\(\int \frac {(e \sec (c+d x))^{2/3}}{\sqrt {a+a \sec (c+d x)}} \, dx\) [280]

Optimal result

Integrand size = 27, antiderivative size = 78 \[ \int \frac {(e \sec (c+d x))^{2/3}}{\sqrt {a+a \sec (c+d x)}} \, dx=-\frac {3 \operatorname {AppellF1}\left (\frac {2}{3},\frac {1}{2},1,\frac {5}{3},\sec (c+d x),-\sec (c+d x)\right ) (e \sec (c+d x))^{2/3} \tan (c+d x)}{2 d \sqrt {1-\sec (c+d x)} \sqrt {a+a \sec (c+d x)}} \]

[Out]

-3/2*AppellF1(2/3,1,1/2,5/3,-sec(d*x+c),sec(d*x+c))*(e*sec(d*x+c))^(2/3)*tan(d*x+c)/d/(1-sec(d*x+c))^(1/2)/(a+
a*sec(d*x+c))^(1/2)

Rubi [A] (verified)

Time = 0.21 (sec) , antiderivative size = 78, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.148, Rules used = {3913, 3912, 129, 524} \[ \int \frac {(e \sec (c+d x))^{2/3}}{\sqrt {a+a \sec (c+d x)}} \, dx=-\frac {3 \tan (c+d x) \operatorname {AppellF1}\left (\frac {2}{3},\frac {1}{2},1,\frac {5}{3},\sec (c+d x),-\sec (c+d x)\right ) (e \sec (c+d x))^{2/3}}{2 d \sqrt {1-\sec (c+d x)} \sqrt {a \sec (c+d x)+a}} \]

[In]

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

[Out]

(-3*AppellF1[2/3, 1/2, 1, 5/3, Sec[c + d*x], -Sec[c + d*x]]*(e*Sec[c + d*x])^(2/3)*Tan[c + d*x])/(2*d*Sqrt[1 -
Sec[c + d*x]]*Sqrt[a + a*Sec[c + d*x]])

Rule 129

Int[((e_.)*(x_))^(p_)*((a_) + (b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_), x_Symbol] :> With[{k = Denominator[p]
}, Dist[k/e, Subst[Int[x^(k*(p + 1) - 1)*(a + b*(x^k/e))^m*(c + d*(x^k/e))^n, x], x, (e*x)^(1/k)], x]] /; Free
Q[{a, b, c, d, e, m, n}, x] && NeQ[b*c - a*d, 0] && FractionQ[p] && IntegerQ[m]

Rule 524

Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> Simp[a^p*c^q*
((e*x)^(m + 1)/(e*(m + 1)))*AppellF1[(m + 1)/n, -p, -q, 1 + (m + 1)/n, (-b)*(x^n/a), (-d)*(x^n/c)], x] /; Free
Q[{a, b, c, d, e, m, n, p, q}, x] && NeQ[b*c - a*d, 0] && NeQ[m, -1] && NeQ[m, n - 1] && (IntegerQ[p] || GtQ[a
, 0]) && (IntegerQ[q] || GtQ[c, 0])

Rule 3912

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

Rule 3913

Int[(csc[(e_.) + (f_.)*(x_)]*(d_.))^(n_.)*(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_))^(m_), x_Symbol] :> Dist[a^Int
Part[m]*((a + b*Csc[e + f*x])^FracPart[m]/(1 + (b/a)*Csc[e + f*x])^FracPart[m]), Int[(1 + (b/a)*Csc[e + f*x])^
m*(d*Csc[e + f*x])^n, x], x] /; FreeQ[{a, b, d, e, f, m, n}, x] && EqQ[a^2 - b^2, 0] &&  !IntegerQ[m] &&  !GtQ
[a, 0]

