Integrand size = 16, antiderivative size = 162 \[ \int (c+d x)^4 \sinh ^2(a+b x) \, dx=-\frac {3 d^4 x}{4 b^4}-\frac {d (c+d x)^3}{2 b^2}-\frac {(c+d x)^5}{10 d}+\frac {3 d^4 \cosh (a+b x) \sinh (a+b x)}{4 b^5}+\frac {3 d^2 (c+d x)^2 \cosh (a+b x) \sinh (a+b x)}{2 b^3}+\frac {(c+d x)^4 \cosh (a+b x) \sinh (a+b x)}{2 b}-\frac {3 d^3 (c+d x) \sinh ^2(a+b x)}{2 b^4}-\frac {d (c+d x)^3 \sinh ^2(a+b x)}{b^2} \]
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Time = 0.07 (sec) , antiderivative size = 162, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {3392, 32, 2715, 8} \[ \int (c+d x)^4 \sinh ^2(a+b x) \, dx=\frac {3 d^4 \sinh (a+b x) \cosh (a+b x)}{4 b^5}-\frac {3 d^3 (c+d x) \sinh ^2(a+b x)}{2 b^4}+\frac {3 d^2 (c+d x)^2 \sinh (a+b x) \cosh (a+b x)}{2 b^3}-\frac {d (c+d x)^3 \sinh ^2(a+b x)}{b^2}+\frac {(c+d x)^4 \sinh (a+b x) \cosh (a+b x)}{2 b}-\frac {3 d^4 x}{4 b^4}-\frac {d (c+d x)^3}{2 b^2}-\frac {(c+d x)^5}{10 d} \]
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Rule 8
Rule 32
Rule 2715
Rule 3392
Rubi steps \begin{align*} \text {integral}& = \frac {(c+d x)^4 \cosh (a+b x) \sinh (a+b x)}{2 b}-\frac {d (c+d x)^3 \sinh ^2(a+b x)}{b^2}-\frac {1}{2} \int (c+d x)^4 \, dx+\frac {\left (3 d^2\right ) \int (c+d x)^2 \sinh ^2(a+b x) \, dx}{b^2} \\ & = -\frac {(c+d x)^5}{10 d}+\frac {3 d^2 (c+d x)^2 \cosh (a+b x) \sinh (a+b x)}{2 b^3}+\frac {(c+d x)^4 \cosh (a+b x) \sinh (a+b x)}{2 b}-\frac {3 d^3 (c+d x) \sinh ^2(a+b x)}{2 b^4}-\frac {d (c+d x)^3 \sinh ^2(a+b x)}{b^2}-\frac {\left (3 d^2\right ) \int (c+d x)^2 \, dx}{2 b^2}+\frac {\left (3 d^4\right ) \int \sinh ^2(a+b x) \, dx}{2 b^4} \\ & = -\frac {d (c+d x)^3}{2 b^2}-\frac {(c+d x)^5}{10 d}+\frac {3 d^4 \cosh (a+b x) \sinh (a+b x)}{4 b^5}+\frac {3 d^2 (c+d x)^2 \cosh (a+b x) \sinh (a+b x)}{2 b^3}+\frac {(c+d x)^4 \cosh (a+b x) \sinh (a+b x)}{2 b}-\frac {3 d^3 (c+d x) \sinh ^2(a+b x)}{2 b^4}-\frac {d (c+d x)^3 \sinh ^2(a+b x)}{b^2}-\frac {\left (3 d^4\right ) \int 1 \, dx}{4 b^4} \\ & = -\frac {3 d^4 x}{4 b^4}-\frac {d (c+d x)^3}{2 b^2}-\frac {(c+d x)^5}{10 d}+\frac {3 d^4 \cosh (a+b x) \sinh (a+b x)}{4 b^5}+\frac {3 d^2 (c+d x)^2 \cosh (a+b x) \sinh (a+b x)}{2 b^3}+\frac {(c+d x)^4 \cosh (a+b x) \sinh (a+b x)}{2 b}-\frac {3 d^3 (c+d x) \sinh ^2(a+b x)}{2 b^4}-\frac {d (c+d x)^3 \sinh ^2(a+b x)}{b^2} \\ \end{align*}
