Optimal. Leaf size=416 \[ \frac {\cosh \left (\frac {\sqrt {-a} d}{\sqrt {b}}+c\right ) \text {Chi}\left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right )}{4 \sqrt {-a} b^{3/2}}-\frac {\cosh \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Chi}\left (x d+\frac {\sqrt {-a} d}{\sqrt {b}}\right )}{4 \sqrt {-a} b^{3/2}}-\frac {\sinh \left (\frac {\sqrt {-a} d}{\sqrt {b}}+c\right ) \text {Shi}\left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right )}{4 \sqrt {-a} b^{3/2}}-\frac {\sinh \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Shi}\left (x d+\frac {\sqrt {-a} d}{\sqrt {b}}\right )}{4 \sqrt {-a} b^{3/2}}+\frac {d \sinh \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Chi}\left (x d+\frac {\sqrt {-a} d}{\sqrt {b}}\right )}{4 b^2}+\frac {d \sinh \left (\frac {\sqrt {-a} d}{\sqrt {b}}+c\right ) \text {Chi}\left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right )}{4 b^2}-\frac {d \cosh \left (\frac {\sqrt {-a} d}{\sqrt {b}}+c\right ) \text {Shi}\left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right )}{4 b^2}+\frac {d \cosh \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Shi}\left (x d+\frac {\sqrt {-a} d}{\sqrt {b}}\right )}{4 b^2}-\frac {x \cosh (c+d x)}{2 b \left (a+b x^2\right )} \]
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Rubi [A] time = 0.61, antiderivative size = 416, normalized size of antiderivative = 1.00, number of steps used = 17, number of rules used = 6, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.316, Rules used = {5291, 5281, 3303, 3298, 3301, 5292} \[ \frac {d \sinh \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Chi}\left (x d+\frac {\sqrt {-a} d}{\sqrt {b}}\right )}{4 b^2}+\frac {d \sinh \left (\frac {\sqrt {-a} d}{\sqrt {b}}+c\right ) \text {Chi}\left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right )}{4 b^2}+\frac {\cosh \left (\frac {\sqrt {-a} d}{\sqrt {b}}+c\right ) \text {Chi}\left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right )}{4 \sqrt {-a} b^{3/2}}-\frac {\cosh \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Chi}\left (x d+\frac {\sqrt {-a} d}{\sqrt {b}}\right )}{4 \sqrt {-a} b^{3/2}}-\frac {\sinh \left (\frac {\sqrt {-a} d}{\sqrt {b}}+c\right ) \text {Shi}\left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right )}{4 \sqrt {-a} b^{3/2}}-\frac {\sinh \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Shi}\left (x d+\frac {\sqrt {-a} d}{\sqrt {b}}\right )}{4 \sqrt {-a} b^{3/2}}-\frac {d \cosh \left (\frac {\sqrt {-a} d}{\sqrt {b}}+c\right ) \text {Shi}\left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right )}{4 b^2}+\frac {d \cosh \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Shi}\left (x d+\frac {\sqrt {-a} d}{\sqrt {b}}\right )}{4 b^2}-\frac {x \cosh (c+d x)}{2 b \left (a+b x^2\right )} \]
Antiderivative was successfully verified.
