\(\int \frac {1}{(a+a \sec (e+f x))^{5/2} (c+d \sec (e+f x))} \, dx\) [181]

Optimal result

Integrand size = 27, antiderivative size = 592 \[ \int \frac {1}{(a+a \sec (e+f x))^{5/2} (c+d \sec (e+f x))} \, dx=-\frac {\tan (e+f x)}{4 a^2 (c-d) f (1+\sec (e+f x))^2 \sqrt {a+a \sec (e+f x)}}-\frac {(c-2 d) \tan (e+f x)}{2 a^2 (c-d)^2 f (1+\sec (e+f x)) \sqrt {a+a \sec (e+f x)}}-\frac {3 \tan (e+f x)}{16 a^2 (c-d) f (1+\sec (e+f x)) \sqrt {a+a \sec (e+f x)}}+\frac {2 \text {arctanh}\left (\frac {\sqrt {a-a \sec (e+f x)}}{\sqrt {a}}\right ) \tan (e+f x)}{a^{3/2} c f \sqrt {a-a \sec (e+f x)} \sqrt {a+a \sec (e+f x)}}-\frac {(c-2 d) \text {arctanh}\left (\frac {\sqrt {a-a \sec (e+f x)}}{\sqrt {2} \sqrt {a}}\right ) \tan (e+f x)}{2 \sqrt {2} a^{3/2} (c-d)^2 f \sqrt {a-a \sec (e+f x)} \sqrt {a+a \sec (e+f x)}}-\frac {3 \text {arctanh}\left (\frac {\sqrt {a-a \sec (e+f x)}}{\sqrt {2} \sqrt {a}}\right ) \tan (e+f x)}{16 \sqrt {2} a^{3/2} (c-d) f \sqrt {a-a \sec (e+f x)} \sqrt {a+a \sec (e+f x)}}-\frac {\sqrt {2} \left (c^2-3 c d+3 d^2\right ) \text {arctanh}\left (\frac {\sqrt {a-a \sec (e+f x)}}{\sqrt {2} \sqrt {a}}\right ) \tan (e+f x)}{a^{3/2} (c-d)^3 f \sqrt {a-a \sec (e+f x)} \sqrt {a+a \sec (e+f x)}}+\frac {2 d^{7/2} \text {arctanh}\left (\frac {\sqrt {d} \sqrt {a-a \sec (e+f x)}}{\sqrt {a} \sqrt {c+d}}\right ) \tan (e+f x)}{a^{3/2} c (c-d)^3 \sqrt {c+d} f \sqrt {a-a \sec (e+f x)} \sqrt {a+a \sec (e+f x)}} \]

[Out]

