Optimal. Leaf size=70 \[ -\frac {a+b \sinh ^{-1}(c x)}{c^2 d \sqrt {d+c^2 d x^2}}+\frac {b \sqrt {1+c^2 x^2} \text {ArcTan}(c x)}{c^2 d \sqrt {d+c^2 d x^2}} \]
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Rubi [A]
time = 0.05, antiderivative size = 70, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 2, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.083, Rules used = {5798, 209}
\begin {gather*} \frac {b \sqrt {c^2 x^2+1} \text {ArcTan}(c x)}{c^2 d \sqrt {c^2 d x^2+d}}-\frac {a+b \sinh ^{-1}(c x)}{c^2 d \sqrt {c^2 d x^2+d}} \end {gather*}
Antiderivative was successfully verified.
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Rule 209
Rule 5798
Rubi steps
\begin {align*} \int \frac {x \left (a+b \sinh ^{-1}(c x)\right )}{\left (d+c^2 d x^2\right )^{3/2}} \, dx &=-\frac {a+b \sinh ^{-1}(c x)}{c^2 d \sqrt {d+c^2 d x^2}}+\frac {\left (b \sqrt {1+c^2 x^2}\right ) \int \frac {1}{1+c^2 x^2} \, dx}{c d \sqrt {d+c^2 d x^2}}\\ &=-\frac {a+b \sinh ^{-1}(c x)}{c^2 d \sqrt {d+c^2 d x^2}}+\frac {b \sqrt {1+c^2 x^2} \tan ^{-1}(c x)}{c^2 d \sqrt {d+c^2 d x^2}}\\ \end {align*}
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Mathematica [A]
time = 0.10, size = 82, normalized size = 1.17 \begin {gather*} -\frac {\sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{c^2 d^2 \left (1+c^2 x^2\right )}+\frac {b \sqrt {d \left (1+c^2 x^2\right )} \text {ArcTan}(c x)}{c^2 d^2 \sqrt {1+c^2 x^2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [C] Result contains complex when optimal does not.
time = 0.92, size = 164, normalized size = 2.34
method | result | size |
default | \(-\frac {a}{c^{2} d \sqrt {c^{2} d \,x^{2}+d}}-\frac {b \sqrt {d \left (c^{2} x^{2}+1\right )}\, \arcsinh \left (c x \right )}{c^{2} d^{2} \left (c^{2} x^{2}+1\right )}+\frac {i b \sqrt {d \left (c^{2} x^{2}+1\right )}\, \ln \left (c x +\sqrt {c^{2} x^{2}+1}+i\right )}{\sqrt {c^{2} x^{2}+1}\, c^{2} d^{2}}-\frac {i b \sqrt {d \left (c^{2} x^{2}+1\right )}\, \ln \left (c x +\sqrt {c^{2} x^{2}+1}-i\right )}{\sqrt {c^{2} x^{2}+1}\, c^{2} d^{2}}\) | \(164\) |
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 [A]
time = 0.45, size = 128, normalized size = 1.83 \begin {gather*} -\frac {{\left (b c^{2} x^{2} + b\right )} \sqrt {d} \arctan \left (\frac {2 \, \sqrt {c^{2} d x^{2} + d} \sqrt {c^{2} x^{2} + 1} c \sqrt {d} x}{c^{4} d x^{4} - d}\right ) + 2 \, \sqrt {c^{2} d x^{2} + d} b \log \left (c x + \sqrt {c^{2} x^{2} + 1}\right ) + 2 \, \sqrt {c^{2} d x^{2} + d} a}{2 \, {\left (c^{4} d^{2} x^{2} + c^{2} d^{2}\right )}} \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*} \int \frac {x \left (a + b \operatorname {asinh}{\left (c x \right )}\right )}{\left (d \left (c^{2} x^{2} + 1\right )\right )^{\frac {3}{2}}}\, dx \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 {x\,\left (a+b\,\mathrm {asinh}\left (c\,x\right )\right )}{{\left (d\,c^2\,x^2+d\right )}^{3/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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