Optimal. Leaf size=169 \[ -\frac {b^2}{3 d e^4 (c+d x)}-\frac {b \sqrt {1+(c+d x)^2} \left (a+b \sinh ^{-1}(c+d x)\right )}{3 d e^4 (c+d x)^2}-\frac {\left (a+b \sinh ^{-1}(c+d x)\right )^2}{3 d e^4 (c+d x)^3}+\frac {2 b \left (a+b \sinh ^{-1}(c+d x)\right ) \tanh ^{-1}\left (e^{\sinh ^{-1}(c+d x)}\right )}{3 d e^4}+\frac {b^2 \text {PolyLog}\left (2,-e^{\sinh ^{-1}(c+d x)}\right )}{3 d e^4}-\frac {b^2 \text {PolyLog}\left (2,e^{\sinh ^{-1}(c+d x)}\right )}{3 d e^4} \]
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Rubi [A]
time = 0.17, antiderivative size = 169, 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 = {5859, 12,
5776, 5809, 5816, 4267, 2317, 2438, 30} \begin {gather*} -\frac {b \sqrt {(c+d x)^2+1} \left (a+b \sinh ^{-1}(c+d x)\right )}{3 d e^4 (c+d x)^2}-\frac {\left (a+b \sinh ^{-1}(c+d x)\right )^2}{3 d e^4 (c+d x)^3}+\frac {2 b \tanh ^{-1}\left (e^{\sinh ^{-1}(c+d x)}\right ) \left (a+b \sinh ^{-1}(c+d x)\right )}{3 d e^4}+\frac {b^2 \text {Li}_2\left (-e^{\sinh ^{-1}(c+d x)}\right )}{3 d e^4}-\frac {b^2 \text {Li}_2\left (e^{\sinh ^{-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 4267
Rule 5776
Rule 5809
Rule 5816
Rule 5859
Rubi steps
\begin {align*} \int \frac {\left (a+b \sinh ^{-1}(c+d x)\right )^2}{(c e+d e x)^4} \, dx &=\frac {\text {Subst}\left (\int \frac {\left (a+b \sinh ^{-1}(x)\right )^2}{e^4 x^4} \, dx,x,c+d x\right )}{d}\\ &=\frac {\text {Subst}\left (\int \frac {\left (a+b \sinh ^{-1}(x)\right )^2}{x^4} \, dx,x,c+d x\right )}{d e^4}\\ &=-\frac {\left (a+b \sinh ^{-1}(c+d x)\right )^2}{3 d e^4 (c+d x)^3}+\frac {(2 b) \text {Subst}\left (\int \frac {a+b \sinh ^{-1}(x)}{x^3 \sqrt {1+x^2}} \, dx,x,c+d x\right )}{3 d e^4}\\ &=-\frac {b \sqrt {1+(c+d x)^2} \left (a+b \sinh ^{-1}(c+d x)\right )}{3 d e^4 (c+d x)^2}-\frac {\left (a+b \sinh ^{-1}(c+d x)\right )^2}{3 d e^4 (c+d x)^3}-\frac {b \text {Subst}\left (\int \frac {a+b \sinh ^{-1}(x)}{x \sqrt {1+x^2}} \, 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)^2} \left (a+b \sinh ^{-1}(c+d x)\right )}{3 d e^4 (c+d x)^2}-\frac {\left (a+b \sinh ^{-1}(c+d x)\right )^2}{3 d e^4 (c+d x)^3}-\frac {b \text {Subst}\left (\int (a+b x) \text {csch}(x) \, dx,x,\sinh ^{-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)^2} \left (a+b \sinh ^{-1}(c+d x)\right )}{3 d e^4 (c+d x)^2}-\frac {\left (a+b \sinh ^{-1}(c+d x)\right )^2}{3 d e^4 (c+d x)^3}+\frac {2 b \left (a+b \sinh ^{-1}(c+d x)\right ) \tanh ^{-1}\left (e^{\sinh ^{-1}(c+d x)}\right )}{3 d e^4}+\frac {b^2 \text {Subst}\left (\int \log \left (1-e^x\right ) \, dx,x,\sinh ^{-1}(c+d x)\right )}{3 d e^4}-\frac {b^2 \text {Subst}\left (\int \log \left (1+e^x\right ) \, dx,x,\sinh ^{-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)^2} \left (a+b \sinh ^{-1}(c+d x)\right )}{3 d e^4 (c+d x)^2}-\frac {\left (a+b \sinh ^{-1}(c+d x)\right )^2}{3 d e^4 (c+d x)^3}+\frac {2 b \left (a+b \sinh ^{-1}(c+d x)\right ) \tanh ^{-1}\left (e^{\sinh ^{-1}(c+d x)}\right )}{3 d e^4}+\frac {b^2 \text {Subst}\left (\int \frac {\log (1-x)}{x} \, dx,x,e^{\sinh ^{-1}(c+d x)}\right )}{3 d e^4}-\frac {b^2 \text {Subst}\left (\int \frac {\log (1+x)}{x} \, dx,x,e^{\sinh ^{-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)^2} \left (a+b \sinh ^{-1}(c+d x)\right )}{3 d e^4 (c+d x)^2}-\frac {\left (a+b \sinh ^{-1}(c+d x)\right )^2}{3 d e^4 (c+d x)^3}+\frac {2 b \left (a+b \sinh ^{-1}(c+d x)\right ) \tanh ^{-1}\left (e^{\sinh ^{-1}(c+d x)}\right )}{3 d e^4}+\frac {b^2 \text {Li}_2\left (-e^{\sinh ^{-1}(c+d x)}\right )}{3 d e^4}-\frac {b^2 \text {Li}_2\left (e^{\sinh ^{-1}(c+d x)}\right )}{3 d e^4}\\ \end {align*}
