Optimal. Leaf size=221 \[ -\frac {19}{48} b c d^3 x \sqrt {1+c^2 x^2}-\frac {7}{72} b c d^3 x \left (1+c^2 x^2\right )^{3/2}-\frac {1}{36} b c d^3 x \left (1+c^2 x^2\right )^{5/2}-\frac {19}{48} b d^3 \sinh ^{-1}(c x)+\frac {1}{2} d^3 \left (1+c^2 x^2\right ) \left (a+b \sinh ^{-1}(c x)\right )+\frac {1}{4} d^3 \left (1+c^2 x^2\right )^2 \left (a+b \sinh ^{-1}(c x)\right )+\frac {1}{6} d^3 \left (1+c^2 x^2\right )^3 \left (a+b \sinh ^{-1}(c x)\right )+\frac {d^3 \left (a+b \sinh ^{-1}(c x)\right )^2}{2 b}+d^3 \left (a+b \sinh ^{-1}(c x)\right ) \log \left (1-e^{-2 \sinh ^{-1}(c x)}\right )-\frac {1}{2} b d^3 \text {PolyLog}\left (2,e^{-2 \sinh ^{-1}(c x)}\right ) \]
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Rubi [A]
time = 0.23, antiderivative size = 221, normalized size of antiderivative = 1.00, number of steps
used = 17, number of rules used = 8, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {5801, 5775,
3797, 2221, 2317, 2438, 201, 221} \begin {gather*} \frac {1}{6} d^3 \left (c^2 x^2+1\right )^3 \left (a+b \sinh ^{-1}(c x)\right )+\frac {1}{4} d^3 \left (c^2 x^2+1\right )^2 \left (a+b \sinh ^{-1}(c x)\right )+\frac {1}{2} d^3 \left (c^2 x^2+1\right ) \left (a+b \sinh ^{-1}(c x)\right )+\frac {d^3 \left (a+b \sinh ^{-1}(c x)\right )^2}{2 b}+d^3 \log \left (1-e^{-2 \sinh ^{-1}(c x)}\right ) \left (a+b \sinh ^{-1}(c x)\right )-\frac {1}{36} b c d^3 x \left (c^2 x^2+1\right )^{5/2}-\frac {7}{72} b c d^3 x \left (c^2 x^2+1\right )^{3/2}-\frac {19}{48} b c d^3 x \sqrt {c^2 x^2+1}-\frac {1}{2} b d^3 \text {Li}_2\left (e^{-2 \sinh ^{-1}(c x)}\right )-\frac {19}{48} b d^3 \sinh ^{-1}(c x) \end {gather*}
Antiderivative was successfully verified.
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Rule 201
Rule 221
Rule 2221
Rule 2317
Rule 2438
Rule 3797
Rule 5775
Rule 5801
Rubi steps
\begin {align*} \int \frac {\left (d+c^2 d x^2\right )^3 \left (a+b \sinh ^{-1}(c x)\right )}{x} \, dx &=\frac {1}{6} d^3 \left (1+c^2 x^2\right )^3 \left (a+b \sinh ^{-1}(c x)\right )+d \int \frac {\left (d+c^2 d x^2\right )^2 \left (a+b \sinh ^{-1}(c x)\right )}{x} \, dx-\frac {1}{6} \left (b c d^3\right ) \int \left (1+c^2 x^2\right )^{5/2} \, dx\\ &=-\frac {1}{36} b c d^3 x \left (1+c^2 x^2\right )^{5/2}+\frac {1}{4} d^3 \left (1+c^2 x^2\right )^2 \left (a+b \sinh ^{-1}(c x)\right )+\frac {1}{6} d^3 \left (1+c^2 x^2\right )^3 \left (a+b \sinh ^{-1}(c x)\right )+d^2 \int \frac {\left (d+c^2 d x^2\right ) \left (a+b \sinh ^{-1}(c x)\right )}{x} \, dx-\frac {1}{36} \left (5 b c d^3\right ) \int \left (1+c^2 x^2\right )^{3/2} \, dx-\frac {1}{4} \left (b c d^3\right ) \int \left (1+c^2 x^2\right )^{3/2} \, dx\\ &=-\frac {7}{72} b c d^3 x \left (1+c^2 x^2\right )^{3/2}-\frac {1}{36} b c d^3 x \left (1+c^2 x^2\right )^{5/2}+\frac {1}{2} d^3 \left (1+c^2 x^2\right ) \left (a+b \sinh ^{-1}(c x)\right )+\frac {1}{4} d^3 \left (1+c^2 x^2\right )^2 \left (a+b \sinh ^{-1}(c x)\right )+\frac {1}{6} d^3 \left (1+c^2 x^2\right )^3 \left (a+b \sinh ^{-1}(c x)\right )+d^3 \int \frac {a+b \sinh ^{-1}(c x)}{x} \, dx-\frac {1}{48} \left (5 b c d^3\right ) \int \sqrt {1+c^2 x^2} \, dx-\frac {1}{16} \left (3 b c d^3\right ) \int \sqrt {1+c^2 x^2} \, dx-\frac {1}{2} \left (b c d^3\right ) \int \sqrt {1+c^2 x^2} \, dx\\ &=-\frac {19}{48} b c d^3 x \sqrt {1+c^2 x^2}-\frac {7}{72} b c d^3 x \left (1+c^2 x^2\right )^{3/2}-\frac {1}{36} b c d^3 x \left (1+c^2 x^2\right )^{5/2}+\frac {1}{2} d^3 \left (1+c^2 x^2\right ) \left (a+b \sinh ^{-1}(c x)\right )+\frac {1}{4} d^3 \left (1+c^2 x^2\right )^2 \left (a+b \sinh ^{-1}(c x)\right )+\frac {1}{6} d^3 \left (1+c^2 x^2\right )^3 \left (a+b \sinh ^{-1}(c x)\right )+d^3 \text {Subst}\left (\int (a+b x) \coth (x) \, dx,x,\sinh ^{-1}(c x)\right )-\frac {1}{96} \left (5 b c d^3\right ) \int \frac {1}{\sqrt {1+c^2 x^2}} \, dx-\frac {1}{32} \left (3 b c d^3\right ) \int \frac {1}{\sqrt {1+c^2 x^2}} \, dx-\frac {1}{4} \left (b c d^3\right ) \int \frac {1}{\sqrt {1+c^2 x^2}} \, dx\\ &=-\frac {19}{48} b c d^3 x \sqrt {1+c^2 x^2}-\frac {7}{72} b c d^3 x \left (1+c^2 x^2\right )^{3/2}-\frac {1}{36} b c d^3 x \left (1+c^2 x^2\right )^{5/2}-\frac {19}{48} b d^3 \sinh ^{-1}(c x)+\frac {1}{2} d^3 \left (1+c^2 x^2\right ) \left (a+b \sinh ^{-1}(c x)\right )+\frac {1}{4} d^3 \left (1+c^2 x^2\right )^2 \left (a+b \sinh ^{-1}(c x)\right )+\frac {1}{6} d^3 \left (1+c^2 x^2\right )^3 \left (a+b \sinh ^{-1}(c x)\right )-\frac {d^3 \left (a+b \sinh ^{-1}(c x)\right )^2}{2 b}-\left (2 d^3\right ) \text {Subst}\left (\int \frac {e^{2 x} (a+b x)}{1-e^{2 x}} \, dx,x,\sinh ^{-1}(c x)\right )\\ &=-\frac {19}{48} b c d^3 x \sqrt {1+c^2 x^2}-\frac {7}{72} b c d^3 x \left (1+c^2 x^2\right )^{3/2}-\frac {1}{36} b c d^3 x \left (1+c^2 x^2\right )^{5/2}-\frac {19}{48} b d^3 \sinh ^{-1}(c x)+\frac {1}{2} d^3 \left (1+c^2 x^2\right ) \left (a+b \sinh ^{-1}(c x)\right )+\frac {1}{4} d^3 \left (1+c^2 x^2\right )^2 \left (a+b \sinh ^{-1}(c x)\right )+\frac {1}{6} d^3 \left (1+c^2 x^2\right )^3 \left (a+b \sinh ^{-1}(c x)\right )-\frac {d^3 \left (a+b \sinh ^{-1}(c x)\right )^2}{2 b}+d^3 \left (a+b \sinh ^{-1}(c x)\right ) \log \left (1-e^{2 \sinh ^{-1}(c x)}\right )-\left (b d^3\right ) \text {Subst}\left (\int \log \left (1-e^{2 x}\right ) \, dx,x,\sinh ^{-1}(c x)\right )\\ &=-\frac {19}{48} b c d^3 x \sqrt {1+c^2 x^2}-\frac {7}{72} b c d^3 x \left (1+c^2 x^2\right )^{3/2}-\frac {1}{36} b c d^3 x \left (1+c^2 x^2\right )^{5/2}-\frac {19}{48} b d^3 \sinh ^{-1}(c x)+\frac {1}{2} d^3 \left (1+c^2 x^2\right ) \left (a+b \sinh ^{-1}(c x)\right )+\frac {1}{4} d^3 \left (1+c^2 x^2\right )^2 \left (a+b \sinh ^{-1}(c x)\right )+\frac {1}{6} d^3 \left (1+c^2 x^2\right )^3 \left (a+b \sinh ^{-1}(c x)\right )-\frac {d^3 \left (a+b \sinh ^{-1}(c x)\right )^2}{2 b}+d^3 \left (a+b \sinh ^{-1}(c x)\right ) \log \left (1-e^{2 \sinh ^{-1}(c x)}\right )-\frac {1}{2} \left (b d^3\right ) \text {Subst}\left (\int \frac {\log (1-x)}{x} \, dx,x,e^{2 \sinh ^{-1}(c x)}\right )\\ &=-\frac {19}{48} b c d^3 x \sqrt {1+c^2 x^2}-\frac {7}{72} b c d^3 x \left (1+c^2 x^2\right )^{3/2}-\frac {1}{36} b c d^3 x \left (1+c^2 x^2\right )^{5/2}-\frac {19}{48} b d^3 \sinh ^{-1}(c x)+\frac {1}{2} d^3 \left (1+c^2 x^2\right ) \left (a+b \sinh ^{-1}(c x)\right )+\frac {1}{4} d^3 \left (1+c^2 x^2\right )^2 \left (a+b \sinh ^{-1}(c x)\right )+\frac {1}{6} d^3 \left (1+c^2 x^2\right )^3 \left (a+b \sinh ^{-1}(c x)\right )-\frac {d^3 \left (a+b \sinh ^{-1}(c x)\right )^2}{2 b}+d^3 \left (a+b \sinh ^{-1}(c x)\right ) \log \left (1-e^{2 \sinh ^{-1}(c x)}\right )+\frac {1}{2} b d^3 \text {Li}_2\left (e^{2 \sinh ^{-1}(c x)}\right )\\ \end {align*}
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Mathematica [A]
time = 0.09, size = 189, normalized size = 0.86 \begin {gather*} \frac {1}{144} d^3 \left (216 a c^2 x^2+108 a c^4 x^4+24 a c^6 x^6-75 b c x \sqrt {1+c^2 x^2}-22 b c^3 x^3 \sqrt {1+c^2 x^2}-4 b c^5 x^5 \sqrt {1+c^2 x^2}-72 b \sinh ^{-1}(c x)^2+144 a \log \left (1-e^{2 \sinh ^{-1}(c x)}\right )+3 \sinh ^{-1}(c x) \left (-48 a+b \left (25+72 c^2 x^2+36 c^4 x^4+8 c^6 x^6\right )+48 b \log \left (1-e^{2 \sinh ^{-1}(c x)}\right )\right )+72 b \text {PolyLog}\left (2,e^{2 \sinh ^{-1}(c x)}\right )\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 3.60, size = 284, normalized size = 1.29
method | result | size |
