Optimal. Leaf size=76 \[ \frac {x \left (a+b \sinh ^{-1}(c x)\right )}{d \sqrt {c^2 d x^2+d}}-\frac {b \sqrt {c^2 x^2+1} \log \left (c^2 x^2+1\right )}{2 c d \sqrt {c^2 d x^2+d}} \]
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Rubi [A] time = 0.04, antiderivative size = 76, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.087, Rules used = {5687, 260} \[ \frac {x \left (a+b \sinh ^{-1}(c x)\right )}{d \sqrt {c^2 d x^2+d}}-\frac {b \sqrt {c^2 x^2+1} \log \left (c^2 x^2+1\right )}{2 c d \sqrt {c^2 d x^2+d}} \]
Antiderivative was successfully verified.
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Rule 260
Rule 5687
Rubi steps
\begin {align*} \int \frac {a+b \sinh ^{-1}(c x)}{\left (d+c^2 d x^2\right )^{3/2}} \, dx &=\frac {x \left (a+b \sinh ^{-1}(c x)\right )}{d \sqrt {d+c^2 d x^2}}-\frac {\left (b c \sqrt {1+c^2 x^2}\right ) \int \frac {x}{1+c^2 x^2} \, dx}{d \sqrt {d+c^2 d x^2}}\\ &=\frac {x \left (a+b \sinh ^{-1}(c x)\right )}{d \sqrt {d+c^2 d x^2}}-\frac {b \sqrt {1+c^2 x^2} \log \left (1+c^2 x^2\right )}{2 c d \sqrt {d+c^2 d x^2}}\\ \end {align*}
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Mathematica [A] time = 0.17, size = 100, normalized size = 1.32 \[ \frac {\sqrt {c^2 d x^2+d} \left (2 a c x \sqrt {c^2 x^2+1}-\left (b c^2 x^2+b\right ) \log \left (c^2 x^2+1\right )+2 b c x \sqrt {c^2 x^2+1} \sinh ^{-1}(c x)\right )}{2 c d^2 \left (c^2 x^2+1\right )^{3/2}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.65, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {c^{2} d x^{2} + d} {\left (b \operatorname {arsinh}\left (c x\right ) + a\right )}}{c^{4} d^{2} x^{4} + 2 \, c^{2} d^{2} x^{2} + d^{2}}, x\right ) \]
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 \[ \int \frac {b \operatorname {arsinh}\left (c x\right ) + a}{{\left (c^{2} d x^{2} + d\right )}^{\frac {3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.11, size = 143, normalized size = 1.88 \[ \frac {a x}{d \sqrt {c^{2} d \,x^{2}+d}}+\frac {b \sqrt {d \left (c^{2} x^{2}+1\right )}\, \arcsinh \left (c x \right )}{\sqrt {c^{2} x^{2}+1}\, c \,d^{2}}+\frac {b \sqrt {d \left (c^{2} x^{2}+1\right )}\, \arcsinh \left (c x \right ) x}{d^{2} \left (c^{2} x^{2}+1\right )}-\frac {b \sqrt {d \left (c^{2} x^{2}+1\right )}\, \ln \left (1+\left (c x +\sqrt {c^{2} x^{2}+1}\right )^{2}\right )}{\sqrt {c^{2} x^{2}+1}\, c \,d^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.35, size = 58, normalized size = 0.76 \[ \frac {b x \operatorname {arsinh}\left (c x\right )}{\sqrt {c^{2} d x^{2} + d} d} + \frac {a x}{\sqrt {c^{2} d x^{2} + d} d} - \frac {b \log \left (x^{2} + \frac {1}{c^{2}}\right )}{2 \, c d^{\frac {3}{2}}} \]
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 \[ \int \frac {a+b\,\mathrm {asinh}\left (c\,x\right )}{{\left (d\,c^2\,x^2+d\right )}^{3/2}} \,d x \]
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 \[ \int \frac {a + b \operatorname {asinh}{\left (c x \right )}}{\left (d \left (c^{2} x^{2} + 1\right )\right )^{\frac {3}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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