Optimal. Leaf size=310 \[ \frac {35 c^3 \tanh ^{-1}\left (e^{\cosh ^{-1}(c x)}\right ) \left (a+b \cosh ^{-1}(c x)\right )}{4 d^3}-\frac {7 c^2 \left (a+b \cosh ^{-1}(c x)\right )}{3 d^3 x \left (1-c^2 x^2\right )^2}-\frac {a+b \cosh ^{-1}(c x)}{3 d^3 x^3 \left (1-c^2 x^2\right )^2}+\frac {35 c^4 x \left (a+b \cosh ^{-1}(c x)\right )}{8 d^3 \left (1-c^2 x^2\right )}+\frac {35 c^4 x \left (a+b \cosh ^{-1}(c x)\right )}{12 d^3 \left (1-c^2 x^2\right )^2}+\frac {35 b c^3 \text {Li}_2\left (-e^{\cosh ^{-1}(c x)}\right )}{8 d^3}-\frac {35 b c^3 \text {Li}_2\left (e^{\cosh ^{-1}(c x)}\right )}{8 d^3}-\frac {29 b c^3}{24 d^3 \sqrt {c x-1} \sqrt {c x+1}}-\frac {b c^3}{12 d^3 (c x-1)^{3/2} (c x+1)^{3/2}}+\frac {19 b c^3 \tan ^{-1}\left (\sqrt {c x-1} \sqrt {c x+1}\right )}{6 d^3}+\frac {b c}{6 d^3 x^2 (c x-1)^{3/2} (c x+1)^{3/2}} \]
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Rubi [A] time = 0.39, antiderivative size = 310, normalized size of antiderivative = 1.00, number of steps used = 26, number of rules used = 13, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.520, Rules used = {5746, 103, 12, 104, 21, 92, 205, 5689, 74, 5694, 4182, 2279, 2391} \[ \frac {35 b c^3 \text {PolyLog}\left (2,-e^{\cosh ^{-1}(c x)}\right )}{8 d^3}-\frac {35 b c^3 \text {PolyLog}\left (2,e^{\cosh ^{-1}(c x)}\right )}{8 d^3}+\frac {35 c^4 x \left (a+b \cosh ^{-1}(c x)\right )}{8 d^3 \left (1-c^2 x^2\right )}+\frac {35 c^4 x \left (a+b \cosh ^{-1}(c x)\right )}{12 d^3 \left (1-c^2 x^2\right )^2}-\frac {7 c^2 \left (a+b \cosh ^{-1}(c x)\right )}{3 d^3 x \left (1-c^2 x^2\right )^2}-\frac {a+b \cosh ^{-1}(c x)}{3 d^3 x^3 \left (1-c^2 x^2\right )^2}+\frac {35 c^3 \tanh ^{-1}\left (e^{\cosh ^{-1}(c x)}\right ) \left (a+b \cosh ^{-1}(c x)\right )}{4 d^3}-\frac {29 b c^3}{24 d^3 \sqrt {c x-1} \sqrt {c x+1}}-\frac {b c^3}{12 d^3 (c x-1)^{3/2} (c x+1)^{3/2}}+\frac {19 b c^3 \tan ^{-1}\left (\sqrt {c x-1} \sqrt {c x+1}\right )}{6 d^3}+\frac {b c}{6 d^3 x^2 (c x-1)^{3/2} (c x+1)^{3/2}} \]
Antiderivative was successfully verified.
