\(\int \frac {\sec ^2(c+d x)}{\sqrt [3]{a+b \sec (c+d x)}} \, dx\) [701]

Optimal result

Integrand size = 23, antiderivative size = 219 \[ \int \frac {\sec ^2(c+d x)}{\sqrt [3]{a+b \sec (c+d x)}} \, dx=\frac {\sqrt {2} \operatorname {AppellF1}\left (\frac {1}{2},\frac {1}{2},-\frac {2}{3},\frac {3}{2},\frac {1}{2} (1-\sec (c+d x)),\frac {b (1-\sec (c+d x))}{a+b}\right ) (a+b \sec (c+d x))^{2/3} \tan (c+d x)}{b d \sqrt {1+\sec (c+d x)} \left (\frac {a+b \sec (c+d x)}{a+b}\right )^{2/3}}-\frac {\sqrt {2} a \operatorname {AppellF1}\left (\frac {1}{2},\frac {1}{2},\frac {1}{3},\frac {3}{2},\frac {1}{2} (1-\sec (c+d x)),\frac {b (1-\sec (c+d x))}{a+b}\right ) \sqrt [3]{\frac {a+b \sec (c+d x)}{a+b}} \tan (c+d x)}{b d \sqrt {1+\sec (c+d x)} \sqrt [3]{a+b \sec (c+d x)}} \]

[Out]

AppellF1(1/2,-2/3,1/2,3/2,b*(1-sec(d*x+c))/(a+b),1/2-1/2*sec(d*x+c))*(a+b*sec(d*x+c))^(2/3)*2^(1/2)*tan(d*x+c)
/b/d/((a+b*sec(d*x+c))/(a+b))^(2/3)/(1+sec(d*x+c))^(1/2)-a*AppellF1(1/2,1/3,1/2,3/2,b*(1-sec(d*x+c))/(a+b),1/2
-1/2*sec(d*x+c))*((a+b*sec(d*x+c))/(a+b))^(1/3)*2^(1/2)*tan(d*x+c)/b/d/(a+b*sec(d*x+c))^(1/3)/(1+sec(d*x+c))^(
1/2)

Rubi [A] (verified)

Time = 0.28 (sec) , antiderivative size = 219, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.174, Rules used = {3923, 3919, 144, 143} \[ \int \frac {\sec ^2(c+d x)}{\sqrt [3]{a+b \sec (c+d x)}} \, dx=\frac {\sqrt {2} \tan (c+d x) (a+b \sec (c+d x))^{2/3} \operatorname {AppellF1}\left (\frac {1}{2},\frac {1}{2},-\frac {2}{3},\frac {3}{2},\frac {1}{2} (1-\sec (c+d x)),\frac {b (1-\sec (c+d x))}{a+b}\right )}{b d \sqrt {\sec (c+d x)+1} \left (\frac {a+b \sec (c+d x)}{a+b}\right )^{2/3}}-\frac {\sqrt {2} a \tan (c+d x) \sqrt [3]{\frac {a+b \sec (c+d x)}{a+b}} \operatorname {AppellF1}\left (\frac {1}{2},\frac {1}{2},\frac {1}{3},\frac {3}{2},\frac {1}{2} (1-\sec (c+d x)),\frac {b (1-\sec (c+d x))}{a+b}\right )}{b d \sqrt {\sec (c+d x)+1} \sqrt [3]{a+b \sec (c+d x)}} \]

[In]

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

[Out]

(Sqrt[2]*AppellF1[1/2, 1/2, -2/3, 3/2, (1 - Sec[c + d*x])/2, (b*(1 - Sec[c + d*x]))/(a + b)]*(a + b*Sec[c + d*
x])^(2/3)*Tan[c + d*x])/(b*d*Sqrt[1 + Sec[c + d*x]]*((a + b*Sec[c + d*x])/(a + b))^(2/3)) - (Sqrt[2]*a*AppellF
1[1/2, 1/2, 1/3, 3/2, (1 - Sec[c + d*x])/2, (b*(1 - Sec[c + d*x]))/(a + b)]*((a + b*Sec[c + d*x])/(a + b))^(1/
3)*Tan[c + d*x])/(b*d*Sqrt[1 + Sec[c + d*x]]*(a + b*Sec[c + d*x])^(1/3))

