Optimal. Leaf size=198 \[ \frac {x \left (a+b \cosh ^{-1}(c x)\right )^2}{d \sqrt {d-c^2 d x^2}}+\frac {\sqrt {-1+c x} \sqrt {1+c x} \left (a+b \cosh ^{-1}(c x)\right )^2}{c d \sqrt {d-c^2 d x^2}}-\frac {2 b \sqrt {-1+c x} \sqrt {1+c x} \left (a+b \cosh ^{-1}(c x)\right ) \log \left (1-e^{2 \cosh ^{-1}(c x)}\right )}{c d \sqrt {d-c^2 d x^2}}-\frac {b^2 \sqrt {-1+c x} \sqrt {1+c x} \text {PolyLog}\left (2,e^{2 \cosh ^{-1}(c x)}\right )}{c d \sqrt {d-c^2 d x^2}} \]
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Rubi [A]
time = 0.14, antiderivative size = 198, normalized size of antiderivative = 1.00, number of steps
used = 6, number of rules used = 6, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {5899, 5913,
3797, 2221, 2317, 2438} \begin {gather*} \frac {x \left (a+b \cosh ^{-1}(c x)\right )^2}{d \sqrt {d-c^2 d x^2}}+\frac {\sqrt {c x-1} \sqrt {c x+1} \left (a+b \cosh ^{-1}(c x)\right )^2}{c d \sqrt {d-c^2 d x^2}}-\frac {2 b \sqrt {c x-1} \sqrt {c x+1} \log \left (1-e^{2 \cosh ^{-1}(c x)}\right ) \left (a+b \cosh ^{-1}(c x)\right )}{c d \sqrt {d-c^2 d x^2}}-\frac {b^2 \sqrt {c x-1} \sqrt {c x+1} \text {Li}_2\left (e^{2 \cosh ^{-1}(c x)}\right )}{c d \sqrt {d-c^2 d x^2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 2221
Rule 2317
Rule 2438
Rule 3797
Rule 5899
Rule 5913
Rubi steps
\begin {align*} \int \frac {\left (a+b \cosh ^{-1}(c x)\right )^2}{\left (d-c^2 d x^2\right )^{3/2}} \, dx &=-\frac {\left (\sqrt {-1+c x} \sqrt {1+c x}\right ) \int \frac {\left (a+b \cosh ^{-1}(c x)\right )^2}{(-1+c x)^{3/2} (1+c x)^{3/2}} \, dx}{d \sqrt {d-c^2 d x^2}}\\ &=\frac {x \left (a+b \cosh ^{-1}(c x)\right )^2}{d \sqrt {d-c^2 d x^2}}+\frac {\left (2 b c \sqrt {-1+c x} \sqrt {1+c x}\right ) \int \frac {x \left (a+b \cosh ^{-1}(c x)\right )}{1-c^2 x^2} \, dx}{d \sqrt {d-c^2 d x^2}}\\ &=\frac {x \left (a+b \cosh ^{-1}(c x)\right )^2}{d \sqrt {d-c^2 d x^2}}-\frac {\left (2 b \sqrt {-1+c x} \sqrt {1+c x}\right ) \text {Subst}\left (\int (a+b x) \coth (x) \, dx,x,\cosh ^{-1}(c x)\right )}{c d \sqrt {d-c^2 d x^2}}\\ &=\frac {x \left (a+b \cosh ^{-1}(c x)\right )^2}{d \sqrt {d-c^2 d x^2}}+\frac {\sqrt {-1+c x} \sqrt {1+c x} \left (a+b \cosh ^{-1}(c x)\right )^2}{c d \sqrt {d-c^2 d x^2}}+\frac {\left (4 b \sqrt {-1+c x} \sqrt {1+c x}\right ) \text {Subst}\left (\int \frac {e^{2 x} (a+b x)}{1-e^{2 x}} \, dx,x,\cosh ^{-1}(c x)\right )}{c d \sqrt {d-c^2 d x^2}}\\ &=\frac {x \left (a+b \cosh ^{-1}(c x)\right )^2}{d \sqrt {d-c^2 d x^2}}+\frac {\sqrt {-1+c x} \sqrt {1+c x} \left (a+b \cosh ^{-1}(c x)\right )^2}{c d \sqrt {d-c^2 d x^2}}-\frac {2 b \sqrt {-1+c x} \sqrt {1+c x} \left (a+b \cosh ^{-1}(c x)\right ) \log \left (1-e^{2 \cosh ^{-1}(c x)}\right )}{c d \sqrt {d-c^2 d x^2}}+\frac {\left (2 b^2 \sqrt {-1+c x} \sqrt {1+c x}\right ) \text {Subst}\left (\int \log \left (1-e^{2 x}\right ) \, dx,x,\cosh ^{-1}(c x)\right )}{c d \sqrt {d-c^2 d x^2}}\\ &=\frac {x \left (a+b \cosh ^{-1}(c x)\right )^2}{d \sqrt {d-c^2 d x^2}}+\frac {\sqrt {-1+c x} \sqrt {1+c x} \left (a+b \cosh ^{-1}(c x)\right )^2}{c d \sqrt {d-c^2 d x^2}}-\frac {2 b \sqrt {-1+c x} \sqrt {1+c x} \left (a+b \cosh ^{-1}(c x)\right ) \log \left (1-e^{2 \cosh ^{-1}(c x)}\right )}{c d \sqrt {d-c^2 d x^2}}+\frac {\left (b^2 \sqrt {-1+c x} \sqrt {1+c x}\right ) \text {Subst}\left (\int \frac {\log (1-x)}{x} \, dx,x,e^{2 \cosh ^{-1}(c x)}\right )}{c d \sqrt {d-c^2 d x^2}}\\ &=\frac {x \left (a+b \cosh ^{-1}(c x)\right )^2}{d \sqrt {d-c^2 d x^2}}+\frac {\sqrt {-1+c x} \sqrt {1+c x} \left (a+b \cosh ^{-1}(c x)\right )^2}{c d \sqrt {d-c^2 d x^2}}-\frac {2 b \sqrt {-1+c x} \sqrt {1+c x} \left (a+b \cosh ^{-1}(c x)\right ) \log \left (1-e^{2 \cosh ^{-1}(c x)}\right )}{c d \sqrt {d-c^2 d x^2}}-\frac {b^2 \sqrt {-1+c x} \sqrt {1+c x} \text {Li}_2\left (e^{2 \cosh ^{-1}(c x)}\right )}{c d \sqrt {d-c^2 d x^2}}\\ \end {align*}
