Integrand size = 16, antiderivative size = 174 \[ \int \frac {\sinh (a+b x)}{(c+d x)^{7/2}} \, dx=-\frac {4 b \cosh (a+b x)}{15 d^2 (c+d x)^{3/2}}+\frac {4 b^{5/2} e^{-a+\frac {b c}{d}} \sqrt {\pi } \text {erf}\left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {d}}\right )}{15 d^{7/2}}+\frac {4 b^{5/2} e^{a-\frac {b c}{d}} \sqrt {\pi } \text {erfi}\left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {d}}\right )}{15 d^{7/2}}-\frac {2 \sinh (a+b x)}{5 d (c+d x)^{5/2}}-\frac {8 b^2 \sinh (a+b x)}{15 d^3 \sqrt {c+d x}} \]
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Time = 0.22 (sec) , antiderivative size = 174, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.312, Rules used = {3378, 3388, 2211, 2235, 2236} \[ \int \frac {\sinh (a+b x)}{(c+d x)^{7/2}} \, dx=\frac {4 \sqrt {\pi } b^{5/2} e^{\frac {b c}{d}-a} \text {erf}\left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {d}}\right )}{15 d^{7/2}}+\frac {4 \sqrt {\pi } b^{5/2} e^{a-\frac {b c}{d}} \text {erfi}\left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {d}}\right )}{15 d^{7/2}}-\frac {8 b^2 \sinh (a+b x)}{15 d^3 \sqrt {c+d x}}-\frac {4 b \cosh (a+b x)}{15 d^2 (c+d x)^{3/2}}-\frac {2 \sinh (a+b x)}{5 d (c+d x)^{5/2}} \]
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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)}{5 d (c+d x)^{5/2}}+\frac {(2 b) \int \frac {\cosh (a+b x)}{(c+d x)^{5/2}} \, dx}{5 d} \\ & = -\frac {4 b \cosh (a+b x)}{15 d^2 (c+d x)^{3/2}}-\frac {2 \sinh (a+b x)}{5 d (c+d x)^{5/2}}+\frac {\left (4 b^2\right ) \int \frac {\sinh (a+b x)}{(c+d x)^{3/2}} \, dx}{15 d^2} \\ & = -\frac {4 b \cosh (a+b x)}{15 d^2 (c+d x)^{3/2}}-\frac {2 \sinh (a+b x)}{5 d (c+d x)^{5/2}}-\frac {8 b^2 \sinh (a+b x)}{15 d^3 \sqrt {c+d x}}+\frac {\left (8 b^3\right ) \int \frac {\cosh (a+b x)}{\sqrt {c+d x}} \, dx}{15 d^3} \\ & = -\frac {4 b \cosh (a+b x)}{15 d^2 (c+d x)^{3/2}}-\frac {2 \sinh (a+b x)}{5 d (c+d x)^{5/2}}-\frac {8 b^2 \sinh (a+b x)}{15 d^3 \sqrt {c+d x}}+\frac {\left (4 b^3\right ) \int \frac {e^{-i (i a+i b x)}}{\sqrt {c+d x}} \, dx}{15 d^3}+\frac {\left (4 b^3\right ) \int \frac {e^{i (i a+i b x)}}{\sqrt {c+d x}} \, dx}{15 d^3} \\ & = -\frac {4 b \cosh (a+b x)}{15 d^2 (c+d x)^{3/2}}-\frac {2 \sinh (a+b x)}{5 d (c+d x)^{5/2}}-\frac {8 b^2 \sinh (a+b x)}{15 d^3 \sqrt {c+d x}}+\frac {\left (8 b^3\right ) \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 )}{15 d^4}+\frac {\left (8 b^3\right ) \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 )}{15 d^4} \\ & = -\frac {4 b \cosh (a+b x)}{15 d^2 (c+d x)^{3/2}}+\frac {4 b^{5/2} e^{-a+\frac {b c}{d}} \sqrt {\pi } \text {erf}\left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {d}}\right )}{15 d^{7/2}}+\frac {4 b^{5/2} e^{a-\frac {b c}{d}} \sqrt {\pi } \text {erfi}\left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {d}}\right )}{15 d^{7/2}}-\frac {2 \sinh (a+b x)}{5 d (c+d x)^{5/2}}-\frac {8 b^2 \sinh (a+b x)}{15 d^3 \sqrt {c+d x}} \\ \end{align*}
Time = 0.37 (sec) , antiderivative size = 168, normalized size of antiderivative = 0.97 \[ \int \frac {\sinh (a+b x)}{(c+d x)^{7/2}} \, dx=\frac {2 \left (-b (c+d x) \left (e^{a-\frac {b c}{d}} \left (e^{b \left (\frac {c}{d}+x\right )} (d+2 b (c+d x))+2 d \left (-\frac {b (c+d x)}{d}\right )^{3/2} \Gamma \left (\frac {1}{2},-\frac {b (c+d x)}{d}\right )\right )+e^{-a-b x} \left (d-2 b (c+d x)+2 d e^{b \left (\frac {c}{d}+x\right )} \left (\frac {b (c+d x)}{d}\right )^{3/2} \Gamma \left (\frac {1}{2},\frac {b (c+d x)}{d}\right )\right )\right )-3 d^2 \sinh (a+b x)\right )}{15 d^3 (c+d x)^{5/2}} \]
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\[\int \frac {\sinh \left (b x +a \right )}{\left (d x +c \right )^{\frac {7}{2}}}d x\]
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Leaf count of result is larger than twice the leaf count of optimal. 855 vs. \(2 (132) = 264\).
