Optimal. Leaf size=264 \[ -\frac {\left (a+b \cosh ^{-1}(c+d x)\right )^4}{d e^2 (c+d x)}+\frac {8 b \left (a+b \cosh ^{-1}(c+d x)\right )^3 \text {ArcTan}\left (e^{\cosh ^{-1}(c+d x)}\right )}{d e^2}-\frac {12 i b^2 \left (a+b \cosh ^{-1}(c+d x)\right )^2 \text {PolyLog}\left (2,-i e^{\cosh ^{-1}(c+d x)}\right )}{d e^2}+\frac {12 i b^2 \left (a+b \cosh ^{-1}(c+d x)\right )^2 \text {PolyLog}\left (2,i e^{\cosh ^{-1}(c+d x)}\right )}{d e^2}+\frac {24 i b^3 \left (a+b \cosh ^{-1}(c+d x)\right ) \text {PolyLog}\left (3,-i e^{\cosh ^{-1}(c+d x)}\right )}{d e^2}-\frac {24 i b^3 \left (a+b \cosh ^{-1}(c+d x)\right ) \text {PolyLog}\left (3,i e^{\cosh ^{-1}(c+d x)}\right )}{d e^2}-\frac {24 i b^4 \text {PolyLog}\left (4,-i e^{\cosh ^{-1}(c+d x)}\right )}{d e^2}+\frac {24 i b^4 \text {PolyLog}\left (4,i e^{\cosh ^{-1}(c+d x)}\right )}{d e^2} \]
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Rubi [A]
time = 0.28, antiderivative size = 264, normalized size of antiderivative = 1.00, number of steps
used = 13, number of rules used = 9, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.391, Rules used = {5996, 12,
5883, 5947, 4265, 2611, 6744, 2320, 6724} \begin {gather*} \frac {8 b \text {ArcTan}\left (e^{\cosh ^{-1}(c+d x)}\right ) \left (a+b \cosh ^{-1}(c+d x)\right )^3}{d e^2}+\frac {24 i b^3 \text {Li}_3\left (-i e^{\cosh ^{-1}(c+d x)}\right ) \left (a+b \cosh ^{-1}(c+d x)\right )}{d e^2}-\frac {24 i b^3 \text {Li}_3\left (i e^{\cosh ^{-1}(c+d x)}\right ) \left (a+b \cosh ^{-1}(c+d x)\right )}{d e^2}-\frac {12 i b^2 \text {Li}_2\left (-i e^{\cosh ^{-1}(c+d x)}\right ) \left (a+b \cosh ^{-1}(c+d x)\right )^2}{d e^2}+\frac {12 i b^2 \text {Li}_2\left (i e^{\cosh ^{-1}(c+d x)}\right ) \left (a+b \cosh ^{-1}(c+d x)\right )^2}{d e^2}-\frac {\left (a+b \cosh ^{-1}(c+d x)\right )^4}{d e^2 (c+d x)}-\frac {24 i b^4 \text {Li}_4\left (-i e^{\cosh ^{-1}(c+d x)}\right )}{d e^2}+\frac {24 i b^4 \text {Li}_4\left (i e^{\cosh ^{-1}(c+d x)}\right )}{d e^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 2320
Rule 2611
Rule 4265
Rule 5883
Rule 5947
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)^2} \, dx &=\frac {\text {Subst}\left (\int \frac {\left (a+b \cosh ^{-1}(x)\right )^4}{e^2 x^2} \, dx,x,c+d x\right )}{d}\\ &=\frac {\text {Subst}\left (\int \frac {\left (a+b \cosh ^{-1}(x)\right )^4}{x^2} \, dx,x,c+d x\right )}{d e^2}\\ &=-\frac {\left (a+b \cosh ^{-1}(c+d x)\right )^4}{d e^2 (c+d x)}+\frac {(4 b) \text {Subst}\left (\int \frac {\left (a+b \cosh ^{-1}(x)\right )^3}{\sqrt {-1+x} x \sqrt {1+x}} \, dx,x,c+d x\right )}{d e^2}\\ &=-\frac {\left (a+b \cosh ^{-1}(c+d x)\right )^4}{d e^2 (c+d x)}+\frac {(4 b) \text {Subst}\left (\int (a+b x)^3 \text {sech}(x) \, dx,x,\cosh ^{-1}(c+d x)\right )}{d e^2}\\ &=-\frac {\left (a+b \cosh ^{-1}(c+d x)\right )^4}{d e^2 (c+d x)}+\frac {8 b \left (a+b \cosh ^{-1}(c+d x)\right )^3 \tan ^{-1}\left (e^{\cosh ^{-1}(c+d x)}\right )}{d e^2}-\frac {\left (12 i b^2\right ) \text {Subst}\left (\int (a+b x)^2 \log \left (1-i e^x\right ) \, dx,x,\cosh ^{-1}(c+d x)\right )}{d e^2}+\frac {\left (12 i b^2\right ) \text {Subst}\left (\int (a+b x)^2 \log \left (1+i e^x\right ) \, dx,x,\cosh ^{-1}(c+d x)\right )}{d e^2}\\ &=-\frac {\left (a+b \cosh ^{-1}(c+d x)\right )^4}{d e^2 (c+d x)}+\frac {8 b \left (a+b \cosh ^{-1}(c+d x)\right )^3 \tan ^{-1}\left (e^{\cosh ^{-1}(c+d x)}\right )}{d e^2}-\frac {12 i b^2 \left (a+b \cosh ^{-1}(c+d x)\right )^2 \text {Li}_2\left (-i e^{\cosh ^{-1}(c+d x)}\right )}{d e^2}+\frac {12 i b^2 \left (a+b \cosh ^{-1}(c+d x)\right )^2 \text {Li}_2\left (i e^{\cosh ^{-1}(c+d x)}\right )}{d e^2}+\frac {\left (24 i b^3\right ) \text {Subst}\left (\int (a+b x) \text {Li}_2\left (-i e^x\right ) \, dx,x,\cosh ^{-1}(c+d x)\right )}{d e^2}-\frac {\left (24 i b^3\right ) \text {Subst}\left (\int (a+b x) \text {Li}_2\left (i e^x\right ) \, dx,x,\cosh ^{-1}(c+d x)\right )}{d e^2}\\ &=-\frac {\left (a+b \cosh ^{-1}(c+d x)\right )^4}{d e^2 (c+d x)}+\frac {8 b \left (a+b \cosh ^{-1}(c+d x)\right )^3 \tan ^{-1}\left (e^{\cosh ^{-1}(c+d x)}\right )}{d e^2}-\frac {12 i b^2 \left (a+b \cosh ^{-1}(c+d x)\right )^2 \text {Li}_2\left (-i e^{\cosh ^{-1}(c+d x)}\right )}{d e^2}+\frac {12 i b^2 \left (a+b \cosh ^{-1}(c+d x)\right )^2 \text {Li}_2\left (i e^{\cosh ^{-1}(c+d x)}\right )}{d e^2}+\frac {24 i b^3 \left (a+b \cosh ^{-1}(c+d x)\right ) \text {Li}_3\left (-i e^{\cosh ^{-1}(c+d x)}\right )}{d e^2}-\frac {24 i b^3 \left (a+b \cosh ^{-1}(c+d x)\right ) \text {Li}_3\left (i e^{\cosh ^{-1}(c+d x)}\right )}{d e^2}-\frac {\left (24 i b^4\right ) \text {Subst}\left (\int \text {Li}_3\left (-i e^x\right ) \, dx,x,\cosh ^{-1}(c+d x)\right )}{d e^2}+\frac {\left (24 i b^4\right ) \text {Subst}\left (\int \text {Li}_3\left (i e^x\right ) \, dx,x,\cosh ^{-1}(c+d x)\right )}{d e^2}\\ &=-\frac {\left (a+b \cosh ^{-1}(c+d x)\right )^4}{d e^2 (c+d x)}+\frac {8 b \left (a+b \cosh ^{-1}(c+d x)\right )^3 \tan ^{-1}\left (e^{\cosh ^{-1}(c+d x)}\right )}{d e^2}-\frac {12 i b^2 \left (a+b \cosh ^{-1}(c+d x)\right )^2 \text {Li}_2\left (-i e^{\cosh ^{-1}(c+d x)}\right )}{d e^2}+\frac {12 i b^2 \left (a+b \cosh ^{-1}(c+d x)\right )^2 \text {Li}_2\left (i e^{\cosh ^{-1}(c+d x)}\right )}{d e^2}+\frac {24 i b^3 \left (a+b \cosh ^{-1}(c+d x)\right ) \text {Li}_3\left (-i e^{\cosh ^{-1}(c+d x)}\right )}{d e^2}-\frac {24 i b^3 \left (a+b \cosh ^{-1}(c+d x)\right ) \text {Li}_3\left (i e^{\cosh ^{-1}(c+d x)}\right )}{d e^2}-\frac {\left (24 i b^4\right ) \text {Subst}\left (\int \frac {\text {Li}_3(-i x)}{x} \, dx,x,e^{\cosh ^{-1}(c+d x)}\right )}{d e^2}+\frac {\left (24 i b^4\right ) \text {Subst}\left (\int \frac {\text {Li}_3(i x)}{x} \, dx,x,e^{\cosh ^{-1}(c+d x)}\right )}{d e^2}\\ &=-\frac {\left (a+b \cosh ^{-1}(c+d x)\right )^4}{d e^2 (c+d x)}+\frac {8 b \left (a+b \cosh ^{-1}(c+d x)\right )^3 \tan ^{-1}\left (e^{\cosh ^{-1}(c+d x)}\right )}{d e^2}-\frac {12 i b^2 \left (a+b \cosh ^{-1}(c+d x)\right )^2 \text {Li}_2\left (-i e^{\cosh ^{-1}(c+d x)}\right )}{d e^2}+\frac {12 i b^2 \left (a+b \cosh ^{-1}(c+d x)\right )^2 \text {Li}_2\left (i e^{\cosh ^{-1}(c+d x)}\right )}{d e^2}+\frac {24 i b^3 \left (a+b \cosh ^{-1}(c+d x)\right ) \text {Li}_3\left (-i e^{\cosh ^{-1}(c+d x)}\right )}{d e^2}-\frac {24 i b^3 \left (a+b \cosh ^{-1}(c+d x)\right ) \text {Li}_3\left (i e^{\cosh ^{-1}(c+d x)}\right )}{d e^2}-\frac {24 i b^4 \text {Li}_4\left (-i e^{\cosh ^{-1}(c+d x)}\right )}{d e^2}+\frac {24 i b^4 \text {Li}_4\left (i e^{\cosh ^{-1}(c+d x)}\right )}{d e^2}\\ \end {align*}
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Mathematica [B] Both result and optimal contain complex but leaf count is larger than twice
the leaf count of optimal. \(872\) vs. \(2(264)=528\).
time = 1.64, size = 872, normalized size = 3.30 \begin {gather*} \frac {-\frac {a^4}{c+d x}+4 a^3 b \left (-\frac {\cosh ^{-1}(c+d x)}{c+d x}+2 \text {ArcTan}\left (\tanh \left (\frac {1}{2} \cosh ^{-1}(c+d x)\right )\right )\right )-6 i a^2 b^2 \left (\cosh ^{-1}(c+d x) \left (-\frac {i \cosh ^{-1}(c+d x)}{c+d x}+2 \log \left (1-i e^{-\cosh ^{-1}(c+d x)}\right )-2 \log \left (1+i e^{-\cosh ^{-1}(c+d x)}\right )\right )+2 \text {PolyLog}\left (2,-i e^{-\cosh ^{-1}(c+d x)}\right )-2 \text {PolyLog}\left (2,i e^{-\cosh ^{-1}(c+d x)}\right )\right )+4 a b^3 \left (-\frac {\cosh ^{-1}(c+d x)^3}{c+d x}+3 i \left (-\cosh ^{-1}(c+d x)^2 \left (\log \left (1-i e^{-\cosh ^{-1}(c+d x)}\right )-\log \left (1+i e^{-\cosh ^{-1}(c+d x)}\right )\right )-2 \cosh ^{-1}(c+d x) \left (\text {PolyLog}\left (2,-i e^{-\cosh ^{-1}(c+d x)}\right )-\text {PolyLog}\left (2,i e^{-\cosh ^{-1}(c+d x)}\right )\right )-2 \text {PolyLog}\left (3,-i e^{-\cosh ^{-1}(c+d x)}\right )+2 \text {PolyLog}\left (3,i e^{-\cosh ^{-1}(c+d x)}\right )\right )\right )+b^4 \left (-\frac {7 i \pi ^4}{16}+\frac {1}{2} \pi ^3 \cosh ^{-1}(c+d x)-\frac {3}{2} i \pi ^2 \cosh ^{-1}(c+d