Optimal. Leaf size=107 \[ \frac {(a+b x) \sqrt {(a+b x)^2+1}}{4 b}+\frac {\sinh ^{-1}(a+b x)^3}{6 b}+\frac {(a+b x) \sqrt {(a+b x)^2+1} \sinh ^{-1}(a+b x)^2}{2 b}-\frac {(a+b x)^2 \sinh ^{-1}(a+b x)}{2 b}-\frac {\sinh ^{-1}(a+b x)}{4 b} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.12, antiderivative size = 107, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, integrand size = 30, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {5867, 5682, 5675, 5661, 321, 215} \[ \frac {(a+b x) \sqrt {(a+b x)^2+1}}{4 b}+\frac {\sinh ^{-1}(a+b x)^3}{6 b}+\frac {(a+b x) \sqrt {(a+b x)^2+1} \sinh ^{-1}(a+b x)^2}{2 b}-\frac {(a+b x)^2 \sinh ^{-1}(a+b x)}{2 b}-\frac {\sinh ^{-1}(a+b x)}{4 b} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 215
Rule 321
Rule 5661
Rule 5675
Rule 5682
Rule 5867
Rubi steps
\begin {align*} \int \sqrt {1+a^2+2 a b x+b^2 x^2} \sinh ^{-1}(a+b x)^2 \, dx &=\frac {\operatorname {Subst}\left (\int \sqrt {1+x^2} \sinh ^{-1}(x)^2 \, dx,x,a+b x\right )}{b}\\ &=\frac {(a+b x) \sqrt {1+(a+b x)^2} \sinh ^{-1}(a+b x)^2}{2 b}+\frac {\operatorname {Subst}\left (\int \frac {\sinh ^{-1}(x)^2}{\sqrt {1+x^2}} \, dx,x,a+b x\right )}{2 b}-\frac {\operatorname {Subst}\left (\int x \sinh ^{-1}(x) \, dx,x,a+b x\right )}{b}\\ &=-\frac {(a+b x)^2 \sinh ^{-1}(a+b x)}{2 b}+\frac {(a+b x) \sqrt {1+(a+b x)^2} \sinh ^{-1}(a+b x)^2}{2 b}+\frac {\sinh ^{-1}(a+b x)^3}{6 b}+\frac {\operatorname {Subst}\left (\int \frac {x^2}{\sqrt {1+x^2}} \, dx,x,a+b x\right )}{2 b}\\ &=\frac {(a+b x) \sqrt {1+(a+b x)^2}}{4 b}-\frac {(a+b x)^2 \sinh ^{-1}(a+b x)}{2 b}+\frac {(a+b x) \sqrt {1+(a+b x)^2} \sinh ^{-1}(a+b x)^2}{2 b}+\frac {\sinh ^{-1}(a+b x)^3}{6 b}-\frac {\operatorname {Subst}\left (\int \frac {1}{\sqrt {1+x^2}} \, dx,x,a+b x\right )}{4 b}\\ &=\frac {(a+b x) \sqrt {1+(a+b x)^2}}{4 b}-\frac {\sinh ^{-1}(a+b x)}{4 b}-\frac {(a+b x)^2 \sinh ^{-1}(a+b x)}{2 b}+\frac {(a+b x) \sqrt {1+(a+b x)^2} \sinh ^{-1}(a+b x)^2}{2 b}+\frac {\sinh ^{-1}(a+b x)^3}{6 b}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.09, size = 110, normalized size = 1.03 \[ \frac {3 (a+b x) \sqrt {a^2+2 a b x+b^2 x^2+1}+6 (a+b x) \sqrt {a^2+2 a b x+b^2 x^2+1} \sinh ^{-1}(a+b x)^2-3 \left (2 a^2+4 a b x+2 b^2 x^2+1\right ) \sinh ^{-1}(a+b x)+2 \sinh ^{-1}(a+b x)^3}{12 b} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.71, size = 161, normalized size = 1.50 \[ \frac {6 \, \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1} {\left (b x + a\right )} \log \left (b x + a + \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1}\right )^{2} + 2 \, \log \left (b x + a + \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1}\right )^{3} - 3 \, {\left (2 \, b^{2} x^{2} + 4 \, a b x + 2 \, a^{2} + 1\right )} \log \left (b x + a + \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1}\right ) + 3 \, \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1} {\left (b x + a\right )}}{12 \, b} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1} \operatorname {arsinh}\left (b x + a\right )^{2}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.11, size = 167, normalized size = 1.56 \[ \frac {6 \arcsinh \left (b x +a \right )^{2} \sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}\, x b -6 \arcsinh \left (b x +a \right ) x^{2} b^{2}+6 \arcsinh \left (b x +a \right )^{2} \sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}\, a -12 \arcsinh \left (b x +a \right ) x a b +2 \arcsinh \left (b x +a \right )^{3}-6 \arcsinh \left (b x +a \right ) a^{2}+3 \sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}\, x b +3 \sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}\, a -3 \arcsinh \left (b x +a \right )}{12 b} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1} \operatorname {arsinh}\left (b x + a\right )^{2}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int {\mathrm {asinh}\left (a+b\,x\right )}^2\,\sqrt {a^2+2\,a\,b\,x+b^2\,x^2+1} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \sqrt {a^{2} + 2 a b x + b^{2} x^{2} + 1} \operatorname {asinh}^{2}{\left (a + b x \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________