3.242 \(\int \frac{(e \sin (c+d x))^{7/2}}{(a+b \sec (c+d x))^2} \, dx\)

Optimal. Leaf size=1101 \[ \text{result too large to display} \]

[Out]

(5*b^3*(a^2 - b^2)^(1/4)*e^(7/2)*ArcTan[(Sqrt[a]*Sqrt[e*Sin[c + d*x]])/((a^2 - b^2)^(1/4)*Sqrt[e])])/(2*a^(11/
2)*d) - (2*b*(a^2 - b^2)^(5/4)*e^(7/2)*ArcTan[(Sqrt[a]*Sqrt[e*Sin[c + d*x]])/((a^2 - b^2)^(1/4)*Sqrt[e])])/(a^
(11/2)*d) + (5*b^3*(a^2 - b^2)^(1/4)*e^(7/2)*ArcTanh[(Sqrt[a]*Sqrt[e*Sin[c + d*x]])/((a^2 - b^2)^(1/4)*Sqrt[e]
)])/(2*a^(11/2)*d) - (2*b*(a^2 - b^2)^(5/4)*e^(7/2)*ArcTanh[(Sqrt[a]*Sqrt[e*Sin[c + d*x]])/((a^2 - b^2)^(1/4)*
Sqrt[e])])/(a^(11/2)*d) + (10*e^4*EllipticF[(c - Pi/2 + d*x)/2, 2]*Sqrt[Sin[c + d*x]])/(21*a^2*d*Sqrt[e*Sin[c
+ d*x]]) - (5*b^2*(a^2 - 3*b^2)*e^4*EllipticF[(c - Pi/2 + d*x)/2, 2]*Sqrt[Sin[c + d*x]])/(3*a^6*d*Sqrt[e*Sin[c
+ d*x]]) - (4*b^2*(4*a^2 - 3*b^2)*e^4*EllipticF[(c - Pi/2 + d*x)/2, 2]*Sqrt[Sin[c + d*x]])/(3*a^6*d*Sqrt[e*Si
n[c + d*x]]) - (5*b^4*(a^2 - b^2)*e^4*EllipticPi[(2*a)/(a - Sqrt[a^2 - b^2]), (c - Pi/2 + d*x)/2, 2]*Sqrt[Sin[
c + d*x]])/(2*a^6*(a^2 - b^2 - a*Sqrt[a^2 - b^2])*d*Sqrt[e*Sin[c + d*x]]) + (2*b^2*(a^2 - b^2)^2*e^4*EllipticP
i[(2*a)/(a - Sqrt[a^2 - b^2]), (c - Pi/2 + d*x)/2, 2]*Sqrt[Sin[c + d*x]])/(a^6*(a^2 - b^2 - a*Sqrt[a^2 - b^2])
*d*Sqrt[e*Sin[c + d*x]]) - (5*b^4*(a^2 - b^2)*e^4*EllipticPi[(2*a)/(a + Sqrt[a^2 - b^2]), (c - Pi/2 + d*x)/2,
2]*Sqrt[Sin[c + d*x]])/(2*a^6*(a^2 - b^2 + a*Sqrt[a^2 - b^2])*d*Sqrt[e*Sin[c + d*x]]) + (2*b^2*(a^2 - b^2)^2*e
^4*EllipticPi[(2*a)/(a + Sqrt[a^2 - b^2]), (c - Pi/2 + d*x)/2, 2]*Sqrt[Sin[c + d*x]])/(a^6*(a^2 - b^2 + a*Sqrt
[a^2 - b^2])*d*Sqrt[e*Sin[c + d*x]]) - (10*e^3*Cos[c + d*x]*Sqrt[e*Sin[c + d*x]])/(21*a^2*d) - (5*b^2*e^3*(3*b
- a*Cos[c + d*x])*Sqrt[e*Sin[c + d*x]])/(3*a^5*d) + (4*b*e^3*(3*(a^2 - b^2) + a*b*Cos[c + d*x])*Sqrt[e*Sin[c
+ d*x]])/(3*a^5*d) + (4*b*e*(e*Sin[c + d*x])^(5/2))/(5*a^3*d) - (2*e*Cos[c + d*x]*(e*Sin[c + d*x])^(5/2))/(7*a
^2*d) + (b^2*e*(e*Sin[c + d*x])^(5/2))/(a^3*d*(b + a*Cos[c + d*x]))

