Integrand size = 14, antiderivative size = 371 \[ \int x \cosh (a+b x) \text {Chi}(c+d x) \, dx=-\frac {\cosh (a-c+(b-d) x)}{2 b (b-d)}-\frac {\cosh (a+c+(b+d) x)}{2 b (b+d)}+\frac {\cosh \left (a-\frac {b c}{d}\right ) \text {Chi}\left (\frac {c (b-d)}{d}+(b-d) x\right )}{2 b^2}-\frac {\cosh (a+b x) \text {Chi}(c+d x)}{b^2}+\frac {\cosh \left (a-\frac {b c}{d}\right ) \text {Chi}\left (\frac {c (b+d)}{d}+(b+d) x\right )}{2 b^2}+\frac {c \text {Chi}\left (\frac {c (b-d)}{d}+(b-d) x\right ) \sinh \left (a-\frac {b c}{d}\right )}{2 b d}+\frac {c \text {Chi}\left (\frac {c (b+d)}{d}+(b+d) x\right ) \sinh \left (a-\frac {b c}{d}\right )}{2 b d}+\frac {x \text {Chi}(c+d x) \sinh (a+b x)}{b}+\frac {c \cosh \left (a-\frac {b c}{d}\right ) \text {Shi}\left (\frac {c (b-d)}{d}+(b-d) x\right )}{2 b d}+\frac {\sinh \left (a-\frac {b c}{d}\right ) \text {Shi}\left (\frac {c (b-d)}{d}+(b-d) x\right )}{2 b^2}+\frac {c \cosh \left (a-\frac {b c}{d}\right ) \text {Shi}\left (\frac {c (b+d)}{d}+(b+d) x\right )}{2 b d}+\frac {\sinh \left (a-\frac {b c}{d}\right ) \text {Shi}\left (\frac {c (b+d)}{d}+(b+d) x\right )}{2 b^2} \]
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Time = 0.64 (sec) , antiderivative size = 371, normalized size of antiderivative = 1.00, number of steps used = 24, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.714, Rules used = {6678, 6874, 5737, 2718, 5580, 3384, 3379, 3382, 6682, 5579} \[ \int x \cosh (a+b x) \text {Chi}(c+d x) \, dx=\frac {\cosh \left (a-\frac {b c}{d}\right ) \text {Chi}\left (x (b-d)+\frac {c (b-d)}{d}\right )}{2 b^2}-\frac {\cosh (a+b x) \text {Chi}(c+d x)}{b^2}+\frac {\cosh \left (a-\frac {b c}{d}\right ) \text {Chi}\left (x (b+d)+\frac {c (b+d)}{d}\right )}{2 b^2}+\frac {\sinh \left (a-\frac {b c}{d}\right ) \text {Shi}\left (x (b-d)+\frac {c (b-d)}{d}\right )}{2 b^2}+\frac {\sinh \left (a-\frac {b c}{d}\right ) \text {Shi}\left (x (b+d)+\frac {c (b+d)}{d}\right )}{2 b^2}+\frac {c \sinh \left (a-\frac {b c}{d}\right ) \text {Chi}\left (x (b-d)+\frac {c (b-d)}{d}\right )}{2 b d}+\frac {c \sinh \left (a-\frac {b c}{d}\right ) \text {Chi}\left (x (b+d)+\frac {c (b+d)}{d}\right )}{2 b d}+\frac {x \sinh (a+b x) \text {Chi}(c+d x)}{b}+\frac {c \cosh \left (a-\frac {b c}{d}\right ) \text {Shi}\left (x (b-d)+\frac {c (b-d)}{d}\right )}{2 b d}+\frac {c \cosh \left (a-\frac {b c}{d}\right ) \text {Shi}\left (x (b+d)+\frac {c (b+d)}{d}\right )}{2 b d}-\frac {\cosh (a+x (b-d)-c)}{2 b (b-d)}-\frac {\cosh (a+x (b+d)+c)}{2 b (b+d)} \]
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Rule 2718
Rule 3379
Rule 3382
Rule 3384
Rule 5579
Rule 5580
Rule 5737
Rule 6678
Rule 6682
Rule 6874
