\(\int \frac {1}{\sqrt {a+a \sec (e+f x)} (c+d \sec (e+f x))^3} \, dx\) [171]

Optimal result

Integrand size = 27, antiderivative size = 653 \[ \int \frac {1}{\sqrt {a+a \sec (e+f x)} (c+d \sec (e+f x))^3} \, dx=\frac {2 \sqrt {a} \text {arctanh}\left (\frac {\sqrt {a-a \sec (e+f x)}}{\sqrt {a}}\right ) \tan (e+f x)}{c^3 f \sqrt {a-a \sec (e+f x)} \sqrt {a+a \sec (e+f x)}}-\frac {\sqrt {2} \sqrt {a} \text {arctanh}\left (\frac {\sqrt {a-a \sec (e+f x)}}{\sqrt {2} \sqrt {a}}\right ) \tan (e+f x)}{(c-d)^3 f \sqrt {a-a \sec (e+f x)} \sqrt {a+a \sec (e+f x)}}+\frac {3 \sqrt {a} d^{3/2} \text {arctanh}\left (\frac {\sqrt {d} \sqrt {a-a \sec (e+f x)}}{\sqrt {a} \sqrt {c+d}}\right ) \tan (e+f x)}{4 c (c-d) (c+d)^{5/2} f \sqrt {a-a \sec (e+f x)} \sqrt {a+a \sec (e+f x)}}+\frac {\sqrt {a} (2 c-d) d^{3/2} \text {arctanh}\left (\frac {\sqrt {d} \sqrt {a-a \sec (e+f x)}}{\sqrt {a} \sqrt {c+d}}\right ) \tan (e+f x)}{c^2 (c-d)^2 (c+d)^{3/2} f \sqrt {a-a \sec (e+f x)} \sqrt {a+a \sec (e+f x)}}+\frac {2 \sqrt {a} d^{3/2} \left (3 c^2-3 c d+d^2\right ) \text {arctanh}\left (\frac {\sqrt {d} \sqrt {a-a \sec (e+f x)}}{\sqrt {a} \sqrt {c+d}}\right ) \tan (e+f x)}{c^3 (c-d)^3 \sqrt {c+d} f \sqrt {a-a \sec (e+f x)} \sqrt {a+a \sec (e+f x)}}+\frac {d^2 \tan (e+f x)}{2 c \left (c^2-d^2\right ) f \sqrt {a+a \sec (e+f x)} (c+d \sec (e+f x))^2}+\frac {3 d^2 \tan (e+f x)}{4 c (c-d) (c+d)^2 f \sqrt {a+a \sec (e+f x)} (c+d \sec (e+f x))}+\frac {(2 c-d) d^2 \tan (e+f x)}{c^2 (c-d)^2 (c+d) f \sqrt {a+a \sec (e+f x)} (c+d \sec (e+f x))} \]

[Out]

1/2*d^2*tan(f*x+e)/c/(c^2-d^2)/f/(c+d*sec(f*x+e))^2/(a+a*sec(f*x+e))^(1/2)+3/4*d^2*tan(f*x+e)/c/(c-d)/(c+d)^2/
f/(c+d*sec(f*x+e))/(a+a*sec(f*x+e))^(1/2)+(2*c-d)*d^2*tan(f*x+e)/c^2/(c-d)^2/(c+d)/f/(c+d*sec(f*x+e))/(a+a*sec
(f*x+e))^(1/2)+2*arctanh((a-a*sec(f*x+e))^(1/2)/a^(1/2))*a^(1/2)*tan(f*x+e)/c^3/f/(a-a*sec(f*x+e))^(1/2)/(a+a*
sec(f*x+e))^(1/2)+3/4*d^(3/2)*arctanh(d^(1/2)*(a-a*sec(f*x+e))^(1/2)/a^(1/2)/(c+d)^(1/2))*a^(1/2)*tan(f*x+e)/c
/(c-d)/(c+d)^(5/2)/f/(a-a*sec(f*x+e))^(1/2)/(a+a*sec(f*x+e))^(1/2)+(2*c-d)*d^(3/2)*arctanh(d^(1/2)*(a-a*sec(f*
x+e))^(1/2)/a^(1/2)/(c+d)^(1/2))*a^(1/2)*tan(f*x+e)/c^2/(c-d)^2/(c+d)^(3/2)/f/(a-a*sec(f*x+e))^(1/2)/(a+a*sec(
f*x+e))^(1/2)-arctanh(1/2*(a-a*sec(f*x+e))^(1/2)*2^(1/2)/a^(1/2))*2^(1/2)*a^(1/2)*tan(f*x+e)/(c-d)^3/f/(a-a*se
c(f*x+e))^(1/2)/(a+a*sec(f*x+e))^(1/2)+2*d^(3/2)*(3*c^2-3*c*d+d^2)*arctanh(d^(1/2)*(a-a*sec(f*x+e))^(1/2)/a^(1
/2)/(c+d)^(1/2))*a^(1/2)*tan(f*x+e)/c^3/(c-d)^3/f/(c+d)^(1/2)/(a-a*sec(f*x+e))^(1/2)/(a+a*sec(f*x+e))^(1/2)

