Optimal. Leaf size=237 \[ -\frac {x^3 \sqrt {-1+c x}}{b c \sqrt {1-c x} \left (a+b \cosh ^{-1}(c x)\right )}-\frac {3 \sqrt {-1+c x} \text {Chi}\left (\frac {a+b \cosh ^{-1}(c x)}{b}\right ) \sinh \left (\frac {a}{b}\right )}{4 b^2 c^4 \sqrt {1-c x}}-\frac {3 \sqrt {-1+c x} \text {Chi}\left (\frac {3 \left (a+b \cosh ^{-1}(c x)\right )}{b}\right ) \sinh \left (\frac {3 a}{b}\right )}{4 b^2 c^4 \sqrt {1-c x}}+\frac {3 \sqrt {-1+c x} \cosh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a+b \cosh ^{-1}(c x)}{b}\right )}{4 b^2 c^4 \sqrt {1-c x}}+\frac {3 \sqrt {-1+c x} \cosh \left (\frac {3 a}{b}\right ) \text {Shi}\left (\frac {3 \left (a+b \cosh ^{-1}(c x)\right )}{b}\right )}{4 b^2 c^4 \sqrt {1-c x}} \]
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Rubi [A]
time = 0.25, antiderivative size = 237, normalized size of antiderivative = 1.00, number of steps
used = 10, number of rules used = 6, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.214, Rules used = {5950, 5887,
5556, 3384, 3379, 3382} \begin {gather*} -\frac {3 \sqrt {c x-1} \sinh \left (\frac {a}{b}\right ) \text {Chi}\left (\frac {a+b \cosh ^{-1}(c x)}{b}\right )}{4 b^2 c^4 \sqrt {1-c x}}-\frac {3 \sqrt {c x-1} \sinh \left (\frac {3 a}{b}\right ) \text {Chi}\left (\frac {3 \left (a+b \cosh ^{-1}(c x)\right )}{b}\right )}{4 b^2 c^4 \sqrt {1-c x}}+\frac {3 \sqrt {c x-1} \cosh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a+b \cosh ^{-1}(c x)}{b}\right )}{4 b^2 c^4 \sqrt {1-c x}}+\frac {3 \sqrt {c x-1} \cosh \left (\frac {3 a}{b}\right ) \text {Shi}\left (\frac {3 \left (a+b \cosh ^{-1}(c x)\right )}{b}\right )}{4 b^2 c^4 \sqrt {1-c x}}-\frac {x^3 \sqrt {c x-1}}{b c \sqrt {1-c x} \left (a+b \cosh ^{-1}(c x)\right )} \end {gather*}
Antiderivative was successfully verified.
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Rule 3379
Rule 3382
Rule 3384
Rule 5556
Rule 5887
Rule 5950
Rubi steps
\begin {align*} \int \frac {x^3}{\sqrt {1-c^2 x^2} \left (a+b \cosh ^{-1}(c x)\right )^2} \, dx &=\frac {\left (\sqrt {-1+c x} \sqrt {1+c x}\right ) \int \frac {x^3}{\sqrt {-1+c x} \sqrt {1+c x} \left (a+b \cosh ^{-1}(c x)\right )^2} \, dx}{\sqrt {1-c^2 x^2}}\\ &=-\frac {x^3 \sqrt {-1+c x} \sqrt {1+c x}}{b c \sqrt {1-c^2 x^2} \left (a+b \cosh ^{-1}(c x)\right )}+\frac {\left (3 \sqrt {-1+c x} \sqrt {1+c x}\right ) \int \frac {x^2}{a+b \cosh ^{-1}(c x)} \, dx}{b c \sqrt {1-c^2 x^2}}\\ &=-\frac {x^3 \sqrt {-1+c x} \sqrt {1+c x}}{b c \sqrt {1-c^2 x^2} \left (a+b \cosh ^{-1}(c x)\right )}+\frac {\left (3 \sqrt {-1+c x} \sqrt {1+c x}\right ) \text {Subst}\left (\int \frac {\cosh ^2(x) \sinh (x)}{a+b x} \, dx,x,\cosh ^{-1}(c x)\right )}{b c^4 \sqrt {1-c^2 x^2}}\\ &=-\frac {x^3 \sqrt {-1+c x} \sqrt {1+c x}}{b c \sqrt {1-c^2 