Optimal. Leaf size=444 \[ -\frac{\sqrt{\pi } e^4 e^{a/b} \text{Erf}\left (\frac{\sqrt{a+b \cosh ^{-1}(c+d x)}}{\sqrt{b}}\right )}{12 b^{5/2} d}-\frac{3 \sqrt{3 \pi } e^4 e^{\frac{3 a}{b}} \text{Erf}\left (\frac{\sqrt{3} \sqrt{a+b \cosh ^{-1}(c+d x)}}{\sqrt{b}}\right )}{8 b^{5/2} d}-\frac{5 \sqrt{5 \pi } e^4 e^{\frac{5 a}{b}} \text{Erf}\left (\frac{\sqrt{5} \sqrt{a+b \cosh ^{-1}(c+d x)}}{\sqrt{b}}\right )}{24 b^{5/2} d}+\frac{\sqrt{\pi } e^4 e^{-\frac{a}{b}} \text{Erfi}\left (\frac{\sqrt{a+b \cosh ^{-1}(c+d x)}}{\sqrt{b}}\right )}{12 b^{5/2} d}+\frac{3 \sqrt{3 \pi } e^4 e^{-\frac{3 a}{b}} \text{Erfi}\left (\frac{\sqrt{3} \sqrt{a+b \cosh ^{-1}(c+d x)}}{\sqrt{b}}\right )}{8 b^{5/2} d}+\frac{5 \sqrt{5 \pi } e^4 e^{-\frac{5 a}{b}} \text{Erfi}\left (\frac{\sqrt{5} \sqrt{a+b \cosh ^{-1}(c+d x)}}{\sqrt{b}}\right )}{24 b^{5/2} d}-\frac{20 e^4 (c+d x)^5}{3 b^2 d \sqrt{a+b \cosh ^{-1}(c+d x)}}+\frac{16 e^4 (c+d x)^3}{3 b^2 d \sqrt{a+b \cosh ^{-1}(c+d x)}}-\frac{2 e^4 \sqrt{c+d x-1} \sqrt{c+d x+1} (c+d x)^4}{3 b d \left (a+b \cosh ^{-1}(c+d x)\right )^{3/2}} \]
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Rubi [A] time = 1.76501, antiderivative size = 444, normalized size of antiderivative = 1., number of steps used = 36, number of rules used = 10, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.4, Rules used = {5866, 12, 5668, 5775, 5670, 5448, 3308, 2180, 2204, 2205} \[ -\frac{\sqrt{\pi } e^4 e^{a/b} \text{Erf}\left (\frac{\sqrt{a+b \cosh ^{-1}(c+d x)}}{\sqrt{b}}\right )}{12 b^{5/2} d}-\frac{3 \sqrt{3 \pi } e^4 e^{\frac{3 a}{b}} \text{Erf}\left (\frac{\sqrt{3} \sqrt{a+b \cosh ^{-1}(c+d x)}}{\sqrt{b}}\right )}{8 b^{5/2} d}-\frac{5 \sqrt{5 \pi } e^4 e^{\frac{5 a}{b}} \text{Erf}\left (\frac{\sqrt{5} \sqrt{a+b \cosh ^{-1}(c+d x)}}{\sqrt{b}}\right )}{24 b^{5/2} d}+\frac{\sqrt{\pi } e^4 e^{-\frac{a}{b}} \text{Erfi}\left (\frac{\sqrt{a+b \cosh ^{-1}(c+d x)}}{\sqrt{b}}\right )}{12 b^{5/2} d}+\frac{3 \sqrt{3 \pi } e^4 e^{-\frac{3 a}{b}} \text{Erfi}\left (\frac{\sqrt{3} \sqrt{a+b \cosh ^{-1}(c+d x)}}{\sqrt{b}}\right )}{8 b^{5/2} d}+\frac{5 \sqrt{5 \pi } e^4 e^{-\frac{5 a}{b}} \text{Erfi}\left (\frac{\sqrt{5} \sqrt{a+b \cosh ^{-1}(c+d x)}}{\sqrt{b}}\right )}{24 b^{5/2} d}-\frac{20 e^4 (c+d x)^5}{3 b^2 d \sqrt{a+b \cosh ^{-1}(c+d x)}}+\frac{16 e^4 (c+d x)^3}{3 b^2 d \sqrt{a+b \cosh ^{-1}(c+d x)}}-\frac{2 e^4 \sqrt{c+d x-1} \sqrt{c+d x+1} (c+d x)^4}{3 b d \left (a+b \cosh ^{-1}(c+d x)\right )^{3/2}} \]
Antiderivative was successfully verified.
