Optimal. Leaf size=359 \[ -\frac {\sqrt {\pi } a^3 \text {erfi}\left (\frac {1}{2} \left (2 \sinh ^{-1}(a+b x)-1\right )\right )}{4 \sqrt [4]{e} b^4}-\frac {\sqrt {\pi } a^3 \text {erfi}\left (\frac {1}{2} \left (2 \sinh ^{-1}(a+b x)+1\right )\right )}{4 \sqrt [4]{e} b^4}+\frac {3 \sqrt {\pi } a^2 \text {erfi}\left (1-\sinh ^{-1}(a+b x)\right )}{8 e b^4}+\frac {3 \sqrt {\pi } a^2 \text {erfi}\left (\sinh ^{-1}(a+b x)+1\right )}{8 e b^4}-\frac {3 \sqrt {\pi } a \text {erfi}\left (\frac {1}{2} \left (2 \sinh ^{-1}(a+b x)-3\right )\right )}{16 e^{9/4} b^4}+\frac {3 \sqrt {\pi } a \text {erfi}\left (\frac {1}{2} \left (2 \sinh ^{-1}(a+b x)-1\right )\right )}{16 \sqrt [4]{e} b^4}+\frac {3 \sqrt {\pi } a \text {erfi}\left (\frac {1}{2} \left (2 \sinh ^{-1}(a+b x)+1\right )\right )}{16 \sqrt [4]{e} b^4}-\frac {3 \sqrt {\pi } a \text {erfi}\left (\frac {1}{2} \left (2 \sinh ^{-1}(a+b x)+3\right )\right )}{16 e^{9/4} b^4}-\frac {\sqrt {\pi } \text {erfi}\left (1-\sinh ^{-1}(a+b x)\right )}{16 e b^4}+\frac {\sqrt {\pi } \text {erfi}\left (2-\sinh ^{-1}(a+b x)\right )}{32 e^4 b^4}-\frac {\sqrt {\pi } \text {erfi}\left (\sinh ^{-1}(a+b x)+1\right )}{16 e b^4}+\frac {\sqrt {\pi } \text {erfi}\left (\sinh ^{-1}(a+b x)+2\right )}{32 e^4 b^4} \]
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Rubi [A] time = 0.74, antiderivative size = 359, normalized size of antiderivative = 1.00, number of steps used = 37, number of rules used = 8, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.571, Rules used = {5898, 6741, 12, 6742, 5513, 2234, 2204, 5514} \[ -\frac {\sqrt {\pi } a^3 \text {Erfi}\left (\frac {1}{2} \left (2 \sinh ^{-1}(a+b x)-1\right )\right )}{4 \sqrt [4]{e} b^4}-\frac {\sqrt {\pi } a^3 \text {Erfi}\left (\frac {1}{2} \left (2 \sinh ^{-1}(a+b x)+1\right )\right )}{4 \sqrt [4]{e} b^4}+\frac {3 \sqrt {\pi } a^2 \text {Erfi}\left (1-\sinh ^{-1}(a+b x)\right )}{8 e b^4}+\frac {3 \sqrt {\pi } a^2 \text {Erfi}\left (\sinh ^{-1}(a+b x)+1\right )}{8 e b^4}-\frac {3 \sqrt {\pi } a \text {Erfi}\left (\frac {1}{2} \left (2 \sinh ^{-1}(a+b x)-3\right )\right )}{16 e^{9/4} b^4}+\frac {3 \sqrt {\pi } a \text {Erfi}\left (\frac {1}{2} \left (2 \sinh ^{-1}(a+b x)-1\right )\right )}{16 \sqrt [4]{e} b^4}+\frac {3 \sqrt {\pi } a \text {Erfi}\left (\frac {1}{2} \left (2 \sinh ^{-1}(a+b x)+1\right )\right )}{16 \sqrt [4]{e} b^4}-\frac {3 \sqrt {\pi } a \text {Erfi}\left (\frac {1}{2} \left (2 \sinh ^{-1}(a+b x)+3\right )\right )}{16 e^{9/4} b^4}-\frac {\sqrt {\pi } \text {Erfi}\left (1-\sinh ^{-1}(a+b x)\right )}{16 e b^4}+\frac {\sqrt {\pi } \text {Erfi}\left (2-\sinh ^{-1}(a+b x)\right )}{32 e^4 b^4}-\frac {\sqrt {\pi } \text {Erfi}\left (\sinh ^{-1}(a+b x)+1\right )}{16 e b^4}+\frac {\sqrt {\pi } \text {Erfi}\left (\sinh ^{-1}(a+b x)+2\right )}{32 e^4 b^4} \]
Antiderivative was successfully verified.
