Integrand size = 16, antiderivative size = 184 \[ \int \frac {\sinh ^3(a+b x)}{(c+d x)^3} \, dx=\frac {9 b^2 \text {Chi}\left (\frac {3 b c}{d}+3 b x\right ) \sinh \left (3 a-\frac {3 b c}{d}\right )}{8 d^3}-\frac {3 b^2 \text {Chi}\left (\frac {b c}{d}+b x\right ) \sinh \left (a-\frac {b c}{d}\right )}{8 d^3}-\frac {3 b \cosh (a+b x) \sinh ^2(a+b x)}{2 d^2 (c+d x)}-\frac {\sinh ^3(a+b x)}{2 d (c+d x)^2}-\frac {3 b^2 \cosh \left (a-\frac {b c}{d}\right ) \text {Shi}\left (\frac {b c}{d}+b x\right )}{8 d^3}+\frac {9 b^2 \cosh \left (3 a-\frac {3 b c}{d}\right ) \text {Shi}\left (\frac {3 b c}{d}+3 b x\right )}{8 d^3} \]
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Time = 0.32 (sec) , antiderivative size = 184, normalized size of antiderivative = 1.00, number of steps used = 12, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.312, Rules used = {3395, 3384, 3379, 3382, 3393} \[ \int \frac {\sinh ^3(a+b x)}{(c+d x)^3} \, dx=\frac {9 b^2 \sinh \left (3 a-\frac {3 b c}{d}\right ) \text {Chi}\left (\frac {3 b c}{d}+3 b x\right )}{8 d^3}-\frac {3 b^2 \sinh \left (a-\frac {b c}{d}\right ) \text {Chi}\left (\frac {b c}{d}+b x\right )}{8 d^3}-\frac {3 b^2 \cosh \left (a-\frac {b c}{d}\right ) \text {Shi}\left (\frac {b c}{d}+b x\right )}{8 d^3}+\frac {9 b^2 \cosh \left (3 a-\frac {3 b c}{d}\right ) \text {Shi}\left (\frac {3 b c}{d}+3 b x\right )}{8 d^3}-\frac {3 b \sinh ^2(a+b x) \cosh (a+b x)}{2 d^2 (c+d x)}-\frac {\sinh ^3(a+b x)}{2 d (c+d x)^2} \]
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Rule 3379
Rule 3382
Rule 3384
Rule 3393
Rule 3395
Rubi steps \begin{align*} \text {integral}& = -\frac {3 b \cosh (a+b x) \sinh ^2(a+b x)}{2 d^2 (c+d x)}-\frac {\sinh ^3(a+b x)}{2 d (c+d x)^2}+\frac {\left (3 b^2\right ) \int \frac {\sinh (a+b x)}{c+d x} \, dx}{d^2}+\frac {\left (9 b^2\right ) \int \frac {\sinh ^3(a+b x)}{c+d x} \, dx}{2 d^2} \\ & = -\frac {3 b \cosh (a+b x) \sinh ^2(a+b x)}{2 d^2 (c+d x)}-\frac {\sinh ^3(a+b x)}{2 d (c+d x)^2}+\frac {\left (9 i b^2\right ) \int \left (\frac {3 i \sinh (a+b x)}{4 (c+d x)}-\frac {i \sinh (3 a+3 b x)}{4 (c+d x)}\right ) \, dx}{2 d^2}+\frac {\left (3 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}{d^2}+\frac {\left (3 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}{d^2} \\ & = \frac {3 b^2 \text {Chi}\left (\frac {b c}{d}+b x\right ) \sinh \left (a-\frac {b c}{d}\right )}{d^3}-\frac {3 b \cosh (a+b x) \sinh ^2(a+b x)}{2 d^2 (c+d x)}-\frac {\sinh ^3(a+b x)}{2 d (c+d x)^2}+\frac {3 b^2 \cosh \left (a-\frac {b c}{d}\right ) \text {Shi}\left (\frac {b c}{d}+b x\right )}{d^3}+\frac {\left (9 b^2\right ) \int \frac {\sinh (3 a+3 b x)}{c+d x} \, dx}{8 d^2}-\frac {\left (27 b^2\right ) \int \frac {\sinh (a+b x)}{c+d x} \, dx}{8 d^2} \\ & = \frac {3 b^2 \text {Chi}\left (\frac {b c}{d}+b x\right ) \sinh \left (a-\frac {b c}{d}\right )}{d^3}-\frac {3 b \cosh (a+b x) \sinh ^2(a+b x)}{2 d^2 (c+d x)}-\frac {\sinh ^3(a+b x)}{2 d (c+d x)^2}+\frac {3 b^2 \cosh \left (a-\frac {b c}{d}\right ) \text {Shi}\left (\frac {b c}{d}+b x\right )}{d^3}+\frac {\left (9 b^2 \cosh \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}{8 d^2}-\frac {\left (27 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}{8 d^2}+\frac {\left (9 b^2 \sinh \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}{8 d^2}-\frac {\left (27 