Optimal. Leaf size=279 \[ \frac {3 i b \text {Li}_2\left (-i e^{\sinh ^{-1}(c x)}\right ) \left (a+b \sinh ^{-1}(c x)\right )}{c^5 d^2}-\frac {3 i b \text {Li}_2\left (i e^{\sinh ^{-1}(c x)}\right ) \left (a+b \sinh ^{-1}(c x)\right )}{c^5 d^2}-\frac {3 \tan ^{-1}\left (e^{\sinh ^{-1}(c x)}\right ) \left (a+b \sinh ^{-1}(c x)\right )^2}{c^5 d^2}+\frac {3 x \left (a+b \sinh ^{-1}(c x)\right )^2}{2 c^4 d^2}-\frac {x^3 \left (a+b \sinh ^{-1}(c x)\right )^2}{2 c^2 d^2 \left (c^2 x^2+1\right )}-\frac {2 b \sqrt {c^2 x^2+1} \left (a+b \sinh ^{-1}(c x)\right )}{c^5 d^2}+\frac {b \left (a+b \sinh ^{-1}(c x)\right )}{c^5 d^2 \sqrt {c^2 x^2+1}}-\frac {3 i b^2 \text {Li}_3\left (-i e^{\sinh ^{-1}(c x)}\right )}{c^5 d^2}+\frac {3 i b^2 \text {Li}_3\left (i e^{\sinh ^{-1}(c x)}\right )}{c^5 d^2}-\frac {b^2 \tan ^{-1}(c x)}{c^5 d^2}+\frac {2 b^2 x}{c^4 d^2} \]
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Rubi [A] time = 0.53, antiderivative size = 279, normalized size of antiderivative = 1.00, number of steps used = 15, number of rules used = 14, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.538, Rules used = {5751, 5767, 5693, 4180, 2531, 2282, 6589, 5717, 8, 266, 43, 5732, 388, 205} \[ \frac {3 i b \text {PolyLog}\left (2,-i e^{\sinh ^{-1}(c x)}\right ) \left (a+b \sinh ^{-1}(c x)\right )}{c^5 d^2}-\frac {3 i b \text {PolyLog}\left (2,i e^{\sinh ^{-1}(c x)}\right ) \left (a+b \sinh ^{-1}(c x)\right )}{c^5 d^2}-\frac {3 i b^2 \text {PolyLog}\left (3,-i e^{\sinh ^{-1}(c x)}\right )}{c^5 d^2}+\frac {3 i b^2 \text {PolyLog}\left (3,i e^{\sinh ^{-1}(c x)}\right )}{c^5 d^2}-\frac {x^3 \left (a+b \sinh ^{-1}(c x)\right )^2}{2 c^2 d^2 \left (c^2 x^2+1\right )}-\frac {2 b \sqrt {c^2 x^2+1} \left (a+b \sinh ^{-1}(c x)\right )}{c^5 d^2}+\frac {b \left (a+b \sinh ^{-1}(c x)\right )}{c^5 d^2 \sqrt {c^2 x^2+1}}+\frac {3 x \left (a+b \sinh ^{-1}(c x)\right )^2}{2 c^4 d^2}-\frac {3 \tan ^{-1}\left (e^{\sinh ^{-1}(c x)}\right ) \left (a+b \sinh ^{-1}(c x)\right )^2}{c^5 d^2}+\frac {2 b^2 x}{c^4 d^2}-\frac {b^2 \tan ^{-1}(c x)}{c^5 d^2} \]
Antiderivative was successfully verified.
