Integrand size = 18, antiderivative size = 209 \[ \int \sinh ^{\frac {5}{2}}\left (a+\frac {2 \log \left (c x^n\right )}{n}\right ) \, dx=-\frac {1}{4} x \sinh ^{\frac {5}{2}}\left (a+\frac {2 \log \left (c x^n\right )}{n}\right )-\frac {5 e^{-2 a} x \left (c x^n\right )^{-4/n} \sinh ^{\frac {5}{2}}\left (a+\frac {2 \log \left (c x^n\right )}{n}\right )}{4 \left (1-e^{-2 a} \left (c x^n\right )^{-4/n}\right )^2}+\frac {5 x \sinh ^{\frac {5}{2}}\left (a+\frac {2 \log \left (c x^n\right )}{n}\right )}{12 \left (1-e^{-2 a} \left (c x^n\right )^{-4/n}\right )}-\frac {5 e^{-3 a} x \left (c x^n\right )^{-6/n} \csc ^{-1}\left (e^a \left (c x^n\right )^{2/n}\right ) \sinh ^{\frac {5}{2}}\left (a+\frac {2 \log \left (c x^n\right )}{n}\right )}{4 \left (1-e^{-2 a} \left (c x^n\right )^{-4/n}\right )^{5/2}} \] Output:
-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*x*sinh(a+2*ln(c*x^n )/n)^(5/2)/(12-12/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)
Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.
Time = 0.21 (sec) , antiderivative size = 86, normalized size of antiderivative = 0.41 \[ \int \sinh ^{\frac {5}{2}}\left (a+\frac {2 \log \left (c x^n\right )}{n}\right ) \, dx=\frac {1}{14} e^{2 a} x \left (c x^n\right )^{4/n} \left (-1+e^{2 a} \left (c x^n\right )^{4/n}\right ) \operatorname {Hypergeometric2F1}\left (2,\frac {7}{2},\frac {9}{2},1-e^{2 a} \left (c x^n\right )^{4/n}\right ) \sinh ^{\frac {5}{2}}\left (a+\frac {2 \log \left (c x^n\right )}{n}\right ) \] Input:
Integrate[Sinh[a + (2*Log[c*x^n])/n]^(5/2),x]
Output:
(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
Time = 0.52 (sec) , antiderivative size = 208, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.444, Rules used = {6051, 6059, 876, 872, 868, 773, 247, 223}
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.
| \(\displaystyle \int \sinh ^{\frac {5}{2}}\left (a+\frac {2 \log \left (c x^n\right )}{n}\right ) \, dx\) |
| \(\Big \downarrow \) 6051 |
| \(\displaystyle \frac {x \left (c x^n\right )^{-1/n} \int \left (c x^n\right )^{\frac {1}{n}-1} \sinh ^{\frac {5}{2}}\left (a+\frac {2 \log \left (c x^n\right )}{n}\right )d\left (c x^n\right )}{n}\) |
| \(\Big \downarrow \) 6059 |
| \(\displaystyle \frac {x \left (c x^n\right )^{-6/n} \sinh ^{\frac {5}{2}}\left (a+\frac {2 \log \left (c x^n\right )}{n}\right ) \int \left (c x^n\right )^{\frac {6}{n}-1} \left (1-e^{-2 a} \left (c x^n\right )^{-4/n}\right )^{5/2}d\left (c x^n\right )}{n \left (1-e^{-2 a} \left (c x^n\right )^{-4/n}\right )^{5/2}}\) |
| \(\Big \downarrow \) 876 |
| \(\displaystyle \frac {x \left (c x^n\right )^{-6/n} \sinh ^{\frac {5}{2}}\left (a+\frac {2 \log \left (c x^n\right )}{n}\right ) \left (\frac {5}{2} \int \left (c x^n\right )^{\frac {6}{n}-1} \left (1-e^{-2 a} \left (c x^n\right )^{-4/n}\right )^{3/2}d\left (c x^n\right )-\frac {1}{4} n \left (c x^n\right )^{6/n} \left (1-e^{-2 a} \left (c x^n\right )^{-4/n}\right )^{5/2}\right )}{n \left (1-e^{-2 a} \left (c x^n\right )^{-4/n}\right )^{5/2}}\) |
