Optimal. Leaf size=163 \[ -\frac {a+b \text {csch}^{-1}(c x)}{2 e (d+e x)^2}-\frac {b c e \sqrt {\frac {1}{c^2 x^2}+1}}{2 d \left (c^2 d^2+e^2\right ) \left (\frac {d}{x}+e\right )}+\frac {b \left (2 c^2 d^2+e^2\right ) \tanh ^{-1}\left (\frac {c^2 d-\frac {e}{x}}{c \sqrt {\frac {1}{c^2 x^2}+1} \sqrt {c^2 d^2+e^2}}\right )}{2 d^2 \left (c^2 d^2+e^2\right )^{3/2}}+\frac {b \text {csch}^{-1}(c x)}{2 d^2 e} \]
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Rubi [A] time = 0.29, antiderivative size = 163, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 8, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {6290, 1568, 1475, 1651, 844, 215, 725, 206} \[ -\frac {a+b \text {csch}^{-1}(c x)}{2 e (d+e x)^2}-\frac {b c e \sqrt {\frac {1}{c^2 x^2}+1}}{2 d \left (c^2 d^2+e^2\right ) \left (\frac {d}{x}+e\right )}+\frac {b \left (2 c^2 d^2+e^2\right ) \tanh ^{-1}\left (\frac {c^2 d-\frac {e}{x}}{c \sqrt {\frac {1}{c^2 x^2}+1} \sqrt {c^2 d^2+e^2}}\right )}{2 d^2 \left (c^2 d^2+e^2\right )^{3/2}}+\frac {b \text {csch}^{-1}(c x)}{2 d^2 e} \]
Antiderivative was successfully verified.
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Rule 206
Rule 215
Rule 725
Rule 844
Rule 1475
Rule 1568
Rule 1651
Rule 6290
Rubi steps
\begin {align*} \int \frac {a+b \text {csch}^{-1}(c x)}{(d+e x)^3} \, dx &=-\frac {a+b \text {csch}^{-1}(c x)}{2 e (d+e x)^2}-\frac {b \int \frac {1}{\sqrt {1+\frac {1}{c^2 x^2}} x^2 (d+e x)^2} \, dx}{2 c e}\\ &=-\frac {a+b \text {csch}^{-1}(c x)}{2 e (d+e x)^2}-\frac {b \int \frac {1}{\sqrt {1+\frac {1}{c^2 x^2}} \left (e+\frac {d}{x}\right )^2 x^4} \, dx}{2 c e}\\ &=-\frac {a+b \text {csch}^{-1}(c x)}{2 e (d+e x)^2}+\frac {b \operatorname {Subst}\left (\int \frac {x^2}{(e+d x)^2 \sqrt {1+\frac {x^2}{c^2}}} \, dx,x,\frac {1}{x}\right )}{2 c e}\\ &=-\frac {b c e \sqrt {1+\frac {1}{c^2 x^2}}}{2 d \left (c^2 d^2+e^2\right ) \left (e+\frac {d}{x}\right )}-\frac {a+b \text {csch}^{-1}(c x)}{2 e (d+e x)^2}-\frac {(b c) \operatorname {Subst}\left (\int \frac {e-\left (d+\frac {e^2}{c^2 d}\right ) x}{(e+d x) \sqrt {1+\frac {x^2}{c^2}}} \, dx,x,\frac {1}{x}\right )}{2 e \left (c^2 d^2+e^2\right )}\\ &=-\frac {b c e \sqrt {1+\frac {1}{c^2 x^2}}}{2 d \left (c^2 d^2+e^2\right ) \left (e+\frac {d}{x}\right )}-\frac {a+b \text {csch}^{-1}(c x)}{2 e (d+e x)^2}+\frac {b \operatorname {Subst}\left (\int \frac {1}{\sqrt {1+\frac {x^2}{c^2}}} \, dx,x,\frac {1}{x}\right )}{2 c d^2 e}-\frac {\left (b c \left (2+\frac {e^2}{c^2 d^2}\right )\right ) \operatorname {Subst}\left (\int \frac {1}{(e+d x) \sqrt {1+\frac {x^2}{c^2}}} \, dx,x,\frac {1}{x}\right )}{2 \left (c^2 d^2+e^2\right )}\\ &=-\frac {b c e \sqrt {1+\frac {1}{c^2 x^2}}}{2 d \left (c^2 d^2+e^2\right ) \left (e+\frac {d}{x}\right )}+\frac {b \text {csch}^{-1}(c x)}{2 d^2 