Integrand size = 29, antiderivative size = 139 \[ \int (b \cos (c+d x))^n (A+B \cos (c+d x)) \sec ^3(c+d x) \, dx=\frac {A b^2 (b \cos (c+d x))^{-2+n} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {1}{2} (-2+n),\frac {n}{2},\cos ^2(c+d x)\right ) \sin (c+d x)}{d (2-n) \sqrt {\sin ^2(c+d x)}}+\frac {b B (b \cos (c+d x))^{-1+n} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {1}{2} (-1+n),\frac {1+n}{2},\cos ^2(c+d x)\right ) \sin (c+d x)}{d (1-n) \sqrt {\sin ^2(c+d x)}} \]
A*b^2*(b*cos(d*x+c))^(-2+n)*hypergeom([1/2, -1+1/2*n],[1/2*n],cos(d*x+c)^2 )*sin(d*x+c)/d/(2-n)/(sin(d*x+c)^2)^(1/2)+b*B*(b*cos(d*x+c))^(-1+n)*hyperg eom([1/2, -1/2+1/2*n],[1/2+1/2*n],cos(d*x+c)^2)*sin(d*x+c)/d/(1-n)/(sin(d* x+c)^2)^(1/2)
Time = 0.12 (sec) , antiderivative size = 118, normalized size of antiderivative = 0.85 \[ \int (b \cos (c+d x))^n (A+B \cos (c+d x)) \sec ^3(c+d x) \, dx=-\frac {(b \cos (c+d x))^n \csc (c+d x) \left (A (-1+n) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {1}{2} (-2+n),\frac {n}{2},\cos ^2(c+d x)\right )+B (-2+n) \cos (c+d x) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {1}{2} (-1+n),\frac {1+n}{2},\cos ^2(c+d x)\right )\right ) \sec ^2(c+d x) \sqrt {\sin ^2(c+d x)}}{d (-2+n) (-1+n)} \]
-(((b*Cos[c + d*x])^n*Csc[c + d*x]*(A*(-1 + n)*Hypergeometric2F1[1/2, (-2 + n)/2, n/2, Cos[c + d*x]^2] + B*(-2 + n)*Cos[c + d*x]*Hypergeometric2F1[1 /2, (-1 + n)/2, (1 + n)/2, Cos[c + d*x]^2])*Sec[c + d*x]^2*Sqrt[Sin[c + d* x]^2])/(d*(-2 + n)*(-1 + n)))
Time = 0.41 (sec) , antiderivative size = 145, normalized size of antiderivative = 1.04, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.172, Rules used = {3042, 2030, 3227, 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.
\(\displaystyle \int \sec ^3(c+d x) (A+B \cos (c+d x)) (b \cos (c+d x))^n \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {\left (A+B \sin \left (c+d x+\frac {\pi }{2}\right )\right ) \left (b \sin \left (c+d x+\frac {\pi }{2}\right )\right )^n}{\sin \left (c+d x+\frac {\pi }{2}\right )^3}dx\) |
\(\Big \downarrow \) 2030 |
\(\displaystyle b^3 \int \left (b \sin \left (\frac {1}{2} (2 c+\pi )+d x\right )\right )^{n-3} \left (A+B \sin \left (\frac {1}{2} (2 c+\pi )+d x\right )\right )dx\) |
\(\Big \downarrow \) 3227 |
\(\displaystyle b^3 \left (A \int (b \cos (c+d x))^{n-3}dx+\frac {B \int (b \cos (c+d x))^{n-2}dx}{b}\right )\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle b^3 \left (A \int \left (b \sin \left (c+d x+\frac {\pi }{2}\right )\right )^{n-3}dx+\frac {B \int \left (b \sin \left (c+d x+\frac {\pi }{2}\right )\right )^{n-2}dx}{b}\right )\) |
\(\Big \downarrow \) 3122 |
\(\displaystyle b^3 \left (\frac {A \sin (c+d x) (b \cos (c+d x))^{n-2} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {n-2}{2},\frac {n}{2},\cos ^2(c+d x)\right )}{b d (2-n) \sqrt {\sin ^2(c+d x)}}+\frac {B \sin (c+d x) (b \cos (c+d x))^{n-1} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {n-1}{2},\frac {n+1}{2},\cos ^2(c+d x)\right )}{b^2 d (1-n) \sqrt {\sin ^2(c+d x)}}\right )\) |
b^3*((A*(b*Cos[c + d*x])^(-2 + n)*Hypergeometric2F1[1/2, (-2 + n)/2, n/2, Cos[c + d*x]^2]*Sin[c + d*x])/(b*d*(2 - n)*Sqrt[Sin[c + d*x]^2]) + (B*(b*C os[c + d*x])^(-1 + n)*Hypergeometric2F1[1/2, (-1 + n)/2, (1 + n)/2, Cos[c + d*x]^2]*Sin[c + d*x])/(b^2*d*(1 - n)*Sqrt[Sin[c + d*x]^2]))
3.10.17.3.1 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[((b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_) + (d_.)*sin[(e_.) + (f_.)*(x _)]), x_Symbol] :> Simp[c Int[(b*Sin[e + f*x])^m, x], x] + Simp[d/b Int [(b*Sin[e + f*x])^(m + 1), x], x] /; FreeQ[{b, c, d, e, f, m}, x]
\[\int \left (\cos \left (d x +c \right ) b \right )^{n} \left (A +B \cos \left (d x +c \right )\right ) \left (\sec ^{3}\left (d x +c \right )\right )d x\]
\[ \int (b \cos (c+d x))^n (A+B \cos (c+d x)) \sec ^3(c+d x) \, dx=\int { {\left (B \cos \left (d x + c\right ) + A\right )} \left (b \cos \left (d x + c\right )\right )^{n} \sec \left (d x + c\right )^{3} \,d x } \]
Timed out. \[ \int (b \cos (c+d x))^n (A+B \cos (c+d x)) \sec ^3(c+d x) \, dx=\text {Timed out} \]
\[ \int (b \cos (c+d x))^n (A+B \cos (c+d x)) \sec ^3(c+d x) \, dx=\int { {\left (B \cos \left (d x + c\right ) + A\right )} \left (b \cos \left (d x + c\right )\right )^{n} \sec \left (d x + c\right )^{3} \,d x } \]
\[ \int (b \cos (c+d x))^n (A+B \cos (c+d x)) \sec ^3(c+d x) \, dx=\int { {\left (B \cos \left (d x + c\right ) + A\right )} \left (b \cos \left (d x + c\right )\right )^{n} \sec \left (d x + c\right )^{3} \,d x } \]
Timed out. \[ \int (b \cos (c+d x))^n (A+B \cos (c+d x)) \sec ^3(c+d x) \, dx=\int \frac {{\left (b\,\cos \left (c+d\,x\right )\right )}^n\,\left (A+B\,\cos \left (c+d\,x\right )\right )}{{\cos \left (c+d\,x\right )}^3} \,d x \]