Optimal. Leaf size=388 \[ -\frac {d^2 \text {Chi}\left (\frac {a+b \cosh ^{-1}(c x)}{b}\right ) \sinh \left (\frac {a}{b}\right )}{b c}-\frac {d e \text {Chi}\left (\frac {a+b \cosh ^{-1}(c x)}{b}\right ) \sinh \left (\frac {a}{b}\right )}{2 b c^3}-\frac {e^2 \text {Chi}\left (\frac {a+b \cosh ^{-1}(c x)}{b}\right ) \sinh \left (\frac {a}{b}\right )}{8 b c^5}-\frac {d e \text {Chi}\left (\frac {3 \left (a+b \cosh ^{-1}(c x)\right )}{b}\right ) \sinh \left (\frac {3 a}{b}\right )}{2 b c^3}-\frac {3 e^2 \text {Chi}\left (\frac {3 \left (a+b \cosh ^{-1}(c x)\right )}{b}\right ) \sinh \left (\frac {3 a}{b}\right )}{16 b c^5}-\frac {e^2 \text {Chi}\left (\frac {5 \left (a+b \cosh ^{-1}(c x)\right )}{b}\right ) \sinh \left (\frac {5 a}{b}\right )}{16 b c^5}+\frac {d^2 \cosh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a+b \cosh ^{-1}(c x)}{b}\right )}{b c}+\frac {d e \cosh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a+b \cosh ^{-1}(c x)}{b}\right )}{2 b c^3}+\frac {e^2 \cosh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a+b \cosh ^{-1}(c x)}{b}\right )}{8 b c^5}+\frac {d e \cosh \left (\frac {3 a}{b}\right ) \text {Shi}\left (\frac {3 \left (a+b \cosh ^{-1}(c x)\right )}{b}\right )}{2 b c^3}+\frac {3 e^2 \cosh \left (\frac {3 a}{b}\right ) \text {Shi}\left (\frac {3 \left (a+b \cosh ^{-1}(c x)\right )}{b}\right )}{16 b c^5}+\frac {e^2 \cosh \left (\frac {5 a}{b}\right ) \text {Shi}\left (\frac {5 \left (a+b \cosh ^{-1}(c x)\right )}{b}\right )}{16 b c^5} \]
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Rubi [A]
time = 0.52, antiderivative size = 388, normalized size of antiderivative = 1.00, number of steps
used = 27, number of rules used = 7, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.350, Rules used = {5909, 5881,
3384, 3379, 3382, 5887, 5556} \begin {gather*} -\frac {e^2 \sinh \left (\frac {a}{b}\right ) \text {Chi}\left (\frac {a+b \cosh ^{-1}(c x)}{b}\right )}{8 b c^5}-\frac {3 e^2 \sinh \left (\frac {3 a}{b}\right ) \text {Chi}\left (\frac {3 \left (a+b \cosh ^{-1}(c x)\right )}{b}\right )}{16 b c^5}-\frac {e^2 \sinh \left (\frac {5 a}{b}\right ) \text {Chi}\left (\frac {5 \left (a+b \cosh ^{-1}(c x)\right )}{b}\right )}{16 b c^5}+\frac {e^2 \cosh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a+b \cosh ^{-1}(c x)}{b}\right )}{8 b c^5}+\frac {3 e^2 \cosh \left (\frac {3 a}{b}\right ) \text {Shi}\left (\frac {3 \left (a+b \cosh ^{-1}(c x)\right )}{b}\right )}{16 b c^5}+\frac {e^2 \cosh \left (\frac {5 a}{b}\right ) \text {Shi}\left (\frac {5 \left (a+b \cosh ^{-1}(c x)\right )}{b}\right )}{16 b c^5}-\frac {d e \sinh \left (\frac {a}{b}\right ) \text {Chi}\left (\frac {a+b \cosh ^{-1}(c x)}{b}\right )}{2 b c^3}-\frac {d e \sinh \left (\frac {3 a}{b}\right ) \text {Chi}\left (\frac {3 \left (a+b \cosh ^{-1}(c x)\right )}{b}\right )}{2 b c^3}+\frac {d e \cosh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a+b \cosh ^{-1}(c x)}{b}\right )}{2 b c^3}+\frac {d e \cosh \left (\frac {3 a}{b}\right ) \text {Shi}\left (\frac {3 \left (a+b \cosh ^{-1}(c x)\right )}{b}\right )}{2 b c^3}-\frac {d^2 \sinh \left (\frac {a}{b}\right ) \text {Chi}\left (\frac {a+b \cosh ^{-1}(c x)}{b}\right )}{b c}+\frac {d^2 \cosh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a+b \cosh ^{-1}(c x)}{b}\right )}{b c} \end {gather*}
Antiderivative was successfully verified.
