\(\int \sec ^{\frac {4}{3}}(c+d x) (a+a \sec (c+d x))^{2/3} \, dx\) [285]

Optimal result

Integrand size = 25, antiderivative size = 79 \[ \int \sec ^{\frac {4}{3}}(c+d x) (a+a \sec (c+d x))^{2/3} \, dx=\frac {2 \sqrt [6]{2} \operatorname {AppellF1}\left (\frac {1}{2},-\frac {1}{3},-\frac {1}{6},\frac {3}{2},1-\sec (c+d x),\frac {1}{2} (1-\sec (c+d x))\right ) (a+a \sec (c+d x))^{2/3} \tan (c+d x)}{d (1+\sec (c+d x))^{7/6}} \]

[Out]

2*2^(1/6)*AppellF1(1/2,-1/3,-1/6,3/2,1-sec(d*x+c),1/2-1/2*sec(d*x+c))*(a+a*sec(d*x+c))^(2/3)*tan(d*x+c)/d/(1+s
ec(d*x+c))^(7/6)

Rubi [A] (verified)

Time = 0.16 (sec) , antiderivative size = 79, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.120, Rules used = {3913, 3910, 138} \[ \int \sec ^{\frac {4}{3}}(c+d x) (a+a \sec (c+d x))^{2/3} \, dx=\frac {2 \sqrt [6]{2} \tan (c+d x) (a \sec (c+d x)+a)^{2/3} \operatorname {AppellF1}\left (\frac {1}{2},-\frac {1}{3},-\frac {1}{6},\frac {3}{2},1-\sec (c+d x),\frac {1}{2} (1-\sec (c+d x))\right )}{d (\sec (c+d x)+1)^{7/6}} \]

[In]

Int[Sec[c + d*x]^(4/3)*(a + a*Sec[c + d*x])^(2/3),x]

[Out]

(2*2^(1/6)*AppellF1[1/2, -1/3, -1/6, 3/2, 1 - Sec[c + d*x], (1 - Sec[c + d*x])/2]*(a + a*Sec[c + d*x])^(2/3)*T
an[c + d*x])/(d*(1 + Sec[c + d*x])^(7/6))

Rule 138

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

Rule 3910

Int[(csc[(e_.) + (f_.)*(x_)]*(d_.))^(n_)*(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_))^(m_), x_Symbol] :> Dist[(-(a*(
d/b))^n)*(Cot[e + f*x]/(a^(n - 2)*f*Sqrt[a + b*Csc[e + f*x]]*Sqrt[a - b*Csc[e + f*x]])), Subst[Int[(a - x)^(n
- 1)*((2*a - x)^(m - 1/2)/Sqrt[x]), x], x, a - b*Csc[e + f*x]], x] /; FreeQ[{a, b, d, e, f, m, n}, x] && EqQ[a
^2 - b^2, 0] &&  !IntegerQ[m] && GtQ[a, 0] &&  !IntegerQ[n] && GtQ[a*(d/b), 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 {(a+a \sec (c+d x))^{2/3} \int \sec ^{\frac {4}{3}}(c+d x) (1+\sec (c+d x))^{2/3} \, dx}{(1+\sec (c+d x))^{2/3}} \\ & = \frac {\left ((a+a \sec (c+d x))^{2/3} \tan (c+d x)\right ) \text {Subst}\left (\int \frac {\sqrt [3]{1-x} \sqrt [6]{2-x}}{\sqrt {x}} \, dx,x,1-\sec (c+d x)\right )}{d \sqrt {1-\sec (c+d x)} (1+\sec (c+d x))^{7/6}} \\ & = \frac {2 \sqrt [6]{2} \operatorname {AppellF1}\left (\frac {1}{2},-\frac {1}{3},-\frac {1}{6},\frac {3}{2},1-\sec (c+d x),\frac {1}{2} (1-\sec (c+d x))\right ) (a+a \sec (c+d x))^{2/3} \tan (c+d x)}{d (1+\sec (c+d x))^{7/6}} \\ \end{align*}

Mathematica [C] (warning: unable to verify)

Result contains complex when optimal does not.

