Integrand size = 17, antiderivative size = 190 \[ \int \frac {\cosh (c+d x)}{x^3 (a+b x)} \, dx=-\frac {\cosh (c+d x)}{2 a x^2}+\frac {b \cosh (c+d x)}{a^2 x}+\frac {b^2 \cosh (c) \text {Chi}(d x)}{a^3}+\frac {d^2 \cosh (c) \text {Chi}(d x)}{2 a}-\frac {b^2 \cosh \left (c-\frac {a d}{b}\right ) \text {Chi}\left (\frac {a d}{b}+d x\right )}{a^3}-\frac {b d \text {Chi}(d x) \sinh (c)}{a^2}-\frac {d \sinh (c+d x)}{2 a x}-\frac {b d \cosh (c) \text {Shi}(d x)}{a^2}+\frac {b^2 \sinh (c) \text {Shi}(d x)}{a^3}+\frac {d^2 \sinh (c) \text {Shi}(d x)}{2 a}-\frac {b^2 \sinh \left (c-\frac {a d}{b}\right ) \text {Shi}\left (\frac {a d}{b}+d x\right )}{a^3} \]
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Time = 0.36 (sec) , antiderivative size = 190, normalized size of antiderivative = 1.00, number of steps used = 17, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.294, Rules used = {6874, 3378, 3384, 3379, 3382} \[ \int \frac {\cosh (c+d x)}{x^3 (a+b x)} \, dx=\frac {b^2 \cosh (c) \text {Chi}(d x)}{a^3}-\frac {b^2 \cosh \left (c-\frac {a d}{b}\right ) \text {Chi}\left (x d+\frac {a d}{b}\right )}{a^3}+\frac {b^2 \sinh (c) \text {Shi}(d x)}{a^3}-\frac {b^2 \sinh \left (c-\frac {a d}{b}\right ) \text {Shi}\left (x d+\frac {a d}{b}\right )}{a^3}-\frac {b d \sinh (c) \text {Chi}(d x)}{a^2}-\frac {b d \cosh (c) \text {Shi}(d x)}{a^2}+\frac {b \cosh (c+d x)}{a^2 x}+\frac {d^2 \cosh (c) \text {Chi}(d x)}{2 a}+\frac {d^2 \sinh (c) \text {Shi}(d x)}{2 a}-\frac {\cosh (c+d x)}{2 a x^2}-\frac {d \sinh (c+d x)}{2 a x} \]
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Rule 3378
Rule 3379
Rule 3382
Rule 3384
Rule 6874
Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {\cosh (c+d x)}{a x^3}-\frac {b \cosh (c+d x)}{a^2 x^2}+\frac {b^2 \cosh (c+d x)}{a^3 x}-\frac {b^3 \cosh (c+d x)}{a^3 (a+b x)}\right ) \, dx \\ & = \frac {\int \frac {\cosh (c+d x)}{x^3} \, dx}{a}-\frac {b \int \frac {\cosh (c+d x)}{x^2} \, dx}{a^2}+\frac {b^2 \int \frac {\cosh (c+d x)}{x} \, dx}{a^3}-\frac {b^3 \int \frac {\cosh (c+d x)}{a+b x} \, dx}{a^3} \\ & = -\frac {\cosh (c+d x)}{2 a x^2}+\frac {b \cosh (c+d x)}{a^2 x}+\frac {d \int \frac {\sinh (c+d x)}{x^2} \, dx}{2 a}-\frac {(b d) \int \frac {\sinh (c+d x)}{x} \, dx}{a^2}+\frac {\left (b^2 \cosh (c)\right ) \int \frac {\cosh (d x)}{x} \, dx}{a^3}-\frac {\left (b^3 \cosh \left (c-\frac {a d}{b}\right )\right ) \int \frac {\cosh \left (\frac {a d}{b}+d x\right )}{a+b x} \, dx}{a^3}+\frac {\left (b^2 \sinh (c)\right ) \int \frac {\sinh (d x)}{x} \, dx}{a^3}-\frac {\left (b^3 \sinh \left (c-\frac {a d}{b}\right )\right ) \int \frac {\sinh \left (\frac {a d}{b}+d x\right )}{a+b x} \, dx}{a^3} \\ & = -\frac {\cosh (c+d x)}{2 a x^2}+\frac {b \cosh (c+d x)}{a^2 x}+\frac {b^2 \cosh (c) \text {Chi}(d x)}{a^3}-\frac {b^2 \cosh \left (c-\frac {a d}{b}\right ) \text {Chi}\left (\frac {a d}{b}+d x\right )}{a^3}-\frac {d \sinh (c+d x)}{2 a x}+\frac {b^2 \sinh (c) \text {Shi}(d x)}{a^3}-\frac {b^2 \sinh \left (c-\frac {a d}{b}\right ) \text {Shi}\left (\frac {a d}{b}+d x\right )}{a^3}+\frac {d^2 \int \frac {\cosh (c+d x)}{x} \, dx}{2 a}-\frac {(b d \cosh (c)) \int \frac {\sinh (d x)}{x} \, dx}{a^2}-\frac {(b d \sinh (c)) \int \frac {\cosh (d x)}{x} \, dx}{a^2} \\ & = -\frac {\cosh (c+d x)}{2 a x^2}+\frac {b \cosh (c+d x)}{a^2 x}+\frac {b^2 \cosh (c) \text {Chi}(d x)}{a^3}-\frac {b^2 \cosh \left (c-\frac {a d}{b}\right ) \text {Chi}\left (\frac {a d}{b}+d x\right )}{a^3}-\frac {b d \text {Chi}(d x) \sinh (c)}{a^2}-\frac {d \sinh (c+d x)}{2 a x}-\frac {b d \cosh (c) \text {Shi}(d x)}{a^2}+\frac {b^2 \sinh (c) \text {Shi}(d x)}{a^3}-\frac {b^2 \sinh \left (c-\frac {a d}{b}\right ) \text {Shi}\left (\frac {a d}{b}+d x\right )}{a^3}+\frac {\left (d^2 \cosh (c)\right ) \int \frac {\cosh (d x)}{x} \, dx}{2 a}+\frac {\left (d^2 \sinh (c)\right ) \int \frac {\sinh (d x)}{x} \, dx}{2 a} \\ & = -\frac {\cosh (c+d x)}{2 a x^2}+\frac {b \cosh (c+d x)}{a^2 x}+\frac {b^2 \cosh (c) \text {Chi}(d x)}{a^3}+\frac {d^2 \cosh (c) \text {Chi}(d x)}{2 a}-\frac {b^2 \cosh \left (c-\frac {a d}{b}\right ) \text {Chi}\left (\frac {a d}{b}+d x\right )}{a^3}-\frac {b d \text {Chi}(d x) \sinh (c)}{a^2}-\frac {d \sinh (c+d x)}{2 a x}-\frac {b d \cosh (c) \text {Shi}(d x)}{a^2}+\frac {b^2 \sinh (c) \text {Shi}(d x)}{a^3}+\frac {d^2 \sinh (c) \text {Shi}(d x)}{2 a}-\frac {b^2 \sinh \left (c-\frac {a d}{b}\right ) \text {Shi}\left (\frac {a d}{b}+d x\right )}{a^3} \\ \end{align*}
Time = 0.31 (sec) , antiderivative size = 178, normalized size of antiderivative = 0.94 \[ \int \frac {\cosh (c+d x)}{x^3 (a+b x)} \, dx=\frac {-a^2 \cosh (c+d x)+2 a b x \cosh (c+d x)-2 b^2 x^2 \cosh \left (c-\frac {a d}{b}\right ) \text {Chi}\left (d \left (\frac {a}{b}+x\right )\right )+x^2 \text {Chi}(d x) \left (\left (2 b^2+a^2 d^2\right ) \cosh (c)-2 a b d \sinh (c)\right )-a^2 d x \sinh (c+d x)-2 a b d x^2 \cosh (c) \text {Shi}(d x)+2 b^2 x^2 \sinh (c) \text {Shi}(d x)+a^2 d^2 x^2 \sinh (c) \text {Shi}(d x)-2 b^2 x^2 \sinh \left (c-\frac {a d}{b}\right ) \text {Shi}\left (d \left (\frac {a}{b}+x\right )\right )}{2 a^3 x^2} \]