-1/4*tan(f*x+e)/a^2/(c-d)/f/(1+sec(f*x+e))^2/(a+a*sec(f*x+e))^(1/2)-1/2*(c-2*d)*tan(f*x+e)/a^2/(c-d)^2/f/(1+se
c(f*x+e))/(a+a*sec(f*x+e))^(1/2)-3/16*tan(f*x+e)/a^2/(c-d)/f/(1+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))*tan(f*x+e)/a^(3/2)/c/f/(a-a*sec(f*x+e))^(1/2)/(a+a*sec(f*x+e))^(1/2)-1/4*(c-2*d
)*arctanh(1/2*(a-a*sec(f*x+e))^(1/2)*2^(1/2)/a^(1/2))*tan(f*x+e)/a^(3/2)/(c-d)^2/f*2^(1/2)/(a-a*sec(f*x+e))^(1
/2)/(a+a*sec(f*x+e))^(1/2)-3/32*arctanh(1/2*(a-a*sec(f*x+e))^(1/2)*2^(1/2)/a^(1/2))*tan(f*x+e)/a^(3/2)/(c-d)/f
*2^(1/2)/(a-a*sec(f*x+e))^(1/2)/(a+a*sec(f*x+e))^(1/2)-(c^2-3*c*d+3*d^2)*arctanh(1/2*(a-a*sec(f*x+e))^(1/2)*2^
(1/2)/a^(1/2))*2^(1/2)*tan(f*x+e)/a^(3/2)/(c-d)^3/f/(a-a*sec(f*x+e))^(1/2)/(a+a*sec(f*x+e))^(1/2)+2*d^(7/2)*ar
ctanh(d^(1/2)*(a-a*sec(f*x+e))^(1/2)/a^(1/2)/(c+d)^(1/2))*tan(f*x+e)/a^(3/2)/c/(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 = 0.55 (sec) , antiderivative size = 592, 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} \[ \int \frac {1}{(a+a \sec (e+f x))^{5/2} (c+d \sec (e+f x))} \, dx=-\frac {\sqrt {2} \left (c^2-3 c d+3 d^2\right ) \tan (e+f x) \text {arctanh}\left (\frac {\sqrt {a-a \sec (e+f x)}}{\sqrt {2} \sqrt {a}}\right )}{a^{3/2} f (c-d)^3 \sqrt {a-a \sec (e+f x)} \sqrt {a \sec (e+f x)+a}}+\frac {2 d^{7/2} \tan (e+f x) \text {arctanh}\left (\frac {\sqrt {d} \sqrt {a-a \sec (e+f x)}}{\sqrt {a} \sqrt {c+d}}\right )}{a^{3/2} c f (c-d)^3 \sqrt {c+d} \sqrt {a-a \sec (e+f x)} \sqrt {a \sec (e+f x)+a}}-\frac {3 \tan (e+f x) \text {arctanh}\left (\frac {\sqrt {a-a \sec (e+f x)}}{\sqrt {2} \sqrt {a}}\right )}{16 \sqrt {2} a^{3/2} f (c-d) \sqrt {a-a \sec (e+f x)} \sqrt {a \sec (e+f x)+a}}-\frac {(c-2 d) \tan (e+f x) \text {arctanh}\left (\frac {\sqrt {a-a \sec (e+f x)}}{\sqrt {2} \sqrt {a}}\right )}{2 \sqrt {2} a^{3/2} f (c-d)^2 \sqrt {a-a \sec (e+f x)} \sqrt {a \sec (e+f x)+a}}+\frac {2 \tan (e+f x) \text {arctanh}\left (\frac {\sqrt {a-a \sec (e+f x)}}{\sqrt {a}}\right )}{a^{3/2} c f \sqrt {a-a \sec (e+f x)} \sqrt {a \sec (e+f x)+a}}-\frac {3 \tan (e+f x)}{16 a^2 f (c-d) (\sec (e+f x)+1) \sqrt {a \sec (e+f x)+a}}-\frac {(c-2 d) \tan (e+f x)}{2 a^2 f (c-d)^2 (\sec (e+f x)+1) \sqrt {a \sec (e+f x)+a}}-\frac {\tan (e+f x)}{4 a^2 f (c-d) (\sec (e+f x)+1)^2 \sqrt {a \sec (e+f x)+a}} \]

[In]

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

[Out]

