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 = 0.575408, antiderivative size = 164, normalized size of antiderivative = 1., 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 5866
Rule 12
Rule 5664
Rule 5781
Rule 3312
Rule 3307
Rule 2180
Rule 2204
Rule 2205
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*}
Mathematica [B] time = 2.3693, size = 437, normalized size = 2.66 \[ \frac{e \left (16 c e^{-\frac{a}{b}} \sqrt{a+b \cosh ^{-1}(c+d x)} \left (\frac{e^{\frac{2 a}{b}} \text{Gamma}\left (\frac{3}{2},\frac{a}{b}+\cosh ^{-1}(c+d x)\right )}{\sqrt{\frac{a}{b}+\cosh ^{-1}(c+d x)}}+\frac{\text{Gamma}\left (\frac{3}{2},-\frac{a+b \cosh ^{-1}(c+d x)}{b}\right )}{\sqrt{-\frac{a+b \cosh ^{-1}(c+d x)}{b}}}\right )+8 \sqrt{\pi } \sqrt{b} c \left (\sinh \left (\frac{a}{b}\right )+\cosh \left (\frac{a}{b}\right )\right ) \text{Erf}\left (\frac{\sqrt{a+b \cosh ^{-1}(c+d x)}}{\sqrt{b}}\right )-\sqrt{2 \pi } \sqrt{b} \left (\sinh \left (\frac{2 a}{b}\right )+\cosh \left (\frac{2 a}{b}\right )\right ) \text{Erf}\left (\frac{\sqrt{2} \sqrt{a+b \cosh ^{-1}(c+d x)}}{\sqrt{b}}\right )+8 \sqrt{\pi } \sqrt{b} c \cosh \left (\frac{a}{b}\right ) \text{Erfi}\left (\frac{\sqrt{a+b \cosh ^{-1}(c+d x)}}{\sqrt{b}}\right )-\sqrt{2 \pi } \sqrt{b} \cosh \left (\frac{2 a}{b}\right ) \text{Erfi}\left (\frac{\sqrt{2} \sqrt{a+b \cosh ^{-1}(c+d x)}}{\sqrt{b}}\right )-8 \sqrt{\pi } \sqrt{b} c \sinh \left (\frac{a}{b}\right ) \text{Erfi}\left (\frac{\sqrt{a+b \cosh ^{-1}(c+d x)}}{\sqrt{b}}\right )+\sqrt{2 \pi } \sqrt{b} \sinh \left (\frac{2 a}{b}\right ) \text{Erfi}\left (\frac{\sqrt{2} \sqrt{a+b \cosh ^{-1}(c+d x)}}{\sqrt{b}}\right )-32 c (c+d x) \sqrt{a+b \cosh ^{-1}(c+d x)}+8 \cosh \left (2 \cosh ^{-1}(c+d x)\right ) \sqrt{a+b \cosh ^{-1}(c+d x)}\right )}{32 d} \]
Warning: Unable to verify antiderivative.
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Maple [F] time = 0.128, size = 0, normalized size = 0. \begin{align*} \int \left ( dex+ce \right ) \sqrt{a+b{\rm arccosh} \left (dx+c\right )}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (d e x + c e\right )} \sqrt{b \operatorname{arcosh}\left (d x + c\right ) + a}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: UnboundLocalError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} 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 ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \mathit{sage}_{0} x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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