Integrand size = 23, antiderivative size = 336 \[ \int \frac {\sec ^3(c+d x)}{\sqrt [3]{a+a \sec (c+d x)}} \, dx=-\frac {9 \tan (c+d x)}{10 d \sqrt [3]{a+a \sec (c+d x)}}+\frac {3 (a+a \sec (c+d x))^{2/3} \tan (c+d x)}{5 a d}-\frac {7\ 3^{3/4} \operatorname {EllipticF}\left (\arccos \left (\frac {\sqrt [3]{2}-\left (1-\sqrt {3}\right ) \sqrt [3]{1+\sec (c+d x)}}{\sqrt [3]{2}-\left (1+\sqrt {3}\right ) \sqrt [3]{1+\sec (c+d x)}}\right ),\frac {1}{4} \left (2+\sqrt {3}\right )\right ) \left (\sqrt [3]{2}-\sqrt [3]{1+\sec (c+d x)}\right ) \sqrt {\frac {2^{2/3}+\sqrt [3]{2} \sqrt [3]{1+\sec (c+d x)}+(1+\sec (c+d x))^{2/3}}{\left (\sqrt [3]{2}-\left (1+\sqrt {3}\right ) \sqrt [3]{1+\sec (c+d x)}\right )^2}} \tan (c+d x)}{10 \sqrt [3]{2} d (1-\sec (c+d x)) \sqrt [3]{a+a \sec (c+d x)} \sqrt {-\frac {\sqrt [3]{1+\sec (c+d x)} \left (\sqrt [3]{2}-\sqrt [3]{1+\sec (c+d x)}\right )}{\left (\sqrt [3]{2}-\left (1+\sqrt {3}\right ) \sqrt [3]{1+\sec (c+d x)}\right )^2}}} \]
-9/10*tan(d*x+c)/d/(a+a*sec(d*x+c))^(1/3)+3/5*(a+a*sec(d*x+c))^(2/3)*tan(d *x+c)/a/d-7/20*3^(3/4)*((2^(1/3)-(1+sec(d*x+c))^(1/3)*(1-3^(1/2)))^2/(2^(1 /3)-(1+sec(d*x+c))^(1/3)*(1+3^(1/2)))^2)^(1/2)/(2^(1/3)-(1+sec(d*x+c))^(1/ 3)*(1-3^(1/2)))*(2^(1/3)-(1+sec(d*x+c))^(1/3)*(1+3^(1/2)))*EllipticF((1-(2 ^(1/3)-(1+sec(d*x+c))^(1/3)*(1-3^(1/2)))^2/(2^(1/3)-(1+sec(d*x+c))^(1/3)*( 1+3^(1/2)))^2)^(1/2),1/4*6^(1/2)+1/4*2^(1/2))*(2^(1/3)-(1+sec(d*x+c))^(1/3 ))*((2^(2/3)+2^(1/3)*(1+sec(d*x+c))^(1/3)+(1+sec(d*x+c))^(2/3))/(2^(1/3)-( 1+sec(d*x+c))^(1/3)*(1+3^(1/2)))^2)^(1/2)*tan(d*x+c)*2^(2/3)/d/(1-sec(d*x+ c))/(a+a*sec(d*x+c))^(1/3)/(-(1+sec(d*x+c))^(1/3)*(2^(1/3)-(1+sec(d*x+c))^ (1/3))/(2^(1/3)-(1+sec(d*x+c))^(1/3)*(1+3^(1/2)))^2)^(1/2)
Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
Time = 0.20 (sec) , antiderivative size = 95, normalized size of antiderivative = 0.28 \[ \int \frac {\sec ^3(c+d x)}{\sqrt [3]{a+a \sec (c+d x)}} \, dx=\frac {\left (7 \sqrt [6]{2} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {5}{6},\frac {3}{2},\frac {1}{2} (1-\sec (c+d x))\right )+3 \sqrt [6]{1+\sec (c+d x)} (-1+2 \sec (c+d x))\right ) \tan (c+d x)}{10 d \sqrt [6]{1+\sec (c+d x)} \sqrt [3]{a (1+\sec (c+d x))}} \]
((7*2^(1/6)*Hypergeometric2F1[1/2, 5/6, 3/2, (1 - Sec[c + d*x])/2] + 3*(1 + Sec[c + d*x])^(1/6)*(-1 + 2*Sec[c + d*x]))*Tan[c + d*x])/(10*d*(1 + Sec[ c + d*x])^(1/6)*(a*(1 + Sec[c + d*x]))^(1/3))
Time = 0.75 (sec) , antiderivative size = 346, normalized size of antiderivative = 1.03, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.478, Rules used = {3042, 4287, 27, 3042, 4489, 3042, 4315, 3042, 4314, 73, 766}
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.
