Optimal. Leaf size=273 \[ \frac {(-a)^{3/2} \cosh \left (\frac {\sqrt {-a} d}{\sqrt {b}}+c\right ) \text {Chi}\left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right )}{2 b^{5/2}}-\frac {(-a)^{3/2} \cosh \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Chi}\left (x d+\frac {\sqrt {-a} d}{\sqrt {b}}\right )}{2 b^{5/2}}-\frac {(-a)^{3/2} \sinh \left (\frac {\sqrt {-a} d}{\sqrt {b}}+c\right ) \text {Shi}\left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right )}{2 b^{5/2}}-\frac {(-a)^{3/2} \sinh \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Shi}\left (x d+\frac {\sqrt {-a} d}{\sqrt {b}}\right )}{2 b^{5/2}}-\frac {a \sinh (c+d x)}{b^2 d}+\frac {2 \sinh (c+d x)}{b d^3}-\frac {2 x \cosh (c+d x)}{b d^2}+\frac {x^2 \sinh (c+d x)}{b d} \]
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Rubi [A] time = 0.73, antiderivative size = 273, normalized size of antiderivative = 1.00, number of steps used = 14, number of rules used = 7, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.368, Rules used = {5293, 2637, 3296, 5281, 3303, 3298, 3301} \[ \frac {(-a)^{3/2} \cosh \left (\frac {\sqrt {-a} d}{\sqrt {b}}+c\right ) \text {Chi}\left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right )}{2 b^{5/2}}-\frac {(-a)^{3/2} \cosh \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Chi}\left (x d+\frac {\sqrt {-a} d}{\sqrt {b}}\right )}{2 b^{5/2}}-\frac {(-a)^{3/2} \sinh \left (\frac {\sqrt {-a} d}{\sqrt {b}}+c\right ) \text {Shi}\left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right )}{2 b^{5/2}}-\frac {(-a)^{3/2} \sinh \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Shi}\left (x d+\frac {\sqrt {-a} d}{\sqrt {b}}\right )}{2 b^{5/2}}-\frac {a \sinh (c+d x)}{b^2 d}+\frac {2 \sinh (c+d x)}{b d^3}-\frac {2 x \cosh (c+d x)}{b d^2}+\frac {x^2 \sinh (c+d x)}{b d} \]
Antiderivative was successfully verified.
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Rule 2637
Rule 3296
Rule 3298
Rule 3301
Rule 3303
Rule 5281
Rule 5293
Rubi steps
\begin {align*} \int \frac {x^4 \cosh (c+d x)}{a+b x^2} \, dx &=\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*}
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Mathematica [C] time = 0.45, size = 274, normalized size = 1.00 \[ \frac {i a^{3/2} d^3 \cosh \left (c-\frac {i \sqrt {a} d}{\sqrt {b}}\right ) \text {Ci}\left (i d x-\frac {\sqrt {a} d}{\sqrt {b}}\right )-i a^{3/2} d^3 \cosh \left (c+\frac {i \sqrt {a} d}{\sqrt {b}}\right ) \text {Ci}\left (i x d+\frac {\sqrt {a} d}{\sqrt {b}}\right )-a^{3/2} d^3 \sinh \left (c-\frac {i \sqrt {a} d}{\sqrt {b}}\right ) \text {Si}\left (\frac {\sqrt {a} d}{\sqrt {b}}-i d x\right )-a^{3/2} d^3 \sinh \left (c+\frac {i \sqrt {a} d}{\sqrt {b}}\right ) \text {Si}\left (i x d+\frac {\sqrt {a} d}{\sqrt {b}}\right )-2 a \sqrt {b} d^2 \sinh (c+d x)+2 b^{3/2} d^2 x^2 \sinh (c+d x)+4 b^{3/2} \sinh (c+d x)-4 b^{3/2} d x \cosh (c+d x)}{2 b^{5/2} d^3} \]
Warning: Unable to verify antiderivative.
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fricas [B] time = 0.80, size = 605, normalized size = 2.22 \[ -\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 )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x^{4} \cosh \left (d x + c\right )}{b x^{2} + a}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.29, size = 369, normalized size = 1.35 \[ \frac {{\mathrm e}^{-\frac {-d \sqrt {-a b}+c b}{b}} \Ei \left (1, \frac {d \sqrt {-a b}+\left (d x +c \right ) b -c b}{b}\right ) a^{2}}{4 b^{2} \sqrt {-a b}}-\frac {{\mathrm e}^{-\frac {d \sqrt {-a b}+c b}{b}} \Ei \left (1, -\frac {d \sqrt {-a b}-\left (d x +c \right ) b +c b}{b}\right ) a^{2}}{4 b^{2} \sqrt {-a b}}-\frac {{\mathrm e}^{-d x -c} x^{2}}{2 d b}+\frac {{\mathrm e}^{-d x -c} a}{2 d \,b^{2}}-\frac {{\mathrm e}^{-d x -c} x}{d^{2} b}-\frac {{\mathrm e}^{-d x -c}}{d^{3} b}-\frac {{\mathrm e}^{\frac {d \sqrt {-a b}+c b}{b}} \Ei \left (1, \frac {d \sqrt {-a b}-\left (d x +c \right ) b +c b}{b}\right ) a^{2}}{4 b^{2} \sqrt {-a b}}+\frac {{\mathrm e}^{\frac {-d \sqrt {-a b}+c b}{b}} \Ei \left (1, -\frac {d \sqrt {-a b}+\left (d x +c \right ) b -c b}{b}\right ) a^{2}}{4 b^{2} \sqrt {-a b}}+\frac {{\mathrm e}^{d x +c} x^{2}}{2 d b}-\frac {a \,{\mathrm e}^{d x +c}}{2 d \,b^{2}}-\frac {{\mathrm e}^{d x +c} x}{d^{2} b}+\frac {{\mathrm e}^{d x +c}}{d^{3} b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \frac {{\left (b d^{2} x^{4} e^{\left (2 \, c\right )} - 2 \, b d x^{3} e^{\left (2 \, c\right )} - 2 \, a d x e^{\left (2 \, c\right )} + 2 \, b x^{2} e^{\left (2 \, c\right )}\right )} e^{\left (d x\right )} - {\left (b d^{2} x^{4} + 2 \, b d x^{3} + 2 \, a d x + 2 \, b x^{2}\right )} e^{\left (-d x\right )}}{2 \, {\left (b^{2} d^{3} x^{2} e^{c} + a b d^{3} e^{c}\right )}} + \frac {1}{2} \, \int \frac {2 \, {\left (a b d x^{2} e^{c} + a^{2} d e^{c} + {\left (a^{2} d^{2} e^{c} - 2 \, a b e^{c}\right )} x\right )} e^{\left (d x\right )}}{b^{3} d^{3} x^{4} + 2 \, a b^{2} d^{3} x^{2} + a^{2} b d^{3}}\,{d x} + \frac {1}{2} \, \int \frac {2 \, {\left (a b d x^{2} + a^{2} d - {\left (a^{2} d^{2} - 2 \, a b\right )} x\right )} e^{\left (-d x\right )}}{b^{3} d^{3} x^{4} e^{c} + 2 \, a b^{2} d^{3} x^{2} e^{c} + a^{2} b d^{3} e^{c}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {x^4\,\mathrm {cosh}\left (c+d\,x\right )}{b\,x^2+a} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x^{4} \cosh {\left (c + d x \right )}}{a + b x^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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