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Mathematica [A]
time = 0.32, size = 126, normalized size = 0.64 \begin {gather*} \frac {x \left (a+b \cosh ^{-1}(c x)\right )^2+\frac {\sqrt {-1+c x} \sqrt {1+c x} \left (\left (a+b \cosh ^{-1}(c x)\right ) \left (a+b \cosh ^{-1}(c x)-2 b \log \left (1-e^{\cosh ^{-1}(c x)}\right )-2 b \log \left (1+e^{\cosh ^{-1}(c x)}\right )\right )-2 b^2 \text {PolyLog}\left (2,-e^{\cosh ^{-1}(c x)}\right )-2 b^2 \text {PolyLog}\left (2,e^{\cosh ^{-1}(c x)}\right )\right )}{c}}{d \sqrt {d-c^2 d x^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. \(577\) vs.
\(2(204)=408\).
time = 1.56, size = 578, normalized size = 2.92
method | result | size |
default | \(\frac {a^{2} x}{d \sqrt {-c^{2} d \,x^{2}+d}}-\frac {b^{2} \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {c x -1}\, \sqrt {c x +1}\, \mathrm {arccosh}\left (c x \right )^{2}}{d^{2} c \left (c^{2} x^{2}-1\right )}-\frac {b^{2} \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \mathrm {arccosh}\left (c x \right )^{2} x}{d^{2} \left (c^{2} x^{2}-1\right )}+\frac {2 b^{2} \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {c x -1}\, \sqrt {c x +1}\, \mathrm {arccosh}\left (c x \right ) \ln \left (1+c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )}{d^{2} c \left (c^{2} x^{2}-1\right )}+\frac {2 b^{2} \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {c x -1}\, \sqrt {c x +1}\, \polylog \left (2, -c x -\sqrt {c x -1}\, \sqrt {c x +1}\right )}{d^{2} c \left (c^{2} x^{2}-1\right )}+\frac {2 b^{2} \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {c x -1}\, \sqrt {c x +1}\, \mathrm {arccosh}\left (c x \right ) \ln \left (1-c x -\sqrt {c x -1}\, \sqrt {c x +1}\right )}{d^{2} c \left (c^{2} x^{2}-1\right )}+\frac {2 b^{2} \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {c x -1}\, \sqrt {c x +1}\, \polylog \left (2, c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )}{d^{2} c \left (c^{2} x^{2}-1\right )}-\frac {2 a b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {c x -1}\, \sqrt {c x +1}\, \mathrm {arccosh}\left (c x \right )}{d^{2} c \left (c^{2} x^{2}-1\right )}-\frac {2 a b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \mathrm {arccosh}\left (c x \right ) x}{d^{2} \left (c^{2} x^{2}-1\right )}+\frac {2 a b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {c x -1}\, \sqrt {c x +1}\, \ln \left (\left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )^{2}-1\right )}{d^{2} c \left (c^{2} x^{2}-1\right )}\) | \(578\) |
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 (a + b \operatorname {acosh}{\left (c x \right )}\right )^{2}}{\left (- d \left (c x - 1\right ) \left (c x + 1\right )\right )^{\frac {3}{2}}}\, 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 = -0.01 \begin {gather*} \int \frac {{\left (a+b\,\mathrm {acosh}\left (c\,x\right )\right )}^2}{{\left (d-c^2\,d\,x^2\right )}^{3/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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