Integrand size = 14, antiderivative size = 101 \[ \int \sinh \left (\frac {a+b x}{c+d x}\right ) \, dx=\frac {(b c-a d) \cosh \left (\frac {b}{d}\right ) \text {Chi}\left (\frac {b c-a d}{d (c+d x)}\right )}{d^2}+\frac {(c+d x) \sinh \left (\frac {a+b x}{c+d x}\right )}{d}-\frac {(b c-a d) \sinh \left (\frac {b}{d}\right ) \text {Shi}\left (\frac {b c-a d}{d (c+d x)}\right )}{d^2} \]
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Time = 0.13 (sec) , antiderivative size = 101, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.357, Rules used = {5726, 3378, 3384, 3379, 3382} \[ \int \sinh \left (\frac {a+b x}{c+d x}\right ) \, dx=\frac {\cosh \left (\frac {b}{d}\right ) (b c-a d) \text {Chi}\left (\frac {b c-a d}{d (c+d x)}\right )}{d^2}-\frac {\sinh \left (\frac {b}{d}\right ) (b c-a d) \text {Shi}\left (\frac {b c-a d}{d (c+d x)}\right )}{d^2}+\frac {(c+d x) \sinh \left (\frac {a+b x}{c+d x}\right )}{d} \]
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Rule 3378
Rule 3379
Rule 3382
Rule 3384
Rule 5726
Rubi steps \begin{align*} \text {integral}& = -\frac {\text {Subst}\left (\int \frac {\sinh \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 \left (\frac {a+b x}{c+d x}\right )}{d}+\frac {(b c-a d) \text {Subst}\left (\int \frac {\cosh \left (\frac {b}{d}-\frac {(b c-a d) x}{d}\right )}{x} \, dx,x,\frac {1}{c+d x}\right )}{d^2} \\ & = \frac {(c+d x) \sinh \left (\frac {a+b x}{c+d x}\right )}{d}+\frac {\left ((b c-a d) \cosh \left (\frac {b}{d}\right )\right ) \text {Subst}\left (\int \frac {\cosh \left (\frac {(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 {b}{d}\right )\right ) \text {Subst}\left (\int \frac {\sinh \left (\frac {(b c-a d) x}{d}\right )}{x} \, dx,x,\frac {1}{c+d x}\right )}{d^2} \\ & = \frac {(b c-a d) \cosh \left (\frac {b}{d}\right ) \text {Chi}\left (\frac {b c-a d}{d (c+d x)}\right )}{d^2}+\frac {(c+d x) \sinh \left (\frac {a+b x}{c+d x}\right )}{d}-\frac {(b c-a d) \sinh \left (\frac {b}{d}\right ) \text {Shi}\left (\frac {b c-a d}{d (c+d x)}\right )}{d^2} \\ \end{align*}
Leaf count is larger than twice the leaf count of optimal. \(302\) vs. \(2(101)=202\).
Time = 0.51 (sec) , antiderivative size = 302, normalized size of antiderivative = 2.99 \[ \int \sinh \left (\frac {a+b x}{c+d x}\right ) \, dx=\frac {-c d e^{-\frac {a+b x}{c+d x}}+c d e^{\frac {a+b x}{c+d x}}+2 d^2 x \cosh \left (\frac {-b c+a d}{d (c+d x)}\right ) \sinh \left (\frac {b}{d}\right )+2 d^2 x \cosh \left (\frac {b}{d}\right ) \sinh \left (\frac {-b c+a d}{d (c+d x)}\right )+(b c-a d) \left (\text {Chi}\left (\frac {b c-a d}{c d+d^2 x}\right ) \left (\cosh \left (\frac {b}{d}\right )-\sinh \left (\frac {b}{d}\right )\right )+\text {Chi}\left (\frac {-b c+a d}{d (c+d x)}\right ) \left (\cosh \left (\frac {b}{d}\right )+\sinh \left (\frac {b}{d}\right )\right )+\cosh \left (\frac {b}{d}\right ) \text {Shi}\left (\frac {-b c+a d}{d (c+d x)}\right )+\sinh \left (\frac {b}{d}\right ) \text {Shi}\left (\frac {-b c+a d}{d (c+d x)}\right )+\cosh \left (\frac {b}{d}\right ) \text {Shi}\left (\frac {b c-a d}{c d+d^2 x}\right )-\sinh \left (\frac {b}{d}\right ) \text {Shi}\left (\frac {b c-a d}{c d+d^2 x}\right )\right )}{2 d^2} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(346\) vs. \(2(101)=202\).