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Time = 0.26 (sec) , antiderivative size = 281, normalized size of antiderivative = 1.48
method | result | size |
risch | \(\frac {d \,{\mathrm e}^{-d x -c}}{4 a x}+\frac {{\mathrm e}^{-d x -c} b}{2 a^{2} x}-\frac {{\mathrm e}^{-d x -c}}{4 a \,x^{2}}-\frac {d^{2} {\mathrm e}^{-c} \operatorname {Ei}_{1}\left (d x \right )}{4 a}-\frac {d \,{\mathrm e}^{-c} \operatorname {Ei}_{1}\left (d x \right ) b}{2 a^{2}}-\frac {{\mathrm e}^{-c} \operatorname {Ei}_{1}\left (d x \right ) b^{2}}{2 a^{3}}+\frac {b^{2} {\mathrm e}^{\frac {d a -c b}{b}} \operatorname {Ei}_{1}\left (d x +c +\frac {d a -c b}{b}\right )}{2 a^{3}}+\frac {b \,{\mathrm e}^{d x +c}}{2 a^{2} x}+\frac {d b \,{\mathrm e}^{c} \operatorname {Ei}_{1}\left (-d x \right )}{2 a^{2}}-\frac {b^{2} {\mathrm e}^{c} \operatorname {Ei}_{1}\left (-d x \right )}{2 a^{3}}+\frac {b^{2} {\mathrm e}^{-\frac {d a -c b}{b}} \operatorname {Ei}_{1}\left (-d x -c -\frac {d a -c b}{b}\right )}{2 a^{3}}-\frac {{\mathrm e}^{d x +c}}{4 a \,x^{2}}-\frac {d \,{\mathrm e}^{d x +c}}{4 a x}-\frac {d^{2} {\mathrm e}^{c} \operatorname {Ei}_{1}\left (-d x \right )}{4 a}\) | \(281\) |
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Time = 0.26 (sec) , antiderivative size = 275, normalized size of antiderivative = 1.45 \[ \int \frac {\cosh (c+d x)}{x^3 (a+b x)} \, dx=-\frac {2 \, a^{2} d x \sinh \left (d x + c\right ) - 2 \, {\left (2 \, a b x - a^{2}\right )} \cosh \left (d x + c\right ) - {\left ({\left (a^{2} d^{2} - 2 \, a b d + 2 \, b^{2}\right )} x^{2} {\rm Ei}\left (d x\right ) + {\left (a^{2} d^{2} + 2 \, a b d + 2 \, b^{2}\right )} x^{2} {\rm Ei}\left (-d x\right )\right )} \cosh \left (c\right ) + 2 \, {\left (b^{2} x^{2} {\rm Ei}\left (\frac {b d x + a d}{b}\right ) + b^{2} x^{2} {\rm Ei}\left (-\frac {b d x + a d}{b}\right )\right )} \cosh \left (-\frac {b c - a d}{b}\right ) - {\left ({\left (a^{2} d^{2} - 2 \, a b d + 2 \, b^{2}\right )} x^{2} {\rm Ei}\left (d x\right ) - {\left (a^{2} d^{2} + 2 \, a b d + 2 \, b^{2}\right )} x^{2} {\rm Ei}\left (-d x\right )\right )} \sinh \left (c\right ) - 2 \, {\left (b^{2} x^{2} {\rm Ei}\left (\frac {b d x + a d}{b}\right ) - b^{2} x^{2} {\rm Ei}\left (-\frac {b d x + a d}{b}\right )\right )} \sinh \left (-\frac {b c - a d}{b}\right )}{4 \, a^{3} x^{2}} \]
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\[ \int \frac {\cosh (c+d x)}{x^3 (a+b x)} \, dx=\int \frac {\cosh {\left (c + d x \right )}}{x^{3} \left (a + b x\right )}\, dx \]
