Optimal. Leaf size=104 \[ -\frac {b \cosh (a+b x)}{2 d^2 (c+d x)}+\frac {b^2 \text {Chi}\left (\frac {b c}{d}+b x\right ) \sinh \left (a-\frac {b c}{d}\right )}{2 d^3}-\frac {\sinh (a+b x)}{2 d (c+d x)^2}+\frac {b^2 \cosh \left (a-\frac {b c}{d}\right ) \text {Shi}\left (\frac {b c}{d}+b x\right )}{2 d^3} \]
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Rubi [A]
time = 0.13, antiderivative size = 104, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 4, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {3378, 3384,
3379, 3382} \begin {gather*} \frac {b^2 \sinh \left (a-\frac {b c}{d}\right ) \text {Chi}\left (\frac {b c}{d}+b x\right )}{2 d^3}+\frac {b^2 \cosh \left (a-\frac {b c}{d}\right ) \text {Shi}\left (\frac {b c}{d}+b x\right )}{2 d^3}-\frac {b \cosh (a+b x)}{2 d^2 (c+d x)}-\frac {\sinh (a+b x)}{2 d (c+d x)^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 3378
Rule 3379
Rule 3382
Rule 3384
Rubi steps
\begin {align*} \int \frac {\sinh (a+b x)}{(c+d x)^3} \, dx &=-\frac {\sinh (a+b x)}{2 d (c+d x)^2}+\frac {b \int \frac {\cosh (a+b x)}{(c+d x)^2} \, dx}{2 d}\\ &=-\frac {b \cosh (a+b x)}{2 d^2 (c+d x)}-\frac {\sinh (a+b x)}{2 d (c+d x)^2}+\frac {b^2 \int \frac {\sinh (a+b x)}{c+d x} \, dx}{2 d^2}\\ &=-\frac {b \cosh (a+b x)}{2 d^2 (c+d x)}-\frac {\sinh (a+b x)}{2 d (c+d x)^2}+\frac {\left (b^2 \cosh \left (a-\frac {b c}{d}\right )\right ) \int \frac {\sinh \left (\frac {b c}{d}+b x\right )}{c+d x} \, dx}{2 d^2}+\frac {\left (b^2 \sinh \left (a-\frac {b c}{d}\right )\right ) \int \frac {\cosh \left (\frac {b c}{d}+b x\right )}{c+d x} \, dx}{2 d^2}\\ &=-\frac {b \cosh (a+b x)}{2 d^2 (c+d x)}+\frac {b^2 \text {Chi}\left (\frac {b c}{d}+b x\right ) \sinh \left (a-\frac {b c}{d}\right )}{2 d^3}-\frac {\sinh (a+b x)}{2 d (c+d x)^2}+\frac {b^2 \cosh \left (a-\frac {b c}{d}\right ) \text {Shi}\left (\frac {b c}{d}+b x\right )}{2 d^3}\\ \end {align*}
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Mathematica [A]
time = 0.38, size = 88, normalized size = 0.85 \begin {gather*} \frac {b^2 \text {Chi}\left (b \left (\frac {c}{d}+x\right )\right ) \sinh \left (a-\frac {b c}{d}\right )-\frac {d (b (c+d x) \cosh (a+b x)+d \sinh (a+b x))}{(c+d x)^2}+b^2 \cosh \left (a-\frac {b c}{d}\right ) \text {Shi}\left (b \left (\frac {c}{d}+x\right )\right )}{2 d^3} \end {gather*}
Antiderivative was successfully verified.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(276\) vs.
\(2(96)=192\).