Time = 1.08 (sec) , antiderivative size = 347, normalized size of antiderivative = 3.44
method | result | size |
risch | \(-\frac {{\mathrm e}^{-\frac {b x +a}{d x +c}} a}{2 \left (\frac {d a}{d x +c}-\frac {b c}{d x +c}\right )}+\frac {{\mathrm e}^{-\frac {b x +a}{d x +c}} b c}{2 d \left (\frac {d a}{d x +c}-\frac {b c}{d x +c}\right )}+\frac {{\mathrm e}^{-\frac {b}{d}} \operatorname {Ei}_{1}\left (\frac {a d -b c}{d \left (d x +c \right )}\right ) a}{2 d}-\frac {{\mathrm e}^{-\frac {b}{d}} \operatorname {Ei}_{1}\left (\frac {a d -b c}{d \left (d x +c \right )}\right ) b c}{2 d^{2}}+\frac {d \,{\mathrm e}^{\frac {b x +a}{d x +c}} x a}{2 a d -2 b c}-\frac {{\mathrm e}^{\frac {b x +a}{d x +c}} x b c}{2 \left (a d -b c \right )}+\frac {{\mathrm e}^{\frac {b x +a}{d x +c}} c a}{2 a d -2 b c}-\frac {{\mathrm e}^{\frac {b x +a}{d x +c}} c^{2} b}{2 d \left (a d -b c \right )}+\frac {{\mathrm e}^{\frac {b}{d}} \operatorname {Ei}_{1}\left (-\frac {a d -b c}{d \left (d x +c \right )}\right ) a}{2 d}-\frac {{\mathrm e}^{\frac {b}{d}} \operatorname {Ei}_{1}\left (-\frac {a d -b c}{d \left (d x +c \right )}\right ) b c}{2 d^{2}}\) | \(347\) |
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Time = 0.28 (sec) , antiderivative size = 171, normalized size of antiderivative = 1.69 \[ \int \sinh \left (\frac {a+b x}{c+d x}\right ) \, dx=\frac {{\left ({\left (b c - a d\right )} {\rm Ei}\left (\frac {b c - a d}{d^{2} x + c d}\right ) + {\left (b c - a d\right )} {\rm Ei}\left (-\frac {b c - a d}{d^{2} x + c d}\right )\right )} \cosh \left (\frac {b}{d}\right ) + 2 \, {\left (d^{2} x + c d\right )} \sinh \left (\frac {b x + a}{d x + c}\right ) - {\left ({\left (b c - a d\right )} {\rm Ei}\left (\frac {b c - a d}{d^{2} x + c d}\right ) - {\left (b c - a d\right )} {\rm Ei}\left (-\frac {b c - a d}{d^{2} x + c d}\right )\right )} \sinh \left (\frac {b}{d}\right )}{2 \, d^{2}} \]
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\[ \int \sinh \left (\frac {a+b x}{c+d x}\right ) \, dx=\int \sinh {\left (\frac {a + b x}{c + d x} \right )}\, dx \]
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\[ \int \sinh \left (\frac {a+b x}{c+d x}\right ) \, dx=\int { \sinh \left (\frac {b x + a}{d x + c}\right ) \,d x } \]
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Leaf count of result is larger than twice the leaf count of optimal. 764 vs. \(2 (101) = 202\).
