Optimal. Leaf size=167 \[ \frac {(d+e x)^4 \left (a+b \text {csch}^{-1}(c x)\right )}{4 e}+\frac {b d e^2 x^2 \sqrt {\frac {1}{c^2 x^2}+1}}{2 c}+\frac {b e^3 x^3 \sqrt {\frac {1}{c^2 x^2}+1}}{12 c}+\frac {b e x \sqrt {\frac {1}{c^2 x^2}+1} \left (9 c^2 d^2-e^2\right )}{6 c^3}+\frac {b d \left (2 c^2 d^2-e^2\right ) \tanh ^{-1}\left (\sqrt {\frac {1}{c^2 x^2}+1}\right )}{2 c^3}-\frac {b d^4 \text {csch}^{-1}(c x)}{4 e} \]
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Rubi [A] time = 0.38, antiderivative size = 167, normalized size of antiderivative = 1.00, number of steps used = 11, 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} \[ \frac {(d+e x)^4 \left (a+b \text {csch}^{-1}(c x)\right )}{4 e}+\frac {b e x \sqrt {\frac {1}{c^2 x^2}+1} \left (9 c^2 d^2-e^2\right )}{6 c^3}+\frac {b d \left (2 c^2 d^2-e^2\right ) \tanh ^{-1}\left (\sqrt {\frac {1}{c^2 x^2}+1}\right )}{2 c^3}+\frac {b d e^2 x^2 \sqrt {\frac {1}{c^2 x^2}+1}}{2 c}+\frac {b e^3 x^3 \sqrt {\frac {1}{c^2 x^2}+1}}{12 c}-\frac {b d^4 \text {csch}^{-1}(c x)}{4 e} \]
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)^3 \left (a+b \text {csch}^{-1}(c x)\right ) \, dx &=\frac {(d+e x)^4 \left (a+b \text {csch}^{-1}(c x)\right )}{4 e}+\frac {b \int \frac {(d+e x)^4}{\sqrt {1+\frac {1}{c^2 x^2}} x^2} \, dx}{4 c e}\\ &=\frac {(d+e x)^4 \left (a+b \text {csch}^{-1}(c x)\right )}{4 e}+\frac {b \int \frac {\left (e+\frac {d}{x}\right )^4 x^2}{\sqrt {1+\frac {1}{c^2 x^2}}} \, dx}{4 c e}\\ &=\frac {(d+e x)^4 \left (a+b \text {csch}^{-1}(c x)\right )}{4 e}-\frac {b \operatorname {Subst}\left (\int \frac {(e+d x)^4}{x^4 \sqrt {1+\frac {x^2}{c^2}}} \, dx,x,\frac {1}{x}\right )}{4 c e}\\ &=\frac {b e^3 \sqrt {1+\frac {1}{c^2 x^2}} x^3}{12 c}+\frac {(d+e x)^4 \left (a+b \text {csch}^{-1}(c x)\right )}{4 e}+\frac {b \operatorname {Subst}\left (\int \frac {-12 d e^3-2 e^2 \left (9 d^2-\frac {e^2}{c^2}\right ) x-12 d^3 e x^2-3 d^4 x^3}{x^3 \sqrt {1+\frac {x^2}{c^2}}} \, dx,x,\frac {1}{x}\right )}{12 c e}\\ &=\frac {b d e^2 \sqrt {1+\frac {1}{c^2 x^2}} x^2}{2 c}+\frac {b e^3 \sqrt {1+\frac {1}{c^2 x^2}} x^3}{12 c}+\frac {(d+e x)^4 \left (a+b \text {csch}^{-1}(c x)\right )}{4 e}-\frac {b \operatorname {Subst}\left (\int \frac {4 e^2 \left (9 d^2-\frac {e^2}{c^2}\right )+12 d e \left (2 d^2-\frac {e^2}{c^2}\right ) x+6 d^4 x^2}{x^2 \sqrt {1+\frac {x^2}{c^2}}} \, dx,x,\frac {1}{x}\right )}{24 c e}\\ &=\frac {b e \left (9 c^2 d^2-e^2\right ) \sqrt {1+\frac {1}{c^2 x^2}} x}{6 c^3}+\frac {b d e^2 \sqrt {1+\frac {1}{c^2 x^2}} x^2}{2 c}+\frac {b e^3 \sqrt {1+\frac {1}{c^2 x^2}} x^3}{12 c}+\frac {(d+e x)^4 \left (a+b \text {csch}^{-1}(c x)\right )}{4 e}+\frac {b \operatorname {Subst}\left (\int \frac {-12 d e \left (2 d^2-\frac {e^2}{c^2}\right )-6 d^4 x}{x \sqrt {1+\frac {x^2}{c^2}}} \, dx,x,\frac {1}{x}\right )}{24 c e}\\ &=\frac {b e \left (9 c^2 d^2-e^2\right ) \sqrt {1+\frac {1}{c^2 x^2}} x}{6 c^3}+\frac {b d e^2 \sqrt {1+\frac {1}{c^2 x^2}} x^2}{2 c}+\frac {b e^3 \sqrt {1+\frac {1}{c^2 x^2}} x^3}{12 c}+\frac {(d+e x)^4 \left (a+b \text {csch}^{-1}(c x)\right )}{4 e}-\frac {\left (b d^4\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {1+\frac {x^2}{c^2}}} \, dx,x,\frac {1}{x}\right )}{4 c e}-\frac {\left (b d \left (2 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 )}{2 c}\\ &=\frac {b e \left (9 c^2 d^2-e^2\right ) \sqrt {1+\frac {1}{c^2 x^2}} x}{6 c^3}+\frac {b d e^2 \sqrt {1+\frac {1}{c^2 x^2}} x^2}{2 c}+\frac {b e^3 \sqrt {1+\frac {1}{c^2 x^2}} x^3}{12 c}-\frac {b d^4 \text {csch}^{-1}(c x)}{4 e}+\frac {(d+e x)^4 \left (a+b \text {csch}^{-1}(c x)\right )}{4 e}-\frac {\left (b d \left (2 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 )}{4 c^3}\\ &=\frac {b e \left (9 c^2 d^2-e^2\right ) \sqrt {1+\frac {1}{c^2 x^2}} x}{6 c^3}+\frac {b d e^2 \sqrt {1+\frac {1}{c^2 x^2}} x^2}{2 c}+\frac {b e^3 \sqrt {1+\frac {1}{c^2 x^2}} x^3}{12 c}-\frac {b d^4 \text {csch}^{-1}(c x)}{4 e}+\frac {(d+e x)^4 \left (a+b \text {csch}^{-1}(c x)\right )}{4 e}-\frac {\left (b d \left (2 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 )}{2 c}\\ &=\frac {b e \left (9 c^2 d^2-e^2\right ) \sqrt {1+\frac {1}{c^2 x^2}} x}{6 c^3}+\frac {b d e^2 \sqrt {1+\frac {1}{c^2 x^2}} x^2}{2 c}+\frac {b e^3 \sqrt {1+\frac {1}{c^2 x^2}} x^3}{12 c}-\frac {b d^4 \text {csch}^{-1}(c x)}{4 e}+\frac {(d+e x)^4 \left (a+b \text {csch}^{-1}(c x)\right )}{4 e}+\frac {b d \left (2 c^2 d^2-e^2\right ) \tanh ^{-1}\left (\sqrt {1+\frac {1}{c^2 x^2}}\right )}{2 c^3}\\ \end {align*}
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Mathematica [A] time = 0.29, size = 165, normalized size = 0.99 \[ \frac {3 a c^3 x \left (4 d^3+6 d^2 e x+4 d e^2 x^2+e^3 x^3\right )+3 b c^3 x \text {csch}^{-1}(c x) \left (4 d^3+6 d^2 e x+4 d e^2 x^2+e^3 x^3\right )+b e x \sqrt {\frac {1}{c^2 x^2}+1} \left (c^2 \left (18 d^2+6 d e x+e^2 x^2\right )-2 e^2\right )+6 b d \left (2 c^2 d^2-e^2\right ) \log \left (x \left (\sqrt {\frac {1}{c^2 x^2}+1}+1\right )\right )}{12 c^3} \]
Antiderivative was successfully verified.
