\(\int (e x)^{-1+n} (a+b \cosh (c+d x^n))^p \, dx\) [45]

Optimal result

Integrand size = 22, antiderivative size = 131 \[ \int (e x)^{-1+n} \left (a+b \cosh \left (c+d x^n\right )\right )^p \, dx=\frac {\sqrt {2} x^{-n} (e x)^n \operatorname {AppellF1}\left (\frac {1}{2},\frac {1}{2},-p,\frac {3}{2},\frac {1}{2} \left (1-\cosh \left (c+d x^n\right )\right ),\frac {b \left (1-\cosh \left (c+d x^n\right )\right )}{a+b}\right ) \left (a+b \cosh \left (c+d x^n\right )\right )^p \left (\frac {a+b \cosh \left (c+d x^n\right )}{a+b}\right )^{-p} \sinh \left (c+d x^n\right )}{d e n \sqrt {1+\cosh \left (c+d x^n\right )}} \]

[Out]

(e*x)^n*AppellF1(1/2,-p,1/2,3/2,b*(1-cosh(c+d*x^n))/(a+b),1/2-1/2*cosh(c+d*x^n))*(a+b*cosh(c+d*x^n))^p*sinh(c+
d*x^n)*2^(1/2)/d/e/n/(x^n)/(((a+b*cosh(c+d*x^n))/(a+b))^p)/(1+cosh(c+d*x^n))^(1/2)

Rubi [A] (verified)

Time = 0.14 (sec) , antiderivative size = 131, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.227, Rules used = {5431, 5429, 2744, 144, 143} \[ \int (e x)^{-1+n} \left (a+b \cosh \left (c+d x^n\right )\right )^p \, dx=\frac {\sqrt {2} x^{-n} (e x)^n \sinh \left (c+d x^n\right ) \left (a+b \cosh \left (c+d x^n\right )\right )^p \left (\frac {a+b \cosh \left (c+d x^n\right )}{a+b}\right )^{-p} \operatorname {AppellF1}\left (\frac {1}{2},\frac {1}{2},-p,\frac {3}{2},\frac {1}{2} \left (1-\cosh \left (d x^n+c\right )\right ),\frac {b \left (1-\cosh \left (d x^n+c\right )\right )}{a+b}\right )}{d e n \sqrt {\cosh \left (c+d x^n\right )+1}} \]

[In]

Int[(e*x)^(-1 + n)*(a + b*Cosh[c + d*x^n])^p,x]

[Out]

(Sqrt[2]*(e*x)^n*AppellF1[1/2, 1/2, -p, 3/2, (1 - Cosh[c + d*x^n])/2, (b*(1 - Cosh[c + d*x^n]))/(a + b)]*(a +
b*Cosh[c + d*x^n])^p*Sinh[c + d*x^n])/(d*e*n*x^n*Sqrt[1 + Cosh[c + d*x^n]]*((a + b*Cosh[c + d*x^n])/(a + b))^p
)

Rule 143

Int[((a_) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_))^(p_), x_Symbol] :> Simp[((a + b*x)
^(m + 1)/(b*(m + 1)*(b/(b*c - a*d))^n*(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] &&  !Inte
gerQ[n] &&  !IntegerQ[p] && GtQ[b/(b*c - a*d), 0] && GtQ[b/(b*e - a*f), 0] &&  !(GtQ[d/(d*a - c*b), 0] && GtQ[
d/(d*e - c*f), 0] && SimplerQ[c + d*x, a + b*x]) &&  !(GtQ[f/(f*a - e*b), 0] && GtQ[f/(f*c - e*d), 0] && Simpl
erQ[e + f*x, a + b*x])

Rule 144

Int[((a_) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_))^(p_), x_Symbol] :> Dist[(e + f*x)^
FracPart[p]/((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*
(b*(e/(b*e - a*f)) + b*f*(x/(b*e - a*f)))^p, x], x] /; FreeQ[{a, b, c, d, e, f, m, n, p}, x] &&  !IntegerQ[m]
&&  !IntegerQ[n] &&  !IntegerQ[p] && GtQ[b/(b*c - a*d), 0] &&  !GtQ[b/(b*e - a*f), 0]

