Integrand size = 31, antiderivative size = 92 \[ \int \frac {\sec (c+d x) \left (A+C \sec ^2(c+d x)\right )}{\sqrt [3]{b \sec (c+d x)}} \, dx=-\frac {3 (5 A+2 C) \operatorname {Hypergeometric2F1}\left (\frac {1}{6},\frac {1}{2},\frac {7}{6},\cos ^2(c+d x)\right ) \sin (c+d x)}{5 d \sqrt [3]{b \sec (c+d x)} \sqrt {\sin ^2(c+d x)}}+\frac {3 C (b \sec (c+d x))^{2/3} \tan (c+d x)}{5 b d} \] Output:
-3/5*(5*A+2*C)*hypergeom([1/6, 1/2],[7/6],cos(d*x+c)^2)*sin(d*x+c)/d/(b*se c(d*x+c))^(1/3)/(sin(d*x+c)^2)^(1/2)+3/5*C*(b*sec(d*x+c))^(2/3)*tan(d*x+c) /b/d
Time = 0.46 (sec) , antiderivative size = 80, normalized size of antiderivative = 0.87 \[ \int \frac {\sec (c+d x) \left (A+C \sec ^2(c+d x)\right )}{\sqrt [3]{b \sec (c+d x)}} \, dx=\frac {3 \cot (c+d x) (b \sec (c+d x))^{2/3} \left (2 C \tan ^2(c+d x)+(5 A+2 C) \operatorname {Hypergeometric2F1}\left (\frac {1}{3},\frac {1}{2},\frac {4}{3},\sec ^2(c+d x)\right ) \sqrt {-\tan ^2(c+d x)}\right )}{10 b d} \] Input:
Integrate[(Sec[c + d*x]*(A + C*Sec[c + d*x]^2))/(b*Sec[c + d*x])^(1/3),x]
Output:
(3*Cot[c + d*x]*(b*Sec[c + d*x])^(2/3)*(2*C*Tan[c + d*x]^2 + (5*A + 2*C)*H ypergeometric2F1[1/3, 1/2, 4/3, Sec[c + d*x]^2]*Sqrt[-Tan[c + d*x]^2]))/(1 0*b*d)
Time = 0.43 (sec) , antiderivative size = 94, normalized size of antiderivative = 1.02, number of steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.226, Rules used = {2030, 3042, 4534, 3042, 4259, 3042, 3122}
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.
\(\displaystyle \int \frac {\sec (c+d x) \left (A+C \sec ^2(c+d x)\right )}{\sqrt [3]{b \sec (c+d x)}} \, dx\) |
\(\Big \downarrow \) 2030 |
\(\displaystyle \frac {\int (b \sec (c+d x))^{2/3} \left (C \sec ^2(c+d x)+A\right )dx}{b}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {\int \left (b \csc \left (c+d x+\frac {\pi }{2}\right )\right )^{2/3} \left (C \csc \left (c+d x+\frac {\pi }{2}\right )^2+A\right )dx}{b}\) |
\(\Big \downarrow \) 4534 |
\(\displaystyle \frac {\frac {1}{5} (5 A+2 C) \int (b \sec (c+d x))^{2/3}dx+\frac {3 C \tan (c+d x) (b \sec (c+d x))^{2/3}}{5 d}}{b}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {\frac {1}{5} (5 A+2 C) \int \left (b \csc \left (c+d x+\frac {\pi }{2}\right )\right )^{2/3}dx+\frac {3 C \tan (c+d x) (b \sec (c+d x))^{2/3}}{5 d}}{b}\) |
\(\Big \downarrow \) 4259 |
