Optimal. Leaf size=186 \[ \frac {3 \tan ^{-1}\left (e^{\sinh ^{-1}(c x)}\right ) \left (a+b \sinh ^{-1}(c x)\right )}{4 c^5 d^3}-\frac {x^3 \left (a+b \sinh ^{-1}(c x)\right )}{4 c^2 d^3 \left (c^2 x^2+1\right )^2}-\frac {3 x \left (a+b \sinh ^{-1}(c x)\right )}{8 c^4 d^3 \left (c^2 x^2+1\right )}-\frac {3 i b \text {Li}_2\left (-i e^{\sinh ^{-1}(c x)}\right )}{8 c^5 d^3}+\frac {3 i b \text {Li}_2\left (i e^{\sinh ^{-1}(c x)}\right )}{8 c^5 d^3}-\frac {5 b}{8 c^5 d^3 \sqrt {c^2 x^2+1}}+\frac {b}{12 c^5 d^3 \left (c^2 x^2+1\right )^{3/2}} \]
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Rubi [A] time = 0.23, antiderivative size = 186, normalized size of antiderivative = 1.00, number of steps used = 12, number of rules used = 8, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {5751, 5693, 4180, 2279, 2391, 261, 266, 43} \[ -\frac {3 i b \text {PolyLog}\left (2,-i e^{\sinh ^{-1}(c x)}\right )}{8 c^5 d^3}+\frac {3 i b \text {PolyLog}\left (2,i e^{\sinh ^{-1}(c x)}\right )}{8 c^5 d^3}-\frac {x^3 \left (a+b \sinh ^{-1}(c x)\right )}{4 c^2 d^3 \left (c^2 x^2+1\right )^2}-\frac {3 x \left (a+b \sinh ^{-1}(c x)\right )}{8 c^4 d^3 \left (c^2 x^2+1\right )}+\frac {3 \tan ^{-1}\left (e^{\sinh ^{-1}(c x)}\right ) \left (a+b \sinh ^{-1}(c x)\right )}{4 c^5 d^3}-\frac {5 b}{8 c^5 d^3 \sqrt {c^2 x^2+1}}+\frac {b}{12 c^5 d^3 \left (c^2 x^2+1\right )^{3/2}} \]
Antiderivative was successfully verified.
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Rule 43
Rule 261
Rule 266
Rule 2279
Rule 2391
Rule 4180
Rule 5693
Rule 5751
Rubi steps
\begin {align*} \int \frac {x^4 \left (a+b \sinh ^{-1}(c x)\right )}{\left (d+c^2 d x^2\right )^3} \, dx &=-\frac {x^3 \left (a+b \sinh ^{-1}(c x)\right )}{4 c^2 d^3 \left (1+c^2 x^2\right )^2}+\frac {b \int \frac {x^3}{\left (1+c^2 x^2\right )^{5/2}} \, dx}{4 c d^3}+\frac {3 \int \frac {x^2 \left (a+b \sinh ^{-1}(c x)\right )}{\left (d+c^2 d x^2\right )^2} \, dx}{4 c^2 d}\\ &=-\frac {x^3 \left (a+b \sinh ^{-1}(c x)\right )}{4 c^2 d^3 \left (1+c^2 x^2\right )^2}-\frac {3 x \left (a+b \sinh ^{-1}(c x)\right )}{8 c^4 d^3 \left (1+c^2 x^2\right )}+\frac {(3 b) \int \frac {x}{\left (1+c^2 x^2\right )^{3/2}} \, dx}{8 c^3 d^3}+\frac {b \operatorname {Subst}\left (\int \frac {x}{\left (1+c^2 x\right )^{5/2}} \, dx,x,x^2\right )}{8 c d^3}+\frac {3 \int \frac {a+b \sinh ^{-1}(c x)}{d+c^2 d x^2} \, dx}{8 c^4 d^2}\\ &=-\frac {3 b}{8 c^5 d^3 \sqrt {1+c^2 x^2}}-\frac {x^3 \left (a+b \sinh ^{-1}(c x)\right )}{4 c^2 d^3 \left (1+c^2 x^2\right )^2}-\frac {3 x \left (a+b \sinh ^{-1}(c x)\right )}{8 c^4 d^3 \left (1+c^2 x^2\right )}+\frac {3 \operatorname {Subst}\left (\int (a+b x) \text {sech}(x) \, dx,x,\sinh ^{-1}(c x)\right )}{8 c^5 d^3}+\frac {b \operatorname {Subst}\left (\int \left (-\frac {1}{c^2 \left (1+c^2 x\right )^{5/2}}+\frac {1}{c^2 \left (1+c^2 x\right )^{3/2}}\right ) \, dx,x,x^2\right )}{8 c d^3}\\ &=\frac {b}{12 c^5 d^3 \left (1+c^2 x^2\right )^{3/2}}-\frac {5 b}{8 c^5 d^3 \sqrt {1+c^2 x^2}}-\frac {x^3 \left (a+b \sinh ^{-1}(c x)\right )}{4 c^2 d^3 \left (1+c^2 x^2\right )^2}-\frac {3 x \left (a+b \sinh ^{-1}(c x)\right )}{8 c^4 d^3 \left (1+c^2 x^2\right )}+\frac {3 \left (a+b \sinh ^{-1}(c x)\right ) \tan ^{-1}\left (e^{\sinh ^{-1}(c x)}\right )}{4 c^5 d^3}-\frac {(3 i b) \operatorname {Subst}\left (\int \log \left (1-i e^x\right ) \, dx,x,\sinh ^{-1}(c x)\right )}{8 c^5 d^3}+\frac {(3 i b) \operatorname {Subst}\left (\int \log \left (1+i e^x\right ) \, dx,x,\sinh ^{-1}(c x)\right )}{8 c^5 d^3}\\ &=\frac {b}{12 c^5 d^3 \left (1+c^2 x^2\right )^{3/2}}-\frac {5 b}{8 c^5 d^3 \sqrt {1+c^2 x^2}}-\frac {x^3 \left (a+b \sinh ^{-1}(c x)\right )}{4 c^2 d^3 \left (1+c^2 x^2\right )^2}-\frac {3 x \left (a+b \sinh ^{-1}(c x)\right )}{8 c^4 d^3 \left (1+c^2 x^2\right )}+\frac {3 \left (a+b \sinh ^{-1}(c x)\right ) \tan ^{-1}\left (e^{\sinh ^{-1}(c x)}\right )}{4 c^5 d^3}-\frac {(3 i b) \operatorname {Subst}\left (\int \frac {\log (1-i x)}{x} \, dx,x,e^{\sinh ^{-1}(c x)}\right )}{8 c^5 d^3}+\frac {(3 i b) \operatorname {Subst}\left (\int \frac {\log (1+i x)}{x} \, dx,x,e^{\sinh ^{-1}(c x)}\right )}{8 c^5 d^3}\\ &=\frac {b}{12 c^5 d^3 \left (1+c^2 x^2\right )^{3/2}}-\frac {5 b}{8 c^5 d^3 \sqrt {1+c^2 x^2}}-\frac {x^3 \left (a+b \sinh ^{-1}(c x)\right )}{4 c^2 d^3 \left (1+c^2 x^2\right )^2}-\frac {3 x \left (a+b \sinh ^{-1}(c x)\right )}{8 c^4 d^3 \left (1+c^2 x^2\right )}+\frac {3 \left (a+b \sinh ^{-1}(c x)\right ) \tan ^{-1}\left (e^{\sinh ^{-1}(c x)}\right )}{4 c^5 d^3}-\frac {3 i b \text {Li}_2\left (-i e^{\sinh ^{-1}(c x)}\right )}{8 c^5 d^3}+\frac {3 i b \text {Li}_2\left (i e^{\sinh ^{-1}(c x)}\right )}{8 c^5 d^3}\\ \end {align*}
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Mathematica [A] time = 0.68, size = 341, normalized size = 1.83 \[ -\frac {-9 a c^4 x^4 \tan ^{-1}(c x)+15 a c^3 x^3-18 a c^2 x^2 \tan ^{-1}(c x)+9 a c x-9 a \tan ^{-1}(c x)-9 i b c^4 x^4 \sinh ^{-1}(c x) \log \left (1-i e^{\sinh ^{-1}(c x)}\right )+9 i b c^4 x^4 \sinh ^{-1}(c x) \log \left (1+i e^{\sinh ^{-1}(c x)}\right )+15 b c^3 x^3 \sinh ^{-1}(c x)+9 i b \left (c^2 x^2+1\right )^2 \text {Li}_2\left (-i e^{\sinh ^{-1}(c x)}\right )-9 i b \left (c^2 x^2+1\right )^2 \text {Li}_2\left (i e^{\sinh ^{-1}(c x)}\right )+15 b c^2 x^2 \sqrt {c^2 x^2+1}+13 b \sqrt {c^2 x^2+1}-18 i b c^2 x^2 \sinh ^{-1}(c x) \log \left (1-i e^{\sinh ^{-1}(c x)}\right )+18 i b c^2 x^2 \sinh ^{-1}(c x) \log \left (1+i e^{\sinh ^{-1}(c x)}\right )+9 b c x \sinh ^{-1}(c x)-9 i b \sinh ^{-1}(c x) \log \left (1-i e^{\sinh ^{-1}(c x)}\right )+9 i b \sinh ^{-1}(c x) \log \left (1+i e^{\sinh ^{-1}(c x)}\right )}{24 c^5 d^3 \left (c^2 x^2+1\right )^2} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.52, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {b x^{4} \operatorname {arsinh}\left (c x\right ) + a x^{4}}{c^{6} d^{3} x^{6} + 3 \, c^{4} d^{3} x^{4} + 3 \, c^{2} d^{3} x^{2} + d^{3}}, x\right ) \]
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 \[ \int \frac {{\left (b \operatorname {arsinh}\left (c x\right ) + a\right )} x^{4}}{{\left (c^{2} d x^{2} + d\right )}^{3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.02, size = 313, normalized size = 1.68 \[ -\frac {5 a \,x^{3}}{8 c^{2} d^{3} \left (c^{2} x^{2}+1\right )^{2}}-\frac {3 a x}{8 c^{4} d^{3} \left (c^{2} x^{2}+1\right )^{2}}+\frac {3 a \arctan \left (c x \right )}{8 c^{5} d^{3}}-\frac {5 b \arcsinh \left (c x \right ) x^{3}}{8 c^{2} d^{3} \left (c^{2} x^{2}+1\right )^{2}}-\frac {3 b \arcsinh \left (c x \right ) x}{8 c^{4} d^{3} \left (c^{2} x^{2}+1\right )^{2}}+\frac {3 b \arcsinh \left (c x \right ) \arctan \left (c x \right )}{8 c^{5} d^{3}}+\frac {3 b \arctan \left (c x \right ) \ln \left (1+\frac {i \left (i c x +1\right )}{\sqrt {c^{2} x^{2}+1}}\right )}{8 c^{5} d^{3}}-\frac {3 b \arctan \left (c x \right ) \ln \left (1-\frac {i \left (i c x +1\right )}{\sqrt {c^{2} x^{2}+1}}\right )}{8 c^{5} d^{3}}-\frac {3 i b \dilog \left (1+\frac {i \left (i c x +1\right )}{\sqrt {c^{2} x^{2}+1}}\right )}{8 c^{5} d^{3}}+\frac {3 i b \dilog \left (1-\frac {i \left (i c x +1\right )}{\sqrt {c^{2} x^{2}+1}}\right )}{8 c^{5} d^{3}}-\frac {5 b \,x^{2}}{8 c^{3} d^{3} \left (c^{2} x^{2}+1\right )^{\frac {3}{2}}}-\frac {13 b}{24 c^{5} d^{3} \left (c^{2} x^{2}+1\right )^{\frac {3}{2}}} \]
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 \[ -\frac {1}{8} \, a {\left (\frac {5 \, c^{2} x^{3} + 3 \, x}{c^{8} d^{3} x^{4} + 2 \, c^{6} d^{3} x^{2} + c^{4} d^{3}} - \frac {3 \, \arctan \left (c x\right )}{c^{5} d^{3}}\right )} + b \int \frac {x^{4} \log \left (c x + \sqrt {c^{2} x^{2} + 1}\right )}{c^{6} d^{3} x^{6} + 3 \, c^{4} d^{3} x^{4} + 3 \, c^{2} d^{3} x^{2} + d^{3}}\,{d x} \]
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 \[ \int \frac {x^4\,\left (a+b\,\mathrm {asinh}\left (c\,x\right )\right )}{{\left (d\,c^2\,x^2+d\right )}^3} \,d x \]
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 \[ \frac {\int \frac {a x^{4}}{c^{6} x^{6} + 3 c^{4} x^{4} + 3 c^{2} x^{2} + 1}\, dx + \int \frac {b x^{4} \operatorname {asinh}{\left (c x \right )}}{c^{6} x^{6} + 3 c^{4} x^{4} + 3 c^{2} x^{2} + 1}\, dx}{d^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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