| \(\displaystyle \int \frac {\sec ^3(c+d x)}{\sqrt [3]{a \sec (c+d x)+a}} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\csc \left (c+d x+\frac {\pi }{2}\right )^3}{\sqrt [3]{a \csc \left (c+d x+\frac {\pi }{2}\right )+a}}dx\) |
| \(\Big \downarrow \) 4287 |
| \(\displaystyle \frac {3 \int \frac {\sec (c+d x) (2 a-3 a \sec (c+d x))}{3 \sqrt [3]{\sec (c+d x) a+a}}dx}{5 a}+\frac {3 \tan (c+d x) (a \sec (c+d x)+a)^{2/3}}{5 a d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int \frac {\sec (c+d x) (2 a-3 a \sec (c+d x))}{\sqrt [3]{\sec (c+d x) a+a}}dx}{5 a}+\frac {3 \tan (c+d x) (a \sec (c+d x)+a)^{2/3}}{5 a d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\int \frac {\csc \left (c+d x+\frac {\pi }{2}\right ) \left (2 a-3 a \csc \left (c+d x+\frac {\pi }{2}\right )\right )}{\sqrt [3]{\csc \left (c+d x+\frac {\pi }{2}\right ) a+a}}dx}{5 a}+\frac {3 \tan (c+d x) (a \sec (c+d x)+a)^{2/3}}{5 a d}\) |
| \(\Big \downarrow \) 4489 |
| \(\displaystyle \frac {\frac {7}{2} a \int \frac {\sec (c+d x)}{\sqrt [3]{\sec (c+d x) a+a}}dx-\frac {9 a \tan (c+d x)}{2 d \sqrt [3]{a \sec (c+d x)+a}}}{5 a}+\frac {3 \tan (c+d x) (a \sec (c+d x)+a)^{2/3}}{5 a d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {7}{2} a \int \frac {\csc \left (c+d x+\frac {\pi }{2}\right )}{\sqrt [3]{\csc \left (c+d x+\frac {\pi }{2}\right ) a+a}}dx-\frac {9 a \tan (c+d x)}{2 d \sqrt [3]{a \sec (c+d x)+a}}}{5 a}+\frac {3 \tan (c+d x) (a \sec (c+d x)+a)^{2/3}}{5 a d}\) |
| \(\Big \downarrow \) 4315 |
| \(\displaystyle \frac {\frac {7 a \sqrt [3]{\sec (c+d x)+1} \int \frac {\sec (c+d x)}{\sqrt [3]{\sec (c+d x)+1}}dx}{2 \sqrt [3]{a \sec (c+d x)+a}}-\frac {9 a \tan (c+d x)}{2 d \sqrt [3]{a \sec (c+d x)+a}}}{5 a}+\frac {3 \tan (c+d x) (a \sec (c+d x)+a)^{2/3}}{5 a d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {7 a \sqrt [3]{\sec (c+d x)+1} \int \frac {\csc \left (c+d x+\frac {\pi }{2}\right )}{\sqrt [3]{\csc \left (c+d x+\frac {\pi }{2}\right )+1}}dx}{2 \sqrt [3]{a \sec (c+d x)+a}}-\frac {9 a \tan (c+d x)}{2 d \sqrt [3]{a \sec (c+d x)+a}}}{5 a}+\frac {3 \tan (c+d x) (a \sec (c+d x)+a)^{2/3}}{5 a d}\) |
| \(\Big \downarrow \) 4314 |
| \(\displaystyle \frac {-\frac {7 a \tan (c+d x) \int \frac {1}{\sqrt {1-\sec (c+d x)} (\sec (c+d x)+1)^{5/6}}d\sec (c+d x)}{2 d \sqrt {1-\sec (c+d x)} \sqrt [6]{\sec (c+d x)+1} \sqrt [3]{a \sec (c+d x)+a}}-\frac {9 a \tan (c+d x)}{2 d \sqrt [3]{a \sec (c+d x)+a}}}{5 a}+\frac {3 \tan (c+d x) (a \sec (c+d x)+a)^{2/3}}{5 a d}\) |