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Rule 12
Rule 21
Rule 74
Rule 92
Rule 103
Rule 104
Rule 205
Rule 2279
Rule 2391
Rule 4182
Rule 5689
Rule 5694
Rule 5746
Rubi steps
\begin {align*} \int \frac {a+b \cosh ^{-1}(c x)}{x^4 \left (d-c^2 d x^2\right )^3} \, dx &=-\frac {a+b \cosh ^{-1}(c x)}{3 d^3 x^3 \left (1-c^2 x^2\right )^2}+\frac {1}{3} \left (7 c^2\right ) \int \frac {a+b \cosh ^{-1}(c x)}{x^2 \left (d-c^2 d x^2\right )^3} \, dx+\frac {(b c) \int \frac {1}{x^3 (-1+c x)^{5/2} (1+c x)^{5/2}} \, dx}{3 d^3}\\ &=\frac {b c}{6 d^3 x^2 (-1+c x)^{3/2} (1+c x)^{3/2}}-\frac {a+b \cosh ^{-1}(c x)}{3 d^3 x^3 \left (1-c^2 x^2\right )^2}-\frac {7 c^2 \left (a+b \cosh ^{-1}(c x)\right )}{3 d^3 x \left (1-c^2 x^2\right )^2}+\frac {1}{3} \left (35 c^4\right ) \int \frac {a+b \cosh ^{-1}(c x)}{\left (d-c^2 d x^2\right )^3} \, dx+\frac {(b c) \int \frac {5 c^2}{x (-1+c x)^{5/2} (1+c x)^{5/2}} \, dx}{6 d^3}+\frac {\left (7 b c^3\right ) \int \frac {1}{x (-1+c x)^{5/2} (1+c x)^{5/2}} \, dx}{3 d^3}\\ &=-\frac {7 b c^3}{9 d^3 (-1+c x)^{3/2} (1+c x)^{3/2}}+\frac {b c}{6 d^3 x^2 (-1+c x)^{3/2} (1+c x)^{3/2}}-\frac {a+b \cosh ^{-1}(c x)}{3 d^3 x^3 \left (1-c^2 x^2\right )^2}-\frac {7 c^2 \left (a+b \cosh ^{-1}(c x)\right )}{3 d^3 x \left (1-c^2 x^2\right )^2}+\frac {35 c^4 x \left (a+b \cosh ^{-1}(c x)\right )}{12 d^3 \left (1-c^2 x^2\right )^2}-\frac {\left (7 b c^2\right ) \int \frac {3 c+3 c^2 x}{x (-1+c x)^{3/2} (1+c x)^{5/2}} \, dx}{9 d^3}+\frac {\left (5 b c^3\right ) \int \frac {1}{x (-1+c x)^{5/2} (1+c x)^{5/2}} \, dx}{6 d^3}-\frac {\left (35 b c^5\right ) \int \frac {x}{(-1+c x)^{5/2} (1+c x)^{5/2}} \, dx}{12 d^3}+\frac {\left (35 c^4\right ) \int \frac {a+b \cosh ^{-1}(c x)}{\left (d-c^2 d x^2\right )^2} \, dx}{4 d}\\ &=-\frac {b c^3}{12 d^3 (-1+c x)^{3/2} (1+c x)^{3/2}}+\frac {b c}{6 d^3 x^2 (-1+c x)^{3/2} (1+c x)^{3/2}}-\frac {a+b \cosh ^{-1}(c x)}{3 d^3 x^3 \left (1-c^2 x^2\right )^2}-\frac {7 c^2 \left (a+b \cosh ^{-1}(c x)\right )}{3 d^3 x \left (1-c^2 x^2\right )^2}+\frac {35 c^4 x \left (a+b \cosh ^{-1}(c x)\right )}{12 d^3 \left (1-c^2 x^2\right )^2}+\frac {35 c^4 x \left (a+b \cosh ^{-1}(c x)\right )}{8 d^3 \left (1-c^2 x^2\right )}-\frac {\left (5 b c^2\right ) \int \frac {3 c+3 c^2 x}{x (-1+c x)^{3/2} (1+c x)^{5/2}} \, dx}{18 d^3}-\frac {\left (7 b c^3\right ) \int \frac {1}{x (-1+c x)^{3/2} (1+c x)^{3/2}} \, dx}{3 d^3}+\frac {\left (35 b c^5\right ) \int \frac {x}{(-1+c x)^{3/2} (1+c x)^{3/2}} \, dx}{8 d^3}+\frac {\left (35 c^4\right ) \int \frac {a+b \cosh ^{-1}(c x)}{d-c^2 d x^2} \, dx}{8 d^2}\\ &=-\frac {b c^3}{12 d^3 (-1+c x)^{3/2} (1+c x)^{3/2}}+\frac {b c}{6 d^3 x^2 (-1+c x)^{3/2} (1+c x)^{3/2}}-\frac {49 b c^3}{24 d^3 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {a+b \cosh ^{-1}(c x)}{3 d^3 x^3 \left (1-c^2 x^2\right )^2}-\frac {7 c^2 \left (a+b \cosh ^{-1}(c x)\right )}{3 d^3 