Optimal. Leaf size=322 \[ -\frac {\sqrt {\pi } \sqrt {b} e e^{a/b} \text {erf}\left (\frac {\sqrt {a+b \cosh ^{-1}(c x)}}{\sqrt {b}}\right )}{16 c^3}-\frac {\sqrt {\frac {\pi }{3}} \sqrt {b} e e^{\frac {3 a}{b}} \text {erf}\left (\frac {\sqrt {3} \sqrt {a+b \cosh ^{-1}(c x)}}{\sqrt {b}}\right )}{48 c^3}-\frac {\sqrt {\pi } \sqrt {b} e e^{-\frac {a}{b}} \text {erfi}\left (\frac {\sqrt {a+b \cosh ^{-1}(c x)}}{\sqrt {b}}\right )}{16 c^3}-\frac {\sqrt {\frac {\pi }{3}} \sqrt {b} e e^{-\frac {3 a}{b}} \text {erfi}\left (\frac {\sqrt {3} \sqrt {a+b \cosh ^{-1}(c x)}}{\sqrt {b}}\right )}{48 c^3}-\frac {\sqrt {\pi } \sqrt {b} d e^{a/b} \text {erf}\left (\frac {\sqrt {a+b \cosh ^{-1}(c x)}}{\sqrt {b}}\right )}{4 c}-\frac {\sqrt {\pi } \sqrt {b} d e^{-\frac {a}{b}} \text {erfi}\left (\frac {\sqrt {a+b \cosh ^{-1}(c x)}}{\sqrt {b}}\right )}{4 c}+d x \sqrt {a+b \cosh ^{-1}(c x)}+\frac {1}{3} e x^3 \sqrt {a+b \cosh ^{-1}(c x)} \]
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Rubi [A] time = 1.29, antiderivative size = 322, normalized size of antiderivative = 1.00, number of steps used = 23, number of rules used = 9, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.450, Rules used = {5707, 5654, 5781, 3307, 2180, 2204, 2205, 5664, 3312} \[ -\frac {\sqrt {\pi } \sqrt {b} e e^{a/b} \text {Erf}\left (\frac {\sqrt {a+b \cosh ^{-1}(c x)}}{\sqrt {b}}\right )}{16 c^3}-\frac {\sqrt {\frac {\pi }{3}} \sqrt {b} e e^{\frac {3 a}{b}} \text {Erf}\left (\frac {\sqrt {3} \sqrt {a+b \cosh ^{-1}(c x)}}{\sqrt {b}}\right )}{48 c^3}-\frac {\sqrt {\pi } \sqrt {b} e e^{-\frac {a}{b}} \text {Erfi}\left (\frac {\sqrt {a+b \cosh ^{-1}(c x)}}{\sqrt {b}}\right )}{16 c^3}-\frac {\sqrt {\frac {\pi }{3}} \sqrt {b} e e^{-\frac {3 a}{b}} \text {Erfi}\left (\frac {\sqrt {3} \sqrt {a+b \cosh ^{-1}(c x)}}{\sqrt {b}}\right )}{48 c^3}-\frac {\sqrt {\pi } \sqrt {b} d e^{a/b} \text {Erf}\left (\frac {\sqrt {a+b \cosh ^{-1}(c x)}}{\sqrt {b}}\right )}{4 c}-\frac {\sqrt {\pi } \sqrt {b} d e^{-\frac {a}{b}} \text {Erfi}\left (\frac {\sqrt {a+b \cosh ^{-1}(c x)}}{\sqrt {b}}\right )}{4 c}+d x \sqrt {a+b \cosh ^{-1}(c x)}+\frac {1}{3} e x^3 \sqrt {a+b \cosh ^{-1}(c x)} \]
Antiderivative was successfully verified.
