Optimal. Leaf size=164 \[ -\frac {\sqrt {\frac {\pi }{2}} \sqrt {b} e e^{\frac {2 a}{b}} \text {erf}\left (\frac {\sqrt {2} \sqrt {a+b \cosh ^{-1}(c+d x)}}{\sqrt {b}}\right )}{16 d}-\frac {\sqrt {\frac {\pi }{2}} \sqrt {b} e e^{-\frac {2 a}{b}} \text {erfi}\left (\frac {\sqrt {2} \sqrt {a+b \cosh ^{-1}(c+d x)}}{\sqrt {b}}\right )}{16 d}+\frac {e (c+d x)^2 \sqrt {a+b \cosh ^{-1}(c+d x)}}{2 d}-\frac {e \sqrt {a+b \cosh ^{-1}(c+d x)}}{4 d} \]
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Rubi [A] time = 0.58, antiderivative size = 164, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 9, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.391, Rules used = {5866, 12, 5664, 5781, 3312, 3307, 2180, 2204, 2205} \[ -\frac {\sqrt {\frac {\pi }{2}} \sqrt {b} e e^{\frac {2 a}{b}} \text {Erf}\left (\frac {\sqrt {2} \sqrt {a+b \cosh ^{-1}(c+d x)}}{\sqrt {b}}\right )}{16 d}-\frac {\sqrt {\frac {\pi }{2}} \sqrt {b} e e^{-\frac {2 a}{b}} \text {Erfi}\left (\frac {\sqrt {2} \sqrt {a+b \cosh ^{-1}(c+d x)}}{\sqrt {b}}\right )}{16 d}+\frac {e (c+d x)^2 \sqrt {a+b \cosh ^{-1}(c+d x)}}{2 d}-\frac {e \sqrt {a+b \cosh ^{-1}(c+d x)}}{4 d} \]
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) \sqrt {a+b \cosh ^{-1}(c+d x)} \, dx &=\frac {\operatorname {Subst}\left (\int e x \sqrt {a+b \cosh ^{-1}(x)} \, dx,x,c+d x\right )}{d}\\ &=\frac {e \operatorname {Subst}\left (\int x \sqrt {a+b \cosh ^{-1}(x)} \, dx,x,c+d x\right )}{d}\\ &=\frac {e (c+d x)^2 \sqrt {a+b \cosh ^{-1}(c+d x)}}{2 d}-\frac {(b e) \operatorname {Subst}\left (\int \frac {x^2}{\sqrt {-1+x} \sqrt {1+x} \sqrt {a+b \cosh ^{-1}(x)}} \, dx,x,c+d x\right )}{4 d}\\ &=\frac {e (c+d x)^2 \sqrt {a+b \cosh ^{-1}(c+d x)}}{2 d}-\frac {(b e) \operatorname {Subst}\left (\int \frac {\cosh ^2(x)}{\sqrt {a+b x}} \, dx,x,\cosh ^{-1}(c+d x)\right )}{4 d}\\ &=\frac {e (c+d x)^2 \sqrt {a+b \cosh ^{-1}(c+d x)}}{2 d}-\frac {(b e) \operatorname {Subst}\left (\int \left (\frac {1}{2 \sqrt {a+b x}}+\frac {\cosh (2 x)}{2 \sqrt {a+b x}}\right ) \, dx,x,\cosh ^{-1}(c+d x)\right )}{4 d}\\ &=-\frac {e \sqrt {a+b \cosh ^{-1}(c+d x)}}{4 d}+\frac {e (c+d x)^2 \sqrt {a+b \cosh ^{-1}(c+d x)}}{2 d}-\frac {(b e) \operatorname {Subst}\left (\int \frac {\cosh (2 x)}{\sqrt {a+b x}} \, dx,x,\cosh ^{-1}(c+d x)\right )}{8 d}\\ &=-\frac {e \sqrt {a+b \cosh ^{-1}(c+d x)}}{4 d}+\frac {e (c+d x)^2 \sqrt {a+b \cosh ^{-1}(c+d x)}}{2 d}-\frac {(b e) \operatorname {Subst}\left (\int \frac {e^{-2 x}}{\sqrt {a+b x}} \, dx,x,\cosh ^{-1}(c+d x)\right )}{16 d}-\frac {(b e) \operatorname {Subst}\left (\int \frac {e^{2 x}}{\sqrt {a+b x}} \, dx,x,\cosh ^{-1}(c+d x)\right )}{16 d}\\ &=-\frac {e \sqrt {a+b \cosh ^{-1}(c+d x)}}{4 d}+\frac {e (c+d x)^2 \sqrt {a+b \cosh ^{-1}(c+d x)}}{2 d}-\frac {e \operatorname {Subst}\left (\int e^{\frac {2 a}{b}-\frac {2 x^2}{b}} \, dx,x,\sqrt {a+b \cosh ^{-1}(c+d x)}\right )}{8 d}-\frac {e \operatorname {Subst}\left (\int e^{-\frac {2 a}{b}+\frac {2 x^2}{b}} \, dx,x,\sqrt {a+b \cosh ^{-1}(c+d x)}\right )}{8 d}\\ &=-\frac {e \sqrt {a+b \cosh ^{-1}(c+d x)}}{4 d}+\frac {e (c+d x)^2 \sqrt {a+b \cosh ^{-1}(c+d x)}}{2 d}-\frac {\sqrt {b} e e^{\frac {2 a}{b}} \sqrt {\frac {\pi }{2}} \text {erf}\left (\frac {\sqrt {2} \sqrt {a+b \cosh ^{-1}(c+d x)}}{\sqrt {b}}\right )}{16 d}-\frac {\sqrt {b} e e^{-\frac {2 a}{b}} \sqrt {\frac {\pi }{2}} \text {erfi}\left (\frac {\sqrt {2} \sqrt {a+b \cosh ^{-1}(c+d x)}}{\sqrt {b}}\right )}{16 d}\\ \end {align*}
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Mathematica [B] time = 2.52, size = 437, normalized size = 2.66 \[ \frac {e \left (8 \sqrt {\pi } \sqrt {b} c \left (\sinh \left (\frac {a}{b}\right )+\cosh \left (\frac {a}{b}\right )\right ) \text {erf}\left (\frac {\sqrt {a+b \cosh ^{-1}(c+d x)}}{\sqrt {b}}\right )-\sqrt {2 \pi } \sqrt {b} \left (\sinh \left (\frac {2 a}{b}\right )+\cosh \left (\frac {2 a}{b}\right )\right ) \text {erf}\left (\frac {\sqrt {2} \sqrt {a+b \cosh ^{-1}(c+d x)}}{\sqrt {b}}\right )+8 \sqrt {\pi } \sqrt {b} c \cosh \left (\frac {a}{b}\right ) \text {erfi}\left (\frac {\sqrt {a+b \cosh ^{-1}(c+d x)}}{\sqrt {b}}\right )-\sqrt {2 \pi } \sqrt {b} \cosh \left (\frac {2 a}{b}\right ) \text {erfi}\left (\frac {\sqrt {2} \sqrt {a+b \cosh ^{-1}(c+d x)}}{\sqrt {b}}\right )-8 \sqrt {\pi } \sqrt {b} c \sinh \left (\frac {a}{b}\right ) \text {erfi}\left (\frac {\sqrt {a+b \cosh ^{-1}(c+d x)}}{\sqrt {b}}\right )+\sqrt {2 \pi } \sqrt {b} \sinh \left (\frac {2 a}{b}\right ) \text {erfi}\left (\frac {\sqrt {2} \sqrt {a+b \cosh ^{-1}(c+d x)}}{\sqrt {b}}\right )-32 c (c+d x) \sqrt {a+b \cosh ^{-1}(c+d x)}+8 \cosh \left (2 \cosh ^{-1}(c+d x)\right ) \sqrt {a+b \cosh ^{-1}(c+d x)}+16 c e^{-\frac {a}{b}} \sqrt {a+b \cosh ^{-1}(c+d x)} \left (\frac {e^{\frac {2 a}{b}} \Gamma \left (\frac {3}{2},\frac {a}{b}+\cosh ^{-1}(c+d x)\right )}{\sqrt {\frac {a}{b}+\cosh ^{-1}(c+d x)}}+\frac {\Gamma \left (\frac {3}{2},-\frac {a+b \cosh ^{-1}(c+d x)}{b}\right )}{\sqrt {-\frac {a+b \cosh ^{-1}(c+d x)}{b}}}\right )\right )}{32 d} \]
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] time = 0.16, size = 0, normalized size = 0.00 \[ \int \left (d e x +c e \right ) \sqrt {a +b \,\mathrm {arccosh}\left (d x +c \right )}\, dx \]
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 \[ \int {\left (d e x + c e\right )} \sqrt {b \operatorname {arcosh}\left (d x + c\right ) + a}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \left (c\,e+d\,e\,x\right )\,\sqrt {a+b\,\mathrm {acosh}\left (c+d\,x\right )} \,d x \]
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 \[ e \left (\int c \sqrt {a + b \operatorname {acosh}{\left (c + d x \right )}}\, dx + \int d x \sqrt {a + b \operatorname {acosh}{\left (c + d x \right )}}\, dx\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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