________________________________________________________________________________________

Rubi [A]  time = 2.93047, antiderivative size = 1101, normalized size of antiderivative = 1., number of steps used = 35, number of rules used = 16, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.64, Rules used = {3872, 2912, 2635, 2642, 2641, 2693, 2865, 2867, 2702, 2807, 2805, 329, 212, 208, 205, 2695} \[ -\frac{5 b^2 \left (a^2-3 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)} e^4}{3 a^6 d \sqrt{e \sin (c+d x)}}-\frac{4 b^2 \left (4 a^2-3 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)} e^4}{3 a^6 d \sqrt{e \sin (c+d x)}}+\frac{10 F\left (\left .\frac{1}{2} \left (c+d x-\frac{\pi }{2}\right )\right |2\right ) \sqrt{\sin (c+d x)} e^4}{21 a^2 d \sqrt{e \sin (c+d x)}}+\frac{2 b^2 \left (a^2-b^2\right )^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)} e^4}{a^6 \left (a^2-\sqrt{a^2-b^2} a-b^2\right ) d \sqrt{e \sin (c+d x)}}-\frac{5 b^4 \left (a^2-b^2\right ) \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)} e^4}{2 a^6 \left (a^2-\sqrt{a^2-b^2} a-b^2\right ) d \sqrt{e \sin (c+d x)}}+\frac{2 b^2 \left (a^2-b^2\right )^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)} e^4}{a^6 \left (a^2+\sqrt{a^2-b^2} a-b^2\right ) d \sqrt{e \sin (c+d x)}}-\frac{5 b^4 \left (a^2-b^2\right ) \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)} e^4}{2 a^6 \left (a^2+\sqrt{a^2-b^2} a-b^2\right ) d \sqrt{e \sin (c+d x)}}-\frac{2 b \left (a^2-b^2\right )^{5/4} \tan ^{-1}\left (\frac{\sqrt{a} \sqrt{e \sin (c+d x)}}{\sqrt [4]{a^2-b^2} \sqrt{e}}\right ) e^{7/2}}{a^{11/2} d}+\frac{5 b^3 \sqrt [4]{a^2-b^2} \tan ^{-1}\left (\frac{\sqrt{a} \sqrt{e \sin (c+d x)}}{\sqrt [4]{a^2-b^2} \sqrt{e}}\right ) e^{7/2}}{2 a^{11/2} d}-\frac{2 b \left (a^2-b^2\right )^{5/4} \tanh ^{-1}\left (\frac{\sqrt{a} \sqrt{e \sin (c+d x)}}{\sqrt [4]{a^2-b^2} \sqrt{e}}\right ) e^{7/2}}{a^{11/2} d}+\frac{5 b^3 \sqrt [4]{a^2-b^2} \tanh ^{-1}\left (\frac{\sqrt{a} \sqrt{e \sin (c+d x)}}{\sqrt [4]{a^2-b^2} \sqrt{e}}\right ) e^{7/2}}{2 a^{11/2} d}-\frac{10 \cos (c+d x) \sqrt{e \sin (c+d x)} e^3}{21 a^2 d}-\frac{5 b^2 (3 b-a \cos (c+d x)) \sqrt{e \sin (c+d x)} e^3}{3 a^5 d}+\frac{4 b \left (3 \left (a^2-b^2\right )+a b \cos (c+d x)\right ) \sqrt{e \sin (c+d x)} e^3}{3 a^5 d}-\frac{2 \cos (c+d x) (e \sin (c+d x))^{5/2} e}{7 a^2 d}+\frac{4 b (e \sin (c+d x))^{5/2} e}{5 a^3 d}+\frac{b^2 (e \sin (c+d x))^{5/2} e}{a^3 d (b+a \cos (c+d x))} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(5*b^3*(a^2 - b^2)^(1/4)*e^(7/2)*ArcTan[(Sqrt[a]*Sqrt[e*Sin[c + d*x]])/((a^2 - b^2)^(1/4)*Sqrt[e])])/(2*a^(11/
2)*d) - (2*b*(a^2 - b^2)^(5/4)*e^(7/2)*ArcTan[(Sqrt[a]*Sqrt[e*Sin[c + d*x]])/((a^2 - b^2)^(1/4)*Sqrt[e])])/(a^
(11/2)*d) + (5*b^3*(a^2 - b^2)^(1/4)*e^(7/2)*ArcTanh[(Sqrt[a]*Sqrt[e*Sin[c + d*x]])/((a^2 - b^2)^(1/4)*Sqrt[e]
)])/(2*a^(11/2)*d) - (2*b*(a^2 - b^2)^(5/4)*e^(7/2)*ArcTanh[(Sqrt[a]*Sqrt[e*Sin[c + d*x]])/((a^2 - b^2)^(1/4)*
Sqrt[e])])/(a^(11/2)*d) + (10*e^4*EllipticF[(c - Pi/2 + d*x)/2, 2]*Sqrt[Sin[c + d*x]])/(21*a^2*d*Sqrt[e*Sin[c
+ d*x]]) - (5*b^2*(a^2 - 3*b^2)*e^4*EllipticF[(c - Pi/2 + d*x)/2, 2]*Sqrt[Sin[c + d*x]])/(3*a^6*d*Sqrt[e*Sin[c
+ d*x]]) - (4*b^2*(4*a^2 - 3*b^2)*e^4*EllipticF[(c - Pi/2 + d*x)/2, 2]*Sqrt[Sin[c + d*x]])/(3*a^6*d*Sqrt[e*Si
n[c + d*x]]) - (5*b^4*(a^2 - b^2)*e^4*EllipticPi[(2*a)/(a - Sqrt[a^2 - b^2]), (c - Pi/2 + d*x)/2, 2]*Sqrt[Sin[
c + d*x]])/(2*a^6*(a^2 - b^2 - a*Sqrt[a^2 - b^2])*d*Sqrt[e*Sin[c + d*x]]) + (2*b^2*(a^2 - b^2)^2*e^4*EllipticP
i[(2*a)/(a - Sqrt[a^2 - b^2]), (c - Pi/2 + d*x)/2, 2]*Sqrt[Sin[c + d*x]])/(a^6*(a^2 - b^2 - a*Sqrt[a^2 - b^2])
*d*Sqrt[e*Sin[c + d*x]]) - (5*b^4*(a^2 - b^2)*e^4*EllipticPi[(2*a)/(a + Sqrt[a^2 - b^2]), (c - Pi/2 + d*x)/2,
2]*Sqrt[Sin[c + d*x]])/(2*a^6*(a^2 - b^2 + a*Sqrt[a^2 - b^2])*d*Sqrt[e*Sin[c + d*x]]) + (2*b^2*(a^2 - b^2)^2*e
^4*EllipticPi[(2*a)/(a + Sqrt[a^2 - b^2]), (c - Pi/2 + d*x)/2, 2]*Sqrt[Sin[c + d*x]])/(a^6*(a^2 - b^2 + a*Sqrt
[a^2 - b^2])*d*Sqrt[e*Sin[c + d*x]]) - (10*e^3*Cos[c + d*x]*Sqrt[e*Sin[c + d*x]])/(21*a^2*d) - (5*b^2*e^3*(3*b
- a*Cos[c + d*x])*Sqrt[e*Sin[c + d*x]])/(3*a^5*d) + (4*b*e^3*(3*(a^2 - b^2) + a*b*Cos[c + d*x])*Sqrt[e*Sin[c
+ d*x]])/(3*a^5*d) + (4*b*e*(e*Sin[c + d*x])^(5/2))/(5*a^3*d) - (2*e*Cos[c + d*x]*(e*Sin[c + d*x])^(5/2))/(7*a
^2*d) + (b^2*e*(e*Sin[c + d*x])^(5/2))/(a^3*d*(b + a*Cos[c + d*x]))