Time = 1.90 (sec) , antiderivative size = 764, normalized size of antiderivative = 7.56 \[ \int \sinh \left (\frac {a+b x}{c+d x}\right ) \, dx=\frac {{\left (b^{3} c^{2} {\rm Ei}\left (-\frac {b - \frac {{\left (b x + a\right )} d}{d x + c}}{d}\right ) e^{\frac {b}{d}} - 2 \, a b^{2} c d {\rm Ei}\left (-\frac {b - \frac {{\left (b x + a\right )} d}{d x + c}}{d}\right ) e^{\frac {b}{d}} - \frac {{\left (b x + a\right )} b^{2} c^{2} d {\rm Ei}\left (-\frac {b - \frac {{\left (b x + a\right )} d}{d x + c}}{d}\right ) e^{\frac {b}{d}}}{d x + c} + a^{2} b d^{2} {\rm Ei}\left (-\frac {b - \frac {{\left (b x + a\right )} d}{d x + c}}{d}\right ) e^{\frac {b}{d}} + \frac {2 \, {\left (b x + a\right )} a b c d^{2} {\rm Ei}\left (-\frac {b - \frac {{\left (b x + a\right )} d}{d x + c}}{d}\right ) e^{\frac {b}{d}}}{d x + c} - \frac {{\left (b x + a\right )} a^{2} d^{3} {\rm Ei}\left (-\frac {b - \frac {{\left (b x + a\right )} d}{d x + c}}{d}\right ) e^{\frac {b}{d}}}{d x + c} + b^{2} c^{2} d e^{\left (\frac {b x + a}{d x + c}\right )} - 2 \, a b c d^{2} e^{\left (\frac {b x + a}{d x + c}\right )} + a^{2} d^{3} e^{\left (\frac {b x + a}{d x + c}\right )}\right )} {\left (\frac {b c}{{\left (b c - a d\right )}^{2}} - \frac {a d}{{\left (b c - a d\right )}^{2}}\right )}}{2 \, {\left (b d^{2} - \frac {{\left (b x + a\right )} d^{3}}{d x + c}\right )}} + \frac {{\left (b^{3} c^{2} {\rm Ei}\left (\frac {b - \frac {{\left (b x + a\right )} d}{d x + c}}{d}\right ) e^{\left (-\frac {b}{d}\right )} - 2 \, a b^{2} c d {\rm Ei}\left (\frac {b - \frac {{\left (b x + a\right )} d}{d x + c}}{d}\right ) e^{\left (-\frac {b}{d}\right )} - \frac {{\left (b x + a\right )} b^{2} c^{2} d {\rm Ei}\left (\frac {b - \frac {{\left (b x + a\right )} d}{d x + c}}{d}\right ) e^{\left (-\frac {b}{d}\right )}}{d x + c} + a^{2} b d^{2} {\rm Ei}\left (\frac {b - \frac {{\left (b x + a\right )} d}{d x + c}}{d}\right ) e^{\left (-\frac {b}{d}\right )} + \frac {2 \, {\left (b x + a\right )} a b c d^{2} {\rm Ei}\left (\frac {b - \frac {{\left (b x + a\right )} d}{d x + c}}{d}\right ) e^{\left (-\frac {b}{d}\right )}}{d x + c} - \frac {{\left (b x + a\right )} a^{2} d^{3} {\rm Ei}\left (\frac {b - \frac {{\left (b x + a\right )} d}{d x + c}}{d}\right ) e^{\left (-\frac {b}{d}\right )}}{d x + c} - b^{2} c^{2} d e^{\left (-\frac {b x + a}{d x + c}\right )} + 2 \, a b c d^{2} e^{\left (-\frac {b x + a}{d x + c}\right )} - a^{2} d^{3} e^{\left (-\frac {b x + a}{d x + c}\right )}\right )} {\left (\frac {b c}{{\left (b c - a d\right )}^{2}} - \frac {a d}{{\left (b c - a d\right )}^{2}}\right )}}{2 \, {\left (b d^{2} - \frac {{\left (b x + a\right )} d^{3}}{d x + c}\right )}} \]
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Timed out. \[ \int \sinh \left (\frac {a+b x}{c+d x}\right ) \, dx=\int \mathrm {sinh}\left (\frac {a+b\,x}{c+d\,x}\right ) \,d x \]
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