Integrand size = 22, antiderivative size = 131 \[ \int (e x)^{-1+n} \left (a+b \cos \left (c+d x^n\right )\right )^p \, dx=\frac {\sqrt {2} x^{-n} (e x)^n \operatorname {AppellF1}\left (\frac {1}{2},\frac {1}{2},-p,\frac {3}{2},\frac {1}{2} \left (1-\cos \left (c+d x^n\right )\right ),\frac {b \left (1-\cos \left (c+d x^n\right )\right )}{a+b}\right ) \left (a+b \cos \left (c+d x^n\right )\right )^p \left (\frac {a+b \cos \left (c+d x^n\right )}{a+b}\right )^{-p} \sin \left (c+d x^n\right )}{d e n \sqrt {1+\cos \left (c+d x^n\right )}} \]
(e*x)^n*AppellF1(1/2,-p,1/2,3/2,b*(1-cos(c+d*x^n))/(a+b),1/2-1/2*cos(c+d*x ^n))*(a+b*cos(c+d*x^n))^p*sin(c+d*x^n)*2^(1/2)/d/e/n/(x^n)/(((a+b*cos(c+d* x^n))/(a+b))^p)/(1+cos(c+d*x^n))^(1/2)
Time = 0.82 (sec) , antiderivative size = 149, normalized size of antiderivative = 1.14 \[ \int (e x)^{-1+n} \left (a+b \cos \left (c+d x^n\right )\right )^p \, dx=-\frac {x^{-n} (e x)^n \operatorname {AppellF1}\left (1+p,\frac {1}{2},\frac {1}{2},2+p,\frac {a+b \cos \left (c+d x^n\right )}{a-b},\frac {a+b \cos \left (c+d x^n\right )}{a+b}\right ) \sqrt {-\frac {b \left (-1+\cos \left (c+d x^n\right )\right )}{a+b}} \sqrt {\frac {b \left (1+\cos \left (c+d x^n\right )\right )}{-a+b}} \left (a+b \cos \left (c+d x^n\right )\right )^{1+p} \csc \left (c+d x^n\right )}{b d e n (1+p)} \]
-(((e*x)^n*AppellF1[1 + p, 1/2, 1/2, 2 + p, (a + b*Cos[c + d*x^n])/(a - b) , (a + b*Cos[c + d*x^n])/(a + b)]*Sqrt[-((b*(-1 + Cos[c + d*x^n]))/(a + b) )]*Sqrt[(b*(1 + Cos[c + d*x^n]))/(-a + b)]*(a + b*Cos[c + d*x^n])^(1 + p)* Csc[c + d*x^n])/(b*d*e*n*(1 + p)*x^n))
Time = 0.42 (sec) , antiderivative size = 131, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.273, Rules used = {3863, 3861, 3042, 3144, 156, 155}
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.
\(\displaystyle \int (e x)^{n-1} \left (a+b \cos \left (c+d x^n\right )\right )^p \, dx\) |
\(\Big \downarrow \) 3863 |
\(\displaystyle \frac {x^{-n} (e x)^n \int x^{n-1} \left (a+b \cos \left (d x^n+c\right )\right )^pdx}{e}\) |
\(\Big \downarrow \) 3861 |
\(\displaystyle \frac {x^{-n} (e x)^n \int \left (a+b \cos \left (d x^n+c\right )\right )^pdx^n}{e n}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {x^{-n} (e x)^n \int \left (a+b \sin \left (d x^n+c+\frac {\pi }{2}\right )\right )^pdx^n}{e n}\) |
\(\Big \downarrow \) 3144 |
\(\displaystyle -\frac {x^{-n} (e x)^n \sin \left (c+d x^n\right ) \int \frac {\left (a+b \cos \left (d x^n+c\right )\right )^p}{\sqrt {1-\cos \left (d x^n+c\right )} \sqrt {\cos \left (d x^n+c\right )+1}}d\cos \left (d x^n+c\right )}{d e n \sqrt {1-\cos \left (c+d x^n\right )} \sqrt {\cos \left (c+d x^n\right )+1}}\) |
\(\Big \downarrow \) 156 |
\(\displaystyle -\frac {x^{-n} (e x)^n \sin \left (c+d x^n\right ) \left (a+b \cos \left (c+d x^n\right )\right )^p \left (\frac {a+b \cos \left (c+d x^n\right )}{a+b}\right )^{-p} \int \frac {\left (\frac {a}{a+b}+\frac {b \cos \left (d x^n+c\right )}{a+b}\right )^p}{\sqrt {1-\cos \left (d x^n+c\right )} \sqrt {\cos \left (d x^n+c\right )+1}}d\cos \left (d x^n+c\right )}{d e n \sqrt {1-\cos \left (c+d x^n\right )} \sqrt {\cos \left (c+d x^n\right )+1}}\) |
\(\Big \downarrow \) 155 |
\(\displaystyle \frac {\sqrt {2} x^{-n} (e x)^n \sin \left (c+d x^n\right ) \left (a+b \cos \left (c+d x^n\right )\right )^p \left (\frac {a+b \cos \left (c+d x^n\right )}{a+b}\right )^{-p} \operatorname {AppellF1}\left (\frac {1}{2},\frac {1}{2},-p,\frac {3}{2},\frac {1}{2} \left (1-\cos \left (d x^n+c\right )\right ),\frac {b \left (1-\cos \left (d x^n+c\right )\right )}{a+b}\right )}{d e n \sqrt {\cos \left (c+d x^n\right )+1}}\) |
