Optimal. Leaf size=275 \[ -\frac {b^2}{12 d^3 \left (1+c^2 x^2\right )}-\frac {b c x \left (a+b \sinh ^{-1}(c x)\right )}{6 d^3 \left (1+c^2 x^2\right )^{3/2}}-\frac {4 b c x \left (a+b \sinh ^{-1}(c x)\right )}{3 d^3 \sqrt {1+c^2 x^2}}+\frac {\left (a+b \sinh ^{-1}(c x)\right )^2}{4 d^3 \left (1+c^2 x^2\right )^2}+\frac {\left (a+b \sinh ^{-1}(c x)\right )^2}{2 d^3 \left (1+c^2 x^2\right )}-\frac {2 \left (a+b \sinh ^{-1}(c x)\right )^2 \tanh ^{-1}\left (e^{2 \sinh ^{-1}(c x)}\right )}{d^3}+\frac {2 b^2 \log \left (1+c^2 x^2\right )}{3 d^3}-\frac {b \left (a+b \sinh ^{-1}(c x)\right ) \text {PolyLog}\left (2,-e^{2 \sinh ^{-1}(c x)}\right )}{d^3}+\frac {b \left (a+b \sinh ^{-1}(c x)\right ) \text {PolyLog}\left (2,e^{2 \sinh ^{-1}(c x)}\right )}{d^3}+\frac {b^2 \text {PolyLog}\left (3,-e^{2 \sinh ^{-1}(c x)}\right )}{2 d^3}-\frac {b^2 \text {PolyLog}\left (3,e^{2 \sinh ^{-1}(c x)}\right )}{2 d^3} \]
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Rubi [A]
time = 0.36, antiderivative size = 275, normalized size of antiderivative = 1.00, number of steps
used = 17, number of rules used = 11, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.423, Rules used = {5811, 5799,
5569, 4267, 2611, 2320, 6724, 5787, 266, 5788, 267} \begin {gather*} -\frac {4 b c x \left (a+b \sinh ^{-1}(c x)\right )}{3 d^3 \sqrt {c^2 x^2+1}}-\frac {b c x \left (a+b \sinh ^{-1}(c x)\right )}{6 d^3 \left (c^2 x^2+1\right )^{3/2}}+\frac {\left (a+b \sinh ^{-1}(c x)\right )^2}{2 d^3 \left (c^2 x^2+1\right )}+\frac {\left (a+b \sinh ^{-1}(c x)\right )^2}{4 d^3 \left (c^2 x^2+1\right )^2}-\frac {b \text {Li}_2\left (-e^{2 \sinh ^{-1}(c x)}\right ) \left (a+b \sinh ^{-1}(c x)\right )}{d^3}+\frac {b \text {Li}_2\left (e^{2 \sinh ^{-1}(c x)}\right ) \left (a+b \sinh ^{-1}(c x)\right )}{d^3}-\frac {2 \tanh ^{-1}\left (e^{2 \sinh ^{-1}(c x)}\right ) \left (a+b \sinh ^{-1}(c x)\right )^2}{d^3}-\frac {b^2}{12 d^3 \left (c^2 x^2+1\right )}+\frac {2 b^2 \log \left (c^2 x^2+1\right )}{3 d^3}+\frac {b^2 \text {Li}_3\left (-e^{2 \sinh ^{-1}(c x)}\right )}{2 d^3}-\frac {b^2 \text {Li}_3\left (e^{2 \sinh ^{-1}(c x)}\right )}{2 d^3} \end {gather*}
Antiderivative was successfully verified.
