Optimal. Leaf size=122 \[ \frac {(d+e x)^3 \left (a+b \text {csch}^{-1}(c x)\right )}{3 e}+\frac {b d e x \sqrt {\frac {1}{c^2 x^2}+1}}{c}+\frac {b e^2 x^2 \sqrt {\frac {1}{c^2 x^2}+1}}{6 c}+\frac {b \left (6 c^2 d^2-e^2\right ) \tanh ^{-1}\left (\sqrt {\frac {1}{c^2 x^2}+1}\right )}{6 c^3}-\frac {b d^3 \text {csch}^{-1}(c x)}{3 e} \]
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Rubi [A] time = 0.26, antiderivative size = 122, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 9, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.562, Rules used = {6290, 1568, 1475, 1807, 844, 215, 266, 63, 208} \[ \frac {(d+e x)^3 \left (a+b \text {csch}^{-1}(c x)\right )}{3 e}+\frac {b \left (6 c^2 d^2-e^2\right ) \tanh ^{-1}\left (\sqrt {\frac {1}{c^2 x^2}+1}\right )}{6 c^3}+\frac {b d e x \sqrt {\frac {1}{c^2 x^2}+1}}{c}+\frac {b e^2 x^2 \sqrt {\frac {1}{c^2 x^2}+1}}{6 c}-\frac {b d^3 \text {csch}^{-1}(c x)}{3 e} \]
Antiderivative was successfully verified.
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Rule 63
Rule 208
Rule 215
Rule 266
Rule 844
Rule 1475
Rule 1568
Rule 1807
Rule 6290
Rubi steps
\begin {align*} \int (d+e x)^2 \left (a+b \text {csch}^{-1}(c x)\right ) \, dx &=\frac {(d+e x)^3 \left (a+b \text {csch}^{-1}(c x)\right )}{3 e}+\frac {b \int \frac {(d+e x)^3}{\sqrt {1+\frac {1}{c^2 x^2}} x^2} \, dx}{3 c e}\\ &=\frac {(d+e x)^3 \left (a+b \text {csch}^{-1}(c x)\right )}{3 e}+\frac {b \int \frac {\left (e+\frac {d}{x}\right )^3 x}{\sqrt {1+\frac {1}{c^2 x^2}}} \, dx}{3 c e}\\ &=\frac {(d+e x)^3 \left (a+b \text {csch}^{-1}(c x)\right )}{3 e}-\frac {b \operatorname {Subst}\left (\int \frac {(e+d x)^3}{x^3 \sqrt {1+\frac {x^2}{c^2}}} \, dx,x,\frac {1}{x}\right )}{3 c e}\\ &=\frac {b e^2 \sqrt {1+\frac {1}{c^2 x^2}} x^2}{6 c}+\frac {(d+e x)^3 \left (a+b \text {csch}^{-1}(c x)\right )}{3 e}+\frac {b \operatorname {Subst}\left (\int \frac {-6 d e^2-e \left (6 d^2-\frac {e^2}{c^2}\right ) x-2 d^3 x^2}{x^2 \sqrt {1+\frac {x^2}{c^2}}} \, dx,x,\frac {1}{x}\right )}{6 c e}\\ &=\frac {b d e \sqrt {1+\frac {1}{c^2 x^2}} x}{c}+\frac {b e^2 \sqrt {1+\frac {1}{c^2 x^2}} x^2}{6 c}+\frac {(d+e x)^3 \left (a+b \text {csch}^{-1}(c x)\right )}{3 e}-\frac {b \operatorname {Subst}\left (\int \frac {e \left (6 d^2-\frac {e^2}{c^2}\right )+2 d^3 x}{x \sqrt {1+\frac {x^2}{c^2}}} \, dx,x,\frac {1}{x}\right )}{6 c e}\\ &=\frac {b d e \sqrt {1+\frac {1}{c^2 x^2}} x}{c}+\frac {b e^2 \sqrt {1+\frac {1}{c^2 x^2}} x^2}{6 c}+\frac {(d+e x)^3 \left (a+b \text {csch}^{-1}(c x)\right )}{3 e}-\frac {\left (b d^3\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {1+\frac {x^2}{c^2}}} \, dx,x,\frac {1}{x}\right )}{3 c e}-\frac {\left (b \left (6 d^2-\frac {e^2}{c^2}\right )\right ) \operatorname {Subst}\left (\int \frac {1}{x \sqrt {1+\frac {x^2}{c^2}}} \, dx,x,\frac {1}{x}\right )}{6 c}\\ &=\frac {b d e \sqrt {1+\frac {1}{c^2 x^2}} x}{c}+\frac {b e^2 \sqrt {1+\frac {1}{c^2 x^2}} x^2}{6 c}-\frac {b d^3 \text {csch}^{-1}(c x)}{3 e}+\frac {(d+e x)^3 \left (a+b \text {csch}^{-1}(c x)\right )}{3 e}-\frac {\left (b \left (6 c^2 d^2-e^2\right )\right ) \operatorname {Subst}\left (\int \frac {1}{x \sqrt {1+\frac {x}{c^2}}} \, dx,x,\frac {1}{x^2}\right )}{12 c^3}\\ &=\frac {b d e \sqrt {1+\frac {1}{c^2 x^2}} x}{c}+\frac {b e^2 \sqrt {1+\frac {1}{c^2 x^2}} x^2}{6 c}-\frac {b d^3 \text {csch}^{-1}(c x)}{3 e}+\frac {(d+e x)^3 \left (a+b \text {csch}^{-1}(c x)\right )}{3 e}-\frac {\left (b \left (6 c^2 d^2-e^2\right )\right ) \operatorname {Subst}\left (\int \frac {1}{-c^2+c^2 x^2} \, dx,x,\sqrt {1+\frac {1}{c^2 x^2}}\right )}{6 c}\\ &=\frac {b d e \sqrt {1+\frac {1}{c^2 x^2}} x}{c}+\frac {b e^2 \sqrt {1+\frac {1}{c^2 x^2}} x^2}{6 c}-\frac {b d^3 \text {csch}^{-1}(c x)}{3 e}+\frac {(d+e x)^3 \left (a+b \text {csch}^{-1}(c x)\right )}{3 e}+\frac {b \left (6 c^2 d^2-e^2\right ) \tanh ^{-1}\left (\sqrt {1+\frac {1}{c^2 x^2}}\right )}{6 c^3}\\ \end {align*}
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Mathematica [A] time = 0.19, size = 122, normalized size = 1.00 \[ \frac {c^2 x \left (2 a c \left (3 d^2+3 d e x+e^2 x^2\right )+b e \sqrt {\frac {1}{c^2 x^2}+1} (6 d+e x)\right )+2 b c^3 x \text {csch}^{-1}(c x) \left (3 d^2+3 d e x+e^2 x^2\right )+b \left (6 c^2 d^2-e^2\right ) \log \left (x \left (\sqrt {\frac {1}{c^2 x^2}+1}+1\right )\right )}{6 c^3} \]
Antiderivative was successfully verified.