Time = 27.27 (sec) , antiderivative size = 1965, normalized size of antiderivative = 24.87 \[ \int \sec ^{\frac {4}{3}}(c+d x) (a+a \sec (c+d x))^{2/3} \, dx=\frac {2^{2/3} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {2}{3},\frac {3}{2},-\tan ^2\left (\frac {1}{2} (c+d x)\right )\right ) (a (1+\sec (c+d x)))^{2/3} \tan \left (\frac {1}{2} (c+d x)\right )}{3 d (1+\sec (c+d x))^{2/3}}+\frac {\sqrt [3]{\sec (c+d x)} ((1+\cos (c+d x)) \sec (c+d x))^{2/3} (a (1+\sec (c+d x)))^{2/3} \tan \left (\frac {1}{2} (c+d x)\right )}{d (1+\sec (c+d x))^{2/3}}-\frac {\sec ^2\left (\frac {1}{2} (c+d x)\right )^{4/3} \sqrt [3]{\sec (c+d x)} (a (1+\sec (c+d x)))^{2/3} \left (\frac {\operatorname {AppellF1}\left (-\frac {2}{3},-\frac {1}{3},-\frac {1}{3},\frac {1}{3},-\frac {1+i}{-1+\tan \left (\frac {1}{2} (c+d x)\right )},-\frac {1-i}{-1+\tan \left (\frac {1}{2} (c+d x)\right )}\right )}{\sqrt [3]{\frac {-i+\tan \left (\frac {1}{2} (c+d x)\right )}{-1+\tan \left (\frac {1}{2} (c+d x)\right )}} \sqrt [3]{\frac {i+\tan \left (\frac {1}{2} (c+d x)\right )}{-1+\tan \left (\frac {1}{2} (c+d x)\right )}}}-\frac {\operatorname {AppellF1}\left (-\frac {2}{3},-\frac {1}{3},-\frac {1}{3},\frac {1}{3},\frac {1-i}{1+\tan \left (\frac {1}{2} (c+d x)\right )},\frac {1+i}{1+\tan \left (\frac {1}{2} (c+d x)\right )}\right )}{\sqrt [3]{\frac {-i+\tan \left (\frac {1}{2} (c+d x)\right )}{1+\tan \left (\frac {1}{2} (c+d x)\right )}} \sqrt [3]{\frac {i+\tan \left (\frac {1}{2} (c+d x)\right )}{1+\tan \left (\frac {1}{2} (c+d x)\right )}}}\right )}{\sqrt [3]{2} d \left (-\frac {\sqrt [3]{\sec ^2\left (\frac {1}{2} (c+d x)\right )} \tan \left (\frac {1}{2} (c+d x)\right ) \left (\frac {\operatorname {AppellF1}\left (-\frac {2}{3},-\frac {1}{3},-\frac {1}{3},\frac {1}{3},-\frac {1+i}{-1+\tan \left (\frac {1}{2} (c+d x)\right )},-\frac {1-i}{-1+\tan \left (\frac {1}{2} (c+d x)\right )}\right )}{\sqrt [3]{\frac {-i+\tan \left (\frac {1}{2} (c+d x)\right )}{-1+\tan \left (\frac {1}{2} (c+d x)\right )}} \sqrt [3]{\frac {i+\tan \left (\frac {1}{2} (c+d x)\right )}{-1+\tan \left (\frac {1}{2} (c+d x)\right )}}}-\frac {\operatorname {AppellF1}\left (-\frac {2}{3},-\frac {1}{3},-\frac {1}{3},\frac {1}{3},\frac {1-i}{1+\tan \left (\frac {1}{2} (c+d x)\right )},\frac {1+i}{1+\tan \left (\frac {1}{2} (c+d x)\right )}\right )}{\sqrt [3]{\frac {-i+\tan \left (\frac {1}{2} (c+d x)\right )}{1+\tan \left (\frac {1}{2} (c+d x)\right )}} \sqrt [3]{\frac {i+\tan \left (\frac {1}{2} (c+d x)\right )}{1+\tan \left (\frac {1}{2} (c+d x)\right )}}}\right )}{\sqrt [3]{2}}-\frac {3 \sqrt [3]{\sec ^2\left (\frac {1}{2} (c+d x)\right )} \left (\frac {\frac {\left (\frac {1}{3}-\frac {i}{3}\right ) \operatorname {AppellF1}\left (\frac {1}{3},-\frac {1}{3},\frac {2}{3},\frac {4}{3},-\frac {1+i}{-1+\tan \left (\frac {1}{2} (c+d x)\right )},-\frac {1-i}{-1+\tan \left (\frac {1}{2} (c+d x)\right )}\right ) \sec ^2\left (\frac {1}{2} (c+d x)\right )}{\left (-1+\tan \left (\frac {1}{2} (c+d x)\right )\right )^2}+\frac {\left (\frac {1}{3}+\frac {i}{3}\right ) \operatorname {AppellF1}\left (\frac {1}{3},\frac {2}{3},-\frac {1}{3},\frac {4}{3},-\frac {1+i}{-1+\tan \left (\frac {1}{2} (c+d x)\right )},-\frac {1-i}{-1+\tan \left (\frac {1}{2} (c+d x)\right )}\right ) \sec ^2\left (\frac {1}{2} (c+d x)\right )}{\left (-1+\tan \left (\frac {1}{2} (c+d x)\right )\right )^2}}{\sqrt [3]{\frac {-i+\tan \left (\frac {1}{2} (c+d x)\right )}{-1+\tan \left (\frac {1}{2} (c+d x)\right )}} \sqrt [3]{\frac {i+\tan \left (\frac {1}{2} (c+d x)\right )}{-1+\tan \left (\frac {1}{2} (c+d x)\right )}}}-\frac {\operatorname {AppellF1}\left (-\frac {2}{3},-\frac {1}{3},-\frac {1}{3},\frac {1}{3},-\frac {1+i}{-1+\tan \left (\frac {1}{2} (c+d x)\right )},-\frac {1-i}{-1+\tan \left (\frac {1}{2} (c+d x)\right )}\right ) \left (\frac {\sec ^2\left (\frac {1}{2} (c+d x)\right )}{2 \left (-1+\tan \left (\frac {1}{2} (c+d x)\right )\right )}-\frac {\sec ^2\left (\frac {1}{2} (c+d x)\right ) \left (-i+\tan \left (\frac {1}{2} (c+d x)\right )\right )}{2 \left (-1+\tan \left (\frac {1}{2} (c+d x)\right )\right )^2}\right )}{3 \left (\frac {-i+\tan \left (\frac {1}{2} (c+d x)\right )}{-1+\tan \left (\frac {1}{2} (c+d x)\right )}\right )^{4/3} \sqrt [3]{\frac {i+\tan \left (\frac {1}{2} (c+d x)\right )}{-1+\tan \left (\frac {1}{2} (c+d x)\right )}}}-\frac {\operatorname {AppellF1}\left (-\frac {2}{3},-\frac {1}{3},-\frac {1}{3},\frac {1}{3},-\frac {1+i}{-1+\tan \left (\frac {1}{2} (c+d x)\right )},-\frac {1-i}{-1+\tan \left (\frac {1}{2} (c+d x)\right )}\right ) \left (\frac {\sec ^2\left (\frac {1}{2} (c+d x)\right )}{2 \left (-1+\tan \left (\frac {1}{2} (c+d x)\right )\right )}-\frac {\sec ^2\left (\frac {1}{2} (c+d x)\right ) \left (i+\tan \left (\frac {1}{2} (c+d x)\right )\right )}{2 \left (-1+\tan \left (\frac {1}{2} (c+d x)\right )\right )^2}\right )}{3 \sqrt [3]{\frac {-i+\tan \left (\frac {1}{2} (c+d x)\right )}{-1+\tan \left (\frac {1}{2} (c+d x)\right )}} \left (\frac {i+\tan \left (\frac {1}{2} (c+d x)\right )}{-1+\tan \left (\frac {1}{2} (c+d x)\right )}\right )^{4/3}}-\frac {-\frac {\left (\frac {1}{3}+\frac {i}{3}\right ) \operatorname {AppellF1}\left (\frac {1}{3},-\frac {1}{3},\frac {2}{3},\frac {4}{3},\frac {1-i}{1+\tan \left (\frac {1}{2} (c+d x)\right )},\frac {1+i}{1+\tan \left (\frac {1}{2} (c+d x)\right )}\right ) \sec ^2\left (\frac {1}{2} (c+d x)\right )}{\left (1+\tan \left (\frac {1}{2} (c+d x)\right )\right )^2}-\frac {\left (\frac {1}{3}-\frac {i}{3}\right ) \operatorname {AppellF1}\left (\frac {1}{3},\frac {2}{3},-\frac {1}{3},\frac {4}{3},\frac {1-i}{1+\tan \left (\frac {1}{2} (c+d x)\right )},\frac {1+i}{1+\tan \left (\frac {1}{2} (c+d x)\right )}\right ) \sec ^2\left (\frac {1}{2} (c+d x)\right )}{\left (1+\tan \left (\frac {1}{2} (c+d x)\right )\right )^2}}{\sqrt [3]{\frac {-i+\tan \left (\frac {1}{2} (c+d x)\right )}{1+\tan \left (\frac {1}{2} (c+d x)\right )}} \sqrt [3]{\frac {i+\tan \left (\frac {1}{2} (c+d x)\right )}{1+\tan \left (\frac {1}{2} (c+d x)\right )}}}+\frac {\operatorname {AppellF1}\left (-\frac {2}{3},-\frac {1}{3},-\frac {1}{3},\frac {1}{3},\frac {1-i}{1+\tan \left (\frac {1}{2} (c+d x)\right )},\frac {1+i}{1+\tan \left (\frac {1}{2} (c+d x)\right )}\right ) \left (-\frac {\sec ^2\left (\frac {1}{2} (c+d x)\right ) \left (-i+\tan \left (\frac {1}{2} (c+d x)\right )\right )}{2 \left (1+\tan \left (\frac {1}{2} (c+d x)\right )\right )^2}+\frac {\sec ^2\left (\frac {1}{2} (c+d x)\right )}{2 \left (1+\tan \left (\frac {1}{2} (c+d x)\right )\right )}\right )}{3 \left (\frac {-i+\tan \left (\frac {1}{2} (c+d x)\right )}{1+\tan \left (\frac {1}{2} (c+d x)\right )}\right )^{4/3} \sqrt [3]{\frac {i+\tan \left (\frac {1}{2} (c+d x)\right )}{1+\tan \left (\frac {1}{2} (c+d x)\right )}}}+\frac {\operatorname {AppellF1}\left (-\frac {2}{3},-\frac {1}{3},-\frac {1}{3},\frac {1}{3},\frac {1-i}{1+\tan \left (\frac {1}{2} (c+d x)\right )},\frac {1+i}{1+\tan \left (\frac {1}{2} (c+d x)\right )}\right ) \left (-\frac {\sec ^2\left (\frac {1}{2} (c+d x)\right ) \left (i+\tan \left (\frac {1}{2} (c+d x)\right )\right )}{2 \left (1+\tan \left (\frac {1}{2} (c+d x)\right )\right )^2}+\frac {\sec ^2\left (\frac {1}{2} (c+d x)\right )}{2 \left (1+\tan \left (\frac {1}{2} (c+d x)\right )\right )}\right )}{3 \sqrt [3]{\frac {-i+\tan \left (\frac {1}{2} (c+d x)\right )}{1+\tan \left (\frac {1}{2} (c+d x)\right )}} \left (\frac {i+\tan \left (\frac {1}{2} (c+d x)\right )}{1+\tan \left (\frac {1}{2} (c+d x)\right )}\right )^{4/3}}\right )}{\sqrt [3]{2}}\right )} \]