Time = 0.37 (sec) , antiderivative size = 132, normalized size of antiderivative = 0.81 \[ \int (c+d x)^4 \sinh ^2(a+b x) \, dx=\frac {-8 b^5 x \left (5 c^4+10 c^3 d x+10 c^2 d^2 x^2+5 c d^3 x^3+d^4 x^4\right )-20 b d (c+d x) \left (3 d^2+2 b^2 (c+d x)^2\right ) \cosh (2 (a+b x))+10 \left (3 d^4+6 b^2 d^2 (c+d x)^2+2 b^4 (c+d x)^4\right ) \sinh (2 (a+b x))}{80 b^5} \]
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Time = 1.02 (sec) , antiderivative size = 145, normalized size of antiderivative = 0.90
method | result | size |
parallelrisch | \(\frac {\left (2 \left (d x +c \right )^{4} b^{4}+6 d^{2} \left (d x +c \right )^{2} b^{2}+3 d^{4}\right ) \sinh \left (2 b x +2 a \right )-4 \left (d \left (\left (d x +c \right )^{2} b^{2}+\frac {3 d^{2}}{2}\right ) \left (d x +c \right ) \cosh \left (2 b x +2 a \right )+x \left (\frac {1}{5} d^{4} x^{4}+c \,d^{3} x^{3}+2 c^{2} d^{2} x^{2}+2 c^{3} d x +c^{4}\right ) b^{4}-b^{2} c^{3} d -\frac {3 d^{3} c}{2}\right ) b}{8 b^{5}}\) | \(145\) |
risch | \(-\frac {d^{4} x^{5}}{10}-\frac {d^{3} c \,x^{4}}{2}-d^{2} c^{2} x^{3}-c^{3} d \,x^{2}-\frac {x \,c^{4}}{2}-\frac {c^{5}}{10 d}+\frac {\left (2 d^{4} x^{4} b^{4}+8 b^{4} c \,d^{3} x^{3}+12 b^{4} c^{2} d^{2} x^{2}-4 b^{3} d^{4} x^{3}+8 b^{4} c^{3} d x -12 b^{3} c \,d^{3} x^{2}+2 b^{4} c^{4}-12 b^{3} c^{2} d^{2} x +6 b^{2} d^{4} x^{2}-4 b^{3} c^{3} d +12 b^{2} c \,d^{3} x +6 b^{2} c^{2} d^{2}-6 b \,d^{4} x -6 b c \,d^{3}+3 d^{4}\right ) {\mathrm e}^{2 b x +2 a}}{16 b^{5}}-\frac {\left (2 d^{4} x^{4} b^{4}+8 b^{4} c \,d^{3} x^{3}+12 b^{4} c^{2} d^{2} x^{2}+4 b^{3} d^{4} x^{3}+8 b^{4} c^{3} d x +12 b^{3} c \,d^{3} x^{2}+2 b^{4} c^{4}+12 b^{3} c^{2} d^{2} x +6 b^{2} d^{4} x^{2}+4 b^{3} c^{3} d +12 b^{2} c \,d^{3} x +6 b^{2} c^{2} d^{2}+6 b \,d^{4} x +6 b c \,d^{3}+3 d^{4}\right ) {\mathrm e}^{-2 b x -2 a}}{16 b^{5}}\) | \(383\) |
derivativedivides | \(\text {Expression too large to display}\) | \(910\) |
default | \(\text {Expression too large to display}\) | \(910\) |
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Leaf count of result is larger than twice the leaf count of optimal. 312 vs. \(2 (148) = 296\).
