Integrand size = 16, antiderivative size = 107 \[ \int \sinh ^2\left (\frac {a+b x}{c+d x}\right ) \, dx=\frac {(b c-a d) \text {Chi}\left (\frac {2 (b c-a d)}{d (c+d x)}\right ) \sinh \left (\frac {2 b}{d}\right )}{d^2}+\frac {(c+d x) \sinh ^2\left (\frac {a+b x}{c+d x}\right )}{d}-\frac {(b c-a d) \cosh \left (\frac {2 b}{d}\right ) \text {Shi}\left (\frac {2 (b c-a d)}{d (c+d x)}\right )}{d^2} \]
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Time = 0.14 (sec) , antiderivative size = 107, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.375, Rules used = {5726, 3394, 12, 3384, 3379, 3382} \[ \int \sinh ^2\left (\frac {a+b x}{c+d x}\right ) \, dx=\frac {\sinh \left (\frac {2 b}{d}\right ) (b c-a d) \text {Chi}\left (\frac {2 (b c-a d)}{d (c+d x)}\right )}{d^2}-\frac {\cosh \left (\frac {2 b}{d}\right ) (b c-a d) \text {Shi}\left (\frac {2 (b c-a d)}{d (c+d x)}\right )}{d^2}+\frac {(c+d x) \sinh ^2\left (\frac {a+b x}{c+d x}\right )}{d} \]
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Rule 12
Rule 3379
Rule 3382
Rule 3384
Rule 3394
Rule 5726
Rubi steps \begin{align*} \text {integral}& = -\frac {\text {Subst}\left (\int \frac {\sinh ^2\left (\frac {b}{d}-\frac {(b c-a d) x}{d}\right )}{x^2} \, dx,x,\frac {1}{c+d x}\right )}{d} \\ & = \frac {(c+d x) \sinh ^2\left (\frac {a+b x}{c+d x}\right )}{d}-\frac {(2 i (b c-a d)) \text {Subst}\left (\int \frac {i \sinh \left (\frac {2 b}{d}-\frac {2 (b c-a d) x}{d}\right )}{2 x} \, dx,x,\frac {1}{c+d x}\right )}{d^2} \\ & = \frac {(c+d x) \sinh ^2\left (\frac {a+b x}{c+d x}\right )}{d}+\frac {(b c-a d) \text {Subst}\left (\int \frac {\sinh \left (\frac {2 b}{d}-\frac {2 (b c-a d) x}{d}\right )}{x} \, dx,x,\frac {1}{c+d x}\right )}{d^2} \\ & = \frac {(c+d x) \sinh ^2\left (\frac {a+b x}{c+d x}\right )}{d}-\frac {\left ((b c-a d) \cosh \left (\frac {2 b}{d}\right )\right ) \text {Subst}\left (\int \frac {\sinh \left (\frac {2 (b c-a d) x}{d}\right )}{x} \, dx,x,\frac {1}{c+d x}\right )}{d^2}+\frac {\left ((b c-a d) \sinh \left (\frac {2 b}{d}\right )\right ) \text {Subst}\left (\int \frac {\cosh \left (\frac {2 (b c-a d) x}{d}\right )}{x} \, dx,x,\frac {1}{c+d x}\right )}{d^2} \\ & = \frac {(b c-a d) \text {Chi}\left (\frac {2 (b c-a d)}{d (c+d x)}\right ) \sinh \left (\frac {2 b}{d}\right )}{d^2}+\frac {(c+d x) \sinh ^2\left (\frac {a+b x}{c+d x}\right )}{d}-\frac {(b c-a d) \cosh \left (\frac {2 b}{d}\right ) \text {Shi}\left (\frac {2 (b c-a d)}{d (c+d x)}\right )}{d^2} \\ \end{align*}
Leaf count is larger than twice the leaf count of optimal. \(475\) vs. \(2(107)=214\).
