Optimal. Leaf size=254 \[ -\frac {e^3 \sqrt {-1+c+d x} (c+d x)^3 \sqrt {1+c+d x}}{2 b d \left (a+b \cosh ^{-1}(c+d x)\right )^2}+\frac {3 e^3 (c+d x)^2}{2 b^2 d \left (a+b \cosh ^{-1}(c+d x)\right )}-\frac {2 e^3 (c+d x)^4}{b^2 d \left (a+b \cosh ^{-1}(c+d x)\right )}-\frac {e^3 \text {Chi}\left (\frac {2 \left (a+b \cosh ^{-1}(c+d x)\right )}{b}\right ) \sinh \left (\frac {2 a}{b}\right )}{2 b^3 d}-\frac {e^3 \text {Chi}\left (\frac {4 \left (a+b \cosh ^{-1}(c+d x)\right )}{b}\right ) \sinh \left (\frac {4 a}{b}\right )}{b^3 d}+\frac {e^3 \cosh \left (\frac {2 a}{b}\right ) \text {Shi}\left (\frac {2 \left (a+b \cosh ^{-1}(c+d x)\right )}{b}\right )}{2 b^3 d}+\frac {e^3 \cosh \left (\frac {4 a}{b}\right ) \text {Shi}\left (\frac {4 \left (a+b \cosh ^{-1}(c+d x)\right )}{b}\right )}{b^3 d} \]
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Rubi [A]
time = 0.58, antiderivative size = 254, normalized size of antiderivative = 1.00, number of steps
used = 20, number of rules used = 9, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.391, Rules used = {5996, 12,
5886, 5951, 5887, 5556, 3384, 3379, 3382} \begin {gather*} -\frac {e^3 \sinh \left (\frac {2 a}{b}\right ) \text {Chi}\left (\frac {2 \left (a+b \cosh ^{-1}(c+d x)\right )}{b}\right )}{2 b^3 d}-\frac {e^3 \sinh \left (\frac {4 a}{b}\right ) \text {Chi}\left (\frac {4 \left (a+b \cosh ^{-1}(c+d x)\right )}{b}\right )}{b^3 d}+\frac {e^3 \cosh \left (\frac {2 a}{b}\right ) \text {Shi}\left (\frac {2 \left (a+b \cosh ^{-1}(c+d x)\right )}{b}\right )}{2 b^3 d}+\frac {e^3 \cosh \left (\frac {4 a}{b}\right ) \text {Shi}\left (\frac {4 \left (a+b \cosh ^{-1}(c+d x)\right )}{b}\right )}{b^3 d}-\frac {2 e^3 (c+d x)^4}{b^2 d \left (a+b \cosh ^{-1}(c+d x)\right )}+\frac {3 e^3 (c+d x)^2}{2 b^2 d \left (a+b \cosh ^{-1}(c+d x)\right )}-\frac {e^3 \sqrt {c+d x-1} \sqrt {c+d x+1} (c+d x)^3}{2 b d \left (a+b \cosh ^{-1}(c+d x)\right )^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 3379
Rule 3382
Rule 3384
Rule 5556
Rule 5886
Rule 5887
Rule 5951
Rule 5996
Rubi steps
\begin {align*} \int \frac {(c e+d e x)^3}{\left (a+b \cosh ^{-1}(c+d x)\right )^3} \, dx &=\frac {\text {Subst}\left (\int \frac {e^3 x^3}{\left (a+b \cosh ^{-1}(x)\right )^3} \, dx,x,c+d x\right )}{d}\\ &=\frac {e^3 \text {Subst}\left (\int \frac {x^3}{\left (a+b \cosh ^{-1}(x)\right )^3} \, dx,x,c+d x\right )}{d}\\ &=-\frac {e^3 \sqrt {-1+c+d x} (c+d x)^3 \sqrt {1+c+d x}}{2 b d \left (a+b \cosh ^{-1}(c+d x)\right )^2}-\frac {\left (3 e^3\right ) \text {Subst}\left (\int \frac {x^2}{\sqrt {-1+x} \sqrt {1+x} \left (a+b \cosh ^{-1}(x)\right )^2} \, dx,x,c+d x\right )}{2 b d}+\frac {\left (2 e^3\right ) \text {Subst}\left (\int \frac {x^4}{\sqrt {-1+x} \sqrt {1+x} \left (a+b \cosh ^{-1}(x)\right )^2} \, dx,x,c+d x\right )}{b d}\\ &=-\frac {e^3 \sqrt {-1+c+d x} (c+d x)^3 \sqrt {1+c+d x}}{2 b d \left (a+b \cosh ^{-1}(c+d x)\right )^2}+\frac {3 e^3 (c+d x)^2}{2 b^2 d \left (a+b \cosh ^{-1}(c+d x)\right )}-\frac {2 e^3 (c+d x)^4}{b^2 d \left (a+b \cosh ^{-1}(c+d x)\right )}-\frac {\left (3 e^3\right ) \text {Subst}\left (\int \frac {x}{a+b \cosh ^{-1}(x)} \, dx,x,c+d x\right )}{b^2 d}+\frac {\left (8 e^3\right ) \text {Subst}\left (\int \frac {x^3}{a+b \cosh ^{-1}(x)} \, dx,x,c+d x\right )}{b^2 d}\\ &=-\frac {e^3 \sqrt {-1+c+d x} (c+d x)^3 \sqrt {1+c+d x}}{2 b d \left (a+b \cosh ^{-1}(c+d x)\right )^2}+\frac {3 e^3 (c+d x)^2}{2 b^2 d \left (a+b \cosh ^{-1}(c+d x)\right )}-\frac {2 e^3 (c+d x)^4}{b^2 d \left (a+b \cosh ^{-1}(c+d x)\right )}-\frac {\left (3 e^3\right ) \text {Subst}\left (\int \frac {\cosh (x) \sinh (x)}{a+b x} \, dx,x,\cosh ^{-1}(c+d x)\right )}{b^2 d}+\frac {\left (8 e^3\right ) \text {Subst}\left (\int \frac {\cosh ^3(x) \sinh (x)}{a+b x} \, dx,x,\cosh ^{-1}(c+d x)\right )}{b^2 d}\\ &=-\frac {e^3 \sqrt {-1+c+d x} (c+d x)^3 \sqrt {1+c+d x}}{2 b d \left (a+b \cosh ^{-1}(c+d x)\right )^2}+\frac {3 e^3 (c+d x)^2}{2 b^2 d \left (a+b \cosh ^{-1}(c+d x)\right )}-\frac {2 e^3 (c+d x)^4}{b^2 d \left (a+b \cosh ^{-1}(c+d x)\right )}-\frac {\left (3 e^3\right ) \text {Subst}\left (\int \frac {\sinh (2 x)}{2 (a+b x)} \, dx,x,\cosh ^{-1}(c+d x)\right )}{b^2 d}+\frac {\left (8 e^3\right ) \text {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+d x)\right )}{b^2 d}\\ &=-\frac {e^3 \sqrt {-1+c+d x} (c+d x)^3 \sqrt {1+c+d x}}{2 b d \left (a+b \cosh ^{-1}(c+d x)\right )^2}+\frac {3 e^3 (c+d x)^2}{2 b^2 d \left (a+b \cosh ^{-1}(c+d x)\right )}-\frac {2 e^3 (c+d x)^4}{b^2 d \left (a+b \cosh ^{-1}(c+d x)\right )}+\frac {e^3 \text {Subst}\left (\int \frac {\sinh (4 x)}{a+b x} \, dx,x,\cosh ^{-1}(c+d x)\right )}{b^2 d}-\frac {\left (3 e^3\right ) \text {Subst}\left (\int \frac {\sinh (2 x)}{a+b x} \, dx,x,\cosh ^{-1}(c+d x)\right )}{2 b^2 d}+\frac {\left (2 e^3\right ) \text {Subst}\left (\int \frac {\sinh (2 x)}{a+b x} \, dx,x,\cosh ^{-1}(c+d x)\right )}{b^2 d}\\ &=-\frac {e^3 \sqrt {-1+c+d x} (c+d x)^3 \sqrt {1+c+d x}}{2 b d \left (a+b \cosh ^{-1}(c+d x)\right )^2}+\frac {3 e^3 (c+d x)^2}{2 b^2 d \left (a+b \cosh ^{-1}(c+d x)\right )}-\frac {2 e^3 (c+d x)^4}{b^2 d \left (a+b \cosh ^{-1}(c+d x)\right )}-\frac {\left (3 e^3 \cosh \left (\frac {2 a}{b}\right )\right ) \text {Subst}\left (\int \frac {\sinh \left (\frac {2 a}{b}+2 x\right )}{a+b x} \, dx,x,\cosh ^{-1}(c+d x)\right )}{2 b^2 d}+\frac {\left (2 e^3 \cosh \left (\frac {2 a}{b}\right )\right ) \text {Subst}\left (\int \frac {\sinh \left (\frac {2 a}{b}+2 x\right )}{a+b x} \, dx,x,\cosh ^{-1}(c+d x)\right )}{b^2 d}+\frac {\left (e^3 \cosh \left (\frac {4 a}{b}\right )\right ) \text {Subst}\left (\int \frac {\sinh \left (\frac {4 a}{b}+4 x\right )}{a+b x} \, dx,x,\cosh ^{-1}(c+d x)\right )}{b^2 d}+\frac {\left (3 e^3 \sinh \left (\frac {2 a}{b}\right )\right ) \text {Subst}\left (\int \frac {\cosh \left (\frac {2 a}{b}+2 x\right )}{a+b x} \, dx,x,\cosh ^{-1}(c+d x)\right )}{2 b^2 d}-\frac {\left (2 e^3 \sinh \left (\frac {2 a}{b}\right )\right ) \text {Subst}\left (\int \frac {\cosh \left (\frac {2 a}{b}+2 x\right )}{a+b x} \, dx,x,\cosh ^{-1}(c+d x)\right )}{b^2 d}-\frac {\left (e^3 \sinh \left (\frac {4 a}{b}\right )\right ) \text {Subst}\left (\int \frac {\cosh \left (\frac {4 a}{b}+4 x\right )}{a+b x} \, dx,x,\cosh ^{-1}(c+d x)\right )}{b^2 d}\\ &=-\frac {e^3 \sqrt {-1+c+d x} (c+d x)^3 \sqrt {1+c+d x}}{2 b d \left (a+b \cosh ^{-1}(c+d x)\right )^2}+\frac {3 e^3 (c+d x)^2}{2 b^2 d \left (a+b \cosh ^{-1}(c+d x)\right )}-\frac {2 e^3 (c+d x)^4}{b^2 d \left (a+b \cosh ^{-1}(c+d x)\right )}-\frac {e^3 \text {Chi}\left (\frac {2 a}{b}+2 \cosh ^{-1}(c+d x)\right ) \sinh \left (\frac {2 a}{b}\right )}{2 b^3 d}-\frac {e^3 \text {Chi}\left (\frac {4 a}{b}+4 \cosh ^{-1}(c+d x)\right ) \sinh \left (\frac {4 a}{b}\right )}{b^3 d}+\frac {e^3 \cosh \left (\frac {2 a}{b}\right ) \text {Shi}\left (\frac {2 a}{b}+2 \cosh ^{-1}(c+d x)\right )}{2 b^3 d}+\frac {e^3 \cosh \left (\frac {4 a}{b}\right ) \text {Shi}\left (\frac {4 a}{b}+4 \cosh ^{-1}(c+d x)\right )}{b^3 d}\\ \end {align*}
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Mathematica [A]
time = 0.50, size = 186, normalized size = 0.73 \begin {gather*} \frac {e^3 \left (-\frac {b^2 \sqrt {-1+c+d x} (c+d x)^3 \sqrt {1+c+d x}}{\left (a+b \cosh ^{-1}(c+d x)\right )^2}+\frac {b \left (3 (c+d x)^2-4 (c+d x)^4\right )}{a+b \cosh ^{-1}(c+d x)}-\text {Chi}\left (2 \left (\frac {a}{b}+\cosh ^{-1}(c+d x)\right )\right ) \sinh \left (\frac {2 a}{b}\right )-2 \text {Chi}\left (4 \left (\frac {a}{b}+\cosh ^{-1}(c+d x)\right )\right ) \sinh \left (\frac {4 a}{b}\right )+\cosh \left (\frac {2 a}{b}\right ) \text {Shi}\left (2 \left (\frac {a}{b}+\cosh ^{-1}(c+d x)\right )\right )+2 \cosh \left (\frac {4 a}{b}\right ) \text {Shi}\left (4 \left (\frac {a}{b}+\cosh ^{-1}(c+d x)\right )\right )\right )}{2 b^3 d} \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. \(623\) vs.
