Optimal. Leaf size=257 \[ \frac {a^2 \text {Chi}\left (\sinh ^{-1}(a+b x)\right )}{2 b^3}-\frac {a^2 (a+b x)}{2 b^3 \sinh ^{-1}(a+b x)}-\frac {a^2 \sqrt {(a+b x)^2+1}}{2 b^3 \sinh ^{-1}(a+b x)^2}-\frac {\text {Chi}\left (\sinh ^{-1}(a+b x)\right )}{8 b^3}+\frac {9 \text {Chi}\left (3 \sinh ^{-1}(a+b x)\right )}{8 b^3}-\frac {2 a \text {Shi}\left (2 \sinh ^{-1}(a+b x)\right )}{b^3}-\frac {3 (a+b x)^3}{2 b^3 \sinh ^{-1}(a+b x)}+\frac {2 a (a+b x)^2}{b^3 \sinh ^{-1}(a+b x)}-\frac {\sqrt {(a+b x)^2+1} (a+b x)^2}{2 b^3 \sinh ^{-1}(a+b x)^2}-\frac {a+b x}{b^3 \sinh ^{-1}(a+b x)}+\frac {a \sqrt {(a+b x)^2+1} (a+b x)}{b^3 \sinh ^{-1}(a+b x)^2}+\frac {a}{b^3 \sinh ^{-1}(a+b x)} \]
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Rubi [A] time = 0.50, antiderivative size = 257, normalized size of antiderivative = 1.00, number of steps used = 24, number of rules used = 12, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.000, Rules used = {5865, 5803, 5655, 5774, 5657, 3301, 5667, 5669, 5448, 12, 3298, 5675} \[ \frac {a^2 \text {Chi}\left (\sinh ^{-1}(a+b x)\right )}{2 b^3}-\frac {a^2 (a+b x)}{2 b^3 \sinh ^{-1}(a+b x)}-\frac {a^2 \sqrt {(a+b x)^2+1}}{2 b^3 \sinh ^{-1}(a+b x)^2}-\frac {\text {Chi}\left (\sinh ^{-1}(a+b x)\right )}{8 b^3}+\frac {9 \text {Chi}\left (3 \sinh ^{-1}(a+b x)\right )}{8 b^3}-\frac {2 a \text {Shi}\left (2 \sinh ^{-1}(a+b x)\right )}{b^3}-\frac {3 (a+b x)^3}{2 b^3 \sinh ^{-1}(a+b x)}+\frac {2 a (a+b x)^2}{b^3 \sinh ^{-1}(a+b x)}-\frac {\sqrt {(a+b x)^2+1} (a+b x)^2}{2 b^3 \sinh ^{-1}(a+b x)^2}-\frac {a+b x}{b^3 \sinh ^{-1}(a+b x)}+\frac {a \sqrt {(a+b x)^2+1} (a+b x)}{b^3 \sinh ^{-1}(a+b x)^2}+\frac {a}{b^3 \sinh ^{-1}(a+b x)} \]
Antiderivative was successfully verified.