Time = 2.96 (sec) , antiderivative size = 475, normalized size of antiderivative = 4.44 \[ \int \sinh ^2\left (\frac {a+b x}{c+d x}\right ) \, dx=\frac {c d e^{-\frac {2 (a+b x)}{c+d x}}+c d e^{\frac {2 (a+b x)}{c+d x}}-2 d^2 x+2 d^2 x \cosh \left (\frac {2 b}{d}\right ) \cosh \left (\frac {2 (-b c+a d)}{d (c+d x)}\right )-2 (b c-a d) \text {Chi}\left (\frac {2 b c-2 a d}{c d+d^2 x}\right ) \left (\cosh \left (\frac {2 b}{d}\right )-\sinh \left (\frac {2 b}{d}\right )\right )+2 (b c-a d) \text {Chi}\left (\frac {2 (-b c+a d)}{d (c+d x)}\right ) \left (\cosh \left (\frac {2 b}{d}\right )+\sinh \left (\frac {2 b}{d}\right )\right )+2 d^2 x \sinh \left (\frac {2 b}{d}\right ) \sinh \left (\frac {2 (-b c+a d)}{d (c+d x)}\right )+2 b c \cosh \left (\frac {2 b}{d}\right ) \text {Shi}\left (\frac {2 (-b c+a d)}{d (c+d x)}\right )-2 a d \cosh \left (\frac {2 b}{d}\right ) \text {Shi}\left (\frac {2 (-b c+a d)}{d (c+d x)}\right )+2 b c \sinh \left (\frac {2 b}{d}\right ) \text {Shi}\left (\frac {2 (-b c+a d)}{d (c+d x)}\right )-2 a d \sinh \left (\frac {2 b}{d}\right ) \text {Shi}\left (\frac {2 (-b c+a d)}{d (c+d x)}\right )-2 b c \cosh \left (\frac {2 b}{d}\right ) \text {Shi}\left (\frac {2 b c-2 a d}{c d+d^2 x}\right )+2 a d \cosh \left (\frac {2 b}{d}\right ) \text {Shi}\left (\frac {2 b c-2 a d}{c d+d^2 x}\right )+2 b c \sinh \left (\frac {2 b}{d}\right ) \text {Shi}\left (\frac {2 b c-2 a d}{c d+d^2 x}\right )-2 a d \sinh \left (\frac {2 b}{d}\right ) \text {Shi}\left (\frac {2 b c-2 a d}{c d+d^2 x}\right )}{4 d^2} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(357\) vs. \(2(107)=214\).
Time = 7.67 (sec) , antiderivative size = 358, normalized size of antiderivative = 3.35
method | result | size |
risch | \(-\frac {x}{2}+\frac {{\mathrm e}^{-\frac {2 \left (b x +a \right )}{d x +c}} a}{\frac {4 d a}{d x +c}-\frac {4 b c}{d x +c}}-\frac {{\mathrm e}^{-\frac {2 \left (b x +a \right )}{d x +c}} b c}{4 d \left (\frac {d a}{d x +c}-\frac {b c}{d x +c}\right )}-\frac {{\mathrm e}^{-\frac {2 b}{d}} \operatorname {Ei}_{1}\left (\frac {2 a d -2 b c}{\left (d x +c \right ) d}\right ) a}{2 d}+\frac {{\mathrm e}^{-\frac {2 b}{d}} \operatorname {Ei}_{1}\left (\frac {2 a d -2 b c}{\left (d x +c \right ) d}\right ) b c}{2 d^{2}}+\frac {d \,{\mathrm e}^{\frac {2 b x +2 a}{d x +c}} x a}{4 a d -4 b c}-\frac {{\mathrm e}^{\frac {2 b x +2 a}{d x +c}} x b c}{4 \left (a d -b c \right )}+\frac {{\mathrm e}^{\frac {2 b x +2 a}{d x +c}} c a}{4 a d -4 b c}-\frac {{\mathrm e}^{\frac {2 b x +2 a}{d x +c}} c^{2} b}{4 d \left (a d -b c \right )}+\frac {{\mathrm e}^{\frac {2 b}{d}} \operatorname {Ei}_{1}\left (-\frac {2 \left (a d -b c \right )}{d \left (d x +c \right )}\right ) a}{2 d}-\frac {{\mathrm e}^{\frac {2 b}{d}} \operatorname {Ei}_{1}\left (-\frac {2 \left (a d -b c \right )}{d \left (d x +c \right )}\right ) b c}{2 d^{2}}\) | \(358\) |
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Leaf count of result is larger than twice the leaf count of optimal. 370 vs. \(2 (107) = 214\).
