3.175 \(\int \frac{1}{(a+a \sec (e+f x))^{3/2} (c+d \sec (e+f x))} \, dx\)

Optimal. Leaf size=394 \[ -\frac{2 d^{5/2} \tan (e+f x) \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a-a \sec (e+f x)}}{\sqrt{a} \sqrt{c+d}}\right )}{\sqrt{a} c f (c-d)^2 \sqrt{c+d} \sqrt{a-a \sec (e+f x)} \sqrt{a \sec (e+f x)+a}}-\frac{\tan (e+f x)}{2 a f (c-d) (\sec (e+f x)+1) \sqrt{a \sec (e+f x)+a}}-\frac{\tan (e+f x) \tanh ^{-1}\left (\frac{\sqrt{a-a \sec (e+f x)}}{\sqrt{2} \sqrt{a}}\right )}{2 \sqrt{2} \sqrt{a} f (c-d) \sqrt{a-a \sec (e+f x)} \sqrt{a \sec (e+f x)+a}}-\frac{\sqrt{2} (c-2 d) \tan (e+f x) \tanh ^{-1}\left (\frac{\sqrt{a-a \sec (e+f x)}}{\sqrt{2} \sqrt{a}}\right )}{\sqrt{a} f (c-d)^2 \sqrt{a-a \sec (e+f x)} \sqrt{a \sec (e+f x)+a}}+\frac{2 \tan (e+f x) \tanh ^{-1}\left (\frac{\sqrt{a-a \sec (e+f x)}}{\sqrt{a}}\right )}{\sqrt{a} c f \sqrt{a-a \sec (e+f x)} \sqrt{a \sec (e+f x)+a}} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.334993, antiderivative size = 394, normalized size of antiderivative = 1., number of steps used = 12, 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} \[ -\frac{2 d^{5/2} \tan (e+f x) \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a-a \sec (e+f x)}}{\sqrt{a} \sqrt{c+d}}\right )}{\sqrt{a} c f (c-d)^2 \sqrt{c+d} \sqrt{a-a \sec (e+f x)} \sqrt{a \sec (e+f x)+a}}-\frac{\tan (e+f x)}{2 a f (c-d) (\sec (e+f x)+1) \sqrt{a \sec (e+f x)+a}}-\frac{\tan (e+f x) \tanh ^{-1}\left (\frac{\sqrt{a-a \sec (e+f x)}}{\sqrt{2} \sqrt{a}}\right )}{2 \sqrt{2} \sqrt{a} f (c-d) \sqrt{a-a \sec (e+f x)} \sqrt{a \sec (e+f x)+a}}-\frac{\sqrt{2} (c-2 d) \tan (e+f x) \tanh ^{-1}\left (\frac{\sqrt{a-a \sec (e+f x)}}{\sqrt{2} \sqrt{a}}\right )}{\sqrt{a} f (c-d)^2 \sqrt{a-a \sec (e+f x)} \sqrt{a \sec (e+f x)+a}}+\frac{2 \tan (e+f x) \tanh ^{-1}\left (\frac{\sqrt{a-a \sec (e+f x)}}{\sqrt{a}}\right )}{\sqrt{a} c f \sqrt{a-a \sec (e+f x)} \sqrt{a \sec (e+f x)+a}} \]

Antiderivative was successfully verified.

[In]

Int[1/((a + a*Sec[e + f*x])^(3/2)*(c + d*Sec[e + f*x])),x]

[Out]

