Optimal. Leaf size=143 \[ -\frac {3 b c \cosh \left (\frac {b}{d}\right ) \text {Chi}\left (\frac {b c}{d (c+d x)}\right )}{4 d^2}+\frac {3 b c \cosh \left (\frac {3 b}{d}\right ) \text {Chi}\left (\frac {3 b c}{d (c+d x)}\right )}{4 d^2}+\frac {(c+d x) \sinh ^3\left (\frac {b x}{c+d x}\right )}{d}+\frac {3 b c \sinh \left (\frac {b}{d}\right ) \text {Shi}\left (\frac {b c}{d (c+d x)}\right )}{4 d^2}-\frac {3 b c \sinh \left (\frac {3 b}{d}\right ) \text {Shi}\left (\frac {3 b c}{d (c+d x)}\right )}{4 d^2} \]
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Rubi [A]
time = 0.18, antiderivative size = 143, 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 = {5726, 3394,
3384, 3379, 3382} \begin {gather*} -\frac {3 b c \cosh \left (\frac {b}{d}\right ) \text {Chi}\left (\frac {b c}{d (c+d x)}\right )}{4 d^2}+\frac {3 b c \cosh \left (\frac {3 b}{d}\right ) \text {Chi}\left (\frac {3 b c}{d (c+d x)}\right )}{4 d^2}+\frac {3 b c \sinh \left (\frac {b}{d}\right ) \text {Shi}\left (\frac {b c}{d (c+d x)}\right )}{4 d^2}-\frac {3 b c \sinh \left (\frac {3 b}{d}\right ) \text {Shi}\left (\frac {3 b c}{d (c+d x)}\right )}{4 d^2}+\frac {(c+d x) \sinh ^3\left (\frac {b x}{c+d x}\right )}{d} \end {gather*}
Antiderivative was successfully verified.
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Rule 3379
Rule 3382
Rule 3384
Rule 3394
Rule 5726
Rubi steps
\begin {align*} \int \sinh ^3\left (\frac {b x}{c+d x}\right ) \, dx &=-\frac {\text {Subst}\left (\int \frac {\sinh ^3\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 ^3\left (\frac {b x}{c+d x}\right )}{d}-\frac {(3 b c) \text {Subst}\left (\int \left (-\frac {\cosh \left (\frac {3 b}{d}-\frac {3 b c x}{d}\right )}{4 x}+\frac {\cosh \left (\frac {b}{d}-\frac {b c x}{d}\right )}{4 x}\right ) \, dx,x,\frac {1}{c+d x}\right )}{d^2}\\ &=\frac {(c+d x) \sinh ^3\left (\frac {b x}{c+d x}\right )}{d}+\frac {(3 b c) \text {Subst}\left (\int \frac {\cosh \left (\frac {3 b}{d}-\frac {3 b c x}{d}\right )}{x} \, dx,x,\frac {1}{c+d x}\right )}{4 d^2}-\frac {(3 b c) \text {Subst}\left (\int \frac {\cosh \left (\frac {b}{d}-\frac {b c x}{d}\right )}{x} \, dx,x,\frac {1}{c+d x}\right )}{4 d^2}\\ &=\frac {(c+d x) \sinh ^3\left (\frac {b x}{c+d x}\right )}{d}-\frac {\left (3 b c \cosh \left (\frac {b}{d}\right )\right ) \text {Subst}\left (\int \frac {\cosh \left (\frac {b c x}{d}\right )}{x} \, dx,x,\frac {1}{c+d x}\right )}{4 d^2}+\frac {\left (3 b c \cosh \left (\frac {3 b}{d}\right )\right ) \text {Subst}\left (\int \frac {\cosh \left (\frac {3 b c x}{d}\right )}{x} \, dx,x,\frac {1}{c+d x}\right )}{4 d^2}+\frac {\left (3 b c \sinh \left (\frac {b}{d}\right )\right ) \text {Subst}\left (\int \frac {\sinh \left (\frac {b c x}{d}\right )}{x} \, dx,x,\frac {1}{c+d x}\right )}{4 d^2}-\frac {\left (3 b c \sinh \left (\frac {3 b}{d}\right )\right ) \text {Subst}\left (\int \frac {\sinh \left (\frac {3 b c x}{d}\right )}{x} \, dx,x,\frac {1}{c+d x}\right )}{4 d^2}\\ &=-\frac {3 b c \cosh \left (\frac {b}{d}\right ) \text {Chi}\left (\frac {b c}{d (c+d x)}\right )}{4 d^2}+\frac {3 b c \cosh \left (\frac {3 b}{d}\right ) \text {Chi}\left (\frac {3 b c}{d (c+d x)}\right )}{4 d^2}+\frac {(c+d x) \sinh ^3\left (\frac {b x}{c+d x}\right )}{d}+\frac {3 b c \sinh \left (\frac {b}{d}\right ) \text {Shi}\left (\frac {b c}{d (c+d x)}\right )}{4 d^2}-\frac {3 b c \sinh \left (\frac {3 b}{d}\right ) \text {Shi}\left (\frac {3 b c}{d (c+d x)}\right )}{4 d^2}\\ \end {align*}
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Mathematica [A]
time = 0.35, size = 172, normalized size = 1.20 \begin {gather*} \frac {-3 b c \cosh \left (\frac {b}{d}\right ) \text {Chi}\left (\frac {b c}{d (c+d x)}\right )+3 b c \cosh \left (\frac {3 b}{d}\right ) \text {Chi}\left (\frac {3 b c}{d (c+d x)}\right )-3 c d \sinh \left (\frac {b x}{c+d x}\right )-3 d^2 x \sinh \left (\frac {b x}{c+d x}\right )+c d \sinh \left (\frac {3 b x}{c+d x}\right )+d^2 x \sinh \left (\frac {3 b x}{c+d x}\right )+3 b c \sinh \left (\frac {b}{d}\right ) \text {Shi}\left (\frac {b c}{d (c+d x)}\right )-3 b c \sinh \left (\frac {3 b}{d}\right ) \text {Shi}\left (\frac {3 b c}{d (c+d x)}\right )}{4 d^2} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 8.99, size = 228, normalized size = 1.59
method | result | size |
risch | \(-\frac {{\mathrm e}^{-\frac {3 b x}{d x +c}} \left (d x +c \right )}{8 d}-\frac {3 b c \,{\mathrm e}^{-\frac {3 b}{d}} \expIntegral \left (1, -\frac {3 b c}{d \left (d x +c \right )}\right )}{8 d^{2}}+\frac {3 \,{\mathrm e}^{-\frac {b x}{d x +c}} \left (d x +c \right )}{8 d}+\frac {3 b c \,{\mathrm e}^{-\frac {b}{d}} \expIntegral \left (1, -\frac {b c}{d \left (d x +c \right )}\right )}{8 d^{2}}+\frac {{\mathrm e}^{\frac {3 b x}{d x +c}} x}{8}+\frac {c \,{\mathrm e}^{\frac {3 b x}{d x +c}}}{8 d}-\frac {3 b c \,{\mathrm e}^{\frac {3 b}{d}} \expIntegral \left (1, \frac {3 b c}{d \left (d x +c \right )}\right )}{8 d^{2}}-\frac {3 \,{\mathrm e}^{\frac {b x}{d x +c}} x}{8}-\frac {3 c \,{\mathrm e}^{\frac {b x}{d x +c}}}{8 d}+\frac {3 b c \,{\mathrm e}^{\frac {b}{d}} \expIntegral \left (1, \frac {b c}{d \left (d x +c \right )}\right )}{8 d^{2}}\) | \(228\) |
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 [B] Leaf count of result is larger than twice the leaf count of optimal. 701 vs.