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Rule 12
Rule 2204
Rule 2234
Rule 5513
Rule 5514
Rule 5898
Rule 6741
Rule 6742
Rubi steps
\begin {align*} \int e^{\sinh ^{-1}(a+b x)^2} x^3 \, dx &=\frac {\operatorname {Subst}\left (\int e^{x^2} \cosh (x) \left (-\frac {a}{b}+\frac {\sinh (x)}{b}\right )^3 \, dx,x,\sinh ^{-1}(a+b x)\right )}{b}\\ &=\frac {\operatorname {Subst}\left (\int \frac {e^{x^2} \cosh (x) (-a+\sinh (x))^3}{b^3} \, dx,x,\sinh ^{-1}(a+b x)\right )}{b}\\ &=\frac {\operatorname {Subst}\left (\int e^{x^2} \cosh (x) (-a+\sinh (x))^3 \, dx,x,\sinh ^{-1}(a+b x)\right )}{b^4}\\ &=\frac {\operatorname {Subst}\left (\int \left (-a^3 e^{x^2} \cosh (x)+3 a^2 e^{x^2} \cosh (x) \sinh (x)-3 a e^{x^2} \cosh (x) \sinh ^2(x)+e^{x^2} \cosh (x) \sinh ^3(x)\right ) \, dx,x,\sinh ^{-1}(a+b x)\right )}{b^4}\\ &=\frac {\operatorname {Subst}\left (\int e^{x^2} \cosh (x) \sinh ^3(x) \, dx,x,\sinh ^{-1}(a+b x)\right )}{b^4}-\frac {(3 a) \operatorname {Subst}\left (\int e^{x^2} \cosh (x) \sinh ^2(x) \, dx,x,\sinh ^{-1}(a+b x)\right )}{b^4}+\frac {\left (3 a^2\right ) \operatorname {Subst}\left (\int e^{x^2} \cosh (x) \sinh (x) \, dx,x,\sinh ^{-1}(a+b x)\right )}{b^4}-\frac {a^3 \operatorname {Subst}\left (\int e^{x^2} \cosh (x) \, dx,x,\sinh ^{-1}(a+b x)\right )}{b^4}\\ &=\frac {\operatorname {Subst}\left (\int \left (-\frac {1}{16} e^{-4 x+x^2}+\frac {1}{8} e^{-2 x+x^2}-\frac {1}{8} e^{2 x+x^2}+\frac {1}{16} e^{4 x+x^2}\right ) \, dx,x,\sinh ^{-1}(a+b x)\right )}{b^4}-\frac {(3 a) \operatorname {Subst}\left (\int \left (\frac {1}{8} e^{-3 x+x^2}-\frac {1}{8} e^{-x+x^2}-\frac {e^{x+x^2}}{8}+\frac {1}{8} e^{3 x+x^2}\right ) \, dx,x,\sinh ^{-1}(a+b x)\right )}{b^4}+\frac {\left (3 a^2\right ) \operatorname {Subst}\left (\int \left (-\frac {1}{4} e^{-2 x+x^2}+\frac {1}{4} e^{2 x+x^2}\right ) \, dx,x,\sinh ^{-1}(a+b x)\right )}{b^4}-\frac {a^3 \operatorname {Subst}\left (\int \left (\frac {1}{2} e^{-x+x^2}+\frac {e^{x+x^2}}{2}\right ) \, dx,x,\sinh ^{-1}(a+b x)\right )}{b^4}\\ &=-\frac {\operatorname {Subst}\left (\int e^{-4 x+x^2} \, dx,x,\sinh ^{-1}(a+b x)\right )}{16 b^4}+\frac {\operatorname {Subst}\left (\int e^{4 x+x^2} \, dx,x,\sinh ^{-1}(a+b x)\right )}{16 b^4}+\frac {\operatorname {Subst}\left (\int e^{-2 x+x^2} \, dx,x,\sinh ^{-1}(a+b x)\right )}{8 b^4}-\frac {\operatorname {Subst}\left (\int e^{2 x+x^2} \, dx,x,\sinh ^{-1}(a+b x)\right )}{8 b^4}-\frac {(3 a) \operatorname {Subst}\left (\int e^{-3 x+x^2} \, dx,x,\sinh ^{-1}(a+b x)\right )}{8 b^4}+\frac {(3 a) \operatorname {Subst}\left (\int e^{-x+x^2} \, dx,x,\sinh ^{-1}(a+b x)\right )}{8 b^4}+\frac {(3 a) \operatorname {Subst}\left (\int e^{x+x^2} \, dx,x,\sinh ^{-1}(a+b x)\right )}{8 b^4}-\frac {(3 a) \operatorname {Subst}\left (\int e^{3 x+x^2} \, dx,x,\sinh ^{-1}(a+b x)\right )}{8 b^4}-\frac {\left (3 a^2\right ) \operatorname {Subst}\left (\int e^{-2 x+x^2} \, dx,x,\sinh ^{-1}(a+b x)\right )}{4 b^4}+\frac {\left (3 a^2\right ) \operatorname {Subst}\left (\int e^{2 x+x^2} \, dx,x,\sinh ^{-1}(a+b x)\right )}{4 b^4}-\frac {a^3 \operatorname {Subst}\left (\int e^{-x+x^2} \, dx,x,\sinh ^{-1}(a+b x)\right )}{2 b^4}-\frac {a^3 \operatorname {Subst}\left (\int e^{x+x^2} \, dx,x,\sinh ^{-1}(a+b x)\right )}{2 b^4}\\ &=-\frac {\operatorname {Subst}\left (\int e^{\frac {1}{4} (-4+2 x)^2} \, dx,x,\sinh ^{-1}(a+b x)\right )}{16 b^4 e^4}+\frac {\operatorname {Subst}\left (\int e^{\frac {1}{4} (4+2 x)^2} \, dx,x,\sinh ^{-1}(a+b x)\right )}{16 b^4 e^4}-\frac {(3 a) \operatorname {Subst}\left (\int e^{\frac {1}{4} (-3+2 x)^2} \, dx,x,\sinh ^{-1}(a+b x)\right )}{8 b^4 e^{9/4}}-\frac {(3 a) \operatorname {Subst}\left (\int e^{\frac {1}{4} (3+2 x)^2} \, dx,x,\sinh ^{-1}(a+b x)\right )}{8 b^4 e^{9/4}}+\frac {\operatorname {Subst}\left (\int e^{\frac {1}{4} (-2+2 x)^2} \, dx,x,\sinh ^{-1}(a+b x)\right )}{8 b^4 e}-\frac {\operatorname {Subst}\left (\int e^{\frac {1}{4} (2+2 x)^2} \, dx,x,\sinh ^{-1}(a+b x)\right )}{8 b^4 e}-\frac {\left (3 a^2\right ) \operatorname {Subst}\left (\int e^{\frac {1}{4} (-2+2 x)^2} \, dx,x,\sinh ^{-1}(a+b x)\right )}{4 b^4 e}+\frac {\left (3 a^2\right ) \operatorname {Subst}\left (\int e^{\frac {1}{4} (2+2 x)^2} \, dx,x,\sinh ^{-1}(a+b x)\right )}{4 b^4 e}+\frac {(3 a) \operatorname {Subst}\left (\int e^{\frac {1}{4} (-1+2 x)^2} \, dx,x,\sinh ^{-1}(a+b x)\right )}{8 b^4 \sqrt [4]{e}}+\frac {(3 a) \operatorname {Subst}\left (\int e^{\frac {1}{4} (1+2 x)^2} \, dx,x,\sinh ^{-1}(a+b x)\right )}{8 b^4 \sqrt [4]{e}}-\frac {a^3 \operatorname {Subst}\left (\int e^{\frac {1}{4} (-1+2 x)^2} \, dx,x,\sinh ^{-1}(a+b x)\right )}{2 b^4 \sqrt [4]{e}}-\frac {a^3 \operatorname {Subst}\left (\int e^{\frac {1}{4} (1+2 x)^2} \, dx,x,\sinh ^{-1}(a+b x)\right )}{2 b^4 \sqrt [4]{e}}\\ &=-\frac {\sqrt {\pi } \text {erfi}\left (1-\sinh ^{-1}(a+b x)\right )}{16 b^4 e}+\frac {3 a^2 \sqrt {\pi } \text {erfi}\left (1-\sinh ^{-1}(a+b x)\right )}{8 b^4 e}+\frac {\sqrt {\pi } \text {erfi}\left (2-\sinh ^{-1}(a+b x)\right )}{32 b^4 e^4}-\frac {\sqrt {\pi } \text {erfi}\left (1+\sinh ^{-1}(a+b x)\right )}{16 b^4 e}+\frac {3 a^2 \sqrt {\pi } \text {erfi}\left (1+\sinh ^{-1}(a+b x)\right )}{8 b^4 e}+\frac {\sqrt {\pi } \text {erfi}\left (2+\sinh ^{-1}(a+b x)\right )}{32 