Integrand size = 10, antiderivative size = 118 \[ \int x^2 \text {Shi}(a+b x) \, dx=-\frac {2 \cosh (a+b x)}{3 b^3}-\frac {a^2 \cosh (a+b x)}{3 b^3}+\frac {a x \cosh (a+b x)}{3 b^2}-\frac {x^2 \cosh (a+b x)}{3 b}-\frac {a \sinh (a+b x)}{3 b^3}+\frac {2 x \sinh (a+b x)}{3 b^2}+\frac {a^3 \text {Shi}(a+b x)}{3 b^3}+\frac {1}{3} x^3 \text {Shi}(a+b x) \] Output:
-2/3*cosh(b*x+a)/b^3-1/3*a^2*cosh(b*x+a)/b^3+1/3*a*x*cosh(b*x+a)/b^2-1/3*x ^2*cosh(b*x+a)/b-1/3*a*sinh(b*x+a)/b^3+2/3*x*sinh(b*x+a)/b^2+1/3*a^3*Shi(b *x+a)/b^3+1/3*x^3*Shi(b*x+a)
Time = 0.12 (sec) , antiderivative size = 64, normalized size of antiderivative = 0.54 \[ \int x^2 \text {Shi}(a+b x) \, dx=-\frac {\left (2+a^2-a b x+b^2 x^2\right ) \cosh (a+b x)+(a-2 b x) \sinh (a+b x)-\left (a^3+b^3 x^3\right ) \text {Shi}(a+b x)}{3 b^3} \] Input:
Integrate[x^2*SinhIntegral[a + b*x],x]
Output:
-1/3*((2 + a^2 - a*b*x + b^2*x^2)*Cosh[a + b*x] + (a - 2*b*x)*Sinh[a + b*x ] - (a^3 + b^3*x^3)*SinhIntegral[a + b*x])/b^3
Time = 0.56 (sec) , antiderivative size = 107, normalized size of antiderivative = 0.91, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {7086, 7293, 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.
\(\displaystyle \int x^2 \text {Shi}(a+b x) \, dx\) |
\(\Big \downarrow \) 7086 |
\(\displaystyle \frac {1}{3} x^3 \text {Shi}(a+b x)-\frac {1}{3} b \int \frac {x^3 \sinh (a+b x)}{a+b x}dx\) |
\(\Big \downarrow \) 7293 |
\(\displaystyle \frac {1}{3} x^3 \text {Shi}(a+b x)-\frac {1}{3} b \int \left (-\frac {\sinh (a+b x) a^3}{b^3 (a+b x)}+\frac {\sinh (a+b x) a^2}{b^3}-\frac {x \sinh (a+b x) a}{b^2}+\frac {x^2 \sinh (a+b x)}{b}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {1}{3} x^3 \text {Shi}(a+b x)-\frac {1}{3} b \left (-\frac {a^3 \text {Shi}(a+b x)}{b^4}+\frac {a^2 \cosh (a+b x)}{b^4}+\frac {a \sinh (a+b x)}{b^4}+\frac {2 \cosh (a+b x)}{b^4}-\frac {2 x \sinh (a+b x)}{b^3}-\frac {a x \cosh (a+b x)}{b^3}+\frac {x^2 \cosh (a+b x)}{b^2}\right )\) |
Input:
Int[x^2*SinhIntegral[a + b*x],x]
Output:
(x^3*SinhIntegral[a + b*x])/3 - (b*((2*Cosh[a + b*x])/b^4 + (a^2*Cosh[a + b*x])/b^4 - (a*x*Cosh[a + b*x])/b^3 + (x^2*Cosh[a + b*x])/b^2 + (a*Sinh[a + b*x])/b^4 - (2*x*Sinh[a + b*x])/b^3 - (a^3*SinhIntegral[a + b*x])/b^4))/ 3
Int[((c_.) + (d_.)*(x_))^(m_.)*SinhIntegral[(a_.) + (b_.)*(x_)], x_Symbol] :> Simp[(c + d*x)^(m + 1)*(SinhIntegral[a + b*x]/(d*(m + 1))), x] - Simp[b/ (d*(m + 1)) Int[(c + d*x)^(m + 1)*(Sinh[a + b*x]/(a + b*x)), x], x] /; Fr eeQ[{a, b, c, d, m}, x] && NeQ[m, -1]
Time = 0.51 (sec) , antiderivative size = 100, normalized size of antiderivative = 0.85
method | result | size |
parts | \(\frac {x^{3} \operatorname {Shi}\left (b x +a \right )}{3}-\frac {-a^{3} \operatorname {Shi}\left (b x +a \right )+3 a^{2} \cosh \left (b x +a \right )-3 a \left (\left (b x +a \right ) \cosh \left (b x +a \right )-\sinh \left (b x +a \right )\right )+\left (b x +a \right )^{2} \cosh \left (b x +a \right )-2 \left (b x +a \right ) \sinh \left (b x +a \right )+2 \cosh \left (b x +a \right )}{3 b^{3}}\) | \(100\) |
