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

Optimal. Leaf size=516 \[ \frac{2 e^4 \left (-28 a^2 b^2+5 a^4+21 b^4\right ) \sqrt{\sin (c+d x)} \text{EllipticF}\left (\frac{1}{2} \left (c+d x-\frac{\pi }{2}\right ),2\right )}{21 a^5 d \sqrt{e \sin (c+d x)}}-\frac{b e^{7/2} \left (a^2-b^2\right )^{5/4} \tan ^{-1}\left (\frac{\sqrt{a} \sqrt{e \sin (c+d x)}}{\sqrt{e} \sqrt [4]{a^2-b^2}}\right )}{a^{9/2} d}-\frac{b e^{7/2} \left (a^2-b^2\right )^{5/4} \tanh ^{-1}\left (\frac{\sqrt{a} \sqrt{e \sin (c+d x)}}{\sqrt{e} \sqrt [4]{a^2-b^2}}\right )}{a^{9/2} d}+\frac{2 e^3 \sqrt{e \sin (c+d x)} \left (21 b \left (a^2-b^2\right )-a \left (5 a^2-7 b^2\right ) \cos (c+d x)\right )}{21 a^4 d}+\frac{b^2 e^4 \left (a^2-b^2\right )^2 \sqrt{\sin (c+d x)} \Pi \left (\frac{2 a}{a-\sqrt{a^2-b^2}};\left .\frac{1}{2} \left (c+d x-\frac{\pi }{2}\right )\right |2\right )}{a^5 d \left (-a \sqrt{a^2-b^2}+a^2-b^2\right ) \sqrt{e \sin (c+d x)}}+\frac{b^2 e^4 \left (a^2-b^2\right )^2 \sqrt{\sin (c+d x)} \Pi \left (\frac{2 a}{a+\sqrt{a^2-b^2}};\left .\frac{1}{2} \left (c+d x-\frac{\pi }{2}\right )\right |2\right )}{a^5 d \left (a \sqrt{a^2-b^2}+a^2-b^2\right ) \sqrt{e \sin (c+d x)}}+\frac{2 e (e \sin (c+d x))^{5/2} (7 b-5 a \cos (c+d x))}{35 a^2 d} \]

[Out]

-((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^(9/2)*d)
) - (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^(9/2)
*d) + (2*(5*a^4 - 28*a^2*b^2 + 21*b^4)*e^4*EllipticF[(c - Pi/2 + d*x)/2, 2]*Sqrt[Sin[c + d*x]])/(21*a^5*d*Sqrt
[e*Sin[c + d*x]]) + (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^5*(a^2 - b^2 - a*Sqrt[a^2 - b^2])*d*Sqrt[e*Sin[c + d*x]]) + (b^2*(a^2 - b^2)^2*e^4*Elliptic
Pi[(2*a)/(a + Sqrt[a^2 - b^2]), (c - Pi/2 + d*x)/2, 2]*Sqrt[Sin[c + d*x]])/(a^5*(a^2 - b^2 + a*Sqrt[a^2 - b^2]
)*d*Sqrt[e*Sin[c + d*x]]) + (2*e^3*(21*b*(a^2 - b^2) - a*(5*a^2 - 7*b^2)*Cos[c + d*x])*Sqrt[e*Sin[c + d*x]])/(
21*a^4*d) + (2*e*(7*b - 5*a*Cos[c + d*x])*(e*Sin[c + d*x])^(5/2))/(35*a^2*d)

