Optimal. Leaf size=145 \[ -\frac {3 b \cosh \left (a-\frac {b c}{d}\right ) \text {Chi}\left (\frac {b c}{d}+b x\right )}{4 d^2}+\frac {3 b \cosh \left (3 a-\frac {3 b c}{d}\right ) \text {Chi}\left (\frac {3 b c}{d}+3 b x\right )}{4 d^2}-\frac {\sinh ^3(a+b x)}{d (c+d x)}-\frac {3 b \sinh \left (a-\frac {b c}{d}\right ) \text {Shi}\left (\frac {b c}{d}+b x\right )}{4 d^2}+\frac {3 b \sinh \left (3 a-\frac {3 b c}{d}\right ) \text {Shi}\left (\frac {3 b c}{d}+3 b x\right )}{4 d^2} \]
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Rubi [A]
time = 0.19, antiderivative size = 145, normalized size of antiderivative = 1.00, number of steps
used = 8, number of rules used = 4, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {3394, 3384,
3379, 3382} \begin {gather*} -\frac {3 b \cosh \left (a-\frac {b c}{d}\right ) \text {Chi}\left (\frac {b c}{d}+b x\right )}{4 d^2}+\frac {3 b \cosh \left (3 a-\frac {3 b c}{d}\right ) \text {Chi}\left (\frac {3 b c}{d}+3 b x\right )}{4 d^2}-\frac {3 b \sinh \left (a-\frac {b c}{d}\right ) \text {Shi}\left (\frac {b c}{d}+b x\right )}{4 d^2}+\frac {3 b \sinh \left (3 a-\frac {3 b c}{d}\right ) \text {Shi}\left (\frac {3 b c}{d}+3 b x\right )}{4 d^2}-\frac {\sinh ^3(a+b x)}{d (c+d x)} \end {gather*}
Antiderivative was successfully verified.
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Rule 3379
Rule 3382
Rule 3384
Rule 3394
Rubi steps
\begin {align*} \int \frac {\sinh ^3(a+b x)}{(c+d x)^2} \, dx &=-\frac {\sinh ^3(a+b x)}{d (c+d x)}-\frac {(3 b) \int \left (\frac {\cosh (a+b x)}{4 (c+d x)}-\frac {\cosh (3 a+3 b x)}{4 (c+d x)}\right ) \, dx}{d}\\ &=-\frac {\sinh ^3(a+b x)}{d (c+d x)}-\frac {(3 b) \int \frac {\cosh (a+b x)}{c+d x} \, dx}{4 d}+\frac {(3 b) \int \frac {\cosh (3 a+3 b x)}{c+d x} \, dx}{4 d}\\ &=-\frac {\sinh ^3(a+b x)}{d (c+d x)}+\frac {\left (3 b \cosh \left (3 a-\frac {3 b c}{d}\right )\right ) \int \frac {\cosh \left (\frac {3 b c}{d}+3 b x\right )}{c+d x} \, dx}{4 d}-\frac {\left (3 b \cosh \left (a-\frac {b c}{d}\right )\right ) \int \frac {\cosh \left (\frac {b c}{d}+b x\right )}{c+d x} \, dx}{4 d}+\frac {\left (3 b \sinh \left (3 a-\frac {3 b c}{d}\right )\right ) \int \frac {\sinh \left (\frac {3 b c}{d}+3 b x\right )}{c+d x} \, dx}{4 d}-\frac {\left (3 b \sinh \left (a-\frac {b c}{d}\right )\right ) \int \frac {\sinh \left (\frac {b c}{d}+b x\right )}{c+d x} \, dx}{4 d}\\ &=-\frac {3 b \cosh \left (a-\frac {b c}{d}\right ) \text {Chi}\left (\frac {b c}{d}+b x\right )}{4 d^2}+\frac {3 b \cosh \left (3 a-\frac {3 b c}{d}\right ) \text {Chi}\left (\frac {3 b c}{d}+3 b x\right )}{4 d^2}-\frac {\sinh ^3(a+b x)}{d (c+d x)}-\frac {3 b \sinh \left (a-\frac {b c}{d}\right ) \text {Shi}\left (\frac {b c}{d}+b x\right )}{4 d^2}+\frac {3 b \sinh \left (3 a-\frac {3 b c}{d}\right ) \text {Shi}\left (\frac {3 b c}{d}+3 b x\right )}{4 d^2}\\ \end {align*}
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Mathematica [A]
time = 0.82, size = 160, normalized size = 1.10 \begin {gather*} \frac {6 d \cosh (b x) \sinh (a)-2 d \cosh (3 b x) \sinh (3 a)+6 d \cosh (a) \sinh (b x)-2 d \cosh (3 a) \sinh (3 b x)+6 b (c+d x) \left (-\cosh \left (a-\frac {b c}{d}\right ) \text {Chi}\left (b \left (\frac {c}{d}+x\right )\right )+\cosh \left (3 a-\frac {3 b c}{d}\right ) \text {Chi}\left (\frac {3 b (c+d x)}{d}\right )-\sinh \left (a-\frac {b c}{d}\right ) \text {Shi}\left (b \left (\frac {c}{d}+x\right )\right )+\sinh \left (3 a-\frac {3 b c}{d}\right ) \text {Shi}\left (\frac {3 b (c+d x)}{d}\right )\right )}{8 d^2 (c+d x)} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 1.62, size = 271, normalized size = 1.87
