Optimal. Leaf size=190 \[ \frac {e \cosh \left (\frac {2 a}{b}\right ) \text {Chi}\left (\frac {2 \left (a+b \cosh ^{-1}(c x)\right )}{b}\right )}{b^2 c^2}-\frac {e \sinh \left (\frac {2 a}{b}\right ) \text {Shi}\left (\frac {2 \left (a+b \cosh ^{-1}(c x)\right )}{b}\right )}{b^2 c^2}+\frac {d \cosh \left (\frac {a}{b}\right ) \text {Chi}\left (\frac {a+b \cosh ^{-1}(c x)}{b}\right )}{b^2 c}-\frac {d \sinh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a+b \cosh ^{-1}(c x)}{b}\right )}{b^2 c}-\frac {d \sqrt {c x-1} \sqrt {c x+1}}{b c \left (a+b \cosh ^{-1}(c x)\right )}-\frac {e x \sqrt {c x-1} \sqrt {c x+1}}{b c \left (a+b \cosh ^{-1}(c x)\right )} \]
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Rubi [A] time = 0.47, antiderivative size = 186, normalized size of antiderivative = 0.98, 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 = {5804, 5656, 5781, 3303, 3298, 3301, 5666} \[ \frac {e \cosh \left (\frac {2 a}{b}\right ) \text {Chi}\left (\frac {2 a}{b}+2 \cosh ^{-1}(c x)\right )}{b^2 c^2}-\frac {e \sinh \left (\frac {2 a}{b}\right ) \text {Shi}\left (\frac {2 a}{b}+2 \cosh ^{-1}(c x)\right )}{b^2 c^2}+\frac {d \cosh \left (\frac {a}{b}\right ) \text {Chi}\left (\frac {a}{b}+\cosh ^{-1}(c x)\right )}{b^2 c}-\frac {d \sinh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a}{b}+\cosh ^{-1}(c x)\right )}{b^2 c}-\frac {d \sqrt {c x-1} \sqrt {c x+1}}{b c \left (a+b \cosh ^{-1}(c x)\right )}-\frac {e x \sqrt {c x-1} \sqrt {c x+1}}{b c \left (a+b \cosh ^{-1}(c x)\right )} \]
Antiderivative was successfully verified.
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Rule 3298
Rule 3301
Rule 3303
Rule 5656
Rule 5666
Rule 5781
Rule 5804
Rubi steps
\begin {align*} \int \frac {d+e x}{\left (a+b \cosh ^{-1}(c x)\right )^2} \, dx &=\int \left (\frac {d}{\left (a+b \cosh ^{-1}(c x)\right )^2}+\frac {e x}{\left (a+b \cosh ^{-1}(c x)\right )^2}\right ) \, dx\\ &=d \int \frac {1}{\left (a+b \cosh ^{-1}(c x)\right )^2} \, dx+e \int \frac {x}{\left (a+b \cosh ^{-1}(c x)\right )^2} \, dx\\ &=-\frac {d \sqrt {-1+c x} \sqrt {1+c x}}{b c \left (a+b \cosh ^{-1}(c x)\right )}-\frac {e x \sqrt {-1+c x} \sqrt {1+c x}}{b c \left (a+b \cosh ^{-1}(c x)\right )}+\frac {(c d) \int \frac {x}{\sqrt {-1+c x} \sqrt {1+c x} \left (a+b \cosh ^{-1}(c x)\right )} \, dx}{b}+\frac {e \operatorname {Subst}\left (\int \frac {\cosh (2 x)}{a+b x} \, dx,x,\cosh ^{-1}(c x)\right )}{b c^2}\\ &=-\frac {d \sqrt {-1+c x} \sqrt {1+c x}}{b c \left (a+b \cosh ^{-1}(c x)\right )}-\frac {e x \sqrt {-1+c x} \sqrt {1+c x}}{b c \left (a+b \cosh ^{-1}(c x)\right )}+\frac {d \operatorname {Subst}\left (\int \frac {\cosh (x)}{a+b x} \, dx,x,\cosh ^{-1}(c x)\right )}{b c}+\frac {\left (e \cosh \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 )}{b c^2}-\frac {\left (e \sinh \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 )}{b c^2}\\ &=-\frac {d \sqrt {-1+c x} \sqrt {1+c x}}{b c \left (a+b \cosh ^{-1}(c x)\right )}-\frac {e x \sqrt {-1+c x} \sqrt {1+c x}}{b c \left (a+b \cosh ^{-1}(c x)\right )}+\frac {e \cosh \left (\frac {2 a}{b}\right ) \text {Chi}\left (\frac {2 a}{b}+2 \cosh ^{-1}(c x)\right )}{b^2 c^2}-\frac {e \sinh \left (\frac {2 a}{b}\right ) \text {Shi}\left (\frac {2 a}{b}+2 \cosh ^{-1}(c x)\right )}{b^2 c^2}+\frac {\left (d \cosh \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 )}{b c}-\frac {\left (d \sinh \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 )}{b c}\\ &=-\frac {d \sqrt {-1+c x} \sqrt {1+c x}}{b c \left (a+b \cosh ^{-1}(c x)\right )}-\frac {e x \sqrt {-1+c x} \sqrt {1+c x}}{b c \left (a+b \cosh ^{-1}(c x)\right )}+\frac {d \cosh \left (\frac {a}{b}\right ) \text {Chi}\left (\frac {a}{b}+\cosh ^{-1}(c x)\right )}{b^2 c}+\frac {e \cosh \left (\frac {2 a}{b}\right ) \text {Chi}\left (\frac {2 a}{b}+2 \cosh ^{-1}(c x)\right )}{b^2 c^2}-\frac {d \sinh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a}{b}+\cosh ^{-1}(c x)\right )}{b^2 c}-\frac {e \sinh \left (\frac {2 a}{b}\right ) \text {Shi}\left (\frac {2 a}{b}+2 \cosh ^{-1}(c x)\right )}{b^2 c^2}\\ \end {align*}
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Mathematica [A] time = 0.79, size = 268, normalized size = 1.41 \[ -\frac {-c d \cosh \left (\frac {a}{b}\right ) \left (a+b \cosh ^{-1}(c x)\right ) \text {Chi}\left (\frac {a}{b}+\cosh ^{-1}(c x)\right )-e \cosh \left (\frac {2 a}{b}\right ) \left (a+b \cosh ^{-1}(c x)\right ) \text {Chi}\left (2 \left (\frac {a}{b}+\cosh ^{-1}(c x)\right )\right )+a c d \sinh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a}{b}+\cosh ^{-1}(c x)\right )+b c d \sinh \left (\frac {a}{b}\right ) \cosh ^{-1}(c x) \text {Shi}\left (\frac {a}{b}+\cosh ^{-1}(c x)\right )+a e \sinh \left (\frac {2 a}{b}\right ) \text {Shi}\left (2 \left (\frac {a}{b}+\cosh ^{-1}(c x)\right )\right )+b e \sinh \left (\frac {2 a}{b}\right ) \cosh ^{-1}(c x) \text {Shi}\left (2 \left (\frac {a}{b}+\cosh ^{-1}(c x)\right )\right )+b c^2 d x \sqrt {\frac {c x-1}{c x+1}}+b c^2 e x^2 \sqrt {\frac {c x-1}{c x+1}}+b c d \sqrt {\frac {c x-1}{c x+1}}+b c e x \sqrt {\frac {c x-1}{c x+1}}}{b^2 c^2 \left (a+b \cosh ^{-1}(c x)\right )} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.58, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {e x + d}{b^{2} \operatorname {arcosh}\left (c x\right )^{2} + 2 \, a b \operatorname {arcosh}\left (c x\right ) + a^{2}}, x\right ) \]
