Integrand size = 23, antiderivative size = 362 \[ \int \sec ^4(c+d x) (a+b \sec (c+d x))^{2/3} \, dx=\frac {3 \left (9 a^2+32 b^2\right ) (a+b \sec (c+d x))^{2/3} \tan (c+d x)}{220 b^2 d}-\frac {9 a (a+b \sec (c+d x))^{5/3} \tan (c+d x)}{44 b^2 d}+\frac {3 \sec (c+d x) (a+b \sec (c+d x))^{5/3} \tan (c+d x)}{11 b d}+\frac {a \left (18 a^2+49 b^2\right ) \operatorname {AppellF1}\left (\frac {1}{2},\frac {1}{2},-\frac {2}{3},\frac {3}{2},\frac {1}{2} (1-\sec (c+d x)),\frac {b (1-\sec (c+d x))}{a+b}\right ) (a+b \sec (c+d x))^{2/3} \tan (c+d x)}{110 \sqrt {2} b^3 d \sqrt {1+\sec (c+d x)} \left (\frac {a+b \sec (c+d x)}{a+b}\right )^{2/3}}-\frac {\left (9 a^4+23 a^2 b^2-32 b^4\right ) \operatorname {AppellF1}\left (\frac {1}{2},\frac {1}{2},\frac {1}{3},\frac {3}{2},\frac {1}{2} (1-\sec (c+d x)),\frac {b (1-\sec (c+d x))}{a+b}\right ) \sqrt [3]{\frac {a+b \sec (c+d x)}{a+b}} \tan (c+d x)}{55 \sqrt {2} b^3 d \sqrt {1+\sec (c+d x)} \sqrt [3]{a+b \sec (c+d x)}} \]
3/220*(9*a^2+32*b^2)*(a+b*sec(d*x+c))^(2/3)*tan(d*x+c)/b^2/d-9/44*a*(a+b*s ec(d*x+c))^(5/3)*tan(d*x+c)/b^2/d+3/11*sec(d*x+c)*(a+b*sec(d*x+c))^(5/3)*t an(d*x+c)/b/d+1/220*a*(18*a^2+49*b^2)*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)*tan(d*x+c)/b^3/d/( (a+b*sec(d*x+c))/(a+b))^(2/3)*2^(1/2)/(1+sec(d*x+c))^(1/2)-1/110*(9*a^4+23 *a^2*b^2-32*b^4)*AppellF1(1/2,1/3,1/2,3/2,b*(1-sec(d*x+c))/(a+b),1/2-1/2*s ec(d*x+c))*((a+b*sec(d*x+c))/(a+b))^(1/3)*tan(d*x+c)/b^3/d/(a+b*sec(d*x+c) )^(1/3)*2^(1/2)/(1+sec(d*x+c))^(1/2)
Leaf count is larger than twice the leaf count of optimal. \(21877\) vs. \(2(362)=724\).
Time = 46.27 (sec) , antiderivative size = 21877, normalized size of antiderivative = 60.43 \[ \int \sec ^4(c+d x) (a+b \sec (c+d x))^{2/3} \, dx=\text {Result too large to show} \]
Time = 1.34 (sec) , antiderivative size = 376, normalized size of antiderivative = 1.04, number of steps used = 16, number of rules used = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.652, Rules used = {3042, 4352, 27, 3042, 4570, 27, 3042, 4490, 27, 3042, 4495, 3042, 4321, 156, 155}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int \sec ^4(c+d x) (a+b \sec (c+d x))^{2/3} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \csc \left (c+d x+\frac {\pi }{2}\right )^4 \left (a+b \csc \left (c+d x+\frac {\pi }{2}\right )\right )^{2/3}dx\) |
| \(\Big \downarrow \) 4352 |