________________________________________________________________________________________

Rubi [A]  time = 1.70264, antiderivative size = 516, normalized size of antiderivative = 1., number of steps used = 15, number of rules used = 12, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.48, Rules used = {3872, 2865, 2867, 2642, 2641, 2702, 2807, 2805, 329, 212, 208, 205} \[ -\frac{b e^{7/2} \left (a^2-b^2\right )^{5/4} \tan ^{-1}\left (\frac{\sqrt{a} \sqrt{e \sin (c+d x)}}{\sqrt{e} \sqrt [4]{a^2-b^2}}\right )}{a^{9/2} d}-\frac{b e^{7/2} \left (a^2-b^2\right )^{5/4} \tanh ^{-1}\left (\frac{\sqrt{a} \sqrt{e \sin (c+d x)}}{\sqrt{e} \sqrt [4]{a^2-b^2}}\right )}{a^{9/2} d}+\frac{2 e^3 \sqrt{e \sin (c+d x)} \left (21 b \left (a^2-b^2\right )-a \left (5 a^2-7 b^2\right ) \cos (c+d x)\right )}{21 a^4 d}+\frac{2 e^4 \left (-28 a^2 b^2+5 a^4+21 b^4\right ) \sqrt{\sin (c+d x)} F\left (\left .\frac{1}{2} \left (c+d x-\frac{\pi }{2}\right )\right |2\right )}{21 a^5 d \sqrt{e \sin (c+d x)}}+\frac{b^2 e^4 \left (a^2-b^2\right )^2 \sqrt{\sin (c+d x)} \Pi \left (\frac{2 a}{a-\sqrt{a^2-b^2}};\left .\frac{1}{2} \left (c+d x-\frac{\pi }{2}\right )\right |2\right )}{a^5 d \left (-a \sqrt{a^2-b^2}+a^2-b^2\right ) \sqrt{e \sin (c+d x)}}+\frac{b^2 e^4 \left (a^2-b^2\right )^2 \sqrt{\sin (c+d x)} \Pi \left (\frac{2 a}{a+\sqrt{a^2-b^2}};\left .\frac{1}{2} \left (c+d x-\frac{\pi }{2}\right )\right |2\right )}{a^5 d \left (a \sqrt{a^2-b^2}+a^2-b^2\right ) \sqrt{e \sin (c+d x)}}+\frac{2 e (e \sin (c+d x))^{5/2} (7 b-5 a \cos (c+d x))}{35 a^2 d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-((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^(9/2)*d)
) - (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^(9/2)
*d) + (2*(5*a^4 - 28*a^2*b^2 + 21*b^4)*e^4*EllipticF[(c - Pi/2 + d*x)/2, 2]*Sqrt[Sin[c + d*x]])/(21*a^5*d*Sqrt
[e*Sin[c + d*x]]) + (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^5*(a^2 - b^2 - a*Sqrt[a^2 - b^2])*d*Sqrt[e*Sin[c + d*x]]) + (b^2*(a^2 - b^2)^2*e^4*Elliptic
Pi[(2*a)/(a + Sqrt[a^2 - b^2]), (c - Pi/2 + d*x)/2, 2]*Sqrt[Sin[c + d*x]])/(a^5*(a^2 - b^2 + a*Sqrt[a^2 - b^2]
)*d*Sqrt[e*Sin[c + d*x]]) + (2*e^3*(21*b*(a^2 - b^2) - a*(5*a^2 - 7*b^2)*Cos[c + d*x])*Sqrt[e*Sin[c + d*x]])/(
21*a^4*d) + (2*e*(7*b - 5*a*Cos[c + d*x])*(e*Sin[c + d*x])^(5/2))/(35*a^2*d)

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 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 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 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]