-1/4*Tan[e + f*x]/(a^2*(c - d)*f*(1 + Sec[e + f*x])^2*Sqrt[a + a*Sec[e + f*x]]) - ((c - 2*d)*Tan[e + f*x])/(2*
a^2*(c - d)^2*f*(1 + Sec[e + f*x])*Sqrt[a + a*Sec[e + f*x]]) - (3*Tan[e + f*x])/(16*a^2*(c - d)*f*(1 + Sec[e +
f*x])*Sqrt[a + a*Sec[e + f*x]]) + (2*ArcTanh[Sqrt[a - a*Sec[e + f*x]]/Sqrt[a]]*Tan[e + f*x])/(a^(3/2)*c*f*Sqr
t[a - a*Sec[e + f*x]]*Sqrt[a + a*Sec[e + f*x]]) - ((c - 2*d)*ArcTanh[Sqrt[a - a*Sec[e + f*x]]/(Sqrt[2]*Sqrt[a]
)]*Tan[e + f*x])/(2*Sqrt[2]*a^(3/2)*(c - d)^2*f*Sqrt[a - a*Sec[e + f*x]]*Sqrt[a + a*Sec[e + f*x]]) - (3*ArcTan
h[Sqrt[a - a*Sec[e + f*x]]/(Sqrt[2]*Sqrt[a])]*Tan[e + f*x])/(16*Sqrt[2]*a^(3/2)*(c - d)*f*Sqrt[a - a*Sec[e + f
*x]]*Sqrt[a + a*Sec[e + f*x]]) - (Sqrt[2]*(c^2 - 3*c*d + 3*d^2)*ArcTanh[Sqrt[a - a*Sec[e + f*x]]/(Sqrt[2]*Sqrt
[a])]*Tan[e + f*x])/(a^(3/2)*(c - d)^3*f*Sqrt[a - a*Sec[e + f*x]]*Sqrt[a + a*Sec[e + f*x]]) + (2*d^(7/2)*ArcTa
nh[(Sqrt[d]*Sqrt[a - a*Sec[e + f*x]])/(Sqrt[a]*Sqrt[c + d])]*Tan[e + f*x])/(a^(3/2)*c*(c - d)^3*Sqrt[c + d]*f*
Sqrt[a - a*Sec[e + f*x]]*Sqrt[a + a*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)^3 (c+d x)} \, 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^3 c x \sqrt {a-a x}}-\frac {1}{a^3 (c-d) (1+x)^3 \sqrt {a-a x}}+\frac {-c+2 d}{a^3 (c-d)^2 (1+x)^2 \sqrt {a-a x}}+\frac {-c^2+3 c d-3 d^2}{a^3 (c-d)^3 (1+x) \sqrt {a-a x}}+\frac {d^4}{a^3 c (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 {\tan (e+f x) \text {Subst}\left (\int \frac {1}{x \sqrt {a-a x}} \, dx,x,\sec (e+f x)\right )}{a c f \sqrt {a-a \sec (e+f x)} \sqrt {a+a \sec (e+f x)}}+\frac {((c-2 d) \tan (e+f x)) \text {Subst}\left (\int \frac {1}{(1+x)^2 \sqrt {a-a x}} \, dx,x,\sec (e+f x)\right )}{a (c-d)^2 f \sqrt {a-a \sec (e+f x)} \sqrt {a+a \sec (e+f x)}}+\frac {\tan (e+f x) \text {Subst}\left (\int \frac {1}{(1+x)^3 \sqrt {a-a x}} \, dx,x,\sec (e+f x)\right )}{a (c-d) f \sqrt {a-a \sec (e+f x)} \sqrt {a+a \sec (e+f x)}}-\frac {\left (d^4 \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 )}{a c (c-d)^3 f \sqrt {a-a \sec (e+f x)} \sqrt {a+a \sec (e+f x)}}+\frac {\left (\left (c^2-3 c d+3 d^2\right ) \tan (e+f x)\right ) \text {Subst}\left (\int \frac {1}{(1+x) \sqrt {a-a x}} \, dx,x,\sec (e+f x)\right )}{a (c-d)^3 f \sqrt {a-a \sec (e+f x)} \sqrt {a+a \sec (e+f x)}} \\ & = -\frac {\tan (e+f x)}{4 a^2 (c-d) f (1+\sec (e+f x))^2 \sqrt {a+a \sec (e+f x)}}-\frac {(c-2 d) \tan (e+f x)}{2 a^2 (c-d)^2 f (1+\sec (e+f x)) \sqrt {a+a \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 )}{a^2 c f \sqrt {a-a \sec (e+f x)} \sqrt {a+a \sec (e+f x)}}+\frac {((c-2 d) \tan (e+f x)) \text {Subst}\left (\int \frac {1}{(1+x) \sqrt {a-a x}} \, dx,x,\sec (e+f x)\right )}{4 a (c-d)^2 f \sqrt {a-a \sec (e+f x)} \sqrt {a+a \sec (e+f x)}}+\frac {(3 \tan (e+f x)) \text {Subst}\left (\int \frac {1}{(1+x)^2 \sqrt {a-a x}} \, dx,x,\sec (e+f x)\right )}{8 a (c-d) f \sqrt {a-a \sec (e+f x)} \sqrt {a+a \sec (e+f x)}}+\frac {\left (2 d^4 \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 )}{a^2 c (c-d)^3 f \sqrt {a-a \sec (e+f x)} \sqrt {a+a \sec (e+f x)}}-\frac {\left (2 \left (c^2-3 c d+3 d^2\right ) \tan (e+f x)\right ) \text {Subst}\left (\int \frac {1}{2-\frac {x^2}{a}} \, dx,x,\sqrt {a-a \sec (e+f x)}\right )}{a^2 (c-d)^3 f \sqrt {a-a \sec (e+f x)} \sqrt {a+a \sec (e+f x)}} \\ & = -\frac {\tan (e+f x)}{4 a^2 (c-d) f (1+\sec (e+f x))^2 \sqrt {a+a \sec (e+f x)}}-\frac {(c-2 d) \tan (e+f x)}{2 a^2 (c-d)^2 f (1+\sec (e+f