Optimal. Leaf size=249 \[ -\frac {3}{32} b c^3 d^3 x \sqrt {1+c^2 x^2}+\frac {7}{16} b c^3 d^3 x \left (1+c^2 x^2\right )^{3/2}-\frac {b c d^3 \left (1+c^2 x^2\right )^{5/2}}{2 x}-\frac {3}{32} b c^2 d^3 \sinh ^{-1}(c x)+\frac {3}{2} c^2 d^3 \left (1+c^2 x^2\right ) \left (a+b \sinh ^{-1}(c x)\right )+\frac {3}{4} c^2 d^3 \left (1+c^2 x^2\right )^2 \left (a+b \sinh ^{-1}(c x)\right )-\frac {d^3 \left (1+c^2 x^2\right )^3 \left (a+b \sinh ^{-1}(c x)\right )}{2 x^2}+\frac {3 c^2 d^3 \left (a+b \sinh ^{-1}(c x)\right )^2}{2 b}+3 c^2 d^3 \left (a+b \sinh ^{-1}(c x)\right ) \log \left (1-e^{-2 \sinh ^{-1}(c x)}\right )-\frac {3}{2} b c^2 d^3 \text {PolyLog}\left (2,e^{-2 \sinh ^{-1}(c x)}\right ) \]
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Rubi [A]
time = 0.23, antiderivative size = 249, normalized size of antiderivative = 1.00, number of steps
used = 17, number of rules used = 10, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.417, Rules used = {5802, 283,
201, 221, 5801, 5775, 3797, 2221, 2317, 2438} \begin {gather*} -\frac {d^3 \left (c^2 x^2+1\right )^3 \left (a+b \sinh ^{-1}(c x)\right )}{2 x^2}+\frac {3}{4} c^2 d^3 \left (c^2 x^2+1\right )^2 \left (a+b \sinh ^{-1}(c x)\right )+\frac {3}{2} c^2 d^3 \left (c^2 x^2+1\right ) \left (a+b \sinh ^{-1}(c x)\right )+\frac {3 c^2 d^3 \left (a+b \sinh ^{-1}(c x)\right )^2}{2 b}+3 c^2 d^3 \log \left (1-e^{-2 \sinh ^{-1}(c x)}\right ) \left (a+b \sinh ^{-1}(c x)\right )-\frac {3}{2} b c^2 d^3 \text {Li}_2\left (e^{-2 \sinh ^{-1}(c x)}\right )-\frac {b c d^3 \left (c^2 x^2+1\right )^{5/2}}{2 x}-\frac {3}{32} b c^2 d^3 \sinh ^{-1}(c x)+\frac {7}{16} b c^3 d^3 x \left (c^2 x^2+1\right )^{3/2}-\frac {3}{32} b c^3 d^3 x \sqrt {c^2 x^2+1} \end {gather*}
Antiderivative was successfully verified.
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Rule 201
Rule 221
Rule 283
Rule 2221
Rule 2317
Rule 2438
Rule 3797
Rule 5775
Rule 5801
Rule 5802
Rubi steps
\begin {align*} \int \frac {\left (d+c^2 d x^2\right )^3 \left (a+b \sinh ^{-1}(c x)\right )}{x^3} \, dx &=-\frac {d^3 \left (1+c^2 x^2\right )^3 \left (a+b \sinh ^{-1}(c x)\right )}{2 x^2}+\left (3 c^2 d\right ) \int \frac {\left (d+c^2 d x^2\right )^2 \left (a+b \sinh ^{-1}(c x)\right )}{x} \, dx+\frac {1}{2} \left (b c d^3\right ) \int \frac {\left (1+c^2 x^2\right )^{5/2}}{x^2} \, dx\\ &=-\frac {b c d^3 \left (1+c^2 x^2\right )^{5/2}}{2 x}+\frac {3}{4} c^2 d^3 \left (1+c^2 x^2\right )^2 \left (a+b \sinh ^{-1}(c x)\right )-\frac {d^3 \left (1+c^2 x^2\right )^3 \left (a+b \sinh ^{-1}(c x)\right )}{2 x^2}+\left (3 c^2 d^2\right ) \int \frac {\left (d+c^2 d x^2\right ) \left (a+b \sinh ^{-1}(c x)\right )}{x} \, dx-\frac {1}{4} \left (3 b c^3 d^3\right ) \int \left (1+c^2 x^2\right )^{3/2} \, dx+\frac {1}{2} \left (5 b c^3 d^3\right ) \int \left (1+c^2 x^2\right )^{3/2} \, dx\\ &=\frac {7}{16} b c^3 d^3 x \left (1+c^2 x^2\right )^{3/2}-\frac {b c d^3 \left (1+c^2 x^2\right )^{5/2}}{2 x}+\frac {3}{2} c^2 d^3 \left (1+c^2 x^2\right ) \left (a+b \sinh ^{-1}(c x)\right )+\frac {3}{4} c^2 d^3 \left (1+c^2 x^2\right )^2 \left (a+b \sinh ^{-1}(c x)\right )-\frac {d^3 \left (1+c^2 x^2\right )^3 \left (a+b \sinh ^{-1}(c x)\right )}{2 x^2}+\left (3 c^2 d^3\right ) \int \frac {a+b \sinh ^{-1}(c x)}{x} \, dx-\frac {1}{16} \left (9 b c^3 d^3\right ) \int \sqrt {1+c^2 x^2} \, dx-\frac {1}{2} \left (3 b c^3 d^3\right ) \int \sqrt {1+c^2 x^2} \, dx+\frac {1}{8} \left (15 b c^3 d^3\right ) \int \sqrt {1+c^2 x^2} \, dx\\ &=-\frac {3}{32} b c^3 d^3 x \sqrt {1+c^2 x^2}+\frac {7}{16} b c^3 d^3 x \left (1+c^2 x^2\right )^{3/2}-\frac {b c d^3 \left (1+c^2 x^2\right )^{5/2}}{2 x}+\frac {3}{2} c^2 d^3 \left (1+c^2 x^2\right ) \left (a+b \sinh ^{-1}(c x)\right )+\frac {3}{4} c^2 d^3 \left (1+c^2 x^2\right )^2 \left (a+b \sinh ^{-1}(c x)\right )-\frac {d^3 \left (1+c^2 x^2\right )^3 \left (a+b \sinh ^{-1}(c x)\right )}{2 x^2}+\left (3 c^2 d^3\right ) \text {Subst}\left (\int (a+b x) \coth (x) \, dx,x,\sinh ^{-1}(c x)\right )-\frac {1}{32} \left (9 b c^3 d^3\right ) \int \frac {1}{\sqrt {1+c^2 x^2}} \, dx-\frac {1}{4} \left (3 b c^3 d^3\right ) \int \frac {1}{\sqrt {1+c^2 x^2}} \, dx+\frac {1}{16} \left (15 b c^3 d^3\right ) \int \frac {1}{\sqrt {1+c^2 x^2}} \, dx\\ &=-\frac {3}{32} b c^3 d^3 x \sqrt {1+c^2 x^2}+\frac {7}{16} b c^3 d^3 x \left (1+c^2 x^2\right )^{3/2}-\frac {b c d^3 \left (1+c^2 x^2\right )^{5/2}}{2 x}-\frac {3}{32} b c^2 d^3 \sinh ^{-1}(c x)+\frac {3}{2} c^2 d^3 \left (1+c^2 x^2\right ) \left (a+b \sinh ^{-1}(c x)\right )+\frac {3}{4} c^2 d^3 \left (1+c^2 x^2\right )^2 \left (a+b \sinh ^{-1}(c x)\right )-\frac {d^3 \left (1+c^2 x^2\right )^3 \left (a+b \sinh ^{-1}(c x)\right )}{2 x^2}-\frac {3 c^2 d^3 \left (a+b \sinh ^{-1}(c x)\right )^2}{2 b}-\left (6 c^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 {3}{32} b c^3 d^3 x \sqrt {1+c^2 x^2}+\frac {7}{16} b c^3 d^3 x \left (1+c^2 x^2\right )^{3/2}-\frac {b c d^3 \left (1+c^2 x^2\right )^{5/2}}{2 x}-\frac {3}{32} b c^2 d^3 \sinh ^{-1}(c x)+\frac {3}{2} c^2 d^3 \left (1+c^2 x^2\right ) \left (a+b \sinh ^{-1}(c x)\right )+\frac {3}{4} c^2 d^3 \left (1+c^2 x^2\right )^2 \left (a+b \sinh ^{-1}(c x)\right )-\frac {d^3 \left (1+c^2 x^2\right )^3 \left (a+b \sinh ^{-1}(c x)\right )}{2 x^2}-\frac {3 c^2 d^3 \left (a+b \sinh ^{-1}(c x)\right )^2}{2 b}+3 c^2 d^3 \left (a+b \sinh ^{-1}(c x)\right ) \log \left (1-e^{2 \sinh ^{-1}(c x)}\right )-\left (3 b c^2 d^3\right ) \text {Subst}\left (\int \log \left (1-e^{2 x}\right ) \, dx,x,\sinh ^{-1}(c x)\right )\\ &=-\frac {3}{32} b c^3 d^3 x \sqrt {1+c^2 x^2}+\frac {7}{16} b c^3 d^3 x \left (1+c^2 x^2\right )^{3/2}-\frac {b c d^3 \left (1+c^2 x^2\right )^{5/2}}{2 x}-\frac {3}{32} b c^2 d^3 \sinh ^{-1}(c x)+\frac {3}{2} c^2 d^3 \left (1+c^2 x^2\right ) \left (a+b \sinh ^{-1}(c x)\right )+\frac {3}{4} c^2 d^3 \left (1+c^2 x^2\right )^2 \left (a+b \sinh ^{-1}(c x)\right )-\frac {d^3 \left (1+c^2 x^2\right )^3 \left (a+b \sinh ^{-1}(c x)\right )}{2 x^2}-\frac {3 c^2 d^3 \left (a+b \sinh ^{-1}(c x)\right )^2}{2 b}+3 c^2 d^3 \left (a+b \sinh ^{-1}(c x)\right ) \log \left (1-e^{2 \sinh ^{-1}(c x)}\right )-\frac {1}{2} \left (3 b c^2 d^3\right ) \text {Subst}\left (\int \frac {\log (1-x)}{x} \, dx,x,e^{2 \sinh ^{-1}(c x)}\right )\\ &=-\frac {3}{32} b c^3 d^3 x \sqrt {1+c^2 x^2}+\frac {7}{16} b c^3 d^3 x \left (1+c^2 x^2\right )^{3/2}-\frac {b c d^3 \left (1+c^2 x^2\right )^{5/2}}{2 x}-\frac {3}{32} b c^2 d^3 \sinh ^{-1}(c x)+\frac {3}{2} c^2 d^3 \left (1+c^2 x^2\right ) \left (a+b \sinh ^{-1}(c x)\right )+\frac {3}{4} c^2 d^3 \left (1+c^2 x^2\right )^2 \left (a+b \sinh ^{-1}(c x)\right )-\frac {d^3 \left (1+c^2 x^2\right )^3 \left (a+b \sinh ^{-1}(c x)\right )}{2 x^2}-\frac {3 c^2 d^3 \left (a+b \sinh ^{-1}(c x)\right )^2}{2 b}+3 c^2 d^3 \left (a+b \sinh ^{-1}(c x)\right ) \log \left (1-e^{2 \sinh ^{-1}(c x)}\right )+\frac {3}{2} b c^2 d^3 \text {Li}_2\left (e^{2 \sinh ^{-1}(c x)}\right )\\ \end {align*}
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Mathematica [A]
time = 0.17, size = 210, normalized size = 0.84 \begin {gather*} \frac {d^3 \left (-16 a+48 a c^4 x^4+8 a c^6 x^6-16 b c x \sqrt {1+c^2 x^2}-21 b c^3 x^3 \sqrt {1+c^2 x^2}-2 b c^5 x^5 \sqrt {1+c^2 x^2}+48 b c^2 x^2 \sinh ^{-1}(c x)^2+21 b c^2 x^2 \tanh ^{-1}\left (\frac {c x}{\sqrt {1+c^2 x^2}}\right )+8 b \sinh ^{-1}(c x) \left (-2+6 c^4 x^4+c^6 x^6+12 c^2 x^2 \log \left (1-e^{-2 \sinh ^{-1}(c x)}\right )\right )+96 a c^2 x^2 \log (x)-48 b c^2 x^2 \text {PolyLog}\left (2,e^{-2 \sinh ^{-1}(c x)}\right )\right )}{32 x^2} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 6.02, size = 299, normalized size = 1.20