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}{8 d^2} \\ & = \frac {9 b^2 \text {Chi}\left (\frac {3 b c}{d}+3 b x\right ) \sinh \left (3 a-\frac {3 b c}{d}\right )}{8 d^3}-\frac {3 b^2 \text {Chi}\left (\frac {b c}{d}+b x\right ) \sinh \left (a-\frac {b c}{d}\right )}{8 d^3}-\frac {3 b \cosh (a+b x) \sinh ^2(a+b x)}{2 d^2 (c+d x)}-\frac {\sinh ^3(a+b x)}{2 d (c+d x)^2}-\frac {3 b^2 \cosh \left (a-\frac {b c}{d}\right ) \text {Shi}\left (\frac {b c}{d}+b x\right )}{8 d^3}+\frac {9 b^2 \cosh \left (3 a-\frac {3 b c}{d}\right ) \text {Shi}\left (\frac {3 b c}{d}+3 b x\right )}{8 d^3} \\ \end{align*}
Time = 0.57 (sec) , antiderivative size = 220, normalized size of antiderivative = 1.20 \[ \int \frac {\sinh ^3(a+b x)}{(c+d x)^3} \, dx=\frac {6 d \cosh (b x) (b (c+d x) \cosh (a)+d \sinh (a))-2 d \cosh (3 b x) (3 b (c+d x) \cosh (3 a)+d \sinh (3 a))+6 d (d \cosh (a)+b (c+d x) \sinh (a)) \sinh (b x)-2 d (d \cosh (3 a)+3 b (c+d x) \sinh (3 a)) \sinh (3 b x)+6 b^2 (c+d x)^2 \left (3 \text {Chi}\left (\frac {3 b (c+d x)}{d}\right ) \sinh \left (3 a-\frac {3 b c}{d}\right )-\text {Chi}\left (b \left (\frac {c}{d}+x\right )\right ) \sinh \left (a-\frac {b c}{d}\right )-\cosh \left (a-\frac {b c}{d}\right ) \text {Shi}\left (b \left (\frac {c}{d}+x\right )\right )+3 \cosh \left (3 a-\frac {3 b c}{d}\right ) \text {Shi}\left (\frac {3 b (c+d x)}{d}\right )\right )}{16 d^3 (c+d x)^2} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(561\) vs. \(2(172)=344\).
Time = 2.38 (sec) , antiderivative size = 562, normalized size of antiderivative = 3.05
method | result | size |
risch | \(-\frac {3 b^{3} {\mathrm e}^{-3 b x -3 a} x}{16 d \left (b^{2} d^{2} x^{2}+2 b^{2} c d x +b^{2} c^{2}\right )}-\frac {3 b^{3} {\mathrm e}^{-3 b x -3 a} c}{16 d^{2} \left (b^{2} d^{2} x^{2}+2 b^{2} c d x +b^{2} c^{2}\right )}+\frac {b^{2} {\mathrm e}^{-3 b x -3 a}}{16 d \left (b^{2} d^{2} x^{2}+2 b^{2} c d x +b^{2} c^{2}\right )}+\frac {9 b^{2} {\mathrm e}^{-\frac {3 \left (a d -b c \right )}{d}} \operatorname {Ei}_{1}\left (3 b x +3 a -\frac {3 \left (a d -b c \right )}{d}\right )}{16 d^{3}}+\frac {3 b^{3} {\mathrm e}^{-b x -a} x}{16 d \left (b^{2} d^{2} x^{2}+2 b^{2} c d x +b^{2} c^{2}\right )}+\frac {3 b^{3} {\mathrm e}^{-b x -a} c}{16 d^{2} \left (b^{2} d^{2} x^{2}+2 b^{2} c d x +b^{2} c^{2}\right )}-\frac {3 b^{2} {\mathrm e}^{-b x -a}}{16 d \left (b^{2} d^{2} x^{2}+2 b^{2} c d x +b^{2} c^{2}\right )}-\frac {3 b^{2} {\mathrm e}^{-\frac {a d -b c}{d}} \operatorname {Ei}_{1}\left (b x +a -\frac {a d -b c}{d}\right )}{16 d^{3}}+\frac {3 b^{2} {\mathrm e}^{b x +a}}{16 d^{3} \left (\frac {b c}{d}+b x \right )^{2}}+\frac {3 b^{2} {\mathrm e}^{b x +a}}{16 d^{3} \left (\frac {b c}{d}+b x \right )}+\frac {3 b^{2} {\mathrm e}^{\frac {a d -b c}{d}} \operatorname {Ei}_{1}\left (-b x -a -\frac {-a d +b c}{d}\right )}{16 d^{3}}-\frac {b^{2} {\mathrm e}^{3 b x +3 a}}{16 d^{3} \left (\frac {b c}{d}+b x \right )^{2}}-\frac {3 b^{2} {\mathrm e}^{3 b x +3 a}}{16 d^{3} \left (\frac {b c}{d}+b x \right )}-\frac {9 b^{2} {\mathrm e}^{\frac {3 a d -3 b c}{d}} \operatorname {Ei}_{1}\left (-3 b x -3 a -\frac {3 \left (-a d +b c \right )}{d}\right )}{16 d^{3}}\) | \(562\) |
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Leaf count of result is larger than twice the leaf count of optimal. 529 vs. \(2 (172) = 344\).