| \(\Big \downarrow \) 73 |
| \(\displaystyle \frac {-\frac {21 a \tan (c+d x) \int \frac {1}{\sqrt {1-\sec (c+d x)}}d\sqrt [6]{\sec (c+d x)+1}}{d \sqrt {1-\sec (c+d x)} \sqrt [6]{\sec (c+d x)+1} \sqrt [3]{a \sec (c+d x)+a}}-\frac {9 a \tan (c+d x)}{2 d \sqrt [3]{a \sec (c+d x)+a}}}{5 a}+\frac {3 \tan (c+d x) (a \sec (c+d x)+a)^{2/3}}{5 a d}\) |
| \(\Big \downarrow \) 766 |
| \(\displaystyle \frac {-\frac {7\ 3^{3/4} a \tan (c+d x) \left (\sqrt [3]{2}-\sqrt [3]{\sec (c+d x)+1}\right ) \sqrt {\frac {(\sec (c+d x)+1)^{2/3}+\sqrt [3]{2} \sqrt [3]{\sec (c+d x)+1}+2^{2/3}}{\left (\sqrt [3]{2}-\left (1+\sqrt {3}\right ) \sqrt [3]{\sec (c+d x)+1}\right )^2}} \operatorname {EllipticF}\left (\arccos \left (\frac {\sqrt [3]{2}-\left (1-\sqrt {3}\right ) \sqrt [3]{\sec (c+d x)+1}}{\sqrt [3]{2}-\left (1+\sqrt {3}\right ) \sqrt [3]{\sec (c+d x)+1}}\right ),\frac {1}{4} \left (2+\sqrt {3}\right )\right )}{2 \sqrt [3]{2} d (1-\sec (c+d x)) \sqrt {-\frac {\sqrt [3]{\sec (c+d x)+1} \left (\sqrt [3]{2}-\sqrt [3]{\sec (c+d x)+1}\right )}{\left (\sqrt [3]{2}-\left (1+\sqrt {3}\right ) \sqrt [3]{\sec (c+d x)+1}\right )^2}} \sqrt [3]{a \sec (c+d x)+a}}-\frac {9 a \tan (c+d x)}{2 d \sqrt [3]{a \sec (c+d x)+a}}}{5 a}+\frac {3 \tan (c+d x) (a \sec (c+d x)+a)^{2/3}}{5 a d}\) |
(3*(a + a*Sec[c + d*x])^(2/3)*Tan[c + d*x])/(5*a*d) + ((-9*a*Tan[c + d*x]) /(2*d*(a + a*Sec[c + d*x])^(1/3)) - (7*3^(3/4)*a*EllipticF[ArcCos[(2^(1/3) - (1 - Sqrt[3])*(1 + Sec[c + d*x])^(1/3))/(2^(1/3) - (1 + Sqrt[3])*(1 + S ec[c + d*x])^(1/3))], (2 + Sqrt[3])/4]*(2^(1/3) - (1 + Sec[c + d*x])^(1/3) )*Sqrt[(2^(2/3) + 2^(1/3)*(1 + Sec[c + d*x])^(1/3) + (1 + Sec[c + d*x])^(2 /3))/(2^(1/3) - (1 + Sqrt[3])*(1 + Sec[c + d*x])^(1/3))^2]*Tan[c + d*x])/( 2*2^(1/3)*d*(1 - Sec[c + d*x])*(a + a*Sec[c + d*x])^(1/3)*Sqrt[-(((1 + Sec [c + d*x])^(1/3)*(2^(1/3) - (1 + Sec[c + d*x])^(1/3)))/(2^(1/3) - (1 + Sqr t[3])*(1 + Sec[c + d*x])^(1/3))^2)]))/(5*a)
3.2.54.3.1 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_), x_Symbol] :> With[ {p = Denominator[m]}, Simp[p/b Subst[Int[x^(p*(m + 1) - 1)*(c - a*(d/b) + d*(x^p/b))^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] && Lt Q[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntL inearQ[a, b, c, d, m, n, x]
