Optimal. Leaf size=162 \[ -\frac {b^2}{3 d^3 (c+d x)}+\frac {2 b^3 \text {Chi}\left (\frac {2 b c}{d}+2 b x\right ) \sinh \left (2 a-\frac {2 b c}{d}\right )}{3 d^4}-\frac {b \cosh (a+b x) \sinh (a+b x)}{3 d^2 (c+d x)^2}-\frac {\sinh ^2(a+b x)}{3 d (c+d x)^3}-\frac {2 b^2 \sinh ^2(a+b x)}{3 d^3 (c+d x)}+\frac {2 b^3 \cosh \left (2 a-\frac {2 b c}{d}\right ) \text {Shi}\left (\frac {2 b c}{d}+2 b x\right )}{3 d^4} \]
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Rubi [A]
time = 0.13, antiderivative size = 162, normalized size of antiderivative = 1.00, number of steps
used = 7, number of rules used = 7, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.438, Rules used = {3395, 32, 3394,
12, 3384, 3379, 3382} \begin {gather*} \frac {2 b^3 \sinh \left (2 a-\frac {2 b c}{d}\right ) \text {Chi}\left (\frac {2 b c}{d}+2 b x\right )}{3 d^4}+\frac {2 b^3 \cosh \left (2 a-\frac {2 b c}{d}\right ) \text {Shi}\left (\frac {2 b c}{d}+2 b x\right )}{3 d^4}-\frac {2 b^2 \sinh ^2(a+b x)}{3 d^3 (c+d x)}-\frac {b \sinh (a+b x) \cosh (a+b x)}{3 d^2 (c+d x)^2}-\frac {\sinh ^2(a+b x)}{3 d (c+d x)^3}-\frac {b^2}{3 d^3 (c+d x)} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 32
Rule 3379
Rule 3382
Rule 3384
Rule 3394
Rule 3395
Rubi steps
\begin {align*} \int \frac {\sinh ^2(a+b x)}{(c+d x)^4} \, dx &=-\frac {b \cosh (a+b x) \sinh (a+b x)}{3 d^2 (c+d x)^2}-\frac {\sinh ^2(a+b x)}{3 d (c+d x)^3}+\frac {b^2 \int \frac {1}{(c+d x)^2} \, dx}{3 d^2}+\frac {\left (2 b^2\right ) \int \frac {\sinh ^2(a+b x)}{(c+d x)^2} \, dx}{3 d^2}\\ &=-\frac {b^2}{3 d^3 (c+d x)}-\frac {b \cosh (a+b x) \sinh (a+b x)}{3 d^2 (c+d x)^2}-\frac {\sinh ^2(a+b x)}{3 d (c+d x)^3}-\frac {2 b^2 \sinh ^2(a+b x)}{3 d^3 (c+d x)}-\frac {\left (4 i b^3\right ) \int \frac {i \sinh (2 a+2 b x)}{2 (c+d x)} \, dx}{3 d^3}\\ &=-\frac {b^2}{3 d^3 (c+d x)}-\frac {b \cosh (a+b x) \sinh (a+b x)}{3 d^2 (c+d x)^2}-\frac {\sinh ^2(a+b x)}{3 d (c+d x)^3}-\frac {2 b^2 \sinh ^2(a+b x)}{3 d^3 (c+d x)}+\frac {\left (2 b^3\right ) \int \frac {\sinh (2 a+2 b x)}{c+d x} \, dx}{3 d^3}\\ &=-\frac {b^2}{3 d^3 (c+d x)}-\frac {b \cosh (a+b x) \sinh (a+b x)}{3 d^2 (c+d x)^2}-\frac {\sinh ^2(a+b x)}{3 d (c+d x)^3}-\frac {2 b^2 \sinh ^2(a+b x)}{3 d^3 (c+d x)}+\frac {\left (2 b^3 \cosh \left (2 a-\frac {2 b c}{d}\right )\right ) \int \frac {\sinh \left (\frac {2 b c}{d}+2 b x\right )}{c+d x} \, dx}{3 d^3}+\frac {\left (2 b^3 \sinh \left (2 a-\frac {2 b c}{d}\right )\right ) \int \frac {\cosh \left (\frac {2 b c}{d}+2 b x\right )}{c+d x} \, dx}{3 d^3}\\ &=-\frac {b^2}{3 d^3 (c+d x)}+\frac {2 b^3 \text {Chi}\left (\frac {2 b c}{d}+2 b x\right ) \sinh \left (2 a-\frac {2 b c}{d}\right )}{3 d^4}-\frac {b \cosh (a+b x) \sinh (a+b x)}{3 d^2 (c+d x)^2}-\frac {\sinh ^2(a+b x)}{3 d (c+d x)^3}-\frac {2 b^2 \sinh ^2(a+b x)}{3 d^3 (c+d x)}+\frac {2 b^3 \cosh \left (2 a-\frac {2 b c}{d}\right ) \text {Shi}\left (\frac {2 b c}{d}+2 b x\right )}{3 d^4}\\ \end {align*}
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Mathematica [A]
time = 0.61, size = 123, normalized size = 0.76 \begin {gather*} \frac {4 b^3 \text {Chi}\left (\frac {2 b (c+d x)}{d}\right ) \sinh \left (2 a-\frac {2 b c}{d}\right )-\frac {d \left (\left (d^2+2 b^2 (c+d x)^2\right ) \cosh (2 (a+b x))+d (-d+b (c+d x) \sinh (2 (a+b x)))\right )}{(c+d x)^3}+4 b^3 \cosh \left (2 a-\frac {2 b c}{d}\right ) \text {Shi}\left (\frac {2 b (c+d x)}{d}\right )}{6 d^4} \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. \(554\) vs.