Time = 0.26 (sec) , antiderivative size = 370, normalized size of antiderivative = 3.46 \[ \int \sinh ^2\left (\frac {a+b x}{c+d x}\right ) \, dx=-\frac {d^{2} x - {\left (d^{2} x + c d\right )} \cosh \left (\frac {b x + a}{d x + c}\right )^{2} - {\left (d^{2} x - {\left (b c - a d\right )} {\rm Ei}\left (-\frac {2 \, {\left (b c - a d\right )}}{d^{2} x + c d}\right ) \cosh \left (\frac {2 \, b}{d}\right ) + c d\right )} \sinh \left (\frac {b x + a}{d x + c}\right )^{2} - {\left ({\left (b c - a d\right )} {\rm Ei}\left (-\frac {2 \, {\left (b c - a d\right )}}{d^{2} x + c d}\right ) \cosh \left (\frac {b x + a}{d x + c}\right )^{2} - {\left (b c - a d\right )} {\rm Ei}\left (\frac {2 \, {\left (b c - a d\right )}}{d^{2} x + c d}\right )\right )} \cosh \left (\frac {2 \, b}{d}\right ) - {\left ({\left (b c - a d\right )} {\rm Ei}\left (-\frac {2 \, {\left (b c - a d\right )}}{d^{2} x + c d}\right ) \cosh \left (\frac {b x + a}{d x + c}\right )^{2} - {\left (b c - a d\right )} {\rm Ei}\left (-\frac {2 \, {\left (b c - a d\right )}}{d^{2} x + c d}\right ) \sinh \left (\frac {b x + a}{d x + c}\right )^{2} + {\left (b c - a d\right )} {\rm Ei}\left (\frac {2 \, {\left (b c - a d\right )}}{d^{2} x + c d}\right )\right )} \sinh \left (\frac {2 \, b}{d}\right )}{2 \, {\left (d^{2} \cosh \left (\frac {b x + a}{d x + c}\right )^{2} - d^{2} \sinh \left (\frac {b x + a}{d x + c}\right )^{2}\right )}} \]
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\[ \int \sinh ^2\left (\frac {a+b x}{c+d x}\right ) \, dx=\int \sinh ^{2}{\left (\frac {a + b x}{c + d x} \right )}\, dx \]
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\[ \int \sinh ^2\left (\frac {a+b x}{c+d x}\right ) \, dx=\int { \sinh \left (\frac {b x + a}{d x + c}\right )^{2} \,d x } \]
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Leaf count of result is larger than twice the leaf count of optimal. 749 vs. \(2 (107) = 214\).
