Integrand size = 17, antiderivative size = 219 \[ \int \frac {x^4 \cosh (c+d x)}{a+b x} \, dx=-\frac {6 \cosh (c+d x)}{b d^4}-\frac {a^2 \cosh (c+d x)}{b^3 d^2}+\frac {2 a x \cosh (c+d x)}{b^2 d^2}-\frac {3 x^2 \cosh (c+d x)}{b d^2}+\frac {a^4 \cosh \left (c-\frac {a d}{b}\right ) \text {Chi}\left (\frac {a d}{b}+d x\right )}{b^5}-\frac {2 a \sinh (c+d x)}{b^2 d^3}-\frac {a^3 \sinh (c+d x)}{b^4 d}+\frac {6 x \sinh (c+d x)}{b d^3}+\frac {a^2 x \sinh (c+d x)}{b^3 d}-\frac {a x^2 \sinh (c+d x)}{b^2 d}+\frac {x^3 \sinh (c+d x)}{b d}+\frac {a^4 \sinh \left (c-\frac {a d}{b}\right ) \text {Shi}\left (\frac {a d}{b}+d x\right )}{b^5} \]
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Time = 0.39 (sec) , antiderivative size = 219, normalized size of antiderivative = 1.00, number of steps used = 15, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.412, Rules used = {6874, 2717, 3377, 2718, 3384, 3379, 3382} \[ \int \frac {x^4 \cosh (c+d x)}{a+b x} \, dx=\frac {a^4 \cosh \left (c-\frac {a d}{b}\right ) \text {Chi}\left (x d+\frac {a d}{b}\right )}{b^5}+\frac {a^4 \sinh \left (c-\frac {a d}{b}\right ) \text {Shi}\left (x d+\frac {a d}{b}\right )}{b^5}-\frac {a^3 \sinh (c+d x)}{b^4 d}-\frac {a^2 \cosh (c+d x)}{b^3 d^2}+\frac {a^2 x \sinh (c+d x)}{b^3 d}-\frac {2 a \sinh (c+d x)}{b^2 d^3}+\frac {2 a x \cosh (c+d x)}{b^2 d^2}-\frac {a x^2 \sinh (c+d x)}{b^2 d}-\frac {6 \cosh (c+d x)}{b d^4}+\frac {6 x \sinh (c+d x)}{b d^3}-\frac {3 x^2 \cosh (c+d x)}{b d^2}+\frac {x^3 \sinh (c+d x)}{b d} \]
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Rule 2717
Rule 2718
Rule 3377
Rule 3379
Rule 3382
Rule 3384
Rule 6874
Rubi steps \begin{align*} \text {integral}& = \int \left (-\frac {a^3 \cosh (c+d x)}{b^4}+\frac {a^2 x \cosh (c+d x)}{b^3}-\frac {a x^2 \cosh (c+d x)}{b^2}+\frac {x^3 \cosh (c+d x)}{b}+\frac {a^4 \cosh (c+d x)}{b^4 (a+b x)}\right ) \, dx \\ & = -\frac {a^3 \int \cosh (c+d x) \, dx}{b^4}+\frac {a^4 \int \frac {\cosh (c+d x)}{a+b x} \, dx}{b^4}+\frac {a^2 \int x \cosh (c+d x) \, dx}{b^3}-\frac {a \int x^2 \cosh (c+d x) \, dx}{b^2}+\frac {\int x^3 \cosh (c+d x) \, dx}{b} \\ & = -\frac {a^3 \sinh (c+d x)}{b^4 d}+\frac {a^2 x \sinh (c+d x)}{b^3 d}-\frac {a x^2 \sinh (c+d x)}{b^2 d}+\frac {x^3 \sinh (c+d x)}{b d}-\frac {a^2 \int \sinh (c+d x) \, dx}{b^3 d}+\frac {(2 a) \int x \sinh (c+d x) \, dx}{b^2 d}-\frac {3 \int x^2 \sinh (c+d x) \, dx}{b d}+\frac {\left (a^4 \cosh \left (c-\frac {a d}{b}\right )\right ) \int \frac {\cosh \left (\frac {a d}{b}+d x\right )}{a+b x} \, dx}{b^4}+\frac {\left (a^4 \sinh \left (c-\frac {a d}{b}\right )\right ) \int \frac {\sinh \left (\frac {a d}{b}+d x\right )}{a+b x} \, dx}{b^4} \\ & = -\frac {a^2 \cosh (c+d