Optimal. Leaf size=84 \[ \frac {\text {Chi}\left (2 \sinh ^{-1}(a+b x)\right )}{b^2}-\frac {a \text {Shi}\left (\sinh ^{-1}(a+b x)\right )}{b^2}+\frac {a \sqrt {(a+b x)^2+1}}{b^2 \sinh ^{-1}(a+b x)}-\frac {(a+b x) \sqrt {(a+b x)^2+1}}{b^2 \sinh ^{-1}(a+b x)} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.13, antiderivative size = 84, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 7, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.700, Rules used = {5865, 5803, 5655, 5779, 3298, 5665, 3301} \[ \frac {\text {Chi}\left (2 \sinh ^{-1}(a+b x)\right )}{b^2}-\frac {a \text {Shi}\left (\sinh ^{-1}(a+b x)\right )}{b^2}+\frac {a \sqrt {(a+b x)^2+1}}{b^2 \sinh ^{-1}(a+b x)}-\frac {(a+b x) \sqrt {(a+b x)^2+1}}{b^2 \sinh ^{-1}(a+b x)} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 3298
Rule 3301
Rule 5655
Rule 5665
Rule 5779
Rule 5803
Rule 5865
Rubi steps
\begin {align*} \int \frac {x}{\sinh ^{-1}(a+b x)^2} \, dx &=\frac {\operatorname {Subst}\left (\int \frac {-\frac {a}{b}+\frac {x}{b}}{\sinh ^{-1}(x)^2} \, dx,x,a+b x\right )}{b}\\ &=\frac {\operatorname {Subst}\left (\int \left (-\frac {a}{b \sinh ^{-1}(x)^2}+\frac {x}{b \sinh ^{-1}(x)^2}\right ) \, dx,x,a+b x\right )}{b}\\ &=\frac {\operatorname {Subst}\left (\int \frac {x}{\sinh ^{-1}(x)^2} \, dx,x,a+b x\right )}{b^2}-\frac {a \operatorname {Subst}\left (\int \frac {1}{\sinh ^{-1}(x)^2} \, dx,x,a+b x\right )}{b^2}\\ &=\frac {a \sqrt {1+(a+b x)^2}}{b^2 \sinh ^{-1}(a+b x)}-\frac {(a+b x) \sqrt {1+(a+b x)^2}}{b^2 \sinh ^{-1}(a+b x)}+\frac {\operatorname {Subst}\left (\int \frac {\cosh (2 x)}{x} \, dx,x,\sinh ^{-1}(a+b x)\right )}{b^2}-\frac {a \operatorname {Subst}\left (\int \frac {x}{\sqrt {1+x^2} \sinh ^{-1}(x)} \, dx,x,a+b x\right )}{b^2}\\ &=\frac {a \sqrt {1+(a+b x)^2}}{b^2 \sinh ^{-1}(a+b x)}-\frac {(a+b x) \sqrt {1+(a+b x)^2}}{b^2 \sinh ^{-1}(a+b x)}+\frac {\text {Chi}\left (2 \sinh ^{-1}(a+b x)\right )}{b^2}-\frac {a \operatorname {Subst}\left (\int \frac {\sinh (x)}{x} \, dx,x,\sinh ^{-1}(a+b x)\right )}{b^2}\\ &=\frac {a \sqrt {1+(a+b x)^2}}{b^2 \sinh ^{-1}(a+b x)}-\frac {(a+b x) \sqrt {1+(a+b x)^2}}{b^2 \sinh ^{-1}(a+b x)}+\frac {\text {Chi}\left (2 \sinh ^{-1}(a+b x)\right )}{b^2}-\frac {a \text {Shi}\left (\sinh ^{-1}(a+b x)\right )}{b^2}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.17, size = 62, normalized size = 0.74 \[ -\frac {-\sinh ^{-1}(a+b x) \text {Chi}\left (2 \sinh ^{-1}(a+b x)\right )+a \sinh ^{-1}(a+b x) \text {Shi}\left (\sinh ^{-1}(a+b x)\right )+b x \sqrt {(a+b x)^2+1}}{b^2 \sinh ^{-1}(a+b x)} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [F] time = 0.58, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {x}{\operatorname {arsinh}\left (b x + a\right )^{2}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x}{\operatorname {arsinh}\left (b x + a\right )^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.08, size = 73, normalized size = 0.87 \[ \frac {-\frac {\sinh \left (2 \arcsinh \left (b x +a \right )\right )}{2 \arcsinh \left (b x +a \right )}+\Chi \left (2 \arcsinh \left (b x +a \right )\right )-\frac {a \left (\Shi \left (\arcsinh \left (b x +a \right )\right ) \arcsinh \left (b x +a \right )-\sqrt {1+\left (b x +a \right )^{2}}\right )}{\arcsinh \left (b x +a \right )}}{b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ -\frac {b^{3} x^{4} + 3 \, a b^{2} x^{3} + {\left (3 \, a^{2} b + b\right )} x^{2} + {\left (a^{3} + a\right )} x + {\left (b^{2} x^{3} + 2 \, a b x^{2} + {\left (a^{2} + 1\right )} x\right )} \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1}}{{\left (b^{3} x^{2} + 2 \, a b^{2} x + a^{2} b + \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1} {\left (b^{2} x + a b\right )} + b\right )} \log \left (b x + a + \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1}\right )} + \int \frac {2 \, b^{5} x^{5} + 9 \, a b^{4} x^{4} + a^{5} + 4 \, {\left (4 \, a^{2} b^{3} + b^{3}\right )} x^{3} + 2 \, a^{3} + 2 \, {\left (7 \, a^{3} b^{2} + 5 \, a b^{2}\right )} x^{2} + {\left (2 \, b^{3} x^{3} + 5 \, a b^{2} x^{2} + 4 \, a^{2} b x + a^{3} + a\right )} {\left (b^{2} x^{2} + 2 \, a b x + a^{2} + 1\right )} + 2 \, {\left (3 \, a^{4} b + 4 \, a^{2} b + b\right )} x + {\left (4 \, b^{4} x^{4} + 14 \, a b^{3} x^{3} + 2 \, a^{4} + 2 \, {\left (9 \, a^{2} b^{2} + 2 \, b^{2}\right )} x^{2} + 3 \, a^{2} + {\left (10 \, a^{3} b + 7 \, a b\right )} x + 1\right )} \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1} + a}{{\left (b^{5} x^{4} + 4 \, a b^{4} x^{3} + a^{4} b + 2 \, a^{2} b + 2 \, {\left (3 \, a^{2} b^{3} + b^{3}\right )} x^{2} + {\left (b^{3} x^{2} + 2 \, a b^{2} x + a^{2} b\right )} {\left (b^{2} x^{2} + 2 \, a b x + a^{2} + 1\right )} + 4 \, {\left (a^{3} b^{2} + a b^{2}\right )} x + 2 \, {\left (b^{4} x^{3} + 3 \, a b^{3} x^{2} + a^{3} b + a b + {\left (3 \, a^{2} b^{2} + b^{2}\right )} x\right )} \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1} + b\right )} \log \left (b x + a + \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1}\right )}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {x}{{\mathrm {asinh}\left (a+b\,x\right )}^2} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x}{\operatorname {asinh}^{2}{\left (a + b x \right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________