Integrand size = 33, antiderivative size = 145 \[ \int \frac {\sec ^m(c+d x) \left (A+C \sec ^2(c+d x)\right )}{(b \sec (c+d x))^{2/3}} \, dx=\frac {3 C \sec ^{1+m}(c+d x) \sin (c+d x)}{d (1+3 m) (b \sec (c+d x))^{2/3}}-\frac {3 (A-C (2-3 m)+3 A m) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {1}{6} (5-3 m),\frac {1}{6} (11-3 m),\cos ^2(c+d x)\right ) \sec ^{-1+m}(c+d x) \sin (c+d x)}{d (5-3 m) (1+3 m) (b \sec (c+d x))^{2/3} \sqrt {\sin ^2(c+d x)}} \]
3*C*sec(d*x+c)^(1+m)*sin(d*x+c)/d/(1+3*m)/(b*sec(d*x+c))^(2/3)-3*(A-C*(2-3 *m)+3*A*m)*hypergeom([1/2, 5/6-1/2*m],[11/6-1/2*m],cos(d*x+c)^2)*sec(d*x+c )^(-1+m)*sin(d*x+c)/d/(-9*m^2+12*m+5)/(b*sec(d*x+c))^(2/3)/(sin(d*x+c)^2)^ (1/2)
Time = 0.54 (sec) , antiderivative size = 143, normalized size of antiderivative = 0.99 \[ \int \frac {\sec ^m(c+d x) \left (A+C \sec ^2(c+d x)\right )}{(b \sec (c+d x))^{2/3}} \, dx=\frac {3 b \csc (c+d x) \sec ^m(c+d x) \left (A (4+3 m) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {1}{6} (-2+3 m),\frac {1}{6} (4+3 m),\sec ^2(c+d x)\right )+C (-2+3 m) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {1}{6} (4+3 m),\frac {5}{3}+\frac {m}{2},\sec ^2(c+d x)\right ) \sec ^2(c+d x)\right ) \sqrt {-\tan ^2(c+d x)}}{d (-2+3 m) (4+3 m) (b \sec (c+d x))^{5/3}} \]
(3*b*Csc[c + d*x]*Sec[c + d*x]^m*(A*(4 + 3*m)*Hypergeometric2F1[1/2, (-2 + 3*m)/6, (4 + 3*m)/6, Sec[c + d*x]^2] + C*(-2 + 3*m)*Hypergeometric2F1[1/2 , (4 + 3*m)/6, 5/3 + m/2, Sec[c + d*x]^2]*Sec[c + d*x]^2)*Sqrt[-Tan[c + d* x]^2])/(d*(-2 + 3*m)*(4 + 3*m)*(b*Sec[c + d*x])^(5/3))
Time = 0.54 (sec) , antiderivative size = 148, normalized size of antiderivative = 1.02, number of steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.212, Rules used = {2034, 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.
\(\displaystyle \int \frac {\sec ^m(c+d x) \left (A+C \sec ^2(c+d x)\right )}{(b \sec (c+d x))^{2/3}} \, dx\) |
\(\Big \downarrow \) 2034 |
\(\displaystyle \frac {\sec ^{\frac {2}{3}}(c+d x) \int \sec ^{m-\frac {2}{3}}(c+d x) \left (C \sec ^2(c+d x)+A\right )dx}{(b \sec (c+d x))^{2/3}}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {\sec ^{\frac {2}{3}}(c+d x) \int \csc \left (c+d x+\frac {\pi }{2}\right )^{m-\frac {2}{3}} \left (C \csc \left (c+d x+\frac {\pi }{2}\right )^2+A\right )dx}{(b \sec (c+d x))^{2/3}}\) |
\(\Big \downarrow \) 4534 |
\(\displaystyle \frac {\sec ^{\frac {2}{3}}(c+d x) \left (\frac {(3 A m+A-C (2-3 m)) \int \sec ^{m-\frac {2}{3}}(c+d x)dx}{3 m+1}+\frac {3 C \sin (c+d x) \sec ^{m+\frac {1}{3}}(c+d x)}{d (3 m+1)}\right )}{(b \sec (c+d x))^{2/3}}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {\sec ^{\frac {2}{3}}(c+d x) \left (\frac {(3 A m+A-C (2-3 m)) \int \csc \left (c+d x+\frac {\pi }{2}\right )^{m-\frac {2}{3}}dx}{3 m+1}+\frac {3 C \sin (c+d x) \sec ^{m+\frac {1}{3}}(c+d x)}{d (3 m+1)}\right )}{(b \sec (c+d x))^{2/3}}\) |
\(\Big \downarrow \) 4259 |
