\(\int \sec (c+d x) \sqrt [3]{a+b \sec (c+d x)} \, dx\) [685]

Optimal result

Integrand size = 21, antiderivative size = 105 \[ \int \sec (c+d x) \sqrt [3]{a+b \sec (c+d x)} \, dx=\frac {\sqrt {2} \operatorname {AppellF1}\left (\frac {1}{2},\frac {1}{2},-\frac {1}{3},\frac {3}{2},\frac {1}{2} (1-\sec (c+d x)),\frac {b (1-\sec (c+d x))}{a+b}\right ) \sqrt [3]{a+b \sec (c+d x)} \tan (c+d x)}{d \sqrt {1+\sec (c+d x)} \sqrt [3]{\frac {a+b \sec (c+d x)}{a+b}}} \]

[Out]

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

Rubi [A] (verified)

Time = 0.14 (sec) , antiderivative size = 105, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {3919, 144, 143} \[ \int \sec (c+d x) \sqrt [3]{a+b \sec (c+d x)} \, dx=\frac {\sqrt {2} \tan (c+d x) \sqrt [3]{a+b \sec (c+d x)} \operatorname {AppellF1}\left (\frac {1}{2},\frac {1}{2},-\frac {1}{3},\frac {3}{2},\frac {1}{2} (1-\sec (c+d x)),\frac {b (1-\sec (c+d x))}{a+b}\right )}{d \sqrt {\sec (c+d x)+1} \sqrt [3]{\frac {a+b \sec (c+d x)}{a+b}}} \]

[In]

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

[Out]

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

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

Mathematica [B] (warning: unable to verify)

Leaf count is larger than twice the leaf count of optimal. \(7160\) vs. \(2(105)=210\).

Time = 49.64 (sec) , antiderivative size = 7160, normalized size of antiderivative = 68.19 \[ \int \sec (c+d x) \sqrt [3]{a+b \sec (c+d x)} \, dx=\text {Result too large to show} \]

[In]

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

[Out]

Result too large to show

Maple [F]

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

[In]

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

[Out]

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

Fricas [F]

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

[In]

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

[Out]

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

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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