Integrand size = 16, antiderivative size = 134 \[ \int (c+d x)^3 \cosh ^2(a+b x) \, dx=\frac {3 c d^2 x}{4 b^2}+\frac {3 d^3 x^2}{8 b^2}+\frac {(c+d x)^4}{8 d}-\frac {3 d^3 \cosh ^2(a+b x)}{8 b^4}-\frac {3 d (c+d x)^2 \cosh ^2(a+b x)}{4 b^2}+\frac {3 d^2 (c+d x) \cosh (a+b x) \sinh (a+b x)}{4 b^3}+\frac {(c+d x)^3 \cosh (a+b x) \sinh (a+b x)}{2 b} \]
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Time = 0.06 (sec) , antiderivative size = 134, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.188, Rules used = {3392, 32, 3391} \[ \int (c+d x)^3 \cosh ^2(a+b x) \, dx=-\frac {3 d^3 \cosh ^2(a+b x)}{8 b^4}+\frac {3 d^2 (c+d x) \sinh (a+b x) \cosh (a+b x)}{4 b^3}-\frac {3 d (c+d x)^2 \cosh ^2(a+b x)}{4 b^2}+\frac {(c+d x)^3 \sinh (a+b x) \cosh (a+b x)}{2 b}+\frac {3 c d^2 x}{4 b^2}+\frac {3 d^3 x^2}{8 b^2}+\frac {(c+d x)^4}{8 d} \]
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Rule 32
Rule 3391
Rule 3392
Rubi steps \begin{align*} \text {integral}& = -\frac {3 d (c+d x)^2 \cosh ^2(a+b x)}{4 b^2}+\frac {(c+d x)^3 \cosh (a+b x) \sinh (a+b x)}{2 b}+\frac {1}{2} \int (c+d x)^3 \, dx+\frac {\left (3 d^2\right ) \int (c+d x) \cosh ^2(a+b x) \, dx}{2 b^2} \\ & = \frac {(c+d x)^4}{8 d}-\frac {3 d^3 \cosh ^2(a+b x)}{8 b^4}-\frac {3 d (c+d x)^2 \cosh ^2(a+b x)}{4 b^2}+\frac {3 d^2 (c+d x) \cosh (a+b x) \sinh (a+b x)}{4 b^3}+\frac {(c+d x)^3 \cosh (a+b x) \sinh (a+b x)}{2 b}+\frac {\left (3 d^2\right ) \int (c+d x) \, dx}{4 b^2} \\ & = \frac {3 c d^2 x}{4 b^2}+\frac {3 d^3 x^2}{8 b^2}+\frac {(c+d x)^4}{8 d}-\frac {3 d^3 \cosh ^2(a+b x)}{8 b^4}-\frac {3 d (c+d x)^2 \cosh ^2(a+b x)}{4 b^2}+\frac {3 d^2 (c+d x) \cosh (a+b x) \sinh (a+b x)}{4 b^3}+\frac {(c+d x)^3 \cosh (a+b x) \sinh (a+b x)}{2 b} \\ \end{align*}
Time = 0.27 (sec) , antiderivative size = 104, normalized size of antiderivative = 0.78 \[ \int (c+d x)^3 \cosh ^2(a+b x) \, dx=\frac {2 b^4 x \left (4 c^3+6 c^2 d x+4 c d^2 x^2+d^3 x^3\right )-3 d \left (d^2+2 b^2 (c+d x)^2\right ) \cosh (2 (a+b x))+2 b (c+d x) \left (3 d^2+2 b^2 (c+d x)^2\right ) \sinh (2 (a+b x))}{16 b^4} \]
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Time = 0.28 (sec) , antiderivative size = 121, normalized size of antiderivative = 0.90
method | result | size |
parallelrisch | \(\frac {4 \left (d x +c \right ) \sinh \left (2 b x +2 a \right ) b \left (\left (d x +c \right )^{2} b^{2}+\frac {3 d^{2}}{2}\right )-6 \left (\left (d x +c \right )^{2} b^{2}+\frac {d^{2}}{2}\right ) d \cosh \left (2 b x +2 a \right )+2 \left (d^{3} x^{4}+4 d^{2} c \,x^{3}+6 d \,c^{2} x^{2}+4 c^{3} x \right ) b^{4}+6 b^{2} c^{2} d +3 d^{3}}{16 b^{4}}\) | \(121\) |
risch | \(\frac {d^{3} x^{4}}{8}+\frac {d^{2} c \,x^{3}}{2}+\frac {3 d \,c^{2} x^{2}}{4}+\frac {c^{3} x}{2}+\frac {c^{4}}{8 d}+\frac {\left (4 d^{3} x^{3} b^{3}+12 b^{3} c \,d^{2} x^{2}+12 b^{3} c^{2} d x -6 b^{2} d^{3} x^{2}+4 b^{3} c^{3}-12 b^{2} c \,d^{2} x -6 b^{2} c^{2} d +6 b \,d^{3} x +6 b c \,d^{2}-3 d^{3}\right ) {\mathrm e}^{2 b x +2 a}}{32 b^{4}}-\frac {\left (4 d^{3} x^{3} b^{3}+12 b^{3} c \,d^{2} x^{2}+12 b^{3} c^{2} d x +6 b^{2} d^{3} x^{2}+4 b^{3} c^{3}+12 b^{2} c \,d^{2} x +6 b^{2} c^{2} d +6 b \,d^{3} x +6 b c \,d^{2}+3 d^{3}\right ) {\mathrm e}^{-2 b x -2 a}}{32 b^{4}}\) | \(252\) |