Rubi steps

\begin{align*} \int \frac{(e \sin (c+d x))^{7/2}}{a+b \sec (c+d x)} \, dx &=-\int \frac{\cos (c+d x) (e \sin (c+d x))^{7/2}}{-b-a \cos (c+d x)} \, dx\\ &=\frac{2 e (7 b-5 a \cos (c+d x)) (e \sin (c+d x))^{5/2}}{35 a^2 d}-\frac{\left (2 e^2\right ) \int \frac{\left (-a b+\frac{1}{2} \left (5 a^2-7 b^2\right ) \cos (c+d x)\right ) (e \sin (c+d x))^{3/2}}{-b-a \cos (c+d x)} \, dx}{7 a^2}\\ &=\frac{2 e^3 \left (21 b \left (a^2-b^2\right )-a \left (5 a^2-7 b^2\right ) \cos (c+d x)\right ) \sqrt{e \sin (c+d x)}}{21 a^4 d}+\frac{2 e (7 b-5 a \cos (c+d x)) (e \sin (c+d x))^{5/2}}{35 a^2 d}-\frac{\left (4 e^4\right ) \int \frac{-\frac{1}{2} a b \left (8 a^2-7 b^2\right )+\frac{1}{4} \left (5 a^4-28 a^2 b^2+21 b^4\right ) \cos (c+d x)}{(-b-a \cos (c+d x)) \sqrt{e \sin (c+d x)}} \, dx}{21 a^4}\\ &=\frac{2 e^3 \left (21 b \left (a^2-b^2\right )-a \left (5 a^2-7 b^2\right ) \cos (c+d x)\right ) \sqrt{e \sin (c+d x)}}{21 a^4 d}+\frac{2 e (7 b-5 a \cos (c+d x)) (e \sin (c+d x))^{5/2}}{35 a^2 d}+\frac{\left (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^5}+\frac{\left (\left (5 a^4-28 a^2 b^2+21 b^4\right ) e^4\right ) \int \frac{1}{\sqrt{e \sin (c+d x)}} \, dx}{21 a^5}\\ &=\frac{2 e^3 \left (21 b \left (a^2-b^2\right )-a \left (5 a^2-7 b^2\right ) \cos (c+d x)\right ) \sqrt{e \sin (c+d x)}}{21 a^4 d}+\frac{2 e (7 b-5 a \cos (c+d x)) (e \sin (c+d x))^{5/2}}{35 a^2 d}+\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}{2 a^5}+\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}{2 a^5}+\frac{\left (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^4 d}+\frac{\left (\left (5 a^4-28 a^2 b^2+21 b^4\right ) e^4 \sqrt{\sin (c+d x)}\right ) \int \frac{1}{\sqrt{\sin (c+d x)}} \, dx}{21 a^5 \sqrt{e \sin (c+d x)}}\\ &=\frac{2 \left (5 a^4-28 a^2 b^2+21 b^4\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)}}{21 a^5 d \sqrt{e \sin (c+d x)}}+\frac{2 e^3 \left (21 b \left (a^2-b^2\right )-a \left (5 a^2-7 b^2\right ) \cos (c+d x)\right ) \sqrt{e \sin (c+d x)}}{21 a^4 d}+\frac{2 e (7 b-5 a \cos (c+d x)) (e \sin (c+d x))^{5/2}}{35 a^2 d}+\frac{\left (2 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^4 d}+\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}{2 a^5 \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}{2 a^5 \sqrt{e \sin (c+d x)}}\\ &=\frac{2 \left (5 a^4-28 a^2 b^2+21 b^4\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)}}{21 a^5 d \sqrt{e \sin (c+d x)}}-\frac{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^5 \left (a-\sqrt{a^2-b^2}\right ) d \sqrt{e \sin (c+d x)}}+\frac{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^5 \left (a+\sqrt{a^2-b^2}\right ) d \sqrt{e \sin (c+d x)}}+\frac{2 e^3 \left (21 b \left (a^2-b^2\right )-a \left (5 a^2-7 b^2\right ) \cos (c+d x)\right ) \sqrt{e \sin (c+d x)}}{21 a^4 d}+\frac{2 e (7 b-5 a \cos (c+d x)) (e \sin (c+d x))^{5/2}}{35 a^2 d}-\frac{\left (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^4 d}-\frac{\left (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^4 d}\\ &=-\frac{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^{9/2} d}-\frac{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^{9/2} d}+\frac{2 \left (5 a^4-28 a^2 b^2+21 b^4\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)}}{21 a^5 d \sqrt{e \sin (c+d x)}}-\frac{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^5 \left (a-\sqrt{a^2-b^2}\right ) d \sqrt{e \sin (c+d x)}}+\frac{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^5 \left (a+\sqrt{a^2-b^2}\right ) d \sqrt{e \sin (c+d x)}}+\frac{2 e^3 \left (21 b \left (a^2-b^2\right )-a \left (5 a^2-7 b^2\right ) \cos (c+d x)\right ) \sqrt{e \sin (c+d x)}}{21 a^4 d}+\frac{2 e (7 b-5 a \cos (c+d x)) (e \sin (c+d x))^{5/2}}{35 a^2 d}\\ \end{align*}

