Optimal. Leaf size=70 \[ \frac{6 d^2 (c+d x) \cosh (a+b x)}{b^3}-\frac{3 d (c+d x)^2 \sinh (a+b x)}{b^2}-\frac{6 d^3 \sinh (a+b x)}{b^4}+\frac{(c+d x)^3 \cosh (a+b x)}{b} \]
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Rubi [A] time = 0.0790137, antiderivative size = 70, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 2, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143, Rules used = {3296, 2637} \[ \frac{6 d^2 (c+d x) \cosh (a+b x)}{b^3}-\frac{3 d (c+d x)^2 \sinh (a+b x)}{b^2}-\frac{6 d^3 \sinh (a+b x)}{b^4}+\frac{(c+d x)^3 \cosh (a+b x)}{b} \]
Antiderivative was successfully verified.
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Rule 3296
Rule 2637
Rubi steps
\begin{align*} \int (c+d x)^3 \sinh (a+b x) \, dx &=\frac{(c+d x)^3 \cosh (a+b x)}{b}-\frac{(3 d) \int (c+d x)^2 \cosh (a+b x) \, dx}{b}\\ &=\frac{(c+d x)^3 \cosh (a+b x)}{b}-\frac{3 d (c+d x)^2 \sinh (a+b x)}{b^2}+\frac{\left (6 d^2\right ) \int (c+d x) \sinh (a+b x) \, dx}{b^2}\\ &=\frac{6 d^2 (c+d x) \cosh (a+b x)}{b^3}+\frac{(c+d x)^3 \cosh (a+b x)}{b}-\frac{3 d (c+d x)^2 \sinh (a+b x)}{b^2}-\frac{\left (6 d^3\right ) \int \cosh (a+b x) \, dx}{b^3}\\ &=\frac{6 d^2 (c+d x) \cosh (a+b x)}{b^3}+\frac{(c+d x)^3 \cosh (a+b x)}{b}-\frac{6 d^3 \sinh (a+b x)}{b^4}-\frac{3 d (c+d x)^2 \sinh (a+b x)}{b^2}\\ \end{align*}
Mathematica [A] time = 0.21966, size = 61, normalized size = 0.87 \[ \frac{b (c+d x) \cosh (a+b x) \left (b^2 (c+d x)^2+6 d^2\right )-3 d \sinh (a+b x) \left (b^2 (c+d x)^2+2 d^2\right )}{b^4} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.007, size = 308, normalized size = 4.4 \begin{align*}{\frac{1}{b} \left ({\frac{{d}^{3} \left ( \left ( bx+a \right ) ^{3}\cosh \left ( bx+a \right ) -3\, \left ( bx+a \right ) ^{2}\sinh \left ( bx+a \right ) +6\, \left ( bx+a \right ) \cosh \left ( bx+a \right ) -6\,\sinh \left ( bx+a \right ) \right ) }{{b}^{3}}}-3\,{\frac{{d}^{3}a \left ( \left ( bx+a \right ) ^{2}\cosh \left ( bx+a \right ) -2\, \left ( bx+a \right ) \sinh \left ( bx+a \right ) +2\,\cosh \left ( bx+a \right ) \right ) }{{b}^{3}}}+3\,{\frac{{d}^{2}c \left ( \left ( bx+a \right ) ^{2}\cosh \left ( bx+a \right ) -2\, \left ( bx+a \right ) \sinh \left ( bx+a \right ) +2\,\cosh \left ( bx+a \right ) \right ) }{{b}^{2}}}+3\,{\frac{{d}^{3}{a}^{2} \left ( \left ( bx+a \right ) \cosh \left ( bx+a \right ) -\sinh \left ( bx+a \right ) \right ) }{{b}^{3}}}-6\,{\frac{{d}^{2}ac \left ( \left ( bx+a \right ) \cosh \left ( bx+a \right ) -\sinh \left ( bx+a \right ) \right ) }{{b}^{2}}}+3\,{\frac{d{c}^{2} \left ( \left ( bx+a \right ) \cosh \left ( bx+a \right ) -\sinh \left ( bx+a \right ) \right ) }{b}}-{\frac{{d}^{3}{a}^{3}\cosh \left ( bx+a \right ) }{{b}^{3}}}+3\,{\frac{{d}^{2}{a}^{2}c\cosh \left ( bx+a \right ) }{{b}^{2}}}-3\,{\frac{da{c}^{2}\cosh \left ( bx+a \right ) }{b}}+{c}^{3}\cosh \left ( bx+a \right ) \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.0586, size = 300, normalized size = 4.29 \begin{align*} \frac{c^{3} e^{\left (b x + a\right )}}{2 \, b} + \frac{3 \,{\left (b x e^{a} - e^{a}\right )} c^{2} d e^{\left (b x\right )}}{2 \, b^{2}} + \frac{c^{3} e^{\left (-b x - a\right )}}{2 \, b} + \frac{3 \,{\left (b x + 1\right )} c^{2} d e^{\left (-b x - a\right )}}{2 \, b^{2}} + \frac{3 \,{\left (b^{2} x^{2} e^{a} - 2 \, b x e^{a} + 2 \, e^{a}\right )} c d^{2} e^{\left (b x\right )}}{2 \, b^{3}} + \frac{3 \,{\left (b^{2} x^{2} + 2 \, b x + 2\right )} c d^{2} e^{\left (-b x - a\right )}}{2 \, b^{3}} + \frac{{\left (b^{3} x^{3} e^{a} - 3 \, b^{2} x^{2} e^{a} + 6 \, b x e^{a} - 6 \, e^{a}\right )} d^{3} e^{\left (b x\right )}}{2 \, b^{4}} + \frac{{\left (b^{3} x^{3} + 3 \, b^{2} x^{2} + 6 \, b x + 6\right )} d^{3} e^{\left (-b x - a\right )}}{2 \, b^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.53654, size = 231, normalized size = 3.3 \begin{align*} \frac{{\left (b^{3} d^{3} x^{3} + 3 \, b^{3} c d^{2} x^{2} + b^{3} c^{3} + 6 \, b c d^{2} + 3 \,{\left (b^{3} c^{2} d + 2 \, b d^{3}\right )} x\right )} \cosh \left (b x + a\right ) - 3 \,{\left (b^{2} d^{3} x^{2} + 2 \, b^{2} c d^{2} x + b^{2} c^{2} d + 2 \, d^{3}\right )} \sinh \left (b x + a\right )}{b^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 1.50237, size = 202, normalized size = 2.89 \begin{align*} \begin{cases} \frac{c^{3} \cosh{\left (a + b x \right )}}{b} + \frac{3 c^{2} d x \cosh{\left (a + b x \right )}}{b} + \frac{3 c d^{2} x^{2} \cosh{\left (a + b x \right )}}{b} + \frac{d^{3} x^{3} \cosh{\left (a + b x \right )}}{b} - \frac{3 c^{2} d \sinh{\left (a + b x \right )}}{b^{2}} - \frac{6 c d^{2} x \sinh{\left (a + b x \right )}}{b^{2}} - \frac{3 d^{3} x^{2} \sinh{\left (a + b x \right )}}{b^{2}} + \frac{6 c d^{2} \cosh{\left (a + b x \right )}}{b^{3}} + \frac{6 d^{3} x \cosh{\left (a + b x \right )}}{b^{3}} - \frac{6 d^{3} \sinh{\left (a + b x \right )}}{b^{4}} & \text{for}\: b \neq 0 \\\left (c^{3} x + \frac{3 c^{2} d x^{2}}{2} + c d^{2} x^{3} + \frac{d^{3} x^{4}}{4}\right ) \sinh{\left (a \right )} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.19993, size = 275, normalized size = 3.93 \begin{align*} \frac{{\left (b^{3} d^{3} x^{3} + 3 \, b^{3} c d^{2} x^{2} + 3 \, b^{3} c^{2} d x - 3 \, b^{2} d^{3} x^{2} + b^{3} c^{3} - 6 \, b^{2} c d^{2} x - 3 \, b^{2} c^{2} d + 6 \, b d^{3} x + 6 \, b c d^{2} - 6 \, d^{3}\right )} e^{\left (b x + a\right )}}{2 \, b^{4}} + \frac{{\left (b^{3} d^{3} x^{3} + 3 \, b^{3} c d^{2} x^{2} + 3 \, b^{3} c^{2} d x + 3 \, b^{2} d^{3} x^{2} + b^{3} c^{3} + 6 \, b^{2} c d^{2} x + 3 \, b^{2} c^{2} d + 6 \, b d^{3} x + 6 \, b c d^{2} + 6 \, d^{3}\right )} e^{\left (-b x - a\right )}}{2 \, b^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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