Rule 143

Int[((a_) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_))^(p_), x_Symbol] :> Simp[((a + b*x)
^(m + 1)/(b*(m + 1)*(b/(b*c - a*d))^n*(b/(b*e - a*f))^p))*AppellF1[m + 1, -n, -p, m + 2, (-d)*((a + b*x)/(b*c
- a*d)), (-f)*((a + b*x)/(b*e - a*f))], x] /; FreeQ[{a, b, c, d, e, f, m, n, p}, x] &&  !IntegerQ[m] &&  !Inte
gerQ[n] &&  !IntegerQ[p] && GtQ[b/(b*c - a*d), 0] && GtQ[b/(b*e - a*f), 0] &&  !(GtQ[d/(d*a - c*b), 0] && GtQ[
d/(d*e - c*f), 0] && SimplerQ[c + d*x, a + b*x]) &&  !(GtQ[f/(f*a - e*b), 0] && GtQ[f/(f*c - e*d), 0] && Simpl
erQ[e + f*x, a + b*x])

Rule 144

Int[((a_) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_))^(p_), x_Symbol] :> Dist[(e + f*x)^
FracPart[p]/((b/(b*e - a*f))^IntPart[p]*(b*((e + f*x)/(b*e - a*f)))^FracPart[p]), Int[(a + b*x)^m*(c + d*x)^n*
(b*(e/(b*e - a*f)) + b*f*(x/(b*e - a*f)))^p, x], x] /; FreeQ[{a, b, c, d, e, f, m, n, p}, x] &&  !IntegerQ[m]
&&  !IntegerQ[n] &&  !IntegerQ[p] && GtQ[b/(b*c - a*d), 0] &&  !GtQ[b/(b*e - a*f), 0]

Rule 3919

Int[csc[(e_.) + (f_.)*(x_)]*(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_))^(m_), x_Symbol] :> Dist[Cot[e + f*x]/(f*Sqr
t[1 + Csc[e + f*x]]*Sqrt[1 - Csc[e + f*x]]), Subst[Int[(a + b*x)^m/(Sqrt[1 + x]*Sqrt[1 - x]), x], x, Csc[e + f
*x]], x] /; FreeQ[{a, b, e, f, m}, x] && NeQ[a^2 - b^2, 0] &&  !IntegerQ[2*m]

Rule 3923

Int[csc[(e_.) + (f_.)*(x_)]^2*(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_))^(m_), x_Symbol] :> Dist[-a/b, Int[Csc[e +
f*x]*(a + b*Csc[e + f*x])^m, x], x] + Dist[1/b, Int[Csc[e + f*x]*(a + b*Csc[e + f*x])^(m + 1), x], x] /; Free
Q[{a, b, e, f, m}, x] && NeQ[a^2 - b^2, 0]

