Optimal. Leaf size=361 \[ -\frac{\sqrt{\pi } \sqrt{b} e^4 e^{a/b} \text{Erf}\left (\frac{\sqrt{a+b \cosh ^{-1}(c+d x)}}{\sqrt{b}}\right )}{32 d}-\frac{\sqrt{\frac{\pi }{3}} \sqrt{b} e^4 e^{\frac{3 a}{b}} \text{Erf}\left (\frac{\sqrt{3} \sqrt{a+b \cosh ^{-1}(c+d x)}}{\sqrt{b}}\right )}{64 d}-\frac{\sqrt{\frac{\pi }{5}} \sqrt{b} e^4 e^{\frac{5 a}{b}} \text{Erf}\left (\frac{\sqrt{5} \sqrt{a+b \cosh ^{-1}(c+d x)}}{\sqrt{b}}\right )}{320 d}-\frac{\sqrt{\pi } \sqrt{b} e^4 e^{-\frac{a}{b}} \text{Erfi}\left (\frac{\sqrt{a+b \cosh ^{-1}(c+d x)}}{\sqrt{b}}\right )}{32 d}-\frac{\sqrt{\frac{\pi }{3}} \sqrt{b} e^4 e^{-\frac{3 a}{b}} \text{Erfi}\left (\frac{\sqrt{3} \sqrt{a+b \cosh ^{-1}(c+d x)}}{\sqrt{b}}\right )}{64 d}-\frac{\sqrt{\frac{\pi }{5}} \sqrt{b} e^4 e^{-\frac{5 a}{b}} \text{Erfi}\left (\frac{\sqrt{5} \sqrt{a+b \cosh ^{-1}(c+d x)}}{\sqrt{b}}\right )}{320 d}+\frac{e^4 (c+d x)^5 \sqrt{a+b \cosh ^{-1}(c+d x)}}{5 d} \]
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Rubi [A] time = 0.951133, antiderivative size = 361, normalized size of antiderivative = 1., number of steps used = 21, number of rules used = 9, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.36, Rules used = {5866, 12, 5664, 5781, 3312, 3307, 2180, 2204, 2205} \[ -\frac{\sqrt{\pi } \sqrt{b} e^4 e^{a/b} \text{Erf}\left (\frac{\sqrt{a+b \cosh ^{-1}(c+d x)}}{\sqrt{b}}\right )}{32 d}-\frac{\sqrt{\frac{\pi }{3}} \sqrt{b} e^4 e^{\frac{3 a}{b}} \text{Erf}\left (\frac{\sqrt{3} \sqrt{a+b \cosh ^{-1}(c+d x)}}{\sqrt{b}}\right )}{64 d}-\frac{\sqrt{\frac{\pi }{5}} \sqrt{b} e^4 e^{\frac{5 a}{b}} \text{Erf}\left (\frac{\sqrt{5} \sqrt{a+b \cosh ^{-1}(c+d x)}}{\sqrt{b}}\right )}{320 d}-\frac{\sqrt{\pi } \sqrt{b} e^4 e^{-\frac{a}{b}} \text{Erfi}\left (\frac{\sqrt{a+b \cosh ^{-1}(c+d x)}}{\sqrt{b}}\right )}{32 d}-\frac{\sqrt{\frac{\pi }{3}} \sqrt{b} e^4 e^{-\frac{3 a}{b}} \text{Erfi}\left (\frac{\sqrt{3} \sqrt{a+b \cosh ^{-1}(c+d x)}}{\sqrt{b}}\right )}{64 d}-\frac{\sqrt{\frac{\pi }{5}} \sqrt{b} e^4 e^{-\frac{5 a}{b}} \text{Erfi}\left (\frac{\sqrt{5} \sqrt{a+b \cosh ^{-1}(c+d x)}}{\sqrt{b}}\right )}{320 d}+\frac{e^4 (c+d x)^5 \sqrt{a+b \cosh ^{-1}(c+d x)}}{5 d} \]
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)^4 \sqrt{a+b \cosh ^{-1}(c+d x)} \, dx &=\frac{\operatorname{Subst}\left (\int e^4 x^4 \sqrt{a+b \cosh ^{-1}(x)} \, dx,x,c+d x\right )}{d}\\ &=\frac{e^4 \operatorname{Subst}\left (\int x^4 \sqrt{a+b \cosh ^{-1}(x)} \, dx,x,c+d x\right )}{d}\\ &=\frac{e^4 (c+d x)^5 \sqrt{a+b \cosh ^{-1}(c+d x)}}{5 d}-\frac{\left (b e^4\right ) \operatorname{Subst}\left (\int \frac{x^5}{\sqrt{-1+x} \sqrt{1+x} \sqrt{a+b \cosh ^{-1}(x)}} \, dx,x,c+d x\right )}{10 d}\\ &=\frac{e^4 (c+d x)^5 \sqrt{a+b \cosh ^{-1}(c+d x)}}{5 d}-\frac{\left (b e^4\right ) \operatorname{Subst}\left (\int \frac{\cosh ^5(x)}{\sqrt{a+b x}} \, dx,x,\cosh ^{-1}(c+d x)\right )}{10 d}\\ &=\frac{e^4 (c+d x)^5 \sqrt{a+b \cosh ^{-1}(c+d x)}}{5 d}-\frac{\left (b e^4\right ) \operatorname{Subst}\left (\int \left (\frac{5 \cosh (x)}{8 \sqrt{a+b x}}+\frac{5 \cosh (3 x)}{16 \sqrt{a+b x}}+\frac{\cosh (5 x)}{16 \sqrt{a+b x}}\right ) \, dx,x,\cosh ^{-1}(c+d x)\right )}{10 d}\\ &=\frac{e^4 (c+d x)^5 \sqrt{a+b \cosh ^{-1}(c+d x)}}{5 d}-\frac{\left (b e^4\right ) \operatorname{Subst}\left (\int \frac{\cosh (5 x)}{\sqrt{a+b x}} \, dx,x,\cosh ^{-1}(c+d x)\right )}{160 d}-\frac{\left (b e^4\right ) \operatorname{Subst}\left (\int \frac{\cosh (3 x)}{\sqrt{a+b x}} \, dx,x,\cosh ^{-1}(c+d x)\right )}{32 d}-\frac{\left (b e^4\right ) \operatorname{Subst}\left (\int \frac{\cosh (x)}{\sqrt{a+b x}} \, dx,x,\cosh ^{-1}(c+d x)\right )}{16 d}\\ &=\frac{e^4 (c+d x)^5 \sqrt{a+b \cosh ^{-1}(c+d x)}}{5 d}-\frac{\left (b e^4\right ) \operatorname{Subst}\left (\int \frac{e^{-5 x}}{\sqrt{a+b x}} \, dx,x,\cosh ^{-1}(c+d x)\right )}{320 d}-\frac{\left (b e^4\right ) \operatorname{Subst}\left (\int \frac{e^{5 x}}{\sqrt{a+b x}} \, dx,x,\cosh ^{-1}(c+d x)\right )}{320 d}-\frac{\left (b e^4\right ) \operatorname{Subst}\left (\int \frac{e^{-3 x}}{\sqrt{a+b x}} \, dx,x,\cosh ^{-1}(c+d x)\right )}{64 d}-\frac{\left (b e^4\right ) \operatorname{Subst}\left (\int \frac{e^{3 x}}{\sqrt{a+b x}} \, dx,x,\cosh ^{-1}(c+d x)\right )}{64 d}-\frac{\left (b e^4\right ) \operatorname{Subst}\left (\int \frac{e^{-x}}{\sqrt{a+b x}} \, dx,x,\cosh ^{-1}(c+d x)\right )}{32 d}-\frac{\left (b e^4\right ) \operatorname{Subst}\left (\int \frac{e^x}{\sqrt{a+b x}} \, dx,x,\cosh ^{-1}(c+d x)\right )}{32 d}\\ &=\frac{e^4 (c+d x)^5 \sqrt{a+b \cosh ^{-1}(c+d x)}}{5 d}-\frac{e^4 \operatorname{Subst}\left (\int e^{\frac{5 a}{b}-\frac{5 x^2}{b}} \, dx,x,\sqrt{a+b \cosh ^{-1}(c+d x)}\right )}{160 d}-\frac{e^4 \operatorname{Subst}\left (\int e^{-\frac{5 a}{b}+\frac{5 x^2}{b}} \, dx,x,\sqrt{a+b \cosh ^{-1}(c+d x)}\right )}{160 d}-\frac{e^4 \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 )}{32 d}-\frac{e^4 \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 )}{32 d}-\frac{e^4 \operatorname{Subst}\left (\int e^{\frac{a}{b}-\frac{x^2}{b}} \, dx,x,\sqrt{a+b \cosh ^{-1}(c+d x)}\right )}{16 d}-\frac{e^4 \operatorname{Subst}\left (\int e^{-\frac{a}{b}+\frac{x^2}{b}} \, dx,x,\sqrt{a+b \cosh ^{-1}(c+d x)}\right )}{16 d}\\ &=\frac{e^4 (c+d x)^5 \sqrt{a+b \cosh ^{-1}(c+d x)}}{5 