Time = 0.27 (sec) , antiderivative size = 855, normalized size of antiderivative = 4.91 \[ \int \frac {\sinh (a+b x)}{(c+d x)^{7/2}} \, dx=\frac {4 \, \sqrt {\pi } {\left ({\left (b^{2} d^{3} x^{3} + 3 \, b^{2} c d^{2} x^{2} + 3 \, b^{2} c^{2} d x + b^{2} c^{3}\right )} \cosh \left (b x + a\right ) \cosh \left (-\frac {b c - a d}{d}\right ) - {\left (b^{2} d^{3} x^{3} + 3 \, b^{2} c d^{2} x^{2} + 3 \, b^{2} c^{2} d x + b^{2} c^{3}\right )} \cosh \left (b x + a\right ) \sinh \left (-\frac {b c - a d}{d}\right ) + {\left ({\left (b^{2} d^{3} x^{3} + 3 \, b^{2} c d^{2} x^{2} + 3 \, b^{2} c^{2} d x + b^{2} c^{3}\right )} \cosh \left (-\frac {b c - a d}{d}\right ) - {\left (b^{2} d^{3} x^{3} + 3 \, b^{2} c d^{2} x^{2} + 3 \, b^{2} c^{2} d x + b^{2} c^{3}\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 ) - 4 \, \sqrt {\pi } {\left ({\left (b^{2} d^{3} x^{3} + 3 \, b^{2} c d^{2} x^{2} + 3 \, b^{2} c^{2} d x + b^{2} c^{3}\right )} \cosh \left (b x + a\right ) \cosh \left (-\frac {b c - a d}{d}\right ) + {\left (b^{2} d^{3} x^{3} + 3 \, b^{2} c d^{2} x^{2} + 3 \, b^{2} c^{2} d x + b^{2} c^{3}\right )} \cosh \left (b x + a\right ) \sinh \left (-\frac {b c - a d}{d}\right ) + {\left ({\left (b^{2} d^{3} x^{3} + 3 \, b^{2} c d^{2} x^{2} + 3 \, b^{2} c^{2} d x + b^{2} c^{3}\right )} \cosh \left (-\frac {b c - a d}{d}\right ) + {\left (b^{2} d^{3} x^{3} + 3 \, b^{2} c d^{2} x^{2} + 3 \, b^{2} c^{2} d x + b^{2} c^{3}\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 ) + {\left (4 \, b^{2} d^{2} x^{2} + 4 \, b^{2} c^{2} - 2 \, b c d - {\left (4 \, b^{2} d^{2} x^{2} + 4 \, b^{2} c^{2} + 2 \, b c d + 3 \, d^{2} + 2 \, {\left (4 \, b^{2} c d + b d^{2}\right )} x\right )} \cosh \left (b x + a\right )^{2} - 2 \, {\left (4 \, b^{2} d^{2} x^{2} + 4 \, b^{2} c^{2} + 2 \, b c d + 3 \, d^{2} + 2 \, {\left (4 \, b^{2} c d + b d^{2}\right )} x\right )} \cosh \left (b x + a\right ) \sinh \left (b x + a\right ) - {\left (4 \, b^{2} d^{2} x^{2} + 4 \, b^{2} c^{2} + 2 \, b c d + 3 \, d^{2} + 2 \, {\left (4 \, b^{2} c d + b d^{2}\right )} x\right )} \sinh \left (b x + a\right )^{2} + 3 \, d^{2} + 2 \, {\left (4 \, b^{2} c d - b d^{2}\right )} x\right )} \sqrt {d x + c}}{15 \, {\left ({\left (d^{6} x^{3} + 3 \, c d^{5} x^{2} + 3 \, c^{2} d^{4} x + c^{3} d^{3}\right )} \cosh \left (b x + a\right ) + {\left (d^{6} x^{3} + 3 \, c d^{5} x^{2} + 3 \, c^{2} d^{4} x + c^{3} d^{3}\right )} \sinh \left (b x + a\right )\right )}} \]
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\[ \int \frac {\sinh (a+b x)}{(c+d x)^{7/2}} \, dx=\int \frac {\sinh {\left (a + b x \right )}}{\left (c + d x\right )^{\frac {7}{2}}}\, dx \]
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Time = 0.27 (sec) , antiderivative size = 114, normalized size of antiderivative = 0.66 \[ \int \frac {\sinh (a+b x)}{(c+d x)^{7/2}} \, dx=-\frac {\frac {{\left (\frac {\left (\frac {{\left (d x + c\right )} b}{d}\right )^{\frac {3}{2}} e^{\left (-a + \frac {b c}{d}\right )} \Gamma \left (-\frac {3}{2}, \frac {{\left (d x + c\right )} b}{d}\right )}{{\left (d x + c\right )}^{\frac {3}{2}}} + \frac {\left (-\frac {{\left (d x + c\right )} b}{d}\right )^{\frac {3}{2}} e^{\left (a - \frac {b c}{d}\right )} \Gamma \left (-\frac {3}{2}, -\frac {{\left (d x + c\right )} b}{d}\right )}{{\left (d x + c\right )}^{\frac {3}{2}}}\right )} b}{d} + \frac {2 \, \sinh \left (b x + a\right )}{{\left (d x + c\right )}^{\frac {5}{2}}}}{5 \, d} \]
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\[ \int \frac {\sinh (a+b x)}{(c+d x)^{7/2}} \, dx=\int { \frac {\sinh \left (b x + a\right )}{{\left (d x + c\right )}^{\frac {7}{2}}} \,d x } \]
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Timed out. \[ \int \frac {\sinh (a+b x)}{(c+d x)^{7/2}} \, dx=\int \frac {\mathrm {sinh}\left (a+b\,x\right )}{{\left (c+d\,x\right )}^{7/2}} \,d x \]
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