Optimal. Leaf size=192 \[ \frac {\left (a+b \cosh ^{-1}(c+d x)\right )^5}{5 b d e}+\frac {\left (a+b \cosh ^{-1}(c+d x)\right )^4 \log \left (1+e^{-2 \cosh ^{-1}(c+d x)}\right )}{d e}-\frac {2 b \left (a+b \cosh ^{-1}(c+d x)\right )^3 \text {PolyLog}\left (2,-e^{-2 \cosh ^{-1}(c+d x)}\right )}{d e}-\frac {3 b^2 \left (a+b \cosh ^{-1}(c+d x)\right )^2 \text {PolyLog}\left (3,-e^{-2 \cosh ^{-1}(c+d x)}\right )}{d e}-\frac {3 b^3 \left (a+b \cosh ^{-1}(c+d x)\right ) \text {PolyLog}\left (4,-e^{-2 \cosh ^{-1}(c+d x)}\right )}{d e}-\frac {3 b^4 \text {PolyLog}\left (5,-e^{-2 \cosh ^{-1}(c+d x)}\right )}{2 d e} \]
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Rubi [A]
time = 0.21, antiderivative size = 192, normalized size of antiderivative = 1.00, number of steps
used = 10, number of rules used = 9, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.391, Rules used = {5996, 12,
5882, 3799, 2221, 2611, 6744, 2320, 6724} \begin {gather*} -\frac {3 b^3 \text {Li}_4\left (-e^{-2 \cosh ^{-1}(c+d x)}\right ) \left (a+b \cosh ^{-1}(c+d x)\right )}{d e}-\frac {3 b^2 \text {Li}_3\left (-e^{-2 \cosh ^{-1}(c+d x)}\right ) \left (a+b \cosh ^{-1}(c+d x)\right )^2}{d e}-\frac {2 b \text {Li}_2\left (-e^{-2 \cosh ^{-1}(c+d x)}\right ) \left (a+b \cosh ^{-1}(c+d x)\right )^3}{d e}+\frac {\left (a+b \cosh ^{-1}(c+d x)\right )^5}{5 b d e}+\frac {\log \left (e^{-2 \cosh ^{-1}(c+d x)}+1\right ) \left (a+b \cosh ^{-1}(c+d x)\right )^4}{d e}-\frac {3 b^4 \text {Li}_5\left (-e^{-2 \cosh ^{-1}(c+d x)}\right )}{2 d e} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 2221
Rule 2320
Rule 2611
Rule 3799
Rule 5882
Rule 5996
Rule 6724
Rule 6744
Rubi steps
\begin {align*} \int \frac {\left (a+b \cosh ^{-1}(c+d x)\right )^4}{c e+d e x} \, dx &=\frac {\text {Subst}\left (\int \frac {\left (a+b \cosh ^{-1}(x)\right )^4}{e x} \, dx,x,c+d x\right )}{d}\\ &=\frac {\text {Subst}\left (\int \frac {\left (a+b \cosh ^{-1}(x)\right )^4}{x} \, dx,x,c+d x\right )}{d e}\\ &=\frac {\text {Subst}\left (\int (a+b x)^4 \tanh (x) \, dx,x,\cosh ^{-1}(c+d x)\right )}{d e}\\ &=-\frac {\left (a+b \cosh ^{-1}(c+d x)\right )^5}{5 b d e}+\frac {2 \text {Subst}\left (\int \frac {e^{2 x} (a+b x)^4}{1+e^{2 x}} \, dx,x,\cosh ^{-1}(c+d x)\right )}{d e}\\ &=-\frac {\left (a+b \cosh ^{-1}(c+d x)\right )^5}{5 b d e}+\frac {\left (a+b \cosh ^{-1}(c+d x)\right )^4 \log \left (1+e^{2 \cosh ^{-1}(c+d x)}\right )}{d e}-\frac {(4 b) \text {Subst}\left (\int (a+b x)^3 \log \left (1+e^{2 x}\right ) \, dx,x,\cosh ^{-1}(c+d x)\right )}{d e}\\ &=-\frac {\left (a+b \cosh ^{-1}(c+d x)\right )^5}{5 b d e}+\frac {\left (a+b \cosh ^{-1}(c+d x)\right )^4 \log \left (1+e^{2 \cosh ^{-1}(c+d x)}\right )}{d e}+\frac {2 b \left (a+b \cosh ^{-1}(c+d x)\right )^3 \text {Li}_2\left (-e^{2 \cosh ^{-1}(c+d x)}\right )}{d e}-\frac {\left (6 b^2\right ) \text {Subst}\left (\int (a+b x)^2 \text {Li}_2\left (-e^{2 x}\right ) \, dx,x,\cosh ^{-1}(c+d x)\right )}{d e}\\ &=-\frac {\left (a+b \cosh ^{-1}(c+d x)\right )^5}{5 b d e}+\frac {\left (a+b \cosh ^{-1}(c+d x)\right )^4 \log \left (1+e^{2 \cosh ^{-1}(c+d x)}\right )}{d e}+\frac {2 b \left (a+b \cosh ^{-1}(c+d x)\right )^3 \text {Li}_2\left (-e^{2 \cosh ^{-1}(c+d