Optimal. Leaf size=449 \[ \frac {3 \sqrt {-a} \cosh \left (\frac {\sqrt {-a} d}{\sqrt {b}}+c\right ) \text {Chi}\left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right )}{4 b^{5/2}}-\frac {3 \sqrt {-a} \cosh \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Chi}\left (x d+\frac {\sqrt {-a} d}{\sqrt {b}}\right )}{4 b^{5/2}}-\frac {3 \sqrt {-a} \sinh \left (\frac {\sqrt {-a} d}{\sqrt {b}}+c\right ) \text {Shi}\left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right )}{4 b^{5/2}}-\frac {3 \sqrt {-a} \sinh \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Shi}\left (x d+\frac {\sqrt {-a} d}{\sqrt {b}}\right )}{4 b^{5/2}}-\frac {a d \sinh \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Chi}\left (x d+\frac {\sqrt {-a} d}{\sqrt {b}}\right )}{4 b^3}-\frac {a d \sinh \left (\frac {\sqrt {-a} d}{\sqrt {b}}+c\right ) \text {Chi}\left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right )}{4 b^3}+\frac {a d \cosh \left (\frac {\sqrt {-a} d}{\sqrt {b}}+c\right ) \text {Shi}\left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right )}{4 b^3}-\frac {a d \cosh \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Shi}\left (x d+\frac {\sqrt {-a} d}{\sqrt {b}}\right )}{4 b^3}-\frac {x^3 \cosh (c+d x)}{2 b \left (a+b x^2\right )}+\frac {\sinh (c+d x)}{b^2 d}+\frac {x \cosh (c+d x)}{2 b^2} \]
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Rubi [A] time = 0.86, antiderivative size = 449, normalized size of antiderivative = 1.00, number of steps used = 24, number of rules used = 9, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.474, Rules used = {5291, 5293, 2637, 5281, 3303, 3298, 3301, 5292, 3296} \[ -\frac {a d \sinh \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Chi}\left (x d+\frac {\sqrt {-a} d}{\sqrt {b}}\right )}{4 b^3}-\frac {a d \sinh \left (\frac {\sqrt {-a} d}{\sqrt {b}}+c\right ) \text {Chi}\left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right )}{4 b^3}+\frac {3 \sqrt {-a} \cosh \left (\frac {\sqrt {-a} d}{\sqrt {b}}+c\right ) \text {Chi}\left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right )}{4 b^{5/2}}-\frac {3 \sqrt {-a} \cosh \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Chi}\left (x d+\frac {\sqrt {-a} d}{\sqrt {b}}\right )}{4 b^{5/2}}-\frac {3 \sqrt {-a} \sinh \left (\frac {\sqrt {-a} d}{\sqrt {b}}+c\right ) \text {Shi}\left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right )}{4 b^{5/2}}-\frac {3 \sqrt {-a} \sinh \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Shi}\left (x d+\frac {\sqrt {-a} d}{\sqrt {b}}\right )}{4 b^{5/2}}+\frac {a d \cosh \left (\frac {\sqrt {-a} d}{\sqrt {b}}+c\right ) \text {Shi}\left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right )}{4 b^3}-\frac {a d \cosh \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Shi}\left (x d+\frac {\sqrt {-a} d}{\sqrt {b}}\right )}{4 b^3}-\frac {x^3 \cosh (c+d x)}{2 b \left (a+b x^2\right )}+\frac {\sinh (c+d x)}{b^2 d}+\frac {x \cosh (c+d x)}{2 b^2} \]
Antiderivative was successfully verified.