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Rule 8
Rule 43
Rule 205
Rule 266
Rule 388
Rule 2282
Rule 2531
Rule 4180
Rule 5693
Rule 5717
Rule 5732
Rule 5751
Rule 5767
Rule 6589
Rubi steps
\begin {align*} \int \frac {x^4 \left (a+b \sinh ^{-1}(c x)\right )^2}{\left (d+c^2 d x^2\right )^2} \, dx &=-\frac {x^3 \left (a+b \sinh ^{-1}(c x)\right )^2}{2 c^2 d^2 \left (1+c^2 x^2\right )}+\frac {b \int \frac {x^3 \left (a+b \sinh ^{-1}(c x)\right )}{\left (1+c^2 x^2\right )^{3/2}} \, dx}{c d^2}+\frac {3 \int \frac {x^2 \left (a+b \sinh ^{-1}(c x)\right )^2}{d+c^2 d x^2} \, dx}{2 c^2 d}\\ &=\frac {b \left (a+b \sinh ^{-1}(c x)\right )}{c^5 d^2 \sqrt {1+c^2 x^2}}+\frac {b \sqrt {1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )}{c^5 d^2}+\frac {3 x \left (a+b \sinh ^{-1}(c x)\right )^2}{2 c^4 d^2}-\frac {x^3 \left (a+b \sinh ^{-1}(c x)\right )^2}{2 c^2 d^2 \left (1+c^2 x^2\right )}-\frac {b^2 \int \frac {2+c^2 x^2}{c^4+c^6 x^2} \, dx}{d^2}-\frac {(3 b) \int \frac {x \left (a+b \sinh ^{-1}(c x)\right )}{\sqrt {1+c^2 x^2}} \, dx}{c^3 d^2}-\frac {3 \int \frac {\left (a+b \sinh ^{-1}(c x)\right )^2}{d+c^2 d x^2} \, dx}{2 c^4 d}\\ &=-\frac {b^2 x}{c^4 d^2}+\frac {b \left (a+b \sinh ^{-1}(c x)\right )}{c^5 d^2 \sqrt {1+c^2 x^2}}-\frac {2 b \sqrt {1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )}{c^5 d^2}+\frac {3 x \left (a+b \sinh ^{-1}(c x)\right )^2}{2 c^4 d^2}-\frac {x^3 \left (a+b \sinh ^{-1}(c x)\right )^2}{2 c^2 d^2 \left (1+c^2 x^2\right )}-\frac {b^2 \int \frac {1}{c^4+c^6 x^2} \, dx}{d^2}-\frac {3 \operatorname {Subst}\left (\int (a+b x)^2 \text {sech}(x) \, dx,x,\sinh ^{-1}(c x)\right )}{2 c^5 d^2}+\frac {\left (3 b^2\right ) \int 1 \, dx}{c^4 d^2}\\ &=\frac {2 b^2 x}{c^4 d^2}+\frac {b \left (a+b \sinh ^{-1}(c x)\right )}{c^5 d^2 \sqrt {1+c^2 x^2}}-\frac {2 b \sqrt {1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )}{c^5 d^2}+\frac {3 x \left (a+b \sinh ^{-1}(c x)\right )^2}{2 c^4 d^2}-\frac {x^3 \left (a+b \sinh ^{-1}(c x)\right )^2}{2 c^2 d^2 \left (1+c^2 x^2\right )}-\frac {3 \left (a+b \sinh ^{-1}(c x)\right )^2 \tan ^{-1}\left (e^{\sinh ^{-1}(c x)}\right )}{c^5 d^2}-\frac {b^2 \tan ^{-1}(c x)}{c^5 d^2}+\frac {(3 i b) \operatorname {Subst}\left (\int (a+b x) \log \left (1-i e^x\right ) \, dx,x,\sinh ^{-1}(c x)\right )}{c^5 d^2}-\frac {(3 i b) \operatorname {Subst}\left (\int (a+b x) \log \left (1+i e^x\right ) \, dx,x,\sinh ^{-1}(c x)\right )}{c^5 d^2}\\ &=\frac {2 b^2 x}{c^4 d^2}+\frac {b \left (a+b \sinh ^{-1}(c x)\right )}{c^5 d^2 \sqrt {1+c^2 x^2}}-\frac {2 b \sqrt {1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )}{c^5 d^2}+\frac {3 x \left (a+b \sinh ^{-1}(c x)\right )^2}{2 c^4 d^2}-\frac {x^3 \left (a+b \sinh ^{-1}(c x)\right )^2}{2 c^2 d^2 \left (1+c^2 x^2\right )}-\frac {3 \left (a+b \sinh ^{-1}(c x)\right )^2 \tan ^{-1}\left (e^{\sinh ^{-1}(c x)}\right )}{c^5 d^2}-\frac {b^2 \tan ^{-1}(c x)}{c^5 d^2}+\frac {3 i b \left (a+b \sinh ^{-1}(c x)\right ) \text {Li}_2\left (-i e^{\sinh ^{-1}(c x)}\right )}{c^5 d^2}-\frac {3 i b \left (a+b \sinh ^{-1}(c x)\right ) \text {Li}_2\left (i e^{\sinh ^{-1}(c x)}\right )}{c^5 d^2}-\frac {\left (3 i b^2\right ) \operatorname {Subst}\left (\int \text {Li}_2\left (-i e^x\right ) \, dx,x,\sinh ^{-1}(c x)\right )}{c^5 d^2}+\frac {\left (3 i b^2\right ) \operatorname {Subst}\left (\int \text {Li}_2\left (i e^x\right ) \, dx,x,\sinh ^{-1}(c x)\right )}{c^5 d^2}\\ &=\frac {2 b^2 x}{c^4 d^2}+\frac {b \left (a+b \sinh ^{-1}(c x)\right )}{c^5 d^2 \sqrt {1+c^2 x^2}}-\frac {2 b \sqrt {1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )}{c^5 d^2}+\frac {3 x \left (a+b \sinh ^{-1}(c x)\right )^2}{2 c^4 d^2}-\frac {x^3 \left (a+b \sinh ^{-1}(c x)\right )^2}{2 c^2 d^2 \left (1+c^2 x^2\right )}-\frac {3 \left (a+b \sinh ^{-1}(c x)\right )^2 \tan ^{-1}\left (e^{\sinh ^{-1}(c x)}\right )}{c^5 d^2}-\frac {b^2 \tan ^{-1}(c x)}{c^5 d^2}+\frac {3 i b \left (a+b \sinh ^{-1}(c x)\right ) \text {Li}_2\left (-i e^{\sinh ^{-1}(c x)}\right )}{c^5 d^2}-\frac {3 i b \left (a+b \sinh ^{-1}(c x)\right ) \text {Li}_2\left (i e^{\sinh ^{-1}(c x)}\right )}{c^5 d^2}-\frac {\left (3 i b^2\right ) \operatorname {Subst}\left (\int \frac {\text {Li}_2(-i x)}{x} \, dx,x,e^{\sinh ^{-1}(c x)}\right )}{c^5 d^2}+\frac {\left (3 i b^2\right ) \operatorname {Subst}\left (\int \frac {\text {Li}_2(i x)}{x} \, dx,x,e^{\sinh ^{-1}(c x)}\right )}{c^5 d^2}\\ &=\frac {2 b^2 x}{c^4 d^2}+\frac {b \left (a+b \sinh ^{-1}(c x)\right )}{c^5 d^2 \sqrt {1+c^2 x^2}}-\frac {2 b \sqrt {1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )}{c^5 d^2}+\frac {3 x \left (a+b \sinh ^{-1}(c x)\right )^2}{2 c^4 d^2}-\frac {x^3 \left (a+b \sinh ^{-1}(c x)\right )^2}{2 c^2 d^2 \left (1+c^2 x^2\right )}-\frac {3 \left (a+b \sinh ^{-1}(c x)\right )^2 \tan ^{-1}\left (e^{\sinh ^{-1}(c x)}\right )}{c^5 d^2}-\frac {b^2 \tan ^{-1}(c x)}{c^5 d^2}+\frac {3 i b \left (a+b \sinh ^{-1}(c x)\right ) \text {Li}_2\left (-i e^{\sinh ^{-1}(c x)}\right )}{c^5 d^2}-\frac {3 i b \left (a+b \sinh ^{-1}(c x)\right ) \text {Li}_2\left (i e^{\sinh ^{-1}(c x)}\right )}{c^5 d^2}-\frac {3 i b^2 \text {Li}_3\left (-i e^{\sinh ^{-1}(c x)}\right )}{c^5 d^2}+\frac {3 i b^2 \text {Li}_3\left (i e^{\sinh ^{-1}(c x)}\right )}{c^5 d^2}\\ \end {align*}