x)}\right )}{d e}-\frac {3 b^2 \left (a+b \cosh ^{-1}(c+d x)\right )^2 \text {Li}_3\left (-e^{2 \cosh ^{-1}(c+d x)}\right )}{d e}+\frac {\left (6 b^3\right ) \text {Subst}\left (\int (a+b x) \text {Li}_3\left (-e^{2 x}\right ) \, dx,x,\cosh ^{-1}(c+d x)\right )}{d e}\\ &=-\frac {\left (a+b \cosh ^{-1}(c+d x)\right )^5}{5 b d e}+\frac {\left (a+b \cosh ^{-1}(c+d x)\right )^4 \log \left (1+e^{2 \cosh ^{-1}(c+d x)}\right )}{d e}+\frac {2 b \left (a+b \cosh ^{-1}(c+d x)\right )^3 \text {Li}_2\left (-e^{2 \cosh ^{-1}(c+d x)}\right )}{d e}-\frac {3 b^2 \left (a+b \cosh ^{-1}(c+d x)\right )^2 \text {Li}_3\left (-e^{2 \cosh ^{-1}(c+d x)}\right )}{d e}+\frac {3 b^3 \left (a+b \cosh ^{-1}(c+d x)\right ) \text {Li}_4\left (-e^{2 \cosh ^{-1}(c+d x)}\right )}{d e}-\frac {\left (3 b^4\right ) \text {Subst}\left (\int \text {Li}_4\left (-e^{2 x}\right ) \, dx,x,\cosh ^{-1}(c+d x)\right )}{d e}\\ &=-\frac {\left (a+b \cosh ^{-1}(c+d x)\right )^5}{5 b d e}+\frac {\left (a+b \cosh ^{-1}(c+d x)\right )^4 \log \left (1+e^{2 \cosh ^{-1}(c+d x)}\right )}{d e}+\frac {2 b \left (a+b \cosh ^{-1}(c+d x)\right )^3 \text {Li}_2\left (-e^{2 \cosh ^{-1}(c+d x)}\right )}{d e}-\frac {3 b^2 \left (a+b \cosh ^{-1}(c+d x)\right )^2 \text {Li}_3\left (-e^{2 \cosh ^{-1}(c+d x)}\right )}{d e}+\frac {3 b^3 \left (a+b \cosh ^{-1}(c+d x)\right ) \text {Li}_4\left (-e^{2 \cosh ^{-1}(c+d x)}\right )}{d e}-\frac {\left (3 b^4\right ) \text {Subst}\left (\int \frac {\text {Li}_4(-x)}{x} \, dx,x,e^{2 \cosh ^{-1}(c+d x)}\right )}{2 d e}\\ &=-\frac {\left (a+b \cosh ^{-1}(c+d x)\right )^5}{5 b d e}+\frac {\left (a+b \cosh ^{-1}(c+d x)\right )^4 \log \left (1+e^{2 \cosh ^{-1}(c+d x)}\right )}{d e}+\frac {2 b \left (a+b \cosh ^{-1}(c+d x)\right )^3 \text {Li}_2\left (-e^{2 \cosh ^{-1}(c+d x)}\right )}{d e}-\frac {3 b^2 \left (a+b \cosh ^{-1}(c+d x)\right )^2 \text {Li}_3\left (-e^{2 \cosh ^{-1}(c+d x)}\right )}{d e}+\frac {3 b^3 \left (a+b \cosh ^{-1}(c+d x)\right ) \text {Li}_4\left (-e^{2 \cosh ^{-1}(c+d x)}\right )}{d e}-\frac {3 b^4 \text {Li}_5\left (-e^{2 \cosh ^{-1}(c+d x)}\right )}{2 d e}\\ \end {align*}
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Mathematica [A]
time = 0.52, size = 308, normalized size = 1.60 \begin {gather*} \frac {2 a^3 b \cosh ^{-1}(c+d x)^2+2 a^2 b^2 \cosh ^{-1}(c+d x)^3+a b^3 \cosh ^{-1}(c+d x)^4+\frac {1}{5} b^4 \cosh ^{-1}(c+d x)^5+4 a^3 b \cosh ^{-1}(c+d x) \log \left (1+e^{-2 \cosh ^{-1}(c+d x)}\right )+6 a^2 b^2 \cosh ^{-1}(c+d x)^2 \log \left (1+e^{-2 \cosh ^{-1}(c+d x)}\right )+4 a b^3 \cosh ^{-1}(c+d x)^3 \log \left (1+e^{-2 \cosh ^{-1}(c+d x)}\right )+b^4 \cosh ^{-1}(c+d x)^4 \log \left (1+e^{-2 \cosh ^{-1}(c+d x)}\right )+a^4 \log (c+d x)-2 b \left (a+b \cosh ^{-1}(c+d x)\right )^3 \text {PolyLog}\left (2,-e^{-2 \cosh ^{-1}(c+d x)}\right )-3 b^2 \left (a+b \cosh ^{-1}(c+d x)\right )^2 \text {PolyLog}\left (3,-e^{-2 \cosh ^{-1}(c+d x)}\right )-3 a b^3 \text {PolyLog}\left (4,-e^{-2 \cosh ^{-1}(c+d x)}\right )-3 b^4 \cosh ^{-1}(c+d x) \text {PolyLog}\left (4,-e^{-2 \cosh ^{-1}(c+d x)}\right )-\frac {3}{2} b^4 \text {PolyLog}\left (5,-e^{-2 \cosh ^{-1}(c+d x)}\right )}{d e} \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. \(673\) vs.
