3.171 \(\int \frac{1}{\sqrt{a+a \sec (e+f x)} (c+d \sec (e+f x))^3} \, dx\)

Optimal. Leaf size=653 \[ \frac{d^2 (2 c-d) \tan (e+f x)}{c^2 f (c-d)^2 (c+d) \sqrt{a \sec (e+f x)+a} (c+d \sec (e+f x))}+\frac{d^2 \tan (e+f x)}{2 c f \left (c^2-d^2\right ) \sqrt{a \sec (e+f x)+a} (c+d \sec (e+f x))^2}+\frac{2 \sqrt{a} d^{3/2} \left (3 c^2-3 c d+d^2\right ) \tan (e+f x) \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a-a \sec (e+f x)}}{\sqrt{a} \sqrt{c+d}}\right )}{c^3 f (c-d)^3 \sqrt{c+d} \sqrt{a-a \sec (e+f x)} \sqrt{a \sec (e+f x)+a}}+\frac{\sqrt{a} d^{3/2} (2 c-d) \tan (e+f x) \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a-a \sec (e+f x)}}{\sqrt{a} \sqrt{c+d}}\right )}{c^2 f (c-d)^2 (c+d)^{3/2} \sqrt{a-a \sec (e+f x)} \sqrt{a \sec (e+f x)+a}}+\frac{2 \sqrt{a} \tan (e+f x) \tanh ^{-1}\left (\frac{\sqrt{a-a \sec (e+f x)}}{\sqrt{a}}\right )}{c^3 f \sqrt{a-a \sec (e+f x)} \sqrt{a \sec (e+f x)+a}}+\frac{3 d^2 \tan (e+f x)}{4 c f (c-d) (c+d)^2 \sqrt{a \sec (e+f x)+a} (c+d \sec (e+f x))}+\frac{3 \sqrt{a} d^{3/2} \tan (e+f x) \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a-a \sec (e+f x)}}{\sqrt{a} \sqrt{c+d}}\right )}{4 c f (c-d) (c+d)^{5/2} \sqrt{a-a \sec (e+f x)} \sqrt{a \sec (e+f x)+a}}-\frac{\sqrt{2} \sqrt{a} \tan (e+f x) \tanh ^{-1}\left (\frac{\sqrt{a-a \sec (e+f x)}}{\sqrt{2} \sqrt{a}}\right )}{f (c-d)^3 \sqrt{a-a \sec (e+f x)} \sqrt{a \sec (e+f x)+a}} \]

[Out]

(2*Sqrt[a]*ArcTanh[Sqrt[a - a*Sec[e + f*x]]/Sqrt[a]]*Tan[e + f*x])/(c^3*f*Sqrt[a - a*Sec[e + f*x]]*Sqrt[a + a*
Sec[e + f*x]]) - (Sqrt[2]*Sqrt[a]*ArcTanh[Sqrt[a - a*Sec[e + f*x]]/(Sqrt[2]*Sqrt[a])]*Tan[e + f*x])/((c - d)^3
*f*Sqrt[a - a*Sec[e + f*x]]*Sqrt[a + a*Sec[e + f*x]]) + (3*Sqrt[a]*d^(3/2)*ArcTanh[(Sqrt[d]*Sqrt[a - a*Sec[e +
f*x]])/(Sqrt[a]*Sqrt[c + d])]*Tan[e + f*x])/(4*c*(c - d)*(c + d)^(5/2)*f*Sqrt[a - a*Sec[e + f*x]]*Sqrt[a + a*
Sec[e + f*x]]) + (Sqrt[a]*(2*c - d)*d^(3/2)*ArcTanh[(Sqrt[d]*Sqrt[a - a*Sec[e + f*x]])/(Sqrt[a]*Sqrt[c + d])]*
Tan[e + f*x])/(c^2*(c - d)^2*(c + d)^(3/2)*f*Sqrt[a - a*Sec[e + f*x]]*Sqrt[a + a*Sec[e + f*x]]) + (2*Sqrt[a]*d
^(3/2)*(3*c^2 - 3*c*d + d^2)*ArcTanh[(Sqrt[d]*Sqrt[a - a*Sec[e + f*x]])/(Sqrt[a]*Sqrt[c + d])]*Tan[e + f*x])/(
c^3*(c - d)^3*Sqrt[c + d]*f*Sqrt[a - a*Sec[e + f*x]]*Sqrt[a + a*Sec[e + f*x]]) + (d^2*Tan[e + f*x])/(2*c*(c^2
- d^2)*f*Sqrt[a + a*Sec[e + f*x]]*(c + d*Sec[e + f*x])^2) + (3*d^2*Tan[e + f*x])/(4*c*(c - d)*(c + d)^2*f*Sqrt
[a + a*Sec[e + f*x]]*(c + d*Sec[e + f*x])) + ((2*c - d)*d^2*Tan[e + f*x])/(c^2*(c - d)^2*(c + d)*f*Sqrt[a + a*
Sec[e + f*x]]*(c + d*Sec[e + f*x]))