Rubi [A] (verified)

Time = 0.76 (sec) , antiderivative size = 653, normalized size of antiderivative = 1.00, number of steps used = 16, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {4025, 186, 65, 212, 44, 214} \[ \int \frac {1}{\sqrt {a+a \sec (e+f x)} (c+d \sec (e+f x))^3} \, dx=\frac {2 \sqrt {a} \tan (e+f x) \text {arctanh}\left (\frac {\sqrt {a-a \sec (e+f x)}}{\sqrt {a}}\right )}{c^3 f \sqrt {a-a \sec (e+f x)} \sqrt {a \sec (e+f x)+a}}+\frac {\sqrt {a} d^{3/2} (2 c-d) \tan (e+f x) \text {arctanh}\left (\frac {\sqrt {d} \sqrt {a-a \sec (e+f x)}}{\sqrt {a} \sqrt {c+d}}\right )}{c^2 f (c-d)^2 (c+d)^{3/2} \sqrt {a-a \sec (e+f x)} \sqrt {a \sec (e+f x)+a}}+\frac {2 \sqrt {a} d^{3/2} \left (3 c^2-3 c d+d^2\right ) \tan (e+f x) \text {arctanh}\left (\frac {\sqrt {d} \sqrt {a-a \sec (e+f x)}}{\sqrt {a} \sqrt {c+d}}\right )}{c^3 f (c-d)^3 \sqrt {c+d} \sqrt {a-a \sec (e+f x)} \sqrt {a \sec (e+f x)+a}}+\frac {3 \sqrt {a} d^{3/2} \tan (e+f x) \text {arctanh}\left (\frac {\sqrt {d} \sqrt {a-a \sec (e+f x)}}{\sqrt {a} \sqrt {c+d}}\right )}{4 c f (c-d) (c+d)^{5/2} \sqrt {a-a \sec (e+f x)} \sqrt {a \sec (e+f x)+a}}-\frac {\sqrt {2} \sqrt {a} \tan (e+f x) \text {arctanh}\left (\frac {\sqrt {a-a \sec (e+f x)}}{\sqrt {2} \sqrt {a}}\right )}{f (c-d)^3 \sqrt {a-a \sec (e+f x)} \sqrt {a \sec (e+f x)+a}}+\frac {d^2 (2 c-d) \tan (e+f x)}{c^2 f (c-d)^2 (c+d) \sqrt {a \sec (e+f x)+a} (c+d \sec (e+f x))}+\frac {d^2 \tan (e+f x)}{2 c f \left (c^2-d^2\right ) \sqrt {a \sec (e+f x)+a} (c+d \sec (e+f x))^2}+\frac {3 d^2 \tan (e+f x)}{4 c f (c-d) (c+d)^2 \sqrt {a \sec (e+f x)+a} (c+d \sec (e+f x))} \]

[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 44

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] && ILtQ[m, -1] &&  !IntegerQ[n] && LtQ[n, 0]

Rule 65

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 186

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 212

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

Rule 214

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

Rule 4025

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]

Rubi steps \begin{align*} \text {integral}& = -\frac {\left (a^2 \tan (e+f x)\right ) \text {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 ) \text {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)) \text {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)) \text {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 ) \text {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 ) \text {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 ) \text {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)) \text {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)) \text {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 ) \text {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 ) \text {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 ) \text {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} \text {arctanh}\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} \text {arctanh}\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 ) \text {arctanh}\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 ) \text {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 ) \text {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} \text {arctanh}\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} \text {arctanh}\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} \text {arctanh}\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 ) \text {arctanh}\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 ) \text {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} \text {arctanh}\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} \text {arctanh}\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} \text {arctanh}\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} \text {arctanh}\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 ) \text {arctanh}\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 [B] (warning: unable to verify)

Leaf count is larger than twice the leaf count of optimal. \(2940\) vs. \(2(653)=1306\).

