Optimal. Leaf size=218 \[ -\frac {e \sqrt {-1+c+d x} (c+d x) \sqrt {1+c+d x}}{3 b d \left (a+b \cosh ^{-1}(c+d x)\right )^3}+\frac {e}{6 b^2 d \left (a+b \cosh ^{-1}(c+d x)\right )^2}-\frac {e (c+d x)^2}{3 b^2 d \left (a+b \cosh ^{-1}(c+d x)\right )^2}-\frac {2 e \sqrt {-1+c+d x} (c+d x) \sqrt {1+c+d x}}{3 b^3 d \left (a+b \cosh ^{-1}(c+d x)\right )}+\frac {2 e \cosh \left (\frac {2 a}{b}\right ) \text {Chi}\left (\frac {2 \left (a+b \cosh ^{-1}(c+d x)\right )}{b}\right )}{3 b^4 d}-\frac {2 e \sinh \left (\frac {2 a}{b}\right ) \text {Shi}\left (\frac {2 \left (a+b \cosh ^{-1}(c+d x)\right )}{b}\right )}{3 b^4 d} \]
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Rubi [A]
time = 0.34, antiderivative size = 218, normalized size of antiderivative = 1.00, number of steps
used = 9, number of rules used = 9, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.429, Rules used = {5996, 12, 5886,
5951, 5885, 3384, 3379, 3382, 5893} \begin {gather*} \frac {2 e \cosh \left (\frac {2 a}{b}\right ) \text {Chi}\left (\frac {2 \left (a+b \cosh ^{-1}(c+d x)\right )}{b}\right )}{3 b^4 d}-\frac {2 e \sinh \left (\frac {2 a}{b}\right ) \text {Shi}\left (\frac {2 \left (a+b \cosh ^{-1}(c+d x)\right )}{b}\right )}{3 b^4 d}-\frac {2 e \sqrt {c+d x-1} \sqrt {c+d x+1} (c+d x)}{3 b^3 d \left (a+b \cosh ^{-1}(c+d x)\right )}-\frac {e (c+d x)^2}{3 b^2 d \left (a+b \cosh ^{-1}(c+d x)\right )^2}+\frac {e}{6 b^2 d \left (a+b \cosh ^{-1}(c+d x)\right )^2}-\frac {e \sqrt {c+d x-1} \sqrt {c+d x+1} (c+d x)}{3 b d \left (a+b \cosh ^{-1}(c+d x)\right )^3} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 3379
Rule 3382
Rule 3384
Rule 5885
Rule 5886
Rule 5893
Rule 5951
Rule 5996
Rubi steps
\begin {align*} \int \frac {c e+d e x}{\left (a+b \cosh ^{-1}(c+d x)\right )^4} \, dx &=\frac {\text {Subst}\left (\int \frac {e x}{\left (a+b \cosh ^{-1}(x)\right )^4} \, dx,x,c+d x\right )}{d}\\ &=\frac {e \text {Subst}\left (\int \frac {x}{\left (a+b \cosh ^{-1}(x)\right )^4} \, dx,x,c+d x\right )}{d}\\ &=-\frac {e \sqrt {-1+c+d x} (c+d x) \sqrt {1+c+d x}}{3 b d \left (a+b \cosh ^{-1}(c+d x)\right )^3}-\frac {e \text {Subst}\left (\int \frac {1}{\sqrt {-1+x} \sqrt {1+x} \left (a+b \cosh ^{-1}(x)\right )^3} \, dx,x,c+d x\right )}{3 b d}+\frac {(2 e) \text {Subst}\left (\int \frac {x^2}{\sqrt {-1+x} \sqrt {1+x} \left (a+b \cosh ^{-1}(x)\right )^3} \, dx,x,c+d x\right )}{3 b d}\\ &=-\frac {e \sqrt {-1+c+d x} (c+d x) \sqrt {1+c+d x}}{3 b d \left (a+b \cosh ^{-1}(c+d x)\right )^3}+\frac {e}{6 b^2 d \left (a+b \cosh ^{-1}(c+d x)\right )^2}-\frac {e (c+d x)^2}{3 b^2 d \left (a+b \cosh ^{-1}(c+d x)\right )^2}+\frac {(2 e) \text {Subst}\left (\int \frac {x}{\left (a+b \cosh ^{-1}(x)\right )^2} \, dx,x,c+d x\right )}{3 b^2 d}\\ &=-\frac {e \sqrt {-1+c+d x} (c+d x) \sqrt {1+c+d x}}{3 b d \left (a+b \cosh ^{-1}(c+d x)\right )^3}+\frac {e}{6 b^2 d \left (a+b \cosh ^{-1}(c+d x)\right )^2}-\frac {e (c+d x)^2}{3 b^2 d \left (a+b \cosh ^{-1}(c+d x)\right )^2}-\frac {2 e \sqrt {-1+c+d x} (c+d x) \sqrt {1+c+d x}}{3 b^3 d \left (a+b \cosh ^{-1}(c+d x)\right )}+\frac {(2 e) \text {Subst}\left (\int \frac {\cosh (2 x)}{a+b x} \, dx,x,\cosh ^{-1}(c+d x)\right )}{3 b^3 d}\\ &=-\frac {e \sqrt {-1+c+d x} (c+d x) \sqrt {1+c+d x}}{3 b d \left (a+b \cosh ^{-1}(c+d x)\right )^3}+\frac {e}{6 b^2 d \left (a+b \cosh ^{-1}(c+d x)\right )^2}-\frac {e (c+d x)^2}{3 b^2 d \left (a+b \cosh ^{-1}(c+d x)\right )^2}-\frac {2 e \sqrt {-1+c+d x} (c+d x) \sqrt {1+c+d x}}{3 b^3 d \left (a+b \cosh ^{-1}(c+d x)\right )}+\frac {\left (2 e \cosh \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 )}{3 b^3 d}-\frac {\left (2 e \sinh \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 )}{3 b^3 d}\\ &=-\frac {e \sqrt {-1+c+d x} (c+d x) \sqrt {1+c+d x}}{3 b d \left (a+b \cosh ^{-1}(c+d x)\right )^3}+\frac {e}{6 b^2 d \left (a+b \cosh ^{-1}(c+d x)\right )^2}-\frac {e (c+d x)^2}{3 b^2 d \left (a+b \cosh ^{-1}(c+d x)\right )^2}-\frac {2 e \sqrt {-1+c+d x} (c+d x) \sqrt {1+c+d x}}{3 b^3 d \left (a+b \cosh ^{-1}(c+d x)\right )}+\frac {2 e \cosh \left (\frac {2 a}{b}\right ) \text {Chi}\left (\frac {2 a}{b}+2 \cosh ^{-1}(c+d x)\right )}{3 b^4 d}-\frac {2 e \sinh \left (\frac {2 a}{b}\right ) \text {Shi}\left (\frac {2 a}{b}+2 \cosh ^{-1}(c+d x)\right )}{3 b^4 d}\\ \end {align*}
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Mathematica [A]
time = 0.76, size = 195, normalized size = 0.89 \begin {gather*} \frac {e \left (-\frac {2 b^3 \sqrt {-1+c+d x} (c+d x) \sqrt {1+c+d x}}{\left (a+b \cosh ^{-1}(c+d x)\right )^3}+\frac {b^2 \left (1-2 (c+d x)^2\right )}{\left (a+b \cosh ^{-1}(c+d x)\right )^2}-\frac {4 b \sqrt {-1+c+d x} (c+d x) \sqrt {1+c+d x}}{a+b \cosh ^{-1}(c+d x)}-4 \log \left (a+b \cosh ^{-1}(c+d x)\right )+4 \left (\cosh \left (\frac {2 a}{b}\right ) \text {Chi}\left (2 \left (\frac {a}{b}+\cosh ^{-1}(c+d x)\right )\right )+\log \left (a+b \cosh ^{-1}(c+d x)\right )-\sinh \left (\frac {2 a}{b}\right ) \text {Shi}\left (2 \left (\frac {a}{b}+\cosh ^{-1}(c+d x)\right )\right )\right )\right )}{6 b^4 d} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.05, size = 353, normalized size = 1.62
method | result | size |
derivativedivides | \(\frac {\frac {\left (-2 \sqrt {d x +c +1}\, \sqrt {d x +c -1}\, \left (d x +c \right )+2 \left (d x +c \right )^{2}-1\right ) e \left (2 b^{2} \mathrm {arccosh}\left (d x +c \right )^{2}+4 a b \,\mathrm {arccosh}\left (d x +c \right )-b^{2} \mathrm {arccosh}\left (d x +c \right )+2 a^{2}-a b +b^{2}\right )}{12 b^{3} \left (b^{3} \mathrm {arccosh}\left (d x +c \right )^{3}+3 a \,b^{2} \mathrm {arccosh}\left (d x +c \right )^{2}+3 a^{2} b \,\mathrm {arccosh}\left (d x +c \right )+a^{3}\right )}-\frac {e \,{\mathrm e}^{\frac {2 a}{b}} \expIntegral \left (1, 2 \,\mathrm {arccosh}\left (d x +c \right )+\frac {2 a}{b}\right )}{3 b^{4}}-\frac {e \left (2 \left (d x +c \right )^{2}-1+2 \sqrt {d x +c +1}\, \sqrt {d x +c -1}\, \left (d x +c \right )\right )}{12 b \left (a +b \,\mathrm {arccosh}\left (d x +c \right )\right )^{3}}-\frac {e \left (2 \left (d x +c \right )^{2}-1+2 \sqrt {d x +c +1}\, \sqrt {d x +c -1}\, \left (d x +c \right )\right )}{12 b^{2} \left (a +b \,\mathrm {arccosh}\left (d x +c \right )\right )^{2}}-\frac {e \left (2 \left (d x +c \right )^{2}-1+2 \sqrt {d x +c +1}\, \sqrt {d x +c -1}\, \left (d x +c \right )\right )}{6 b^{3} \left (a +b \,\mathrm {arccosh}\left (d x +c \right )\right )}-\frac {e \,{\mathrm e}^{-\frac {2 a}{b}} \expIntegral \left (1, -2 \,\mathrm {arccosh}\left (d x +c \right )-\frac {2 a}{b}\right )}{3 b^{4}}}{d}\) | \(353\) |
default | \(\frac {\frac {\left (-2 \sqrt {d x +c +1}\, \sqrt {d x +c -1}\, \left (d x +c \right )+2 \left (d x +c \right )^{2}-1\right ) e \left (2 b^{2} \mathrm {arccosh}\left (d x +c \right )^{2}+4 a b \,\mathrm {arccosh}\left (d x +c \right )-b^{2} \mathrm {arccosh}\left (d x +c \right )+2 a^{2}-a b +b^{2}\right )}{12 b^{3} \left (b^{3} \mathrm {arccosh}\left (d x +c \right )^{3}+3 a \,b^{2} \mathrm {arccosh}\left (d x +c \right )^{2}+3 a^{2} b \,\mathrm {arccosh}\left (d x +c \right )+a^{3}\right )}-\frac {e \,{\mathrm e}^{\frac {2 a}{b}} \expIntegral \left (1, 2 \,\mathrm {arccosh}\left (d x +c \right )+\frac {2 a}{b}\right )}{3 b^{4}}-\frac {e \left (2 \left (d x +c \right )^{2}-1+2 \sqrt {d x +c +1}\, \sqrt {d x +c -1}\, \left (d x +c \right )\right )}{12 b \left (a +b \,\mathrm {arccosh}\left (d x +c \right )\right )^{3}}-\frac {e \left (2 \left (d x +c \right )^{2}-1+2 \sqrt {d x +c +1}\, \sqrt {d x +c -1}\, \left (d x +c \right )\right )}{12 b^{2} \left (a +b \,\mathrm {arccosh}\left (d x +c \right )\right )^{2}}-\frac {e \left (2 \left (d x +c \right )^{2}-1+2 \sqrt {d x +c +1}\, \sqrt {d x +c -1}\, \left (d x +c \right )\right )}{6 b^{3} \left (a +b \,\mathrm {arccosh}\left (d x +c \right )\right )}-\frac {e \,{\mathrm e}^{-\frac {2 a}{b}} \expIntegral \left (1, -2 \,\mathrm {arccosh}\left (d x +c \right )-\frac {2 a}{b}\right )}{3 b^{4}}}{d}\) | \(353\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \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 \left (\int \frac {c}{a^{4} + 4 a^{3} b \operatorname {acosh}{\left (c + d x \right )} + 6 a^{2} b^{2} \operatorname {acosh}^{2}{\left (c + d x \right )} + 4 a b^{3} \operatorname {acosh}^{3}{\left (c + d x \right )} + b^{4} \operatorname {acosh}^{4}{\left (c + d x \right )}}\, dx + \int \frac {d x}{a^{4} + 4 a^{3} b \operatorname {acosh}{\left (c + d x \right )} + 6 a^{2} b^{2} \operatorname {acosh}^{2}{\left (c + d x \right )} + 4 a b^{3} \operatorname {acosh}^{3}{\left (c + d x \right )} + b^{4} \operatorname {acosh}^{4}{\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 {c\,e+d\,e\,x}{{\left (a+b\,\mathrm {acosh}\left (c+d\,x\right )\right )}^4} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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