3.180 \(\int \frac{\sec ^{\frac{5}{2}}(e+f x)}{\sqrt{a+a \sec (e+f x)} (c-c \sec (e+f x))} \, dx\)

Optimal. Leaf size=140 \[ \frac{\csc (e+f x) \sqrt{a \sec (e+f x)+a}}{a c f \sqrt{\sec (e+f x)}}+\frac{\tanh ^{-1}\left (\frac{\sqrt{a} \sin (e+f x) \sqrt{\sec (e+f x)}}{\sqrt{2} \sqrt{a \sec (e+f x)+a}}\right )}{\sqrt{2} \sqrt{a} c f}-\frac{2 \sinh ^{-1}\left (\frac{\sqrt{a} \tan (e+f x)}{\sqrt{a \sec (e+f x)+a}}\right )}{\sqrt{a} c f} \]

[Out]

(-2*ArcSinh[(Sqrt[a]*Tan[e + f*x])/Sqrt[a + a*Sec[e + f*x]]])/(Sqrt[a]*c*f) + ArcTanh[(Sqrt[a]*Sqrt[Sec[e + f*
x]]*Sin[e + f*x])/(Sqrt[2]*Sqrt[a + a*Sec[e + f*x]])]/(Sqrt[2]*Sqrt[a]*c*f) + (Csc[e + f*x]*Sqrt[a + a*Sec[e +
f*x]])/(a*c*f*Sqrt[Sec[e + f*x]])

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Rubi [A]  time = 0.281189, antiderivative size = 213, normalized size of antiderivative = 1.52, number of steps used = 8, number of rules used = 8, integrand size = 38, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.21, Rules used = {3964, 98, 157, 63, 217, 203, 93, 205} \[ -\frac{\sin (e+f x) \sec ^{\frac{3}{2}}(e+f x)}{f \sqrt{a \sec (e+f x)+a} (c-c \sec (e+f x))}+\frac{2 \tan (e+f x) \tan ^{-1}\left (\frac{\sqrt{c} \sqrt{\sec (e+f x)}}{\sqrt{c-c \sec (e+f x)}}\right )}{\sqrt{c} f \sqrt{a \sec (e+f x)+a} \sqrt{c-c \sec (e+f x)}}-\frac{\tan (e+f x) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{c} \sqrt{\sec (e+f x)}}{\sqrt{c-c \sec (e+f x)}}\right )}{\sqrt{2} \sqrt{c} f \sqrt{a \sec (e+f x)+a} \sqrt{c-c \sec (e+f x)}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-((Sec[e + f*x]^(3/2)*Sin[e + f*x])/(f*Sqrt[a + a*Sec[e + f*x]]*(c - c*Sec[e + f*x]))) + (2*ArcTan[(Sqrt[c]*Sq
rt[Sec[e + f*x]])/Sqrt[c - c*Sec[e + f*x]]]*Tan[e + f*x])/(Sqrt[c]*f*Sqrt[a + a*Sec[e + f*x]]*Sqrt[c - c*Sec[e
+ f*x]]) - (ArcTan[(Sqrt[2]*Sqrt[c]*Sqrt[Sec[e + f*x]])/Sqrt[c - c*Sec[e + f*x]]]*Tan[e + f*x])/(Sqrt[2]*Sqrt
[c]*f*Sqrt[a + a*Sec[e + f*x]]*Sqrt[c - c*Sec[e + f*x]])