x)) \sqrt {a+a \sec (e+f x)}}-\frac {3 \tan (e+f x)}{16 a^2 (c-d) f (1+\sec (e+f x)) \sqrt {a+a \sec (e+f x)}}+\frac {2 \text {arctanh}\left (\frac {\sqrt {a-a \sec (e+f x)}}{\sqrt {a}}\right ) \tan (e+f x)}{a^{3/2} c f \sqrt {a-a \sec (e+f x)} \sqrt {a+a \sec (e+f x)}}-\frac {\sqrt {2} \left (c^2-3 c d+3 d^2\right ) \text {arctanh}\left (\frac {\sqrt {a-a \sec (e+f x)}}{\sqrt {2} \sqrt {a}}\right ) \tan (e+f x)}{a^{3/2} (c-d)^3 f \sqrt {a-a \sec (e+f x)} \sqrt {a+a \sec (e+f x)}}+\frac {2 d^{7/2} \text {arctanh}\left (\frac {\sqrt {d} \sqrt {a-a \sec (e+f x)}}{\sqrt {a} \sqrt {c+d}}\right ) \tan (e+f x)}{a^{3/2} c (c-d)^3 \sqrt {c+d} f \sqrt {a-a \sec (e+f x)} \sqrt {a+a \sec (e+f x)}}-\frac {((c-2 d) \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 )}{2 a^2 (c-d)^2 f \sqrt {a-a \sec (e+f x)} \sqrt {a+a \sec (e+f x)}}+\frac {(3 \tan (e+f x)) \text {Subst}\left (\int \frac {1}{(1+x) \sqrt {a-a x}} \, dx,x,\sec (e+f x)\right )}{32 a (c-d) f \sqrt {a-a \sec (e+f x)} \sqrt {a+a \sec (e+f x)}} \\ & = -\frac {\tan (e+f x)}{4 a^2 (c-d) f (1+\sec (e+f x))^2 \sqrt {a+a \sec (e+f x)}}-\frac {(c-2 d) \tan (e+f x)}{2 a^2 (c-d)^2 f (1+\sec (e+f x)) \sqrt {a+a \sec (e+f x)}}-\frac {3 \tan (e+f x)}{16 a^2 (c-d) f (1+\sec (e+f x)) \sqrt {a+a \sec (e+f x)}}+\frac {2 \text {arctanh}\left (\frac {\sqrt {a-a \sec (e+f x)}}{\sqrt {a}}\right ) \tan (e+f x)}{a^{3/2} c f \sqrt {a-a \sec (e+f x)} \sqrt {a+a \sec (e+f x)}}-\frac {(c-2 d) \text {arctanh}\left (\frac {\sqrt {a-a \sec (e+f x)}}{\sqrt {2} \sqrt {a}}\right ) \tan (e+f x)}{2 \sqrt {2} a^{3/2} (c-d)^2 f \sqrt {a-a \sec (e+f x)} \sqrt {a+a \sec (e+f x)}}-\frac {\sqrt {2} \left (c^2-3 c d+3 d^2\right ) \text {arctanh}\left (\frac {\sqrt {a-a \sec (e+f x)}}{\sqrt {2} \sqrt {a}}\right ) \tan (e+f x)}{a^{3/2} (c-d)^3 f \sqrt {a-a \sec (e+f x)} \sqrt {a+a \sec (e+f x)}}+\frac {2 d^{7/2} \text {arctanh}\left (\frac {\sqrt {d} \sqrt {a-a \sec (e+f x)}}{\sqrt {a} \sqrt {c+d}}\right ) \tan (e+f x)}{a^{3/2} c (c-d)^3 \sqrt {c+d} f \sqrt {a-a \sec (e+f x)} \sqrt {a+a \sec (e+f x)}}-\frac {(3 \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 )}{16 a^2 (c-d) f \sqrt {a-a \sec (e+f x)} \sqrt {a+a \sec (e+f x)}} \\ & = -\frac {\tan (e+f x)}{4 a^2 (c-d) f (1+\sec (e+f x))^2 \sqrt {a+a \sec (e+f x)}}-\frac {(c-2 d) \tan (e+f x)}{2 a^2 (c-d)^2 f (1+\sec (e+f x)) \sqrt {a+a \sec (e+f x)}}-\frac {3 \tan (e+f x)}{16 a^2 (c-d) f (1+\sec (e+f x)) \sqrt {a+a \sec (e+f x)}}+\frac {2 \text {arctanh}\left (\frac {\sqrt {a-a \sec (e+f x)}}{\sqrt {a}}\right ) \tan (e+f x)}{a^{3/2} c f \sqrt {a-a \sec (e+f x)} \sqrt {a+a \sec (e+f x)}}-\frac {(c-2 d) \text {arctanh}\left (\frac {\sqrt {a-a \sec (e+f x)}}{\sqrt {2} \sqrt {a}}\right ) \tan (e+f x)}{2 \sqrt {2} a^{3/2} (c-d)^2 f \sqrt {a-a \sec (e+f x)} \sqrt {a+a \sec (e+f x)}}-\frac {3 \text {arctanh}\left (\frac {\sqrt {a-a \sec (e+f x)}}{\sqrt {2} \sqrt {a}}\right ) \tan (e+f x)}{16 \sqrt {2} a^{3/2} (c-d) f \sqrt {a-a \sec (e+f x)} \sqrt {a+a \sec (e+f x)}}-\frac {\sqrt {2} \left (c^2-3 c d+3 d^2\right ) \text {arctanh}\left (\frac {\sqrt {a-a \sec (e+f x)}}{\sqrt {2} \sqrt {a}}\right ) \tan (e+f x)}{a^{3/2} (c-d)^3 f \sqrt {a-a \sec (e+f x)} \sqrt {a+a \sec (e+f x)}}+\frac {2 d^{7/2} \text {arctanh}\left (\frac {\sqrt {d} \sqrt {a-a \sec (e+f x)}}{\sqrt {a} \sqrt {c+d}}\right ) \tan (e+f x)}{a^{3/2} c (c-d)^3 \sqrt {c+d} f \sqrt {a-a \sec (e+f x)} \sqrt {a+a \sec (e+f x)}} \\ \end{align*}