x^2} \left (a+b \cosh ^{-1}(c x)\right )}+\frac {\left (3 \sqrt {-1+c x} \sqrt {1+c x}\right ) \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 )}{b c^4 \sqrt {1-c^2 x^2}}\\ &=-\frac {x^3 \sqrt {-1+c x} \sqrt {1+c x}}{b c \sqrt {1-c^2 x^2} \left (a+b \cosh ^{-1}(c x)\right )}+\frac {\left (3 \sqrt {-1+c x} \sqrt {1+c x}\right ) \text {Subst}\left (\int \frac {\sinh (x)}{a+b x} \, dx,x,\cosh ^{-1}(c x)\right )}{4 b c^4 \sqrt {1-c^2 x^2}}+\frac {\left (3 \sqrt {-1+c x} \sqrt {1+c x}\right ) \text {Subst}\left (\int \frac {\sinh (3 x)}{a+b x} \, dx,x,\cosh ^{-1}(c x)\right )}{4 b c^4 \sqrt {1-c^2 x^2}}\\ &=-\frac {x^3 \sqrt {-1+c x} \sqrt {1+c x}}{b c \sqrt {1-c^2 x^2} \left (a+b \cosh ^{-1}(c x)\right )}+\frac {\left (3 \sqrt {-1+c x} \sqrt {1+c x} \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 )}{4 b c^4 \sqrt {1-c^2 x^2}}+\frac {\left (3 \sqrt {-1+c x} \sqrt {1+c x} \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 )}{4 b c^4 \sqrt {1-c^2 x^2}}-\frac {\left (3 \sqrt {-1+c x} \sqrt {1+c x} \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 )}{4 b c^4 \sqrt {1-c^2 x^2}}-\frac {\left (3 \sqrt {-1+c x} \sqrt {1+c x} \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 )}{4 b c^4 \sqrt {1-c^2 x^2}}\\ &=-\frac {x^3 \sqrt {-1+c x} \sqrt {1+c x}}{b c \sqrt {1-c^2 x^2} \left (a+b \cosh ^{-1}(c x)\right )}-\frac {3 \sqrt {-1+c x} \sqrt {1+c x} \text {Chi}\left (\frac {a}{b}+\cosh ^{-1}(c x)\right ) \sinh \left (\frac {a}{b}\right )}{4 b^2 c^4 \sqrt {1-c^2 x^2}}-\frac {3 \sqrt {-1+c x} \sqrt {1+c x} \text {Chi}\left (\frac {3 a}{b}+3 \cosh ^{-1}(c x)\right ) \sinh \left (\frac {3 a}{b}\right )}{4 b^2 c^4 \sqrt {1-c^2 x^2}}+\frac {3 \sqrt {-1+c x} \sqrt {1+c x} \cosh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a}{b}+\cosh ^{-1}(c x)\right )}{4 b^2 c^4 \sqrt {1-c^2 x^2}}+\frac {3 \sqrt {-1+c x} \sqrt {1+c x} \cosh \left (\frac {3 a}{b}\right ) \text {Shi}\left (\frac {3 a}{b}+3 \cosh ^{-1}(c x)\right )}{4 b^2 c^4 \sqrt {1-c^2 x^2}}\\ \end {align*}
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Mathematica [A]
time = 0.33, size = 144, normalized size = 0.61 \begin {gather*} \frac {\sqrt {1-c^2 x^2} \left (\frac {4 b c^3 x^3}{a+b \cosh ^{-1}(c x)}+3 \text {Chi}\left (\frac {a}{b}+\cosh ^{-1}(c x)\right ) \sinh \left (\frac {a}{b}\right )+3 \text {Chi}\left (3 \left (\frac {a}{b}+\cosh ^{-1}(c x)\right )\right ) \sinh \left (\frac {3 a}{b}\right )-3 \cosh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a}{b}+\cosh ^{-1}(c x)\right )-3 \cosh \left (\frac {3 a}{b}\right ) \text {Shi}\left (3 \left (\frac {a}{b}+\cosh ^{-1}(c x)\right )\right )\right )}{4 b^2 c^4 \sqrt {-1+c x} \sqrt {1+c x}} \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. \(631\) vs.