Time = 0.24 (sec) , antiderivative size = 312, normalized size of antiderivative = 1.93 \[ \int (c+d x)^4 \sinh ^2(a+b x) \, dx=-\frac {2 \, b^{5} d^{4} x^{5} + 10 \, b^{5} c d^{3} x^{4} + 20 \, b^{5} c^{2} d^{2} x^{3} + 20 \, b^{5} c^{3} d x^{2} + 10 \, b^{5} c^{4} x + 5 \, {\left (2 \, b^{3} d^{4} x^{3} + 6 \, b^{3} c d^{3} x^{2} + 2 \, b^{3} c^{3} d + 3 \, b c d^{3} + 3 \, {\left (2 \, b^{3} c^{2} d^{2} + b d^{4}\right )} x\right )} \cosh \left (b x + a\right )^{2} - 5 \, {\left (2 \, b^{4} d^{4} x^{4} + 8 \, b^{4} c d^{3} x^{3} + 2 \, b^{4} c^{4} + 6 \, b^{2} c^{2} d^{2} + 3 \, d^{4} + 6 \, {\left (2 \, b^{4} c^{2} d^{2} + b^{2} d^{4}\right )} x^{2} + 4 \, {\left (2 \, b^{4} c^{3} d + 3 \, b^{2} c d^{3}\right )} x\right )} \cosh \left (b x + a\right ) \sinh \left (b x + a\right ) + 5 \, {\left (2 \, b^{3} d^{4} x^{3} + 6 \, b^{3} c d^{3} x^{2} + 2 \, b^{3} c^{3} d + 3 \, b c d^{3} + 3 \, {\left (2 \, b^{3} c^{2} d^{2} + b d^{4}\right )} x\right )} \sinh \left (b x + a\right )^{2}}{20 \, b^{5}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 660 vs. \(2 (156) = 312\).
Time = 0.51 (sec) , antiderivative size = 660, normalized size of antiderivative = 4.07 \[ \int (c+d x)^4 \sinh ^2(a+b x) \, dx=\begin {cases} \frac {c^{4} x \sinh ^{2}{\left (a + b x \right )}}{2} - \frac {c^{4} x \cosh ^{2}{\left (a + b x \right )}}{2} + c^{3} d x^{2} \sinh ^{2}{\left (a + b x \right )} - c^{3} d x^{2} \cosh ^{2}{\left (a + b x \right )} + c^{2} d^{2} x^{3} \sinh ^{2}{\left (a + b x \right )} - c^{2} d^{2} x^{3} \cosh ^{2}{\left (a + b x \right )} + \frac {c d^{3} x^{4} \sinh ^{2}{\left (a + b x \right )}}{2} - \frac {c d^{3} x^{4} \cosh ^{2}{\left (a + b x \right )}}{2} + \frac {d^{4} x^{5} \sinh ^{2}{\left (a + b x \right )}}{10} - \frac {d^{4} x^{5} \cosh ^{2}{\left (a + b x \right )}}{10} + \frac {c^{4} \sinh {\left (a + b x \right )} \cosh {\left (a + b x \right )}}{2 b} + \frac {2 c^{3} d x \sinh {\left (a + b x \right )} \cosh {\left (a + b x \right )}}{b} + \frac {3 c^{2} d^{2} x^{2} \sinh {\left (a + b x \right )} \cosh {\left (a + b x \right )}}{b} + \frac {2 c d^{3} x^{3} \sinh {\left (a + b x \right )} \cosh {\left (a + b x \right )}}{b} + \frac {d^{4} x^{4} \sinh {\left (a + b x \right )} \cosh {\left (a + b x \right )}}{2 b} - \frac {c^{3} d \cosh ^{2}{\left (a + b x \right )}}{b^{2}} - \frac {3 c^{2} d^{2} x \sinh ^{2}{\left (a + b x \right )}}{2 b^{2}} - \frac {3 c^{2} d^{2} x \cosh ^{2}{\left (a + b x \right )}}{2 b^{2}} - \frac {3 c d^{3} x^{2} \sinh ^{2}{\left (a + b x \right )}}{2 b^{2}} - \frac {3 c d^{3} x^{2} \cosh ^{2}{\left (a + b x \right )}}{2 b^{2}} - \frac {d^{4} x^{3} \sinh ^{2}{\left (a + b x \right )}}{2 b^{2}} - \frac {d^{4} x^{3} \cosh ^{2}{\left (a + b x \right )}}{2 b^{2}} + \frac {3 c^{2} d^{2} \sinh {\left (a + b x \right )} \cosh {\left (a + b x \right )}}{2 b^{3}} + \frac {3 c d^{3} x \sinh {\left (a + b x \right )} \cosh {\left (a + b x \right )}}{b^{3}} + \frac {3 d^{4} x^{2} \sinh {\left (a + b x \right )} \cosh {\left (a + b x \right )}}{2 b^{3}} - \frac {3 c d^{3} \cosh ^{2}{\left (a + b x \right )}}{2 b^{4}} - \frac {3 d^{4} x \sinh ^{2}{\left (a + b x \right )}}{4 b^{4}} - \frac {3 d^{4} x \cosh ^{2}{\left (a + b x \right )}}{4 b^{4}} + \frac {3 d^{4} \sinh {\left (a + b x \right )} \cosh {\left (a + b x \right )}}{4 b^{5}} & \text {for}\: b \neq 0 \\\left (c^{4} x + 2 c^{3} d x^{2} + 2 c^{2} d^{2} x^{3} + c d^{3} x^{4} + \frac {d^{4} x^{5}}{5}\right ) \sinh ^{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. 382 vs. \(2 (148) = 296\).
