Integrand size = 14, antiderivative size = 70 \[ \int (c+d x)^3 \sinh (a+b x) \, dx=\frac {6 d^2 (c+d x) \cosh (a+b x)}{b^3}+\frac {(c+d x)^3 \cosh (a+b x)}{b}-\frac {6 d^3 \sinh (a+b x)}{b^4}-\frac {3 d (c+d x)^2 \sinh (a+b x)}{b^2} \]
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Time = 0.07 (sec) , antiderivative size = 70, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {3377, 2717} \[ \int (c+d x)^3 \sinh (a+b x) \, dx=-\frac {6 d^3 \sinh (a+b x)}{b^4}+\frac {6 d^2 (c+d x) \cosh (a+b x)}{b^3}-\frac {3 d (c+d x)^2 \sinh (a+b x)}{b^2}+\frac {(c+d x)^3 \cosh (a+b x)}{b} \]
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Rule 2717
Rule 3377
Rubi steps \begin{align*} \text {integral}& = \frac {(c+d x)^3 \cosh (a+b x)}{b}-\frac {(3 d) \int (c+d x)^2 \cosh (a+b x) \, dx}{b} \\ & = \frac {(c+d x)^3 \cosh (a+b x)}{b}-\frac {3 d (c+d x)^2 \sinh (a+b x)}{b^2}+\frac {\left (6 d^2\right ) \int (c+d x) \sinh (a+b x) \, dx}{b^2} \\ & = \frac {6 d^2 (c+d x) \cosh (a+b x)}{b^3}+\frac {(c+d x)^3 \cosh (a+b x)}{b}-\frac {3 d (c+d x)^2 \sinh (a+b x)}{b^2}-\frac {\left (6 d^3\right ) \int \cosh (a+b x) \, dx}{b^3} \\ & = \frac {6 d^2 (c+d x) \cosh (a+b x)}{b^3}+\frac {(c+d x)^3 \cosh (a+b x)}{b}-\frac {6 d^3 \sinh (a+b x)}{b^4}-\frac {3 d (c+d x)^2 \sinh (a+b x)}{b^2} \\ \end{align*}
Time = 0.12 (sec) , antiderivative size = 61, normalized size of antiderivative = 0.87 \[ \int (c+d x)^3 \sinh (a+b x) \, dx=\frac {b (c+d x) \left (6 d^2+b^2 (c+d x)^2\right ) \cosh (a+b x)-3 d \left (2 d^2+b^2 (c+d x)^2\right ) \sinh (a+b x)}{b^4} \]
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Time = 1.14 (sec) , antiderivative size = 126, normalized size of antiderivative = 1.80
method | result | size |
parallelrisch | \(\frac {-3 x d \left (\left (\frac {1}{3} d^{2} x^{2}+c d x +c^{2}\right ) b^{2}+2 d^{2}\right ) b \tanh \left (\frac {b x}{2}+\frac {a}{2}\right )^{2}+6 d \left (\left (d x +c \right )^{2} b^{2}+2 d^{2}\right ) \tanh \left (\frac {b x}{2}+\frac {a}{2}\right )-2 b \left (\frac {d x}{2}+c \right ) \left (\left (d^{2} x^{2}+c d x +c^{2}\right ) b^{2}+6 d^{2}\right )}{b^{4} \left (\tanh \left (\frac {b x}{2}+\frac {a}{2}\right )^{2}-1\right )}\) | \(126\) |
risch | \(\frac {\left (d^{3} x^{3} b^{3}+3 b^{3} c \,d^{2} x^{2}+3 b^{3} c^{2} d x -3 b^{2} d^{3} x^{2}+b^{3} c^{3}-6 b^{2} c \,d^{2} x -3 b^{2} c^{2} d +6 b \,d^{3} x +6 b c \,d^{2}-6 d^{3}\right ) {\mathrm e}^{b x +a}}{2 b^{4}}+\frac {\left (d^{3} x^{3} b^{3}+3 b^{3} c \,d^{2} x^{2}+3 b^{3} c^{2} d x +3 b^{2} d^{3} x^{2}+b^{3} c^{3}+6 b^{2} c \,d^{2} x +3 b^{2} c^{2} d +6 b \,d^{3} x +6 b c \,d^{2}+6 d^{3}\right ) {\mathrm e}^{-b x -a}}{2 b^{4}}\) | \(205\) |