Rule 2744

Int[((a_) + (b_.)*sin[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Dist[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]

Rule 5429

Int[((a_.) + Cosh[(c_.) + (d_.)*(x_)^(n_)]*(b_.))^(p_.)*(x_)^(m_.), x_Symbol] :> Dist[1/n, Subst[Int[x^(Simpli
fy[(m + 1)/n] - 1)*(a + b*Cosh[c + d*x])^p, x], x, x^n], x] /; FreeQ[{a, b, c, d, m, n, p}, x] && IntegerQ[Sim
plify[(m + 1)/n]] && (EqQ[p, 1] || EqQ[m, n - 1] || (IntegerQ[p] && GtQ[Simplify[(m + 1)/n], 0]))

Rule 5431

Int[((a_.) + Cosh[(c_.) + (d_.)*(x_)^(n_)]*(b_.))^(p_.)*((e_)*(x_))^(m_), x_Symbol] :> Dist[e^IntPart[m]*((e*x
)^FracPart[m]/x^FracPart[m]), Int[x^m*(a + b*Cosh[c + d*x^n])^p, x], x] /; FreeQ[{a, b, c, d, e, m, n, p}, x]
&& IntegerQ[Simplify[(m + 1)/n]]

Rubi steps \begin{align*} \text {integral}& = \frac {\left (x^{-n} (e x)^n\right ) \int x^{-1+n} \left (a+b \cosh \left (c+d x^n\right )\right )^p \, dx}{e} \\ & = \frac {\left (x^{-n} (e x)^n\right ) \text {Subst}\left (\int (a+b \cosh (c+d x))^p \, dx,x,x^n\right )}{e n} \\ & = -\frac {\left (x^{-n} (e x)^n \sinh \left (c+d x^n\right )\right ) \text {Subst}\left (\int \frac {(a+b x)^p}{\sqrt {1-x} \sqrt {1+x}} \, dx,x,\cosh \left (c+d x^n\right )\right )}{d e n \sqrt {1-\cosh \left (c+d x^n\right )} \sqrt {1+\cosh \left (c+d x^n\right )}} \\ & = -\frac {\left (x^{-n} (e x)^n \left (a+b \cosh \left (c+d x^n\right )\right )^p \left (-\frac {a+b \cosh \left (c+d x^n\right )}{-a-b}\right )^{-p} \sinh \left (c+d x^n\right )\right ) \text {Subst}\left (\int \frac {\left (-\frac {a}{-a-b}-\frac {b x}{-a-b}\right )^p}{\sqrt {1-x} \sqrt {1+x}} \, dx,x,\cosh \left (c+d x^n\right )\right )}{d e n \sqrt {1-\cosh \left (c+d x^n\right )} \sqrt {1+\cosh \left (c+d x^n\right )}} \\ & = \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-\cosh \left (c+d x^n\right )\right ),\frac {b \left (1-\cosh \left (c+d x^n\right )\right )}{a+b}\right ) \left (a+b \cosh \left (c+d x^n\right )\right )^p \left (\frac {a+b \cosh \left (c+d x^n\right )}{a+b}\right )^{-p} \sinh \left (c+d x^n\right )}{d e n \sqrt {1+\cosh \left (c+d x^n\right )}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.37 (sec) , antiderivative size = 148, normalized size of antiderivative = 1.13 \[ \int (e x)^{-1+n} \left (a+b \cosh \left (c+d x^n\right )\right )^p \, dx=\frac {x^{-n} (e x)^n \operatorname {AppellF1}\left (1+p,\frac {1}{2},\frac {1}{2},2+p,\frac {a+b \cosh \left (c+d x^n\right )}{a+b},\frac {a+b \cosh \left (c+d x^n\right )}{a-b}\right ) \sqrt {-\frac {b \left (-1+\cosh \left (c+d x^n\right )\right )}{a+b}} \sqrt {\frac {b \left (1+\cosh \left (c+d x^n\right )\right )}{-a+b}} \left (a+b \cosh \left (c+d x^n\right )\right )^{1+p} \text {csch}\left (c+d x^n\right )}{b d e n (1+p)} \]