Time = 8.41 (sec) , antiderivative size = 749, normalized size of antiderivative = 7.00 \[ \int \sinh ^2\left (\frac {a+b x}{c+d x}\right ) \, dx=\frac {{\left (2 \, b^{3} c^{2} {\rm Ei}\left (-\frac {2 \, {\left (b - \frac {{\left (b x + a\right )} d}{d x + c}\right )}}{d}\right ) e^{\left (\frac {2 \, b}{d}\right )} - 4 \, a b^{2} c d {\rm Ei}\left (-\frac {2 \, {\left (b - \frac {{\left (b x + a\right )} d}{d x + c}\right )}}{d}\right ) e^{\left (\frac {2 \, b}{d}\right )} - \frac {2 \, {\left (b x + a\right )} b^{2} c^{2} d {\rm Ei}\left (-\frac {2 \, {\left (b - \frac {{\left (b x + a\right )} d}{d x + c}\right )}}{d}\right ) e^{\left (\frac {2 \, b}{d}\right )}}{d x + c} + 2 \, a^{2} b d^{2} {\rm Ei}\left (-\frac {2 \, {\left (b - \frac {{\left (b x + a\right )} d}{d x + c}\right )}}{d}\right ) e^{\left (\frac {2 \, b}{d}\right )} + \frac {4 \, {\left (b x + a\right )} a b c d^{2} {\rm Ei}\left (-\frac {2 \, {\left (b - \frac {{\left (b x + a\right )} d}{d x + c}\right )}}{d}\right ) e^{\left (\frac {2 \, b}{d}\right )}}{d x + c} - \frac {2 \, {\left (b x + a\right )} a^{2} d^{3} {\rm Ei}\left (-\frac {2 \, {\left (b - \frac {{\left (b x + a\right )} d}{d x + c}\right )}}{d}\right ) e^{\left (\frac {2 \, b}{d}\right )}}{d x + c} - 2 \, b^{3} c^{2} {\rm Ei}\left (\frac {2 \, {\left (b - \frac {{\left (b x + a\right )} d}{d x + c}\right )}}{d}\right ) e^{\left (-\frac {2 \, b}{d}\right )} + 4 \, a b^{2} c d {\rm Ei}\left (\frac {2 \, {\left (b - \frac {{\left (b x + a\right )} d}{d x + c}\right )}}{d}\right ) e^{\left (-\frac {2 \, b}{d}\right )} + \frac {2 \, {\left (b x + a\right )} b^{2} c^{2} d {\rm Ei}\left (\frac {2 \, {\left (b - \frac {{\left (b x + a\right )} d}{d x + c}\right )}}{d}\right ) e^{\left (-\frac {2 \, b}{d}\right )}}{d x + c} - 2 \, a^{2} b d^{2} {\rm Ei}\left (\frac {2 \, {\left (b - \frac {{\left (b x + a\right )} d}{d x + c}\right )}}{d}\right ) e^{\left (-\frac {2 \, b}{d}\right )} - \frac {4 \, {\left (b x + a\right )} a b c d^{2} {\rm Ei}\left (\frac {2 \, {\left (b - \frac {{\left (b x + a\right )} d}{d x + c}\right )}}{d}\right ) e^{\left (-\frac {2 \, b}{d}\right )}}{d x + c} + \frac {2 \, {\left (b x + a\right )} a^{2} d^{3} {\rm Ei}\left (\frac {2 \, {\left (b - \frac {{\left (b x + a\right )} d}{d x + c}\right )}}{d}\right ) e^{\left (-\frac {2 \, b}{d}\right )}}{d x + c} + b^{2} c^{2} d e^{\left (\frac {2 \, {\left (b x + a\right )}}{d x + c}\right )} - 2 \, a b c d^{2} e^{\left (\frac {2 \, {\left (b x + a\right )}}{d x + c}\right )} + a^{2} d^{3} e^{\left (\frac {2 \, {\left (b x + a\right )}}{d x + c}\right )} + b^{2} c^{2} d e^{\left (-\frac {2 \, {\left (b x + a\right )}}{d x + c}\right )} - 2 \, a b c d^{2} e^{\left (-\frac {2 \, {\left (b x + a\right )}}{d x + c}\right )} + a^{2} d^{3} e^{\left (-\frac {2 \, {\left (b x + a\right )}}{d x + c}\right )} - 2 \, b^{2} c^{2} d + 4 \, a b c d^{2} - 2 \, a^{2} d^{3}\right )} {\left (\frac {b c}{{\left (b c - a d\right )}^{2}} - \frac {a d}{{\left (b c - a d\right )}^{2}}\right )}}{4 \, {\left (b d^{2} - \frac {{\left (b x + a\right )} d^{3}}{d x + c}\right )}} \]
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Timed out. \[ \int \sinh ^2\left (\frac {a+b x}{c+d x}\right ) \, dx=\int {\mathrm {sinh}\left (\frac {a+b\,x}{c+d\,x}\right )}^2 \,d x \]
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