Time = 22.45 (sec) , antiderivative size = 2940, normalized size of antiderivative = 4.50 \[ \int \frac {1}{\sqrt {a+a \sec (e+f x)} (c+d \sec (e+f x))^3} \, dx=\text {Result too large to show} \]

[In]

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

[Out]

(Cos[(e + f*x)/2]*(d + c*Cos[e + f*x])^3*Sec[e + f*x]^4*(-1/2*(d^2*(-13*c^2 - c*d + 6*d^2)*Sin[(e + f*x)/2])/(
c^3*(-c + d)^2*(c + d)^2) - (d^4*Sin[(e + f*x)/2])/(c^3*(-c + d)*(c + d)*(d + c*Cos[e + f*x])^2) + (-15*c^2*d^
3*Sin[(e + f*x)/2] - c*d^4*Sin[(e + f*x)/2] + 8*d^5*Sin[(e + f*x)/2])/(2*c^3*(-c + d)^2*(c + d)^2*(d + c*Cos[e
+ f*x]))))/(f*Sqrt[a*(1 + Sec[e + f*x])]*(c + d*Sec[e + f*x])^3) - (Cos[(e + f*x)/2]*(d + c*Cos[e + f*x])^3*(
(Sqrt[2]*d^(3/2)*(35*c^4 + 14*c^3*d - 21*c^2*d^2 - 4*c*d^3 + 8*d^4)*ArcTan[(Sqrt[d]*Tan[(e + f*x)/2])/(Sqrt[-c
- d]*Sqrt[-(Cos[e + f*x]/(1 + Cos[e + f*x]))])])/(Sqrt[-c - d]*(c - d)) - 2*Sqrt[2]*(c^2 - d^2)^2*Log[Sec[(e
+ f*x)/2]^2*(-1 + 2*Cos[e + f*x] - 2*Sqrt[-(Cos[e + f*x]/(1 + Cos[e + f*x]))]*Sin[e + f*x])] + 2*Sqrt[2]*(c^2
- d^2)^2*Log[Sec[(e + f*x)/2]^2*(-1 + 2*Cos[e + f*x] + 2*Sqrt[-(Cos[e + f*x]/(1 + Cos[e + f*x]))]*Sin[e + f*x]
)] + (8*c^3*(c + d)^2*Log[Tan[(e + f*x)/2] + Sqrt[-1 + Tan[(e + f*x)/2]^2]])/(c - d))*((-2*c*d*Sec[(e + f*x)/2
])/((-c + d)^2*(c + d)^2*(d + c*Cos[e + f*x])*Sqrt[Sec[e + f*x]]) - (13*d^2*Sec[(e + f*x)/2])/(8*(-c + d)^2*(c
+ d)^2*(d + c*Cos[e + f*x])*Sqrt[Sec[e + f*x]]) + (d^3*Sec[(e + f*x)/2])/(8*c*(-c + d)^2*(c + d)^2*(d + c*Cos
[e + f*x])*Sqrt[Sec[e + f*x]]) + (d^4*Sec[(e + f*x)/2])/(2*c^2*(-c + d)^2*(c + d)^2*(d + c*Cos[e + f*x])*Sqrt[
Sec[e + f*x]]) + (c^2*Sec[(e + f*x)/2]*Sqrt[Sec[e + f*x]])/(2*(-c + d)^2*(c + d)^2*(d + c*Cos[e + f*x])) + (3*
d^2*Sec[(e + f*x)/2]*Sqrt[Sec[e + f*x]])/(8*(-c + d)^2*(c + d)^2*(d + c*Cos[e + f*x])) + (d^3*Sec[(e + f*x)/2]
*Sqrt[Sec[e + f*x]])/(8*c*(-c + d)^2*(c + d)^2*(d + c*Cos[e + f*x])) + (c^2*Cos[2*(e + f*x)]*Sec[(e + f*x)/2]*
Sqrt[Sec[e + f*x]])/(2*(-c + d)^2*(c + d)^2*(d + c*Cos[e + f*x])) - (d^2*Cos[2*(e + f*x)]*Sec[(e + f*x)/2]*Sqr
t[Sec[e + f*x]])/((-c + d)^2*(c + d)^2*(d + c*Cos[e + f*x])) + (d^4*Cos[2*(e + f*x)]*Sec[(e + f*x)/2]*Sqrt[Sec
[e + f*x]])/(2*c^2*(-c + d)^2*(c + d)^2*(d + c*Cos[e + f*x])))*Sec[e + f*x]^(7/2)*Sqrt[Cos[(e + f*x)/2]^2*Sec[
e + f*x]]*Sqrt[-1 + Tan[(e + f*x)/2]^2])/(4*c^3*(c - d)^2*(c + d)^2*f*Sqrt[a*(1 + Sec[e + f*x])]*(c + d*Sec[e
+ f*x])^3*(-1/8*(((Sqrt[2]*d^(3/2)*(35*c^4 + 14*c^3*d - 21*c^2*d^2 - 4*c*d^3 + 8*d^4)*ArcTan[(Sqrt[d]*Tan[(e +
f*x)/2])/(Sqrt[-c - d]*Sqrt[-(Cos[e + f*x]/(1 + Cos[e + f*x]))])])/(Sqrt[-c - d]*(c - d)) - 2*Sqrt[2]*(c^2 -
d^2)^2*Log[Sec[(e + f*x)/2]^2*(-1 + 2*Cos[e + f*x] - 2*Sqrt[-(Cos[e + f*x]/(1 + Cos[e + f*x]))]*Sin[e + f*x])]
