Integrand size = 31, antiderivative size = 119 \[ \int \cos ^2(c+d x) (b \sec (c+d x))^{4/3} (A+B \sec (c+d x)) \, dx=-\frac {3 A b^3 \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {5}{6},\frac {11}{6},\cos ^2(c+d x)\right ) \sin (c+d x)}{5 d (b \sec (c+d x))^{5/3} \sqrt {\sin ^2(c+d x)}}-\frac {3 b^2 B \operatorname {Hypergeometric2F1}\left (\frac {1}{3},\frac {1}{2},\frac {4}{3},\cos ^2(c+d x)\right ) \sin (c+d x)}{2 d (b \sec (c+d x))^{2/3} \sqrt {\sin ^2(c+d x)}} \]
-3/5*A*b^3*hypergeom([1/2, 5/6],[11/6],cos(d*x+c)^2)*sin(d*x+c)/d/(b*sec(d *x+c))^(5/3)/(sin(d*x+c)^2)^(1/2)-3/2*b^2*B*hypergeom([1/3, 1/2],[4/3],cos (d*x+c)^2)*sin(d*x+c)/d/(b*sec(d*x+c))^(2/3)/(sin(d*x+c)^2)^(1/2)
Time = 0.08 (sec) , antiderivative size = 88, normalized size of antiderivative = 0.74 \[ \int \cos ^2(c+d x) (b \sec (c+d x))^{4/3} (A+B \sec (c+d x)) \, dx=-\frac {3 b \cot (c+d x) \left (A \cos (c+d x) \operatorname {Hypergeometric2F1}\left (-\frac {1}{3},\frac {1}{2},\frac {2}{3},\sec ^2(c+d x)\right )-2 B \operatorname {Hypergeometric2F1}\left (\frac {1}{6},\frac {1}{2},\frac {7}{6},\sec ^2(c+d x)\right )\right ) \sqrt [3]{b \sec (c+d x)} \sqrt {-\tan ^2(c+d x)}}{2 d} \]
(-3*b*Cot[c + d*x]*(A*Cos[c + d*x]*Hypergeometric2F1[-1/3, 1/2, 2/3, Sec[c + d*x]^2] - 2*B*Hypergeometric2F1[1/6, 1/2, 7/6, Sec[c + d*x]^2])*(b*Sec[ c + d*x])^(1/3)*Sqrt[-Tan[c + d*x]^2])/(2*d)
Time = 0.48 (sec) , antiderivative size = 118, normalized size of antiderivative = 0.99, number of steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.226, Rules used = {3042, 2030, 4274, 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.
\(\displaystyle \int \cos ^2(c+d x) (b \sec (c+d x))^{4/3} (A+B \sec (c+d x)) \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {\left (b \csc \left (c+d x+\frac {\pi }{2}\right )\right )^{4/3} \left (A+B \csc \left (c+d x+\frac {\pi }{2}\right )\right )}{\csc \left (c+d x+\frac {\pi }{2}\right )^2}dx\) |
\(\Big \downarrow \) 2030 |
\(\displaystyle b^2 \int \frac {A+B \csc \left (\frac {1}{2} (2 c+\pi )+d x\right )}{\left (b \csc \left (\frac {1}{2} (2 c+\pi )+d x\right )\right )^{2/3}}dx\) |
\(\Big \downarrow \) 4274 |
\(\displaystyle b^2 \left (A \int \frac {1}{(b \sec (c+d x))^{2/3}}dx+\frac {B \int \sqrt [3]{b \sec (c+d x)}dx}{b}\right )\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle b^2 \left (A \int \frac {1}{\left (b \csc \left (c+d x+\frac {\pi }{2}\right )\right )^{2/3}}dx+\frac {B \int \sqrt [3]{b \csc \left (c+d x+\frac {\pi }{2}\right )}dx}{b}\right )\) |
\(\Big \downarrow \) 4259 |