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Rule 12
Rule 3298
Rule 3301
Rule 5448
Rule 5655
Rule 5657
Rule 5667
Rule 5669
Rule 5675
Rule 5774
Rule 5803
Rule 5865
Rubi steps
\begin {align*} \int \frac {x^2}{\sinh ^{-1}(a+b x)^3} \, dx &=\frac {\operatorname {Subst}\left (\int \frac {\left (-\frac {a}{b}+\frac {x}{b}\right )^2}{\sinh ^{-1}(x)^3} \, dx,x,a+b x\right )}{b}\\ &=\frac {\operatorname {Subst}\left (\int \left (\frac {a^2}{b^2 \sinh ^{-1}(x)^3}-\frac {2 a x}{b^2 \sinh ^{-1}(x)^3}+\frac {x^2}{b^2 \sinh ^{-1}(x)^3}\right ) \, dx,x,a+b x\right )}{b}\\ &=\frac {\operatorname {Subst}\left (\int \frac {x^2}{\sinh ^{-1}(x)^3} \, dx,x,a+b x\right )}{b^3}-\frac {(2 a) \operatorname {Subst}\left (\int \frac {x}{\sinh ^{-1}(x)^3} \, dx,x,a+b x\right )}{b^3}+\frac {a^2 \operatorname {Subst}\left (\int \frac {1}{\sinh ^{-1}(x)^3} \, dx,x,a+b x\right )}{b^3}\\ &=-\frac {a^2 \sqrt {1+(a+b x)^2}}{2 b^3 \sinh ^{-1}(a+b x)^2}+\frac {a (a+b x) \sqrt {1+(a+b x)^2}}{b^3 \sinh ^{-1}(a+b x)^2}-\frac {(a+b x)^2 \sqrt {1+(a+b x)^2}}{2 b^3 \sinh ^{-1}(a+b x)^2}+\frac {\operatorname {Subst}\left (\int \frac {x}{\sqrt {1+x^2} \sinh ^{-1}(x)^2} \, dx,x,a+b x\right )}{b^3}+\frac {3 \operatorname {Subst}\left (\int \frac {x^3}{\sqrt {1+x^2} \sinh ^{-1}(x)^2} \, dx,x,a+b x\right )}{2 b^3}-\frac {a \operatorname {Subst}\left (\int \frac {1}{\sqrt {1+x^2} \sinh ^{-1}(x)^2} \, dx,x,a+b x\right )}{b^3}-\frac {(2 a) \operatorname {Subst}\left (\int \frac {x^2}{\sqrt {1+x^2} \sinh ^{-1}(x)^2} \, dx,x,a+b x\right )}{b^3}+\frac {a^2 \operatorname {Subst}\left (\int \frac {x}{\sqrt {1+x^2} \sinh ^{-1}(x)^2} \, dx,x,a+b x\right )}{2 b^3}\\ &=-\frac {a^2 \sqrt {1+(a+b x)^2}}{2 b^3 \sinh ^{-1}(a+b x)^2}+\frac {a (a+b x) \sqrt {1+(a+b x)^2}}{b^3 \sinh ^{-1}(a+b x)^2}-\frac {(a+b x)^2 \sqrt {1+(a+b x)^2}}{2 b^3 \sinh ^{-1}(a+b x)^2}+\frac {a}{b^3 \sinh ^{-1}(a+b x)}-\frac {a+b x}{b^3 \sinh ^{-1}(a+b x)}-\frac {a^2 (a+b x)}{2 b^3 \sinh ^{-1}(a+b x)}+\frac {2 a (a+b x)^2}{b^3 \sinh ^{-1}(a+b x)}-\frac {3 (a+b x)^3}{2 b^3 \sinh ^{-1}(a+b x)}+\frac {\operatorname {Subst}\left (\int \frac {1}{\sinh ^{-1}(x)} \, dx,x,a+b x\right )}{b^3}+\frac {9 \operatorname {Subst}\left (\int \frac {x^2}{\sinh ^{-1}(x)} \, dx,x,a+b x\right )}{2 b^3}-\frac {(4 a) \operatorname {Subst}\left (\int \frac {x}{\sinh ^{-1}(x)} \, dx,x,a+b x\right )}{b^3}+\frac {a^2 \operatorname {Subst}\left (\int \frac {1}{\sinh ^{-1}(x)} \, dx,x,a+b x\right )}{2 b^3}\\ &=-\frac {a^2 \sqrt {1+(a+b x)^2}}{2 b^3 \sinh ^{-1}(a+b x)^2}+\frac {a (a+b x) \sqrt {1+(a+b x)^2}}{b^3 \sinh ^{-1}(a+b x)^2}-\frac {(a+b x)^2 \sqrt {1+(a+b x)^2}}{2 b^3 \sinh ^{-1}(a+b x)^2}+\frac {a}{b^3 \sinh ^{-1}(a+b x)}-\frac {a+b x}{b^3 \sinh ^{-1}(a+b x)}-\frac {a^2 (a+b x)}{2 b^3 \sinh ^{-1}(a+b x)}+\frac {2 a (a+b x)^2}{b^3 \sinh ^{-1}(a+b x)}-\frac {3 (a+b x)^3}{2 b^3 \sinh ^{-1}(a+b x)}+\frac {\operatorname {Subst}\left (\int \frac {\cosh (x)}{x} \, dx,x,\sinh ^{-1}(a+b x)\right )}{b^3}+\frac {9 \operatorname {Subst}\left (\int \frac {\cosh (x) \sinh ^2(x)}{x} \, dx,x,\sinh ^{-1}(a+b x)\right )}{2 b^3}-\frac {(4 a) \operatorname {Subst}\left (\int \frac {\cosh (x) \sinh (x)}{x} \, dx,x,\sinh ^{-1}(a+b x)\right )}{b^3}+\frac {a^2 \operatorname {Subst}\left (\int \frac {\cosh (x)}{x} \, dx,x,\sinh ^{-1}(a+b x)\right )}{2 b^3}\\ &=-\frac {a^2 \sqrt {1+(a+b x)^2}}{2 b^3 \sinh ^{-1}(a+b x)^2}+\frac {a (a+b x) \sqrt {1+(a+b x)^2}}{b^3 \sinh ^{-1}(a+b x)^2}-\frac {(a+b x)^2 \sqrt {1+(a+b x)^2}}{2 b^3 \sinh ^{-1}(a+b x)^2}+\frac {a}{b^3 \sinh ^{-1}(a+b x)}-\frac {a+b x}{b^3 \sinh ^{-1}(a+b x)}-\frac {a^2 (a+b x)}{2 b^3 \sinh ^{-1}(a+b x)}+\frac {2 a (a+b x)^2}{b^3 \sinh ^{-1}(a+b x)}-\frac {3 (a+b x)^3}{2 b^3 \sinh ^{-1}(a+b x)}+\frac {\text {Chi}\left (\sinh ^{-1}(a+b x)\right )}{b^3}+\frac {a^2 \text {Chi}\left (\sinh ^{-1}(a+b x)\right )}{2 b^3}+\frac {9 \operatorname {Subst}\left (\int \left (-\frac {\cosh (x)}{4 x}+\frac {\cosh (3 x)}{4 x}\right ) \, dx,x,\sinh ^{-1}(a+b x)\right )}{2 b^3}-\frac {(4 a) \operatorname {Subst}\left (\int \frac {\sinh (2 x)}{2 x} \, dx,x,\sinh ^{-1}(a+b x)\right )}{b^3}\\ &=-\frac {a^2 \sqrt {1+(a+b x)^2}}{2 b^3 \sinh ^{-1}(a+b x)^2}+\frac {a (a+b x) \sqrt {1+(a+b x)^2}}{b^3 \sinh ^{-1}(a+b x)^2}-\frac {(a+b x)^2 \sqrt {1+(a+b x)^2}}{2 b^3 \sinh ^{-1}(a+b x)^2}+\frac {a}{b^3 \sinh ^{-1}(a+b x)}-\frac {a+b x}{b^3 \sinh ^{-1}(a+b x)}-\frac {a^2 (a+b x)}{2 b^3 \sinh ^{-1}(a+b x)}+\frac {2 a (a+b x)^2}{b^3 \sinh ^{-1}(a+b x)}-\frac {3 (a+b x)^3}{2 b^3 \sinh ^{-1}(a+b x)}+\frac {\text {Chi}\left (\sinh ^{-1}(a+b x)\right )}{b^3}+\frac {a^2 \text {Chi}\left (\sinh ^{-1}(a+b x)\right )}{2 b^3}-\frac {9 \operatorname {Subst}\left (\int \frac {\cosh (x)}{x} \, dx,x,\sinh ^{-1}(a+b x)\right )}{8 b^3}+\frac {9 \operatorname {Subst}\left (\int \frac {\cosh (3 x)}{x} \, dx,x,\sinh ^{-1}(a+b x)\right )}{8 b^3}-\frac {(2 a) \operatorname {Subst}\left (\int \frac {\sinh (2 x)}{x} \, dx,x,\sinh ^{-1}(a+b x)\right )}{b^3}\\ &=-\frac {a^2 \sqrt {1+(a+b x)^2}}{2 b^3 \sinh ^{-1}(a+b x)^2}+\frac {a (a+b x) \sqrt {1+(a+b x)^2}}{b^3 \sinh ^{-1}(a+b x)^2}-\frac {(a+b x)^2 \sqrt {1+(a+b x)^2}}{2 b^3 \sinh ^{-1}(a+b x)^2}+\frac {a}{b^3 \sinh ^{-1}(a+b x)}-\frac {a+b