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Rule 5866
Rule 12
Rule 5668
Rule 5775
Rule 5670
Rule 5448
Rule 3308
Rule 2180
Rule 2204
Rule 2205
Rubi steps
\begin{align*} \int \frac{(c e+d e x)^4}{\left (a+b \cosh ^{-1}(c+d x)\right )^{5/2}} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{e^4 x^4}{\left (a+b \cosh ^{-1}(x)\right )^{5/2}} \, dx,x,c+d x\right )}{d}\\ &=\frac{e^4 \operatorname{Subst}\left (\int \frac{x^4}{\left (a+b \cosh ^{-1}(x)\right )^{5/2}} \, dx,x,c+d x\right )}{d}\\ &=-\frac{2 e^4 \sqrt{-1+c+d x} (c+d x)^4 \sqrt{1+c+d x}}{3 b d \left (a+b \cosh ^{-1}(c+d x)\right )^{3/2}}-\frac{\left (8 e^4\right ) \operatorname{Subst}\left (\int \frac{x^3}{\sqrt{-1+x} \sqrt{1+x} \left (a+b \cosh ^{-1}(x)\right )^{3/2}} \, dx,x,c+d x\right )}{3 b d}+\frac{\left (10 e^4\right ) \operatorname{Subst}\left (\int \frac{x^5}{\sqrt{-1+x} \sqrt{1+x} \left (a+b \cosh ^{-1}(x)\right )^{3/2}} \, dx,x,c+d x\right )}{3 b d}\\ &=-\frac{2 e^4 \sqrt{-1+c+d x} (c+d x)^4 \sqrt{1+c+d x}}{3 b d \left (a+b \cosh ^{-1}(c+d x)\right )^{3/2}}+\frac{16 e^4 (c+d x)^3}{3 b^2 d \sqrt{a+b \cosh ^{-1}(c+d x)}}-\frac{20 e^4 (c+d x)^5}{3 b^2 d \sqrt{a+b \cosh ^{-1}(c+d x)}}-\frac{\left (16 e^4\right ) \operatorname{Subst}\left (\int \frac{x^2}{\sqrt{a+b \cosh ^{-1}(x)}} \, dx,x,c+d x\right )}{b^2 d}+\frac{\left (100 e^4\right ) \operatorname{Subst}\left (\int \frac{x^4}{\sqrt{a+b \cosh ^{-1}(x)}} \, dx,x,c+d x\right )}{3 b^2 d}\\ &=-\frac{2 e^4 \sqrt{-1+c+d x} (c+d x)^4 \sqrt{1+c+d x}}{3 b d \left (a+b \cosh ^{-1}(c+d x)\right )^{3/2}}+\frac{16 e^4 (c+d x)^3}{3 b^2 d \sqrt{a+b \cosh ^{-1}(c+d x)}}-\frac{20 e^4 (c+d x)^5}{3 b^2 d \sqrt{a+b \cosh ^{-1}(c+d x)}}-\frac{\left (16 e^4\right ) \operatorname{Subst}\left (\int \frac{\cosh ^2(x) \sinh (x)}{\sqrt{a+b x}} \, dx,x,\cosh ^{-1}(c+d x)\right )}{b^2 d}+\frac{\left (100 e^4\right ) \operatorname{Subst}\left (\int \frac{\cosh ^4(x) \sinh (x)}{\sqrt{a+b x}} \, dx,x,\cosh ^{-1}(c+d x)\right )}{3 b^2 d}\\ &=-\frac{2 e^4 \sqrt{-1+c+d x} (c+d x)^4 \sqrt{1+c+d x}}{3 b d \left (a+b \cosh ^{-1}(c+d x)\right )^{3/2}}+\frac{16 e^4 (c+d x)^3}{3 b^2 d \sqrt{a+b \cosh ^{-1}(c+d x)}}-\frac{20 e^4 (c+d x)^5}{3 b^2 d \sqrt{a+b \cosh ^{-1}(c+d x)}}-\frac{\left (16 e^4\right ) \operatorname{Subst}\left (\int \left (\frac{\sinh (x)}{4 \sqrt{a+b x}}+\frac{\sinh (3 x)}{4 \sqrt{a+b x}}\right ) \, dx,x,\cosh ^{-1}(c+d x)\right )}{b^2 d}+\frac{\left (100 e^4\right ) \operatorname{Subst}\left (\int \left (\frac{\sinh (x)}{8 \sqrt{a+b x}}+\frac{3 \sinh (3 x)}{16 \sqrt{a+b x}}+\frac{\sinh (5 x)}{16 \sqrt{a+b x}}\right ) \, dx,x,\cosh ^{-1}(c+d x)\right )}{3 b^2 d}\\ &=-\frac{2 e^4 \sqrt{-1+c+d x} (c+d x)^4 \sqrt{1+c+d x}}{3 b d \left (a+b \cosh ^{-1}(c+d x)\right )^{3/2}}+\frac{16 e^4 (c+d x)^3}{3 b^2 d \sqrt{a+b \cosh ^{-1}(c+d x)}}-\frac{20 e^4 (c+d x)^5}{3 b^2 d \sqrt{a+b \cosh ^{-1}(c+d x)}}+\frac{\left (25 e^4\right ) \operatorname{Subst}\left (\int \frac{\sinh (5 x)}{\sqrt{a+b x}} \, dx,x,\cosh ^{-1}(c+d x)\right )}{12 b^2 d}-\frac{\left (4 e^4\right ) \operatorname{Subst}\left (\int \frac{\sinh (x)}{\sqrt{a+b x}} \, dx,x,\cosh ^{-1}(c+d x)\right )}{b^2 d}-\frac{\left (4 e^4\right ) \operatorname{Subst}\left (\int \frac{\sinh (3 x)}{\sqrt{a+b x}} \, dx,x,\cosh ^{-1}(c+d x)\right )}{b^2 d}+\frac{\left (25 e^4\right ) \operatorname{Subst}\left (\int \frac{\sinh (x)}{\sqrt{a+b x}} \, dx,x,\cosh ^{-1}(c+d x)\right )}{6 b^2 d}+\frac{\left (25 e^4\right ) \operatorname{Subst}\left (\int \frac{\sinh (3 x)}{\sqrt{a+b x}} \, dx,x,\cosh ^{-1}(c+d x)\right )}{4 b^2 d}\\ &=-\frac{2 e^4 \sqrt{-1+c+d x} (c+d x)^4 \sqrt{1+c+d x}}{3 b d \left (a+b \cosh ^{-1}(c+d x)\right )^{3/2}}+\frac{16 e^4 (c+d x)^3}{3 b^2 d \sqrt{a+b \cosh ^{-1}(c+d x)}}-\frac{20 e^4 (c+d x)^5}{3 b^2 d \sqrt{a+b \cosh ^{-1}(c+d x)}}-\frac{\left (25 e^4\right ) \operatorname{Subst}\left (\int \frac{e^{-5 x}}{\sqrt{a+b x}} \, dx,x,\cosh ^{-1}(c+d x)\right )}{24 b^2 d}+\frac{\left (25 e^4\right ) \operatorname{Subst}\left (\int \frac{e^{5 x}}{\sqrt{a+b x}} \, dx,x,\cosh ^{-1}(c+d x)\right )}{24 b^2 d}+\frac{\left (2 e^4\right ) \operatorname{Subst}\left (\int \frac{e^{-3 x}}{\sqrt{a+b x}} \, dx,x,\cosh ^{-1}(c+d x)\right )}{b^2 d}+\frac{\left (2 e^4\right ) \operatorname{Subst}\left (\int \frac{e^{-x}}{\sqrt{a+b x}} \, dx,x,\cosh ^{-1}(c+d x)\right )}{b^2 d}-\frac{\left (2 e^4\right ) \operatorname{Subst}\left (\int \frac{e^x}{\sqrt{a+b x}} \, dx,x,\cosh ^{-1}(c+d x)\right )}{b^2 d}-\frac{\left (2 e^4\right ) \operatorname{Subst}\left (\int \frac{e^{3 x}}{\sqrt{a+b x}} \, dx,x,\cosh ^{-1}(c+d x)\right )}{b^2 d}-\frac{\left (25 e^4\right ) \operatorname{Subst}\left (\int \frac{e^{-x}}{\sqrt{a+b x}} \, dx,x,\cosh ^{-1}(c+d x)\right )}{12 b^2 d}+\frac{\left (25 e^4\right ) \operatorname{Subst}\left (\int \frac{e^x}{\sqrt{a+b x}} \, dx,x,\cosh ^{-1}(c+d x)\right )}{12 b^2 d}-\frac{\left (25 e^4\right ) \operatorname{Subst}\left (\int \frac{e^{-3 x}}{\sqrt{a+b x}} \, dx,x,\cosh ^{-1}(c+d x)\right )}{8 b^2 d}+\frac{\left (25 e^4\right ) \operatorname{Subst}\left (\int \frac{e^{3 x}}{\sqrt{a+b x}} \, dx,x,\cosh ^{-1}(c+d x)\right )}{8 b^2 d}\\ &=-\frac{2 