Optimal. Leaf size=400 \[ \frac {b^2 x \left (1+c^2 x^2\right )}{4 c^4 d \sqrt {d+c^2 d x^2}}-\frac {b^2 \sqrt {1+c^2 x^2} \sinh ^{-1}(c x)}{4 c^5 d \sqrt {d+c^2 d x^2}}-\frac {b x^2 \sqrt {1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )}{2 c^3 d \sqrt {d+c^2 d x^2}}-\frac {x^3 \left (a+b \sinh ^{-1}(c x)\right )^2}{c^2 d \sqrt {d+c^2 d x^2}}+\frac {\sqrt {1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )^2}{c^5 d \sqrt {d+c^2 d x^2}}+\frac {3 x \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2}{2 c^4 d^2}-\frac {\sqrt {1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )^3}{2 b c^5 d \sqrt {d+c^2 d x^2}}-\frac {2 b \sqrt {1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right ) \log \left (1+e^{2 \sinh ^{-1}(c x)}\right )}{c^5 d \sqrt {d+c^2 d x^2}}-\frac {b^2 \sqrt {1+c^2 x^2} \text {PolyLog}\left (2,-e^{2 \sinh ^{-1}(c x)}\right )}{c^5 d \sqrt {d+c^2 d x^2}} \]
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Rubi [A]
time = 0.42, antiderivative size = 400, normalized size of antiderivative = 1.00, number of steps
used = 14, number of rules used = 11, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.393, Rules used = {5810, 5812,
5783, 5776, 327, 221, 5797, 3799, 2221, 2317, 2438} \begin {gather*} -\frac {x^3 \left (a+b \sinh ^{-1}(c x)\right )^2}{c^2 d \sqrt {c^2 d x^2+d}}-\frac {\sqrt {c^2 x^2+1} \left (a+b \sinh ^{-1}(c x)\right )^3}{2 b c^5 d \sqrt {c^2 d x^2+d}}+\frac {\sqrt {c^2 x^2+1} \left (a+b \sinh ^{-1}(c x)\right )^2}{c^5 d \sqrt {c^2 d x^2+d}}-\frac {2 b \sqrt {c^2 x^2+1} \log \left (e^{2 \sinh ^{-1}(c x)}+1\right ) \left (a+b \sinh ^{-1}(c x)\right )}{c^5 d \sqrt {c^2 d x^2+d}}+\frac {3 x \sqrt {c^2 d x^2+d} \left (a+b \sinh ^{-1}(c x)\right )^2}{2 c^4 d^2}-\frac {b x^2 \sqrt {c^2 x^2+1} \left (a+b \sinh ^{-1}(c x)\right )}{2 c^3 d \sqrt {c^2 d x^2+d}}-\frac {b^2 \sqrt {c^2 x^2+1} \text {Li}_2\left (-e^{2 \sinh ^{-1}(c x)}\right )}{c^5 d \sqrt {c^2 d x^2+d}}-\frac {b^2 \sqrt {c^2 x^2+1} \sinh ^{-1}(c x)}{4 c^5 d \sqrt {c^2 d x^2+d}}+\frac {b^2 x \left (c^2 x^2+1\right )}{4 c^4 d \sqrt {c^2 d x^2+d}} \end {gather*}
Antiderivative was successfully verified.