[In]

Integrate[Sec[c + d*x]^(4/3)*(a + a*Sec[c + d*x])^(2/3),x]

[Out]

(2^(2/3)*Hypergeometric2F1[1/2, 2/3, 3/2, -Tan[(c + d*x)/2]^2]*(a*(1 + Sec[c + d*x]))^(2/3)*Tan[(c + d*x)/2])/
(3*d*(1 + Sec[c + d*x])^(2/3)) + (Sec[c + d*x]^(1/3)*((1 + Cos[c + d*x])*Sec[c + d*x])^(2/3)*(a*(1 + Sec[c + d
*x]))^(2/3)*Tan[(c + d*x)/2])/(d*(1 + Sec[c + d*x])^(2/3)) - ((Sec[(c + d*x)/2]^2)^(4/3)*Sec[c + d*x]^(1/3)*(a
*(1 + Sec[c + d*x]))^(2/3)*(AppellF1[-2/3, -1/3, -1/3, 1/3, (-1 - I)/(-1 + Tan[(c + d*x)/2]), (-1 + I)/(-1 + T
an[(c + d*x)/2])]/(((-I + Tan[(c + d*x)/2])/(-1 + Tan[(c + d*x)/2]))^(1/3)*((I + Tan[(c + d*x)/2])/(-1 + Tan[(
c + d*x)/2]))^(1/3)) - AppellF1[-2/3, -1/3, -1/3, 1/3, (1 - I)/(1 + Tan[(c + d*x)/2]), (1 + I)/(1 + Tan[(c + d
*x)/2])]/(((-I + Tan[(c + d*x)/2])/(1 + Tan[(c + d*x)/2]))^(1/3)*((I + Tan[(c + d*x)/2])/(1 + Tan[(c + d*x)/2]
))^(1/3))))/(2^(1/3)*d*(-(((Sec[(c + d*x)/2]^2)^(1/3)*Tan[(c + d*x)/2]*(AppellF1[-2/3, -1/3, -1/3, 1/3, (-1 -
I)/(-1 + Tan[(c + d*x)/2]), (-1 + I)/(-1 + Tan[(c + d*x)/2])]/(((-I + Tan[(c + d*x)/2])/(-1 + Tan[(c + d*x)/2]
))^(1/3)*((I + Tan[(c + d*x)/2])/(-1 + Tan[(c + d*x)/2]))^(1/3)) - AppellF1[-2/3, -1/3, -1/3, 1/3, (1 - I)/(1
+ Tan[(c + d*x)/2]), (1 + I)/(1 + Tan[(c + d*x)/2])]/(((-I + Tan[(c + d*x)/2])/(1 + Tan[(c + d*x)/2]))^(1/3)*(
(I + Tan[(c + d*x)/2])/(1 + Tan[(c + d*x)/2]))^(1/3))))/2^(1/3)) - (3*(Sec[(c + d*x)/2]^2)^(1/3)*((((1/3 - I/3
)*AppellF1[1/3, -1/3, 2/3, 4/3, (-1 - I)/(-1 + Tan[(c + d*x)/2]), (-1 + I)/(-1 + Tan[(c + d*x)/2])]*Sec[(c + d
*x)/2]^2)/(-1 + Tan[(c + d*x)/2])^2 + ((1/3 + I/3)*AppellF1[1/3, 2/3, -1/3, 4/3, (-1 - I)/(-1 + Tan[(c + d*x)/
2]), (-1 + I)/(-1 + Tan[(c + d*x)/2])]*Sec[(c + d*x)/2]^2)/(-1 + Tan[(c + d*x)/2])^2)/(((-I + Tan[(c + d*x)/2]
)/(-1 + Tan[(c + d*x)/2]))^(1/3)*((I + Tan[(c + d*x)/2])/(-1 + Tan[(c + d*x)/2]))^(1/3)) - (AppellF1[-2/3, -1/