Mathematica [A] (warning: unable to verify)

Time = 11.41 (sec) , antiderivative size = 360, normalized size of antiderivative = 0.61 \[ \int \frac {1}{(a+a \sec (e+f x))^{5/2} (c+d \sec (e+f x))} \, dx=\frac {\cos ^4\left (\frac {1}{2} (e+f x)\right ) (d+c \cos (e+f x)) \sec ^{\frac {7}{2}}(e+f x) \left (-\frac {4 \left (\sqrt {-c-d} \left (c \left (43 c^2-126 c d+115 d^2\right ) \arcsin \left (\tan \left (\frac {1}{2} (e+f x)\right )\right )-32 \sqrt {2} (c-d)^3 \arctan \left (\frac {\tan \left (\frac {1}{2} (e+f x)\right )}{\sqrt {\frac {\cos (e+f x)}{1+\cos (e+f x)}}}\right )\right )+32 \sqrt {2} d^{7/2} \text {arctanh}\left (\frac {\sqrt {d} \tan \left (\frac {1}{2} (e+f x)\right )}{\sqrt {-c-d} \sqrt {\frac {\cos (e+f x)}{1+\cos (e+f x)}}}\right )\right ) \sqrt {\frac {\cos (e+f x)}{1+\cos (e+f x)}} \sqrt {1+\sec (e+f x)}}{c \sqrt {-c-d} \sqrt {\sec ^2\left (\frac {1}{2} (e+f x)\right )}}+(c-d) (11 c-19 d+(15 c-23 d) \cos (e+f x)) \sec ^3\left (\frac {1}{2} (e+f x)\right ) \sqrt {\sec (e+f x)} \left (\sin \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {3}{2} (e+f x)\right )\right )\right )}{16 (c-d)^3 f (a (1+\sec (e+f x)))^{5/2} (c+d \sec (e+f x))} \]

