Optimal. Leaf size=116 \[ -\frac {e \sinh \left (\frac {2 a}{b}\right ) \text {Chi}\left (\frac {2 a}{b}+2 \cosh ^{-1}(c x)\right )}{2 b c^2}+\frac {e \cosh \left (\frac {2 a}{b}\right ) \text {Shi}\left (\frac {2 a}{b}+2 \cosh ^{-1}(c x)\right )}{2 b c^2}-\frac {d \sinh \left (\frac {a}{b}\right ) \text {Chi}\left (\frac {a}{b}+\cosh ^{-1}(c x)\right )}{b c}+\frac {d \cosh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a}{b}+\cosh ^{-1}(c x)\right )}{b c} \]
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Rubi [A] time = 0.34, antiderivative size = 116, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 7, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.438, Rules used = {5806, 6742, 3303, 3298, 3301, 5448, 12} \[ -\frac {e \sinh \left (\frac {2 a}{b}\right ) \text {Chi}\left (\frac {2 a}{b}+2 \cosh ^{-1}(c x)\right )}{2 b c^2}+\frac {e \cosh \left (\frac {2 a}{b}\right ) \text {Shi}\left (\frac {2 a}{b}+2 \cosh ^{-1}(c x)\right )}{2 b c^2}-\frac {d \sinh \left (\frac {a}{b}\right ) \text {Chi}\left (\frac {a}{b}+\cosh ^{-1}(c x)\right )}{b c}+\frac {d \cosh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a}{b}+\cosh ^{-1}(c x)\right )}{b c} \]
Antiderivative was successfully verified.
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Rule 12
Rule 3298
Rule 3301
Rule 3303
Rule 5448
Rule 5806
Rule 6742
Rubi steps
\begin {align*} \int \frac {d+e x}{a+b \cosh ^{-1}(c x)} \, dx &=\frac {\operatorname {Subst}\left (\int \frac {(c d+e \cosh (x)) \sinh (x)}{a+b x} \, dx,x,\cosh ^{-1}(c x)\right )}{c^2}\\ &=\frac {\operatorname {Subst}\left (\int \left (\frac {c d \sinh (x)}{a+b x}+\frac {e \cosh (x) \sinh (x)}{a+b x}\right ) \, dx,x,\cosh ^{-1}(c x)\right )}{c^2}\\ &=\frac {d \operatorname {Subst}\left (\int \frac {\sinh (x)}{a+b x} \, dx,x,\cosh ^{-1}(c x)\right )}{c}+\frac {e \operatorname {Subst}\left (\int \frac {\cosh (x) \sinh (x)}{a+b x} \, dx,x,\cosh ^{-1}(c x)\right )}{c^2}\\ &=\frac {e \operatorname {Subst}\left (\int \frac {\sinh (2 x)}{2 (a+b x)} \, dx,x,\cosh ^{-1}(c x)\right )}{c^2}+\frac {\left (d \cosh \left (\frac {a}{b}\right )\right ) \operatorname {Subst}\left (\int \frac {\sinh \left (\frac {a}{b}+x\right )}{a+b x} \, dx,x,\cosh ^{-1}(c x)\right )}{c}-\frac {\left (d \sinh \left (\frac {a}{b}\right )\right ) \operatorname {Subst}\left (\int \frac {\cosh \left (\frac {a}{b}+x\right )}{a+b x} \, dx,x,\cosh ^{-1}(c x)\right )}{c}\\ &=-\frac {d \text {Chi}\left (\frac {a}{b}+\cosh ^{-1}(c x)\right ) \sinh \left (\frac {a}{b}\right )}{b c}+\frac {d \cosh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a}{b}+\cosh ^{-1}(c x)\right )}{b c}+\frac {e \operatorname {Subst}\left (\int \frac {\sinh (2 x)}{a+b x} \, dx,x,\cosh ^{-1}(c x)\right )}{2 c^2}\\ &=-\frac {d \text {Chi}\left (\frac {a}{b}+\cosh ^{-1}(c x)\right ) \sinh \left (\frac {a}{b}\right )}{b c}+\frac {d \cosh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a}{b}+\cosh ^{-1}(c x)\right )}{b c}+\frac {\left (e \cosh \left (\frac {2 a}{b}\right )\right ) \operatorname {Subst}\left (\int \frac {\sinh \left (\frac {2 a}{b}+2 x\right )}{a+b x} \, dx,x,\cosh ^{-1}(c x)\right )}{2 c^2}-\frac {\left (e \sinh \left (\frac {2 a}{b}\right )\right ) \operatorname {Subst}\left (\int \frac {\cosh \left (\frac {2 a}{b}+2 x\right )}{a+b x} \, dx,x,\cosh ^{-1}(c x)\right )}{2 c^2}\\ &=-\frac {d \text {Chi}\left (\frac {a}{b}+\cosh ^{-1}(c x)\right ) \sinh \left (\frac {a}{b}\right )}{b c}-\frac {e \text {Chi}\left (\frac {2 a}{b}+2 \cosh ^{-1}(c x)\right ) \sinh \left (\frac {2 a}{b}\right )}{2 b c^2}+\frac {d \cosh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a}{b}+\cosh ^{-1}(c x)\right )}{b c}+\frac {e \cosh \left (\frac {2 a}{b}\right ) \text {Shi}\left (\frac {2 a}{b}+2 \cosh ^{-1}(c x)\right )}{2 b c^2}\\ \end {align*}
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Mathematica [A] time = 0.14, size = 98, normalized size = 0.84 \[ \frac {-2 c d \sinh \left (\frac {a}{b}\right ) \text {Chi}\left (\frac {a}{b}+\cosh ^{-1}(c x)\right )-e \sinh \left (\frac {2 a}{b}\right ) \text {Chi}\left (2 \left (\frac {a}{b}+\cosh ^{-1}(c x)\right )\right )+2 c d \cosh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a}{b}+\cosh ^{-1}(c x)\right )+e \cosh \left (\frac {2 a}{b}\right ) \text {Shi}\left (2 \left (\frac {a}{b}+\cosh ^{-1}(c x)\right )\right )}{2 b c^2} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.48, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {e x + d}{b \operatorname {arcosh}\left (c x\right ) + a}, x\right ) \]
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 \[ \int \frac {e x + d}{b \operatorname {arcosh}\left (c x\right ) + a}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.29, size = 120, normalized size = 1.03 \[ \frac {\frac {{\mathrm e}^{\frac {a}{b}} \Ei \left (1, \mathrm {arccosh}\left (c x \right )+\frac {a}{b}\right ) d}{2 b}-\frac {{\mathrm e}^{-\frac {a}{b}} \Ei \left (1, -\mathrm {arccosh}\left (c x \right )-\frac {a}{b}\right ) d}{2 b}-\frac {e \,{\mathrm e}^{-\frac {2 a}{b}} \Ei \left (1, -2 \,\mathrm {arccosh}\left (c x \right )-\frac {2 a}{b}\right )}{4 c b}+\frac {e \,{\mathrm e}^{\frac {2 a}{b}} \Ei \left (1, 2 \,\mathrm {arccosh}\left (c x \right )+\frac {2 a}{b}\right )}{4 c b}}{c} \]
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 \[ \int \frac {e x + d}{b \operatorname {arcosh}\left (c x\right ) + a}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {d+e\,x}{a+b\,\mathrm {acosh}\left (c\,x\right )} \,d x \]
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 \[ \int \frac {d + e x}{a + b \operatorname {acosh}{\left (c x \right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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