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 = 0.61, antiderivative size = 73, normalized size of antiderivative = 1.00, 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 2251
Rule 2252
Rule 2253
Rule 5476
Rule 5482
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*}
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Mathematica [C] time = 1.32, 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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fricas [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int e^{\left (b x + a\right )} \sinh \left (d x + c\right )^{\frac {3}{2}} - \frac {3 \, d^{2} e^{\left (b x + a\right )}}{{\left (4 \, b^{2} - 9 \, d^{2}\right )} \sqrt {\sinh \left (d x + c\right )}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 1.03, size = 0, normalized size = 0.00 \[ \int {\mathrm e}^{b x +a} \left (\sinh ^{\frac {3}{2}}\left (d x +c \right )\right )-\frac {3 d^{2} {\mathrm e}^{b x +a}}{4 \left (b^{2}-\frac {9 d^{2}}{4}\right ) \sqrt {\sinh \left (d x +c \right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int e^{\left (b x + a\right )} \sinh \left (d x + c\right )^{\frac {3}{2}} - \frac {3 \, d^{2} e^{\left (b x + a\right )}}{{\left (4 \, b^{2} - 9 \, d^{2}\right )} \sqrt {\sinh \left (d x + c\right )}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int {\mathrm {e}}^{a+b\,x}\,{\mathrm {sinh}\left (c+d\,x\right )}^{3/2}-\frac {3\,d^2\,{\mathrm {e}}^{a+b\,x}}{4\,\sqrt {\mathrm {sinh}\left (c+d\,x\right )}\,\left (b^2-\frac {9\,d^2}{4}\right )} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \frac {\left (\int 4 b^{2} e^{b x} \sinh ^{\frac {3}{2}}{\left (c + d x \right )}\, dx + \int \left (- \frac {3 d^{2} e^{b x}}{\sqrt {\sinh {\left (c + d x \right )}}}\right )\, dx + \int \left (- 9 d^{2} e^{b x} \sinh ^{\frac {3}{2}}{\left (c + d x \right )}\right )\, dx\right ) e^{a}}{\left (2 b - 3 d\right ) \left (2 b + 3 d\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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