Time = 0.21 (sec) , antiderivative size = 382, normalized size of antiderivative = 2.36 \[ \int (c+d x)^4 \sinh ^2(a+b x) \, dx=-\frac {1}{4} \, {\left (4 \, x^{2} - \frac {{\left (2 \, b x e^{\left (2 \, a\right )} - e^{\left (2 \, a\right )}\right )} e^{\left (2 \, b x\right )}}{b^{2}} + \frac {{\left (2 \, b x + 1\right )} e^{\left (-2 \, b x - 2 \, a\right )}}{b^{2}}\right )} c^{3} d - \frac {1}{8} \, {\left (8 \, x^{3} - \frac {3 \, {\left (2 \, b^{2} x^{2} e^{\left (2 \, a\right )} - 2 \, b x e^{\left (2 \, a\right )} + e^{\left (2 \, a\right )}\right )} e^{\left (2 \, b x\right )}}{b^{3}} + \frac {3 \, {\left (2 \, b^{2} x^{2} + 2 \, b x + 1\right )} e^{\left (-2 \, b x - 2 \, a\right )}}{b^{3}}\right )} c^{2} d^{2} - \frac {1}{8} \, {\left (4 \, x^{4} - \frac {{\left (4 \, b^{3} x^{3} e^{\left (2 \, a\right )} - 6 \, b^{2} x^{2} e^{\left (2 \, a\right )} + 6 \, b x e^{\left (2 \, a\right )} - 3 \, e^{\left (2 \, a\right )}\right )} e^{\left (2 \, b x\right )}}{b^{4}} + \frac {{\left (4 \, b^{3} x^{3} + 6 \, b^{2} x^{2} + 6 \, b x + 3\right )} e^{\left (-2 \, b x - 2 \, a\right )}}{b^{4}}\right )} c d^{3} - \frac {1}{80} \, {\left (8 \, x^{5} - \frac {5 \, {\left (2 \, b^{4} x^{4} e^{\left (2 \, a\right )} - 4 \, b^{3} x^{3} e^{\left (2 \, a\right )} + 6 \, b^{2} x^{2} e^{\left (2 \, a\right )} - 6 \, b x e^{\left (2 \, a\right )} + 3 \, e^{\left (2 \, a\right )}\right )} e^{\left (2 \, b x\right )}}{b^{5}} + \frac {5 \, {\left (2 \, b^{4} x^{4} + 4 \, b^{3} x^{3} + 6 \, b^{2} x^{2} + 6 \, b x + 3\right )} e^{\left (-2 \, b x - 2 \, a\right )}}{b^{5}}\right )} d^{4} - \frac {1}{8} \, c^{4} {\left (4 \, x - \frac {e^{\left (2 \, b x + 2 \, a\right )}}{b} + \frac {e^{\left (-2 \, b x - 2 \, a\right )}}{b}\right )} \]
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Leaf count of result is larger than twice the leaf count of optimal. 374 vs. \(2 (148) = 296\).
