Integrand size = 25, antiderivative size = 245 \[ \int \frac {x \left (1+c^2 x^2\right )^{5/2}}{a+b \text {arcsinh}(c x)} \, dx=-\frac {5 \text {Chi}\left (\frac {a+b \text {arcsinh}(c x)}{b}\right ) \sinh \left (\frac {a}{b}\right )}{64 b c^2}-\frac {9 \text {Chi}\left (\frac {3 (a+b \text {arcsinh}(c x))}{b}\right ) \sinh \left (\frac {3 a}{b}\right )}{64 b c^2}-\frac {5 \text {Chi}\left (\frac {5 (a+b \text {arcsinh}(c x))}{b}\right ) \sinh \left (\frac {5 a}{b}\right )}{64 b c^2}-\frac {\text {Chi}\left (\frac {7 (a+b \text {arcsinh}(c x))}{b}\right ) \sinh \left (\frac {7 a}{b}\right )}{64 b c^2}+\frac {5 \cosh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a+b \text {arcsinh}(c x)}{b}\right )}{64 b c^2}+\frac {9 \cosh \left (\frac {3 a}{b}\right ) \text {Shi}\left (\frac {3 (a+b \text {arcsinh}(c x))}{b}\right )}{64 b c^2}+\frac {5 \cosh \left (\frac {5 a}{b}\right ) \text {Shi}\left (\frac {5 (a+b \text {arcsinh}(c x))}{b}\right )}{64 b c^2}+\frac {\cosh \left (\frac {7 a}{b}\right ) \text {Shi}\left (\frac {7 (a+b \text {arcsinh}(c x))}{b}\right )}{64 b c^2} \] Output:
-5/64*Chi((a+b*arcsinh(c*x))/b)*sinh(a/b)/b/c^2-9/64*Chi(3*(a+b*arcsinh(c* x))/b)*sinh(3*a/b)/b/c^2-5/64*Chi(5*(a+b*arcsinh(c*x))/b)*sinh(5*a/b)/b/c^ 2-1/64*Chi(7*(a+b*arcsinh(c*x))/b)*sinh(7*a/b)/b/c^2+5/64*cosh(a/b)*Shi((a +b*arcsinh(c*x))/b)/b/c^2+9/64*cosh(3*a/b)*Shi(3*(a+b*arcsinh(c*x))/b)/b/c ^2+5/64*cosh(5*a/b)*Shi(5*(a+b*arcsinh(c*x))/b)/b/c^2+1/64*cosh(7*a/b)*Shi (7*(a+b*arcsinh(c*x))/b)/b/c^2
Time = 0.62 (sec) , antiderivative size = 180, normalized size of antiderivative = 0.73 \[ \int \frac {x \left (1+c^2 x^2\right )^{5/2}}{a+b \text {arcsinh}(c x)} \, dx=\frac {-5 \text {Chi}\left (\frac {a}{b}+\text {arcsinh}(c x)\right ) \sinh \left (\frac {a}{b}\right )-9 \text {Chi}\left (3 \left (\frac {a}{b}+\text {arcsinh}(c x)\right )\right ) \sinh \left (\frac {3 a}{b}\right )-5 \text {Chi}\left (5 \left (\frac {a}{b}+\text {arcsinh}(c x)\right )\right ) \sinh \left (\frac {5 a}{b}\right )-\text {Chi}\left (7 \left (\frac {a}{b}+\text {arcsinh}(c x)\right )\right ) \sinh \left (\frac {7 a}{b}\right )+5 \cosh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a}{b}+\text {arcsinh}(c x)\right )+9 \cosh \left (\frac {3 a}{b}\right ) \text {Shi}\left (3 \left (\frac {a}{b}+\text {arcsinh}(c x)\right )\right )+5 \cosh \left (\frac {5 a}{b}\right ) \text {Shi}\left (5 \left (\frac {a}{b}+\text {arcsinh}(c x)\right )\right )+\cosh \left (\frac {7 a}{b}\right ) \text {Shi}\left (7 \left (\frac {a}{b}+\text {arcsinh}(c x)\right )\right )}{64 b c^2} \] Input:
Integrate[(x*(1 + c^2*x^2)^(5/2))/(a + b*ArcSinh[c*x]),x]
Output:
(-5*CoshIntegral[a/b + ArcSinh[c*x]]*Sinh[a/b] - 9*CoshIntegral[3*(a/b + A rcSinh[c*x])]*Sinh[(3*a)/b] - 5*CoshIntegral[5*(a/b + ArcSinh[c*x])]*Sinh[ (5*a)/b] - CoshIntegral[7*(a/b + ArcSinh[c*x])]*Sinh[(7*a)/b] + 5*Cosh[a/b ]*SinhIntegral[a/b + ArcSinh[c*x]] + 9*Cosh[(3*a)/b]*SinhIntegral[3*(a/b + ArcSinh[c*x])] + 5*Cosh[(5*a)/b]*SinhIntegral[5*(a/b + ArcSinh[c*x])] + C osh[(7*a)/b]*SinhIntegral[7*(a/b + ArcSinh[c*x])])/(64*b*c^2)
Time = 1.05 (sec) , antiderivative size = 204, normalized size of antiderivative = 0.83, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.160, Rules used = {6234, 25, 5971, 2009}
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.