x \left (1-c^2 x^2\right )^2}+\frac {35 c^4 x \left (a+b \cosh ^{-1}(c x)\right )}{12 d^3 \left (1-c^2 x^2\right )^2}+\frac {35 c^4 x \left (a+b \cosh ^{-1}(c x)\right )}{8 d^3 \left (1-c^2 x^2\right )}+\frac {\left (7 b c^2\right ) \int \frac {c+c^2 x}{x \sqrt {-1+c x} (1+c x)^{3/2}} \, dx}{3 d^3}-\frac {\left (35 c^3\right ) \operatorname {Subst}\left (\int (a+b x) \text {csch}(x) \, dx,x,\cosh ^{-1}(c x)\right )}{8 d^3}-\frac {\left (5 b c^3\right ) \int \frac {1}{x (-1+c x)^{3/2} (1+c x)^{3/2}} \, dx}{6 d^3}\\ &=-\frac {b c^3}{12 d^3 (-1+c x)^{3/2} (1+c x)^{3/2}}+\frac {b c}{6 d^3 x^2 (-1+c x)^{3/2} (1+c x)^{3/2}}-\frac {29 b c^3}{24 d^3 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {a+b \cosh ^{-1}(c x)}{3 d^3 x^3 \left (1-c^2 x^2\right )^2}-\frac {7 c^2 \left (a+b \cosh ^{-1}(c x)\right )}{3 d^3 x \left (1-c^2 x^2\right )^2}+\frac {35 c^4 x \left (a+b \cosh ^{-1}(c x)\right )}{12 d^3 \left (1-c^2 x^2\right )^2}+\frac {35 c^4 x \left (a+b \cosh ^{-1}(c x)\right )}{8 d^3 \left (1-c^2 x^2\right )}+\frac {35 c^3 \left (a+b \cosh ^{-1}(c x)\right ) \tanh ^{-1}\left (e^{\cosh ^{-1}(c x)}\right )}{4 d^3}+\frac {\left (5 b c^2\right ) \int \frac {c+c^2 x}{x \sqrt {-1+c x} (1+c x)^{3/2}} \, dx}{6 d^3}+\frac {\left (7 b c^3\right ) \int \frac {1}{x \sqrt {-1+c x} \sqrt {1+c x}} \, dx}{3 d^3}+\frac {\left (35 b c^3\right ) \operatorname {Subst}\left (\int \log \left (1-e^x\right ) \, dx,x,\cosh ^{-1}(c x)\right )}{8 d^3}-\frac {\left (35 b c^3\right ) \operatorname {Subst}\left (\int \log \left (1+e^x\right ) \, dx,x,\cosh ^{-1}(c x)\right )}{8 d^3}\\ &=-\frac {b c^3}{12 d^3 (-1+c x)^{3/2} (1+c x)^{3/2}}+\frac {b c}{6 d^3 x^2 (-1+c x)^{3/2} (1+c x)^{3/2}}-\frac {29 b c^3}{24 d^3 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {a+b \cosh ^{-1}(c x)}{3 d^3 x^3 \left (1-c^2 x^2\right )^2}-\frac {7 c^2 \left (a+b \cosh ^{-1}(c x)\right )}{3 d^3 x \left (1-c^2 x^2\right )^2}+\frac {35 c^4 x \left (a+b \cosh ^{-1}(c x)\right )}{12 d^3 \left (1-c^2 x^2\right )^2}+\frac {35 c^4 x \left (a+b \cosh ^{-1}(c x)\right )}{8 d^3 \left (1-c^2 x^2\right )}+\frac {35 c^3 \left (a+b \cosh ^{-1}(c x)\right ) \tanh ^{-1}\left (e^{\cosh ^{-1}(c x)}\right )}{4 d^3}+\frac {\left (5 b c^3\right ) \int \frac {1}{x \sqrt {-1+c x} \sqrt {1+c x}} \, dx}{6 d^3}+\frac {\left (35 b c^3\right ) \operatorname {Subst}\left (\int \frac {\log (1-x)}{x} \, dx,x,e^{\cosh ^{-1}(c x)}\right )}{8 d^3}-\frac {\left (35 b c^3\right ) \operatorname {Subst}\left (\int \frac {\log (1+x)}{x} \, dx,x,e^{\cosh ^{-1}(c x)}\right )}{8 d^3}+\frac {\left (7 b c^4\right ) \operatorname {Subst}\left (\int \frac {1}{c+c x^2} \, dx,x,\sqrt {-1+c x} \sqrt {1+c x}\right )}{3 d^3}\\ &=-\frac {b c^3}{12 d^3 (-1+c x)^{3/2} (1+c x)^{3/2}}+\frac {b c}{6 d^3 x^2 (-1+c x)^{3/2} (1+c x)^{3/2}}-\frac {29 b c^3}{24 d^3 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {a+b \cosh ^{-1}(c x)}{3 d^3 x^3 \left (1-c^2 x^2\right )^2}-\frac {7 c^2 \left (a+b \cosh ^{-1}(c x)\right )}{3 d^3 x \left (1-c^2 x^2\right )^2}+\frac {35 c^4 x \left (a+b \cosh ^{-1}(c x)\right )}{12 d^3 \left (1-c^2 x^2\right )^2}+\frac {35 c^4 x \left (a+b \cosh ^{-1}(c x)\right )}{8 d^3 \left (1-c^2 x^2\right )}+\frac {7 b c^3 \tan ^{-1}\left (\sqrt {-1+c x} \sqrt {1+c x}\right )}{3 d^3}+\frac {35 c^3 \left (a+b \cosh ^{-1}(c x)\right ) \tanh ^{-1}\left (e^{\cosh ^{-1}(c x)}\right )}{4 d^3}+\frac {35 b c^3 \text {Li}_2\left (-e^{\cosh ^{-1}(c x)}\right )}{8 d^3}-\frac {35 b c^3 \text {Li}_2\left (e^{\cosh ^{-1}(c x)}\right )}{8 d^3}+\frac {\left (5 b c^4\right ) \operatorname {Subst}\left (\int \frac {1}{c+c x^2} \, dx,x,\sqrt {-1+c x} \sqrt {1+c x}\right )}{6 d^3}\\ &=-\frac {b c^3}{12 d^3 (-1+c x)^{3/2} (1+c x)^{3/2}}+\frac {b c}{6 d^3 x^2 (-1+c x)^{3/2} (1+c x)^{3/2}}-\frac {29 b c^3}{24 d^3 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {a+b \cosh ^{-1}(c x)}{3 d^3 x^3 \left (1-c^2 x^2\right )^2}-\frac {7 c^2 \left (a+b \cosh ^{-1}(c x)\right )}{3 d^3 x \left (1-c^2 x^2\right )^2}+\frac {35 c^4 x \left (a+b \cosh ^{-1}(c x)\right )}{12 d^3 \left (1-c^2 x^2\right )^2}+\frac {35 c^4 x \left (a+b \cosh ^{-1}(c x)\right )}{8 d^3 \left (1-c^2 x^2\right )}+\frac {19 b c^3 \tan ^{-1}\left (\sqrt {-1+c x} \sqrt {1+c x}\right )}{6 d^3}+\frac {35 c^3 \left (a+b \cosh ^{-1}(c x)\right ) \tanh ^{-1}\left (e^{\cosh ^{-1}(c x)}\right )}{4 d^3}+\frac {35 b c^3 \text {Li}_2\left (-e^{\cosh ^{-1}(c x)}\right )}{8 d^3}-\frac {35 b c^3 \text {Li}_2\left (e^{\cosh ^{-1}(c x)}\right )}{8 d^3}\\ \end {align*}
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Mathematica [A] time = 1.93, size = 471, normalized size = 1.52 \[ \frac {-105 a c^3 \log (1-c x)+105 a c^3 \log (c x+1)-\frac {144 a c^2}{x}-\frac {66 a c^4 x}{c^2 x^2-1}+\frac {12 a c^4 x}{\left (c^2 x^2-1\right )^2}-\frac {16 a}{x^3}-\frac {105}{2} b c^3 \left (\cosh ^{-1}(c x) \left (\cosh ^{-1}(c x)-4 \log \left (e^{\cosh ^{-1}(c x)}+1\right )\right )-4 \text {Li}_2\left (-e^{\cosh ^{-1}(c x)}\right )\right )+\frac {105}{2} b c^3 \left (\cosh ^{-1}(c x) \left (\cosh ^{-1}(c x)-4 \log \left (1-e^{\cosh ^{-1}(c x)}\right )\right )-4 \text {Li}_2\left (e^{\cosh ^{-1}(c x)}\right )\right )-\frac {b c^3 \left ((c x-2) \sqrt {c x-1} \sqrt {c x+1}-3 \cosh ^{-1}(c x)\right )}{(c x-1)^2}+\frac {b c^3 \left (\sqrt {c x-1} \sqrt {c x+1} (c x+2)-3 \cosh ^{-1}(c x)\right )}{(c x+1)^2}+33 b c^3 \left (\frac {\cosh ^{-1}(c x)}{1-c x}-\frac {1}{\sqrt {\frac {c x-1}{c x+1}}}\right )+33 b c^3 \left (\sqrt {\frac {c x-1}{c x+1}}-\frac {\cosh ^{-1}(c x)}{c x+1}\right )+144 b c^2 \left (\frac {c \sqrt {c^2 x^2-1} \tan ^{-1}\left (\sqrt {c^2 x^2-1}\right )}{\sqrt {c x-1} \sqrt {c x+1}}-\frac {\cosh ^{-1}(c x)}{x}\right )+\frac {8 b \left (\frac {c x \left (c^2 x^2+c^2 x^2 \sqrt {c^2 x^2-1} \tan ^{-1}\left (\sqrt {c^2 x^2-1}\right )-1\right )}{\sqrt {c x-1} \sqrt {c x+1}}-2 \cosh ^{-1}(c x)\right )}{x^3}}{48 d^3} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.57, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-\frac {b \operatorname {arcosh}\left (c x\right ) + a}{c^{6} d^{3} x^{10} - 3 \, c^{4} d^{3} x^{8} + 3 \, c^{2} d^{3} x^{6} - d^{3} x^{4}}, x\right ) \]