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Rule 3298
Rule 3301
Rule 3303
Rule 5281
Rule 5291
Rule 5292
Rubi steps
\begin {align*} \int \frac {x^2 \cosh (c+d x)}{\left (a+b x^2\right )^2} \, dx &=-\frac {x \cosh (c+d x)}{2 b \left (a+b x^2\right )}+\frac {\int \frac {\cosh (c+d x)}{a+b x^2} \, dx}{2 b}+\frac {d \int \frac {x \sinh (c+d x)}{a+b x^2} \, dx}{2 b}\\ &=-\frac {x \cosh (c+d x)}{2 b \left (a+b x^2\right )}+\frac {\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}{2 b}+\frac {d \int \left (-\frac {\sinh (c+d x)}{2 \sqrt {b} \left (\sqrt {-a}-\sqrt {b} x\right )}+\frac {\sinh (c+d x)}{2 \sqrt {b} \left (\sqrt {-a}+\sqrt {b} x\right )}\right ) \, dx}{2 b}\\ &=-\frac {x \cosh (c+d x)}{2 b \left (a+b x^2\right )}-\frac {\int \frac {\cosh (c+d x)}{\sqrt {-a}-\sqrt {b} x} \, dx}{4 \sqrt {-a} b}-\frac {\int \frac {\cosh (c+d x)}{\sqrt {-a}+\sqrt {b} x} \, dx}{4 \sqrt {-a} b}-\frac {d \int \frac {\sinh (c+d x)}{\sqrt {-a}-\sqrt {b} x} \, dx}{4 b^{3/2}}+\frac {d \int \frac {\sinh (c+d x)}{\sqrt {-a}+\sqrt {b} x} \, dx}{4 b^{3/2}}\\ &=-\frac {x \cosh (c+d x)}{2 b \left (a+b x^2\right )}-\frac {\cosh \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \int \frac {\cosh \left (\frac {\sqrt {-a} d}{\sqrt {b}}+d x\right )}{\sqrt {-a}+\sqrt {b} x} \, dx}{4 \sqrt {-a} b}+\frac {\left (d \cosh \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}{4 b^{3/2}}-\frac {\cosh \left (c+\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \int \frac {\cosh \left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right )}{\sqrt {-a}-\sqrt {b} x} \, dx}{4 \sqrt {-a} b}+\frac {\left (d \cosh \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}{4 b^{3/2}}-\frac {\sinh \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \int \frac {\sinh \left (\frac {\sqrt {-a} d}{\sqrt {b}}+d x\right )}{\sqrt {-a}+\sqrt {b} x} \, dx}{4 \sqrt {-a} b}+\frac {\left (d \sinh \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}{4 b^{3/2}}+\frac {\sinh \left (c+\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \int \frac {\sinh \left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right )}{\sqrt {-a}-\sqrt {b} x} \, dx}{4 \sqrt {-a} b}-\frac {\left (d \sinh \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}{4 b^{3/2}}\\ &=-\frac {x \cosh (c+d x)}{2 b \left (a+b x^2\right )}+\frac {\cosh \left (c+\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Chi}\left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right )}{4 \sqrt {-a} b^{3/2}}-\frac {\cosh \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Chi}\left (\frac {\sqrt {-a} d}{\sqrt {b}}+d x\right )}{4 \sqrt {-a} b^{3/2}}+\frac {d \text {Chi}\left (\frac {\sqrt {-a} d}{\sqrt {b}}+d x\right ) \sinh \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right )}{4 b^2}+\frac {d \text {Chi}\left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right ) \sinh \left (c+\frac {\sqrt {-a} d}{\sqrt {b}}\right )}{4 b^2}-\frac {d \cosh \left (c+\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Shi}\left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right )}{4 b^2}-\frac {\sinh \left (c+\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Shi}\left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right )}{4 \sqrt {-a} b^{3/2}}+\frac {d \cosh \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Shi}\left (\frac {\sqrt {-a} d}{\sqrt {b}}+d x\right )}{4 b^2}-\frac {\sinh \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Shi}\left (\frac {\sqrt {-a} d}{\sqrt {b}}+d x\right )}{4 \sqrt {-a} b^{3/2}}\\ \end {align*}
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Mathematica [C] time = 1.06, size = 364, normalized size = 0.88 \[ \frac {\left (a+b x^2\right ) \text {Ci}\left (i d x-\frac {\sqrt {a} d}{\sqrt {b}}\right ) \left (\sqrt {a} d \sinh \left (c-\frac {i \sqrt {a} d}{\sqrt {b}}\right )+i \sqrt {b} \cosh \left (c-\frac {i \sqrt {a} d}{\sqrt {b}}\right )\right )+\left (a+b x^2\right ) \text {Ci}\left (i x d+\frac {\sqrt {a} d}{\sqrt {b}}\right ) \left (\sqrt {a} d \sinh \left (c+\frac {i \sqrt {a} d}{\sqrt {b}}\right )-i \sqrt {b} \cosh \left (c+\frac {i \sqrt {a} d}{\sqrt {b}}\right )\right )+\left (a+b x^2\right ) \text {Si}\left (\frac {\sqrt {a} d}{\sqrt {b}}-i d x\right ) \left (i \sqrt {a} d \cosh \left (c-\frac {i \sqrt {a} d}{\sqrt {b}}\right )-\sqrt {b} \sinh \left (c-\frac {i \sqrt {a} d}{\sqrt {b}}\right )\right )-\left (a+b x^2\right ) \text {Si}\left (i x d+\frac {\sqrt {a} d}{\sqrt {b}}\right ) \left (\sqrt {b} \sinh \left (c+\frac {i \sqrt {a} d}{\sqrt {b}}\right )+i \sqrt {a} d \cosh \left (c+\frac {i \sqrt {a} d}{\sqrt {b}}\right )\right )-2 \sqrt {a} b x \cosh (c+d x)}{4 \sqrt {a} b^2 \left (a+b x^2\right )} \]
Warning: Unable to verify antiderivative.
