Optimal. Leaf size=81 \[ \frac {(d+e x)^2 \left (a+b \text {csch}^{-1}(c x)\right )}{2 e}+\frac {b d \tanh ^{-1}\left (\sqrt {\frac {1}{c^2 x^2}+1}\right )}{c}+\frac {b e x \sqrt {\frac {1}{c^2 x^2}+1}}{2 c}-\frac {b d^2 \text {csch}^{-1}(c x)}{2 e} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.16, antiderivative size = 81, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 9, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.643, Rules used = {6290, 1568, 1396, 1807, 844, 215, 266, 63, 208} \[ \frac {(d+e x)^2 \left (a+b \text {csch}^{-1}(c x)\right )}{2 e}+\frac {b d \tanh ^{-1}\left (\sqrt {\frac {1}{c^2 x^2}+1}\right )}{c}+\frac {b e x \sqrt {\frac {1}{c^2 x^2}+1}}{2 c}-\frac {b d^2 \text {csch}^{-1}(c x)}{2 e} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 63
Rule 208
Rule 215
Rule 266
Rule 844
Rule 1396
Rule 1568
Rule 1807
Rule 6290
Rubi steps
\begin {align*} \int (d+e x) \left (a+b \text {csch}^{-1}(c x)\right ) \, dx &=\frac {(d+e x)^2 \left (a+b \text {csch}^{-1}(c x)\right )}{2 e}+\frac {b \int \frac {(d+e x)^2}{\sqrt {1+\frac {1}{c^2 x^2}} x^2} \, dx}{2 c e}\\ &=\frac {(d+e x)^2 \left (a+b \text {csch}^{-1}(c x)\right )}{2 e}+\frac {b \int \frac {\left (e+\frac {d}{x}\right )^2}{\sqrt {1+\frac {1}{c^2 x^2}}} \, dx}{2 c e}\\ &=\frac {(d+e x)^2 \left (a+b \text {csch}^{-1}(c x)\right )}{2 e}-\frac {b \operatorname {Subst}\left (\int \frac {(e+d x)^2}{x^2 \sqrt {1+\frac {x^2}{c^2}}} \, dx,x,\frac {1}{x}\right )}{2 c e}\\ &=\frac {b e \sqrt {1+\frac {1}{c^2 x^2}} x}{2 c}+\frac {(d+e x)^2 \left (a+b \text {csch}^{-1}(c x)\right )}{2 e}+\frac {b \operatorname {Subst}\left (\int \frac {-2 d e-d^2 x}{x \sqrt {1+\frac {x^2}{c^2}}} \, dx,x,\frac {1}{x}\right )}{2 c e}\\ &=\frac {b e \sqrt {1+\frac {1}{c^2 x^2}} x}{2 c}+\frac {(d+e x)^2 \left (a+b \text {csch}^{-1}(c x)\right )}{2 e}-\frac {(b d) \operatorname {Subst}\left (\int \frac {1}{x \sqrt {1+\frac {x^2}{c^2}}} \, dx,x,\frac {1}{x}\right )}{c}-\frac {\left (b d^2\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {1+\frac {x^2}{c^2}}} \, dx,x,\frac {1}{x}\right )}{2 c e}\\ &=\frac {b e \sqrt {1+\frac {1}{c^2 x^2}} x}{2 c}-\frac {b d^2 \text {csch}^{-1}(c x)}{2 e}+\frac {(d+e x)^2 \left (a+b \text {csch}^{-1}(c x)\right )}{2 e}-\frac {(b d) \operatorname {Subst}\left (\int \frac {1}{x \sqrt {1+\frac {x}{c^2}}} \, dx,x,\frac {1}{x^2}\right )}{2 c}\\ &=\frac {b e \sqrt {1+\frac {1}{c^2 x^2}} x}{2 c}-\frac {b d^2 \text {csch}^{-1}(c x)}{2 e}+\frac {(d+e x)^2 \left (a+b \text {csch}^{-1}(c x)\right )}{2 e}-(b c d) \operatorname {Subst}\left (\int \frac {1}{-c^2+c^2 x^2} \, dx,x,\sqrt {1+\frac {1}{c^2 x^2}}\right )\\ &=\frac {b