3, -1/3, 1/3, (-1 - I)/(-1 + Tan[(c + d*x)/2]), (-1 + I)/(-1 + Tan[(c + d*x)/2])]*(Sec[(c + d*x)/2]^2/(2*(-1 +
Tan[(c + d*x)/2])) - (Sec[(c + d*x)/2]^2*(-I + Tan[(c + d*x)/2]))/(2*(-1 + Tan[(c + d*x)/2])^2)))/(3*((-I + T
an[(c + d*x)/2])/(-1 + Tan[(c + d*x)/2]))^(4/3)*((I + Tan[(c + d*x)/2])/(-1 + Tan[(c + d*x)/2]))^(1/3)) - (App
ellF1[-2/3, -1/3, -1/3, 1/3, (-1 - I)/(-1 + Tan[(c + d*x)/2]), (-1 + I)/(-1 + Tan[(c + d*x)/2])]*(Sec[(c + d*x
)/2]^2/(2*(-1 + Tan[(c + d*x)/2])) - (Sec[(c + d*x)/2]^2*(I + Tan[(c + d*x)/2]))/(2*(-1 + Tan[(c + d*x)/2])^2)
))/(3*((-I + Tan[(c + d*x)/2])/(-1 + Tan[(c + d*x)/2]))^(1/3)*((I + Tan[(c + d*x)/2])/(-1 + Tan[(c + d*x)/2]))
^(4/3)) - (((-1/3 - I/3)*AppellF1[1/3, -1/3, 2/3, 4/3, (1 - I)/(1 + Tan[(c + d*x)/2]), (1 + I)/(1 + Tan[(c + d
*x)/2])]*Sec[(c + d*x)/2]^2)/(1 + Tan[(c + d*x)/2])^2 - ((1/3 - I/3)*AppellF1[1/3, 2/3, -1/3, 4/3, (1 - I)/(1
+ Tan[(c + d*x)/2]), (1 + I)/(1 + Tan[(c + d*x)/2])]*Sec[(c + d*x)/2]^2)/(1 + Tan[(c + d*x)/2])^2)/(((-I + Tan
[(c + d*x)/2])/(1 + Tan[(c + d*x)/2]))^(1/3)*((I + Tan[(c + d*x)/2])/(1 + Tan[(c + d*x)/2]))^(1/3)) + (AppellF
1[-2/3, -1/3, -1/3, 1/3, (1 - I)/(1 + Tan[(c + d*x)/2]), (1 + I)/(1 + Tan[(c + d*x)/2])]*(-1/2*(Sec[(c + d*x)/
2]^2*(-I + Tan[(c + d*x)/2]))/(1 + Tan[(c + d*x)/2])^2 + Sec[(c + d*x)/2]^2/(2*(1 + Tan[(c + d*x)/2]))))/(3*((
-I + Tan[(c + d*x)/2])/(1 + Tan[(c + d*x)/2]))^(4/3)*((I + Tan[(c + d*x)/2])/(1 + Tan[(c + d*x)/2]))^(1/3)) +
(AppellF1[-2/3, -1/3, -1/3, 1/3, (1 - I)/(1 + Tan[(c + d*x)/2]), (1 + I)/(1 + Tan[(c + d*x)/2])]*(-1/2*(Sec[(c
+ d*x)/2]^2*(I + Tan[(c + d*x)/2]))/(1 + Tan[(c + d*x)/2])^2 + Sec[(c + d*x)/2]^2/(2*(1 + Tan[(c + d*x)/2])))
)/(3*((-I + Tan[(c + d*x)/2])/(1 + Tan[(c + d*x)/2]))^(1/3)*((I + Tan[(c + d*x)/2])/(1 + Tan[(c + d*x)/2]))^(4
/3))))/2^(1/3)))