________________________________________________________________________________________

Rubi [A]  time = 0.646345, antiderivative size = 653, normalized size of antiderivative = 1., number of steps used = 16, number of rules used = 6, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.222, Rules used = {3940, 180, 63, 206, 51, 208} \[ \frac{d^2 (2 c-d) \tan (e+f x)}{c^2 f (c-d)^2 (c+d) \sqrt{a \sec (e+f x)+a} (c+d \sec (e+f x))}+\frac{d^2 \tan (e+f x)}{2 c f \left (c^2-d^2\right ) \sqrt{a \sec (e+f x)+a} (c+d \sec (e+f x))^2}+\frac{2 \sqrt{a} d^{3/2} \left (3 c^2-3 c d+d^2\right ) \tan (e+f x) \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a-a \sec (e+f x)}}{\sqrt{a} \sqrt{c+d}}\right )}{c^3 f (c-d)^3 \sqrt{c+d} \sqrt{a-a \sec (e+f x)} \sqrt{a \sec (e+f x)+a}}+\frac{\sqrt{a} d^{3/2} (2 c-d) \tan (e+f x) \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a-a \sec (e+f x)}}{\sqrt{a} \sqrt{c+d}}\right )}{c^2 f (c-d)^2 (c+d)^{3/2} \sqrt{a-a \sec (e+f x)} \sqrt{a \sec (e+f x)+a}}+\frac{2 \sqrt{a} \tan (e+f x) \tanh ^{-1}\left (\frac{\sqrt{a-a \sec (e+f x)}}{\sqrt{a}}\right )}{c^3 f \sqrt{a-a \sec (e+f x)} \sqrt{a \sec (e+f x)+a}}+\frac{3 d^2 \tan (e+f x)}{4 c f (c-d) (c+d)^2 \sqrt{a \sec (e+f x)+a} (c+d \sec (e+f x))}+\frac{3 \sqrt{a} d^{3/2} \tan (e+f x) \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a-a \sec (e+f x)}}{\sqrt{a} \sqrt{c+d}}\right )}{4 c f (c-d) (c+d)^{5/2} \sqrt{a-a \sec (e+f x)} \sqrt{a \sec (e+f x)+a}}-\frac{\sqrt{2} \sqrt{a} \tan (e+f x) \tanh ^{-1}\left (\frac{\sqrt{a-a \sec (e+f x)}}{\sqrt{2} \sqrt{a}}\right )}{f (c-d)^3 \sqrt{a-a \sec (e+f x)} \sqrt{a \sec (e+f x)+a}} \]

Antiderivative was successfully verified.

[In]

Int[1/(Sqrt[a + a*Sec[e + f*x]]*(c + d*Sec[e + f*x])^3),x]

[Out]

(2*Sqrt[a]*ArcTanh[Sqrt[a - a*Sec[e + f*x]]/Sqrt[a]]*Tan[e + f*x])/(c^3*f*Sqrt[a - a*Sec[e + f*x]]*Sqrt[a + a*
Sec[e + f*x]]) - (Sqrt[2]*Sqrt[a]*ArcTanh[Sqrt[a - a*Sec[e + f*x]]/(Sqrt[2]*Sqrt[a])]*Tan[e + f*x])/((c - d)^3
*f*Sqrt[a - a*Sec[e + f*x]]*Sqrt[a + a*Sec[e + f*x]]) + (3*Sqrt[a]*d^(3/2)*ArcTanh[(Sqrt[d]*Sqrt[a - a*Sec[e +
f*x]])/(Sqrt[a]*Sqrt[c + d])]*Tan[e + f*x])/(4*c*(c - d)*(c + d)^(5/2)*f*Sqrt[a - a*Sec[e + f*x]]*Sqrt[a + a*
Sec[e + f*x]]) + (Sqrt[a]*(2*c - d)*d^(3/2)*ArcTanh[(Sqrt[d]*Sqrt[a - a*Sec[e + f*x]])/(Sqrt[a]*Sqrt[c + d])]*
Tan[e + f*x])/(c^2*(c - d)^2*(c + d)^(3/2)*f*Sqrt[a - a*Sec[e + f*x]]*Sqrt[a + a*Sec[e + f*x]]) + (2*Sqrt[a]*d
^(3/2)*(3*c^2 - 3*c*d + d^2)*ArcTanh[(Sqrt[d]*Sqrt[a - a*Sec[e + f*x]])/(Sqrt[a]*Sqrt[c + d])]*Tan[e + f*x])/(
c^3*(c - d)^3*Sqrt[c + d]*f*Sqrt[a - a*Sec[e + f*x]]*Sqrt[a + a*Sec[e + f*x]]) + (d^2*Tan[e + f*x])/(2*c*(c^2
- d^2)*f*Sqrt[a + a*Sec[e + f*x]]*(c + d*Sec[e + f*x])^2) + (3*d^2*Tan[e + f*x])/(4*c*(c - d)*(c + d)^2*f*Sqrt
[a + a*Sec[e + f*x]]*(c + d*Sec[e + f*x])) + ((2*c - d)*d^2*Tan[e + f*x])/(c^2*(c - d)^2*(c + d)*f*Sqrt[a + a*
Sec[e + f*x]]*(c + d*Sec[e + f*x]))

Rule 3940

Int[(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_))^(m_.)*(csc[(e_.) + (f_.)*(x_)]*(d_.) + (c_))^(n_.), x_Symbol] :> Di
st[(a^2*Cot[e + f*x])/(f*Sqrt[a + b*Csc[e + f*x]]*Sqrt[a - b*Csc[e + f*x]]), Subst[Int[((a + b*x)^(m - 1/2)*(c
+ d*x)^n)/(x*Sqrt[a - b*x]), x], x, Csc[e + f*x]], x] /; FreeQ[{a, b, c, d, e, f, m, n}, x] && NeQ[b*c - a*d,
0] && EqQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && IntegerQ[m - 1/2]