\(2(258)=516\).
time = 47.61, size = 674, normalized size = 3.51
method | result | size |
derivativedivides | \(\frac {\frac {a^{4} \ln \left (d x +c \right )}{e}-\frac {b^{4} \mathrm {arccosh}\left (d x +c \right )^{5}}{5 e}+\frac {b^{4} \mathrm {arccosh}\left (d x +c \right )^{4} \ln \left (1+\left (d x +c +\sqrt {d x +c -1}\, \sqrt {d x +c +1}\right )^{2}\right )}{e}+\frac {2 b^{4} \mathrm {arccosh}\left (d x +c \right )^{3} \polylog \left (2, -\left (d x +c +\sqrt {d x +c -1}\, \sqrt {d x +c +1}\right )^{2}\right )}{e}-\frac {3 b^{4} \mathrm {arccosh}\left (d x +c \right )^{2} \polylog \left (3, -\left (d x +c +\sqrt {d x +c -1}\, \sqrt {d x +c +1}\right )^{2}\right )}{e}+\frac {3 b^{4} \mathrm {arccosh}\left (d x +c \right ) \polylog \left (4, -\left (d x +c +\sqrt {d x +c -1}\, \sqrt {d x +c +1}\right )^{2}\right )}{e}-\frac {3 b^{4} \polylog \left (5, -\left (d x +c +\sqrt {d x +c -1}\, \sqrt {d x +c +1}\right )^{2}\right )}{2 e}-\frac {a \,b^{3} \mathrm {arccosh}\left (d x +c \right )^{4}}{e}+\frac {4 a \,b^{3} \mathrm {arccosh}\left (d x +c \right )^{3} \ln \left (1+\left (d x +c +\sqrt {d x +c -1}\, \sqrt {d x +c +1}\right )^{2}\right )}{e}+\frac {6 a \,b^{3} \mathrm {arccosh}\left (d x +c \right )^{2} \polylog \left (2, -\left (d x +c +\sqrt {d x +c -1}\, \sqrt {d x +c +1}\right )^{2}\right )}{e}-\frac {6 a \,b^{3} \mathrm {arccosh}\left (d x +c \right ) \polylog \left (3, -\left (d x +c +\sqrt {d x +c -1}\, \sqrt {d x +c +1}\right )^{2}\right )}{e}+\frac {3 a \,b^{3} \polylog \left (4, -\left (d x +c +\sqrt {d x +c -1}\, \sqrt {d x +c +1}\right )^{2}\right )}{e}-\frac {2 a^{2} b^{2} \mathrm {arccosh}\left (d x +c \right )^{3}}{e}+\frac {6 a^{2} b^{2} \mathrm {arccosh}\left (d x +c \right )^{2} \ln \left (1+\left (d x +c +\sqrt {d x +c -1}\, \sqrt {d x +c +1}\right )^{2}\right )}{e}+\frac {6 a^{2} b^{2} \mathrm {arccosh}\left (d x +c \right ) \polylog \left (2, -\left (d x +c +\sqrt {d x +c -1}\, \sqrt {d x +c +1}\right )^{2}\right )}{e}-\frac {3 a^{2} b^{2} \polylog \left (3, -\left (d x +c +\sqrt {d x +c -1}\, \sqrt {d x +c +1}\right )^{2}\right )}{e}-\frac {2 a^{3} b \mathrm {arccosh}\left (d x +c \right )^{2}}{e}+\frac {4 a^{3} b \,\mathrm {arccosh}\left (d x +c \right ) \ln \left (1+\left (d x +c +\sqrt {d x +c -1}\, \sqrt {d x +c +1}\right )^{2}\right )}{e}+\frac {2 a^{3} b \polylog \left (2, -\left (d x +c +\sqrt {d x +c -1}\, \sqrt {d x +c +1}\right )^{2}\right )}{e}}{d}\) | \(674\) |