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Rule 2637
Rule 3296
Rule 3298
Rule 3301
Rule 3303
Rule 5281
Rule 5291
Rule 5292
Rule 5293
Rubi steps
\begin {align*} \int \frac {x^4 \cosh (c+d x)}{\left (a+b x^2\right )^2} \, dx &=-\frac {x^3 \cosh (c+d x)}{2 b \left (a+b x^2\right )}+\frac {3 \int \frac {x^2 \cosh (c+d x)}{a+b x^2} \, dx}{2 b}+\frac {d \int \frac {x^3 \sinh (c+d x)}{a+b x^2} \, dx}{2 b}\\ &=-\frac {x^3 \cosh (c+d x)}{2 b \left (a+b x^2\right )}+\frac {3 \int \left (\frac {\cosh (c+d x)}{b}-\frac {a \cosh (c+d x)}{b \left (a+b x^2\right )}\right ) \, dx}{2 b}+\frac {d \int \left (\frac {x \sinh (c+d x)}{b}-\frac {a x \sinh (c+d x)}{b \left (a+b x^2\right )}\right ) \, dx}{2 b}\\ &=-\frac {x^3 \cosh (c+d x)}{2 b \left (a+b x^2\right )}+\frac {3 \int \cosh (c+d x) \, dx}{2 b^2}-\frac {(3 a) \int \frac {\cosh (c+d x)}{a+b x^2} \, dx}{2 b^2}+\frac {d \int x \sinh (c+d x) \, dx}{2 b^2}-\frac {(a d) \int \frac {x \sinh (c+d x)}{a+b x^2} \, dx}{2 b^2}\\ &=\frac {x \cosh (c+d x)}{2 b^2}-\frac {x^3 \cosh (c+d x)}{2 b \left (a+b x^2\right )}+\frac {3 \sinh (c+d x)}{2 b^2 d}-\frac {\int \cosh (c+d x) \, dx}{2 b^2}-\frac {(3 a) \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^2}-\frac {(a 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^2}\\ &=\frac {x \cosh (c+d x)}{2 b^2}-\frac {x^3 \cosh (c+d x)}{2 b \left (a+b x^2\right )}+\frac {\sinh (c+d x)}{b^2 d}-\frac {\left (3 \sqrt {-a}\right ) \int \frac {\cosh (c+d x)}{\sqrt {-a}-\sqrt {b} x} \, dx}{4 b^2}-\frac {\left (3 \sqrt {-a}\right ) \int \frac {\cosh (c+d x)}{\sqrt {-a}+\sqrt {b} x} \, dx}{4 b^2}+\frac {(a d) \int \frac {\sinh (c+d x)}{\sqrt {-a}-\sqrt {b} x} \, dx}{4 b^{5/2}}-\frac {(a d) \int \frac {\sinh (c+d x)}{\sqrt {-a}+\sqrt {b} x} \, dx}{4 b^{5/2}}\\ &=\frac {x \cosh (c+d x)}{2 b^2}-\frac {x^3 \cosh (c+d x)}{2 b \left (a+b x^2\right )}+\frac {\sinh (c+d x)}{b^2 d}-\frac {\left (3 \sqrt {-a} \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}{4 b^2}-\frac {\left (a 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^{5/2}}-\frac {\left (3 \sqrt {-a} \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}{4 b^2}-\frac {\left (a 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^{5/2}}-\frac {\left (3 \sqrt {-a} \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}{4 b^2}-\frac {\left (a 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^{5/2}}+\frac {\left (3 \sqrt {-a} \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}{4 b^2}+\frac {\left (a 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^{5/2}}\\ &=\frac {x \cosh (c+d x)}{2 b^2}-\frac {x^3 \cosh (c+d x)}{2 b \left (a+b x^2\right )}+\frac {3 \sqrt {-a} \cosh \left (c+\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Chi}\left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right )}{4 b^{5/2}}-\frac {3 \sqrt {-a} \cosh \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Chi}\left (\frac {\sqrt {-a} d}{\sqrt {b}}+d x\right )}{4 b^{5/2}}-\frac {a 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^3}-\frac {a 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^3}+\frac {\sinh (c+d x)}{b^2 d}+\frac {a 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^3}-\frac {3 \sqrt {-a} \sinh \left (c+\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Shi}\left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right )}{4 b^{5/2}}-\frac {a 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^3}-\frac {3 \sqrt {-a} \sinh \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Shi}\left (\frac {\sqrt {-a} d}{\sqrt {b}}+d x\right )}{4 b^{5/2}}\\ \end {align*}
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Mathematica [C] time = 1.63, size = 621, normalized size = 1.38 \[ \frac {-\frac {3 \sqrt {a} \sinh (c) \left (\sin \left (\frac {\sqrt {a} d}{\sqrt {b}}\right ) \text {Ci}\left (i d x-\frac {\sqrt {a} d}{\sqrt {b}}\right )+\sin \left (\frac {\sqrt {a} d}{\sqrt {b}}\right ) \text {Ci}\left (i x d+\frac {\sqrt {a} d}{\sqrt {b}}\right )-\cos \left (\frac {\sqrt {a} d}{\sqrt {b}}\right ) \left (\text {Si}\left (\frac {\sqrt {a} d}{\sqrt {b}}-i d x\right )+\text {Si}\left (i x d+\frac {\sqrt {a} d}{\sqrt {b}}\right )\right )\right )}{\sqrt {b}}-\frac {a d \sinh (c) \left (\cos \left (\frac {\sqrt {a} d}{\sqrt {b}}\right ) \text {Ci}\left (i d x-\frac {\sqrt {a} d}{\sqrt {b}}\right )+\cos \left (\frac {\sqrt {a} d}{\sqrt {b}}\right ) \text {Ci}\left (i x d+\frac {\sqrt {a} d}{\sqrt {b}}\right )+\sin \left (\frac {\sqrt {a} d}{\sqrt {b}}\right ) \left (\text {Si}\left (\frac {\sqrt {a} d}{\sqrt {b}}-i d x\right )+\text {Si}\left (i x d+\frac {\sqrt {a} d}{\sqrt {b}}\right )\right )\right )}{b}-\frac {3 i \sqrt {a} \cosh (c) \left (\cos \left (\frac {\sqrt {a} d}{\sqrt {b}}\right ) \text {Ci}\left (i d x-\frac {\sqrt {a} d}{\sqrt {b}}\right )-\cos \left (\frac {\sqrt {a} d}{\sqrt {b}}\right ) \text {Ci}\left (i x d+\frac {\sqrt {a} d}{\sqrt {b}}\right )+\sin \left (\frac {\sqrt {a} d}{\sqrt {b}}\right ) \left (\text {Si}\left (\frac {\sqrt {a} d}{\sqrt {b}}-i d x\right )-\text {Si}\left (i x d+\frac {\sqrt {a} d}{\sqrt {b}}\right )\right )\right )}{\sqrt {b}}+\frac {i a d \cosh (c) \left (\sin \left (\frac {\sqrt {a} d}{\sqrt {b}}\right ) \text {Ci}\left (i d x-\frac {\sqrt {a} d}{\sqrt {b}}\right )-\sin \left (\frac {\sqrt {a} d}{\sqrt {b}}\right ) \text {Ci}\left (i x d+\frac {\sqrt {a} d}{\sqrt {b}}\right )+\cos \left (\frac {\sqrt {a} d}{\sqrt {b}}\right ) \left (\text {Si}\left (i x d+\frac {\sqrt {a} d}{\sqrt {b}}\right )-\text {Si}\left (\frac {\sqrt {a} d}{\sqrt {b}}-i d x\right )\right )\right )}{b}+2 \cosh (d x) \left (\frac {a x \cosh (c)}{a+b x^2}+\frac {2 \sinh (c)}{d}\right )+2 \sinh (d x) \left (\frac {a x \sinh (c)}{a+b x^2}+\frac {2 \cosh (c)}{d}\right )}{4 b^2} \]
Warning: Unable to verify antiderivative.