\(2(150)=300\).
time = 3.25, size = 555, normalized size = 3.43
method | result | size |
risch | \(\frac {1}{6 \left (d x +c \right )^{3} d}-\frac {b^{5} {\mathrm e}^{-2 b x -2 a} x^{2}}{6 d \left (d^{3} x^{3} b^{3}+3 b^{3} c \,d^{2} x^{2}+3 b^{3} c^{2} d x +b^{3} c^{3}\right )}-\frac {b^{5} {\mathrm e}^{-2 b x -2 a} c x}{3 d^{2} \left (d^{3} x^{3} b^{3}+3 b^{3} c \,d^{2} x^{2}+3 b^{3} c^{2} d x +b^{3} c^{3}\right )}-\frac {b^{5} {\mathrm e}^{-2 b x -2 a} c^{2}}{6 d^{3} \left (d^{3} x^{3} b^{3}+3 b^{3} c \,d^{2} x^{2}+3 b^{3} c^{2} d x +b^{3} c^{3}\right )}+\frac {b^{4} {\mathrm e}^{-2 b x -2 a} x}{12 d \left (d^{3} x^{3} b^{3}+3 b^{3} c \,d^{2} x^{2}+3 b^{3} c^{2} d x +b^{3} c^{3}\right )}+\frac {b^{4} {\mathrm e}^{-2 b x -2 a} c}{12 d^{2} \left (d^{3} x^{3} b^{3}+3 b^{3} c \,d^{2} x^{2}+3 b^{3} c^{2} d x +b^{3} c^{3}\right )}-\frac {b^{3} {\mathrm e}^{-2 b x -2 a}}{12 d \left (d^{3} x^{3} b^{3}+3 b^{3} c \,d^{2} x^{2}+3 b^{3} c^{2} d x +b^{3} c^{3}\right )}+\frac {b^{3} {\mathrm e}^{-\frac {2 \left (a d -b c \right )}{d}} \expIntegral \left (1, 2 b x +2 a -\frac {2 \left (a d -b c \right )}{d}\right )}{3 d^{4}}-\frac {b^{3} {\mathrm e}^{2 b x +2 a}}{12 d^{4} \left (\frac {b c}{d}+b x \right )^{3}}-\frac {b^{3} {\mathrm e}^{2 b x +2 a}}{12 d^{4} \left (\frac {b c}{d}+b x \right )^{2}}-\frac {b^{3} {\mathrm e}^{2 b x +2 a}}{6 d^{4} \left (\frac {b c}{d}+b x \right )}-\frac {b^{3} {\mathrm e}^{\frac {2 a d -2 b c}{d}} \expIntegral \left (1, -2 b x -2 a -\frac {2 \left (-a d +b c \right )}{d}\right )}{3 d^{4}}\) | \(555\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.31, size = 110, normalized size = 0.68 \begin {gather*} \frac {1}{6 \, {\left (d^{4} x^{3} + 3 \, c d^{3} x^{2} + 3 \, c^{2} d^{2} x + c^{3} d\right )}} - \frac {e^{\left (-2 \, a + \frac {2 \, b c}{d}\right )} E_{4}\left (\frac {2 \, {\left (d x + c\right )} b}{d}\right )}{4 \, {\left (d x + c\right )}^{3} d} - \frac {e^{\left (2 \, a - \frac {2 \, b c}{d}\right )} E_{4}\left (-\frac {2 \, {\left (d x + c\right )} b}{d}\right )}{4 \, {\left (d x + c\right )}^{3} 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. 411 vs.
