Optimal. Leaf size=153 \[ -\frac {\cosh \left (a-\frac {b c}{d}\right ) \text {Chi}\left (x (b-d)+\frac {c (b-d)}{d}\right )}{2 b}+\frac {\cosh (a+b x) \text {Chi}(c+d x)}{b}-\frac {\cosh \left (a-\frac {b c}{d}\right ) \text {Chi}\left (x (b+d)+\frac {c (b+d)}{d}\right )}{2 b}-\frac {\sinh \left (a-\frac {b c}{d}\right ) \text {Shi}\left (x (b-d)+\frac {c (b-d)}{d}\right )}{2 b}-\frac {\sinh \left (a-\frac {b c}{d}\right ) \text {Shi}\left (x (b+d)+\frac {c (b+d)}{d}\right )}{2 b} \]
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Rubi [A] time = 0.24, antiderivative size = 153, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 5, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.385, Rules used = {6547, 5471, 3303, 3298, 3301} \[ -\frac {\cosh \left (a-\frac {b c}{d}\right ) \text {Chi}\left (x (b-d)+\frac {c (b-d)}{d}\right )}{2 b}+\frac {\cosh (a+b x) \text {Chi}(c+d x)}{b}-\frac {\cosh \left (a-\frac {b c}{d}\right ) \text {Chi}\left (x (b+d)+\frac {c (b+d)}{d}\right )}{2 b}-\frac {\sinh \left (a-\frac {b c}{d}\right ) \text {Shi}\left (x (b-d)+\frac {c (b-d)}{d}\right )}{2 b}-\frac {\sinh \left (a-\frac {b c}{d}\right ) \text {Shi}\left (x (b+d)+\frac {c (b+d)}{d}\right )}{2 b} \]
Antiderivative was successfully verified.
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Rule 3298
Rule 3301
Rule 3303
Rule 5471
Rule 6547
Rubi steps
\begin {align*} \int \text {Chi}(c+d x) \sinh (a+b x) \, dx &=\frac {\cosh (a+b x) \text {Chi}(c+d x)}{b}-\frac {d \int \frac {\cosh (a+b x) \cosh (c+d x)}{c+d x} \, dx}{b}\\ &=\frac {\cosh (a+b x) \text {Chi}(c+d x)}{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}\\ &=\frac {\cosh (a+b x) \text {Chi}(c+d x)}{b}-\frac {d \int \frac {\cosh (a-c+(b-d) x)}{c+d x} \, dx}{2 b}-\frac {d \int \frac {\cosh (a+c+(b+d) x)}{c+d x} \, dx}{2 b}\\ &=\frac {\cosh (a+b x) \text {Chi}(c+d x)}{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}-\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}-\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}-\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}\\ &=-\frac {\cosh \left (a-\frac {b c}{d}\right ) \text {Chi}\left (\frac {c (b-d)}{d}+(b-d) x\right )}{2 b}+\frac {\cosh (a+b x) \text {Chi}(c+d x)}{b}-\frac {\cosh \left (a-\frac {b c}{d}\right ) \text {Chi}\left (\frac {c (b+d)}{d}+(b+d) x\right )}{2 b}-\frac {\sinh \left (a-\frac {b c}{d}\right ) \text {Shi}\left (\frac {c (b-d)}{d}+(b-d) x\right )}{2 b}-\frac {\sinh \left (a-\frac {b c}{d}\right ) \text {Shi}\left (\frac {c (b+d)}{d}+(b+d) x\right )}{2 b}\\ \end {align*}
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Mathematica [A] time = 0.59, size = 209, normalized size = 1.37 \[ -\frac {-4 \cosh (a+b x) \text {Chi}(c+d x)+2 \cosh \left (a-\frac {b c}{d}\right ) \text {Chi}\left (-\frac {(b-d) (c+d x)}{d}\right )+2 \cosh \left (a-\frac {b c}{d}\right ) \text {Chi}\left (\frac {(b+d) (c+d x)}{d}\right )+\sinh \left (a-\frac {b c}{d}\right ) \text {Shi}\left (\frac {(b-d) (c+d x)}{d}\right )+2 \sinh \left (a-\frac {b c}{d}\right ) \text {Shi}\left (\frac {(b+d) (c+d x)}{d}\right )-\sinh \left (a-\frac {b c}{d}\right ) \text {Shi}\left (-\frac {b c}{d}+c-b x+d x\right )+\cosh \left (a-\frac {b c}{d}\right ) \text {Shi}\left (\frac {(b-d) (c+d x)}{d}\right )+\cosh \left (a-\frac {b c}{d}\right ) \text {Shi}\left (-\frac {b c}{d}+c-b x+d x\right )}{4 b} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.61, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\operatorname {Chi}\left (d x + c\right ) \sinh \left (b x + a\right ), x\right ) \]
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 \[ \int {\rm Chi}\left (d x + c\right ) \sinh \left (b x + a\right )\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.23, size = 0, normalized size = 0.00 \[ \int \Chi \left (d x +c \right ) \sinh \left (b x +a \right )\, dx \]
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 \[ \int {\rm Chi}\left (d x + c\right ) \sinh \left (b x + a\right )\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \mathrm {coshint}\left (c+d\,x\right )\,\mathrm {sinh}\left (a+b\,x\right ) \,d x \]
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 \[ \int \sinh {\left (a + b x \right )} \operatorname {Chi}\left (c + d x\right )\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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