Optimal. Leaf size=107 \[ -\frac {a^2 \text {sech}^{-1}(a+b x)}{2 b^2}-\frac {\sqrt {\frac {-a-b x+1}{a+b x+1}} (a+b x+1)}{2 b^2}+\frac {a \tan ^{-1}\left (\frac {\sqrt {\frac {-a-b x+1}{a+b x+1}} (a+b x+1)}{a+b x}\right )}{b^2}+\frac {1}{2} x^2 \text {sech}^{-1}(a+b x) \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.06, antiderivative size = 107, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, integrand size = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.750, Rules used = {6321, 5468, 3773, 3770, 3767, 8} \[ -\frac {a^2 \text {sech}^{-1}(a+b x)}{2 b^2}-\frac {\sqrt {\frac {-a-b x+1}{a+b x+1}} (a+b x+1)}{2 b^2}+\frac {a \tan ^{-1}\left (\frac {\sqrt {\frac {-a-b x+1}{a+b x+1}} (a+b x+1)}{a+b x}\right )}{b^2}+\frac {1}{2} x^2 \text {sech}^{-1}(a+b x) \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 8
Rule 3767
Rule 3770
Rule 3773
Rule 5468
Rule 6321
Rubi steps
\begin {align*} \int x \text {sech}^{-1}(a+b x) \, dx &=-\frac {\operatorname {Subst}\left (\int x \text {sech}(x) (-a+\text {sech}(x)) \tanh (x) \, dx,x,\text {sech}^{-1}(a+b x)\right )}{b^2}\\ &=\frac {1}{2} x^2 \text {sech}^{-1}(a+b x)-\frac {\operatorname {Subst}\left (\int (-a+\text {sech}(x))^2 \, dx,x,\text {sech}^{-1}(a+b x)\right )}{2 b^2}\\ &=-\frac {a^2 \text {sech}^{-1}(a+b x)}{2 b^2}+\frac {1}{2} x^2 \text {sech}^{-1}(a+b x)-\frac {\operatorname {Subst}\left (\int \text {sech}^2(x) \, dx,x,\text {sech}^{-1}(a+b x)\right )}{2 b^2}+\frac {a \operatorname {Subst}\left (\int \text {sech}(x) \, dx,x,\text {sech}^{-1}(a+b x)\right )}{b^2}\\ &=-\frac {a^2 \text {sech}^{-1}(a+b x)}{2 b^2}+\frac {1}{2} x^2 \text {sech}^{-1}(a+b x)+\frac {a \tan ^{-1}\left (\frac {\sqrt {\frac {1-a-b x}{1+a+b x}} (1+a+b x)}{a+b x}\right )}{b^2}-\frac {i \operatorname {Subst}\left (\int 1 \, dx,x,-i \sqrt {\frac {1-a-b x}{1+a+b x}} (1+a+b x)\right )}{2 b^2}\\ &=-\frac {\sqrt {\frac {1-a-b x}{1+a+b x}} (1+a+b x)}{2 b^2}-\frac {a^2 \text {sech}^{-1}(a+b x)}{2 b^2}+\frac {1}{2} x^2 \text {sech}^{-1}(a+b x)+\frac {a \tan ^{-1}\left (\frac {\sqrt {\frac {1-a-b x}{1+a+b x}} (1+a+b x)}{a+b x}\right )}{b^2}\\ \end {align*}
________________________________________________________________________________________
Mathematica [C] time = 0.17, size = 176, normalized size = 1.64 \[ \frac {a^2 \log (a+b x)-a^2 \log \left (a \sqrt {-\frac {a+b x-1}{a+b x+1}}+b x \sqrt {-\frac {a+b x-1}{a+b x+1}}+\sqrt {-\frac {a+b x-1}{a+b x+1}}+1\right )+b^2 x^2 \text {sech}^{-1}(a+b x)-\sqrt {-\frac {a+b x-1}{a+b x+1}} (a+b x+1)-2 i a \log \left (2 \sqrt {-\frac {a+b x-1}{a+b x+1}} (a+b x+1)-2 i (a+b x)\right )}{2 b^2} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [B] time = 1.39, size = 308, normalized size = 2.88 \[ \frac {2 \, b^{2} x^{2} \log \left (\frac {{\left (b x + a\right )} \sqrt {-\frac {b^{2} x^{2} + 2 \, a b x + a^{2} - 1}{b^{2} x^{2} + 2 \, a b x + a^{2}}} + 1}{b x + a}\right ) - a^{2} \log \left (\frac {{\left (b x + a\right )} \sqrt {-\frac {b^{2} x^{2} + 2 \, a b x + a^{2} - 1}{b^{2} x^{2} + 2 \, a b x + a^{2}}} + 1}{x}\right ) + a^{2} \log \left (\frac {{\left (b x + a\right )} \sqrt {-\frac {b^{2} x^{2} + 2 \, a b x + a^{2} - 1}{b^{2} x^{2} + 2 \, a b x + a^{2}}} - 1}{x}\right ) + 4 \, a \arctan \left (\frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )} \sqrt {-\frac {b^{2} x^{2} + 2 \, a b x + a^{2} - 1}{b^{2} x^{2} + 2 \, a b x + a^{2}}}}{b^{2} x^{2} + 2 \, a b x + a^{2} - 1}\right ) - 2 \, {\left (b x + a\right )} \sqrt {-\frac {b^{2} x^{2} + 2 \, a b x + a^{2} - 1}{b^{2} x^{2} + 2 \, a b x + a^{2}}}}{4 \, b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int x \operatorname {arsech}\left (b x + a\right )\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.07, size = 111, normalized size = 1.04 \[ \frac {\frac {\mathrm {arcsech}\left (b x +a \right ) \left (b x +a \right )^{2}}{2}-\mathrm {arcsech}\left (b x +a \right ) a \left (b x +a \right )-\frac {\sqrt {-\frac {b x +a -1}{b x +a}}\, \left (b x +a \right ) \sqrt {\frac {b x +a +1}{b x +a}}\, \left (2 a \arcsin \left (b x +a \right )+\sqrt {1-\left (b x +a \right )^{2}}\right )}{2 \sqrt {1-\left (b x +a \right )^{2}}}}{b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \frac {2 \, b^{2} x^{2} \log \left (\sqrt {b x + a + 1} \sqrt {-b x - a + 1} b x + \sqrt {b x + a + 1} \sqrt {-b x - a + 1} a + b x + a\right ) - 2 \, b^{2} x^{2} \log \left (b x + a\right ) - {\left (a^{2} + 2 \, a + 1\right )} \log \left (b x + a + 1\right ) - 2 \, {\left (b^{2} x^{2} - a^{2}\right )} \log \left (b x + a\right ) - {\left (a^{2} - 2 \, a + 1\right )} \log \left (-b x - a + 1\right )}{4 \, b^{2}} + \int \frac {b^{2} x^{3} + a b x^{2}}{2 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2} + {\left (b^{2} x^{2} + 2 \, a b x + a^{2} - 1\right )} e^{\left (\frac {1}{2} \, \log \left (b x + a + 1\right ) + \frac {1}{2} \, \log \left (-b x - a + 1\right )\right )} - 1\right )}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int x\,\mathrm {acosh}\left (\frac {1}{a+b\,x}\right ) \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int x \operatorname {asech}{\left (a + b x \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________