-Tan[e + f*x]/(2*a*(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])/(Sqrt[a]*c*f*Sqrt[a - a*Sec[e + f*x]]*Sqrt[a + a*Sec[e + f*x]]) - (Sqrt[2]*(c - 2*d)*
ArcTanh[Sqrt[a - a*Sec[e + f*x]]/(Sqrt[2]*Sqrt[a])]*Tan[e + f*x])/(Sqrt[a]*(c - d)^2*f*Sqrt[a - a*Sec[e + f*x]
]*Sqrt[a + a*Sec[e + f*x]]) - (ArcTanh[Sqrt[a - a*Sec[e + f*x]]/(Sqrt[2]*Sqrt[a])]*Tan[e + f*x])/(2*Sqrt[2]*Sq
rt[a]*(c - d)*f*Sqrt[a - a*Sec[e + f*x]]*Sqrt[a + a*Sec[e + f*x]]) - (2*d^(5/2)*ArcTanh[(Sqrt[d]*Sqrt[a - a*Se
c[e + f*x]])/(Sqrt[a]*Sqrt[c + d])]*Tan[e + f*x])/(Sqrt[a]*c*(c - d)^2*Sqrt[c + d]*f*Sqrt[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))^{3/2} (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)^2 (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^2 c x \sqrt{a-a x}}-\frac{1}{a^2 (c-d) (1+x)^2 \sqrt{a-a x}}+\frac{-c+2 d}{a^2 (c-d)^2 (1+x) \sqrt{a-a x}}-\frac{d^3}{a^2 c (c-d)^2 \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 )}{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 )}{(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)^2 \sqrt{a-a x}} \, dx,x,\sec (e+f x)\right )}{(c-d) f \sqrt{a-a \sec (e+f x)} \sqrt{a+a \sec (e+f x)}}+\frac{\left (d^3 \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 (c-d)^2 f \sqrt{a-a \sec (e+f x)} \sqrt{a+a \sec (e+f x)}}\\ &=-\frac{\tan (e+f x)}{2 a (c-d) 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 c f \sqrt{a-a \sec (e+f x)} \sqrt{a+a \sec (e+f x)}}-\frac{(2 (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 )}{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) \sqrt{a-a x}} \, dx,x,\sec (e+f x)\right )}{4 (c-d) f \sqrt{a-a \sec (e+f x)} \sqrt{a+a \sec (e+f x)}}-\frac{\left (2 d^3 \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 c (c-d)^2 f \sqrt{a-a \sec (e+f x)} \sqrt{a+a \sec (e+f x)}}\\ &=-\frac{\tan (e+f x)}{2 a (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)}{\sqrt{a} c f \sqrt{a-a \sec (e+f x)} \sqrt{a+a \sec (e+f x)}}-\frac{\sqrt{2} (c-2 d) \tanh ^{-1}\left (\frac{\sqrt{a-a \sec (e+f x)}}{\sqrt{2} \sqrt{a}}\right ) \tan (e+f x)}{\sqrt{a} (c-d)^2 f \sqrt{a-a \sec (e+f x)} \sqrt{a+a \sec (e+f x)}}-\frac{2 d^{5/2} \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a-a \sec (e+f x)}}{\sqrt{a} \sqrt{c+d}}\right ) \tan (e+f x)}{\sqrt{a} c (c-d)^2 \sqrt{c+d} 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}{2-\frac{x^2}{a}} \, dx,x,\sqrt{a-a \sec (e+f x)}\right )}{2 a (c-d) f \sqrt{a-a \sec (e+f x)} \sqrt{a+a \sec (e+f x)}}\\ &=-\frac{\tan (e+f x)}{2 a (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)}{\sqrt{a} c f \sqrt{a-a \sec (e+f x)} \sqrt{a+a \sec (e+f x)}}-\frac{\sqrt{2} (c-2 d) \tanh ^{-1}\left (\frac{\sqrt{a-a \sec (e+f x)}}{\sqrt{2} \sqrt{a}}\right ) \tan (e+f x)}{\sqrt{a} (c-d)^2 f \sqrt{a-a \sec (e+f x)} \sqrt{a+a \sec (e+f x)}}-\frac{\tanh ^{-1}\left (\frac{\sqrt{a-a \sec (e+f x)}}{\sqrt{2} \sqrt{a}}\right ) \tan (e+f x)}{2 \sqrt{2} \sqrt{a} (c-d) f \sqrt{a-a \sec (e+f x)} \sqrt{a+a \sec (e+f x)}}-\frac{2 d^{5/2} \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a-a \sec (e+f x)}}{\sqrt{a} \sqrt{c+d}}\right ) \tan (e+f x)}{\sqrt{a} c (c-d)^2 \sqrt{c+d} f \sqrt{a-a \sec (e+f x)} \sqrt{a+a \sec (e+f x)}}\\ \end{align*}

