Optimal. Leaf size=261 \[ -\frac {b^2 \left (a+b \sinh ^{-1}(c+d x)\right )}{d e^4 (c+d x)}-\frac {b \sqrt {1+(c+d x)^2} \left (a+b \sinh ^{-1}(c+d x)\right )^2}{2 d e^4 (c+d x)^2}-\frac {\left (a+b \sinh ^{-1}(c+d x)\right )^3}{3 d e^4 (c+d x)^3}+\frac {b \left (a+b \sinh ^{-1}(c+d x)\right )^2 \tanh ^{-1}\left (e^{\sinh ^{-1}(c+d x)}\right )}{d e^4}-\frac {b^3 \tanh ^{-1}\left (\sqrt {1+(c+d x)^2}\right )}{d e^4}+\frac {b^2 \left (a+b \sinh ^{-1}(c+d x)\right ) \text {PolyLog}\left (2,-e^{\sinh ^{-1}(c+d x)}\right )}{d e^4}-\frac {b^2 \left (a+b \sinh ^{-1}(c+d x)\right ) \text {PolyLog}\left (2,e^{\sinh ^{-1}(c+d x)}\right )}{d e^4}-\frac {b^3 \text {PolyLog}\left (3,-e^{\sinh ^{-1}(c+d x)}\right )}{d e^4}+\frac {b^3 \text {PolyLog}\left (3,e^{\sinh ^{-1}(c+d x)}\right )}{d e^4} \]
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Rubi [A]
time = 0.27, antiderivative size = 261, normalized size of antiderivative = 1.00, number of steps
used = 16, number of rules used = 12, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.522, Rules used = {5859, 12,
5776, 5809, 5816, 4267, 2611, 2320, 6724, 272, 65, 213} \begin {gather*} \frac {b^2 \text {Li}_2\left (-e^{\sinh ^{-1}(c+d x)}\right ) \left (a+b \sinh ^{-1}(c+d x)\right )}{d e^4}-\frac {b^2 \text {Li}_2\left (e^{\sinh ^{-1}(c+d x)}\right ) \left (a+b \sinh ^{-1}(c+d x)\right )}{d e^4}-\frac {b^2 \left (a+b \sinh ^{-1}(c+d x)\right )}{d e^4 (c+d x)}-\frac {b \sqrt {(c+d x)^2+1} \left (a+b \sinh ^{-1}(c+d x)\right )^2}{2 d e^4 (c+d x)^2}-\frac {\left (a+b \sinh ^{-1}(c+d x)\right )^3}{3 d e^4 (c+d x)^3}+\frac {b \tanh ^{-1}\left (e^{\sinh ^{-1}(c+d x)}\right ) \left (a+b \sinh ^{-1}(c+d x)\right )^2}{d e^4}-\frac {b^3 \text {Li}_3\left (-e^{\sinh ^{-1}(c+d x)}\right )}{d e^4}+\frac {b^3 \text {Li}_3\left (e^{\sinh ^{-1}(c+d x)}\right )}{d e^4}-\frac {b^3 \tanh ^{-1}\left (\sqrt {(c+d x)^2+1}\right )}{d e^4} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 65
Rule 213
Rule 272
Rule 2320
Rule 2611
Rule 4267
Rule 5776
Rule 5809
Rule 5816
Rule 5859
Rule 6724
Rubi steps
\begin {align*} \int \frac {\left (a+b \sinh ^{-1}(c+d x)\right )^3}{(c e+d e x)^4} \, dx &=\frac {\text {Subst}\left (\int \frac {\left (a+b \sinh ^{-1}(x)\right )^3}{e^4 x^4} \, dx,x,c+d x\right )}{d}\\ &=\frac {\text {Subst}\left (\int \frac {\left (a+b \sinh ^{-1}(x)\right )^3}{x^4} \, dx,x,c+d x\right )}{d e^4}\\ &=-\frac {\left (a+b \sinh ^{-1}(c+d x)\right )^3}{3 d e^4 (c+d x)^3}+\frac {b \text {Subst}\left (\int \frac {\left (a+b \sinh ^{-1}(x)\right )^2}{x^3 \sqrt {1+x^2}} \, dx,x,c+d x\right )}{d e^4}\\ &=-\frac {b \sqrt {1+(c+d x)^2} \left (a+b \sinh ^{-1}(c+d x)\right )^2}{2 d e^4 (c+d