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

Optimal. Leaf size=592 \[ -\frac{\sqrt{2} \left (c^2-3 c d+3 d^2\right ) \tan (e+f x) \tanh ^{-1}\left (\frac{\sqrt{a-a \sec (e+f x)}}{\sqrt{2} \sqrt{a}}\right )}{a^{3/2} f (c-d)^3 \sqrt{a-a \sec (e+f x)} \sqrt{a \sec (e+f x)+a}}+\frac{2 d^{7/2} \tan (e+f x) \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a-a \sec (e+f x)}}{\sqrt{a} \sqrt{c+d}}\right )}{a^{3/2} c f (c-d)^3 \sqrt{c+d} \sqrt{a-a \sec (e+f x)} \sqrt{a \sec (e+f x)+a}}-\frac{3 \tan (e+f x)}{16 a^2 f (c-d) (\sec (e+f x)+1) \sqrt{a \sec (e+f x)+a}}-\frac{(c-2 d) \tan (e+f x)}{2 a^2 f (c-d)^2 (\sec (e+f x)+1) \sqrt{a \sec (e+f x)+a}}-\frac{\tan (e+f x)}{4 a^2 f (c-d) (\sec (e+f x)+1)^2 \sqrt{a \sec (e+f x)+a}}-\frac{3 \tan (e+f x) \tanh ^{-1}\left (\frac{\sqrt{a-a \sec (e+f x)}}{\sqrt{2} \sqrt{a}}\right )}{16 \sqrt{2} a^{3/2} f (c-d) \sqrt{a-a \sec (e+f x)} \sqrt{a \sec (e+f x)+a}}-\frac{(c-2 d) \tan (e+f x) \tanh ^{-1}\left (\frac{\sqrt{a-a \sec (e+f x)}}{\sqrt{2} \sqrt{a}}\right )}{2 \sqrt{2} a^{3/2} f (c-d)^2 \sqrt{a-a \sec (e+f x)} \sqrt{a \sec (e+f x)+a}}+\frac{2 \tan (e+f x) \tanh ^{-1}\left (\frac{\sqrt{a-a \sec (e+f x)}}{\sqrt{a}}\right )}{a^{3/2} c f \sqrt{a-a \sec (e+f x)} \sqrt{a \sec (e+f x)+a}} \]

[Out]

-Tan[e + f*x]/(4*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*Sqrt[
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*ArcTanh[
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)*ArcTanh
[(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*Sq
rt[a - a*Sec[e + f*x]]*Sqrt[a + a*Sec[e + f*x]])

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Rubi [A]  time = 0.456727, antiderivative size = 592, 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} \[ -\frac{\sqrt{2} \left (c^2-3 c d+3 d^2\right ) \tan (e+f x) \tanh ^{-1}\left (\frac{\sqrt{a-a \sec (e+f x)}}{\sqrt{2} \sqrt{a}}\right )}{a^{3/2} f (c-d)^3 \sqrt{a-a \sec (e+f x)} \sqrt{a \sec (e+f x)+a}}+\frac{2 d^{7/2} \tan (e+f x) \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a-a \sec (e+f x)}}{\sqrt{a} \sqrt{c+d}}\right )}{a^{3/2} c f (c-d)^3 \sqrt{c+d} \sqrt{a-a \sec (e+f x)} \sqrt{a \sec (e+f x)+a}}-\frac{3 \tan (e+f x)}{16 a^2 f (c-d) (\sec (e+f x)+1) \sqrt{a \sec (e+f x)+a}}-\frac{(c-2 d) \tan (e+f x)}{2 a^2 f (c-d)^2 (\sec (e+f x)+1) \sqrt{a \sec (e+f x)+a}}-\frac{\tan (e+f x)}{4 a^2 f (c-d) (\sec (e+f x)+1)^2 \sqrt{a \sec (e+f x)+a}}-\frac{3 \tan (e+f x) \tanh ^{-1}\left (\frac{\sqrt{a-a \sec (e+f x)}}{\sqrt{2} \sqrt{a}}\right )}{16 \sqrt{2} a^{3/2} f (c-d) \sqrt{a-a \sec (e+f x)} \sqrt{a \sec (e+f x)+a}}-\frac{(c-2 d) \tan (e+f x) \tanh ^{-1}\left (\frac{\sqrt{a-a \sec (e+f x)}}{\sqrt{2} \sqrt{a}}\right )}{2 \sqrt{2} a^{3/2} f (c-d)^2 \sqrt{a-a \sec (e+f x)} \sqrt{a \sec (e+f x)+a}}+\frac{2 \tan (e+f x) \tanh ^{-1}\left (\frac{\sqrt{a-a \sec (e+f x)}}{\sqrt{a}}\right )}{a^{3/2} c f \sqrt{a-a \sec (e+f x)} \sqrt{a \sec (e+f x)+a}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-Tan[e + f*x]/(4*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*Sqrt[
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*ArcTanh[
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)*ArcTanh
[(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*Sq
rt[a - a*Sec[e + f*x]]*Sqrt[a + a*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}{(a+a \sec (e+f x))^{5/2} (c+d \sec (e+f x))} \, dx &=-\frac{\left (a^2 \tan (e+f x)\right ) \operatorname{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 ) \operatorname{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) \operatorname{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)) \operatorname{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) \operatorname{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 ) \operatorname{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 ) \operatorname{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)) \operatorname{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)) \operatorname{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)) \operatorname{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 ) \operatorname{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 ) \operatorname{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 \tanh ^{-1}\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 ) \tanh ^{-1}\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} \tanh ^{-1}\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)) \operatorname{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)) \operatorname{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 \tanh ^{-1}\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) \tanh ^{-1}\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 ) \tanh ^{-1}\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} \tanh ^{-1}\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)) \operatorname{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 \tanh ^{-1}\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) \tanh ^{-1}\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 \tanh ^{-1}\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 ) \tanh ^{-1}\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} \tanh ^{-1}\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 [C]  time = 36.9869, size = 484869, normalized size = 819.04 \[ \text{Result too large to show} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