Mathematica [C]  time = 35.0797, size = 377837, normalized size = 958.98 \[ \text{Result too large to show} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[1/((a + a*Sec[e + f*x])^(3/2)*(c + d*Sec[e + f*x])),x]

[Out]

Result too large to show

________________________________________________________________________________________

Maple [B]  time = 0.294, size = 2076, normalized size = 5.3 \begin{align*} \text{result 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))^(3/2)/(c+d*sec(f*x+e)),x)

[Out]

1/4/f/(d/(c-d))^(1/2)/(c-d)^2/c/((c+d)*(c-d))^(1/2)/a^2*(-1+cos(f*x+e))*(4*2^(1/2)*((c+d)*(c-d))^(1/2)*sin(f*x
+e)*cos(f*x+e)*(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))*c^2-8*2^(1/2)*((c+d)*(c-d))^(1/2)*sin(f*x+e)*cos(f*x+e)*(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))*c*d+4*2^(1/2)*((c+d)*(c-d))^(1/2)*sin(f*x+e)*cos(f*x+e)*(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))*d^2+4*2^(1/2)*((c+d)*(c-
d))^(1/2)*sin(f*x+e)*(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))*c^2-8*2^(1/2)*((c+d)*(c-d))^(1/2)*sin(f*x+e)*(d/(c-d))^(1/2)*(-2*c
os(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)
)*c*d+4*2^(1/2)*((c+d)*(c-d))^(1/2)*sin(f*x+e)*(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))*d^2-2*2^(1/2)*sin(f*x+e)*cos(f*x+e)*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*cos(f*x+e)/(1+cos(f*x+e)))^(1/2)*d
^3+2*2^(1/2)*sin(f*x+e)*cos(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))*d^3+5*((c+d)*(c-d))^(1/2)*sin(f*x+e)*cos(f*x+e)*(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^2-9*((c+d)*(c-d))^(1/2)*sin(f*x+e)*cos(f*x+e)*(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*2^(1/2)*sin(f*x+e)*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+co
s(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*cos(f*x+e)/(1+cos(f*x+e)))^(1/2)*d^3+2*2^(1/
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))*d^3+5*((c+d)*(c-d))^(1/2)*sin(f*x+e)*(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^2-9*((c+d)*(c-d))^(1/2)*sin(f*
x+e)*(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*((c+d)*(c-d))^(1/2)*(d/(c-d))^(1/2)*cos(f*x+e)^2*c^2+2*((c+d)*(c-d))^(1/2)*c
os(f*x+e)^2*(d/(c-d))^(1/2)*c*d+2*((c+d)*(c-d))^(1/2)*(d/(c-d))^(1/2)*cos(f*x+e)*c^2-2*((c+d)*(c-d))^(1/2)*(d/
(c-d))^(1/2)*cos(f*x+e)*c*d)*(1/cos(f*x+e)*a*(1+cos(f*x+e)))^(1/2)/sin(f*x+e)^3

