Optimal. Leaf size=378 \[ -\frac {c^2 d \sqrt {c^2 d x^2+d} \left (a+b \sinh ^{-1}(c x)\right )^2}{x}-\frac {b c d \sqrt {c^2 x^2+1} \sqrt {c^2 d x^2+d} \left (a+b \sinh ^{-1}(c x)\right )}{3 x^2}-\frac {\left (c^2 d x^2+d\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )^2}{3 x^3}+\frac {c^3 d \sqrt {c^2 d x^2+d} \left (a+b \sinh ^{-1}(c x)\right )^3}{3 b \sqrt {c^2 x^2+1}}+\frac {4 c^3 d \sqrt {c^2 d x^2+d} \left (a+b \sinh ^{-1}(c x)\right )^2}{3 \sqrt {c^2 x^2+1}}+\frac {8 b c^3 d \sqrt {c^2 d x^2+d} \log \left (1-e^{-2 \sinh ^{-1}(c x)}\right ) \left (a+b \sinh ^{-1}(c x)\right )}{3 \sqrt {c^2 x^2+1}}-\frac {b^2 c^2 d \sqrt {c^2 d x^2+d}}{3 x}-\frac {4 b^2 c^3 d \sqrt {c^2 d x^2+d} \text {Li}_2\left (e^{-2 \sinh ^{-1}(c x)}\right )}{3 \sqrt {c^2 x^2+1}}+\frac {b^2 c^3 d \sqrt {c^2 d x^2+d} \sinh ^{-1}(c x)}{3 \sqrt {c^2 x^2+1}} \]
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Rubi [A] time = 0.58, antiderivative size = 378, normalized size of antiderivative = 1.00, number of steps used = 16, number of rules used = 11, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.393, Rules used = {5739, 5737, 5659, 3716, 2190, 2279, 2391, 5675, 5728, 277, 215} \[ \frac {4 b^2 c^3 d \sqrt {c^2 d x^2+d} \text {PolyLog}\left (2,e^{2 \sinh ^{-1}(c x)}\right )}{3 \sqrt {c^2 x^2+1}}+\frac {c^3 d \sqrt {c^2 d x^2+d} \left (a+b \sinh ^{-1}(c x)\right )^3}{3 b \sqrt {c^2 x^2+1}}-\frac {4 c^3 d \sqrt {c^2 d x^2+d} \left (a+b \sinh ^{-1}(c x)\right )^2}{3 \sqrt {c^2 x^2+1}}-\frac {c^2 d \sqrt {c^2 d x^2+d} \left (a+b \sinh ^{-1}(c x)\right )^2}{x}-\frac {b c d \sqrt {c^2 x^2+1} \sqrt {c^2 d x^2+d} \left (a+b \sinh ^{-1}(c x)\right )}{3 x^2}-\frac {\left (c^2 d x^2+d\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )^2}{3 x^3}+\frac {8 b c^3 d \sqrt {c^2 d x^2+d} \log \left (1-e^{2 \sinh ^{-1}(c x)}\right ) \left (a+b \sinh ^{-1}(c x)\right )}{3 \sqrt {c^2 x^2+1}}-\frac {b^2 c^2 d \sqrt {c^2 d x^2+d}}{3 x}+\frac {b^2 c^3 d \sqrt {c^2 d x^2+d} \sinh ^{-1}(c x)}{3 \sqrt {c^2 x^2+1}} \]
Warning: Unable to verify antiderivative.