parts | \(\frac {\cosh \left (b x +a \right ) d^{3} x^{3}}{b}+\frac {3 \cosh \left (b x +a \right ) c \,d^{2} x^{2}}{b}+\frac {3 \cosh \left (b x +a \right ) d x \,c^{2}}{b}+\frac {\cosh \left (b x +a \right ) c^{3}}{b}-\frac {3 d \left (\frac {d^{2} \left (\left (b x +a \right )^{2} \sinh \left (b x +a \right )-2 \left (b x +a \right ) \cosh \left (b x +a \right )+2 \sinh \left (b x +a \right )\right )}{b^{2}}-\frac {2 d^{2} a \left (\left (b x +a \right ) \sinh \left (b x +a \right )-\cosh \left (b x +a \right )\right )}{b^{2}}+\frac {2 d c \left (\left (b x +a \right ) \sinh \left (b x +a \right )-\cosh \left (b x +a \right )\right )}{b}+\frac {d^{2} a^{2} \sinh \left (b x +a \right )}{b^{2}}-\frac {2 d a c \sinh \left (b x +a \right )}{b}+c^{2} \sinh \left (b x +a \right )\right )}{b^{2}}\) | \(213\) |
derivativedivides | \(\frac {\frac {d^{3} \left (\left (b x +a \right )^{3} \cosh \left (b x +a \right )-3 \left (b x +a \right )^{2} \sinh \left (b x +a \right )+6 \left (b x +a \right ) \cosh \left (b x +a \right )-6 \sinh \left (b x +a \right )\right )}{b^{3}}-\frac {3 d^{3} a \left (\left (b x +a \right )^{2} \cosh \left (b x +a \right )-2 \left (b x +a \right ) \sinh \left (b x +a \right )+2 \cosh \left (b x +a \right )\right )}{b^{3}}+\frac {3 d^{2} c \left (\left (b x +a \right )^{2} \cosh \left (b x +a \right )-2 \left (b x +a \right ) \sinh \left (b x +a \right )+2 \cosh \left (b x +a \right )\right )}{b^{2}}+\frac {3 d^{3} a^{2} \left (\left (b x +a \right ) \cosh \left (b x +a \right )-\sinh \left (b x +a \right )\right )}{b^{3}}-\frac {6 d^{2} a c \left (\left (b x +a \right ) \cosh \left (b x +a \right )-\sinh \left (b x +a \right )\right )}{b^{2}}+\frac {3 d \,c^{2} \left (\left (b x +a \right ) \cosh \left (b x +a \right )-\sinh \left (b x +a \right )\right )}{b}-\frac {d^{3} a^{3} \cosh \left (b x +a \right )}{b^{3}}+\frac {3 d^{2} a^{2} c \cosh \left (b x +a \right )}{b^{2}}-\frac {3 d a \,c^{2} \cosh \left (b x +a \right )}{b}+c^{3} \cosh \left (b x +a \right )}{b}\) | \(308\) |
default | \(\frac {\frac {d^{3} \left (\left (b x +a \right )^{3} \cosh \left (b x +a \right )-3 \left (b x +a \right )^{2} \sinh \left (b x +a \right )+6 \left (b x +a \right ) \cosh \left (b x +a \right )-6 \sinh \left (b x +a \right )\right )}{b^{3}}-\frac {3 d^{3} a \left (\left (b x +a \right )^{2} \cosh \left (b x +a \right )-2 \left (b x +a \right ) \sinh \left (b x +a \right )+2 \cosh \left (b x +a \right )\right )}{b^{3}}+\frac {3 d^{2} c \left (\left (b x +a \right )^{2} \cosh \left (b x +a \right )-2 \left (b x +a \right ) \sinh \left (b x +a \right )+2 \cosh \left (b x +a \right )\right )}{b^{2}}+\frac {3 d^{3} a^{2} \left (\left (b x +a \right ) \cosh \left (b x +a \right )-\sinh \left (b x +a \right )\right )}{b^{3}}-\frac {6 d^{2} a c \left (\left (b x +a \right ) \cosh \left (b x +a \right )-\sinh \left (b x +a \right )\right )}{b^{2}}+\frac {3 d \,c^{2} \left (\left (b x +a \right ) \cosh \left (b x +a \right )-\sinh \left (b x +a \right )\right )}{b}-\frac {d^{3} a^{3} \cosh \left (b x +a \right )}{b^{3}}+\frac {3 d^{2} a^{2} c \cosh \left (b x +a \right )}{b^{2}}-\frac {3 d a \,c^{2} \cosh \left (b x +a \right )}{b}+c^{3} \cosh \left (b x +a \right )}{b}\) | \(308\) |
