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

Optimal. Leaf size=258 \[ \frac{2 \sqrt{a} c^3 \tan (e+f x) \tanh ^{-1}\left (\frac{\sqrt{a-a \sec (e+f x)}}{\sqrt{a}}\right )}{f \sqrt{a-a \sec (e+f x)} \sqrt{a \sec (e+f x)+a}}+\frac{2 d^2 (3 c-d) \tan (e+f x)}{f \sqrt{a \sec (e+f x)+a}}-\frac{\sqrt{2} \sqrt{a} (c-d)^3 \tan (e+f x) \tanh ^{-1}\left (\frac{\sqrt{a-a \sec (e+f x)}}{\sqrt{2} \sqrt{a}}\right )}{f \sqrt{a-a \sec (e+f x)} \sqrt{a \sec (e+f x)+a}}-\frac{2 d^3 \tan (e+f x) (1-\sec (e+f x))}{3 f \sqrt{a \sec (e+f x)+a}}+\frac{2 d^3 \tan (e+f x)}{f \sqrt{a \sec (e+f x)+a}} \]

[Out]

(2*(3*c - d)*d^2*Tan[e + f*x])/(f*Sqrt[a + a*Sec[e + f*x]]) + (2*d^3*Tan[e + f*x])/(f*Sqrt[a + a*Sec[e + f*x]]
) - (2*d^3*(1 - Sec[e + f*x])*Tan[e + f*x])/(3*f*Sqrt[a + a*Sec[e + f*x]]) + (2*Sqrt[a]*c^3*ArcTanh[Sqrt[a - a
*Sec[e + f*x]]/Sqrt[a]]*Tan[e + f*x])/(f*Sqrt[a - a*Sec[e + f*x]]*Sqrt[a + a*Sec[e + f*x]]) - (Sqrt[2]*Sqrt[a]
*(c - d)^3*ArcTanh[Sqrt[a - a*Sec[e + f*x]]/(Sqrt[2]*Sqrt[a])]*Tan[e + f*x])/(f*Sqrt[a - a*Sec[e + f*x]]*Sqrt[
a + a*Sec[e + f*x]])

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Rubi [A]  time = 0.207306, antiderivative size = 258, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 5, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.185, Rules used = {3940, 180, 63, 206, 43} \[ \frac{2 \sqrt{a} c^3 \tan (e+f x) \tanh ^{-1}\left (\frac{\sqrt{a-a \sec (e+f x)}}{\sqrt{a}}\right )}{f \sqrt{a-a \sec (e+f x)} \sqrt{a \sec (e+f x)+a}}+\frac{2 d^2 (3 c-d) \tan (e+f x)}{f \sqrt{a \sec (e+f x)+a}}-\frac{\sqrt{2} \sqrt{a} (c-d)^3 \tan (e+f x) \tanh ^{-1}\left (\frac{\sqrt{a-a \sec (e+f x)}}{\sqrt{2} \sqrt{a}}\right )}{f \sqrt{a-a \sec (e+f x)} \sqrt{a \sec (e+f x)+a}}-\frac{2 d^3 \tan (e+f x) (1-\sec (e+f x))}{3 f \sqrt{a \sec (e+f x)+a}}+\frac{2 d^3 \tan (e+f x)}{f \sqrt{a \sec (e+f x)+a}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*(3*c - d)*d^2*Tan[e + f*x])/(f*Sqrt[a + a*Sec[e + f*x]]) + (2*d^3*Tan[e + f*x])/(f*Sqrt[a + a*Sec[e + f*x]]
) - (2*d^3*(1 - Sec[e + f*x])*Tan[e + f*x])/(3*f*Sqrt[a + a*Sec[e + f*x]]) + (2*Sqrt[a]*c^3*ArcTanh[Sqrt[a - a
*Sec[e + f*x]]/Sqrt[a]]*Tan[e + f*x])/(f*Sqrt[a - a*Sec[e + f*x]]*Sqrt[a + a*Sec[e + f*x]]) - (Sqrt[2]*Sqrt[a]
*(c - d)^3*ArcTanh[Sqrt[a - a*Sec[e + f*x]]/(Sqrt[2]*Sqrt[a])]*Tan[e + f*x])/(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 43

