Optimal. Leaf size=91 \[ \frac {24 d^4 \cosh (a+b x)}{b^5}-\frac {24 d^3 (c+d x) \sinh (a+b x)}{b^4}+\frac {12 d^2 (c+d x)^2 \cosh (a+b x)}{b^3}-\frac {4 d (c+d x)^3 \sinh (a+b x)}{b^2}+\frac {(c+d x)^4 \cosh (a+b x)}{b} \]
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Rubi [A] time = 0.12, antiderivative size = 91, normalized size of antiderivative = 1.00, 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 2638
Rule 3296
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*}
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Mathematica [A] time = 0.35, size = 76, normalized size = 0.84 \[ \frac {\cosh (a+b x) \left (b^4 (c+d x)^4+12 b^2 d^2 (c+d x)^2+24 d^4\right )-4 b d (c+d x) \sinh (a+b x) \left (b^2 (c+d x)^2+6 d^2\right )}{b^5} \]
Antiderivative was successfully verified.
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fricas [A] time = 2.19, size = 169, normalized size = 1.86 \[ \frac {{\left (b^{4} d^{4} x^{4} + 4 \, b^{4} c d^{3} x^{3} + b^{4} c^{4} + 12 \, b^{2} c^{2} d^{2} + 24 \, d^{4} + 6 \, {\left (b^{4} c^{2} d^{2} + 2 \, b^{2} d^{4}\right )} x^{2} + 4 \, {\left (b^{4} c^{3} d + 6 \, b^{2} c d^{3}\right )} x\right )} \cosh \left (b x + a\right ) - 4 \, {\left (b^{3} d^{4} x^{3} + 3 \, b^{3} c d^{3} x^{2} + b^{3} c^{3} d + 6 \, b c d^{3} + 3 \, {\left (b^{3} c^{2} d^{2} + 2 \, b d^{4}\right )} x\right )} \sinh \left (b x + a\right )}{b^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.55, size = 324, normalized size = 3.56 \[ \frac {{\left (b^{4} d^{4} x^{4} + 4 \, b^{4} c d^{3} x^{3} + 6 \, b^{4} c^{2} d^{2} x^{2} - 4 \, b^{3} d^{4} x^{3} + 4 \, b^{4} c^{3} d x - 12 \, b^{3} c d^{3} x^{2} + b^{4} c^{4} - 12 \, b^{3} c^{2} d^{2} x + 12 \, b^{2} d^{4} x^{2} - 4 \, b^{3} c^{3} d + 24 \, b^{2} c d^{3} x + 12 \, b^{2} c^{2} d^{2} - 24 \, b d^{4} x - 24 \, b c d^{3} + 24 \, d^{4}\right )} e^{\left (b x + a\right )}}{2 \, b^{5}} + \frac {{\left (b^{4} d^{4} x^{4} + 4 \, b^{4} c d^{3} x^{3} + 6 \, b^{4} c^{2} d^{2} x^{2} + 4 \, b^{3} d^{4} x^{3} + 4 \, b^{4} c^{3} d x + 12 \, b^{3} c d^{3} x^{2} + b^{4} c^{4} + 12 \, b^{3} c^{2} d^{2} x + 12 \, b^{2} d^{4} x^{2} + 4 \, b^{3} c^{3} d + 24 \, b^{2} c d^{3} x + 12 \, b^{2} c^{2} d^{2} + 24 \, b d^{4} x + 24 \, b c d^{3} + 24 \, d^{4}\right )} e^{\left (-b x - a\right )}}{2 \, b^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.06, size = 547, normalized size = 6.01 \[ \frac {-\frac {12 d^{3} a c \left (\left (b x +a \right )^{2} \cosh \left (b x +a \right )-2 \left (b x +a \right ) \sinh \left (b x +a \right )+2 \cosh \left (b x +a \right )\right )}{b^{3}}+\frac {12 d^{3} a^{2} c \left (\left (b x +a \right ) \cosh \left (b x +a \right )-\sinh \left (b x +a \right )\right )}{b^{3}}-\frac {12 d^{2} a \,c^{2} \left (\left (b x +a \right ) \cosh \left (b x +a \right )-\sinh \left (b x +a \right )\right )}{b^{2}}+\frac {6 d^{2} a^{2} c^{2} \cosh \left (b x +a \right )}{b^{2}}-\frac {4 d a \,c^{3} \cosh \left (b x +a \right )}{b}+\frac {4 d \,c^{3} \left (\left (b x +a \right ) \cosh \left (b x +a \right )-\sinh \left (b x +a \right )\right )}{b}-\frac {4 d^{3} a^{3} c \cosh \left (b x +a \right )}{b^{3}}-\frac {4 d^{4} a \left (\left (b x +a \right )^{3} \cosh \left (b x +a \right )-3 \left (b x +a \right )^{2} \sinh \left (b x +a \right )+6 \left (b x +a \right ) \cosh \left (b x +a \right )-6 \sinh \left (b x +a \right )\right )}{b^{4}}+\frac {4 d^{3} c \left (\left (b x +a \right )^{3} \cosh \left (b x +a \right )-3 \left (b x +a \right )^{2} \sinh \left (b x +a \right )+6 \left (b x +a \right ) \cosh \left (b x +a \right )-6 \sinh \left (b x +a \right )\right )}{b^{3}}+\frac {6 d^{4} a^{2} \left (\left (b x +a \right )^{2} \cosh \left (b x +a \right )-2 \left (b x +a \right ) \sinh \left (b x +a \right )+2 \cosh \left (b x +a \right )\right )}{b^{4}}+\frac {6 d^{2} c^{2} \left (\left (b x +a \right )^{2} \cosh \left (b x +a \right )-2 \left (b x +a \right ) \sinh \left (b x +a \right )+2 \cosh \left (b x +a \right )\right )}{b^{2}}-\frac {4 d^{4} a^{3} \left (\left (b x +a \right ) \cosh \left (b x +a \right )-\sinh \left (b x +a \right )\right )}{b^{4}}+\frac {d^{4} \left (\left (b x +a \right )^{4} \cosh \left (b x +a \right )-4 \left (b x +a \right )^{3} \sinh \left (b x +a \right )+12 \left (b x +a \right )^{2} \cosh \left (b x +a \right )-24 \left (b x +a \right ) \sinh \left (b x +a \right )+24 \cosh \left (b x +a \right )\right )}{b^{4}}+\frac {d^{4} a^{4} \cosh \left (b x +a \right )}{b^{4}}+c^{4} \cosh \left (b x +a \right )}{b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.69, size = 326, normalized size = 3.58 \[ \frac {c^{4} e^{\left (b x + a\right )}}{2 \, b} + \frac {2 \, {\left (b x e^{a} - e^{a}\right )} c^{3} d e^{\left (b x\right )}}{b^{2}} + \frac {c^{4} e^{\left (-b x - a\right )}}{2 \, b} + \frac {2 \, {\left (b x + 1\right )} c^{3} d e^{\left (-b x - a\right )}}{b^{2}} + \frac {3 \, {\left (b^{2} x^{2} e^{a} - 2 \, b x e^{a} + 2 \, e^{a}\right )} c^{2} d^{2} e^{\left (b x\right )}}{b^{3}} + \frac {3 \, {\left (b^{2} x^{2} + 2 \, b x + 2\right )} c^{2} d^{2} e^{\left (-b x - a\right )}}{b^{3}} + \frac {2 \, {\left (b^{3} x^{3} e^{a} - 3 \, b^{2} x^{2} e^{a} + 6 \, b x