default | \(\frac {\frac {a^{4} \ln \left (d x +c \right )}{e}-\frac {b^{4} \mathrm {arccosh}\left (d x +c \right )^{5}}{5 e}+\frac {b^{4} \mathrm {arccosh}\left (d x +c \right )^{4} \ln \left (1+\left (d x +c +\sqrt {d x +c -1}\, \sqrt {d x +c +1}\right )^{2}\right )}{e}+\frac {2 b^{4} \mathrm {arccosh}\left (d x +c \right )^{3} \polylog \left (2, -\left (d x +c +\sqrt {d x +c -1}\, \sqrt {d x +c +1}\right )^{2}\right )}{e}-\frac {3 b^{4} \mathrm {arccosh}\left (d x +c \right )^{2} \polylog \left (3, -\left (d x +c +\sqrt {d x +c -1}\, \sqrt {d x +c +1}\right )^{2}\right )}{e}+\frac {3 b^{4} \mathrm {arccosh}\left (d x +c \right ) \polylog \left (4, -\left (d x +c +\sqrt {d x +c -1}\, \sqrt {d x +c +1}\right )^{2}\right )}{e}-\frac {3 b^{4} \polylog \left (5, -\left (d x +c +\sqrt {d x +c -1}\, \sqrt {d x +c +1}\right )^{2}\right )}{2 e}-\frac {a \,b^{3} \mathrm {arccosh}\left (d x +c \right )^{4}}{e}+\frac {4 a \,b^{3} \mathrm {arccosh}\left (d x +c \right )^{3} \ln \left (1+\left (d x +c +\sqrt {d x +c -1}\, \sqrt {d x +c +1}\right )^{2}\right )}{e}+\frac {6 a \,b^{3} \mathrm {arccosh}\left (d x +c \right )^{2} \polylog \left (2, -\left (d x +c +\sqrt {d x +c -1}\, \sqrt {d x +c +1}\right )^{2}\right )}{e}-\frac {6 a \,b^{3} \mathrm {arccosh}\left (d x +c \right ) \polylog \left (3, -\left (d x +c +\sqrt {d x +c -1}\, \sqrt {d x +c +1}\right )^{2}\right )}{e}+\frac {3 a \,b^{3} \polylog \left (4, -\left (d x +c +\sqrt {d x +c -1}\, \sqrt {d x +c +1}\right )^{2}\right )}{e}-\frac {2 a^{2} b^{2} \mathrm {arccosh}\left (d x +c \right )^{3}}{e}+\frac {6 a^{2} b^{2} \mathrm {arccosh}\left (d x +c \right )^{2} \ln \left (1+\left (d x +c +\sqrt {d x +c -1}\, \sqrt {d x +c +1}\right )^{2}\right )}{e}+\frac {6 a^{2} b^{2} \mathrm {arccosh}\left (d x +c \right ) \polylog \left (2, -\left (d x +c +\sqrt {d x +c -1}\, \sqrt {d x +c +1}\right )^{2}\right )}{e}-\frac {3 a^{2} b^{2} \polylog \left (3, -\left (d x +c +\sqrt {d x +c -1}\, \sqrt {d x +c +1}\right )^{2}\right )}{e}-\frac {2 a^{3} b \mathrm {arccosh}\left (d x +c \right )^{2}}{e}+\frac {4 a^{3} b \,\mathrm {arccosh}\left (d x +c \right ) \ln \left (1+\left (d x +c +\sqrt {d x +c -1}\, \sqrt {d x +c +1}\right )^{2}\right )}{e}+\frac {2 a^{3} b \polylog \left (2, -\left (d x +c +\sqrt {d x +c -1}\, \sqrt {d x +c +1}\right )^{2}\right )}{e}}{d}\) | \(674\) |
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*} \frac {\int \frac {a^{4}}{c + d x}\, dx + \int \frac {b^{4} \operatorname {acosh}^{4}{\left (c + d x \right )}}{c + d x}\, dx + \int \frac {4 a b^{3} \operatorname {acosh}^{3}{\left (c + d x \right )}}{c + d x}\, dx + \int \frac {6 a^{2} b^{2} \operatorname {acosh}^{2}{\left (c + d x \right )}}{c + d x}\, dx + \int \frac {4 a^{3} b \operatorname {acosh}{\left (c + d x \right )}}{c + d x}\, dx}{e} \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 = -0.01 \begin {gather*} \int \frac {{\left (a+b\,\mathrm {acosh}\left (c+d\,x\right )\right )}^4}{c\,e+d\,e\,x} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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