Rubi steps \begin{align*} \text {integral}& = \frac {\int \sec (c+d x) (a+b \sec (c+d x))^{2/3} \, dx}{b}-\frac {a \int \frac {\sec (c+d x)}{\sqrt [3]{a+b \sec (c+d x)}} \, dx}{b} \\ & = -\frac {\tan (c+d x) \text {Subst}\left (\int \frac {(a+b x)^{2/3}}{\sqrt {1-x} \sqrt {1+x}} \, dx,x,\sec (c+d x)\right )}{b d \sqrt {1-\sec (c+d x)} \sqrt {1+\sec (c+d x)}}+\frac {(a \tan (c+d x)) \text {Subst}\left (\int \frac {1}{\sqrt {1-x} \sqrt {1+x} \sqrt [3]{a+b x}} \, dx,x,\sec (c+d x)\right )}{b d \sqrt {1-\sec (c+d x)} \sqrt {1+\sec (c+d x)}} \\ & = -\frac {\left ((a+b \sec (c+d x))^{2/3} \tan (c+d x)\right ) \text {Subst}\left (\int \frac {\left (-\frac {a}{-a-b}-\frac {b x}{-a-b}\right )^{2/3}}{\sqrt {1-x} \sqrt {1+x}} \, dx,x,\sec (c+d x)\right )}{b d \sqrt {1-\sec (c+d x)} \sqrt {1+\sec (c+d x)} \left (-\frac {a+b \sec (c+d x)}{-a-b}\right )^{2/3}}+\frac {\left (a \sqrt [3]{-\frac {a+b \sec (c+d x)}{-a-b}} \tan (c+d x)\right ) \text {Subst}\left (\int \frac {1}{\sqrt {1-x} \sqrt {1+x} \sqrt [3]{-\frac {a}{-a-b}-\frac {b x}{-a-b}}} \, dx,x,\sec (c+d x)\right )}{b d \sqrt {1-\sec (c+d x)} \sqrt {1+\sec (c+d x)} \sqrt [3]{a+b \sec (c+d x)}} \\ & = \frac {\sqrt {2} \operatorname {AppellF1}\left (\frac {1}{2},\frac {1}{2},-\frac {2}{3},\frac {3}{2},\frac {1}{2} (1-\sec (c+d x)),\frac {b (1-\sec (c+d x))}{a+b}\right ) (a+b \sec (c+d x))^{2/3} \tan (c+d x)}{b d \sqrt {1+\sec (c+d x)} \left (\frac {a+b \sec (c+d x)}{a+b}\right )^{2/3}}-\frac {\sqrt {2} a \operatorname {AppellF1}\left (\frac {1}{2},\frac {1}{2},\frac {1}{3},\frac {3}{2},\frac {1}{2} (1-\sec (c+d x)),\frac {b (1-\sec (c+d x))}{a+b}\right ) \sqrt [3]{\frac {a+b \sec (c+d x)}{a+b}} \tan (c+d x)}{b d \sqrt {1+\sec (c+d x)} \sqrt [3]{a+b \sec (c+d x)}} \\ \end{align*}

Mathematica [B] (warning: unable to verify)

Leaf count is larger than twice the leaf count of optimal. \(1748\) vs. \(2(219)=438\).

