\(\int \sinh ^{\frac {5}{2}}(a+\frac {2 \log (c x^n)}{n}) \, dx\) [285]

Optimal result

Integrand size = 18, antiderivative size = 209 \[ \int \sinh ^{\frac {5}{2}}\left (a+\frac {2 \log \left (c x^n\right )}{n}\right ) \, dx=-\frac {1}{4} x \sinh ^{\frac {5}{2}}\left (a+\frac {2 \log \left (c x^n\right )}{n}\right )-\frac {5 e^{-2 a} x \left (c x^n\right )^{-4/n} \sinh ^{\frac {5}{2}}\left (a+\frac {2 \log \left (c x^n\right )}{n}\right )}{4 \left (1-e^{-2 a} \left (c x^n\right )^{-4/n}\right )^2}+\frac {5 x \sinh ^{\frac {5}{2}}\left (a+\frac {2 \log \left (c x^n\right )}{n}\right )}{12 \left (1-e^{-2 a} \left (c x^n\right )^{-4/n}\right )}-\frac {5 e^{-3 a} x \left (c x^n\right )^{-6/n} \csc ^{-1}\left (e^a \left (c x^n\right )^{2/n}\right ) \sinh ^{\frac {5}{2}}\left (a+\frac {2 \log \left (c x^n\right )}{n}\right )}{4 \left (1-e^{-2 a} \left (c x^n\right )^{-4/n}\right )^{5/2}} \]

[Out]

-1/4*x*sinh(a+2*ln(c*x^n)/n)^(5/2)-5/4*x*sinh(a+2*ln(c*x^n)/n)^(5/2)/exp(2*a)/((c*x^n)^(4/n))/(1-1/exp(2*a)/((
c*x^n)^(4/n)))^2+5/12*x*sinh(a+2*ln(c*x^n)/n)^(5/2)/(1-1/exp(2*a)/((c*x^n)^(4/n)))-5/4*x*arccsc(exp(a)*(c*x^n)
^(2/n))*sinh(a+2*ln(c*x^n)/n)^(5/2)/exp(3*a)/((c*x^n)^(6/n))/(1-1/exp(2*a)/((c*x^n)^(4/n)))^(5/2)

Rubi [A] (verified)

Time = 0.11 (sec) , antiderivative size = 209, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.444, Rules used = {5636, 5644, 360, 356, 352, 248, 283, 222} \[ \int \sinh ^{\frac {5}{2}}\left (a+\frac {2 \log \left (c x^n\right )}{n}\right ) \, dx=-\frac {5 e^{-2 a} x \left (c x^n\right )^{-4/n} \sinh ^{\frac {5}{2}}\left (a+\frac {2 \log \left (c x^n\right )}{n}\right )}{4 \left (1-e^{-2 a} \left (c x^n\right )^{-4/n}\right )^2}-\frac {1}{4} x \sinh ^{\frac {5}{2}}\left (a+\frac {2 \log \left (c x^n\right )}{n}\right )+\frac {5 x \sinh ^{\frac {5}{2}}\left (a+\frac {2 \log \left (c x^n\right )}{n}\right )}{12 \left (1-e^{-2 a} \left (c x^n\right )^{-4/n}\right )}-\frac {5 e^{-3 a} x \left (c x^n\right )^{-6/n} \csc ^{-1}\left (e^a \left (c x^n\right )^{2/n}\right ) \sinh ^{\frac {5}{2}}\left (a+\frac {2 \log \left (c x^n\right )}{n}\right )}{4 \left (1-e^{-2 a} \left (c x^n\right )^{-4/n}\right )^{5/2}} \]

[In]

Int[Sinh[a + (2*Log[c*x^n])/n]^(5/2),x]

[Out]

-1/4*(x*Sinh[a + (2*Log[c*x^n])/n]^(5/2)) - (5*x*Sinh[a + (2*Log[c*x^n])/n]^(5/2))/(4*E^(2*a)*(c*x^n)^(4/n)*(1
- 1/(E^(2*a)*(c*x^n)^(4/n)))^2) + (5*x*Sinh[a + (2*Log[c*x^n])/n]^(5/2))/(12*(1 - 1/(E^(2*a)*(c*x^n)^(4/n))))
- (5*x*ArcCsc[E^a*(c*x^n)^(2/n)]*Sinh[a + (2*Log[c*x^n])/n]^(5/2))/(4*E^(3*a)*(c*x^n)^(6/n)*(1 - 1/(E^(2*a)*(
c*x^n)^(4/n)))^(5/2))

