Integrand size = 16, antiderivative size = 118 \[ \int \frac {\sinh (a+b x)}{(c+d x)^{3/2}} \, dx=\frac {\sqrt {b} e^{-a+\frac {b c}{d}} \sqrt {\pi } \text {erf}\left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {d}}\right )}{d^{3/2}}+\frac {\sqrt {b} e^{a-\frac {b c}{d}} \sqrt {\pi } \text {erfi}\left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {d}}\right )}{d^{3/2}}-\frac {2 \sinh (a+b x)}{d \sqrt {c+d x}} \]
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Time = 0.14 (sec) , antiderivative size = 118, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.312, Rules used = {3378, 3388, 2211, 2235, 2236} \[ \int \frac {\sinh (a+b x)}{(c+d x)^{3/2}} \, dx=\frac {\sqrt {\pi } \sqrt {b} e^{\frac {b c}{d}-a} \text {erf}\left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {d}}\right )}{d^{3/2}}+\frac {\sqrt {\pi } \sqrt {b} e^{a-\frac {b c}{d}} \text {erfi}\left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {d}}\right )}{d^{3/2}}-\frac {2 \sinh (a+b x)}{d \sqrt {c+d x}} \]
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Rule 2211
Rule 2235
Rule 2236
Rule 3378
Rule 3388
Rubi steps \begin{align*} \text {integral}& = -\frac {2 \sinh (a+b x)}{d \sqrt {c+d x}}+\frac {(2 b) \int \frac {\cosh (a+b x)}{\sqrt {c+d x}} \, dx}{d} \\ & = -\frac {2 \sinh (a+b x)}{d \sqrt {c+d x}}+\frac {b \int \frac {e^{-i (i a+i b x)}}{\sqrt {c+d x}} \, dx}{d}+\frac {b \int \frac {e^{i (i a+i b x)}}{\sqrt {c+d x}} \, dx}{d} \\ & = -\frac {2 \sinh (a+b x)}{d \sqrt {c+d x}}+\frac {(2 b) \text {Subst}\left (\int e^{i \left (i a-\frac {i b c}{d}\right )-\frac {b x^2}{d}} \, dx,x,\sqrt {c+d x}\right )}{d^2}+\frac {(2 b) \text {Subst}\left (\int e^{-i \left (i a-\frac {i b c}{d}\right )+\frac {b x^2}{d}} \, dx,x,\sqrt {c+d x}\right )}{d^2} \\ & = \frac {\sqrt {b} e^{-a+\frac {b c}{d}} \sqrt {\pi } \text {erf}\left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {d}}\right )}{d^{3/2}}+\frac {\sqrt {b} e^{a-\frac {b c}{d}} \sqrt {\pi } \text {erfi}\left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {d}}\right )}{d^{3/2}}-\frac {2 \sinh (a+b x)}{d \sqrt {c+d x}} \\ \end{align*}
Time = 0.10 (sec) , antiderivative size = 120, normalized size of antiderivative = 1.02 \[ \int \frac {\sinh (a+b x)}{(c+d x)^{3/2}} \, dx=\frac {e^{-a-\frac {b c}{d}} \left (e^{2 a} \sqrt {-\frac {b (c+d x)}{d}} \Gamma \left (\frac {1}{2},-\frac {b (c+d x)}{d}\right )-e^{\frac {2 b c}{d}} \sqrt {\frac {b (c+d x)}{d}} \Gamma \left (\frac {1}{2},\frac {b (c+d x)}{d}\right )-2 e^{a+\frac {b c}{d}} \sinh (a+b x)\right )}{d \sqrt {c+d x}} \]
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\[\int \frac {\sinh \left (b x +a \right )}{\left (d x +c \right )^{\frac {3}{2}}}d x\]
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Leaf count of result is larger than twice the leaf count of optimal. 339 vs. \(2 (90) = 180\).
Time = 0.25 (sec) , antiderivative size = 339, normalized size of antiderivative = 2.87 \[ \int \frac {\sinh (a+b x)}{(c+d x)^{3/2}} \, dx=\frac {\sqrt {\pi } {\left ({\left (d x + c\right )} \cosh \left (b x + a\right ) \cosh \left (-\frac {b c - a d}{d}\right ) - {\left (d x + c\right )} \cosh \left (b x + a\right ) \sinh \left (-\frac {b c - a d}{d}\right ) + {\left ({\left (d x + c\right )} \cosh \left (-\frac {b c - a d}{d}\right ) - {\left (d x + c\right )} \sinh \left (-\frac {b c - a d}{d}\right )\right )} \sinh \left (b x + a\right )\right )} \sqrt {\frac {b}{d}} \operatorname {erf}\left (\sqrt {d x + c} \sqrt {\frac {b}{d}}\right ) - \sqrt {\pi } {\left ({\left (d x + c\right )} \cosh \left (b x + a\right ) \cosh \left (-\frac {b c - a d}{d}\right ) + {\left (d x + c\right )} \cosh \left (b x + a\right ) \sinh \left (-\frac {b c - a d}{d}\right ) + {\left ({\left (d x + c\right )} \cosh \left (-\frac {b c - a d}{d}\right ) + {\left (d x + c\right )} \sinh \left (-\frac {b c - a d}{d}\right )\right )} \sinh \left (b x + a\right )\right )} \sqrt {-\frac {b}{d}} \operatorname {erf}\left (\sqrt {d x + c} \sqrt {-\frac {b}{d}}\right ) - \sqrt {d x + c} {\left (\cosh \left (b x + a\right )^{2} + 2 \, \cosh \left (b x + a\right ) \sinh \left (b x + a\right ) + \sinh \left (b x + a\right )^{2} - 1\right )}}{{\left (d^{2} x + c d\right )} \cosh \left (b x + a\right ) + {\left (d^{2} x + c d\right )} \sinh \left (b x + a\right )} \]
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\[ \int \frac {\sinh (a+b x)}{(c+d x)^{3/2}} \, dx=\int \frac {\sinh {\left (a + b x \right )}}{\left (c + d x\right )^{\frac {3}{2}}}\, dx \]
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none
Time = 0.21 (sec) , antiderivative size = 103, normalized size of antiderivative = 0.87 \[ \int \frac {\sinh (a+b x)}{(c+d x)^{3/2}} \, dx=\frac {\frac {{\left (\frac {\sqrt {\pi } \operatorname {erf}\left (\sqrt {d x + c} \sqrt {-\frac {b}{d}}\right ) e^{\left (a - \frac {b c}{d}\right )}}{\sqrt {-\frac {b}{d}}} + \frac {\sqrt {\pi } \operatorname {erf}\left (\sqrt {d x + c} \sqrt {\frac {b}{d}}\right ) e^{\left (-a + \frac {b c}{d}\right )}}{\sqrt {\frac {b}{d}}}\right )} b}{d} - \frac {2 \, \sinh \left (b x + a\right )}{\sqrt {d x + c}}}{d} \]
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\[ \int \frac {\sinh (a+b x)}{(c+d x)^{3/2}} \, dx=\int { \frac {\sinh \left (b x + a\right )}{{\left (d x + c\right )}^{\frac {3}{2}}} \,d x } \]
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Timed out. \[ \int \frac {\sinh (a+b x)}{(c+d x)^{3/2}} \, dx=\int \frac {\mathrm {sinh}\left (a+b\,x\right )}{{\left (c+d\,x\right )}^{3/2}} \,d x \]
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