Optimal. Leaf size=371 \[ \frac {\sinh \left (a-\frac {b c}{d}\right ) \text {Chi}\left (x (b-d)+\frac {c (b-d)}{d}\right )}{2 b^2}+\frac {\sinh \left (a-\frac {b c}{d}\right ) \text {Chi}\left (x (b+d)+\frac {c (b+d)}{d}\right )}{2 b^2}-\frac {\sinh (a+b x) \text {Chi}(c+d x)}{b^2}+\frac {\cosh \left (a-\frac {b c}{d}\right ) \text {Shi}\left (x (b-d)+\frac {c (b-d)}{d}\right )}{2 b^2}+\frac {\cosh \left (a-\frac {b c}{d}\right ) \text {Shi}\left (x (b+d)+\frac {c (b+d)}{d}\right )}{2 b^2}+\frac {c \cosh \left (a-\frac {b c}{d}\right ) \text {Chi}\left (x (b-d)+\frac {c (b-d)}{d}\right )}{2 b d}+\frac {x \cosh (a+b x) \text {Chi}(c+d x)}{b}+\frac {c \cosh \left (a-\frac {b c}{d}\right ) \text {Chi}\left (x (b+d)+\frac {c (b+d)}{d}\right )}{2 b d}+\frac {c \sinh \left (a-\frac {b c}{d}\right ) \text {Shi}\left (x (b-d)+\frac {c (b-d)}{d}\right )}{2 b d}+\frac {c \sinh \left (a-\frac {b c}{d}\right ) \text {Shi}\left (x (b+d)+\frac {c (b+d)}{d}\right )}{2 b d}-\frac {\sinh (a+x (b-d)-c)}{2 b (b-d)}-\frac {\sinh (a+x (b+d)+c)}{2 b (b+d)} \]
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Rubi [A] time = 0.99, antiderivative size = 371, normalized size of antiderivative = 1.00, number of steps used = 24, number of rules used = 9, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.643, Rules used = {6549, 5643, 6742, 2637, 3303, 3298, 3301, 6541, 5472} \[ \frac {\sinh \left (a-\frac {b c}{d}\right ) \text {Chi}\left (x (b-d)+\frac {c (b-d)}{d}\right )}{2 b^2}+\frac {\sinh \left (a-\frac {b c}{d}\right ) \text {Chi}\left (x (b+d)+\frac {c (b+d)}{d}\right )}{2 b^2}-\frac {\sinh (a+b x) \text {Chi}(c+d x)}{b^2}+\frac {\cosh \left (a-\frac {b c}{d}\right ) \text {Shi}\left (x (b-d)+\frac {c (b-d)}{d}\right )}{2 b^2}+\frac {\cosh \left (a-\frac {b c}{d}\right ) \text {Shi}\left (x (b+d)+\frac {c (b+d)}{d}\right )}{2 b^2}+\frac {c \cosh \left (a-\frac {b c}{d}\right ) \text {Chi}\left (x (b-d)+\frac {c (b-d)}{d}\right )}{2 b d}+\frac {x \cosh (a+b x) \text {Chi}(c+d x)}{b}+\frac {c \cosh \left (a-\frac {b c}{d}\right ) \text {Chi}\left (x (b+d)+\frac {c (b+d)}{d}\right )}{2 b d}+\frac {c \sinh \left (a-\frac {b c}{d}\right ) \text {Shi}\left (x (b-d)+\frac {c (b-d)}{d}\right )}{2 b d}+\frac {c \sinh \left (a-\frac {b c}{d}\right ) \text {Shi}\left (x (b+d)+\frac {c (b+d)}{d}\right )}{2 b d}-\frac {\sinh (a+x (b-d)-c)}{2 b (b-d)}-\frac {\sinh (a+x (b+d)+c)}{2 b (b+d)} \]
Antiderivative was successfully verified.