derivativedivides | \(\frac {\frac {d^{3} \left (\frac {\left (b x +a \right )^{3} \cosh \left (b x +a \right ) \sinh \left (b x +a \right )}{2}+\frac {\left (b x +a \right )^{4}}{8}-\frac {3 \left (b x +a \right )^{2} \cosh \left (b x +a \right )^{2}}{4}+\frac {3 \left (b x +a \right ) \cosh \left (b x +a \right ) \sinh \left (b x +a \right )}{4}+\frac {3 \left (b x +a \right )^{2}}{8}-\frac {3 \cosh \left (b x +a \right )^{2}}{8}\right )}{b^{3}}-\frac {3 d^{3} a \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 ) \cosh \left (b x +a \right )^{2}}{2}+\frac {\cosh \left (b x +a \right ) \sinh \left (b x +a \right )}{4}+\frac {b x}{4}+\frac {a}{4}\right )}{b^{3}}+\frac {3 d^{2} c \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 ) \cosh \left (b x +a \right )^{2}}{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 {3 d^{3} a^{2} \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 {\cosh \left (b x +a \right )^{2}}{4}\right )}{b^{3}}-\frac {6 d^{2} a c \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 {\cosh \left (b x +a \right )^{2}}{4}\right )}{b^{2}}+\frac {3 d \,c^{2} \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 {\cosh \left (b x +a \right )^{2}}{4}\right )}{b}-\frac {d^{3} a^{3} \left (\frac {\cosh \left (b x +a \right ) \sinh \left (b x +a \right )}{2}+\frac {b x}{2}+\frac {a}{2}\right )}{b^{3}}+\frac {3 d^{2} a^{2} c \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 {3 d a \,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}+c^{3} \left (\frac {\cosh \left (b x +a \right ) \sinh \left (b x +a \right )}{2}+\frac {b x}{2}+\frac {a}{2}\right )}{b}\) | \(523\) |
default | \(\frac {\frac {d^{3} \left (\frac {\left (b x +a \right )^{3} \cosh \left (b x +a \right ) \sinh \left (b x +a \right )}{2}+\frac {\left (b x +a \right )^{4}}{8}-\frac {3 \left (b x +a \right )^{2} \cosh \left (b x +a \right )^{2}}{4}+\frac {3 \left (b x +a \right ) \cosh \left (b x +a \right ) \sinh \left (b x +a \right )}{4}+\frac {3 \left (b x +a \right )^{2}}{8}-\frac {3 \cosh \left (b x +a \right )^{2}}{8}\right )}{b^{3}}-\frac {3 d^{3} a \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 ) \cosh \left (b x +a \right )^{2}}{2}+\frac {\cosh \left (b x +a \right ) \sinh \left (b x +a \right )}{4}+\frac {b x}{4}+\frac {a}{4}\right )}{b^{3}}+\frac {3 d^{2} c \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 ) \cosh \left (b x +a \right )^{2}}{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 {3 d^{3} a^{2} \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 {\cosh \left (b x +a \right )^{2}}{4}\right )}{b^{3}}-\frac {6 d^{2} a c \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 {\cosh \left (b x +a \right )^{2}}{4}\right )}{b^{2}}+\frac {3 d \,c^{2} \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 {\cosh \left (b x +a \right )^{2}}{4}\right )}{b}-\frac {d^{3} a^{3} \left (\frac {\cosh \left (b x +a \right ) \sinh \left (b x +a \right )}{2}+\frac {b x}{2}+\frac {a}{2}\right )}{b^{3}}+\frac {3 d^{2} a^{2} c \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 {3 d a \,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}+c^{3} \left (\frac {\cosh \left (b x +a \right ) \sinh \left (b x +a \right )}{2}+\frac {b x}{2}+\frac {a}{2}\right )}{b}\) | \(523\) |
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Time = 0.26 (sec) , antiderivative size = 209, normalized size of antiderivative = 1.56 \[ \int (c+d x)^3 \cosh ^2(a+b x) \, dx=\frac {2 \, b^{4} d^{3} x^{4} + 8 \, b^{4} c d^{2} x^{3} + 12 \, b^{4} c^{2} d x^{2} + 8 \, b^{4} c^{3} x - 3 \, {\left (2 \, b^{2} d^{3} x^{2} + 4 \, b^{2} c d^{2} x + 2 \, b^{2} c^{2} d + d^{3}\right )} \cosh \left (b x + a\right )^{2} + 4 \, {\left (2 \, b^{3} d^{3} x^{3} + 6 \, b^{3} c d^{2} x^{2} + 2 \, b^{3} c^{3} + 3 \, b c d^{2} + 3 \, {\left (2 \, b^{3} c^{2} d + b d^{3}\right )} x\right )} \cosh \left (b x + a\right ) \sinh \left (b x + a\right ) - 3 \, {\left (2 \, b^{2} d^{3} x^{2} + 4 \, b^{2} c d^{2} x + 2 \, b^{2} c^{2} d + d^{3}\right )} \sinh \left (b x + a\right )^{2}}{16 \, b^{4}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 456 vs. \(2 (131) = 262\).