\(\displaystyle \int \frac {x \left (c^2 x^2+1\right )^{5/2}}{a+b \text {arcsinh}(c x)} \, dx\) |
\(\Big \downarrow \) 6234 |
\(\displaystyle \frac {\int -\frac {\cosh ^6\left (\frac {a}{b}-\frac {a+b \text {arcsinh}(c x)}{b}\right ) \sinh \left (\frac {a}{b}-\frac {a+b \text {arcsinh}(c x)}{b}\right )}{a+b \text {arcsinh}(c x)}d(a+b \text {arcsinh}(c x))}{b c^2}\) |
\(\Big \downarrow \) 25 |
\(\displaystyle -\frac {\int \frac {\cosh ^6\left (\frac {a}{b}-\frac {a+b \text {arcsinh}(c x)}{b}\right ) \sinh \left (\frac {a}{b}-\frac {a+b \text {arcsinh}(c x)}{b}\right )}{a+b \text {arcsinh}(c x)}d(a+b \text {arcsinh}(c x))}{b c^2}\) |
\(\Big \downarrow \) 5971 |
\(\displaystyle -\frac {\int \left (\frac {\sinh \left (\frac {7 a}{b}-\frac {7 (a+b \text {arcsinh}(c x))}{b}\right )}{64 (a+b \text {arcsinh}(c x))}+\frac {5 \sinh \left (\frac {5 a}{b}-\frac {5 (a+b \text {arcsinh}(c x))}{b}\right )}{64 (a+b \text {arcsinh}(c x))}+\frac {9 \sinh \left (\frac {3 a}{b}-\frac {3 (a+b \text {arcsinh}(c x))}{b}\right )}{64 (a+b \text {arcsinh}(c x))}+\frac {5 \sinh \left (\frac {a}{b}-\frac {a+b \text {arcsinh}(c x)}{b}\right )}{64 (a+b \text {arcsinh}(c x))}\right )d(a+b \text {arcsinh}(c x))}{b c^2}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {-\frac {5}{64} \sinh \left (\frac {a}{b}\right ) \text {Chi}\left (\frac {a+b \text {arcsinh}(c x)}{b}\right )-\frac {9}{64} \sinh \left (\frac {3 a}{b}\right ) \text {Chi}\left (\frac {3 (a+b \text {arcsinh}(c x))}{b}\right )-\frac {5}{64} \sinh \left (\frac {5 a}{b}\right ) \text {Chi}\left (\frac {5 (a+b \text {arcsinh}(c x))}{b}\right )-\frac {1}{64} \sinh \left (\frac {7 a}{b}\right ) \text {Chi}\left (\frac {7 (a+b \text {arcsinh}(c x))}{b}\right )+\frac {5}{64} \cosh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a+b \text {arcsinh}(c x)}{b}\right )+\frac {9}{64} \cosh \left (\frac {3 a}{b}\right ) \text {Shi}\left (\frac {3 (a+b \text {arcsinh}(c x))}{b}\right )+\frac {5}{64} \cosh \left (\frac {5 a}{b}\right ) \text {Shi}\left (\frac {5 (a+b \text {arcsinh}(c x))}{b}\right )+\frac {1}{64} \cosh \left (\frac {7 a}{b}\right ) \text {Shi}\left (\frac {7 (a+b \text {arcsinh}(c x))}{b}\right )}{b c^2}\) |
Input:
Int[(x*(1 + c^2*x^2)^(5/2))/(a + b*ArcSinh[c*x]),x]
Output:
((-5*CoshIntegral[(a + b*ArcSinh[c*x])/b]*Sinh[a/b])/64 - (9*CoshIntegral[ (3*(a + b*ArcSinh[c*x]))/b]*Sinh[(3*a)/b])/64 - (5*CoshIntegral[(5*(a + b* ArcSinh[c*x]))/b]*Sinh[(5*a)/b])/64 - (CoshIntegral[(7*(a + b*ArcSinh[c*x] ))/b]*Sinh[(7*a)/b])/64 + (5*Cosh[a/b]*SinhIntegral[(a + b*ArcSinh[c*x])/b ])/64 + (9*Cosh[(3*a)/b]*SinhIntegral[(3*(a + b*ArcSinh[c*x]))/b])/64 + (5 *Cosh[(5*a)/b]*SinhIntegral[(5*(a + b*ArcSinh[c*x]))/b])/64 + (Cosh[(7*a)/ b]*SinhIntegral[(7*(a + b*ArcSinh[c*x]))/b])/64)/(b*c^2)
Int[Cosh[(a_.) + (b_.)*(x_)]^(p_.)*((c_.) + (d_.)*(x_))^(m_.)*Sinh[(a_.) + (b_.)*(x_)]^(n_.), x_Symbol] :> Int[ExpandTrigReduce[(c + d*x)^m, Sinh[a + b*x]^n*Cosh[a + b*x]^p, x], x] /; FreeQ[{a, b, c, d, m}, x] && IGtQ[n, 0] & & IGtQ[p, 0]
Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_.)*(x_)^(m_.)*((d_) + (e_.)*(x_) ^2)^(p_.), x_Symbol] :> Simp[(1/(b*c^(m + 1)))*Simp[(d + e*x^2)^p/(1 + c^2* x^2)^p] Subst[Int[x^n*Sinh[-a/b + x/b]^m*Cosh[-a/b + x/b]^(2*p + 1), x], x, a + b*ArcSinh[c*x]], x] /; FreeQ[{a, b, c, d, e, n}, x] && EqQ[e, c^2*d] && IGtQ[2*p + 2, 0] && IGtQ[m, 0]
Time = 1.98 (sec) , antiderivative size = 197, normalized size of antiderivative = 0.80
method | result | size |
default | \(\frac {{\mathrm e}^{\frac {7 a}{b}} \operatorname {expIntegral}_{1}\left (7 \,\operatorname {arcsinh}\left (x c \right )+\frac {7 a}{b}\right )+5 \,{\mathrm e}^{\frac {5 a}{b}} \operatorname {expIntegral}_{1}\left (5 \,\operatorname {arcsinh}\left (x c \right )+\frac {5 a}{b}\right )+9 \,{\mathrm e}^{\frac {3 a}{b}} \operatorname {expIntegral}_{1}\left (3 \,\operatorname {arcsinh}\left (x c \right )+\frac {3 a}{b}\right )+5 \,{\mathrm e}^{\frac {a}{b}} \operatorname {expIntegral}_{1}\left (\operatorname {arcsinh}\left (x c \right )+\frac {a}{b}\right )-5 \,{\mathrm e}^{-\frac {a}{b}} \operatorname {expIntegral}_{1}\left (-\operatorname {arcsinh}\left (x c \right )-\frac {a}{b}\right )-9 \,{\mathrm e}^{-\frac {3 a}{b}} \operatorname {expIntegral}_{1}\left (-3 \,\operatorname {arcsinh}\left (x c \right )-\frac {3 a}{b}\right )-5 \,{\mathrm e}^{-\frac {5 a}{b}} \operatorname {expIntegral}_{1}\left (-5 \,\operatorname {arcsinh}\left (x c \right )-\frac {5 a}{b}\right )-{\mathrm e}^{-\frac {7 a}{b}} \operatorname {expIntegral}_{1}\left (-7 \,\operatorname {arcsinh}\left (x c \right )-\frac {7 a}{b}\right )}{128 c^{2} b}\) | \(197\) |
Input:
int(x*(c^2*x^2+1)^(5/2)/(a+b*arcsinh(x*c)),x,method=_RETURNVERBOSE)
Output:
1/128*(exp(7*a/b)*Ei(1,7*arcsinh(x*c)+7*a/b)+5*exp(5*a/b)*Ei(1,5*arcsinh(x *c)+5*a/b)+9*exp(3*a/b)*Ei(1,3*arcsinh(x*c)+3*a/b)+5*exp(a/b)*Ei(1,arcsinh (x*c)+a/b)-5*exp(-a/b)*Ei(1,-arcsinh(x*c)-a/b)-9*exp(-3*a/b)*Ei(1,-3*arcsi nh(x*c)-3*a/b)-5*exp(-5*a/b)*Ei(1,-5*arcsinh(x*c)-5*a/b)-exp(-7*a/b)*Ei(1, -7*arcsinh(x*c)-7*a/b))/c^2/b
