Optimal. Leaf size=98 \[ -\frac {a+b \text {csch}^{-1}(c x)}{e (d+e x)}+\frac {b \tanh ^{-1}\left (\frac {c^2 d-\frac {e}{x}}{c \sqrt {\frac {1}{c^2 x^2}+1} \sqrt {c^2 d^2+e^2}}\right )}{d \sqrt {c^2 d^2+e^2}}+\frac {b \text {csch}^{-1}(c x)}{d e} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.15, antiderivative size = 98, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 7, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.438, Rules used = {6290, 1568, 1475, 844, 215, 725, 206} \[ -\frac {a+b \text {csch}^{-1}(c x)}{e (d+e x)}+\frac {b \tanh ^{-1}\left (\frac {c^2 d-\frac {e}{x}}{c \sqrt {\frac {1}{c^2 x^2}+1} \sqrt {c^2 d^2+e^2}}\right )}{d \sqrt {c^2 d^2+e^2}}+\frac {b \text {csch}^{-1}(c x)}{d e} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 206
Rule 215
Rule 725
Rule 844
Rule 1475
Rule 1568
Rule 6290
Rubi steps
\begin {align*} \int \frac {a+b \text {csch}^{-1}(c x)}{(d+e x)^2} \, dx &=-\frac {a+b \text {csch}^{-1}(c x)}{e (d+e x)}-\frac {b \int \frac {1}{\sqrt {1+\frac {1}{c^2 x^2}} x^2 (d+e x)} \, dx}{c e}\\ &=-\frac {a+b \text {csch}^{-1}(c x)}{e (d+e x)}-\frac {b \int \frac {1}{\sqrt {1+\frac {1}{c^2 x^2}} \left (e+\frac {d}{x}\right ) x^3} \, dx}{c e}\\ &=-\frac {a+b \text {csch}^{-1}(c x)}{e (d+e x)}+\frac {b \operatorname {Subst}\left (\int \frac {x}{(e+d x) \sqrt {1+\frac {x^2}{c^2}}} \, dx,x,\frac {1}{x}\right )}{c e}\\ &=-\frac {a+b \text {csch}^{-1}(c x)}{e (d+e x)}-\frac {b \operatorname {Subst}\left (\int \frac {1}{(e+d x) \sqrt {1+\frac {x^2}{c^2}}} \, dx,x,\frac {1}{x}\right )}{c d}+\frac {b \operatorname {Subst}\left (\int \frac {1}{\sqrt {1+\frac {x^2}{c^2}}} \, dx,x,\frac {1}{x}\right )}{c d e}\\ &=\frac {b \text {csch}^{-1}(c x)}{d e}-\frac {a+b \text {csch}^{-1}(c x)}{e (d+e x)}+\frac {b \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 )}{c d}\\ &=\frac {b \text {csch}^{-1}(c x)}{d e}-\frac {a+b \text {csch}^{-1}(c x)}{e (d+e x)}+\frac {b \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 )}{d \sqrt {c^2 d^2+e^2}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.22, size = 134, normalized size = 1.37 \[ -\frac {a}{e (d+e x)}-\frac {b \log \left (c x \left (\sqrt {\frac {1}{c^2 x^2}+1} \sqrt {c^2 d^2+e^2}-c d\right )+e\right )}{d \sqrt {c^2 d^2+e^2}}+\frac {b \log (d+e x)}{d \sqrt {c^2 d^2+e^2}}+\frac {b \sinh ^{-1}\left (\frac {1}{c x}\right )}{d e}-\frac {b \text {csch}^{-1}(c x)}{e (d+e x)} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [B] time = 0.61, size = 354, normalized size = 3.61 \[ -\frac {a c^{2} d^{3} + a d e^{2} - \sqrt {c^{2} d^{2} + e^{2}} {\left (b e^{2} x + b d e\right )} \log \left (-\frac {c^{3} d^{2} x - c d e + {\left (c^{3} d^{2} + c e^{2}\right )} x \sqrt {\frac {c^{2} x^{2} + 1}{c^{2} x^{2}}} + {\left (c^{2} d x \sqrt {\frac {c^{2} x^{2} + 1}{c^{2} x^{2}}} + c^{2} d x - e\right )} \sqrt {c^{2} d^{2} + e^{2}}}{e x + d}\right ) - {\left (b c^{2} d^{3} + b d e^{2} + {\left (b c^{2} d^{2} e + b e^{3}\right )} x\right )} \log \left (c x \sqrt {\frac {c^{2} x^{2} + 1}{c^{2} x^{2}}} - c x + 1\right ) + {\left (b c^{2} d^{3} + b d e^{2} + {\left (b c^{2} d^{2} e + b e^{3}\right )} x\right )} \log \left (c x \sqrt {\frac {c^{2} x^{2} + 1}{c^{2} x^{2}}} - c x - 1\right ) + {\left (b c^{2} d^{3} + b d e^{2}\right )} \log \left (\frac {c x \sqrt {\frac {c^{2} x^{2} + 1}{c^{2} x^{2}}} + 1}{c x}\right )}{c^{2} d^{4} e + d^{2} e^{3} + {\left (c^{2} d^{3} e^{2} + d e^{4}\right )} x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {b \operatorname {arcsch}\left (c x\right ) + a}{{\left (e x + d\right )}^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [B] time = 0.09, size = 208, normalized size = 2.12 \[ -\frac {c a}{\left (c x e +c d \right ) e}-\frac {c b \,\mathrm {arccsch}\left (c x \right )}{\left (c x e +c d \right ) e}+\frac {b \sqrt {c^{2} x^{2}+1}\, \arctanh \left (\frac {1}{\sqrt {c^{2} x^{2}+1}}\right )}{c e \sqrt {\frac {c^{2} x^{2}+1}{c^{2} x^{2}}}\, x d}-\frac {b \sqrt {c^{2} x^{2}+1}\, \ln \left (\frac {2 \sqrt {c^{2} x^{2}+1}\, \sqrt {\frac {c^{2} d^{2}+e^{2}}{e^{2}}}\, e -2 c^{2} d x +2 e}{c x e +c d}\right )}{c e \sqrt {\frac {c^{2} x^{2}+1}{c^{2} x^{2}}}\, x d \sqrt {\frac {c^{2} d^{2}+e^{2}}{e^{2}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ -\frac {1}{2} \, {\left (2 \, c^{2} \int \frac {x}{c^{2} e^{2} x^{3} + c^{2} d e x^{2} + e^{2} x + d e + {\left (c^{2} e^{2} x^{3} + c^{2} d e x^{2} + e^{2} x + d e\right )} \sqrt {c^{2} x^{2} + 1}}\,{d x} + \frac {i \, c {\left (\log \left (i \, c x + 1\right ) - \log \left (-i \, c x + 1\right )\right )}}{c^{2} d^{2} + e^{2}} - \frac {2 \, e \log \left (e x + d\right )}{c^{2} d^{3} + d e^{2}} - \frac {2 \, c^{2} d^{3} \log \relax (c) + 2 \, d e^{2} \log \relax (c) - 2 \, {\left (c^{2} d^{2} e + e^{3}\right )} x \log \relax (x) + {\left (c^{2} d^{2} e x + c^{2} d^{3}\right )} \log \left (c^{2} x^{2} + 1\right ) - 2 \, {\left (c^{2} d^{3} + d e^{2}\right )} \log \left (\sqrt {c^{2} x^{2} + 1} + 1\right )}{c^{2} d^{4} e + d^{2} e^{3} + {\left (c^{2} d^{3} e^{2} + d e^{4}\right )} x}\right )} b - \frac {a}{e^{2} x + d e} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {a+b\,\mathrm {asinh}\left (\frac {1}{c\,x}\right )}{{\left (d+e\,x\right )}^2} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {a + b \operatorname {acsch}{\left (c x \right )}}{\left (d + e x\right )^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________