3.2.33 \(\int (e x)^{-1+n} (a+b \sin (c+d x^n))^p \, dx\) [133]

3.2.33.1 Optimal result

Integrand size = 22, antiderivative size = 132 \[ \int (e x)^{-1+n} \left (a+b \sin \left (c+d x^n\right )\right )^p \, dx=-\frac {\sqrt {2} x^{-n} (e x)^n \operatorname {AppellF1}\left (\frac {1}{2},\frac {1}{2},-p,\frac {3}{2},\frac {1}{2} \left (1-\sin \left (c+d x^n\right )\right ),\frac {b \left (1-\sin \left (c+d x^n\right )\right )}{a+b}\right ) \cos \left (c+d x^n\right ) \left (a+b \sin \left (c+d x^n\right )\right )^p \left (\frac {a+b \sin \left (c+d x^n\right )}{a+b}\right )^{-p}}{d e n \sqrt {1+\sin \left (c+d x^n\right )}} \]

output

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

3.2.33.2 Mathematica [A] (verified)

Time = 0.58 (sec) , antiderivative size = 148, normalized size of antiderivative = 1.12 \[ \int (e x)^{-1+n} \left (a+b \sin \left (c+d x^n\right )\right )^p \, dx=\frac {x^{-n} (e x)^n \operatorname {AppellF1}\left (1+p,\frac {1}{2},\frac {1}{2},2+p,\frac {a+b \sin \left (c+d x^n\right )}{a-b},\frac {a+b \sin \left (c+d x^n\right )}{a+b}\right ) \sec \left (c+d x^n\right ) \sqrt {-\frac {b \left (-1+\sin \left (c+d x^n\right )\right )}{a+b}} \sqrt {\frac {b \left (1+\sin \left (c+d x^n\right )\right )}{-a+b}} \left (a+b \sin \left (c+d x^n\right )\right )^{1+p}}{b d e n (1+p)} \]

input

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

output

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

3.2.33.3 Rubi [A] (verified)

Time = 0.42 (sec) , antiderivative size = 132, 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 = {3862, 3860, 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.

\(\displaystyle \int (e x)^{n-1} \left (a+b \sin \left (c+d x^n\right )\right )^p \, dx\)

\(\Big \downarrow \) 3862

\(\displaystyle \frac {x^{-n} (e x)^n \int x^{n-1} \left (a+b \sin \left (d x^n+c\right )\right )^pdx}{e}\)

\(\Big \downarrow \) 3860

\(\displaystyle \frac {x^{-n} (e x)^n \int \left (a+b \sin \left (d x^n+c\right )\right )^pdx^n}{e n}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {x^{-n} (e x)^n \int \left (a+b \sin \left (d x^n+c\right )\right )^pdx^n}{e n}\)

\(\Big \downarrow \) 3144

\(\displaystyle \frac {x^{-n} (e x)^n \cos \left (c+d x^n\right ) \int \frac {\left (a+b \sin \left (d x^n+c\right )\right )^p}{\sqrt {1-\sin \left (d x^n+c\right )} \sqrt {\sin \left (d x^n+c\right )+1}}d\sin \left (d x^n+c\right )}{d e n \sqrt {1-\sin \left (c+d x^n\right )} \sqrt {\sin \left (c+d x^n\right )+1}}\)

\(\Big \downarrow \) 156

\(\displaystyle \frac {x^{-n} (e x)^n \cos \left (c+d x^n\right ) \left (a+b \sin \left (c+d x^n\right )\right )^p \left (\frac {a+b \sin \left (c+d x^n\right )}{a+b}\right )^{-p} \int \frac {\left (\frac {a}{a+b}+\frac {b \sin \left (d x^n+c\right )}{a+b}\right )^p}{\sqrt {1-\sin \left (d x^n+c\right )} \sqrt {\sin \left (d x^n+c\right )+1}}d\sin \left (d x^n+c\right )}{d e n \sqrt {1-\sin \left (c+d x^n\right )} \sqrt {\sin \left (c+d x^n\right )+1}}\)

\(\Big \downarrow \) 155

\(\displaystyle -\frac {\sqrt {2} x^{-n} (e x)^n \cos \left (c+d x^n\right ) \left (a+b \sin \left (c+d x^n\right )\right )^p \left (\frac {a+b \sin \left (c+d x^n\right )}{a+b}\right )^{-p} \operatorname {AppellF1}\left (\frac {1}{2},\frac {1}{2},-p,\frac {3}{2},\frac {1}{2} \left (1-\sin \left (d x^n+c\right )\right ),\frac {b \left (1-\sin \left (d x^n+c\right )\right )}{a+b}\right )}{d e n \sqrt {\sin \left (c+d x^n\right )+1}}\)

input

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

output

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

3.2.33.3.1 Defintions of rubi rules used

rule 155

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])
 

rule 156

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]
 

rule 3042

Int[u_, x_Symbol] :> Int[DeactivateTrig[u, x], x] /; FunctionOfTrigOfLinear 
Q[u, x]
 

rule 3144

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]
 

rule 3860

Int[(x_)^(m_.)*((a_.) + (b_.)*Sin[(c_.) + (d_.)*(x_)^(n_)])^(p_.), x_Symbol 
] :> Simp[1/n   Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a + b*Sin[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]))
 

rule 3862

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

3.2.33.4 Maple [F]

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

input

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

output

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

3.2.33.5 Fricas [F]

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

input

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

output

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

3.2.33.6 Sympy [F]

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

input

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

output

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

3.2.33.7 Maxima [F]

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

input

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

output

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

3.2.33.8 Giac [F]

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

input

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

output

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

3.2.33.9 Mupad [F(-1)]

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

input

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

output

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