Maple [F]

\[\int \sec \left (d x +c \right )^{\frac {4}{3}} \left (a +a \sec \left (d x +c \right )\right )^{\frac {2}{3}}d x\]

[In]

int(sec(d*x+c)^(4/3)*(a+a*sec(d*x+c))^(2/3),x)

[Out]

int(sec(d*x+c)^(4/3)*(a+a*sec(d*x+c))^(2/3),x)

Fricas [F(-1)]

Timed out. \[ \int \sec ^{\frac {4}{3}}(c+d x) (a+a \sec (c+d x))^{2/3} \, dx=\text {Timed out} \]

[In]

integrate(sec(d*x+c)^(4/3)*(a+a*sec(d*x+c))^(2/3),x, algorithm="fricas")

[Out]

Timed out

Sympy [F(-1)]

Timed out. \[ \int \sec ^{\frac {4}{3}}(c+d x) (a+a \sec (c+d x))^{2/3} \, dx=\text {Timed out} \]

[In]

integrate(sec(d*x+c)**(4/3)*(a+a*sec(d*x+c))**(2/3),x)

[Out]

Timed out

Maxima [F]

\[ \int \sec ^{\frac {4}{3}}(c+d x) (a+a \sec (c+d x))^{2/3} \, dx=\int { {\left (a \sec \left (d x + c\right ) + a\right )}^{\frac {2}{3}} \sec \left (d x + c\right )^{\frac {4}{3}} \,d x } \]

[In]

integrate(sec(d*x+c)^(4/3)*(a+a*sec(d*x+c))^(2/3),x, algorithm="maxima")

[Out]

integrate((a*sec(d*x + c) + a)^(2/3)*sec(d*x + c)^(4/3), x)

Giac [F]

\[ \int \sec ^{\frac {4}{3}}(c+d x) (a+a \sec (c+d x))^{2/3} \, dx=\int { {\left (a \sec \left (d x + c\right ) + a\right )}^{\frac {2}{3}} \sec \left (d x + c\right )^{\frac {4}{3}} \,d x } \]

[In]

integrate(sec(d*x+c)^(4/3)*(a+a*sec(d*x+c))^(2/3),x, algorithm="giac")

[Out]

integrate((a*sec(d*x + c) + a)^(2/3)*sec(d*x + c)^(4/3), x)

Mupad [F(-1)]

Timed out. \[ \int \sec ^{\frac {4}{3}}(c+d x) (a+a \sec (c+d x))^{2/3} \, dx=\int {\left (a+\frac {a}{\cos \left (c+d\,x\right )}\right )}^{2/3}\,{\left (\frac {1}{\cos \left (c+d\,x\right )}\right )}^{4/3} \,d x \]

[In]

int((a + a/cos(c + d*x))^(2/3)*(1/cos(c + d*x))^(4/3),x)

[Out]

int((a + a/cos(c + d*x))^(2/3)*(1/cos(c + d*x))^(4/3), x)