\(\displaystyle \frac {\sec ^{\frac {2}{3}}(c+d x) \left (\frac {(3 A m+A-C (2-3 m)) \cos ^{m+\frac {1}{3}}(c+d x) \sec ^{m+\frac {1}{3}}(c+d x) \int \cos ^{\frac {2}{3}-m}(c+d x)dx}{3 m+1}+\frac {3 C \sin (c+d x) \sec ^{m+\frac {1}{3}}(c+d x)}{d (3 m+1)}\right )}{(b \sec (c+d x))^{2/3}}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {\sec ^{\frac {2}{3}}(c+d x) \left (\frac {(3 A m+A-C (2-3 m)) \cos ^{m+\frac {1}{3}}(c+d x) \sec ^{m+\frac {1}{3}}(c+d x) \int \sin \left (c+d x+\frac {\pi }{2}\right )^{\frac {2}{3}-m}dx}{3 m+1}+\frac {3 C \sin (c+d x) \sec ^{m+\frac {1}{3}}(c+d x)}{d (3 m+1)}\right )}{(b \sec (c+d x))^{2/3}}\) |
\(\Big \downarrow \) 3122 |
\(\displaystyle \frac {\sec ^{\frac {2}{3}}(c+d x) \left (\frac {3 C \sin (c+d x) \sec ^{m+\frac {1}{3}}(c+d x)}{d (3 m+1)}-\frac {3 (3 A m+A-C (2-3 m)) \sin (c+d x) \sec ^{m-\frac {5}{3}}(c+d x) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {1}{6} (5-3 m),\frac {1}{6} (11-3 m),\cos ^2(c+d x)\right )}{d (5-3 m) (3 m+1) \sqrt {\sin ^2(c+d x)}}\right )}{(b \sec (c+d x))^{2/3}}\) |
(Sec[c + d*x]^(2/3)*((3*C*Sec[c + d*x]^(1/3 + m)*Sin[c + d*x])/(d*(1 + 3*m )) - (3*(A - C*(2 - 3*m) + 3*A*m)*Hypergeometric2F1[1/2, (5 - 3*m)/6, (11 - 3*m)/6, Cos[c + d*x]^2]*Sec[c + d*x]^(-5/3 + m)*Sin[c + d*x])/(d*(5 - 3* m)*(1 + 3*m)*Sqrt[Sin[c + d*x]^2])))/(b*Sec[c + d*x])^(2/3)
3.1.25.3.1 Defintions of rubi rules used
Int[(Fx_.)*((a_.)*(v_))^(m_)*((b_.)*(v_))^(n_), x_Symbol] :> Simp[b^IntPart [n]*((b*v)^FracPart[n]/(a^IntPart[n]*(a*v)^FracPart[n])) Int[(a*v)^(m + n )*Fx, x], x] /; FreeQ[{a, b, m, n}, x] && !IntegerQ[m] && !IntegerQ[n] && !IntegerQ[m + n]
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]
\[\int \frac {\sec \left (d x +c \right )^{m} \left (A +C \sec \left (d x +c \right )^{2}\right )}{\left (b \sec \left (d x +c \right )\right )^{\frac {2}{3}}}d x\]
\[ \int \frac {\sec ^m(c+d x) \left (A+C \sec ^2(c+d x)\right )}{(b \sec (c+d x))^{2/3}} \, dx=\int { \frac {{\left (C \sec \left (d x + c\right )^{2} + A\right )} \sec \left (d x + c\right )^{m}}{\left (b \sec \left (d x + c\right )\right )^{\frac {2}{3}}} \,d x } \]
\[ \int \frac {\sec ^m(c+d x) \left (A+C \sec ^2(c+d x)\right )}{(b \sec (c+d x))^{2/3}} \, dx=\int \frac {\left (A + C \sec ^{2}{\left (c + d x \right )}\right ) \sec ^{m}{\left (c + d x \right )}}{\left (b \sec {\left (c + d x \right )}\right )^{\frac {2}{3}}}\, dx \]
\[ \int \frac {\sec ^m(c+d x) \left (A+C \sec ^2(c+d x)\right )}{(b \sec (c+d x))^{2/3}} \, dx=\int { \frac {{\left (C \sec \left (d x + c\right )^{2} + A\right )} \sec \left (d x + c\right )^{m}}{\left (b \sec \left (d x + c\right )\right )^{\frac {2}{3}}} \,d x } \]
\[ \int \frac {\sec ^m(c+d x) \left (A+C \sec ^2(c+d x)\right )}{(b \sec (c+d x))^{2/3}} \, dx=\int { \frac {{\left (C \sec \left (d x + c\right )^{2} + A\right )} \sec \left (d x + c\right )^{m}}{\left (b \sec \left (d x + c\right )\right )^{\frac {2}{3}}} \,d x } \]
Timed out. \[ \int \frac {\sec ^m(c+d x) \left (A+C \sec ^2(c+d x)\right )}{(b \sec (c+d x))^{2/3}} \, dx=\int \frac {\left (A+\frac {C}{{\cos \left (c+d\,x\right )}^2}\right )\,{\left (\frac {1}{\cos \left (c+d\,x\right )}\right )}^m}{{\left (\frac {b}{\cos \left (c+d\,x\right )}\right )}^{2/3}} \,d x \]