\(2(242)=484\).
time = 0.18, size = 624, normalized size = 2.46
method | result | size |
derivativedivides | \(\frac {-\frac {\left (-8 \sqrt {d x +c +1}\, \sqrt {d x +c -1}\, \left (d x +c \right )^{3}+4 \sqrt {d x +c +1}\, \sqrt {d x +c -1}\, \left (d x +c \right )+8 \left (d x +c \right )^{4}-8 \left (d x +c \right )^{2}+1\right ) e^{3} \left (4 b \,\mathrm {arccosh}\left (d x +c \right )+4 a -b \right )}{32 b^{2} \left (b^{2} \mathrm {arccosh}\left (d x +c \right )^{2}+2 a b \,\mathrm {arccosh}\left (d x +c \right )+a^{2}\right )}+\frac {e^{3} {\mathrm e}^{\frac {4 a}{b}} \expIntegral \left (1, 4 \,\mathrm {arccosh}\left (d x +c \right )+\frac {4 a}{b}\right )}{2 b^{3}}-\frac {\left (-2 \sqrt {d x +c +1}\, \sqrt {d x +c -1}\, \left (d x +c \right )+2 \left (d x +c \right )^{2}-1\right ) e^{3} \left (2 b \,\mathrm {arccosh}\left (d x +c \right )+2 a -b \right )}{16 b^{2} \left (b^{2} \mathrm {arccosh}\left (d x +c \right )^{2}+2 a b \,\mathrm {arccosh}\left (d x +c \right )+a^{2}\right )}+\frac {e^{3} {\mathrm e}^{\frac {2 a}{b}} \expIntegral \left (1, 2 \,\mathrm {arccosh}\left (d x +c \right )+\frac {2 a}{b}\right )}{4 b^{3}}-\frac {e^{3} \left (2 \left (d x +c \right )^{2}-1+2 \sqrt {d x +c +1}\, \sqrt {d x +c -1}\, \left (d x +c \right )\right )}{16 b \left (a +b \,\mathrm {arccosh}\left (d x +c \right )\right )^{2}}-\frac {e^{3} \left (2 \left (d x +c \right )^{2}-1+2 \sqrt {d x +c +1}\, \sqrt {d x +c -1}\, \left (d x +c \right )\right )}{8 b^{2} \left (a +b \,\mathrm {arccosh}\left (d x +c \right )\right )}-\frac {e^{3} {\mathrm e}^{-\frac {2 a}{b}} \expIntegral \left (1, -2 \,\mathrm {arccosh}\left (d x +c \right )-\frac {2 a}{b}\right )}{4 b^{3}}-\frac {e^{3} \left (8 \left (d x +c \right )^{4}-8 \left (d x +c \right )^{2}+8 \sqrt {d x +c +1}\, \sqrt {d x +c -1}\, \left (d x +c \right )^{3}-4 \sqrt {d x +c +1}\, \sqrt {d x +c -1}\, \left (d x +c \right )+1\right )}{32 b \left (a +b \,\mathrm {arccosh}\left (d x +c \right )\right )^{2}}-\frac {e^{3} \left (8 \left (d x +c \right )^{4}-8 \left (d x +c \right )^{2}+8 \sqrt {d x +c +1}\, \sqrt {d x +c -1}\, \left (d x +c \right )^{3}-4 \sqrt {d x +c +1}\, \sqrt {d x +c -1}\, \left (d x +c \right )+1\right )}{8 b^{2} \left (a +b \,\mathrm {arccosh}\left (d x +c \right )\right )}-\frac {e^{3} {\mathrm e}^{-\frac {4 a}{b}} \expIntegral \left (1, -4 \,\mathrm {arccosh}\left (d x +c \right )-\frac {4 a}{b}\right )}{2 b^{3}}}{d}\) | \(624\) |
default | \(\frac {-\frac {\left (-8 \sqrt {d x +c +1}\, \sqrt {d x +c -1}\, \left (d x +c \right )^{3}+4 \sqrt {d x +c +1}\, \sqrt {d x +c -1}\, \left (d x +c \right )+8 \left (d x +c \right )^{4}-8 \left (d x +c \right )^{2}+1\right ) e^{3} \left (4 b \,\mathrm {arccosh}\left (d x +c \right )+4 a -b \right )}{32 b^{2} \left (b^{2} \mathrm {arccosh}\left (d x +c \right )^{2}+2 a b \,\mathrm {arccosh}\left (d x +c \right )+a^{2}\right )}+\frac {e^{3} {\mathrm e}^{\frac {4 a}{b}} \expIntegral \left (1, 4 \,\mathrm {arccosh}\left (d x +c \right )+\frac {4 a}{b}\right )}{2 b^{3}}-\frac {\left (-2 \sqrt {d x +c +1}\, \sqrt {d x +c -1}\, \left (d x +c \right )+2 \left (d x +c \right )^{2}-1\right ) e^{3} \left (2 b \,\mathrm {arccosh}\left (d x +c \right )+2 a -b \right )}{16 b^{2} \left (b^{2} \mathrm {arccosh}\left (d x +c \right )^{2}+2 a b \,\mathrm {arccosh}\left (d x +c \right )+a^{2}\right )}+\frac {e^{3} {\mathrm e}^{\frac {2 a}{b}} \expIntegral \left (1, 2 \,\mathrm {arccosh}\left (d x +c \right )+\frac {2 a}{b}\right )}{4 b^{3}}-\frac {e^{3} \left (2 \left (d x +c \right )^{2}-1+2 \sqrt {d x +c +1}\, \sqrt {d x +c -1}\, \left (d x +c \right )\right )}{16 b \left (a +b \,\mathrm {arccosh}\left (d x +c \right )\right )^{2}}-\frac {e^{3} \left (2 \left (d x +c \right )^{2}-1+2 \sqrt {d x +c +1}\, \sqrt {d x +c -1}\, \left (d x +c \right )\right )}{8 b^{2} \left (a +b \,\mathrm {arccosh}\left (d x +c \right )\right )}-\frac {e^{3} {\mathrm e}^{-\frac {2 a}{b}} \expIntegral \left (1, -2 \,\mathrm {arccosh}\left (d x +c \right )-\frac {2 a}{b}\right )}{4 b^{3}}-\frac {e^{3} \left (8 \left (d x +c \right )^{4}-8 \left (d x +c \right )^{2}+8 \sqrt {d x +c +1}\, \sqrt {d x +c -1}\, \left (d x +c \right )^{3}-4 \sqrt {d x +c +1}\, \sqrt {d x +c -1}\, \left (d x +c \right )+1\right )}{32 b \left (a +b \,\mathrm {arccosh}\left (d x +c \right )\right )^{2}}-\frac {e^{3} \left (8 \left (d x +c \right )^{4}-8 \left (d x +c \right )^{2}+8 \sqrt {d x +c +1}\, \sqrt {d x +c -1}\, \left (d x +c \right )^{3}-4 \sqrt {d x +c +1}\, \sqrt {d x +c -1}\, \left (d x +c \right )+1\right )}{8 b^{2} \left (a +b \,\mathrm {arccosh}\left (d x +c \right )\right )}-\frac {e^{3} {\mathrm e}^{-\frac {4 a}{b}} \expIntegral \left (1, -4 \,\mathrm {arccosh}\left (d x +c \right )-\frac {4 a}{b}\right )}{2 b^{3}}}{d}\) | \(624\) |
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*} e^{3} \left (\int \frac {c^{3}}{a^{3} + 3 a^{2} b \operatorname {acosh}{\left (c + d x \right )} + 3 a b^{2} \operatorname {acosh}^{2}{\left (c + d x \right )} + b^{3} \operatorname {acosh}^{3}{\left (c + d x \right )}}\, dx + \int \frac {d^{3} x^{3}}{a^{3} + 3 a^{2} b \operatorname {acosh}{\left (c + d x \right )} + 3 a b^{2} \operatorname {acosh}^{2}{\left (c + d x \right )} + b^{3} \operatorname {acosh}^{3}{\left (c + d x \right )}}\, dx + \int \frac {3 c d^{2} x^{2}}{a^{3} + 3 a^{2} b \operatorname {acosh}{\left (c + d x \right )} + 3 a b^{2} \operatorname {acosh}^{2}{\left (c + d x \right )} + b^{3} \operatorname {acosh}^{3}{\left (c + d x \right )}}\, dx + \int \frac {3 c^{2} d x}{a^{3} + 3 a^{2} b \operatorname {acosh}{\left (c + d x \right )} + 3 a b^{2} \operatorname {acosh}^{2}{\left (c + d x \right )} + b^{3} \operatorname {acosh}^{3}{\left (c + d x \right )}}\, dx\right ) \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 = -0.00 \begin {gather*} \int \frac {{\left (c\,e+d\,e\,x\right )}^3}{{\left (a+b\,\mathrm {acosh}\left (c+d\,x\right )\right )}^3} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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