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]

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 2635

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

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 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 2693

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

Rule 2865

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

Rule 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 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 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 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 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 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 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 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 2695

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

Rubi steps

\begin{align*} \int \frac{(e \sin (c+d x))^{7/2}}{(a+b \sec (c+d x))^2} \, dx &=\int \frac{\cos ^2(c+d x) (e \sin (c+d x))^{7/2}}{(-b-a \cos (c+d x))^2} \, dx\\ &=\int \left (\frac{(e \sin (c+d x))^{7/2}}{a^2}+\frac{b^2 (e \sin (c+d x))^{7/2}}{a^2 (b+a \cos (c+d x))^2}-\frac{2 b (e \sin (c+d x))^{7/2}}{a^2 (b+a \cos (c+d x))}\right ) \, dx\\ &=\frac{\int (e \sin (c+d x))^{7/2} \, dx}{a^2}-\frac{(2 b) \int \frac{(e \sin (c+d x))^{7/2}}{b+a \cos (c+d x)} \, dx}{a^2}+\frac{b^2 \int \frac{(e \sin (c+d x))^{7/2}}{(b+a \cos (c+d x))^2} \, dx}{a^2}\\ &=\frac{4 b e (e \sin (c+d x))^{5/2}}{5 a^3 d}-\frac{2 e \cos (c+d x) (e \sin (c+d x))^{5/2}}{7 a^2 d}+\frac{b^2 e (e \sin (c+d x))^{5/2}}{a^3 d (b+a \cos (c+d x))}+\frac{\left (5 e^2\right ) \int (e \sin (c+d x))^{3/2} \, dx}{7 a^2}+\frac{\left (2 b e^2\right ) \int \frac{(-a-b \cos (c+d x)) (e \sin (c+d x))^{3/2}}{b+a \cos (c+d x)} \, dx}{a^3}-\frac{\left (5 b^2 e^2\right ) \int \frac{\cos (c+d x) (e \sin (c+d x))^{3/2}}{b+a \cos (c+d x)} \, dx}{2 a^3}\\ &=-\frac{10 e^3 \cos (c+d x) \sqrt{e \sin (c+d x)}}{21 a^2 d}-\frac{5 b^2 e^3 (3 b-a \cos (c+d x)) \sqrt{e \sin (c+d x)}}{3 a^5 d}+\frac{4 b e^3 \left (3 \left (a^2-b^2\right )+a b \cos (c+d x)\right ) \sqrt{e \sin (c+d x)}}{3 a^5 d}+\frac{4 b e (e \sin (c+d x))^{5/2}}{5 a^3 d}-\frac{2 e \cos (c+d x) (e \sin (c+d x))^{5/2}}{7 a^2 d}+\frac{b^2 e (e \sin (c+d x))^{5/2}}{a^3 d (b+a \cos (c+d x))}+\frac{\left (5 e^4\right ) \int \frac{1}{\sqrt{e \sin (c+d x)}} \, dx}{21 a^2}+\frac{\left (4 b e^4\right ) \int \frac{-\frac{1}{2} a \left (3 a^2-2 b^2\right )-\frac{1}{2} b \left (4 a^2-3 b^2\right ) \cos (c+d x)}{(b+a \cos (c+d x)) \sqrt{e \sin (c+d x)}} \, dx}{3 a^5}-\frac{\left (5 b^2 e^4\right ) \int \frac{-a b+\frac{1}{2} \left (a^2-3 b^2\right ) \cos (c+d x)}{(b+a \cos (c+d x)) \sqrt{e \sin (c+d x)}} \, dx}{3 a^5}\\ &=-\frac{10 e^3 \cos (c+d x) \sqrt{e \sin (c+d x)}}{21 a^2 d}-\frac{5 b^2 e^3 (3 b-a \cos (c+d x)) \sqrt{e \sin (c+d x)}}{3 a^5 d}+\frac{4 b e^3 \left (3 \left (a^2-b^2\right )+a b \cos (c+d x)\right ) \sqrt{e \sin (c+d x)}}{3 a^5 d}+\frac{4 b e (e \sin (c+d x))^{5/2}}{5 a^3 d}-\frac{2 e \cos (c+d x) (e \sin (c+d x))^{5/2}}{7 a^2 d}+\frac{b^2 e (e \sin (c+d x))^{5/2}}{a^3 d (b+a \cos (c+d x))}-\frac{\left (5 b^2 \left (a^2-3 b^2\right ) e^4\right ) \int \frac{1}{\sqrt{e \sin (c+d x)}} \, dx}{6 a^6}-\frac{\left (2 b^2 \left (4 a^2-3 b^2\right ) e^4\right ) \int \frac{1}{\sqrt{e \sin (c+d x)}} \, dx}{3 a^6}+\frac{\left (5 b^3 \left (a^2-b^2\right ) e^4\right ) \int \frac{1}{(b+a \cos (c+d x)) \sqrt{e \sin (c+d x)}} \, dx}{2 a^6}-\frac{\left (2 b \left (a^2-b^2\right )^2 e^4\right ) \int \frac{1}{(b+a \cos (c+d x)) \sqrt{e \sin (c+d x)}} \, dx}{a^6}+\frac{\left (5 e^4 \sqrt{\sin (c+d x)}\right ) \int \frac{1}{\sqrt{\sin (c+d x)}} \, dx}{21 a^2 \sqrt{e \sin (c+d x)}}\\ &=\frac{10 e^4 F\left (\left .