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Rule 3379
Rule 3382
Rule 3384
Rule 5556
Rule 5881
Rule 5887
Rule 5909
Rubi steps
\begin {align*} \int \frac {\left (d+e x^2\right )^2}{a+b \cosh ^{-1}(c x)} \, dx &=\int \left (\frac {d^2}{a+b \cosh ^{-1}(c x)}+\frac {2 d e x^2}{a+b \cosh ^{-1}(c x)}+\frac {e^2 x^4}{a+b \cosh ^{-1}(c x)}\right ) \, dx\\ &=d^2 \int \frac {1}{a+b \cosh ^{-1}(c x)} \, dx+(2 d e) \int \frac {x^2}{a+b \cosh ^{-1}(c x)} \, dx+e^2 \int \frac {x^4}{a+b \cosh ^{-1}(c x)} \, dx\\ &=-\frac {d^2 \text {Subst}\left (\int \frac {\sinh \left (\frac {a}{b}-\frac {x}{b}\right )}{x} \, dx,x,a+b \cosh ^{-1}(c x)\right )}{b c}+\frac {(2 d e) \text {Subst}\left (\int \frac {\cosh ^2(x) \sinh (x)}{a+b x} \, dx,x,\cosh ^{-1}(c x)\right )}{c^3}+\frac {e^2 \text {Subst}\left (\int \frac {\cosh ^4(x) \sinh (x)}{a+b x} \, dx,x,\cosh ^{-1}(c x)\right )}{c^5}\\ &=\frac {(2 d e) \text {Subst}\left (\int \left (\frac {\sinh (x)}{4 (a+b x)}+\frac {\sinh (3 x)}{4 (a+b x)}\right ) \, dx,x,\cosh ^{-1}(c x)\right )}{c^3}+\frac {e^2 \text {Subst}\left (\int \left (\frac {\sinh (x)}{8 (a+b x)}+\frac {3 \sinh (3 x)}{16 (a+b x)}+\frac {\sinh (5 x)}{16 (a+b x)}\right ) \, dx,x,\cosh ^{-1}(c x)\right )}{c^5}+\frac {\left (d^2 \cosh \left (\frac {a}{b}\right )\right ) \text {Subst}\left (\int \frac {\sinh \left (\frac {x}{b}\right )}{x} \, dx,x,a+b \cosh ^{-1}(c x)\right )}{b c}-\frac {\left (d^2 \sinh \left (\frac {a}{b}\right )\right ) \text {Subst}\left (\int \frac {\cosh \left (\frac {x}{b}\right )}{x} \, dx,x,a+b \cosh ^{-1}(c x)\right )}{b c}\\ &=-\frac {d^2 \text {Chi}\left (\frac {a+b \cosh ^{-1}(c x)}{b}\right ) \sinh \left (\frac {a}{b}\right )}{b c}+\frac {d^2 \cosh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a+b \cosh ^{-1}(c x)}{b}\right )}{b c}+\frac {(d e) \text {Subst}\left (\int \frac {\sinh (x)}{a+b x} \, dx,x,\cosh ^{-1}(c x)\right )}{2 c^3}+\frac {(d e) \text {Subst}\left (\int \frac {\sinh (3 x)}{a+b x} \, dx,x,\cosh ^{-1}(c x)\right )}{2 c^3}+\frac {e^2 \text {Subst}\left (\int \frac {\sinh (5 x)}{a+b x} \, dx,x,\cosh ^{-1}(c x)\right )}{16 c^5}+\frac {e^2 \text {Subst}\left (\int \frac {\sinh (x)}{a+b x} \, dx,x,\cosh ^{-1}(c x)\right )}{8 c^5}+\frac {\left (3 e^2\right ) \text {Subst}\left (\int \frac {\sinh (3 x)}{a+b x} \, dx,x,\cosh ^{-1}(c x)\right )}{16 c^5}\\ &=-\frac {d^2 \text {Chi}\left (\frac {a+b \cosh ^{-1}(c x)}{b}\right ) \sinh \left (\frac {a}{b}\right )}{b c}+\frac {d^2 \cosh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a+b \cosh ^{-1}(c x)}{b}\right )}{b c}+\frac {\left (d e \cosh \left (\frac {a}{b}\right )\right ) \text {Subst}\left (\int \frac {\sinh \left (\frac {a}{b}+x\right )}{a+b x} \, dx,x,\cosh ^{-1}(c x)\right )}{2 c^3}+\frac {\left (e^2 \cosh \left (\frac {a}{b}\right )\right ) \text {Subst}\left (\int \frac {\sinh \left (\frac {a}{b}+x\right )}{a+b x} \, dx,x,\cosh ^{-1}(c x)\right )}{8 c^5}+\frac {\left (d e \cosh \left (\frac {3 a}{b}\right )\right ) \text {Subst}\left (\int \frac {\sinh \left (\frac {3 a}{b}+3 x\right )}{a+b x} \, dx,x,\cosh ^{-1}(c x)\right )}{2 c^3}+\frac {\left (3 e^2 \cosh \left (\frac {3 a}{b}\right )\right ) \text {Subst}\left (\int \frac {\sinh \left (\frac {3 a}{b}+3 x\right )}{a+b x} \, dx,x,\cosh ^{-1}(c x)\right )}{16 c^5}+\frac {\left (e^2 \cosh \left (\frac {5 a}{b}\right )\right ) \text {Subst}\left (\int \frac {\sinh \left (\frac {5 a}{b}+5 x\right )}{a+b x} \, dx,x,\cosh ^{-1}(c x)\right )}{16 c^5}-\frac {\left (d e \sinh \left (\frac {a}{b}\right )\right ) \text {Subst}\left (\int \frac {\cosh \left (\frac {a}{b}+x\right )}{a+b x} \, dx,x,\cosh ^{-1}(c x)\right )}{2 c^3}-\frac {\left (e^2 \sinh \left (\frac {a}{b}\right )\right ) \text {Subst}\left (\int \frac {\cosh \left (\frac {a}{b}+x\right )}{a+b x} \, dx,x,\cosh ^{-1}(c x)\right )}{8 c^5}-\frac {\left (d e \sinh \left (\frac {3 a}{b}\right )\right ) \text {Subst}\left (\int \frac {\cosh \left (\frac {3 a}{b}+3 x\right )}{a+b x} \, dx,x,\cosh ^{-1}(c x)\right )}{2 c^3}-\frac {\left (3 e^2 \sinh \left (\frac {3 a}{b}\right )\right ) \text {Subst}\left (\int \frac {\cosh \left (\frac {3 a}{b}+3 x\right )}{a+b x} \, dx,x,\cosh ^{-1}(c x)\right )}{16 c^5}-\frac {\left (e^2 \sinh \left (\frac {5 a}{b}\right )\right ) \text {Subst}\left (\int \frac {\cosh \left (\frac {5 a}{b}+5 x\right )}{a+b x} \, dx,x,\cosh ^{-1}(c x)\right )}{16 c^5}\\ &=-\frac {d e \text {Chi}\left (\frac {a}{b}+\cosh ^{-1}(c x)\right ) \sinh \left (\frac {a}{b}\right )}{2 b c^3}-\frac {e^2 \text {Chi}\left (\frac {a}{b}+\cosh ^{-1}(c x)\right ) \sinh \left (\frac {a}{b}\right )}{8 b c^5}-\frac {d^2 \text {Chi}\left (\frac {a+b \cosh ^{-1}(c x)}{b}\right ) \sinh \left (\frac {a}{b}\right )}{b c}-\frac {d e \text {Chi}\left (\frac {3 a}{b}+3 \cosh ^{-1}(c x)\right ) \sinh \left (\frac {3 a}{b}\right )}{2 b c^3}-\frac {3 e^2 \text {Chi}\left (\frac {3 a}{b}+3 \cosh ^{-1}(c x)\right ) \sinh \left (\frac {3 a}{b}\right )}{16 b c^5}-\frac {e^2 \text {Chi}\left (\frac {5 a}{b}+5 \cosh ^{-1}(c x)\right ) \sinh \left (\frac {5 a}{b}\right )}{16 b c^5}+\frac {d e \cosh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a}{b}+\cosh ^{-1}(c x)\right )}{2 b c^3}+\frac {e^2 \cosh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a}{b}+\cosh ^{-1}(c x)\right )}{8 b c^5}+\frac {d e \cosh \left (\frac {3 a}{b}\right ) \text {Shi}\left (\frac {3 a}{b}+3 \cosh ^{-1}(c x)\right )}{2 b c^3}+\frac {3 e^2 \cosh \left (\frac {3 a}{b}\right ) \text {Shi}\left (\frac {3 a}{b}+3 \cosh ^{-1}(c x)\right )}{16 b c^5}+\frac {e^2 \cosh \left (\frac {5 a}{b}\right ) \text {Shi}\left (\frac {5 a}{b}+5 \cosh ^{-1}(c x)\right )}{16 b c^5}+\frac {d^2 \cosh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a+b \cosh ^{-1}(c x)}{b}\right )}{b c}\\ \end {align*}
