Optimal. Leaf size=476 \[ -\frac {x^2 \cosh (c+d x)}{4 b \left (a+b x^2\right )^2}-\frac {\cosh (c+d x)}{4 b^2 \left (a+b x^2\right )}+\frac {d^2 \cosh \left (c+\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Chi}\left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right )}{16 b^3}+\frac {d^2 \cosh \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Chi}\left (\frac {\sqrt {-a} d}{\sqrt {b}}+d x\right )}{16 b^3}-\frac {3 d \text {Chi}\left (\frac {\sqrt {-a} d}{\sqrt {b}}+d x\right ) \sinh \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right )}{16 \sqrt {-a} b^{5/2}}+\frac {3 d \text {Chi}\left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right ) \sinh \left (c+\frac {\sqrt {-a} d}{\sqrt {b}}\right )}{16 \sqrt {-a} b^{5/2}}-\frac {d x \sinh (c+d x)}{8 b^2 \left (a+b x^2\right )}-\frac {3 d \cosh \left (c+\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Shi}\left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right )}{16 \sqrt {-a} b^{5/2}}-\frac {d^2 \sinh \left (c+\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Shi}\left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right )}{16 b^3}-\frac {3 d \cosh \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Shi}\left (\frac {\sqrt {-a} d}{\sqrt {b}}+d x\right )}{16 \sqrt {-a} b^{5/2}}+\frac {d^2 \sinh \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Shi}\left (\frac {\sqrt {-a} d}{\sqrt {b}}+d x\right )}{16 b^3} \]
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Rubi [A]
time = 0.75, antiderivative size = 476, normalized size of antiderivative = 1.00, number of steps
used = 27, number of rules used = 8, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.421, Rules used = {5399, 5397,
5388, 3384, 3379, 3382, 5398, 5401} \begin {gather*} -\frac {3 d \sinh \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Chi}\left (x d+\frac {\sqrt {-a} d}{\sqrt {b}}\right )}{16 \sqrt {-a} b^{5/2}}+\frac {3 d \sinh \left (\frac {\sqrt {-a} d}{\sqrt {b}}+c\right ) \text {Chi}\left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right )}{16 \sqrt {-a} b^{5/2}}-\frac {3 d \cosh \left (\frac {\sqrt {-a} d}{\sqrt {b}}+c\right ) \text {Shi}\left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right )}{16 \sqrt {-a} b^{5/2}}-\frac {3 d \cosh \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Shi}\left (x d+\frac {\sqrt {-a} d}{\sqrt {b}}\right )}{16 \sqrt {-a} b^{5/2}}+\frac {d^2 \cosh \left (\frac {\sqrt {-a} d}{\sqrt {b}}+c\right ) \text {Chi}\left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right )}{16 b^3}+\frac {d^2 \cosh \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Chi}\left (x d+\frac {\sqrt {-a} d}{\sqrt {b}}\right )}{16 b^3}-\frac {d^2 \sinh \left (\frac {\sqrt {-a} d}{\sqrt {b}}+c\right ) \text {Shi}\left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right )}{16 b^3}+\frac {d^2 \sinh \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Shi}\left (x d+\frac {\sqrt {-a} d}{\sqrt {b}}\right )}{16 b^3}-\frac {d x \sinh (c+d x)}{8 b^2 \left (a+b x^2\right )}-\frac {\cosh (c+d x)}{4 b^2 \left (a+b x^2\right )}-\frac {x^2 \cosh (c+d x)}{4 b \left (a+b x^2\right )^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 3379
Rule 3382
Rule 3384
Rule 5388
Rule 5397
Rule 5398
Rule 5399
Rule 5401
Rubi steps