e^4 \sqrt{-1+c+d x} (c+d x)^4 \sqrt{1+c+d x}}{3 b d \left (a+b \cosh ^{-1}(c+d x)\right )^{3/2}}+\frac{16 e^4 (c+d x)^3}{3 b^2 d \sqrt{a+b \cosh ^{-1}(c+d x)}}-\frac{20 e^4 (c+d x)^5}{3 b^2 d \sqrt{a+b \cosh ^{-1}(c+d x)}}-\frac{\left (25 e^4\right ) \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 )}{12 b^3 d}+\frac{\left (25 e^4\right ) \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 )}{12 b^3 d}+\frac{\left (4 e^4\right ) \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 )}{b^3 d}+\frac{\left (4 e^4\right ) \operatorname{Subst}\left (\int e^{\frac{a}{b}-\frac{x^2}{b}} \, dx,x,\sqrt{a+b \cosh ^{-1}(c+d x)}\right )}{b^3 d}-\frac{\left (4 e^4\right ) \operatorname{Subst}\left (\int e^{-\frac{a}{b}+\frac{x^2}{b}} \, dx,x,\sqrt{a+b \cosh ^{-1}(c+d x)}\right )}{b^3 d}-\frac{\left (4 e^4\right ) \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 )}{b^3 d}-\frac{\left (25 e^4\right ) \operatorname{Subst}\left (\int e^{\frac{a}{b}-\frac{x^2}{b}} \, dx,x,\sqrt{a+b \cosh ^{-1}(c+d x)}\right )}{6 b^3 d}+\frac{\left (25 e^4\right ) \operatorname{Subst}\left (\int e^{-\frac{a}{b}+\frac{x^2}{b}} \, dx,x,\sqrt{a+b \cosh ^{-1}(c+d x)}\right )}{6 b^3 d}-\frac{\left (25 e^4\right ) \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 )}{4 b^3 d}+\frac{\left (25 e^4\right ) \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 )}{4 b^3 d}\\ &=-\frac{2 e^4 \sqrt{-1+c+d x} (c+d x)^4 \sqrt{1+c+d x}}{3 b d \left (a+b \cosh ^{-1}(c+d x)\right )^{3/2}}+\frac{16 e^4 (c+d x)^3}{3 b^2 d \sqrt{a+b \cosh ^{-1}(c+d x)}}-\frac{20 e^4 (c+d x)^5}{3 b^2 d \sqrt{a+b \cosh ^{-1}(c+d x)}}-\frac{e^4 e^{a/b} \sqrt{\pi } \text{erf}\left (\frac{\sqrt{a+b \cosh ^{-1}(c+d x)}}{\sqrt{b}}\right )}{12 b^{5/2} d}-\frac{3 e^4 e^{\frac{3 a}{b}} \sqrt{3 \pi } \text{erf}\left (\frac{\sqrt{3} \sqrt{a+b \cosh ^{-1}(c+d x)}}{\sqrt{b}}\right )}{8 b^{5/2} d}-\frac{5 e^4 e^{\frac{5 a}{b}} \sqrt{5 \pi } \text{erf}\left (\frac{\sqrt{5} \sqrt{a+b \cosh ^{-1}(c+d x)}}{\sqrt{b}}\right )}{24 b^{5/2} d}+\frac{e^4 e^{-\frac{a}{b}} \sqrt{\pi } \text{erfi}\left (\frac{\sqrt{a+b \cosh ^{-1}(c+d x)}}{\sqrt{b}}\right )}{12 b^{5/2} d}+\frac{3 e^4 e^{-\frac{3 a}{b}} \sqrt{3 \pi } \text{erfi}\left (\frac{\sqrt{3} \sqrt{a+b \cosh ^{-1}(c+d x)}}{\sqrt{b}}\right )}{8 b^{5/2} d}+\frac{5 e^4 e^{-\frac{5 a}{b}} \sqrt{5 \pi } \text{erfi}\left (\frac{\sqrt{5} \sqrt{a+b \cosh ^{-1}(c+d x)}}{\sqrt{b}}\right )}{24 b^{5/2} d}\\ \end{align*}