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Mathematica [A]
time = 0.39, size = 254, normalized size = 0.65 \begin {gather*} \frac {-2 \left (8 c^4 d^2+4 c^2 d e+e^2\right ) \text {Chi}\left (\frac {a}{b}+\cosh ^{-1}(c x)\right ) \sinh \left (\frac {a}{b}\right )-e \left (8 c^2 d+3 e\right ) \text {Chi}\left (3 \left (\frac {a}{b}+\cosh ^{-1}(c x)\right )\right ) \sinh \left (\frac {3 a}{b}\right )-e^2 \text {Chi}\left (5 \left (\frac {a}{b}+\cosh ^{-1}(c x)\right )\right ) \sinh \left (\frac {5 a}{b}\right )+16 c^4 d^2 \cosh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a}{b}+\cosh ^{-1}(c x)\right )+8 c^2 d e \cosh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a}{b}+\cosh ^{-1}(c x)\right )+2 e^2 \cosh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a}{b}+\cosh ^{-1}(c x)\right )+8 c^2 d e \cosh \left (\frac {3 a}{b}\right ) \text {Shi}\left (3 \left (\frac {a}{b}+\cosh ^{-1}(c x)\right )\right )+3 e^2 \cosh \left (\frac {3 a}{b}\right ) \text {Shi}\left (3 \left (\frac {a}{b}+\cosh ^{-1}(c x)\right )\right )+e^2 \cosh \left (\frac {5 a}{b}\right ) \text {Shi}\left (5 \left (\frac {a}{b}+\cosh ^{-1}(c x)\right )\right )}{16 b c^5} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 9.42, size = 380, normalized size = 0.98
method | result | size |
derivativedivides | \(\frac {-\frac {e^{2} {\mathrm e}^{-\frac {5 a}{b}} \expIntegral \left (1, -5 \,\mathrm {arccosh}\left (c x \right )-\frac {5 a}{b}\right )}{32 c^{4} b}+\frac {e^{2} {\mathrm e}^{\frac {5 a}{b}} \expIntegral \left (1, 5 \,\mathrm {arccosh}\left (c x \right )+\frac {5 a}{b}\right )}{32 c^{4} b}+\frac {{\mathrm e}^{\frac {a}{b}} \expIntegral \left (1, \mathrm {arccosh}\left (c x \right )+\frac {a}{b}\right ) d^{2}}{2 b}+\frac {{\mathrm e}^{\frac {a}{b}} \expIntegral \left (1, \mathrm {arccosh}\left (c x \right )+\frac {a}{b}\right ) d e}{4 c^{2} b}+\frac {{\mathrm e}^{\frac {a}{b}} \expIntegral \left (1, \mathrm {arccosh}\left (c x \right )+\frac {a}{b}\right ) e^{2}}{16 c^{4} b}-\frac {{\mathrm e}^{-\frac {a}{b}} \expIntegral \left (1, -\mathrm {arccosh}\left (c x \right )-\frac {a}{b}\right ) d^{2}}{2 b}-\frac {{\mathrm e}^{-\frac {a}{b}} \expIntegral \left (1, -\mathrm {arccosh}\left (c x \right )-\frac {a}{b}\right ) d e}{4 c^{2} b}-\frac {{\mathrm e}^{-\frac {a}{b}} \expIntegral \left (1, -\mathrm {arccosh}\left (c x \right )-\frac {a}{b}\right ) e^{2}}{16 c^{4} b}+\frac {e \,{\mathrm e}^{\frac {3 a}{b}} \expIntegral \left (1, 3 \,\mathrm {arccosh}\left (c x \right )+\frac {3 a}{b}\right ) d}{4 c^{2} b}+\frac {3 e^{2} {\mathrm e}^{\frac {3 a}{b}} \expIntegral \left (1, 3 \,\mathrm {arccosh}\left (c x \right )+\frac {3 a}{b}\right )}{32 c^{4} b}-\frac {e \,{\mathrm e}^{-\frac {3 a}{b}} \expIntegral \left (1, -3 \,\mathrm {arccosh}\left (c x \right )-\frac {3 a}{b}\right ) d}{4 c^{2} b}-\frac {3 e^{2} {\mathrm e}^{-\frac {3 a}{b}} \expIntegral \left (1, -3 \,\mathrm {arccosh}\left (c x \right )-\frac {3 a}{b}\right )}{32 c^{4} b}}{c}\) | \(380\) |
default | \(\frac {-\frac {e^{2} {\mathrm e}^{-\frac {5 a}{b}} \expIntegral \left (1, -5 \,\mathrm {arccosh}\left (c x \right )-\frac {5 a}{b}\right )}{32 c^{4} b}+\frac {e^{2} {\mathrm e}^{\frac {5 a}{b}} \expIntegral \left (1, 5 \,\mathrm {arccosh}\left (c x \right )+\frac {5 a}{b}\right )}{32 c^{4} b}+\frac {{\mathrm e}^{\frac {a}{b}} \expIntegral \left (1, \mathrm {arccosh}\left (c x \right )+\frac {a}{b}\right ) d^{2}}{2 b}+\frac {{\mathrm e}^{\frac {a}{b}} \expIntegral \left (1, \mathrm {arccosh}\left (c x \right )+\frac {a}{b}\right ) d e}{4 c^{2} b}+\frac {{\mathrm e}^{\frac {a}{b}} \expIntegral \left (1, \mathrm {arccosh}\left (c x \right )+\frac {a}{b}\right ) e^{2}}{16 c^{4} b}-\frac {{\mathrm e}^{-\frac {a}{b}} \expIntegral \left (1, -\mathrm {arccosh}\left (c x \right )-\frac {a}{b}\right ) d^{2}}{2 b}-\frac {{\mathrm e}^{-\frac {a}{b}} \expIntegral \left (1, -\mathrm {arccosh}\left (c x \right )-\frac {a}{b}\right ) d e}{4 c^{2} b}-\frac {{\mathrm e}^{-\frac {a}{b}} \expIntegral \left (1, -\mathrm {arccosh}\left (c x \right )-\frac {a}{b}\right ) e^{2}}{16 c^{4} b}+\frac {e \,{\mathrm e}^{\frac {3 a}{b}} \expIntegral \left (1, 3 \,\mathrm {arccosh}\left (c x \right )+\frac {3 a}{b}\right ) d}{4 c^{2} b}+\frac {3 e^{2} {\mathrm e}^{\frac {3 a}{b}} \expIntegral \left (1, 3 \,\mathrm {arccosh}\left (c x \right )+\frac {3 a}{b}\right )}{32 c^{4} b}-\frac {e \,{\mathrm e}^{-\frac {3 a}{b}} \expIntegral \left (1, -3 \,\mathrm {arccosh}\left (c x \right )-\frac {3 a}{b}\right ) d}{4 c^{2} b}-\frac {3 e^{2} {\mathrm e}^{-\frac {3 a}{b}} \expIntegral \left (1, -3 \,\mathrm {arccosh}\left (c x \right )-\frac {3 a}{b}\right )}{32 c^{4} b}}{c}\) | \(380\) |
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 [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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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (d + e x^{2}\right )^{2}}{a + b \operatorname {acosh}{\left (c x \right )}}\, dx \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 {{\left (e\,x^2+d\right )}^2}{a+b\,\mathrm {acosh}\left (c\,x\right )} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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