\frac{1}{2} \left (c-\frac{\pi }{2}+d x\right )\right |2\right ) \sqrt{\sin (c+d x)}}{21 a^2 d \sqrt{e \sin (c+d x)}}-\frac{10 e^3 \cos (c+d x) \sqrt{e \sin (c+d x)}}{21 a^2 d}-\frac{5 b^2 e^3 (3 b-a \cos (c+d x)) \sqrt{e \sin (c+d x)}}{3 a^5 d}+\frac{4 b e^3 \left (3 \left (a^2-b^2\right )+a b \cos (c+d x)\right ) \sqrt{e \sin (c+d x)}}{3 a^5 d}+\frac{4 b e (e \sin (c+d x))^{5/2}}{5 a^3 d}-\frac{2 e \cos (c+d x) (e \sin (c+d x))^{5/2}}{7 a^2 d}+\frac{b^2 e (e \sin (c+d x))^{5/2}}{a^3 d (b+a \cos (c+d x))}-\frac{\left (5 b^4 \sqrt{a^2-b^2} e^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^6}-\frac{\left (5 b^4 \sqrt{a^2-b^2} e^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^6}+\frac{\left (b^2 \left (a^2-b^2\right )^{3/2} e^4\right ) \int \frac{1}{\sqrt{e \sin (c+d x)} \left (\sqrt{a^2-b^2}-a \sin (c+d x)\right )} \, dx}{a^6}+\frac{\left (b^2 \left (a^2-b^2\right )^{3/2} e^4\right ) \int \frac{1}{\sqrt{e \sin (c+d x)} \left (\sqrt{a^2-b^2}+a \sin (c+d x)\right )} \, dx}{a^6}-\frac{\left (5 b^3 \left (a^2-b^2\right ) e^5\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 a^5 d}+\frac{\left (2 b \left (a^2-b^2\right )^2 e^5\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 )}{a^5 d}-\frac{\left (5 b^2 \left (a^2-3 b^2\right ) e^4 \sqrt{\sin (c+d x)}\right ) \int \frac{1}{\sqrt{\sin (c+d x)}} \, dx}{6 a^6 \sqrt{e \sin (c+d x)}}-\frac{\left (2 b^2 \left (4 a^2-3 b^2\right ) e^4 \sqrt{\sin (c+d x)}\right ) \int \frac{1}{\sqrt{\sin (c+d x)}} \, dx}{3 a^6 \sqrt{e \sin (c+d x)}}\\ &=\frac{10 e^4 F\left (\left .\frac{1}{2} \left (c-\frac{\pi }{2}+d x\right )\right |2\right ) \sqrt{\sin (c+d x)}}{21 a^2 d \sqrt{e \sin (c+d x)}}-\frac{5 b^2 \left (a^2-3 b^2\right ) e^4 F\left (\left .\frac{1}{2} \left (c-\frac{\pi }{2}+d x\right )\right |2\right ) \sqrt{\sin (c+d x)}}{3 a^6 d \sqrt{e \sin (c+d x)}}-\frac{4 b^2 \left (4 a^2-3 b^2\right ) e^4 F\left (\left .\frac{1}{2} \left (c-\frac{\pi }{2}+d x\right )\right |2\right ) \sqrt{\sin (c+d x)}}{3 a^6 d \sqrt{e \sin (c+d x)}}-\frac{10 e^3 \cos (c+d x) \sqrt{e \sin (c+d x)}}{21 a^2 d}-\frac{5 b^2 e^3 (3 b-a \cos (c+d x)) \sqrt{e \sin (c+d x)}}{3 a^5 d}+\frac{4 b e^3 \left (3 \left (a^2-b^2\right )+a b \cos (c+d x)\right ) \sqrt{e \sin (c+d x)}}{3 a^5 d}+\frac{4 b e (e \sin (c+d x))^{5/2}}{5 a^3 d}-\frac{2 e \cos (c+d x) (e \sin (c+d x))^{5/2}}{7 a^2 d}+\frac{b^2 e (e \sin (c+d x))^{5/2}}{a^3 d (b+a \cos (c+d x))}-\frac{\left (5 b^3 \left (a^2-b^2\right ) e^5\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 )}{a^5 d}+\frac{\left (4 b \left (a^2-b^2\right )^2 e^5\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 )}{a^5 d}-\frac{\left (5 b^4 \sqrt{a^2-b^2} e^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^6 \sqrt{e \sin (c+d x)}}-\frac{\left (5 b^4 \sqrt{a^2-b^2} e^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^6 \sqrt{e \sin (c+d x)}}+\frac{\left (b^2 \left (a^2-b^2\right )^{3/2} e^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}{a^6 \sqrt{e \sin (c+d x)}}+\frac{\left (b^2 \left (a^2-b^2\right )^{3/2} e^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}{a^6 \sqrt{e \sin (c+d x)}}\\ &=\frac{10 e^4 F\left (\left .\frac{1}{2} \left (c-\frac{\pi }{2}+d x\right )\right |2\right ) \sqrt{\sin (c+d x)}}{21 a^2 d \sqrt{e \sin (c+d x)}}-\frac{5 b^2 \left (a^2-3 b^2\right ) e^4 F\left (\left .\frac{1}{2} \left (c-\frac{\pi }{2}+d x\right )\right |2\right ) \sqrt{\sin (c+d x)}}{3 a^6 d \sqrt{e \sin (c+d x)}}-\frac{4 b^2 \left (4 a^2-3 b^2\right ) e^4 F\left (\left .