\begin {align*} \int \frac {x^3 \cosh (c+d x)}{\left (a+b x^2\right )^3} \, dx &=-\frac {x^2 \cosh (c+d x)}{4 b \left (a+b x^2\right )^2}+\frac {\int \frac {x \cosh (c+d x)}{\left (a+b x^2\right )^2} \, dx}{2 b}+\frac {d \int \frac {x^2 \sinh (c+d x)}{\left (a+b x^2\right )^2} \, dx}{4 b}\\ &=-\frac {x^2 \cosh (c+d x)}{4 b \left (a+b x^2\right )^2}-\frac {\cosh (c+d x)}{4 b^2 \left (a+b x^2\right )}-\frac {d x \sinh (c+d x)}{8 b^2 \left (a+b x^2\right )}+\frac {d \int \frac {\sinh (c+d x)}{a+b x^2} \, dx}{8 b^2}+\frac {d \int \frac {\sinh (c+d x)}{a+b x^2} \, dx}{4 b^2}+\frac {d^2 \int \frac {x \cosh (c+d x)}{a+b x^2} \, dx}{8 b^2}\\ &=-\frac {x^2 \cosh (c+d x)}{4 b \left (a+b x^2\right )^2}-\frac {\cosh (c+d x)}{4 b^2 \left (a+b x^2\right )}-\frac {d x \sinh (c+d x)}{8 b^2 \left (a+b x^2\right )}+\frac {d \int \left (\frac {\sqrt {-a} \sinh (c+d x)}{2 a \left (\sqrt {-a}-\sqrt {b} x\right )}+\frac {\sqrt {-a} \sinh (c+d x)}{2 a \left (\sqrt {-a}+\sqrt {b} x\right )}\right ) \, dx}{8 b^2}+\frac {d \int \left (\frac {\sqrt {-a} \sinh (c+d x)}{2 a \left (\sqrt {-a}-\sqrt {b} x\right )}+\frac {\sqrt {-a} \sinh (c+d x)}{2 a \left (\sqrt {-a}+\sqrt {b} x\right )}\right ) \, dx}{4 b^2}+\frac {d^2 \int \left (-\frac {\cosh (c+d x)}{2 \sqrt {b} \left (\sqrt {-a}-\sqrt {b} x\right )}+\frac {\cosh (c+d x)}{2 \sqrt {b} \left (\sqrt {-a}+\sqrt {b} x\right )}\right ) \, dx}{8 b^2}\\ &=-\frac {x^2 \cosh (c+d x)}{4 b \left (a+b x^2\right )^2}-\frac {\cosh (c+d x)}{4 b^2 \left (a+b x^2\right )}-\frac {d x \sinh (c+d x)}{8 b^2 \left (a+b x^2\right )}-\frac {d \int \frac {\sinh (c+d x)}{\sqrt {-a}-\sqrt {b} x} \, dx}{16 \sqrt {-a} b^2}-\frac {d \int \frac {\sinh (c+d x)}{\sqrt {-a}+\sqrt {b} x} \, dx}{16 \sqrt {-a} b^2}-\frac {d \int \frac {\sinh (c+d x)}{\sqrt {-a}-\sqrt {b} x} \, dx}{8 \sqrt {-a} b^2}-\frac {d \int \frac {\sinh (c+d x)}{\sqrt {-a}+\sqrt {b} x} \, dx}{8 \sqrt {-a} b^2}-\frac {d^2 \int \frac {\cosh (c+d x)}{\sqrt {-a}-\sqrt {b} x} \, dx}{16 b^{5/2}}+\frac {d^2 \int \frac {\cosh (c+d x)}{\sqrt {-a}+\sqrt {b} x} \, dx}{16 b^{5/2}}\\ &=-\frac {x^2 \cosh (c+d x)}{4 b \left (a+b x^2\right )^2}-\frac {\cosh (c+d x)}{4 b^2 \left (a+b x^2\right )}-\frac {d x \sinh (c+d x)}{8 b^2 \left (a+b x^2\right )}-\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}{16 \sqrt {-a} b^2}-\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}{8 \sqrt {-a} b^2}+\frac {\left (d^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}{16 b^{5/2}}+\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}{16 \sqrt {-a} b^2}+\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}{8 \sqrt {-a} b^2}-\frac {\left (d^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}{16 b^{5/2}}-\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}{16 \sqrt {-a} b^2}-\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}{8 \sqrt {-a} b^2}+\frac {\left (d^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}{16 b^{5/2}}-\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}{16 \sqrt {-a} b^2}-\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}{8 \sqrt {-a} b^2}+\frac {\left (d^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}{16 b^{5/2}}\\ &=-\frac {x^2 \cosh (c+d x)}{4 b \left (a+b x^2\right )^2}-\frac {\cosh (c+d x)}{4 b^2 \left (a+b x^2\right )}+\frac {d^2 \cosh \left (c+\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Chi}\left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right )}{16 b^3}+\frac {d^2 \cosh \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Chi}\left (\frac {\sqrt {-a} d}{\sqrt {b}}+d x\right )}{16 b^3}-\frac {3 d \text {Chi}\left (\frac {\sqrt {-a} d}{\sqrt {b}}+d x\right ) \sinh \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right )}{16 \sqrt {-a} b^{5/2}}+\frac {3 d \text {Chi}\left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right ) \sinh \left (c+\frac {\sqrt {-a} d}{\sqrt {b}}\right )}{16 \sqrt {-a} b^{5/2}}-\frac {d x \sinh (c+d x)}{8 b^2 \left (a+b x^2\right )}-\frac {3 d \cosh \left (c+\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Shi}\left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right )}{16 \sqrt {-a} b^{5/2}}-\frac {d^2 \sinh \left (c+\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Shi}\left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right )}{16 b^3}-\frac {3 d \cosh \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Shi}\left (\frac {\sqrt {-a} d}{\sqrt {b}}+d x\right )}{16 \sqrt {-a} b^{5/2}}+\frac {d^2 \sinh \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Shi}\left (\frac {\sqrt {-a} d}{\sqrt {b}}+d x\right )}{16 b^3}\\ \end {align*}
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Mathematica [C] Result contains complex when optimal does not.
time = 1.42, size = 648, normalized size = 1.36 \begin {gather*} \frac {-\frac {2 \cosh (d x) \left (2 \left (a+2 b x^2\right ) \cosh (c)+d x \left (a+b x^2\right ) \sinh (c)\right )}{\left (a+b x^2\right )^2}-\frac {2 \left (d x \left (a+b x^2\right ) \cosh (c)+2 \left (a+2 b x^2\right ) \sinh (c)\right ) \sinh (d x)}{\left (a+b x^2\right )^2}+\frac {3 i d \sinh (c) \left (\cos \left (\frac {\sqrt {a} d}{\sqrt {b}}\right ) \text {CosIntegral}\left (-\frac {\sqrt {a} d}{\sqrt {b}}+i d x\right )-\cos \left (\frac {\sqrt {a} d}{\sqrt {b}}\right ) \text {CosIntegral}\left (\frac {\sqrt {a} d}{\sqrt {b}}+i d x\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 (\frac {\sqrt {a} d}{\sqrt {b}}+i d x\right )\right )\right )}{\sqrt {a} \sqrt {b}}-\frac {i d^2 \sinh (c) \left (\text {CosIntegral}\left (-\frac {\sqrt {a} d}{\sqrt {b}}+i d x\right ) \sin \left (\frac {\sqrt {a} d}{\sqrt {b}}\right )-\text {CosIntegral}\left (\frac {\sqrt {a} d}{\sqrt {b}}+i d x\right ) \sin \left (\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 (\frac {\sqrt {a} d}{\sqrt {b}}+i d x\right )\right )\right )}{b}+\frac {3 d \cosh (c) \left (\text {CosIntegral}\left (-\frac {\sqrt {a} d}{\sqrt {b}}+i d x\right ) \sin \left (\frac {\sqrt {a} d}{\sqrt {b}}\right )+\text {CosIntegral}\left (\frac {\sqrt {a} d}{\sqrt {b}}+i d x\right ) \sin \left (\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 (\frac {\sqrt {a} d}{\sqrt {b}}+i d x\right )\right )\right )}{\sqrt {a} \sqrt {b}}+\frac {d^2 \cosh (c) \left (\cos \left (\frac {\sqrt {a} d}{\sqrt {b}}\right ) \text {CosIntegral}\left (-\frac {\sqrt {a} d}{\sqrt {b}}+i d x\right )+\cos \left (\frac {\sqrt {a} d}{\sqrt {b}}\right ) \text {CosIntegral}\left (\frac {\sqrt {a} d}{\sqrt {b}}+i d x\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 (\frac {\sqrt {a} d}{\sqrt {b}}+i d x\right )\right )\right )}{b}}{16 b^2} \end {gather*}
Antiderivative was successfully verified.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(819\) vs.