| \(\displaystyle \frac {3 \int \frac {1}{3} \sec (c+d x) (a+b \sec (c+d x))^{2/3} \left (-6 a \sec ^2(c+d x)+8 b \sec (c+d x)+3 a\right )dx}{11 b}+\frac {3 \tan (c+d x) \sec (c+d x) (a+b \sec (c+d x))^{5/3}}{11 b d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int \sec (c+d x) (a+b \sec (c+d x))^{2/3} \left (-6 a \sec ^2(c+d x)+8 b \sec (c+d x)+3 a\right )dx}{11 b}+\frac {3 \tan (c+d x) \sec (c+d x) (a+b \sec (c+d x))^{5/3}}{11 b d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\int \csc \left (c+d x+\frac {\pi }{2}\right ) \left (a+b \csc \left (c+d x+\frac {\pi }{2}\right )\right )^{2/3} \left (-6 a \csc \left (c+d x+\frac {\pi }{2}\right )^2+8 b \csc \left (c+d x+\frac {\pi }{2}\right )+3 a\right )dx}{11 b}+\frac {3 \tan (c+d x) \sec (c+d x) (a+b \sec (c+d x))^{5/3}}{11 b d}\) |
| \(\Big \downarrow \) 4570 |
| \(\displaystyle \frac {\frac {3 \int -\frac {2}{3} \sec (c+d x) (a+b \sec (c+d x))^{2/3} \left (3 a b-\left (9 a^2+32 b^2\right ) \sec (c+d x)\right )dx}{8 b}-\frac {9 a \tan (c+d x) (a+b \sec (c+d x))^{5/3}}{4 b d}}{11 b}+\frac {3 \tan (c+d x) \sec (c+d x) (a+b \sec (c+d x))^{5/3}}{11 b d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {-\frac {\int \sec (c+d x) (a+b \sec (c+d x))^{2/3} \left (3 a b-\left (9 a^2+32 b^2\right ) \sec (c+d x)\right )dx}{4 b}-\frac {9 a \tan (c+d x) (a+b \sec (c+d x))^{5/3}}{4 b d}}{11 b}+\frac {3 \tan (c+d x) \sec (c+d x) (a+b \sec (c+d x))^{5/3}}{11 b d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {-\frac {\int \csc \left (c+d x+\frac {\pi }{2}\right ) \left (a+b \csc \left (c+d x+\frac {\pi }{2}\right )\right )^{2/3} \left (3 a b+\left (-9 a^2-32 b^2\right ) \csc \left (c+d x+\frac {\pi }{2}\right )\right )dx}{4 b}-\frac {9 a \tan (c+d x) (a+b \sec (c+d x))^{5/3}}{4 b d}}{11 b}+\frac {3 \tan (c+d x) \sec (c+d x) (a+b \sec (c+d x))^{5/3}}{11 b d}\) |
| \(\Big \downarrow \) 4490 |
| \(\displaystyle \frac {-\frac {\frac {3}{5} \int -\frac {\sec (c+d x) \left (b \left (3 a^2+64 b^2\right )+a \left (18 a^2+49 b^2\right ) \sec (c+d x)\right )}{3 \sqrt [3]{a+b \sec (c+d x)}}dx-\frac {3 \left (9 a^2+32 b^2\right ) \tan (c+d x) (a+b \sec (c+d x))^{2/3}}{5 d}}{4 b}-\frac {9 a \tan (c+d x) (a+b \sec (c+d x))^{5/3}}{4 b d}}{11 b}+\frac {3 \tan (c+d x) \sec (c+d x) (a+b \sec (c+d x))^{5/3}}{11 b d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {-\frac {-\frac {1}{5} \int \frac {\sec (c+d x) \left (b \left (3 a^2+64 b^2\right )+a \left (18 a^2+49 b^2\right ) \sec (c+d x)\right )}{\sqrt [3]{a+b \sec (c+d x)}}dx-\frac {3 \left (9 a^2+32 b^2\right ) \tan (c+d x) (a+b \sec (c+d x))^{2/3}}{5 d}}{4 b}-\frac {9 a \tan (c+d x) (a+b \sec (c+d x))^{5/3}}{4 b d}}{11 b}+\frac {3 \tan (c+d x) \sec (c+d x) (a+b \sec (c+d x))^{5/3}}{11 b d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {-\frac {-\frac {1}{5} \int \frac {\csc \left (c+d x+\frac {\pi }{2}\right ) \left (b \left (3 a^2+64 b^2\right )+a \left (18 a^2+49 b^2\right ) \csc \left (c+d x+\frac {\pi }{2}\right )\right )}{\sqrt [3]{a+b \csc \left (c+d x+\frac {\pi }{2}\right )}}dx-\frac {3 \left (9 a^2+32 b^2\right ) \tan (c+d x) (a+b \sec (c+d x))^{2/3}}{5 d}}{4 b}-\frac {9 a \tan (c+d x) (a+b \sec (c+d x))^{5/3}}{4 b d}}{11 b}+\frac {3 \tan (c+d x) \sec (c+d x) (a+b \sec (c+d x))^{5/3}}{11 b d}\) |