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 \[ \int -\frac {b \operatorname {arcosh}\left (c x\right ) + a}{{\left (c^{2} d x^{2} - d\right )}^{3} x^{4}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.58, size = 504, normalized size = 1.63 \[ -\frac {a}{3 d^{3} x^{3}}-\frac {3 c^{2} a}{d^{3} x}+\frac {c^{3} a}{16 d^{3} \left (c x -1\right )^{2}}-\frac {11 c^{3} a}{16 d^{3} \left (c x -1\right )}-\frac {35 c^{3} a \ln \left (c x -1\right )}{16 d^{3}}-\frac {c^{3} a}{16 d^{3} \left (c x +1\right )^{2}}-\frac {11 c^{3} a}{16 d^{3} \left (c x +1\right )}+\frac {35 c^{3} a \ln \left (c x +1\right )}{16 d^{3}}-\frac {35 c^{6} b \,\mathrm {arccosh}\left (c x \right ) x^{3}}{8 d^{3} \left (c^{4} x^{4}-2 c^{2} x^{2}+1\right )}-\frac {29 c^{5} b \sqrt {c x +1}\, \sqrt {c x -1}\, x^{2}}{24 d^{3} \left (c^{4} x^{4}-2 c^{2} x^{2}+1\right )}+\frac {175 c^{4} b \,\mathrm {arccosh}\left (c x \right ) x}{24 d^{3} \left (c^{4} x^{4}-2 c^{2} x^{2}+1\right )}+\frac {9 c^{3} b \sqrt {c x +1}\, \sqrt {c x -1}}{8 d^{3} \left (c^{4} x^{4}-2 c^{2} x^{2}+1\right )}-\frac {7 c^{2} b \,\mathrm {arccosh}\left (c x \right )}{3 d^{3} x \left (c^{4} x^{4}-2 c^{2} x^{2}+1\right )}+\frac {c b \sqrt {c x +1}\, \sqrt {c x -1}}{6 d^{3} \left (c^{4} x^{4}-2 c^{2} x^{2}+1\right ) x^{2}}-\frac {b \,\mathrm {arccosh}\left (c x \right )}{3 d^{3} \left (c^{4} x^{4}-2 c^{2} x^{2}+1\right ) x^{3}}+\frac {19 c^{3} b \arctan \left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )}{3 d^{3}}+\frac {35 c^{3} b \dilog \left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )}{8 d^{3}}+\frac {35 c^{3} b \dilog \left (1+c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )}{8 d^{3}}+\frac {35 c^{3} b \,\mathrm {arccosh}\left (c x \right ) \ln \left (1+c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )}{8 d^{3}} \]
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 \[ \text {result too large to display} \]
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 \[ \int \frac {a+b\,\mathrm {acosh}\left (c\,x\right )}{x^4\,{\left (d-c^2\,d\,x^2\right )}^3} \,d x \]
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 \[ - \frac {\int \frac {a}{c^{6} x^{10} - 3 c^{4} x^{8} + 3 c^{2} x^{6} - x^{4}}\, dx + \int \frac {b \operatorname {acosh}{\left (c x \right )}}{c^{6} x^{10} - 3 c^{4} x^{8} + 3 c^{2} x^{6} - x^{4}}\, dx}{d^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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