| \(\Big \downarrow \) 872 |
| \(\displaystyle \frac {x \left (c x^n\right )^{-6/n} \sinh ^{\frac {5}{2}}\left (a+\frac {2 \log \left (c x^n\right )}{n}\right ) \left (\frac {5}{2} \left (\frac {1}{6} n \left (c x^n\right )^{6/n} \left (1-e^{-2 a} \left (c x^n\right )^{-4/n}\right )^{3/2}-e^{-2 a} \int \left (c x^n\right )^{\frac {2}{n}-1} \sqrt {1-e^{-2 a} \left (c x^n\right )^{-4/n}}d\left (c x^n\right )\right )-\frac {1}{4} n \left (c x^n\right )^{6/n} \left (1-e^{-2 a} \left (c x^n\right )^{-4/n}\right )^{5/2}\right )}{n \left (1-e^{-2 a} \left (c x^n\right )^{-4/n}\right )^{5/2}}\) |
| \(\Big \downarrow \) 868 |
| \(\displaystyle \frac {x \left (c x^n\right )^{-6/n} \sinh ^{\frac {5}{2}}\left (a+\frac {2 \log \left (c x^n\right )}{n}\right ) \left (\frac {5}{2} \left (\frac {1}{6} n \left (c x^n\right )^{6/n} \left (1-e^{-2 a} \left (c x^n\right )^{-4/n}\right )^{3/2}-\frac {1}{2} e^{-2 a} n \int \sqrt {1-\frac {e^{-2 a} x^{-2 n}}{c^2}}d\left (c x^n\right )^{2/n}\right )-\frac {1}{4} n \left (c x^n\right )^{6/n} \left (1-e^{-2 a} \left (c x^n\right )^{-4/n}\right )^{5/2}\right )}{n \left (1-e^{-2 a} \left (c x^n\right )^{-4/n}\right )^{5/2}}\) |
| \(\Big \downarrow \) 773 |
| \(\displaystyle \frac {x \left (c x^n\right )^{-6/n} \sinh ^{\frac {5}{2}}\left (a+\frac {2 \log \left (c x^n\right )}{n}\right ) \left (\frac {5}{2} \left (\frac {1}{2} e^{-2 a} n \int \frac {x^{-2 n} \sqrt {1-c^2 e^{-2 a} x^{2 n}}}{c^2}d\frac {x^{-n}}{c}+\frac {1}{6} n \left (c x^n\right )^{6/n} \left (1-e^{-2 a} \left (c x^n\right )^{-4/n}\right )^{3/2}\right )-\frac {1}{4} n \left (c x^n\right )^{6/n} \left (1-e^{-2 a} \left (c x^n\right )^{-4/n}\right )^{5/2}\right )}{n \left (1-e^{-2 a} \left (c x^n\right )^{-4/n}\right )^{5/2}}\) |
| \(\Big \downarrow \) 247 |
| \(\displaystyle \frac {x \left (c x^n\right )^{-6/n} \sinh ^{\frac {5}{2}}\left (a+\frac {2 \log \left (c x^n\right )}{n}\right ) \left (\frac {5}{2} \left (\frac {1}{2} e^{-2 a} n \left (-e^{-2 a} \int \frac {1}{\sqrt {1-c^2 e^{-2 a} x^{2 n}}}d\frac {x^{-n}}{c}-\frac {x^{-n} \sqrt {1-e^{-2 a} c^2 x^{2 n}}}{c}\right )+\frac {1}{6} n \left (c x^n\right )^{6/n} \left (1-e^{-2 a} \left (c x^n\right )^{-4/n}\right )^{3/2}\right )-\frac {1}{4} n \left (c x^n\right )^{6/n} \left (1-e^{-2 a} \left (c x^n\right )^{-4/n}\right )^{5/2}\right )}{n \left (1-e^{-2 a} \left (c x^n\right )^{-4/n}\right )^{5/2}}\) |
| \(\Big \downarrow \) 223 |
| \(\displaystyle \frac {x \left (c x^n\right )^{-6/n} \left (\frac {5}{2} \left (\frac {1}{2} e^{-2 a} n \left (-e^{-a} \arcsin \left (\frac {e^{-a} x^{-n}}{c}\right )-\frac {x^{-n} \sqrt {1-e^{-2 a} c^2 x^{2 n}}}{c}\right )+\frac {1}{6} n \left (c x^n\right )^{6/n} \left (1-e^{-2 a} \left (c x^n\right )^{-4/n}\right )^{3/2}\right )-\frac {1}{4} n \left (c x^n\right )^{6/n} \left (1-e^{-2 a} \left (c x^n\right )^{-4/n}\right )^{5/2}\right ) \sinh ^{\frac {5}{2}}\left (a+\frac {2 \log \left (c x^n\right )}{n}\right )}{n \left (1-e^{-2 a} \left (c x^n\right )^{-4/n}\right )^{5/2}}\) |