Rule 180

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_))^(p_)*((g_.) + (h_.)*(x_))^(q_), x
_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d*x)^n*(e + f*x)^p*(g + h*x)^q, x], x] /; FreeQ[{a, b, c, d,
e, f, g, h, m, n}, x] && IntegersQ[p, q]

Rule 63

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[{p = Denominator[m]}, Dist[p/b, Sub
st[Int[x^(p*(m + 1) - 1)*(c - (a*d)/b + (d*x^p)/b)^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] &
& NeQ[b*c - a*d, 0] && LtQ[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntLinearQ[a,
b, c, d, m, n, x]

Rule 206

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTanh[(Rt[-b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[-b, 2]), x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rule 51

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)*(c + d*x)^(n + 1
))/((b*c - a*d)*(m + 1)), x] - Dist[(d*(m + n + 2))/((b*c - a*d)*(m + 1)), Int[(a + b*x)^(m + 1)*(c + d*x)^n,
x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && LtQ[m, -1] &&  !(LtQ[n, -1] && (EqQ[a, 0] || (NeQ[
c, 0] && LtQ[m - n, 0] && IntegerQ[n]))) && IntLinearQ[a, b, c, d, m, n, x]

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]

Rubi steps

\begin{align*} \int \frac{1}{\sqrt{a+a \sec (e+f x)} (c+d \sec (e+f x))^3} \, dx &=-\frac{\left (a^2 \tan (e+f x)\right ) \operatorname{Subst}\left (\int \frac{1}{x \sqrt{a-a x} (a+a x) (c+d x)^3} \, dx,x,\sec (e+f x)\right )}{f \sqrt{a-a \sec (e+f x)} \sqrt{a+a \sec (e+f x)}}\\ &=-\frac{\left (a^2 \tan (e+f x)\right ) \operatorname{Subst}\left (\int \left (\frac{1}{a c^3 x \sqrt{a-a x}}-\frac{1}{a (c-d)^3 (1+x) \sqrt{a-a x}}+\frac{d^2}{a c (c-d) \sqrt{a-a x} (c+d x)^3}+\frac{(2 c-d) d^2}{a c^2 (c-d)^2 \sqrt{a-a x} (c+d x)^2}+\frac{d^2 \left (3 c^2-3 c d+d^2\right )}{a c^3 (c-d)^3 \sqrt{a-a x} (c+d x)}\right ) \, dx,x,\sec (e+f x)\right )}{f \sqrt{a-a \sec (e+f x)} \sqrt{a+a \sec (e+f x)}}\\ &=-\frac{(a \tan (e+f x)) \operatorname{Subst}\left (\int \frac{1}{x \sqrt{a-a x}} \, dx,x,\sec (e+f x)\right )}{c^3 f \sqrt{a-a \sec (e+f x)} \sqrt{a+a \sec (e+f x)}}+\frac{(a \tan (e+f x)) \operatorname{Subst}\left (\int \frac{1}{(1+x) \sqrt{a-a x}} \, dx,x,\sec (e+f x)\right )}{(c-d)^3 f \sqrt{a-a \sec (e+f x)} \sqrt{a+a \sec (e+f x)}}-\frac{\left (a d^2 \tan (e+f x)\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{a-a x} (c+d x)^3} \, dx,x,\sec (e+f x)\right )}{c (c-d) f \sqrt{a-a \sec (e+f x)} \sqrt{a+a \sec (e+f x)}}-\frac{\left (a (2 c-d) d^2 \tan (e+f x)\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{a-a x} (c+d x)^2} \, dx,x,\sec (e+f x)\right )}{c^2 (c-d)^2 f \sqrt{a-a \sec (e+f x)} \sqrt{a+a \sec (e+f x)}}-\frac{\left (a d^2 \left (3 c^2-3 c d+d^2\right ) \tan (e+f x)\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{a-a x} (c+d x)} \, dx,x,\sec (e+f x)\right )}{c^3 (c-d)^3 f \sqrt{a-a \sec (e+f x)} \sqrt{a+a \sec (e+f x)}}\\ &=\frac{d^2 \tan (e+f x)}{2 c \left (c^2-d^2\right ) f \sqrt{a+a \sec (e+f x)} (c+d \sec (e+f x))^2}+\frac{(2 c-d) d^2 \tan (e+f x)}{c^2 (c-d)^2 (c+d) f \sqrt{a+a \sec (e+f x)} (c+d \sec (e+f x))}+\frac{(2 \tan (e+f x)) \operatorname{Subst}\left (\int \frac{1}{1-\frac{x^2}{a}} \, dx,x,\sqrt{a-a \sec (e+f x)}\right )}{c^3 f \sqrt{a-a \sec (e+f x)} \sqrt{a+a \sec (e+f x)}}-\frac{(2 \tan (e+f x)) \operatorname{Subst}\left (\int \frac{1}{2-\frac{x^2}{a}} \, dx,x,\sqrt{a-a \sec (e+f x)}\right )}{(c-d)^3 f \sqrt{a-a \sec (e+f x)} \sqrt{a+a \sec (e+f x)}}-\frac{\left (3 a d^2 \tan (e+f x)\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{a-a x} (c+d x)^2} \, dx,x,\sec (e+f x)\right )}{4 c (c-d) (c+d) f \sqrt{a-a \sec (e+f x)} \sqrt{a+a \sec (e+f x)}}-\frac{\left (a (2 c-d) d^2 \tan (e+f x)\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{a-a x} (c+d x)} \, dx,x,\sec (e+f x)\right )}{2 c^2 (c-d)^2 (c+d) f \sqrt{a-a \sec (e+f x)} \sqrt{a+a \sec (e+f x)}}+\frac{\left (2 d^2 \left (3 c^2-3 c d+d^2\right ) \tan (e+f x)\right ) \operatorname{Subst}\left (\int \frac{1}{c+d-\frac{d x^2}{a}} \, dx,x,\sqrt{a-a \sec (e+f x)}\right )}{c^3 (c-d)^3 f \sqrt{a-a \sec (e+f x)} \sqrt{a+a \sec (e+f x)}}\\ &=\frac{2 \sqrt{a} \tanh ^{-1}\left (\frac{\sqrt{a-a \sec (e+f x)}}{\sqrt{a}}\right ) \tan (e+f x)}{c^3 f \sqrt{a-a \sec (e+f x)} \sqrt{a+a \sec (e+f x)}}-\frac{\sqrt{2} \sqrt{a} \tanh ^{-1}\left (\frac{\sqrt{a-a \sec (e+f x)}}{\sqrt{2} \sqrt{a}}\right ) \tan (e+f x)}{(c-d)^3 f \sqrt{a-a \sec (e+f x)} \sqrt{a+a \sec (e+f x)}}+\frac{2 \sqrt{a} d^{3/2} \left (3 c^2-3 c d+d^2\right ) \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a-a \sec (e+f x)}}{\sqrt{a} \sqrt{c+d}}\right ) \tan (e+f x)}{c^3 (c-d)^3 \sqrt{c+d} f \sqrt{a-a \sec (e+f x)} \sqrt{a+a \sec (e+f x)}}+\frac{d^2 \tan (e+f x)}{2 c \left (c^2-d^2\right ) f \sqrt{a+a \sec (e+f x)} (c+d \sec (e+f x))^2}+\frac{3 d^2 \tan (e+f x)}{4 c (c-d) (c+d)^2 f \sqrt{a+a \sec (e+f x)} (c+d \sec (e+f x))}+\frac{(2 c-d) d^2 \tan (e+f x)}{c^2 (c-d)^2 (c+d) f \sqrt{a+a \sec (e+f x)} (c+d \sec (e+f x))}-\frac{\left (3 a d^2 \tan (e+f x)\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{a-a x} (c+d x)} \, dx,x,\sec (e+f x)\right )}{8 c (c-d) (c+d)^2 f \sqrt{a-a \sec (e+f x)} \sqrt{a+a \sec (e+f x)}}+\frac{\left ((2 c-d) d^2 \tan (e+f x)\right ) \operatorname{Subst}\left (\int \frac{1}{c+d-\frac{d x^2}{a}} \, dx,x,\sqrt{a-a \sec (e+f x)}\right )}{c^2 (c-d)^2 (c+d) f \sqrt{a-a \sec (e+f x)} \sqrt{a+a \sec (e+f x)}}\\ &=\frac{2 \sqrt{a} \tanh ^{-1}\left (\frac{\sqrt{a-a \sec (e+f x)}}{\sqrt{a}}\right ) \tan (e+f x)}{c^3 f \sqrt{a-a \sec (e+f x)} \sqrt{a+a \sec (e+f x)}}-\frac{\sqrt{2} \sqrt{a} \tanh ^{-1}\left (\frac{\sqrt{a-a \sec (e+f x)}}{\sqrt{2} \sqrt{a}}\right ) \tan (e+f x)}{(c-d)^3 f \sqrt{a-a \sec (e+f x)} \sqrt{a+a \sec (e+f x)}}+\frac{\sqrt{a} (2 c-d) d^{3/2} \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a-a \sec (e+f x)}}{\sqrt{a} \sqrt{c+d}}\right ) \tan (e+f x)}{c^2 (c-d)^2 (c+d)^{3/2} f \sqrt{a-a \sec (e+f x)} \sqrt{a+a \sec (e+f x)}}+\frac{2 \sqrt{a} d^{3/2} \left (3 c^2-3 c d+d^2\right ) \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a-a \sec (e+f x)}}{\sqrt{a} \sqrt{c+d}}\right ) \tan (e+f x)}{c^3 (c-d)^3 \sqrt{c+d} f \sqrt{a-a \sec (e+f x)} \sqrt{a+a \sec (e+f x)}}+\frac{d^2 \tan (e+f x)}{2 c \left (c^2-d^2\right ) f \sqrt{a+a \sec (e+f x)} (c+d \sec (e+f x))^2}+\frac{3 d^2 \tan (e+f x)}{4 c (c-d) (c+d)^2 f \sqrt{a+a \sec (e+f x)} (c+d \sec (e+f x))}+\frac{(2 c-d) d^2 \tan (e+f x)}{c^2 (c-d)^2 (c+d) f \sqrt{a+a \sec (e+f x)} (c+d \sec (e+f x))}+\frac{\left (3 d^2 \tan (e+f x)\right ) \operatorname{Subst}\left (\int \frac{1}{c+d-\frac{d x^2}{a}} \, dx,x,\sqrt{a-a \sec (e+f x)}\right )}{4 c (c-d) (c+d)^2 f \sqrt{a-a \sec (e+f x)} \sqrt{a+a \sec (e+f x)}}\\ &=\frac{2 \sqrt{a} \tanh ^{-1}\left (\frac{\sqrt{a-a \sec (e+f x)}}{\sqrt{a}}\right ) \tan (e+f x)}{c^3 f \sqrt{a-a \sec (e+f x)} \sqrt{a+a \sec (e+f x)}}-\frac{\sqrt{2} \sqrt{a} \tanh ^{-1}\left (\frac{\sqrt{a-a \sec (e+f x)}}{\sqrt{2} \sqrt{a}}\right ) \tan (e+f x)}{(c-d)^3 f \sqrt{a-a \sec (e+f x)} \sqrt{a+a \sec (e+f x)}}+\frac{3 \sqrt{a} d^{3/2} \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a-a \sec (e+f x)}}{\sqrt{a} \sqrt{c+d}}\right ) \tan (e+f x)}{4 c (c-d) (c+d)^{5/2} f \sqrt{a-a \sec (e+f x)} \sqrt{a+a \sec (e+f x)}}+\frac{\sqrt{a} (2 c-d) d^{3/2} \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a-a \sec (e+f x)}}{\sqrt{a} \sqrt{c+d}}\right ) \tan (e+f x)}{c^2 (c-d)^2 (c+d)^{3/2} f \sqrt{a-a \sec (e+f x)} \sqrt{a+a \sec (e+f x)}}+\frac{2 \sqrt{a} d^{3/2} \left (3 c^2-3 c d+d^2\right ) \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a-a \sec (e+f x)}}{\sqrt{a} \sqrt{c+d}}\right ) \tan (e+f x)}{c^3 (c-d)^3 \sqrt{c+d} f \sqrt{a-a \sec (e+f x)} \sqrt{a+a \sec (e+f x)}}+\frac{d^2 \tan (e+f x)}{2 c \left (c^2-d^2\right ) f \sqrt{a+a \sec (e+f x)} (c+d \sec (e+f x))^2}+\frac{3 d^2 \tan (e+f x)}{4 c (c-d) (c+d)^2 f \sqrt{a+a \sec (e+f x)} (c+d \sec (e+f x))}+\frac{(2 c-d) d^2 \tan (e+f x)}{c^2 (c-d)^2 (c+d) f \sqrt{a+a \sec (e+f x)} (c+d \sec (e+f x))}\\ \end{align*}