Time = 0.39 (sec) , antiderivative size = 456, normalized size of antiderivative = 3.40 \[ \int (c+d x)^3 \cosh ^2(a+b x) \, dx=\begin {cases} - \frac {c^{3} x \sinh ^{2}{\left (a + b x \right )}}{2} + \frac {c^{3} x \cosh ^{2}{\left (a + b x \right )}}{2} - \frac {3 c^{2} d x^{2} \sinh ^{2}{\left (a + b x \right )}}{4} + \frac {3 c^{2} d x^{2} \cosh ^{2}{\left (a + b x \right )}}{4} - \frac {c d^{2} x^{3} \sinh ^{2}{\left (a + b x \right )}}{2} + \frac {c d^{2} x^{3} \cosh ^{2}{\left (a + b x \right )}}{2} - \frac {d^{3} x^{4} \sinh ^{2}{\left (a + b x \right )}}{8} + \frac {d^{3} x^{4} \cosh ^{2}{\left (a + b x \right )}}{8} + \frac {c^{3} \sinh {\left (a + b x \right )} \cosh {\left (a + b x \right )}}{2 b} + \frac {3 c^{2} d x \sinh {\left (a + b x \right )} \cosh {\left (a + b x \right )}}{2 b} + \frac {3 c d^{2} x^{2} \sinh {\left (a + b x \right )} \cosh {\left (a + b x \right )}}{2 b} + \frac {d^{3} x^{3} \sinh {\left (a + b x \right )} \cosh {\left (a + b x \right )}}{2 b} - \frac {3 c^{2} d \cosh ^{2}{\left (a + b x \right )}}{4 b^{2}} - \frac {3 c d^{2} x \sinh ^{2}{\left (a + b x \right )}}{4 b^{2}} - \frac {3 c d^{2} x \cosh ^{2}{\left (a + b x \right )}}{4 b^{2}} - \frac {3 d^{3} x^{2} \sinh ^{2}{\left (a + b x \right )}}{8 b^{2}} - \frac {3 d^{3} x^{2} \cosh ^{2}{\left (a + b x \right )}}{8 b^{2}} + \frac {3 c d^{2} \sinh {\left (a + b x \right )} \cosh {\left (a + b x \right )}}{4 b^{3}} + \frac {3 d^{3} x \sinh {\left (a + b x \right )} \cosh {\left (a + b x \right )}}{4 b^{3}} - \frac {3 d^{3} \cosh ^{2}{\left (a + b x \right )}}{8 b^{4}} & \text {for}\: b \neq 0 \\\left (c^{3} x + \frac {3 c^{2} d x^{2}}{2} + c d^{2} x^{3} + \frac {d^{3} x^{4}}{4}\right ) \cosh ^{2}{\left (a \right )} & \text {otherwise} \end {cases} \]
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Leaf count of result is larger than twice the leaf count of optimal. 263 vs. \(2 (120) = 240\).