Rule 3964

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

Rule 98

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[((b*c -
a*d)*(a + b*x)^(m + 1)*(c + d*x)^(n - 1)*(e + f*x)^(p + 1))/(b*(b*e - a*f)*(m + 1)), x] + Dist[1/(b*(b*e - a*
f)*(m + 1)), Int[(a + b*x)^(m + 1)*(c + d*x)^(n - 2)*(e + f*x)^p*Simp[a*d*(d*e*(n - 1) + c*f*(p + 1)) + b*c*(d
*e*(m - n + 2) - c*f*(m + p + 2)) + d*(a*d*f*(n + p) + b*(d*e*(m + 1) - c*f*(m + n + p + 1)))*x, x], x], x] /;
FreeQ[{a, b, c, d, e, f, p}, x] && LtQ[m, -1] && GtQ[n, 1] && (IntegersQ[2*m, 2*n, 2*p] || IntegersQ[m, n + p
] || IntegersQ[p, m + n])

Rule 157

Int[(((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_))^(p_)*((g_.) + (h_.)*(x_)))/((a_.) + (b_.)*(x_)), x_Symbol]
:> Dist[h/b, Int[(c + d*x)^n*(e + f*x)^p, x], x] + Dist[(b*g - a*h)/b, Int[((c + d*x)^n*(e + f*x)^p)/(a + b*x
), x], x] /; FreeQ[{a, b, c, d, e, f, g, h, n, p}, x]

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 217

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Subst[Int[1/(1 - b*x^2), x], x, x/Sqrt[a + b*x^2]] /; FreeQ[{a,
b}, x] &&  !GtQ[a, 0]

Rule 203

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

Rule 93

Int[(((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_))/((e_.) + (f_.)*(x_)), x_Symbol] :> With[{q = Denomin
ator[m]}, Dist[q, Subst[Int[x^(q*(m + 1) - 1)/(b*e - a*f - (d*e - c*f)*x^q), x], x, (a + b*x)^(1/q)/(c + d*x)^
(1/q)], x]] /; FreeQ[{a, b, c, d, e, f}, x] && EqQ[m + n + 1, 0] && RationalQ[n] && LtQ[-1, m, 0] && SimplerQ[
a + b*x, c + d*x]

