Optimal. Leaf size=145 \[ \frac {b x \left (a+b \sinh ^{-1}(c x)\right )}{3 c d^3 \sqrt {c^2 x^2+1}}+\frac {b x \left (a+b \sinh ^{-1}(c x)\right )}{6 c d^3 \left (c^2 x^2+1\right )^{3/2}}-\frac {\left (a+b \sinh ^{-1}(c x)\right )^2}{4 c^2 d^3 \left (c^2 x^2+1\right )^2}+\frac {b^2}{12 c^2 d^3 \left (c^2 x^2+1\right )}-\frac {b^2 \log \left (c^2 x^2+1\right )}{6 c^2 d^3} \]
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Rubi [A] time = 0.15, antiderivative size = 145, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.208, Rules used = {5717, 5690, 5687, 260, 261} \[ \frac {b x \left (a+b \sinh ^{-1}(c x)\right )}{3 c d^3 \sqrt {c^2 x^2+1}}+\frac {b x \left (a+b \sinh ^{-1}(c x)\right )}{6 c d^3 \left (c^2 x^2+1\right )^{3/2}}-\frac {\left (a+b \sinh ^{-1}(c x)\right )^2}{4 c^2 d^3 \left (c^2 x^2+1\right )^2}+\frac {b^2}{12 c^2 d^3 \left (c^2 x^2+1\right )}-\frac {b^2 \log \left (c^2 x^2+1\right )}{6 c^2 d^3} \]
Antiderivative was successfully verified.
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Rule 260
Rule 261
Rule 5687
Rule 5690
Rule 5717
Rubi steps
\begin {align*} \int \frac {x \left (a+b \sinh ^{-1}(c x)\right )^2}{\left (d+c^2 d x^2\right )^3} \, dx &=-\frac {\left (a+b \sinh ^{-1}(c x)\right )^2}{4 c^2 d^3 \left (1+c^2 x^2\right )^2}+\frac {b \int \frac {a+b \sinh ^{-1}(c x)}{\left (1+c^2 x^2\right )^{5/2}} \, dx}{2 c d^3}\\ &=\frac {b x \left (a+b \sinh ^{-1}(c x)\right )}{6 c d^3 \left (1+c^2 x^2\right )^{3/2}}-\frac {\left (a+b \sinh ^{-1}(c x)\right )^2}{4 c^2 d^3 \left (1+c^2 x^2\right )^2}-\frac {b^2 \int \frac {x}{\left (1+c^2 x^2\right )^2} \, dx}{6 d^3}+\frac {b \int \frac {a+b \sinh ^{-1}(c x)}{\left (1+c^2 x^2\right )^{3/2}} \, dx}{3 c d^3}\\ &=\frac {b^2}{12 c^2 d^3 \left (1+c^2 x^2\right )}+\frac {b x \left (a+b \sinh ^{-1}(c x)\right )}{6 c d^3 \left (1+c^2 x^2\right )^{3/2}}+\frac {b x \left (a+b \sinh ^{-1}(c x)\right )}{3 c d^3 \sqrt {1+c^2 x^2}}-\frac {\left (a+b \sinh ^{-1}(c x)\right )^2}{4 c^2 d^3 \left (1+c^2 x^2\right )^2}-\frac {b^2 \int \frac {x}{1+c^2 x^2} \, dx}{3 d^3}\\ &=\frac {b^2}{12 c^2 d^3 \left (1+c^2 x^2\right )}+\frac {b x \left (a+b \sinh ^{-1}(c x)\right )}{6 c d^3 \left (1+c^2 x^2\right )^{3/2}}+\frac {b x \left (a+b \sinh ^{-1}(c x)\right )}{3 c d^3 \sqrt {1+c^2 x^2}}-\frac {\left (a+b \sinh ^{-1}(c x)\right )^2}{4 c^2 d^3 \left (1+c^2 x^2\right )^2}-\frac {b^2 \log \left (1+c^2 x^2\right )}{6 c^2 d^3}\\ \end {align*}
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Mathematica [A] time = 0.22, size = 152, normalized size = 1.05 \[ \frac {-3 a^2+6 a b c x \sqrt {c^2 x^2+1}+2 b \sinh ^{-1}(c x) \left (b c x \sqrt {c^2 x^2+1} \left (2 c^2 x^2+3\right )-3 a\right )+4 a b c^3 x^3 \sqrt {c^2 x^2+1}+b^2 c^2 x^2-3 b^2 \sinh ^{-1}(c x)^2+b^2-2 \left (b c^2 x^2+b\right )^2 \log \left (c^2 x^2+1\right )}{12 d^3 \left (c^3 x^2+c\right )^2} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.58, size = 273, normalized size = 1.88 \[ \frac {4 \, a b c^{4} x^{4} + {\left (8 \, a b + b^{2}\right )} c^{2} x^{2} - 3 \, b^{2} \log \left (c x + \sqrt {c^{2} x^{2} + 1}\right )^{2} - 3 \, a^{2} + 4 \, a b + b^{2} - 2 \, {\left (b^{2} c^{4} x^{4} + 2 \, b^{2} c^{2} x^{2} + b^{2}\right )} \log \left (c^{2} x^{2} + 1\right ) + 2 \, {\left (3 \, a b c^{4} x^{4} + 6 \, a b c^{2} x^{2} + {\left (2 \, b^{2} c^{3} x^{3} + 3 \, b^{2} c x\right )} \sqrt {c^{2} x^{2} + 1}\right )} \log \left (c x + \sqrt {c^{2} x^{2} + 1}\right ) + 6 \, {\left (a b c^{4} x^{4} + 2 \, a b c^{2} x^{2} + a b\right )} \log \left (-c x + \sqrt {c^{2} x^{2} + 1}\right ) + 2 \, {\left (2 \, a b c^{3} x^{3} + 3 \, a b c x\right )} \sqrt {c^{2} x^{2} + 1}}{12 \, {\left (c^{6} d^{3} x^{4} + 2 \, c^{4} d^{3} x^{2} + c^{2} d^{3}\right )}} \]