Rule 222

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Simp[ArcSin[Rt[-b, 2]*(x/Sqrt[a])]/Rt[-b, 2], x] /; FreeQ[{a, b}
, x] && GtQ[a, 0] && NegQ[b]

Rule 248

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> -Subst[Int[(a + b/x^n)^p/x^2, x], x, 1/x] /; FreeQ[{a, b, p},
x] && ILtQ[n, 0]

Rule 283

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(c*x)^(m + 1)*((a + b*x^n)^p/(c*(m + 1
))), x] - Dist[b*n*(p/(c^n*(m + 1))), Int[(c*x)^(m + n)*(a + b*x^n)^(p - 1), x], x] /; FreeQ[{a, b, c}, x] &&
IGtQ[n, 0] && GtQ[p, 0] && LtQ[m, -1] &&  !ILtQ[(m + n*p + n + 1)/n, 0] && IntBinomialQ[a, b, c, n, m, p, x]

Rule 352

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[1/(m + 1), Subst[Int[(a + b*x^Simplify[n/(m +
1)])^p, x], x, x^(m + 1)], x] /; FreeQ[{a, b, m, n, p}, x] && IntegerQ[Simplify[n/(m + 1)]] &&  !IntegerQ[n]

Rule 356

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[x^(m + 1)*((a + b*x^n)^p/(m + 1)), x] - Dist[b
*n*(p/(m + 1)), Int[x^(m + n)*(a + b*x^n)^(p - 1), x], x] /; FreeQ[{a, b, m, n}, x] && EqQ[(m + 1)/n + p, 0] &
& GtQ[p, 0]

Rule 360

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(c*x)^(m + 1)*((a + b*x^n)^p/(c*(m + n
*p + 1))), x] + Dist[a*n*(p/(m + n*p + 1)), Int[(c*x)^m*(a + b*x^n)^(p - 1), x], x] /; FreeQ[{a, b, c, m, n},
x] && IntegerQ[p + Simplify[(m + 1)/n]] && GtQ[p, 0] && NeQ[m + n*p + 1, 0]

Rule 5636

Int[Sinh[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))*(d_.)]^(p_.), x_Symbol] :> Dist[x/(n*(c*x^n)^(1/n)), Subst[Int[
x^(1/n - 1)*Sinh[d*(a + b*Log[x])]^p, x], x, c*x^n], x] /; FreeQ[{a, b, c, d, n, p}, x] && (NeQ[c, 1] || NeQ[n
, 1])

Rule 5644

Int[((e_.)*(x_))^(m_.)*Sinh[((a_.) + Log[x_]*(b_.))*(d_.)]^(p_), x_Symbol] :> Dist[Sinh[d*(a + b*Log[x])]^p/(x
^(b*d*p)*(1 - 1/(E^(2*a*d)*x^(2*b*d)))^p), Int[(e*x)^m*x^(b*d*p)*(1 - 1/(E^(2*a*d)*x^(2*b*d)))^p, x], x] /; Fr
eeQ[{a, b, d, e, m, p}, x] &&  !IntegerQ[p]