Time = 23.82 (sec) , antiderivative size = 1748, normalized size of antiderivative = 7.98 \[ \int \frac {\sec ^2(c+d x)}{\sqrt [3]{a+b \sec (c+d x)}} \, dx=\frac {3 (b+a \cos (c+d x)) \tan (c+d x)}{2 b d \sqrt [3]{a+b \sec (c+d x)}}-\frac {3 (b+3 a \cos (c+d x)) (a+b \sec (c+d x))^{2/3} \left (5 \left (a^2-b^2\right )+3 b \operatorname {AppellF1}\left (\frac {5}{3},\frac {3}{2},\frac {3}{2},\frac {8}{3},\frac {a+b \sec (c+d x)}{a-b},\frac {a+b \sec (c+d x)}{a+b}\right ) \sec (c+d x) \sqrt {\frac {b (1+\sec (c+d x))}{-a+b}} \sqrt {\frac {b-b \sec (c+d x)}{a+b}} (a+b \sec (c+d x))\right )}{10 b \left (-a^2+b^2\right ) d \sqrt [3]{b+a \cos (c+d x)} \sqrt {1-\cos ^2(c+d x)} \sec ^{\frac {7}{3}}(c+d x) \left (\frac {3 b \left (5 \left (a^2-b^2\right )+3 b \operatorname {AppellF1}\left (\frac {5}{3},\frac {3}{2},\frac {3}{2},\frac {8}{3},\frac {a+b \sec (c+d x)}{a-b},\frac {a+b \sec (c+d x)}{a+b}\right ) \sec (c+d x) \sqrt {\frac {b (1+\sec (c+d x))}{-a+b}} \sqrt {\frac {b-b \sec (c+d x)}{a+b}} (a+b \sec (c+d x))\right ) \sin (c+d x)}{5 \left (-a^2+b^2\right ) \sqrt [3]{b+a \cos (c+d x)} \sqrt {1-\cos ^2(c+d x)} \sqrt [3]{\sec (c+d x)}}-\frac {3 (a+b \sec (c+d x)) \left (5 \left (a^2-b^2\right )+3 b \operatorname {AppellF1}\left (\frac {5}{3},\frac {3}{2},\frac {3}{2},\frac {8}{3},\frac {a+b \sec (c+d x)}{a-b},\frac {a+b \sec (c+d x)}{a+b}\right ) \sec (c+d x) \sqrt {\frac {b (1+\sec (c+d x))}{-a+b}} \sqrt {\frac {b-b \sec (c+d x)}{a+b}} (a+b \sec (c+d x))\right ) \sin (c+d x)}{5 \left (-a^2+b^2\right ) \sqrt [3]{b+a \cos (c+d x)} \left (1-\cos ^2(c+d x)\right )^{3/2} \sec ^{\frac {10}{3}}(c+d x)}+\frac {a (a+b \sec (c+d x)) \left (5 \left (a^2-b^2\right )+3 b \operatorname {AppellF1}\left (\frac {5}{3},\frac {3}{2},\frac {3}{2},\frac {8}{3},\frac {a+b \sec (c+d x)}{a-b},\frac {a+b \sec (c+d x)}{a+b}\right ) \sec (c+d x) \sqrt {\frac {b (1+\sec (c+d x))}{-a+b}} \sqrt {\frac {b-b \sec (c+d x)}{a+b}} (a+b \sec (c+d x))\right ) \sin (c+d x)}{5 \left (-a^2+b^2\right ) (b+a \cos (c+d x))^{4/3} \sqrt {1-\cos ^2(c+d x)} \sec ^{\frac {7}{3}}(c+d x)}-\frac {7 (a+b \sec (c+d x)) \left (5 \left (a^2-b^2\right )+3 b \operatorname {AppellF1}\left (\frac {5}{3},\frac {3}{2},\frac {3}{2},\frac {8}{3},\frac {a+b \sec (c+d x)}{a-b},\frac {a+b \sec (c+d x)}{a+b}\right ) \sec (c+d x) \sqrt {\frac {b (1+\sec (c+d x))}{-a+b}} \sqrt {\frac {b-b \sec (c+d x)}{a+b}} (a+b \sec (c+d x))\right ) \sin (c+d x)}{5 \left (-a^2+b^2\right ) \sqrt [3]{b+a \cos (c+d x)} \sqrt {1-\cos ^2(c+d x)} \sec ^{\frac {4}{3}}(c+d x)}+\frac {3 (a+b \sec (c+d x)) \left (3 b^2 \operatorname {AppellF1}\left (\frac {5}{3},\frac {3}{2},\frac {3}{2},\frac {8}{3},\frac {a+b \sec (c+d x)}{a-b},\frac {a+b \sec (c+d x)}{a+b}\right ) \sec ^2(c+d x) \sqrt {\frac {b (1+\sec (c+d x))}{-a+b}} \sqrt {\frac {b-b \sec (c+d x)}{a+b}} \tan (c+d x)-\frac {3 b^2 \operatorname {AppellF1}\left (\frac {5}{3},\frac {3}{2},\frac {3}{2},\frac {8}{3},\frac {a+b \sec (c+d x)}{a-b},\frac {a+b \sec (c+d x)}{a+b}\right ) \sec ^2(c+d x) \sqrt {\frac {b (1+\sec (c+d x))}{-a+b}} (a+b \sec (c+d x)) \tan (c+d x)}{2 (a+b) \sqrt {\frac {b-b \sec (c+d x)}{a+b}}}+\frac {3 b^2 \operatorname {AppellF1}\left (\frac {5}{3},\frac {3}{2},\frac {3}{2},\frac {8}{3},\frac {a+b \sec (c+d x)}{a-b},\frac {a+b \sec (c+d x)}{a+b}\right ) \sec ^2(c+d x) \sqrt {\frac {b-b \sec (c+d x)}{a+b}} (a+b \sec (c+d x)) \tan (c+d x)}{2 (-a+b) \sqrt {\frac {b (1+\sec (c+d x))}{-a+b}}}+3 b \operatorname {AppellF1}\left (\frac {5}{3},\frac {3}{2},\frac {3}{2},\frac {8}{3},\frac {a+b \sec (c+d x)}{a-b},\frac {a+b \sec (c+d x)}{a+b}\right ) \sec (c+d x) \sqrt {\frac {b (1+\sec (c+d x))}{-a+b}} \sqrt {\frac {b-b \sec (c+d x)}{a+b}} (a+b \sec (c+d x)) \tan (c+d x)+3 b \sec (c+d x) \sqrt {\frac {b (1+\sec (c+d x))}{-a+b}} \sqrt {\frac {b-b \sec (c+d x)}{a+b}} (a+b \sec (c+d x)) \left (\frac {15 b \operatorname {AppellF1}\left (\frac {8}{3},\frac {3}{2},\frac {5}{2},\frac {11}{3},\frac {a+b \sec (c+d x)}{a-b},\frac {a+b \sec (c+d x)}{a+b}\right ) \sec (c+d x) \tan (c+d x)}{16 (a+b)}+\frac {15 b \operatorname {AppellF1}\left (\frac {8}{3},\frac {5}{2},\frac {3}{2},\frac {11}{3},\frac {a+b \sec (c+d x)}{a-b},\frac {a+b \sec (c+d x)}{a+b}\right ) \sec (c+d x) \tan (c+d x)}{16 (a-b)}\right )\right )}{5 \left (-a^2+b^2\right ) \sqrt [3]{b+a \cos (c+d x)} \sqrt {1-\cos ^2(c+d x)} \sec ^{\frac {7}{3}}(c+d x)}\right )} \]