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Time = 0.32 (sec) , antiderivative size = 242, normalized size of antiderivative = 1.27 \[ \int \frac {\cosh (c+d x)}{x^3 (a+b x)} \, dx=\frac {1}{4} \, d {\left (\frac {d e^{\left (-c\right )} \Gamma \left (-1, d x\right ) + d e^{c} \Gamma \left (-1, -d x\right )}{a} + \frac {2 \, {\left ({\rm Ei}\left (-d x\right ) e^{\left (-c\right )} - {\rm Ei}\left (d x\right ) e^{c}\right )} b}{a^{2}} + \frac {2 \, b^{3} {\left (\frac {e^{\left (-c + \frac {a d}{b}\right )} E_{1}\left (\frac {{\left (b x + a\right )} d}{b}\right )}{b} + \frac {e^{\left (c - \frac {a d}{b}\right )} E_{1}\left (-\frac {{\left (b x + a\right )} d}{b}\right )}{b}\right )}}{a^{3} d} + \frac {4 \, b^{2} \cosh \left (d x + c\right ) \log \left (b x + a\right )}{a^{3} d} - \frac {4 \, b^{2} \cosh \left (d x + c\right ) \log \left (x\right )}{a^{3} d} + \frac {2 \, {\left ({\rm Ei}\left (-d x\right ) e^{\left (-c\right )} + {\rm Ei}\left (d x\right ) e^{c}\right )} b^{2}}{a^{3} d}\right )} - \frac {1}{2} \, {\left (\frac {2 \, b^{2} \log \left (b x + a\right )}{a^{3}} - \frac {2 \, b^{2} \log \left (x\right )}{a^{3}} - \frac {2 \, b x - a}{a^{2} x^{2}}\right )} \cosh \left (d x + c\right ) \]
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Time = 0.27 (sec) , antiderivative size = 248, normalized size of antiderivative = 1.31 \[ \int \frac {\cosh (c+d x)}{x^3 (a+b x)} \, dx=\frac {a^{2} d^{2} x^{2} {\rm Ei}\left (-d x\right ) e^{\left (-c\right )} + a^{2} d^{2} x^{2} {\rm Ei}\left (d x\right ) e^{c} + 2 \, a b d x^{2} {\rm Ei}\left (-d x\right ) e^{\left (-c\right )} - 2 \, a b d x^{2} {\rm Ei}\left (d x\right ) e^{c} + 2 \, b^{2} x^{2} {\rm Ei}\left (-d x\right ) e^{\left (-c\right )} - 2 \, b^{2} x^{2} {\rm Ei}\left (\frac {b d x + a d}{b}\right ) e^{\left (c - \frac {a d}{b}\right )} + 2 \, b^{2} x^{2} {\rm Ei}\left (d x\right ) e^{c} - 2 \, b^{2} x^{2} {\rm Ei}\left (-\frac {b d x + a d}{b}\right ) e^{\left (-c + \frac {a d}{b}\right )} - a^{2} d x e^{\left (d x + c\right )} + a^{2} d x e^{\left (-d x - c\right )} + 2 \, a b x e^{\left (d x + c\right )} + 2 \, a b x e^{\left (-d x - c\right )} - a^{2} e^{\left (d x + c\right )} - a^{2} e^{\left (-d x - c\right )}}{4 \, a^{3} x^{2}} \]
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Timed out. \[ \int \frac {\cosh (c+d x)}{x^3 (a+b x)} \, dx=\int \frac {\mathrm {cosh}\left (c+d\,x\right )}{x^3\,\left (a+b\,x\right )} \,d x \]
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