Mathematica [C]  time = 17.3005, size = 2049, normalized size = 3.97 \[ \text{Result too large to show} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

((b + a*Cos[c + d*x])*(-((23*a^2 - 28*b^2)*Cos[c + d*x])/(42*a^3) - (b*Cos[2*(c + d*x)])/(5*a^2) + Cos[3*(c +
d*x)]/(14*a))*Csc[c + d*x]^3*Sec[c + d*x]*(e*Sin[c + d*x])^(7/2))/(d*(a + b*Sec[c + d*x])) - ((b + a*Cos[c + d
*x])*Sec[c + d*x]*(e*Sin[c + d*x])^(7/2)*((2*(-100*a^3 + 98*a*b^2)*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[Si
n[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]]))/(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)*Appe
llF1[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)])*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*(89*a^2*b - 70*b^3)*Cos[c + d*x]*(b + a*Sqrt[1 - Sin[c + d*x]^2])*(((-1/8 + I/8)*S
qrt[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]*S
qrt[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*Si
n[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^2*b + 210*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)*Sqr
t[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)*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])*(1 - 2*Sin[c + d*x]^2)*Sqrt[1 - Sin[c + d*
x]^2])))/(420*a^3*d*(a + b*Sec[c + d*x])*Sin[c + d*x]^(7/2))

________________________________________________________________________________________

Maple [B]  time = 5.92, size = 1776, normalized size = 3.4 \begin{align*} \text{result 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)),x)

[Out]

2/5/d*b*e/a^2*(e*sin(d*x+c))^(5/2)+2/d*b*e^3/a^2*(e*sin(d*x+c))^(1/2)-2/d*b^3*e^3/a^4*(e*sin(d*x+c))^(1/2)+1/d
*b*e^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))-2/d
*b^3*e^5/a^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*b^5*e^5/a^4*(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))+1/2/d*b*e^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)))-1/d*b^3*e^5/a^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)))
+1/2/d*b^5*e^5/a^4*(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)))+2/7/d/a*e^4*cos(d*x+c)^3/(e*sin(d*x+c))^(1/2)*sin(d*x+
c)-5/21/d/a*e^4/cos(d*x+c)/(e*sin(d*x+c))^(1/2)*(-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))+4/3/d/a^3*e^4/cos(d*x+c)/(e*sin(d*x+c))^(1/2)*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/a^5*e^4/cos(d*
x+c)/(e*sin(d*x+c))^(1/2)*b^4*(-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))-16/21/d/a*e^4*cos(d*x+c)/(e*sin(d*x+c))^(1/2)*sin(d*x+c)+2/3/d/a^3*e^4*cos(d*x+c)/(
e*sin(d*x+c))^(1/2)*b^2*sin(d*x+c)+1/2/d/a^2*e^4/cos(d*x+c)/(e*sin(d*x+c))^(1/2)*b^2/(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/d/a^4*e^4/cos(d*x+c)/(e*sin(d*x+c))^(1/2)*b^4/(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/a^6*e^4/cos(d*x+c)/(e*sin(d*x+c))^(1/2)*b^6/(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/a^2*e^4/cos(d*x+c)/(e*sin(d*x+c))^(1/2)*b^2/(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/d/a^4*e^4/cos(d*x+c)/(e*sin(d*x+c))^(1/2)*b^4/(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/a^6*e^4/cos(d*x+c)/(e*sin(d*x+c))^(1/2)*b^6/(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))

________________________________________________________________________________________

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

[Out]

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

________________________________________________________________________________________

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)),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)),x)

[Out]

Timed out

________________________________________________________________________________________

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}}}{b \sec \left (d x + c\right ) + a}\,{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)),x, algorithm="giac")

[Out]

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