d}-\frac{\sqrt{b} e^4 e^{a/b} \sqrt{\pi } \text{erf}\left (\frac{\sqrt{a+b \cosh ^{-1}(c+d x)}}{\sqrt{b}}\right )}{32 d}-\frac{\sqrt{b} e^4 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 )}{64 d}-\frac{\sqrt{b} e^4 e^{\frac{5 a}{b}} \sqrt{\frac{\pi }{5}} \text{erf}\left (\frac{\sqrt{5} \sqrt{a+b \cosh ^{-1}(c+d x)}}{\sqrt{b}}\right )}{320 d}-\frac{\sqrt{b} e^4 e^{-\frac{a}{b}} \sqrt{\pi } \text{erfi}\left (\frac{\sqrt{a+b \cosh ^{-1}(c+d x)}}{\sqrt{b}}\right )}{32 d}-\frac{\sqrt{b} e^4 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 )}{64 d}-\frac{\sqrt{b} e^4 e^{-\frac{5 a}{b}} \sqrt{\frac{\pi }{5}} \text{erfi}\left (\frac{\sqrt{5} \sqrt{a+b \cosh ^{-1}(c+d x)}}{\sqrt{b}}\right )}{320 d}\\ \end{align*}
Mathematica [A] time = 0.707211, size = 342, normalized size = 0.95 \[ \frac{e^4 e^{-\frac{5 a}{b}} \sqrt{a+b \cosh ^{-1}(c+d x)} \left (150 e^{\frac{6 a}{b}} \sqrt{-\frac{a+b \cosh ^{-1}(c+d x)}{b}} \text{Gamma}\left (\frac{3}{2},\frac{a}{b}+\cosh ^{-1}(c+d x)\right )+3 \sqrt{5} \sqrt{\frac{a}{b}+\cosh ^{-1}(c+d x)} \text{Gamma}\left (\frac{3}{2},-\frac{5 \left (a+b \cosh ^{-1}(c+d x)\right )}{b}\right )+25 \sqrt{3} e^{\frac{2 a}{b}} \sqrt{\frac{a}{b}+\cosh ^{-1}(c+d x)} \text{Gamma}\left (\frac{3}{2},-\frac{3 \left (a+b \cosh ^{-1}(c+d x)\right )}{b}\right )+150 e^{\frac{4 a}{b}} \sqrt{\frac{a}{b}+\cosh ^{-1}(c+d x)} \text{Gamma}\left (\frac{3}{2},-\frac{a+b \cosh ^{-1}(c+d x)}{b}\right )+25 \sqrt{3} e^{\frac{8 a}{b}} \sqrt{-\frac{a+b \cosh ^{-1}(c+d x)}{b}} \text{Gamma}\left (\frac{3}{2},\frac{3 \left (a+b \cosh ^{-1}(c+d x)\right )}{b}\right )+3 \sqrt{5} e^{\frac{10 a}{b}} \sqrt{-\frac{a+b \cosh ^{-1}(c+d x)}{b}} \text{Gamma}\left (\frac{3}{2},\frac{5 \left (a+b \cosh ^{-1}(c+d x)\right )}{b}\right )\right )}{2400 d \sqrt{-\frac{\left (a+b \cosh ^{-1}(c+d x)\right )^2}{b^2}}} \]
Warning: Unable to verify antiderivative.
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Maple [F] time = 0.346, size = 0, normalized size = 0. \begin{align*} \int \left ( dex+ce \right ) ^{4}\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 )}^{4} \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^{4} \left (\int c^{4} \sqrt{a + b \operatorname{acosh}{\left (c + d x \right )}}\, dx + \int d^{4} x^{4} \sqrt{a + b \operatorname{acosh}{\left (c + d x \right )}}\, dx + \int 4 c d^{3} x^{3} \sqrt{a + b \operatorname{acosh}{\left (c + d x \right )}}\, dx + \int 6 c^{2} d^{2} x^{2} \sqrt{a + b \operatorname{acosh}{\left (c + d x \right )}}\, dx + \int 4 c^{3} 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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