Rubi steps \begin{align*} \text {integral}& = \frac {\left (x \left (c x^n\right )^{-1/n}\right ) \text {Subst}\left (\int x^{-1+\frac {1}{n}} \sinh ^{\frac {5}{2}}\left (a+\frac {2 \log (x)}{n}\right ) \, dx,x,c x^n\right )}{n} \\ & = \frac {\left (x \left (c x^n\right )^{-6/n} \sinh ^{\frac {5}{2}}\left (a+\frac {2 \log \left (c x^n\right )}{n}\right )\right ) \text {Subst}\left (\int x^{-1+\frac {6}{n}} \left (1-e^{-2 a} x^{-4/n}\right )^{5/2} \, dx,x,c x^n\right )}{n \left (1-e^{-2 a} \left (c x^n\right )^{-4/n}\right )^{5/2}} \\ & = -\frac {1}{4} x \sinh ^{\frac {5}{2}}\left (a+\frac {2 \log \left (c x^n\right )}{n}\right )+\frac {\left (5 x \left (c x^n\right )^{-6/n} \sinh ^{\frac {5}{2}}\left (a+\frac {2 \log \left (c x^n\right )}{n}\right )\right ) \text {Subst}\left (\int x^{-1+\frac {6}{n}} \left (1-e^{-2 a} x^{-4/n}\right )^{3/2} \, dx,x,c x^n\right )}{2 n \left (1-e^{-2 a} \left (c x^n\right )^{-4/n}\right )^{5/2}} \\ & = -\frac {1}{4} x \sinh ^{\frac {5}{2}}\left (a+\frac {2 \log \left (c x^n\right )}{n}\right )+\frac {5 x \sinh ^{\frac {5}{2}}\left (a+\frac {2 \log \left (c x^n\right )}{n}\right )}{12 \left (1-e^{-2 a} \left (c x^n\right )^{-4/n}\right )}-\frac {\left (5 e^{-2 a} x \left (c x^n\right )^{-6/n} \sinh ^{\frac {5}{2}}\left (a+\frac {2 \log \left (c x^n\right )}{n}\right )\right ) \text {Subst}\left (\int x^{-1+\frac {2}{n}} \sqrt {1-e^{-2 a} x^{-4/n}} \, dx,x,c x^n\right )}{2 n \left (1-e^{-2 a} \left (c x^n\right )^{-4/n}\right )^{5/2}} \\ & = -\frac {1}{4} x \sinh ^{\frac {5}{2}}\left (a+\frac {2 \log \left (c x^n\right )}{n}\right )+\frac {5 x \sinh ^{\frac {5}{2}}\left (a+\frac {2 \log \left (c x^n\right )}{n}\right )}{12 \left (1-e^{-2 a} \left (c x^n\right )^{-4/n}\right )}-\frac {\left (5 e^{-2 a} x \left (c x^n\right )^{-6/n} \sinh ^{\frac {5}{2}}\left (a+\frac {2 \log \left (c x^n\right )}{n}\right )\right ) \text {Subst}\left (\int \sqrt {1-\frac {e^{-2 a}}{x^2}} \, dx,x,\left (c x^n\right )^{2/n}\right )}{4 \left (1-e^{-2 a} \left (c x^n\right )^{-4/n}\right )^{5/2}} \\ & = -\frac {1}{4} x \sinh ^{\frac {5}{2}}\left (a+\frac {2 \log \left (c x^n\right )}{n}\right )+\frac {5 x \sinh ^{\frac {5}{2}}\left (a+\frac {2 \log \left (c x^n\right )}{n}\right )}{12 \left (1-e^{-2 a} \left (c x^n\right )^{-4/n}\right )}+\frac {\left (5 e^{-2 a} x \left (c x^n\right )^{-6/n} \sinh ^{\frac {5}{2}}\left (a+\frac {2 \log \left (c x^n\right )}{n}\right )\right ) \text {Subst}\left (\int \frac {\sqrt {1-e^{-2 a} x^2}}{x^2} \, dx,x,\left (c x^n\right )^{-2/n}\right )}{4 \left (1-e^{-2 a} \left (c x^n\right )^{-4/n}\right )^{5/2}} \\ & = -\frac {1}{4} x \sinh ^{\frac {5}{2}}\left (a+\frac {2 \log \left (c x^n\right )}{n}\right )-\frac {5 e^{-2 a} x \left (c x^n\right )^{-4/n} \sinh ^{\frac {5}{2}}\left (a+\frac {2 \log \left (c x^n\right )}{n}\right )}{4 \left (1-e^{-2 a} \left (c x^n\right )^{-4/n}\right )^2}+\frac {5 x \sinh ^{\frac {5}{2}}\left (a+\frac {2 \log \left (c x^n\right )}{n}\right )}{12 \left (1-e^{-2 a} \left (c x^n\right )^{-4/n}\right )}-\frac {\left (5 e^{-4 a} x \left (c x^n\right )^{-6/n} \sinh ^{\frac {5}{2}}\left (a+\frac {2 \log \left (c x^n\right )}{n}\right )\right ) \text {Subst}\left (\int \frac {1}{\sqrt {1-e^{-2 a} x^2}} \, dx,x,\left (c x^n\right )^{-2/n}\right )}{4 \left (1-e^{-2 a} \left (c x^n\right )^{-4/n}\right )^{5/2}} \\ & = -\frac {1}{4} x \sinh ^{\frac {5}{2}}\left (a+\frac {2 \log \left (c x^n\right )}{n}\right )-\frac {5 e^{-2 a} x \left (c x^n\right )^{-4/n} \sinh ^{\frac {5}{2}}\left (a+\frac {2 \log \left (c x^n\right )}{n}\right )}{4 \left (1-e^{-2 a} \left (c x^n\right )^{-4/n}\right )^2}+\frac {5 x \sinh ^{\frac {5}{2}}\left (a+\frac {2 \log \left (c x^n\right )}{n}\right )}{12 \left (1-e^{-2 a} \left (c x^n\right )^{-4/n}\right )}-\frac {5 e^{-3 a} x \left (c x^n\right )^{-6/n} \arcsin \left (e^{-a} \left (c x^n\right )^{-2/n}\right ) \sinh ^{\frac {5}{2}}\left (a+\frac {2 \log \left (c x^n\right )}{n}\right )}{4 \left (1-e^{-2 a} \left (c x^n\right )^{-4/n}\right )^{5/2}} \\ \end{align*}

