Integrand size = 16, antiderivative size = 167 \[ \int x \cosh \left (a+b \sqrt {c+d x}\right ) \, dx=-\frac {12 \cosh \left (a+b \sqrt {c+d x}\right )}{b^4 d^2}+\frac {2 c \cosh \left (a+b \sqrt {c+d x}\right )}{b^2 d^2}-\frac {6 (c+d x) \cosh \left (a+b \sqrt {c+d x}\right )}{b^2 d^2}+\frac {12 \sqrt {c+d x} \sinh \left (a+b \sqrt {c+d x}\right )}{b^3 d^2}-\frac {2 c \sqrt {c+d x} \sinh \left (a+b \sqrt {c+d x}\right )}{b d^2}+\frac {2 (c+d x)^{3/2} \sinh \left (a+b \sqrt {c+d x}\right )}{b d^2} \]
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Time = 0.14 (sec) , antiderivative size = 167, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {5473, 5395, 3377, 2718} \[ \int x \cosh \left (a+b \sqrt {c+d x}\right ) \, dx=-\frac {12 \cosh \left (a+b \sqrt {c+d x}\right )}{b^4 d^2}+\frac {12 \sqrt {c+d x} \sinh \left (a+b \sqrt {c+d x}\right )}{b^3 d^2}-\frac {6 (c+d x) \cosh \left (a+b \sqrt {c+d x}\right )}{b^2 d^2}+\frac {2 c \cosh \left (a+b \sqrt {c+d x}\right )}{b^2 d^2}+\frac {2 (c+d x)^{3/2} \sinh \left (a+b \sqrt {c+d x}\right )}{b d^2}-\frac {2 c \sqrt {c+d x} \sinh \left (a+b \sqrt {c+d x}\right )}{b d^2} \]
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Rule 2718
Rule 3377
Rule 5395
Rule 5473
Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int (-c+x) \cosh \left (a+b \sqrt {x}\right ) \, dx,x,c+d x\right )}{d^2} \\ & = \frac {2 \text {Subst}\left (\int x \left (-c+x^2\right ) \cosh (a+b x) \, dx,x,\sqrt {c+d x}\right )}{d^2} \\ & = \frac {2 \text {Subst}\left (\int \left (-c x \cosh (a+b x)+x^3 \cosh (a+b x)\right ) \, dx,x,\sqrt {c+d x}\right )}{d^2} \\ & = \frac {2 \text {Subst}\left (\int x^3 \cosh (a+b x) \, dx,x,\sqrt {c+d x}\right )}{d^2}-\frac {(2 c) \text {Subst}\left (\int x \cosh (a+b x) \, dx,x,\sqrt {c+d x}\right )}{d^2} \\ & = -\frac {2 c \sqrt {c+d x} \sinh \left (a+b \sqrt {c+d x}\right )}{b d^2}+\frac {2 (c+d x)^{3/2} \sinh \left (a+b \sqrt {c+d x}\right )}{b d^2}-\frac {6 \text {Subst}\left (\int x^2 \sinh (a+b x) \, dx,x,\sqrt {c+d x}\right )}{b d^2}+\frac {(2 c) \text {Subst}\left (\int \sinh (a+b x) \, dx,x,\sqrt {c+d x}\right )}{b d^2} \\ & = \frac {2 c \cosh \left (a+b \sqrt {c+d x}\right )}{b^2 d^2}-\frac {6 (c+d x) \cosh \left (a+b \sqrt {c+d x}\right )}{b^2 d^2}-\frac {2 c \sqrt {c+d x} \sinh \left (a+b \sqrt {c+d x}\right )}{b d^2}+\frac {2 (c+d x)^{3/2} \sinh \left (a+b \sqrt {c+d x}\right )}{b d^2}+\frac {12 \text {Subst}\left (\int x \cosh (a+b x) \, dx,x,\sqrt {c+d x}\right )}{b^2 d^2} \\ & = \frac {2 c \cosh \left (a+b \sqrt {c+d x}\right )}{b^2 d^2}-\frac {6 (c+d x) \cosh \left (a+b \sqrt {c+d x}\right )}{b^2 d^2}+\frac {12 \sqrt {c+d x} \sinh \left (a+b \sqrt {c+d x}\right )}{b^3 d^2}-\frac {2 c \sqrt {c+d x} \sinh \left (a+b \sqrt {c+d x}\right )}{b d^2}+\frac {2 (c+d x)^{3/2} \sinh \left (a+b \sqrt {c+d x}\right )}{b d^2}-\frac {12 \text {Subst}\left (\int \sinh (a+b x) \, dx,x,\sqrt {c+d x}\right )}{b^3 d^2} \\ & = -\frac {12 \cosh \left (a+b \sqrt {c+d x}\right )}{b^4 d^2}+\frac {2 c \cosh \left (a+b \sqrt {c+d x}\right )}{b^2 d^2}-\frac {6 (c+d x) \cosh \left (a+b \sqrt {c+d x}\right )}{b^2 d^2}+\frac {12 \sqrt {c+d x} \sinh \left (a+b \sqrt {c+d x}\right )}{b^3 d^2}-\frac {2 c \sqrt {c+d x} \sinh \left (a+b \sqrt {c+d x}\right )}{b d^2}+\frac {2 (c+d x)^{3/2} \sinh \left (a+b \sqrt {c+d x}\right )}{b d^2} \\ \end{align*}
Time = 0.15 (sec) , antiderivative size = 72, normalized size of antiderivative = 0.43 \[ \int x \cosh \left (a+b \sqrt {c+d x}\right ) \, dx=\frac {-2 \left (6+b^2 (2 c+3 d x)\right ) \cosh \left (a+b \sqrt {c+d x}\right )+2 b \sqrt {c+d x} \left (6+b^2 d x\right ) \sinh \left (a+b \sqrt {c+d x}\right )}{b^4 d^2} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(300\) vs. \(2(149)=298\).
Time = 0.12 (sec) , antiderivative size = 301, normalized size of antiderivative = 1.80
method | result | size |
parts | \(\frac {2 x \sqrt {d x +c}\, \sinh \left (a +b \sqrt {d x +c}\right )}{d b}-\frac {2 x \cosh \left (a +b \sqrt {d x +c}\right )}{d \,b^{2}}-\frac {2 \left (\frac {2 \left (a +b \sqrt {d x +c}\right )^{2} \cosh \left (a +b \sqrt {d x +c}\right )-4 \left (a +b \sqrt {d x +c}\right ) \sinh \left (a +b \sqrt {d x +c}\right )+4 \cosh \left (a +b \sqrt {d x +c}\right )-2 a \left (\left (a +b \sqrt {d x +c}\right ) \cosh \left (a +b \sqrt {d x +c}\right )-\sinh \left (a +b \sqrt {d x +c}\right )\right )}{b^{2} d}-\frac {2 a \left (\left (a +b \sqrt {d x +c}\right ) \cosh \left (a +b \sqrt {d x +c}\right )-\sinh \left (a +b \sqrt {d x +c}\right )-a \cosh \left (a +b \sqrt {d x +c}\right )\right )}{d \,b^{2}}-\frac {2 \left (\left (a +b \sqrt {d x +c}\right ) \sinh \left (a +b \sqrt {d x +c}\right )-\cosh \left (a +b \sqrt {d x +c}\right )-a \sinh \left (a +b \sqrt {d x +c}\right )\right )}{d \,b^{2}}\right )}{d \,b^{2}}\) | \(301\) |
derivativedivides | \(\frac {\frac {6 a^{2} \left (\left (a +b \sqrt {d x +c}\right ) \sinh \left (a +b \sqrt {d x +c}\right )-\cosh \left (a +b \sqrt {d x +c}\right )\right )}{b^{2}}-\frac {2 a^{3} \sinh \left (a +b \sqrt {d x +c}\right )}{b^{2}}-\frac {6 a \left (\left (a +b \sqrt {d x +c}\right )^{2} \sinh \left (a +b \sqrt {d x +c}\right )-2 \left (a +b \sqrt {d x +c}\right ) \cosh \left (a +b \sqrt {d x +c}\right )+2 \sinh \left (a +b \sqrt {d x +c}\right )\right )}{b^{2}}+\frac {2 \left (\left (a +b \sqrt {d x +c}\right )^{3} \sinh \left (a +b \sqrt {d x +c}\right )-3 \left (a +b \sqrt {d x +c}\right )^{2} \cosh \left (a +b \sqrt {d x +c}\right )+6 \left (a +b \sqrt {d x +c}\right ) \sinh \left (a +b \sqrt {d x +c}\right )-6 \cosh \left (a +b \sqrt {d x +c}\right )\right )}{b^{2}}-2 c \left (\left (a +b \sqrt {d x +c}\right ) \sinh \left (a +b \sqrt {d x +c}\right )-\cosh \left (a +b \sqrt {d x +c}\right )\right )+2 c a \sinh \left (a +b \sqrt {d x +c}\right )}{d^{2} b^{2}}\) | \(303\) |
default | \(\frac {\frac {6 a^{2} \left (\left (a +b \sqrt {d x +c}\right ) \sinh \left (a +b \sqrt {d x +c}\right )-\cosh \left (a +b \sqrt {d x +c}\right )\right )}{b^{2}}-\frac {2 a^{3} \sinh \left (a +b \sqrt {d x +c}\right )}{b^{2}}-\frac {6 a \left (\left (a +b \sqrt {d x +c}\right )^{2} \sinh \left (a +b \sqrt {d x +c}\right )-2 \left (a +b \sqrt {d x +c}\right ) \cosh \left (a +b \sqrt {d x +c}\right )+2 \sinh \left (a +b \sqrt {d x +c}\right )\right )}{b^{2}}+\frac {2 \left (\left (a +b \sqrt {d x +c}\right )^{3} \sinh \left (a +b \sqrt {d x +c}\right )-3 \left (a +b \sqrt {d x +c}\right )^{2} \cosh \left (a +b \sqrt {d x +c}\right )+6 \left (a +b \sqrt {d x +c}\right ) \sinh \left (a +b \sqrt {d x +c}\right )-6 \cosh \left (a +b \sqrt {d x +c}\right )\right )}{b^{2}}-2 c \left (\left (a +b \sqrt {d x +c}\right ) \sinh \left (a +b \sqrt {d x +c}\right )-\cosh \left (a +b \sqrt {d x +c}\right )\right )+2 c a \sinh \left (a +b \sqrt {d x +c}\right )}{d^{2} b^{2}}\) | \(303\) |
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Time = 0.28 (sec) , antiderivative size = 68, normalized size of antiderivative = 0.41 \[ \int x \cosh \left (a+b \sqrt {c+d x}\right ) \, dx=\frac {2 \, {\left ({\left (b^{3} d x + 6 \, b\right )} \sqrt {d x + c} \sinh \left (\sqrt {d x + c} b + a\right ) - {\left (3 \, b^{2} d x + 2 \, b^{2} c + 6\right )} \cosh \left (\sqrt {d x + c} b + a\right )\right )}}{b^{4} d^{2}} \]
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Time = 0.25 (sec) , antiderivative size = 151, normalized size of antiderivative = 0.90 \[ \int x \cosh \left (a+b \sqrt {c+d x}\right ) \, dx=\begin {cases} \frac {x^{2} \cosh {\left (a \right )}}{2} & \text {for}\: b = 0 \wedge \left (b = 0 \vee d = 0\right ) \\\frac {x^{2} \cosh {\left (a + b \sqrt {c} \right )}}{2} & \text {for}\: d = 0 \\\frac {2 x \sqrt {c + d x} \sinh {\left (a + b \sqrt {c + d x} \right )}}{b d} - \frac {4 c \cosh {\left (a + b \sqrt {c + d x} \right )}}{b^{2} d^{2}} - \frac {6 x \cosh {\left (a + b \sqrt {c + d x} \right )}}{b^{2} d} + \frac {12 \sqrt {c + d x} \sinh {\left (a + b \sqrt {c + d x} \right )}}{b^{3} d^{2}} - \frac {12 \cosh {\left (a + b \sqrt {c + d x} \right )}}{b^{4} d^{2}} & \text {otherwise} \end {cases} \]
