Optimal. Leaf size=124 \[ \frac {b}{2 c d^2 \sqrt {1+c^2 x^2}}+\frac {x \left (a+b \sinh ^{-1}(c x)\right )}{2 d^2 \left (1+c^2 x^2\right )}+\frac {\left (a+b \sinh ^{-1}(c x)\right ) \text {ArcTan}\left (e^{\sinh ^{-1}(c x)}\right )}{c d^2}-\frac {i b \text {PolyLog}\left (2,-i e^{\sinh ^{-1}(c x)}\right )}{2 c d^2}+\frac {i b \text {PolyLog}\left (2,i e^{\sinh ^{-1}(c x)}\right )}{2 c d^2} \]
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Rubi [A]
time = 0.07, antiderivative size = 124, normalized size of antiderivative = 1.00, number of steps
used = 8, number of rules used = 6, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {5788, 5789,
4265, 2317, 2438, 267} \begin {gather*} \frac {\text {ArcTan}\left (e^{\sinh ^{-1}(c x)}\right ) \left (a+b \sinh ^{-1}(c x)\right )}{c d^2}+\frac {x \left (a+b \sinh ^{-1}(c x)\right )}{2 d^2 \left (c^2 x^2+1\right )}+\frac {b}{2 c d^2 \sqrt {c^2 x^2+1}}-\frac {i b \text {Li}_2\left (-i e^{\sinh ^{-1}(c x)}\right )}{2 c d^2}+\frac {i b \text {Li}_2\left (i e^{\sinh ^{-1}(c x)}\right )}{2 c d^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 267
Rule 2317
Rule 2438
Rule 4265
Rule 5788
Rule 5789
Rubi steps
\begin {align*} \int \frac {a+b \sinh ^{-1}(c x)}{\left (d+c^2 d x^2\right )^2} \, dx &=\frac {x \left (a+b \sinh ^{-1}(c x)\right )}{2 d^2 \left (1+c^2 x^2\right )}-\frac {(b c) \int \frac {x}{\left (1+c^2 x^2\right )^{3/2}} \, dx}{2 d^2}+\frac {\int \frac {a+b \sinh ^{-1}(c x)}{d+c^2 d x^2} \, dx}{2 d}\\ &=\frac {b}{2 c d^2 \sqrt {1+c^2 x^2}}+\frac {x \left (a+b \sinh ^{-1}(c x)\right )}{2 d^2 \left (1+c^2 x^2\right )}+\frac {\text {Subst}\left (\int (a+b x) \text {sech}(x) \, dx,x,\sinh ^{-1}(c x)\right )}{2 c d^2}\\ &=\frac {b}{2 c d^2 \sqrt {1+c^2 x^2}}+\frac {x \left (a+b \sinh ^{-1}(c x)\right )}{2 d^2 \left (1+c^2 x^2\right )}+\frac {\left (a+b \sinh ^{-1}(c x)\right ) \tan ^{-1}\left (e^{\sinh ^{-1}(c x)}\right )}{c d^2}-\frac {(i b) \text {Subst}\left (\int \log \left (1-i e^x\right ) \, dx,x,\sinh ^{-1}(c x)\right )}{2 c d^2}+\frac {(i b) \text {Subst}\left (\int \log \left (1+i e^x\right ) \, dx,x,\sinh ^{-1}(c x)\right )}{2 c d^2}\\ &=\frac {b}{2 c d^2 \sqrt {1+c^2 x^2}}+\frac {x \left (a+b \sinh ^{-1}(c x)\right )}{2 d^2 \left (1+c^2 x^2\right )}+\frac {\left (a+b \sinh ^{-1}(c x)\right ) \tan ^{-1}\left (e^{\sinh ^{-1}(c x)}\right )}{c d^2}-\frac {(i b) \text {Subst}\left (\int \frac {\log (1-i x)}{x} \, dx,x,e^{\sinh ^{-1}(c x)}\right )}{2 c d^2}+\frac {(i b) \text {Subst}\left (\int \frac {\log (1+i x)}{x} \, dx,x,e^{\sinh ^{-1}(c x)}\right )}{2 c d^2}\\ &=\frac {b}{2 c d^2 \sqrt {1+c^2 x^2}}+\frac {x \left (a+b \sinh ^{-1}(c x)\right )}{2 d^2 \left (1+c^2 x^2\right )}+\frac {\left (a+b \sinh ^{-1}(c x)\right ) \tan ^{-1}\left (e^{\sinh ^{-1}(c x)}\right )}{c d^2}-\frac {i b \text {Li}_2\left (-i e^{\sinh ^{-1}(c x)}\right )}{2 c d^2}+\frac {i b \text {Li}_2\left (i e^{\sinh ^{-1}(c x)}\right )}{2 c d^2}\\ \end {align*}