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rubi steps

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

Mathematica [C]  time = 8.11463, size = 787, normalized size = 3.05 \[ \frac{2 \sqrt{\frac{1}{1-2 \sin ^2\left (\frac{1}{2} (e+f x)\right )}} \sqrt{1-2 \sin ^2\left (\frac{1}{2} (e+f x)\right )} \cos \left (\frac{1}{2} (e+f x)\right ) (c+d \sec (e+f x))^3 \left (-\frac{(c-d)^3 \csc ^5\left (\frac{1}{2} (e+f x)\right ) \left (-12 \sin ^8\left (\frac{1}{2} (e+f x)\right ) \cos ^4\left (\frac{1}{2} (e+f x)\right ) \text{HypergeometricPFQ}\left (\left \{2,2,\frac{7}{2}\right \},\left \{1,\frac{9}{2}\right \},-\frac{\sin ^2\left (\frac{1}{2} (e+f x)\right )}{1-2 \sin ^2\left (\frac{1}{2} (e+f x)\right )}\right )-12 \left (3 \sin ^4\left (\frac{1}{2} (e+f x)\right )-7 \sin ^2\left (\frac{1}{2} (e+f x)\right )+4\right ) \sin ^8\left (\frac{1}{2} (e+f x)\right ) \text{Hypergeometric2F1}\left (2,\frac{7}{2},\frac{9}{2},-\frac{\sin ^2\left (\frac{1}{2} (e+f x)\right )}{1-2 \sin ^2\left (\frac{1}{2} (e+f x)\right )}\right )+7 \sqrt{-\frac{\sin ^2\left (\frac{1}{2} (e+f x)\right )}{1-2 \sin ^2\left (\frac{1}{2} (e+f x)\right )}} \left (1-2 \sin ^2\left (\frac{1}{2} (e+f x)\right )\right )^3 \left (8 \sin ^4\left (\frac{1}{2} (e+f x)\right )-20 \sin ^2\left (\frac{1}{2} (e+f x)\right )+15\right ) \left (\left (3-7 \sin ^2\left (\frac{1}{2} (e+f x)\right )\right ) \sqrt{-\frac{\sin ^2\left (\frac{1}{2} (e+f x)\right )}{1-2 \sin ^2\left (\frac{1}{2} (e+f x)\right )}}-3 \left (1-2 \sin ^2\left (\frac{1}{2} (e+f x)\right )\right ) \tanh ^{-1}\left (\sqrt{-\frac{\sin ^2\left (\frac{1}{2} (e+f x)\right )}{1-2 \sin ^2\left (\frac{1}{2} (e+f x)\right )}}\right )\right )\right )}{63 \left (1-2 \sin ^2\left (\frac{1}{2} (e+f x)\right )\right )^{7/2}}+\frac{4 c \left (c^2+3 d^2\right ) \sin \left (\frac{1}{2} (e+f x)\right )}{3 \sqrt{1-2 \sin ^2\left (\frac{1}{2} (e+f x)\right )}}+\frac{2 c \left (c^2+3 d^2\right ) \sin \left (\frac{1}{2} (e+f x)\right )}{3 \left (1-2 \sin ^2\left (\frac{1}{2} (e+f x)\right )\right )^{3/2}}-\frac{4 c^2 (c+3 d) \sin ^3\left (\frac{1}{2} (e+f x)\right )}{3 \left (1-2 \sin ^2\left (\frac{1}{2} (e+f x)\right )\right )^{3/2}}+\frac{1}{3} c^3 \sqrt{1-2 \sin ^2\left (\frac{1}{2} (e+f x)\right )} \left (\frac{4 \sin ^4\left (\frac{1}{2} (e+f x)\right )}{\left (1-2 \sin ^2\left (\frac{1}{2} (e+f x)\right )\right )^2}-\frac{6 \sin ^2\left (\frac{1}{2} (e+f x)\right )}{1-2 \sin ^2\left (\frac{1}{2} (e+f x)\right )}+\frac{3 \sqrt{2} \sin ^{-1}\left (\sqrt{2} \sin \left (\frac{1}{2} (e+f x)\right )\right ) \sin \left (\frac{1}{2} (e+f x)\right )}{\sqrt{1-2 \sin ^2\left (\frac{1}{2} (e+f x)\right )}}\right ) \csc \left (\frac{1}{2} (e+f x)\right )\right )}{f \sec ^{\frac{5}{2}}(e+f x) \sqrt{a (\sec (e+f x)+1)} (c \cos (e+f x)+d)^3} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(2*Cos[(e + f*x)/2]*(c + d*Sec[e + f*x])^3*Sqrt[(1 - 2*Sin[(e + f*x)/2]^2)^(-1)]*Sqrt[1 - 2*Sin[(e + f*x)/2]^2
]*((2*c*(c^2 + 3*d^2)*Sin[(e + f*x)/2])/(3*(1 - 2*Sin[(e + f*x)/2]^2)^(3/2)) - (4*c^2*(c + 3*d)*Sin[(e + f*x)/
2]^3)/(3*(1 - 2*Sin[(e + f*x)/2]^2)^(3/2)) + (4*c*(c^2 + 3*d^2)*Sin[(e + f*x)/2])/(3*Sqrt[1 - 2*Sin[(e + f*x)/
2]^2]) + (c^3*Csc[(e + f*x)/2]*Sqrt[1 - 2*Sin[(e + f*x)/2]^2]*((4*Sin[(e + f*x)/2]^4)/(1 - 2*Sin[(e + f*x)/2]^
2)^2 - (6*Sin[(e + f*x)/2]^2)/(1 - 2*Sin[(e + f*x)/2]^2) + (3*Sqrt[2]*ArcSin[Sqrt[2]*Sin[(e + f*x)/2]]*Sin[(e
+ f*x)/2])/Sqrt[1 - 2*Sin[(e + f*x)/2]^2]))/3 - ((c - d)^3*Csc[(e + f*x)/2]^5*(-12*Cos[(e + f*x)/2]^4*Hypergeo
metricPFQ[{2, 2, 7/2}, {1, 9/2}, -(Sin[(e + f*x)/2]^2/(1 - 2*Sin[(e + f*x)/2]^2))]*Sin[(e + f*x)/2]^8 - 12*Hyp
ergeometric2F1[2, 7/2, 9/2, -(Sin[(e + f*x)/2]^2/(1 - 2*Sin[(e + f*x)/2]^2))]*Sin[(e + f*x)/2]^8*(4 - 7*Sin[(e
+ f*x)/2]^2 + 3*Sin[(e + f*x)/2]^4) + 7*Sqrt[-(Sin[(e + f*x)/2]^2/(1 - 2*Sin[(e + f*x)/2]^2))]*(1 - 2*Sin[(e
+ f*x)/2]^2)^3*(15 - 20*Sin[(e + f*x)/2]^2 + 8*Sin[(e + f*x)/2]^4)*((3 - 7*Sin[(e + f*x)/2]^2)*Sqrt[-(Sin[(e +
f*x)/2]^2/(1 - 2*Sin[(e + f*x)/2]^2))] - 3*ArcTanh[Sqrt[-(Sin[(e + f*x)/2]^2/(1 - 2*Sin[(e + f*x)/2]^2))]]*(1
- 2*Sin[(e + f*x)/2]^2))))/(63*(1 - 2*Sin[(e + f*x)/2]^2)^(7/2))))/(f*(d + c*Cos[e + f*x])^3*Sec[e + f*x]^(5/
2)*Sqrt[a*(1 + Sec[e + f*x])])

