Integrand size = 16, antiderivative size = 95 \[ \int (c+d x)^2 \sinh ^2(a+b x) \, dx=-\frac {d^2 x}{4 b^2}-\frac {(c+d x)^3}{6 d}+\frac {d^2 \cosh (a+b x) \sinh (a+b x)}{4 b^3}+\frac {(c+d x)^2 \cosh (a+b x) \sinh (a+b x)}{2 b}-\frac {d (c+d x) \sinh ^2(a+b x)}{2 b^2} \]
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Time = 0.04 (sec) , antiderivative size = 95, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {3392, 32, 2715, 8} \[ \int (c+d x)^2 \sinh ^2(a+b x) \, dx=\frac {d^2 \sinh (a+b x) \cosh (a+b x)}{4 b^3}-\frac {d (c+d x) \sinh ^2(a+b x)}{2 b^2}+\frac {(c+d x)^2 \sinh (a+b x) \cosh (a+b x)}{2 b}-\frac {d^2 x}{4 b^2}-\frac {(c+d x)^3}{6 d} \]
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Rule 8
Rule 32
Rule 2715
Rule 3392
Rubi steps \begin{align*} \text {integral}& = \frac {(c+d x)^2 \cosh (a+b x) \sinh (a+b x)}{2 b}-\frac {d (c+d x) \sinh ^2(a+b x)}{2 b^2}-\frac {1}{2} \int (c+d x)^2 \, dx+\frac {d^2 \int \sinh ^2(a+b x) \, dx}{2 b^2} \\ & = -\frac {(c+d x)^3}{6 d}+\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 (c+d x) \sinh ^2(a+b x)}{2 b^2}-\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^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 (c+d x) \sinh ^2(a+b x)}{2 b^2} \\ \end{align*}
Time = 0.19 (sec) , antiderivative size = 75, normalized size of antiderivative = 0.79 \[ \int (c+d x)^2 \sinh ^2(a+b x) \, dx=\frac {-4 b^3 x \left (3 c^2+3 c d x+d^2 x^2\right )-6 b d (c+d x) \cosh (2 (a+b x))+3 \left (d^2+2 b^2 (c+d x)^2\right ) \sinh (2 (a+b x))}{24 b^3} \]
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Time = 0.88 (sec) , antiderivative size = 79, normalized size of antiderivative = 0.83
method | result | size |
parallelrisch | \(\frac {\left (2 \left (d x +c \right )^{2} b^{2}+d^{2}\right ) \sinh \left (2 b x +2 a \right )-4 b \left (\frac {d \left (d x +c \right ) \cosh \left (2 b x +2 a \right )}{2}+x \left (\frac {1}{3} d^{2} x^{2}+c d x +c^{2}\right ) b^{2}-\frac {c d}{2}\right )}{8 b^{3}}\) | \(79\) |
risch | \(-\frac {x^{3} d^{2}}{6}-\frac {d c \,x^{2}}{2}-\frac {c^{2} x}{2}-\frac {c^{3}}{6 d}+\frac {\left (2 b^{2} d^{2} x^{2}+4 b^{2} c d x +2 b^{2} c^{2}-2 b \,d^{2} x -2 b c d +d^{2}\right ) {\mathrm e}^{2 b x +2 a}}{16 b^{3}}-\frac {\left (2 b^{2} d^{2} x^{2}+4 b^{2} c d x +2 b^{2} c^{2}+2 b \,d^{2} x +2 b c d +d^{2}\right ) {\mathrm e}^{-2 b x -2 a}}{16 b^{3}}\) | \(145\) |