Mathematica [C] (verified)

Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.

Time = 0.23 (sec) , antiderivative size = 86, normalized size of antiderivative = 0.41 \[ \int \sinh ^{\frac {5}{2}}\left (a+\frac {2 \log \left (c x^n\right )}{n}\right ) \, dx=\frac {1}{14} e^{2 a} x \left (c x^n\right )^{4/n} \left (-1+e^{2 a} \left (c x^n\right )^{4/n}\right ) \operatorname {Hypergeometric2F1}\left (2,\frac {7}{2},\frac {9}{2},1-e^{2 a} \left (c x^n\right )^{4/n}\right ) \sinh ^{\frac {5}{2}}\left (a+\frac {2 \log \left (c x^n\right )}{n}\right ) \]

[In]

Integrate[Sinh[a + (2*Log[c*x^n])/n]^(5/2),x]

[Out]

(E^(2*a)*x*(c*x^n)^(4/n)*(-1 + E^(2*a)*(c*x^n)^(4/n))*Hypergeometric2F1[2, 7/2, 9/2, 1 - E^(2*a)*(c*x^n)^(4/n)
]*Sinh[a + (2*Log[c*x^n])/n]^(5/2))/14

Maple [F]

\[\int {\sinh \left (a +\frac {2 \ln \left (c \,x^{n}\right )}{n}\right )}^{\frac {5}{2}}d x\]

[In]

int(sinh(a+2*ln(c*x^n)/n)^(5/2),x)

[Out]

int(sinh(a+2*ln(c*x^n)/n)^(5/2),x)

Fricas [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 162, normalized size of antiderivative = 0.78 \[ \int \sinh ^{\frac {5}{2}}\left (a+\frac {2 \log \left (c x^n\right )}{n}\right ) \, dx=\frac {{\left (15 \, \sqrt {2} x^{3} \arctan \left (\sqrt {2} \sqrt {\frac {1}{2}} x \sqrt {\frac {x^{4} e^{\left (\frac {2 \, {\left (a n + 2 \, \log \left (c\right )\right )}}{n}\right )} - 1}{x^{2}}}\right ) e^{\left (\frac {3 \, {\left (a n + 2 \, \log \left (c\right )\right )}}{2 \, n}\right )} + 2 \, \sqrt {\frac {1}{2}} {\left (2 \, x^{8} e^{\left (\frac {4 \, {\left (a n + 2 \, \log \left (c\right )\right )}}{n}\right )} - 14 \, x^{4} e^{\left (\frac {2 \, {\left (a n + 2 \, \log \left (c\right )\right )}}{n}\right )} - 3\right )} \sqrt {\frac {x^{4} e^{\left (\frac {2 \, {\left (a n + 2 \, \log \left (c\right )\right )}}{n}\right )} - 1}{x^{2}}} e^{\left (-\frac {a n + 2 \, \log \left (c\right )}{2 \, n}\right )}\right )} e^{\left (-\frac {2 \, {\left (a n + 2 \, \log \left (c\right )\right )}}{n}\right )}}{96 \, x^{3}} \]

[In]

integrate(sinh(a+2*log(c*x^n)/n)^(5/2),x, algorithm="fricas")

[Out]