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Maple [B]  time = 0.307, size = 907, normalized size = 3.5 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/6/f/a*(1/cos(f*x+e)*a*(1+cos(f*x+e)))^(1/2)*(3*sin(f*x+e)*cos(f*x+e)*2^(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*cos(f*x+e)/(1+cos(f*x+e)))^(3/2)*c^3+3*sin(f*x+e)*cos(f*x
+e)*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))*(-2*cos(f*x+e)/(1+cos(f*x+
e)))^(3/2)*c^3-9*sin(f*x+e)*cos(f*x+e)*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))*(-2*cos(f*x+e)/(1+cos(f*x+e)))^(3/2)*c^2*d+9*sin(f*x+e)*cos(f*x+e)*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))*(-2*cos(f*x+e)/(1+cos(f*x+e)))^(3/2)*c*d^2-3*sin(f*x+e)*cos(f*x+
e)*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))*(-2*cos(f*x+e)/(1+cos(f*x+e
)))^(3/2)*d^3+3*2^(1/2)*(-2*cos(f*x+e)/(1+cos(f*x+e)))^(3/2)*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^3*sin(f*x+e)+3*(-2*cos(f*x+e)/(1+cos(f*x+e)))^(3/2)*ln(-(-(-2*cos(f*x+e)/(1+c
os(f*x+e)))^(1/2)*sin(f*x+e)+cos(f*x+e)-1)/sin(f*x+e))*c^3*sin(f*x+e)-9*(-2*cos(f*x+e)/(1+cos(f*x+e)))^(3/2)*l
n(-(-(-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*sin(f*x+e)+9*(-2*cos(f*x+
e)/(1+cos(f*x+e)))^(3/2)*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
*sin(f*x+e)-3*(-2*cos(f*x+e)/(1+cos(f*x+e)))^(3/2)*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))*d^3*sin(f*x+e)-36*cos(f*x+e)^2*c*d^2+4*cos(f*x+e)^2*d^3+36*cos(f*x+e)*c*d^2-8*cos(f*x+e)*
d^3+4*d^3)/sin(f*x+e)/cos(f*x+e)