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Mathematica [A]
time = 1.13, size = 212, normalized size = 1.25 \begin {gather*} -\frac {4 a^2+a b \left (8 \sinh ^{-1}(c+d x)+2 \sinh \left (2 \sinh ^{-1}(c+d x)\right )+\log \left (\tanh \left (\frac {1}{2} \sinh ^{-1}(c+d x)\right )\right ) \left (-3 (c+d x)+\sinh \left (3 \sinh ^{-1}(c+d x)\right )\right )\right )+b^2 \left (4 (c+d x)^2+4 \sinh ^{-1}(c+d x)^2+4 (c+d x)^3 \text {PolyLog}\left (2,-e^{-\sinh ^{-1}(c+d x)}\right )-4 (c+d x)^3 \text {PolyLog}\left (2,e^{-\sinh ^{-1}(c+d x)}\right )+\sinh ^{-1}(c+d x) \left (2 \sinh \left (2 \sinh ^{-1}(c+d x)\right )+\left (\log \left (1-e^{-\sinh ^{-1}(c+d x)}\right )-\log \left (1+e^{-\sinh ^{-1}(c+d x)}\right )\right ) \left (-3 (c+d x)+\sinh \left (3 \sinh ^{-1}(c+d x)\right )\right )\right )\right )}{12 d e^4 (c+d x)^3} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 4.79, size = 274, normalized size = 1.62
method | result | size |
derivativedivides | \(\frac {-\frac {a^{2}}{3 e^{4} \left (d x +c \right )^{3}}-\frac {b^{2} \arcsinh \left (d x +c \right ) \sqrt {1+\left (d x +c \right )^{2}}}{3 e^{4} \left (d x +c \right )^{2}}-\frac {b^{2} \arcsinh \left (d x +c \right )^{2}}{3 e^{4} \left (d x +c \right )^{3}}-\frac {b^{2}}{3 e^{4} \left (d x +c \right )}+\frac {b^{2} \arcsinh \left (d x +c \right ) \ln \left (1+d x +c +\sqrt {1+\left (d x +c \right )^{2}}\right )}{3 e^{4}}+\frac {b^{2} \polylog \left (2, -d x -c -\sqrt {1+\left (d x +c \right )^{2}}\right )}{3 e^{4}}-\frac {b^{2} \arcsinh \left (d x +c \right ) \ln \left (1-d x -c -\sqrt {1+\left (d x +c \right )^{2}}\right )}{3 e^{4}}-\frac {b^{2} \polylog \left (2, d x +c +\sqrt {1+\left (d x +c \right )^{2}}\right )}{3 e^{4}}+\frac {2 a b \left (-\frac {\arcsinh \left (d x +c \right )}{3 \left (d x +c \right )^{3}}-\frac {\sqrt {1+\left (d x +c \right )^{2}}}{6 \left (d x +c \right )^{2}}+\frac {\arctanh \left (\frac {1}{\sqrt {1+\left (d x +c \right )^{2}}}\right )}{6}\right )}{e^{4}}}{d}\) | \(274\) |
default | \(\frac {-\frac {a^{2}}{3 e^{4} \left (d x +c \right )^{3}}-\frac {b^{2} \arcsinh \left (d x +c \right ) \sqrt {1+\left (d x +c \right )^{2}}}{3 e^{4} \left (d x +c \right )^{2}}-\frac {b^{2} \arcsinh \left (d x +c \right )^{2}}{3 e^{4} \left (d x +c \right )^{3}}-\frac {b^{2}}{3 e^{4} \left (d x +c \right )}+\frac {b^{2} \arcsinh \left (d x +c \right ) \ln \left (1+d x +c +\sqrt {1+\left (d x +c \right )^{2}}\right )}{3 e^{4}}+\frac {b^{2} \polylog \left (2, -d x -c -\sqrt {1+\left (d x +c \right )^{2}}\right )}{3 e^{4}}-\frac {b^{2} \arcsinh \left (d x +c \right ) \ln \left (1-d x -c -\sqrt {1+\left (d x +c \right )^{2}}\right )}{3 e^{4}}-\frac {b^{2} \polylog \left (2, d x +c +\sqrt {1+\left (d x +c \right )^{2}}\right )}{3 e^{4}}+\frac {2 a b \left (-\frac {\arcsinh \left (d x +c \right )}{3 \left (d x +c \right )^{3}}-\frac {\sqrt {1+\left (d x +c \right )^{2}}}{6 \left (d x +c \right )^{2}}+\frac {\arctanh \left (\frac {1}{\sqrt {1+\left (d x +c \right )^{2}}}\right )}{6}\right )}{e^{4}}}{d}\) | \(274\) |
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 {asinh}^{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 {asinh}{\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 {asinh}\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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