1/96*(15*sqrt(2)*x^3*arctan(sqrt(2)*sqrt(1/2)*x*sqrt((x^4*e^(2*(a*n + 2*log(c))/n) - 1)/x^2))*e^(3/2*(a*n + 2*
log(c))/n) + 2*sqrt(1/2)*(2*x^8*e^(4*(a*n + 2*log(c))/n) - 14*x^4*e^(2*(a*n + 2*log(c))/n) - 3)*sqrt((x^4*e^(2
*(a*n + 2*log(c))/n) - 1)/x^2)*e^(-1/2*(a*n + 2*log(c))/n))*e^(-2*(a*n + 2*log(c))/n)/x^3

Sympy [F(-1)]

Timed out. \[ \int \sinh ^{\frac {5}{2}}\left (a+\frac {2 \log \left (c x^n\right )}{n}\right ) \, dx=\text {Timed out} \]

[In]

integrate(sinh(a+2*ln(c*x**n)/n)**(5/2),x)

[Out]

Timed out

Maxima [F]

\[ \int \sinh ^{\frac {5}{2}}\left (a+\frac {2 \log \left (c x^n\right )}{n}\right ) \, dx=\int { \sinh \left (a + \frac {2 \, \log \left (c x^{n}\right )}{n}\right )^{\frac {5}{2}} \,d x } \]

[In]

integrate(sinh(a+2*log(c*x^n)/n)^(5/2),x, algorithm="maxima")

[Out]

integrate(sinh(a + 2*log(c*x^n)/n)^(5/2), x)

Giac [A] (verification not implemented)

none

Time = 0.42 (sec) , antiderivative size = 203, normalized size of antiderivative = 0.97 \[ \int \sinh ^{\frac {5}{2}}\left (a+\frac {2 \log \left (c x^n\right )}{n}\right ) \, dx=\frac {1}{48} \, \sqrt {2} \sqrt {c^{\frac {6}{n}} x^{6} e^{\left (3 \, a\right )} - c^{\frac {2}{n}} x^{2} e^{a}} c^{\frac {2}{n}} x^{3} e^{a} + \frac {\sqrt {2} {\left (15 \, c^{\frac {8}{n}} \arctan \left (\sqrt {c^{\frac {4}{n}} x^{4} e^{\left (3 \, a\right )} - e^{a}} e^{\left (-\frac {1}{2} \, a\right )}\right ) e^{\left (\frac {9}{2} \, a\right )} - 14 \, \sqrt {c^{\frac {4}{n}} x^{4} e^{\left (3 \, a\right )} - e^{a}} c^{\frac {8}{n}} e^{\left (4 \, a\right )} - \frac {3 \, \sqrt {c^{\frac {4}{n}} x^{4} e^{\left (3 \, a\right )} - e^{a}} c^{\frac {8}{n}} e^{\left (2 \, a\right )}}{c^{\frac {4}{n}} x^{4}}\right )} e^{\left (-5 \, a\right )}}{96 \, c^{\frac {8}{n}} c^{\left (\frac {1}{n}\right )} \mathrm {sgn}\left (x\right )} \]

[In]

integrate(sinh(a+2*log(c*x^n)/n)^(5/2),x, algorithm="giac")

[Out]

1/48*sqrt(2)*sqrt(c^(6/n)*x^6*e^(3*a) - c^(2/n)*x^2*e^a)*c^(2/n)*x^3*e^a + 1/96*sqrt(2)*(15*c^(8/n)*arctan(sqr
t(c^(4/n)*x^4*e^(3*a) - e^a)*e^(-1/2*a))*e^(9/2*a) - 14*sqrt(c^(4/n)*x^4*e^(3*a) - e^a)*c^(8/n)*e^(4*a) - 3*sq
rt(c^(4/n)*x^4*e^(3*a) - e^a)*c^(8/n)*e^(2*a)/(c^(4/n)*x^4))*e^(-5*a)/(c^(8/n)*c^(1/n)*sgn(x))

Mupad [F(-1)]

Timed out. \[ \int \sinh ^{\frac {5}{2}}\left (a+\frac {2 \log \left (c x^n\right )}{n}\right ) \, dx=\int {\mathrm {sinh}\left (a+\frac {2\,\ln \left (c\,x^n\right )}{n}\right )}^{5/2} \,d x \]

[In]

int(sinh(a + (2*log(c*x^n))/n)^(5/2),x)

[Out]

int(sinh(a + (2*log(c*x^n))/n)^(5/2), x)