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fricas [B] time = 1.16, size = 419, normalized size = 2.51 \[ \frac {3 \, a c^{3} e^{3} x^{4} + 12 \, a c^{3} d e^{2} x^{3} + 18 \, a c^{3} d^{2} e x^{2} + 12 \, a c^{3} d^{3} x + 3 \, {\left (4 \, b c^{3} d^{3} + 6 \, b c^{3} d^{2} e + 4 \, b c^{3} d e^{2} + b c^{3} e^{3}\right )} \log \left (c x \sqrt {\frac {c^{2} x^{2} + 1}{c^{2} x^{2}}} - c x + 1\right ) - 6 \, {\left (2 \, b c^{2} d^{3} - b d e^{2}\right )} \log \left (c x \sqrt {\frac {c^{2} x^{2} + 1}{c^{2} x^{2}}} - c x\right ) - 3 \, {\left (4 \, b c^{3} d^{3} + 6 \, b c^{3} d^{2} e + 4 \, b c^{3} d e^{2} + b c^{3} e^{3}\right )} \log \left (c x \sqrt {\frac {c^{2} x^{2} + 1}{c^{2} x^{2}}} - c x - 1\right ) + 3 \, {\left (b c^{3} e^{3} x^{4} + 4 \, b c^{3} d e^{2} x^{3} + 6 \, b c^{3} d^{2} e x^{2} + 4 \, b c^{3} d^{3} x - 4 \, b c^{3} d^{3} - 6 \, b c^{3} d^{2} e - 4 \, b c^{3} d e^{2} - b c^{3} e^{3}\right )} \log \left (\frac {c x \sqrt {\frac {c^{2} x^{2} + 1}{c^{2} x^{2}}} + 1}{c x}\right ) + {\left (b c^{2} e^{3} x^{3} + 6 \, b c^{2} d e^{2} x^{2} + 2 \, {\left (9 \, b c^{2} d^{2} e - b e^{3}\right )} x\right )} \sqrt {\frac {c^{2} x^{2} + 1}{c^{2} x^{2}}}}{12 \, c^{3}} \]
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 \[ \int {\left (e x + d\right )}^{3} {\left (b \operatorname {arcsch}\left (c x\right ) + a\right )}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.06, size = 269, normalized size = 1.61 \[ \frac {\frac {\left (c x e +c d \right )^{4} a}{4 c^{3} e}+\frac {b \left (\frac {e^{3} \mathrm {arccsch}\left (c x \right ) c^{4} x^{4}}{4}+e^{2} \mathrm {arccsch}\left (c x \right ) c^{4} x^{3} d +\frac {3 e \,\mathrm {arccsch}\left (c x \right ) c^{4} x^{2} d^{2}}{2}+\mathrm {arccsch}\left (c x \right ) c^{4} x \,d^{3}+\frac {\mathrm {arccsch}\left (c x \right ) c^{4} d^{4}}{4 e}+\frac {\sqrt {c^{2} x^{2}+1}\, \left (-3 c^{4} d^{4} \arctanh \left (\frac {1}{\sqrt {c^{2} x^{2}+1}}\right )+12 c^{3} d^{3} e \arcsinh \left (c x \right )+e^{4} c^{2} x^{2} \sqrt {c^{2} x^{2}+1}+6 c^{2} d \,e^{3} x \sqrt {c^{2} x^{2}+1}+18 c^{2} d^{2} e^{2} \sqrt {c^{2} x^{2}+1}-6 c d \,e^{3} \arcsinh \left (c x \right )-2 e^{4} \sqrt {c^{2} x^{2}+1}\right )}{12 e \sqrt {\frac {c^{2} x^{2}+1}{c^{2} x^{2}}}\, c x}\right )}{c^{3}}}{c} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.32, size = 261, normalized size = 1.56 \[ \frac {1}{4} \, a e^{3} x^{4} + a d e^{2} x^{3} + \frac {3}{2} \, a d^{2} e x^{2} + \frac {3}{2} \, {\left (x^{2} \operatorname {arcsch}\left (c x\right ) + \frac {x \sqrt {\frac {1}{c^{2} x^{2}} + 1}}{c}\right )} b d^{2} e + \frac {1}{4} \, {\left (4 \, x^{3} \operatorname {arcsch}\left (c x\right ) + \frac {\frac {2 \, \sqrt {\frac {1}{c^{2} x^{2}} + 1}}{c^{2} {\left (\frac {1}{c^{2} x^{2}} + 1\right )} - c^{2}} - \frac {\log \left (\sqrt {\frac {1}{c^{2} x^{2}} + 1} + 1\right )}{c^{2}} + \frac {\log \left (\sqrt {\frac {1}{c^{2} x^{2}} + 1} - 1\right )}{c^{2}}}{c}\right )} b d e^{2} + \frac {1}{12} \, {\left (3 \, x^{4} \operatorname {arcsch}\left (c x\right ) + \frac {c^{2} x^{3} {\left (\frac {1}{c^{2} x^{2}} + 1\right )}^{\frac {3}{2}} - 3 \, x \sqrt {\frac {1}{c^{2} x^{2}} + 1}}{c^{3}}\right )} b e^{3} + a d^{3} x + \frac {{\left (2 \, c x \operatorname {arcsch}\left (c x\right ) + \log \left (\sqrt {\frac {1}{c^{2} x^{2}} + 1} + 1\right ) - \log \left (\sqrt {\frac {1}{c^{2} x^{2}} + 1} - 1\right )\right )} b d^{3}}{2 \, c} \]
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 \[ \int \left (a+b\,\mathrm {asinh}\left (\frac {1}{c\,x}\right )\right )\,{\left (d+e\,x\right )}^3 \,d x \]
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 \[ \int \left (a + b \operatorname {acsch}{\left (c x \right )}\right ) \left (d + e x\right )^{3}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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