[In]

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

[Out]

(3*(b + a*Cos[c + d*x])*Tan[c + d*x])/(2*b*d*(a + b*Sec[c + d*x])^(1/3)) - (3*(b + 3*a*Cos[c + d*x])*(a + b*Se
c[c + d*x])^(2/3)*(5*(a^2 - b^2) + 3*b*AppellF1[5/3, 3/2, 3/2, 8/3, (a + b*Sec[c + d*x])/(a - b), (a + b*Sec[c
+ d*x])/(a + b)]*Sec[c + d*x]*Sqrt[(b*(1 + Sec[c + d*x]))/(-a + b)]*Sqrt[(b - b*Sec[c + d*x])/(a + b)]*(a + b
*Sec[c + d*x])))/(10*b*(-a^2 + b^2)*d*(b + a*Cos[c + d*x])^(1/3)*Sqrt[1 - Cos[c + d*x]^2]*Sec[c + d*x]^(7/3)*(
(3*b*(5*(a^2 - b^2) + 3*b*AppellF1[5/3, 3/2, 3/2, 8/3, (a + b*Sec[c + d*x])/(a - b), (a + b*Sec[c + d*x])/(a +
b)]*Sec[c + d*x]*Sqrt[(b*(1 + Sec[c + d*x]))/(-a + b)]*Sqrt[(b - b*Sec[c + d*x])/(a + b)]*(a + b*Sec[c + d*x]
))*Sin[c + d*x])/(5*(-a^2 + b^2)*(b + a*Cos[c + d*x])^(1/3)*Sqrt[1 - Cos[c + d*x]^2]*Sec[c + d*x]^(1/3)) - (3*
(a + b*Sec[c + d*x])*(5*(a^2 - b^2) + 3*b*AppellF1[5/3, 3/2, 3/2, 8/3, (a + b*Sec[c + d*x])/(a - b), (a + b*Se
c[c + d*x])/(a + b)]*Sec[c + d*x]*Sqrt[(b*(1 + Sec[c + d*x]))/(-a + b)]*Sqrt[(b - b*Sec[c + d*x])/(a + b)]*(a
+ b*Sec[c + d*x]))*Sin[c + d*x])/(5*(-a^2 + b^2)*(b + a*Cos[c + d*x])^(1/3)*(1 - Cos[c + d*x]^2)^(3/2)*Sec[c +
d*x]^(10/3)) + (a*(a + b*Sec[c + d*x])*(5*(a^2 - b^2) + 3*b*AppellF1[5/3, 3/2, 3/2, 8/3, (a + b*Sec[c + d*x])
/(a - b), (a + b*Sec[c + d*x])/(a + b)]*Sec[c + d*x]*Sqrt[(b*(1 + Sec[c + d*x]))/(-a + b)]*Sqrt[(b - b*Sec[c +
d*x])/(a + b)]*(a + b*Sec[c + d*x]))*Sin[c + d*x])/(5*(-a^2 + b^2)*(b + a*Cos[c + d*x])^(4/3)*Sqrt[1 - Cos[c
+ d*x]^2]*Sec[c + d*x]^(7/3)) - (7*(a + b*Sec[c + d*x])*(5*(a^2 - b^2) + 3*b*AppellF1[5/3, 3/2, 3/2, 8/3, (a +
b*Sec[c + d*x])/(a - b), (a + b*Sec[c + d*x])/(a + b)]*Sec[c + d*x]*Sqrt[(b*(1 + Sec[c + d*x]))/(-a + b)]*Sqr
t[(b - b*Sec[c + d*x])/(a + b)]*(a + b*Sec[c + d*x]))*Sin[c + d*x])/(5*(-a^2 + b^2)*(b + a*Cos[c + d*x])^(1/3)
*Sqrt[1 - Cos[c + d*x]^2]*Sec[c + d*x]^(4/3)) + (3*(a + b*Sec[c + d*x])*(3*b^2*AppellF1[5/3, 3/2, 3/2, 8/3, (a
+ b*Sec[c + d*x])/(a - b), (a + b*Sec[c + d*x])/(a + b)]*Sec[c + d*x]^2*Sqrt[(b*(1 + Sec[c + d*x]))/(-a + b)]
*Sqrt[(b - b*Sec[c + d*x])/(a + b)]*Tan[c + d*x] - (3*b^2*AppellF1[5/3, 3/2, 3/2, 8/3, (a + b*Sec[c + d*x])/(a
- b), (a + b*Sec[c + d*x])/(a + b)]*Sec[c + d*x]^2*Sqrt[(b*(1 + Sec[c + d*x]))/(-a + b)]*(a + b*Sec[c + d*x])
*Tan[c + d*x])/(2*(a + b)*Sqrt[(b - b*Sec[c + d*x])/(a + b)]) + (3*b^2*AppellF1[5/3, 3/2, 3/2, 8/3, (a + b*Sec
[c + d*x])/(a - b), (a + b*Sec[c + d*x])/(a + b)]*Sec[c + d*x]^2*Sqrt[(b - b*Sec[c + d*x])/(a + b)]*(a + b*Sec
[c + d*x])*Tan[c + d*x])/(2*(-a + b)*Sqrt[(b*(1 + Sec[c + d*x]))/(-a + b)]) + 3*b*AppellF1[5/3, 3/2, 3/2, 8/3,
(a + b*Sec[c + d*x])/(a - b), (a + b*Sec[c + d*x])/(a + b)]*Sec[c + d*x]*Sqrt[(b*(1 + Sec[c + d*x]))/(-a + b)
]*Sqrt[(b - b*Sec[c + d*x])/(a + b)]*(a + b*Sec[c + d*x])*Tan[c + d*x] + 3*b*Sec[c + d*x]*Sqrt[(b*(1 + Sec[c +
d*x]))/(-a + b)]*Sqrt[(b - b*Sec[c + d*x])/(a + b)]*(a + b*Sec[c + d*x])*((15*b*AppellF1[8/3, 3/2, 5/2, 11/3,
(a + b*Sec[c + d*x])/(a - b), (a + b*Sec[c + d*x])/(a + b)]*Sec[c + d*x]*Tan[c + d*x])/(16*(a + b)) + (15*b*A
ppellF1[8/3, 5/2, 3/2, 11/3, (a + b*Sec[c + d*x])/(a - b), (a + b*Sec[c + d*x])/(a + b)]*Sec[c + d*x]*Tan[c +
d*x])/(16*(a - b)))))/(5*(-a^2 + b^2)*(b + a*Cos[c + d*x])^(1/3)*Sqrt[1 - Cos[c + d*x]^2]*Sec[c + d*x]^(7/3)))
)