\frac{1}{2} \left (c-\frac{\pi }{2}+d x\right )\right |2\right ) \sqrt{\sin (c+d x)}}{3 a^6 d \sqrt{e \sin (c+d x)}}+\frac{5 b^4 \sqrt{a^2-b^2} e^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^6 \left (a-\sqrt{a^2-b^2}\right ) d \sqrt{e \sin (c+d x)}}-\frac{2 b^2 \left (a^2-b^2\right )^{3/2} e^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)}}{a^6 \left (a-\sqrt{a^2-b^2}\right ) d \sqrt{e \sin (c+d x)}}-\frac{5 b^4 \sqrt{a^2-b^2} e^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^6 \left (a+\sqrt{a^2-b^2}\right ) d \sqrt{e \sin (c+d x)}}+\frac{2 b^2 \left (a^2-b^2\right )^{3/2} e^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)}}{a^6 \left (a+\sqrt{a^2-b^2}\right ) d \sqrt{e \sin (c+d x)}}-\frac{10 e^3 \cos (c+d x) \sqrt{e \sin (c+d x)}}{21 a^2 d}-\frac{5 b^2 e^3 (3 b-a \cos (c+d x)) \sqrt{e \sin (c+d x)}}{3 a^5 d}+\frac{4 b e^3 \left (3 \left (a^2-b^2\right )+a b \cos (c+d x)\right ) \sqrt{e \sin (c+d x)}}{3 a^5 d}+\frac{4 b e (e \sin (c+d x))^{5/2}}{5 a^3 d}-\frac{2 e \cos (c+d x) (e \sin (c+d x))^{5/2}}{7 a^2 d}+\frac{b^2 e (e \sin (c+d x))^{5/2}}{a^3 d (b+a \cos (c+d x))}+\frac{\left (5 b^3 \sqrt{a^2-b^2} e^4\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 a^5 d}+\frac{\left (5 b^3 \sqrt{a^2-b^2} e^4\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 a^5 d}-\frac{\left (2 b \left (a^2-b^2\right )^{3/2} e^4\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 )}{a^5 d}-\frac{\left (2 b \left (a^2-b^2\right )^{3/2} e^4\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 )}{a^5 d}\\ &=\frac{5 b^3 \sqrt [4]{a^2-b^2} e^{7/2} \tan ^{-1}\left (\frac{\sqrt{a} \sqrt{e \sin (c+d x)}}{\sqrt [4]{a^2-b^2} \sqrt{e}}\right )}{2 a^{11/2} d}-\frac{2 b \left (a^2-b^2\right )^{5/4} e^{7/2} \tan ^{-1}\left (\frac{\sqrt{a} \sqrt{e \sin (c+d x)}}{\sqrt [4]{a^2-b^2} \sqrt{e}}\right )}{a^{11/2} d}+\frac{5 b^3 \sqrt [4]{a^2-b^2} e^{7/2} \tanh ^{-1}\left (\frac{\sqrt{a} \sqrt{e \sin (c+d x)}}{\sqrt [4]{a^2-b^2} \sqrt{e}}\right )}{2 a^{11/2} d}-\frac{2 b \left (a^2-b^2\right )^{5/4} e^{7/2} \tanh ^{-1}\left (\frac{\sqrt{a} \sqrt{e \sin (c+d x)}}{\sqrt [4]{a^2-b^2} \sqrt{e}}\right )}{a^{11/2} d}+\frac{10 e^4 F\left (\left .\frac{1}{2} \left (c-\frac{\pi }{2}+d x\right )\right |2\right ) \sqrt{\sin (c+d x)}}{21 a^2 d \sqrt{e \sin (c+d x)}}-\frac{5 b^2 \left (a^2-3 b^2\right ) e^4 F\left (\left .\frac{1}{2} \left (c-\frac{\pi }{2}+d x\right )\right |2\right ) \sqrt{\sin (c+d x)}}{3 a^6 d \sqrt{e \sin (c+d x)}}-\frac{4 b^2 \left (4 a^2-3 b^2\right ) e^4 F\left (\left .\frac{1}{2} \left (c-\frac{\pi }{2}+d x\right )\right |2\right ) \sqrt{\sin (c+d x)}}{3 a^6 d \sqrt{e \sin (c+d x)}}+\frac{5 b^4 \sqrt{a^2-b^2} e^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^6 \left (a-\sqrt{a^2-b^2}\right ) d \sqrt{e \sin (c+d x)}}-\frac{2 b^2 \left (a^2-b^2\right )^{3/2} e^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)}}{a^6 \left (a-\sqrt{a^2-b^2}\right ) d \sqrt{e \sin (c+d x)}}-\frac{5 b^4 \sqrt{a^2-b^2} e^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^6 \left (a+\sqrt{a^2-b^2}\right ) d \sqrt{e \sin (c+d x)}}+\frac{2 b^2 \left (a^2-b^2\right )^{3/2} e^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)}}{a^6 \left (a+\sqrt{a^2-b^2}\right ) d \sqrt{e \sin (c+d x)}}-\frac{10 e^3 \cos (c+d x) \sqrt{e \sin (c+d x)}}{21 a^2 d}-\frac{5 b^2 e^3 (3 b-a \cos (c+d x)) \sqrt{e \sin (c+d x)}}{3 a^5 d}+\frac{4 b e^3 \left (3 \left (a^2-b^2\right )+a b \cos (c+d x)\right ) \sqrt{e \sin (c+d x)}}{3 a^5 d}+\frac{4 b e (e \sin (c+d x))^{5/2}}{5 a^3 d}-\frac{2 e \cos (c+d x) (e \sin (c+d x))^{5/2}}{7 a^2 d}+\frac{b^2 e (e \sin (c+d x))^{5/2}}{a^3 d (b+a \cos (c+d x))}\\ \end{align*}

