Optimal. Leaf size=73 \[ \frac {\cosh \left (\frac {a}{b}\right ) \text {Chi}\left (\frac {a+b \sinh ^{-1}(c x)}{b}\right )}{b^2 c^2}-\frac {\sinh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a+b \sinh ^{-1}(c x)}{b}\right )}{b^2 c^2}-\frac {x}{b c \left (a+b \sinh ^{-1}(c x)\right )} \]
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Rubi [A] time = 0.16, antiderivative size = 73, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {5774, 5657, 3303, 3298, 3301} \[ \frac {\cosh \left (\frac {a}{b}\right ) \text {Chi}\left (\frac {a+b \sinh ^{-1}(c x)}{b}\right )}{b^2 c^2}-\frac {\sinh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a+b \sinh ^{-1}(c x)}{b}\right )}{b^2 c^2}-\frac {x}{b c \left (a+b \sinh ^{-1}(c x)\right )} \]
Antiderivative was successfully verified.
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Rule 3298
Rule 3301
Rule 3303
Rule 5657
Rule 5774
Rubi steps
\begin {align*} \int \frac {x}{\sqrt {1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )^2} \, dx &=-\frac {x}{b c \left (a+b \sinh ^{-1}(c x)\right )}+\frac {\int \frac {1}{a+b \sinh ^{-1}(c x)} \, dx}{b c}\\ &=-\frac {x}{b c \left (a+b \sinh ^{-1}(c x)\right )}+\frac {\operatorname {Subst}\left (\int \frac {\cosh \left (\frac {a}{b}-\frac {x}{b}\right )}{x} \, dx,x,a+b \sinh ^{-1}(c x)\right )}{b^2 c^2}\\ &=-\frac {x}{b c \left (a+b \sinh ^{-1}(c x)\right )}+\frac {\cosh \left (\frac {a}{b}\right ) \operatorname {Subst}\left (\int \frac {\cosh \left (\frac {x}{b}\right )}{x} \, dx,x,a+b \sinh ^{-1}(c x)\right )}{b^2 c^2}-\frac {\sinh \left (\frac {a}{b}\right ) \operatorname {Subst}\left (\int \frac {\sinh \left (\frac {x}{b}\right )}{x} \, dx,x,a+b \sinh ^{-1}(c x)\right )}{b^2 c^2}\\ &=-\frac {x}{b c \left (a+b \sinh ^{-1}(c x)\right )}+\frac {\cosh \left (\frac {a}{b}\right ) \text {Chi}\left (\frac {a+b \sinh ^{-1}(c x)}{b}\right )}{b^2 c^2}-\frac {\sinh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a+b \sinh ^{-1}(c x)}{b}\right )}{b^2 c^2}\\ \end {align*}
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Mathematica [A] time = 0.12, size = 60, normalized size = 0.82 \[ \frac {\cosh \left (\frac {a}{b}\right ) \text {Chi}\left (\frac {a}{b}+\sinh ^{-1}(c x)\right )-\sinh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a}{b}+\sinh ^{-1}(c x)\right )-\frac {b c x}{a+b \sinh ^{-1}(c x)}}{b^2 c^2} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.56, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {c^{2} x^{2} + 1} x}{a^{2} c^{2} x^{2} + {\left (b^{2} c^{2} x^{2} + b^{2}\right )} \operatorname {arsinh}\left (c x\right )^{2} + a^{2} + 2 \, {\left (a b c^{2} x^{2} + a b\right )} \operatorname {arsinh}\left (c x\right )}, x\right ) \]
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 \[ \int \frac {x}{\sqrt {c^{2} x^{2} + 1} {\left (b \operatorname {arsinh}\left (c x\right ) + a\right )}^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.08, size = 151, normalized size = 2.07 \[ -\frac {c x -\sqrt {c^{2} x^{2}+1}}{2 c^{2} b \left (a +b \arcsinh \left (c x \right )\right )}-\frac {{\mathrm e}^{\frac {a}{b}} \Ei \left (1, \arcsinh \left (c x \right )+\frac {a}{b}\right )}{2 c^{2} b^{2}}-\frac {\arcsinh \left (c x \right ) {\mathrm e}^{-\frac {a}{b}} \Ei \left (1, -\arcsinh \left (c x \right )-\frac {a}{b}\right ) b +{\mathrm e}^{-\frac {a}{b}} \Ei \left (1, -\arcsinh \left (c x \right )-\frac {a}{b}\right ) a +x b c +\sqrt {c^{2} x^{2}+1}\, b}{2 c^{2} b^{2} \left (a +b \arcsinh \left (c x \right )\right )} \]
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 \[ -\frac {c^{3} x^{4} + c x^{2} + {\left (c^{2} x^{3} + x\right )} \sqrt {c^{2} x^{2} + 1}}{{\left (c^{2} x^{2} + 1\right )} a b c^{2} x + {\left ({\left (c^{2} x^{2} + 1\right )} b^{2} c^{2} x + {\left (b^{2} c^{3} x^{2} + b^{2} c\right )} \sqrt {c^{2} x^{2} + 1}\right )} \log \left (c x + \sqrt {c^{2} x^{2} + 1}\right ) + {\left (a b c^{3} x^{2} + a b c\right )} \sqrt {c^{2} x^{2} + 1}} + \int \frac {c^{5} x^{5} + {\left (c^{2} x^{2} + 1\right )} c^{3} x^{3} + 3 \, c^{3} x^{3} + 2 \, c x + {\left (2 \, c^{4} x^{4} + 3 \, c^{2} x^{2} + 1\right )} \sqrt {c^{2} x^{2} + 1}}{{\left (c^{2} x^{2} + 1\right )}^{\frac {3}{2}} a b c^{3} x^{2} + 2 \, {\left (a b c^{4} x^{3} + a b c^{2} x\right )} {\left (c^{2} x^{2} + 1\right )} + {\left ({\left (c^{2} x^{2} + 1\right )}^{\frac {3}{2}} b^{2} c^{3} x^{2} + 2 \, {\left (b^{2} c^{4} x^{3} + b^{2} c^{2} x\right )} {\left (c^{2} x^{2} + 1\right )} + {\left (b^{2} c^{5} x^{4} + 2 \, b^{2} c^{3} x^{2} + b^{2} c\right )} \sqrt {c^{2} x^{2} + 1}\right )} \log \left (c x + \sqrt {c^{2} x^{2} + 1}\right ) + {\left (a b c^{5} x^{4} + 2 \, a b c^{3} x^{2} + a b c\right )} \sqrt {c^{2} x^{2} + 1}}\,{d x} \]
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 \[ \int \frac {x}{{\left (a+b\,\mathrm {asinh}\left (c\,x\right )\right )}^2\,\sqrt {c^2\,x^2+1}} \,d x \]
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 \[ \int \frac {x}{\left (a + b \operatorname {asinh}{\left (c x \right )}\right )^{2} \sqrt {c^{2} x^{2} + 1}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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