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 72.6639, size = 1553, normalized size = 6.02 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[-1/6*(3*sqrt(2)*((a*c^3 - 3*a*c^2*d + 3*a*c*d^2 - a*d^3)*cos(f*x + e)^2 + (a*c^3 - 3*a*c^2*d + 3*a*c*d^2 - a*
d^3)*cos(f*x + e))*sqrt(-1/a)*log(-(2*sqrt(2)*sqrt((a*cos(f*x + e) + a)/cos(f*x + e))*sqrt(-1/a)*cos(f*x + e)*
sin(f*x + e) - 3*cos(f*x + e)^2 - 2*cos(f*x + e) + 1)/(cos(f*x + e)^2 + 2*cos(f*x + e) + 1)) + 6*(c^3*cos(f*x
+ e)^2 + c^3*cos(f*x + e))*sqrt(-a)*log((2*a*cos(f*x + e)^2 + 2*sqrt(-a)*sqrt((a*cos(f*x + e) + a)/cos(f*x + e
))*cos(f*x + e)*sin(f*x + e) + a*cos(f*x + e) - a)/(cos(f*x + e) + 1)) - 4*(d^3 + (9*c*d^2 - d^3)*cos(f*x + e)
)*sqrt((a*cos(f*x + e) + a)/cos(f*x + e))*sin(f*x + e))/(a*f*cos(f*x + e)^2 + a*f*cos(f*x + e)), -1/3*(6*(c^3*
cos(f*x + e)^2 + c^3*cos(f*x + e))*sqrt(a)*arctan(sqrt((a*cos(f*x + e) + a)/cos(f*x + e))*cos(f*x + e)/(sqrt(a
)*sin(f*x + e))) - 2*(d^3 + (9*c*d^2 - d^3)*cos(f*x + e))*sqrt((a*cos(f*x + e) + a)/cos(f*x + e))*sin(f*x + e)
- 3*sqrt(2)*((a*c^3 - 3*a*c^2*d + 3*a*c*d^2 - a*d^3)*cos(f*x + e)^2 + (a*c^3 - 3*a*c^2*d + 3*a*c*d^2 - a*d^3)
*cos(f*x + e))*arctan(sqrt(2)*sqrt((a*cos(f*x + e) + a)/cos(f*x + e))*cos(f*x + e)/(sqrt(a)*sin(f*x + e)))/sqr
t(a))/(a*f*cos(f*x + e)^2 + a*f*cos(f*x + e))]

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

Timed out