Optimal. Leaf size=71 \[ \frac {\cosh (a+b x)}{2 b^2}-\frac {a^2 \text {Chi}(a+b x)}{2 b^2}+\frac {1}{2} x^2 \text {Chi}(a+b x)+\frac {a \sinh (a+b x)}{2 b^2}-\frac {x \sinh (a+b x)}{2 b} \]
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Rubi [A]
time = 0.15, antiderivative size = 71, normalized size of antiderivative = 1.00, number of steps
used = 7, number of rules used = 6, integrand size = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.750, Rules used = {6668, 6874,
2717, 3377, 2718, 3382} \begin {gather*} -\frac {a^2 \text {Chi}(a+b x)}{2 b^2}+\frac {a \sinh (a+b x)}{2 b^2}+\frac {\cosh (a+b x)}{2 b^2}+\frac {1}{2} x^2 \text {Chi}(a+b x)-\frac {x \sinh (a+b x)}{2 b} \end {gather*}
Antiderivative was successfully verified.
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Rule 2717
Rule 2718
Rule 3377
Rule 3382
Rule 6668
Rule 6874
Rubi steps
\begin {align*} \int x \text {Chi}(a+b x) \, dx &=\frac {1}{2} x^2 \text {Chi}(a+b x)-\frac {1}{2} b \int \frac {x^2 \cosh (a+b x)}{a+b x} \, dx\\ &=\frac {1}{2} x^2 \text {Chi}(a+b x)-\frac {1}{2} b \int \left (-\frac {a \cosh (a+b x)}{b^2}+\frac {x \cosh (a+b x)}{b}+\frac {a^2 \cosh (a+b x)}{b^2 (a+b x)}\right ) \, dx\\ &=\frac {1}{2} x^2 \text {Chi}(a+b x)-\frac {1}{2} \int x \cosh (a+b x) \, dx+\frac {a \int \cosh (a+b x) \, dx}{2 b}-\frac {a^2 \int \frac {\cosh (a+b x)}{a+b x} \, dx}{2 b}\\ &=-\frac {a^2 \text {Chi}(a+b x)}{2 b^2}+\frac {1}{2} x^2 \text {Chi}(a+b x)+\frac {a \sinh (a+b x)}{2 b^2}-\frac {x \sinh (a+b x)}{2 b}+\frac {\int \sinh (a+b x) \, dx}{2 b}\\ &=\frac {\cosh (a+b x)}{2 b^2}-\frac {a^2 \text {Chi}(a+b x)}{2 b^2}+\frac {1}{2} x^2 \text {Chi}(a+b x)+\frac {a \sinh (a+b x)}{2 b^2}-\frac {x \sinh (a+b x)}{2 b}\\ \end {align*}
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Mathematica [A]
time = 0.07, size = 47, normalized size = 0.66 \begin {gather*} \frac {\cosh (a+b x)+\left (-a^2+b^2 x^2\right ) \text {Chi}(a+b x)+(a-b x) \sinh (a+b x)}{2 b^2} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.27, size = 60, normalized size = 0.85
method | result | size |
derivativedivides | \(\frac {\hyperbolicCosineIntegral \left (b x +a \right ) \left (-a \left (b x +a \right )+\frac {\left (b x +a \right )^{2}}{2}\right )+a \sinh \left (b x +a \right )-\frac {\left (b x +a \right ) \sinh \left (b x +a \right )}{2}+\frac {\cosh \left (b x +a \right )}{2}}{b^{2}}\) | \(60\) |
default | \(\frac {\hyperbolicCosineIntegral \left (b x +a \right ) \left (-a \left (b x +a \right )+\frac {\left (b x +a \right )^{2}}{2}\right )+a \sinh \left (b x +a \right )-\frac {\left (b x +a \right ) \sinh \left (b x +a \right )}{2}+\frac {\cosh \left (b x +a \right )}{2}}{b^{2}}\) | \(60\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int x \operatorname {Chi}\left (a + b x\right )\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \frac {x^2\,\mathrm {coshint}\left (a+b\,x\right )}{2}+\frac {\frac {{\mathrm {e}}^{-a-b\,x}\,\left ({\mathrm {e}}^{2\,a+2\,b\,x}-a+a\,{\mathrm {e}}^{2\,a+2\,b\,x}-2\,a^2\,\mathrm {coshint}\left (a+b\,x\right )\,{\mathrm {e}}^{a+b\,x}+1\right )}{4}+\frac {b\,{\mathrm {e}}^{-a-b\,x}\,\left (x-x\,{\mathrm {e}}^{2\,a+2\,b\,x}\right )}{4}}{b^2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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