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{{\left (a \sec \left (f x + e\right ) + a\right )}^{\frac{3}{2}}{\left (d \sec \left (f x + e\right ) + c\right )}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 54.4344, size = 5720, normalized size = 14.52 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[-1/16*(8*(c^2 - c*d)*sqrt((a*cos(f*x + e) + a)/cos(f*x + e))*cos(f*x + e)*sin(f*x + e) - sqrt(2)*((5*c^2 - 9*
c*d)*cos(f*x + e)^2 + 5*c^2 - 9*c*d + 2*(5*c^2 - 9*c*d)*cos(f*x + e))*sqrt(-a)*log((17*a*cos(f*x + e)^3 + 4*sq
rt(2)*(3*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) + 3*a*co
s(f*x + e)^2 - 13*a*cos(f*x + e) + a)/(cos(f*x + e)^3 + 3*cos(f*x + e)^2 + 3*cos(f*x + e) + 1)) - 8*(a*d^2*cos
(f*x + e)^2 + 2*a*d^2*cos(f*x + e) + a*d^2)*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))) + 8*((c^2 - 2*c*d + d^2)*cos(f*x
+ e)^2 + c^2 - 2*c*d + d^2 + 2*(c^2 - 2*c*d + d^2)*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)))/((a^2*c^3 - 2*a^2*c^2*d + a^2*c*d^2)*f*cos(f*x + e)^2 + 2*(a^2*c^3 - 2*a^2*c^2*d + a^2
*c*d^2)*f*cos(f*x + e) + (a^2*c^3 - 2*a^2*c^2*d + a^2*c*d^2)*f), -1/16*(8*(c^2 - c*d)*sqrt((a*cos(f*x + e) + a
)/cos(f*x + e))*cos(f*x + e)*sin(f*x + e) - sqrt(2)*((5*c^2 - 9*c*d)*cos(f*x + e)^2 + 5*c^2 - 9*c*d + 2*(5*c^2
- 9*c*d)*cos(f*x + e))*sqrt(-a)*log((17*a*cos(f*x + e)^3 + 4*sqrt(2)*(3*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) + 3*a*cos(f*x + e)^2 - 13*a*cos(f*x + e) + a)/(cos(f*x
+ e)^3 + 3*cos(f*x + e)^2 + 3*cos(f*x + e) + 1)) - 16*(a*d^2*cos(f*x + e)^2 + 2*a*d^2*cos(f*x + e) + a*d^2)*s
qrt(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^2 - 2*c*d + d^2)*cos(f*x + e)^2 + c^2 - 2*c*d + d^2 + 2*(c^2 - 2*c*d + d^2)
*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)))/((a^2*c^3 - 2*a^2*c^2*d + a
^2*c*d^2)*f*cos(f*x + e)^2 + 2*(a^2*c^3 - 2*a^2*c^2*d + a^2*c*d^2)*f*cos(f*x + e) + (a^2*c^3 - 2*a^2*c^2*d + a
^2*c*d^2)*f), -1/8*(4*(c^2 - c*d)*sqrt((a*cos(f*x + e) + a)/cos(f*x + e))*cos(f*x + e)*sin(f*x + e) - sqrt(2)*
((5*c^2 - 9*c*d)*cos(f*x + e)^2 + 5*c^2 - 9*c*d + 2*(5*c^2 - 9*c*d)*cos(f*x + e))*sqrt(a)*arctan(1/4*sqrt(2)*s
qrt((a*cos(f*x + e) + a)/cos(f*x + e))*(3*cos(f*x + e) - 1)/(sqrt(a)*sin(f*x + e))) + 8*((c^2 - 2*c*d + d^2)*c
os(f*x + e)^2 + c^2 - 2*c*d + d^2 + 2*(c^2 - 2*c*d + d^2)*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))) - 4*(a*d^2*cos(f*x + e)^2 + 2*a*d^2*cos(f*x
+ e) + a*d^2)*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))))/((a^2*c^3 - 2*a^2*c^2*d + a^2*c*d^2)*f*cos(f*x + e)^2 + 2*(a^
2*c^3 - 2*a^2*c^2*d + a^2*c*d^2)*f*cos(f*x + e) + (a^2*c^3 - 2*a^2*c^2*d + a^2*c*d^2)*f), -1/8*(4*(c^2 - c*d)*
sqrt((a*cos(f*x + e) + a)/cos(f*x + e))*cos(f*x + e)*sin(f*x + e) - sqrt(2)*((5*c^2 - 9*c*d)*cos(f*x + e)^2 +
5*c^2 - 9*c*d + 2*(5*c^2 - 9*c*d)*cos(f*x + e))*sqrt(a)*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))) - 8*(a*d^2*cos(f*x + e)^2 + 2*a*d^2*cos(f*x + e) + a*d^2)*sq
rt(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^2 - 2*c*d + d^2)*cos(f*x + e)^2 + c^2 - 2*c*d + d^2 + 2*(c^2 - 2*c*d + d^2)*
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))))/((a^2*c^3 - 2*a^2*c^2*d + a^2*c*d^2)*f*cos(f*x + e)^2 + 2*(a^2*c^3 - 2*a^2*c^2*d + a^2*c*d^2)*f*cos(f
*x + e) + (a^2*c^3 - 2*a^2*c^2*d + a^2*c*d^2)*f)]

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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{3}{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))**(3/2)/(c+d*sec(f*x+e)),x)

[Out]

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

[Out]

Timed out