Time = 0.25 (sec) , antiderivative size = 529, normalized size of antiderivative = 2.88 \[ \int \frac {\sinh ^3(a+b x)}{(c+d x)^3} \, dx=-\frac {2 \, d^{2} \sinh \left (b x + a\right )^{3} + 6 \, {\left (b d^{2} x + b c d\right )} \cosh \left (b x + a\right )^{3} + 18 \, {\left (b d^{2} x + b c d\right )} \cosh \left (b x + a\right ) \sinh \left (b x + a\right )^{2} - 6 \, {\left (b d^{2} x + b c d\right )} \cosh \left (b x + a\right ) + 3 \, {\left ({\left (b^{2} d^{2} x^{2} + 2 \, b^{2} c d x + b^{2} c^{2}\right )} {\rm Ei}\left (\frac {b d x + b c}{d}\right ) - {\left (b^{2} d^{2} x^{2} + 2 \, b^{2} c d x + b^{2} c^{2}\right )} {\rm Ei}\left (-\frac {b d x + b c}{d}\right )\right )} \cosh \left (-\frac {b c - a d}{d}\right ) - 9 \, {\left ({\left (b^{2} d^{2} x^{2} + 2 \, b^{2} c d x + b^{2} c^{2}\right )} {\rm Ei}\left (\frac {3 \, {\left (b d x + b c\right )}}{d}\right ) - {\left (b^{2} d^{2} x^{2} + 2 \, b^{2} c d x + b^{2} c^{2}\right )} {\rm Ei}\left (-\frac {3 \, {\left (b d x + b c\right )}}{d}\right )\right )} \cosh \left (-\frac {3 \, {\left (b c - a d\right )}}{d}\right ) + 6 \, {\left (d^{2} \cosh \left (b x + a\right )^{2} - d^{2}\right )} \sinh \left (b x + a\right ) + 3 \, {\left ({\left (b^{2} d^{2} x^{2} + 2 \, b^{2} c d x + b^{2} c^{2}\right )} {\rm Ei}\left (\frac {b d x + b c}{d}\right ) + {\left (b^{2} d^{2} x^{2} + 2 \, b^{2} c d x + b^{2} c^{2}\right )} {\rm Ei}\left (-\frac {b d x + b c}{d}\right )\right )} \sinh \left (-\frac {b c - a d}{d}\right ) - 9 \, {\left ({\left (b^{2} d^{2} x^{2} + 2 \, b^{2} c d x + b^{2} c^{2}\right )} {\rm Ei}\left (\frac {3 \, {\left (b d x + b c\right )}}{d}\right ) + {\left (b^{2} d^{2} x^{2} + 2 \, b^{2} c d x + b^{2} c^{2}\right )} {\rm Ei}\left (-\frac {3 \, {\left (b d x + b c\right )}}{d}\right )\right )} \sinh \left (-\frac {3 \, {\left (b c - a d\right )}}{d}\right )}{16 \, {\left (d^{5} x^{2} + 2 \, c d^{4} x + c^{2} d^{3}\right )}} \]
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\[ \int \frac {\sinh ^3(a+b x)}{(c+d x)^3} \, dx=\int \frac {\sinh ^{3}{\left (a + b x \right )}}{\left (c + d x\right )^{3}}\, dx \]
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Time = 0.25 (sec) , antiderivative size = 145, normalized size of antiderivative = 0.79 \[ \int \frac {\sinh ^3(a+b x)}{(c+d x)^3} \, dx=\frac {e^{\left (-3 \, a + \frac {3 \, b c}{d}\right )} E_{3}\left (\frac {3 \, {\left (d x + c\right )} b}{d}\right )}{8 \, {\left (d x + c\right )}^{2} d} - \frac {3 \, e^{\left (-a + \frac {b c}{d}\right )} E_{3}\left (\frac {{\left (d x + c\right )} b}{d}\right )}{8 \, {\left (d x + c\right )}^{2} d} + \frac {3 \, e^{\left (a - \frac {b c}{d}\right )} E_{3}\left (-\frac {{\left (d x + c\right )} b}{d}\right )}{8 \, {\left (d x + c\right )}^{2} d} - \frac {e^{\left (3 \, a - \frac {3 \, b c}{d}\right )} E_{3}\left (-\frac {3 \, {\left (d x + c\right )} b}{d}\right )}{8 \, {\left (d x + c\right )}^{2} d} \]
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Leaf count of result is larger than twice the leaf count of optimal. 601 vs. \(2 (172) = 344\).
