Optimal. Leaf size=91 \[ -\frac{24 d^3 (c+d x) \sinh (a+b x)}{b^4}+\frac{12 d^2 (c+d x)^2 \cosh (a+b x)}{b^3}-\frac{4 d (c+d x)^3 \sinh (a+b x)}{b^2}+\frac{24 d^4 \cosh (a+b x)}{b^5}+\frac{(c+d x)^4 \cosh (a+b x)}{b} \]
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Rubi [A] time = 0.120453, antiderivative size = 91, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 2, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143, Rules used = {3296, 2638} \[ -\frac{24 d^3 (c+d x) \sinh (a+b x)}{b^4}+\frac{12 d^2 (c+d x)^2 \cosh (a+b x)}{b^3}-\frac{4 d (c+d x)^3 \sinh (a+b x)}{b^2}+\frac{24 d^4 \cosh (a+b x)}{b^5}+\frac{(c+d x)^4 \cosh (a+b x)}{b} \]
Antiderivative was successfully verified.
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Rule 3296
Rule 2638
Rubi steps
\begin{align*} \int (c+d x)^4 \sinh (a+b x) \, dx &=\frac{(c+d x)^4 \cosh (a+b x)}{b}-\frac{(4 d) \int (c+d x)^3 \cosh (a+b x) \, dx}{b}\\ &=\frac{(c+d x)^4 \cosh (a+b x)}{b}-\frac{4 d (c+d x)^3 \sinh (a+b x)}{b^2}+\frac{\left (12 d^2\right ) \int (c+d x)^2 \sinh (a+b x) \, dx}{b^2}\\ &=\frac{12 d^2 (c+d x)^2 \cosh (a+b x)}{b^3}+\frac{(c+d x)^4 \cosh (a+b x)}{b}-\frac{4 d (c+d x)^3 \sinh (a+b x)}{b^2}-\frac{\left (24 d^3\right ) \int (c+d x) \cosh (a+b x) \, dx}{b^3}\\ &=\frac{12 d^2 (c+d x)^2 \cosh (a+b x)}{b^3}+\frac{(c+d x)^4 \cosh (a+b x)}{b}-\frac{24 d^3 (c+d x) \sinh (a+b x)}{b^4}-\frac{4 d (c+d x)^3 \sinh (a+b x)}{b^2}+\frac{\left (24 d^4\right ) \int \sinh (a+b x) \, dx}{b^4}\\ &=\frac{24 d^4 \cosh (a+b x)}{b^5}+\frac{12 d^2 (c+d x)^2 \cosh (a+b x)}{b^3}+\frac{(c+d x)^4 \cosh (a+b x)}{b}-\frac{24 d^3 (c+d x) \sinh (a+b x)}{b^4}-\frac{4 d (c+d x)^3 \sinh (a+b x)}{b^2}\\ \end{align*}
Mathematica [A] time = 0.347484, size = 76, normalized size = 0.84 \[ \frac{\cosh (a+b x) \left (12 b^2 d^2 (c+d x)^2+b^4 (c+d x)^4+24 d^4\right )-4 b d (c+d x) \sinh (a+b x) \left (b^2 (c+d x)^2+6 d^2\right )}{b^5} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.033, size = 547, normalized size = 6. \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.1229, size = 440, normalized size = 4.84 \begin{align*} \frac{c^{4} e^{\left (b x + a\right )}}{2 \, b} + \frac{2 \,{\left (b x e^{a} - e^{a}\right )} c^{3} d e^{\left (b x\right )}}{b^{2}} + \frac{c^{4} e^{\left (-b x - a\right )}}{2 \, b} + \frac{2 \,{\left (b x + 1\right )} c^{3} d e^{\left (-b x - a\right )}}{b^{2}} + \frac{3 \,{\left (b^{2} x^{2} e^{a} - 2 \, b x e^{a} + 2 \, e^{a}\right )} c^{2} d^{2} e^{\left (b x\right )}}{b^{3}} + \frac{3 \,{\left (b^{2} x^{2} + 2 \, b x + 2\right )} c^{2} d^{2} e^{\left (-b x - a\right )}}{b^{3}} + \frac{2 \,{\left (b^{3} x^{3} e^{a} - 3 \, b^{2} x^{2} e^{a} + 6 \, b x e^{a} - 6 \, e^{a}\right )} c d^{3} e^{\left (b x\right )}}{b^{4}} + \frac{2 \,{\left (b^{3} x^{3} + 3 \, b^{2} x^{2} + 6 \, b x + 6\right )} c d^{3} e^{\left (-b x - a\right )}}{b^{4}} + \frac{{\left (b^{4} x^{4} e^{a} - 4 \, b^{3} x^{3} e^{a} + 12 \, b^{2} x^{2} e^{a} - 24 \, b