[In]

Integrate[(e*x)^(-1 + n)*(a + b*Cosh[c + d*x^n])^p,x]

[Out]

((e*x)^n*AppellF1[1 + p, 1/2, 1/2, 2 + p, (a + b*Cosh[c + d*x^n])/(a + b), (a + b*Cosh[c + d*x^n])/(a - b)]*Sq
rt[-((b*(-1 + Cosh[c + d*x^n]))/(a + b))]*Sqrt[(b*(1 + Cosh[c + d*x^n]))/(-a + b)]*(a + b*Cosh[c + d*x^n])^(1
+ p)*Csch[c + d*x^n])/(b*d*e*n*(1 + p)*x^n)

Maple [F]

\[\int \left (e x \right )^{-1+n} {\left (a +b \cosh \left (c +d \,x^{n}\right )\right )}^{p}d x\]

[In]

int((e*x)^(-1+n)*(a+b*cosh(c+d*x^n))^p,x)

[Out]

int((e*x)^(-1+n)*(a+b*cosh(c+d*x^n))^p,x)

Fricas [F]

\[ \int (e x)^{-1+n} \left (a+b \cosh \left (c+d x^n\right )\right )^p \, dx=\int { \left (e x\right )^{n - 1} {\left (b \cosh \left (d x^{n} + c\right ) + a\right )}^{p} \,d x } \]

[In]

integrate((e*x)^(-1+n)*(a+b*cosh(c+d*x^n))^p,x, algorithm="fricas")

[Out]

integral((e*x)^(n - 1)*(b*cosh(d*x^n + c) + a)^p, x)

Sympy [F]

\[ \int (e x)^{-1+n} \left (a+b \cosh \left (c+d x^n\right )\right )^p \, dx=\int \left (e x\right )^{n - 1} \left (a + b \cosh {\left (c + d x^{n} \right )}\right )^{p}\, dx \]

[In]

integrate((e*x)**(-1+n)*(a+b*cosh(c+d*x**n))**p,x)

[Out]

Integral((e*x)**(n - 1)*(a + b*cosh(c + d*x**n))**p, x)

Maxima [F]

\[ \int (e x)^{-1+n} \left (a+b \cosh \left (c+d x^n\right )\right )^p \, dx=\int { \left (e x\right )^{n - 1} {\left (b \cosh \left (d x^{n} + c\right ) + a\right )}^{p} \,d x } \]

[In]

integrate((e*x)^(-1+n)*(a+b*cosh(c+d*x^n))^p,x, algorithm="maxima")

[Out]

integrate((e*x)^(n - 1)*(b*cosh(d*x^n + c) + a)^p, x)

Giac [F]

\[ \int (e x)^{-1+n} \left (a+b \cosh \left (c+d x^n\right )\right )^p \, dx=\int { \left (e x\right )^{n - 1} {\left (b \cosh \left (d x^{n} + c\right ) + a\right )}^{p} \,d x } \]

[In]

integrate((e*x)^(-1+n)*(a+b*cosh(c+d*x^n))^p,x, algorithm="giac")

[Out]

integrate((e*x)^(n - 1)*(b*cosh(d*x^n + c) + a)^p, x)

Mupad [F(-1)]

Timed out. \[ \int (e x)^{-1+n} \left (a+b \cosh \left (c+d x^n\right )\right )^p \, dx=\int {\left (e\,x\right )}^{n-1}\,{\left (a+b\,\mathrm {cosh}\left (c+d\,x^n\right )\right )}^p \,d x \]

[In]

int((e*x)^(n - 1)*(a + b*cosh(c + d*x^n))^p,x)

[Out]

int((e*x)^(n - 1)*(a + b*cosh(c + d*x^n))^p, x)