Optimal. Leaf size=204 \[ \frac {b^2 x \left (1+c^2 x^2\right )}{4 c^2 \sqrt {d+c^2 d x^2}}-\frac {b^2 \sqrt {1+c^2 x^2} \sinh ^{-1}(c x)}{4 c^3 \sqrt {d+c^2 d x^2}}-\frac {b x^2 \sqrt {1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )}{2 c \sqrt {d+c^2 d x^2}}+\frac {x \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2}{2 c^2 d}-\frac {\sqrt {1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )^3}{6 b c^3 \sqrt {d+c^2 d x^2}} \]
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Rubi [A]
time = 0.15, antiderivative size = 204, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 5, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.179, Rules used = {5812, 5783,
5776, 327, 221} \begin {gather*} \frac {x \sqrt {c^2 d x^2+d} \left (a+b \sinh ^{-1}(c x)\right )^2}{2 c^2 d}-\frac {b x^2 \sqrt {c^2 x^2+1} \left (a+b \sinh ^{-1}(c x)\right )}{2 c \sqrt {c^2 d x^2+d}}-\frac {\sqrt {c^2 x^2+1} \left (a+b \sinh ^{-1}(c x)\right )^3}{6 b c^3 \sqrt {c^2 d x^2+d}}+\frac {b^2 x \left (c^2 x^2+1\right )}{4 c^2 \sqrt {c^2 d x^2+d}}-\frac {b^2 \sqrt {c^2 x^2+1} \sinh ^{-1}(c x)}{4 c^3 \sqrt {c^2 d x^2+d}} \end {gather*}
Antiderivative was successfully verified.
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Rule 221
Rule 327
Rule 5776
Rule 5783
Rule 5812
Rubi steps
\begin {align*} \int \frac {x^2 \left (a+b \sinh ^{-1}(c x)\right )^2}{\sqrt {d+c^2 d x^2}} \, dx &=\frac {x \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2}{2 c^2 d}-\frac {\int \frac {\left (a+b \sinh ^{-1}(c x)\right )^2}{\sqrt {d+c^2 d x^2}} \, dx}{2 c^2}-\frac {\left (b \sqrt {1+c^2 x^2}\right ) \int x \left (a+b \sinh ^{-1}(c x)\right ) \, dx}{c \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 \sqrt {d+c^2 d x^2}}+\frac {x \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2}{2 c^2 d}+\frac {\left (b^2 \sqrt {1+c^2 x^2}\right ) \int \frac {x^2}{\sqrt {1+c^2 x^2}} \, dx}{2 \sqrt {d+c^2 d x^2}}-\frac {\sqrt {1+c^2 x^2} \int \frac {\left (a+b \sinh ^{-1}(c x)\right )^2}{\sqrt {1+c^2 x^2}} \, dx}{2 c^2 \sqrt {d+c^2 d x^2}}\\ &=\frac {b^2 x \left (1+c^2 x^2\right )}{4 c^2 \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 \sqrt {d+c^2 d x^2}}+\frac {x \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2}{2 c^2 d}-\frac {\sqrt {1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )^3}{6 b c^3 \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}{4 c^2 \sqrt {d+c^2 d x^2}}\\ &=\frac {b^2 x \left (1+c^2 x^2\right )}{4 c^2 \sqrt {d+c^2 d x^2}}-\frac {b^2 \sqrt {1+c^2 x^2} \sinh ^{-1}(c x)}{4 c^3 \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 \sqrt {d+c^2 d x^2}}+\frac {x \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2}{2 c^2 d}-\frac {\sqrt {1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )^3}{6 b c^3 \sqrt {d+c^2 d x^2}}\\ \end {align*}
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Mathematica [A]
time = 0.59, size = 198, normalized size = 0.97 \begin {gather*} \frac {12 a^2 c x \left (d+c^2 d x^2\right )-12 a^2 \sqrt {d} \sqrt {d+c^2 d x^2} \log \left (c d x+\sqrt {d} \sqrt {d+c^2 d x^2}\right )-6 a b d \sqrt {1+c^2 x^2} \left (\cosh \left (2 \sinh ^{-1}(c x)\right )+2 \sinh ^{-1}(c x) \left (\sinh ^{-1}(c x)-\sinh \left (2 \sinh ^{-1}(c x)\right )\right )\right )-b^2 d \sqrt {1+c^2 x^2} \left (4 \sinh ^{-1}(c x)^3+6 \sinh ^{-1}(c x) \cosh \left (2 \sinh ^{-1}(c x)\right )-3 \left (1+2 \sinh ^{-1}(c x)^2\right ) \sinh \left (2 \sinh ^{-1}(c x)\right )\right )}{24 c^3 d \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. \(505\) vs.
