Integrand size = 14, antiderivative size = 91 \[ \int (c+d x)^4 \sinh (a+b x) \, dx=\frac {24 d^4 \cosh (a+b x)}{b^5}+\frac {12 d^2 (c+d x)^2 \cosh (a+b x)}{b^3}+\frac {(c+d x)^4 \cosh (a+b x)}{b}-\frac {24 d^3 (c+d x) \sinh (a+b x)}{b^4}-\frac {4 d (c+d x)^3 \sinh (a+b x)}{b^2} \]
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Time = 0.09 (sec) , antiderivative size = 91, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {3377, 2718} \[ \int (c+d x)^4 \sinh (a+b x) \, dx=\frac {24 d^4 \cosh (a+b x)}{b^5}-\frac {24 d^3 (c+d x) \sinh (a+b x)}{b^4}+\frac {12 d^2 (c+d x)^2 \cosh (a+b x)}{b^3}-\frac {4 d (c+d x)^3 \sinh (a+b x)}{b^2}+\frac {(c+d x)^4 \cosh (a+b x)}{b} \]
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Rule 2718
Rule 3377
Rubi steps \begin{align*} \text {integral}& = \frac {(c+d x)^4 \cosh (a+b x)}{b}-\frac {(4 d) \int (c+d x)^3 \cosh (a+b x) \, dx}{b} \\ & = \frac {(c+d x)^4 \cosh (a+b x)}{b}-\frac {4 d (c+d x)^3 \sinh (a+b x)}{b^2}+\frac {\left (12 d^2\right ) \int (c+d x)^2 \sinh (a+b x) \, dx}{b^2} \\ & = \frac {12 d^2 (c+d x)^2 \cosh (a+b x)}{b^3}+\frac {(c+d x)^4 \cosh (a+b x)}{b}-\frac {4 d (c+d x)^3 \sinh (a+b x)}{b^2}-\frac {\left (24 d^3\right ) \int (c+d x) \cosh (a+b x) \, dx}{b^3} \\ & = \frac {12 d^2 (c+d x)^2 \cosh (a+b x)}{b^3}+\frac {(c+d x)^4 \cosh (a+b x)}{b}-\frac {24 d^3 (c+d x) \sinh (a+b x)}{b^4}-\frac {4 d (c+d x)^3 \sinh (a+b x)}{b^2}+\frac {\left (24 d^4\right ) \int \sinh (a+b x) \, dx}{b^4} \\ & = \frac {24 d^4 \cosh (a+b x)}{b^5}+\frac {12 d^2 (c+d x)^2 \cosh (a+b x)}{b^3}+\frac {(c+d x)^4 \cosh (a+b x)}{b}-\frac {24 d^3 (c+d x) \sinh (a+b x)}{b^4}-\frac {4 d (c+d x)^3 \sinh (a+b x)}{b^2} \\ \end{align*}
Time = 0.18 (sec) , antiderivative size = 76, normalized size of antiderivative = 0.84 \[ \int (c+d x)^4 \sinh (a+b x) \, dx=\frac {\left (24 d^4+12 b^2 d^2 (c+d x)^2+b^4 (c+d x)^4\right ) \cosh (a+b x)-4 b d (c+d x) \left (6 d^2+b^2 (c+d x)^2\right ) \sinh (a+b x)}{b^5} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(184\) vs. \(2(91)=182\).