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fricas [B] time = 1.00, size = 328, normalized size = 2.69 \[ \frac {2 \, a c^{3} e^{2} x^{3} + 6 \, a c^{3} d e x^{2} + 6 \, a c^{3} d^{2} x + 2 \, {\left (3 \, b c^{3} d^{2} + 3 \, b c^{3} d e + b c^{3} e^{2}\right )} \log \left (c x \sqrt {\frac {c^{2} x^{2} + 1}{c^{2} x^{2}}} - c x + 1\right ) - {\left (6 \, b c^{2} d^{2} - b e^{2}\right )} \log \left (c x \sqrt {\frac {c^{2} x^{2} + 1}{c^{2} x^{2}}} - c x\right ) - 2 \, {\left (3 \, b c^{3} d^{2} + 3 \, b c^{3} d e + b c^{3} e^{2}\right )} \log \left (c x \sqrt {\frac {c^{2} x^{2} + 1}{c^{2} x^{2}}} - c x - 1\right ) + 2 \, {\left (b c^{3} e^{2} x^{3} + 3 \, b c^{3} d e x^{2} + 3 \, b c^{3} d^{2} x - 3 \, b c^{3} d^{2} - 3 \, b c^{3} d e - b c^{3} e^{2}\right )} \log \left (\frac {c x \sqrt {\frac {c^{2} x^{2} + 1}{c^{2} x^{2}}} + 1}{c x}\right ) + {\left (b c^{2} e^{2} x^{2} + 6 \, b c^{2} d e x\right )} \sqrt {\frac {c^{2} x^{2} + 1}{c^{2} x^{2}}}}{6 \, c^{3}} \]
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 \[ \int {\left (e x + d\right )}^{2} {\left (b \operatorname {arcsch}\left (c x\right ) + a\right )}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 204, normalized size = 1.67 \[ \frac {\frac {\left (c x e +c d \right )^{3} a}{3 c^{2} e}+\frac {b \left (\frac {e^{2} \mathrm {arccsch}\left (c x \right ) c^{3} x^{3}}{3}+e \,\mathrm {arccsch}\left (c x \right ) c^{3} x^{2} d +\mathrm {arccsch}\left (c x \right ) c^{3} x \,d^{2}+\frac {\mathrm {arccsch}\left (c x \right ) c^{3} d^{3}}{3 e}+\frac {\sqrt {c^{2} x^{2}+1}\, \left (-2 c^{3} d^{3} \arctanh \left (\frac {1}{\sqrt {c^{2} x^{2}+1}}\right )+6 c^{2} d^{2} e \arcsinh \left (c x \right )+e^{3} c x \sqrt {c^{2} x^{2}+1}+6 c d \,e^{2} \sqrt {c^{2} x^{2}+1}-e^{3} \arcsinh \left (c x \right )\right )}{6 e \sqrt {\frac {c^{2} x^{2}+1}{c^{2} x^{2}}}\, c x}\right )}{c^{2}}}{c} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.34, size = 192, normalized size = 1.57 \[ \frac {1}{3} \, a e^{2} x^{3} + a d e x^{2} + {\left (x^{2} \operatorname {arcsch}\left (c x\right ) + \frac {x \sqrt {\frac {1}{c^{2} x^{2}} + 1}}{c}\right )} b d e + \frac {1}{12} \, {\left (4 \, x^{3} \operatorname {arcsch}\left (c x\right ) + \frac {\frac {2 \, \sqrt {\frac {1}{c^{2} x^{2}} + 1}}{c^{2} {\left (\frac {1}{c^{2} x^{2}} + 1\right )} - c^{2}} - \frac {\log \left (\sqrt {\frac {1}{c^{2} x^{2}} + 1} + 1\right )}{c^{2}} + \frac {\log \left (\sqrt {\frac {1}{c^{2} x^{2}} + 1} - 1\right )}{c^{2}}}{c}\right )} b e^{2} + a d^{2} x + \frac {{\left (2 \, c x \operatorname {arcsch}\left (c x\right ) + \log \left (\sqrt {\frac {1}{c^{2} x^{2}} + 1} + 1\right ) - \log \left (\sqrt {\frac {1}{c^{2} x^{2}} + 1} - 1\right )\right )} b d^{2}}{2 \, c} \]
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 \[ \int \left (a+b\,\mathrm {asinh}\left (\frac {1}{c\,x}\right )\right )\,{\left (d+e\,x\right )}^2 \,d x \]
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 \[ \int \left (a + b \operatorname {acsch}{\left (c x \right )}\right ) \left (d + e x\right )^{2}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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