x}{b^3 \sinh ^{-1}(a+b x)}-\frac {a^2 (a+b x)}{2 b^3 \sinh ^{-1}(a+b x)}+\frac {2 a (a+b x)^2}{b^3 \sinh ^{-1}(a+b x)}-\frac {3 (a+b x)^3}{2 b^3 \sinh ^{-1}(a+b x)}-\frac {\text {Chi}\left (\sinh ^{-1}(a+b x)\right )}{8 b^3}+\frac {a^2 \text {Chi}\left (\sinh ^{-1}(a+b x)\right )}{2 b^3}+\frac {9 \text {Chi}\left (3 \sinh ^{-1}(a+b x)\right )}{8 b^3}-\frac {2 a \text {Shi}\left (2 \sinh ^{-1}(a+b x)\right )}{b^3}\\ \end {align*}
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Mathematica [A] time = 0.40, size = 110, normalized size = 0.43 \[ \frac {-\frac {4 b x \left (b x \sqrt {a^2+2 a b x+b^2 x^2+1}+\left (2 a^2+5 a b x+3 b^2 x^2+2\right ) \sinh ^{-1}(a+b x)\right )}{\sinh ^{-1}(a+b x)^2}+\left (4 a^2-1\right ) \text {Chi}\left (\sinh ^{-1}(a+b x)\right )+9 \text {Chi}\left (3 \sinh ^{-1}(a+b x)\right )-16 a \text {Shi}\left (2 \sinh ^{-1}(a+b x)\right )}{8 b^3} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.51, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {x^{2}}{\operatorname {arsinh}\left (b x + a\right )^{3}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x^{2}}{\operatorname {arsinh}\left (b x + a\right )^{3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.16, size = 215, normalized size = 0.84 \[ \frac {-\frac {a \left (4 \Shi \left (2 \arcsinh \left (b x +a \right )\right ) \arcsinh \left (b x +a \right )^{2}-2 \cosh \left (2 \arcsinh \left (b x +a \right )\right ) \arcsinh \left (b x +a \right )-\sinh \left (2 \arcsinh \left (b x +a \right )\right )\right )}{2 \arcsinh \left (b x +a \right )^{2}}+\frac {\sqrt {1+\left (b x +a \right )^{2}}}{8 \arcsinh \left (b x +a \right )^{2}}+\frac {b x +a}{8 \arcsinh \left (b x +a \right )}-\frac {\Chi \left (\arcsinh \left (b x +a \right )\right )}{8}-\frac {\cosh \left (3 \arcsinh \left (b x +a \right )\right )}{8 \arcsinh \left (b x +a \right )^{2}}-\frac {3 \sinh \left (3 \arcsinh \left (b x +a \right )\right )}{8 \arcsinh \left (b x +a \right )}+\frac {9 \Chi \left (3 \arcsinh \left (b x +a \right )\right )}{8}+\frac {a^{2} \left (\Chi \left (\arcsinh \left (b x +a \right )\right ) \arcsinh \left (b x +a \right )^{2}-\left (b x +a \right ) \arcsinh \left (b x +a \right )-\sqrt {1+\left (b x +a \right )^{2}}\right )}{2 \arcsinh \left (b x +a \right )^{2}}}{b^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {x^2}{{\mathrm {asinh}\left (a+b\,x\right )}^3} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x^{2}}{\operatorname {asinh}^{3}{\left (a + b x \right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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