derivativedivides | \(\frac {a \,d^{3} c^{6} x^{6}}{6}+\frac {3 a \,d^{3} c^{4} x^{4}}{4}+\frac {3 a \,d^{3} c^{2} x^{2}}{2}+a \,d^{3} \ln \left (c x \right )+d^{3} b \polylog \left (2, -c x -\sqrt {c^{2} x^{2}+1}\right )+d^{3} b \polylog \left (2, c x +\sqrt {c^{2} x^{2}+1}\right )+\frac {25 b \,d^{3} \arcsinh \left (c x \right )}{48}-\frac {d^{3} b \arcsinh \left (c x \right )^{2}}{2}+\frac {3 d^{3} b \arcsinh \left (c x \right ) c^{4} x^{4}}{4}+\frac {3 d^{3} b \arcsinh \left (c x \right ) c^{2} x^{2}}{2}+\frac {d^{3} b \arcsinh \left (c x \right ) c^{6} x^{6}}{6}+d^{3} b \arcsinh \left (c x \right ) \ln \left (1+c x +\sqrt {c^{2} x^{2}+1}\right )+d^{3} b \arcsinh \left (c x \right ) \ln \left (1-c x -\sqrt {c^{2} x^{2}+1}\right )-\frac {d^{3} b \,c^{5} x^{5} \sqrt {c^{2} x^{2}+1}}{36}-\frac {11 d^{3} b \,c^{3} x^{3} \sqrt {c^{2} x^{2}+1}}{72}-\frac {25 b c \,d^{3} x \sqrt {c^{2} x^{2}+1}}{48}\) | \(284\) |
default | \(\frac {a \,d^{3} c^{6} x^{6}}{6}+\frac {3 a \,d^{3} c^{4} x^{4}}{4}+\frac {3 a \,d^{3} c^{2} x^{2}}{2}+a \,d^{3} \ln \left (c x \right )+d^{3} b \polylog \left (2, -c x -\sqrt {c^{2} x^{2}+1}\right )+d^{3} b \polylog \left (2, c x +\sqrt {c^{2} x^{2}+1}\right )+\frac {25 b \,d^{3} \arcsinh \left (c x \right )}{48}-\frac {d^{3} b \arcsinh \left (c x \right )^{2}}{2}+\frac {3 d^{3} b \arcsinh \left (c x \right ) c^{4} x^{4}}{4}+\frac {3 d^{3} b \arcsinh \left (c x \right ) c^{2} x^{2}}{2}+\frac {d^{3} b \arcsinh \left (c x \right ) c^{6} x^{6}}{6}+d^{3} b \arcsinh \left (c x \right ) \ln \left (1+c x +\sqrt {c^{2} x^{2}+1}\right )+d^{3} b \arcsinh \left (c x \right ) \ln \left (1-c x -\sqrt {c^{2} x^{2}+1}\right )-\frac {d^{3} b \,c^{5} x^{5} \sqrt {c^{2} x^{2}+1}}{36}-\frac {11 d^{3} b \,c^{3} x^{3} \sqrt {c^{2} x^{2}+1}}{72}-\frac {25 b c \,d^{3} x \sqrt {c^{2} x^{2}+1}}{48}\) | \(284\) |
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*} d^{3} \left (\int \frac {a}{x}\, dx + \int 3 a c^{2} x\, dx + \int 3 a c^{4} x^{3}\, dx + \int a c^{6} x^{5}\, dx + \int \frac {b \operatorname {asinh}{\left (c x \right )}}{x}\, dx + \int 3 b c^{2} x \operatorname {asinh}{\left (c x \right )}\, dx + \int 3 b c^{4} x^{3} \operatorname {asinh}{\left (c x \right )}\, dx + \int b c^{6} x^{5} \operatorname {asinh}{\left (c x \right )}\, dx\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \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\,x\right )\right )\,{\left (d\,c^2\,x^2+d\right )}^3}{x} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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