\(2(178)=356\).
time = 4.06, size = 506, normalized size = 2.48
method | result | size |
default | \(\frac {a^{2} x \sqrt {c^{2} d \,x^{2}+d}}{2 c^{2} d}-\frac {a^{2} \ln \left (\frac {x \,c^{2} d}{\sqrt {c^{2} d}}+\sqrt {c^{2} d \,x^{2}+d}\right )}{2 c^{2} \sqrt {c^{2} d}}+b^{2} \left (-\frac {\sqrt {d \left (c^{2} x^{2}+1\right )}\, \arcsinh \left (c x \right )^{3}}{6 \sqrt {c^{2} x^{2}+1}\, c^{3} d}+\frac {\sqrt {d \left (c^{2} x^{2}+1\right )}\, \left (2 c^{3} x^{3}+2 c^{2} x^{2} \sqrt {c^{2} x^{2}+1}+2 c x +\sqrt {c^{2} x^{2}+1}\right ) \left (2 \arcsinh \left (c x \right )^{2}-2 \arcsinh \left (c x \right )+1\right )}{16 c^{3} d \left (c^{2} x^{2}+1\right )}+\frac {\sqrt {d \left (c^{2} x^{2}+1\right )}\, \left (2 c^{3} x^{3}-2 c^{2} x^{2} \sqrt {c^{2} x^{2}+1}+2 c x -\sqrt {c^{2} x^{2}+1}\right ) \left (2 \arcsinh \left (c x \right )^{2}+2 \arcsinh \left (c x \right )+1\right )}{16 c^{3} d \left (c^{2} x^{2}+1\right )}\right )+2 a b \left (-\frac {\sqrt {d \left (c^{2} x^{2}+1\right )}\, \arcsinh \left (c x \right )^{2}}{4 \sqrt {c^{2} x^{2}+1}\, c^{3} d}+\frac {\sqrt {d \left (c^{2} x^{2}+1\right )}\, \left (2 c^{3} x^{3}+2 c^{2} x^{2} \sqrt {c^{2} x^{2}+1}+2 c x +\sqrt {c^{2} x^{2}+1}\right ) \left (-1+2 \arcsinh \left (c x \right )\right )}{16 c^{3} d \left (c^{2} x^{2}+1\right )}+\frac {\sqrt {d \left (c^{2} x^{2}+1\right )}\, \left (2 c^{3} x^{3}-2 c^{2} x^{2} \sqrt {c^{2} x^{2}+1}+2 c x -\sqrt {c^{2} x^{2}+1}\right ) \left (1+2 \arcsinh \left (c x \right )\right )}{16 c^{3} d \left (c^{2} x^{2}+1\right )}\right )\) | \(506\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: RuntimeError} \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^{2} \left (a + b \operatorname {asinh}{\left (c x \right )}\right )^{2}}{\sqrt {d \left (c^{2} x^{2} + 1\right )}}\, dx \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 {x^2\,{\left (a+b\,\mathrm {asinh}\left (c\,x\right )\right )}^2}{\sqrt {d\,c^2\,x^2+d}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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