e \sqrt {1+\frac {1}{c^2 x^2}} x}{2 c}-\frac {b d^2 \text {csch}^{-1}(c x)}{2 e}+\frac {(d+e x)^2 \left (a+b \text {csch}^{-1}(c x)\right )}{2 e}+\frac {b d \tanh ^{-1}\left (\sqrt {1+\frac {1}{c^2 x^2}}\right )}{c}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.20, size = 99, normalized size = 1.22 \[ a d x+\frac {1}{2} a e x^2+\frac {b d x \sqrt {\frac {1}{c^2 x^2}+1} \sinh ^{-1}(c x)}{\sqrt {c^2 x^2+1}}+\frac {b e x \sqrt {\frac {c^2 x^2+1}{c^2 x^2}}}{2 c}+b d x \text {csch}^{-1}(c x)+\frac {1}{2} b e x^2 \text {csch}^{-1}(c x) \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [B] time = 1.70, size = 207, normalized size = 2.56 \[ \frac {a c e x^{2} + 2 \, a c d x + b e x \sqrt {\frac {c^{2} x^{2} + 1}{c^{2} x^{2}}} - 2 \, b d \log \left (c x \sqrt {\frac {c^{2} x^{2} + 1}{c^{2} x^{2}}} - c x\right ) + {\left (2 \, b c d + b c e\right )} \log \left (c x \sqrt {\frac {c^{2} x^{2} + 1}{c^{2} x^{2}}} - c x + 1\right ) - {\left (2 \, b c d + b c e\right )} \log \left (c x \sqrt {\frac {c^{2} x^{2} + 1}{c^{2} x^{2}}} - c x - 1\right ) + {\left (b c e x^{2} + 2 \, b c d x - 2 \, b c d - b c e\right )} \log \left (\frac {c x \sqrt {\frac {c^{2} x^{2} + 1}{c^{2} x^{2}}} + 1}{c x}\right )}{2 \, c} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (e x + d\right )} {\left (b \operatorname {arcsch}\left (c x\right ) + a\right )}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.05, size = 115, normalized size = 1.42 \[ \frac {\frac {a \left (\frac {1}{2} c^{2} x^{2} e +c^{2} d x \right )}{c}+\frac {b \left (\frac {\mathrm {arccsch}\left (c x \right ) c^{2} x^{2} e}{2}+\mathrm {arccsch}\left (c x \right ) c^{2} x d +\frac {\sqrt {c^{2} x^{2}+1}\, \left (e \sqrt {c^{2} x^{2}+1}+2 c d \arcsinh \left (c x \right )\right )}{2 \sqrt {\frac {c^{2} x^{2}+1}{c^{2} x^{2}}}\, c x}\right )}{c}}{c} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 0.39, size = 87, normalized size = 1.07 \[ \frac {1}{2} \, a e x^{2} + \frac {1}{2} \, {\left (x^{2} \operatorname {arcsch}\left (c x\right ) + \frac {x \sqrt {\frac {1}{c^{2} x^{2}} + 1}}{c}\right )} b e + a d x + \frac {{\left (2 \, c x \operatorname {arcsch}\left (c x\right ) + \log \left (\sqrt {\frac {1}{c^{2} x^{2}} + 1} + 1\right ) - \log \left (\sqrt {\frac {1}{c^{2} x^{2}} + 1} - 1\right )\right )} b d}{2 \, c} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \left (a+b\,\mathrm {asinh}\left (\frac {1}{c\,x}\right )\right )\,\left (d+e\,x\right ) \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \left (a + b \operatorname {acsch}{\left (c x \right )}\right ) \left (d + e x\right )\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________