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fricas [B] time = 0.64, size = 1179, normalized size = 2.63 \[ \text {result too large to display} \]
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 \[ \mathit {sage}_{0} x \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.41, size = 532, normalized size = 1.18 \[ \frac {d^{2} {\mathrm e}^{-d x -c} a x}{4 b^{2} \left (b \,d^{2} x^{2}+a \,d^{2}\right )}-\frac {{\mathrm e}^{-d x -c}}{2 d \,b^{2}}-\frac {d a \,{\mathrm e}^{-\frac {d \sqrt {-a b}+c b}{b}} \Ei \left (1, -\frac {d \sqrt {-a b}-\left (d x +c \right ) b +c b}{b}\right )}{8 b^{3}}-\frac {d a \,{\mathrm e}^{-\frac {-d \sqrt {-a b}+c b}{b}} \Ei \left (1, \frac {d \sqrt {-a b}+\left (d x +c \right ) b -c b}{b}\right )}{8 b^{3}}+\frac {3 a \,{\mathrm e}^{-\frac {d \sqrt {-a b}+c b}{b}} \Ei \left (1, -\frac {d \sqrt {-a b}-\left (d x +c \right ) b +c b}{b}\right )}{8 b^{2} \sqrt {-a b}}-\frac {3 a \,{\mathrm e}^{-\frac {-d \sqrt {-a b}+c b}{b}} \Ei \left (1, \frac {d \sqrt {-a b}+\left (d x +c \right ) b -c b}{b}\right )}{8 b^{2} \sqrt {-a b}}+\frac {{\mathrm e}^{d x +c}}{2 d \,b^{2}}+\frac {d^{2} {\mathrm e}^{d x +c} a x}{4 b^{2} \left (b \,d^{2} x^{2}+a \,d^{2}\right )}+\frac {d a \,{\mathrm e}^{\frac {-d \sqrt {-a b}+c b}{b}} \Ei \left (1, -\frac {d \sqrt {-a b}+\left (d x +c \right ) b -c b}{b}\right )}{8 b^{3}}+\frac {d a \,{\mathrm e}^{\frac {d \sqrt {-a b}+c b}{b}} \Ei \left (1, \frac {d \sqrt {-a b}-\left (d x +c \right ) b +c b}{b}\right )}{8 b^{3}}-\frac {3 a \,{\mathrm e}^{\frac {-d \sqrt {-a b}+c b}{b}} \Ei \left (1, -\frac {d \sqrt {-a b}+\left (d x +c \right ) b -c b}{b}\right )}{8 b^{2} \sqrt {-a b}}+\frac {3 a \,{\mathrm e}^{\frac {d \sqrt {-a b}+c b}{b}} \Ei \left (1, \frac {d \sqrt {-a b}-\left (d x +c \right ) b +c b}{b}\right )}{8 b^{2} \sqrt {-a b}} \]
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 \[ \frac {{\left (b d x^{4} e^{\left (2 \, c\right )} - 4 \, a x e^{\left (2 \, c\right )}\right )} e^{\left (d x\right )} - {\left (b d x^{4} + 4 \, a x\right )} e^{\left (-d x\right )}}{2 \, {\left (b^{3} d^{2} x^{4} e^{c} + 2 \, a b^{2} d^{2} x^{2} e^{c} + a^{2} b d^{2} e^{c}\right )}} - \frac {1}{2} \, \int -\frac {4 \, {\left (a^{2} d x e^{c} - 3 \, a b x^{2} e^{c} + a^{2} e^{c}\right )} e^{\left (d x\right )}}{b^{4} d^{2} x^{6} + 3 \, a b^{3} d^{2} x^{4} + 3 \, a^{2} b^{2} d^{2} x^{2} + a^{3} b d^{2}}\,{d x} - \frac {1}{2} \, \int \frac {4 \, {\left (a^{2} d x + 3 \, a b x^{2} - a^{2}\right )} e^{\left (-d x\right )}}{b^{4} d^{2} x^{6} e^{c} + 3 \, a b^{3} d^{2} x^{4} e^{c} + 3 \, a^{2} b^{2} d^{2} x^{2} e^{c} + a^{3} b d^{2} e^{c}}\,{d x} \]
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 \[ \int \frac {x^4\,\mathrm {cosh}\left (c+d\,x\right )}{{\left (b\,x^2+a\right )}^2} \,d x \]
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 \[ \int \frac {x^{4} \cosh {\left (c + d x \right )}}{\left (a + b x^{2}\right )^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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