Optimal. Leaf size=73 \[ -\frac {x}{b c \left (a+b \sinh ^{-1}(c x)\right )}+\frac {\cosh \left (\frac {a}{b}\right ) \text {Chi}\left (\frac {a+b \sinh ^{-1}(c x)}{b}\right )}{b^2 c^2}-\frac {\sinh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a+b \sinh ^{-1}(c x)}{b}\right )}{b^2 c^2} \]
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Rubi [A]
time = 0.11, 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 = {5818, 5774,
3384, 3379, 3382} \begin {gather*} \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 )} \end {gather*}
Antiderivative was successfully verified.
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Rule 3379
Rule 3382
Rule 3384
Rule 5774
Rule 5818
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 {\text {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 ) \text {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 ) \text {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 = 0.10, size = 60, normalized size = 0.82 \begin {gather*} \frac {-\frac {b c x}{a+b \sinh ^{-1}(c x)}+\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 )}{b^2 c^2} \end {gather*}
Antiderivative was successfully verified.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(150\) vs.
\(2(73)=146\).
time = 3.51, size = 151, normalized size = 2.07
method | result | size |
default | \(-\frac {-\sqrt {c^{2} x^{2}+1}+c x}{2 c^{2} b \left (a +b \arcsinh \left (c x \right )\right )}-\frac {{\mathrm e}^{\frac {a}{b}} \expIntegral \left (1, \arcsinh \left (c x \right )+\frac {a}{b}\right )}{2 c^{2} b^{2}}-\frac {\arcsinh \left (c x \right ) {\mathrm e}^{-\frac {a}{b}} \expIntegral \left (1, -\arcsinh \left (c x \right )-\frac {a}{b}\right ) b +{\mathrm e}^{-\frac {a}{b}} \expIntegral \left (1, -\arcsinh \left (c x \right )-\frac {a}{b}\right ) a +b c x +\sqrt {c^{2} x^{2}+1}\, b}{2 c^{2} b^{2} \left (a +b \arcsinh \left (c x \right )\right )}\) | \(151\) |
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*} \int \frac {x}{\left (a + b \operatorname {asinh}{\left (c x \right )}\right )^{2} \sqrt {c^{2} x^{2} + 1}}\, 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 )}^2\,\sqrt {c^2\,x^2+1}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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