Optimal. Leaf size=394 \[ -\frac {e^3 \sinh \left (\frac {2 a}{b}\right ) \text {Chi}\left (\frac {2 a}{b}+2 \cosh ^{-1}(c x)\right )}{4 b c^4}-\frac {e^3 \sinh \left (\frac {4 a}{b}\right ) \text {Chi}\left (\frac {4 a}{b}+4 \cosh ^{-1}(c x)\right )}{8 b c^4}+\frac {e^3 \cosh \left (\frac {2 a}{b}\right ) \text {Shi}\left (\frac {2 a}{b}+2 \cosh ^{-1}(c x)\right )}{4 b c^4}+\frac {e^3 \cosh \left (\frac {4 a}{b}\right ) \text {Shi}\left (\frac {4 a}{b}+4 \cosh ^{-1}(c x)\right )}{8 b c^4}-\frac {3 d e^2 \sinh \left (\frac {a}{b}\right ) \text {Chi}\left (\frac {a}{b}+\cosh ^{-1}(c x)\right )}{4 b c^3}-\frac {3 d e^2 \sinh \left (\frac {3 a}{b}\right ) \text {Chi}\left (\frac {3 a}{b}+3 \cosh ^{-1}(c x)\right )}{4 b c^3}+\frac {3 d e^2 \cosh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a}{b}+\cosh ^{-1}(c x)\right )}{4 b c^3}+\frac {3 d e^2 \cosh \left (\frac {3 a}{b}\right ) \text {Shi}\left (\frac {3 a}{b}+3 \cosh ^{-1}(c x)\right )}{4 b c^3}-\frac {3 d^2 e \sinh \left (\frac {2 a}{b}\right ) \text {Chi}\left (\frac {2 a}{b}+2 \cosh ^{-1}(c x)\right )}{2 b c^2}+\frac {3 d^2 e \cosh \left (\frac {2 a}{b}\right ) \text {Shi}\left (\frac {2 a}{b}+2 \cosh ^{-1}(c x)\right )}{2 b c^2}-\frac {d^3 \sinh \left (\frac {a}{b}\right ) \text {Chi}\left (\frac {a}{b}+\cosh ^{-1}(c x)\right )}{b c}+\frac {d^3 \cosh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a}{b}+\cosh ^{-1}(c x)\right )}{b c} \]
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Rubi [A] time = 1.17, antiderivative size = 394, normalized size of antiderivative = 1.00, number of steps used = 27, number of rules used = 7, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.389, Rules used = {5806, 6742, 3303, 3298, 3301, 5448, 12} \[ -\frac {3 d^2 e \sinh \left (\frac {2 a}{b}\right ) \text {Chi}\left (\frac {2 a}{b}+2 \cosh ^{-1}(c x)\right )}{2 b c^2}-\frac {3 d e^2 \sinh \left (\frac {a}{b}\right ) \text {Chi}\left (\frac {a}{b}+\cosh ^{-1}(c x)\right )}{4 b c^3}-\frac {3 d e^2 \sinh \left (\frac {3 a}{b}\right ) \text {Chi}\left (\frac {3 a}{b}+3 \cosh ^{-1}(c x)\right )}{4 b c^3}-\frac {e^3 \sinh \left (\frac {2 a}{b}\right ) \text {Chi}\left (\frac {2 a}{b}+2 \cosh ^{-1}(c x)\right )}{4 b c^4}-\frac {e^3 \sinh \left (\frac {4 a}{b}\right ) \text {Chi}\left (\frac {4 a}{b}+4 \cosh ^{-1}(c x)\right )}{8 b c^4}+\frac {3 d^2 e \cosh \left (\frac {2 a}{b}\right ) \text {Shi}\left (\frac {2 a}{b}+2 \cosh ^{-1}(c x)\right )}{2 b c^2}+\frac {3 d e^2 \cosh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a}{b}+\cosh ^{-1}(c x)\right )}{4 b c^3}+\frac {3 d e^2 \cosh \left (\frac {3 a}{b}\right ) \text {Shi}\left (\frac {3 a}{b}+3 \cosh ^{-1}(c x)\right )}{4 b c^3}+\frac {e^3 \cosh \left (\frac {2 a}{b}\right ) \text {Shi}\left (\frac {2 a}{b}+2 \cosh ^{-1}(c x)\right )}{4 b c^4}+\frac {e^3 \cosh \left (\frac {4 a}{b}\right ) \text {Shi}\left (\frac {4 a}{b}+4 \cosh ^{-1}(c x)\right )}{8 b c^4}-\frac {d^3 \sinh \left (\frac {a}{b}\right ) \text {Chi}\left (\frac {a}{b}+\cosh ^{-1}(c x)\right )}{b c}+\frac {d^3 \cosh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a}{b}+\cosh ^{-1}(c x)\right )}{b c} \]