Input:
Int[Sinh[a + (2*Log[c*x^n])/n]^(5/2),x]
Output:
(x*(-1/4*(n*(c*x^n)^(6/n)*(1 - 1/(E^(2*a)*(c*x^n)^(4/n)))^(5/2)) + (5*((n* (c*x^n)^(6/n)*(1 - 1/(E^(2*a)*(c*x^n)^(4/n)))^(3/2))/6 + (n*(-(Sqrt[1 - (c ^2*x^(2*n))/E^(2*a)]/(c*x^n)) - ArcSin[1/(c*E^a*x^n)]/E^a))/(2*E^(2*a))))/ 2)*Sinh[a + (2*Log[c*x^n])/n]^(5/2))/(n*(c*x^n)^(6/n)*(1 - 1/(E^(2*a)*(c*x ^n)^(4/n)))^(5/2))
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]
Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(c*x)^ (m + 1)*((a + b*x^2)^p/(c*(m + 1))), x] - Simp[2*b*(p/(c^2*(m + 1))) Int[ (c*x)^(m + 2)*(a + b*x^2)^(p - 1), x], x] /; FreeQ[{a, b, c}, x] && GtQ[p, 0] && LtQ[m, -1] && !ILtQ[(m + 2*p + 3)/2, 0] && IntBinomialQ[a, b, c, 2, m, p, x]
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] && !IntegerQ[p]
Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[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]
Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[x^(m + 1)*( (a + b*x^n)^p/(m + 1)), x] - Simp[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] && Gt Q[p, 0]
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] + Simp[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]
Int[Sinh[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))*(d_.)]^(p_.), x_Symbol] :> S imp[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] )
Int[((e_.)*(x_))^(m_.)*Sinh[((a_.) + Log[x_]*(b_.))*(d_.)]^(p_), x_Symbol] :> Simp[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] /; FreeQ[ {a, b, d, e, m, p}, x] && !IntegerQ[p]
\[\int {\sinh \left (a +\frac {2 \ln \left (c \,x^{n}\right )}{n}\right )}^{\frac {5}{2}}d x\]
Input:
int(sinh(a+2*ln(c*x^n)/n)^(5/2),x)
Output:
int(sinh(a+2*ln(c*x^n)/n)^(5/2),x)
Time = 0.09 (sec) , antiderivative size = 162, normalized size of antiderivative = 0.78 \[ \int \sinh ^{\frac {5}{2}}\left (a+\frac {2 \log \left (c x^n\right )}{n}\right ) \, dx=\frac {{\left (15 \, \sqrt {2} x^{3} \arctan \left (\sqrt {2} \sqrt {\frac {1}{2}} x \sqrt {\frac {x^{4} e^{\left (\frac {2 \, {\left (a n + 2 \, \log \left (c\right )\right )}}{n}\right )} - 1}{x^{2}}}\right ) e^{\left (\frac {3 \, {\left (a n + 2 \, \log \left (c\right )\right )}}{2 \, n}\right )} + 2 \, \sqrt {\frac {1}{2}} {\left (2 \, x^{8} e^{\left (\frac {4 \, {\left (a n + 2 \, \log \left (c\right )\right )}}{n}\right )} - 14 \, x^{4} e^{\left (\frac {2 \, {\left (a n + 2 \, \log \left (c\right )\right )}}{n}\right )} - 3\right )} \sqrt {\frac {x^{4} e^{\left (\frac {2 \, {\left (a n + 2 \, \log \left (c\right )\right )}}{n}\right )} - 1}{x^{2}}} e^{\left (-\frac {a n + 2 \, \log \left (c\right )}{2 \, n}\right )}\right )} e^{\left (-\frac {2 \, {\left (a n + 2 \, \log \left (c\right )\right )}}{n}\right )}}{96 \, x^{3}} \] Input:
integrate(sinh(a+2*log(c*x^n)/n)^(5/2),x, algorithm="fricas")
Output:
1/96*(15*sqrt(2)*x^3*arctan(sqrt(2)*sqrt(1/2)*x*sqrt((x^4*e^(2*(a*n + 2*lo g(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*l og(c))/n)/x^3
Timed out. \[ \int \sinh ^{\frac {5}{2}}\left (a+\frac {2 \log \left (c x^n\right )}{n}\right ) \, dx=\text {Timed out} \] Input:
integrate(sinh(a+2*ln(c*x**n)/n)**(5/2),x)
Output:
Timed out
\[ \int \sinh ^{\frac {5}{2}}\left (a+\frac {2 \log \left (c x^n\right )}{n}\right ) \, dx=\int { \sinh \left (a + \frac {2 \, \log \left (c x^{n}\right )}{n}\right )^{\frac {5}{2}} \,d x } \] Input:
integrate(sinh(a+2*log(c*x^n)/n)^(5/2),x, algorithm="maxima")
Output:
integrate(sinh(a + 2*log(c*x^n)/n)^(5/2), x)
Time = 0.43 (sec) , antiderivative size = 171, normalized size of antiderivative = 0.82 \[ \int \sinh ^{\frac {5}{2}}\left (a+\frac {2 \log \left (c x^n\right )}{n}\right ) \, dx=\frac {1}{48} \, \sqrt {2} \sqrt {c^{\frac {6}{n}} x^{6} e^{\left (3 \, a\right )} - c^{\frac {2}{n}} x^{2} e^{a}} c^{\frac {2}{n}} x^{3} e^{a} + \frac {\sqrt {2} {\left (15 \, \arctan \left (\sqrt {c^{\frac {4}{n}} x^{4} e^{\left (3 \, a\right )} - e^{a}} e^{\left (-\frac {1}{2} \, a\right )}\right ) e^{\left (-\frac {3}{2} \, a\right )} - 14 \, \sqrt {c^{\frac {4}{n}} x^{4} e^{\left (3 \, a\right )} - e^{a}} e^{\left (-2 \, a\right )} - \frac {3 \, \sqrt {c^{\frac {4}{n}} x^{4} e^{\left (3 \, a\right )} - e^{a}} e^{\left (-4 \, a\right )}}{c^{\frac {4}{n}} x^{4}}\right )} e^{a}}{96 \, c^{\left (\frac {1}{n}\right )} \mathrm {sgn}\left (x\right )} \] Input:
integrate(sinh(a+2*log(c*x^n)/n)^(5/2),x, algorithm="giac")
Output:
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*arctan(sqrt(c^(4/n)*x^4*e^(3*a) - e^a)*e^(-1/2*a))*e^(-3 /2*a) - 14*sqrt(c^(4/n)*x^4*e^(3*a) - e^a)*e^(-2*a) - 3*sqrt(c^(4/n)*x^4*e ^(3*a) - e^a)*e^(-4*a)/(c^(4/n)*x^4))*e^a/(c^(1/n)*sgn(x))
Timed out. \[ \int \sinh ^{\frac {5}{2}}\left (a+\frac {2 \log \left (c x^n\right )}{n}\right ) \, dx=\int {\mathrm {sinh}\left (a+\frac {2\,\ln \left (c\,x^n\right )}{n}\right )}^{5/2} \,d x \] Input:
int(sinh(a + (2*log(c*x^n))/n)^(5/2),x)
Output:
int(sinh(a + (2*log(c*x^n))/n)^(5/2), x)
\[ \int \sinh ^{\frac {5}{2}}\left (a+\frac {2 \log \left (c x^n\right )}{n}\right ) \, dx=\sqrt {\sinh \left (\frac {2 \,\mathrm {log}\left (x^{n} c \right )+a n}{n}\right )}\, {\sinh \left (\frac {2 \,\mathrm {log}\left (x^{n} c \right )+a n}{n}\right )}^{2} x -5 \left (\int \sqrt {\sinh \left (\frac {2 \,\mathrm {log}\left (x^{n} c \right )+a n}{n}\right )}\, \cosh \left (\frac {2 \,\mathrm {log}\left (x^{n} c \right )+a n}{n}\right ) \sinh \left (\frac {2 \,\mathrm {log}\left (x^{n} c \right )+a n}{n}\right )d x \right ) \] Input:
int(sinh(a+2*log(c*x^n)/n)^(5/2),x)
Output:
sqrt(sinh((2*log(x**n*c) + a*n)/n))*sinh((2*log(x**n*c) + a*n)/n)**2*x - 5 *int(sqrt(sinh((2*log(x**n*c) + a*n)/n))*cosh((2*log(x**n*c) + a*n)/n)*sin h((2*log(x**n*c) + a*n)/n),x)