Mathematica [C]  time = 38.6795, size = 652560, normalized size = 999.33 \[ \text{Result too large to show} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[1/(Sqrt[a + a*Sec[e + f*x]]*(c + d*Sec[e + f*x])^3),x]

[Out]

Result too large to show

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Maple [B]  time = 20.612, size = 402966, normalized size = 617.1 \begin{align*} \text{output too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

result too large to display

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

[Out]

Timed out

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Fricas [A]  time = 132.514, size = 10966, normalized size = 16.79 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[-1/16*(4*sqrt(2)*(a*c^5*d^2 + 2*a*c^4*d^3 + a*c^3*d^4 + (a*c^7 + 2*a*c^6*d + a*c^5*d^2)*cos(f*x + e)^3 + (a*c
^7 + 4*a*c^6*d + 5*a*c^5*d^2 + 2*a*c^4*d^3)*cos(f*x + e)^2 + (2*a*c^6*d + 5*a*c^5*d^2 + 4*a*c^4*d^3 + a*c^3*d^
4)*cos(f*x + e))*sqrt(-1/a)*log(-(4*sqrt(2)*(3*cos(f*x + e)^2 - cos(f*x + e))*sqrt((a*cos(f*x + e) + a)/cos(f*
x + e))*sqrt(-1/a)*sin(f*x + e) - 17*cos(f*x + e)^3 - 3*cos(f*x + e)^2 + 13*cos(f*x + e) - 1)/(cos(f*x + e)^3
+ 3*cos(f*x + e)^2 + 3*cos(f*x + e) + 1)) + (35*a*c^4*d^3 + 14*a*c^3*d^4 - 21*a*c^2*d^5 - 4*a*c*d^6 + 8*a*d^7
+ (35*a*c^6*d + 14*a*c^5*d^2 - 21*a*c^4*d^3 - 4*a*c^3*d^4 + 8*a*c^2*d^5)*cos(f*x + e)^3 + (35*a*c^6*d + 84*a*c
^5*d^2 + 7*a*c^4*d^3 - 46*a*c^3*d^4 + 16*a*c*d^6)*cos(f*x + e)^2 + (70*a*c^5*d^2 + 63*a*c^4*d^3 - 28*a*c^3*d^4
- 29*a*c^2*d^5 + 12*a*c*d^6 + 8*a*d^7)*cos(f*x + e))*sqrt(-d/(a*c + a*d))*log(((c^2 + 8*c*d + 8*d^2)*cos(f*x
+ e)^3 + (c^2 + 2*c*d)*cos(f*x + e)^2 + 4*((c^2 + 3*c*d + 2*d^2)*cos(f*x + e)^2 - (c*d + d^2)*cos(f*x + e))*sq
rt(-d/(a*c + a*d))*sqrt((a*cos(f*x + e) + a)/cos(f*x + e))*sin(f*x + e) + d^2 - (6*c*d + 7*d^2)*cos(f*x + e))/
(c^2*cos(f*x + e)^3 + (c^2 + 2*c*d)*cos(f*x + e)^2 + d^2 + (2*c*d + d^2)*cos(f*x + e))) + 8*(c^5*d^2 - c^4*d^3
- 2*c^3*d^4 + 2*c^2*d^5 + c*d^6 - d^7 + (c^7 - c^6*d - 2*c^5*d^2 + 2*c^4*d^3 + c^3*d^4 - c^2*d^5)*cos(f*x + e
)^3 + (c^7 + c^6*d - 4*c^5*d^2 - 2*c^4*d^3 + 5*c^3*d^4 + c^2*d^5 - 2*c*d^6)*cos(f*x + e)^2 + (2*c^6*d - c^5*d^
2 - 5*c^4*d^3 + 2*c^3*d^4 + 4*c^2*d^5 - c*d^6 - d^7)*cos(f*x + e))*sqrt(-a)*log((8*a*cos(f*x + e)^3 + 4*(2*cos
(f*x + e)^2 - cos(f*x + e))*sqrt(-a)*sqrt((a*cos(f*x + e) + a)/cos(f*x + e))*sin(f*x + e) - 7*a*cos(f*x + e) +