Rubi steps \begin{align*} \text {integral}& = \frac {\sqrt {1+\sec (c+d x)} \int \frac {(e \sec (c+d x))^{2/3}}{\sqrt {1+\sec (c+d x)}} \, dx}{\sqrt {a+a \sec (c+d x)}} \\ & = -\frac {(e \tan (c+d x)) \text {Subst}\left (\int \frac {1}{\sqrt {1-x} \sqrt [3]{e x} (1+x)} \, dx,x,\sec (c+d x)\right )}{d \sqrt {1-\sec (c+d x)} \sqrt {a+a \sec (c+d x)}} \\ & = -\frac {(3 \tan (c+d x)) \text {Subst}\left (\int \frac {x}{\sqrt {1-\frac {x^3}{e}} \left (1+\frac {x^3}{e}\right )} \, dx,x,\sqrt [3]{e \sec (c+d x)}\right )}{d \sqrt {1-\sec (c+d x)} \sqrt {a+a \sec (c+d x)}} \\ & = -\frac {3 \operatorname {AppellF1}\left (\frac {2}{3},\frac {1}{2},1,\frac {5}{3},\sec (c+d x),-\sec (c+d x)\right ) (e \sec (c+d x))^{2/3} \tan (c+d x)}{2 d \sqrt {1-\sec (c+d x)} \sqrt {a+a \sec (c+d x)}} \\ \end{align*}

Mathematica [B] (warning: unable to verify)

Leaf count is larger than twice the leaf count of optimal. \(760\) vs. \(2(78)=156\).

Time = 7.12 (sec) , antiderivative size = 760, normalized size of antiderivative = 9.74 \[ \int \frac {(e \sec (c+d x))^{2/3}}{\sqrt {a+a \sec (c+d x)}} \, dx=\frac {90 \operatorname {AppellF1}\left (\frac {1}{2},\frac {1}{6},\frac {1}{3},\frac {3}{2},\tan ^2\left (\frac {1}{2} (c+d x)\right ),-\tan ^2\left (\frac {1}{2} (c+d x)\right )\right ) \cos \left (\frac {1}{2} (c+d x)\right ) \cos ^2(c+d x) (e \sec (c+d x))^{2/3} \sqrt {a (1+\sec (c+d x))} \sin \left (\frac {1}{2} (c+d x)\right ) \left (9 \operatorname {AppellF1}\left (\frac {1}{2},\frac {1}{6},\frac {1}{3},\frac {3}{2},\tan ^2\left (\frac {1}{2} (c+d x)\right ),-\tan ^2\left (\frac {1}{2} (c+d x)\right )\right )+\left (-2 \operatorname {AppellF1}\left (\frac {3}{2},\frac {1}{6},\frac {4}{3},\frac {5}{2},\tan ^2\left (\frac {1}{2} (c+d x)\right ),-\tan ^2\left (\frac {1}{2} (c+d x)\right )\right )+\operatorname {AppellF1}\left (\frac {3}{2},\frac {7}{6},\frac {1}{3},\frac {5}{2},\tan ^2\left (\frac {1}{2} (c+d x)\right ),-\tan ^2\left (\frac {1}{2} (c+d x)\right )\right )\right ) \tan ^2\left (\frac {1}{2} (c+d x)\right )\right )}{a d \left (270 \operatorname {AppellF1}\left (\frac {1}{2},\frac {1}{6},\frac {1}{3},\frac {3}{2},\tan ^2\left (\frac {1}{2} (c+d x)\right ),-\tan ^2\left (\frac {1}{2} (c+d x)\right )\right )^2 \cos ^4\left (\frac {1}{2} (c+d x)\right ) (1+2 \cos (c+d x))+10 \left (-2 \operatorname {AppellF1}\left (\frac {3}{2},\frac {1}{6},\frac {4}{3},\frac {5}{2},\tan ^2\left (\frac {1}{2} (c+d x)\right ),-\tan ^2\left (\frac {1}{2} (c+d x)\right )\right )+\operatorname {AppellF1}\left (\frac {3}{2},\frac {7}{6},\frac {1}{3},\frac {5}{2},\tan ^2\left (\frac {1}{2} (c+d x)\right ),-\tan ^2\left (\frac {1}{2} (c+d x)\right )\right )\right )^2 \cos (c+d x) \sin ^2\left (\frac {1}{2} (c+d x)\right ) \tan ^2\left (\frac {1}{2} (c+d x)\right )-3 \operatorname {AppellF1}\left (\frac {1}{2},\frac {1}{6},\frac {1}{3},\frac {3}{2},\tan ^2\left (\frac {1}{2} (c+d x)\right ),-\tan ^2\left (\frac {1}{2} (c+d x)\right )\right ) \left (10 \operatorname {AppellF1}\left (\frac {3}{2},\frac {1}{6},\frac {4}{3},\frac {5}{2},\tan ^2\left (\frac {1}{2} (c+d x)\right ),-\tan ^2\left (\frac {1}{2} (c+d x)\right )\right ) \cos ^2\left (\frac {1}{2} (c+d x)\right ) (2-9 \cos (c+d x)+\cos (2 (c+d x)))-5 \operatorname {AppellF1}\left (\frac {3}{2},\frac {7}{6},\frac {1}{3},\frac {5}{2},\tan ^2\left (\frac {1}{2} (c+d x)\right ),-\tan ^2\left (\frac {1}{2} (c+d x)\right )\right ) \cos ^2\left (\frac {1}{2} (c+d x)\right ) (2-9 \cos (c+d x)+\cos (2 (c+d x)))+6 \left (16 \operatorname {AppellF1}\left (\frac {5}{2},\frac {1}{6},\frac {7}{3},\frac {7}{2},\tan ^2\left (\frac {1}{2} (c+d x)\right ),-\tan ^2\left (\frac {1}{2} (c+d x)\right )\right )-4 \operatorname {AppellF1}\left (\frac {5}{2},\frac {7}{6},\frac {4}{3},\frac {7}{2},\tan ^2\left (\frac {1}{2} (c+d x)\right ),-\tan ^2\left (\frac {1}{2} (c+d x)\right )\right )+7 \operatorname {AppellF1}\left (\frac {5}{2},\frac {13}{6},\frac {1}{3},\frac {7}{2},\tan ^2\left (\frac {1}{2} (c+d x)\right ),-\tan ^2\left (\frac {1}{2} (c+d x)\right )\right )\right ) \cos (c+d x) \sin ^2\left (\frac {1}{2} (c+d x)\right )\right ) \tan ^2\left (\frac {1}{2} (c+d x)\right )\right )} \]