Rubi steps \begin{align*} \text {integral}& = \frac {x \text {Chi}(c+d x) \sinh (a+b x)}{b}-\frac {\int \text {Chi}(c+d x) \sinh (a+b x) \, dx}{b}-\frac {d \int \frac {x \cosh (c+d x) \sinh (a+b x)}{c+d x} \, dx}{b} \\ & = -\frac {\cosh (a+b x) \text {Chi}(c+d x)}{b^2}+\frac {x \text {Chi}(c+d x) \sinh (a+b x)}{b}+\frac {d \int \frac {\cosh (a+b x) \cosh (c+d x)}{c+d x} \, dx}{b^2}-\frac {d \int \left (\frac {\cosh (c+d x) \sinh (a+b x)}{d}-\frac {c \cosh (c+d x) \sinh (a+b x)}{d (c+d x)}\right ) \, dx}{b} \\ & = -\frac {\cosh (a+b x) \text {Chi}(c+d x)}{b^2}+\frac {x \text {Chi}(c+d x) \sinh (a+b x)}{b}-\frac {\int \cosh (c+d x) \sinh (a+b x) \, dx}{b}+\frac {c \int \frac {\cosh (c+d x) \sinh (a+b x)}{c+d x} \, dx}{b}+\frac {d \int \left (\frac {\cosh (a-c+(b-d) x)}{2 (c+d x)}+\frac {\cosh (a+c+(b+d) x)}{2 (c+d x)}\right ) \, dx}{b^2} \\ & = -\frac {\cosh (a+b x) \text {Chi}(c+d x)}{b^2}+\frac {x \text {Chi}(c+d x) \sinh (a+b x)}{b}-\frac {\int \left (\frac {1}{2} \sinh (a-c+(b-d) x)+\frac {1}{2} \sinh (a+c+(b+d) x)\right ) \, dx}{b}+\frac {c \int \left (\frac {\sinh (a-c+(b-d) x)}{2 (c+d x)}+\frac {\sinh (a+c+(b+d) x)}{2 (c+d x)}\right ) \, dx}{b}+\frac {d \int \frac {\cosh (a-c+(b-d) x)}{c+d x} \, dx}{2 b^2}+\frac {d \int \frac {\cosh (a+c+(b+d) x)}{c+d x} \, dx}{2 b^2} \\ & = -\frac {\cosh (a+b x) \text {Chi}(c+d x)}{b^2}+\frac {x \text {Chi}(c+d x) \sinh (a+b x)}{b}-\frac {\int \sinh (a-c+(b-d) x) \, dx}{2 b}-\frac {\int \sinh (a+c+(b+d) x) \, dx}{2 b}+\frac {c \int \frac {\sinh (a-c+(b-d) x)}{c+d x} \, dx}{2 b}+\frac {c \int \frac {\sinh (a+c+(b+d) x)}{c+d x} \, dx}{2 b}+\frac {\left (d \cosh \left (a-\frac {b c}{d}\right )\right ) \int \frac {\cosh \left (\frac {c (b-d)}{d}+(b-d) x\right )}{c+d x} \, dx}{2 b^2}+\frac {\left (d \cosh \left (a-\frac {b c}{d}\right )\right ) \int \frac {\cosh \left (\frac {c (b+d)}{d}+(b+d) x\right )}{c+d x} \, dx}{2 b^2}+\frac {\left (d \sinh \left (a-\frac {b c}{d}\right )\right ) \int \frac {\sinh \left (\frac {c (b-d)}{d}+(b-d) x\right )}{c+d x} \, dx}{2 b^2}+\frac {\left (d \sinh \left (a-\frac {b c}{d}\right )\right ) \int \frac {\sinh \left (\frac {c (b+d)}{d}+(b+d) x\right )}{c+d x} \, dx}{2 b^2} \\ & = -\frac {\cosh (a-c+(b-d) x)}{2 b (b-d)}-\frac {\cosh (a+c+(b+d) x)}{2 b (b+d)}+\frac {\cosh \left (a-\frac {b c}{d}\right ) \text {Chi}\left (\frac {c (b-d)}{d}+(b-d) x\right )}{2 b^2}-\frac {\cosh (a+b x) \text {Chi}(c+d x)}{b^2}+\frac {\cosh \left (a-\frac {b c}{d}\right ) \text {Chi}\left (\frac {c (b+d)}{d}+(b+d) x\right )}{2 b^2}+\frac {x \text {Chi}(c+d x) \sinh (a+b x)}{b}+\frac {\sinh \left (a-\frac {b c}{d}\right ) \text {Shi}\left (\frac {c (b-d)}{d}+(b-d) x\right )}{2 b^2}+\frac {\sinh \left (a-\frac {b c}{d}\right ) \text {Shi}\left (\frac {c (b+d)}{d}+(b+d) x\right )}{2 b^2}+\frac {\left (c \cosh \left (a-\frac {b c}{d}\right )\right ) \int \frac {\sinh \left (\frac {c (b-d)}{d}+(b-d) x\right )}{c+d x} \, dx}{2 b}+\frac {\left (c \cosh \left (a-\frac {b c}{d}\right )\right ) \int \frac {\sinh \left (\frac {c (b+d)}{d}+(b+d) x\right )}{c+d x} \, dx}{2 b}+\frac {\left (c \sinh \left (a-\frac {b c}{d}\right )\right ) \int \frac {\cosh \left (\frac {c (b-d)}{d}+(b-d) x\right )}{c+d x} \, dx}{2 b}+\frac {\left (c \sinh \left (a-\frac {b c}{d}\right )\right ) \int \frac {\cosh \left (\frac {c (b+d)}{d}+(b+d) x\right )}{c+d x} \, dx}{2 b} \\ & = -\frac {\cosh (a-c+(b-d) x)}{2 b (b-d)}-\frac {\cosh (a+c+(b+d) x)}{2 b (b+d)}+\frac {\cosh \left (a-\frac {b c}{d}\right ) \text {Chi}\left (\frac {c (b-d)}{d}+(b-d) x\right )}{2 b^2}-\frac {\cosh (a+b x) \text {Chi}(c+d x)}{b^2}+\frac {\cosh \left (a-\frac {b c}{d}\right ) \text {Chi}\left (\frac {c (b+d)}{d}+(b+d) x\right )}{2 b^2}+\frac {c \text {Chi}\left (\frac {c (b-d)}{d}+(b-d) x\right ) \sinh \left (a-\frac {b c}{d}\right )}{2 b d}+\frac {c \text {Chi}\left (\frac {c (b+d)}{d}+(b+d) x\right ) \sinh \left (a-\frac {b c}{d}\right )}{2 b d}+\frac {x \text {Chi}(c+d x) \sinh (a+b x)}{b}+\frac {c \cosh \left (a-\frac {b c}{d}\right ) \text {Shi}\left (\frac {c (b-d)}{d}+(b-d) x\right )}{2 b d}+\frac {\sinh \left (a-\frac {b c}{d}\right ) \text {Shi}\left (\frac {c (b-d)}{d}+(b-d) x\right )}{2 b^2}+\frac {c \cosh \left (a-\frac {b c}{d}\right ) \text {Shi}\left (\frac {c (b+d)}{d}+(b+d) x\right )}{2 b d}+\frac {\sinh \left (a-\frac {b c}{d}\right ) \text {Shi}\left (\frac {c (b+d)}{d}+(b+d) x\right )}{2 b^2} \\ \end{align*}
Time = 1.97 (sec) , antiderivative size = 270, normalized size of antiderivative = 0.73 \[ \int x \cosh (a+b x) \text {Chi}(c+d x) \, dx=\frac {\frac {e^{-a} \left (b d e^{-c} \left (-\frac {e^{-((b+d) x)}}{b+d}-\frac {e^{2 a+b x-d x}}{b-d}\right )+(b c+d) e^{2 a-\frac {b c}{d}} \operatorname {ExpIntegralEi}\left (\frac {(b-d) (c+d x)}{d}\right )-(b c-d) e^{\frac {b c}{d}} \operatorname {ExpIntegralEi}\left (-\frac {(b+d) (c+d x)}{d}\right )\right )}{d}+\frac {e^{-a} \left (b d e^c \left (\frac {e^{(-b+d) x}}{-b+d}-\frac {e^{2 a+(b+d) x}}{b+d}\right )+(-b c+d) e^{\frac {b c}{d}} \operatorname {ExpIntegralEi}\left (-\frac {(b-d) (c+d x)}{d}\right )+(b c+d) e^{2 a-\frac {b c}{d}} \operatorname {ExpIntegralEi}\left (\frac {(b+d) (c+d x)}{d}\right )\right )}{d}+4 \text {Chi}(c+d x) (-\cosh (a+b x)+b x \sinh (a+b x))}{4 b^2} \]
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\[\int x \,\operatorname {Chi}\left (d x +c \right ) \cosh \left (b x +a \right )d x\]
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\[ \int x \cosh (a+b x) \text {Chi}(c+d x) \, dx=\int { x {\rm Chi}\left (d x + c\right ) \cosh \left (b x + a\right ) \,d x } \]
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\[ \int x \cosh (a+b x) \text {Chi}(c+d x) \, dx=\int x \cosh {\left (a + b x \right )} \operatorname {Chi}\left (c + d x\right )\, dx \]
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\[ \int x \cosh (a+b x) \text {Chi}(c+d x) \, dx=\int { x {\rm Chi}\left (d x + c\right ) \cosh \left (b x + a\right ) \,d x } \]
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\[ \int x \cosh (a+b x) \text {Chi}(c+d x) \, dx=\int { x {\rm Chi}\left (d x + c\right ) \cosh \left (b x + a\right ) \,d x } \]
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Timed out. \[ \int x \cosh (a+b x) \text {Chi}(c+d x) \, dx=\int x\,\mathrm {coshint}\left (c+d\,x\right )\,\mathrm {cosh}\left (a+b\,x\right ) \,d x \]
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