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Mathematica [A] time = 2.23, size = 482, normalized size = 1.73 \[ \frac {-\frac {3 a^2 \tan ^{-1}(c x)}{c^5}+\frac {2 a^2 x}{c^4}+\frac {a^2 x}{c^6 x^2+c^4}-\frac {2 a b \left (-2 c^3 x^3 \sinh ^{-1}(c x)-3 i \left (c^2 x^2+1\right ) \text {Li}_2\left (-i e^{\sinh ^{-1}(c x)}\right )+3 i \left (c^2 x^2+1\right ) \text {Li}_2\left (i e^{\sinh ^{-1}(c x)}\right )+2 c^2 x^2 \sqrt {c^2 x^2+1}+\sqrt {c^2 x^2+1}+3 i c^2 x^2 \sinh ^{-1}(c x) \log \left (1-i e^{\sinh ^{-1}(c x)}\right )-3 i c^2 x^2 \sinh ^{-1}(c x) \log \left (1+i e^{\sinh ^{-1}(c x)}\right )-3 c x \sinh ^{-1}(c x)+3 i \sinh ^{-1}(c x) \log \left (1-i e^{\sinh ^{-1}(c x)}\right )-3 i \sinh ^{-1}(c x) \log \left (1+i e^{\sinh ^{-1}(c x)}\right )\right )}{c^7 x^2+c^5}+\frac {2 b^2 \left (\frac {c x \sinh ^{-1}(c x)^2}{2 c^2 x^2+2}-2 \sqrt {c^2 x^2+1} \sinh ^{-1}(c x)+\frac {\sinh ^{-1}(c x)}{\sqrt {c^2 x^2+1}}+\frac {1}{2} i \left (6 \sinh ^{-1}(c x) \text {Li}_2\left (-i e^{-\sinh ^{-1}(c x)}\right )-6 \sinh ^{-1}(c x) \text {Li}_2\left (i e^{-\sinh ^{-1}(c x)}\right )+6 \text {Li}_3\left (-i e^{-\sinh ^{-1}(c x)}\right )-6 \text {Li}_3\left (i e^{-\sinh ^{-1}(c x)}\right )+3 \sinh ^{-1}(c x)^2 \log \left (1-i e^{-\sinh ^{-1}(c x)}\right )-3 \sinh ^{-1}(c x)^2 \log \left (1+i e^{-\sinh ^{-1}(c x)}\right )+4 i \tan ^{-1}\left (\tanh \left (\frac {1}{2} \sinh ^{-1}(c x)\right )\right )\right )+c x \left (\sinh ^{-1}(c x)^2+2\right )\right )}{c^5}}{2 d^2} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.61, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {b^{2} x^{4} \operatorname {arsinh}\left (c x\right )^{2} + 2 \, a b x^{4} \operatorname {arsinh}\left (c x\right ) + a^{2} x^{4}}{c^{4} d^{2} x^{4} + 2 \, c^{2} d^{2} x^{2} + d^{2}}, x\right ) \]
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 \[ \text {Exception raised: TypeError} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.37, size = 0, normalized size = 0.00 \[ \int \frac {x^{4} \left (a +b \arcsinh \left (c x \right )\right )^{2}}{\left (c^{2} d \,x^{2}+d \right )^{2}}\, dx \]
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 \[ \frac {1}{2} \, a^{2} {\left (\frac {x}{c^{6} d^{2} x^{2} + c^{4} d^{2}} + \frac {2 \, x}{c^{4} d^{2}} - \frac {3 \, \arctan \left (c x\right )}{c^{5} d^{2}}\right )} + \int \frac {b^{2} x^{4} \log \left (c x + \sqrt {c^{2} x^{2} + 1}\right )^{2}}{c^{4} d^{2} x^{4} + 2 \, c^{2} d^{2} x^{2} + d^{2}} + \frac {2 \, a b x^{4} \log \left (c x + \sqrt {c^{2} x^{2} + 1}\right )}{c^{4} d^{2} x^{4} + 2 \, c^{2} d^{2} x^{2} + d^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {x^4\,{\left (a+b\,\mathrm {asinh}\left (c\,x\right )\right )}^2}{{\left (d\,c^2\,x^2+d\right )}^2} \,d x \]
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 \[ \frac {\int \frac {a^{2} x^{4}}{c^{4} x^{4} + 2 c^{2} x^{2} + 1}\, dx + \int \frac {b^{2} x^{4} \operatorname {asinh}^{2}{\left (c x \right )}}{c^{4} x^{4} + 2 c^{2} x^{2} + 1}\, dx + \int \frac {2 a b x^{4} \operatorname {asinh}{\left (c x \right )}}{c^{4} x^{4} + 2 c^{2} x^{2} + 1}\, dx}{d^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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