\(2 (135) = 270\).
time = 0.47, size = 701, normalized size = 4.90 \begin {gather*} \frac {3 \, {\left (b c {\rm Ei}\left (-\frac {3 \, b c}{d^{2} x + c d}\right ) \cosh \left (\frac {3 \, b}{d}\right ) - b c {\rm Ei}\left (-\frac {b c}{d^{2} x + c d}\right ) \cosh \left (\frac {b}{d}\right )\right )} \sinh \left (\frac {b x}{d x + c}\right )^{4} + 2 \, {\left (d^{2} x + c d\right )} \sinh \left (\frac {b x}{d x + c}\right )^{3} - 6 \, {\left (b c {\rm Ei}\left (-\frac {3 \, b c}{d^{2} x + c d}\right ) \cosh \left (\frac {b x}{d x + c}\right )^{2} \cosh \left (\frac {3 \, b}{d}\right ) - b c {\rm Ei}\left (-\frac {b c}{d^{2} x + c d}\right ) \cosh \left (\frac {b x}{d x + c}\right )^{2} \cosh \left (\frac {b}{d}\right )\right )} \sinh \left (\frac {b x}{d x + c}\right )^{2} + 3 \, {\left (b c {\rm Ei}\left (-\frac {3 \, b c}{d^{2} x + c d}\right ) \cosh \left (\frac {b x}{d x + c}\right )^{4} + b c {\rm Ei}\left (\frac {3 \, b c}{d^{2} x + c d}\right )\right )} \cosh \left (\frac {3 \, b}{d}\right ) - 3 \, {\left (b c {\rm Ei}\left (-\frac {b c}{d^{2} x + c d}\right ) \cosh \left (\frac {b x}{d x + c}\right )^{4} + b c {\rm Ei}\left (\frac {b c}{d^{2} x + c d}\right )\right )} \cosh \left (\frac {b}{d}\right ) - 6 \, {\left (d^{2} x - {\left (d^{2} x + c d\right )} \cosh \left (\frac {b x}{d x + c}\right )^{2} + c d\right )} \sinh \left (\frac {b x}{d x + c}\right ) + 3 \, {\left (b c {\rm Ei}\left (-\frac {3 \, b c}{d^{2} x + c d}\right ) \cosh \left (\frac {b x}{d x + c}\right )^{4} - 2 \, b c {\rm Ei}\left (-\frac {3 \, b c}{d^{2} x + c d}\right ) \cosh \left (\frac {b x}{d x + c}\right )^{2} \sinh \left (\frac {b x}{d x + c}\right )^{2} + b c {\rm Ei}\left (-\frac {3 \, b c}{d^{2} x + c d}\right ) \sinh \left (\frac {b x}{d x + c}\right )^{4} - b c {\rm Ei}\left (\frac {3 \, b c}{d^{2} x + c d}\right )\right )} \sinh \left (\frac {3 \, b}{d}\right ) - 3 \, {\left (b c {\rm Ei}\left (-\frac {b c}{d^{2} x + c d}\right ) \cosh \left (\frac {b x}{d x + c}\right )^{4} - 2 \, b c {\rm Ei}\left (-\frac {b c}{d^{2} x + c d}\right ) \cosh \left (\frac {b x}{d x + c}\right )^{2} \sinh \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 )^{4} - b c {\rm Ei}\left (\frac {b c}{d^{2} x + c d}\right )\right )} \sinh \left (\frac {b}{d}\right )}{8 \, {\left (d^{2} \cosh \left (\frac {b x}{d x + c}\right )^{4} - 2 \, d^{2} \cosh \left (\frac {b x}{d x + c}\right )^{2} \sinh \left (\frac {b x}{d x + c}\right )^{2} + d^{2} \sinh \left (\frac {b x}{d x + c}\right )^{4}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \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*} \int {\mathrm {sinh}\left (\frac {b\,x}{c+d\,x}\right )}^3 \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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