Optimal. Leaf size=70 \[ \frac {6 d^2 (c+d x) \cosh (a+b x)}{b^3}+\frac {(c+d x)^3 \cosh (a+b x)}{b}-\frac {6 d^3 \sinh (a+b x)}{b^4}-\frac {3 d (c+d x)^2 \sinh (a+b x)}{b^2} \]
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Rubi [A]
time = 0.06, antiderivative size = 70, normalized size of antiderivative = 1.00, 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 = {3377, 2717}
\begin {gather*} -\frac {6 d^3 \sinh (a+b x)}{b^4}+\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 {(c+d x)^3 \cosh (a+b x)}{b} \end {gather*}
Antiderivative was successfully verified.
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Rule 2717
Rule 3377
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*}
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Mathematica [A]
time = 0.11, size = 61, normalized size = 0.87 \begin {gather*} \frac {b (c+d x) \left (6 d^2+b^2 (c+d x)^2\right ) \cosh (a+b x)-3 d \left (2 d^2+b^2 (c+d x)^2\right ) \sinh (a+b x)}{b^4} \end {gather*}
Antiderivative was successfully verified.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(307\) vs.
\(2(70)=140\).
time = 0.30, size = 308, normalized size = 4.40
method | result | size |
risch | \(\frac {\left (d^{3} x^{3} b^{3}+3 b^{3} c \,d^{2} x^{2}+3 b^{3} c^{2} d x -3 b^{2} d^{3} x^{2}+b^{3} c^{3}-6 b^{2} c \,d^{2} x -3 b^{2} c^{2} d +6 b \,d^{3} x +6 b c \,d^{2}-6 d^{3}\right ) {\mathrm e}^{b x +a}}{2 b^{4}}+\frac {\left (d^{3} x^{3} b^{3}+3 b^{3} c \,d^{2} x^{2}+3 b^{3} c^{2} d x +3 b^{2} d^{3} x^{2}+b^{3} c^{3}+6 b^{2} c \,d^{2} x +3 b^{2} c^{2} d +6 b \,d^{3} x +6 b c \,d^{2}+6 d^{3}\right ) {\mathrm e}^{-b x -a}}{2 b^{4}}\) | \(205\) |
derivativedivides | \(\frac {\frac {d^{3} \left (\left (b x +a \right )^{3} \cosh \left (b x +a \right )-3 \left (b x +a \right )^{2} \sinh \left (b x +a \right )+6 \left (b x +a \right ) \cosh \left (b x +a \right )-6 \sinh \left (b x +a \right )\right )}{b^{3}}-\frac {3 d^{3} a \left (\left (b x +a \right )^{2} \cosh \left (b x +a \right )-2 \left (b x +a \right ) \sinh \left (b x +a \right )+2 \cosh \left (b x +a \right )\right )}{b^{3}}+\frac {3 d^{2} c \left (\left (b x +a \right )^{2} \cosh \left (b x +a \right )-2 \left (b x +a \right ) \sinh \left (b x +a \right )+2 \cosh \left (b x +a \right )\right )}{b^{2}}+\frac {3 d^{3} a^{2} \left (\left (b x +a \right ) \cosh \left (b x +a \right )-\sinh \left (b x +a \right )\right )}{b^{3}}-\frac {6 d^{2} a c \left (\left (b x +a \right ) \cosh \left (b x +a \right )-\sinh \left (b x +a \right )\right )}{b^{2}}+\frac {3 d \,c^{2} \left (\left (b x +a \right ) \cosh \left (b x +a \right )-\sinh \left (b x +a \right )\right )}{b}-\frac {d^{3} a^{3} \cosh \left (b x +a \right )}{b^{3}}+\frac {3 d^{2} a^{2} c \cosh \left (b x +a \right )}{b^{2}}-\frac {3 d a \,c^{2} \cosh \left (b x +a \right )}{b}+c^{3} \cosh \left (b x +a \right )}{b}\) | \(308\) |
default | \(\frac {\frac {d^{3} \left (\left (b x +a \right )^{3} \cosh \left (b x +a \right )-3 \left (b x +a \right )^{2} \sinh \left (b x +a \right )+6 \left (b x +a \right ) \cosh \left (b x +a \right )-6 \sinh \left (b x +a \right )\right )}{b^{3}}-\frac {3 d^{3} a \left (\left (b x +a \right )^{2} \cosh \left (b x +a \right )-2 \left (b x +a \right ) \sinh \left (b x +a \right )+2 \cosh \left (b x +a \right )\right )}{b^{3}}+\frac {3 d^{2} c \left (\left (b x +a \right )^{2} \cosh \left (b x +a \right )-2 \left (b x +a \right ) \sinh \left (b x +a \right )+2 \cosh \left (b x +a \right )\right )}{b^{2}}+\frac {3 d^{3} a^{2} \left (\left (b x +a \right ) \cosh \left (b x +a \right )-\sinh \left (b x +a \right )\right )}{b^{3}}-\frac {6 d^{2} a c \left (\left (b x +a \right ) \cosh \left (b x +a \right )-\sinh \left (b x +a \right )\right )}{b^{2}}+\frac {3 d \,c^{2} \left (\left (b x +a \right ) \cosh \left (b x +a \right )-\sinh \left (b x +a \right )\right )}{b}-\frac {d^{3} a^{3} \cosh \left (b x +a \right )}{b^{3}}+\frac {3 d^{2} a^{2} c \cosh \left (b x +a \right )}{b^{2}}-\frac {3 d a \,c^{2} \cosh \left (b x +a \right )}{b}+c^{3} \cosh \left (b x +a \right )}{b}\) | \(308\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 222 vs.
\(2 (70) = 140\).
time = 0.27, size = 222, normalized size = 3.17 \begin {gather*} \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 {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.34, size = 109, normalized size = 1.56 \begin {gather*} \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 {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 202 vs.
\(2 (70) = 140\).
time = 0.25, size = 202, normalized size = 2.89 \begin {gather*} \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 {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 204 vs.
\(2 (70) = 140\).
time = 0.44, size = 204, normalized size = 2.91 \begin {gather*} \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 {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.14, size = 143, normalized size = 2.04 \begin {gather*} \frac {\mathrm {cosh}\left (a+b\,x\right )\,\left (b^2\,c^3+6\,c\,d^2\right )}{b^3}-\frac {3\,\mathrm {sinh}\left (a+b\,x\right )\,\left (b^2\,c^2\,d+2\,d^3\right )}{b^4}+\frac {d^3\,x^3\,\mathrm {cosh}\left (a+b\,x\right )}{b}-\frac {3\,d^3\,x^2\,\mathrm {sinh}\left (a+b\,x\right )}{b^2}+\frac {3\,x\,\mathrm {cosh}\left (a+b\,x\right )\,\left (b^2\,c^2\,d+2\,d^3\right )}{b^3}-\frac {6\,c\,d^2\,x\,\mathrm {sinh}\left (a+b\,x\right )}{b^2}+\frac {3\,c\,d^2\,x^2\,\mathrm {cosh}\left (a+b\,x\right )}{b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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