Optimal. Leaf size=186 \[ \frac {b^2}{3 d e^4 (c+d x)}+\frac {b \sqrt {-1+c+d x} \sqrt {1+c+d x} \left (a+b \cosh ^{-1}(c+d x)\right )}{3 d e^4 (c+d x)^2}-\frac {\left (a+b \cosh ^{-1}(c+d x)\right )^2}{3 d e^4 (c+d x)^3}+\frac {2 b \left (a+b \cosh ^{-1}(c+d x)\right ) \text {ArcTan}\left (e^{\cosh ^{-1}(c+d x)}\right )}{3 d e^4}-\frac {i b^2 \text {PolyLog}\left (2,-i e^{\cosh ^{-1}(c+d x)}\right )}{3 d e^4}+\frac {i b^2 \text {PolyLog}\left (2,i e^{\cosh ^{-1}(c+d x)}\right )}{3 d e^4} \]
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Rubi [A]
time = 0.25, antiderivative size = 186, normalized size of antiderivative = 1.00, number of steps
used = 11, number of rules used = 9, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.391, Rules used = {5996, 12,
5883, 5933, 5947, 4265, 2317, 2438, 30} \begin {gather*} \frac {2 b \text {ArcTan}\left (e^{\cosh ^{-1}(c+d x)}\right ) \left (a+b \cosh ^{-1}(c+d x)\right )}{3 d e^4}+\frac {b \sqrt {c+d x-1} \sqrt {c+d x+1} \left (a+b \cosh ^{-1}(c+d x)\right )}{3 d e^4 (c+d x)^2}-\frac {\left (a+b \cosh ^{-1}(c+d x)\right )^2}{3 d e^4 (c+d x)^3}-\frac {i b^2 \text {Li}_2\left (-i e^{\cosh ^{-1}(c+d x)}\right )}{3 d e^4}+\frac {i b^2 \text {Li}_2\left (i e^{\cosh ^{-1}(c+d x)}\right )}{3 d e^4}+\frac {b^2}{3 d e^4 (c+d x)} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 30
Rule 2317
Rule 2438
Rule 4265
Rule 5883
Rule 5933
Rule 5947
Rule 5996
Rubi steps
\begin {align*} \int \frac {\left (a+b \cosh ^{-1}(c+d x)\right )^2}{(c e+d e x)^4} \, dx &=\frac {\text {Subst}\left (\int \frac {\left (a+b \cosh ^{-1}(x)\right )^2}{e^4 x^4} \, dx,x,c+d x\right )}{d}\\ &=\frac {\text {Subst}\left (\int \frac {\left (a+b \cosh ^{-1}(x)\right )^2}{x^4} \, dx,x,c+d x\right )}{d e^4}\\ &=-\frac {\left (a+b \cosh ^{-1}(c+d x)\right )^2}{3 d e^4 (c+d x)^3}+\frac {(2 b) \text {Subst}\left (\int \frac {a+b \cosh ^{-1}(x)}{\sqrt {-1+x} x^3 \sqrt {1+x}} \, dx,x,c+d x\right )}{3 d e^4}\\ &=\frac {b \sqrt {-1+c+d x} \sqrt {1+c+d x} \left (a+b \cosh ^{-1}(c+d x)\right )}{3 d e^4 (c+d x)^2}-\frac {\left (a+b \cosh ^{-1}(c+d x)\right )^2}{3 d e^4 (c+d x)^3}+\frac {b \text {Subst}\left (\int \frac {a+b \cosh ^{-1}(x)}{\sqrt {-1+x} x \sqrt {1+x}} \, dx,x,c+d x\right )}{3 d e^4}-\frac {b^2 \text {Subst}\left (\int \frac {1}{x^2} \, dx,x,c+d x\right )}{3 d e^4}\\ &=\frac {b^2}{3 d e^4 (c+d x)}+\frac {b \sqrt {-1+c+d x} \sqrt {1+c+d x} \left (a+b \cosh ^{-1}(c+d x)\right )}{3 d e^4 (c+d x)^2}-\frac {\left (a+b \cosh ^{-1}(c+d x)\right )^2}{3 d e^4 (c+d x)^3}+\frac {b \text {Subst}\left (\int (a+b x) \text {sech}(x) \, dx,x,\cosh ^{-1}(c+d x)\right )}{3 d e^4}\\ &=\frac {b^2}{3 d e^4 (c+d x)}+\frac {b \sqrt {-1+c+d x} \sqrt {1+c+d x} \left (a+b \cosh ^{-1}(c+d x)\right )}{3 d e^4 (c+d x)^2}-\frac {\left (a+b \cosh ^{-1}(c+d x)\right )^2}{3 d e^4 (c+d x)^3}+\frac {2 b \left (a+b \cosh ^{-1}(c+d x)\right ) \tan ^{-1}\left (e^{\cosh ^{-1}(c+d x)}\right )}{3 d e^4}-\frac {\left (i b^2\right ) \text {Subst}\left (\int \log \left (1-i e^x\right ) \, dx,x,\cosh ^{-1}(c+d x)\right )}{3 d e^4}+\frac {\left (i b^2\right ) \text {Subst}\left (\int \log \left (1+i e^x\right ) \, dx,x,\cosh ^{-1}(c+d x)\right )}{3 d e^4}\\ &=\frac {b^2}{3 d e^4 (c+d x)}+\frac {b \sqrt {-1+c+d x} \sqrt {1+c+d x} \left (a+b \cosh ^{-1}(c+d x)\right )}{3 d e^4 (c+d x)^2}-\frac {\left (a+b \cosh ^{-1}(c+d x)\right )^2}{3 d e^4 (c+d x)^3}+\frac {2 b \left (a+b \cosh ^{-1}(c+d x)\right ) \tan ^{-1}\left (e^{\cosh ^{-1}(c+d x)}\right )}{3 d e^4}-\frac {\left (i b^2\right ) \text {Subst}\left (\int \frac {\log (1-i x)}{x} \, dx,x,e^{\cosh ^{-1}(c+d x)}\right )}{3 d e^4}+\frac {\left (i b^2\right ) \text {Subst}\left (\int \frac {\log (1+i x)}{x} \, dx,x,e^{\cosh ^{-1}(c+d x)}\right )}{3 d e^4}\\ &=\frac {b^2}{3 d e^4 (c+d x)}+\frac {b \sqrt {-1+c+d x} \sqrt {1+c+d x} \left (a+b \cosh ^{-1}(c+d x)\right )}{3 d e^4 (c+d x)^2}-\frac {\left (a+b \cosh ^{-1}(c+d x)\right )^2}{3 d e^4 (c+d x)^3}+\frac {2 b \left (a+b \cosh ^{-1}(c+d x)\right ) \tan ^{-1}\left (e^{\cosh ^{-1}(c+d x)}\right )}{3 d e^4}-\frac {i b^2 \text {Li}_2\left (-i e^{\cosh ^{-1}(c+d x)}\right )}{3 d e^4}+\frac {i b^2 \text {Li}_2\left (i e^{\cosh ^{-1}(c+d x)}\right )}{3 d e^4}\\ \end {align*}
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Mathematica [A]
time = 0.69, size = 251, normalized size = 1.35 \begin {gather*} \frac {-\frac {a^2}{(c+d x)^3}+a b \left (\frac {\sqrt {\frac {-1+c+d x}{1+c+d x}} (1+c+d x)}{(c+d x)^2}-\frac {2 \cosh ^{-1}(c+d x)}{(c+d x)^3}+2 \text {ArcTan}\left (\tanh \left (\frac {1}{2} \cosh ^{-1}(c+d x)\right )\right )\right )+b^2 \left (\frac {1}{c+d x}+\frac {\sqrt {\frac {-1+c+d x}{1+c+d x}} (1+c+d x) \cosh ^{-1}(c+d x)}{(c+d x)^2}-\frac {\cosh ^{-1}(c+d x)^2}{(c+d x)^3}-i \cosh ^{-1}(c+d x) \log \left (1-i e^{-\cosh ^{-1}(c+d x)}\right )+i \cosh ^{-1}(c+d x) \log \left (1+i e^{-\cosh ^{-1}(c+d x)}\right )-i \text {PolyLog}\left (2,-i e^{-\cosh ^{-1}(c+d x)}\right )+i \text {PolyLog}\left (2,i e^{-\cosh ^{-1}(c+d x)}\right )\right )}{3 d e^4} \end {gather*}
Warning: Unable to verify antiderivative.