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Rule 266
Rule 267
Rule 2320
Rule 2611
Rule 4267
Rule 5569
Rule 5787
Rule 5788
Rule 5799
Rule 5811
Rule 6724
Rubi steps
\begin {align*} \int \frac {\left (a+b \sinh ^{-1}(c x)\right )^2}{x \left (d+c^2 d x^2\right )^3} \, dx &=\frac {\left (a+b \sinh ^{-1}(c x)\right )^2}{4 d^3 \left (1+c^2 x^2\right )^2}-\frac {(b c) \int \frac {a+b \sinh ^{-1}(c x)}{\left (1+c^2 x^2\right )^{5/2}} \, dx}{2 d^3}+\frac {\int \frac {\left (a+b \sinh ^{-1}(c x)\right )^2}{x \left (d+c^2 d x^2\right )^2} \, dx}{d}\\ &=-\frac {b c x \left (a+b \sinh ^{-1}(c x)\right )}{6 d^3 \left (1+c^2 x^2\right )^{3/2}}+\frac {\left (a+b \sinh ^{-1}(c x)\right )^2}{4 d^3 \left (1+c^2 x^2\right )^2}+\frac {\left (a+b \sinh ^{-1}(c x)\right )^2}{2 d^3 \left (1+c^2 x^2\right )}-\frac {(b c) \int \frac {a+b \sinh ^{-1}(c x)}{\left (1+c^2 x^2\right )^{3/2}} \, dx}{3 d^3}-\frac {(b c) \int \frac {a+b \sinh ^{-1}(c x)}{\left (1+c^2 x^2\right )^{3/2}} \, dx}{d^3}+\frac {\left (b^2 c^2\right ) \int \frac {x}{\left (1+c^2 x^2\right )^2} \, dx}{6 d^3}+\frac {\int \frac {\left (a+b \sinh ^{-1}(c x)\right )^2}{x \left (d+c^2 d x^2\right )} \, dx}{d^2}\\ &=-\frac {b^2}{12 d^3 \left (1+c^2 x^2\right )}-\frac {b c x \left (a+b \sinh ^{-1}(c x)\right )}{6 d^3 \left (1+c^2 x^2\right )^{3/2}}-\frac {4 b c x \left (a+b \sinh ^{-1}(c x)\right )}{3 d^3 \sqrt {1+c^2 x^2}}+\frac {\left (a+b \sinh ^{-1}(c x)\right )^2}{4 d^3 \left (1+c^2 x^2\right )^2}+\frac {\left (a+b \sinh ^{-1}(c x)\right )^2}{2 d^3 \left (1+c^2 x^2\right )}+\frac {\text {Subst}\left (\int (a+b x)^2 \text {csch}(x) \text {sech}(x) \, dx,x,\sinh ^{-1}(c x)\right )}{d^3}+\frac {\left (b^2 c^2\right ) \int \frac {x}{1+c^2 x^2} \, dx}{3 d^3}+\frac {\left (b^2 c^2\right ) \int \frac {x}{1+c^2 x^2} \, dx}{d^3}\\ &=-\frac {b^2}{12 d^3 \left (1+c^2 x^2\right )}-\frac {b c x \left (a+b \sinh ^{-1}(c x)\right )}{6 d^3 \left (1+c^2 x^2\right )^{3/2}}-\frac {4 b c x \left (a+b \sinh ^{-1}(c x)\right )}{3 d^3 \sqrt {1+c^2 x^2}}+\frac {\left (a+b \sinh ^{-1}(c x)\right )^2}{4 d^3 \left (1+c^2 x^2\right )^2}+\frac {\left (a+b \sinh ^{-1}(c x)\right )^2}{2 d^3 \left (1+c^2 x^2\right )}+\frac {2 b^2 \log \left (1+c^2 x^2\right )}{3 d^3}+\frac {2 \text {Subst}\left (\int (a+b x)^2 \text {csch}(2 x) \, dx,x,\sinh ^{-1}(c x)\right )}{d^3}\\ &=-\frac {b^2}{12 d^3 \left (1+c^2 x^2\right )}-\frac {b c x \left (a+b \sinh ^{-1}(c x)\right )}{6 d^3 \left (1+c^2 x^2\right )^{3/2}}-\frac {4 b c x \left (a+b \sinh ^{-1}(c x)\right )}{3 d^3 \sqrt {1+c^2 x^2}}+\frac {\left (a+b \sinh ^{-1}(c x)\right )^2}{4 d^3 \left (1+c^2 x^2\right )^2}+\frac {\left (a+b \sinh ^{-1}(c x)\right )^2}{2 d^3 \left (1+c^2 x^2\right )}-\frac {2 \left (a+b \sinh ^{-1}(c x)\right )^2 \tanh ^{-1}\left (e^{2 \sinh ^{-1}(c x)}\right )}{d^3}+\frac {2 b^2 \log \left (1+c^2 x^2\right )}{3 d^3}-\frac {(2 b) \text {Subst}\left (\int (a+b x) \log \left (1-e^{2 x}\right ) \, dx,x,\sinh ^{-1}(c x)\right )}{d^3}+\frac {(2 b) \text {Subst}\left (\int (a+b x) \log \left (1+e^{2 x}\right ) \, dx,x,\sinh ^{-1}(c x)\right )}{d^3}\\ &=-\frac {b^2}{12 d^3 \left (1+c^2 x^2\right )}-\frac {b c x \left (a+b \sinh ^{-1}(c x)\right )}{6 d^3 \left (1+c^2 x^2\right )^{3/2}}-\frac {4 b c x \left (a+b \sinh ^{-1}(c x)\right )}{3 d^3 \sqrt {1+c^2 x^2}}+\frac {\left (a+b \sinh ^{-1}(c x)\right )^2}{4 d^3 \left (1+c^2 x^2\right )^2}+\frac {\left (a+b \sinh ^{-1}(c x)\right )^2}{2 d^3 \left (1+c^2 x^2\right )}-\frac {2 \left (a+b \sinh ^{-1}(c x)\right )^2 \tanh ^{-1}\left (e^{2 \sinh ^{-1}(c x)}\right )}{d^3}+\frac {2 b^2 \log \left (1+c^2 x^2\right )}{3 d^3}-\frac {b \left (a+b \sinh ^{-1}(c x)\right ) \text {Li}_2\left (-e^{2 \sinh ^{-1}(c x)}\right )}{d^3}+\frac {b \left (a+b \sinh ^{-1}(c x)\right ) \text {Li}_2\left (e^{2 \sinh ^{-1}(c x)}\right )}{d^3}+\frac {b^2 \text {Subst}\left (\int \text {Li}_2\left (-e^{2 x}\right ) \, dx,x,\sinh ^{-1}(c x)\right )}{d^3}-\frac {b^2 \text {Subst}\left (\int \text {Li}_2\left (e^{2 x}\right ) \, dx,x,\sinh ^{-1}(c x)\right )}{d^3}\\ &=-\frac {b^2}{12 d^3 \left (1+c^2 x^2\right )}-\frac {b c x \left (a+b \sinh ^{-1}(c x)\right )}{6 d^3 \left (1+c^2 x^2\right )^{3/2}}-\frac {4 b c x \left (a+b \sinh ^{-1}(c x)\right )}{3 d^3 \sqrt {1+c^2 x^2}}+\frac {\left (a+b \sinh ^{-1}(c x)\right )^2}{4 d^3 \left (1+c^2 x^2\right )^2}+\frac {\left (a+b \sinh ^{-1}(c x)\right )^2}{2 d^3 \left (1+c^2 x^2\right )}-\frac {2 \left (a+b \sinh ^{-1}(c x)\right )^2 \tanh ^{-1}\left (e^{2 \sinh ^{-1}(c x)}\right )}{d^3}+\frac {2 b^2 \log \left (1+c^2 x^2\right )}{3 d^3}-\frac {b \left (a+b \sinh ^{-1}(c x)\right ) \text {Li}_2\left (-e^{2 \sinh ^{-1}(c x)}\right )}{d^3}+\frac {b \left (a+b \sinh ^{-1}(c x)\right ) \text {Li}_2\left (e^{2 \sinh ^{-1}(c x)}\right )}{d^3}+\frac {b^2 \text {Subst}\left (\int \frac {\text {Li}_2(-x)}{x} \, dx,x,e^{2 \sinh ^{-1}(c x)}\right )}{2 d^3}-\frac {b^2 \text {Subst}\left (\int \frac {\text {Li}_2(x)}{x} \, dx,x,e^{2 \sinh ^{-1}(c x)}\right )}{2 d^3}\\ &=-\frac {b^2}{12 d^3 \left (1+c^2 x^2\right )}-\frac {b c x \left (a+b \sinh ^{-1}(c x)\right )}{6 d^3 \left (1+c^2 x^2\right )^{3/2}}-\frac {4 b c x \left (a+b \sinh ^{-1}(c x)\right )}{3 d^3 \sqrt {1+c^2 x^2}}+\frac {\left (a+b \sinh ^{-1}(c x)\right )^2}{4 d^3 \left (1+c^2 x^2\right )^2}+\frac {\left (a+b \sinh ^{-1}(c x)\right )^2}{2 d^3 \left (1+c^2 x^2\right )}-\frac {2 \left (a+b \sinh ^{-1}(c x)\right )^2 \tanh ^{-1}\left (e^{2 \sinh ^{-1}(c x)}\right )}{d^3}+\frac {2 b^2 \log \left (1+c^2 x^2\right )}{3 d^3}-\frac {b \left (a+b \sinh ^{-1}(c x)\right ) \text {Li}_2\left (-e^{2 \sinh ^{-1}(c x)}\right )}{d^3}+\frac {b \left (a+b \sinh ^{-1}(c x)\right ) \text {Li}_2\left (e^{2 \sinh ^{-1}(c x)}\right )}{d^3}+\frac {b^2 \text {Li}_3\left (-e^{2 \sinh ^{-1}(c x)}\right )}{2 d^3}-\frac {b^2 \text {Li}_3\left (e^{2 \sinh ^{-1}(c x)}\right )}{2 d^3}\\ \end {align*}
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Mathematica [C] Result contains complex when optimal does not.