b^4 e^4}-\frac {3 a \sqrt {\pi } \text {erfi}\left (\frac {1}{2} \left (-3+2 \sinh ^{-1}(a+b x)\right )\right )}{16 b^4 e^{9/4}}+\frac {3 a \sqrt {\pi } \text {erfi}\left (\frac {1}{2} \left (-1+2 \sinh ^{-1}(a+b x)\right )\right )}{16 b^4 \sqrt [4]{e}}-\frac {a^3 \sqrt {\pi } \text {erfi}\left (\frac {1}{2} \left (-1+2 \sinh ^{-1}(a+b x)\right )\right )}{4 b^4 \sqrt [4]{e}}+\frac {3 a \sqrt {\pi } \text {erfi}\left (\frac {1}{2} \left (1+2 \sinh ^{-1}(a+b x)\right )\right )}{16 b^4 \sqrt [4]{e}}-\frac {a^3 \sqrt {\pi } \text {erfi}\left (\frac {1}{2} \left (1+2 \sinh ^{-1}(a+b x)\right )\right )}{4 b^4 \sqrt [4]{e}}-\frac {3 a \sqrt {\pi } \text {erfi}\left (\frac {1}{2} \left (3+2 \sinh ^{-1}(a+b x)\right )\right )}{16 b^4 e^{9/4}}\\ \end {align*}
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Mathematica [A] time = 0.33, size = 198, normalized size = 0.55 \[ \frac {\sqrt {\pi } \left (-8 e^{15/4} a^3 \text {erfi}\left (\sinh ^{-1}(a+b x)+\frac {1}{2}\right )+12 e^3 a^2 \text {erfi}\left (\sinh ^{-1}(a+b x)+1\right )+2 e^{15/4} \left (4 a^2-3\right ) a \text {erfi}\left (\frac {1}{2}-\sinh ^{-1}(a+b x)\right )+2 e^3 \left (6 a^2-1\right ) \text {erfi}\left (1-\sinh ^{-1}(a+b x)\right )+6 e^{7/4} a \text {erfi}\left (\frac {3}{2}-\sinh ^{-1}(a+b x)\right )+6 e^{15/4} a \text {erfi}\left (\sinh ^{-1}(a+b x)+\frac {1}{2}\right )-6 e^{7/4} a \text {erfi}\left (\sinh ^{-1}(a+b x)+\frac {3}{2}\right )+\text {erfi}\left (2-\sinh ^{-1}(a+b x)\right )-2 e^3 \text {erfi}\left (\sinh ^{-1}(a+b x)+1\right )+\text {erfi}\left (\sinh ^{-1}(a+b x)+2\right )\right )}{32 e^4 b^4} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.63, size = 0, normalized size = 0.00 \[ {\rm integral}\left (x^{3} e^{\left (\operatorname {arsinh}\left (b x + a\right )^{2}\right )}, x\right ) \]
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 \[ \int x^{3} e^{\left (\operatorname {arsinh}\left (b x + a\right )^{2}\right )}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.01, size = 0, normalized size = 0.00 \[ \int {\mathrm e}^{\arcsinh \left (b x +a \right )^{2}} x^{3}\, dx \]
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 \[ \int x^{3} e^{\left (\operatorname {arsinh}\left (b x + a\right )^{2}\right )}\,{d x} \]
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 \[ \int x^3\,{\mathrm {e}}^{{\mathrm {asinh}\left (a+b\,x\right )}^2} \,d x \]
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 \[ \int x^{3} e^{\operatorname {asinh}^{2}{\left (a + b x \right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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