| \(\Big \downarrow \) 4495 |
| \(\displaystyle \frac {-\frac {\frac {1}{5} \left (\frac {2 \left (9 a^4+23 a^2 b^2-32 b^4\right ) \int \frac {\sec (c+d x)}{\sqrt [3]{a+b \sec (c+d x)}}dx}{b}-\frac {a \left (18 a^2+49 b^2\right ) \int \sec (c+d x) (a+b \sec (c+d x))^{2/3}dx}{b}\right )-\frac {3 \left (9 a^2+32 b^2\right ) \tan (c+d x) (a+b \sec (c+d x))^{2/3}}{5 d}}{4 b}-\frac {9 a \tan (c+d x) (a+b \sec (c+d x))^{5/3}}{4 b d}}{11 b}+\frac {3 \tan (c+d x) \sec (c+d x) (a+b \sec (c+d x))^{5/3}}{11 b d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {-\frac {\frac {1}{5} \left (\frac {2 \left (9 a^4+23 a^2 b^2-32 b^4\right ) \int \frac {\csc \left (c+d x+\frac {\pi }{2}\right )}{\sqrt [3]{a+b \csc \left (c+d x+\frac {\pi }{2}\right )}}dx}{b}-\frac {a \left (18 a^2+49 b^2\right ) \int \csc \left (c+d x+\frac {\pi }{2}\right ) \left (a+b \csc \left (c+d x+\frac {\pi }{2}\right )\right )^{2/3}dx}{b}\right )-\frac {3 \left (9 a^2+32 b^2\right ) \tan (c+d x) (a+b \sec (c+d x))^{2/3}}{5 d}}{4 b}-\frac {9 a \tan (c+d x) (a+b \sec (c+d x))^{5/3}}{4 b d}}{11 b}+\frac {3 \tan (c+d x) \sec (c+d x) (a+b \sec (c+d x))^{5/3}}{11 b d}\) |
| \(\Big \downarrow \) 4321 |
| \(\displaystyle \frac {-\frac {\frac {1}{5} \left (\frac {a \left (18 a^2+49 b^2\right ) \tan (c+d x) \int \frac {(a+b \sec (c+d x))^{2/3}}{\sqrt {1-\sec (c+d x)} \sqrt {\sec (c+d x)+1}}d\sec (c+d x)}{b d \sqrt {1-\sec (c+d x)} \sqrt {\sec (c+d x)+1}}-\frac {2 \left (9 a^4+23 a^2 b^2-32 b^4\right ) \tan (c+d x) \int \frac {1}{\sqrt {1-\sec (c+d x)} \sqrt {\sec (c+d x)+1} \sqrt [3]{a+b \sec (c+d x)}}d\sec (c+d x)}{b d \sqrt {1-\sec (c+d x)} \sqrt {\sec (c+d x)+1}}\right )-\frac {3 \left (9 a^2+32 b^2\right ) \tan (c+d x) (a+b \sec (c+d x))^{2/3}}{5 d}}{4 b}-\frac {9 a \tan (c+d x) (a+b \sec (c+d x))^{5/3}}{4 b d}}{11 b}+\frac {3 \tan (c+d x) \sec (c+d x) (a+b \sec (c+d x))^{5/3}}{11 b d}\) |
| \(\Big \downarrow \) 156 |
| \(\displaystyle \frac {-\frac {\frac {1}{5} \left (\frac {a \left (18 a^2+49 b^2\right ) \tan (c+d x) (a+b \sec (c+d x))^{2/3} \int \frac {\left (\frac {a}{a+b}+\frac {b \sec (c+d x)}{a+b}\right )^{2/3}}{\sqrt {1-\sec (c+d x)} \sqrt {\sec (c+d x)+1}}d\sec (c+d x)}{b d \sqrt {1-\sec (c+d x)} \sqrt {\sec (c+d x)+1} \left (\frac {a+b \sec (c+d x)}{a+b}\right )^{2/3}}-\frac {2 \left (9 a^4+23 a^2 b^2-32 b^4\right ) \tan (c+d x) \sqrt [3]{\frac {a+b \sec (c+d x)}{a+b}} \int \frac {1}{\sqrt {1-\sec (c+d x)} \sqrt {\sec (c+d x)+1} \sqrt [3]{\frac {a}{a+b}+\frac {b \sec (c+d x)}{a+b}}}d\sec (c+d x)}{b d \sqrt {1-\sec (c+d x)} \sqrt {\sec (c+d x)+1} \sqrt [3]{a+b \sec (c+d x)}}\right )-\frac {3 \left (9 a^2+32 b^2\right ) \tan (c+d x) (a+b \sec (c+d x))^{2/3}}{5 d}}{4 b}-\frac {9 a \tan (c+d x) (a+b \sec (c+d x))^{5/3}}{4 b d}}{11 b}+\frac {3 \tan (c+d x) \sec (c+d x) (a+b \sec (c+d x))^{5/3}}{11 b d}\) |