\(2(374)=748\).
time = 1.10, size = 820, normalized size = 1.72
method | result | size |
risch | \(-\frac {d^{4} {\mathrm e}^{-d x -c} x^{2}}{4 b \left (x^{4} b^{2} d^{4}+2 a b \,d^{4} x^{2}+a^{2} d^{4}\right )}-\frac {d^{2} {\mathrm e}^{-\frac {-d \sqrt {-a b}+b c}{b}} \expIntegral \left (1, \frac {d \sqrt {-a b}+b \left (d x +c \right )-b c}{b}\right )}{32 b^{3}}-\frac {d^{2} {\mathrm e}^{-\frac {d \sqrt {-a b}+b c}{b}} \expIntegral \left (1, -\frac {d \sqrt {-a b}-b \left (d x +c \right )+b c}{b}\right )}{32 b^{3}}-\frac {3 d \,{\mathrm e}^{-\frac {-d \sqrt {-a b}+b c}{b}} \expIntegral \left (1, \frac {d \sqrt {-a b}+b \left (d x +c \right )-b c}{b}\right )}{32 b^{2} \sqrt {-a b}}+\frac {3 d \,{\mathrm e}^{-\frac {d \sqrt {-a b}+b c}{b}} \expIntegral \left (1, -\frac {d \sqrt {-a b}-b \left (d x +c \right )+b c}{b}\right )}{32 b^{2} \sqrt {-a b}}+\frac {d^{5} {\mathrm e}^{-d x -c} x^{3}}{16 b \left (x^{4} b^{2} d^{4}+2 a b \,d^{4} x^{2}+a^{2} d^{4}\right )}+\frac {d^{5} {\mathrm e}^{-d x -c} a x}{16 b^{2} \left (x^{4} b^{2} d^{4}+2 a b \,d^{4} x^{2}+a^{2} d^{4}\right )}-\frac {d^{4} {\mathrm e}^{-d x -c} a}{8 b^{2} \left (x^{4} b^{2} d^{4}+2 a b \,d^{4} x^{2}+a^{2} d^{4}\right )}-\frac {d^{4} {\mathrm e}^{d x +c} x^{2}}{4 b \left (x^{4} b^{2} d^{4}+2 a b \,d^{4} x^{2}+a^{2} d^{4}\right )}-\frac {d^{2} {\mathrm e}^{\frac {d \sqrt {-a b}+b c}{b}} \expIntegral \left (1, \frac {d \sqrt {-a b}-b \left (d x +c \right )+b c}{b}\right )}{32 b^{3}}-\frac {d^{2} {\mathrm e}^{\frac {-d \sqrt {-a b}+b c}{b}} \expIntegral \left (1, -\frac {d \sqrt {-a b}+b \left (d x +c \right )-b c}{b}\right )}{32 b^{3}}-\frac {3 d \,{\mathrm e}^{\frac {d \sqrt {-a b}+b c}{b}} \expIntegral \left (1, \frac {d \sqrt {-a b}-b \left (d x +c \right )+b c}{b}\right )}{32 b^{2} \sqrt {-a b}}+\frac {3 d \,{\mathrm e}^{\frac {-d \sqrt {-a b}+b c}{b}} \expIntegral \left (1, -\frac {d \sqrt {-a b}+b \left (d x +c \right )-b c}{b}\right )}{32 b^{2} \sqrt {-a b}}-\frac {d^{5} {\mathrm e}^{d x +c} x^{3}}{16 b \left (x^{4} b^{2} d^{4}+2 a b \,d^{4} x^{2}+a^{2} d^{4}\right )}-\frac {d^{5} {\mathrm e}^{d x +c} a x}{16 b^{2} \left (x^{4} b^{2} d^{4}+2 a b \,d^{4} x^{2}+a^{2} d^{4}\right )}-\frac {d^{4} {\mathrm e}^{d x +c} a}{8 b^{2} \left (x^{4} b^{2} d^{4}+2 a b \,d^{4} x^{2}+a^{2} d^{4}\right )}\) | \(820\) |
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 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 1620 vs.
