\(\int \frac {\sinh ^{\frac {5}{2}}(a+b \log (c x^n))}{x} \, dx\) [279]

Optimal result

Integrand size = 19, antiderivative size = 109 \[ \int \frac {\sinh ^{\frac {5}{2}}\left (a+b \log \left (c x^n\right )\right )}{x} \, dx=\frac {6 i E\left (\left .\frac {1}{4} \left (2 i a-\pi +2 i b \log \left (c x^n\right )\right )\right |2\right ) \sqrt {\sinh \left (a+b \log \left (c x^n\right )\right )}}{5 b n \sqrt {i \sinh \left (a+b \log \left (c x^n\right )\right )}}+\frac {2 \cosh \left (a+b \log \left (c x^n\right )\right ) \sinh ^{\frac {3}{2}}\left (a+b \log \left (c x^n\right )\right )}{5 b n} \] Output:

-6/5*I*EllipticE(cos(1/2*I*a+1/4*Pi+1/2*I*b*ln(c*x^n)),2^(1/2))*sinh(a+b*l 
n(c*x^n))^(1/2)/b/n/(I*sinh(a+b*ln(c*x^n)))^(1/2)+2/5*cosh(a+b*ln(c*x^n))* 
sinh(a+b*ln(c*x^n))^(3/2)/b/n
 

Mathematica [A] (verified)

Time = 0.05 (sec) , antiderivative size = 96, normalized size of antiderivative = 0.88 \[ \int \frac {\sinh ^{\frac {5}{2}}\left (a+b \log \left (c x^n\right )\right )}{x} \, dx=\frac {-6 E\left (\left .\frac {1}{4} \left (-2 i a+\pi -2 i b \log \left (c x^n\right )\right )\right |2\right ) \sqrt {i \sinh \left (a+b \log \left (c x^n\right )\right )}+\sinh \left (a+b \log \left (c x^n\right )\right ) \sinh \left (2 \left (a+b \log \left (c x^n\right )\right )\right )}{5 b n \sqrt {\sinh \left (a+b \log \left (c x^n\right )\right )}} \] Input:

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

Output:

(-6*EllipticE[((-2*I)*a + Pi - (2*I)*b*Log[c*x^n])/4, 2]*Sqrt[I*Sinh[a + b 
*Log[c*x^n]]] + Sinh[a + b*Log[c*x^n]]*Sinh[2*(a + b*Log[c*x^n])])/(5*b*n* 
Sqrt[Sinh[a + b*Log[c*x^n]]])
 

Rubi [A] (verified)

Time = 0.39 (sec) , antiderivative size = 109, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.368, Rules used = {3039, 3042, 3115, 3042, 3121, 3042, 3119}

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.

\(\displaystyle \int \frac {\sinh ^{\frac {5}{2}}\left (a+b \log \left (c x^n\right )\right )}{x} \, dx\)

\(\Big \downarrow \) 3039

\(\displaystyle \frac {\int \sinh ^{\frac {5}{2}}\left (a+b \log \left (c x^n\right )\right )d\log \left (c x^n\right )}{n}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\int \left (-i \sin \left (i a+i b \log \left (c x^n\right )\right )\right )^{5/2}d\log \left (c x^n\right )}{n}\)

\(\Big \downarrow \) 3115

\(\displaystyle \frac {\frac {2 \sinh ^{\frac {3}{2}}\left (a+b \log \left (c x^n\right )\right ) \cosh \left (a+b \log \left (c x^n\right )\right )}{5 b}-\frac {3}{5} \int \sqrt {\sinh \left (a+b \log \left (c x^n\right )\right )}d\log \left (c x^n\right )}{n}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\frac {2 \sinh ^{\frac {3}{2}}\left (a+b \log \left (c x^n\right )\right ) \cosh \left (a+b \log \left (c x^n\right )\right )}{5 b}-\frac {3}{5} \int \sqrt {-i \sin \left (i a+i b \log \left (c x^n\right )\right )}d\log \left (c x^n\right )}{n}\)

\(\Big \downarrow \) 3121

\(\displaystyle \frac {\frac {2 \sinh ^{\frac {3}{2}}\left (a+b \log \left (c x^n\right )\right ) \cosh \left (a+b \log \left (c x^n\right )\right )}{5 b}-\frac {3 \sqrt {\sinh \left (a+b \log \left (c x^n\right )\right )} \int \sqrt {i \sinh \left (a+b \log \left (c x^n\right )\right )}d\log \left (c x^n\right )}{5 \sqrt {i \sinh \left (a+b \log \left (c x^n\right )\right )}}}{n}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\frac {2 \sinh ^{\frac {3}{2}}\left (a+b \log \left (c x^n\right )\right ) \cosh \left (a+b \log \left (c x^n\right )\right )}{5 b}-\frac {3 \sqrt {\sinh \left (a+b \log \left (c x^n\right )\right )} \int \sqrt {\sin \left (i a+i b \log \left (c x^n\right )\right )}d\log \left (c x^n\right )}{5 \sqrt {i \sinh \left (a+b \log \left (c x^n\right )\right )}}}{n}\)