Time = 0.27 (sec) , antiderivative size = 374, normalized size of antiderivative = 2.31 \[ \int (c+d x)^4 \sinh ^2(a+b x) \, dx=-\frac {1}{10} \, d^{4} x^{5} - \frac {1}{2} \, c d^{3} x^{4} - c^{2} d^{2} x^{3} - c^{3} d x^{2} - \frac {1}{2} \, c^{4} x + \frac {{\left (2 \, b^{4} d^{4} x^{4} + 8 \, b^{4} c d^{3} x^{3} + 12 \, b^{4} c^{2} d^{2} x^{2} - 4 \, b^{3} d^{4} x^{3} + 8 \, b^{4} c^{3} d x - 12 \, b^{3} c d^{3} x^{2} + 2 \, b^{4} c^{4} - 12 \, b^{3} c^{2} d^{2} x + 6 \, b^{2} d^{4} x^{2} - 4 \, b^{3} c^{3} d + 12 \, b^{2} c d^{3} x + 6 \, b^{2} c^{2} d^{2} - 6 \, b d^{4} x - 6 \, b c d^{3} + 3 \, d^{4}\right )} e^{\left (2 \, b x + 2 \, a\right )}}{16 \, b^{5}} - \frac {{\left (2 \, b^{4} d^{4} x^{4} + 8 \, b^{4} c d^{3} x^{3} + 12 \, b^{4} c^{2} d^{2} x^{2} + 4 \, b^{3} d^{4} x^{3} + 8 \, b^{4} c^{3} d x + 12 \, b^{3} c d^{3} x^{2} + 2 \, b^{4} c^{4} + 12 \, b^{3} c^{2} d^{2} x + 6 \, b^{2} d^{4} x^{2} + 4 \, b^{3} c^{3} d + 12 \, b^{2} c d^{3} x + 6 \, b^{2} c^{2} d^{2} + 6 \, b d^{4} x + 6 \, b c d^{3} + 3 \, d^{4}\right )} e^{\left (-2 \, b x - 2 \, a\right )}}{16 \, b^{5}} \]
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Time = 1.43 (sec) , antiderivative size = 334, normalized size of antiderivative = 2.06 \[ \int (c+d x)^4 \sinh ^2(a+b x) \, dx=\frac {c^4\,\mathrm {sinh}\left (2\,a+2\,b\,x\right )}{4\,b}-\frac {d^4\,x^5}{10}-c^3\,d\,x^2-\frac {c\,d^3\,x^4}{2}-\frac {c^4\,x}{2}+\frac {3\,d^4\,\mathrm {sinh}\left (2\,a+2\,b\,x\right )}{8\,b^5}-c^2\,d^2\,x^3-\frac {c^3\,d\,\mathrm {cosh}\left (2\,a+2\,b\,x\right )}{2\,b^2}-\frac {3\,c\,d^3\,\mathrm {cosh}\left (2\,a+2\,b\,x\right )}{4\,b^4}-\frac {3\,d^4\,x\,\mathrm {cosh}\left (2\,a+2\,b\,x\right )}{4\,b^4}+\frac {3\,c^2\,d^2\,\mathrm {sinh}\left (2\,a+2\,b\,x\right )}{4\,b^3}-\frac {d^4\,x^3\,\mathrm {cosh}\left (2\,a+2\,b\,x\right )}{2\,b^2}+\frac {d^4\,x^4\,\mathrm {sinh}\left (2\,a+2\,b\,x\right )}{4\,b}+\frac {3\,d^4\,x^2\,\mathrm {sinh}\left (2\,a+2\,b\,x\right )}{4\,b^3}+\frac {3\,c^2\,d^2\,x^2\,\mathrm {sinh}\left (2\,a+2\,b\,x\right )}{2\,b}+\frac {c^3\,d\,x\,\mathrm {sinh}\left (2\,a+2\,b\,x\right )}{b}+\frac {3\,c\,d^3\,x\,\mathrm {sinh}\left (2\,a+2\,b\,x\right )}{2\,b^3}-\frac {3\,c^2\,d^2\,x\,\mathrm {cosh}\left (2\,a+2\,b\,x\right )}{2\,b^2}-\frac {3\,c\,d^3\,x^2\,\mathrm {cosh}\left (2\,a+2\,b\,x\right )}{2\,b^2}+\frac {c\,d^3\,x^3\,\mathrm {sinh}\left (2\,a+2\,b\,x\right )}{b} \]
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