^2-b^2}\right ) d e^2 \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)}}{\left (a^2-b^2\right )^{3/2} \left (a+\sqrt {a^2-b^2}\right ) d e^2 \sqrt {e \sin (c+d x)}}\\ \end {align*}

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Mathematica [C]  time = 15.77, size = 1320, normalized size = 1.21 \[ \frac {(b+a \cos (c+d x))^2 \left (\frac {a b^2}{\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 ^2(c+d x)}{3 \left (b^2-a^2\right )^2}\right ) \sin (c+d x) \tan ^2(c+d x)}{d (a+b \sec (c+d x))^2 (e \sin (c+d x))^{5/2}}-\frac {(b+a \cos (c+d x))^2 \sec ^2(c+d x) \sin ^{\frac {5}{2}}(c+d x) \left (\frac {2 \left (-2 a^3-5 b^2 a\right ) \left (\sqrt {1-\sin ^2(c+d x)} a+b\right ) \left (\frac {b \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 )}{4 \sqrt {2} \sqrt {a} \left (b^2-a^2\right )^{3/4}}-\frac {5 a \left (a^2-b^2\right ) F_1\left (\frac {1}{4};-\frac {1}{2},1;\frac {5}{4};\sin ^2(c+d x),\frac {a^2 \sin ^2(c+d x)}{a^2-b^2}\right ) \sqrt {\sin (c+d x)} \sqrt {1-\sin ^2(c+d x)}}{\left (2 \left (2 F_1\left (\frac {5}{4};-\frac {1}{2},2;\frac {9}{4};\sin ^2(c+d x),\frac {a^2 \sin ^2(c+d x)}{a^2-b^2}\right ) a^2+\left (b^2-a^2\right ) F_1\left (\frac {5}{4};\frac {1}{2},1;\frac {9}{4};\sin ^2(c+d x),\frac {a^2 \sin ^2(c+d x)}{a^2-b^2}\right )\right ) \sin ^2(c+d x)+5 \left (a^2-b^2\right ) F_1\left (\frac {1}{4};-\frac {1}{2},1;\frac {5}{4};\sin ^2(c+d x),\frac {a^2 \sin ^2(c+d x)}{a^2-b^2}\right )\right ) \left (\left (\sin ^2(c+d x)-1\right ) a^2+b^2\right )}\right ) \cos ^2(c+d x)}{(b+a \cos (c+d x)) \left (1-\sin ^2(c+d x)\right )}+\frac {2 \left (4 b^3+10 a^2 b\right ) \left (\sqrt {1-\sin ^2(c+d x)} a+b\right ) \left (\frac {5 b \left (a^2-b^2\right ) F_1\left (\frac {1}{4};\frac {1}{2},1;\frac {5}{4};\sin ^2(c+d x),\frac {a^2 \sin ^2(c+d x)}{a^2-b^2}\right ) \sqrt {\sin (c+d x)}}{\sqrt {1-\sin ^2(c+d x)} \left (2 \left (2 F_1\left (\frac {5}{4};\frac {1}{2},2;\frac {9}{4};\sin ^2(c+d x),\frac {a^2 \sin ^2(c+d x)}{a^2-b^2}\right ) a^2+\left (a^2-b^2\right ) F_1\left (\frac {5}{4};\frac {3}{2},1;\frac {9}{4};\sin ^2(c+d x),\frac {a^2 \sin ^2(c+d x)}{a^2-b^2}\right )\right ) \sin ^2(c+d x)+5 \left (a^2-b^2\right ) F_1\left (\frac {1}{4};\frac {1}{2},1;\frac {5}{4};\sin ^2(c+d x),\frac {a^2 \sin ^2(c+d x)}{a^2-b^2}\right )\right ) \left (\left (\sin ^2(c+d x)-1\right ) a^2+b^2\right )}-\frac {\left (\frac {1}{8}-\frac {i}{8}\right ) \sqrt {a} \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 )}{\left (a^2-b^2\right )^{3/4}}\right ) \cos (c+d x)}{(b+a \cos (c+d x)) \sqrt {1-\sin ^2(c+d x)}}\right )}{6 (a-b)^2 (a+b)^2 d (a+b \sec (c+d x))^2 (e \sin (c+d x))^{5/2}} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[1/((a + b*Sec[c + d*x])^2*(e*Sin[c + d*x])^(5/2)),x]

[Out]