(Sqrt[2]*(e*x)^n*AppellF1[1/2, 1/2, -p, 3/2, (1 - Cos[c + d*x^n])/2, (b*(1 - Cos[c + d*x^n]))/(a + b)]*(a + b*Cos[c + d*x^n])^p*Sin[c + d*x^n])/(d*e *n*x^n*Sqrt[1 + Cos[c + d*x^n]]*((a + b*Cos[c + d*x^n])/(a + b))^p)
3.1.67.3.1 Defintions of rubi rules used
Int[((a_) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_)) ^(p_), x_] :> Simp[((a + b*x)^(m + 1)/(b*(m + 1)*Simplify[b/(b*c - a*d)]^n* Simplify[b/(b*e - a*f)]^p))*AppellF1[m + 1, -n, -p, m + 2, (-d)*((a + b*x)/ (b*c - a*d)), (-f)*((a + b*x)/(b*e - a*f))], x] /; FreeQ[{a, b, c, d, e, f, m, n, p}, x] && !IntegerQ[m] && !IntegerQ[n] && !IntegerQ[p] && GtQ[Sim plify[b/(b*c - a*d)], 0] && GtQ[Simplify[b/(b*e - a*f)], 0] && !(GtQ[Simpl ify[d/(d*a - c*b)], 0] && GtQ[Simplify[d/(d*e - c*f)], 0] && SimplerQ[c + d *x, a + b*x]) && !(GtQ[Simplify[f/(f*a - e*b)], 0] && GtQ[Simplify[f/(f*c - e*d)], 0] && SimplerQ[e + f*x, a + b*x])
Int[((a_) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_)) ^(p_), x_] :> Simp[(e + f*x)^FracPart[p]/(Simplify[b/(b*e - a*f)]^IntPart[p ]*(b*((e + f*x)/(b*e - a*f)))^FracPart[p]) Int[(a + b*x)^m*(c + d*x)^n*Si mp[b*(e/(b*e - a*f)) + b*f*(x/(b*e - a*f)), x]^p, x], x] /; FreeQ[{a, b, c, d, e, f, m, n, p}, x] && !IntegerQ[m] && !IntegerQ[n] && !IntegerQ[p] & & GtQ[Simplify[b/(b*c - a*d)], 0] && !GtQ[Simplify[b/(b*e - a*f)], 0]
Int[((a_) + (b_.)*sin[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[Cos[c + d*x]/(d*Sqrt[1 + Sin[c + d*x]]*Sqrt[1 - Sin[c + d*x]]) Subst[Int[(a + b*x )^n/(Sqrt[1 + x]*Sqrt[1 - x]), x], x, Sin[c + d*x]], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[a^2 - b^2, 0] && !IntegerQ[2*n]
Int[((a_.) + Cos[(c_.) + (d_.)*(x_)^(n_)]*(b_.))^(p_.)*(x_)^(m_.), x_Symbol ] :> Simp[1/n Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a + b*Cos[c + d*x])^ p, x], x, x^n], x] /; FreeQ[{a, b, c, d, m, n, p}, x] && IntegerQ[Simplify[ (m + 1)/n]] && (EqQ[p, 1] || EqQ[m, n - 1] || (IntegerQ[p] && GtQ[Simplify[ (m + 1)/n], 0]))
Int[((a_.) + Cos[(c_.) + (d_.)*(x_)^(n_)]*(b_.))^(p_.)*((e_)*(x_))^(m_), x_ Symbol] :> Simp[e^IntPart[m]*((e*x)^FracPart[m]/x^FracPart[m]) Int[x^m*(a + b*Cos[c + d*x^n])^p, x], x] /; FreeQ[{a, b, c, d, e, m, n, p}, x] && Int egerQ[Simplify[(m + 1)/n]]
\[\int \left (e x \right )^{-1+n} {\left (a +b \cos \left (c +d \,x^{n}\right )\right )}^{p}d x\]
\[ \int (e x)^{-1+n} \left (a+b \cos \left (c+d x^n\right )\right )^p \, dx=\int { \left (e x\right )^{n - 1} {\left (b \cos \left (d x^{n} + c\right ) + a\right )}^{p} \,d x } \]
\[ \int (e x)^{-1+n} \left (a+b \cos \left (c+d x^n\right )\right )^p \, dx=\int \left (e x\right )^{n - 1} \left (a + b \cos {\left (c + d x^{n} \right )}\right )^{p}\, dx \]
\[ \int (e x)^{-1+n} \left (a+b \cos \left (c+d x^n\right )\right )^p \, dx=\int { \left (e x\right )^{n - 1} {\left (b \cos \left (d x^{n} + c\right ) + a\right )}^{p} \,d x } \]
\[ \int (e x)^{-1+n} \left (a+b \cos \left (c+d x^n\right )\right )^p \, dx=\int { \left (e x\right )^{n - 1} {\left (b \cos \left (d x^{n} + c\right ) + a\right )}^{p} \,d x } \]
Timed out. \[ \int (e x)^{-1+n} \left (a+b \cos \left (c+d x^n\right )\right )^p \, dx=\int {\left (e\,x\right )}^{n-1}\,{\left (a+b\,\cos \left (c+d\,x^n\right )\right )}^p \,d x \]