Integrand size = 10, antiderivative size = 184 \[ \int x^3 \text {Chi}(a+b x) \, dx=\frac {3 \cosh (a+b x)}{2 b^4}+\frac {a^2 \cosh (a+b x)}{4 b^4}-\frac {a x \cosh (a+b x)}{2 b^3}+\frac {3 x^2 \cosh (a+b x)}{4 b^2}-\frac {a^4 \text {Chi}(a+b x)}{4 b^4}+\frac {1}{4} x^4 \text {Chi}(a+b x)+\frac {a \sinh (a+b x)}{2 b^4}+\frac {a^3 \sinh (a+b x)}{4 b^4}-\frac {3 x \sinh (a+b x)}{2 b^3}-\frac {a^2 x \sinh (a+b x)}{4 b^3}+\frac {a x^2 \sinh (a+b x)}{4 b^2}-\frac {x^3 \sinh (a+b x)}{4 b} \]
-1/4*a^4*Chi(b*x+a)/b^4+1/4*x^4*Chi(b*x+a)+3/2*cosh(b*x+a)/b^4+1/4*a^2*cos h(b*x+a)/b^4-1/2*a*x*cosh(b*x+a)/b^3+3/4*x^2*cosh(b*x+a)/b^2+1/2*a*sinh(b* x+a)/b^4+1/4*a^3*sinh(b*x+a)/b^4-3/2*x*sinh(b*x+a)/b^3-1/4*a^2*x*sinh(b*x+ a)/b^3+1/4*a*x^2*sinh(b*x+a)/b^2-1/4*x^3*sinh(b*x+a)/b
Time = 0.13 (sec) , antiderivative size = 94, normalized size of antiderivative = 0.51 \[ \int x^3 \text {Chi}(a+b x) \, dx=\frac {\left (6+a^2-2 a b x+3 b^2 x^2\right ) \cosh (a+b x)+\left (-a^4+b^4 x^4\right ) \text {Chi}(a+b x)+\left (2 a+a^3-6 b x-a^2 b x+a b^2 x^2-b^3 x^3\right ) \sinh (a+b x)}{4 b^4} \]
((6 + a^2 - 2*a*b*x + 3*b^2*x^2)*Cosh[a + b*x] + (-a^4 + b^4*x^4)*CoshInte gral[a + b*x] + (2*a + a^3 - 6*b*x - a^2*b*x + a*b^2*x^2 - b^3*x^3)*Sinh[a + b*x])/(4*b^4)
Time = 0.63 (sec) , antiderivative size = 165, normalized size of antiderivative = 0.90, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {7087, 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.
\(\displaystyle \int x^3 \text {Chi}(a+b x) \, dx\) |
\(\Big \downarrow \) 7087 |
\(\displaystyle \frac {1}{4} x^4 \text {Chi}(a+b x)-\frac {1}{4} b \int \frac {x^4 \cosh (a+b x)}{a+b x}dx\) |
\(\Big \downarrow \) 7293 |
\(\displaystyle \frac {1}{4} x^4 \text {Chi}(a+b x)-\frac {1}{4} b \int \left (\frac {\cosh (a+b x) a^4}{b^4 (a+b x)}-\frac {\cosh (a+b x) a^3}{b^4}+\frac {x \cosh (a+b x) a^2}{b^3}-\frac {x^2 \cosh (a+b x) a}{b^2}+\frac {x^3 \cosh (a+b x)}{b}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {1}{4} x^4 \text {Chi}(a+b x)-\frac {1}{4} b \left (\frac {a^4 \text {Chi}(a+b x)}{b^5}-\frac {a^3 \sinh (a+b x)}{b^5}-\frac {a^2 \cosh (a+b x)}{b^5}+\frac {a^2 x \sinh (a+b x)}{b^4}-\frac {2 a \sinh (a+b x)}{b^5}-\frac {6 \cosh (a+b x)}{b^5}+\frac {6 x \sinh (a+b x)}{b^4}+\frac {2 a x \cosh (a+b x)}{b^4}-\frac {a x^2 \sinh (a+b x)}{b^3}-\frac {3 x^2 \cosh (a+b x)}{b^3}+\frac {x^3 \sinh (a+b x)}{b^2}\right )\) |
(x^4*CoshIntegral[a + b*x])/4 - (b*((-6*Cosh[a + b*x])/b^5 - (a^2*Cosh[a + b*x])/b^5 + (2*a*x*Cosh[a + b*x])/b^4 - (3*x^2*Cosh[a + b*x])/b^3 + (a^4* CoshIntegral[a + b*x])/b^5 - (2*a*Sinh[a + b*x])/b^5 - (a^3*Sinh[a + b*x]) /b^5 + (6*x*Sinh[a + b*x])/b^4 + (a^2*x*Sinh[a + b*x])/b^4 - (a*x^2*Sinh[a + b*x])/b^3 + (x^3*Sinh[a + b*x])/b^2))/4
3.1.86.3.1 Defintions of rubi rules used
Int[CoshIntegral[(a_.) + (b_.)*(x_)]*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> Simp[(c + d*x)^(m + 1)*(CoshIntegral[a + b*x]/(d*(m + 1))), x] - Simp[b/ (d*(m + 1)) Int[(c + d*x)^(m + 1)*(Cosh[a + b*x]/(a + b*x)), x], x] /; Fr eeQ[{a, b, c, d, m}, x] && NeQ[m, -1]