Time = 0.28 (sec) , antiderivative size = 601, normalized size of antiderivative = 3.27 \[ \int \frac {\sinh ^3(a+b x)}{(c+d x)^3} \, dx=\frac {9 \, b^{2} d^{2} x^{2} {\rm Ei}\left (\frac {3 \, {\left (b d x + b c\right )}}{d}\right ) e^{\left (3 \, a - \frac {3 \, b c}{d}\right )} - 3 \, 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 )} + 3 \, 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 )} - 9 \, b^{2} d^{2} x^{2} {\rm Ei}\left (-\frac {3 \, {\left (b d x + b c\right )}}{d}\right ) e^{\left (-3 \, a + \frac {3 \, b c}{d}\right )} + 18 \, b^{2} c d x {\rm Ei}\left (\frac {3 \, {\left (b d x + b c\right )}}{d}\right ) e^{\left (3 \, a - \frac {3 \, b c}{d}\right )} - 6 \, b^{2} c d x {\rm Ei}\left (\frac {b d x + b c}{d}\right ) e^{\left (a - \frac {b c}{d}\right )} + 6 \, b^{2} c d x {\rm Ei}\left (-\frac {b d x + b c}{d}\right ) e^{\left (-a + \frac {b c}{d}\right )} - 18 \, b^{2} c d x {\rm Ei}\left (-\frac {3 \, {\left (b d x + b c\right )}}{d}\right ) e^{\left (-3 \, a + \frac {3 \, b c}{d}\right )} + 9 \, b^{2} c^{2} {\rm Ei}\left (\frac {3 \, {\left (b d x + b c\right )}}{d}\right ) e^{\left (3 \, a - \frac {3 \, b c}{d}\right )} - 3 \, b^{2} c^{2} {\rm Ei}\left (\frac {b d x + b c}{d}\right ) e^{\left (a - \frac {b c}{d}\right )} + 3 \, b^{2} c^{2} {\rm Ei}\left (-\frac {b d x + b c}{d}\right ) e^{\left (-a + \frac {b c}{d}\right )} - 9 \, b^{2} c^{2} {\rm Ei}\left (-\frac {3 \, {\left (b d x + b c\right )}}{d}\right ) e^{\left (-3 \, a + \frac {3 \, b c}{d}\right )} - 3 \, b d^{2} x e^{\left (3 \, b x + 3 \, a\right )} + 3 \, b d^{2} x e^{\left (b x + a\right )} + 3 \, b d^{2} x e^{\left (-b x - a\right )} - 3 \, b d^{2} x e^{\left (-3 \, b x - 3 \, a\right )} - 3 \, b c d e^{\left (3 \, b x + 3 \, a\right )} + 3 \, b c d e^{\left (b x + a\right )} + 3 \, b c d e^{\left (-b x - a\right )} - 3 \, b c d e^{\left (-3 \, b x - 3 \, a\right )} - d^{2} e^{\left (3 \, b x + 3 \, a\right )} + 3 \, d^{2} e^{\left (b x + a\right )} - 3 \, d^{2} e^{\left (-b x - a\right )} + d^{2} e^{\left (-3 \, b x - 3 \, a\right )}}{16 \, {\left (d^{5} x^{2} + 2 \, c d^{4} x + c^{2} d^{3}\right )}} \]
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Timed out. \[ \int \frac {\sinh ^3(a+b x)}{(c+d x)^3} \, dx=\int \frac {{\mathrm {sinh}\left (a+b\,x\right )}^3}{{\left (c+d\,x\right )}^3} \,d x \]
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