\(2(209)=418\).
time = 4.91, size = 632, normalized size = 2.67
method | result | size |
default | \(-\frac {\sqrt {-c^{2} x^{2}+1}\, \left (-4 \sqrt {c x +1}\, \sqrt {c x -1}\, x^{3} c^{3}+4 c^{4} x^{4}+3 \sqrt {c x +1}\, \sqrt {c x -1}\, x c -5 c^{2} x^{2}+1\right )}{8 \left (c^{2} x^{2}-1\right ) c^{4} b \left (a +b \,\mathrm {arccosh}\left (c x \right )\right )}-\frac {3 \left (\sqrt {c x +1}\, \sqrt {c x -1}\, x c +c^{2} x^{2}-1\right ) \sqrt {-c^{2} x^{2}+1}\, \expIntegral \left (1, 3 \,\mathrm {arccosh}\left (c x \right )+\frac {3 a}{b}\right ) {\mathrm e}^{\frac {-b \,\mathrm {arccosh}\left (c x \right )+3 a}{b}}}{8 b^{2} \left (c^{2} x^{2}-1\right ) c^{4}}+\frac {\sqrt {-c^{2} x^{2}+1}\, \sqrt {c x -1}\, \sqrt {c x +1}\, \left (4 \sqrt {c x +1}\, \sqrt {c x -1}\, b \,c^{2} x^{2}+4 b \,c^{3} x^{3}+3 \expIntegral \left (1, -3 \,\mathrm {arccosh}\left (c x \right )-\frac {3 a}{b}\right ) \mathrm {arccosh}\left (c x \right ) {\mathrm e}^{-\frac {3 a}{b}} b +3 \expIntegral \left (1, -3 \,\mathrm {arccosh}\left (c x \right )-\frac {3 a}{b}\right ) {\mathrm e}^{-\frac {3 a}{b}} a -\sqrt {c x +1}\, \sqrt {c x -1}\, b -3 b c x \right )}{8 \left (c^{2} x^{2}-1\right ) c^{4} b^{2} \left (a +b \,\mathrm {arccosh}\left (c x \right )\right )}-\frac {3 \sqrt {-c^{2} x^{2}+1}\, \left (-\sqrt {c x +1}\, \sqrt {c x -1}\, x c +c^{2} x^{2}-1\right )}{8 \left (c^{2} x^{2}-1\right ) c^{4} b \left (a +b \,\mathrm {arccosh}\left (c x \right )\right )}-\frac {3 \left (\sqrt {c x +1}\, \sqrt {c x -1}\, x c +c^{2} x^{2}-1\right ) \sqrt {-c^{2} x^{2}+1}\, \expIntegral \left (1, \mathrm {arccosh}\left (c x \right )+\frac {a}{b}\right ) {\mathrm e}^{\frac {-b \,\mathrm {arccosh}\left (c x \right )+a}{b}}}{8 b^{2} \left (c^{2} x^{2}-1\right ) c^{4}}+\frac {3 \sqrt {-c^{2} x^{2}+1}\, \sqrt {c x -1}\, \sqrt {c x +1}\, \left (\mathrm {arccosh}\left (c x \right ) {\mathrm e}^{-\frac {a}{b}} \expIntegral \left (1, -\mathrm {arccosh}\left (c x \right )-\frac {a}{b}\right ) b +\sqrt {c x +1}\, \sqrt {c x -1}\, b +{\mathrm e}^{-\frac {a}{b}} \expIntegral \left (1, -\mathrm {arccosh}\left (c x \right )-\frac {a}{b}\right ) a +b c x \right )}{8 \left (c^{2} x^{2}-1\right ) c^{4} b^{2} \left (a +b \,\mathrm {arccosh}\left (c x \right )\right )}\) | \(632\) |
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 {x^{3}}{\sqrt {- \left (c x - 1\right ) \left (c x + 1\right )} \left (a + b \operatorname {acosh}{\left (c x \right )}\right )^{2}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \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}{{\left (a+b\,\mathrm {acosh}\left (c\,x\right )\right )}^2\,\sqrt {1-c^2\,x^2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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