Rule 205

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

Rubi steps

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

Mathematica [B]  time = 6.45511, size = 724, normalized size = 5.17 \[ \frac{\sin (e+f x) \sin ^2\left (\frac{e}{2}+\frac{f x}{2}\right ) \cos (e+f x) (\sec (e+f x)+1)^{3/2} \sqrt{\sec ^2(e+f x)-1} \left (\log \left (-3 \sec ^2(e+f x)-2 \sqrt{2} \sqrt{\sec (e+f x)+1} \sqrt{\sec ^2(e+f x)-1} \sqrt{\sec (e+f x)}-2 \sec (e+f x)+1\right )-\log \left (-3 \sec ^2(e+f x)+2 \sqrt{2} \sqrt{\sec (e+f x)+1} \sqrt{\sec ^2(e+f x)-1} \sqrt{\sec (e+f x)}-2 \sec (e+f x)+1\right )\right )}{2 f (\cos (e+f x)+1) \sqrt{2-2 \cos ^2(e+f x)} \sqrt{1-\cos ^2(e+f x)} \sqrt{a (\sec (e+f x)+1)} (c-c \sec (e+f x))}+\frac{\sin (e+f x) \sin ^2\left (\frac{e}{2}+\frac{f x}{2}\right ) \cos (e+f x) (\sec (e+f x)+1)^{3/2} \sqrt{\sec ^2(e+f x)-1} \left (8 \log \left (\sec ^{\frac{3}{2}}(e+f x)+\sqrt{\sec (e+f x)+1} \sqrt{\sec ^2(e+f x)-1}+\sqrt{\sec (e+f x)}\right )+\sqrt{2} \left (\log \left (-3 \sec ^2(e+f x)+2 \sqrt{2} \sqrt{\sec (e+f x)+1} \sqrt{\sec ^2(e+f x)-1} \sqrt{\sec (e+f x)}-2 \sec (e+f x)+1\right )-\log \left (-3 \sec ^2(e+f x)-2 \sqrt{2} \sqrt{\sec (e+f x)+1} \sqrt{\sec ^2(e+f x)-1} \sqrt{\sec (e+f x)}-2 \sec (e+f x)+1\right )\right )-8 \log (\sec (e+f x)+1)\right )}{2 f (\cos (e+f x)+1) \left (1-\cos ^2(e+f x)\right ) \sqrt{a (\sec (e+f x)+1)} (c-c \sec (e+f x))}+\frac{\sin ^2\left (\frac{e}{2}+\frac{f x}{2}\right ) \sec ^{\frac{3}{2}}(e+f x) \sqrt{\sec (e+f x)+1} \sqrt{(\cos (e+f x)+1) \sec (e+f x)} \left (\frac{\csc \left (\frac{e}{2}\right ) \sin \left (\frac{f x}{2}\right ) \csc \left (\frac{e}{2}+\frac{f x}{2}\right )}{f}+\frac{\sec \left (\frac{e}{2}\right ) \sin \left (\frac{f x}{2}\right ) \sec \left (\frac{e}{2}+\frac{f x}{2}\right )}{f}-\frac{2 \cot (e)}{f}\right )}{\sqrt{a (\sec (e+f x)+1)} (c-c \sec (e+f x))} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(Sec[e + f*x]^(3/2)*Sqrt[(1 + Cos[e + f*x])*Sec[e + f*x]]*Sqrt[1 + Sec[e + f*x]]*((-2*Cot[e])/f + (Csc[e/2]*Cs
c[e/2 + (f*x)/2]*Sin[(f*x)/2])/f + (Sec[e/2]*Sec[e/2 + (f*x)/2]*Sin[(f*x)/2])/f)*Sin[e/2 + (f*x)/2]^2)/(Sqrt[a
*(1 + Sec[e + f*x])]*(c - c*Sec[e + f*x])) + (Cos[e + f*x]*(Log[1 - 2*Sec[e + f*x] - 3*Sec[e + f*x]^2 - 2*Sqrt
[2]*Sqrt[Sec[e + f*x]]*Sqrt[1 + Sec[e + f*x]]*Sqrt[-1 + Sec[e + f*x]^2]] - Log[1 - 2*Sec[e + f*x] - 3*Sec[e +
f*x]^2 + 2*Sqrt[2]*Sqrt[Sec[e + f*x]]*Sqrt[1 + Sec[e + f*x]]*Sqrt[-1 + Sec[e + f*x]^2]])*(1 + Sec[e + f*x])^(3
/2)*Sqrt[-1 + Sec[e + f*x]^2]*Sin[e/2 + (f*x)/2]^2*Sin[e + f*x])/(2*f*(1 + Cos[e + f*x])*Sqrt[2 - 2*Cos[e + f*
x]^2]*Sqrt[1 - Cos[e + f*x]^2]*Sqrt[a*(1 + Sec[e + f*x])]*(c - c*Sec[e + f*x])) + (Cos[e + f*x]*(-8*Log[1 + Se
c[e + f*x]] + 8*Log[Sqrt[Sec[e + f*x]] + Sec[e + f*x]^(3/2) + Sqrt[1 + Sec[e + f*x]]*Sqrt[-1 + Sec[e + f*x]^2]
] + Sqrt[2]*(-Log[1 - 2*Sec[e + f*x] - 3*Sec[e + f*x]^2 - 2*Sqrt[2]*Sqrt[Sec[e + f*x]]*Sqrt[1 + Sec[e + f*x]]*
Sqrt[-1 + Sec[e + f*x]^2]] + Log[1 - 2*Sec[e + f*x] - 3*Sec[e + f*x]^2 + 2*Sqrt[2]*Sqrt[Sec[e + f*x]]*Sqrt[1 +
Sec[e + f*x]]*Sqrt[-1 + Sec[e + f*x]^2]]))*(1 + Sec[e + f*x])^(3/2)*Sqrt[-1 + Sec[e + f*x]^2]*Sin[e/2 + (f*x)
/2]^2*Sin[e + f*x])/(2*f*(1 + Cos[e + f*x])*(1 - Cos[e + f*x]^2)*Sqrt[a*(1 + Sec[e + f*x])]*(c - c*Sec[e + f*x
]))