Mathematica [A] time = 3.2334, size = 615, normalized size = 1.39 \[ \frac{e^4 e^{-5 \left (\frac{a}{b}+\cosh ^{-1}(c+d x)\right )} \left (-10 \sqrt{5} b e^{5 \cosh ^{-1}(c+d x)} \left (-\frac{a+b \cosh ^{-1}(c+d x)}{b}\right )^{3/2} \text{Gamma}\left (\frac{1}{2},-\frac{5 \left (a+b \cosh ^{-1}(c+d x)\right )}{b}\right )-18 \sqrt{3} b e^{\frac{2 a}{b}+5 \cosh ^{-1}(c+d x)} \left (-\frac{a+b \cosh ^{-1}(c+d x)}{b}\right )^{3/2} \text{Gamma}\left (\frac{1}{2},-\frac{3 \left (a+b \cosh ^{-1}(c+d x)\right )}{b}\right )+2 e^{4 \left (\frac{a}{b}+\cosh ^{-1}(c+d x)\right )} \left (2 e^{\frac{2 a}{b}+\cosh ^{-1}(c+d x)} \sqrt{\frac{a}{b}+\cosh ^{-1}(c+d x)} \left (a+b \cosh ^{-1}(c+d x)\right ) \text{Gamma}\left (\frac{1}{2},\frac{a}{b}+\cosh ^{-1}(c+d x)\right )-2 \left (b e^{\cosh ^{-1}(c+d x)} \left (-\frac{a+b \cosh ^{-1}(c+d x)}{b}\right )^{3/2} \text{Gamma}\left (\frac{1}{2},-\frac{a+b \cosh ^{-1}(c+d x)}{b}\right )+e^{a/b} \left (\left (e^{2 \cosh ^{-1}(c+d x)}+1\right ) \left (a+b \cosh ^{-1}(c+d x)\right )+b \sqrt{\frac{c+d x-1}{c+d x+1}} (c+d x+1) e^{\cosh ^{-1}(c+d x)}\right )\right )\right )+3 e^{\frac{5 a}{b}+2 \cosh ^{-1}(c+d x)} \left (6 \sqrt{3} e^{3 \left (\frac{a}{b}+\cosh ^{-1}(c+d x)\right )} \sqrt{\frac{a}{b}+\cosh ^{-1}(c+d x)} \left (a+b \cosh ^{-1}(c+d x)\right ) \text{Gamma}\left (\frac{1}{2},\frac{3 \left (a+b \cosh ^{-1}(c+d x)\right )}{b}\right )-6 a \left (e^{6 \cosh ^{-1}(c+d x)}+1\right )-6 b \cosh ^{-1}(c+d x)-b e^{6 \cosh ^{-1}(c+d x)} \left (6 \cosh ^{-1}(c+d x)+1\right )+b\right )+2 e^{\frac{5 a}{b}} \left (5 \sqrt{5} e^{5 \left (\frac{a}{b}+\cosh ^{-1}(c+d x)\right )} \sqrt{\frac{a}{b}+\cosh ^{-1}(c+d x)} \left (a+b \cosh ^{-1}(c+d x)\right ) \text{Gamma}\left (\frac{1}{2},\frac{5 \left (a+b \cosh ^{-1}(c+d x)\right )}{b}\right )-5 \left (e^{10 \cosh ^{-1}(c+d x)}+1\right ) \left (a+b \cosh ^{-1}(c+d x)\right )-\frac{1}{2} b \left (e^{10 \cosh ^{-1}(c+d x)}-1\right )\right )\right )}{48 b^2 d \left (a+b \cosh ^{-1}(c+d x)\right )^{3/2}} \]
Warning: Unable to verify antiderivative.
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Maple [F] time = 0.331, size = 0, normalized size = 0. \begin{align*} \int{ \left ( dex+ce \right ) ^{4} \left ( a+b{\rm arccosh} \left (dx+c\right ) \right ) ^{-{\frac{5}{2}}}}\, 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 \frac{{\left (d e x + c e\right )}^{4}}{{\left (b \operatorname{arcosh}\left (d x + c\right ) + a\right )}^{\frac{5}{2}}}\,{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(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \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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