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fricas [B] time = 0.58, size = 1162, normalized size = 2.79 \[ \text {result too large to display} \]
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 \[ \int \frac {x^{2} \cosh \left (d x + c\right )}{{\left (b x^{2} + a\right )}^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.23, size = 491, normalized size = 1.18 \[ -\frac {d^{2} {\mathrm e}^{-d x -c} x}{4 b \left (b \,d^{2} x^{2}+a \,d^{2}\right )}+\frac {d \,{\mathrm e}^{-\frac {d \sqrt {-a b}+c b}{b}} \Ei \left (1, -\frac {d \sqrt {-a b}-\left (d x +c \right ) b +c b}{b}\right )}{8 b^{2}}+\frac {d \,{\mathrm e}^{-\frac {-d \sqrt {-a b}+c b}{b}} \Ei \left (1, \frac {d \sqrt {-a b}+\left (d x +c \right ) b -c b}{b}\right )}{8 b^{2}}-\frac {{\mathrm e}^{-\frac {d \sqrt {-a b}+c b}{b}} \Ei \left (1, -\frac {d \sqrt {-a b}-\left (d x +c \right ) b +c b}{b}\right )}{8 b \sqrt {-a b}}+\frac {{\mathrm e}^{-\frac {-d \sqrt {-a b}+c b}{b}} \Ei \left (1, \frac {d \sqrt {-a b}+\left (d x +c \right ) b -c b}{b}\right )}{8 b \sqrt {-a b}}-\frac {d^{2} {\mathrm e}^{d x +c} x}{4 b \left (b \,d^{2} x^{2}+a \,d^{2}\right )}-\frac {d \,{\mathrm e}^{\frac {d \sqrt {-a b}+c b}{b}} \Ei \left (1, \frac {d \sqrt {-a b}-\left (d x +c \right ) b +c b}{b}\right )}{8 b^{2}}-\frac {d \,{\mathrm e}^{\frac {-d \sqrt {-a b}+c b}{b}} \Ei \left (1, -\frac {d \sqrt {-a b}+\left (d x +c \right ) b -c b}{b}\right )}{8 b^{2}}-\frac {{\mathrm e}^{\frac {d \sqrt {-a b}+c b}{b}} \Ei \left (1, \frac {d \sqrt {-a b}-\left (d x +c \right ) b +c b}{b}\right )}{8 b \sqrt {-a b}}+\frac {{\mathrm e}^{\frac {-d \sqrt {-a b}+c b}{b}} \Ei \left (1, -\frac {d \sqrt {-a b}+\left (d x +c \right ) b -c b}{b}\right )}{8 b \sqrt {-a b}} \]
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 \[ \frac {{\left (d x^{2} e^{\left (2 \, c\right )} + 2 \, x e^{\left (2 \, c\right )}\right )} e^{\left (d x\right )} - {\left (d x^{2} - 2 \, x\right )} e^{\left (-d x\right )}}{2 \, {\left (b^{2} d^{2} x^{4} e^{c} + 2 \, a b d^{2} x^{2} e^{c} + a^{2} d^{2} e^{c}\right )}} + \frac {1}{2} \, \int -\frac {2 \, {\left (2 \, a d x e^{c} - 3 \, b x^{2} e^{c} + a e^{c}\right )} e^{\left (d x\right )}}{b^{3} d^{2} x^{6} + 3 \, a b^{2} d^{2} x^{4} + 3 \, a^{2} b d^{2} x^{2} + a^{3} d^{2}}\,{d x} + \frac {1}{2} \, \int \frac {2 \, {\left (2 \, a d x + 3 \, b x^{2} - a\right )} e^{\left (-d x\right )}}{b^{3} d^{2} x^{6} e^{c} + 3 \, a b^{2} d^{2} x^{4} e^{c} + 3 \, a^{2} b d^{2} x^{2} e^{c} + a^{3} d^{2} e^{c}}\,{d x} \]
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 \[ \int \frac {x^2\,\mathrm {cosh}\left (c+d\,x\right )}{{\left (b\,x^2+a\right )}^2} \,d x \]
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 \[ \int \frac {x^{2} \cosh {\left (c + d x \right )}}{\left (a + b x^{2}\right )^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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