\(\displaystyle b^2 \left (A \sqrt [3]{\frac {\cos (c+d x)}{b}} \sqrt [3]{b \sec (c+d x)} \int \left (\frac {\cos (c+d x)}{b}\right )^{2/3}dx+\frac {B \sqrt [3]{\frac {\cos (c+d x)}{b}} \sqrt [3]{b \sec (c+d x)} \int \frac {1}{\sqrt [3]{\frac {\cos (c+d x)}{b}}}dx}{b}\right )\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle b^2 \left (A \sqrt [3]{\frac {\cos (c+d x)}{b}} \sqrt [3]{b \sec (c+d x)} \int \left (\frac {\sin \left (c+d x+\frac {\pi }{2}\right )}{b}\right )^{2/3}dx+\frac {B \sqrt [3]{\frac {\cos (c+d x)}{b}} \sqrt [3]{b \sec (c+d x)} \int \frac {1}{\sqrt [3]{\frac {\sin \left (c+d x+\frac {\pi }{2}\right )}{b}}}dx}{b}\right )\) |
\(\Big \downarrow \) 3122 |
\(\displaystyle b^2 \left (-\frac {3 A b \sin (c+d x) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {5}{6},\frac {11}{6},\cos ^2(c+d x)\right )}{5 d \sqrt {\sin ^2(c+d x)} (b \sec (c+d x))^{5/3}}-\frac {3 B \sin (c+d x) \operatorname {Hypergeometric2F1}\left (\frac {1}{3},\frac {1}{2},\frac {4}{3},\cos ^2(c+d x)\right )}{2 d \sqrt {\sin ^2(c+d x)} (b \sec (c+d x))^{2/3}}\right )\) |
b^2*((-3*A*b*Hypergeometric2F1[1/2, 5/6, 11/6, Cos[c + d*x]^2]*Sin[c + d*x ])/(5*d*(b*Sec[c + d*x])^(5/3)*Sqrt[Sin[c + d*x]^2]) - (3*B*Hypergeometric 2F1[1/3, 1/2, 4/3, Cos[c + d*x]^2]*Sin[c + d*x])/(2*d*(b*Sec[c + d*x])^(2/ 3)*Sqrt[Sin[c + d*x]^2]))
3.1.16.3.1 Defintions of rubi rules used
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_)]*(d_.))^(n_.)*(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_)), x_Symbol] :> Simp[a Int[(d*Csc[e + f*x])^n, x], x] + Simp[b/d In t[(d*Csc[e + f*x])^(n + 1), x], x] /; FreeQ[{a, b, d, e, f, n}, x]
\[\int \cos \left (d x +c \right )^{2} \left (b \sec \left (d x +c \right )\right )^{\frac {4}{3}} \left (A +B \sec \left (d x +c \right )\right )d x\]
\[ \int \cos ^2(c+d x) (b \sec (c+d x))^{4/3} (A+B \sec (c+d x)) \, dx=\int { {\left (B \sec \left (d x + c\right ) + A\right )} \left (b \sec \left (d x + c\right )\right )^{\frac {4}{3}} \cos \left (d x + c\right )^{2} \,d x } \]
integral((B*b*cos(d*x + c)^2*sec(d*x + c)^2 + A*b*cos(d*x + c)^2*sec(d*x + c))*(b*sec(d*x + c))^(1/3), x)
Timed out. \[ \int \cos ^2(c+d x) (b \sec (c+d x))^{4/3} (A+B \sec (c+d x)) \, dx=\text {Timed out} \]
\[ \int \cos ^2(c+d x) (b \sec (c+d x))^{4/3} (A+B \sec (c+d x)) \, dx=\int { {\left (B \sec \left (d x + c\right ) + A\right )} \left (b \sec \left (d x + c\right )\right )^{\frac {4}{3}} \cos \left (d x + c\right )^{2} \,d x } \]
\[ \int \cos ^2(c+d x) (b \sec (c+d x))^{4/3} (A+B \sec (c+d x)) \, dx=\int { {\left (B \sec \left (d x + c\right ) + A\right )} \left (b \sec \left (d x + c\right )\right )^{\frac {4}{3}} \cos \left (d x + c\right )^{2} \,d x } \]
Timed out. \[ \int \cos ^2(c+d x) (b \sec (c+d x))^{4/3} (A+B \sec (c+d x)) \, dx=\int {\cos \left (c+d\,x\right )}^2\,\left (A+\frac {B}{\cos \left (c+d\,x\right )}\right )\,{\left (\frac {b}{\cos \left (c+d\,x\right )}\right )}^{4/3} \,d x \]