\[ \int \frac {x \left (1+c^2 x^2\right )^{5/2}}{a+b \text {arcsinh}(c x)} \, dx=\int { \frac {{\left (c^{2} x^{2} + 1\right )}^{\frac {5}{2}} x}{b \operatorname {arsinh}\left (c x\right ) + a} \,d x } \] Input:
integrate(x*(c^2*x^2+1)^(5/2)/(a+b*arcsinh(c*x)),x, algorithm="fricas")
Output:
integral((c^4*x^5 + 2*c^2*x^3 + x)*sqrt(c^2*x^2 + 1)/(b*arcsinh(c*x) + a), x)
\[ \int \frac {x \left (1+c^2 x^2\right )^{5/2}}{a+b \text {arcsinh}(c x)} \, dx=\int \frac {x \left (c^{2} x^{2} + 1\right )^{\frac {5}{2}}}{a + b \operatorname {asinh}{\left (c x \right )}}\, dx \] Input:
integrate(x*(c**2*x**2+1)**(5/2)/(a+b*asinh(c*x)),x)
Output:
Integral(x*(c**2*x**2 + 1)**(5/2)/(a + b*asinh(c*x)), x)
\[ \int \frac {x \left (1+c^2 x^2\right )^{5/2}}{a+b \text {arcsinh}(c x)} \, dx=\int { \frac {{\left (c^{2} x^{2} + 1\right )}^{\frac {5}{2}} x}{b \operatorname {arsinh}\left (c x\right ) + a} \,d x } \] Input:
integrate(x*(c^2*x^2+1)^(5/2)/(a+b*arcsinh(c*x)),x, algorithm="maxima")
Output:
integrate((c^2*x^2 + 1)^(5/2)*x/(b*arcsinh(c*x) + a), x)
Exception generated. \[ \int \frac {x \left (1+c^2 x^2\right )^{5/2}}{a+b \text {arcsinh}(c x)} \, dx=\text {Exception raised: RuntimeError} \] Input:
integrate(x*(c^2*x^2+1)^(5/2)/(a+b*arcsinh(c*x)),x, algorithm="giac")
Output:
Exception raised: RuntimeError >> an error occurred running a Giac command :INPUT:sage2OUTPUT:sym2poly/r2sym(const gen & e,const index_m & i,const ve cteur & l) Error: Bad Argument Value
Timed out. \[ \int \frac {x \left (1+c^2 x^2\right )^{5/2}}{a+b \text {arcsinh}(c x)} \, dx=\int \frac {x\,{\left (c^2\,x^2+1\right )}^{5/2}}{a+b\,\mathrm {asinh}\left (c\,x\right )} \,d x \] Input:
int((x*(c^2*x^2 + 1)^(5/2))/(a + b*asinh(c*x)),x)
Output:
int((x*(c^2*x^2 + 1)^(5/2))/(a + b*asinh(c*x)), x)
\[ \int \frac {x \left (1+c^2 x^2\right )^{5/2}}{a+b \text {arcsinh}(c x)} \, dx=\left (\int \frac {\sqrt {c^{2} x^{2}+1}\, x^{5}}{\mathit {asinh} \left (c x \right ) b +a}d x \right ) c^{4}+2 \left (\int \frac {\sqrt {c^{2} x^{2}+1}\, x^{3}}{\mathit {asinh} \left (c x \right ) b +a}d x \right ) c^{2}+\int \frac {\sqrt {c^{2} x^{2}+1}\, x}{\mathit {asinh} \left (c x \right ) b +a}d x \] Input:
int(x*(c^2*x^2+1)^(5/2)/(a+b*asinh(c*x)),x)
Output:
int((sqrt(c**2*x**2 + 1)*x**5)/(asinh(c*x)*b + a),x)*c**4 + 2*int((sqrt(c* *2*x**2 + 1)*x**3)/(asinh(c*x)*b + a),x)*c**2 + int((sqrt(c**2*x**2 + 1)*x )/(asinh(c*x)*b + a),x)