\(\Big \downarrow \) 3119

\(\displaystyle \frac {\frac {2 \sinh ^{\frac {3}{2}}\left (a+b \log \left (c x^n\right )\right ) \cosh \left (a+b \log \left (c x^n\right )\right )}{5 b}+\frac {6 i \sqrt {\sinh \left (a+b \log \left (c x^n\right )\right )} E\left (\left .\frac {1}{2} \left (i a+i b \log \left (c x^n\right )-\frac {\pi }{2}\right )\right |2\right )}{5 b \sqrt {i \sinh \left (a+b \log \left (c x^n\right )\right )}}}{n}\)

Input:

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

Output:

((((6*I)/5)*EllipticE[(I*a - Pi/2 + I*b*Log[c*x^n])/2, 2]*Sqrt[Sinh[a + b* 
Log[c*x^n]]])/(b*Sqrt[I*Sinh[a + b*Log[c*x^n]]]) + (2*Cosh[a + b*Log[c*x^n 
]]*Sinh[a + b*Log[c*x^n]]^(3/2))/(5*b))/n
 

Defintions of rubi rules used

rule 3039

Int[u_, x_Symbol] :> With[{lst = FunctionOfLog[Cancel[x*u], x]}, Simp[1/lst 
[[3]]   Subst[Int[lst[[1]], x], x, Log[lst[[2]]]], x] /;  !FalseQ[lst]] /; 
NonsumQ[u]
 

rule 3042

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

rule 3115

Int[((b_.)*sin[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[(-b)*Cos[c + d* 
x]*((b*Sin[c + d*x])^(n - 1)/(d*n)), x] + Simp[b^2*((n - 1)/n)   Int[(b*Sin 
[c + d*x])^(n - 2), x], x] /; FreeQ[{b, c, d}, x] && GtQ[n, 1] && IntegerQ[ 
2*n]
 

rule 3119

Int[Sqrt[sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[(2/d)*EllipticE[(1/2)* 
(c - Pi/2 + d*x), 2], x] /; FreeQ[{c, d}, x]
 

rule 3121

Int[((b_)*sin[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[(b*Sin[c + d*x]) 
^n/Sin[c + d*x]^n   Int[Sin[c + d*x]^n, x], x] /; FreeQ[{b, c, d}, x] && Lt 
Q[-1, n, 1] && IntegerQ[2*n]
 

Maple [B] (verified)

Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 226 vs. \(2 (94 ) = 188\).

Time = 2.04 (sec) , antiderivative size = 227, normalized size of antiderivative = 2.08

method	result	size
derivativedivides	\(\frac {-\frac {6 \sqrt {1-i \sinh \left (a +b \ln \left (c \,x^{n}\right )\right )}\, \sqrt {2}\, \sqrt {1+i \sinh \left (a +b \ln \left (c \,x^{n}\right )\right )}\, \sqrt {i \sinh \left (a +b \ln \left (c \,x^{n}\right )\right )}\, \operatorname {EllipticE}\left (\sqrt {1-i \sinh \left (a +b \ln \left (c \,x^{n}\right )\right )}, \frac {\sqrt {2}}{2}\right )}{5}+\frac {3 \sqrt {1-i \sinh \left (a +b \ln \left (c \,x^{n}\right )\right )}\, \sqrt {2}\, \sqrt {1+i \sinh \left (a +b \ln \left (c \,x^{n}\right )\right )}\, \sqrt {i \sinh \left (a +b \ln \left (c \,x^{n}\right )\right )}\, \operatorname {EllipticF}\left (\sqrt {1-i \sinh \left (a +b \ln \left (c \,x^{n}\right )\right )}, \frac {\sqrt {2}}{2}\right )}{5}+\frac {2 {\cosh \left (a +b \ln \left (c \,x^{n}\right )\right )}^{4}}{5}-\frac {2 {\cosh \left (a +b \ln \left (c \,x^{n}\right )\right )}^{2}}{5}}{n \cosh \left (a +b \ln \left (c \,x^{n}\right )\right ) \sqrt {\sinh \left (a +b \ln \left (c \,x^{n}\right )\right )}\, b}\)	\(227\)
default	\(\frac {-\frac {6 \sqrt {1-i \sinh \left (a +b \ln \left (c \,x^{n}\right )\right )}\, \sqrt {2}\, \sqrt {1+i \sinh \left (a +b \ln \left (c \,x^{n}\right )\right )}\, \sqrt {i \sinh \left (a +b \ln \left (c \,x^{n}\right )\right )}\, \operatorname {EllipticE}\left (\sqrt {1-i \sinh \left (a +b \ln \left (c \,x^{n}\right )\right )}, \frac {\sqrt {2}}{2}\right )}{5}+\frac {3 \sqrt {1-i \sinh \left (a +b \ln \left (c \,x^{n}\right )\right )}\, \sqrt {2}\, \sqrt {1+i \sinh \left (a +b \ln \left (c \,x^{n}\right )\right )}\, \sqrt {i \sinh \left (a +b \ln \left (c \,x^{n}\right )\right )}\, \operatorname {EllipticF}\left (\sqrt {1-i \sinh \left (a +b \ln \left (c \,x^{n}\right )\right )}, \frac {\sqrt {2}}{2}\right )}{5}+\frac {2 {\cosh \left (a +b \ln \left (c \,x^{n}\right )\right )}^{4}}{5}-\frac {2 {\cosh \left (a +b \ln \left (c \,x^{n}\right )\right )}^{2}}{5}}{n \cosh \left (a +b \ln \left (c \,x^{n}\right )\right ) \sqrt {\sinh \left (a +b \ln \left (c \,x^{n}\right )\right )}\, b}\)	\(227\)