a)/(cos(f*x + e) + 1)) - 4*((13*c^5*d^2 - 12*c^4*d^3 - 7*c^3*d^4 + 6*c^2*d^5)*cos(f*x + e)^2 + (11*c^4*d^3 -
10*c^3*d^4 - 5*c^2*d^5 + 4*c*d^6)*cos(f*x + e))*sqrt((a*cos(f*x + e) + a)/cos(f*x + e))*sin(f*x + e))/((a*c^10
- a*c^9*d - 2*a*c^8*d^2 + 2*a*c^7*d^3 + a*c^6*d^4 - a*c^5*d^5)*f*cos(f*x + e)^3 + (a*c^10 + a*c^9*d - 4*a*c^8
*d^2 - 2*a*c^7*d^3 + 5*a*c^6*d^4 + a*c^5*d^5 - 2*a*c^4*d^6)*f*cos(f*x + e)^2 + (2*a*c^9*d - a*c^8*d^2 - 5*a*c^
7*d^3 + 2*a*c^6*d^4 + 4*a*c^5*d^5 - a*c^4*d^6 - a*c^3*d^7)*f*cos(f*x + e) + (a*c^8*d^2 - a*c^7*d^3 - 2*a*c^6*d
^4 + 2*a*c^5*d^5 + a*c^4*d^6 - a*c^3*d^7)*f), -1/8*(2*sqrt(2)*(a*c^5*d^2 + 2*a*c^4*d^3 + a*c^3*d^4 + (a*c^7 +
2*a*c^6*d + a*c^5*d^2)*cos(f*x + e)^3 + (a*c^7 + 4*a*c^6*d + 5*a*c^5*d^2 + 2*a*c^4*d^3)*cos(f*x + e)^2 + (2*a*
c^6*d + 5*a*c^5*d^2 + 4*a*c^4*d^3 + a*c^3*d^4)*cos(f*x + e))*sqrt(-1/a)*log(-(4*sqrt(2)*(3*cos(f*x + e)^2 - co
s(f*x + e))*sqrt((a*cos(f*x + e) + a)/cos(f*x + e))*sqrt(-1/a)*sin(f*x + e) - 17*cos(f*x + e)^3 - 3*cos(f*x +
e)^2 + 13*cos(f*x + e) - 1)/(cos(f*x + e)^3 + 3*cos(f*x + e)^2 + 3*cos(f*x + e) + 1)) + (35*a*c^4*d^3 + 14*a*c
^3*d^4 - 21*a*c^2*d^5 - 4*a*c*d^6 + 8*a*d^7 + (35*a*c^6*d + 14*a*c^5*d^2 - 21*a*c^4*d^3 - 4*a*c^3*d^4 + 8*a*c^
2*d^5)*cos(f*x + e)^3 + (35*a*c^6*d + 84*a*c^5*d^2 + 7*a*c^4*d^3 - 46*a*c^3*d^4 + 16*a*c*d^6)*cos(f*x + e)^2 +
(70*a*c^5*d^2 + 63*a*c^4*d^3 - 28*a*c^3*d^4 - 29*a*c^2*d^5 + 12*a*c*d^6 + 8*a*d^7)*cos(f*x + e))*sqrt(d/(a*c
+ a*d))*arctan(1/2*((c + 2*d)*cos(f*x + e) - d)*sqrt(d/(a*c + a*d))*sqrt((a*cos(f*x + e) + a)/cos(f*x + e))/(d
*sin(f*x + e))) + 4*(c^5*d^2 - c^4*d^3 - 2*c^3*d^4 + 2*c^2*d^5 + c*d^6 - d^7 + (c^7 - c^6*d - 2*c^5*d^2 + 2*c^
4*d^3 + c^3*d^4 - c^2*d^5)*cos(f*x + e)^3 + (c^7 + c^6*d - 4*c^5*d^2 - 2*c^4*d^3 + 5*c^3*d^4 + c^2*d^5 - 2*c*d
^6)*cos(f*x + e)^2 + (2*c^6*d - c^5*d^2 - 5*c^4*d^3 + 2*c^3*d^4 + 4*c^2*d^5 - c*d^6 - d^7)*cos(f*x + e))*sqrt(
-a)*log((8*a*cos(f*x + e)^3 + 4*(2*cos(f*x + e)^2 - cos(f*x + e))*sqrt(-a)*sqrt((a*cos(f*x + e) + a)/cos(f*x +
e))*sin(f*x + e) - 7*a*cos(f*x + e) + a)/(cos(f*x + e) + 1)) - 2*((13*c^5*d^2 - 12*c^4*d^3 - 7*c^3*d^4 + 6*c^
2*d^5)*cos(f*x + e)^2 + (11*c^4*d^3 - 10*c^3*d^4 - 5*c^2*d^5 + 4*c*d^6)*cos(f*x + e))*sqrt((a*cos(f*x + e) + a
)/cos(f*x + e))*sin(f*x + e))/((a*c^10 - a*c^9*d - 2*a*c^8*d^2 + 2*a*c^7*d^3 + a*c^6*d^4 - a*c^5*d^5)*f*cos(f*