x e^{a} + 24 \, e^{a}\right )} d^{4} e^{\left (b x\right )}}{2 \, b^{5}} + \frac{{\left (b^{4} x^{4} + 4 \, b^{3} x^{3} + 12 \, b^{2} x^{2} + 24 \, b x + 24\right )} d^{4} e^{\left (-b x - a\right )}}{2 \, b^{5}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.59911, size = 350, normalized size = 3.85 \begin{align*} \frac{{\left (b^{4} d^{4} x^{4} + 4 \, b^{4} c d^{3} x^{3} + b^{4} c^{4} + 12 \, b^{2} c^{2} d^{2} + 24 \, d^{4} + 6 \,{\left (b^{4} c^{2} d^{2} + 2 \, b^{2} d^{4}\right )} x^{2} + 4 \,{\left (b^{4} c^{3} d + 6 \, b^{2} c d^{3}\right )} x\right )} \cosh \left (b x + a\right ) - 4 \,{\left (b^{3} d^{4} x^{3} + 3 \, b^{3} c d^{3} x^{2} + b^{3} c^{3} d + 6 \, b c d^{3} + 3 \,{\left (b^{3} c^{2} d^{2} + 2 \, b d^{4}\right )} x\right )} \sinh \left (b x + a\right )}{b^{5}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 3.23735, size = 311, normalized size = 3.42 \begin{align*} \begin{cases} \frac{c^{4} \cosh{\left (a + b x \right )}}{b} + \frac{4 c^{3} d x \cosh{\left (a + b x \right )}}{b} + \frac{6 c^{2} d^{2} x^{2} \cosh{\left (a + b x \right )}}{b} + \frac{4 c d^{3} x^{3} \cosh{\left (a + b x \right )}}{b} + \frac{d^{4} x^{4} \cosh{\left (a + b x \right )}}{b} - \frac{4 c^{3} d \sinh{\left (a + b x \right )}}{b^{2}} - \frac{12 c^{2} d^{2} x \sinh{\left (a + b x \right )}}{b^{2}} - \frac{12 c d^{3} x^{2} \sinh{\left (a + b x \right )}}{b^{2}} - \frac{4 d^{4} x^{3} \sinh{\left (a + b x \right )}}{b^{2}} + \frac{12 c^{2} d^{2} \cosh{\left (a + b x \right )}}{b^{3}} + \frac{24 c d^{3} x \cosh{\left (a + b x \right )}}{b^{3}} + \frac{12 d^{4} x^{2} \cosh{\left (a + b x \right )}}{b^{3}} - \frac{24 c d^{3} \sinh{\left (a + b x \right )}}{b^{4}} - \frac{24 d^{4} x \sinh{\left (a + b x \right )}}{b^{4}} + \frac{24 d^{4} \cosh{\left (a + b x \right )}}{b^{5}} & \text{for}\: b \neq 0 \\\left (c^{4} x + 2 c^{3} d x^{2} + 2 c^{2} d^{2} x^{3} + c d^{3} x^{4} + \frac{d^{4} x^{5}}{5}\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 = 1.16781, size = 437, normalized size = 4.8 \begin{align*} \frac{{\left (b^{4} d^{4} x^{4} + 4 \, b^{4} c d^{3} x^{3} + 6 \, b^{4} c^{2} d^{2} x^{2} - 4 \, b^{3} d^{4} x^{3} + 4 \, b^{4} c^{3} d x - 12 \, b^{3} c d^{3} x^{2} + b^{4} c^{4} - 12 \, b^{3} c^{2} d^{2} x + 12 \, b^{2} d^{4} x^{2} - 4 \, b^{3} c^{3} d + 24 \, b^{2} c d^{3} x + 12 \, b^{2} c^{2} d^{2} - 24 \, b d^{4} x - 24 \, b c d^{3} + 24 \, d^{4}\right )} e^{\left (b x + a\right )}}{2 \, b^{5}} + \frac{{\left (b^{4} d^{4} x^{4} + 4 \, b^{4} c d^{3} x^{3} + 6 \, b^{4} c^{2} d^{2} x^{2} + 4 \, b^{3} d^{4} x^{3} + 4 \, b^{4} c^{3} d x + 12 \, b^{3} c d^{3} x^{2} + b^{4} c^{4} + 12 \, b^{3} c^{2} d^{2} x + 12 \, b^{2} d^{4} x^{2} + 4 \, b^{3} c^{3} d + 24 \, b^{2} c d^{3} x + 12 \, b^{2} c^{2} d^{2} + 24 \, b d^{4} x + 24 \, b c d^{3} + 24 \, d^{4}\right )} e^{\left (-b x - a\right )}}{2 \, b^{5}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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