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Rule 2637
Rule 3298
Rule 3301
Rule 3303
Rule 5472
Rule 5643
Rule 6541
Rule 6549
Rule 6742
Rubi steps
\begin {align*} \int x \text {Chi}(c+d x) \sinh (a+b x) \, dx &=\frac {x \cosh (a+b x) \text {Chi}(c+d x)}{b}-\frac {\int \cosh (a+b x) \text {Chi}(c+d x) \, dx}{b}-\frac {d \int \frac {x \cosh (a+b x) \cosh (c+d x)}{c+d x} \, dx}{b}\\ &=\frac {x \cosh (a+b x) \text {Chi}(c+d x)}{b}-\frac {\text {Chi}(c+d x) \sinh (a+b x)}{b^2}+\frac {d \int \frac {\cosh (c+d x) \sinh (a+b x)}{c+d x} \, dx}{b^2}-\frac {d \int \left (\frac {x \cosh (a-c+(b-d) x)}{2 (c+d x)}+\frac {x \cosh (a+c+(b+d) x)}{2 (c+d x)}\right ) \, dx}{b}\\ &=\frac {x \cosh (a+b x) \text {Chi}(c+d x)}{b}-\frac {\text {Chi}(c+d x) \sinh (a+b x)}{b^2}+\frac {d \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^2}-\frac {d \int \frac {x \cosh (a-c+(b-d) x)}{c+d x} \, dx}{2 b}-\frac {d \int \frac {x \cosh (a+c+(b+d) x)}{c+d x} \, dx}{2 b}\\ &=\frac {x \cosh (a+b x) \text {Chi}(c+d x)}{b}-\frac {\text {Chi}(c+d x) \sinh (a+b x)}{b^2}+\frac {d \int \frac {\sinh (a-c+(b-d) x)}{c+d x} \, dx}{2 b^2}+\frac {d \int \frac {\sinh (a+c+(b+d) x)}{c+d x} \, dx}{2 b^2}-\frac {d \int \left (\frac {\cosh (a-c+(b-d) x)}{d}-\frac {c \cosh (a-c+(b-d) x)}{d (c+d x)}\right ) \, dx}{2 b}-\frac {d \int \left (\frac {\cosh (a+c+(b+d) x)}{d}-\frac {c \cosh (a+c+(b+d) x)}{d (c+d x)}\right ) \, dx}{2 b}\\ &=\frac {x \cosh (a+b x) \text {Chi}(c+d x)}{b}-\frac {\text {Chi}(c+d x) \sinh (a+b x)}{b^2}-\frac {\int \cosh (a-c+(b-d) x) \, dx}{2 b}-\frac {\int \cosh (a+c+(b+d) x) \, dx}{2 b}+\frac {c \int \frac {\cosh (a-c+(b-d) x)}{c+d x} \, dx}{2 b}+\frac {c \int \frac {\cosh (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 {\sinh \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 {\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 {\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 {\cosh \left (\frac {c (b+d)}{d}+(b+d) x\right )}{c+d x} \, dx}{2 b^2}\\ &=\frac {x \cosh (a+b x) \text {Chi}(c+d x)}{b}+\frac {\text {Chi}\left (\frac {c (b-d)}{d}+(b-d) x\right ) \sinh \left (a-\frac {b c}{d}\right )}{2 b^2}+\frac {\text {Chi}\left (\frac {c (b+d)}{d}+(b+d) x\right ) \sinh \left (a-\frac {b c}{d}\right )}{2 b^2}-\frac {\text {Chi}(c+d x) \sinh (a+b x)}{b^2}-\frac {\sinh (a-c+(b-d) x)}{2 b (b-d)}-\frac {\sinh (a+c+(b+d) x)}{2 b (b+d)}+\frac {\cosh \left (a-\frac {b c}{d}\right ) \text {Shi}\left (\frac {c (b-d)}{d}+(b-d) x\right )}{2 b^2}+\frac {\cosh \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 {\cosh \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 {\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 {\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 {\sinh \left (\frac {c (b+d)}{d}+(b+d) x\right )}{c+d x} \, dx}{2 b}\\ &=\frac {c \cosh \left (a-\frac {b c}{d}\right ) \text {Chi}\left (\frac {c (b-d)}{d}+(b-d) x\right )}{2 b d}+\frac {x \cosh (a+b x) \text {Chi}(c+d x)}{b}+\frac {c \cosh \left (a-\frac {b c}{d}\right ) \text {Chi}\left (\frac {c (b+d)}{d}+(b+d) x\right )}{2 b d}+\frac {\text {Chi}\left (\frac {c (b-d)}{d}+(b-d) x\right ) \sinh \left (a-\frac {b c}{d}\right )}{2 b^2}+\frac {\text {Chi}\left (\frac {c (b+d)}{d}+(b+d) x\right ) \sinh \left (a-\frac {b c}{d}\right )}{2 b^2}-\frac {\text {Chi}(c+d x) \sinh (a+b x)}{b^2}-\frac {\sinh (a-c+(b-d) x)}{2 b (b-d)}-\frac {\sinh (a+c+(b+d) x)}{2 b (b+d)}+\frac {\cosh \left (a-\frac {b c}{d}\right ) \text {Shi}\left (\frac {c (b-d)}{d}+(b-d) x\right )}{2 b^2}+\frac {c \sinh \left (a-\frac {b c}{d}\right ) \text {Shi}\left (\frac {c (b-d)}{d}+(b-d) x\right )}{2 b d}+\frac {\cosh \left (a-\frac {b c}{d}\right ) \text {Shi}\left (\frac {c (b+d)}{d}+(b+d) x\right )}{2 b^2}+\frac {c \sinh \left (a-\frac {b c}{d}\right ) \text {Shi}\left (\frac {c (b+d)}{d}+(b+d) x\right )}{2 b d}\\ \end {align*}
