Integrand size = 23, antiderivative size = 174 \[ \int \frac {1}{\sec ^{\frac {2}{3}}(c+d x) (a+b \sec (c+d x))} \, dx=-\frac {b \operatorname {AppellF1}\left (\frac {1}{2},-\frac {1}{3},1,\frac {3}{2},\sin ^2(c+d x),\frac {a^2 \sin ^2(c+d x)}{a^2-b^2}\right ) \sin (c+d x)}{\left (a^2-b^2\right ) d \sqrt [3]{\cos ^2(c+d x)} \sec ^{\frac {2}{3}}(c+d x)}+\frac {a \operatorname {AppellF1}\left (\frac {1}{2},-\frac {5}{6},1,\frac {3}{2},\sin ^2(c+d x),\frac {a^2 \sin ^2(c+d x)}{a^2-b^2}\right ) \sqrt [6]{\cos ^2(c+d x)} \sqrt [3]{\sec (c+d x)} \sin (c+d x)}{\left (a^2-b^2\right ) d} \]
-b*AppellF1(1/2,-1/3,1,3/2,sin(d*x+c)^2,a^2*sin(d*x+c)^2/(a^2-b^2))*sin(d* x+c)/(a^2-b^2)/d/(cos(d*x+c)^2)^(1/3)/sec(d*x+c)^(2/3)+a*AppellF1(1/2,-5/6 ,1,3/2,sin(d*x+c)^2,a^2*sin(d*x+c)^2/(a^2-b^2))*(cos(d*x+c)^2)^(1/6)*sec(d *x+c)^(1/3)*sin(d*x+c)/(a^2-b^2)/d
Leaf count is larger than twice the leaf count of optimal. \(7588\) vs. \(2(174)=348\).
Time = 82.40 (sec) , antiderivative size = 7588, normalized size of antiderivative = 43.61 \[ \int \frac {1}{\sec ^{\frac {2}{3}}(c+d x) (a+b \sec (c+d x))} \, dx=\text {Result too large to show} \]
Time = 0.60 (sec) , antiderivative size = 195, normalized size of antiderivative = 1.12, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.348, Rules used = {3042, 4356, 3042, 3302, 3042, 3668, 25, 333}
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.
\(\displaystyle \int \frac {1}{\sec ^{\frac {2}{3}}(c+d x) (a+b \sec (c+d x))} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {1}{\csc \left (c+d x+\frac {\pi }{2}\right )^{2/3} \left (a+b \csc \left (c+d x+\frac {\pi }{2}\right )\right )}dx\) |
\(\Big \downarrow \) 4356 |
\(\displaystyle \sqrt [3]{\cos (c+d x)} \sqrt [3]{\sec (c+d x)} \int \frac {\cos ^{\frac {5}{3}}(c+d x)}{b+a \cos (c+d x)}dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \sqrt [3]{\cos (c+d x)} \sqrt [3]{\sec (c+d x)} \int \frac {\sin \left (c+d x+\frac {\pi }{2}\right )^{5/3}}{b+a \sin \left (c+d x+\frac {\pi }{2}\right )}dx\) |
\(\Big \downarrow \) 3302 |
\(\displaystyle \sqrt [3]{\cos (c+d x)} \sqrt [3]{\sec (c+d x)} \left (b \int \frac {\cos ^{\frac {5}{3}}(c+d x)}{b^2-a^2 \cos ^2(c+d x)}dx-a \int \frac {\cos ^{\frac {8}{3}}(c+d x)}{b^2-a^2 \cos ^2(c+d x)}dx\right )\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \sqrt [3]{\cos (c+d x)} \sqrt [3]{\sec (c+d x)} \left (b \int \frac {\sin \left (c+d x+\frac {\pi }{2}\right )^{5/3}}{b^2-a^2 \sin \left (c+d x+\frac {\pi }{2}\right )^2}dx-a \int \frac {\sin \left (c+d x+\frac {\pi }{2}\right )^{8/3}}{b^2-a^2 \sin \left (c+d x+\frac {\pi }{2}\right )^2}dx\right )\) |
\(\Big \downarrow \) 3668 |
\(\displaystyle \sqrt [3]{\cos (c+d x)} \sqrt [3]{\sec (c+d x)} \left (\frac {b \cos ^{\frac {2}{3}}(c+d x) \int -\frac {\sqrt [3]{1-\sin ^2(c+d x)}}{-\sin ^2(c+d x) a^2+a^2-b^2}d\sin (c+d x)}{d \sqrt [3]{\cos ^2(c+d x)}}-\frac {a \cos ^{\frac {5}{3}}(c+d x) \int -\frac {\left (1-\sin ^2(c+d x)\right )^{5/6}}{-\sin ^2(c+d x) a^2+a^2-b^2}d\sin (c+d x)}{d \cos ^2(c+d x)^{5/6}}\right )\) |
\(\Big \downarrow \) 25 |
\(\displaystyle \sqrt [3]{\cos (c+d x)} \sqrt [3]{\sec (c+d x)} \left (\frac {a \cos ^{\frac {5}{3}}(c+d x) \int \frac {\left (1-\sin ^2(c+d x)\right )^{5/6}}{-\sin ^2(c+d x) a^2+a^2-b^2}d\sin (c+d x)}{d \cos ^2(c+d x)^{5/6}}-\frac {b \cos ^{\frac {2}{3}}(c+d x) \int \frac {\sqrt [3]{1-\sin ^2(c+d x)}}{-\sin ^2(c+d x) a^2+a^2-b^2}d\sin (c+d x)}{d \sqrt [3]{\cos ^2(c+d x)}}\right )\) |
