Optimal. Leaf size=143 \[ -\frac {2 c^2 x \left (a+b \sinh ^{-1}(c x)\right )}{d \sqrt {c^2 d x^2+d}}-\frac {a+b \sinh ^{-1}(c x)}{d x \sqrt {c^2 d x^2+d}}+\frac {b c \log (x) \sqrt {c^2 d x^2+d}}{d^2 \sqrt {c^2 x^2+1}}+\frac {b c \sqrt {c^2 d x^2+d} \log \left (c^2 x^2+1\right )}{2 d^2 \sqrt {c^2 x^2+1}} \]
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Rubi [A] time = 0.15, antiderivative size = 143, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 7, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.269, Rules used = {5747, 5687, 260, 266, 36, 29, 31} \[ -\frac {2 c^2 x \left (a+b \sinh ^{-1}(c x)\right )}{d \sqrt {c^2 d x^2+d}}-\frac {a+b \sinh ^{-1}(c x)}{d x \sqrt {c^2 d x^2+d}}+\frac {b c \sqrt {c^2 x^2+1} \log (x)}{d \sqrt {c^2 d x^2+d}}+\frac {b c \sqrt {c^2 x^2+1} \log \left (c^2 x^2+1\right )}{2 d \sqrt {c^2 d x^2+d}} \]
Antiderivative was successfully verified.
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Rule 29
Rule 31
Rule 36
Rule 260
Rule 266
Rule 5687
Rule 5747
Rubi steps
\begin {align*} \int \frac {a+b \sinh ^{-1}(c x)}{x^2 \left (d+c^2 d x^2\right )^{3/2}} \, dx &=-\frac {a+b \sinh ^{-1}(c x)}{d x \sqrt {d+c^2 d x^2}}-\left (2 c^2\right ) \int \frac {a+b \sinh ^{-1}(c x)}{\left (d+c^2 d x^2\right )^{3/2}} \, dx+\frac {\left (b c \sqrt {1+c^2 x^2}\right ) \int \frac {1}{x \left (1+c^2 x^2\right )} \, dx}{d \sqrt {d+c^2 d x^2}}\\ &=-\frac {a+b \sinh ^{-1}(c x)}{d x \sqrt {d+c^2 d x^2}}-\frac {2 c^2 x \left (a+b \sinh ^{-1}(c x)\right )}{d \sqrt {d+c^2 d x^2}}+\frac {\left (b c \sqrt {1+c^2 x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{x \left (1+c^2 x\right )} \, dx,x,x^2\right )}{2 d \sqrt {d+c^2 d x^2}}+\frac {\left (2 b c^3 \sqrt {1+c^2 x^2}\right ) \int \frac {x}{1+c^2 x^2} \, dx}{d \sqrt {d+c^2 d x^2}}\\ &=-\frac {a+b \sinh ^{-1}(c x)}{d x \sqrt {d+c^2 d x^2}}-\frac {2 c^2 x \left (a+b \sinh ^{-1}(c x)\right )}{d \sqrt {d+c^2 d x^2}}+\frac {b c \sqrt {1+c^2 x^2} \log \left (1+c^2 x^2\right )}{d \sqrt {d+c^2 d x^2}}+\frac {\left (b c \sqrt {1+c^2 x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{x} \, dx,x,x^2\right )}{2 d \sqrt {d+c^2 d x^2}}-\frac {\left (b c^3 \sqrt {1+c^2 x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{1+c^2 x} \, dx,x,x^2\right )}{2 d \sqrt {d+c^2 d x^2}}\\ &=-\frac {a+b \sinh ^{-1}(c x)}{d x \sqrt {d+c^2 d x^2}}-\frac {2 c^2 x \left (a+b \sinh ^{-1}(c x)\right )}{d \sqrt {d+c^2 d x^2}}+\frac {b c \sqrt {1+c^2 x^2} \log (x)}{d \sqrt {d+c^2 d x^2}}+\frac {b c \sqrt {1+c^2 x^2} \log \left (1+c^2 x^2\right )}{2 d \sqrt {d+c^2 d x^2}}\\ \end {align*}
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Mathematica [A] time = 0.28, size = 163, normalized size = 1.14 \[ -\frac {\sqrt {c^2 d x^2+d} \left (4 a c^2 x^2 \sqrt {c^2 x^2+1}+2 a \sqrt {c^2 x^2+1}+b c x \left (c^2 x^2+1\right ) \log \left (\frac {1}{c^2 x^2}+1\right )-2 b c x \log \left (c^2 x^2+1\right )+2 b \sqrt {c^2 x^2+1} \left (2 c^2 x^2+1\right ) \sinh ^{-1}(c x)-2 b c^3 x^3 \log \left (c^2 x^2+1\right )\right )}{2 d^2 x \left (c^2 x^2+1\right )^{3/2}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.79, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {c^{2} d x^{2} + d} {\left (b \operatorname {arsinh}\left (c x\right ) + a\right )}}{c^{4} d^{2} x^{6} + 2 \, c^{2} d^{2} x^{4} + d^{2} x^{2}}, x\right ) \]
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 \[ \int \frac {b \operatorname {arsinh}\left (c x\right ) + a}{{\left (c^{2} d x^{2} + d\right )}^{\frac {3}{2}} x^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.17, size = 205, normalized size = 1.43 \[ -\frac {a}{d x \sqrt {c^{2} d \,x^{2}+d}}-\frac {2 a \,c^{2} x}{d \sqrt {c^{2} d \,x^{2}+d}}-\frac {2 b \sqrt {d \left (c^{2} x^{2}+1\right )}\, \arcsinh \left (c x \right ) c}{\sqrt {c^{2} x^{2}+1}\, d^{2}}-\frac {2 b \sqrt {d \left (c^{2} x^{2}+1\right )}\, \arcsinh \left (c x \right ) x \,c^{2}}{\left (c^{2} x^{2}+1\right ) d^{2}}-\frac {b \sqrt {d \left (c^{2} x^{2}+1\right )}\, \arcsinh \left (c x \right )}{\left (c^{2} x^{2}+1\right ) d^{2} x}+\frac {b \sqrt {d \left (c^{2} x^{2}+1\right )}\, \ln \left (\left (c x +\sqrt {c^{2} x^{2}+1}\right )^{4}-1\right ) c}{\sqrt {c^{2} x^{2}+1}\, d^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.41, size = 119, normalized size = 0.83 \[ \frac {1}{2} \, b c {\left (\frac {\log \left (c^{2} x^{2} + 1\right )}{d^{\frac {3}{2}}} + \frac {2 \, \log \relax (x)}{d^{\frac {3}{2}}}\right )} - {\left (\frac {2 \, c^{2} x}{\sqrt {c^{2} d x^{2} + d} d} + \frac {1}{\sqrt {c^{2} d x^{2} + d} d x}\right )} b \operatorname {arsinh}\left (c x\right ) - {\left (\frac {2 \, c^{2} x}{\sqrt {c^{2} d x^{2} + d} d} + \frac {1}{\sqrt {c^{2} d x^{2} + d} d x}\right )} a \]
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 \[ \int \frac {a+b\,\mathrm {asinh}\left (c\,x\right )}{x^2\,{\left (d\,c^2\,x^2+d\right )}^{3/2}} \,d x \]
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 \[ \int \frac {a + b \operatorname {asinh}{\left (c x \right )}}{x^{2} \left (d \left (c^{2} x^{2} + 1\right )\right )^{\frac {3}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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