Mathematica [C]  time = 16.6283, size = 2095, normalized size = 1.9 \[ \text{Result too large to show} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

((b + a*Cos[c + d*x])^2*(-((23*a^2 - 84*b^2)*Cos[c + d*x])/(42*a^4) - (b^2*(-a^2 + b^2))/(a^5*(b + a*Cos[c + d
*x])) - (2*b*Cos[2*(c + d*x)])/(5*a^3) + Cos[3*(c + d*x)]/(14*a^2))*Csc[c + d*x]^3*Sec[c + d*x]^2*(e*Sin[c + d
*x])^(7/2))/(d*(a + b*Sec[c + d*x])^2) + ((b + a*Cos[c + d*x])^2*Sec[c + d*x]^2*(e*Sin[c + d*x])^(7/2)*((2*(50
*a^4 - 273*a^2*b^2 + 105*b^4)*Cos[c + d*x]^2*(b + a*Sqrt[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[S
qrt[-a^2 + b^2] + Sqrt[2]*Sqrt[a]*(-a^2 + b^2)^(1/4)*Sqrt[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*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, 1/2, 1, 9/4, Sin[c + d*x]^2, (a^2*Sin[c + d*x]^2)/(a^2 - b^2)])*S
in[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*(-139*a^3
*b + 210*a*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)*Sq
rt[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[S
qrt[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 - Sin[c + d*x]^2]) + ((231*a^3*b - 420*a*b^3)*Cos[c +
d*x]*Cos[2*(c + d*x)]*(b + a*Sqrt[1 - Sin[c + d*x]^2])*(((1/2 - I/2)*(a^2 - 2*b^2)*ArcTan[1 - ((1 + I)*Sqrt[a]
*Sqrt[Sin[c + d*x]])/(a^2 - b^2)^(1/4)])/(a^(3/2)*(a^2 - b^2)^(3/4)) - ((1/2 - I/2)*(a^2 - 2*b^2)*ArcTan[1 + (
(1 + I)*Sqrt[a]*Sqrt[Sin[c + d*x]])/(a^2 - b^2)^(1/4)])/(a^(3/2)*(a^2 - b^2)^(3/4)) + ((1/4 - I/4)*(a^2 - 2*b^
2)*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^(3/2)*(a
^2 - b^2)^(3/4)) - ((1/4 - I/4)*(a^2 - 2*b^2)*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^(3/2)*(a^2 - b^2)^(3/4)) + (4*Sqrt[Sin[c + d*x]])/a + (4*b*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]^(5/2))/(5*(a^2 - b^2)) + (10*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)*Appell
F1[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])*(1 - 2*Sin[c + d*x]^2)*Sqrt[1 - Sin[c + d*x]^2])))/(210*a^5*d*(a + b*Sec[
c + d*x])^2*Sin[c + d*x]^(7/2))

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Maple [B]  time = 9.819, size = 3412, normalized size = 3.1 \begin{align*} \text{output too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