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Maple [B]  time = 0.346, size = 317, normalized size = 2.3 \begin{align*}{\frac{ \left ( \cos \left ( fx+e \right ) \right ) ^{3}}{cfa \left ( \sin \left ( fx+e \right ) \right ) ^{2}} \left ( \sqrt{2}\arctan \left ({\frac{\sqrt{2} \left ( -\cos \left ( fx+e \right ) -1+\sin \left ( fx+e \right ) \right ) }{4}\sqrt{-2\, \left ( 1+\cos \left ( fx+e \right ) \right ) ^{-1}}} \right ) \cos \left ( fx+e \right ) +\sqrt{2}\arctan \left ({\frac{\sqrt{2} \left ( \cos \left ( fx+e \right ) +1+\sin \left ( fx+e \right ) \right ) }{4}\sqrt{-2\, \left ( 1+\cos \left ( fx+e \right ) \right ) ^{-1}}} \right ) \cos \left ( fx+e \right ) -\sqrt{2}\arctan \left ({\frac{\sqrt{2} \left ( -\cos \left ( fx+e \right ) -1+\sin \left ( fx+e \right ) \right ) }{4}\sqrt{-2\, \left ( 1+\cos \left ( fx+e \right ) \right ) ^{-1}}} \right ) -\sqrt{2}\arctan \left ({\frac{\sqrt{2} \left ( \cos \left ( fx+e \right ) +1+\sin \left ( fx+e \right ) \right ) }{4}\sqrt{-2\, \left ( 1+\cos \left ( fx+e \right ) \right ) ^{-1}}} \right ) +\sin \left ( fx+e \right ) \sqrt{-2\, \left ( 1+\cos \left ( fx+e \right ) \right ) ^{-1}}-\arctan \left ({\frac{\sin \left ( fx+e \right ) }{2}\sqrt{-2\, \left ( 1+\cos \left ( fx+e \right ) \right ) ^{-1}}} \right ) \cos \left ( fx+e \right ) +\arctan \left ({\frac{\sin \left ( fx+e \right ) }{2}\sqrt{-2\, \left ( 1+\cos \left ( fx+e \right ) \right ) ^{-1}}} \right ) \right ) \sqrt{{\frac{a \left ( 1+\cos \left ( fx+e \right ) \right ) }{\cos \left ( fx+e \right ) }}} \left ( \left ( \cos \left ( fx+e \right ) \right ) ^{-1} \right ) ^{{\frac{5}{2}}}{\frac{1}{\sqrt{-2\, \left ( 1+\cos \left ( fx+e \right ) \right ) ^{-1}}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/c/f/a*(2^(1/2)*arctan(1/4*2^(1/2)*(-2/(1+cos(f*x+e)))^(1/2)*(-cos(f*x+e)-1+sin(f*x+e)))*cos(f*x+e)+2^(1/2)*a
rctan(1/4*2^(1/2)*(-2/(1+cos(f*x+e)))^(1/2)*(cos(f*x+e)+1+sin(f*x+e)))*cos(f*x+e)-2^(1/2)*arctan(1/4*2^(1/2)*(
-2/(1+cos(f*x+e)))^(1/2)*(-cos(f*x+e)-1+sin(f*x+e)))-2^(1/2)*arctan(1/4*2^(1/2)*(-2/(1+cos(f*x+e)))^(1/2)*(cos
(f*x+e)+1+sin(f*x+e)))+sin(f*x+e)*(-2/(1+cos(f*x+e)))^(1/2)-arctan(1/2*sin(f*x+e)*(-2/(1+cos(f*x+e)))^(1/2))*c
os(f*x+e)+arctan(1/2*sin(f*x+e)*(-2/(1+cos(f*x+e)))^(1/2)))*(1/cos(f*x+e)*a*(1+cos(f*x+e)))^(1/2)*(1/cos(f*x+e
))^(5/2)*cos(f*x+e)^3/sin(f*x+e)^2/(-2/(1+cos(f*x+e)))^(1/2)