x + e)^3 + (a*c^10 + a*c^9*d - 4*a*c^8*d^2 - 2*a*c^7*d^3 + 5*a*c^6*d^4 + a*c^5*d^5 - 2*a*c^4*d^6)*f*cos(f*x +
e)^2 + (2*a*c^9*d - a*c^8*d^2 - 5*a*c^7*d^3 + 2*a*c^6*d^4 + 4*a*c^5*d^5 - a*c^4*d^6 - a*c^3*d^7)*f*cos(f*x + e
) + (a*c^8*d^2 - a*c^7*d^3 - 2*a*c^6*d^4 + 2*a*c^5*d^5 + a*c^4*d^6 - a*c^3*d^7)*f), -1/16*(16*(c^5*d^2 - c^4*d
^3 - 2*c^3*d^4 + 2*c^2*d^5 + c*d^6 - d^7 + (c^7 - c^6*d - 2*c^5*d^2 + 2*c^4*d^3 + c^3*d^4 - c^2*d^5)*cos(f*x +
e)^3 + (c^7 + c^6*d - 4*c^5*d^2 - 2*c^4*d^3 + 5*c^3*d^4 + c^2*d^5 - 2*c*d^6)*cos(f*x + e)^2 + (2*c^6*d - c^5*
d^2 - 5*c^4*d^3 + 2*c^3*d^4 + 4*c^2*d^5 - c*d^6 - d^7)*cos(f*x + e))*sqrt(a)*arctan(1/2*sqrt((a*cos(f*x + e) +
a)/cos(f*x + e))*(2*cos(f*x + e) - 1)/(sqrt(a)*sin(f*x + e))) + (35*a*c^4*d^3 + 14*a*c^3*d^4 - 21*a*c^2*d^5 -
4*a*c*d^6 + 8*a*d^7 + (35*a*c^6*d + 14*a*c^5*d^2 - 21*a*c^4*d^3 - 4*a*c^3*d^4 + 8*a*c^2*d^5)*cos(f*x + e)^3 +
(35*a*c^6*d + 84*a*c^5*d^2 + 7*a*c^4*d^3 - 46*a*c^3*d^4 + 16*a*c*d^6)*cos(f*x + e)^2 + (70*a*c^5*d^2 + 63*a*c
^4*d^3 - 28*a*c^3*d^4 - 29*a*c^2*d^5 + 12*a*c*d^6 + 8*a*d^7)*cos(f*x + e))*sqrt(-d/(a*c + a*d))*log(((c^2 + 8*
c*d + 8*d^2)*cos(f*x + e)^3 + (c^2 + 2*c*d)*cos(f*x + e)^2 + 4*((c^2 + 3*c*d + 2*d^2)*cos(f*x + e)^2 - (c*d +
d^2)*cos(f*x + e))*sqrt(-d/(a*c + a*d))*sqrt((a*cos(f*x + e) + a)/cos(f*x + e))*sin(f*x + e) + d^2 - (6*c*d +
7*d^2)*cos(f*x + e))/(c^2*cos(f*x + e)^3 + (c^2 + 2*c*d)*cos(f*x + e)^2 + d^2 + (2*c*d + d^2)*cos(f*x + e))) -
4*((13*c^5*d^2 - 12*c^4*d^3 - 7*c^3*d^4 + 6*c^2*d^5)*cos(f*x + e)^2 + (11*c^4*d^3 - 10*c^3*d^4 - 5*c^2*d^5 +
4*c*d^6)*cos(f*x + e))*sqrt((a*cos(f*x + e) + a)/cos(f*x + e))*sin(f*x + e) - 8*sqrt(2)*(a*c^5*d^2 + 2*a*c^4*d
^3 + a*c^3*d^4 + (a*c^7 + 2*a*c^6*d + a*c^5*d^2)*cos(f*x + e)^3 + (a*c^7 + 4*a*c^6*d + 5*a*c^5*d^2 + 2*a*c^4*d
^3)*cos(f*x + e)^2 + (2*a*c^6*d + 5*a*c^5*d^2 + 4*a*c^4*d^3 + a*c^3*d^4)*cos(f*x + e))*arctan(1/4*sqrt(2)*sqrt
((a*cos(f*x + e) + a)/cos(f*x + e))*(3*cos(f*x + e) - 1)/(sqrt(a)*sin(f*x + e)))/sqrt(a))/((a*c^10 - a*c^9*d -
2*a*c^8*d^2 + 2*a*c^7*d^3 + a*c^6*d^4 - a*c^5*d^5)*f*cos(f*x + e)^3 + (a*c^10 + a*c^9*d - 4*a*c^8*d^2 - 2*a*c
^7*d^3 + 5*a*c^6*d^4 + a*c^5*d^5 - 2*a*c^4*d^6)*f*cos(f*x + e)^2 + (2*a*c^9*d - a*c^8*d^2 - 5*a*c^7*d^3 + 2*a*
c^6*d^4 + 4*a*c^5*d^5 - a*c^4*d^6 - a*c^3*d^7)*f*cos(f*x + e) + (a*c^8*d^2 - a*c^7*d^3 - 2*a*c^6*d^4 + 2*a*c^5