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)^3}{a+b \cosh ^{-1}(c x)} \, dx &=\frac {\operatorname {Subst}\left (\int \frac {(c d+e \cosh (x))^3 \sinh (x)}{a+b x} \, dx,x,\cosh ^{-1}(c x)\right )}{c^4}\\ &=\frac {\operatorname {Subst}\left (\int \left (\frac {c^3 d^3 \sinh (x)}{a+b x}+\frac {3 c^2 d^2 e \cosh (x) \sinh (x)}{a+b x}+\frac {3 c d e^2 \cosh ^2(x) \sinh (x)}{a+b x}+\frac {e^3 \cosh ^3(x) \sinh (x)}{a+b x}\right ) \, dx,x,\cosh ^{-1}(c x)\right )}{c^4}\\ &=\frac {d^3 \operatorname {Subst}\left (\int \frac {\sinh (x)}{a+b x} \, dx,x,\cosh ^{-1}(c x)\right )}{c}+\frac {\left (3 d^2 e\right ) \operatorname {Subst}\left (\int \frac {\cosh (x) \sinh (x)}{a+b x} \, dx,x,\cosh ^{-1}(c x)\right )}{c^2}+\frac {\left (3 d e^2\right ) \operatorname {Subst}\left (\int \frac {\cosh ^2(x) \sinh (x)}{a+b x} \, dx,x,\cosh ^{-1}(c x)\right )}{c^3}+\frac {e^3 \operatorname {Subst}\left (\int \frac {\cosh ^3(x) \sinh (x)}{a+b x} \, dx,x,\cosh ^{-1}(c x)\right )}{c^4}\\ &=\frac {\left (3 d^2 e\right ) \operatorname {Subst}\left (\int \frac {\sinh (2 x)}{2 (a+b x)} \, dx,x,\cosh ^{-1}(c x)\right )}{c^2}+\frac {\left (3 d e^2\right ) \operatorname {Subst}\left (\int \left (\frac {\sinh (x)}{4 (a+b x)}+\frac {\sinh (3 x)}{4 (a+b x)}\right ) \, dx,x,\cosh ^{-1}(c x)\right )}{c^3}+\frac {e^3 \operatorname {Subst}\left (\int \left (\frac {\sinh (2 x)}{4 (a+b x)}+\frac {\sinh (4 x)}{8 (a+b x)}\right ) \, dx,x,\cosh ^{-1}(c x)\right )}{c^4}+\frac {\left (d^3 \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^3 \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^3 \text {Chi}\left (\frac {a}{b}+\cosh ^{-1}(c x)\right ) \sinh \left (\frac {a}{b}\right )}{b c}+\frac {d^3 \cosh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a}{b}+\cosh ^{-1}(c x)\right )}{b c}+\frac {\left (3 d^2 e\right ) \operatorname {Subst}\left (\int \frac {\sinh (2 x)}{a+b x} \, dx,x,\cosh ^{-1}(c x)\right )}{2 c^2}+\frac {\left (3 d e^2\right ) \operatorname {Subst}\left (\int \frac {\sinh (x)}{a+b x} \, dx,x,\cosh ^{-1}(c x)\right )}{4 c^3}+\frac {\left (3 d e^2\right ) \operatorname {Subst}\left (\int \frac {\sinh (3 x)}{a+b x} \, dx,x,\cosh ^{-1}(c x)\right )}{4 c^3}+\frac {e^3 \operatorname {Subst}\left (\int \frac {\sinh (4 x)}{a+b x} \, dx,x,\cosh ^{-1}(c x)\right )}{8 c^4}+\frac {e^3 \operatorname {Subst}\left (\int \frac {\sinh (2 x)}{a+b x} \, dx,x,\cosh ^{-1}(c x)\right )}{4 c^4}\\ &=-\frac {d^3 \text {Chi}\left (\frac {a}{b}+\cosh ^{-1}(c x)\right ) \sinh \left (\frac {a}{b}\right )}{b c}+\frac {d^3 \cosh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a}{b}+\cosh ^{-1}(c x)\right )}{b c}+\frac {\left (3 d e^2 \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 )}{4 c^3}+\frac {\left (3 d^2 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^3 \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 )}{4 c^4}+\frac {\left (3 d e^2 \cosh \left (\frac {3 a}{b}\right )\right ) \operatorname {Subst}\left (\int \frac {\sinh \left (\frac {3 a}{b}+3 x\right )}{a+b x} \, dx,x,\cosh ^{-1}(c x)\right )}{4 c^3}+\frac {\left (e^3 \cosh \left (\frac {4 a}{b}\right )\right ) \operatorname {Subst}\left (\int \frac {\sinh \left (\frac {4 a}{b}+4 x\right )}{a+b x} \, dx,x,\cosh ^{-1}(c x)\right )}{8 c^4}-\frac {\left (3 d e^2 \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 )}{4 c^3}-\frac {\left (3 d^2 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 {\left (e^3 \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 )}{4 c^4}-\frac {\left (3 d e^2 \sinh \left (\frac {3 a}{b}\right )\right ) \operatorname {Subst}\left (\int \frac {\cosh \left (\frac {3 a}{b}+3 x\right )}{a+b x} \, dx,x,\cosh ^{-1}(c x)\right )}{4 c^3}-\frac {\left (e^3 \sinh \left (\frac {4 a}{b}\right )\right ) \operatorname {Subst}\left (\int \frac {\cosh \left (\frac {4 a}{b}+4 x\right )}{a+b x} \, dx,x,\cosh ^{-1}(c x)\right )}{8 c^4}\\ &=-\frac {d^3 \text {Chi}\left (\frac {a}{b}+\cosh ^{-1}(c x)\right ) \sinh \left (\frac {a}{b}\right )}{b c}-\frac {3 d e^2 \text {Chi}\left (\frac {a}{b}+\cosh ^{-1}(c x)\right ) \sinh \left (\frac {a}{b}\right )}{4 b c^3}-\frac {3 d^2 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 {e^3 \text {Chi}\left (\frac {2 a}{b}+2 \cosh ^{-1}(c x)\right ) \sinh \left (\frac {2 a}{b}\right )}{4 b c^4}-\frac {3 d e^2 \text {Chi}\left (\frac {3 a}{b}+3 \cosh ^{-1}(c x)\right ) \sinh \left (\frac {3 a}{b}\right )}{4 b c^3}-\frac {e^3 \text {Chi}\left (\frac {4 a}{b}+4 \cosh ^{-1}(c x)\right ) \sinh \left (\frac {4 a}{b}\right )}{8 b c^4}+\frac {d^3 \cosh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a}{b}+\cosh ^{-1}(c x)\right )}{b c}+\frac {3 d e^2 \cosh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a}{b}+\cosh ^{-1}(c x)\right )}{4 b c^3}+\frac {3 d^2 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 {e^3 \cosh \left (\frac {2 a}{b}\right ) \text {Shi}\left (\frac {2 a}{b}+2 \cosh ^{-1}(c x)\right )}{4 b c^4}+\frac {3 d e^2 \cosh \left (\frac {3 a}{b}\right ) \text {Shi}\left (\frac {3 a}{b}+3 \cosh ^{-1}(c x)\right )}{4 b c^3}+\frac {e^3 \cosh \left (\frac {4 a}{b}\right ) \text {Shi}\left (\frac {4 a}{b}+4 \cosh ^{-1}(c x)\right )}{8 b c^4}\\ \end {align*}
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Mathematica [A] time = 0.61, size = 287, normalized size = 0.73 \[ \frac {8 c^3 d^3 \cosh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a}{b}+\cosh ^{-1}(c x)\right )-2 c d \sinh \left (\frac {a}{b}\right ) \left (4 c^2 d^2+3 e^2\right ) \text {Chi}\left (\frac {a}{b}+\cosh ^{-1}(c x)\right )-2 e \sinh \left (\frac {2 a}{b}\right ) \left (6 c^2 d^2+e^2\right ) \text {Chi}\left (2 \left (\frac {a}{b}+\cosh ^{-1}(c x)\right )\right )+12 c^2 d^2 e \cosh \left (\frac {2 a}{b}\right ) \text {Shi}\left (2 \left (\frac {a}{b}+\cosh ^{-1}(c x)\right )\right )-6 c d e^2 \sinh \left (\frac {3 a}{b}\right ) \text {Chi}\left (3 \left (\frac {a}{b}+\cosh ^{-1}(c x)\right )\right )-e^3 \sinh \left (\frac {4 a}{b}\right ) \text {Chi}\left (4 \left (\frac {a}{b}+\cosh ^{-1}(c x)\right )\right )+6 