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Maxima [B]  time = 2.07162, size = 1769, normalized size = 12.64 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*((sqrt(2)*cos(1/2*arctan2(sin(2*f*x + 2*e), cos(2*f*x + 2*e)))^2 + sqrt(2)*sin(1/2*arctan2(sin(2*f*x + 2*
e), cos(2*f*x + 2*e)))^2 - 2*sqrt(2)*cos(1/2*arctan2(sin(2*f*x + 2*e), cos(2*f*x + 2*e))) + sqrt(2))*log(2*cos
(1/4*arctan2(sin(2*f*x + 2*e), cos(2*f*x + 2*e)))^2 + 2*sin(1/4*arctan2(sin(2*f*x + 2*e), cos(2*f*x + 2*e)))^2
+ 2*sqrt(2)*cos(1/4*arctan2(sin(2*f*x + 2*e), cos(2*f*x + 2*e))) + 2*sqrt(2)*sin(1/4*arctan2(sin(2*f*x + 2*e)
, cos(2*f*x + 2*e))) + 2) - (sqrt(2)*cos(1/2*arctan2(sin(2*f*x + 2*e), cos(2*f*x + 2*e)))^2 + sqrt(2)*sin(1/2*
arctan2(sin(2*f*x + 2*e), cos(2*f*x + 2*e)))^2 - 2*sqrt(2)*cos(1/2*arctan2(sin(2*f*x + 2*e), cos(2*f*x + 2*e))
) + sqrt(2))*log(2*cos(1/4*arctan2(sin(2*f*x + 2*e), cos(2*f*x + 2*e)))^2 + 2*sin(1/4*arctan2(sin(2*f*x + 2*e)
, cos(2*f*x + 2*e)))^2 + 2*sqrt(2)*cos(1/4*arctan2(sin(2*f*x + 2*e), cos(2*f*x + 2*e))) - 2*sqrt(2)*sin(1/4*ar
ctan2(sin(2*f*x + 2*e), cos(2*f*x + 2*e))) + 2) + (sqrt(2)*cos(1/2*arctan2(sin(2*f*x + 2*e), cos(2*f*x + 2*e))
)^2 + sqrt(2)*sin(1/2*arctan2(sin(2*f*x + 2*e), cos(2*f*x + 2*e)))^2 - 2*sqrt(2)*cos(1/2*arctan2(sin(2*f*x + 2
*e), cos(2*f*x + 2*e))) + sqrt(2))*log(2*cos(1/4*arctan2(sin(2*f*x + 2*e), cos(2*f*x + 2*e)))^2 + 2*sin(1/4*ar
ctan2(sin(2*f*x + 2*e), cos(2*f*x + 2*e)))^2 - 2*sqrt(2)*cos(1/4*arctan2(sin(2*f*x + 2*e), cos(2*f*x + 2*e)))
+ 2*sqrt(2)*sin(1/4*arctan2(sin(2*f*x + 2*e), cos(2*f*x + 2*e))) + 2) - (sqrt(2)*cos(1/2*arctan2(sin(2*f*x + 2