4/d/a^3*e^3*b*(e*sin(d*x+c))^(1/2)-8/d/a^5*e^3*b^3*(e*sin(d*x+c))^(1/2)-5/d*e^4/cos(d*x+c)/(e*sin(d*x+c))^(1/2
)/a^6*(-sin(d*x+c)+1)^(1/2)*(2+2*sin(d*x+c))^(1/2)*sin(d*x+c)^(1/2)*EllipticF((-sin(d*x+c)+1)^(1/2),1/2*2^(1/2
))*b^4-1/2/d*e^4/cos(d*x+c)/(e*sin(d*x+c))^(1/2)*b^2/a^2/(a^2-b^2)*(-sin(d*x+c)+1)^(1/2)*(2+2*sin(d*x+c))^(1/2
)*sin(d*x+c)^(1/2)*EllipticF((-sin(d*x+c)+1)^(1/2),1/2*2^(1/2))+1/d*e^4/cos(d*x+c)/(e*sin(d*x+c))^(1/2)*b^4/a^
4/(a^2-b^2)*(-sin(d*x+c)+1)^(1/2)*(2+2*sin(d*x+c))^(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^4/cos(d*x+c)/(e*sin(d*x+c))^(1/2)*b^6/a^6/(a^2-b^2)*(-sin(d*x+c)+1)^(1/2)*(2+2*sin(d*x+c))^(1
/2)*sin(d*x+c)^(1/2)*EllipticF((-sin(d*x+c)+1)^(1/2),1/2*2^(1/2))-5/d*e^4/cos(d*x+c)/(e*sin(d*x+c))^(1/2)*b^4/
a^5/(a^2-b^2)^(1/2)*(-sin(d*x+c)+1)^(1/2)*(2+2*sin(d*x+c))^(1/2)*sin(d*x+c)^(1/2)/(1-(a^2-b^2)^(1/2)/a)*Ellipt
icPi((-sin(d*x+c)+1)^(1/2),1/(1-(a^2-b^2)^(1/2)/a),1/2*2^(1/2))+7/2/d*e^4/cos(d*x+c)/(e*sin(d*x+c))^(1/2)*b^6/
a^7/(a^2-b^2)^(1/2)*(-sin(d*x+c)+1)^(1/2)*(2+2*sin(d*x+c))^(1/2)*sin(d*x+c)^(1/2)/(1-(a^2-b^2)^(1/2)/a)*Ellipt
icPi((-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^4/cos(d*x+c)/(e*sin(d*x+c))^(1/2)*b^2/
a^3/(a^2-b^2)^(1/2)*(-sin(d*x+c)+1)^(1/2)*(2+2*sin(d*x+c))^(1/2)*sin(d*x+c)^(1/2)/(1+(a^2-b^2)^(1/2)/a)*Ellipt
icPi((-sin(d*x+c)+1)^(1/2),1/(1+(a^2-b^2)^(1/2)/a),1/2*2^(1/2))+5/d*e^4/cos(d*x+c)/(e*sin(d*x+c))^(1/2)*b^4/a^
5/(a^2-b^2)^(1/2)*(-sin(d*x+c)+1)^(1/2)*(2+2*sin(d*x+c))^(1/2)*sin(d*x+c)^(1/2)/(1+(a^2-b^2)^(1/2)/a)*Elliptic
Pi((-sin(d*x+c)+1)^(1/2),1/(1+(a^2-b^2)^(1/2)/a),1/2*2^(1/2))-3/d*e^4/cos(d*x+c)/(e*sin(d*x+c))^(1/2)*b^6/a^5/
(a^2-b^2)^(3/2)*(-sin(d*x+c)+1)^(1/2)*(2+2*sin(d*x+c))^(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^4/cos(d*x+c)/(e*sin(d*x+c))^(1/2)*b^8/a^7/
(a^2-b^2)^(3/2)*(-sin(d*x+c)+1)^(1/2)*(2+2*sin(d*x+c))^(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^4/cos(d*x+c)/(e*sin(d*x+c))^(1/2)*b^2/a/(a
^2-b^2)^(3/2)*(-sin(d*x+c)+1)^(1/2)*(2+2*sin(d*x+c))^(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))-7/2/d*e^4/cos(d*x+c)/(e*sin(d*x+c))^(1/2)*b^6/a^7/(a
^2-b^2)^(1/2)*(-sin(d*x+c)+1)^(1/2)*(2+2*sin(d*x+c))^(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^4/cos(d*x+c)/(e*sin(d*x+c))^(1/2)*b^2/a/(a^2
-b^2)^(3/2)*(-sin(d*x+c)+1)^(1/2)*(2+2*sin(d*x+c))^(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))+9/4/d*e^4/cos(d*x+c)/(e*sin(d*x+c))^(1/2)*b^4/a^3/(a^2
-b^2)^(3/2)*(-sin(d*x+c)+1)^(1/2)*(2+2*sin(d*x+c))^(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))-5/21/d*e^4/cos(d*x+c)/(e*sin(d*x+c))^(1/2)/a^2*(-sin(d