[In]

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

[Out]

(Cos[(e + f*x)/2]^4*(d + c*Cos[e + f*x])*Sec[e + f*x]^(7/2)*((-4*(Sqrt[-c - d]*(c*(43*c^2 - 126*c*d + 115*d^2)
*ArcSin[Tan[(e + f*x)/2]] - 32*Sqrt[2]*(c - d)^3*ArcTan[Tan[(e + f*x)/2]/Sqrt[Cos[e + f*x]/(1 + Cos[e + f*x])]
]) + 32*Sqrt[2]*d^(7/2)*ArcTanh[(Sqrt[d]*Tan[(e + f*x)/2])/(Sqrt[-c - d]*Sqrt[Cos[e + f*x]/(1 + Cos[e + f*x])]
)])*Sqrt[Cos[e + f*x]/(1 + Cos[e + f*x])]*Sqrt[1 + Sec[e + f*x]])/(c*Sqrt[-c - d]*Sqrt[Sec[(e + f*x)/2]^2]) +
(c - d)*(11*c - 19*d + (15*c - 23*d)*Cos[e + f*x])*Sec[(e + f*x)/2]^3*Sqrt[Sec[e + f*x]]*(Sin[(e + f*x)/2] - S
in[(3*(e + f*x))/2])))/(16*(c - d)^3*f*(a*(1 + Sec[e + f*x]))^(5/2)*(c + d*Sec[e + f*x]))

Maple [B] (warning: unable to verify)

Leaf count of result is larger than twice the leaf count of optimal. \(2344\) vs. \(2(509)=1018\).

Time = 16.76 (sec) , antiderivative size = 2345, normalized size of antiderivative = 3.96

method	result	size
default	\(\text {Expression too large to display}\)	\(2345\)

[In]

int(1/(a+a*sec(f*x+e))^(5/2)/(c+d*sec(f*x+e)),x,method=_RETURNVERBOSE)

[Out]

