Optimal. Leaf size=123 \[ \frac {3 b^3 c \sqrt {1+\frac {1}{c^2 x^2}}}{8 x}-\frac {3}{8} b^3 c^2 \text {csch}^{-1}(c x)-\frac {3 b^2 \left (a+b \text {csch}^{-1}(c x)\right )}{4 x^2}+\frac {3 b c \sqrt {1+\frac {1}{c^2 x^2}} \left (a+b \text {csch}^{-1}(c x)\right )^2}{4 x}-\frac {1}{4} c^2 \left (a+b \text {csch}^{-1}(c x)\right )^3-\frac {\left (a+b \text {csch}^{-1}(c x)\right )^3}{2 x^2} \]
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Rubi [A]
time = 0.08, antiderivative size = 123, normalized size of antiderivative = 1.00, number of steps
used = 6, number of rules used = 6, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.429, Rules used = {6421, 5554,
3392, 32, 2715, 8} \begin {gather*} -\frac {3 b^2 \left (a+b \text {csch}^{-1}(c x)\right )}{4 x^2}+\frac {3 b c \sqrt {\frac {1}{c^2 x^2}+1} \left (a+b \text {csch}^{-1}(c x)\right )^2}{4 x}-\frac {1}{4} c^2 \left (a+b \text {csch}^{-1}(c x)\right )^3-\frac {\left (a+b \text {csch}^{-1}(c x)\right )^3}{2 x^2}+\frac {3 b^3 c \sqrt {\frac {1}{c^2 x^2}+1}}{8 x}-\frac {3}{8} b^3 c^2 \text {csch}^{-1}(c x) \end {gather*}
Antiderivative was successfully verified.
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Rule 8
Rule 32
Rule 2715
Rule 3392
Rule 5554
Rule 6421
Rubi steps
\begin {align*} \int \frac {\left (a+b \text {csch}^{-1}(c x)\right )^3}{x^3} \, dx &=-\left (c^2 \text {Subst}\left (\int (a+b x)^3 \cosh (x) \sinh (x) \, dx,x,\text {csch}^{-1}(c x)\right )\right )\\ &=-\frac {\left (a+b \text {csch}^{-1}(c x)\right )^3}{2 x^2}+\frac {1}{2} \left (3 b c^2\right ) \text {Subst}\left (\int (a+b x)^2 \sinh ^2(x) \, dx,x,\text {csch}^{-1}(c x)\right )\\ &=-\frac {3 b^2 \left (a+b \text {csch}^{-1}(c x)\right )}{4 x^2}+\frac {3 b c \sqrt {1+\frac {1}{c^2 x^2}} \left (a+b \text {csch}^{-1}(c x)\right )^2}{4 x}-\frac {\left (a+b \text {csch}^{-1}(c x)\right )^3}{2 x^2}-\frac {1}{4} \left (3 b c^2\right ) \text {Subst}\left (\int (a+b x)^2 \, dx,x,\text {csch}^{-1}(c x)\right )+\frac {1}{4} \left (3 b^3 c^2\right ) \text {Subst}\left (\int \sinh ^2(x) \, dx,x,\text {csch}^{-1}(c x)\right )\\ &=\frac {3 b^3 c \sqrt {1+\frac {1}{c^2 x^2}}}{8 x}-\frac {3 b^2 \left (a+b \text {csch}^{-1}(c x)\right )}{4 x^2}+\frac {3 b c \sqrt {1+\frac {1}{c^2 x^2}} \left (a+b \text {csch}^{-1}(c x)\right )^2}{4 x}-\frac {1}{4} c^2 \left (a+b \text {csch}^{-1}(c x)\right )^3-\frac {\left (a+b \text {csch}^{-1}(c x)\right )^3}{2 x^2}-\frac {1}{8} \left (3 b^3 c^2\right ) \text {Subst}\left (\int 1 \, dx,x,\text {csch}^{-1}(c x)\right )\\ &=\frac {3 b^3 c \sqrt {1+\frac {1}{c^2 x^2}}}{8 x}-\frac {3}{8} b^3 c^2 \text {csch}^{-1}(c x)-\frac {3 b^2 \left (a+b \text {csch}^{-1}(c x)\right )}{4 x^2}+\frac {3 b c \sqrt {1+\frac {1}{c^2 x^2}} \left (a+b \text {csch}^{-1}(c x)\right )^2}{4 x}-\frac {1}{4} c^2 \left (a+b \text {csch}^{-1}(c x)\right )^3-\frac {\left (a+b \text {csch}^{-1}(c x)\right )^3}{2 x^2}\\ \end {align*}
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Mathematica [A]
time = 0.20, size = 182, normalized size = 1.48 \begin {gather*} -\frac {4 a^3+6 a b^2-6 a^2 b c \sqrt {1+\frac {1}{c^2 x^2}} x-3 b^3 c \sqrt {1+\frac {1}{c^2 x^2}} x+6 b \left (2 a^2+b^2-2 a b c \sqrt {1+\frac {1}{c^2 x^2}} x\right ) \text {csch}^{-1}(c x)+6 b^2 \left (-b c \sqrt {1+\frac {1}{c^2 x^2}} x+a \left (2+c^2 x^2\right )\right ) \text {csch}^{-1}(c x)^2+2 b^3 \left (2+c^2 x^2\right ) \text {csch}^{-1}(c x)^3+3 b \left (2 a^2+b^2\right ) c^2 x^2 \sinh ^{-1}\left (\frac {1}{c x}\right )}{8 x^2} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.02, size = 0, normalized size = 0.00 \[\int \frac {\left (a +b \,\mathrm {arccsch}\left (c x \right )\right )^{3}}{x^{3}}\, dx\]
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 [B] Leaf count of result is larger than twice the leaf count of optimal. 267 vs.
\(2 (107) = 214\).
time = 0.39, size = 267, normalized size = 2.17 \begin {gather*} -\frac {2 \, {\left (b^{3} c^{2} x^{2} + 2 \, b^{3}\right )} \log \left (\frac {c x \sqrt {\frac {c^{2} x^{2} + 1}{c^{2} x^{2}}} + 1}{c x}\right )^{3} - 3 \, {\left (2 \, a^{2} b + b^{3}\right )} c x \sqrt {\frac {c^{2} x^{2} + 1}{c^{2} x^{2}}} + 4 \, a^{3} + 6 \, a b^{2} + 6 \, {\left (a b^{2} c^{2} x^{2} - b^{3} c x \sqrt {\frac {c^{2} x^{2} + 1}{c^{2} x^{2}}} + 2 \, a b^{2}\right )} \log \left (\frac {c x \sqrt {\frac {c^{2} x^{2} + 1}{c^{2} x^{2}}} + 1}{c x}\right )^{2} - 3 \, {\left (4 \, a b^{2} c x \sqrt {\frac {c^{2} x^{2} + 1}{c^{2} x^{2}}} - {\left (2 \, a^{2} b + b^{3}\right )} c^{2} x^{2} - 4 \, a^{2} b - 2 \, b^{3}\right )} \log \left (\frac {c x \sqrt {\frac {c^{2} x^{2} + 1}{c^{2} x^{2}}} + 1}{c x}\right )}{8 \, x^{2}} \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 {\left (a + b \operatorname {acsch}{\left (c x \right )}\right )^{3}}{x^{3}}\, 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 = -0.01 \begin {gather*} \int \frac {{\left (a+b\,\mathrm {asinh}\left (\frac {1}{c\,x}\right )\right )}^3}{x^3} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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