e}-\frac {a+b \text {csch}^{-1}(c x)}{2 e (d+e x)^2}+\frac {\left (b c \left (2+\frac {e^2}{c^2 d^2}\right )\right ) \operatorname {Subst}\left (\int \frac {1}{d^2+\frac {e^2}{c^2}-x^2} \, dx,x,\frac {d-\frac {e}{c^2 x}}{\sqrt {1+\frac {1}{c^2 x^2}}}\right )}{2 \left (c^2 d^2+e^2\right )}\\ &=-\frac {b c e \sqrt {1+\frac {1}{c^2 x^2}}}{2 d \left (c^2 d^2+e^2\right ) \left (e+\frac {d}{x}\right )}+\frac {b \text {csch}^{-1}(c x)}{2 d^2 e}-\frac {a+b \text {csch}^{-1}(c x)}{2 e (d+e x)^2}+\frac {b \left (2 c^2 d^2+e^2\right ) \tanh ^{-1}\left (\frac {c^2 d-\frac {e}{x}}{c \sqrt {c^2 d^2+e^2} \sqrt {1+\frac {1}{c^2 x^2}}}\right )}{2 d^2 \left (c^2 d^2+e^2\right )^{3/2}}\\ \end {align*}
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Mathematica [A] time = 0.48, size = 204, normalized size = 1.25 \[ \frac {1}{2} \left (-\frac {a}{e (d+e x)^2}-\frac {b c e x \sqrt {\frac {1}{c^2 x^2}+1}}{d \left (c^2 d^2+e^2\right ) (d+e x)}-\frac {b \left (2 c^2 d^2+e^2\right ) \log \left (c x \left (\sqrt {\frac {1}{c^2 x^2}+1} \sqrt {c^2 d^2+e^2}-c d\right )+e\right )}{d^2 \left (c^2 d^2+e^2\right )^{3/2}}+\frac {b \left (2 c^2 d^2+e^2\right ) \log (d+e x)}{d^2 \left (c^2 d^2+e^2\right )^{3/2}}+\frac {b \sinh ^{-1}\left (\frac {1}{c x}\right )}{d^2 e}-\frac {b \text {csch}^{-1}(c x)}{e (d+e x)^2}\right ) \]
Antiderivative was successfully verified.
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fricas [B] time = 1.34, size = 745, normalized size = 4.57 \[ -\frac {a c^{4} d^{6} + b c^{3} d^{5} e + 2 \, a c^{2} d^{4} e^{2} + b c d^{3} e^{3} + a d^{2} e^{4} + {\left (b c^{3} d^{3} e^{3} + b c d e^{5}\right )} x^{2} - {\left (2 \, b c^{2} d^{4} e + b d^{2} e^{3} + {\left (2 \, b c^{2} d^{2} e^{3} + b e^{5}\right )} x^{2} + 2 \, {\left (2 \, b c^{2} d^{3} e^{2} + b d e^{4}\right )} x\right )} \sqrt {c^{2} d^{2} + e^{2}} \log \left (-\frac {c^{3} d^{2} x - c d e + {\left (c^{3} d^{2} + c e^{2}\right )} x \sqrt {\frac {c^{2} x^{2} + 1}{c^{2} x^{2}}} + {\left (c^{2} d x \sqrt {\frac {c^{2} x^{2} + 1}{c^{2} x^{2}}} + c^{2} d x - e\right )} \sqrt {c^{2} d^{2} + e^{2}}}{e x + d}\right ) + 2 \, {\left (b c^{3} d^{4} e^{2} + b c d^{2} e^{4}\right )} x - {\left (b c^{4} d^{6} + 2 \, b c^{2} d^{4} e^{2} + b d^{2} e^{4} + {\left (b c^{4} d^{4} e^{2} + 2 \, b c^{2} d^{2} e^{4} + b e^{6}\right )} x^{2} + 2 \, {\left (b c^{4} d^{5} e + 2 \, b c^{2} d^{3} e^{3} + b d e^{5}\right )} x\right )} \log \left (c x \sqrt {\frac {c^{2} x^{2} + 1}{c^{2} x^{2}}} - c x + 1\right ) + {\left (b c^{4} d^{6} + 2 \, b c^{2} d^{4} e^{2} + b d^{2} e^{4} + {\left (b c^{4} d^{4} e^{2} + 2 \, b c^{2} d^{2} e^{4} + b e^{6}\right )} x^{2} + 2 \, {\left (b c^{4} d^{5} e + 2 \, b c^{2} d^{3} e^{3} + b d e^{5}\right )} x\right )} \log \left (c x \sqrt {\frac {c^{2} x^{2} + 