*e), cos(2*f*x + 2*e)))^2 + sqrt(2)*sin(1/2*arctan2(sin(2*f*x + 2*e), cos(2*f*x + 2*e)))^2 - 2*sqrt(2)*cos(1/2
*arctan2(sin(2*f*x + 2*e), cos(2*f*x + 2*e))) + sqrt(2))*log(2*cos(1/4*arctan2(sin(2*f*x + 2*e), cos(2*f*x + 2
*e)))^2 + 2*sin(1/4*arctan2(sin(2*f*x + 2*e), cos(2*f*x + 2*e)))^2 - 2*sqrt(2)*cos(1/4*arctan2(sin(2*f*x + 2*e
), cos(2*f*x + 2*e))) - 2*sqrt(2)*sin(1/4*arctan2(sin(2*f*x + 2*e), cos(2*f*x + 2*e))) + 2) - (cos(1/2*arctan2
(sin(2*f*x + 2*e), cos(2*f*x + 2*e)))^2 + sin(1/2*arctan2(sin(2*f*x + 2*e), cos(2*f*x + 2*e)))^2 - 2*cos(1/2*a
rctan2(sin(2*f*x + 2*e), cos(2*f*x + 2*e))) + 1)*log(cos(1/4*arctan2(sin(2*f*x + 2*e), cos(2*f*x + 2*e)))^2 +
sin(1/4*arctan2(sin(2*f*x + 2*e), cos(2*f*x + 2*e)))^2 + 2*sin(1/4*arctan2(sin(2*f*x + 2*e), cos(2*f*x + 2*e))
) + 1) + (cos(1/2*arctan2(sin(2*f*x + 2*e), cos(2*f*x + 2*e)))^2 + sin(1/2*arctan2(sin(2*f*x + 2*e), cos(2*f*x
+ 2*e)))^2 - 2*cos(1/2*arctan2(sin(2*f*x + 2*e), cos(2*f*x + 2*e))) + 1)*log(cos(1/4*arctan2(sin(2*f*x + 2*e)
, cos(2*f*x + 2*e)))^2 + sin(1/4*arctan2(sin(2*f*x + 2*e), cos(2*f*x + 2*e)))^2 - 2*sin(1/4*arctan2(sin(2*f*x
+ 2*e), cos(2*f*x + 2*e))) + 1) - 4*cos(1/4*arctan2(sin(2*f*x + 2*e), cos(2*f*x + 2*e)))*sin(1/2*arctan2(sin(2
*f*x + 2*e), cos(2*f*x + 2*e))) + 4*cos(1/2*arctan2(sin(2*f*x + 2*e), cos(2*f*x + 2*e)))*sin(1/4*arctan2(sin(2
*f*x + 2*e), cos(2*f*x + 2*e))) - 4*sin(1/4*arctan2(sin(2*f*x + 2*e), cos(2*f*x + 2*e))))/((sqrt(2)*c*cos(1/2*
arctan2(sin(2*f*x + 2*e), cos(2*f*x + 2*e)))^2 + sqrt(2)*c*sin(1/2*arctan2(sin(2*f*x + 2*e), cos(2*f*x + 2*e))
)^2 - 2*sqrt(2)*c*cos(1/2*arctan2(sin(2*f*x + 2*e), cos(2*f*x + 2*e))) + sqrt(2)*c)*sqrt(a)*f)