Int[1/Sqrt[(a_) + (b_.)*(x_)^6], x_Symbol] :> With[{r = Numer[Rt[b/a, 3]], s = Denom[Rt[b/a, 3]]}, Simp[x*(s + r*x^2)*(Sqrt[(s^2 - r*s*x^2 + r^2*x^4)/ (s + (1 + Sqrt[3])*r*x^2)^2]/(2*3^(1/4)*s*Sqrt[a + b*x^6]*Sqrt[r*x^2*((s + r*x^2)/(s + (1 + Sqrt[3])*r*x^2)^2)]))*EllipticF[ArcCos[(s + (1 - Sqrt[3])* r*x^2)/(s + (1 + Sqrt[3])*r*x^2)], (2 + Sqrt[3])/4], x]] /; FreeQ[{a, b}, x ]
Int[csc[(e_.) + (f_.)*(x_)]^3*(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_))^(m_), x_Symbol] :> Simp[(-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*(b*( m + 1) - a*Csc[e + f*x]), x], x] /; FreeQ[{a, b, e, f, m}, x] && EqQ[a^2 - b^2, 0] && !LtQ[m, -2^(-1)]
Int[(csc[(e_.) + (f_.)*(x_)]*(d_.))^(n_.)*(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_))^(m_), x_Symbol] :> Simp[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] && !IntegerQ[m] && GtQ[a, 0]
Int[(csc[(e_.) + (f_.)*(x_)]*(d_.))^(n_.)*(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_))^(m_), x_Symbol] :> Simp[a^IntPart[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]
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[(a*B*m + A*b*(m + 1))/(b*(m + 1)) Int[Csc[e + f*x]*(a + b*Csc[e + f*x])^m, x], x] /; FreeQ[{a, b, A, B , e, f, m}, x] && NeQ[A*b - a*B, 0] && EqQ[a^2 - b^2, 0] && NeQ[a*B*m + A*b *(m + 1), 0] && !LtQ[m, -2^(-1)]
\[\int \frac {\sec \left (d x +c \right )^{3}}{\left (a +a \sec \left (d x +c \right )\right )^{\frac {1}{3}}}d x\]
\[ \int \frac {\sec ^3(c+d x)}{\sqrt [3]{a+a \sec (c+d x)}} \, dx=\int { \frac {\sec \left (d x + c\right )^{3}}{{\left (a \sec \left (d x + c\right ) + a\right )}^{\frac {1}{3}}} \,d x } \]
\[ \int \frac {\sec ^3(c+d x)}{\sqrt [3]{a+a \sec (c+d x)}} \, dx=\int \frac {\sec ^{3}{\left (c + d x \right )}}{\sqrt [3]{a \left (\sec {\left (c + d x \right )} + 1\right )}}\, dx \]
\[ \int \frac {\sec ^3(c+d x)}{\sqrt [3]{a+a \sec (c+d x)}} \, dx=\int { \frac {\sec \left (d x + c\right )^{3}}{{\left (a \sec \left (d x + c\right ) + a\right )}^{\frac {1}{3}}} \,d x } \]
\[ \int \frac {\sec ^3(c+d x)}{\sqrt [3]{a+a \sec (c+d x)}} \, dx=\int { \frac {\sec \left (d x + c\right )^{3}}{{\left (a \sec \left (d x + c\right ) + a\right )}^{\frac {1}{3}}} \,d x } \]
Timed out. \[ \int \frac {\sec ^3(c+d x)}{\sqrt [3]{a+a \sec (c+d x)}} \, dx=\int \frac {1}{{\cos \left (c+d\,x\right )}^3\,{\left (a+\frac {a}{\cos \left (c+d\,x\right )}\right )}^{1/3}} \,d x \]