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Mathematica [A]
time = 0.07, size = 216, normalized size = 1.74 \begin {gather*} \frac {a c x+b \sqrt {1+c^2 x^2}+b c x \sinh ^{-1}(c x)+a \text {ArcTan}(c x)+a c^2 x^2 \text {ArcTan}(c x)+i b \sinh ^{-1}(c x) \log \left (1-i e^{\sinh ^{-1}(c x)}\right )+i b c^2 x^2 \sinh ^{-1}(c x) \log \left (1-i e^{\sinh ^{-1}(c x)}\right )-i b \sinh ^{-1}(c x) \log \left (1+i e^{\sinh ^{-1}(c x)}\right )-i b c^2 x^2 \sinh ^{-1}(c x) \log \left (1+i e^{\sinh ^{-1}(c x)}\right )-i b \left (1+c^2 x^2\right ) \text {PolyLog}\left (2,-i e^{\sinh ^{-1}(c x)}\right )+i b \left (1+c^2 x^2\right ) \text {PolyLog}\left (2,i e^{\sinh ^{-1}(c x)}\right )}{2 d^2 \left (c+c^3 x^2\right )} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.64, size = 219, normalized size = 1.77
method | result | size |
derivativedivides | \(\frac {\frac {a c x}{2 d^{2} \left (c^{2} x^{2}+1\right )}+\frac {a \arctan \left (c x \right )}{2 d^{2}}+\frac {b \arcsinh \left (c x \right ) c x}{2 d^{2} \left (c^{2} x^{2}+1\right )}+\frac {b \arcsinh \left (c x \right ) \arctan \left (c x \right )}{2 d^{2}}+\frac {b}{2 d^{2} \sqrt {c^{2} x^{2}+1}}+\frac {b \arctan \left (c x \right ) \ln \left (1+\frac {i \left (i c x +1\right )}{\sqrt {c^{2} x^{2}+1}}\right )}{2 d^{2}}-\frac {b \arctan \left (c x \right ) \ln \left (1-\frac {i \left (i c x +1\right )}{\sqrt {c^{2} x^{2}+1}}\right )}{2 d^{2}}-\frac {i b \dilog \left (1+\frac {i \left (i c x +1\right )}{\sqrt {c^{2} x^{2}+1}}\right )}{2 d^{2}}+\frac {i b \dilog \left (1-\frac {i \left (i c x +1\right )}{\sqrt {c^{2} x^{2}+1}}\right )}{2 d^{2}}}{c}\) | \(219\) |
default | \(\frac {\frac {a c x}{2 d^{2} \left (c^{2} x^{2}+1\right )}+\frac {a \arctan \left (c x \right )}{2 d^{2}}+\frac {b \arcsinh \left (c x \right ) c x}{2 d^{2} \left (c^{2} x^{2}+1\right )}+\frac {b \arcsinh \left (c x \right ) \arctan \left (c x \right )}{2 d^{2}}+\frac {b}{2 d^{2} \sqrt {c^{2} x^{2}+1}}+\frac {b \arctan \left (c x \right ) \ln \left (1+\frac {i \left (i c x +1\right )}{\sqrt {c^{2} x^{2}+1}}\right )}{2 d^{2}}-\frac {b \arctan \left (c x \right ) \ln \left (1-\frac {i \left (i c x +1\right )}{\sqrt {c^{2} x^{2}+1}}\right )}{2 d^{2}}-\frac {i b \dilog \left (1+\frac {i \left (i c x +1\right )}{\sqrt {c^{2} x^{2}+1}}\right )}{2 d^{2}}+\frac {i b \dilog \left (1-\frac {i \left (i c x +1\right )}{\sqrt {c^{2} x^{2}+1}}\right )}{2 d^{2}}}{c}\) | \(219\) |
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}{c^{4} x^{4} + 2 c^{2} x^{2} + 1}\, dx + \int \frac {b \operatorname {asinh}{\left (c x \right )}}{c^{4} x^{4} + 2 c^{2} x^{2} + 1}\, dx}{d^{2}} \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 {a+b\,\mathrm {asinh}\left (c\,x\right )}{{\left (d\,c^2\,x^2+d\right )}^2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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