Result too large to show

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Maple [B]  time = 0.29, size = 3860, normalized size = 6.5 \begin{align*} \text{output too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/32/f/(d/(c-d))^(1/2)/(c-d)^3/c/((c+d)*(c-d))^(1/2)/a^3*(-1+cos(f*x+e))^2*(-16*sin(f*x+e)*(-2*cos(f*x+e)/(1+c
os(f*x+e)))^(1/2)*ln(2*((-2*cos(f*x+e)/(1+cos(f*x+e)))^(1/2)*(d/(c-d))^(1/2)*2^(1/2)*c*sin(f*x+e)-2^(1/2)*(d/(
c-d))^(1/2)*(-2*cos(f*x+e)/(1+cos(f*x+e)))^(1/2)*d*sin(f*x+e)+((c+d)*(c-d))^(1/2)*cos(f*x+e)-c*sin(f*x+e)+d*si
n(f*x+e)-((c+d)*(c-d))^(1/2))/(((c+d)*(c-d))^(1/2)*sin(f*x+e)-c*cos(f*x+e)+d*cos(f*x+e)+c-d))*2^(1/2)*d^4+16*(
d/(c-d))^(1/2)*cos(f*x+e)^2*((c+d)*(c-d))^(1/2)*c^2*d-8*(d/(c-d))^(1/2)*cos(f*x+e)^2*((c+d)*(c-d))^(1/2)*c*d^2
+60*(d/(c-d))^(1/2)*((c+d)*(c-d))^(1/2)*cos(f*x+e)*c^2*d-38*(d/(c-d))^(1/2)*((c+d)*(c-d))^(1/2)*cos(f*x+e)*c*d
^2-76*(d/(c-d))^(1/2)*cos(f*x+e)^3*((c+d)*(c-d))^(1/2)*c^2*d+46*(d/(c-d))^(1/2)*cos(f*x+e)^3*((c+d)*(c-d))^(1/
2)*c*d^2+16*sin(f*x+e)*(-2*cos(f*x+e)/(1+cos(f*x+e)))^(1/2)*ln(-2*((-2*cos(f*x+e)/(1+cos(f*x+e)))^(1/2)*(d/(c-
d))^(1/2)*2^(1/2)*c*sin(f*x+e)-2^(1/2)*(d/(c-d))^(1/2)*(-2*cos(f*x+e)/(1+cos(f*x+e)))^(1/2)*d*sin(f*x+e)-((c+d
)*(c-d))^(1/2)*cos(f*x+e)-c*sin(f*x+e)+d*sin(f*x+e)+((c+d)*(c-d))^(1/2))/(((c+d)*(c-d))^(1/2)*sin(f*x+e)+c*cos
(f*x+e)-d*cos(f*x+e)-c+d))*2^(1/2)*d^4+30*(d/(c-d))^(1/2)*cos(f*x+e)^3*((c+d)*(c-d))^(1/2)*c^3-8*(d/(c-d))^(1/
2)*cos(f*x+e)^2*((c+d)*(c-d))^(1/2)*c^3-22*(d/(c-d))^(1/2)*((c+d)*(c-d))^(1/2)*cos(f*x+e)*c^3-192*(d/(c-d))^(1
/2)*cos(f*x+e)*sin(f*x+e)*((c+d)*(c-d))^(1/2)*(-2*cos(f*x+e)/(1+cos(f*x+e)))^(1/2)*arctanh(1/2*2^(1/2)*(-2*cos
(f*x+e)/(1+cos(f*x+e)))^(1/2)*sin(f*x+e)/cos(f*x+e))*2^(1/2)*c*d^2+64*(d/(c-d))^(1/2)*cos(f*x+e)*sin(f*x+e)*((
