Optimal. Leaf size=89 \[ \frac {a (c+d x)^4}{4 d}+\frac {6 b d^2 (c+d x) \sinh (e+f x)}{f^3}-\frac {3 b d (c+d x)^2 \cosh (e+f x)}{f^2}+\frac {b (c+d x)^3 \sinh (e+f x)}{f}-\frac {6 b d^3 \cosh (e+f x)}{f^4} \]
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Rubi [A] time = 0.13, antiderivative size = 89, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 3, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {3317, 3296, 2638} \[ \frac {a (c+d x)^4}{4 d}+\frac {6 b d^2 (c+d x) \sinh (e+f x)}{f^3}-\frac {3 b d (c+d x)^2 \cosh (e+f x)}{f^2}+\frac {b (c+d x)^3 \sinh (e+f x)}{f}-\frac {6 b d^3 \cosh (e+f x)}{f^4} \]
Antiderivative was successfully verified.
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Rule 2638
Rule 3296
Rule 3317
Rubi steps
\begin {align*} \int (c+d x)^3 (a+b \cosh (e+f x)) \, dx &=\int \left (a (c+d x)^3+b (c+d x)^3 \cosh (e+f x)\right ) \, dx\\ &=\frac {a (c+d x)^4}{4 d}+b \int (c+d x)^3 \cosh (e+f x) \, dx\\ &=\frac {a (c+d x)^4}{4 d}+\frac {b (c+d x)^3 \sinh (e+f x)}{f}-\frac {(3 b d) \int (c+d x)^2 \sinh (e+f x) \, dx}{f}\\ &=\frac {a (c+d x)^4}{4 d}-\frac {3 b d (c+d x)^2 \cosh (e+f x)}{f^2}+\frac {b (c+d x)^3 \sinh (e+f x)}{f}+\frac {\left (6 b d^2\right ) \int (c+d x) \cosh (e+f x) \, dx}{f^2}\\ &=\frac {a (c+d x)^4}{4 d}-\frac {3 b d (c+d x)^2 \cosh (e+f x)}{f^2}+\frac {6 b d^2 (c+d x) \sinh (e+f x)}{f^3}+\frac {b (c+d x)^3 \sinh (e+f x)}{f}-\frac {\left (6 b d^3\right ) \int \sinh (e+f x) \, dx}{f^3}\\ &=\frac {a (c+d x)^4}{4 d}-\frac {6 b d^3 \cosh (e+f x)}{f^4}-\frac {3 b d (c+d x)^2 \cosh (e+f x)}{f^2}+\frac {6 b d^2 (c+d x) \sinh (e+f x)}{f^3}+\frac {b (c+d x)^3 \sinh (e+f x)}{f}\\ \end {align*}
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Mathematica [A] time = 0.51, size = 123, normalized size = 1.38 \[ \frac {1}{4} a x \left (4 c^3+6 c^2 d x+4 c d^2 x^2+d^3 x^3\right )-\frac {3 b d \left (c^2 f^2+2 c d f^2 x+d^2 \left (f^2 x^2+2\right )\right ) \cosh (e+f x)}{f^4}+\frac {b (c+d x) \left (c^2 f^2+2 c d f^2 x+d^2 \left (f^2 x^2+6\right )\right ) \sinh (e+f x)}{f^3} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.58, size = 168, normalized size = 1.89 \[ \frac {a d^{3} f^{4} x^{4} + 4 \, a c d^{2} f^{4} x^{3} + 6 \, a c^{2} d f^{4} x^{2} + 4 \, a c^{3} f^{4} x - 12 \, {\left (b d^{3} f^{2} x^{2} + 2 \, b c d^{2} f^{2} x + b c^{2} d f^{2} + 2 \, b d^{3}\right )} \cosh \left (f x + e\right ) + 4 \, {\left (b d^{3} f^{3} x^{3} + 3 \, b c d^{2} f^{3} x^{2} + b c^{3} f^{3} + 6 \, b c d^{2} f + 3 \, {\left (b c^{2} d f^{3} + 2 \, b d^{3} f\right )} x\right )} \sinh \left (f x + e\right )}{4 \, f^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.12, size = 260, normalized size = 2.92 \[ \frac {1}{4} \, a d^{3} x^{4} + a c d^{2} x^{3} + \frac {3}{2} \, a c^{2} d x^{2} + a c^{3} x + \frac {{\left (b d^{3} f^{3} x^{3} + 3 \, b c d^{2} f^{3} x^{2} + 3 \, b c^{2} d f^{3} x - 3 \, b d^{3} f^{2} x^{2} + b c^{3} f^{3} - 6 \, b c d^{2} f^{2} x - 3 \, b c^{2} d f^{2} + 6 \, b d^{3} f x + 6 \, b c d^{2} f - 6 \, b d^{3}\right )} e^{\left (f x + e\right )}}{2 \, f^{4}} - \frac {{\left (b d^{3} f^{3} x^{3} + 3 \, b c d^{2} f^{3} x^{2} + 3 \, b c^{2} d f^{3} x + 3 \, b d^{3} f^{2} x^{2} + b c^{3} f^{3} + 6 \, b c d^{2} f^{2} x + 3 \, b c^{2} d f^{2} + 6 \, b d^{3} f x + 6 \, b c d^{2} f + 6 \, b d^{3}\right )} e^{\left (-f x - e\right )}}{2 \, f^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.07, size = 482, normalized size = 5.42 \[ \frac {\frac {d^{3} a \left (f x +e \right )^{4}}{4 f^{3}}+\frac {d^{3} b \left (\left (f x +e \right )^{3} \sinh \left (f x +e \right )-3 \left (f x +e \right )^{2} \cosh \left (f x +e \right )+6 \left (f x +e \right ) \sinh \left (f x +e \right )-6 \cosh \left (f x +e \right )\right )}{f^{3}}-\frac {d^{3} e a \left (f x +e \right )^{3}}{f^{3}}-\frac {3 d^{3} e b \left (\left (f x +e \right )^{2} \sinh \left (f x +e \right )-2 \left (f x +e \right ) \cosh \left (f x +e \right )+2 \sinh \left (f x +e \right )\right )}{f^{3}}+\frac {d^{2} c a \left (f x +e \right )^{3}}{f^{2}}+\frac {3 d^{2} c b \left (\left (f x +e \right )^{2} \sinh \left (f x +e \right )-2 \left (f x +e \right ) \cosh \left (f x +e \right )+2 \sinh \left (f x +e \right )\right )}{f^{2}}+\frac {3 d^{3} e^{2} a \left (f x +e \right )^{2}}{2 f^{3}}+\frac {3 d^{3} e^{2} b \left (\left (f x +e \right ) \sinh \left (f x +e \right )-\cosh \left (f x +e \right )\right )}{f^{3}}-\frac {3 d^{2} e c a \left (f x +e \right )^{2}}{f^{2}}-\frac {6 d^{2} e c b \left (\left (f x +e \right ) \sinh \left (f x +e \right )-\cosh \left (f x +e \right )\right )}{f^{2}}+\frac {3 d \,c^{2} a \left (f x +e \right )^{2}}{2 f}+\frac {3 d \,c^{2} b \left (\left (f x +e \right ) \sinh \left (f x +e \right )-\cosh \left (f x +e \right )\right )}{f}-\frac {d^{3} e^{3} a \left (f x +e \right )}{f^{3}}-\frac {d^{3} e^{3} b \sinh \left (f x +e \right )}{f^{3}}+\frac {3 d^{2} e^{2} c a \left (f x +e \right )}{f^{2}}+\frac {3 d^{2} e^{2} c b \sinh \left (f x +e \right )}{f^{2}}-\frac {3 d e \,c^{2} a \left (f x +e \right )}{f}-\frac {3 d e \,c^{2} b \sinh \left (f x +e \right )}{f}+c^{3} a \left (f x +e \right )+b \,c^{3} \sinh \left (f x +e \right )}{f} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.35, size = 237, normalized size = 2.66 \[ \frac {1}{4} \, a d^{3} x^{4} + a c d^{2} x^{3} + \frac {3}{2} \, a c^{2} d x^{2} + a c^{3} x + \frac {3}{2} \, b c^{2} d {\left (\frac {{\left (f x e^{e} - e^{e}\right )} e^{\left (f x\right )}}{f^{2}} - \frac {{\left (f x + 1\right )} e^{\left (-f x - e\right )}}{f^{2}}\right )} + \frac {3}{2} \, b c d^{2} {\left (\frac {{\left (f^{2} x^{2} e^{e} - 2 \, f x e^{e} + 2 \, e^{e}\right )} e^{\left (f x\right )}}{f^{3}} - \frac {{\left (f^{2} x^{2} + 2 \, f x + 2\right )} e^{\left (-f x - e\right )}}{f^{3}}\right )} + \frac {1}{2} \, b d^{3} {\left (\frac {{\left (f^{3} x^{3} e^{e} - 3 \, f^{2} x^{2} e^{e} + 6 \, f x e^{e} - 6 \, e^{e}\right )} e^{\left (f x\right )}}{f^{4}} - \frac {{\left (f^{3} x^{3} + 3 \, f^{2} x^{2} + 6 \, f x + 6\right )} e^{\left (-f x - e\right )}}{f^{4}}\right )} + \frac {b c^{3} \sinh \left (f x + e\right )}{f} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.99, size = 187, normalized size = 2.10 \[ \frac {\mathrm {sinh}\left (e+f\,x\right )\,\left (b\,c^3\,f^2+6\,b\,c\,d^2\right )}{f^3}-\frac {3\,\mathrm {cosh}\left (e+f\,x\right )\,\left (b\,c^2\,d\,f^2+2\,b\,d^3\right )}{f^4}+\frac {a\,d^3\,x^4}{4}+a\,c^3\,x+\frac {3\,x\,\mathrm {sinh}\left (e+f\,x\right )\,\left (b\,c^2\,d\,f^2+2\,b\,d^3\right )}{f^3}+\frac {3\,a\,c^2\,d\,x^2}{2}+a\,c\,d^2\,x^3-\frac {3\,b\,d^3\,x^2\,\mathrm {cosh}\left (e+f\,x\right )}{f^2}+\frac {b\,d^3\,x^3\,\mathrm {sinh}\left (e+f\,x\right )}{f}-\frac {6\,b\,c\,d^2\,x\,\mathrm {cosh}\left (e+f\,x\right )}{f^2}+\frac {3\,b\,c\,d^2\,x^2\,\mathrm {sinh}\left (e+f\,x\right )}{f} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 1.41, size = 264, normalized size = 2.97 \[ \begin {cases} a c^{3} x + \frac {3 a c^{2} d x^{2}}{2} + a c d^{2} x^{3} + \frac {a d^{3} x^{4}}{4} + \frac {b c^{3} \sinh {\left (e + f x \right )}}{f} + \frac {3 b c^{2} d x \sinh {\left (e + f x \right )}}{f} - \frac {3 b c^{2} d \cosh {\left (e + f x \right )}}{f^{2}} + \frac {3 b c d^{2} x^{2} \sinh {\left (e + f x \right )}}{f} - \frac {6 b c d^{2} x \cosh {\left (e + f x \right )}}{f^{2}} + \frac {6 b c d^{2} \sinh {\left (e + f x \right )}}{f^{3}} + \frac {b d^{3} x^{3} \sinh {\left (e + f x \right )}}{f} - \frac {3 b d^{3} x^{2} \cosh {\left (e + f x \right )}}{f^{2}} + \frac {6 b d^{3} x \sinh {\left (e + f x \right )}}{f^{3}} - \frac {6 b d^{3} \cosh {\left (e + f x \right )}}{f^{4}} & \text {for}\: f \neq 0 \\\left (a + b \cosh {\relax (e )}\right ) \left (c^{3} x + \frac {3 c^{2} d x^{2}}{2} + c d^{2} x^{3} + \frac {d^{3} x^{4}}{4}\right ) & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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