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 \[ \int \frac {e x + d}{{\left (b \operatorname {arcosh}\left (c x\right ) + a\right )}^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.33, size = 285, normalized size = 1.50 \[ \frac {\frac {\left (-\sqrt {c x -1}\, \sqrt {c x +1}+c x \right ) d}{2 b \left (a +b \,\mathrm {arccosh}\left (c x \right )\right )}-\frac {{\mathrm e}^{\frac {a}{b}} \Ei \left (1, \mathrm {arccosh}\left (c x \right )+\frac {a}{b}\right ) d}{2 b^{2}}-\frac {\left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right ) d}{2 b \left (a +b \,\mathrm {arccosh}\left (c x \right )\right )}-\frac {{\mathrm e}^{-\frac {a}{b}} \Ei \left (1, -\mathrm {arccosh}\left (c x \right )-\frac {a}{b}\right ) d}{2 b^{2}}+\frac {\left (-2 \sqrt {c x +1}\, \sqrt {c x -1}\, x c +2 c^{2} x^{2}-1\right ) e}{4 c \left (a +b \,\mathrm {arccosh}\left (c x \right )\right ) b}-\frac {e \,{\mathrm e}^{\frac {2 a}{b}} \Ei \left (1, 2 \,\mathrm {arccosh}\left (c x \right )+\frac {2 a}{b}\right )}{2 c \,b^{2}}-\frac {e \left (2 c^{2} x^{2}-1+2 \sqrt {c x +1}\, \sqrt {c x -1}\, x c \right )}{4 c b \left (a +b \,\mathrm {arccosh}\left (c x \right )\right )}-\frac {e \,{\mathrm e}^{-\frac {2 a}{b}} \Ei \left (1, -2 \,\mathrm {arccosh}\left (c x \right )-\frac {2 a}{b}\right )}{2 c \,b^{2}}}{c} \]
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 \[ -\frac {c^{3} e x^{4} + c^{3} d x^{3} - c e x^{2} - c d x + {\left (c^{2} e x^{3} + c^{2} d x^{2} - e x - d\right )} \sqrt {c x + 1} \sqrt {c x - 1}}{a b c^{3} x^{2} + \sqrt {c x + 1} \sqrt {c x - 1} a b c^{2} x - a b c + {\left (b^{2} c^{3} x^{2} + \sqrt {c x + 1} \sqrt {c x - 1} b^{2} c^{2} x - b^{2} c\right )} \log \left (c x + \sqrt {c x + 1} \sqrt {c x - 1}\right )} + \int \frac {2 \, c^{5} e x^{5} + c^{5} d x^{4} - 4 \, c^{3} e x^{3} - 2 \, c^{3} d x^{2} + {\left (2 \, c^{3} e x^{3} + c^{3} d x^{2} + c d\right )} {\left (c x + 1\right )} {\left (c x - 1\right )} + 2 \, c e x + {\left (4 \, c^{4} e x^{4} + 2 \, c^{4} d x^{3} - 4 \, c^{2} e x^{2} - c^{2} d x + e\right )} \sqrt {c x + 1} \sqrt {c x - 1} + c d}{a b c^{5} x^{4} + {\left (c x + 1\right )} {\left (c x - 1\right )} a b c^{3} x^{2} - 2 \, a b c^{3} x^{2} + a b c + 2 \, {\left (a b c^{4} x^{3} - a b c^{2} x\right )} \sqrt {c x + 1} \sqrt {c x - 1} + {\left (b^{2} c^{5} x^{4} + {\left (c x + 1\right )} {\left (c x - 1\right )} b^{2} c^{3} x^{2} - 2 \, b^{2} c^{3} x^{2} + b^{2} c + 2 \, {\left (b^{2} c^{4} x^{3} - b^{2} c^{2} x\right )} \sqrt {c x + 1} \sqrt {c x - 1}\right )} \log \left (c x + \sqrt {c x + 1} \sqrt {c x - 1}\right )}\,{d x} \]
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 \[ \int \frac {d+e\,x}{{\left (a+b\,\mathrm {acosh}\left (c\,x\right )\right )}^2} \,d x \]
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 \[ \int \frac {d + e x}{\left (a + b \operatorname {acosh}{\left (c x \right )}\right )^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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