+ 2*Sqrt[2]*(c^2 - d^2)^2*Log[Sec[(e + f*x)/2]^2*(-1 + 2*Cos[e + f*x] + 2*Sqrt[-(Cos[e + f*x]/(1 + Cos[e + f*
x]))]*Sin[e + f*x])] + (8*c^3*(c + d)^2*Log[Tan[(e + f*x)/2] + Sqrt[-1 + Tan[(e + f*x)/2]^2]])/(c - d))*Sec[(e
+ f*x)/2]^2*Sqrt[Cos[(e + f*x)/2]^2*Sec[e + f*x]]*Tan[(e + f*x)/2])/(c^3*(c - d)^2*(c + d)^2*Sqrt[-1 + Tan[(e
+ f*x)/2]^2]) - (Sqrt[Cos[(e + f*x)/2]^2*Sec[e + f*x]]*Sqrt[-1 + Tan[(e + f*x)/2]^2]*((-2*Sqrt[2]*(c^2 - d^2)
^2*Cos[(e + f*x)/2]^2*(Sec[(e + f*x)/2]^2*(-2*Cos[e + f*x]*Sqrt[-(Cos[e + f*x]/(1 + Cos[e + f*x]))] - 2*Sin[e
+ f*x] - (Sin[e + f*x]*(-((Cos[e + f*x]*Sin[e + f*x])/(1 + Cos[e + f*x])^2) + Sin[e + f*x]/(1 + Cos[e + f*x]))
)/Sqrt[-(Cos[e + f*x]/(1 + Cos[e + f*x]))]) + Sec[(e + f*x)/2]^2*(-1 + 2*Cos[e + f*x] - 2*Sqrt[-(Cos[e + f*x]/
(1 + Cos[e + f*x]))]*Sin[e + f*x])*Tan[(e + f*x)/2]))/(-1 + 2*Cos[e + f*x] - 2*Sqrt[-(Cos[e + f*x]/(1 + Cos[e
+ f*x]))]*Sin[e + f*x]) + (2*Sqrt[2]*(c^2 - d^2)^2*Cos[(e + f*x)/2]^2*(Sec[(e + f*x)/2]^2*(2*Cos[e + f*x]*Sqrt
[-(Cos[e + f*x]/(1 + Cos[e + f*x]))] - 2*Sin[e + f*x] + (Sin[e + f*x]*(-((Cos[e + f*x]*Sin[e + f*x])/(1 + Cos[
e + f*x])^2) + Sin[e + f*x]/(1 + Cos[e + f*x])))/Sqrt[-(Cos[e + f*x]/(1 + Cos[e + f*x]))]) + Sec[(e + f*x)/2]^
2*(-1 + 2*Cos[e + f*x] + 2*Sqrt[-(Cos[e + f*x]/(1 + Cos[e + f*x]))]*Sin[e + f*x])*Tan[(e + f*x)/2]))/(-1 + 2*C
os[e + f*x] + 2*Sqrt[-(Cos[e + f*x]/(1 + Cos[e + f*x]))]*Sin[e + f*x]) + (Sqrt[2]*d^(3/2)*(35*c^4 + 14*c^3*d -
21*c^2*d^2 - 4*c*d^3 + 8*d^4)*((Sqrt[d]*Sec[(e + f*x)/2]^2)/(2*Sqrt[-c - d]*Sqrt[-(Cos[e + f*x]/(1 + Cos[e +
f*x]))]) - (Sqrt[d]*(-((Cos[e + f*x]*Sin[e + f*x])/(1 + Cos[e + f*x])^2) + Sin[e + f*x]/(1 + Cos[e + f*x]))*Ta
n[(e + f*x)/2])/(2*Sqrt[-c - d]*(-(Cos[e + f*x]/(1 + Cos[e + f*x])))^(3/2))))/(Sqrt[-c - d]*(c - d)*(1 - (d*(1
+ Cos[e + f*x])*Sec[e + f*x]*Tan[(e + f*x)/2]^2)/(-c - d))) + (8*c^3*(c + d)^2*(Sec[(e + f*x)/2]^2/2 + (Sec[(
e + f*x)/2]^2*Tan[(e + f*x)/2])/(2*Sqrt[-1 + Tan[(e + f*x)/2]^2])))/((c - d)*(Tan[(e + f*x)/2] + Sqrt[-1 + Tan
[(e + f*x)/2]^2]))))/(4*c^3*(c - d)^2*(c + d)^2) - (((Sqrt[2]*d^(3/2)*(35*c^4 + 14*c^3*d - 21*c^2*d^2 - 4*c*d^
3 + 8*d^4)*ArcTan[(Sqrt[d]*Tan[(e + f*x)/2])/(Sqrt[-c - d]*Sqrt[-(Cos[e + f*x]/(1 + Cos[e + f*x]))])])/(Sqrt[-
c - d]*(c - d)) - 2*Sqrt[2]*(c^2 - d^2)^2*Log[Sec[(e + f*x)/2]^2*(-1 + 2*Cos[e + f*x] - 2*Sqrt[-(Cos[e + f*x]/
(1 + Cos[e + f*x]))]*Sin[e + f*x])] + 2*Sqrt[2]*(c^2 - d^2)^2*Log[Sec[(e + f*x)/2]^2*(-1 + 2*Cos[e + f*x] + 2*
Sqrt[-(Cos[e + f*x]/(1 + Cos[e + f*x]))]*Sin[e + f*x])] + (8*c^3*(c + d)^2*Log[Tan[(e + f*x)/2] + Sqrt[-1 + Ta
n[(e + f*x)/2]^2]])/(c - d))*Sqrt[-1 + Tan[(e + f*x)/2]^2]*(-(Cos[(e + f*x)/2]*Sec[e + f*x]*Sin[(e + f*x)/2])
+ Cos[(e + f*x)/2]^2*Sec[e + f*x]*Tan[e + f*x]))/(8*c^3*(c - d)^2*(c + d)^2*Sqrt[Cos[(e + f*x)/2]^2*Sec[e + f*
x]])))