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Maple [A]
time = 16.74, size = 352, normalized size = 1.89
method | result | size |
derivativedivides | \(\frac {-\frac {a^{2}}{3 e^{4} \left (d x +c \right )^{3}}+\frac {b^{2} \sqrt {d x +c -1}\, \sqrt {d x +c +1}\, \mathrm {arccosh}\left (d x +c \right )}{3 e^{4} \left (d x +c \right )^{2}}-\frac {b^{2} \mathrm {arccosh}\left (d x +c \right )^{2}}{3 e^{4} \left (d x +c \right )^{3}}+\frac {b^{2}}{3 e^{4} \left (d x +c \right )}-\frac {i b^{2} \mathrm {arccosh}\left (d x +c \right ) \ln \left (1+i \left (d x +c +\sqrt {d x +c -1}\, \sqrt {d x +c +1}\right )\right )}{3 e^{4}}+\frac {i b^{2} \mathrm {arccosh}\left (d x +c \right ) \ln \left (1-i \left (d x +c +\sqrt {d x +c -1}\, \sqrt {d x +c +1}\right )\right )}{3 e^{4}}-\frac {i b^{2} \dilog \left (1+i \left (d x +c +\sqrt {d x +c -1}\, \sqrt {d x +c +1}\right )\right )}{3 e^{4}}+\frac {i b^{2} \dilog \left (1-i \left (d x +c +\sqrt {d x +c -1}\, \sqrt {d x +c +1}\right )\right )}{3 e^{4}}-\frac {2 a b \,\mathrm {arccosh}\left (d x +c \right )}{3 e^{4} \left (d x +c \right )^{3}}-\frac {a b \sqrt {d x +c -1}\, \sqrt {d x +c +1}\, \arctan \left (\frac {1}{\sqrt {\left (d x +c \right )^{2}-1}}\right )}{3 e^{4} \sqrt {\left (d x +c \right )^{2}-1}}+\frac {a b \sqrt {d x +c -1}\, \sqrt {d x +c +1}}{3 e^{4} \left (d x +c \right )^{2}}}{d}\) | \(352\) |
default | \(\frac {-\frac {a^{2}}{3 e^{4} \left (d x +c \right )^{3}}+\frac {b^{2} \sqrt {d x +c -1}\, \sqrt {d x +c +1}\, \mathrm {arccosh}\left (d x +c \right )}{3 e^{4} \left (d x +c \right )^{2}}-\frac {b^{2} \mathrm {arccosh}\left (d x +c \right )^{2}}{3 e^{4} \left (d x +c \right )^{3}}+\frac {b^{2}}{3 e^{4} \left (d x +c \right )}-\frac {i b^{2} \mathrm {arccosh}\left (d x +c \right ) \ln \left (1+i \left (d x +c +\sqrt {d x +c -1}\, \sqrt {d x +c +1}\right )\right )}{3 e^{4}}+\frac {i b^{2} \mathrm {arccosh}\left (d x +c \right ) \ln \left (1-i \left (d x +c +\sqrt {d x +c -1}\, \sqrt {d x +c +1}\right )\right )}{3 e^{4}}-\frac {i b^{2} \dilog \left (1+i \left (d x +c +\sqrt {d x +c -1}\, \sqrt {d x +c +1}\right )\right )}{3 e^{4}}+\frac {i b^{2} \dilog \left (1-i \left (d x +c +\sqrt {d x +c -1}\, \sqrt {d x +c +1}\right )\right )}{3 e^{4}}-\frac {2 a b \,\mathrm {arccosh}\left (d x +c \right )}{3 e^{4} \left (d x +c \right )^{3}}-\frac {a b \sqrt {d x +c -1}\, \sqrt {d x +c +1}\, \arctan \left (\frac {1}{\sqrt {\left (d x +c \right )^{2}-1}}\right )}{3 e^{4} \sqrt {\left (d x +c \right )^{2}-1}}+\frac {a b \sqrt {d x +c -1}\, \sqrt {d x +c +1}}{3 e^{4} \left (d x +c \right )^{2}}}{d}\) | \(352\) |
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*} \frac {\int \frac {a^{2}}{c^{4} + 4 c^{3} d x + 6 c^{2} d^{2} x^{2} + 4 c d^{3} x^{3} + d^{4} x^{4}}\, dx + \int \frac {b^{2} \operatorname {acosh}^{2}{\left (c + d x \right )}}{c^{4} + 4 c^{3} d x + 6 c^{2} d^{2} x^{2} + 4 c d^{3} x^{3} + d^{4} x^{4}}\, dx + \int \frac {2 a b \operatorname {acosh}{\left (c + d x \right )}}{c^{4} + 4 c^{3} d x + 6 c^{2} d^{2} x^{2} + 4 c d^{3} x^{3} + d^{4} x^{4}}\, dx}{e^{4}} \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+d\,x\right )\right )}^2}{{\left (c\,e+d\,e\,x\right )}^4} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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