x)^2-2 \pi \cosh ^{-1}(c+d x)^3+i \cosh ^{-1}(c+d x)^4-\frac {\cosh ^{-1}(c+d x)^4}{c+d x}+\frac {1}{2} \pi ^3 \log \left (1+i e^{-\cosh ^{-1}(c+d x)}\right )-3 i \pi ^2 \cosh ^{-1}(c+d x) \log \left (1+i e^{-\cosh ^{-1}(c+d x)}\right )-6 \pi \cosh ^{-1}(c+d x)^2 \log \left (1+i e^{-\cosh ^{-1}(c+d x)}\right )+4 i \cosh ^{-1}(c+d x)^3 \log \left (1+i e^{-\cosh ^{-1}(c+d x)}\right )+3 i \pi ^2 \cosh ^{-1}(c+d x) \log \left (1-i e^{\cosh ^{-1}(c+d x)}\right )+6 \pi \cosh ^{-1}(c+d x)^2 \log \left (1-i e^{\cosh ^{-1}(c+d x)}\right )-\frac {1}{2} \pi ^3 \log \left (1+i e^{\cosh ^{-1}(c+d x)}\right )-4 i \cosh ^{-1}(c+d x)^3 \log \left (1+i e^{\cosh ^{-1}(c+d x)}\right )+\frac {1}{2} \pi ^3 \log \left (\tan \left (\frac {1}{4} \left (\pi +2 i \cosh ^{-1}(c+d x)\right )\right )\right )+3 i \left (\pi -2 i \cosh ^{-1}(c+d x)\right )^2 \text {PolyLog}\left (2,-i e^{-\cosh ^{-1}(c+d x)}\right )-12 i \cosh ^{-1}(c+d x)^2 \text {PolyLog}\left (2,-i e^{\cosh ^{-1}(c+d x)}\right )+3 i \pi ^2 \text {PolyLog}\left (2,i e^{\cosh ^{-1}(c+d x)}\right )+12 \pi \cosh ^{-1}(c+d x) \text {PolyLog}\left (2,i e^{\cosh ^{-1}(c+d x)}\right )+12 \pi \text {PolyLog}\left (3,-i e^{-\cosh ^{-1}(c+d x)}\right )-24 i \cosh ^{-1}(c+d x) \text {PolyLog}\left (3,-i e^{-\cosh ^{-1}(c+d x)}\right )+24 i \cosh ^{-1}(c+d x) \text {PolyLog}\left (3,-i e^{\cosh ^{-1}(c+d x)}\right )-12 \pi \text {PolyLog}\left (3,i e^{\cosh ^{-1}(c+d x)}\right )-24 i \text {PolyLog}\left (4,-i e^{-\cosh ^{-1}(c+d x)}\right )-24 i \text {PolyLog}\left (4,-i e^{\cosh ^{-1}(c+d x)}\right )\right )}{d e^2} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.22, size = 0, normalized size = 0.00 \[\int \frac {\left (a +b \,\mathrm {arccosh}\left (d x +c \right )\right )^{4}}{\left (d e x +c e \right )^{2}}\, dx\]
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^{2} + 2 c d x + d^{2} x^{2}}\, dx + \int \frac {b^{4} \operatorname {acosh}^{4}{\left (c + d x \right )}}{c^{2} + 2 c d x + d^{2} x^{2}}\, dx + \int \frac {4 a b^{3} \operatorname {acosh}^{3}{\left (c + d x \right )}}{c^{2} + 2 c d x + d^{2} x^{2}}\, dx + \int \frac {6 a^{2} b^{2} \operatorname {acosh}^{2}{\left (c + d x \right )}}{c^{2} + 2 c d x + d^{2} x^{2}}\, dx + \int \frac {4 a^{3} b \operatorname {acosh}{\left (c + d x \right )}}{c^{2} + 2 c d x + d^{2} x^{2}}\, dx}{e^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \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 (a+b\,\mathrm {acosh}\left (c+d\,x\right )\right )}^4}{{\left (c\,e+d\,e\,x\right )}^2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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