Optimal. Leaf size=74 \[ \frac {b c \cosh \left (\frac {b}{d}\right ) \text {Chi}\left (\frac {b c}{d (c+d x)}\right )}{d^2}-\frac {b c \sinh \left (\frac {b}{d}\right ) \text {Shi}\left (\frac {b c}{d (c+d x)}\right )}{d^2}+\frac {(c+d x) \sinh \left (\frac {b x}{c+d x}\right )}{d} \]
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Rubi [A] time = 0.13, antiderivative size = 74, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.454, Rules used = {5607, 3297, 3303, 3298, 3301} \[ \frac {b c \cosh \left (\frac {b}{d}\right ) \text {Chi}\left (\frac {b c}{d (c+d x)}\right )}{d^2}-\frac {b c \sinh \left (\frac {b}{d}\right ) \text {Shi}\left (\frac {b c}{d (c+d x)}\right )}{d^2}+\frac {(c+d x) \sinh \left (\frac {b x}{c+d x}\right )}{d} \]
Antiderivative was successfully verified.
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Rule 3297
Rule 3298
Rule 3301
Rule 3303
Rule 5607
Rubi steps
\begin {align*} \int \sinh \left (\frac {b x}{c+d x}\right ) \, dx &=-\frac {\operatorname {Subst}\left (\int \frac {\sinh \left (\frac {b}{d}-\frac {b c x}{d}\right )}{x^2} \, dx,x,\frac {1}{c+d x}\right )}{d}\\ &=\frac {(c+d x) \sinh \left (\frac {b x}{c+d x}\right )}{d}+\frac {(b c) \operatorname {Subst}\left (\int \frac {\cosh \left (\frac {b}{d}-\frac {b c x}{d}\right )}{x} \, dx,x,\frac {1}{c+d x}\right )}{d^2}\\ &=\frac {(c+d x) \sinh \left (\frac {b x}{c+d x}\right )}{d}+\frac {\left (b c \cosh \left (\frac {b}{d}\right )\right ) \operatorname {Subst}\left (\int \frac {\cosh \left (\frac {b c x}{d}\right )}{x} \, dx,x,\frac {1}{c+d x}\right )}{d^2}-\frac {\left (b c \sinh \left (\frac {b}{d}\right )\right ) \operatorname {Subst}\left (\int \frac {\sinh \left (\frac {b c x}{d}\right )}{x} \, dx,x,\frac {1}{c+d x}\right )}{d^2}\\ &=\frac {b c \cosh \left (\frac {b}{d}\right ) \text {Chi}\left (\frac {b c}{d (c+d x)}\right )}{d^2}+\frac {(c+d x) \sinh \left (\frac {b x}{c+d x}\right )}{d}-\frac {b c \sinh \left (\frac {b}{d}\right ) \text {Shi}\left (\frac {b c}{d (c+d x)}\right )}{d^2}\\ \end {align*}
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Mathematica [A] time = 0.30, size = 70, normalized size = 0.95 \[ \frac {b c \cosh \left (\frac {b}{d}\right ) \text {Chi}\left (\frac {b c}{d (c+d x)}\right )-b c \sinh \left (\frac {b}{d}\right ) \text {Shi}\left (\frac {b c}{d (c+d x)}\right )+d (c+d x) \sinh \left (\frac {b x}{c+d x}\right )}{d^2} \]
Antiderivative was successfully verified.
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fricas [B] time = 1.12, size = 253, normalized size = 3.42 \[ -\frac {b c {\rm Ei}\left (-\frac {b c}{d^{2} x + c d}\right ) \cosh \left (\frac {b}{d}\right ) \sinh \left (\frac {b x}{d x + c}\right )^{2} - {\left (b c {\rm Ei}\left (-\frac {b c}{d^{2} x + c d}\right ) \cosh \left (\frac {b x}{d x + c}\right )^{2} + b c {\rm Ei}\left (\frac {b c}{d^{2} x + c d}\right )\right )} \cosh \left (\frac {b}{d}\right ) - 2 \, {\left (d^{2} x + c d\right )} \sinh \left (\frac {b x}{d x + c}\right ) - {\left (b c {\rm Ei}\left (-\frac {b c}{d^{2} x + c d}\right ) \cosh \left (\frac {b x}{d x + c}\right )^{2} - b c {\rm Ei}\left (-\frac {b c}{d^{2} x + c d}\right ) \sinh \left (\frac {b x}{d x + c}\right )^{2} - b c {\rm Ei}\left (\frac {b c}{d^{2} x + c d}\right )\right )} \sinh \left (\frac {b}{d}\right )}{2 \, {\left (d^{2} \cosh \left (\frac {b x}{d x + c}\right )^{2} - d^{2} \sinh \left (\frac {b x}{d x + c}\right )^{2}\right )}} \]
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 \[ \int \sinh \left (\frac {b x}{d x + c}\right )\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 113, normalized size = 1.53 \[ -\frac {{\mathrm e}^{-\frac {b x}{d x +c}} \left (d x +c \right )}{2 d}-\frac {c b \,{\mathrm e}^{-\frac {b}{d}} \Ei \left (1, -\frac {b c}{d \left (d x +c \right )}\right )}{2 d^{2}}+\frac {{\mathrm e}^{\frac {b x}{d x +c}} x}{2}+\frac {c \,{\mathrm e}^{\frac {b x}{d x +c}}}{2 d}-\frac {c b \,{\mathrm e}^{\frac {b}{d}} \Ei \left (1, \frac {b c}{d \left (d x +c \right )}\right )}{2 d^{2}} \]
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 \[ -\frac {1}{2} \, b c \int \frac {x e^{\left (\frac {b c}{d^{2} x + c d}\right )}}{d^{2} x^{2} e^{\frac {b}{d}} + 2 \, c d x e^{\frac {b}{d}} + c^{2} e^{\frac {b}{d}}}\,{d x} - \frac {1}{2} \, b c \int \frac {x e^{\left (-\frac {b c}{d^{2} x + c d} + \frac {b}{d}\right )}}{d^{2} x^{2} + 2 \, c d x + c^{2}}\,{d x} - \frac {1}{2} \, {\left (x e^{\left (\frac {b c}{d^{2} x + c d}\right )} - x e^{\left (-\frac {b c}{d^{2} x + c d} + \frac {2 \, b}{d}\right )}\right )} e^{\left (-\frac {b}{d}\right )} \]
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 \[ \int \mathrm {sinh}\left (\frac {b\,x}{c+d\,x}\right ) \,d x \]
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 \[ \int \sinh {\left (\frac {b x}{c + d x} \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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