c+d)*(c-d))^(1/2)*(-2*cos(f*x+e)/(1+cos(f*x+e)))^(1/2)*arctanh(1/2*2^(1/2)*(-2*cos(f*x+e)/(1+cos(f*x+e)))^(1/2
)*sin(f*x+e)/cos(f*x+e))*2^(1/2)*d^3+252*(d/(c-d))^(1/2)*cos(f*x+e)*sin(f*x+e)*((c+d)*(c-d))^(1/2)*(-2*cos(f*x
+e)/(1+cos(f*x+e)))^(1/2)*ln(-(-(-2*cos(f*x+e)/(1+cos(f*x+e)))^(1/2)*sin(f*x+e)+cos(f*x+e)-1)/sin(f*x+e))*c^2*
d-230*(d/(c-d))^(1/2)*cos(f*x+e)*sin(f*x+e)*((c+d)*(c-d))^(1/2)*(-2*cos(f*x+e)/(1+cos(f*x+e)))^(1/2)*ln(-(-(-2
*cos(f*x+e)/(1+cos(f*x+e)))^(1/2)*sin(f*x+e)+cos(f*x+e)-1)/sin(f*x+e))*c*d^2+96*(d/(c-d))^(1/2)*sin(f*x+e)*((c
+d)*(c-d))^(1/2)*(-2*cos(f*x+e)/(1+cos(f*x+e)))^(1/2)*arctanh(1/2*2^(1/2)*(-2*cos(f*x+e)/(1+cos(f*x+e)))^(1/2)
*sin(f*x+e)/cos(f*x+e))*2^(1/2)*c^2*d-96*(d/(c-d))^(1/2)*sin(f*x+e)*((c+d)*(c-d))^(1/2)*(-2*cos(f*x+e)/(1+cos(
f*x+e)))^(1/2)*arctanh(1/2*2^(1/2)*(-2*cos(f*x+e)/(1+cos(f*x+e)))^(1/2)*sin(f*x+e)/cos(f*x+e))*2^(1/2)*c*d^2+3
2*(d/(c-d))^(1/2)*cos(f*x+e)^2*sin(f*x+e)*((c+d)*(c-d))^(1/2)*(-2*cos(f*x+e)/(1+cos(f*x+e)))^(1/2)*arctanh(1/2
*2^(1/2)*(-2*cos(f*x+e)/(1+cos(f*x+e)))^(1/2)*sin(f*x+e)/cos(f*x+e))*2^(1/2)*d^3+126*(d/(c-d))^(1/2)*cos(f*x+e
)^2*sin(f*x+e)*((c+d)*(c-d))^(1/2)*(-2*cos(f*x+e)/(1+cos(f*x+e)))^(1/2)*ln(-(-(-2*cos(f*x+e)/(1+cos(f*x+e)))^(
1/2)*sin(f*x+e)+cos(f*x+e)-1)/sin(f*x+e))*c^2*d-115*(d/(c-d))^(1/2)*cos(f*x+e)^2*sin(f*x+e)*((c+d)*(c-d))^(1/2
)*(-2*cos(f*x+e)/(1+cos(f*x+e)))^(1/2)*ln(-(-(-2*cos(f*x+e)/(1+cos(f*x+e)))^(1/2)*sin(f*x+e)+cos(f*x+e)-1)/sin
(f*x+e))*c*d^2-32*(d/(c-d))^(1/2)*cos(f*x+e)^2*sin(f*x+e)*((c+d)*(c-d))^(1/2)*(-2*cos(f*x+e)/(1+cos(f*x+e)))^(
1/2)*arctanh(1/2*2^(1/2)*(-2*cos(f*x+e)/(1+cos(f*x+e)))^(1/2)*sin(f*x+e)/cos(f*x+e))*2^(1/2)*c^3+16*cos(f*x+e)
^2*sin(f*x+e)*(-2*cos(f*x+e)/(1+cos(f*x+e)))^(1/2)*ln(-2*((-2*cos(f*x+e)/(1+cos(f*x+e)))^(1/2)*(d/(c-d))^(1/2)
*2^(1/2)*c*sin(f*x+e)-2^(1/2)*(d/(c-d))^(1/2)*(-2*cos(f*x+e)/(1+cos(f*x+e)))^(1/2)*d*sin(f*x+e)-((c+d)*(c-d))^