Input:

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

Output:

1/n*(-6/5*(1-I*sinh(a+b*ln(c*x^n)))^(1/2)*2^(1/2)*(1+I*sinh(a+b*ln(c*x^n)) 
)^(1/2)*(I*sinh(a+b*ln(c*x^n)))^(1/2)*EllipticE((1-I*sinh(a+b*ln(c*x^n)))^ 
(1/2),1/2*2^(1/2))+3/5*(1-I*sinh(a+b*ln(c*x^n)))^(1/2)*2^(1/2)*(1+I*sinh(a 
+b*ln(c*x^n)))^(1/2)*(I*sinh(a+b*ln(c*x^n)))^(1/2)*EllipticF((1-I*sinh(a+b 
*ln(c*x^n)))^(1/2),1/2*2^(1/2))+2/5*cosh(a+b*ln(c*x^n))^4-2/5*cosh(a+b*ln( 
c*x^n))^2)/cosh(a+b*ln(c*x^n))/sinh(a+b*ln(c*x^n))^(1/2)/b
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 331 vs. \(2 (90) = 180\).

Time = 0.10 (sec) , antiderivative size = 331, normalized size of antiderivative = 3.04 \[ \int \frac {\sinh ^{\frac {5}{2}}\left (a+b \log \left (c x^n\right )\right )}{x} \, dx=\frac {12 \, {\left (\sqrt {2} \cosh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )^{2} + 2 \, \sqrt {2} \cosh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right ) \sinh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right ) + \sqrt {2} \sinh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )^{2}\right )} {\rm weierstrassZeta}\left (4, 0, {\rm weierstrassPInverse}\left (4, 0, \cosh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right ) + \sinh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )\right )\right ) + {\left (\cosh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )^{4} + 4 \, \cosh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right ) \sinh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )^{3} + \sinh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )^{4} + 6 \, {\left (\cosh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )^{2} + 2\right )} \sinh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )^{2} + 12 \, \cosh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )^{2} + 4 \, {\left (\cosh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )^{3} + 6 \, \cosh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )\right )} \sinh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right ) - 1\right )} \sqrt {\sinh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )}}{10 \, {\left (b n \cosh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )^{2} + 2 \, b n \cosh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right ) \sinh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right ) + b n \sinh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )^{2}\right )}} \] Input:

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

Output:

1/10*(12*(sqrt(2)*cosh(b*n*log(x) + b*log(c) + a)^2 + 2*sqrt(2)*cosh(b*n*l 
og(x) + b*log(c) + a)*sinh(b*n*log(x) + b*log(c) + a) + sqrt(2)*sinh(b*n*l 
og(x) + b*log(c) + a)^2)*weierstrassZeta(4, 0, weierstrassPInverse(4, 0, c 
osh(b*n*log(x) + b*log(c) + a) + sinh(b*n*log(x) + b*log(c) + a))) + (cosh 
(b*n*log(x) + b*log(c) + a)^4 + 4*cosh(b*n*log(x) + b*log(c) + a)*sinh(b*n 
*log(x) + b*log(c) + a)^3 + sinh(b*n*log(x) + b*log(c) + a)^4 + 6*(cosh(b* 
n*log(x) + b*log(c) + a)^2 + 2)*sinh(b*n*log(x) + b*log(c) + a)^2 + 12*cos 
h(b*n*log(x) + b*log(c) + a)^2 + 4*(cosh(b*n*log(x) + b*log(c) + a)^3 + 6* 
cosh(b*n*log(x) + b*log(c) + a))*sinh(b*n*log(x) + b*log(c) + a) - 1)*sqrt 
(sinh(b*n*log(x) + b*log(c) + a)))/(b*n*cosh(b*n*log(x) + b*log(c) + a)^2 
+ 2*b*n*cosh(b*n*log(x) + b*log(c) + a)*sinh(b*n*log(x) + b*log(c) + a) + 
b*n*sinh(b*n*log(x) + b*log(c) + a)^2)
 

Sympy [F(-1)]

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

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

Output:

Timed out
 

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

int((sqrt(sinh(log(x**n*c)*b + a))*sinh(log(x**n*c)*b + a)**2)/x,x)