Time = 0.99 (sec) , antiderivative size = 185, normalized size of antiderivative = 2.03
method | result | size |
parallelrisch | \(\frac {-4 \left (\left (\frac {1}{2} d^{2} x^{2}+c d x +c^{2}\right ) b^{2}+6 d^{2}\right ) x d \,b^{2} \left (\frac {d x}{2}+c \right ) \tanh \left (\frac {b x}{2}+\frac {a}{2}\right )^{2}+8 d \left (\left (d x +c \right )^{2} b^{2}+6 d^{2}\right ) b \left (d x +c \right ) \tanh \left (\frac {b x}{2}+\frac {a}{2}\right )+\left (-d^{4} x^{4}-4 c \,d^{3} x^{3}-6 c^{2} d^{2} x^{2}-4 c^{3} d x -2 c^{4}\right ) b^{4}+12 \left (-d^{4} x^{2}-2 c \,d^{3} x -2 c^{2} d^{2}\right ) b^{2}-48 d^{4}}{b^{5} \left (\tanh \left (\frac {b x}{2}+\frac {a}{2}\right )^{2}-1\right )}\) | \(185\) |
risch | \(\frac {\left (d^{4} x^{4} b^{4}+4 b^{4} c \,d^{3} x^{3}+6 b^{4} c^{2} d^{2} x^{2}-4 b^{3} d^{4} x^{3}+4 b^{4} c^{3} d x -12 b^{3} c \,d^{3} x^{2}+b^{4} c^{4}-12 b^{3} c^{2} d^{2} x +12 b^{2} d^{4} x^{2}-4 b^{3} c^{3} d +24 b^{2} c \,d^{3} x +12 b^{2} c^{2} d^{2}-24 b \,d^{4} x -24 b c \,d^{3}+24 d^{4}\right ) {\mathrm e}^{b x +a}}{2 b^{5}}+\frac {\left (d^{4} x^{4} b^{4}+4 b^{4} c \,d^{3} x^{3}+6 b^{4} c^{2} d^{2} x^{2}+4 b^{3} d^{4} x^{3}+4 b^{4} c^{3} d x +12 b^{3} c \,d^{3} x^{2}+b^{4} c^{4}+12 b^{3} c^{2} d^{2} x +12 b^{2} d^{4} x^{2}+4 b^{3} c^{3} d +24 b^{2} c \,d^{3} x +12 b^{2} c^{2} d^{2}+24 b \,d^{4} x +24 b c \,d^{3}+24 d^{4}\right ) {\mathrm e}^{-b x -a}}{2 b^{5}}\) | \(325\) |
parts | \(\frac {\cosh \left (b x +a \right ) d^{4} x^{4}}{b}+\frac {4 \cosh \left (b x +a \right ) c \,d^{3} x^{3}}{b}+\frac {6 \cosh \left (b x +a \right ) c^{2} d^{2} x^{2}}{b}+\frac {4 \cosh \left (b x +a \right ) c^{3} d x}{b}+\frac {\cosh \left (b x +a \right ) c^{4}}{b}-\frac {4 d \left (\frac {d^{3} \left (\left (b x +a \right )^{3} \sinh \left (b x +a \right )-3 \left (b x +a \right )^{2} \cosh \left (b x +a \right )+6 \left (b x +a \right ) \sinh \left (b x +a \right )-6 \cosh \left (b x +a \right )\right )}{b^{3}}-\frac {3 d^{3} a \left (\left (b x +a \right )^{2} \sinh \left (b x +a \right )-2 \left (b x +a \right ) \cosh \left (b x +a \right )+2 \sinh \left (b x +a \right )\right )}{b^{3}}+\frac {3 d^{2} c \left (\left (b x +a \right )^{2} \sinh \left (b x +a \right )-2 \left (b x +a \right ) \cosh \left (b x +a \right )+2 \sinh \left (b x +a \right )\right )}{b^{2}}+\frac {3 d^{3} a^{2} \left (\left (b x +a \right ) \sinh \left (b x +a \right )-\cosh \left (b x +a \right )\right )}{b^{3}}-\frac {6 d^{2} a c \left (\left (b x +a \right ) \sinh \left (b x +a \right )-\cosh \left (b x +a \right )\right )}{b^{2}}+\frac {3 d \,c^{2} \left (\left (b x +a \right ) \sinh \left (b x +a \right )-\cosh \left (b x +a \right )\right )}{b}-\frac {d^{3} a^{3} \sinh \left (b x +a \right )}{b^{3}}+\frac {3 d^{2} a^{2} c \sinh \left (b x +a \right )}{b^{2}}-\frac {3 d a \,c^{2} \sinh \left (b x +a \right )}{b}+c^{3} \sinh \left (b x +a \right )\right )}{b^{2}}\) | \(394\) |