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Fricas [A]  time = 0.647244, size = 1229, normalized size = 8.78 \begin{align*} \left [\frac{\sqrt{2} \sqrt{a} \log \left (-\frac{\cos \left (f x + e\right )^{2} - \frac{2 \, \sqrt{2} \sqrt{\frac{a \cos \left (f x + e\right ) + a}{\cos \left (f x + e\right )}} \sqrt{\cos \left (f x + e\right )} \sin \left (f x + e\right )}{\sqrt{a}} - 2 \, \cos \left (f x + e\right ) - 3}{\cos \left (f x + e\right )^{2} + 2 \, \cos \left (f x + e\right ) + 1}\right ) \sin \left (f x + e\right ) + 2 \, \sqrt{a} \log \left (\frac{a \cos \left (f x + e\right )^{3} - 7 \, a \cos \left (f x + e\right )^{2} + \frac{4 \,{\left (\cos \left (f x + e\right )^{2} - 2 \, \cos \left (f x + e\right )\right )} \sqrt{a} \sqrt{\frac{a \cos \left (f x + e\right ) + a}{\cos \left (f x + e\right )}} \sin \left (f x + e\right )}{\sqrt{\cos \left (f x + e\right )}} + 8 \, a}{\cos \left (f x + e\right )^{3} + \cos \left (f x + e\right )^{2}}\right ) \sin \left (f x + e\right ) + 4 \, \sqrt{\frac{a \cos \left (f x + e\right ) + a}{\cos \left (f x + e\right )}} \sqrt{\cos \left (f x + e\right )}}{4 \, a c f \sin \left (f x + e\right )}, -\frac{\sqrt{2} a \sqrt{-\frac{1}{a}} \arctan \left (\frac{\sqrt{2} \sqrt{\frac{a \cos \left (f x + e\right ) + a}{\cos \left (f x + e\right )}} \sqrt{-\frac{1}{a}} \sqrt{\cos \left (f x + e\right )}}{\sin \left (f x + e\right )}\right ) \sin \left (f x + e\right ) + 2 \, \sqrt{-a} \arctan \left (\frac{2 \, \sqrt{-a} \sqrt{\frac{a \cos \left (f x + e\right ) + a}{\cos \left (f x + e\right )}} \sqrt{\cos \left (f x + e\right )} \sin \left (f x + e\right )}{a \cos \left (f x + e\right )^{2} - a \cos \left (f x + e\right ) - 2 \, a}\right ) \sin \left (f x + e\right ) - 2 \, \sqrt{\frac{a \cos \left (f x + e\right ) + a}{\cos \left (f x + e\right )}} \sqrt{\cos \left (f x + e\right )}}{2 \, a c f \sin \left (f x + e\right )}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[1/4*(sqrt(2)*sqrt(a)*log(-(cos(f*x + e)^2 - 2*sqrt(2)*sqrt((a*cos(f*x + e) + a)/cos(f*x + e))*sqrt(cos(f*x +
e))*sin(f*x + e)/sqrt(a) - 2*cos(f*x + e) - 3)/(cos(f*x + e)^2 + 2*cos(f*x + e) + 1))*sin(f*x + e) + 2*sqrt(a)
*log((a*cos(f*x + e)^3 - 7*a*cos(f*x + e)^2 + 4*(cos(f*x + e)^2 - 2*cos(f*x + e))*sqrt(a)*sqrt((a*cos(f*x + e)
+ a)/cos(f*x + e))*sin(f*x + e)/sqrt(cos(f*x + e)) + 8*a)/(cos(f*x + e)^3 + cos(f*x + e)^2))*sin(f*x + e) + 4
*sqrt((a*cos(f*x + e) + a)/cos(f*x + e))*sqrt(cos(f*x + e)))/(a*c*f*sin(f*x + e)), -1/2*(sqrt(2)*a*sqrt(-1/a)*
arctan(sqrt(2)*sqrt((a*cos(f*x + e) + a)/cos(f*x + e))*sqrt(-1/a)*sqrt(cos(f*x + e))/sin(f*x + e))*sin(f*x + e
) + 2*sqrt(-a)*arctan(2*sqrt(-a)*sqrt((a*cos(f*x + e) + a)/cos(f*x + e))*sqrt(cos(f*x + e))*sin(f*x + e)/(a*co
s(f*x + e)^2 - a*cos(f*x + e) - 2*a))*sin(f*x + e) - 2*sqrt((a*cos(f*x + e) + a)/cos(f*x + e))*sqrt(cos(f*x +
e)))/(a*c*f*sin(f*x + e))]

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Sympy [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(sec(f*x+e)**(5/2)/(c-c*sec(f*x+e))/(a+a*sec(f*x+e))**(1/2),x)

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int -\frac{\sec \left (f x + e\right )^{\frac{5}{2}}}{\sqrt{a \sec \left (f x + e\right ) + a}{\left (c \sec \left (f x + e\right ) - c\right )}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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