time = 2.62, size = 560, normalized size = 2.04 \begin {gather*} \frac {\frac {6 a^2}{\left (1+c^2 x^2\right )^2}+\frac {12 a^2}{1+c^2 x^2}+24 a^2 \log (c x)-12 a^2 \log \left (1+c^2 x^2\right )+a b \left (-\frac {15 \left (\sqrt {1+c^2 x^2}-i \sinh ^{-1}(c x)\right )}{i+c x}-\frac {15 \left (\sqrt {1+c^2 x^2}+i \sinh ^{-1}(c x)\right )}{-i+c x}-24 \sinh ^{-1}(c x)^2-\frac {(-2 i+c x) \sqrt {1+c^2 x^2}+3 \sinh ^{-1}(c x)}{(-i+c x)^2}-\frac {(2 i+c x) \sqrt {1+c^2 x^2}+3 \sinh ^{-1}(c x)}{(i+c x)^2}+48 \sinh ^{-1}(c x) \log \left (1-e^{2 \sinh ^{-1}(c x)}\right )+12 \left (\sinh ^{-1}(c x) \left (\sinh ^{-1}(c x)-4 \log \left (1+i e^{\sinh ^{-1}(c x)}\right )\right )-4 \text {PolyLog}\left (2,-i e^{\sinh ^{-1}(c x)}\right )\right )+12 \left (\sinh ^{-1}(c x) \left (\sinh ^{-1}(c x)-4 \log \left (1-i e^{\sinh ^{-1}(c x)}\right )\right )-4 \text {PolyLog}\left (2,i e^{\sinh ^{-1}(c x)}\right )\right )+24 \text {PolyLog}\left (2,e^{2 \sinh ^{-1}(c x)}\right )\right )+b^2 \left (i \pi ^3-\frac {2}{1+c^2 x^2}-\frac {4 c x \sinh ^{-1}(c x)}{\left (1+c^2 x^2\right )^{3/2}}-\frac {32 c x \sinh ^{-1}(c x)}{\sqrt {1+c^2 x^2}}+\frac {6 \sinh ^{-1}(c x)^2}{\left (1+c^2 x^2\right )^2}+\frac {12 \sinh ^{-1}(c x)^2}{1+c^2 x^2}-16 \sinh ^{-1}(c x)^3-24 \sinh ^{-1}(c x)^2 \log \left (1+e^{-2 \sinh ^{-1}(c x)}\right )+24 \sinh ^{-1}(c x)^2 \log \left (1-e^{2 \sinh ^{-1}(c x)}\right )+16 \log \left (1+c^2 x^2\right )+24 \sinh ^{-1}(c x) \text {PolyLog}\left (2,-e^{-2 \sinh ^{-1}(c x)}\right )+24 \sinh ^{-1}(c x) \text {PolyLog}\left (2,e^{2 \sinh ^{-1}(c x)}\right )+12 \text {PolyLog}\left (3,-e^{-2 \sinh ^{-1}(c x)}\right )-12 \text {PolyLog}\left (3,e^{2 \sinh ^{-1}(c x)}\right )\right )}{24 d^3} \end {gather*}
Antiderivative was successfully verified.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(1128\) vs.
\(2(300)=600\).
time = 5.57, size = 1129, normalized size = 4.11
method | result | size |
derivativedivides | \(\text {Expression too large to display}\) | \(1129\) |
default | \(\text {Expression too large to display}\) | \(1129\) |
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^{2}}{c^{6} x^{7} + 3 c^{4} x^{5} + 3 c^{2} x^{3} + x}\, dx + \int \frac {b^{2} \operatorname {asinh}^{2}{\left (c x \right )}}{c^{6} x^{7} + 3 c^{4} x^{5} + 3 c^{2} x^{3} + x}\, dx + \int \frac {2 a b \operatorname {asinh}{\left (c x \right )}}{c^{6} x^{7} + 3 c^{4} x^{5} + 3 c^{2} x^{3} + x}\, dx}{d^{3}} \end {gather*}
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 \begin {gather*} \text {could not integrate} \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 {asinh}\left (c\,x\right )\right )}^2}{x\,{\left (d\,c^2\,x^2+d\right )}^3} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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