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Time = 0.20 (sec) , antiderivative size = 291, normalized size of antiderivative = 1.74 \[ \int x \cosh \left (a+b \sqrt {c+d x}\right ) \, dx=\frac {2 \, d^{2} x^{2} \cosh \left (\sqrt {d x + c} b + a\right ) - {\left (\frac {c^{2} e^{\left (\sqrt {d x + c} b + a\right )}}{b} + \frac {c^{2} e^{\left (-\sqrt {d x + c} b - a\right )}}{b} - \frac {2 \, {\left ({\left (d x + c\right )} b^{2} e^{a} - 2 \, \sqrt {d x + c} b e^{a} + 2 \, e^{a}\right )} c e^{\left (\sqrt {d x + c} b\right )}}{b^{3}} - \frac {2 \, {\left ({\left (d x + c\right )} b^{2} + 2 \, \sqrt {d x + c} b + 2\right )} c e^{\left (-\sqrt {d x + c} b - a\right )}}{b^{3}} + \frac {{\left ({\left (d x + c\right )}^{2} b^{4} e^{a} - 4 \, {\left (d x + c\right )}^{\frac {3}{2}} b^{3} e^{a} + 12 \, {\left (d x + c\right )} b^{2} e^{a} - 24 \, \sqrt {d x + c} b e^{a} + 24 \, e^{a}\right )} e^{\left (\sqrt {d x + c} b\right )}}{b^{5}} + \frac {{\left ({\left (d x + c\right )}^{2} b^{4} + 4 \, {\left (d x + c\right )}^{\frac {3}{2}} b^{3} + 12 \, {\left (d x + c\right )} b^{2} + 24 \, \sqrt {d x + c} b + 24\right )} e^{\left (-\sqrt {d x + c} b - a\right )}}{b^{5}}\right )} b}{4 \, d^{2}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 300 vs. \(2 (149) = 298\).
Time = 0.26 (sec) , antiderivative size = 300, normalized size of antiderivative = 1.80 \[ \int x \cosh \left (a+b \sqrt {c+d x}\right ) \, dx=-\frac {\frac {{\left ({\left (\sqrt {d x + c} b + a\right )} b^{2} c - a b^{2} c - {\left (\sqrt {d x + c} b + a\right )}^{3} + 3 \, {\left (\sqrt {d x + c} b + a\right )}^{2} a - 3 \, {\left (\sqrt {d x + c} b + a\right )} a^{2} + a^{3} - b^{2} c + 3 \, {\left (\sqrt {d x + c} b + a\right )}^{2} - 6 \, {\left (\sqrt {d x + c} b + a\right )} a + 3 \, a^{2} - 6 \, \sqrt {d x + c} b + 6\right )} e^{\left (\sqrt {d x + c} b + a\right )}}{b^{3} d} - \frac {{\left ({\left (\sqrt {d x + c} b + a\right )} b^{2} c - a b^{2} c - {\left (\sqrt {d x + c} b + a\right )}^{3} + 3 \, {\left (\sqrt {d x + c} b + a\right )}^{2} a - 3 \, {\left (\sqrt {d x + c} b + a\right )} a^{2} + a^{3} + b^{2} c - 3 \, {\left (\sqrt {d x + c} b + a\right )}^{2} + 6 \, {\left (\sqrt {d x + c} b + a\right )} a - 3 \, a^{2} - 6 \, \sqrt {d x + c} b - 6\right )} e^{\left (-\sqrt {d x + c} b - a\right )}}{b^{3} d}}{b d} \]
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Timed out. \[ \int x \cosh \left (a+b \sqrt {c+d x}\right ) \, dx=\int x\,\mathrm {cosh}\left (a+b\,\sqrt {c+d\,x}\right ) \,d x \]
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