Optimal. Leaf size=245 \[ -\frac {\sqrt {\pi } \sqrt {b} e^2 e^{a/b} \text {erf}\left (\frac {\sqrt {a+b \cosh ^{-1}(c+d x)}}{\sqrt {b}}\right )}{16 d}-\frac {\sqrt {\frac {\pi }{3}} \sqrt {b} e^2 e^{\frac {3 a}{b}} \text {erf}\left (\frac {\sqrt {3} \sqrt {a+b \cosh ^{-1}(c+d x)}}{\sqrt {b}}\right )}{48 d}-\frac {\sqrt {\pi } \sqrt {b} e^2 e^{-\frac {a}{b}} \text {erfi}\left (\frac {\sqrt {a+b \cosh ^{-1}(c+d x)}}{\sqrt {b}}\right )}{16 d}-\frac {\sqrt {\frac {\pi }{3}} \sqrt {b} e^2 e^{-\frac {3 a}{b}} \text {erfi}\left (\frac {\sqrt {3} \sqrt {a+b \cosh ^{-1}(c+d x)}}{\sqrt {b}}\right )}{48 d}+\frac {e^2 (c+d x)^3 \sqrt {a+b \cosh ^{-1}(c+d x)}}{3 d} \]
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Rubi [A] time = 0.76, antiderivative size = 245, normalized size of antiderivative = 1.00, number of steps used = 16, number of rules used = 9, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.360, Rules used = {5866, 12, 5664, 5781, 3312, 3307, 2180, 2204, 2205} \[ -\frac {\sqrt {\pi } \sqrt {b} e^2 e^{a/b} \text {Erf}\left (\frac {\sqrt {a+b \cosh ^{-1}(c+d x)}}{\sqrt {b}}\right )}{16 d}-\frac {\sqrt {\frac {\pi }{3}} \sqrt {b} e^2 e^{\frac {3 a}{b}} \text {Erf}\left (\frac {\sqrt {3} \sqrt {a+b \cosh ^{-1}(c+d x)}}{\sqrt {b}}\right )}{48 d}-\frac {\sqrt {\pi } \sqrt {b} e^2 e^{-\frac {a}{b}} \text {Erfi}\left (\frac {\sqrt {a+b \cosh ^{-1}(c+d x)}}{\sqrt {b}}\right )}{16 d}-\frac {\sqrt {\frac {\pi }{3}} \sqrt {b} e^2 e^{-\frac {3 a}{b}} \text {Erfi}\left (\frac {\sqrt {3} \sqrt {a+b \cosh ^{-1}(c+d x)}}{\sqrt {b}}\right )}{48 d}+\frac {e^2 (c+d x)^3 \sqrt {a+b \cosh ^{-1}(c+d x)}}{3 d} \]
Antiderivative was successfully verified.
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Rule 12
Rule 2180
Rule 2204
Rule 2205
Rule 3307
Rule 3312
Rule 5664
Rule 5781
Rule 5866
Rubi steps
\begin {align*} \int (c e+d e x)^2 \sqrt {a+b \cosh ^{-1}(c+d x)} \, dx &=\frac {\operatorname {Subst}\left (\int e^2 x^2 \sqrt {a+b \cosh ^{-1}(x)} \, dx,x,c+d x\right )}{d}\\ &=\frac {e^2 \operatorname {Subst}\left (\int x^2 \sqrt {a+b \cosh ^{-1}(x)} \, dx,x,c+d x\right )}{d}\\ &=\frac {e^2 (c+d x)^3 \sqrt {a+b \cosh ^{-1}(c+d x)}}{3 d}-\frac {\left (b e^2\right ) \operatorname {Subst}\left (\int \frac {x^3}{\sqrt {-1+x} \sqrt {1+x} \sqrt {a+b \cosh ^{-1}(x)}} \, dx,x,c+d x\right )}{6 d}\\ &=\frac {e^2 (c+d x)^3 \sqrt {a+b \cosh ^{-1}(c+d x)}}{3 d}-\frac {\left (b e^2\right ) \operatorname {Subst}\left (\int \frac {\cosh ^3(x)}{\sqrt {a+b x}} \, dx,x,\cosh ^{-1}(c+d x)\right )}{6 d}\\ &=\frac {e^2 (c+d x)^3 \sqrt {a+b \cosh ^{-1}(c+d x)}}{3 d}-\frac {\left (b e^2\right ) \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+d x)\right )}{6 d}\\ &=\frac {e^2 (c+d x)^3 \sqrt {a+b \cosh ^{-1}(c+d x)}}{3 d}-\frac {\left (b e^2\right ) \operatorname {Subst}\left (\int \frac {\cosh (3 x)}{\sqrt {a+b x}} \, dx,x,\cosh ^{-1}(c+d x)\right )}{24 d}-\frac {\left (b e^2\right ) \operatorname {Subst}\left (\int \frac {\cosh (x)}{\sqrt {a+b x}} \, dx,x,\cosh ^{-1}(c+d x)\right )}{8 d}\\ &=\frac {e^2 (c+d x)^3 \sqrt {a+b \cosh ^{-1}(c+d x)}}{3 d}-\frac {\left (b e^2\right ) \operatorname {Subst}\left (\int \frac {e^{-3 x}}{\sqrt {a+b x}} \, dx,x,\cosh ^{-1}(c+d x)\right )}{48 d}-\frac {\left (b e^2\right ) \operatorname {Subst}\left (\int \frac {e^{3 x}}{\sqrt {a+b x}} \, dx,x,\cosh ^{-1}(c+d x)\right )}{48 d}-\frac {\left (b e^2\right ) \operatorname {Subst}\left (\int \frac {e^{-x}}{\sqrt {a+b x}} \, dx,x,\cosh ^{-1}(c+d x)\right )}{16 d}-\frac {\left (b e^2\right ) \operatorname {Subst}\left (\int \frac {e^x}{\sqrt {a+b x}} \, dx,x,\cosh ^{-1}(c+d x)\right )}{16 d}\\ &=\frac {e^2 (c+d x)^3 \sqrt {a+b \cosh ^{-1}(c+d x)}}{3 d}-\frac {e^2 \operatorname {Subst}\left (\int e^{\frac {3 a}{b}-\frac {3 x^2}{b}} \, dx,x,\sqrt {a+b \cosh ^{-1}(c+d x)}\right )}{24 d}-\frac {e^2 \operatorname {Subst}\left (\int e^{-\frac {3 a}{b}+\frac {3 x^2}{b}} \, dx,x,\sqrt {a+b \cosh ^{-1}(c+d x)}\right )}{24 d}-\frac {e^2 \operatorname {Subst}\left (\int e^{\frac {a}{b}-\frac {x^2}{b}} \, dx,x,\sqrt {a+b \cosh ^{-1}(c+d x)}\right )}{8 d}-\frac {e^2 \operatorname {Subst}\left (\int e^{-\frac {a}{b}+\frac {x^2}{b}} \, dx,x,\sqrt {a+b \cosh ^{-1}(c+d x)}\right )}{8 d}\\ &=\frac {e^2 (c+d x)^3 \sqrt {a+b \cosh ^{-1}(c+d x)}}{3 d}-\frac {\sqrt {b} e^2 e^{a/b} \sqrt {\pi } \text {erf}\left (\frac {\sqrt {a+b \cosh ^{-1}(c+d x)}}{\sqrt {b}}\right )}{16 d}-\frac {\sqrt {b} e^2 e^{\frac {3 a}{b}} \sqrt {\frac {\pi }{3}} \text {erf}\left (\frac {\sqrt {3} \sqrt {a+b \cosh ^{-1}(c+d x)}}{\sqrt {b}}\right )}{48 d}-\frac {\sqrt {b} e^2 e^{-\frac {a}{b}} \sqrt {\pi } \text {erfi}\left (\frac {\sqrt {a+b \cosh ^{-1}(c+d x)}}{\sqrt {b}}\right )}{16 d}-\frac {\sqrt {b} e^2 e^{-\frac {3 a}{b}} \sqrt {\frac {\pi }{3}} \text {erfi}\left (\frac {\sqrt {3} \sqrt {a+b \cosh ^{-1}(c+d x)}}{\sqrt {b}}\right )}{48 d}\\ \end {align*}
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Mathematica [A] time = 0.51, size = 237, normalized size = 0.97 \[ \frac {e^2 e^{-\frac {3 a}{b}} \sqrt {a+b \cosh ^{-1}(c+d x)} \left (9 e^{\frac {4 a}{b}} \sqrt {-\frac {a+b \cosh ^{-1}(c+d x)}{b}} \Gamma \left (\frac {3}{2},\frac {a}{b}+\cosh ^{-1}(c+d x)\right )+\sqrt {3} \sqrt {\frac {a}{b}+\cosh ^{-1}(c+d x)} \Gamma \left (\frac {3}{2},-\frac {3 \left (a+b \cosh ^{-1}(c+d x)\right )}{b}\right )+9 e^{\frac {2 a}{b}} \sqrt {\frac {a}{b}+\cosh ^{-1}(c+d x)} \Gamma \left (\frac {3}{2},-\frac {a+b \cosh ^{-1}(c+d x)}{b}\right )+\sqrt {3} e^{\frac {6 a}{b}} \sqrt {-\frac {a+b \cosh ^{-1}(c+d x)}{b}} \Gamma \left (\frac {3}{2},\frac {3 \left (a+b \cosh ^{-1}(c+d x)\right )}{b}\right )\right )}{72 d \sqrt {-\frac {\left (a+b \cosh ^{-1}(c+d x)\right )^2}{b^2}}} \]
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(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
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 \[ \int \left (d e x +c e \right )^{2} \sqrt {a +b \,\mathrm {arccosh}\left (d x +c \right )}\, dx \]
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 \[ \int {\left (d e x + c e\right )}^{2} \sqrt {b \operatorname {arcosh}\left (d x + c\right ) + a}\,{d x} \]
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 \[ \int {\left (c\,e+d\,e\,x\right )}^2\,\sqrt {a+b\,\mathrm {acosh}\left (c+d\,x\right )} \,d x \]
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 \[ e^{2} \left (\int c^{2} \sqrt {a + b \operatorname {acosh}{\left (c + d x \right )}}\, dx + \int d^{2} x^{2} \sqrt {a + b \operatorname {acosh}{\left (c + d x \right )}}\, dx + \int 2 c d x \sqrt {a + b \operatorname {acosh}{\left (c + d x \right )}}\, dx\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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