| \(\Big \downarrow \) 155 |
| \(\displaystyle \frac {-\frac {\frac {1}{5} \left (\frac {2 \sqrt {2} \left (9 a^4+23 a^2 b^2-32 b^4\right ) \tan (c+d x) \sqrt [3]{\frac {a+b \sec (c+d x)}{a+b}} \operatorname {AppellF1}\left (\frac {1}{2},\frac {1}{2},\frac {1}{3},\frac {3}{2},\frac {1}{2} (1-\sec (c+d x)),\frac {b (1-\sec (c+d x))}{a+b}\right )}{b d \sqrt {\sec (c+d x)+1} \sqrt [3]{a+b \sec (c+d x)}}-\frac {\sqrt {2} a \left (18 a^2+49 b^2\right ) \tan (c+d x) (a+b \sec (c+d x))^{2/3} \operatorname {AppellF1}\left (\frac {1}{2},\frac {1}{2},-\frac {2}{3},\frac {3}{2},\frac {1}{2} (1-\sec (c+d x)),\frac {b (1-\sec (c+d x))}{a+b}\right )}{b d \sqrt {\sec (c+d x)+1} \left (\frac {a+b \sec (c+d x)}{a+b}\right )^{2/3}}\right )-\frac {3 \left (9 a^2+32 b^2\right ) \tan (c+d x) (a+b \sec (c+d x))^{2/3}}{5 d}}{4 b}-\frac {9 a \tan (c+d x) (a+b \sec (c+d x))^{5/3}}{4 b d}}{11 b}+\frac {3 \tan (c+d x) \sec (c+d x) (a+b \sec (c+d x))^{5/3}}{11 b d}\) |
(3*Sec[c + d*x]*(a + b*Sec[c + d*x])^(5/3)*Tan[c + d*x])/(11*b*d) + ((-9*a *(a + b*Sec[c + d*x])^(5/3)*Tan[c + d*x])/(4*b*d) - ((-3*(9*a^2 + 32*b^2)* (a + b*Sec[c + d*x])^(2/3)*Tan[c + d*x])/(5*d) + (-((Sqrt[2]*a*(18*a^2 + 4 9*b^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 + Se c[c + d*x]]*((a + b*Sec[c + d*x])/(a + b))^(2/3))) + (2*Sqrt[2]*(9*a^4 + 2 3*a^2*b^2 - 32*b^4)*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])/(a + b))^(1/3)*Tan[c + d*x])/(b*d*Sqrt[1 + Sec[c + d*x]]*(a + b*Sec[c + d*x])^(1/3)))/5)/(4*b))/( 11*b)
3.7.87.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((a_) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_)) ^(p_), x_] :> Simp[((a + b*x)^(m + 1)/(b*(m + 1)*Simplify[b/(b*c - a*d)]^n* Simplify[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] && !IntegerQ[n] && !IntegerQ[p] && GtQ[Sim plify[b/(b*c - a*d)], 0] && GtQ[Simplify[b/(b*e - a*f)], 0] && !(GtQ[Simpl ify[d/(d*a - c*b)], 0] && GtQ[Simplify[d/(d*e - c*f)], 0] && SimplerQ[c + d *x, a + b*x]) && !(GtQ[Simplify[f/(f*a - e*b)], 0] && GtQ[Simplify[f/(f*c - e*d)], 0] && SimplerQ[e + f*x, a + b*x])
Int[((a_) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_)) ^(p_), x_] :> Simp[(e + f*x)^FracPart[p]/(Simplify[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*Si mp[b*(e/(b*e - a*f)) + b*f*(x/(b*e - a*f)), x]^p, x], x] /; FreeQ[{a, b, c, d, e, f, m, n, p}, x] && !IntegerQ[m] && !IntegerQ[n] && !IntegerQ[p] & & GtQ[Simplify[b/(b*c - a*d)], 0] && !GtQ[Simplify[b/(b*e - a*f)], 0]
Int[csc[(e_.) + (f_.)*(x_)]*(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_))^(m_), x_ Symbol] :> Simp[Cot[e + f*x]/(f*Sqrt[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]