\(2 (150) = 300\).
time = 0.36, size = 411, normalized size = 2.54 \begin {gather*} \frac {d^{3} - {\left (2 \, b^{2} d^{3} x^{2} + 4 \, b^{2} c d^{2} x + 2 \, b^{2} c^{2} d + d^{3}\right )} \cosh \left (b x + a\right )^{2} - 2 \, {\left (b d^{3} x + b c d^{2}\right )} \cosh \left (b x + a\right ) \sinh \left (b x + a\right ) - {\left (2 \, b^{2} d^{3} x^{2} + 4 \, b^{2} c d^{2} x + 2 \, b^{2} c^{2} d + d^{3}\right )} \sinh \left (b x + a\right )^{2} + 2 \, {\left ({\left (b^{3} d^{3} x^{3} + 3 \, b^{3} c d^{2} x^{2} + 3 \, b^{3} c^{2} d x + b^{3} c^{3}\right )} {\rm Ei}\left (\frac {2 \, {\left (b d x + b c\right )}}{d}\right ) - {\left (b^{3} d^{3} x^{3} + 3 \, b^{3} c d^{2} x^{2} + 3 \, b^{3} c^{2} d x + b^{3} c^{3}\right )} {\rm Ei}\left (-\frac {2 \, {\left (b d x + b c\right )}}{d}\right )\right )} \cosh \left (-\frac {2 \, {\left (b c - a d\right )}}{d}\right ) + 2 \, {\left ({\left (b^{3} d^{3} x^{3} + 3 \, b^{3} c d^{2} x^{2} + 3 \, b^{3} c^{2} d x + b^{3} c^{3}\right )} {\rm Ei}\left (\frac {2 \, {\left (b d x + b c\right )}}{d}\right ) + {\left (b^{3} d^{3} x^{3} + 3 \, b^{3} c d^{2} x^{2} + 3 \, b^{3} c^{2} d x + b^{3} c^{3}\right )} {\rm Ei}\left (-\frac {2 \, {\left (b d x + b c\right )}}{d}\right )\right )} \sinh \left (-\frac {2 \, {\left (b c - a d\right )}}{d}\right )}{6 \, {\left (d^{7} x^{3} + 3 \, c d^{6} x^{2} + 3 \, c^{2} d^{5} x + c^{3} d^{4}\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 ^{2}{\left (a + b x \right )}}{\left (c + d x\right )^{4}}\, 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. 537 vs.
\(2 (150) = 300\).
time = 0.42, size = 537, normalized size = 3.31 \begin {gather*} \frac {4 \, b^{3} d^{3} x^{3} {\rm Ei}\left (\frac {2 \, {\left (b d x + b c\right )}}{d}\right ) e^{\left (2 \, a - \frac {2 \, b c}{d}\right )} - 4 \, b^{3} d^{3} x^{3} {\rm Ei}\left (-\frac {2 \, {\left (b d x + b c\right )}}{d}\right ) e^{\left (-2 \, a + \frac {2 \, b c}{d}\right )} + 12 \, b^{3} c d^{2} x^{2} {\rm Ei}\left (\frac {2 \, {\left (b d x + b c\right )}}{d}\right ) e^{\left (2 \, a - \frac {2 \, b c}{d}\right )} - 12 \, b^{3} c d^{2} x^{2} {\rm Ei}\left (-\frac {2 \, {\left (b d x + b c\right )}}{d}\right ) e^{\left (-2 \, a + \frac {2 \, b c}{d}\right )} + 12 \, b^{3} c^{2} d x {\rm Ei}\left (\frac {2 \, {\left (b d x + b c\right )}}{d}\right ) e^{\left (2 \, a - \frac {2 \, b c}{d}\right )} - 12 \, b^{3} c^{2} d x {\rm Ei}\left (-\frac {2 \, {\left (b d x + b c\right )}}{d}\right ) e^{\left (-2 \, a + \frac {2 \, b c}{d}\right )} - 2 \, b^{2} d^{3} x^{2} e^{\left (2 \, b x + 2 \, a\right )} - 2 \, b^{2} d^{3} x^{2} e^{\left (-2 \, b x - 2 \, a\right )} + 4 \, b^{3} c^{3} {\rm Ei}\left (\frac {2 \, {\left (b d x + b c\right )}}{d}\right ) e^{\left (2 \, a - \frac {2 \, b c}{d}\right )} - 4 \, b^{3} c^{3} {\rm Ei}\left (-\frac {2 \, {\left (b d x + b c\right )}}{d}\right ) e^{\left (-2 \, a + \frac {2 \, b c}{d}\right )} - 4 \, b^{2} c d^{2} x e^{\left (2 \, b x + 2 \, a\right )} - 4 \, b^{2} c d^{2} x e^{\left (-2 \, b x - 2 \, a\right )} - 2 \, b^{2} c^{2} d e^{\left (2 \, b x + 2 \, a\right )} - b d^{3} x e^{\left (2 \, b x + 2 \, a\right )} - 2 \, b^{2} c^{2} d e^{\left (-2 \, b x - 2 \, a\right )} + b d^{3} x e^{\left (-2 \, b x - 2 \, a\right )} - b c d^{2} e^{\left (2 \, b x + 2 \, a\right )} + b c d^{2} e^{\left (-2 \, b x - 2 \, a\right )} - d^{3} e^{\left (2 \, b x + 2 \, a\right )} - d^{3} e^{\left (-2 \, b x - 2 \, a\right )} + 2 \, d^{3}}{12 \, {\left (d^{7} x^{3} + 3 \, c d^{6} x^{2} + 3 \, c^{2} d^{5} x + c^{3} d^{4}\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 )}^2}{{\left (c+d\,x\right )}^4} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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