Integrand size = 19, antiderivative size = 273 \[ \int \frac {x^4 \cosh (c+d x)}{a+b x^2} \, dx=-\frac {2 x \cosh (c+d x)}{b d^2}+\frac {(-a)^{3/2} \cosh \left (c+\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Chi}\left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right )}{2 b^{5/2}}-\frac {(-a)^{3/2} \cosh \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Chi}\left (\frac {\sqrt {-a} d}{\sqrt {b}}+d x\right )}{2 b^{5/2}}+\frac {2 \sinh (c+d x)}{b d^3}-\frac {a \sinh (c+d x)}{b^2 d}+\frac {x^2 \sinh (c+d x)}{b d}-\frac {(-a)^{3/2} \sinh \left (c+\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Shi}\left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right )}{2 b^{5/2}}-\frac {(-a)^{3/2} \sinh \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Shi}\left (\frac {\sqrt {-a} d}{\sqrt {b}}+d x\right )}{2 b^{5/2}} \]
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Time = 0.57 (sec) , antiderivative size = 273, normalized size of antiderivative = 1.00, number of steps used = 14, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.368, Rules used = {5401, 2717, 3377, 5389, 3384, 3379, 3382} \[ \int \frac {x^4 \cosh (c+d x)}{a+b x^2} \, dx=\frac {(-a)^{3/2} \cosh \left (\frac {\sqrt {-a} d}{\sqrt {b}}+c\right ) \text {Chi}\left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right )}{2 b^{5/2}}-\frac {(-a)^{3/2} \cosh \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Chi}\left (x d+\frac {\sqrt {-a} d}{\sqrt {b}}\right )}{2 b^{5/2}}-\frac {(-a)^{3/2} \sinh \left (\frac {\sqrt {-a} d}{\sqrt {b}}+c\right ) \text {Shi}\left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right )}{2 b^{5/2}}-\frac {(-a)^{3/2} \sinh \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Shi}\left (x d+\frac {\sqrt {-a} d}{\sqrt {b}}\right )}{2 b^{5/2}}-\frac {a \sinh (c+d x)}{b^2 d}+\frac {2 \sinh (c+d x)}{b d^3}-\frac {2 x \cosh (c+d x)}{b d^2}+\frac {x^2 \sinh (c+d x)}{b d} \]
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Rule 2717
Rule 3377
Rule 3379
Rule 3382
Rule 3384
Rule 5389
Rule 5401
Rubi steps \begin{align*} \text {integral}& = \int \left (-\frac {a \cosh (c+d x)}{b^2}+\frac {x^2 \cosh (c+d x)}{b}+\frac {a^2 \cosh (c+d x)}{b^2 \left (a+b x^2\right )}\right ) \, dx \\ & = -\frac {a \int \cosh (c+d x) \, dx}{b^2}+\frac {a^2 \int \frac {\cosh (c+d x)}{a+b x^2} \, dx}{b^2}+\frac {\int x^2 \cosh (c+d x) \, dx}{b} \\ & = -\frac {a \sinh (c+d x)}{b^2 d}+\frac {x^2 \sinh (c+d x)}{b d}+\frac {a^2 \int \left (\frac {\sqrt {-a} \cosh (c+d x)}{2 a \left (\sqrt {-a}-\sqrt {b} x\right )}+\frac {\sqrt {-a} \cosh (c+d x)}{2 a \left (\sqrt {-a}+\sqrt {b} x\right )}\right ) \, dx}{b^2}-\frac {2 \int x \sinh (c+d x) \, dx}{b d} \\ & = -\frac {2 x \cosh (c+d x)}{b d^2}-\frac {a \sinh (c+d x)}{b^2 d}+\frac {x^2 \sinh (c+d x)}{b d}-\frac {(-a)^{3/2} \int \frac {\cosh (c+d x)}{\sqrt {-a}-\sqrt {b} x} \, dx}{2 b^2}-\frac {(-a)^{3/2} \int \frac {\cosh (c+d x)}{\sqrt {-a}+\sqrt {b} x} \, dx}{2 b^2}+\frac {2 \int \cosh (c+d x) \, dx}{b d^2} \\ & = -\frac {2 x \cosh (c+d x)}{b d^2}+\frac {2 \sinh (c+d x)}{b d^3}-\frac {a \sinh (c+d x)}{b^2 d}+\frac {x^2 \sinh (c+d x)}{b d}-\frac {\left ((-a)^{3/2} \cosh \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right )\right ) \int \frac {\cosh \left (\frac {\sqrt {-a} d}{\sqrt {b}}+d x\right )}{\sqrt {-a}+\sqrt {b} x} \, dx}{2 b^2}-\frac {\left ((-a)^{3/2} \cosh \left (c+\frac {\sqrt {-a} d}{\sqrt {b}}\right )\right ) \int \frac {\cosh \left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right )}{\sqrt {-a}-\sqrt {b} x} \, dx}{2 b^2}-\frac {\left ((-a)^{3/2} \sinh \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right )\right ) \int \frac {\sinh \left (\frac {\sqrt {-a} d}{\sqrt {b}}+d x\right )}{\sqrt {-a}+\sqrt {b} x} \, dx}{2 b^2}+\frac {\left ((-a)^{3/2} \sinh \left (c+\frac {\sqrt {-a} d}{\sqrt {b}}\right )\right ) \int \frac {\sinh \left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right )}{\sqrt {-a}-\sqrt {b} x} \, dx}{2 b^2} \\ & = -\frac {2 x \cosh (c+d x)}{b d^2}+\frac {(-a)^{3/2} \cosh \left (c+\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Chi}\left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right )}{2 b^{5/2}}-\frac {(-a)^{3/2} \cosh \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Chi}\left (\frac {\sqrt {-a} d}{\sqrt {b}}+d x\right )}{2 b^{5/2}}+\frac {2 \sinh (c+d x)}{b d^3}-\frac {a \sinh (c+d x)}{b^2 d}+\frac {x^2 \sinh (c+d x)}{b d}-\frac {(-a)^{3/2} \sinh \left (c+\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Shi}\left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right )}{2 b^{5/2}}-\frac {(-a)^{3/2} \sinh \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Shi}\left (\frac {\sqrt {-a} d}{\sqrt {b}}+d x\right )}{2 b^{5/2}} \\ \end{align*}
Result contains complex when optimal does not.
Time = 1.08 (sec) , antiderivative size = 276, normalized size of antiderivative = 1.01 \[ \int \frac {x^4 \cosh (c+d x)}{a+b x^2} \, dx=\frac {-i a^{3/2} e^{c-\frac {i \sqrt {a} d}{\sqrt {b}}} \left (e^{\frac {2 i \sqrt {a} d}{\sqrt {b}}} \operatorname {ExpIntegralEi}\left (d \left (-\frac {i \sqrt {a}}{\sqrt {b}}+x\right )\right )-\operatorname {ExpIntegralEi}\left (d \left (\frac {i \sqrt {a}}{\sqrt {b}}+x\right )\right )\right )+i a^{3/2} e^{-c-\frac {i \sqrt {a} d}{\sqrt {b}}} \left (e^{\frac {2 i \sqrt {a} d}{\sqrt {b}}} \operatorname {ExpIntegralEi}\left (-\frac {i \sqrt {a} d}{\sqrt {b}}-d x\right )-\operatorname {ExpIntegralEi}\left (\frac {i \sqrt {a} d}{\sqrt {b}}-d x\right )\right )+\frac {4 \sqrt {b} \cosh (d x) \left (-2 b d x \cosh (c)+\left (-a d^2+b \left (2+d^2 x^2\right )\right ) \sinh (c)\right )}{d^3}+\frac {4 \sqrt {b} \left (\left (-a d^2+b \left (2+d^2 x^2\right )\right ) \cosh (c)-2 b d x \sinh (c)\right ) \sinh (d x)}{d^3}}{4 b^{5/2}} \]
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Time = 0.33 (sec) , antiderivative size = 369, normalized size of antiderivative = 1.35
method | result | size |
risch | \(-\frac {{\mathrm e}^{\frac {d \sqrt {-a b}+c b}{b}} \operatorname {Ei}_{1}\left (\frac {d \sqrt {-a b}-\left (d x +c \right ) b +c b}{b}\right ) a^{2}}{4 b^{2} \sqrt {-a b}}-\frac {{\mathrm e}^{-\frac {d \sqrt {-a b}+c b}{b}} \operatorname {Ei}_{1}\left (-\frac {d \sqrt {-a b}-\left (d x +c \right ) b +c b}{b}\right ) a^{2}}{4 b^{2} \sqrt {-a b}}+\frac {{\mathrm e}^{d x +c} x^{2}}{2 d b}-\frac {{\mathrm e}^{-d x -c} x^{2}}{2 d b}+\frac {{\mathrm e}^{\frac {-d \sqrt {-a b}+c b}{b}} \operatorname {Ei}_{1}\left (-\frac {d \sqrt {-a b}+\left (d x +c \right ) b -c b}{b}\right ) a^{2}}{4 b^{2} \sqrt {-a b}}+\frac {{\mathrm e}^{-\frac {-d \sqrt {-a b}+c b}{b}} \operatorname {Ei}_{1}\left (\frac {d \sqrt {-a b}+\left (d x +c \right ) b -c b}{b}\right ) a^{2}}{4 b^{2} \sqrt {-a b}}-\frac {a \,{\mathrm e}^{d x +c}}{2 d \,b^{2}}-\frac {{\mathrm e}^{d x +c} x}{d^{2} b}+\frac {{\mathrm e}^{-d x -c} a}{2 d \,b^{2}}-\frac {{\mathrm e}^{-d x -c} x}{d^{2} b}+\frac {{\mathrm e}^{d x +c}}{d^{3} b}-\frac {{\mathrm e}^{-d x -c}}{d^{3} b}\) | \(369\) |
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Leaf count of result is larger than twice the leaf count of optimal. 605 vs. \(2 (217) = 434\).