x)}{b^3 d^2}+\frac {2 a x \cosh (c+d x)}{b^2 d^2}-\frac {3 x^2 \cosh (c+d x)}{b d^2}+\frac {a^4 \cosh \left (c-\frac {a d}{b}\right ) \text {Chi}\left (\frac {a d}{b}+d x\right )}{b^5}-\frac {a^3 \sinh (c+d x)}{b^4 d}+\frac {a^2 x \sinh (c+d x)}{b^3 d}-\frac {a x^2 \sinh (c+d x)}{b^2 d}+\frac {x^3 \sinh (c+d x)}{b d}+\frac {a^4 \sinh \left (c-\frac {a d}{b}\right ) \text {Shi}\left (\frac {a d}{b}+d x\right )}{b^5}-\frac {(2 a) \int \cosh (c+d x) \, dx}{b^2 d^2}+\frac {6 \int x \cosh (c+d x) \, dx}{b d^2} \\ & = -\frac {a^2 \cosh (c+d x)}{b^3 d^2}+\frac {2 a x \cosh (c+d x)}{b^2 d^2}-\frac {3 x^2 \cosh (c+d x)}{b d^2}+\frac {a^4 \cosh \left (c-\frac {a d}{b}\right ) \text {Chi}\left (\frac {a d}{b}+d x\right )}{b^5}-\frac {2 a \sinh (c+d x)}{b^2 d^3}-\frac {a^3 \sinh (c+d x)}{b^4 d}+\frac {6 x \sinh (c+d x)}{b d^3}+\frac {a^2 x \sinh (c+d x)}{b^3 d}-\frac {a x^2 \sinh (c+d x)}{b^2 d}+\frac {x^3 \sinh (c+d x)}{b d}+\frac {a^4 \sinh \left (c-\frac {a d}{b}\right ) \text {Shi}\left (\frac {a d}{b}+d x\right )}{b^5}-\frac {6 \int \sinh (c+d x) \, dx}{b d^3} \\ & = -\frac {6 \cosh (c+d x)}{b d^4}-\frac {a^2 \cosh (c+d x)}{b^3 d^2}+\frac {2 a x \cosh (c+d x)}{b^2 d^2}-\frac {3 x^2 \cosh (c+d x)}{b d^2}+\frac {a^4 \cosh \left (c-\frac {a d}{b}\right ) \text {Chi}\left (\frac {a d}{b}+d x\right )}{b^5}-\frac {2 a \sinh (c+d x)}{b^2 d^3}-\frac {a^3 \sinh (c+d x)}{b^4 d}+\frac {6 x \sinh (c+d x)}{b d^3}+\frac {a^2 x \sinh (c+d x)}{b^3 d}-\frac {a x^2 \sinh (c+d x)}{b^2 d}+\frac {x^3 \sinh (c+d x)}{b d}+\frac {a^4 \sinh \left (c-\frac {a d}{b}\right ) \text {Shi}\left (\frac {a d}{b}+d x\right )}{b^5} \\ \end{align*}
Time = 0.40 (sec) , antiderivative size = 159, normalized size of antiderivative = 0.73 \[ \int \frac {x^4 \cosh (c+d x)}{a+b x} \, dx=\frac {a^4 d^4 \cosh \left (c-\frac {a d}{b}\right ) \text {Chi}\left (d \left (\frac {a}{b}+x\right )\right )-b \left (b \left (a^2 d^2-2 a b d^2 x+3 b^2 \left (2+d^2 x^2\right )\right ) \cosh (c+d x)+d \left (a^3 d^2-a^2 b d^2 x+a b^2 \left (2+d^2 x^2\right )-b^3 x \left (6+d^2 x^2\right )\right ) \sinh (c+d x)\right )+a^4 d^4 \sinh \left (c-\frac {a d}{b}\right ) \text {Shi}\left (d \left (\frac {a}{b}+x\right )\right )}{b^5 d^4} \]
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Time = 0.31 (sec) , antiderivative size = 442, normalized size of antiderivative = 2.02
method | result | size |