(1/2)*cos(f*x+e)-c*sin(f*x+e)+d*sin(f*x+e)+((c+d)*(c-d))^(1/2))/(((c+d)*(c-d))^(1/2)*sin(f*x+e)+c*cos(f*x+e)-d
*cos(f*x+e)-c+d))*2^(1/2)*d^4-16*cos(f*x+e)^2*sin(f*x+e)*(-2*cos(f*x+e)/(1+cos(f*x+e)))^(1/2)*ln(2*((-2*cos(f*
x+e)/(1+cos(f*x+e)))^(1/2)*(d/(c-d))^(1/2)*2^(1/2)*c*sin(f*x+e)-2^(1/2)*(d/(c-d))^(1/2)*(-2*cos(f*x+e)/(1+cos(
f*x+e)))^(1/2)*d*sin(f*x+e)+((c+d)*(c-d))^(1/2)*cos(f*x+e)-c*sin(f*x+e)+d*sin(f*x+e)-((c+d)*(c-d))^(1/2))/(((c
+d)*(c-d))^(1/2)*sin(f*x+e)-c*cos(f*x+e)+d*cos(f*x+e)+c-d))*2^(1/2)*d^4+32*cos(f*x+e)*sin(f*x+e)*(-2*cos(f*x+e
)/(1+cos(f*x+e)))^(1/2)*ln(-2*((-2*cos(f*x+e)/(1+cos(f*x+e)))^(1/2)*(d/(c-d))^(1/2)*2^(1/2)*c*sin(f*x+e)-2^(1/
2)*(d/(c-d))^(1/2)*(-2*cos(f*x+e)/(1+cos(f*x+e)))^(1/2)*d*sin(f*x+e)-((c+d)*(c-d))^(1/2)*cos(f*x+e)-c*sin(f*x+
e)+d*sin(f*x+e)+((c+d)*(c-d))^(1/2))/(((c+d)*(c-d))^(1/2)*sin(f*x+e)+c*cos(f*x+e)-d*cos(f*x+e)-c+d))*2^(1/2)*d
^4-32*cos(f*x+e)*sin(f*x+e)*(-2*cos(f*x+e)/(1+cos(f*x+e)))^(1/2)*ln(2*((-2*cos(f*x+e)/(1+cos(f*x+e)))^(1/2)*(d
/(c-d))^(1/2)*2^(1/2)*c*sin(f*x+e)-2^(1/2)*(d/(c-d))^(1/2)*(-2*cos(f*x+e)/(1+cos(f*x+e)))^(1/2)*d*sin(f*x+e)+(
(c+d)*(c-d))^(1/2)*cos(f*x+e)-c*sin(f*x+e)+d*sin(f*x+e)-((c+d)*(c-d))^(1/2))/(((c+d)*(c-d))^(1/2)*sin(f*x+e)-c
*cos(f*x+e)+d*cos(f*x+e)+c-d))*2^(1/2)*d^4-43*(d/(c-d))^(1/2)*sin(f*x+e)*((c+d)*(c-d))^(1/2)*(-2*cos(f*x+e)/(1
+cos(f*x+e)))^(1/2)*ln(-(-(-2*cos(f*x+e)/(1+cos(f*x+e)))^(1/2)*sin(f*x+e)+cos(f*x+e)-1)/sin(f*x+e))*c^3-43*(d/
(c-d))^(1/2)*cos(f*x+e)^2*sin(f*x+e)*((c+d)*(c-d))^(1/2)*(-2*cos(f*x+e)/(1+cos(f*x+e)))^(1/2)*ln(-(-(-2*cos(f*
x+e)/(1+cos(f*x+e)))^(1/2)*sin(f*x+e)+cos(f*x+e)-1)/sin(f*x+e))*c^3-86*(d/(c-d))^(1/2)*cos(f*x+e)*sin(f*x+e)*(
(c+d)*(c-d))^(1/2)*(-2*cos(f*x+e)/(1+cos(f*x+e)))^(1/2)*ln(-(-(-2*cos(f*x+e)/(1+cos(f*x+e)))^(1/2)*sin(f*x+e)+
