Optimal. Leaf size=95 \[ \frac {d^2 \sinh (a+b x) \cosh (a+b x)}{4 b^3}-\frac {d (c+d x) \cosh ^2(a+b x)}{2 b^2}+\frac {(c+d x)^2 \sinh (a+b x) \cosh (a+b x)}{2 b}+\frac {d^2 x}{4 b^2}+\frac {(c+d x)^3}{6 d} \]
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Rubi [A] time = 0.05, antiderivative size = 95, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {3311, 32, 2635, 8} \[ -\frac {d (c+d x) \cosh ^2(a+b x)}{2 b^2}+\frac {d^2 \sinh (a+b x) \cosh (a+b x)}{4 b^3}+\frac {(c+d x)^2 \sinh (a+b x) \cosh (a+b x)}{2 b}+\frac {d^2 x}{4 b^2}+\frac {(c+d x)^3}{6 d} \]
Antiderivative was successfully verified.
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Rule 8
Rule 32
Rule 2635
Rule 3311
Rubi steps
\begin {align*} \int (c+d x)^2 \cosh ^2(a+b x) \, dx &=-\frac {d (c+d x) \cosh ^2(a+b x)}{2 b^2}+\frac {(c+d x)^2 \cosh (a+b x) \sinh (a+b x)}{2 b}+\frac {1}{2} \int (c+d x)^2 \, dx+\frac {d^2 \int \cosh ^2(a+b x) \, dx}{2 b^2}\\ &=\frac {(c+d x)^3}{6 d}-\frac {d (c+d x) \cosh ^2(a+b x)}{2 b^2}+\frac {d^2 \cosh (a+b x) \sinh (a+b x)}{4 b^3}+\frac {(c+d x)^2 \cosh (a+b x) \sinh (a+b x)}{2 b}+\frac {d^2 \int 1 \, dx}{4 b^2}\\ &=\frac {d^2 x}{4 b^2}+\frac {(c+d x)^3}{6 d}-\frac {d (c+d x) \cosh ^2(a+b x)}{2 b^2}+\frac {d^2 \cosh (a+b x) \sinh (a+b x)}{4 b^3}+\frac {(c+d x)^2 \cosh (a+b x) \sinh (a+b x)}{2 b}\\ \end {align*}
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Mathematica [A] time = 0.30, size = 75, normalized size = 0.79 \[ \frac {3 \sinh (2 (a+b x)) \left (2 b^2 (c+d x)^2+d^2\right )-6 b d (c+d x) \cosh (2 (a+b x))+4 b^3 x \left (3 c^2+3 c d x+d^2 x^2\right )}{24 b^3} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.72, size = 123, normalized size = 1.29 \[ \frac {2 \, b^{3} d^{2} x^{3} + 6 \, b^{3} c d x^{2} + 6 \, b^{3} c^{2} x - 3 \, {\left (b d^{2} x + b c d\right )} \cosh \left (b x + a\right )^{2} + 3 \, {\left (2 \, b^{2} d^{2} x^{2} + 4 \, b^{2} c d x + 2 \, b^{2} c^{2} + d^{2}\right )} \cosh \left (b x + a\right ) \sinh \left (b x + a\right ) - 3 \, {\left (b d^{2} x + b c d\right )} \sinh \left (b x + a\right )^{2}}{12 \, b^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.12, size = 136, normalized size = 1.43 \[ \frac {1}{6} \, d^{2} x^{3} + \frac {1}{2} \, c d x^{2} + \frac {1}{2} \, c^{2} x + \frac {{\left (2 \, b^{2} d^{2} x^{2} + 4 \, b^{2} c d x + 2 \, b^{2} c^{2} - 2 \, b d^{2} x - 2 \, b c d + d^{2}\right )} e^{\left (2 \, b x + 2 \, a\right )}}{16 \, b^{3}} - \frac {{\left (2 \, b^{2} d^{2} x^{2} + 4 \, b^{2} c d x + 2 \, b^{2} c^{2} + 2 \, b d^{2} x + 2 \, b c d + d^{2}\right )} e^{\left (-2 \, b x - 2 \, a\right )}}{16 \, b^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.05, size = 262, normalized size = 2.76 \[ \frac {\frac {d^{2} \left (\frac {\left (b x +a \right )^{2} \cosh \left (b x +a \right ) \sinh \left (b x +a \right )}{2}+\frac {\left (b x +a \right )^{3}}{6}-\frac {\left (b x +a \right ) \left (\cosh ^{2}\left (b x +a \right )\right )}{2}+\frac {\cosh \left (b x +a \right ) \sinh \left (b x +a \right )}{4}+\frac {b x}{4}+\frac {a}{4}\right )}{b^{2}}-\frac {2 d^{2} a \left (\frac {\left (b x +a \right ) \cosh \left (b x +a \right ) \sinh \left (b x +a \right )}{2}+\frac {\left (b x +a \right )^{2}}{4}-\frac {\left (\cosh ^{2}\left (b x +a \right )\right )}{4}\right )}{b^{2}}+\frac {d^{2} a^{2} \left (\frac {\cosh \left (b x +a \right ) \sinh \left (b x +a \right )}{2}+\frac {b x}{2}+\frac {a}{2}\right )}{b^{2}}+\frac {2 c d \left (\frac {\left (b x +a \right ) \cosh \left (b x +a \right ) \sinh \left (b x +a \right )}{2}+\frac {\left (b x +a \right )^{2}}{4}-\frac {\left (\cosh ^{2}\left (b x +a \right )\right )}{4}\right )}{b}-\frac {2 c d a \left (\frac {\cosh \left (b x +a \right ) \sinh \left (b x +a \right )}{2}+\frac {b x}{2}+\frac {a}{2}\right )}{b}+c^{2} \left (\frac {\cosh \left (b x +a \right ) \sinh \left (b x +a \right )}{2}+\frac {b x}{2}+\frac {a}{2}\right )}{b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.38, size = 165, normalized size = 1.74 \[ \frac {1}{8} \, {\left (4 \, x^{2} + \frac {{\left (2 \, b x e^{\left (2 \, a\right )} - e^{\left (2 \, a\right )}\right )} e^{\left (2 \, b x\right )}}{b^{2}} - \frac {{\left (2 \, b x + 1\right )} e^{\left (-2 \, b x - 2 \, a\right )}}{b^{2}}\right )} c d + \frac {1}{48} \, {\left (8 \, x^{3} + \frac {3 \, {\left (2 \, b^{2} x^{2} e^{\left (2 \, a\right )} - 2 \, b x e^{\left (2 \, a\right )} + e^{\left (2 \, a\right )}\right )} e^{\left (2 \, b x\right )}}{b^{3}} - \frac {3 \, {\left (2 \, b^{2} x^{2} + 2 \, b x + 1\right )} e^{\left (-2 \, b x - 2 \, a\right )}}{b^{3}}\right )} d^{2} + \frac {1}{8} \, c^{2} {\left (4 \, x + \frac {e^{\left (2 \, b x + 2 \, a\right )}}{b} - \frac {e^{\left (-2 \, b x - 2 \, a\right )}}{b}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.98, size = 127, normalized size = 1.34 \[ \frac {c^2\,x}{2}+\frac {d^2\,x^3}{6}+\frac {c^2\,\mathrm {sinh}\left (2\,a+2\,b\,x\right )}{4\,b}+\frac {d^2\,\mathrm {sinh}\left (2\,a+2\,b\,x\right )}{8\,b^3}+\frac {c\,d\,x^2}{2}-\frac {d^2\,x\,\mathrm {cosh}\left (2\,a+2\,b\,x\right )}{4\,b^2}+\frac {d^2\,x^2\,\mathrm {sinh}\left (2\,a+2\,b\,x\right )}{4\,b}-\frac {c\,d\,\mathrm {cosh}\left (2\,a+2\,b\,x\right )}{4\,b^2}+\frac {c\,d\,x\,\mathrm {sinh}\left (2\,a+2\,b\,x\right )}{2\,b} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 1.22, size = 264, normalized size = 2.78 \[ \begin {cases} - \frac {c^{2} x \sinh ^{2}{\left (a + b x \right )}}{2} + \frac {c^{2} x \cosh ^{2}{\left (a + b x \right )}}{2} - \frac {c d x^{2} \sinh ^{2}{\left (a + b x \right )}}{2} + \frac {c d x^{2} \cosh ^{2}{\left (a + b x \right )}}{2} - \frac {d^{2} x^{3} \sinh ^{2}{\left (a + b x \right )}}{6} + \frac {d^{2} x^{3} \cosh ^{2}{\left (a + b x \right )}}{6} + \frac {c^{2} \sinh {\left (a + b x \right )} \cosh {\left (a + b x \right )}}{2 b} + \frac {c d x \sinh {\left (a + b x \right )} \cosh {\left (a + b x \right )}}{b} + \frac {d^{2} x^{2} \sinh {\left (a + b x \right )} \cosh {\left (a + b x \right )}}{2 b} - \frac {c d \cosh ^{2}{\left (a + b x \right )}}{2 b^{2}} - \frac {d^{2} x \sinh ^{2}{\left (a + b x \right )}}{4 b^{2}} - \frac {d^{2} x \cosh ^{2}{\left (a + b x \right )}}{4 b^{2}} + \frac {d^{2} \sinh {\left (a + b x \right )} \cosh {\left (a + b x \right )}}{4 b^{3}} & \text {for}\: b \neq 0 \\\left (c^{2} x + c d x^{2} + \frac {d^{2} x^{3}}{3}\right ) \cosh ^{2}{\relax (a )} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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