Maple [F]

\[\int \frac {\sec \left (d x +c \right )^{2}}{\left (a +b \sec \left (d x +c \right )\right )^{\frac {1}{3}}}d x\]

[In]

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

[Out]

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

Fricas [F]

\[ \int \frac {\sec ^2(c+d x)}{\sqrt [3]{a+b \sec (c+d x)}} \, dx=\int { \frac {\sec \left (d x + c\right )^{2}}{{\left (b \sec \left (d x + c\right ) + a\right )}^{\frac {1}{3}}} \,d x } \]

[In]

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

[Out]

integral(sec(d*x + c)^2/(b*sec(d*x + c) + a)^(1/3), x)

Sympy [F]

\[ \int \frac {\sec ^2(c+d x)}{\sqrt [3]{a+b \sec (c+d x)}} \, dx=\int \frac {\sec ^{2}{\left (c + d x \right )}}{\sqrt [3]{a + b \sec {\left (c + d x \right )}}}\, dx \]

[In]

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

[Out]

Integral(sec(c + d*x)**2/(a + b*sec(c + d*x))**(1/3), x)

Maxima [F]

\[ \int \frac {\sec ^2(c+d x)}{\sqrt [3]{a+b \sec (c+d x)}} \, dx=\int { \frac {\sec \left (d x + c\right )^{2}}{{\left (b \sec \left (d x + c\right ) + a\right )}^{\frac {1}{3}}} \,d x } \]

[In]

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

[Out]

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

Giac [F]

\[ \int \frac {\sec ^2(c+d x)}{\sqrt [3]{a+b \sec (c+d x)}} \, dx=\int { \frac {\sec \left (d x + c\right )^{2}}{{\left (b \sec \left (d x + c\right ) + a\right )}^{\frac {1}{3}}} \,d x } \]

[In]

integrate(sec(d*x+c)^2/(a+b*sec(d*x+c))^(1/3),x, algorithm="giac")

[Out]

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

Mupad [F(-1)]

Timed out. \[ \int \frac {\sec ^2(c+d x)}{\sqrt [3]{a+b \sec (c+d x)}} \, dx=\int \frac {1}{{\cos \left (c+d\,x\right )}^2\,{\left (a+\frac {b}{\cos \left (c+d\,x\right )}\right )}^{1/3}} \,d x \]

[In]

int(1/(cos(c + d*x)^2*(a + b/cos(c + d*x))^(1/3)),x)

[Out]

int(1/(cos(c + d*x)^2*(a + b/cos(c + d*x))^(1/3)), x)