[In]

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

[Out]

(90*AppellF1[1/2, 1/6, 1/3, 3/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2]*Cos[(c + d*x)/2]*Cos[c + d*x]^2*(e*S
ec[c + d*x])^(2/3)*Sqrt[a*(1 + Sec[c + d*x])]*Sin[(c + d*x)/2]*(9*AppellF1[1/2, 1/6, 1/3, 3/2, Tan[(c + d*x)/2
]^2, -Tan[(c + d*x)/2]^2] + (-2*AppellF1[3/2, 1/6, 4/3, 5/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2] + Appell
F1[3/2, 7/6, 1/3, 5/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2])*Tan[(c + d*x)/2]^2))/(a*d*(270*AppellF1[1/2,
1/6, 1/3, 3/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2]^2*Cos[(c + d*x)/2]^4*(1 + 2*Cos[c + d*x]) + 10*(-2*App
ellF1[3/2, 1/6, 4/3, 5/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2] + AppellF1[3/2, 7/6, 1/3, 5/2, Tan[(c + d*x
)/2]^2, -Tan[(c + d*x)/2]^2])^2*Cos[c + d*x]*Sin[(c + d*x)/2]^2*Tan[(c + d*x)/2]^2 - 3*AppellF1[1/2, 1/6, 1/3,
3/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2]*(10*AppellF1[3/2, 1/6, 4/3, 5/2, Tan[(c + d*x)/2]^2, -Tan[(c +
d*x)/2]^2]*Cos[(c + d*x)/2]^2*(2 - 9*Cos[c + d*x] + Cos[2*(c + d*x)]) - 5*AppellF1[3/2, 7/6, 1/3, 5/2, Tan[(c
+ d*x)/2]^2, -Tan[(c + d*x)/2]^2]*Cos[(c + d*x)/2]^2*(2 - 9*Cos[c + d*x] + Cos[2*(c + d*x)]) + 6*(16*AppellF1[
5/2, 1/6, 7/3, 7/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2] - 4*AppellF1[5/2, 7/6, 4/3, 7/2, Tan[(c + d*x)/2]
^2, -Tan[(c + d*x)/2]^2] + 7*AppellF1[5/2, 13/6, 1/3, 7/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2])*Cos[c + d
*x]*Sin[(c + d*x)/2]^2)*Tan[(c + d*x)/2]^2))

Maple [F]

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

[In]

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

[Out]

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

Fricas [F(-1)]

Timed out. \[ \int \frac {(e \sec (c+d x))^{2/3}}{\sqrt {a+a \sec (c+d x)}} \, dx=\text {Timed out} \]

[In]

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

[Out]

Timed out

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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