meijerg | \(-\frac {16 i d^{4} \sinh \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 )}{10 \sqrt {\pi }}+\frac {i \left (\frac {5}{8} x^{4} b^{4}+\frac {15}{2} b^{2} x^{2}+15\right ) \sinh \left (b x \right )}{10 \sqrt {\pi }}\right )}{b^{5}}-\frac {16 d^{4} \cosh \left (a \right ) \sqrt {\pi }\, \left (\frac {3}{2 \sqrt {\pi }}-\frac {\left (\frac {3}{8} x^{4} b^{4}+\frac {9}{2} b^{2} x^{2}+9\right ) \cosh \left (b x \right )}{6 \sqrt {\pi }}+\frac {b x \left (\frac {3 b^{2} x^{2}}{2}+9\right ) \sinh \left (b x \right )}{6 \sqrt {\pi }}\right )}{b^{5}}+\frac {32 d^{3} c \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 {32 i d^{3} c \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 {24 i c^{2} 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 {24 d^{2} c^{2} \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 {8 d \,c^{3} \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 {4 d \,c^{3} \cosh \left (a \right ) \left (-\cosh \left (b x \right ) b x +\sinh \left (b x \right )\right )}{b^{2}}+\frac {c^{4} \sinh \left (a \right ) \sinh \left (b x \right )}{b}-\frac {c^{4} \cosh \left (a \right ) \sqrt {\pi }\, \left (\frac {1}{\sqrt {\pi }}-\frac {\cosh \left (b x \right )}{\sqrt {\pi }}\right )}{b}\) | \(457\) |
derivativedivides | \(\frac {\frac {d^{4} \left (\left (b x +a \right )^{4} \cosh \left (b x +a \right )-4 \left (b x +a \right )^{3} \sinh \left (b x +a \right )+12 \left (b x +a \right )^{2} \cosh \left (b x +a \right )-24 \left (b x +a \right ) \sinh \left (b x +a \right )+24 \cosh \left (b x +a \right )\right )}{b^{4}}+\frac {d^{4} a^{4} \cosh \left (b x +a \right )}{b^{4}}-\frac {4 d^{3} a^{3} c \cosh \left (b x +a \right )}{b^{3}}+\frac {6 d^{2} a^{2} c^{2} \cosh \left (b x +a \right )}{b^{2}}-\frac {4 d a \,c^{3} \cosh \left (b x +a \right )}{b}-\frac {4 d^{4} a \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^{4}}+\frac {4 d^{3} c \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 {6 d^{4} a^{2} \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^{4}}+\frac {6 d^{2} c^{2} \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 {4 d^{4} a^{3} \left (\left (b x +a \right ) \cosh \left (b x +a \right )-\sinh \left (b x +a \right )\right )}{b^{4}}+\frac {4 d \,c^{3} \left (\left (b x +a \right ) \cosh \left (b x +a \right )-\sinh \left (b x +a \right )\right )}{b}-\frac {12 d^{2} a \,c^{2} \left (\left (b x +a \right ) \cosh \left (b x +a \right )-\sinh \left (b x +a \right )\right )}{b^{2}}-\frac {12 d^{3} a 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^{3}}+\frac {12 d^{3} a^{2} c \left (\left (b x +a \right ) \cosh \left (b x +a \right )-\sinh \left (b x +a \right )\right )}{b^{3}}+c^{4} \cosh \left (b x +a \right )}{b}\) | \(547\) |
default | \(\frac {\frac {d^{4} \left (\left (b x +a \right )^{4} \cosh \left (b x +a \right )-4 \left (b x +a \right )^{3} \sinh \left (b x +a \right )+12 \left (b x +a \right )^{2} \cosh \left (b x +a \right )-24 \left (b x +a \right ) \sinh \left (b x +a \right )+24 \cosh \left (b x +a \right )\right )}{b^{4}}+\frac {d^{4} a^{4} \cosh \left (b x +a \right )}{b^{4}}-\frac {4 d^{3} a^{3} c \cosh \left (b x +a \right )}{b^{3}}+\frac {6 d^{2} a^{2} c^{2} \cosh \left (b x +a \right )}{b^{2}}-\frac {4 d a \,c^{3} \cosh \left (b x +a \right )}{b}-\frac {4 d^{4} a \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^{4}}+\frac {4 d^{3} c \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 {6 d^{4} a^{2} \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^{4}}+\frac {6 d^{2} c^{2} \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 {4 d^{4} a^{3} \left (\left (b x +a \right ) \cosh \left (b x +a \right )-\sinh \left (b x +a \right )\right )}{b^{4}}+\frac {4 d \,c^{3} \left (\left (b x +a \right ) \cosh \left (b x +a \right )-\sinh \left (b x +a \right )\right )}{b}-\frac {12 d^{2} a \,c^{2} \left (\left (b x +a \right ) \cosh \left (b x +a \right )-\sinh \left (b x +a \right )\right )}{b^{2}}-\frac {12 d^{3} a 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^{3}}+\frac {12 d^{3} a^{2} c \left (\left (b x +a \right ) \cosh \left (b x +a \right )-\sinh \left (b x +a \right )\right )}{b^{3}}+c^{4} \cosh \left (b x +a \right )}{b}\) | \(547\) |
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Time = 0.26 (sec) , antiderivative size = 169, normalized size of antiderivative = 1.86 \[ \int (c+d x)^4 \sinh (a+b x) \, dx=\frac {{\left (b^{4} d^{4} x^{4} + 4 \, b^{4} c d^{3} x^{3} + b^{4} c^{4} + 12 \, b^{2} c^{2} d^{2} + 24 \, d^{4} + 6 \, {\left (b^{4} c^{2} d^{2} + 2 \, b^{2} d^{4}\right )} x^{2} + 4 \, {\left (b^{4} c^{3} d + 6 \, b^{2} c d^{3}\right )} x\right )} \cosh \left (b x + a\right ) - 4 \, {\left (b^{3} d^{4} x^{3} + 3 \, b^{3} c d^{3} x^{2} + b^{3} c^{3} d + 6 \, b c d^{3} + 3 \, {\left (b^{3} c^{2} d^{2} + 2 \, b d^{4}\right )} x\right )} \sinh \left (b x + a\right )}{b^{5}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 311 vs. \(2 (92) = 184\).
Time = 0.35 (sec) , antiderivative size = 311, normalized size of antiderivative = 3.42 \[ \int (c+d x)^4 \sinh (a+b x) \, dx=\begin {cases} \frac {c^{4} \cosh {\left (a + b x \right )}}{b} + \frac {4 c^{3} d x \cosh {\left (a + b x \right )}}{b} + \frac {6 c^{2} d^{2} x^{2} \cosh {\left (a + b x \right )}}{b} + \frac {4 c d^{3} x^{3} \cosh {\left (a + b x \right )}}{b} + \frac {d^{4} x^{4} \cosh {\left (a + b x \right )}}{b} - \frac {4 c^{3} d \sinh {\left (a + b x \right )}}{b^{2}} - \frac {12 c^{2} d^{2} x \sinh {\left (a + b x \right )}}{b^{2}} - \frac {12 c d^{3} x^{2} \sinh {\left (a + b x \right )}}{b^{2}} - \frac {4 d^{4} x^{3} \sinh {\left (a + b x \right )}}{b^{2}} + \frac {12 c^{2} d^{2} \cosh {\left (a + b x \right )}}{b^{3}} + \frac {24 c d^{3} x \cosh {\left (a + b x \right )}}{b^{3}} + \frac {12 d^{4} x^{2} \cosh {\left (a + b x \right )}}{b^{3}} - \frac {24 c d^{3} \sinh {\left (a + b x \right )}}{b^{4}} - \frac {24 d^{4} x \sinh {\left (a + b x \right )}}{b^{4}} + \frac {24 d^{4} \cosh {\left (a + b x \right )}}{b^{5}} & \text {for}\: b \neq 0 \\\left (c^{4} x + 2 c^{3} d x^{2} + 2 c^{2} d^{2} x^{3} + c d^{3} x^{4} + \frac {d^{4} x^{5}}{5}\right ) \sinh {\left (a \right )} & \text {otherwise} \end {cases} \]
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Leaf count of result is larger than twice the leaf count of optimal. 326 vs. \(2 (91) = 182\).