-1/6*((b + a*Cos[c + d*x])^2*Sec[c + d*x]^2*Sin[c + d*x]^(5/2)*((2*(-2*a^3 - 5*a*b^2)*Cos[c + d*x]^2*(b + a*Sq
rt[1 - Sin[c + d*x]^2])*((b*(-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]*Sqrt[a]*(-a^2 + b^2)^(1/4)*S
qrt[Sin[c + d*x]] + a*Sin[c + d*x]]))/(4*Sqrt[2]*Sqrt[a]*(-a^2 + b^2)^(3/4)) - (5*a*(a^2 - b^2)*AppellF1[1/4,
-1/2, 1, 5/4, Sin[c + d*x]^2, (a^2*Sin[c + d*x]^2)/(a^2 - b^2)]*Sqrt[Sin[c + d*x]]*Sqrt[1 - Sin[c + d*x]^2])/(
(5*(a^2 - b^2)*AppellF1[1/4, -1/2, 1, 5/4, Sin[c + d*x]^2, (a^2*Sin[c + d*x]^2)/(a^2 - b^2)] + 2*(2*a^2*Appell
F1[5/4, -1/2, 2, 9/4, Sin[c + d*x]^2, (a^2*Sin[c + d*x]^2)/(a^2 - b^2)] + (-a^2 + b^2)*AppellF1[5/4, 1/2, 1, 9
/4, Sin[c + d*x]^2, (a^2*Sin[c + d*x]^2)/(a^2 - b^2)])*Sin[c + d*x]^2)*(b^2 + a^2*(-1 + Sin[c + d*x]^2)))))/((
b + a*Cos[c + d*x])*(1 - Sin[c + d*x]^2)) + (2*(10*a^2*b + 4*b^3)*Cos[c + d*x]*(b + a*Sqrt[1 - Sin[c + d*x]^2]
)*(((-1/8 + I/8)*Sqrt[a]*(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]]))/(a^2 - b^2)^(3/4) + (5*b*(a^2 - b^2)*AppellF1[1/4, 1/2, 1, 5/4, Sin[c + d*x]^2
, (a^2*Sin[c + d*x]^2)/(a^2 - b^2)]*Sqrt[Sin[c + d*x]])/(Sqrt[1 - Sin[c + d*x]^2]*(5*(a^2 - b^2)*AppellF1[1/4,
1/2, 1, 5/4, Sin[c + d*x]^2, (a^2*Sin[c + d*x]^2)/(a^2 - b^2)] + 2*(2*a^2*AppellF1[5/4, 1/2, 2, 9/4, Sin[c +
d*x]^2, (a^2*Sin[c + d*x]^2)/(a^2 - b^2)] + (a^2 - b^2)*AppellF1[5/4, 3/2, 1, 9/4, Sin[c + d*x]^2, (a^2*Sin[c
+ d*x]^2)/(a^2 - b^2)])*Sin[c + d*x]^2)*(b^2 + a^2*(-1 + Sin[c + d*x]^2)))))/((b + a*Cos[c + d*x])*Sqrt[1 - Si
n[c + d*x]^2])))/((a - b)^2*(a + b)^2*d*(a + b*Sec[c + d*x])^2*(e*Sin[c + d*x])^(5/2)) + ((b + a*Cos[c + d*x])
^2*((a*b^2)/((-a^2 + b^2)^2*(b + a*Cos[c + d*x])) - (2*(-2*a*b + a^2*Cos[c + d*x] + b^2*Cos[c + d*x])*Csc[c +
d*x]^2)/(3*(-a^2 + b^2)^2))*Sin[c + d*x]*Tan[c + d*x]^2)/(d*(a + b*Sec[c + d*x])^2*(e*Sin[c + d*x])^(5/2))

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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))^(5/2),x, algorithm="fricas")

[Out]

Timed out

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{{\left (b \sec \left (d x + c\right ) + a\right )}^{2} \left (e \sin \left (d x + c\right )\right )^{\frac {5}{2}}}\,{d x} \]

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))^(5/2),x, algorithm="giac")

[Out]

integrate(1/((b*sec(d*x + c) + a)^2*(e*sin(d*x + c))^(5/2)), x)

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maple [A]  time = 18.90, size = 2159, normalized size = 1.98 \[ \text {result too large to display} \]

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))^(5/2),x)

[Out]