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Rule 2180
Rule 2204
Rule 2205
Rule 3307
Rule 3312
Rule 5654
Rule 5664
Rule 5707
Rule 5781
Rubi steps
\begin {align*} \int \left (d+e x^2\right ) \sqrt {a+b \cosh ^{-1}(c x)} \, dx &=\int \left (d \sqrt {a+b \cosh ^{-1}(c x)}+e x^2 \sqrt {a+b \cosh ^{-1}(c x)}\right ) \, dx\\ &=d \int \sqrt {a+b \cosh ^{-1}(c x)} \, dx+e \int x^2 \sqrt {a+b \cosh ^{-1}(c x)} \, dx\\ &=d x \sqrt {a+b \cosh ^{-1}(c x)}+\frac {1}{3} e x^3 \sqrt {a+b \cosh ^{-1}(c x)}-\frac {1}{2} (b c d) \int \frac {x}{\sqrt {-1+c x} \sqrt {1+c x} \sqrt {a+b \cosh ^{-1}(c x)}} \, dx-\frac {1}{6} (b c e) \int \frac {x^3}{\sqrt {-1+c x} \sqrt {1+c x} \sqrt {a+b \cosh ^{-1}(c x)}} \, dx\\ &=d x \sqrt {a+b \cosh ^{-1}(c x)}+\frac {1}{3} e x^3 \sqrt {a+b \cosh ^{-1}(c x)}-\frac {(b d) \operatorname {Subst}\left (\int \frac {\cosh (x)}{\sqrt {a+b x}} \, dx,x,\cosh ^{-1}(c x)\right )}{2 c}-\frac {(b e) \operatorname {Subst}\left (\int \frac {\cosh ^3(x)}{\sqrt {a+b x}} \, dx,x,\cosh ^{-1}(c x)\right )}{6 c^3}\\ &=d x \sqrt {a+b \cosh ^{-1}(c x)}+\frac {1}{3} e x^3 \sqrt {a+b \cosh ^{-1}(c x)}-\frac {(b d) \operatorname {Subst}\left (\int \frac {e^{-x}}{\sqrt {a+b x}} \, dx,x,\cosh ^{-1}(c x)\right )}{4 c}-\frac {(b d) \operatorname {Subst}\left (\int \frac {e^x}{\sqrt {a+b x}} \, dx,x,\cosh ^{-1}(c x)\right )}{4 c}-\frac {(b e) \operatorname {Subst}\left (\int \left (\frac {3 \cosh (x)}{4 \sqrt {a+b x}}+\frac {\cosh (3 x)}{4 \sqrt {a+b x}}\right ) \, dx,x,\cosh ^{-1}(c x)\right )}{6 c^3}\\ &=d x \sqrt {a+b \cosh ^{-1}(c x)}+\frac {1}{3} e x^3 \sqrt {a+b \cosh ^{-1}(c x)}-\frac {d \operatorname {Subst}\left (\int e^{\frac {a}{b}-\frac {x^2}{b}} \, dx,x,\sqrt {a+b \cosh ^{-1}(c x)}\right )}{2 c}-\frac {d \operatorname {Subst}\left (\int e^{-\frac {a}{b}+\frac {x^2}{b}} \, dx,x,\sqrt {a+b \cosh ^{-1}(c x)}\right )}{2 c}-\frac {(b e) \operatorname {Subst}\left (\int \frac {\cosh (3 x)}{\sqrt {a+b x}} \, dx,x,\cosh ^{-1}(c x)\right )}{24 c^3}-\frac {(b e) \operatorname {Subst}\left (\int \frac {\cosh (x)}{\sqrt {a+b x}} \, dx,x,\cosh ^{-1}(c x)\right )}{8 c^3}\\ &=d x \sqrt {a+b \cosh ^{-1}(c x)}+\frac {1}{3} e x^3 \sqrt {a+b \cosh ^{-1}(c x)}-\frac {\sqrt {b} d e^{a/b} \sqrt {\pi } \text {erf}\left (\frac {\sqrt {a+b \cosh ^{-1}(c x)}}{\sqrt {b}}\right )}{4 c}-\frac {\sqrt {b} d e^{-\frac {a}{b}} \sqrt {\pi } \text {erfi}\left (\frac {\sqrt {a+b \cosh ^{-1}(c x)}}{\sqrt {b}}\right )}{4 c}-\frac {(b e) \operatorname {Subst}\left (\int \frac {e^{-3 x}}{\sqrt {a+b x}} \, dx,x,\cosh ^{-1}(c x)\right )}{48 c^3}-\frac {(b e) \operatorname {Subst}\left (\int \frac {e^{3 x}}{\sqrt {a+b x}} \, dx,x,\cosh ^{-1}(c x)\right )}{48 c^3}-\frac {(b e) \operatorname {Subst}\left (\int \frac {e^{-x}}{\sqrt {a+b x}} \, dx,x,\cosh ^{-1}(c x)\right )}{16 c^3}-\frac {(b e) \operatorname {Subst}\left (\int \frac {e^x}{\sqrt {a+b x}} \, dx,x,\cosh ^{-1}(c x)\right )}{16 c^3}\\ &=d x \sqrt {a+b \cosh ^{-1}(c x)}+\frac {1}{3} e x^3 \sqrt {a+b \cosh ^{-1}(c x)}-\frac {\sqrt {b} d e^{a/b} \sqrt {\pi } \text {erf}\left (\frac {\sqrt {a+b \cosh ^{-1}(c x)}}{\sqrt {b}}\right )}{4 c}-\frac {\sqrt {b} d e^{-\frac {a}{b}} \sqrt {\pi } \text {erfi}\left (\frac {\sqrt {a+b \cosh ^{-1}(c x)}}{\sqrt {b}}\right )}{4 c}-\frac {e \operatorname {Subst}\left (\int e^{\frac {3 a}{b}-\frac {3 x^2}{b}} \, dx,x,\sqrt {a+b \cosh ^{-1}(c x)}\right )}{24 c^3}-\frac {e \operatorname {Subst}\left (\int e^{-\frac {3 a}{b}+\frac {3 x^2}{b}} \, dx,x,\sqrt {a+b \cosh ^{-1}(c x)}\right )}{24 c^3}-\frac {e \operatorname {Subst}\left (\int e^{\frac {a}{b}-\frac {x^2}{b}} \, dx,x,\sqrt {a+b \cosh ^{-1}(c x)}\right )}{8 c^3}-\frac {e \operatorname {Subst}\left (\int e^{-\frac {a}{b}+\frac {x^2}{b}} \, dx,x,\sqrt {a+b \cosh ^{-1}(c x)}\right )}{8 c^3}\\ &=d x \sqrt {a+b \cosh ^{-1}(c x)}+\frac {1}{3} e x^3 \sqrt {a+b \cosh ^{-1}(c x)}-\frac {\sqrt {b} d e^{a/b} \sqrt {\pi } \text {erf}\left (\frac {\sqrt {a+b \cosh ^{-1}(c x)}}{\sqrt {b}}\right )}{4 c}-\frac {\sqrt {b} e e^{a/b} \sqrt {\pi } \text {erf}\left (\frac {\sqrt {a+b \cosh ^{-1}(c x)}}{\sqrt {b}}\right )}{16 c^3}-\frac {\sqrt {b} e e^{\frac {3 a}{b}} \sqrt {\frac {\pi }{3}} \text {erf}\left (\frac {\sqrt {3} \sqrt {a+b \cosh ^{-1}(c x)}}{\sqrt {b}}\right )}{48 c^3}-\frac {\sqrt {b} d e^{-\frac {a}{b}} \sqrt {\pi } \text {erfi}\left (\frac {\sqrt {a+b \cosh ^{-1}(c x)}}{\sqrt {b}}\right )}{4 c}-\frac {\sqrt {b} e e^{-\frac {a}{b}} \sqrt {\pi } \text {erfi}\left (\frac {\sqrt {a+b \cosh ^{-1}(c x)}}{\sqrt {b}}\right )}{16 c^3}-\frac {\sqrt {b} e e^{-\frac {3 a}{b}} \sqrt {\frac {\pi }{3}} \text {erfi}\left (\frac {\sqrt {3} \sqrt {a+b \cosh ^{-1}(c x)}}{\sqrt {b}}\right )}{48 c^3}\\ \end {align*}
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Mathematica [A] time = 2.82, size = 317, normalized size = 0.98 \[ \frac {e e^{-\frac {3 a}{b}} \sqrt {a+b \cosh ^{-1}(c x)} \left (9 e^{\frac {4 a}{b}} \sqrt {-\frac {a+b \cosh ^{-1}(c x)}{b}} \Gamma \left (\frac {3}{2},\frac {a}{b}+\cosh ^{-1}(c x)\right )+\sqrt {3} \sqrt {\frac {a}{b}+\cosh ^{-1}(c x)} \Gamma \left (\frac {3}{2},-\frac {3 \left (a+b \cosh ^{-1}(c x)\right )}{b}\right )+9 e^{\frac {2 a}{b}} \sqrt {\frac {a}{b}+\cosh ^{-1}(c x)} \Gamma \left (\frac {3}{2},-\frac {a+b \cosh ^{-1}(c x)}{b}\right )+\sqrt {3} e^{\frac {6 a}{b}} \sqrt {-\frac {a+b \cosh ^{-1}(c x)}{b}} \Gamma \left (\frac {3}{2},\frac {3 \left (a+b \cosh ^{-1}(c x)\right )}{b}\right )\right )}{72 c^3 \sqrt {-\frac {\left (a+b \cosh ^{-1}(c x)\right )^2}{b^2}}}+\frac {d e^{-\frac {a}{b}} \sqrt {a+b \cosh ^{-1}(c x)} \left (\frac {e^{\frac {2 a}{b}} \Gamma \left (\frac {3}{2},\frac {a}{b}+\cosh ^{-1}(c x)\right )}{\sqrt {\frac {a}{b}+\cosh ^{-1}(c x)}}+\frac {\Gamma \left (\frac {3}{2},-\frac {a+b \cosh ^{-1}(c x)}{b}\right )}{\sqrt {-\frac {a+b \cosh ^{-1}(c x)}{b}}}\right )}{2 c} \]
Warning: Unable to verify antiderivative.
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fricas [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]
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 \[ \int {\left (e x^{2} + d\right )} \sqrt {b \operatorname {arcosh}\left (c x\right ) + a}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F(-2)] time = 180.00, size = 0, normalized size = 0.00 \[ \int \left (e \,x^{2}+d \right ) \sqrt {a +b \,\mathrm {arccosh}\left (c x \right )}\, dx \]
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 \[ \int {\left (e x^{2} + d\right )} \sqrt {b \operatorname {arcosh}\left (c x\right ) + a}\,{d x} \]
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 \[ \int \sqrt {a+b\,\mathrm {acosh}\left (c\,x\right )}\,\left (e\,x^2+d\right ) \,d x \]
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 \[ \int \sqrt {a + b \operatorname {acosh}{\left (c x \right )}} \left (d + e x^{2}\right )\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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