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Rule 221
Rule 327
Rule 2221
Rule 2317
Rule 2438
Rule 3799
Rule 5776
Rule 5783
Rule 5797
Rule 5810
Rule 5812
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 )^{3/2}} \, dx &=-\frac {x^3 \left (a+b \sinh ^{-1}(c x)\right )^2}{c^2 d \sqrt {d+c^2 d x^2}}+\frac {3 \int \frac {x^2 \left (a+b \sinh ^{-1}(c x)\right )^2}{\sqrt {d+c^2 d x^2}} \, dx}{c^2 d}+\frac {\left (2 b \sqrt {1+c^2 x^2}\right ) \int \frac {x^3 \left (a+b \sinh ^{-1}(c x)\right )}{1+c^2 x^2} \, dx}{c d \sqrt {d+c^2 d x^2}}\\ &=\frac {b x^2 \sqrt {1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )}{c^3 d \sqrt {d+c^2 d x^2}}-\frac {x^3 \left (a+b \sinh ^{-1}(c x)\right )^2}{c^2 d \sqrt {d+c^2 d x^2}}+\frac {3 x \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2}{2 c^4 d^2}-\frac {3 \int \frac {\left (a+b \sinh ^{-1}(c x)\right )^2}{\sqrt {d+c^2 d x^2}} \, dx}{2 c^4 d}-\frac {\left (2 b \sqrt {1+c^2 x^2}\right ) \int \frac {x \left (a+b \sinh ^{-1}(c x)\right )}{1+c^2 x^2} \, dx}{c^3 d \sqrt {d+c^2 d x^2}}-\frac {\left (3 b \sqrt {1+c^2 x^2}\right ) \int x \left (a+b \sinh ^{-1}(c x)\right ) \, dx}{c^3 d \sqrt {d+c^2 d x^2}}-\frac {\left (b^2 \sqrt {1+c^2 x^2}\right ) \int \frac {x^2}{\sqrt {1+c^2 x^2}} \, dx}{c^2 d \sqrt {d+c^2 d x^2}}\\ &=-\frac {b^2 x \left (1+c^2 x^2\right )}{2 c^4 d \sqrt {d+c^2 d x^2}}-\frac {b x^2 \sqrt {1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )}{2 c^3 d \sqrt {d+c^2 d x^2}}-\frac {x^3 \left (a+b \sinh ^{-1}(c x)\right )^2}{c^2 d \sqrt {d+c^2 d x^2}}+\frac {3 x \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2}{2 c^4 d^2}-\frac {\left (2 b \sqrt {1+c^2 x^2}\right ) \text {Subst}\left (\int (a+b x) \tanh (x) \, dx,x,\sinh ^{-1}(c x)\right )}{c^5 d \sqrt {d+c^2 d x^2}}-\frac {\left (3 \sqrt {1+c^2 x^2}\right ) \int \frac {\left (a+b \sinh ^{-1}(c x)\right )^2}{\sqrt {1+c^2 x^2}} \, dx}{2 c^4 d \sqrt {d+c^2 d x^2}}+\frac {\left (b^2 \sqrt {1+c^2 x^2}\right ) \int \frac {1}{\sqrt {1+c^2 x^2}} \, dx}{2 c^4 d \sqrt {d+c^2 d x^2}}+\frac {\left (3 b^2 \sqrt {1+c^2 x^2}\right ) \int \frac {x^2}{\sqrt {1+c^2 x^2}} \, dx}{2 c^2 d \sqrt {d+c^2 d x^2}}\\ &=\frac {b^2 x \left (1+c^2 x^2\right )}{4 c^4 d \sqrt {d+c^2 d x^2}}+\frac {b^2 \sqrt {1+c^2 x^2} \sinh ^{-1}(c x)}{2 c^5 d \sqrt {d+c^2 d x^2}}-\frac {b x^2 \sqrt {1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )}{2 c^3 d \sqrt {d+c^2 d x^2}}-\frac {x^3 \left (a+b \sinh ^{-1}(c x)\right )^2}{c^2 d \sqrt {d+c^2 d x^2}}+\frac {\sqrt {1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )^2}{c^5 d \sqrt {d+c^2 d x^2}}+\frac {3 x \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2}{2 c^4 d^2}-\frac {\sqrt {1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )^3}{2 b c^5 d \sqrt {d+c^2 d x^2}}-\frac {\left (4 b \sqrt {1+c^2 x^2}\right ) \text {Subst}\left (\int \frac {e^{2 x} (a+b x)}{1+e^{2 x}} \, dx,x,\sinh ^{-1}(c x)\right )}{c^5 d \sqrt {d+c^2 d x^2}}-\frac {\left (3 b^2 \sqrt {1+c^2 x^2}\right ) \int \frac {1}{\sqrt {1+c^2 x^2}} \, dx}{4 c^4 d \sqrt {d+c^2 d x^2}}\\ &=\frac {b^2 x \left (1+c^2 x^2\right )}{4 c^4 d \sqrt {d+c^2 d x^2}}-\frac {b^2 \sqrt {1+c^2 x^2} \sinh ^{-1}(c x)}{4 c^5 d \sqrt {d+c^2 d x^2}}-\frac {b x^2 \sqrt {1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )}{2 c^3 d \sqrt {d+c^2 d x^2}}-\frac {x^3 \left (a+b \sinh ^{-1}(c x)\right )^2}{c^2 d \sqrt {d+c^2 d x^2}}+\frac {\sqrt {1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )^2}{c^5 d \sqrt {d+c^2 d x^2}}+\frac {3 x \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2}{2 c^4 d^2}-\frac {\sqrt {1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )^3}{2 b c^5 d \sqrt {d+c^2 d x^2}}-\frac {2 b \sqrt {1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right ) \log \left (1+e^{2 \sinh ^{-1}(c x)}\right )}{c^5 d \sqrt {d+c^2 d x^2}}+\frac {\left (2 b^2 \sqrt {1+c^2 x^2}\right ) \text {Subst}\left (\int \log \left (1+e^{2 x}\right ) \, dx,x,\sinh ^{-1}(c x)\right )}{c^5 d \sqrt {d+c^2 d x^2}}\\ &=\frac {b^2 x \left (1+c^2 x^2\right )}{4 c^4 d \sqrt {d+c^2 d x^2}}-\frac {b^2 \sqrt {1+c^2 x^2} \sinh ^{-1}(c x)}{4 c^5 d \sqrt {d+c^2 d x^2}}-\frac {b x^2 \sqrt {1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )}{2 c^3 d \sqrt {d+c^2 d x^2}}-\frac {x^3 \left (a+b \sinh ^{-1}(c x)\right )^2}{c^2 d \sqrt {d+c^2 d x^2}}+\frac {\sqrt {1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )^2}{c^5 d \sqrt {d+c^2 d x^2}}+\frac {3 x \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2}{2 c^4 d^2}-\frac {\sqrt {1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )^3}{2 b c^5 d \sqrt {d+c^2 d x^2}}-\frac {2 b \sqrt {1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right ) \log \left (1+e^{2 \sinh ^{-1}(c x)}\right )}{c^5 d \sqrt {d+c^2 d x^2}}+\frac {\left (b^2 \sqrt {1+c^2 x^2}\right ) \text {Subst}\left (\int \frac {\log (1+x)}{x} \, dx,x,e^{2 \sinh ^{-1}(c x)}\right )}{c^5 d \sqrt {d+c^2 d x^2}}\\ &=\frac {b^2 x \left (1+c^2 x^2\right )}{4 c^4 d \sqrt {d+c^2 d x^2}}-\frac {b^2 \sqrt {1+c^2 x^2} \sinh ^{-1}(c x)}{4 c^5 d \sqrt {d+c^2 d x^2}}-\frac {b x^2 \sqrt {1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )}{2 c^3 d \sqrt {d+c^2 d x^2}}-\frac {x^3 \left (a+b \sinh ^{-1}(c x)\right )^2}{c^2 d \sqrt {d+c^2 d x^2}}+\frac {\sqrt {1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )^2}{c^5 d \sqrt {d+c^2 d x^2}}+\frac {3 x \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2}{2 c^4 d^2}-\frac {\sqrt {1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )^3}{2 b c^5 d \sqrt {d+c^2 d x^2}}-\frac {2 b \sqrt {1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right ) \log \left (1+e^{2 \sinh ^{-1}(c x)}\right )}{c^5 d \sqrt {d+c^2 d x^2}}-\frac {b^2 \sqrt {1+c^2 x^2} \text {Li}_2\left (-e^{2 \sinh ^{-1}(c x)}\right )}{c^5 d \sqrt {d+c^2 d x^2}}\\ \end {align*}
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Mathematica [A]
time = 1.33, size = 288, normalized size = 0.72 \begin {gather*} \frac {4 a^2 c \sqrt {d} x \left (3+c^2 x^2\right )-12 a^2 \sqrt {d+c^2 d x^2} \log \left (c d x+\sqrt {d} \sqrt {d+c^2 d x^2}\right )+b^2 \sqrt {d} \left (8 c x \sinh ^{-1}(c x)^2+8 \sqrt {1+c^2 x^2} \text {PolyLog}\left (2,-e^{-2 \sinh ^{-1}(c x)}\right )+\sqrt {1+c^2 x^2} \left (-4 \sinh ^{-1}(c x)^3-2 \sinh ^{-1}(c x) \left (\cosh \left (2 \sinh ^{-1}(c x)\right )+8 \log \left (1+e^{-2 \sinh ^{-1}(c x)}\right )\right )+2 \sinh ^{-1}(c x)^2 \left (-4+\sinh \left (2 \sinh ^{-1}(c x)\right )\right )+\sinh \left (2 \sinh ^{-1}(c x)\right )\right )\right )+2 a b \sqrt {d} \left (8 c x \sinh ^{-1}(c x)-\sqrt {1+c^2 x^2} \left (6 \sinh ^{-1}(c x)^2+\cosh \left (2 \sinh ^{-1}(c x)\right )+4 \log \left (1+c^2 x^2\right )-2 \sinh ^{-1}(c x) \sinh \left (2 \sinh ^{-1}(c x)\right )\right )\right )}{8 c^5 d^{3/2} \sqrt {d+c^2 d x^2}} \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. \(815\) vs.