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 \[ \int \frac {{\left (b \operatorname {arsinh}\left (c x\right ) + a\right )}^{2} x}{{\left (c^{2} d x^{2} + d\right )}^{3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.15, size = 432, normalized size = 2.98 \[ -\frac {a^{2}}{4 c^{2} d^{3} \left (c^{2} x^{2}+1\right )^{2}}+\frac {2 b^{2} \arcsinh \left (c x \right )}{3 c^{2} d^{3}}+\frac {c \,b^{2} \arcsinh \left (c x \right ) \sqrt {c^{2} x^{2}+1}\, x^{3}}{3 d^{3} \left (c^{4} x^{4}+2 c^{2} x^{2}+1\right )}-\frac {c^{2} b^{2} \arcsinh \left (c x \right ) x^{4}}{3 d^{3} \left (c^{4} x^{4}+2 c^{2} x^{2}+1\right )}+\frac {b^{2} \arcsinh \left (c x \right ) \sqrt {c^{2} x^{2}+1}\, x}{2 c \,d^{3} \left (c^{4} x^{4}+2 c^{2} x^{2}+1\right )}-\frac {2 b^{2} \arcsinh \left (c x \right ) x^{2}}{3 d^{3} \left (c^{4} x^{4}+2 c^{2} x^{2}+1\right )}-\frac {b^{2} \arcsinh \left (c x \right )^{2}}{4 c^{2} d^{3} \left (c^{4} x^{4}+2 c^{2} x^{2}+1\right )}+\frac {b^{2} x^{2}}{12 d^{3} \left (c^{4} x^{4}+2 c^{2} x^{2}+1\right )}-\frac {b^{2} \arcsinh \left (c x \right )}{3 c^{2} d^{3} \left (c^{4} x^{4}+2 c^{2} x^{2}+1\right )}+\frac {b^{2}}{12 c^{2} d^{3} \left (c^{4} x^{4}+2 c^{2} x^{2}+1\right )}-\frac {b^{2} \ln \left (1+\left (c x +\sqrt {c^{2} x^{2}+1}\right )^{2}\right )}{3 c^{2} d^{3}}-\frac {a b \arcsinh \left (c x \right )}{2 c^{2} d^{3} \left (c^{2} x^{2}+1\right )^{2}}+\frac {a b x}{6 c \,d^{3} \left (c^{2} x^{2}+1\right )^{\frac {3}{2}}}+\frac {a b x}{3 c \,d^{3} \sqrt {c^{2} x^{2}+1}} \]
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 \[ -\frac {b^{2} \log \left (c x + \sqrt {c^{2} x^{2} + 1}\right )^{2}}{4 \, {\left (c^{6} d^{3} x^{4} + 2 \, c^{4} d^{3} x^{2} + c^{2} d^{3}\right )}} - \frac {a^{2}}{4 \, {\left (c^{6} d^{3} x^{4} + 2 \, c^{4} d^{3} x^{2} + c^{2} d^{3}\right )}} + \int \frac {{\left ({\left (4 \, a b c^{2} + b^{2} c^{2}\right )} x^{2} + \sqrt {c^{2} x^{2} + 1} {\left (4 \, a b c + b^{2} c\right )} x + b^{2}\right )} \log \left (c x + \sqrt {c^{2} x^{2} + 1}\right )}{2 \, {\left (c^{8} d^{3} x^{7} + 3 \, c^{6} d^{3} x^{5} + 3 \, c^{4} d^{3} x^{3} + c^{2} d^{3} x + {\left (c^{7} d^{3} x^{6} + 3 \, c^{5} d^{3} x^{4} + 3 \, c^{3} d^{3} x^{2} + c d^{3}\right )} \sqrt {c^{2} x^{2} + 1}\right )}}\,{d x} \]
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 \[ \int \frac {x\,{\left (a+b\,\mathrm {asinh}\left (c\,x\right )\right )}^2}{{\left (d\,c^2\,x^2+d\right )}^3} \,d x \]
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 \[ \frac {\int \frac {a^{2} x}{c^{6} x^{6} + 3 c^{4} x^{4} + 3 c^{2} x^{2} + 1}\, dx + \int \frac {b^{2} x \operatorname {asinh}^{2}{\left (c x \right )}}{c^{6} x^{6} + 3 c^{4} x^{4} + 3 c^{2} x^{2} + 1}\, dx + \int \frac {2 a b x \operatorname {asinh}{\left (c x \right )}}{c^{6} x^{6} + 3 c^{4} x^{4} + 3 c^{2} x^{2} + 1}\, dx}{d^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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