Optimal. Leaf size=112 \[ \frac {b^2 \cosh \left (2 a-\frac {2 b c}{d}\right ) \text {Chi}\left (\frac {2 b c}{d}+2 b x\right )}{d^3}+\frac {b^2 \sinh \left (2 a-\frac {2 b c}{d}\right ) \text {Shi}\left (\frac {2 b c}{d}+2 b x\right )}{d^3}-\frac {b \sinh (a+b x) \cosh (a+b x)}{d^2 (c+d x)}-\frac {\sinh ^2(a+b x)}{2 d (c+d x)^2} \]
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Rubi [A] time = 0.20, antiderivative size = 112, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.375, Rules used = {3314, 31, 3312, 3303, 3298, 3301} \[ \frac {b^2 \cosh \left (2 a-\frac {2 b c}{d}\right ) \text {Chi}\left (\frac {2 b c}{d}+2 b x\right )}{d^3}+\frac {b^2 \sinh \left (2 a-\frac {2 b c}{d}\right ) \text {Shi}\left (\frac {2 b c}{d}+2 b x\right )}{d^3}-\frac {b \sinh (a+b x) \cosh (a+b x)}{d^2 (c+d x)}-\frac {\sinh ^2(a+b x)}{2 d (c+d x)^2} \]
Antiderivative was successfully verified.
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Rule 31
Rule 3298
Rule 3301
Rule 3303
Rule 3312
Rule 3314
Rubi steps
\begin {align*} \int \frac {\sinh ^2(a+b x)}{(c+d x)^3} \, dx &=-\frac {b \cosh (a+b x) \sinh (a+b x)}{d^2 (c+d x)}-\frac {\sinh ^2(a+b x)}{2 d (c+d x)^2}+\frac {b^2 \int \frac {1}{c+d x} \, dx}{d^2}+\frac {\left (2 b^2\right ) \int \frac {\sinh ^2(a+b x)}{c+d x} \, dx}{d^2}\\ &=\frac {b^2 \log (c+d x)}{d^3}-\frac {b \cosh (a+b x) \sinh (a+b x)}{d^2 (c+d x)}-\frac {\sinh ^2(a+b x)}{2 d (c+d x)^2}-\frac {\left (2 b^2\right ) \int \left (\frac {1}{2 (c+d x)}-\frac {\cosh (2 a+2 b x)}{2 (c+d x)}\right ) \, dx}{d^2}\\ &=-\frac {b \cosh (a+b x) \sinh (a+b x)}{d^2 (c+d x)}-\frac {\sinh ^2(a+b x)}{2 d (c+d x)^2}+\frac {b^2 \int \frac {\cosh (2 a+2 b x)}{c+d x} \, dx}{d^2}\\ &=-\frac {b \cosh (a+b x) \sinh (a+b x)}{d^2 (c+d x)}-\frac {\sinh ^2(a+b x)}{2 d (c+d x)^2}+\frac {\left (b^2 \cosh \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}{d^2}+\frac {\left (b^2 \sinh \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}{d^2}\\ &=\frac {b^2 \cosh \left (2 a-\frac {2 b c}{d}\right ) \text {Chi}\left (\frac {2 b c}{d}+2 b x\right )}{d^3}-\frac {b \cosh (a+b x) \sinh (a+b x)}{d^2 (c+d x)}-\frac {\sinh ^2(a+b x)}{2 d (c+d x)^2}+\frac {b^2 \sinh \left (2 a-\frac {2 b c}{d}\right ) \text {Shi}\left (\frac {2 b c}{d}+2 b x\right )}{d^3}\\ \end {align*}
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Mathematica [A] time = 0.77, size = 102, normalized size = 0.91 \[ \frac {2 b^2 \cosh \left (2 a-\frac {2 b c}{d}\right ) \text {Chi}\left (\frac {2 b (c+d x)}{d}\right )+2 b^2 \sinh \left (2 a-\frac {2 b c}{d}\right ) \text {Shi}\left (\frac {2 b (c+d x)}{d}\right )-\frac {d \left (b (c+d x) \sinh (2 (a+b x))+d \sinh ^2(a+b x)\right )}{(c+d x)^2}}{2 d^3} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.38, size = 280, normalized size = 2.50 \[ -\frac {d^{2} \cosh \left (b x + a\right )^{2} + d^{2} \sinh \left (b x + a\right )^{2} + 4 \, {\left (b d^{2} x + b c d\right )} \cosh \left (b x + a\right ) \sinh \left (b x + a\right ) - d^{2} - 2 \, {\left ({\left (b^{2} d^{2} x^{2} + 2 \, b^{2} c d x + b^{2} c^{2}\right )} {\rm Ei}\left (\frac {2 \, {\left (b d x + b c\right )}}{d}\right ) + {\left (b^{2} d^{2} x^{2} + 2 \, b^{2} c d x + b^{2} c^{2}\right )} {\rm Ei}\left (-\frac {2 \, {\left (b d x + b c\right )}}{d}\right )\right )} \cosh \left (-\frac {2 \, {\left (b c - a d\right )}}{d}\right ) - 2 \, {\left ({\left (b^{2} d^{2} x^{2} + 2 \, b^{2} c d x + b^{2} c^{2}\right )} {\rm Ei}\left (\frac {2 \, {\left (b d x + b c\right )}}{d}\right ) - {\left (b^{2} d^{2} x^{2} + 2 \, b^{2} c d x + b^{2} c^{2}\right )} {\rm Ei}\left (-\frac {2 \, {\left (b d x + b c\right )}}{d}\right )\right )} \sinh \left (-\frac {2 \, {\left (b c - a d\right )}}{d}\right )}{4 \, {\left (d^{5} x^{2} + 2 \, c d^{4} x + c^{2} d^{3}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.18, size = 330, normalized size = 2.95 \[ \frac {4 \, b^{2} d^{2} x^{2} {\rm Ei}\left (\frac {2 \, {\left (b d x + b c\right )}}{d}\right ) e^{\left (2 \, a - \frac {2 \, b c}{d}\right )} + 4 \, b^{2} d^{2} x^{2} {\rm Ei}\left (-\frac {2 \, {\left (b d x + b c\right )}}{d}\right ) e^{\left (-2 \, a + \frac {2 \, b c}{d}\right )} + 8 \, b^{2} c d x {\rm Ei}\left (\frac {2 \, {\left (b d x + b c\right )}}{d}\right ) e^{\left (2 \, a - \frac {2 \, b c}{d}\right )} + 8 \, b^{2} c d x {\rm Ei}\left (-\frac {2 \, {\left (b d x + b c\right )}}{d}\right ) e^{\left (-2 \, a + \frac {2 \, b c}{d}\right )} + 4 \, b^{2} c^{2} {\rm Ei}\left (\frac {2 \, {\left (b d x + b c\right )}}{d}\right ) e^{\left (2 \, a - \frac {2 \, b c}{d}\right )} + 4 \, b^{2} c^{2} {\rm Ei}\left (-\frac {2 \, {\left (b d x + b c\right )}}{d}\right ) e^{\left (-2 \, a + \frac {2 \, b c}{d}\right )} - 2 \, b d^{2} x e^{\left (2 \, b x + 2 \, a\right )} + 2 \, b d^{2} x e^{\left (-2 \, b x - 2 \, a\right )} - 2 \, b c d e^{\left (2 \, b x + 2 \, a\right )} + 2 \, b c d e^{\left (-2 \, b x - 2 \, a\right )} - d^{2} e^{\left (2 \, b x + 2 \, a\right )} - d^{2} e^{\left (-2 \, b x - 2 \, a\right )} + 2 \, d^{2}}{8 \, {\left (d^{5} x^{2} + 2 \, c d^{4} x + c^{2} d^{3}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.15, size = 299, normalized size = 2.67 \[ \frac {1}{4 d \left (d x +c \right )^{2}}+\frac {b^{3} {\mathrm e}^{-2 b x -2 a} x}{4 d \left (b^{2} d^{2} x^{2}+2 b^{2} c d x +c^{2} b^{2}\right )}+\frac {b^{3} {\mathrm e}^{-2 b x -2 a} c}{4 d^{2} \left (b^{2} d^{2} x^{2}+2 b^{2} c d x +c^{2} b^{2}\right )}-\frac {b^{2} {\mathrm e}^{-2 b x -2 a}}{8 d \left (b^{2} d^{2} x^{2}+2 b^{2} c d x +c^{2} b^{2}\right )}-\frac {b^{2} {\mathrm e}^{-\frac {2 \left (d a -c b \right )}{d}} \Ei \left (1, 2 b x +2 a -\frac {2 \left (d a -c b \right )}{d}\right )}{2 d^{3}}-\frac {b^{2} {\mathrm e}^{2 b x +2 a}}{8 d^{3} \left (\frac {b c}{d}+b x \right )^{2}}-\frac {b^{2} {\mathrm e}^{2 b x +2 a}}{4 d^{3} \left (\frac {b c}{d}+b x \right )}-\frac {b^{2} {\mathrm e}^{\frac {2 d a -2 c b}{d}} \Ei \left (1, -2 b x -2 a -\frac {2 \left (-d a +c b \right )}{d}\right )}{2 d^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.71, size = 99, normalized size = 0.88 \[ \frac {1}{4 \, {\left (d^{3} x^{2} + 2 \, c d^{2} x + c^{2} d\right )}} - \frac {e^{\left (-2 \, a + \frac {2 \, b c}{d}\right )} E_{3}\left (\frac {2 \, {\left (d x + c\right )} b}{d}\right )}{4 \, {\left (d x + c\right )}^{2} d} - \frac {e^{\left (2 \, a - \frac {2 \, b c}{d}\right )} E_{3}\left (-\frac {2 \, {\left (d x + c\right )} b}{d}\right )}{4 \, {\left (d x + c\right )}^{2} d} \]
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 \[ \int \frac {{\mathrm {sinh}\left (a+b\,x\right )}^2}{{\left (c+d\,x\right )}^3} \,d x \]
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 \[ \int \frac {\sinh ^{2}{\left (a + b x \right )}}{\left (c + d x\right )^{3}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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