Optimal. Leaf size=123 \[ \frac {2 d^2 \sinh ^3(a+b x)}{27 b^3}+\frac {14 d^2 \sinh (a+b x)}{9 b^3}-\frac {2 d (c+d x) \cosh ^3(a+b x)}{9 b^2}-\frac {4 d (c+d x) \cosh (a+b x)}{3 b^2}+\frac {2 (c+d x)^2 \sinh (a+b x)}{3 b}+\frac {(c+d x)^2 \sinh (a+b x) \cosh ^2(a+b x)}{3 b} \]
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Rubi [A] time = 0.10, antiderivative size = 123, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 4, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {3311, 3296, 2637, 2633} \[ -\frac {2 d (c+d x) \cosh ^3(a+b x)}{9 b^2}-\frac {4 d (c+d x) \cosh (a+b x)}{3 b^2}+\frac {2 d^2 \sinh ^3(a+b x)}{27 b^3}+\frac {14 d^2 \sinh (a+b x)}{9 b^3}+\frac {2 (c+d x)^2 \sinh (a+b x)}{3 b}+\frac {(c+d x)^2 \sinh (a+b x) \cosh ^2(a+b x)}{3 b} \]
Antiderivative was successfully verified.
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Rule 2633
Rule 2637
Rule 3296
Rule 3311
Rubi steps
\begin {align*} \int (c+d x)^2 \cosh ^3(a+b x) \, dx &=-\frac {2 d (c+d x) \cosh ^3(a+b x)}{9 b^2}+\frac {(c+d x)^2 \cosh ^2(a+b x) \sinh (a+b x)}{3 b}+\frac {2}{3} \int (c+d x)^2 \cosh (a+b x) \, dx+\frac {\left (2 d^2\right ) \int \cosh ^3(a+b x) \, dx}{9 b^2}\\ &=-\frac {2 d (c+d x) \cosh ^3(a+b x)}{9 b^2}+\frac {2 (c+d x)^2 \sinh (a+b x)}{3 b}+\frac {(c+d x)^2 \cosh ^2(a+b x) \sinh (a+b x)}{3 b}-\frac {(4 d) \int (c+d x) \sinh (a+b x) \, dx}{3 b}+\frac {\left (2 i d^2\right ) \operatorname {Subst}\left (\int \left (1-x^2\right ) \, dx,x,-i \sinh (a+b x)\right )}{9 b^3}\\ &=-\frac {4 d (c+d x) \cosh (a+b x)}{3 b^2}-\frac {2 d (c+d x) \cosh ^3(a+b x)}{9 b^2}+\frac {2 d^2 \sinh (a+b x)}{9 b^3}+\frac {2 (c+d x)^2 \sinh (a+b x)}{3 b}+\frac {(c+d x)^2 \cosh ^2(a+b x) \sinh (a+b x)}{3 b}+\frac {2 d^2 \sinh ^3(a+b x)}{27 b^3}+\frac {\left (4 d^2\right ) \int \cosh (a+b x) \, dx}{3 b^2}\\ &=-\frac {4 d (c+d x) \cosh (a+b x)}{3 b^2}-\frac {2 d (c+d x) \cosh ^3(a+b x)}{9 b^2}+\frac {14 d^2 \sinh (a+b x)}{9 b^3}+\frac {2 (c+d x)^2 \sinh (a+b x)}{3 b}+\frac {(c+d x)^2 \cosh ^2(a+b x) \sinh (a+b x)}{3 b}+\frac {2 d^2 \sinh ^3(a+b x)}{27 b^3}\\ \end {align*}
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Mathematica [A] time = 0.55, size = 93, normalized size = 0.76 \[ \frac {2 \sinh (a+b x) \left (\cosh (2 (a+b x)) \left (9 b^2 (c+d x)^2+2 d^2\right )+45 b^2 (c+d x)^2+82 d^2\right )-162 b d (c+d x) \cosh (a+b x)-6 b d (c+d x) \cosh (3 (a+b x))}{108 b^3} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.54, size = 199, normalized size = 1.62 \[ -\frac {6 \, {\left (b d^{2} x + b c d\right )} \cosh \left (b x + a\right )^{3} + 18 \, {\left (b d^{2} x + b c d\right )} \cosh \left (b x + a\right ) \sinh \left (b x + a\right )^{2} - {\left (9 \, b^{2} d^{2} x^{2} + 18 \, b^{2} c d x + 9 \, b^{2} c^{2} + 2 \, d^{2}\right )} \sinh \left (b x + a\right )^{3} + 162 \, {\left (b d^{2} x + b c d\right )} \cosh \left (b x + a\right ) - 3 \, {\left (27 \, b^{2} d^{2} x^{2} + 54 \, b^{2} c d x + 27 \, b^{2} c^{2} + {\left (9 \, b^{2} d^{2} x^{2} + 