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Rule 215
Rule 277
Rule 2190
Rule 2279
Rule 2391
Rule 3716
Rule 5659
Rule 5675
Rule 5728
Rule 5737
Rule 5739
Rubi steps
\begin {align*} \int \frac {\left (d+c^2 d x^2\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )^2}{x^4} \, dx &=-\frac {\left (d+c^2 d x^2\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )^2}{3 x^3}+\left (c^2 d\right ) \int \frac {\sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2}{x^2} \, dx+\frac {\left (2 b c d \sqrt {d+c^2 d x^2}\right ) \int \frac {\left (1+c^2 x^2\right ) \left (a+b \sinh ^{-1}(c x)\right )}{x^3} \, dx}{3 \sqrt {1+c^2 x^2}}\\ &=-\frac {b c d \sqrt {1+c^2 x^2} \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{3 x^2}-\frac {c^2 d \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2}{x}-\frac {\left (d+c^2 d x^2\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )^2}{3 x^3}+\frac {\left (b^2 c^2 d \sqrt {d+c^2 d x^2}\right ) \int \frac {\sqrt {1+c^2 x^2}}{x^2} \, dx}{3 \sqrt {1+c^2 x^2}}+\frac {\left (2 b c^3 d \sqrt {d+c^2 d x^2}\right ) \int \frac {a+b \sinh ^{-1}(c x)}{x} \, dx}{3 \sqrt {1+c^2 x^2}}+\frac {\left (2 b c^3 d \sqrt {d+c^2 d x^2}\right ) \int \frac {a+b \sinh ^{-1}(c x)}{x} \, dx}{\sqrt {1+c^2 x^2}}+\frac {\left (c^4 d \sqrt {d+c^2 d x^2}\right ) \int \frac {\left (a+b \sinh ^{-1}(c x)\right )^2}{\sqrt {1+c^2 x^2}} \, dx}{\sqrt {1+c^2 x^2}}\\ &=-\frac {b^2 c^2 d \sqrt {d+c^2 d x^2}}{3 x}-\frac {b c d \sqrt {1+c^2 x^2} \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{3 x^2}-\frac {c^2 d \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2}{x}-\frac {\left (d+c^2 d x^2\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )^2}{3 x^3}+\frac {c^3 d \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^3}{3 b \sqrt {1+c^2 x^2}}+\frac {\left (2 b c^3 d \sqrt {d+c^2 d x^2}\right ) \operatorname {Subst}\left (\int (a+b x) \coth (x) \, dx,x,\sinh ^{-1}(c x)\right )}{3 \sqrt {1+c^2 x^2}}+\frac {\left (2 b c^3 d \sqrt {d+c^2 d x^2}\right ) \operatorname {Subst}\left (\int (a+b x) \coth (x) \, dx,x,\sinh ^{-1}(c x)\right )}{\sqrt {1+c^2 x^2}}+\frac {\left (b^2 c^4 d \sqrt {d+c^2 d x^2}\right ) \int \frac {1}{\sqrt {1+c^2 x^2}} \, dx}{3 \sqrt {1+c^2 x^2}}\\ &=-\frac {b^2 c^2 d \sqrt {d+c^2 d x^2}}{3 x}+\frac {b^2 c^3 d \sqrt {d+c^2 d x^2} \sinh ^{-1}(c x)}{3 \sqrt {1+c^2 x^2}}-\frac {b c d \sqrt {1+c^2 x^2} \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{3 x^2}-\frac {c^2 d \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2}{x}-\frac {4 c^3 d \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2}{3 \sqrt {1+c^2 x^2}}-\frac {\left (d+c^2 d x^2\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )^2}{3 x^3}+\frac {c^3 d \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^3}{3 b \sqrt {1+c^2 x^2}}-\frac {\left (4 b c^3 d \sqrt {d+c^2 d x^2}\right ) \operatorname {Subst}\left (\int \frac {e^{2 x} (a+b x)}{1-e^{2 x}} \, dx,x,\sinh ^{-1}(c x)\right )}{3 \sqrt {1+c^2 x^2}}-\frac {\left (4 b c^3 d \sqrt {d+c^2 d x^2}\right ) \operatorname {Subst}\left (\int \frac {e^{2 x} (a+b x)}{1-e^{2 x}} \, dx,x,\sinh ^{-1}(c x)\right )}{\sqrt {1+c^2 x^2}}\\ &=-\frac {b^2 c^2 d \sqrt {d+c^2 d x^2}}{3 x}+\frac {b^2 c^3 d \sqrt {d+c^2 d x^2} \sinh ^{-1}(c x)}{3 \sqrt {1+c^2 x^2}}-\frac {b c d \sqrt {1+c^2 x^2} \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{3 x^2}-\frac {c^2 d \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2}{x}-\frac {4 c^3 d \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2}{3 \sqrt {1+c^2 x^2}}-\frac {\left (d+c^2 d x^2\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )^2}{3 x^3}+\frac {c^3 d \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^3}{3 b \sqrt {1+c^2 x^2}}+\frac {8 b c^3 d \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right ) \log \left (1-e^{2 \sinh ^{-1}(c x)}\right )}{3 \sqrt {1+c^2 x^2}}-\frac {\left (2 b^2 c^3 d \sqrt {d+c^2 d x^2}\right ) \operatorname {Subst}\left (\int \log \left (1-e^{2 x}\right ) \, dx,x,\sinh ^{-1}(c x)\right )}{3 \sqrt {1+c^2 x^2}}-\frac {\left (2 b^2 c^3 d \sqrt {d+c^2 d x^2}\right ) \operatorname {Subst}\left (\int \log \left (1-e^{2 x}\right ) \, dx,x,\sinh ^{-1}(c x)\right )}{\sqrt {1+c^2 x^2}}\\ &=-\frac {b^2 c^2 d \sqrt {d+c^2 d x^2}}{3 x}+\frac {b^2 c^3 d \sqrt {d+c^2 d x^2} \sinh ^{-1}(c x)}{3 \sqrt {1+c^2 x^2}}-\frac {b c d \sqrt {1+c^2 x^2} \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{3 x^2}-\frac {c^2 d \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2}{x}-\frac {4 c^3 d \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2}{3 \sqrt {1+c^2 x^2}}-\frac {\left (d+c^2 d x^2\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )^2}{3 x^3}+\frac {c^3 d \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^3}{3 b \sqrt {1+c^2 x^2}}+\frac {8 b c^3 d \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right ) \log \left (1-e^{2 \sinh ^{-1}(c x)}\right )}{3 \sqrt {1+c^2 x^2}}-\frac {\left (b^2 c^3 d \sqrt {d+c^2 d x^2}\right ) \operatorname {Subst}\left (\int \frac {\log (1-x)}{x} \, dx,x,e^{2 \sinh ^{-1}(c x)}\right )}{3 \sqrt {1+c^2 x^2}}-\frac {\left (b^2 c^3 d \sqrt {d+c^2 d x^2}\right ) \operatorname {Subst}\left (\int \frac {\log (1-x)}{x} \, dx,x,e^{2 \sinh ^{-1}(c x)}\right )}{\sqrt {1+c^2 x^2}}\\ &=-\frac {b^2 c^2 d \sqrt {d+c^2 d x^2}}{3 x}+\frac {b^2 c^3 d \sqrt {d+c^2 d x^2} \sinh ^{-1}(c x)}{3 \sqrt {1+c^2 x^2}}-\frac {b c d \sqrt {1+c^2 x^2} \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{3 x^2}-\frac {c^2 d \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2}{x}-\frac {4 c^3 d \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2}{3 \sqrt {1+c^2 x^2}}-\frac {\left (d+c^2 d x^2\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )^2}{3 x^3}+\frac {c^3 d \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^3}{3 b \sqrt {1+c^2 x^2}}+\frac {8 b c^3 d \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right ) \log \left (1-e^{2 \sinh ^{-1}(c x)}\right )}{3 \sqrt {1+c^2 x^2}}+\frac {4 b^2 c^3 d \sqrt {d+c^2 d x^2} \text {Li}_2\left (e^{2 \sinh ^{-1}(c x)}\right )}{3 \sqrt {1+c^2 x^2}}\\ \end {align*}
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Mathematica [A] time = 1.42, size = 458, normalized size = 1.21 \[ \frac {-4 a^2 c^2 d x^2 \sqrt {c^2 x^2+1} \sqrt {c^2 d x^2+d}-a^2 d \sqrt {c^2 x^2+1} \sqrt {c^2 d x^2+d}+3 a^2 c^3 d^{3/2} x^3 \sqrt {c^2 x^2+1} \log \left (\sqrt {d} \sqrt {c^2 d x^2+d}+c d x\right )-a b c d x \sqrt {c^2 d x^2+d}+8 a b c^3 d x^3 \sqrt {c^2 d x^2+d} \log (c x)+b d \sqrt {c^2 d x^2+d} \sinh ^{-1}(c x)^2 \left (3 a c^3 x^3-b \left (-4 c^3 x^3+4 c^2 x^2 \sqrt {c^2 x^2+1}+\sqrt {c^2 x^2+1}\right )\right )+b d \sqrt {c^2 d x^2+d} \sinh ^{-1}(c x) \left (-2 a \sqrt {c^2 x^2+1} \left (4 c^2 x^2+1\right )+8 b c^3 x^3 \log \left (1-e^{-2 \sinh ^{-1}(c x)}\right )-b c x\right )-b^2 c^2 d x^2 \sqrt {c^2 x^2+1} \sqrt {c^2 d x^2+d}-4 b^2 c^3 d x^3 \sqrt {c^2 d x^2+d} \text {Li}_2\left (e^{-2 \sinh ^{-1}(c x)}\right )+b^2 c^3 d x^3 \sqrt {c^2 d x^2+d} \sinh ^{-1}(c x)^3}{3 x^3 \sqrt {c^2 x^2+1}} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.55, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (a^{2} c^{2} d x^{2} + a^{2} d + {\left (b^{2} c^{2} d x^{2} + b^{2} d\right )} \operatorname {arsinh}\left (c x\right )^{2} + 2 \, {\left (a b c^{2} d x^{2} + a b d\right )} \operatorname {arsinh}\left (c x\right )\right )} \sqrt {c^{2} d x^{2} + d}}{x^{4}}, x\right ) \]
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 \[ \text {Exception raised: TypeError} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.46, size = 2796, normalized size = 7.40 \[ \text {Expression too large to display} \]
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 \[ \text {Exception raised: RuntimeError} \]
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 \[ \int \frac {{\left (a+b\,\mathrm {asinh}\left (c\,x\right )\right )}^2\,{\left (d\,c^2\,x^2+d\right )}^{3/2}}{x^4} \,d x \]
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 \[ \int \frac {\left (d \left (c^{2} x^{2} + 1\right )\right )^{\frac {3}{2}} \left (a + b \operatorname {asinh}{\left (c x \right )}\right )^{2}}{x^{4}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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