method | result | size |
risch | \(\frac {b \,{\mathrm e}^{-3 b x -3 a}}{8 d \left (b d x +b c \right )}-\frac {3 b \,{\mathrm e}^{-\frac {3 \left (a d -b c \right )}{d}} \expIntegral \left (1, 3 b x +3 a -\frac {3 \left (a d -b c \right )}{d}\right )}{8 d^{2}}-\frac {3 b \,{\mathrm e}^{-b x -a}}{8 d \left (b d x +b c \right )}+\frac {3 b \,{\mathrm e}^{-\frac {a d -b c}{d}} \expIntegral \left (1, b x +a -\frac {a d -b c}{d}\right )}{8 d^{2}}+\frac {3 b \,{\mathrm e}^{b x +a}}{8 d^{2} \left (\frac {b c}{d}+b x \right )}+\frac {3 b \,{\mathrm e}^{\frac {a d -b c}{d}} \expIntegral \left (1, -b x -a -\frac {-a d +b c}{d}\right )}{8 d^{2}}-\frac {b \,{\mathrm e}^{3 b x +3 a}}{8 d^{2} \left (\frac {b c}{d}+b x \right )}-\frac {3 b \,{\mathrm e}^{\frac {3 a d -3 b c}{d}} \expIntegral \left (1, -3 b x -3 a -\frac {3 \left (-a d +b c \right )}{d}\right )}{8 d^{2}}\) | \(271\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.34, size = 145, normalized size = 1.00 \begin {gather*} \frac {e^{\left (-3 \, a + \frac {3 \, b c}{d}\right )} E_{2}\left (\frac {3 \, {\left (d x + c\right )} b}{d}\right )}{8 \, {\left (d x + c\right )} d} - \frac {3 \, e^{\left (-a + \frac {b c}{d}\right )} E_{2}\left (\frac {{\left (d x + c\right )} b}{d}\right )}{8 \, {\left (d x + c\right )} d} + \frac {3 \, e^{\left (a - \frac {b c}{d}\right )} E_{2}\left (-\frac {{\left (d x + c\right )} b}{d}\right )}{8 \, {\left (d x + c\right )} d} - \frac {e^{\left (3 \, a - \frac {3 \, b c}{d}\right )} E_{2}\left (-\frac {3 \, {\left (d x + c\right )} b}{d}\right )}{8 \, {\left (d x + c\right )} d} \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. 301 vs.
\(2 (137) = 274\).
time = 0.34, size = 301, normalized size = 2.08 \begin {gather*} -\frac {2 \, d \sinh \left (b x + a\right )^{3} + 3 \, {\left ({\left (b d x + b c\right )} {\rm Ei}\left (\frac {b d x + b c}{d}\right ) + {\left (b d x + b c\right )} {\rm Ei}\left (-\frac {b d x + b c}{d}\right )\right )} \cosh \left (-\frac {b c - a d}{d}\right ) - 3 \, {\left ({\left (b d x + b c\right )} {\rm Ei}\left (\frac {3 \, {\left (b d x + b c\right )}}{d}\right ) + {\left (b d x + b c\right )} {\rm Ei}\left (-\frac {3 \, {\left (b d x + b c\right )}}{d}\right )\right )} \cosh \left (-\frac {3 \, {\left (b c - a d\right )}}{d}\right ) + 6 \, {\left (d \cosh \left (b x + a\right )^{2} - d\right )} \sinh \left (b x + a\right ) + 3 \, {\left ({\left (b d x + b c\right )} {\rm Ei}\left (\frac {b d x + b c}{d}\right ) - {\left (b d x + b c\right )} {\rm Ei}\left (-\frac {b d x + b c}{d}\right )\right )} \sinh \left (-\frac {b c - a d}{d}\right ) - 3 \, {\left ({\left (b d x + b c\right )} {\rm Ei}\left (\frac {3 \, {\left (b d x + b c\right )}}{d}\right ) - {\left (b d x + b c\right )} {\rm Ei}\left (-\frac {3 \, {\left (b d x + b c\right )}}{d}\right )\right )} \sinh \left (-\frac {3 \, {\left (b c - a d\right )}}{d}\right )}{8 \, {\left (d^{3} x + c d^{2}\right )}} \end {gather*}
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 \begin {gather*} \int \frac {\sinh ^{3}{\left (a + b x \right )}}{\left (c + d x\right )^{2}}\, dx \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. 1076 vs.