Time = 0.21 (sec) , antiderivative size = 326, normalized size of antiderivative = 3.58 \[ \int (c+d x)^4 \sinh (a+b x) \, dx=\frac {c^{4} e^{\left (b x + a\right )}}{2 \, b} + \frac {2 \, {\left (b x e^{a} - e^{a}\right )} c^{3} d e^{\left (b x\right )}}{b^{2}} + \frac {c^{4} e^{\left (-b x - a\right )}}{2 \, b} + \frac {2 \, {\left (b x + 1\right )} c^{3} d e^{\left (-b x - a\right )}}{b^{2}} + \frac {3 \, {\left (b^{2} x^{2} e^{a} - 2 \, b x e^{a} + 2 \, e^{a}\right )} c^{2} d^{2} e^{\left (b x\right )}}{b^{3}} + \frac {3 \, {\left (b^{2} x^{2} + 2 \, b x + 2\right )} c^{2} d^{2} e^{\left (-b x - a\right )}}{b^{3}} + \frac {2 \, {\left (b^{3} x^{3} e^{a} - 3 \, b^{2} x^{2} e^{a} + 6 \, b x e^{a} - 6 \, e^{a}\right )} c d^{3} e^{\left (b x\right )}}{b^{4}} + \frac {2 \, {\left (b^{3} x^{3} + 3 \, b^{2} x^{2} + 6 \, b x + 6\right )} c d^{3} e^{\left (-b x - a\right )}}{b^{4}} + \frac {{\left (b^{4} x^{4} e^{a} - 4 \, b^{3} x^{3} e^{a} + 12 \, b^{2} x^{2} e^{a} - 24 \, b x e^{a} + 24 \, e^{a}\right )} d^{4} e^{\left (b x\right )}}{2 \, b^{5}} + \frac {{\left (b^{4} x^{4} + 4 \, b^{3} x^{3} + 12 \, b^{2} x^{2} + 24 \, b x + 24\right )} d^{4} e^{\left (-b x - a\right )}}{2 \, b^{5}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 324 vs. \(2 (91) = 182\).
Time = 0.26 (sec) , antiderivative size = 324, normalized size of antiderivative = 3.56 \[ \int (c+d x)^4 \sinh (a+b x) \, dx=\frac {{\left (b^{4} d^{4} x^{4} + 4 \, b^{4} c d^{3} x^{3} + 6 \, b^{4} c^{2} d^{2} x^{2} - 4 \, b^{3} d^{4} x^{3} + 4 \, b^{4} c^{3} d x - 12 \, b^{3} c d^{3} x^{2} + b^{4} c^{4} - 12 \, b^{3} c^{2} d^{2} x + 12 \, b^{2} d^{4} x^{2} - 4 \, b^{3} c^{3} d + 24 \, b^{2} c d^{3} x + 12 \, b^{2} c^{2} d^{2} - 24 \, b d^{4} x - 24 \, b c d^{3} + 24 \, d^{4}\right )} e^{\left (b x + a\right )}}{2 \, b^{5}} + \frac {{\left (b^{4} d^{4} x^{4} + 4 \, b^{4} c d^{3} x^{3} + 6 \, b^{4} c^{2} d^{2} x^{2} + 4 \, b^{3} d^{4} x^{3} + 4 \, b^{4} c^{3} d x + 12 \, b^{3} c d^{3} x^{2} + b^{4} c^{4} + 12 \, b^{3} c^{2} d^{2} x + 12 \, b^{2} d^{4} x^{2} + 4 \, b^{3} c^{3} d + 24 \, b^{2} c d^{3} x + 12 \, b^{2} c^{2} d^{2} + 24 \, b d^{4} x + 24 \, b c d^{3} + 24 \, d^{4}\right )} e^{\left (-b x - a\right )}}{2 \, b^{5}} \]
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Time = 0.20 (sec) , antiderivative size = 215, normalized size of antiderivative = 2.36 \[ \int (c+d x)^4 \sinh (a+b x) \, dx=\frac {\mathrm {cosh}\left (a+b\,x\right )\,\left (b^4\,c^4+12\,b^2\,c^2\,d^2+24\,d^4\right )}{b^5}-\frac {4\,\mathrm {sinh}\left (a+b\,x\right )\,\left (b^2\,c^3\,d+6\,c\,d^3\right )}{b^4}+\frac {d^4\,x^4\,\mathrm {cosh}\left (a+b\,x\right )}{b}+\frac {4\,x\,\mathrm {cosh}\left (a+b\,x\right )\,\left (b^2\,c^3\,d+6\,c\,d^3\right )}{b^3}-\frac {4\,d^4\,x^3\,\mathrm {sinh}\left (a+b\,x\right )}{b^2}-\frac {12\,x\,\mathrm {sinh}\left (a+b\,x\right )\,\left (b^2\,c^2\,d^2+2\,d^4\right )}{b^4}+\frac {6\,x^2\,\mathrm {cosh}\left (a+b\,x\right )\,\left (b^2\,c^2\,d^2+2\,d^4\right )}{b^3}+\frac {4\,c\,d^3\,x^3\,\mathrm {cosh}\left (a+b\,x\right )}{b}-\frac {12\,c\,d^3\,x^2\,\mathrm {sinh}\left (a+b\,x\right )}{b^2} \]
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