time = 0.39, size = 277, normalized size = 2.66
method | result | size |
risch | \(-\frac {b^{3} {\mathrm e}^{-b x -a} x}{4 d \left (b^{2} d^{2} x^{2}+2 b^{2} c d x +b^{2} c^{2}\right )}-\frac {b^{3} {\mathrm e}^{-b x -a} c}{4 d^{2} \left (b^{2} d^{2} x^{2}+2 b^{2} c d x +b^{2} c^{2}\right )}+\frac {b^{2} {\mathrm e}^{-b x -a}}{4 d \left (b^{2} d^{2} x^{2}+2 b^{2} c d x +b^{2} c^{2}\right )}+\frac {b^{2} {\mathrm e}^{-\frac {a d -b c}{d}} \expIntegral \left (1, b x +a -\frac {a d -b c}{d}\right )}{4 d^{3}}-\frac {b^{2} {\mathrm e}^{b x +a}}{4 d^{3} \left (\frac {b c}{d}+b x \right )^{2}}-\frac {b^{2} {\mathrm e}^{b x +a}}{4 d^{3} \left (\frac {b c}{d}+b x \right )}-\frac {b^{2} {\mathrm e}^{\frac {a d -b c}{d}} \expIntegral \left (1, -b x -a -\frac {-a d +b c}{d}\right )}{4 d^{3}}\) | \(277\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.32, size = 94, normalized size = 0.90 \begin {gather*} -\frac {b {\left (\frac {e^{\left (-a + \frac {b c}{d}\right )} E_{2}\left (\frac {{\left (d x + c\right )} b}{d}\right )}{{\left (d x + c\right )} d} + \frac {e^{\left (a - \frac {b c}{d}\right )} E_{2}\left (-\frac {{\left (d x + c\right )} b}{d}\right )}{{\left (d x + c\right )} d}\right )}}{4 \, d} - \frac {\sinh \left (b x + a\right )}{2 \, {\left (d x + c\right )}^{2} 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. 254 vs.
\(2 (96) = 192\).
time = 0.37, size = 254, normalized size = 2.44 \begin {gather*} -\frac {2 \, d^{2} \sinh \left (b x + a\right ) + 2 \, {\left (b d^{2} x + b c d\right )} \cosh \left (b x + a\right ) - {\left ({\left (b^{2} d^{2} x^{2} + 2 \, b^{2} c d x + b^{2} c^{2}\right )} {\rm Ei}\left (\frac {b d x + b c}{d}\right ) - {\left (b^{2} d^{2} x^{2} + 2 \, b^{2} c d x + b^{2} c^{2}\right )} {\rm Ei}\left (-\frac {b d x + b c}{d}\right )\right )} \cosh \left (-\frac {b c - a d}{d}\right ) - {\left ({\left (b^{2} d^{2} x^{2} + 2 \, b^{2} c d x + b^{2} c^{2}\right )} {\rm Ei}\left (\frac {b d x + b c}{d}\right ) + {\left (b^{2} d^{2} x^{2} + 2 \, b^{2} c d x + b^{2} c^{2}\right )} {\rm Ei}\left (-\frac {b d x + b c}{d}\right )\right )} \sinh \left (-\frac {b c - a d}{d}\right )}{4 \, {\left (d^{5} x^{2} + 2 \, c d^{4} x + c^{2} d^{3}\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 [B] Leaf count of result is larger than twice the leaf count of optimal. 301 vs.
\(2 (96) = 192\).
time = 0.41, size = 301, normalized size = 2.89 \begin {gather*} \frac {b^{2} d^{2} x^{2} {\rm Ei}\left (\frac {b d x + b c}{d}\right ) e^{\left (a - \frac {b c}{d}\right )} - b^{2} d^{2} x^{2} {\rm Ei}\left (-\frac {b d x + b c}{d}\right ) e^{\left (-a + \frac {b c}{d}\right )} + 2 \, b^{2} c d x {\rm Ei}\left (\frac {b d x + b c}{d}\right ) e^{\left (a - \frac {b c}{d}\right )} - 2 \, b^{2} c d x {\rm Ei}\left (-\frac {b d x + b c}{d}\right ) e^{\left (-a + \frac {b c}{d}\right )} + b^{2} c^{2} {\rm Ei}\left (\frac {b d x + b c}{d}\right ) e^{\left (a - \frac {b c}{d}\right )} - b^{2} c^{2} {\rm Ei}\left (-\frac {b d x + b c}{d}\right ) e^{\left (-a + \frac {b c}{d}\right )} - b d^{2} x e^{\left (b x + a\right )} - b d^{2} x e^{\left (-b x - a\right )} - b c d e^{\left (b x + a\right )} - b c d e^{\left (-b x - a\right )} - d^{2} e^{\left (b x + a\right )} + d^{2} e^{\left (-b x - a\right )}}{4 \, {\left (d^{5} x^{2} + 2 \, c d^{4} x + c^{2} d^{3}\right )}} \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 )}{{\left (c+d\,x\right )}^3} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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