\(\Big \downarrow \) 333 |
\(\displaystyle \sqrt [3]{\cos (c+d x)} \sqrt [3]{\sec (c+d x)} \left (\frac {a \sin (c+d x) \cos ^{\frac {5}{3}}(c+d x) \operatorname {AppellF1}\left (\frac {1}{2},-\frac {5}{6},1,\frac {3}{2},\sin ^2(c+d x),\frac {a^2 \sin ^2(c+d x)}{a^2-b^2}\right )}{d \left (a^2-b^2\right ) \cos ^2(c+d x)^{5/6}}-\frac {b \sin (c+d x) \cos ^{\frac {2}{3}}(c+d x) \operatorname {AppellF1}\left (\frac {1}{2},-\frac {1}{3},1,\frac {3}{2},\sin ^2(c+d x),\frac {a^2 \sin ^2(c+d x)}{a^2-b^2}\right )}{d \left (a^2-b^2\right ) \sqrt [3]{\cos ^2(c+d x)}}\right )\) |
Cos[c + d*x]^(1/3)*Sec[c + d*x]^(1/3)*((a*AppellF1[1/2, -5/6, 1, 3/2, Sin[ c + d*x]^2, (a^2*Sin[c + d*x]^2)/(a^2 - b^2)]*Cos[c + d*x]^(5/3)*Sin[c + d *x])/((a^2 - b^2)*d*(Cos[c + d*x]^2)^(5/6)) - (b*AppellF1[1/2, -1/3, 1, 3/ 2, Sin[c + d*x]^2, (a^2*Sin[c + d*x]^2)/(a^2 - b^2)]*Cos[c + d*x]^(2/3)*Si n[c + d*x])/((a^2 - b^2)*d*(Cos[c + d*x]^2)^(1/3)))
3.8.16.3.1 Defintions of rubi rules used
Int[((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2)^(q_), x_Symbol] :> Sim p[a^p*c^q*x*AppellF1[1/2, -p, -q, 3/2, (-b)*(x^2/a), (-d)*(x^2/c)], x] /; F reeQ[{a, b, c, d, p, q}, x] && NeQ[b*c - a*d, 0] && (IntegerQ[p] || GtQ[a, 0]) && (IntegerQ[q] || GtQ[c, 0])
Int[((d_.)*sin[(e_.) + (f_.)*(x_)])^(n_.)/((a_) + (b_.)*sin[(e_.) + (f_.)*( x_)]), x_Symbol] :> Simp[a Int[(d*Sin[e + f*x])^n/(a^2 - b^2*Sin[e + f*x] ^2), x], x] - Simp[b/d Int[(d*Sin[e + f*x])^(n + 1)/(a^2 - b^2*Sin[e + f* x]^2), x], x] /; FreeQ[{a, b, d, e, f, n}, x] && NeQ[a^2 - b^2, 0]
Int[((d_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x _)]^2)^(p_.), x_Symbol] :> With[{ff = FreeFactors[Cos[e + f*x], x]}, Simp[( -ff)*d^(2*IntPart[(m - 1)/2] + 1)*((d*Sin[e + f*x])^(2*FracPart[(m - 1)/2]) /(f*(Sin[e + f*x]^2)^FracPart[(m - 1)/2])) Subst[Int[(1 - ff^2*x^2)^((m - 1)/2)*(a + b - b*ff^2*x^2)^p, x], x, Cos[e + f*x]/ff], x]] /; FreeQ[{a, b, d, e, f, m, p}, x] && !IntegerQ[m]
Int[(csc[(e_.) + (f_.)*(x_)]*(d_.))^(n_.)*(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_))^(m_.), x_Symbol] :> Simp[Sin[e + f*x]^n*(d*Csc[e + f*x])^n Int[(b + a*Sin[e + f*x])^m/Sin[e + f*x]^(m + n), x], x] /; FreeQ[{a, b, d, e, f, n} , x] && NeQ[a^2 - b^2, 0] && IntegerQ[m]
\[\int \frac {1}{\sec \left (d x +c \right )^{\frac {2}{3}} \left (a +b \sec \left (d x +c \right )\right )}d x\]
Timed out. \[ \int \frac {1}{\sec ^{\frac {2}{3}}(c+d x) (a+b \sec (c+d x))} \, dx=\text {Timed out} \]
\[ \int \frac {1}{\sec ^{\frac {2}{3}}(c+d x) (a+b \sec (c+d x))} \, dx=\int \frac {1}{\left (a + b \sec {\left (c + d x \right )}\right ) \sec ^{\frac {2}{3}}{\left (c + d x \right )}}\, dx \]
\[ \int \frac {1}{\sec ^{\frac {2}{3}}(c+d x) (a+b \sec (c+d x))} \, dx=\int { \frac {1}{{\left (b \sec \left (d x + c\right ) + a\right )} \sec \left (d x + c\right )^{\frac {2}{3}}} \,d x } \]
\[ \int \frac {1}{\sec ^{\frac {2}{3}}(c+d x) (a+b \sec (c+d x))} \, dx=\int { \frac {1}{{\left (b \sec \left (d x + c\right ) + a\right )} \sec \left (d x + c\right )^{\frac {2}{3}}} \,d x } \]
Timed out. \[ \int \frac {1}{\sec ^{\frac {2}{3}}(c+d x) (a+b \sec (c+d x))} \, dx=\int \frac {1}{\left (a+\frac {b}{\cos \left (c+d\,x\right )}\right )\,{\left (\frac {1}{\cos \left (c+d\,x\right )}\right )}^{2/3}} \,d x \]