\(2 (137) = 274\).
time = 0.48, size = 1076, normalized size = 7.42 \begin {gather*} \frac {{\left (3 \, {\left (d x + c\right )} {\left (b - \frac {b c}{d x + c} + \frac {a d}{d x + c}\right )} b^{2} {\rm Ei}\left (-\frac {3 \, {\left ({\left (d x + c\right )} {\left (b - \frac {b c}{d x + c} + \frac {a d}{d x + c}\right )} + b c - a d\right )}}{d}\right ) e^{\left (\frac {3 \, {\left (b c - a d\right )}}{d}\right )} + 3 \, b^{3} c {\rm Ei}\left (-\frac {3 \, {\left ({\left (d x + c\right )} {\left (b - \frac {b c}{d x + c} + \frac {a d}{d x + c}\right )} + b c - a d\right )}}{d}\right ) e^{\left (\frac {3 \, {\left (b c - a d\right )}}{d}\right )} - 3 \, a b^{2} d {\rm Ei}\left (-\frac {3 \, {\left ({\left (d x + c\right )} {\left (b - \frac {b c}{d x + c} + \frac {a d}{d x + c}\right )} + b c - a d\right )}}{d}\right ) e^{\left (\frac {3 \, {\left (b c - a d\right )}}{d}\right )} - 3 \, {\left (d x + c\right )} {\left (b - \frac {b c}{d x + c} + \frac {a d}{d x + c}\right )} b^{2} {\rm Ei}\left (-\frac {{\left (d x + c\right )} {\left (b - \frac {b c}{d x + c} + \frac {a d}{d x + c}\right )} + b c - a d}{d}\right ) e^{\left (\frac {b c - a d}{d}\right )} - 3 \, b^{3} c {\rm Ei}\left (-\frac {{\left (d x + c\right )} {\left (b - \frac {b c}{d x + c} + \frac {a d}{d x + c}\right )} + b c - a d}{d}\right ) e^{\left (\frac {b c - a d}{d}\right )} + 3 \, a b^{2} d {\rm Ei}\left (-\frac {{\left (d x + c\right )} {\left (b - \frac {b c}{d x + c} + \frac {a d}{d x + c}\right )} + b c - a d}{d}\right ) e^{\left (\frac {b c - a d}{d}\right )} - 3 \, {\left (d x + c\right )} {\left (b - \frac {b c}{d x + c} + \frac {a d}{d x + c}\right )} b^{2} {\rm Ei}\left (\frac {{\left (d x + c\right )} {\left (b - \frac {b c}{d x + c} + \frac {a d}{d x + c}\right )} + b c - a d}{d}\right ) e^{\left (-\frac {b c - a d}{d}\right )} - 3 \, b^{3} c {\rm Ei}\left (\frac {{\left (d x + c\right )} {\left (b - \frac {b c}{d x + c} + \frac {a d}{d x + c}\right )} + b c - a d}{d}\right ) e^{\left (-\frac {b c - a d}{d}\right )} + 3 \, a b^{2} d {\rm Ei}\left (\frac {{\left (d x + c\right )} {\left (b - \frac {b c}{d x + c} + \frac {a d}{d x + c}\right )} + b c - a d}{d}\right ) e^{\left (-\frac {b c - a d}{d}\right )} + 3 \, {\left (d x + c\right )} {\left (b - \frac {b c}{d x + c} + \frac {a d}{d x + c}\right )} b^{2} {\rm Ei}\left (\frac {3 \, {\left ({\left (d x + c\right )} {\left (b - \frac {b c}{d x + c} + \frac {a d}{d x + c}\right )} + b c - a d\right )}}{d}\right ) e^{\left (-\frac {3 \, {\left (b c - a d\right )}}{d}\right )} + 3 \, b^{3} c {\rm Ei}\left (\frac {3 \, {\left ({\left (d x + c\right )} {\left (b - \frac {b c}{d x + c} + \frac {a d}{d x + c}\right )} + b c - a d\right )}}{d}\right ) e^{\left (-\frac {3 \, {\left (b c - a d\right )}}{d}\right )} - 3 \, a b^{2} d {\rm Ei}\left (\frac {3 \, {\left ({\left (d x + c\right )} {\left (b - \frac {b c}{d x + c} + \frac {a d}{d x + c}\right )} + b c - a d\right )}}{d}\right ) e^{\left (-\frac {3 \, {\left (b c - a d\right )}}{d}\right )} - b^{2} d e^{\left (\frac {3 \, {\left (d x + c\right )} {\left (b - \frac {b c}{d x + c} + \frac {a d}{d x + c}\right )}}{d}\right )} + 3 \, b^{2} d e^{\left (\frac {{\left (d x + c\right )} {\left (b - \frac {b c}{d x + c} + \frac {a d}{d x + c}\right )}}{d}\right )} - 3 \, b^{2} d e^{\left (-\frac {{\left (d x + c\right )} {\left (b - \frac {b c}{d x + c} + \frac {a d}{d x + c}\right )}}{d}\right )} + b^{2} d e^{\left (-\frac {3 \, {\left (d x + c\right )} {\left (b - \frac {b c}{d x + c} + \frac {a d}{d x + c}\right )}}{d}\right )}\right )} d^{2}}{8 \, {\left ({\left (d x + c\right )} {\left (b - \frac {b c}{d x + c} + \frac {a d}{d x + c}\right )} d^{4} + b c d^{4} - a d^{5}\right )} b} \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 \frac {{\mathrm {sinh}\left (a+b\,x\right )}^3}{{\left (c+d\,x\right )}^2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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