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Mathematica [B] time = 8.72, size = 916, normalized size = 2.47 \[ \frac {2 c \cosh \left (a-\frac {b c}{d}\right ) \text {Chi}\left (\frac {(b+d) (c+d x)}{d}\right ) b^3+c \cosh \left (a-\frac {b c}{d}\right ) \text {Shi}\left (\frac {(b-d) (c+d x)}{d}\right ) b^3+c \sinh \left (a-\frac {b c}{d}\right ) \text {Shi}\left (\frac {(b-d) (c+d x)}{d}\right ) b^3+2 c \sinh \left (a-\frac {b c}{d}\right ) \text {Shi}\left (\frac {(b+d) (c+d x)}{d}\right ) b^3+c \cosh \left (a-\frac {b c}{d}\right ) \text {Shi}\left (-\frac {b c}{d}+c-b x+d x\right ) b^3-c \sinh \left (a-\frac {b c}{d}\right ) \text {Shi}\left (-\frac {b c}{d}+c-b x+d x\right ) b^3+2 d \text {Chi}\left (\frac {(b+d) (c+d x)}{d}\right ) \sinh \left (a-\frac {b c}{d}\right ) b^2-2 d \sinh (a-c+b x-d x) b^2-2 d \sinh (a+c+(b+d) x) b^2+d \cosh \left (a-\frac {b c}{d}\right ) \text {Shi}\left (\frac {(b-d) (c+d x)}{d}\right ) b^2+d \sinh \left (a-\frac {b c}{d}\right ) \text {Shi}\left (\frac {(b-d) (c+d x)}{d}\right ) b^2+2 d \cosh \left (a-\frac {b c}{d}\right ) \text {Shi}\left (\frac {(b+d) (c+d x)}{d}\right ) b^2-d \cosh \left (a-\frac {b c}{d}\right ) \text {Shi}\left (-\frac {b c}{d}+c-b x+d x\right ) b^2+d \sinh \left (a-\frac {b c}{d}\right ) \text {Shi}\left (-\frac {b c}{d}+c-b x+d x\right ) b^2-2 c d^2 \cosh \left (a-\frac {b c}{d}\right ) \text {Chi}\left (\frac {(b+d) (c+d x)}{d}\right ) b-2 d^2 \sinh (a-c+b x-d x) b+2 d^2 \sinh (a+c+(b+d) x) b-c d^2 \cosh \left (a-\frac {b c}{d}\right ) \text {Shi}\left (\frac {(b-d) (c+d x)}{d}\right ) b-c d^2 \sinh \left (a-\frac {b c}{d}\right ) \text {Shi}\left (\frac {(b-d) (c+d x)}{d}\right ) b-2 c d^2 \sinh \left (a-\frac {b c}{d}\right ) \text {Shi}\left (\frac {(b+d) (c+d x)}{d}\right ) b-c d^2 \cosh \left (a-\frac {b c}{d}\right ) \text {Shi}\left (-\frac {b c}{d}+c-b x+d x\right ) b+c d^2 \sinh \left (a-\frac {b c}{d}\right ) \text {Shi}\left (-\frac {b c}{d}+c-b x+d x\right ) b-2 d^3 \text {Chi}\left (\frac {(b+d) (c+d x)}{d}\right ) \sinh \left (a-\frac {b c}{d}\right )+2 \left (b^2-d^2\right ) \text {Chi}\left (-\frac {(b-d) (c+d x)}{d}\right ) \left (b c \cosh \left (a-\frac {b c}{d}\right )+d \sinh \left (a-\frac {b c}{d}\right )\right )+4 d \left (b^2-d^2\right ) \text {Chi}(c+d x) (b x \cosh (a+b x)-\sinh (a+b x))-d^3 \cosh \left (a-\frac {b c}{d}\right ) \text {Shi}\left (\frac {(b-d) (c+d x)}{d}\right )-d^3 \sinh \left (a-\frac {b c}{d}\right ) \text {Shi}\left (\frac {(b-d) (c+d x)}{d}\right )-2 d^3 \cosh \left (a-\frac {b c}{d}\right ) \text {Shi}\left (\frac {(b+d) (c+d x)}{d}\right )+d^3 \cosh \left (a-\frac {b c}{d}\right ) \text {Shi}\left (-\frac {b c}{d}+c-b x+d x\right )-d^3 \sinh \left (a-\frac {b c}{d}\right ) \text {Shi}\left (-\frac {b c}{d}+c-b x+d x\right )}{4 b^2 (b-d) d (b+d)} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.96, size = 0, normalized size = 0.00 \[ {\rm integral}\left (x \operatorname {Chi}\left (d x + c\right ) \sinh \left (b x + a\right ), x\right ) \]
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 \[ \int x {\rm Chi}\left (d x + c\right ) \sinh \left (b x + a\right )\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.56, size = 0, normalized size = 0.00 \[ \int x \Chi \left (d x +c \right ) \sinh \left (b x +a \right )\, dx \]
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 \[ \int x {\rm Chi}\left (d x + c\right ) \sinh \left (b x + a\right )\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int x\,\mathrm {coshint}\left (c+d\,x\right )\,\mathrm {sinh}\left (a+b\,x\right ) \,d x \]
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 \[ \int x \sinh {\left (a + b x \right )} \operatorname {Chi}\left (c + d x\right )\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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