Optimal. Leaf size=317 \[ \frac {b c x \sqrt {-1+c x} \sqrt {1+c x}}{6 d^2 \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2}}+\frac {a+b \cosh ^{-1}(c x)}{3 d \left (d-c^2 d x^2\right )^{3/2}}+\frac {a+b \cosh ^{-1}(c x)}{d^2 \sqrt {d-c^2 d x^2}}+\frac {2 \sqrt {-1+c x} \sqrt {1+c x} \left (a+b \cosh ^{-1}(c x)\right ) \text {ArcTan}\left (e^{\cosh ^{-1}(c x)}\right )}{d^2 \sqrt {d-c^2 d x^2}}+\frac {7 b \sqrt {-1+c x} \sqrt {1+c x} \tanh ^{-1}(c x)}{6 d^2 \sqrt {d-c^2 d x^2}}-\frac {i b \sqrt {-1+c x} \sqrt {1+c x} \text {PolyLog}\left (2,-i e^{\cosh ^{-1}(c x)}\right )}{d^2 \sqrt {d-c^2 d x^2}}+\frac {i b \sqrt {-1+c x} \sqrt {1+c x} \text {PolyLog}\left (2,i e^{\cosh ^{-1}(c x)}\right )}{d^2 \sqrt {d-c^2 d x^2}} \]
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Rubi [A]
time = 0.27, antiderivative size = 317, normalized size of antiderivative = 1.00, number of steps
used = 13, number of rules used = 9, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {5936, 5946,
4265, 2317, 2438, 35, 213, 41, 205} \begin {gather*} \frac {2 \sqrt {c x-1} \sqrt {c x+1} \text {ArcTan}\left (e^{\cosh ^{-1}(c x)}\right ) \left (a+b \cosh ^{-1}(c x)\right )}{d^2 \sqrt {d-c^2 d x^2}}+\frac {a+b \cosh ^{-1}(c x)}{d^2 \sqrt {d-c^2 d x^2}}+\frac {a+b \cosh ^{-1}(c x)}{3 d \left (d-c^2 d x^2\right )^{3/2}}-\frac {i b \sqrt {c x-1} \sqrt {c x+1} \text {Li}_2\left (-i e^{\cosh ^{-1}(c x)}\right )}{d^2 \sqrt {d-c^2 d x^2}}+\frac {i b \sqrt {c x-1} \sqrt {c x+1} \text {Li}_2\left (i e^{\cosh ^{-1}(c x)}\right )}{d^2 \sqrt {d-c^2 d x^2}}+\frac {b c x \sqrt {c x-1} \sqrt {c x+1}}{6 d^2 \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2}}+\frac {7 b \sqrt {c x-1} \sqrt {c x+1} \tanh ^{-1}(c x)}{6 d^2 \sqrt {d-c^2 d x^2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 35
Rule 41
Rule 205
Rule 213
Rule 2317
Rule 2438
Rule 4265
Rule 5936
Rule 5946
Rubi steps
\begin {align*} \int \frac {a+b \cosh ^{-1}(c x)}{x \left (d-c^2 d x^2\right )^{5/2}} \, dx &=\frac {\left (\sqrt {-1+c x} \sqrt {1+c x}\right ) \int \frac {a+b \cosh ^{-1}(c x)}{x (-1+c x)^{5/2} (1+c x)^{5/2}} \, dx}{d^2 \sqrt {d-c^2 d x^2}}\\ &=\frac {a+b \cosh ^{-1}(c x)}{3 d^2 (1-c x) (1+c x) \sqrt {d-c^2 d x^2}}-\frac {\left (\sqrt {-1+c x} \sqrt {1+c x}\right ) \int \frac {a+b \cosh ^{-1}(c x)}{x (-1+c x)^{3/2} (1+c x)^{3/2}} \, dx}{d^2 \sqrt {d-c^2 d x^2}}+\frac {\left (b c \sqrt {-1+c x} \sqrt {1+c x}\right ) \int \frac {1}{\left (-1+c^2 x^2\right )^2} \, dx}{3 d^2 \sqrt {d-c^2 d x^2}}\\ &=\frac {b c x \sqrt {-1+c x} \sqrt {1+c x}}{6 d^2 \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2}}+\frac {a+b \cosh ^{-1}(c x)}{d^2 \sqrt {d-c^2 d x^2}}+\frac {a+b \cosh ^{-1}(c x)}{3 d^2 (1-c x) (1+c x) \sqrt {d-c^2 d x^2}}+\frac {\left (\sqrt {-1+c x} \sqrt {1+c x}\right ) \int \frac {a+b \cosh ^{-1}(c x)}{x \sqrt {-1+c x} \sqrt {1+c x}} \, dx}{d^2 \sqrt {d-c^2 d x^2}}-\frac {\left (b c \sqrt {-1+c x} \sqrt {1+c x}\right ) \int \frac {1}{-1+c^2 x^2} \, dx}{6 d^2 \sqrt {d-c^2 d x^2}}-\frac {\left (b c \sqrt {-1+c x} \sqrt {1+c x}\right ) \int \frac {1}{-1+c^2 x^2} \, dx}{d^2 \sqrt {d-c^2 d x^2}}\\ &=\frac {b c x \sqrt {-1+c x} \sqrt {1+c x}}{6 d^2 \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2}}+\frac {a+b \cosh ^{-1}(c x)}{d^2 \sqrt {d-c^2 d x^2}}+\frac {a+b \cosh ^{-1}(c x)}{3 d^2 (1-c x) (1+c x) \sqrt {d-c^2 d x^2}}+\frac {7 b \sqrt {-1+c x} \sqrt {1+c x} \tanh ^{-1}(c x)}{6 d^2 \sqrt {d-c^2 d x^2}}+\frac {\left (\sqrt {-1+c x} \sqrt {1+c x}\right ) \text {Subst}\left (\int (a+b x) \text {sech}(x) \, dx,x,\cosh ^{-1}(c x)\right )}{d^2 \sqrt {d-c^2 d x^2}}\\ &=\frac {b c x \sqrt {-1+c x} \sqrt {1+c x}}{6 d^2 \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2}}+\frac {a+b \cosh ^{-1}(c x)}{d^2 \sqrt {d-c^2 d x^2}}+\frac {a+b \cosh ^{-1}(c x)}{3 d^2 (1-c x) (1+c x) \sqrt {d-c^2 d x^2}}+\frac {2 \sqrt {-1+c x} \sqrt {1+c x} \left (a+b \cosh ^{-1}(c x)\right ) \tan ^{-1}\left (e^{\cosh ^{-1}(c x)}\right )}{d^2 \sqrt {d-c^2 d x^2}}+\frac {7 b \sqrt {-1+c x} \sqrt {1+c x} \tanh ^{-1}(c x)}{6 d^2 \sqrt {d-c^2 d x^2}}-\frac {\left (i b \sqrt {-1+c x} \sqrt {1+c x}\right ) \text {Subst}\left (\int \log \left (1-i e^x\right ) \, dx,x,\cosh ^{-1}(c x)\right )}{d^2 \sqrt {d-c^2 d x^2}}+\frac {\left (i b \sqrt {-1+c x} \sqrt {1+c x}\right ) \text {Subst}\left (\int \log \left (1+i e^x\right ) \, dx,x,\cosh ^{-1}(c x)\right )}{d^2 \sqrt {d-c^2 d x^2}}\\ &=\frac {b c x \sqrt {-1+c x} \sqrt {1+c x}}{6 d^2 \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2}}+\frac {a+b \cosh ^{-1}(c x)}{d^2 \sqrt {d-c^2 d x^2}}+\frac {a+b \cosh ^{-1}(c x)}{3 d^2 (1-c x) (1+c x) \sqrt {d-c^2 d x^2}}+\frac {2 \sqrt {-1+c x} \sqrt {1+c x} \left (a+b \cosh ^{-1}(c x)\right ) \tan ^{-1}\left (e^{\cosh ^{-1}(c x)}\right )}{d^2 \sqrt {d-c^2 d x^2}}+\frac {7 b \sqrt {-1+c x} \sqrt {1+c x} \tanh ^{-1}(c x)}{6 d^2 \sqrt {d-c^2 d x^2}}-\frac {\left (i b \sqrt {-1+c x} \sqrt {1+c x}\right ) \text {Subst}\left (\int \frac {\log (1-i x)}{x} \, dx,x,e^{\cosh ^{-1}(c x)}\right )}{d^2 \sqrt {d-c^2 d x^2}}+\frac {\left (i b \sqrt {-1+c x} \sqrt {1+c x}\right ) \text {Subst}\left (\int \frac {\log (1+i x)}{x} \, dx,x,e^{\cosh ^{-1}(c x)}\right )}{d^2 \sqrt {d-c^2 d x^2}}\\ &=\frac {b c x \sqrt {-1+c x} \sqrt {1+c x}}{6 d^2 \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2}}+\frac {a+b \cosh ^{-1}(c x)}{d^2 \sqrt {d-c^2 d x^2}}+\frac {a+b \cosh ^{-1}(c x)}{3 d^2 (1-c x) (1+c x) \sqrt {d-c^2 d x^2}}+\frac {2 \sqrt {-1+c x} \sqrt {1+c x} \left (a+b \cosh ^{-1}(c x)\right ) \tan ^{-1}\left (e^{\cosh ^{-1}(c x)}\right )}{d^2 \sqrt {d-c^2 d x^2}}+\frac {7 b \sqrt {-1+c x} \sqrt {1+c x} \tanh ^{-1}(c x)}{6 d^2 \sqrt {d-c^2 d x^2}}-\frac {i b \sqrt {-1+c x} \sqrt {1+c x} \text {Li}_2\left (-i e^{\cosh ^{-1}(c x)}\right )}{d^2 \sqrt {d-c^2 d x^2}}+\frac {i b \sqrt {-1+c x} \sqrt {1+c x} \text {Li}_2\left (i e^{\cosh ^{-1}(c x)}\right )}{d^2 \sqrt {d-c^2 d x^2}}\\ \end {align*}
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Mathematica [A]
time = 5.12, size = 364, normalized size = 1.15 \begin {gather*} -\frac {a \left (-4+3 c^2 x^2\right ) \sqrt {d-c^2 d x^2}}{3 d^3 \left (-1+c^2 x^2\right )^2}+\frac {a \log (x)}{d^{5/2}}-\frac {a \log \left (d+\sqrt {d} \sqrt {d-c^2 d x^2}\right )}{d^{5/2}}+\frac {b \sqrt {\frac {-1+c x}{1+c x}} (1+c x) \left (14 \cosh ^{-1}(c x) \coth \left (\frac {1}{2} \cosh ^{-1}(c x)\right )-\text {csch}^2\left (\frac {1}{2} \cosh ^{-1}(c x)\right )-\frac {1}{2} \sqrt {\frac {-1+c x}{1+c x}} (1+c x) \cosh ^{-1}(c x) \text {csch}^4\left (\frac {1}{2} \cosh ^{-1}(c x)\right )-24 i \cosh ^{-1}(c x) \log \left (1-i e^{-\cosh ^{-1}(c x)}\right )+24 i \cosh ^{-1}(c x) \log \left (1+i e^{-\cosh ^{-1}(c x)}\right )-28 \log \left (\tanh \left (\frac {1}{2} \cosh ^{-1}(c x)\right )\right )-24 i \text {PolyLog}\left (2,-i e^{-\cosh ^{-1}(c x)}\right )+24 i \text {PolyLog}\left (2,i e^{-\cosh ^{-1}(c x)}\right )-\text {sech}^2\left (\frac {1}{2} \cosh ^{-1}(c x)\right )-\frac {8 \cosh ^{-1}(c x) \sinh ^4\left (\frac {1}{2} \cosh ^{-1}(c x)\right )}{\left (\frac {-1+c x}{1+c x}\right )^{3/2} (1+c x)^3}-14 \cosh ^{-1}(c x) \tanh \left (\frac {1}{2} \cosh ^{-1}(c x)\right )\right )}{24 d^2 \sqrt {d-c^2 d x^2}} \end {gather*}
Warning: Unable to verify antiderivative.
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Maple [A]
time = 2.77, size = 619, normalized size = 1.95
method | result | size |
default | \(\frac {a}{3 d \left (-c^{2} d \,x^{2}+d \right )^{\frac {3}{2}}}+\frac {a}{d^{2} \sqrt {-c^{2} d \,x^{2}+d}}-\frac {a \ln \left (\frac {2 d +2 \sqrt {d}\, \sqrt {-c^{2} d \,x^{2}+d}}{x}\right )}{d^{\frac {5}{2}}}-\frac {b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \mathrm {arccosh}\left (c x \right ) x^{2} c^{2}}{d^{3} \left (c^{2} x^{2}-1\right )^{2}}+\frac {b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {c x +1}\, \sqrt {c x -1}\, x c}{6 d^{3} \left (c^{2} x^{2}-1\right )^{2}}+\frac {4 b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \mathrm {arccosh}\left (c x \right )}{3 d^{3} \left (c^{2} x^{2}-1\right )^{2}}+\frac {7 b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {c x -1}\, \sqrt {c x +1}\, \ln \left (c x +\sqrt {c x -1}\, \sqrt {c x +1}-1\right )}{6 d^{3} \left (c^{2} x^{2}-1\right )}-\frac {7 b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {c x -1}\, \sqrt {c x +1}\, \ln \left (1+c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )}{6 d^{3} \left (c^{2} x^{2}-1\right )}-\frac {i b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {c x -1}\, \sqrt {c x +1}\, \mathrm {arccosh}\left (c x \right ) \ln \left (1-i \left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )\right )}{d^{3} \left (c^{2} x^{2}-1\right )}+\frac {i b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {c x -1}\, \sqrt {c x +1}\, \dilog \left (1+i \left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )\right )}{d^{3} \left (c^{2} x^{2}-1\right )}+\frac {i b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {c x -1}\, \sqrt {c x +1}\, \mathrm {arccosh}\left (c x \right ) \ln \left (1+i \left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )\right )}{d^{3} \left (c^{2} x^{2}-1\right )}-\frac {i b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {c x -1}\, \sqrt {c x +1}\, \dilog \left (1-i \left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )\right )}{d^{3} \left (c^{2} x^{2}-1\right )}\) | \(619\) |
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 {a + b \operatorname {acosh}{\left (c x \right )}}{x \left (- d \left (c x - 1\right ) \left (c x + 1\right )\right )^{\frac {5}{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 = -0.00 \begin {gather*} \int \frac {a+b\,\mathrm {acosh}\left (c\,x\right )}{x\,{\left (d-c^2\,d\,x^2\right )}^{5/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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