\(\displaystyle \frac {\frac {1}{5} (5 A+2 C) \left (\frac {\cos (c+d x)}{b}\right )^{2/3} (b \sec (c+d x))^{2/3} \int \frac {1}{\left (\frac {\cos (c+d x)}{b}\right )^{2/3}}dx+\frac {3 C \tan (c+d x) (b \sec (c+d x))^{2/3}}{5 d}}{b}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {\frac {1}{5} (5 A+2 C) \left (\frac {\cos (c+d x)}{b}\right )^{2/3} (b \sec (c+d x))^{2/3} \int \frac {1}{\left (\frac {\sin \left (c+d x+\frac {\pi }{2}\right )}{b}\right )^{2/3}}dx+\frac {3 C \tan (c+d x) (b \sec (c+d x))^{2/3}}{5 d}}{b}\) |
\(\Big \downarrow \) 3122 |
\(\displaystyle \frac {\frac {3 C \tan (c+d x) (b \sec (c+d x))^{2/3}}{5 d}-\frac {3 b (5 A+2 C) \sin (c+d x) \operatorname {Hypergeometric2F1}\left (\frac {1}{6},\frac {1}{2},\frac {7}{6},\cos ^2(c+d x)\right )}{5 d \sqrt {\sin ^2(c+d x)} \sqrt [3]{b \sec (c+d x)}}}{b}\) |
Input:
Int[(Sec[c + d*x]*(A + C*Sec[c + d*x]^2))/(b*Sec[c + d*x])^(1/3),x]
Output:
((-3*b*(5*A + 2*C)*Hypergeometric2F1[1/6, 1/2, 7/6, Cos[c + d*x]^2]*Sin[c + d*x])/(5*d*(b*Sec[c + d*x])^(1/3)*Sqrt[Sin[c + d*x]^2]) + (3*C*(b*Sec[c + d*x])^(2/3)*Tan[c + d*x])/(5*d))/b
Int[(Fx_.)*(v_)^(m_.)*((b_)*(v_))^(n_), x_Symbol] :> Simp[1/b^m Int[(b*v) ^(m + n)*Fx, x], x] /; FreeQ[{b, n}, x] && IntegerQ[m]
Int[((b_.)*sin[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[Cos[c + d*x]*(( b*Sin[c + d*x])^(n + 1)/(b*d*(n + 1)*Sqrt[Cos[c + d*x]^2]))*Hypergeometric2 F1[1/2, (n + 1)/2, (n + 3)/2, Sin[c + d*x]^2], x] /; FreeQ[{b, c, d, n}, x] && !IntegerQ[2*n]
Int[(csc[(c_.) + (d_.)*(x_)]*(b_.))^(n_), x_Symbol] :> Simp[(b*Csc[c + d*x] )^(n - 1)*((Sin[c + d*x]/b)^(n - 1) Int[1/(Sin[c + d*x]/b)^n, x]), x] /; FreeQ[{b, c, d, n}, x] && !IntegerQ[n]
Int[(csc[(e_.) + (f_.)*(x_)]*(b_.))^(m_.)*(csc[(e_.) + (f_.)*(x_)]^2*(C_.) + (A_)), x_Symbol] :> Simp[(-C)*Cot[e + f*x]*((b*Csc[e + f*x])^m/(f*(m + 1) )), x] + Simp[(C*m + A*(m + 1))/(m + 1) Int[(b*Csc[e + f*x])^m, x], x] /; FreeQ[{b, e, f, A, C, m}, x] && NeQ[C*m + A*(m + 1), 0] && !LeQ[m, -1]
\[\int \frac {\sec \left (d x +c \right ) \left (A +C \sec \left (d x +c \right )^{2}\right )}{\left (b \sec \left (d x +c \right )\right )^{\frac {1}{3}}}d x\]
Input:
int(sec(d*x+c)*(A+C*sec(d*x+c)^2)/(b*sec(d*x+c))^(1/3),x)
Output:
int(sec(d*x+c)*(A+C*sec(d*x+c)^2)/(b*sec(d*x+c))^(1/3),x)
\[ \int \frac {\sec (c+d x) \left (A+C \sec ^2(c+d x)\right )}{\sqrt [3]{b \sec (c+d x)}} \, dx=\int { \frac {{\left (C \sec \left (d x + c\right )^{2} + A\right )} \sec \left (d x + c\right )}{\left (b \sec \left (d x + c\right )\right )^{\frac {1}{3}}} \,d x } \] Input:
integrate(sec(d*x+c)*(A+C*sec(d*x+c)^2)/(b*sec(d*x+c))^(1/3),x, algorithm= "fricas")
Output:
integral((C*sec(d*x + c)^2 + A)*(b*sec(d*x + c))^(2/3)/b, x)
\[ \int \frac {\sec (c+d x) \left (A+C \sec ^2(c+d x)\right )}{\sqrt [3]{b \sec (c+d x)}} \, dx=\int \frac {\left (A + C \sec ^{2}{\left (c + d x \right )}\right ) \sec {\left (c + d x \right )}}{\sqrt [3]{b \sec {\left (c + d x \right )}}}\, dx \] Input:
integrate(sec(d*x+c)*(A+C*sec(d*x+c)**2)/(b*sec(d*x+c))**(1/3),x)
Output:
Integral((A + C*sec(c + d*x)**2)*sec(c + d*x)/(b*sec(c + d*x))**(1/3), x)
\[ \int \frac {\sec (c+d x) \left (A+C \sec ^2(c+d x)\right )}{\sqrt [3]{b \sec (c+d x)}} \, dx=\int { \frac {{\left (C \sec \left (d x + c\right )^{2} + A\right )} \sec \left (d x + c\right )}{\left (b \sec \left (d x + c\right )\right )^{\frac {1}{3}}} \,d x } \] Input:
integrate(sec(d*x+c)*(A+C*sec(d*x+c)^2)/(b*sec(d*x+c))^(1/3),x, algorithm= "maxima")
Output:
integrate((C*sec(d*x + c)^2 + A)*sec(d*x + c)/(b*sec(d*x + c))^(1/3), x)
\[ \int \frac {\sec (c+d x) \left (A+C \sec ^2(c+d x)\right )}{\sqrt [3]{b \sec (c+d x)}} \, dx=\int { \frac {{\left (C \sec \left (d x + c\right )^{2} + A\right )} \sec \left (d x + c\right )}{\left (b \sec \left (d x + c\right )\right )^{\frac {1}{3}}} \,d x } \] Input:
integrate(sec(d*x+c)*(A+C*sec(d*x+c)^2)/(b*sec(d*x+c))^(1/3),x, algorithm= "giac")
Output:
integrate((C*sec(d*x + c)^2 + A)*sec(d*x + c)/(b*sec(d*x + c))^(1/3), x)
Timed out. \[ \int \frac {\sec (c+d x) \left (A+C \sec ^2(c+d x)\right )}{\sqrt [3]{b \sec (c+d x)}} \, dx=\int \frac {A+\frac {C}{{\cos \left (c+d\,x\right )}^2}}{\cos \left (c+d\,x\right )\,{\left (\frac {b}{\cos \left (c+d\,x\right )}\right )}^{1/3}} \,d x \] Input:
int((A + C/cos(c + d*x)^2)/(cos(c + d*x)*(b/cos(c + d*x))^(1/3)),x)
Output:
int((A + C/cos(c + d*x)^2)/(cos(c + d*x)*(b/cos(c + d*x))^(1/3)), x)
\[ \int \frac {\sec (c+d x) \left (A+C \sec ^2(c+d x)\right )}{\sqrt [3]{b \sec (c+d x)}} \, dx=\frac {\left (\int \sec \left (d x +c \right )^{\frac {8}{3}}d x \right ) c +\left (\int \sec \left (d x +c \right )^{\frac {2}{3}}d x \right ) a}{b^{\frac {1}{3}}} \] Input:
int(sec(d*x+c)*(A+C*sec(d*x+c)^2)/(b*sec(d*x+c))^(1/3),x)
Output:
(int(sec(c + d*x)**3/sec(c + d*x)**(1/3),x)*c + int(sec(c + d*x)/sec(c + d *x)**(1/3),x)*a)/b**(1/3)