1/192/f/(d/(c-d))^(1/2)/(c-d)^3/c/((c+d)*(c-d))^(1/2)/a^3*(576*((c+d)*(c-d))^(1/2)*2^(1/2)*arctanh(2^(1/2)/((1
-cos(f*x+e))^2*csc(f*x+e)^2-1)^(1/2)*(-cot(f*x+e)+csc(f*x+e)))*(d/(c-d))^(1/2)*c*d^2+12*((c+d)*(c-d))^(1/2)*((
1-cos(f*x+e))^2*csc(f*x+e)^2-1)^(5/2)*(d/(c-d))^(1/2)*c^2*d*(-cot(f*x+e)+csc(f*x+e))-576*((c+d)*(c-d))^(1/2)*2
^(1/2)*arctanh(2^(1/2)/((1-cos(f*x+e))^2*csc(f*x+e)^2-1)^(1/2)*(-cot(f*x+e)+csc(f*x+e)))*(d/(c-d))^(1/2)*c^2*d
-96*2^(1/2)*ln(-2*(-((1-cos(f*x+e))^2*csc(f*x+e)^2-1)^(1/2)*2^(1/2)*(d/(c-d))^(1/2)*c+2^(1/2)*(d/(c-d))^(1/2)*
((1-cos(f*x+e))^2*csc(f*x+e)^2-1)^(1/2)*d+((c+d)*(c-d))^(1/2)*(-cot(f*x+e)+csc(f*x+e))+c-d)/(c*(-cot(f*x+e)+cs
c(f*x+e))-(-cot(f*x+e)+csc(f*x+e))*d+((c+d)*(c-d))^(1/2)))*d^4+96*2^(1/2)*ln(-2*(((1-cos(f*x+e))^2*csc(f*x+e)^
2-1)^(1/2)*2^(1/2)*(d/(c-d))^(1/2)*c-2^(1/2)*(d/(c-d))^(1/2)*((1-cos(f*x+e))^2*csc(f*x+e)^2-1)^(1/2)*d+((c+d)*
(c-d))^(1/2)*(-cot(f*x+e)+csc(f*x+e))-c+d)/(-c*(-cot(f*x+e)+csc(f*x+e))+(-cot(f*x+e)+csc(f*x+e))*d+((c+d)*(c-d
))^(1/2)))*d^4-12*((c+d)*(c-d))^(1/2)*((1-cos(f*x+e))^2*csc(f*x+e)^2-1)^(1/2)*(d/(c-d))^(1/2)*c^2*d*(1-cos(f*x
+e))^5*csc(f*x+e)^5+5*((c+d)*(c-d))^(1/2)*((1-cos(f*x+e))^2*csc(f*x+e)^2-1)^(3/2)*(d/(c-d))^(1/2)*c^3*(-cot(f*
x+e)+csc(f*x+e))-17*((c+d)*(c-d))^(1/2)*((1-cos(f*x+e))^2*csc(f*x+e)^2-1)^(3/2)*(d/(c-d))^(1/2)*d^3*(-cot(f*x+
e)+csc(f*x+e))+87*((c+d)*(c-d))^(1/2)*((1-cos(f*x+e))^2*csc(f*x+e)^2-1)^(1/2)*(d/(c-d))^(1/2)*c^3*(-cot(f*x+e)
+csc(f*x+e))+192*((c+d)*(c-d))^(1/2)*2^(1/2)*arctanh(2^(1/2)/((1-cos(f*x+e))^2*csc(f*x+e)^2-1)^(1/2)*(-cot(f*x
+e)+csc(f*x+e)))*(d/(c-d))^(1/2)*c^3-192*((c+d)*(c-d))^(1/2)*2^(1/2)*arctanh(2^(1/2)/((1-cos(f*x+e))^2*csc(f*x
+e)^2-1)^(1/2)*(-cot(f*x+e)+csc(f*x+e)))*(d/(c-d))^(1/2)*d^3+756*((c+d)*(c-d))^(1/2)*ln(csc(f*x+e)-cot(f*x+e)+
((1-cos(f*x+e))^2*csc(f*x+e)^2-1)^(1/2))*(d/(c-d))^(1/2)*c^2*d-690*((c+d)*(c-d))^(1/2)*ln(csc(f*x+e)-cot(f*x+e
)+((1-cos(f*x+e))^2*csc(f*x+e)^2-1)^(1/2))*(d/(c-d))^(1/2)*c*d^2-21*((c+d)*(c-d))^(1/2)*((1-cos(f*x+e))^2*csc(
f*x+e)^2-1)^(1/2)*(d/(c-d))^(1/2)*d^3*(-cot(f*x+e)+csc(f*x+e))-4*((c+d)*(c-d))^(1/2)*((1-cos(f*x+e))^2*csc(f*x
+e)^2-1)^(5/2)*(d/(c-d))^(1/2)*c^3*(-cot(f*x+e)+csc(f*x+e))+4*((c+d)*(c-d))^(1/2)*((1-cos(f*x+e))^2*csc(f*x+e)
^2-1)^(5/2)*(d/(c-d))^(1/2)*d^3*(-cot(f*x+e)+csc(f*x+e))-12*((c+d)*(c-d))^(1/2)*((1-cos(f*x+e))^2*csc(f*x+e)^2
-1)^(5/2)*(d/(c-d))^(1/2)*c*d^2*(-cot(f*x+e)+csc(f*x+e))-27*((c+d)*(c-d))^(1/2)*((1-cos(f*x+e))^2*csc(f*x+e)^2
-1)^(3/2)*(d/(c-d))^(1/2)*c^2*d*(-cot(f*x+e)+csc(f*x+e))+39*((c+d)*(c-d))^(1/2)*((1-cos(f*x+e))^2*csc(f*x+e)^2
-1)^(3/2)*(d/(c-d))^(1/2)*c*d^2*(-cot(f*x+e)+csc(f*x+e))-243*((c+d)*(c-d))^(1/2)*((1-cos(f*x+e))^2*csc(f*x+e)^
2-1)^(1/2)*(d/(c-d))^(1/2)*c^2*d*(-cot(f*x+e)+csc(f*x+e))-25*((c+d)*(c-d))^(1/2)*((1-cos(f*x+e))^2*csc(f*x+e)^
2-1)^(1/2)*(d/(c-d))^(1/2)*c^3*(1-cos(f*x+e))^3*csc(f*x+e)^3+25*((c+d)*(c-d))^(1/2)*((1-cos(f*x+e))^2*csc(f*x+
e)^2-1)^(1/2)*(d/(c-d))^(1/2)*d^3*(1-cos(f*x+e))^3*csc(f*x+e)^3+177*((c+d)*(c-d))^(1/2)*((1-cos(f*x+e))^2*csc(
f*x+e)^2-1)^(1/2)*(d/(c-d))^(1/2)*c*d^2*(-cot(f*x+e)+csc(f*x+e))+4*((c+d)*(c-d))^(1/2)*((1-cos(f*x+e))^2*csc(f
*x+e)^2-1)^(1/2)*(d/(c-d))^(1/2)*c^3*(1-cos(f*x+e))^5*csc(f*x+e)^5-4*((c+d)*(c-d))^(1/2)*((1-cos(f*x+e))^2*csc
(f*x+e)^2-1)^(1/2)*(d/(c-d))^(1/2)*d^3*(1-cos(f*x+e))^5*csc(f*x+e)^5+12*((c+d)*(c-d))^(1/2)*((1-cos(f*x+e))^2*
csc(f*x+e)^2-1)^(1/2)*(d/(c-d))^(1/2)*c*d^2*(1-cos(f*x+e))^5*csc(f*x+e)^5+75*((c+d)*(c-d))^(1/2)*((1-cos(f*x+e
))^2*csc(f*x+e)^2-1)^(1/2)*(d/(c-d))^(1/2)*c^2*d*(1-cos(f*x+e))^3*csc(f*x+e)^3-75*((c+d)*(c-d))^(1/2)*((1-cos(
f*x+e))^2*csc(f*x+e)^2-1)^(1/2)*(d/(c-d))^(1/2)*c*d^2*(1-cos(f*x+e))^3*csc(f*x+e)^3-258*((c+d)*(c-d))^(1/2)*ln
(csc(f*x+e)-cot(f*x+e)+((1-cos(f*x+e))^2*csc(f*x+e)^2-1)^(1/2))*(d/(c-d))^(1/2)*c^3)*((1-cos(f*x+e))^2*csc(f*x
+e)^2-1)^(1/2)*(-2*a/((1-cos(f*x+e))^2*csc(f*x+e)^2-1))^(1/2)