*d^5 + a*c^4*d^6 - a*c^3*d^7)*f), -1/8*((35*a*c^4*d^3 + 14*a*c^3*d^4 - 21*a*c^2*d^5 - 4*a*c*d^6 + 8*a*d^7 + (3
5*a*c^6*d + 14*a*c^5*d^2 - 21*a*c^4*d^3 - 4*a*c^3*d^4 + 8*a*c^2*d^5)*cos(f*x + e)^3 + (35*a*c^6*d + 84*a*c^5*d
^2 + 7*a*c^4*d^3 - 46*a*c^3*d^4 + 16*a*c*d^6)*cos(f*x + e)^2 + (70*a*c^5*d^2 + 63*a*c^4*d^3 - 28*a*c^3*d^4 - 2
9*a*c^2*d^5 + 12*a*c*d^6 + 8*a*d^7)*cos(f*x + e))*sqrt(d/(a*c + a*d))*arctan(1/2*((c + 2*d)*cos(f*x + e) - d)*
sqrt(d/(a*c + a*d))*sqrt((a*cos(f*x + e) + a)/cos(f*x + e))/(d*sin(f*x + e))) + 8*(c^5*d^2 - c^4*d^3 - 2*c^3*d
^4 + 2*c^2*d^5 + c*d^6 - d^7 + (c^7 - c^6*d - 2*c^5*d^2 + 2*c^4*d^3 + c^3*d^4 - c^2*d^5)*cos(f*x + e)^3 + (c^7
+ c^6*d - 4*c^5*d^2 - 2*c^4*d^3 + 5*c^3*d^4 + c^2*d^5 - 2*c*d^6)*cos(f*x + e)^2 + (2*c^6*d - c^5*d^2 - 5*c^4*
d^3 + 2*c^3*d^4 + 4*c^2*d^5 - c*d^6 - d^7)*cos(f*x + e))*sqrt(a)*arctan(1/2*sqrt((a*cos(f*x + e) + a)/cos(f*x
+ e))*(2*cos(f*x + e) - 1)/(sqrt(a)*sin(f*x + e))) - 2*((13*c^5*d^2 - 12*c^4*d^3 - 7*c^3*d^4 + 6*c^2*d^5)*cos(
f*x + e)^2 + (11*c^4*d^3 - 10*c^3*d^4 - 5*c^2*d^5 + 4*c*d^6)*cos(f*x + e))*sqrt((a*cos(f*x + e) + a)/cos(f*x +
e))*sin(f*x + e) - 4*sqrt(2)*(a*c^5*d^2 + 2*a*c^4*d^3 + a*c^3*d^4 + (a*c^7 + 2*a*c^6*d + a*c^5*d^2)*cos(f*x +
e)^3 + (a*c^7 + 4*a*c^6*d + 5*a*c^5*d^2 + 2*a*c^4*d^3)*cos(f*x + e)^2 + (2*a*c^6*d + 5*a*c^5*d^2 + 4*a*c^4*d^
3 + a*c^3*d^4)*cos(f*x + e))*arctan(1/4*sqrt(2)*sqrt((a*cos(f*x + e) + a)/cos(f*x + e))*(3*cos(f*x + e) - 1)/(
sqrt(a)*sin(f*x + e)))/sqrt(a))/((a*c^10 - a*c^9*d - 2*a*c^8*d^2 + 2*a*c^7*d^3 + a*c^6*d^4 - a*c^5*d^5)*f*cos(
f*x + e)^3 + (a*c^10 + a*c^9*d - 4*a*c^8*d^2 - 2*a*c^7*d^3 + 5*a*c^6*d^4 + a*c^5*d^5 - 2*a*c^4*d^6)*f*cos(f*x
+ e)^2 + (2*a*c^9*d - a*c^8*d^2 - 5*a*c^7*d^3 + 2*a*c^6*d^4 + 4*a*c^5*d^5 - a*c^4*d^6 - a*c^3*d^7)*f*cos(f*x +
e) + (a*c^8*d^2 - a*c^7*d^3 - 2*a*c^6*d^4 + 2*a*c^5*d^5 + a*c^4*d^6 - a*c^3*d^7)*f)]

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\sqrt{a \left (\sec{\left (e + f x \right )} + 1\right )} \left (c + d \sec{\left (e + f x \right )}\right )^{3}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(1/(sqrt(a*(sec(e + f*x) + 1))*(c + d*sec(e + f*x))**3), x)

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Giac [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(1/(c+d*sec(f*x+e))^3/(a+a*sec(f*x+e))^(1/2),x, algorithm="giac")

[Out]

Timed out