\(2 (374) = 748\).
time = 0.44, size = 1620, normalized size = 3.40 \begin {gather*} -\frac {8 \, {\left (2 \, a b^{2} x^{2} + a^{2} b\right )} \cosh \left (d x + c\right ) - {\left ({\left ({\left (a b^{2} d^{2} x^{4} + 2 \, a^{2} b d^{2} x^{2} + a^{3} d^{2}\right )} \cosh \left (d x + c\right )^{2} - {\left (a b^{2} d^{2} x^{4} + 2 \, a^{2} b d^{2} x^{2} + a^{3} d^{2}\right )} \sinh \left (d x + c\right )^{2} - 3 \, {\left ({\left (b^{3} x^{4} + 2 \, a b^{2} x^{2} + a^{2} b\right )} \cosh \left (d x + c\right )^{2} - {\left (b^{3} x^{4} + 2 \, a b^{2} x^{2} + a^{2} b\right )} \sinh \left (d x + c\right )^{2}\right )} \sqrt {-\frac {a d^{2}}{b}}\right )} {\rm Ei}\left (d x - \sqrt {-\frac {a d^{2}}{b}}\right ) + {\left ({\left (a b^{2} d^{2} x^{4} + 2 \, a^{2} b d^{2} x^{2} + a^{3} d^{2}\right )} \cosh \left (d x + c\right )^{2} - {\left (a b^{2} d^{2} x^{4} + 2 \, a^{2} b d^{2} x^{2} + a^{3} d^{2}\right )} \sinh \left (d x + c\right )^{2} + 3 \, {\left ({\left (b^{3} x^{4} + 2 \, a b^{2} x^{2} + a^{2} b\right )} \cosh \left (d x + c\right )^{2} - {\left (b^{3} x^{4} + 2 \, a b^{2} x^{2} + a^{2} b\right )} \sinh \left (d x + c\right )^{2}\right )} \sqrt {-\frac {a d^{2}}{b}}\right )} {\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 ({\left (a b^{2} d^{2} x^{4} + 2 \, a^{2} b d^{2} x^{2} + a^{3} d^{2}\right )} \cosh \left (d x + c\right )^{2} - {\left (a b^{2} d^{2} x^{4} + 2 \, a^{2} b d^{2} x^{2} + a^{3} d^{2}\right )} \sinh \left (d x + c\right )^{2} + 3 \, {\left ({\left (b^{3} x^{4} + 2 \, a b^{2} x^{2} + a^{2} b\right )} \cosh \left (d x + c\right )^{2} - {\left (b^{3} x^{4} + 2 \, a b^{2} x^{2} + a^{2} b\right )} \sinh \left (d x + c\right )^{2}\right )} \sqrt {-\frac {a d^{2}}{b}}\right )} {\rm Ei}\left (d x + \sqrt {-\frac {a d^{2}}{b}}\right ) + {\left ({\left (a b^{2} d^{2} x^{4} + 2 \, a^{2} b d^{2} x^{2} + a^{3} d^{2}\right )} \cosh \left (d x + c\right )^{2} - {\left (a b^{2} d^{2} x^{4} + 2 \, a^{2} b d^{2} x^{2} + a^{3} d^{2}\right )} \sinh \left (d x + c\right )^{2} - 3 \, {\left ({\left (b^{3} x^{4} + 2 \, a b^{2} x^{2} + a^{2} b\right )} \cosh \left (d x + c\right )^{2} - {\left (b^{3} x^{4} + 2 \, a b^{2} x^{2} + a^{2} b\right )} \sinh \left (d x + c\right )^{2}\right )} \sqrt {-\frac {a d^{2}}{b}}\right )} {\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 (a b^{2} d x^{3} + a^{2} b d x\right )} \sinh \left (d x + c\right ) - {\left ({\left ({\left (a b^{2} d^{2} x^{4} + 2 \, a^{2} b d^{2} x^{2} + a^{3} d^{2}\right )} \cosh \left (d x + c\right )^{2} - {\left (a b^{2} d^{2} x^{4} + 2 \, a^{2} b d^{2} x^{2} + a^{3} d^{2}\right )} \sinh \left (d x + c\right )^{2} - 3 \, {\left ({\left (b^{3} x^{4} + 2 \, a b^{2} x^{2} + a^{2} b\right )} \cosh \left (d x + c\right )^{2} - {\left (b^{3} x^{4} + 2 \, a b^{2} x^{2} + a^{2} b\right )} \sinh \left (d x + c\right )^{2}\right )} \sqrt {-\frac {a d^{2}}{b}}\right )} {\rm Ei}\left (d x - \sqrt {-\frac {a d^{2}}{b}}\right ) - {\left ({\left (a b^{2} d^{2} x^{4} + 2 \, a^{2} b d^{2} x^{2} + a^{3} d^{2}\right )} \cosh \left (d x + c\right )^{2} - {\left (a b^{2} d^{2} x^{4} + 2 \, a^{2} b d^{2} x^{2} + a^{3} d^{2}\right )} \sinh \left (d x + c\right )^{2} + 3 \, {\left ({\left (b^{3} x^{4} + 2 \, a b^{2} x^{2} + a^{2} b\right )} \cosh \left (d x + c\right )^{2} - {\left (b^{3} x^{4} + 2 \, a b^{2} x^{2} + a^{2} b\right )} \sinh \left (d x + c\right )^{2}\right )} \sqrt {-\frac {a d^{2}}{b}}\right )} {\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 ({\left (a b^{2} d^{2} x^{4} + 2 \, a^{2} b d^{2} x^{2} + a^{3} d^{2}\right )} \cosh \left (d x + c\right )^{2} - {\left (a b^{2} d^{2} x^{4} + 2 \, a^{2} b d^{2} x^{2} + a^{3} d^{2}\right )} \sinh \left (d x + c\right )^{2} + 3 \, {\left ({\left (b^{3} x^{4} + 2 \, a b^{2} x^{2} + a^{2} b\right )} \cosh \left (d x + c\right )^{2} - {\left (b^{3} x^{4} + 2 \, a b^{2} x^{2} + a^{2} b\right )} \sinh \left (d x + c\right )^{2}\right )} \sqrt {-\frac {a d^{2}}{b}}\right )} {\rm Ei}\left (d x + \sqrt {-\frac {a d^{2}}{b}}\right ) - {\left ({\left (a b^{2} d^{2} x^{4} + 2 \, a^{2} b d^{2} x^{2} + a^{3} d^{2}\right )} \cosh \left (d x + c\right )^{2} - {\left (a b^{2} d^{2} x^{4} + 2 \, a^{2} b d^{2} x^{2} + a^{3} d^{2}\right )} \sinh \left (d x + c\right )^{2} - 3 \, {\left ({\left (b^{3} x^{4} + 2 \, a b^{2} x^{2} + a^{2} b\right )} \cosh \left (d x + c\right )^{2} - {\left (b^{3} x^{4} + 2 \, a b^{2} x^{2} + a^{2} b\right )} \sinh \left (d x + c\right )^{2}\right )} \sqrt {-\frac {a d^{2}}{b}}\right )} {\rm Ei}\left (-d x - \sqrt {-\frac {a d^{2}}{b}}\right )\right )} \sinh \left (-c + \sqrt {-\frac {a d^{2}}{b}}\right )}{32 \, {\left ({\left (a b^{5} x^{4} + 2 \, a^{2} b^{4} x^{2} + a^{3} b^{3}\right )} \cosh \left (d x + c\right )^{2} - {\left (a b^{5} x^{4} + 2 \, a^{2} b^{4} x^{2} + a^{3} b^{3}\right )} \sinh \left (d x + c\right )^{2}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
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 \begin {gather*} \text {could not integrate} \end {gather*}
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 \begin {gather*} \int \frac {x^3\,\mathrm {cosh}\left (c+d\,x\right )}{{\left (b\,x^2+a\right )}^3} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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