cos(f*x+e)-1)/sin(f*x+e))*c^3-32*(d/(c-d))^(1/2)*sin(f*x+e)*((c+d)*(c-d))^(1/2)*(-2*cos(f*x+e)/(1+cos(f*x+e)))
^(1/2)*arctanh(1/2*2^(1/2)*(-2*cos(f*x+e)/(1+cos(f*x+e)))^(1/2)*sin(f*x+e)/cos(f*x+e))*2^(1/2)*c^3+32*(d/(c-d)
)^(1/2)*sin(f*x+e)*((c+d)*(c-d))^(1/2)*(-2*cos(f*x+e)/(1+cos(f*x+e)))^(1/2)*arctanh(1/2*2^(1/2)*(-2*cos(f*x+e)
/(1+cos(f*x+e)))^(1/2)*sin(f*x+e)/cos(f*x+e))*2^(1/2)*d^3+126*(d/(c-d))^(1/2)*sin(f*x+e)*((c+d)*(c-d))^(1/2)*(
-2*cos(f*x+e)/(1+cos(f*x+e)))^(1/2)*ln(-(-(-2*cos(f*x+e)/(1+cos(f*x+e)))^(1/2)*sin(f*x+e)+cos(f*x+e)-1)/sin(f*
x+e))*c^2*d-115*(d/(c-d))^(1/2)*sin(f*x+e)*((c+d)*(c-d))^(1/2)*(-2*cos(f*x+e)/(1+cos(f*x+e)))^(1/2)*ln(-(-(-2*
cos(f*x+e)/(1+cos(f*x+e)))^(1/2)*sin(f*x+e)+cos(f*x+e)-1)/sin(f*x+e))*c*d^2-64*(d/(c-d))^(1/2)*cos(f*x+e)*sin(
f*x+e)*((c+d)*(c-d))^(1/2)*(-2*cos(f*x+e)/(1+cos(f*x+e)))^(1/2)*arctanh(1/2*2^(1/2)*(-2*cos(f*x+e)/(1+cos(f*x+
e)))^(1/2)*sin(f*x+e)/cos(f*x+e))*2^(1/2)*c^3+96*(d/(c-d))^(1/2)*cos(f*x+e)^2*sin(f*x+e)*((c+d)*(c-d))^(1/2)*(
-2*cos(f*x+e)/(1+cos(f*x+e)))^(1/2)*arctanh(1/2*2^(1/2)*(-2*cos(f*x+e)/(1+cos(f*x+e)))^(1/2)*sin(f*x+e)/cos(f*
x+e))*2^(1/2)*c^2*d-96*(d/(c-d))^(1/2)*cos(f*x+e)^2*sin(f*x+e)*((c+d)*(c-d))^(1/2)*(-2*cos(f*x+e)/(1+cos(f*x+e
)))^(1/2)*arctanh(1/2*2^(1/2)*(-2*cos(f*x+e)/(1+cos(f*x+e)))^(1/2)*sin(f*x+e)/cos(f*x+e))*2^(1/2)*c*d^2+192*(d
/(c-d))^(1/2)*cos(f*x+e)*sin(f*x+e)*((c+d)*(c-d))^(1/2)*(-2*cos(f*x+e)/(1+cos(f*x+e)))^(1/2)*arctanh(1/2*2^(1/
2)*(-2*cos(f*x+e)/(1+cos(f*x+e)))^(1/2)*sin(f*x+e)/cos(f*x+e))*2^(1/2)*c^2*d)*(1/cos(f*x+e)*a*(1+cos(f*x+e)))^
(1/2)/sin(f*x+e)^5

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

[Out]

Timed out

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \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 \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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

[Out]

Timed out