derivativedivides | \(\frac {\frac {d^{2} \left (\frac {\left (b x +a \right )^{2} \cosh \left (b x +a \right ) \sinh \left (b x +a \right )}{2}-\frac {\left (b x +a \right )^{3}}{6}-\frac {\left (b x +a \right ) \cosh \left (b x +a \right )^{2}}{2}+\frac {\cosh \left (b x +a \right ) \sinh \left (b x +a \right )}{4}+\frac {b x}{4}+\frac {a}{4}\right )}{b^{2}}-\frac {2 d^{2} a \left (\frac {\left (b x +a \right ) \cosh \left (b x +a \right ) \sinh \left (b x +a \right )}{2}-\frac {\left (b x +a \right )^{2}}{4}-\frac {\cosh \left (b x +a \right )^{2}}{4}\right )}{b^{2}}+\frac {2 d c \left (\frac {\left (b x +a \right ) \cosh \left (b x +a \right ) \sinh \left (b x +a \right )}{2}-\frac {\left (b x +a \right )^{2}}{4}-\frac {\cosh \left (b x +a \right )^{2}}{4}\right )}{b}+\frac {d^{2} a^{2} \left (\frac {\cosh \left (b x +a \right ) \sinh \left (b x +a \right )}{2}-\frac {b x}{2}-\frac {a}{2}\right )}{b^{2}}-\frac {2 d a c \left (\frac {\cosh \left (b x +a \right ) \sinh \left (b x +a \right )}{2}-\frac {b x}{2}-\frac {a}{2}\right )}{b}+c^{2} \left (\frac {\cosh \left (b x +a \right ) \sinh \left (b x +a \right )}{2}-\frac {b x}{2}-\frac {a}{2}\right )}{b}\) | \(262\) |
default | \(\frac {\frac {d^{2} \left (\frac {\left (b x +a \right )^{2} \cosh \left (b x +a \right ) \sinh \left (b x +a \right )}{2}-\frac {\left (b x +a \right )^{3}}{6}-\frac {\left (b x +a \right ) \cosh \left (b x +a \right )^{2}}{2}+\frac {\cosh \left (b x +a \right ) \sinh \left (b x +a \right )}{4}+\frac {b x}{4}+\frac {a}{4}\right )}{b^{2}}-\frac {2 d^{2} a \left (\frac {\left (b x +a \right ) \cosh \left (b x +a \right ) \sinh \left (b x +a \right )}{2}-\frac {\left (b x +a \right )^{2}}{4}-\frac {\cosh \left (b x +a \right )^{2}}{4}\right )}{b^{2}}+\frac {2 d c \left (\frac {\left (b x +a \right ) \cosh \left (b x +a \right ) \sinh \left (b x +a \right )}{2}-\frac {\left (b x +a \right )^{2}}{4}-\frac {\cosh \left (b x +a \right )^{2}}{4}\right )}{b}+\frac {d^{2} a^{2} \left (\frac {\cosh \left (b x +a \right ) \sinh \left (b x +a \right )}{2}-\frac {b x}{2}-\frac {a}{2}\right )}{b^{2}}-\frac {2 d a c \left (\frac {\cosh \left (b x +a \right ) \sinh \left (b x +a \right )}{2}-\frac {b x}{2}-\frac {a}{2}\right )}{b}+c^{2} \left (\frac {\cosh \left (b x +a \right ) \sinh \left (b x +a \right )}{2}-\frac {b x}{2}-\frac {a}{2}\right )}{b}\) | \(262\) |
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Time = 0.26 (sec) , antiderivative size = 123, normalized size of antiderivative = 1.29 \[ \int (c+d x)^2 \sinh ^2(a+b x) \, dx=-\frac {2 \, b^{3} d^{2} x^{3} + 6 \, b^{3} c d x^{2} + 6 \, b^{3} c^{2} x + 3 \, {\left (b d^{2} x + b c d\right )} \cosh \left (b x + a\right )^{2} - 3 \, {\left (2 \, b^{2} d^{2} x^{2} + 4 \, b^{2} c d x + 2 \, b^{2} c^{2} + d^{2}\right )} \cosh \left (b x + a\right ) \sinh \left (b x + a\right ) + 3 \, {\left (b d^{2} x + b c d\right )} \sinh \left (b x + a\right )^{2}}{12 \, b^{3}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 264 vs. \(2 (85) = 170\).
