Optimal. Leaf size=58 \[ \frac {\cosh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a+b \cosh ^{-1}(c+d x)}{b}\right )}{b d}-\frac {\sinh \left (\frac {a}{b}\right ) \text {Chi}\left (\frac {a+b \cosh ^{-1}(c+d x)}{b}\right )}{b d} \]
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Rubi [A] time = 0.09, antiderivative size = 58, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.417, Rules used = {5864, 5658, 3303, 3298, 3301} \[ \frac {\cosh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a+b \cosh ^{-1}(c+d x)}{b}\right )}{b d}-\frac {\sinh \left (\frac {a}{b}\right ) \text {Chi}\left (\frac {a+b \cosh ^{-1}(c+d x)}{b}\right )}{b d} \]
Antiderivative was successfully verified.
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Rule 3298
Rule 3301
Rule 3303
Rule 5658
Rule 5864
Rubi steps
\begin {align*} \int \frac {1}{a+b \cosh ^{-1}(c+d x)} \, dx &=\frac {\operatorname {Subst}\left (\int \frac {1}{a+b \cosh ^{-1}(x)} \, dx,x,c+d x\right )}{d}\\ &=-\frac {\operatorname {Subst}\left (\int \frac {\sinh \left (\frac {a}{b}-\frac {x}{b}\right )}{x} \, dx,x,a+b \cosh ^{-1}(c+d x)\right )}{b d}\\ &=\frac {\cosh \left (\frac {a}{b}\right ) \operatorname {Subst}\left (\int \frac {\sinh \left (\frac {x}{b}\right )}{x} \, dx,x,a+b \cosh ^{-1}(c+d x)\right )}{b d}-\frac {\sinh \left (\frac {a}{b}\right ) \operatorname {Subst}\left (\int \frac {\cosh \left (\frac {x}{b}\right )}{x} \, dx,x,a+b \cosh ^{-1}(c+d x)\right )}{b d}\\ &=-\frac {\text {Chi}\left (\frac {a+b \cosh ^{-1}(c+d x)}{b}\right ) \sinh \left (\frac {a}{b}\right )}{b d}+\frac {\cosh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a+b \cosh ^{-1}(c+d x)}{b}\right )}{b d}\\ \end {align*}
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Mathematica [A] time = 0.12, size = 49, normalized size = 0.84 \[ \frac {\cosh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a}{b}+\cosh ^{-1}(c+d x)\right )-\sinh \left (\frac {a}{b}\right ) \text {Chi}\left (\frac {a}{b}+\cosh ^{-1}(c+d x)\right )}{b d} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.54, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {1}{b \operatorname {arcosh}\left (d x + c\right ) + a}, x\right ) \]
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 \[ \int \frac {1}{b \operatorname {arcosh}\left (d x + c\right ) + a}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.03, size = 60, normalized size = 1.03 \[ \frac {\frac {{\mathrm e}^{\frac {a}{b}} \Ei \left (1, \mathrm {arccosh}\left (d x +c \right )+\frac {a}{b}\right )}{2 b}-\frac {{\mathrm e}^{-\frac {a}{b}} \Ei \left (1, -\mathrm {arccosh}\left (d x +c \right )-\frac {a}{b}\right )}{2 b}}{d} \]
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 \[ \int \frac {1}{b \operatorname {arcosh}\left (d x + c\right ) + a}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.02 \[ \int \frac {1}{a+b\,\mathrm {acosh}\left (c+d\,x\right )} \,d x \]
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 \[ \int \frac {1}{a + b \operatorname {acosh}{\left (c + d x \right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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