derivativedivides | \(\frac {\frac {\operatorname {Shi}\left (b x +a \right ) b^{3} x^{3}}{3}+\frac {a^{3} \operatorname {Shi}\left (b x +a \right )}{3}-a^{2} \cosh \left (b x +a \right )+a \left (\left (b x +a \right ) \cosh \left (b x +a \right )-\sinh \left (b x +a \right )\right )-\frac {\left (b x +a \right )^{2} \cosh \left (b x +a \right )}{3}+\frac {2 \left (b x +a \right ) \sinh \left (b x +a \right )}{3}-\frac {2 \cosh \left (b x +a \right )}{3}}{b^{3}}\) | \(101\) |
default | \(\frac {\frac {\operatorname {Shi}\left (b x +a \right ) b^{3} x^{3}}{3}+\frac {a^{3} \operatorname {Shi}\left (b x +a \right )}{3}-a^{2} \cosh \left (b x +a \right )+a \left (\left (b x +a \right ) \cosh \left (b x +a \right )-\sinh \left (b x +a \right )\right )-\frac {\left (b x +a \right )^{2} \cosh \left (b x +a \right )}{3}+\frac {2 \left (b x +a \right ) \sinh \left (b x +a \right )}{3}-\frac {2 \cosh \left (b x +a \right )}{3}}{b^{3}}\) | \(101\) |
Input:
int(x^2*Shi(b*x+a),x,method=_RETURNVERBOSE)
Output:
1/3*x^3*Shi(b*x+a)-1/3/b^3*(-a^3*Shi(b*x+a)+3*a^2*cosh(b*x+a)-3*a*((b*x+a) *cosh(b*x+a)-sinh(b*x+a))+(b*x+a)^2*cosh(b*x+a)-2*(b*x+a)*sinh(b*x+a)+2*co sh(b*x+a))
\[ \int x^2 \text {Shi}(a+b x) \, dx=\int { x^{2} {\rm Shi}\left (b x + a\right ) \,d x } \] Input:
integrate(x^2*Shi(b*x+a),x, algorithm="fricas")
Output:
integral(x^2*sinh_integral(b*x + a), x)
\[ \int x^2 \text {Shi}(a+b x) \, dx=\int x^{2} \operatorname {Shi}{\left (a + b x \right )}\, dx \] Input:
integrate(x**2*Shi(b*x+a),x)
Output:
Integral(x**2*Shi(a + b*x), x)
\[ \int x^2 \text {Shi}(a+b x) \, dx=\int { x^{2} {\rm Shi}\left (b x + a\right ) \,d x } \] Input:
integrate(x^2*Shi(b*x+a),x, algorithm="maxima")
Output:
integrate(x^2*Shi(b*x + a), x)
\[ \int x^2 \text {Shi}(a+b x) \, dx=\int { x^{2} {\rm Shi}\left (b x + a\right ) \,d x } \] Input:
integrate(x^2*Shi(b*x+a),x, algorithm="giac")
Output:
integrate(x^2*Shi(b*x + a), x)
Timed out. \[ \int x^2 \text {Shi}(a+b x) \, dx=\int x^2\,\mathrm {sinhint}\left (a+b\,x\right ) \,d x \] Input:
int(x^2*sinhint(a + b*x),x)
Output:
int(x^2*sinhint(a + b*x), x)
Time = 0.21 (sec) , antiderivative size = 91, normalized size of antiderivative = 0.77 \[ \int x^2 \text {Shi}(a+b x) \, dx=\frac {-\cosh \left (b x +a \right ) a^{2}+\cosh \left (b x +a \right ) a b x -\cosh \left (b x +a \right ) b^{2} x^{2}-2 \cosh \left (b x +a \right )+\mathit {shi} \left (b x +a \right ) a^{3}+\mathit {shi} \left (b x +a \right ) b^{3} x^{3}-\sinh \left (b x +a \right ) a +2 \sinh \left (b x +a \right ) b x}{3 b^{3}} \] Input:
int(x^2*Shi(b*x+a),x)
Output:
( - cosh(a + b*x)*a**2 + cosh(a + b*x)*a*b*x - cosh(a + b*x)*b**2*x**2 - 2 *cosh(a + b*x) + shi(a + b*x)*a**3 + shi(a + b*x)*b**3*x**3 - sinh(a + b*x )*a + 2*sinh(a + b*x)*b*x)/(3*b**3)