x)^2}-\frac {\left (a+b \sinh ^{-1}(c+d x)\right )^3}{3 d e^4 (c+d x)^3}-\frac {b \text {Subst}\left (\int \frac {\left (a+b \sinh ^{-1}(x)\right )^2}{x \sqrt {1+x^2}} \, dx,x,c+d x\right )}{2 d e^4}+\frac {b^2 \text {Subst}\left (\int \frac {a+b \sinh ^{-1}(x)}{x^2} \, dx,x,c+d x\right )}{d e^4}\\ &=-\frac {b^2 \left (a+b \sinh ^{-1}(c+d x)\right )}{d e^4 (c+d x)}-\frac {b \sqrt {1+(c+d x)^2} \left (a+b \sinh ^{-1}(c+d x)\right )^2}{2 d e^4 (c+d x)^2}-\frac {\left (a+b \sinh ^{-1}(c+d x)\right )^3}{3 d e^4 (c+d x)^3}-\frac {b \text {Subst}\left (\int (a+b x)^2 \text {csch}(x) \, dx,x,\sinh ^{-1}(c+d x)\right )}{2 d e^4}+\frac {b^3 \text {Subst}\left (\int \frac {1}{x \sqrt {1+x^2}} \, dx,x,c+d x\right )}{d e^4}\\ &=-\frac {b^2 \left (a+b \sinh ^{-1}(c+d x)\right )}{d e^4 (c+d x)}-\frac {b \sqrt {1+(c+d x)^2} \left (a+b \sinh ^{-1}(c+d x)\right )^2}{2 d e^4 (c+d x)^2}-\frac {\left (a+b \sinh ^{-1}(c+d x)\right )^3}{3 d e^4 (c+d x)^3}+\frac {b \left (a+b \sinh ^{-1}(c+d x)\right )^2 \tanh ^{-1}\left (e^{\sinh ^{-1}(c+d x)}\right )}{d e^4}+\frac {b^2 \text {Subst}\left (\int (a+b x) \log \left (1-e^x\right ) \, dx,x,\sinh ^{-1}(c+d x)\right )}{d e^4}-\frac {b^2 \text {Subst}\left (\int (a+b x) \log \left (1+e^x\right ) \, dx,x,\sinh ^{-1}(c+d x)\right )}{d e^4}+\frac {b^3 \text {Subst}\left (\int \frac {1}{x \sqrt {1+x}} \, dx,x,(c+d x)^2\right )}{2 d e^4}\\ &=-\frac {b^2 \left (a+b \sinh ^{-1}(c+d x)\right )}{d e^4 (c+d x)}-\frac {b \sqrt {1+(c+d x)^2} \left (a+b \sinh ^{-1}(c+d x)\right )^2}{2 d e^4 (c+d x)^2}-\frac {\left (a+b \sinh ^{-1}(c+d x)\right )^3}{3 d e^4 (c+d x)^3}+\frac {b \left (a+b \sinh ^{-1}(c+d x)\right )^2 \tanh ^{-1}\left (e^{\sinh ^{-1}(c+d x)}\right )}{d e^4}+\frac {b^2 \left (a+b \sinh ^{-1}(c+d x)\right ) \text {Li}_2\left (-e^{\sinh ^{-1}(c+d x)}\right )}{d e^4}-\frac {b^2 \left (a+b \sinh ^{-1}(c+d x)\right ) \text {Li}_2\left (e^{\sinh ^{-1}(c+d x)}\right )}{d e^4}+\frac {b^3 \text {Subst}\left (\int \frac {1}{-1+x^2} \, dx,x,\sqrt {1+(c+d x)^2}\right )}{d e^4}-\frac {b^3 \text {Subst}\left (\int \text {Li}_2\left (-e^x\right ) \, dx,x,\sinh ^{-1}(c+d x)\right )}{d e^4}+\frac {b^3 \text {Subst}\left (\int \text {Li}_2\left (e^x\right ) \, dx,x,\sinh ^{-1}(c+d x)\right )}{d e^4}\\ &=-\frac {b^2 \left (a+b \sinh ^{-1}(c+d x)\right )}{d e^4 (c+d x)}-\frac {b \sqrt {1+(c+d x)^2} \left (a+b \sinh ^{-1}(c+d x)\right )^2}{2 d e^4 (c+d x)^2}-\frac {\left (a+b \sinh ^{-1}(c+d x)\right )^3}{3 d e^4 (c+d x)^3}+\frac {b \left (a+b \sinh ^{-1}(c+d x)\right )^2 \tanh ^{-1}\left (e^{\sinh ^{-1}(c+d x)}\right )}{d e^4}-\frac {b^3 \tanh ^{-1}\left (\sqrt {1+(c+d x)^2}\right )}{d e^4}+\frac {b^2 \left (a+b \sinh ^{-1}(c+d x)\right ) \text {Li}_2\left (-e^{\sinh ^{-1}(c+d x)}\right )}{d e^4}-\frac {b^2 \left (a+b \sinh ^{-1}(c+d x)\right ) \text {Li}_2\left (e^{\sinh ^{-1}(c+d x)}\right )}{d e^4}-\frac {b^3 \text {Subst}\left (\int \frac {\text {Li}_2(-x)}{x} \, dx,x,e^{\sinh ^{-1}(c+d x)}\right )}{d e^4}+\frac {b^3 \text {Subst}\left (\int \frac {\text {Li}_2(x)}{x} \, dx,x,e^{\sinh ^{-1}(c+d x)}\right )}{d e^4}\\ &=-\frac {b^2 \left (a+b \sinh ^{-1}(c+d x)\right )}{d e^4 (c+d x)}-\frac {b \sqrt {1+(c+d x)^2} \left (a+b \sinh ^{-1}(c+d x)\right )^2}{2 d e^4 (c+d x)^2}-\frac {\left (a+b \sinh ^{-1}(c+d x)\right )^3}{3 d e^4 (c+d x)^3}+\frac {b \left (a+b \sinh ^{-1}(c+d x)\right )^2 \tanh ^{-1}\left (e^{\sinh ^{-1}(c+d x)}\right )}{d e^4}-\frac {b^3 \tanh ^{-1}\left (\sqrt {1+(c+d x)^2}\right )}{d e^4}+\frac {b^2 \left (a+b \sinh ^{-1}(c+d x)\right ) \text {Li}_2\left (-e^{\sinh ^{-1}(c+d x)}\right )}{d e^4}-\frac {b^2 \left (a+b \sinh ^{-1}(c+d x)\right ) \text {Li}_2\left (e^{\sinh ^{-1}(c+d x)}\right )}{d e^4}-\frac {b^3 \text {Li}_3\left (-e^{\sinh ^{-1}(c+d x)}\right )}{d e^4}+\frac {b^3 \text {Li}_3\left (e^{\sinh ^{-1}(c+d x)}\right )}{d e^4}\\ \end {align*}
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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(694\) vs. \(2(261)=522\).
time = 7.03, size = 694, normalized size = 2.66 \begin {gather*} -\frac {a^3}{3 d e^4 (c+d x)^3}-\frac {a^2 b \sqrt {1+c^2+2 c d x+d^2 x^2}}{2 d e^4 (c+d x)^2}-\frac {a^2 b \sinh ^{-1}(c+d x)}{d e^4 (c+d x)^3}-\frac {a^2 b \log (c+d x)}{2 d e^4}+\frac {a^2 b \log \left (1+\sqrt {1+c^2+2 c d x+d^2 x^2}\right )}{2 d e^4}+\frac {a b^2 \left (-8 \text {PolyLog}\left (2,-e^{-\sinh ^{-1}(c+d x)}\right )-\frac {2 \left (-2+4 \sinh ^{-1}(c+d x)^2+2 \cosh \left (2 \sinh ^{-1}(c+d x)\right )-3 (c+d x) \sinh ^{-1}(c+d x) \log \left (1-e^{-\sinh ^{-1}(c+d x)}\right )+3 (c+d x) \sinh ^{-1}(c+d x) \log \left (1+e^{-\sinh ^{-1}(c+d x)}\right )-4 (c+d x)^3 \text {PolyLog}\left (2,e^{-\sinh ^{-1}(c+d x)}\right )+2 \sinh ^{-1}(c+d x) \sinh \left (2 \sinh ^{-1}(c+d x)\right )+\sinh ^{-1}(c+d x) \log \left (1-e^{-\sinh ^{-1}(c+d x)}\right ) \sinh \left (3 \sinh ^{-1}(c+d x)\right )-\sinh ^{-1}(c+d x) \log \left (1+e^{-\sinh ^{-1}(c+d x)}\right ) \sinh \left (3 \sinh ^{-1}(c+d x)\right )\right )}{(c+d x)^3}\right )}{8 d e^4}+\frac {b^3 \left (-24 \sinh ^{-1}(c+d x) \coth \left (\frac {1}{2} \sinh ^{-1}(c+d x)\right )+4 \sinh ^{-1}(c+d x)^3 \coth \left (\frac {1}{2} \sinh ^{-1}(c+d x)\right )-6 \sinh ^{-1}(c+d x)^2 \text {csch}^2\left (\frac {1}{2} \sinh ^{-1}(c+d x)\right )-(c+d x) \sinh ^{-1}(c+d x)^3 \text {csch}^4\left (\frac {1}{2} \sinh ^{-1}(c+d x)\right )-24 \sinh ^{-1}(c+d x)^2 \log \left (1-e^{-\sinh ^{-1}(c+d x)}\right )+24 \sinh ^{-1}(c+d x)^2 \log \left (1+e^{-\sinh ^{-1}(c+d x)}\right )+48 \log \left (\tanh \left (\frac {1}{2} \sinh ^{-1}(c+d x)\right )\right )-48 \sinh ^{-1}(c+d x) \text {PolyLog}\left (2,-e^{-\sinh ^{-1}(c+d x)}\right )+48 \sinh ^{-1}(c+d x) \text {PolyLog}\left (2,e^{-\sinh ^{-1}(c+d x)}\right )-48 \text {PolyLog}\left (3,-e^{-\sinh ^{-1}(c+d x)}\right )+48 \text {PolyLog}\left (3,e^{-\sinh ^{-1}(c+d x)}\right )-6 \sinh ^{-1}(c+d x)^2 \text {sech}^2\left (\frac {1}{2} \sinh ^{-1}(c+d x)\right )-\frac {16 \sinh ^{-1}(c+d x)^3 \sinh ^4\left (\frac {1}{2} \sinh ^{-1}(c+d x)\right )}{(c+d x)^3}+24 \sinh ^{-1}(c+d x) \tanh \left (\frac {1}{2} \sinh ^{-1}(c+d x)\right )-4 \sinh ^{-1}(c+d x)^3 \tanh \left (\frac {1}{2} \sinh ^{-1}(c+d x)\right )\right )}{48 d e^4} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 5.53, size = 581, normalized size = 2.23
method | result | size |
derivativedivides | \(\frac {-\frac {a^{3}}{3 e^{4} \left (d x +c \right )^{3}}-\frac {b^{3} \arcsinh \left (d x +c \right )^{2} \sqrt {1+\left (d x +c \right )^{2}}}{2 e^{4} \left (d x +c \right )^{2}}-\frac {b^{3} \arcsinh \left (d x +c \right )^{3}}{3 e^{4} \left (d x +c \right )^{3}}-\frac {b^{3} \arcsinh \left (d x +c \right )}{e^{4} \left (d x +c \right )}-\frac {b^{3} \arcsinh \left (d x +c \right )^{2} \ln \left (1-d x -c -\sqrt {1+\left (d x +c \right )^{2}}\right )}{2 e^{4}}-\frac {b^{3} \arcsinh \left (d x +c \right ) \polylog \left (2, d x +c +\sqrt {1+\left (d x +c \right )^{2}}\right )}{e^{4}}+\frac {b^{3} \polylog \left (3, d x +c +\sqrt {1+\left (d x +c \right )^{2}}\right )}{e^{4}}+\frac {b^{3} \arcsinh \left (d x +c \right )^{2} \ln \left (1+d x +c +\sqrt {1+\left (d x +c \right )^{2}}\right )}{2 e^{4}}+\frac {b^{3} \arcsinh \left (d x +c \right ) \polylog \left (2, -d x -c -\sqrt {1+\left (d x +c \right )^{2}}\right )}{e^{4}}-\frac {b^{3} \polylog \left (3, -d x -c -\sqrt {1+\left (d x +c \right )^{2}}\right )}{e^{4}}-\frac {2 b^{3} \arctanh \left (d x +c +\sqrt {1+\left (d x +c \right )^{2}}\right )}{e^{4}}-\frac {a \,b^{2} \sqrt {1+\left (d x +c \right )^{2}}\, \arcsinh \left (d x +c \right )}{e^{4} \left (d x +c \right )^{2}}-\frac {a \,b^{2} \arcsinh \left (d x +c \right )^{2}}{e^{4} \left (d x +c \right )^{3}}-\frac {a \,b^{2}}{e^{4} \left (d x +c \right )}-\frac {a \,b^{2} \arcsinh \left (d x +c \right ) \ln \left (1-d x -c -\sqrt {1+\left (d x +c \right )^{2}}\right )}{e^{4}}-\frac {a \,b^{2} \polylog \left (2, d x +c +\sqrt {1+\left (d x +c \right )^{2}}\right )}{e^{4}}+\frac {a \,b^{2} \arcsinh \left (d x +c \right ) \ln \left (1+d x +c +\sqrt {1+\left (d x +c \right )^{2}}\right )}{e^{4}}+\frac {a \,b^{2} \polylog \left (2, -d x -c -\sqrt {1+\left (d x +c \right )^{2}}\right )}{e^{4}}+\frac {3 a^{2} 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}\) | \(581\) |