method | result | size |
derivativedivides | \(c^{2} \left (\frac {a \,d^{3} c^{4} x^{4}}{4}+\frac {3 a \,d^{3} c^{2} x^{2}}{2}-\frac {a \,d^{3}}{2 c^{2} x^{2}}+3 a \,d^{3} \ln \left (c x \right )+3 d^{3} b \arcsinh \left (c x \right ) \ln \left (1-c x -\sqrt {c^{2} x^{2}+1}\right )+3 d^{3} b \arcsinh \left (c x \right ) \ln \left (1+c x +\sqrt {c^{2} x^{2}+1}\right )+\frac {d^{3} b}{2}+3 d^{3} b \polylog \left (2, c x +\sqrt {c^{2} x^{2}+1}\right )+3 d^{3} b \polylog \left (2, -c x -\sqrt {c^{2} x^{2}+1}\right )-\frac {3 d^{3} b \arcsinh \left (c x \right )^{2}}{2}+\frac {21 b \,d^{3} \arcsinh \left (c x \right )}{32}+\frac {d^{3} b \arcsinh \left (c x \right ) c^{4} x^{4}}{4}+\frac {3 d^{3} b \arcsinh \left (c x \right ) c^{2} x^{2}}{2}-\frac {d^{3} b \arcsinh \left (c x \right )}{2 c^{2} x^{2}}-\frac {d^{3} b \sqrt {c^{2} x^{2}+1}}{2 c x}-\frac {d^{3} b \,c^{3} x^{3} \sqrt {c^{2} x^{2}+1}}{16}-\frac {21 b c \,d^{3} x \sqrt {c^{2} x^{2}+1}}{32}\right )\) | \(299\) |
default | \(c^{2} \left (\frac {a \,d^{3} c^{4} x^{4}}{4}+\frac {3 a \,d^{3} c^{2} x^{2}}{2}-\frac {a \,d^{3}}{2 c^{2} x^{2}}+3 a \,d^{3} \ln \left (c x \right )+3 d^{3} b \arcsinh \left (c x \right ) \ln \left (1-c x -\sqrt {c^{2} x^{2}+1}\right )+3 d^{3} b \arcsinh \left (c x \right ) \ln \left (1+c x +\sqrt {c^{2} x^{2}+1}\right )+\frac {d^{3} b}{2}+3 d^{3} b \polylog \left (2, c x +\sqrt {c^{2} x^{2}+1}\right )+3 d^{3} b \polylog \left (2, -c x -\sqrt {c^{2} x^{2}+1}\right )-\frac {3 d^{3} b \arcsinh \left (c x \right )^{2}}{2}+\frac {21 b \,d^{3} \arcsinh \left (c x \right )}{32}+\frac {d^{3} b \arcsinh \left (c x \right ) c^{4} x^{4}}{4}+\frac {3 d^{3} b \arcsinh \left (c x \right ) c^{2} x^{2}}{2}-\frac {d^{3} b \arcsinh \left (c x \right )}{2 c^{2} x^{2}}-\frac {d^{3} b \sqrt {c^{2} x^{2}+1}}{2 c x}-\frac {d^{3} b \,c^{3} x^{3} \sqrt {c^{2} x^{2}+1}}{16}-\frac {21 b c \,d^{3} x \sqrt {c^{2} x^{2}+1}}{32}\right )\) | \(299\) |
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^{3}}\, dx + \int \frac {3 a c^{2}}{x}\, dx + \int 3 a c^{4} x\, dx + \int a c^{6} x^{3}\, dx + \int \frac {b \operatorname {asinh}{\left (c x \right )}}{x^{3}}\, dx + \int \frac {3 b c^{2} \operatorname {asinh}{\left (c x \right )}}{x}\, dx + \int 3 b c^{4} x \operatorname {asinh}{\left (c x \right )}\, dx + \int b c^{6} x^{3} \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^3} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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