meijerg | \(\frac {8 d^{3} \sinh \left (a \right ) \sqrt {\pi }\, \left (\frac {3}{4 \sqrt {\pi }}-\frac {\left (\frac {3 b^{2} x^{2}}{2}+3\right ) \cosh \left (b x \right )}{4 \sqrt {\pi }}+\frac {x b \left (\frac {b^{2} x^{2}}{2}+3\right ) \sinh \left (b x \right )}{4 \sqrt {\pi }}\right )}{b^{4}}-\frac {8 i d^{3} \cosh \left (a \right ) \sqrt {\pi }\, \left (\frac {i x b \left (\frac {5 b^{2} x^{2}}{2}+15\right ) \cosh \left (b x \right )}{20 \sqrt {\pi }}-\frac {i \left (\frac {15 b^{2} x^{2}}{2}+15\right ) \sinh \left (b x \right )}{20 \sqrt {\pi }}\right )}{b^{4}}+\frac {12 i c \,d^{2} \sinh \left (a \right ) \sqrt {\pi }\, \left (\frac {i b x \cosh \left (b x \right )}{2 \sqrt {\pi }}-\frac {i \left (\frac {3 b^{2} x^{2}}{2}+3\right ) \sinh \left (b x \right )}{6 \sqrt {\pi }}\right )}{b^{3}}+\frac {12 d^{2} c \cosh \left (a \right ) \sqrt {\pi }\, \left (-\frac {1}{2 \sqrt {\pi }}+\frac {\left (\frac {b^{2} x^{2}}{2}+1\right ) \cosh \left (b x \right )}{2 \sqrt {\pi }}-\frac {x b \sinh \left (b x \right )}{2 \sqrt {\pi }}\right )}{b^{3}}-\frac {6 d \,c^{2} \sinh \left (a \right ) \sqrt {\pi }\, \left (-\frac {1}{2 \sqrt {\pi }}+\frac {\cosh \left (b x \right )}{2 \sqrt {\pi }}-\frac {x b \sinh \left (b x \right )}{2 \sqrt {\pi }}\right )}{b^{2}}-\frac {3 d \,c^{2} \cosh \left (a \right ) \left (-\cosh \left (b x \right ) b x +\sinh \left (b x \right )\right )}{b^{2}}+\frac {c^{3} \sinh \left (a \right ) \sinh \left (b x \right )}{b}-\frac {c^{3} \cosh \left (a \right ) \sqrt {\pi }\, \left (\frac {1}{\sqrt {\pi }}-\frac {\cosh \left (b x \right )}{\sqrt {\pi }}\right )}{b}\) | \(319\) |
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Time = 0.24 (sec) , antiderivative size = 109, normalized size of antiderivative = 1.56 \[ \int (c+d x)^3 \sinh (a+b x) \, dx=\frac {{\left (b^{3} d^{3} x^{3} + 3 \, b^{3} c d^{2} x^{2} + b^{3} c^{3} + 6 \, b c d^{2} + 3 \, {\left (b^{3} c^{2} d + 2 \, b d^{3}\right )} x\right )} \cosh \left (b x + a\right ) - 3 \, {\left (b^{2} d^{3} x^{2} + 2 \, b^{2} c d^{2} x + b^{2} c^{2} d + 2 \, d^{3}\right )} \sinh \left (b x + a\right )}{b^{4}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 202 vs. \(2 (70) = 140\).
Time = 0.27 (sec) , antiderivative size = 202, normalized size of antiderivative = 2.89 \[ \int (c+d x)^3 \sinh (a+b x) \, dx=\begin {cases} \frac {c^{3} \cosh {\left (a + b x \right )}}{b} + \frac {3 c^{2} d x \cosh {\left (a + b x \right )}}{b} + \frac {3 c d^{2} x^{2} \cosh {\left (a + b x \right )}}{b} + \frac {d^{3} x^{3} \cosh {\left (a + b x \right )}}{b} - \frac {3 c^{2} d \sinh {\left (a + b x \right )}}{b^{2}} - \frac {6 c d^{2} x \sinh {\left (a + b x \right )}}{b^{2}} - \frac {3 d^{3} x^{2} \sinh {\left (a + b x \right )}}{b^{2}} + \frac {6 c d^{2} \cosh {\left (a + b x \right )}}{b^{3}} + \frac {6 d^{3} x \cosh {\left (a + b x \right )}}{b^{3}} - \frac {6 d^{3} \sinh {\left (a + b x \right )}}{b^{4}} & \text {for}\: b \neq 0 \\\left (c^{3} x + \frac {3 c^{2} d x^{2}}{2} + c d^{2} x^{3} + \frac {d^{3} x^{4}}{4}\right ) \sinh {\left (a \right )} & \text {otherwise} \end {cases} \]
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Leaf count of result is larger than twice the leaf count of optimal. 222 vs. \(2 (70) = 140\).