default | \(\frac {-\frac {a^{3}}{3 e^{4} \left (d x +c \right )^{3}}-\frac {b^{3} \arcsinh \left (d x +c \right )^{2} \sqrt {1+\left (d x +c \right )^{2}}}{2 e^{4} \left (d x +c \right )^{2}}-\frac {b^{3} \arcsinh \left (d x +c \right )^{3}}{3 e^{4} \left (d x +c \right )^{3}}-\frac {b^{3} \arcsinh \left (d x +c \right )}{e^{4} \left (d x +c \right )}-\frac {b^{3} \arcsinh \left (d x +c \right )^{2} \ln \left (1-d x -c -\sqrt {1+\left (d x +c \right )^{2}}\right )}{2 e^{4}}-\frac {b^{3} \arcsinh \left (d x +c \right ) \polylog \left (2, d x +c +\sqrt {1+\left (d x +c \right )^{2}}\right )}{e^{4}}+\frac {b^{3} \polylog \left (3, d x +c +\sqrt {1+\left (d x +c \right )^{2}}\right )}{e^{4}}+\frac {b^{3} \arcsinh \left (d x +c \right )^{2} \ln \left (1+d x +c +\sqrt {1+\left (d x +c \right )^{2}}\right )}{2 e^{4}}+\frac {b^{3} \arcsinh \left (d x +c \right ) \polylog \left (2, -d x -c -\sqrt {1+\left (d x +c \right )^{2}}\right )}{e^{4}}-\frac {b^{3} \polylog \left (3, -d x -c -\sqrt {1+\left (d x +c \right )^{2}}\right )}{e^{4}}-\frac {2 b^{3} \arctanh \left (d x +c +\sqrt {1+\left (d x +c \right )^{2}}\right )}{e^{4}}-\frac {a \,b^{2} \sqrt {1+\left (d x +c \right )^{2}}\, \arcsinh \left (d x +c \right )}{e^{4} \left (d x +c \right )^{2}}-\frac {a \,b^{2} \arcsinh \left (d x +c \right )^{2}}{e^{4} \left (d x +c \right )^{3}}-\frac {a \,b^{2}}{e^{4} \left (d x +c \right )}-\frac {a \,b^{2} \arcsinh \left (d x +c \right ) \ln \left (1-d x -c -\sqrt {1+\left (d x +c \right )^{2}}\right )}{e^{4}}-\frac {a \,b^{2} \polylog \left (2, d x +c +\sqrt {1+\left (d x +c \right )^{2}}\right )}{e^{4}}+\frac {a \,b^{2} \arcsinh \left (d x +c \right ) \ln \left (1+d x +c +\sqrt {1+\left (d x +c \right )^{2}}\right )}{e^{4}}+\frac {a \,b^{2} \polylog \left (2, -d x -c -\sqrt {1+\left (d x +c \right )^{2}}\right )}{e^{4}}+\frac {3 a^{2} 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}\) | \(581\) |
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^{3}}{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^{3} \operatorname {asinh}^{3}{\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 {3 a 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 {3 a^{2} 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 = -0.00 \begin {gather*} \int \frac {{\left (a+b\,\mathrm {asinh}\left (c+d\,x\right )\right )}^3}{{\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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