Int[(csc[(e_.) + (f_.)*(x_)]*(d_.))^(n_)*(csc[(e_.) + (f_.)*(x_)]*(b_.) + ( a_))^(m_), x_Symbol] :> Simp[(-d^3)*Cot[e + f*x]*(a + b*Csc[e + f*x])^(m + 1)*((d*Csc[e + f*x])^(n - 3)/(b*f*(m + n - 1))), x] + Simp[d^3/(b*(m + n - 1)) Int[(a + b*Csc[e + f*x])^m*(d*Csc[e + f*x])^(n - 3)*Simp[a*(n - 3) + b*(m + n - 2)*Csc[e + f*x] - a*(n - 2)*Csc[e + f*x]^2, x], x], x] /; FreeQ[ {a, b, d, e, f, m}, x] && NeQ[a^2 - b^2, 0] && GtQ[n, 3] && (IntegerQ[n] || IntegersQ[2*m, 2*n]) && !IGtQ[m, 2]
Int[csc[(e_.) + (f_.)*(x_)]*(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_))^(m_)*(cs c[(e_.) + (f_.)*(x_)]*(B_.) + (A_)), x_Symbol] :> Simp[(-B)*Cot[e + f*x]*(( a + b*Csc[e + f*x])^m/(f*(m + 1))), x] + Simp[1/(m + 1) Int[Csc[e + f*x]* (a + b*Csc[e + f*x])^(m - 1)*Simp[b*B*m + a*A*(m + 1) + (a*B*m + A*b*(m + 1 ))*Csc[e + f*x], x], x], x] /; FreeQ[{a, b, A, B, e, f}, x] && NeQ[A*b - a* B, 0] && NeQ[a^2 - b^2, 0] && GtQ[m, 0]
Int[csc[(e_.) + (f_.)*(x_)]*(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_))^(m_)*(cs c[(e_.) + (f_.)*(x_)]*(B_.) + (A_)), x_Symbol] :> Simp[(A*b - a*B)/b Int[ Csc[e + f*x]*(a + b*Csc[e + f*x])^m, x], x] + Simp[B/b Int[Csc[e + f*x]*( a + b*Csc[e + f*x])^(m + 1), x], x] /; FreeQ[{a, b, A, B, e, f, m}, x] && N eQ[A*b - a*B, 0] && NeQ[a^2 - b^2, 0]
Int[csc[(e_.) + (f_.)*(x_)]*((A_.) + csc[(e_.) + (f_.)*(x_)]*(B_.) + csc[(e _.) + (f_.)*(x_)]^2*(C_.))*(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_))^(m_), x_S ymbol] :> Simp[(-C)*Cot[e + f*x]*((a + b*Csc[e + f*x])^(m + 1)/(b*f*(m + 2) )), x] + Simp[1/(b*(m + 2)) Int[Csc[e + f*x]*(a + b*Csc[e + f*x])^m*Simp[ b*A*(m + 2) + b*C*(m + 1) + (b*B*(m + 2) - a*C)*Csc[e + f*x], x], x], x] /; FreeQ[{a, b, e, f, A, B, C, m}, x] && !LtQ[m, -1]
\[\int \sec \left (d x +c \right )^{4} \left (a +b \sec \left (d x +c \right )\right )^{\frac {2}{3}}d x\]
\[ \int \sec ^4(c+d x) (a+b \sec (c+d x))^{2/3} \, dx=\int { {\left (b \sec \left (d x + c\right ) + a\right )}^{\frac {2}{3}} \sec \left (d x + c\right )^{4} \,d x } \]
\[ \int \sec ^4(c+d x) (a+b \sec (c+d x))^{2/3} \, dx=\int \left (a + b \sec {\left (c + d x \right )}\right )^{\frac {2}{3}} \sec ^{4}{\left (c + d x \right )}\, dx \]
\[ \int \sec ^4(c+d x) (a+b \sec (c+d x))^{2/3} \, dx=\int { {\left (b \sec \left (d x + c\right ) + a\right )}^{\frac {2}{3}} \sec \left (d x + c\right )^{4} \,d x } \]
\[ \int \sec ^4(c+d x) (a+b \sec (c+d x))^{2/3} \, dx=\int { {\left (b \sec \left (d x + c\right ) + a\right )}^{\frac {2}{3}} \sec \left (d x + c\right )^{4} \,d x } \]
Timed out. \[ \int \sec ^4(c+d x) (a+b \sec (c+d x))^{2/3} \, dx=\int \frac {{\left (a+\frac {b}{\cos \left (c+d\,x\right )}\right )}^{2/3}}{{\cos \left (c+d\,x\right )}^4} \,d x \]