Maple [B] (warning: unable to verify)

Leaf count of result is larger than twice the leaf count of optimal. \(97276\) vs. \(2(564)=1128\).

Time = 18.67 (sec) , antiderivative size = 97277, normalized size of antiderivative = 148.97

method	result	size
default	\(\text {Expression too large to display}\)	\(97277\)

[In]

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

[Out]

result too large to display

Fricas [F(-1)]

Timed out. \[ \int \frac {1}{\sqrt {a+a \sec (e+f x)} (c+d \sec (e+f x))^3} \, dx=\text {Timed out} \]

[In]

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

[Out]

Timed out

Sympy [F]

\[ \int \frac {1}{\sqrt {a+a \sec (e+f x)} (c+d \sec (e+f x))^3} \, dx=\int \frac {1}{\sqrt {a \left (\sec {\left (e + f x \right )} + 1\right )} \left (c + d \sec {\left (e + f x \right )}\right )^{3}}\, dx \]

[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)

Maxima [F]

\[ \int \frac {1}{\sqrt {a+a \sec (e+f x)} (c+d \sec (e+f x))^3} \, dx=\int { \frac {1}{\sqrt {a \sec \left (f x + e\right ) + a} {\left (d \sec \left (f x + e\right ) + c\right )}^{3}} \,d x } \]

[In]

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

[Out]

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

Giac [F(-2)]

Exception generated. \[ \int \frac {1}{\sqrt {a+a \sec (e+f x)} (c+d \sec (e+f x))^3} \, dx=\text {Exception raised: TypeError} \]

[In]

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

[Out]

Exception raised: TypeError >> an error occurred running a Giac command:INPUT:sage2:=int(sage0,sageVARx):;OUTP
UT:index.cc index_m i_lex_is_greater Error: Bad Argument Value

Mupad [F(-1)]

Timed out. \[ \int \frac {1}{\sqrt {a+a \sec (e+f x)} (c+d \sec (e+f x))^3} \, dx=\int \frac {1}{\sqrt {a+\frac {a}{\cos \left (e+f\,x\right )}}\,{\left (c+\frac {d}{\cos \left (e+f\,x\right )}\right )}^3} \,d x \]

[In]

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

[Out]

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