Time = 0.21 (sec) , antiderivative size = 222, normalized size of antiderivative = 3.17 \[ \int (c+d x)^3 \sinh (a+b x) \, dx=\frac {c^{3} e^{\left (b x + a\right )}}{2 \, b} + \frac {3 \, {\left (b x e^{a} - e^{a}\right )} c^{2} d e^{\left (b x\right )}}{2 \, b^{2}} + \frac {c^{3} e^{\left (-b x - a\right )}}{2 \, b} + \frac {3 \, {\left (b x + 1\right )} c^{2} d e^{\left (-b x - a\right )}}{2 \, b^{2}} + \frac {3 \, {\left (b^{2} x^{2} e^{a} - 2 \, b x e^{a} + 2 \, e^{a}\right )} c d^{2} e^{\left (b x\right )}}{2 \, b^{3}} + \frac {3 \, {\left (b^{2} x^{2} + 2 \, b x + 2\right )} c d^{2} e^{\left (-b x - a\right )}}{2 \, b^{3}} + \frac {{\left (b^{3} x^{3} e^{a} - 3 \, b^{2} x^{2} e^{a} + 6 \, b x e^{a} - 6 \, e^{a}\right )} d^{3} e^{\left (b x\right )}}{2 \, b^{4}} + \frac {{\left (b^{3} x^{3} + 3 \, b^{2} x^{2} + 6 \, b x + 6\right )} d^{3} e^{\left (-b x - a\right )}}{2 \, b^{4}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 204 vs. \(2 (70) = 140\).
Time = 0.28 (sec) , antiderivative size = 204, normalized size of antiderivative = 2.91 \[ \int (c+d x)^3 \sinh (a+b x) \, dx=\frac {{\left (b^{3} d^{3} x^{3} + 3 \, b^{3} c d^{2} x^{2} + 3 \, b^{3} c^{2} d x - 3 \, b^{2} d^{3} x^{2} + b^{3} c^{3} - 6 \, b^{2} c d^{2} x - 3 \, b^{2} c^{2} d + 6 \, b d^{3} x + 6 \, b c d^{2} - 6 \, d^{3}\right )} e^{\left (b x + a\right )}}{2 \, b^{4}} + \frac {{\left (b^{3} d^{3} x^{3} + 3 \, b^{3} c d^{2} x^{2} + 3 \, b^{3} c^{2} d x + 3 \, b^{2} d^{3} x^{2} + b^{3} c^{3} + 6 \, b^{2} c d^{2} x + 3 \, b^{2} c^{2} d + 6 \, b d^{3} x + 6 \, b c d^{2} + 6 \, d^{3}\right )} e^{\left (-b x - a\right )}}{2 \, b^{4}} \]
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Time = 0.87 (sec) , antiderivative size = 143, normalized size of antiderivative = 2.04 \[ \int (c+d x)^3 \sinh (a+b x) \, dx=\frac {\mathrm {cosh}\left (a+b\,x\right )\,\left (b^2\,c^3+6\,c\,d^2\right )}{b^3}-\frac {3\,\mathrm {sinh}\left (a+b\,x\right )\,\left (b^2\,c^2\,d+2\,d^3\right )}{b^4}+\frac {d^3\,x^3\,\mathrm {cosh}\left (a+b\,x\right )}{b}-\frac {3\,d^3\,x^2\,\mathrm {sinh}\left (a+b\,x\right )}{b^2}+\frac {3\,x\,\mathrm {cosh}\left (a+b\,x\right )\,\left (b^2\,c^2\,d+2\,d^3\right )}{b^3}-\frac {6\,c\,d^2\,x\,\mathrm {sinh}\left (a+b\,x\right )}{b^2}+\frac {3\,c\,d^2\,x^2\,\mathrm {cosh}\left (a+b\,x\right )}{b} \]
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