Optimal. Leaf size=49 \[ a x-\frac {2 b \sqrt {\frac {d x^2}{d x^2+2}} \left (d x^2+2\right )}{d x}+b x \cosh ^{-1}\left (d x^2+1\right ) \]
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Rubi [A] time = 0.04, antiderivative size = 49, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {5901, 12, 6719, 261} \[ a x-\frac {2 b \sqrt {\frac {d x^2}{d x^2+2}} \left (d x^2+2\right )}{d x}+b x \cosh ^{-1}\left (d x^2+1\right ) \]
Antiderivative was successfully verified.
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Rule 12
Rule 261
Rule 5901
Rule 6719
Rubi steps
\begin {align*} \int \left (a+b \cosh ^{-1}\left (1+d x^2\right )\right ) \, dx &=a x+b \int \cosh ^{-1}\left (1+d x^2\right ) \, dx\\ &=a x+b x \cosh ^{-1}\left (1+d x^2\right )-b \int 2 \sqrt {\frac {d x^2}{2+d x^2}} \, dx\\ &=a x+b x \cosh ^{-1}\left (1+d x^2\right )-(2 b) \int \sqrt {\frac {d x^2}{2+d x^2}} \, dx\\ &=a x+b x \cosh ^{-1}\left (1+d x^2\right )-\frac {\left (2 b \sqrt {\frac {d x^2}{2+d x^2}} \sqrt {2+d x^2}\right ) \int \frac {x}{\sqrt {2+d x^2}} \, dx}{x}\\ &=a x-\frac {2 b \sqrt {\frac {d x^2}{2+d x^2}} \left (2+d x^2\right )}{d x}+b x \cosh ^{-1}\left (1+d x^2\right )\\ \end {align*}
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Mathematica [A] time = 0.07, size = 37, normalized size = 0.76 \[ a x-\frac {2 b x}{\sqrt {\frac {d x^2}{d x^2+2}}}+b x \cosh ^{-1}\left (d x^2+1\right ) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.58, size = 63, normalized size = 1.29 \[ \frac {b d x^{2} \log \left (d x^{2} + \sqrt {d^{2} x^{4} + 2 \, d x^{2}} + 1\right ) + a d x^{2} - 2 \, \sqrt {d^{2} x^{4} + 2 \, d x^{2}} b}{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.20, size = 62, normalized size = 1.27 \[ {\left (x \log \left (d x^{2} + \sqrt {{\left (d x^{2} + 1\right )}^{2} - 1} + 1\right ) + \frac {2 \, \sqrt {2} \mathrm {sgn}\relax (x)}{\sqrt {d}} - \frac {2 \, \sqrt {d^{2} x^{2} + 2 \, d}}{d \mathrm {sgn}\relax (x)}\right )} b + a x \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 37, normalized size = 0.76 \[ a x +b \left (x \,\mathrm {arccosh}\left (d \,x^{2}+1\right )-\frac {2 x \sqrt {d \,x^{2}+2}}{\sqrt {d \,x^{2}}}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.95, size = 44, normalized size = 0.90 \[ {\left (x \operatorname {arcosh}\left (d x^{2} + 1\right ) - \frac {2 \, {\left (d^{\frac {3}{2}} x^{2} + 2 \, \sqrt {d}\right )}}{\sqrt {d x^{2} + 2} d}\right )} b + a x \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.59, size = 32, normalized size = 0.65 \[ a\,x+b\,x\,\mathrm {acosh}\left (d\,x^2+1\right )-\frac {2\,b\,\mathrm {sign}\relax (x)\,\sqrt {d\,x^2+2}}{\sqrt {d}} \]
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 \[ \int \left (a + b \operatorname {acosh}{\left (d x^{2} + 1 \right )}\right )\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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