Optimal. Leaf size=186 \[ \frac {b \sqrt {c+d x-1} \sqrt {c+d x+1} \left (a+b \cosh ^{-1}(c+d x)\right )}{3 d e^4 (c+d x)^2}-\frac {\left (a+b \cosh ^{-1}(c+d x)\right )^2}{3 d e^4 (c+d x)^3}+\frac {2 b \tan ^{-1}\left (e^{\cosh ^{-1}(c+d x)}\right ) \left (a+b \cosh ^{-1}(c+d x)\right )}{3 d e^4}-\frac {i b^2 \text {Li}_2\left (-i e^{\cosh ^{-1}(c+d x)}\right )}{3 d e^4}+\frac {i b^2 \text {Li}_2\left (i e^{\cosh ^{-1}(c+d x)}\right )}{3 d e^4}+\frac {b^2}{3 d e^4 (c+d x)} \]
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Rubi [A] time = 0.39, antiderivative size = 186, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 9, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.391, Rules used = {5866, 12, 5662, 5748, 5761, 4180, 2279, 2391, 30} \[ -\frac {i b^2 \text {PolyLog}\left (2,-i e^{\cosh ^{-1}(c+d x)}\right )}{3 d e^4}+\frac {i b^2 \text {PolyLog}\left (2,i e^{\cosh ^{-1}(c+d x)}\right )}{3 d e^4}+\frac {b \sqrt {c+d x-1} \sqrt {c+d x+1} \left (a+b \cosh ^{-1}(c+d x)\right )}{3 d e^4 (c+d x)^2}-\frac {\left (a+b \cosh ^{-1}(c+d x)\right )^2}{3 d e^4 (c+d x)^3}+\frac {2 b \tan ^{-1}\left (e^{\cosh ^{-1}(c+d x)}\right ) \left (a+b \cosh ^{-1}(c+d x)\right )}{3 d e^4}+\frac {b^2}{3 d e^4 (c+d x)} \]
Antiderivative was successfully verified.
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Rule 12
Rule 30
Rule 2279
Rule 2391
Rule 4180
Rule 5662
Rule 5748
Rule 5761
Rule 5866
Rubi steps
\begin {align*} \int \frac {\left (a+b \cosh ^{-1}(c+d x)\right )^2}{(c e+d e x)^4} \, dx &=\frac {\operatorname {Subst}\left (\int \frac {\left (a+b \cosh ^{-1}(x)\right )^2}{e^4 x^4} \, dx,x,c+d x\right )}{d}\\ &=\frac {\operatorname {Subst}\left (\int \frac {\left (a+b \cosh ^{-1}(x)\right )^2}{x^4} \, dx,x,c+d x\right )}{d e^4}\\ &=-\frac {\left (a+b \cosh ^{-1}(c+d x)\right )^2}{3 d e^4 (c+d x)^3}+\frac {(2 b) \operatorname {Subst}\left (\int \frac {a+b \cosh ^{-1}(x)}{\sqrt {-1+x} x^3 \sqrt {1+x}} \, dx,x,c+d x\right )}{3 d e^4}\\ &=\frac {b \sqrt {-1+c+d x} \sqrt {1+c+d x} \left (a+b \cosh ^{-1}(c+d x)\right )}{3 d e^4 (c+d x)^2}-\frac {\left (a+b \cosh ^{-1}(c+d x)\right )^2}{3 d e^4 (c+d x)^3}+\frac {b \operatorname {Subst}\left (\int \frac {a+b \cosh ^{-1}(x)}{\sqrt {-1+x} x \sqrt {1+x}} \, dx,x,c+d x\right )}{3 d e^4}-\frac {b^2 \operatorname {Subst}\left (\int \frac {1}{x^2} \, dx,x,c+d x\right )}{3 d e^4}\\ &=\frac {b^2}{3 d e^4 (c+d x)}+\frac {b \sqrt {-1+c+d x} \sqrt {1+c+d x} \left (a+b \cosh ^{-1}(c+d x)\right )}{3 d e^4 (c+d x)^2}-\frac {\left (a+b \cosh ^{-1}(c+d x)\right )^2}{3 d e^4 (c+d x)^3}+\frac {b \operatorname {Subst}\left (\int (a+b x) \text {sech}(x) \, dx,x,\cosh ^{-1}(c+d x)\right )}{3 d e^4}\\ &=\frac {b^2}{3 d e^4 (c+d x)}+\frac {b \sqrt {-1+c+d x} \sqrt {1+c+d x} \left (a+b \cosh ^{-1}(c+d x)\right )}{3 d e^4 (c+d x)^2}-\frac {\left (a+b \cosh ^{-1}(c+d x)\right )^2}{3 d e^4 (c+d x)^3}+\frac {2 b \left (a+b \cosh ^{-1}(c+d x)\right ) \tan ^{-1}\left (e^{\cosh ^{-1}(c+d x)}\right )}{3 d e^4}-\frac {\left (i b^2\right ) \operatorname {Subst}\left (\int \log \left (1-i e^x\right ) \, dx,x,\cosh ^{-1}(c+d x)\right )}{3 d e^4}+\frac {\left (i b^2\right ) \operatorname {Subst}\left (\int \log \left (1+i e^x\right ) \, dx,x,\cosh ^{-1}(c+d x)\right )}{3 d e^4}\\ &=\frac {b^2}{3 d e^4 (c+d x)}+\frac {b \sqrt {-1+c+d x} \sqrt {1+c+d x} \left (a+b \cosh ^{-1}(c+d x)\right )}{3 d e^4 (c+d x)^2}-\frac {\left (a+b \cosh ^{-1}(c+d x)\right )^2}{3 d e^4 (c+d x)^3}+\frac {2 b \left (a+b \cosh ^{-1}(c+d x)\right ) \tan ^{-1}\left (e^{\cosh ^{-1}(c+d x)}\right )}{3 d e^4}-\frac {\left (i b^2\right ) \operatorname {Subst}\left (\int \frac {\log (1-i x)}{x} \, dx,x,e^{\cosh ^{-1}(c+d x)}\right )}{3 d e^4}+\frac {\left (i b^2\right ) \operatorname {Subst}\left (\int \frac {\log (1+i x)}{x} \, dx,x,e^{\cosh ^{-1}(c+d x)}\right )}{3 d e^4}\\ &=\frac {b^2}{3 d e^4 (c+d x)}+\frac {b \sqrt {-1+c+d x} \sqrt {1+c+d x} \left (a+b \cosh ^{-1}(c+d x)\right )}{3 d e^4 (c+d x)^2}-\frac {\left (a+b \cosh ^{-1}(c+d x)\right )^2}{3 d e^4 (c+d x)^3}+\frac {2 b \left (a+b \cosh ^{-1}(c+d x)\right ) \tan ^{-1}\left (e^{\cosh ^{-1}(c+d x)}\right )}{3 d e^4}-\frac {i b^2 \text {Li}_2\left (-i e^{\cosh ^{-1}(c+d x)}\right )}{3 d e^4}+\frac {i b^2 \text {Li}_2\left (i e^{\cosh ^{-1}(c+d x)}\right )}{3 d e^4}\\ \end {align*}
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Mathematica [A] time = 1.02, size = 251, normalized size = 1.35 \[ \frac {-\frac {a^2}{(c+d x)^3}+a b \left (\frac {\sqrt {\frac {c+d x-1}{c+d x+1}} (c+d x+1)}{(c+d x)^2}-\frac {2 \cosh ^{-1}(c+d x)}{(c+d x)^3}+2 \tan ^{-1}\left (\tanh \left (\frac {1}{2} \cosh ^{-1}(c+d x)\right )\right )\right )+b^2 \left (-i \text {Li}_2\left (-i e^{-\cosh ^{-1}(c+d x)}\right )+i \text {Li}_2\left (i e^{-\cosh ^{-1}(c+d x)}\right )+\frac {1}{c+d x}-\frac {\cosh ^{-1}(c+d x)^2}{(c+d x)^3}+\frac {\sqrt {\frac {c+d x-1}{c+d x+1}} (c+d x+1) \cosh ^{-1}(c+d x)}{(c+d x)^2}-i \cosh ^{-1}(c+d x) \log \left (1-i e^{-\cosh ^{-1}(c+d x)}\right )+i \cosh ^{-1}(c+d x) \log \left (1+i e^{-\cosh ^{-1}(c+d x)}\right )\right )}{3 d e^4} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.64, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {b^{2} \operatorname {arcosh}\left (d x + c\right )^{2} + 2 \, a b \operatorname {arcosh}\left (d x + c\right ) + a^{2}}{d^{4} e^{4} x^{4} + 4 \, c d^{3} e^{4} x^{3} + 6 \, c^{2} d^{2} e^{4} x^{2} + 4 \, c^{3} d e^{4} x + c^{4} e^{4}}, x\right ) \]
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 \[ \int \frac {{\left (b \operatorname {arcosh}\left (d x + c\right ) + a\right )}^{2}}{{\left (d e x + c e\right )}^{4}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.31, size = 381, normalized size = 2.05 \[ -\frac {a^{2}}{3 d \,e^{4} \left (d x +c \right )^{3}}+\frac {b^{2} \mathrm {arccosh}\left (d x +c \right ) \sqrt {d x +c +1}\, \sqrt {d x +c -1}}{3 d \,e^{4} \left (d x +c \right )^{2}}-\frac {b^{2} \mathrm {arccosh}\left (d x +c \right )^{2}}{3 d \,e^{4} \left (d x +c \right )^{3}}+\frac {b^{2}}{3 d \,e^{4} \left (d x +c \right )}-\frac {i b^{2} \mathrm {arccosh}\left (d x +c \right ) \ln \left (1+i \left (d x +c +\sqrt {d x +c -1}\, \sqrt {d x +c +1}\right )\right )}{3 d \,e^{4}}+\frac {i b^{2} \mathrm {arccosh}\left (d x +c \right ) \ln \left (1-i \left (d x +c +\sqrt {d x +c -1}\, \sqrt {d x +c +1}\right )\right )}{3 d \,e^{4}}-\frac {i b^{2} \dilog \left (1+i \left (d x +c +\sqrt {d x +c -1}\, \sqrt {d x +c +1}\right )\right )}{3 d \,e^{4}}+\frac {i b^{2} \dilog \left (1-i \left (d x +c +\sqrt {d x +c -1}\, \sqrt {d x +c +1}\right )\right )}{3 d \,e^{4}}-\frac {2 a b \,\mathrm {arccosh}\left (d x +c \right )}{3 d \,e^{4} \left (d x +c \right )^{3}}-\frac {a b \sqrt {d x +c -1}\, \sqrt {d x +c +1}\, \arctan \left (\frac {1}{\sqrt {\left (d x +c \right )^{2}-1}}\right )}{3 d \,e^{4} \sqrt {\left (d x +c \right )^{2}-1}}+\frac {a b \sqrt {d x +c -1}\, \sqrt {d x +c +1}}{3 d \,e^{4} \left (d x +c \right )^{2}} \]
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 \[ -\frac {b^{2} \log \left (d x + \sqrt {d x + c + 1} \sqrt {d x + c - 1} + c\right )^{2}}{3 \, {\left (d^{4} e^{4} x^{3} + 3 \, c d^{3} e^{4} x^{2} + 3 \, c^{2} d^{2} e^{4} x + c^{3} d e^{4}\right )}} - \frac {a^{2}}{3 \, {\left (d^{4} e^{4} x^{3} + 3 \, c d^{3} e^{4} x^{2} + 3 \, c^{2} d^{2} e^{4} x + c^{3} d e^{4}\right )}} + \int \frac {2 \, {\left ({\left (3 \, a b d^{3} + b^{2} d^{3}\right )} x^{3} + 3 \, {\left (c^{3} - c\right )} a b + {\left (c^{3} - c\right )} b^{2} + 3 \, {\left (3 \, a b c d^{2} + b^{2} c d^{2}\right )} x^{2} + {\left (b^{2} c^{2} + 3 \, {\left (c^{2} - 1\right )} a b + {\left (3 \, a b d^{2} + b^{2} d^{2}\right )} x^{2} + 2 \, {\left (3 \, a b c d + b^{2} c d\right )} x\right )} \sqrt {d x + c + 1} \sqrt {d x + c - 1} + {\left (3 \, {\left (3 \, c^{2} d - d\right )} a b + {\left (3 \, c^{2} d - d\right )} b^{2}\right )} x\right )} \log \left (d x + \sqrt {d x + c + 1} \sqrt {d x + c - 1} + c\right )}{3 \, {\left (d^{7} e^{4} x^{7} + 7 \, c d^{6} e^{4} x^{6} + c^{7} e^{4} - c^{5} e^{4} + {\left (21 \, c^{2} d^{5} e^{4} - d^{5} e^{4}\right )} x^{5} + 5 \, {\left (7 \, c^{3} d^{4} e^{4} - c d^{4} e^{4}\right )} x^{4} + 5 \, {\left (7 \, c^{4} d^{3} e^{4} - 2 \, c^{2} d^{3} e^{4}\right )} x^{3} + {\left (21 \, c^{5} d^{2} e^{4} - 10 \, c^{3} d^{2} e^{4}\right )} x^{2} + {\left (d^{6} e^{4} x^{6} + 6 \, c d^{5} e^{4} x^{5} + c^{6} e^{4} - c^{4} e^{4} + {\left (15 \, c^{2} d^{4} e^{4} - d^{4} e^{4}\right )} x^{4} + 4 \, {\left (5 \, c^{3} d^{3} e^{4} - c d^{3} e^{4}\right )} x^{3} + 3 \, {\left (5 \, c^{4} d^{2} e^{4} - 2 \, c^{2} d^{2} e^{4}\right )} x^{2} + 2 \, {\left (3 \, c^{5} d e^{4} - 2 \, c^{3} d e^{4}\right )} x\right )} \sqrt {d x + c + 1} \sqrt {d x + c - 1} + {\left (7 \, c^{6} d e^{4} - 5 \, c^{4} d e^{4}\right )} x\right )}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {{\left (a+b\,\mathrm {acosh}\left (c+d\,x\right )\right )}^2}{{\left (c\,e+d\,e\,x\right )}^4} \,d x \]
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 \[ \frac {\int \frac {a^{2}}{c^{4} + 4 c^{3} d x + 6 c^{2} d^{2} x^{2} + 4 c d^{3} x^{3} + d^{4} x^{4}}\, dx + \int \frac {b^{2} \operatorname {acosh}^{2}{\left (c + d x \right )}}{c^{4} + 4 c^{3} d x + 6 c^{2} d^{2} x^{2} + 4 c d^{3} x^{3} + d^{4} x^{4}}\, dx + \int \frac {2 a b \operatorname {acosh}{\left (c + d x \right )}}{c^{4} + 4 c^{3} d x + 6 c^{2} d^{2} x^{2} + 4 c d^{3} x^{3} + d^{4} x^{4}}\, dx}{e^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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