*x+c)+1)^(1/2)*(2+2*sin(d*x+c))^(1/2)*sin(d*x+c)^(1/2)*EllipticF((-sin(d*x+c)+1)^(1/2),1/2*2^(1/2))+4/d*e^4/co
s(d*x+c)/(e*sin(d*x+c))^(1/2)/a^4*(-sin(d*x+c)+1)^(1/2)*(2+2*sin(d*x+c))^(1/2)*sin(d*x+c)^(1/2)*EllipticF((-si
n(d*x+c)+1)^(1/2),1/2*2^(1/2))*b^2-9/4/d*e^4/cos(d*x+c)/(e*sin(d*x+c))^(1/2)*b^4/a^3/(a^2-b^2)^(3/2)*(-sin(d*x
+c)+1)^(1/2)*(2+2*sin(d*x+c))^(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/d*e^4/cos(d*x+c)/(e*sin(d*x+c))^(1/2)*b^6/a^5/(a^2-b^2)^(3/2)*(-sin(d*x+c
)+1)^(1/2)*(2+2*sin(d*x+c))^(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^4/cos(d*x+c)/(e*sin(d*x+c))^(1/2)*b^8/a^7/(a^2-b^2)^(3/2)*(-sin(d*x+c
)+1)^(1/2)*(2+2*sin(d*x+c))^(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^4/cos(d*x+c)/(e*sin(d*x+c))^(1/2)*b^2/a^3/(a^2-b^2)^(1/2)*(-sin(d*x+c
)+1)^(1/2)*(2+2*sin(d*x+c))^(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))-13/4/d/a^3*e^5*b^3*(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)))+9/4/d/a^5*e^5*b^5*(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*e^4*cos(d*x+c)/(e*sin(d*x+c))^(1/2)/a^4*b^2*sin(d*x+c)+2/d/a*e^5*b*(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))-13/2/d/a^3*e^5*
b^3*(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))+9/2/d/
a^5*e^5*b^5*(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/a*e^5*b*(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)))-1/d*e^4*sin(d*x+c)*cos(d*x+c)/(e*sin(d*x+c))^(1/2)*b^6/a^4/
(a^2-b^2)/(-cos(d*x+c)^2*a^2+b^2)+2/d*e^4*sin(d*x+c)*cos(d*x+c)/(e*sin(d*x+c))^(1/2)*b^4/a^2/(a^2-b^2)/(-cos(d
*x+c)^2*a^2+b^2)+2/7/d*e^4*cos(d*x+c)^3/(e*sin(d*x+c))^(1/2)/a^2*sin(d*x+c)-16/21/d*e^4*cos(d*x+c)/(e*sin(d*x+
c))^(1/2)/a^2*sin(d*x+c)+1/d/a^3*e^5*b^3*(e*sin(d*x+c))^(1/2)/(-a^2*cos(d*x+c)^2*e^2+b^2*e^2)-1/d/a^5*e^5*b^5*
(e*sin(d*x+c))^(1/2)/(-a^2*cos(d*x+c)^2*e^2+b^2*e^2)+4/5*b*e*(e*sin(d*x+c))^(5/2)/a^3/d-1/d*e^4*sin(d*x+c)*cos
(d*x+c)/(e*sin(d*x+c))^(1/2)*b^2/(a^2-b^2)/(-cos(d*x+c)^2*a^2+b^2)

________________________________________________________________________________________

Maxima [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Fricas [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

________________________________________________________________________________________

Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\left (e \sin \left (d x + c\right )\right )^{\frac{7}{2}}}{{\left (b \sec \left (d x + c\right ) + a\right )}^{2}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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