Time = 0.27 (sec) , antiderivative size = 605, normalized size of antiderivative = 2.22 \[ \int \frac {x^4 \cosh (c+d x)}{a+b x^2} \, dx=-\frac {8 \, b d x \cosh \left (d x + c\right ) + {\left ({\left (a d^{2} \cosh \left (d x + c\right )^{2} - a d^{2} \sinh \left (d x + c\right )^{2}\right )} \sqrt {-\frac {a d^{2}}{b}} {\rm Ei}\left (d x - \sqrt {-\frac {a d^{2}}{b}}\right ) + {\left (a d^{2} \cosh \left (d x + c\right )^{2} - a d^{2} \sinh \left (d x + c\right )^{2}\right )} \sqrt {-\frac {a d^{2}}{b}} {\rm Ei}\left (-d x + \sqrt {-\frac {a d^{2}}{b}}\right )\right )} \cosh \left (c + \sqrt {-\frac {a d^{2}}{b}}\right ) - {\left ({\left (a d^{2} \cosh \left (d x + c\right )^{2} - a d^{2} \sinh \left (d x + c\right )^{2}\right )} \sqrt {-\frac {a d^{2}}{b}} {\rm Ei}\left (d x + \sqrt {-\frac {a d^{2}}{b}}\right ) + {\left (a d^{2} \cosh \left (d x + c\right )^{2} - a d^{2} \sinh \left (d x + c\right )^{2}\right )} \sqrt {-\frac {a d^{2}}{b}} {\rm Ei}\left (-d x - \sqrt {-\frac {a d^{2}}{b}}\right )\right )} \cosh \left (-c + \sqrt {-\frac {a d^{2}}{b}}\right ) - 4 \, {\left (b d^{2} x^{2} - a d^{2} + 2 \, b\right )} \sinh \left (d x + c\right ) + {\left ({\left (a d^{2} \cosh \left (d x + c\right )^{2} - a d^{2} \sinh \left (d x + c\right )^{2}\right )} \sqrt {-\frac {a d^{2}}{b}} {\rm Ei}\left (d x - \sqrt {-\frac {a d^{2}}{b}}\right ) - {\left (a d^{2} \cosh \left (d x + c\right )^{2} - a d^{2} \sinh \left (d x + c\right )^{2}\right )} \sqrt {-\frac {a d^{2}}{b}} {\rm Ei}\left (-d x + \sqrt {-\frac {a d^{2}}{b}}\right )\right )} \sinh \left (c + \sqrt {-\frac {a d^{2}}{b}}\right ) + {\left ({\left (a d^{2} \cosh \left (d x + c\right )^{2} - a d^{2} \sinh \left (d x + c\right )^{2}\right )} \sqrt {-\frac {a d^{2}}{b}} {\rm Ei}\left (d x + \sqrt {-\frac {a d^{2}}{b}}\right ) - {\left (a d^{2} \cosh \left (d x + c\right )^{2} - a d^{2} \sinh \left (d x + c\right )^{2}\right )} \sqrt {-\frac {a d^{2}}{b}} {\rm Ei}\left (-d x - \sqrt {-\frac {a d^{2}}{b}}\right )\right )} \sinh \left (-c + \sqrt {-\frac {a d^{2}}{b}}\right )}{4 \, {\left (b^{2} d^{3} \cosh \left (d x + c\right )^{2} - b^{2} d^{3} \sinh \left (d x + c\right )^{2}\right )}} \]
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\[ \int \frac {x^4 \cosh (c+d x)}{a+b x^2} \, dx=\int \frac {x^{4} \cosh {\left (c + d x \right )}}{a + b x^{2}}\, dx \]
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\[ \int \frac {x^4 \cosh (c+d x)}{a+b x^2} \, dx=\int { \frac {x^{4} \cosh \left (d x + c\right )}{b x^{2} + a} \,d x } \]
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\[ \int \frac {x^4 \cosh (c+d x)}{a+b x^2} \, dx=\int { \frac {x^{4} \cosh \left (d x + c\right )}{b x^{2} + a} \,d x } \]
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Timed out. \[ \int \frac {x^4 \cosh (c+d x)}{a+b x^2} \, dx=\int \frac {x^4\,\mathrm {cosh}\left (c+d\,x\right )}{b\,x^2+a} \,d x \]
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