Fricas [F(-1)]

Timed out. \[ \int \frac {1}{(a+a \sec (e+f x))^{5/2} (c+d \sec (e+f x))} \, dx=\text {Timed out} \]

[In]

integrate(1/(a+a*sec(f*x+e))^(5/2)/(c+d*sec(f*x+e)),x, algorithm="fricas")

[Out]

Timed out

Sympy [F]

\[ \int \frac {1}{(a+a \sec (e+f x))^{5/2} (c+d \sec (e+f x))} \, dx=\int \frac {1}{\left (a \left (\sec {\left (e + f x \right )} + 1\right )\right )^{\frac {5}{2}} \left (c + d \sec {\left (e + f x \right )}\right )}\, dx \]

[In]

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

[Out]

Integral(1/((a*(sec(e + f*x) + 1))**(5/2)*(c + d*sec(e + f*x))), x)

Maxima [F]

\[ \int \frac {1}{(a+a \sec (e+f x))^{5/2} (c+d \sec (e+f x))} \, dx=\int { \frac {1}{{\left (a \sec \left (f x + e\right ) + a\right )}^{\frac {5}{2}} {\left (d \sec \left (f x + e\right ) + c\right )}} \,d x } \]

[In]

integrate(1/(a+a*sec(f*x+e))^(5/2)/(c+d*sec(f*x+e)),x, algorithm="maxima")

[Out]

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

Giac [F(-2)]

Exception generated. \[ \int \frac {1}{(a+a \sec (e+f x))^{5/2} (c+d \sec (e+f x))} \, dx=\text {Exception raised: TypeError} \]

[In]

integrate(1/(a+a*sec(f*x+e))^(5/2)/(c+d*sec(f*x+e)),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. \[ \int \frac {1}{(a+a \sec (e+f x))^{5/2} (c+d \sec (e+f x))} \, dx=\int \frac {1}{{\left (a+\frac {a}{\cos \left (e+f\,x\right )}\right )}^{5/2}\,\left (c+\frac {d}{\cos \left (e+f\,x\right )}\right )} \,d x \]

[In]

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

[Out]

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