e^{a} - 6 \, e^{a}\right )} c d^{3} e^{\left (b x\right )}}{b^{4}} + \frac {2 \, {\left (b^{3} x^{3} + 3 \, b^{2} x^{2} + 6 \, b x + 6\right )} c d^{3} e^{\left (-b x - a\right )}}{b^{4}} + \frac {{\left (b^{4} x^{4} e^{a} - 4 \, b^{3} x^{3} e^{a} + 12 \, b^{2} x^{2} e^{a} - 24 \, b x e^{a} + 24 \, e^{a}\right )} d^{4} e^{\left (b x\right )}}{2 \, b^{5}} + \frac {{\left (b^{4} x^{4} + 4 \, b^{3} x^{3} + 12 \, b^{2} x^{2} + 24 \, b x + 24\right )} d^{4} e^{\left (-b x - a\right )}}{2 \, b^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.45, size = 215, normalized size = 2.36 \[ \frac {\mathrm {cosh}\left (a+b\,x\right )\,\left (b^4\,c^4+12\,b^2\,c^2\,d^2+24\,d^4\right )}{b^5}-\frac {4\,\mathrm {sinh}\left (a+b\,x\right )\,\left (b^2\,c^3\,d+6\,c\,d^3\right )}{b^4}+\frac {d^4\,x^4\,\mathrm {cosh}\left (a+b\,x\right )}{b}+\frac {4\,x\,\mathrm {cosh}\left (a+b\,x\right )\,\left (b^2\,c^3\,d+6\,c\,d^3\right )}{b^3}-\frac {4\,d^4\,x^3\,\mathrm {sinh}\left (a+b\,x\right )}{b^2}-\frac {12\,x\,\mathrm {sinh}\left (a+b\,x\right )\,\left (b^2\,c^2\,d^2+2\,d^4\right )}{b^4}+\frac {6\,x^2\,\mathrm {cosh}\left (a+b\,x\right )\,\left (b^2\,c^2\,d^2+2\,d^4\right )}{b^3}+\frac {4\,c\,d^3\,x^3\,\mathrm {cosh}\left (a+b\,x\right )}{b}-\frac {12\,c\,d^3\,x^2\,\mathrm {sinh}\left (a+b\,x\right )}{b^2} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 2.41, size = 311, normalized size = 3.42 \[ \begin {cases} \frac {c^{4} \cosh {\left (a + b x \right )}}{b} + \frac {4 c^{3} d x \cosh {\left (a + b x \right )}}{b} + \frac {6 c^{2} d^{2} x^{2} \cosh {\left (a + b x \right )}}{b} + \frac {4 c d^{3} x^{3} \cosh {\left (a + b x \right )}}{b} + \frac {d^{4} x^{4} \cosh {\left (a + b x \right )}}{b} - \frac {4 c^{3} d \sinh {\left (a + b x \right )}}{b^{2}} - \frac {12 c^{2} d^{2} x \sinh {\left (a + b x \right )}}{b^{2}} - \frac {12 c d^{3} x^{2} \sinh {\left (a + b x \right )}}{b^{2}} - \frac {4 d^{4} x^{3} \sinh {\left (a + b x \right )}}{b^{2}} + \frac {12 c^{2} d^{2} \cosh {\left (a + b x \right )}}{b^{3}} + \frac {24 c d^{3} x \cosh {\left (a + b x \right )}}{b^{3}} + \frac {12 d^{4} x^{2} \cosh {\left (a + b x \right )}}{b^{3}} - \frac {24 c d^{3} \sinh {\left (a + b x \right )}}{b^{4}} - \frac {24 d^{4} x \sinh {\left (a + b x \right )}}{b^{4}} + \frac {24 d^{4} \cosh {\left (a + b x \right )}}{b^{5}} & \text {for}\: b \neq 0 \\\left (c^{4} x + 2 c^{3} d x^{2} + 2 c^{2} d^{2} x^{3} + c d^{3} x^{4} + \frac {d^{4} x^{5}}{5}\right ) \sinh {\relax (a )} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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