Time = 0.19 (sec) , antiderivative size = 263, normalized size of antiderivative = 1.96 \[ \int (c+d x)^3 \cosh ^2(a+b x) \, dx=\frac {3}{16} \, {\left (4 \, x^{2} + \frac {{\left (2 \, b x e^{\left (2 \, a\right )} - e^{\left (2 \, a\right )}\right )} e^{\left (2 \, b x\right )}}{b^{2}} - \frac {{\left (2 \, b x + 1\right )} e^{\left (-2 \, b x - 2 \, a\right )}}{b^{2}}\right )} c^{2} d + \frac {1}{16} \, {\left (8 \, x^{3} + \frac {3 \, {\left (2 \, b^{2} x^{2} e^{\left (2 \, a\right )} - 2 \, b x e^{\left (2 \, a\right )} + e^{\left (2 \, a\right )}\right )} e^{\left (2 \, b x\right )}}{b^{3}} - \frac {3 \, {\left (2 \, b^{2} x^{2} + 2 \, b x + 1\right )} e^{\left (-2 \, b x - 2 \, a\right )}}{b^{3}}\right )} c d^{2} + \frac {1}{32} \, {\left (4 \, x^{4} + \frac {{\left (4 \, b^{3} x^{3} e^{\left (2 \, a\right )} - 6 \, b^{2} x^{2} e^{\left (2 \, a\right )} + 6 \, b x e^{\left (2 \, a\right )} - 3 \, e^{\left (2 \, a\right )}\right )} e^{\left (2 \, b x\right )}}{b^{4}} - \frac {{\left (4 \, b^{3} x^{3} + 6 \, b^{2} x^{2} + 6 \, b x + 3\right )} e^{\left (-2 \, b x - 2 \, a\right )}}{b^{4}}\right )} d^{3} + \frac {1}{8} \, c^{3} {\left (4 \, x + \frac {e^{\left (2 \, b x + 2 \, a\right )}}{b} - \frac {e^{\left (-2 \, b x - 2 \, a\right )}}{b}\right )} \]
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Leaf count of result is larger than twice the leaf count of optimal. 243 vs. \(2 (120) = 240\).
Time = 0.27 (sec) , antiderivative size = 243, normalized size of antiderivative = 1.81 \[ \int (c+d x)^3 \cosh ^2(a+b x) \, dx=\frac {1}{8} \, d^{3} x^{4} + \frac {1}{2} \, c d^{2} x^{3} + \frac {3}{4} \, c^{2} d x^{2} + \frac {1}{2} \, c^{3} x + \frac {{\left (4 \, b^{3} d^{3} x^{3} + 12 \, b^{3} c d^{2} x^{2} + 12 \, b^{3} c^{2} d x - 6 \, b^{2} d^{3} x^{2} + 4 \, b^{3} c^{3} - 12 \, b^{2} c d^{2} x - 6 \, b^{2} c^{2} d + 6 \, b d^{3} x + 6 \, b c d^{2} - 3 \, d^{3}\right )} e^{\left (2 \, b x + 2 \, a\right )}}{32 \, b^{4}} - \frac {{\left (4 \, b^{3} d^{3} x^{3} + 12 \, b^{3} c d^{2} x^{2} + 12 \, b^{3} c^{2} d x + 6 \, b^{2} d^{3} x^{2} + 4 \, b^{3} c^{3} + 12 \, b^{2} c d^{2} x + 6 \, b^{2} c^{2} d + 6 \, b d^{3} x + 6 \, b c d^{2} + 3 \, d^{3}\right )} e^{\left (-2 \, b x - 2 \, a\right )}}{32 \, b^{4}} \]
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Time = 0.44 (sec) , antiderivative size = 229, normalized size of antiderivative = 1.71 \[ \int (c+d x)^3 \cosh ^2(a+b x) \, dx=\frac {4\,b^4\,c^3\,x-\frac {3\,d^3\,\mathrm {cosh}\left (2\,a+2\,b\,x\right )}{2}+2\,b^3\,c^3\,\mathrm {sinh}\left (2\,a+2\,b\,x\right )+b^4\,d^3\,x^4-3\,b^2\,c^2\,d\,\mathrm {cosh}\left (2\,a+2\,b\,x\right )+6\,b^4\,c^2\,d\,x^2+4\,b^4\,c\,d^2\,x^3-3\,b^2\,d^3\,x^2\,\mathrm {cosh}\left (2\,a+2\,b\,x\right )+2\,b^3\,d^3\,x^3\,\mathrm {sinh}\left (2\,a+2\,b\,x\right )+3\,b\,c\,d^2\,\mathrm {sinh}\left (2\,a+2\,b\,x\right )+3\,b\,d^3\,x\,\mathrm {sinh}\left (2\,a+2\,b\,x\right )-6\,b^2\,c\,d^2\,x\,\mathrm {cosh}\left (2\,a+2\,b\,x\right )+6\,b^3\,c^2\,d\,x\,\mathrm {sinh}\left (2\,a+2\,b\,x\right )+6\,b^3\,c\,d^2\,x^2\,\mathrm {sinh}\left (2\,a+2\,b\,x\right )}{8\,b^4} \]
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