risch | \(\frac {{\mathrm e}^{d x +c} x^{3}}{2 d b}-\frac {{\mathrm e}^{-d x -c} x^{3}}{2 d b}-\frac {{\mathrm e}^{d x +c} a \,x^{2}}{2 d \,b^{2}}-\frac {{\mathrm e}^{-\frac {d a -c b}{b}} \operatorname {Ei}_{1}\left (-d x -c -\frac {d a -c b}{b}\right ) a^{4}}{2 b^{5}}+\frac {{\mathrm e}^{-d x -c} a \,x^{2}}{2 d \,b^{2}}-\frac {{\mathrm e}^{\frac {d a -c b}{b}} \operatorname {Ei}_{1}\left (d x +c +\frac {d a -c b}{b}\right ) a^{4}}{2 b^{5}}+\frac {{\mathrm e}^{d x +c} a^{2} x}{2 d \,b^{3}}-\frac {3 \,{\mathrm e}^{d x +c} x^{2}}{2 d^{2} b}-\frac {{\mathrm e}^{-d x -c} a^{2} x}{2 d \,b^{3}}-\frac {3 \,{\mathrm e}^{-d x -c} x^{2}}{2 d^{2} b}-\frac {a^{3} {\mathrm e}^{d x +c}}{2 d \,b^{4}}+\frac {{\mathrm e}^{d x +c} a x}{d^{2} b^{2}}+\frac {{\mathrm e}^{-d x -c} a^{3}}{2 d \,b^{4}}+\frac {{\mathrm e}^{-d x -c} a x}{d^{2} b^{2}}-\frac {a^{2} {\mathrm e}^{d x +c}}{2 d^{2} b^{3}}+\frac {3 \,{\mathrm e}^{d x +c} x}{d^{3} b}-\frac {{\mathrm e}^{-d x -c} a^{2}}{2 d^{2} b^{3}}-\frac {3 \,{\mathrm e}^{-d x -c} x}{d^{3} b}-\frac {a \,{\mathrm e}^{d x +c}}{d^{3} b^{2}}+\frac {{\mathrm e}^{-d x -c} a}{d^{3} b^{2}}-\frac {3 \,{\mathrm e}^{d x +c}}{d^{4} b}-\frac {3 \,{\mathrm e}^{-d x -c}}{d^{4} b}\) | \(442\) |
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Time = 0.25 (sec) , antiderivative size = 236, normalized size of antiderivative = 1.08 \[ \int \frac {x^4 \cosh (c+d x)}{a+b x} \, dx=-\frac {2 \, {\left (3 \, b^{4} d^{2} x^{2} - 2 \, a b^{3} d^{2} x + a^{2} b^{2} d^{2} + 6 \, b^{4}\right )} \cosh \left (d x + c\right ) - {\left (a^{4} d^{4} {\rm Ei}\left (\frac {b d x + a d}{b}\right ) + a^{4} d^{4} {\rm Ei}\left (-\frac {b d x + a d}{b}\right )\right )} \cosh \left (-\frac {b c - a d}{b}\right ) - 2 \, {\left (b^{4} d^{3} x^{3} - a b^{3} d^{3} x^{2} - a^{3} b d^{3} - 2 \, a b^{3} d + {\left (a^{2} b^{2} d^{3} + 6 \, b^{4} d\right )} x\right )} \sinh \left (d x + c\right ) + {\left (a^{4} d^{4} {\rm Ei}\left (\frac {b d x + a d}{b}\right ) - a^{4} d^{4} {\rm Ei}\left (-\frac {b d x + a d}{b}\right )\right )} \sinh \left (-\frac {b c - a d}{b}\right )}{2 \, b^{5} d^{4}} \]
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\[ \int \frac {x^4 \cosh (c+d x)}{a+b x} \, dx=\int \frac {x^{4} \cosh {\left (c + d x \right )}}{a + b x}\, dx \]
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Time = 0.26 (sec) , antiderivative size = 437, normalized size of antiderivative = 2.00 \[ \int \frac {x^4 \cosh (c+d x)}{a+b x} \, dx=-\frac {1}{24} \, d {\left (\frac {12 \, a^{4} {\left (\frac {e^{\left (-c + \frac {a d}{b}\right )} E_{1}\left (\frac {{\left (b x + a\right )} d}{b}\right )}{b} + \frac {e^{\left (c - \frac {a d}{b}\right )} E_{1}\left (-\frac {{\left (b x + a\right )} d}{b}\right )}{b}\right )}}{b^{4} d} - \frac {12 \, a^{3} {\left (\frac {{\left (d x e^{c} - e^{c}\right )} e^{\left (d x\right )}}{d^{2}} + \frac {{\left (d x + 1\right )} e^{\left (-d x - c\right )}}{d^{2}}\right )}}{b^{4}} + \frac {6 \, a^{2} {\left (\frac {{\left (d^{2} x^{2} e^{c} - 2 \, d x e^{c} + 2 \, e^{c}\right )} e^{\left (d x\right )}}{d^{3}} + \frac {{\left (d^{2} x^{2} + 2 \, d x + 2\right )} e^{\left (-d x - c\right )}}{d^{3}}\right )}}{b^{3}} - \frac {4 \, a {\left (\frac {{\left (d^{3} x^{3} e^{c} - 3 \, d^{2} x^{2} e^{c} + 6 \, d x e^{c} - 6 \, e^{c}\right )} e^{\left (d x\right )}}{d^{4}} + \frac {{\left (d^{3} x^{3} + 3 \, d^{2} x^{2} + 6 \, d x + 6\right )} e^{\left (-d x - c\right )}}{d^{4}}\right )}}{b^{2}} + \frac {3 \, {\left (\frac {{\left (d^{4} x^{4} e^{c} - 4 \, d^{3} x^{3} e^{c} + 12 \, d^{2} x^{2} e^{c} - 24 \, d x e^{c} + 24 \, e^{c}\right )} e^{\left (d x\right )}}{d^{5}} + \frac {{\left (d^{4} x^{4} + 4 \, d^{3} x^{3} + 12 \, d^{2} x^{2} + 24 \, d x + 24\right )} e^{\left (-d x - c\right )}}{d^{5}}\right )}}{b} + \frac {24 \, a^{4} \cosh \left (d x + c\right ) \log \left (b x + a\right )}{b^{5} d}\right )} + \frac {1}{12} \, {\left (\frac {12 \, a^{4} \log \left (b x + a\right )}{b^{5}} + \frac {3 \, b^{3} x^{4} - 4 \, a b^{2} x^{3} + 6 \, a^{2} b x^{2} - 12 \, a^{3} x}{b^{4}}\right )} \cosh \left (d x + c\right ) \]
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Time = 0.27 (sec) , antiderivative size = 407, normalized size of antiderivative = 1.86 \[ \int \frac {x^4 \cosh (c+d x)}{a+b x} \, dx=\frac {b^{4} d^{3} x^{3} e^{\left (d x + c\right )} - b^{4} d^{3} x^{3} e^{\left (-d x - c\right )} - a b^{3} d^{3} x^{2} e^{\left (d x + c\right )} + a b^{3} d^{3} x^{2} e^{\left (-d x - c\right )} + a^{4} d^{4} {\rm Ei}\left (\frac {b d x + a d}{b}\right ) e^{\left (c - \frac {a d}{b}\right )} + a^{4} d^{4} {\rm Ei}\left (-\frac {b d x + a d}{b}\right ) e^{\left (-c + \frac {a d}{b}\right )} + a^{2} b^{2} d^{3} x e^{\left (d x + c\right )} - 3 \, b^{4} d^{2} x^{2} e^{\left (d x + c\right )} - a^{2} b^{2} d^{3} x e^{\left (-d x - c\right )} - 3 \, b^{4} d^{2} x^{2} e^{\left (-d x - c\right )} - a^{3} b d^{3} e^{\left (d x + c\right )} + 2 \, a b^{3} d^{2} x e^{\left (d x + c\right )} + a^{3} b d^{3} e^{\left (-d x - c\right )} + 2 \, a b^{3} d^{2} x e^{\left (-d x - c\right )} - a^{2} b^{2} d^{2} e^{\left (d x + c\right )} + 6 \, b^{4} d x e^{\left (d x + c\right )} - a^{2} b^{2} d^{2} e^{\left (-d x - c\right )} - 6 \, b^{4} d x e^{\left (-d x - c\right )} - 2 \, a b^{3} d e^{\left (d x + c\right )} + 2 \, a b^{3} d e^{\left (-d x - c\right )} - 6 \, b^{4} e^{\left (d x + c\right )} - 6 \, b^{4} e^{\left (-d x - c\right )}}{2 \, b^{5} d^{4}} \]
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Timed out. \[ \int \frac {x^4 \cosh (c+d x)}{a+b x} \, dx=\int \frac {x^4\,\mathrm {cosh}\left (c+d\,x\right )}{a+b\,x} \,d x \]
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