c d e^2 \cosh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a}{b}+\cosh ^{-1}(c x)\right )+6 c d e^2 \cosh \left (\frac {3 a}{b}\right ) \text {Shi}\left (3 \left (\frac {a}{b}+\cosh ^{-1}(c x)\right )\right )+2 e^3 \cosh \left (\frac {2 a}{b}\right ) \text {Shi}\left (2 \left (\frac {a}{b}+\cosh ^{-1}(c x)\right )\right )+e^3 \cosh \left (\frac {4 a}{b}\right ) \text {Shi}\left (4 \left (\frac {a}{b}+\cosh ^{-1}(c x)\right )\right )}{8 b c^4} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.49, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {e^{3} x^{3} + 3 \, d e^{2} x^{2} + 3 \, d^{2} e x + d^{3}}{b \operatorname {arcosh}\left (c x\right ) + a}, x\right ) \]
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 \[ \int \frac {{\left (e x + d\right )}^{3}}{b \operatorname {arcosh}\left (c x\right ) + a}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.56, size = 394, normalized size = 1.00 \[ \frac {-\frac {e^{3} {\mathrm e}^{-\frac {4 a}{b}} \Ei \left (1, -4 \,\mathrm {arccosh}\left (c x \right )-\frac {4 a}{b}\right )}{16 c^{3} b}+\frac {e^{3} {\mathrm e}^{\frac {4 a}{b}} \Ei \left (1, 4 \,\mathrm {arccosh}\left (c x \right )+\frac {4 a}{b}\right )}{16 c^{3} b}+\frac {3 e \,{\mathrm e}^{\frac {2 a}{b}} \Ei \left (1, 2 \,\mathrm {arccosh}\left (c x \right )+\frac {2 a}{b}\right ) d^{2}}{4 c b}+\frac {e^{3} {\mathrm e}^{\frac {2 a}{b}} \Ei \left (1, 2 \,\mathrm {arccosh}\left (c x \right )+\frac {2 a}{b}\right )}{8 c^{3} b}-\frac {3 e \,{\mathrm e}^{-\frac {2 a}{b}} \Ei \left (1, -2 \,\mathrm {arccosh}\left (c x \right )-\frac {2 a}{b}\right ) d^{2}}{4 c b}-\frac {e^{3} {\mathrm e}^{-\frac {2 a}{b}} \Ei \left (1, -2 \,\mathrm {arccosh}\left (c x \right )-\frac {2 a}{b}\right )}{8 c^{3} b}-\frac {3 d \,e^{2} {\mathrm e}^{-\frac {3 a}{b}} \Ei \left (1, -3 \,\mathrm {arccosh}\left (c x \right )-\frac {3 a}{b}\right )}{8 c^{2} b}+\frac {d^{3} {\mathrm e}^{\frac {a}{b}} \Ei \left (1, \mathrm {arccosh}\left (c x \right )+\frac {a}{b}\right )}{2 b}+\frac {3 d \,{\mathrm e}^{\frac {a}{b}} \Ei \left (1, \mathrm {arccosh}\left (c x \right )+\frac {a}{b}\right ) e^{2}}{8 c^{2} b}-\frac {d^{3} {\mathrm e}^{-\frac {a}{b}} \Ei \left (1, -\mathrm {arccosh}\left (c x \right )-\frac {a}{b}\right )}{2 b}-\frac {3 d \,{\mathrm e}^{-\frac {a}{b}} \Ei \left (1, -\mathrm {arccosh}\left (c x \right )-\frac {a}{b}\right ) e^{2}}{8 c^{2} b}+\frac {3 d \,e^{2} {\mathrm e}^{\frac {3 a}{b}} \Ei \left (1, 3 \,\mathrm {arccosh}\left (c x \right )+\frac {3 a}{b}\right )}{8 c^{2} b}}{c} \]
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 \[ \int \frac {{\left (e x + d\right )}^{3}}{b \operatorname {arcosh}\left (c x\right ) + a}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {{\left (d+e\,x\right )}^3}{a+b\,\mathrm {acosh}\left (c\,x\right )} \,d x \]
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 \[ \int \frac {\left (d + e x\right )^{3}}{a + b \operatorname {acosh}{\left (c x \right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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