Optimal. Leaf size=109 \[ \frac {a \text {Chi}(2 a+2 b x)}{2 b^2}-\frac {\text {Chi}(a+b x) \sinh (a+b x)}{b^2}+\frac {\text {Shi}(2 a+2 b x)}{2 b^2}+\frac {a \log (a+b x)}{2 b^2}-\frac {\sinh (a+b x) \cosh (a+b x)}{2 b^2}+\frac {x \text {Chi}(a+b x) \cosh (a+b x)}{b}-\frac {x}{2 b} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.22, antiderivative size = 109, normalized size of antiderivative = 1.00, number of steps used = 12, number of rules used = 10, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.714, Rules used = {6549, 6742, 2635, 8, 3312, 3301, 6541, 5448, 12, 3298} \[ \frac {a \text {Chi}(2 a+2 b x)}{2 b^2}-\frac {\text {Chi}(a+b x) \sinh (a+b x)}{b^2}+\frac {\text {Shi}(2 a+2 b x)}{2 b^2}+\frac {a \log (a+b x)}{2 b^2}-\frac {\sinh (a+b x) \cosh (a+b x)}{2 b^2}+\frac {x \text {Chi}(a+b x) \cosh (a+b x)}{b}-\frac {x}{2 b} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 8
Rule 12
Rule 2635
Rule 3298
Rule 3301
Rule 3312
Rule 5448
Rule 6541
Rule 6549
Rule 6742
Rubi steps
\begin {align*} \int x \text {Chi}(a+b x) \sinh (a+b x) \, dx &=\frac {x \cosh (a+b x) \text {Chi}(a+b x)}{b}-\frac {\int \cosh (a+b x) \text {Chi}(a+b x) \, dx}{b}-\int \frac {x \cosh ^2(a+b x)}{a+b x} \, dx\\ &=\frac {x \cosh (a+b x) \text {Chi}(a+b x)}{b}-\frac {\text {Chi}(a+b x) \sinh (a+b x)}{b^2}+\frac {\int \frac {\cosh (a+b x) \sinh (a+b x)}{a+b x} \, dx}{b}-\int \left (\frac {\cosh ^2(a+b x)}{b}-\frac {a \cosh ^2(a+b x)}{b (a+b x)}\right ) \, dx\\ &=\frac {x \cosh (a+b x) \text {Chi}(a+b x)}{b}-\frac {\text {Chi}(a+b x) \sinh (a+b x)}{b^2}-\frac {\int \cosh ^2(a+b x) \, dx}{b}+\frac {\int \frac {\sinh (2 a+2 b x)}{2 (a+b x)} \, dx}{b}+\frac {a \int \frac {\cosh ^2(a+b x)}{a+b x} \, dx}{b}\\ &=\frac {x \cosh (a+b x) \text {Chi}(a+b x)}{b}-\frac {\cosh (a+b x) \sinh (a+b x)}{2 b^2}-\frac {\text {Chi}(a+b x) \sinh (a+b x)}{b^2}-\frac {\int 1 \, dx}{2 b}+\frac {\int \frac {\sinh (2 a+2 b x)}{a+b x} \, dx}{2 b}+\frac {a \int \left (\frac {1}{2 (a+b x)}+\frac {\cosh (2 a+2 b x)}{2 (a+b x)}\right ) \, dx}{b}\\ &=-\frac {x}{2 b}+\frac {x \cosh (a+b x) \text {Chi}(a+b x)}{b}+\frac {a \log (a+b x)}{2 b^2}-\frac {\cosh (a+b x) \sinh (a+b x)}{2 b^2}-\frac {\text {Chi}(a+b x) \sinh (a+b x)}{b^2}+\frac {\text {Shi}(2 a+2 b x)}{2 b^2}+\frac {a \int \frac {\cosh (2 a+2 b x)}{a+b x} \, dx}{2 b}\\ &=-\frac {x}{2 b}+\frac {x \cosh (a+b x) \text {Chi}(a+b x)}{b}+\frac {a \text {Chi}(2 a+2 b x)}{2 b^2}+\frac {a \log (a+b x)}{2 b^2}-\frac {\cosh (a+b x) \sinh (a+b x)}{2 b^2}-\frac {\text {Chi}(a+b x) \sinh (a+b x)}{b^2}+\frac {\text {Shi}(2 a+2 b x)}{2 b^2}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.19, size = 78, normalized size = 0.72 \[ \frac {2 a \text {Chi}(2 (a+b x))+4 \text {Chi}(a+b x) (b x \cosh (a+b x)-\sinh (a+b x))+2 \text {Shi}(2 (a+b x))+2 a \log (a+b x)-\sinh (2 (a+b x))-2 b x}{4 b^2} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [F] time = 1.53, size = 0, normalized size = 0.00 \[ {\rm integral}\left (x \operatorname {Chi}\left (b x + a\right ) \sinh \left (b x + a\right ), x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int x {\rm Chi}\left (b x + a\right ) \sinh \left (b x + a\right )\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.04, size = 106, normalized size = 0.97 \[ \frac {x \Chi \left (b x +a \right ) \cosh \left (b x +a \right )}{b}-\frac {\Chi \left (b x +a \right ) \sinh \left (b x +a \right )}{b^{2}}-\frac {\cosh \left (b x +a \right ) \sinh \left (b x +a \right )}{2 b^{2}}-\frac {x}{2 b}-\frac {a}{2 b^{2}}+\frac {\Shi \left (2 b x +2 a \right )}{2 b^{2}}+\frac {a \ln \left (b x +a \right )}{2 b^{2}}+\frac {a \Chi \left (2 b x +2 a \right )}{2 b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int x {\rm Chi}\left (b x + a\right ) \sinh \left (b x + a\right )\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int x\,\mathrm {coshint}\left (a+b\,x\right )\,\mathrm {sinh}\left (a+b\,x\right ) \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int x \sinh {\left (a + b x \right )} \operatorname {Chi}\left (a + b x\right )\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________