Time = 0.49 (sec) , antiderivative size = 155, normalized size of antiderivative = 0.84
method | result | size |
parts | \(\frac {x^{4} \operatorname {Chi}\left (b x +a \right )}{4}-\frac {a^{4} \operatorname {Chi}\left (b x +a \right )-4 a^{3} \sinh \left (b x +a \right )+6 a^{2} \left (\left (b x +a \right ) \sinh \left (b x +a \right )-\cosh \left (b x +a \right )\right )-4 a \left (\left (b x +a \right )^{2} \sinh \left (b x +a \right )-2 \left (b x +a \right ) \cosh \left (b x +a \right )+2 \sinh \left (b x +a \right )\right )+\left (b x +a \right )^{3} \sinh \left (b x +a \right )-3 \left (b x +a \right )^{2} \cosh \left (b x +a \right )+6 \left (b x +a \right ) \sinh \left (b x +a \right )-6 \cosh \left (b x +a \right )}{4 b^{4}}\) | \(155\) |
derivativedivides | \(\frac {\frac {\operatorname {Chi}\left (b x +a \right ) b^{4} x^{4}}{4}-\frac {a^{4} \operatorname {Chi}\left (b x +a \right )}{4}+a^{3} \sinh \left (b x +a \right )-\frac {3 a^{2} \left (\left (b x +a \right ) \sinh \left (b x +a \right )-\cosh \left (b x +a \right )\right )}{2}+a \left (\left (b x +a \right )^{2} \sinh \left (b x +a \right )-2 \left (b x +a \right ) \cosh \left (b x +a \right )+2 \sinh \left (b x +a \right )\right )-\frac {\left (b x +a \right )^{3} \sinh \left (b x +a \right )}{4}+\frac {3 \left (b x +a \right )^{2} \cosh \left (b x +a \right )}{4}-\frac {3 \left (b x +a \right ) \sinh \left (b x +a \right )}{2}+\frac {3 \cosh \left (b x +a \right )}{2}}{b^{4}}\) | \(156\) |
default | \(\frac {\frac {\operatorname {Chi}\left (b x +a \right ) b^{4} x^{4}}{4}-\frac {a^{4} \operatorname {Chi}\left (b x +a \right )}{4}+a^{3} \sinh \left (b x +a \right )-\frac {3 a^{2} \left (\left (b x +a \right ) \sinh \left (b x +a \right )-\cosh \left (b x +a \right )\right )}{2}+a \left (\left (b x +a \right )^{2} \sinh \left (b x +a \right )-2 \left (b x +a \right ) \cosh \left (b x +a \right )+2 \sinh \left (b x +a \right )\right )-\frac {\left (b x +a \right )^{3} \sinh \left (b x +a \right )}{4}+\frac {3 \left (b x +a \right )^{2} \cosh \left (b x +a \right )}{4}-\frac {3 \left (b x +a \right ) \sinh \left (b x +a \right )}{2}+\frac {3 \cosh \left (b x +a \right )}{2}}{b^{4}}\) | \(156\) |
1/4*x^4*Chi(b*x+a)-1/4/b^4*(a^4*Chi(b*x+a)-4*a^3*sinh(b*x+a)+6*a^2*((b*x+a )*sinh(b*x+a)-cosh(b*x+a))-4*a*((b*x+a)^2*sinh(b*x+a)-2*(b*x+a)*cosh(b*x+a )+2*sinh(b*x+a))+(b*x+a)^3*sinh(b*x+a)-3*(b*x+a)^2*cosh(b*x+a)+6*(b*x+a)*s inh(b*x+a)-6*cosh(b*x+a))
\[ \int x^3 \text {Chi}(a+b x) \, dx=\int { x^{3} {\rm Chi}\left (b x + a\right ) \,d x } \]
\[ \int x^3 \text {Chi}(a+b x) \, dx=\int x^{3} \operatorname {Chi}\left (a + b x\right )\, dx \]
\[ \int x^3 \text {Chi}(a+b x) \, dx=\int { x^{3} {\rm Chi}\left (b x + a\right ) \,d x } \]
\[ \int x^3 \text {Chi}(a+b x) \, dx=\int { x^{3} {\rm Chi}\left (b x + a\right ) \,d x } \]
Timed out. \[ \int x^3 \text {Chi}(a+b x) \, dx=\int x^3\,\mathrm {coshint}\left (a+b\,x\right ) \,d x \]