Optimal. Leaf size=59 \[ \frac {3 a \text {Chi}\left (\frac {a}{c+d x}\right )}{4 d}-\frac {3 a \text {Chi}\left (\frac {3 a}{c+d x}\right )}{4 d}+\frac {(c+d x) \sinh ^3\left (\frac {a}{c+d x}\right )}{d} \]
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Rubi [A]
time = 0.06, antiderivative size = 59, normalized size of antiderivative = 1.00, number of steps
used = 6, number of rules used = 4, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {5418, 5410,
3394, 3382} \begin {gather*} \frac {3 a \text {Chi}\left (\frac {a}{c+d x}\right )}{4 d}-\frac {3 a \text {Chi}\left (\frac {3 a}{c+d x}\right )}{4 d}+\frac {(c+d x) \sinh ^3\left (\frac {a}{c+d x}\right )}{d} \end {gather*}
Antiderivative was successfully verified.
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Rule 3382
Rule 3394
Rule 5410
Rule 5418
Rubi steps
\begin {align*} \int \sinh ^3\left (\frac {a}{c+d x}\right ) \, dx &=\frac {\text {Subst}\left (\int \sinh ^3\left (\frac {a}{x}\right ) \, dx,x,c+d x\right )}{d}\\ &=-\frac {\text {Subst}\left (\int \frac {\sinh ^3(a x)}{x^2} \, dx,x,\frac {1}{c+d x}\right )}{d}\\ &=\frac {(c+d x) \sinh ^3\left (\frac {a}{c+d x}\right )}{d}+\frac {(3 a) \text {Subst}\left (\int \left (\frac {\cosh (a x)}{4 x}-\frac {\cosh (3 a x)}{4 x}\right ) \, dx,x,\frac {1}{c+d x}\right )}{d}\\ &=\frac {(c+d x) \sinh ^3\left (\frac {a}{c+d x}\right )}{d}+\frac {(3 a) \text {Subst}\left (\int \frac {\cosh (a x)}{x} \, dx,x,\frac {1}{c+d x}\right )}{4 d}-\frac {(3 a) \text {Subst}\left (\int \frac {\cosh (3 a x)}{x} \, dx,x,\frac {1}{c+d x}\right )}{4 d}\\ &=\frac {3 a \text {Chi}\left (\frac {a}{c+d x}\right )}{4 d}-\frac {3 a \text {Chi}\left (\frac {3 a}{c+d x}\right )}{4 d}+\frac {(c+d x) \sinh ^3\left (\frac {a}{c+d x}\right )}{d}\\ \end {align*}
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Mathematica [A]
time = 0.05, size = 54, normalized size = 0.92 \begin {gather*} \frac {3 a \text {Chi}\left (\frac {a}{c+d x}\right )-3 a \text {Chi}\left (\frac {3 a}{c+d x}\right )+4 (c+d x) \sinh ^3\left (\frac {a}{c+d x}\right )}{4 d} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 1.30, size = 74, normalized size = 1.25
method | result | size |
derivativedivides | \(-\frac {a \left (\frac {3 \left (d x +c \right ) \sinh \left (\frac {a}{d x +c}\right )}{4 a}-\frac {3 \hyperbolicCosineIntegral \left (\frac {a}{d x +c}\right )}{4}-\frac {\left (d x +c \right ) \sinh \left (\frac {3 a}{d x +c}\right )}{4 a}+\frac {3 \hyperbolicCosineIntegral \left (\frac {3 a}{d x +c}\right )}{4}\right )}{d}\) | \(74\) |
default | \(-\frac {a \left (\frac {3 \left (d x +c \right ) \sinh \left (\frac {a}{d x +c}\right )}{4 a}-\frac {3 \hyperbolicCosineIntegral \left (\frac {a}{d x +c}\right )}{4}-\frac {\left (d x +c \right ) \sinh \left (\frac {3 a}{d x +c}\right )}{4 a}+\frac {3 \hyperbolicCosineIntegral \left (\frac {3 a}{d x +c}\right )}{4}\right )}{d}\) | \(74\) |
risch | \(-\frac {{\mathrm e}^{-\frac {3 a}{d x +c}} x}{8}-\frac {{\mathrm e}^{-\frac {3 a}{d x +c}} c}{8 d}+\frac {3 a \expIntegral \left (1, \frac {3 a}{d x +c}\right )}{8 d}+\frac {3 \,{\mathrm e}^{-\frac {a}{d x +c}} x}{8}+\frac {3 \,{\mathrm e}^{-\frac {a}{d x +c}} c}{8 d}-\frac {3 a \expIntegral \left (1, \frac {a}{d x +c}\right )}{8 d}+\frac {{\mathrm e}^{\frac {3 a}{d x +c}} x}{8}+\frac {{\mathrm e}^{\frac {3 a}{d x +c}} c}{8 d}+\frac {3 a \expIntegral \left (1, -\frac {3 a}{d x +c}\right )}{8 d}-\frac {3 \,{\mathrm e}^{\frac {a}{d x +c}} x}{8}-\frac {3 \,{\mathrm e}^{\frac {a}{d x +c}} c}{8 d}-\frac {3 a \expIntegral \left (1, -\frac {a}{d x +c}\right )}{8 d}\) | \(195\) |
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. 118 vs.
\(2 (55) = 110\).
time = 0.50, size = 118, normalized size = 2.00 \begin {gather*} \frac {2 \, {\left (d x + c\right )} \sinh \left (\frac {a}{d x + c}\right )^{3} - 3 \, a {\rm Ei}\left (\frac {3 \, a}{d x + c}\right ) + 3 \, a {\rm Ei}\left (\frac {a}{d x + c}\right ) + 3 \, a {\rm Ei}\left (-\frac {a}{d x + c}\right ) - 3 \, a {\rm Ei}\left (-\frac {3 \, a}{d x + c}\right ) + 6 \, {\left ({\left (d x + c\right )} \cosh \left (\frac {a}{d x + c}\right )^{2} - d x - c\right )} \sinh \left (\frac {a}{d x + c}\right )}{8 \, d} \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 [B] Leaf count of result is larger than twice the leaf count of optimal. 167 vs.
\(2 (55) = 110\).
time = 0.44, size = 167, normalized size = 2.83 \begin {gather*} -\frac {{\left (\frac {3 \, a^{3} {\rm Ei}\left (\frac {3 \, a}{d x + c}\right )}{d x + c} - \frac {3 \, a^{3} {\rm Ei}\left (\frac {a}{d x + c}\right )}{d x + c} - \frac {3 \, a^{3} {\rm Ei}\left (-\frac {a}{d x + c}\right )}{d x + c} + \frac {3 \, a^{3} {\rm Ei}\left (-\frac {3 \, a}{d x + c}\right )}{d x + c} - a^{2} e^{\left (\frac {3 \, a}{d x + c}\right )} + 3 \, a^{2} e^{\left (\frac {a}{d x + c}\right )} - 3 \, a^{2} e^{\left (-\frac {a}{d x + c}\right )} + a^{2} e^{\left (-\frac {3 \, a}{d x + c}\right )}\right )} {\left (d x + c\right )}}{8 \, a^{2} d} \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 = -0.02 \begin {gather*} \int {\mathrm {sinh}\left (\frac {a}{c+d\,x}\right )}^3 \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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