4/3/d/e*a*b/(a^2-b^2)^2/(e*sin(d*x+c))^(3/2)+1/d*a/e*b^3/(a-b)^2/(a+b)^2*(e*sin(d*x+c))^(1/2)/(-a^2*cos(d*x+c)
^2*e^2+b^2*e^2)+1/d*a^3/e*b/(a-b)^2/(a+b)^2*(e^2*(a^2-b^2)/a^2)^(1/4)/(-a^2*e^2+b^2*e^2)*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)))+3/4/d*a/e*b^3/(a-b)^2/(a+b)^2
*(e^2*(a^2-b^2)/a^2)^(1/4)/(-a^2*e^2+b^2*e^2)*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)))+2/d*a^3/e*b/(a-b)^2/(a+b)^2*(e^2*(a^2-b^2)/a^2)^(1/4)/(-a^2*e^2+b^2*e^2)
*arctan((e*sin(d*x+c))^(1/2)/(e^2*(a^2-b^2)/a^2)^(1/4))+3/2/d*a/e*b^3/(a-b)^2/(a+b)^2*(e^2*(a^2-b^2)/a^2)^(1/4
)/(-a^2*e^2+b^2*e^2)*arctan((e*sin(d*x+c))^(1/2)/(e^2*(a^2-b^2)/a^2)^(1/4))-1/d/e^2*sin(d*x+c)*cos(d*x+c)/(e*s
in(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)-1/2/d/e^2/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^2/cos(d*x+c)/(e*sin(d*x+c))^(1/2)*b^2/(a-b)/(a+b)/(a^2-b^2)^(3/2)*a*(-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))+5/4/d/e^2/cos(d*x+c)/(e*sin(d*x+c))^(1/2)*b^4/(a-b)/(a+b)/(a^2-b^2)^(3/2)/
a*(-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^2/cos(d*x+c)/(e*sin(d*x+c))^(1/2)*b^2/(a-b)/(a+b)/(a^2-b
^2)^(3/2)*a*(-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((-s
in(d*x+c)+1)^(1/2),1/(1+(a^2-b^2)^(1/2)/a),1/2*2^(1/2))-5/4/d/e^2/cos(d*x+c)/(e*sin(d*x+c))^(1/2)*b^4/(a-b)/(a
+b)/(a^2-b^2)^(3/2)/a*(-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)*Elli
pticPi((-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^2/cos(d*x+c)/(e*sin(d*x+c))^(1/2)*b^
2/(a-b)^2/(a+b)^2*a/(a^2-b^2)^(1/2)*(-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^2/cos(d*x+c)/(e*sin(d*
x+c))^(1/2)*b^4/(a-b)^2/(a+b)^2/a/(a^2-b^2)^(1/2)*(-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^2/cos(d*
x+c)/(e*sin(d*x+c))^(1/2)*b^2/(a-b)^2/(a+b)^2*a/(a^2-b^2)^(1/2)*(-sin(d*x+c)+1)^(1/2)*(2*sin(d*x+c)+2)^(1/2)*s
in(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^2/cos(d*x+c)/(e*sin(d*x+c))^(1/2)*b^4/(a-b)^2/(a+b)^2/a/(a^2-b^2)^(1/2)*(-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))+1/3/d/e^2/cos(d*x+c)/(e*sin(d*x+c))^(1/2)/(a^2-b^2)^2/(cos(d*x+c)^2-1)*(-sin(d*x+c)+1)^(1/2)*(2*si
n(d*x+c)+2)^(1/2)*sin(d*x+c)^(5/2)*EllipticF((-sin(d*x+c)+1)^(1/2),1/2*2^(1/2))*a^2+1/3/d/e^2/cos(d*x+c)/(e*si
n(d*x+c))^(1/2)/(a^2-b^2)^2/(cos(d*x+c)^2-1)*(-sin(d*x+c)+1)^(1/2)*(2*sin(d*x+c)+2)^(1/2)*sin(d*x+c)^(5/2)*Ell
ipticF((-sin(d*x+c)+1)^(1/2),1/2*2^(1/2))*b^2+2/3/d/e^2*cos(d*x+c)/(e*sin(d*x+c))^(1/2)/(a^2-b^2)^2/(cos(d*x+c
)^2-1)*sin(d*x+c)*a^2+2/3/d/e^2*cos(d*x+c)/(e*sin(d*x+c))^(1/2)/(a^2-b^2)^2/(cos(d*x+c)^2-1)*sin(d*x+c)*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))^(5/2),x, algorithm="maxima")

[Out]

Timed out

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mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {{\cos \left (c+d\,x\right )}^2}{{\left (e\,\sin \left (c+d\,x\right )\right )}^{5/2}\,{\left (b+a\,\cos \left (c+d\,x\right )\right )}^2} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/((e*sin(c + d*x))^(5/2)*(a + b/cos(c + d*x))^2),x)

[Out]

int(cos(c + d*x)^2/((e*sin(c + d*x))^(5/2)*(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))**(5/2),x)

[Out]

Timed out

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