Time = 0.26 (sec) , antiderivative size = 264, normalized size of antiderivative = 2.78 \[ \int (c+d x)^2 \sinh ^2(a+b x) \, dx=\begin {cases} \frac {c^{2} x \sinh ^{2}{\left (a + b x \right )}}{2} - \frac {c^{2} x \cosh ^{2}{\left (a + b x \right )}}{2} + \frac {c d x^{2} \sinh ^{2}{\left (a + b x \right )}}{2} - \frac {c d x^{2} \cosh ^{2}{\left (a + b x \right )}}{2} + \frac {d^{2} x^{3} \sinh ^{2}{\left (a + b x \right )}}{6} - \frac {d^{2} x^{3} \cosh ^{2}{\left (a + b x \right )}}{6} + \frac {c^{2} \sinh {\left (a + b x \right )} \cosh {\left (a + b x \right )}}{2 b} + \frac {c d x \sinh {\left (a + b x \right )} \cosh {\left (a + b x \right )}}{b} + \frac {d^{2} x^{2} \sinh {\left (a + b x \right )} \cosh {\left (a + b x \right )}}{2 b} - \frac {c d \cosh ^{2}{\left (a + b x \right )}}{2 b^{2}} - \frac {d^{2} x \sinh ^{2}{\left (a + b x \right )}}{4 b^{2}} - \frac {d^{2} x \cosh ^{2}{\left (a + b x \right )}}{4 b^{2}} + \frac {d^{2} \sinh {\left (a + b x \right )} \cosh {\left (a + b x \right )}}{4 b^{3}} & \text {for}\: b \neq 0 \\\left (c^{2} x + c d x^{2} + \frac {d^{2} x^{3}}{3}\right ) \sinh ^{2}{\left (a \right )} & \text {otherwise} \end {cases} \]
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Time = 0.20 (sec) , antiderivative size = 165, normalized size of antiderivative = 1.74 \[ \int (c+d x)^2 \sinh ^2(a+b x) \, dx=-\frac {1}{8} \, {\left (4 \, x^{2} - \frac {{\left (2 \, b x e^{\left (2 \, a\right )} - e^{\left (2 \, a\right )}\right )} e^{\left (2 \, b x\right )}}{b^{2}} + \frac {{\left (2 \, b x + 1\right )} e^{\left (-2 \, b x - 2 \, a\right )}}{b^{2}}\right )} c d - \frac {1}{48} \, {\left (8 \, x^{3} - \frac {3 \, {\left (2 \, b^{2} x^{2} e^{\left (2 \, a\right )} - 2 \, b x e^{\left (2 \, a\right )} + e^{\left (2 \, a\right )}\right )} e^{\left (2 \, b x\right )}}{b^{3}} + \frac {3 \, {\left (2 \, b^{2} x^{2} + 2 \, b x + 1\right )} e^{\left (-2 \, b x - 2 \, a\right )}}{b^{3}}\right )} d^{2} - \frac {1}{8} \, c^{2} {\left (4 \, x - \frac {e^{\left (2 \, b x + 2 \, a\right )}}{b} + \frac {e^{\left (-2 \, b x - 2 \, a\right )}}{b}\right )} \]
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Time = 0.27 (sec) , antiderivative size = 136, normalized size of antiderivative = 1.43 \[ \int (c+d x)^2 \sinh ^2(a+b x) \, dx=-\frac {1}{6} \, d^{2} x^{3} - \frac {1}{2} \, c d x^{2} - \frac {1}{2} \, c^{2} x + \frac {{\left (2 \, b^{2} d^{2} x^{2} + 4 \, b^{2} c d x + 2 \, b^{2} c^{2} - 2 \, b d^{2} x - 2 \, b c d + d^{2}\right )} e^{\left (2 \, b x + 2 \, a\right )}}{16 \, b^{3}} - \frac {{\left (2 \, b^{2} d^{2} x^{2} + 4 \, b^{2} c d x + 2 \, b^{2} c^{2} + 2 \, b d^{2} x + 2 \, b c d + d^{2}\right )} e^{\left (-2 \, b x - 2 \, a\right )}}{16 \, b^{3}} \]
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Time = 0.17 (sec) , antiderivative size = 127, normalized size of antiderivative = 1.34 \[ \int (c+d x)^2 \sinh ^2(a+b x) \, dx=\frac {c^2\,\mathrm {sinh}\left (2\,a+2\,b\,x\right )}{4\,b}-\frac {d^2\,x^3}{6}-\frac {c^2\,x}{2}+\frac {d^2\,\mathrm {sinh}\left (2\,a+2\,b\,x\right )}{8\,b^3}-\frac {c\,d\,x^2}{2}-\frac {d^2\,x\,\mathrm {cosh}\left (2\,a+2\,b\,x\right )}{4\,b^2}+\frac {d^2\,x^2\,\mathrm {sinh}\left (2\,a+2\,b\,x\right )}{4\,b}-\frac {c\,d\,\mathrm {cosh}\left (2\,a+2\,b\,x\right )}{4\,b^2}+\frac {c\,d\,x\,\mathrm {sinh}\left (2\,a+2\,b\,x\right )}{2\,b} \]
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