18 \, b^{2} c d x + 9 \, b^{2} c^{2} + 2 \, d^{2}\right )} \cosh \left (b x + a\right )^{2} + 54 \, d^{2}\right )} \sinh \left (b x + a\right )}{108 \, b^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.16, size = 230, normalized size = 1.87 \[ \frac {{\left (9 \, b^{2} d^{2} x^{2} + 18 \, b^{2} c d x + 9 \, b^{2} c^{2} - 6 \, b d^{2} x - 6 \, b c d + 2 \, d^{2}\right )} e^{\left (3 \, b x + 3 \, a\right )}}{216 \, b^{3}} + \frac {3 \, {\left (b^{2} d^{2} x^{2} + 2 \, b^{2} c d x + b^{2} c^{2} - 2 \, b d^{2} x - 2 \, b c d + 2 \, d^{2}\right )} e^{\left (b x + a\right )}}{8 \, b^{3}} - \frac {3 \, {\left (b^{2} d^{2} x^{2} + 2 \, b^{2} c d x + b^{2} c^{2} + 2 \, b d^{2} x + 2 \, b c d + 2 \, d^{2}\right )} e^{\left (-b x - a\right )}}{8 \, b^{3}} - \frac {{\left (9 \, b^{2} d^{2} x^{2} + 18 \, b^{2} c d x + 9 \, b^{2} c^{2} + 6 \, b d^{2} x + 6 \, b c d + 2 \, d^{2}\right )} e^{\left (-3 \, b x - 3 \, a\right )}}{216 \, b^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.16, size = 302, normalized size = 2.46 \[ \frac {\frac {d^{2} \left (\frac {2 \left (b x +a \right )^{2} \sinh \left (b x +a \right )}{3}+\frac {\left (b x +a \right )^{2} \sinh \left (b x +a \right ) \left (\cosh ^{2}\left (b x +a \right )\right )}{3}-\frac {4 \left (b x +a \right ) \cosh \left (b x +a \right )}{3}+\frac {40 \sinh \left (b x +a \right )}{27}-\frac {2 \left (b x +a \right ) \left (\cosh ^{3}\left (b x +a \right )\right )}{9}+\frac {2 \left (\cosh ^{2}\left (b x +a \right )\right ) \sinh \left (b x +a \right )}{27}\right )}{b^{2}}-\frac {2 d^{2} a \left (\frac {2 \left (b x +a \right ) \sinh \left (b x +a \right )}{3}+\frac {\left (b x +a \right ) \sinh \left (b x +a \right ) \left (\cosh ^{2}\left (b x +a \right )\right )}{3}-\frac {2 \cosh \left (b x +a \right )}{3}-\frac {\left (\cosh ^{3}\left (b x +a \right )\right )}{9}\right )}{b^{2}}+\frac {d^{2} a^{2} \left (\frac {2}{3}+\frac {\left (\cosh ^{2}\left (b x +a \right )\right )}{3}\right ) \sinh \left (b x +a \right )}{b^{2}}+\frac {2 c d \left (\frac {2 \left (b x +a \right ) \sinh \left (b x +a \right )}{3}+\frac {\left (b x +a \right ) \sinh \left (b x +a \right ) \left (\cosh ^{2}\left (b x +a \right )\right )}{3}-\frac {2 \cosh \left (b x +a \right )}{3}-\frac {\left (\cosh ^{3}\left (b x +a \right )\right )}{9}\right )}{b}-\frac {2 c d a \left (\frac {2}{3}+\frac {\left (\cosh ^{2}\left (b x +a \right )\right )}{3}\right ) \sinh \left (b x +a \right )}{b}+c^{2} \left (\frac {2}{3}+\frac {\left (\cosh ^{2}\left (b x +a \right )\right )}{3}\right ) \sinh \left (b x +a \right )}{b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.42, size = 272, normalized size = 2.21 \[ \frac {1}{36} \, c d {\left (\frac {{\left (3 \, b x e^{\left (3 \, a\right )} - e^{\left (3 \, a\right )}\right )} e^{\left (3 \, b x\right )}}{b^{2}} + \frac {27 \, {\left (b x e^{a} - e^{a}\right )} e^{\left (b x\right )}}{b^{2}} - \frac {27 \, {\left (b x + 1\right )} e^{\left (-b x - a\right )}}{b^{2}} - \frac {{\left (3 \, b x + 1\right )} e^{\left (-3 \, b x - 3 \, a\right )}}{b^{2}}\right )} + \frac {1}{24} \, c^{2} {\left (\frac {e^{\left (3 \, b x + 3 \, a\right )}}{b} + \frac {9 \, e^{\left (b x + a\right )}}{b} - \frac {9 \, e^{\left (-b x - a\right )}}{b} - \frac {e^{\left (-3 \, b x - 3 \, a\right )}}{b}\right )} + \frac {1}{216} \, d^{2} {\left (\frac {{\left (9 \, b^{2} x^{2} e^{\left (3 \, a\right )} - 6 \, b x e^{\left (3 \, a\right )} + 2 \, e^{\left (3 \, a\right )}\right )} e^{\left (3 \, b x\right )}}{b^{3}} + \frac {81 \, {\left (b^{2} x^{2} e^{a} - 2 \, b x e^{a} + 2 \, e^{a}\right )} e^{\left (b x\right )}}{b^{3}} - \frac {81 \, {\left (b^{2} x^{2} + 2 \, b x + 2\right )} e^{\left (-b x - a\right )}}{b^{3}} - \frac {{\left (9 \, b^{2} x^{2} + 6 \, b x + 2\right )} e^{\left (-3 \, b x - 3 \, a\right )}}{b^{3}}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.22, size = 183, normalized size = 1.49 \[ \frac {\frac {3\,d^2\,\mathrm {sinh}\left (a+b\,x\right )}{2}+\frac {d^2\,\mathrm {sinh}\left (3\,a+3\,b\,x\right )}{54}+\frac {3\,b^2\,c^2\,\mathrm {sinh}\left (a+b\,x\right )}{4}+\frac {b^2\,c^2\,\mathrm {sinh}\left (3\,a+3\,b\,x\right )}{12}+\frac {3\,b^2\,d^2\,x^2\,\mathrm {sinh}\left (a+b\,x\right )}{4}-\frac {b\,c\,d\,\mathrm {cosh}\left (3\,a+3\,b\,x\right )}{18}-\frac {3\,b\,d^2\,x\,\mathrm {cosh}\left (a+b\,x\right )}{2}+\frac {b^2\,d^2\,x^2\,\mathrm {sinh}\left (3\,a+3\,b\,x\right )}{12}-\frac {b\,d^2\,x\,\mathrm {cosh}\left (3\,a+3\,b\,x\right )}{18}-\frac {3\,b\,c\,d\,\mathrm {cosh}\left (a+b\,x\right )}{2}+\frac {b^2\,c\,d\,x\,\mathrm {sinh}\left (3\,a+3\,b\,x\right )}{6}+\frac {3\,b^2\,c\,d\,x\,\mathrm {sinh}\left (a+b\,x\right )}{2}}{b^3} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 2.25, size = 284, normalized size = 2.31 \[ \begin {cases} - \frac {2 c^{2} \sinh ^{3}{\left (a + b x \right )}}{3 b} + \frac {c^{2} \sinh {\left (a + b x \right )} \cosh ^{2}{\left (a + b x \right )}}{b} - \frac {4 c d x \sinh ^{3}{\left (a + b x \right )}}{3 b} + \frac {2 c d x \sinh {\left (a + b x \right )} \cosh ^{2}{\left (a + b x \right )}}{b} - \frac {2 d^{2} x^{2} \sinh ^{3}{\left (a + b x \right )}}{3 b} + \frac {d^{2} x^{2} \sinh {\left (a + b x \right )} \cosh ^{2}{\left (a + b x \right )}}{b} + \frac {4 c d \sinh ^{2}{\left (a + b x \right )} \cosh {\left (a + b x \right )}}{3 b^{2}} - \frac {14 c d \cosh ^{3}{\left (a + b x \right )}}{9 b^{2}} + \frac {4 d^{2} x \sinh ^{2}{\left (a + b x \right )} \cosh {\left (a + b x \right )}}{3 b^{2}} - \frac {14 d^{2} x \cosh ^{3}{\left (a + b x \right )}}{9 b^{2}} - \frac {40 d^{2} \sinh ^{3}{\left (a + b x \right )}}{27 b^{3}} + \frac {14 d^{2} \sinh {\left (a + b x \right )} \cosh ^{2}{\left (a + b x \right )}}{9 b^{3}} & \text {for}\: b \neq 0 \\\left (c^{2} x + c d x^{2} + \frac {d^{2} x^{3}}{3}\right ) \cosh ^{3}{\relax (a )} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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