1}{c^{2} x^{2}}} - c x - 1\right ) + {\left (b c^{4} d^{6} + 2 \, b c^{2} d^{4} e^{2} + b d^{2} e^{4}\right )} \log \left (\frac {c x \sqrt {\frac {c^{2} x^{2} + 1}{c^{2} x^{2}}} + 1}{c x}\right ) + {\left ({\left (b c^{3} d^{3} e^{3} + b c d e^{5}\right )} x^{2} + {\left (b c^{3} d^{4} e^{2} + b c d^{2} e^{4}\right )} x\right )} \sqrt {\frac {c^{2} x^{2} + 1}{c^{2} x^{2}}}}{2 \, {\left (c^{4} d^{8} e + 2 \, c^{2} d^{6} e^{3} + d^{4} e^{5} + {\left (c^{4} d^{6} e^{3} + 2 \, c^{2} d^{4} e^{5} + d^{2} e^{7}\right )} x^{2} + 2 \, {\left (c^{4} d^{7} e^{2} + 2 \, c^{2} d^{5} e^{4} + d^{3} e^{6}\right )} x\right )}} \]
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 \[ \int \frac {b \operatorname {arcsch}\left (c x\right ) + a}{{\left (e x + d\right )}^{3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.07, size = 963, normalized size = 5.91 \[ -\frac {c^{2} a}{2 \left (c x e +c d \right )^{2} e}-\frac {c^{2} b \,\mathrm {arccsch}\left (c x \right )}{2 \left (c x e +c d \right )^{2} e}+\frac {c^{2} b \sqrt {c^{2} x^{2}+1}\, \arctanh \left (\frac {1}{\sqrt {c^{2} x^{2}+1}}\right )}{2 \sqrt {\frac {c^{2} x^{2}+1}{c^{2} x^{2}}}\, \left (c^{2} d^{2}+e^{2}\right ) \left (c x e +c d \right )}+\frac {c^{2} b \sqrt {c^{2} x^{2}+1}\, d \arctanh \left (\frac {1}{\sqrt {c^{2} x^{2}+1}}\right )}{2 e \sqrt {\frac {c^{2} x^{2}+1}{c^{2} x^{2}}}\, x \left (c^{2} d^{2}+e^{2}\right ) \left (c x e +c d \right )}-\frac {c^{2} b \sqrt {c^{2} x^{2}+1}\, \ln \left (\frac {2 \sqrt {c^{2} x^{2}+1}\, \sqrt {\frac {c^{2} d^{2}+e^{2}}{e^{2}}}\, e -2 c^{2} d x +2 e}{c x e +c d}\right )}{\sqrt {\frac {c^{2} x^{2}+1}{c^{2} x^{2}}}\, \sqrt {\frac {c^{2} d^{2}+e^{2}}{e^{2}}}\, \left (c^{2} d^{2}+e^{2}\right ) \left (c x e +c d \right )}-\frac {c^{2} b \sqrt {c^{2} x^{2}+1}\, d \ln \left (\frac {2 \sqrt {c^{2} x^{2}+1}\, \sqrt {\frac {c^{2} d^{2}+e^{2}}{e^{2}}}\, e -2 c^{2} d x +2 e}{c x e +c d}\right )}{e \sqrt {\frac {c^{2} x^{2}+1}{c^{2} x^{2}}}\, x \sqrt {\frac {c^{2} d^{2}+e^{2}}{e^{2}}}\, \left (c^{2} d^{2}+e^{2}\right ) \left (c x e +c d \right )}-\frac {c^{2} b e x}{2 \sqrt {\frac {c^{2} x^{2}+1}{c^{2} x^{2}}}\, d \left (c^{2} d^{2}+e^{2}\right ) \left (c x e +c d \right )}-\frac {b e}{2 \sqrt {\frac {c^{2} x^{2}+1}{c^{2} x^{2}}}\, x d \left (c^{2} d^{2}+e^{2}\right ) \left (c x e +c d \right )}+\frac {b \,e^{2} \sqrt {c^{2} x^{2}+1}\, \arctanh \left (\frac {1}{\sqrt {c^{2} x^{2}+1}}\right )}{2 \sqrt {\frac {c^{2} x^{2}+1}{c^{2} x^{2}}}\, d^{2} \left (c^{2} d^{2}+e^{2}\right ) \left (c x e +c d \right )}+\frac {b e \sqrt {c^{2} x^{2}+1}\, \arctanh \left (\frac {1}{\sqrt {c^{2} x^{2}+1}}\right )}{2 \sqrt {\frac {c^{2} x^{2}+1}{c^{2} x^{2}}}\, x d \left (c^{2} d^{2}+e^{2}\right ) \left (c x e +c d \right )}-\frac {b \,e^{2} \sqrt {c^{2} x^{2}+1}\, \ln \left (\frac {2 \sqrt {c^{2} x^{2}+1}\, \sqrt {\frac {c^{2} d^{2}+e^{2}}{e^{2}}}\, e -2 c^{2} d x +2 e}{c x e +c d}\right )}{2 \sqrt {\frac {c^{2} x^{2}+1}{c^{2} x^{2}}}\, d^{2} \sqrt {\frac {c^{2} d^{2}+e^{2}}{e^{2}}}\, \left (c^{2} d^{2}+e^{2}\right ) \left (c x e +c d \right )}-\frac {b e \sqrt {c^{2} x^{2}+1}\, \ln \left (\frac {2 \sqrt {c^{2} x^{2}+1}\, \sqrt {\frac {c^{2} d^{2}+e^{2}}{e^{2}}}\, e -2 c^{2} d x +2 e}{c x e +c d}\right )}{2 \sqrt {\frac {c^{2} x^{2}+1}{c^{2} x^{2}}}\, x d \sqrt {\frac {c^{2} d^{2}+e^{2}}{e^{2}}}\, \left (c^{2} d^{2}+e^{2}\right ) \left (c x e +c d \right )} \]
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 \[ -\frac {1}{4} \, {\left (\frac {2 i \, c^{3} d {\left (\log \left (i \, c x + 1\right ) - \log \left (-i \, c x + 1\right )\right )}}{c^{4} d^{4} + 2 \, c^{2} d^{2} e^{2} + e^{4}} + 4 \, c^{2} \int \frac {x}{2 \, {\left (c^{2} e^{3} x^{4} + 2 \, c^{2} d e^{2} x^{3} + 2 \, d e^{2} x + d^{2} e + {\left (c^{2} d^{2} e + e^{3}\right )} x^{2} + {\left (c^{2} e^{3} x^{4} + 2 \, c^{2} d e^{2} x^{3} + 2 \, d e^{2} x + d^{2} e + {\left (c^{2} d^{2} e + e^{3}\right )} x^{2}\right )} \sqrt {c^{2} x^{2} + 1}\right )}}\,{d x} - \frac {2 \, {\left (3 \, c^{2} d^{2} e + e^{3}\right )} \log \left (e x + d\right )}{c^{4} d^{6} + 2 \, c^{2} d^{4} e^{2} + d^{2} e^{4}} - \frac {2 \, c^{4} d^{6} \log \relax (c) + 2 \, d^{2} e^{4} \log \relax (c) - 2 \, d^{2} e^{4} + 2 \, {\left (2 \, d^{4} e^{2} \log \relax (c) - d^{4} e^{2}\right )} c^{2} - 2 \, {\left (c^{2} d^{3} e^{3} + d e^{5}\right )} x + {\left (c^{4} d^{6} - c^{2} d^{4} e^{2} + {\left (c^{4} d^{4} e^{2} - c^{2} d^{2} e^{4}\right )} x^{2} + 2 \, {\left (c^{4} d^{5} e - c^{2} d^{3} e^{3}\right )} x\right )} \log \left (c^{2} x^{2} + 1\right ) - 2 \, {\left ({\left (c^{4} d^{4} e^{2} + 2 \, c^{2} d^{2} e^{4} + e^{6}\right )} x^{2} + 2 \, {\left (c^{4} d^{5} e + 2 \, c^{2} d^{3} e^{3} + d e^{5}\right )} x\right )} \log \relax (x) - 2 \, {\left (c^{4} d^{6} + 2 \, c^{2} d^{4} e^{2} + d^{2} e^{4}\right )} \log \left (\sqrt {c^{2} x^{2} + 1} + 1\right )}{c^{4} d^{8} e + 2 \, c^{2} d^{6} e^{3} + d^{4} e^{5} + {\left (c^{4} d^{6} e^{3} + 2 \, c^{2} d^{4} e^{5} + d^{2} e^{7}\right )} x^{2} + 2 \, {\left (c^{4} d^{7} e^{2} + 2 \, c^{2} d^{5} e^{4} + d^{3} e^{6}\right )} x}\right )} b - \frac {a}{2 \, {\left (e^{3} x^{2} + 2 \, d e^{2} x + d^{2} e\right )}} \]
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 \[ \int \frac {a+b\,\mathrm {asinh}\left (\frac {1}{c\,x}\right )}{{\left (d+e\,x\right )}^3} \,d x \]
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 \[ \int \frac {a + b \operatorname {acsch}{\left (c x \right )}}{\left (d + e x\right )^{3}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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