Integrand size = 21, antiderivative size = 132 \[ \int (d \cos (e+f x))^n (a+a \sec (e+f x)) \, dx=-\frac {a (d \cos (e+f x))^n \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {n}{2},\frac {2+n}{2},\cos ^2(e+f x)\right ) \sin (e+f x)}{f n \sqrt {\sin ^2(e+f x)}}-\frac {a (d \cos (e+f x))^{1+n} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {1+n}{2},\frac {3+n}{2},\cos ^2(e+f x)\right ) \sin (e+f x)}{d f (1+n) \sqrt {\sin ^2(e+f x)}} \]
-a*(d*cos(f*x+e))^n*hypergeom([1/2, 1/2*n],[1+1/2*n],cos(f*x+e)^2)*sin(f*x +e)/f/n/(sin(f*x+e)^2)^(1/2)-a*(d*cos(f*x+e))^(1+n)*hypergeom([1/2, 1/2+1/ 2*n],[3/2+1/2*n],cos(f*x+e)^2)*sin(f*x+e)/d/f/(1+n)/(sin(f*x+e)^2)^(1/2)
Time = 0.12 (sec) , antiderivative size = 105, normalized size of antiderivative = 0.80 \[ \int (d \cos (e+f x))^n (a+a \sec (e+f x)) \, dx=-\frac {a (d \cos (e+f x))^n \left ((1+n) \csc (e+f x) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {n}{2},\frac {2+n}{2},\cos ^2(e+f x)\right )+n \cot (e+f x) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {1+n}{2},\frac {3+n}{2},\cos ^2(e+f x)\right )\right ) \sqrt {\sin ^2(e+f x)}}{f n (1+n)} \]
-((a*(d*Cos[e + f*x])^n*((1 + n)*Csc[e + f*x]*Hypergeometric2F1[1/2, n/2, (2 + n)/2, Cos[e + f*x]^2] + n*Cot[e + f*x]*Hypergeometric2F1[1/2, (1 + n) /2, (3 + n)/2, Cos[e + f*x]^2])*Sqrt[Sin[e + f*x]^2])/(f*n*(1 + n)))
Time = 0.44 (sec) , antiderivative size = 137, normalized size of antiderivative = 1.04, number of steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {3042, 4713, 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.
\(\displaystyle \int (a \sec (e+f x)+a) (d \cos (e+f x))^n \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \left (a \csc \left (e+f x+\frac {\pi }{2}\right )+a\right ) \left (d \sin \left (e+f x+\frac {\pi }{2}\right )\right )^ndx\) |
\(\Big \downarrow \) 4713 |
\(\displaystyle \int \sec (e+f x) (a \cos (e+f x)+a) (d \cos (e+f x))^ndx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {\left (a \sin \left (e+f x+\frac {\pi }{2}\right )+a\right ) \left (d \sin \left (e+f x+\frac {\pi }{2}\right )\right )^n}{\sin \left (e+f x+\frac {\pi }{2}\right )}dx\) |
\(\Big \downarrow \) 2030 |
\(\displaystyle d \int \left (d \sin \left (\frac {1}{2} (2 e+\pi )+f x\right )\right )^{n-1} \left (\sin \left (\frac {1}{2} (2 e+\pi )+f x\right ) a+a\right )dx\) |
\(\Big \downarrow \) 3227 |
\(\displaystyle d \left (a \int (d \cos (e+f x))^{n-1}dx+\frac {a \int (d \cos (e+f x))^ndx}{d}\right )\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle d \left (a \int \left (d \sin \left (e+f x+\frac {\pi }{2}\right )\right )^{n-1}dx+\frac {a \int \left (d \sin \left (e+f x+\frac {\pi }{2}\right )\right )^ndx}{d}\right )\) |
\(\Big \downarrow \) 3122 |
\(\displaystyle d \left (-\frac {a \sin (e+f x) (d \cos (e+f x))^{n+1} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {n+1}{2},\frac {n+3}{2},\cos ^2(e+f x)\right )}{d^2 f (n+1) \sqrt {\sin ^2(e+f x)}}-\frac {a \sin (e+f x) (d \cos (e+f x))^n \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {n}{2},\frac {n+2}{2},\cos ^2(e+f x)\right )}{d f n \sqrt {\sin ^2(e+f x)}}\right )\) |
d*(-((a*(d*Cos[e + f*x])^n*Hypergeometric2F1[1/2, n/2, (2 + n)/2, Cos[e + f*x]^2]*Sin[e + f*x])/(d*f*n*Sqrt[Sin[e + f*x]^2])) - (a*(d*Cos[e + f*x])^ (1 + n)*Hypergeometric2F1[1/2, (1 + n)/2, (3 + n)/2, Cos[e + f*x]^2]*Sin[e + f*x])/(d^2*f*(1 + n)*Sqrt[Sin[e + f*x]^2]))
3.5.43.3.1 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[(csc[(a_.) + (b_.)*(x_)]*(B_.) + (A_))*(u_), x_Symbol] :> Int[ActivateT rig[u]*((B + A*Sin[a + b*x])/Sin[a + b*x]), x] /; FreeQ[{a, b, A, B}, x] && KnownSineIntegrandQ[u, x]
\[\int \left (d \cos \left (f x +e \right )\right )^{n} \left (a +a \sec \left (f x +e \right )\right )d x\]
\[ \int (d \cos (e+f x))^n (a+a \sec (e+f x)) \, dx=\int { {\left (a \sec \left (f x + e\right ) + a\right )} \left (d \cos \left (f x + e\right )\right )^{n} \,d x } \]
\[ \int (d \cos (e+f x))^n (a+a \sec (e+f x)) \, dx=a \left (\int \left (d \cos {\left (e + f x \right )}\right )^{n}\, dx + \int \left (d \cos {\left (e + f x \right )}\right )^{n} \sec {\left (e + f x \right )}\, dx\right ) \]
\[ \int (d \cos (e+f x))^n (a+a \sec (e+f x)) \, dx=\int { {\left (a \sec \left (f x + e\right ) + a\right )} \left (d \cos \left (f x + e\right )\right )^{n} \,d x } \]
\[ \int (d \cos (e+f x))^n (a+a \sec (e+f x)) \, dx=\int { {\left (a \sec \left (f x + e\right ) + a\right )} \left (d \cos \left (f x + e\right )\right )^{n} \,d x } \]
Timed out. \[ \int (d \cos (e+f x))^n (a+a \sec (e+f x)) \, dx=\int {\left (d\,\cos \left (e+f\,x\right )\right )}^n\,\left (a+\frac {a}{\cos \left (e+f\,x\right )}\right ) \,d x \]