Optimal. Leaf size=73 \[ \frac{4 b e^{a+b x} \sinh ^{\frac{3}{2}}(c+d x)}{4 b^2-9 d^2}-\frac{6 d e^{a+b x} \sqrt{\sinh (c+d x)} \cosh (c+d x)}{4 b^2-9 d^2} \]
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Rubi [A] time = 0.613906, antiderivative size = 73, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 5, integrand size = 56, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.089, Rules used = {5482, 2253, 2252, 2251, 5476} \[ \frac{4 b e^{a+b x} \sinh ^{\frac{3}{2}}(c+d x)}{4 b^2-9 d^2}-\frac{6 d e^{a+b x} \sqrt{\sinh (c+d x)} \cosh (c+d x)}{4 b^2-9 d^2} \]
Antiderivative was successfully verified.
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Rule 5482
Rule 2253
Rule 2252
Rule 2251
Rule 5476
Rubi steps
\begin{align*} \int \left (-\frac{3 d^2 e^{a+b x}}{4 \left (b^2-\frac{9 d^2}{4}\right ) \sqrt{\sinh (c+d x)}}+e^{a+b x} \sinh ^{\frac{3}{2}}(c+d x)\right ) \, dx &=-\frac{\left (3 d^2\right ) \int \frac{e^{a+b x}}{\sqrt{\sinh (c+d x)}} \, dx}{4 b^2-9 d^2}+\int e^{a+b x} \sinh ^{\frac{3}{2}}(c+d x) \, dx\\ &=-\frac{6 d e^{a+b x} \cosh (c+d x) \sqrt{\sinh (c+d x)}}{4 b^2-9 d^2}+\frac{4 b e^{a+b x} \sinh ^{\frac{3}{2}}(c+d x)}{4 b^2-9 d^2}+\frac{\left (3 d^2\right ) \int \frac{e^{a+b x}}{\sqrt{\sinh (c+d x)}} \, dx}{4 b^2-9 d^2}-\frac{\left (3 d^2 e^{\frac{1}{2} (-c-d x)} \sqrt{-1+e^{2 (c+d x)}}\right ) \int \frac{e^{a+b x+\frac{1}{2} (c+d x)}}{\sqrt{-1+e^{2 (c+d x)}}} \, dx}{\left (4 b^2-9 d^2\right ) \sqrt{\sinh (c+d x)}}\\ &=-\frac{6 d e^{a+b x} \cosh (c+d x) \sqrt{\sinh (c+d x)}}{4 b^2-9 d^2}+\frac{4 b e^{a+b x} \sinh ^{\frac{3}{2}}(c+d x)}{4 b^2-9 d^2}-\frac{\left (3 d^2 e^{\frac{1}{2} (-c-d x)} \sqrt{-1+e^{2 (c+d x)}}\right ) \int \frac{e^{\frac{1}{2} (2 a+c)+\frac{1}{2} (2 b+d) x}}{\sqrt{-1+e^{2 (c+d x)}}} \, dx}{\left (4 b^2-9 d^2\right ) \sqrt{\sinh (c+d x)}}+\frac{\left (3 d^2 e^{\frac{1}{2} (-c-d x)} \sqrt{-1+e^{2 (c+d x)}}\right ) \int \frac{e^{a+b x+\frac{1}{2} (c+d x)}}{\sqrt{-1+e^{2 (c+d x)}}} \, dx}{\left (4 b^2-9 d^2\right ) \sqrt{\sinh (c+d x)}}\\ &=-\frac{6 d e^{a+b x} \cosh (c+d x) \sqrt{\sinh (c+d x)}}{4 b^2-9 d^2}+\frac{4 b e^{a+b x} \sinh ^{\frac{3}{2}}(c+d x)}{4 b^2-9 d^2}-\frac{\left (3 d^2 e^{\frac{1}{2} (-c-d x)} \sqrt{1-e^{2 (c+d x)}}\right ) \int \frac{e^{\frac{1}{2} (2 a+c)+\frac{1}{2} (2 b+d) x}}{\sqrt{1-e^{2 (c+d x)}}} \, dx}{\left (4 b^2-9 d^2\right ) \sqrt{\sinh (c+d x)}}+\frac{\left (3 d^2 e^{\frac{1}{2} (-c-d x)} \sqrt{-1+e^{2 (c+d x)}}\right ) \int \frac{e^{\frac{1}{2} (2 a+c)+\frac{1}{2} (2 b+d) x}}{\sqrt{-1+e^{2 (c+d x)}}} \, dx}{\left (4 b^2-9 d^2\right ) \sqrt{\sinh (c+d x)}}\\ &=-\frac{6 d^2 \exp \left (\frac{1}{2} (2 a+c)+\frac{1}{2} (2 b+d) x+\frac{1}{2} (-c-d x)\right ) \sqrt{1-e^{2 (c+d x)}} \, _2F_1\left (\frac{1}{2},\frac{2 b+d}{4 d};\frac{1}{4} \left (5+\frac{2 b}{d}\right );e^{2 (c+d x)}\right )}{(2 b+d) \left (4 b^2-9 d^2\right ) \sqrt{\sinh (c+d x)}}-\frac{6 d e^{a+b x} \cosh (c+d x) \sqrt{\sinh (c+d x)}}{4 b^2-9 d^2}+\frac{4 b e^{a+b x} \sinh ^{\frac{3}{2}}(c+d x)}{4 b^2-9 d^2}+\frac{\left (3 d^2 e^{\frac{1}{2} (-c-d x)} \sqrt{1-e^{2 (c+d x)}}\right ) \int \frac{e^{\frac{1}{2} (2 a+c)+\frac{1}{2} (2 b+d) x}}{\sqrt{1-e^{2 (c+d x)}}} \, dx}{\left (4 b^2-9 d^2\right ) \sqrt{\sinh (c+d x)}}\\ &=-\frac{6 d e^{a+b x} \cosh (c+d x) \sqrt{\sinh (c+d x)}}{4 b^2-9 d^2}+\frac{4 b e^{a+b x} \sinh ^{\frac{3}{2}}(c+d x)}{4 b^2-9 d^2}\\ \end{align*}
Mathematica [C] time = 1.32062, size = 155, normalized size = 2.12 \[ \frac{2 e^{a+b x} \left (e^{2 (c+d x)}-1\right ) \left (\left (4 b^2+8 b d+3 d^2\right ) \sinh ^2(c+d x) \, _2F_1\left (1,\frac{1}{4} \left (\frac{2 b}{d}+7\right );\frac{2 b+d}{4 d};e^{2 (c+d x)}\right )-3 d^2 \, _2F_1\left (1,\frac{1}{4} \left (\frac{2 b}{d}+3\right );\frac{1}{4} \left (\frac{2 b}{d}+5\right );e^{2 (c+d x)}\right )\right )}{(2 b+d) (3 d-2 b) (2 b+3 d) \sqrt{\sinh (c+d x)}} \]
Warning: Unable to verify antiderivative.
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Maple [F] time = 0.214, size = 0, normalized size = 0. \begin{align*} \int{{\rm e}^{bx+a}} \left ( \sinh \left ( dx+c \right ) \right ) ^{{\frac{3}{2}}}-{\frac{3\,{d}^{2}{{\rm e}^{bx+a}}}{4} \left ({b}^{2}-{\frac{9\,{d}^{2}}{4}} \right ) ^{-1}{\frac{1}{\sqrt{\sinh \left ( dx+c \right ) }}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int e^{\left (b x + a\right )} \sinh \left (d x + c\right )^{\frac{3}{2}} - \frac{3 \, d^{2} e^{\left (b x + a\right )}}{{\left (4 \, b^{2} - 9 \, d^{2}\right )} \sqrt{\sinh \left (d x + c\right )}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: UnboundLocalError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int e^{\left (b x + a\right )} \sinh \left (d x + c\right )^{\frac{3}{2}} - \frac{3 \, d^{2} e^{\left (b x + a\right )}}{{\left (4 \, b^{2} - 9 \, d^{2}\right )} \sqrt{\sinh \left (d x + c\right )}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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