\(2(378)=756\).
time = 4.65, size = 816, normalized size = 2.04
method | result | size |
default | \(\frac {a^{2} x^{3}}{2 c^{2} d \sqrt {c^{2} d \,x^{2}+d}}+\frac {3 a^{2} x}{2 c^{4} d \sqrt {c^{2} d \,x^{2}+d}}-\frac {3 a^{2} \ln \left (\frac {x \,c^{2} d}{\sqrt {c^{2} d}}+\sqrt {c^{2} d \,x^{2}+d}\right )}{2 c^{4} d \sqrt {c^{2} d}}-\frac {b^{2} \sqrt {d \left (c^{2} x^{2}+1\right )}\, \arcsinh \left (c x \right ) x^{2}}{2 c^{3} d^{2} \sqrt {c^{2} x^{2}+1}}-\frac {b^{2} \sqrt {d \left (c^{2} x^{2}+1\right )}\, \polylog \left (2, -\left (c x +\sqrt {c^{2} x^{2}+1}\right )^{2}\right )}{\sqrt {c^{2} x^{2}+1}\, c^{5} d^{2}}+\frac {b^{2} \sqrt {d \left (c^{2} x^{2}+1\right )}\, x^{3}}{4 c^{2} d^{2} \left (c^{2} x^{2}+1\right )}+\frac {b^{2} \sqrt {d \left (c^{2} x^{2}+1\right )}\, x}{4 c^{4} d^{2} \left (c^{2} x^{2}+1\right )}-\frac {b^{2} \sqrt {d \left (c^{2} x^{2}+1\right )}\, \arcsinh \left (c x \right )^{3}}{2 \sqrt {c^{2} x^{2}+1}\, c^{5} d^{2}}+\frac {b^{2} \sqrt {d \left (c^{2} x^{2}+1\right )}\, \arcsinh \left (c x \right )^{2} x^{3}}{2 c^{2} d^{2} \left (c^{2} x^{2}+1\right )}+\frac {3 b^{2} \sqrt {d \left (c^{2} x^{2}+1\right )}\, \arcsinh \left (c x \right )^{2} x}{2 c^{4} d^{2} \left (c^{2} x^{2}+1\right )}+\frac {b^{2} \sqrt {d \left (c^{2} x^{2}+1\right )}\, \arcsinh \left (c x \right )^{2}}{c^{5} d^{2} \sqrt {c^{2} x^{2}+1}}-\frac {b^{2} \sqrt {d \left (c^{2} x^{2}+1\right )}\, \arcsinh \left (c x \right )}{4 c^{5} d^{2} \sqrt {c^{2} x^{2}+1}}-\frac {2 b^{2} \sqrt {d \left (c^{2} x^{2}+1\right )}\, \arcsinh \left (c x \right ) \ln \left (1+\left (c x +\sqrt {c^{2} x^{2}+1}\right )^{2}\right )}{\sqrt {c^{2} x^{2}+1}\, c^{5} d^{2}}-\frac {3 a b \sqrt {d \left (c^{2} x^{2}+1\right )}\, \arcsinh \left (c x \right )^{2}}{2 \sqrt {c^{2} x^{2}+1}\, c^{5} d^{2}}+\frac {a b \sqrt {d \left (c^{2} x^{2}+1\right )}\, \arcsinh \left (c x \right ) x^{3}}{c^{2} d^{2} \left (c^{2} x^{2}+1\right )}-\frac {a b \sqrt {d \left (c^{2} x^{2}+1\right )}\, x^{2}}{2 c^{3} d^{2} \sqrt {c^{2} x^{2}+1}}+\frac {3 a b \sqrt {d \left (c^{2} x^{2}+1\right )}\, \arcsinh \left (c x \right ) x}{c^{4} d^{2} \left (c^{2} x^{2}+1\right )}+\frac {2 a b \sqrt {d \left (c^{2} x^{2}+1\right )}\, \arcsinh \left (c x \right )}{c^{5} d^{2} \sqrt {c^{2} x^{2}+1}}-\frac {a b \sqrt {d \left (c^{2} x^{2}+1\right )}}{4 c^{5} d^{2} \sqrt {c^{2} x^{2}+1}}-\frac {2 a b \sqrt {d \left (c^{2} x^{2}+1\right )}\, \ln \left (1+\left (c x +\sqrt {c^{2} x^{2}+1}\right )^{2}\right )}{\sqrt {c^{2} x^{2}+1}\, c^{5} d^{2}}\) | \(816\) |
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*} \int \frac {x^{4} \left (a + b \operatorname {asinh}{\left (c x \right )}\right )^{2}}{\left (d \left (c^{2} x^{2} + 1\right )\right )^{\frac {3}{2}}}\, dx \end {gather*}
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 \begin {gather*} \text {Exception raised: TypeError} \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 {x^4\,{\left (a+b\,\mathrm {asinh}\left (c\,x\right )\right )}^2}{{\left (d\,c^2\,x^2+d\right )}^{3/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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