Optimal. Leaf size=319 \[ -\frac{105 \sqrt{\frac{\pi }{2}} b^{7/2} e e^{\frac{2 a}{b}} \text{Erf}\left (\frac{\sqrt{2} \sqrt{a+b \cosh ^{-1}(c+d x)}}{\sqrt{b}}\right )}{1024 d}+\frac{105 \sqrt{\frac{\pi }{2}} b^{7/2} e e^{-\frac{2 a}{b}} \text{Erfi}\left (\frac{\sqrt{2} \sqrt{a+b \cosh ^{-1}(c+d x)}}{\sqrt{b}}\right )}{1024 d}-\frac{105 b^3 e \sqrt{c+d x-1} (c+d x) \sqrt{c+d x+1} \sqrt{a+b \cosh ^{-1}(c+d x)}}{128 d}+\frac{35 b^2 e (c+d x)^2 \left (a+b \cosh ^{-1}(c+d x)\right )^{3/2}}{32 d}-\frac{35 b^2 e \left (a+b \cosh ^{-1}(c+d x)\right )^{3/2}}{64 d}-\frac{7 b e \sqrt{c+d x-1} (c+d x) \sqrt{c+d x+1} \left (a+b \cosh ^{-1}(c+d x)\right )^{5/2}}{8 d}+\frac{e (c+d x)^2 \left (a+b \cosh ^{-1}(c+d x)\right )^{7/2}}{2 d}-\frac{e \left (a+b \cosh ^{-1}(c+d x)\right )^{7/2}}{4 d} \]
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Rubi [A] time = 1.27648, antiderivative size = 319, normalized size of antiderivative = 1., number of steps used = 16, number of rules used = 11, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.478, Rules used = {5866, 12, 5664, 5759, 5676, 5670, 5448, 3308, 2180, 2204, 2205} \[ -\frac{105 \sqrt{\frac{\pi }{2}} b^{7/2} e e^{\frac{2 a}{b}} \text{Erf}\left (\frac{\sqrt{2} \sqrt{a+b \cosh ^{-1}(c+d x)}}{\sqrt{b}}\right )}{1024 d}+\frac{105 \sqrt{\frac{\pi }{2}} b^{7/2} e e^{-\frac{2 a}{b}} \text{Erfi}\left (\frac{\sqrt{2} \sqrt{a+b \cosh ^{-1}(c+d x)}}{\sqrt{b}}\right )}{1024 d}-\frac{105 b^3 e \sqrt{c+d x-1} (c+d x) \sqrt{c+d x+1} \sqrt{a+b \cosh ^{-1}(c+d x)}}{128 d}+\frac{35 b^2 e (c+d x)^2 \left (a+b \cosh ^{-1}(c+d x)\right )^{3/2}}{32 d}-\frac{35 b^2 e \left (a+b \cosh ^{-1}(c+d x)\right )^{3/2}}{64 d}-\frac{7 b e \sqrt{c+d x-1} (c+d x) \sqrt{c+d x+1} \left (a+b \cosh ^{-1}(c+d x)\right )^{5/2}}{8 d}+\frac{e (c+d x)^2 \left (a+b \cosh ^{-1}(c+d x)\right )^{7/2}}{2 d}-\frac{e \left (a+b \cosh ^{-1}(c+d x)\right )^{7/2}}{4 d} \]
Antiderivative was successfully verified.
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Rule 5866
Rule 12
Rule 5664
Rule 5759
Rule 5676
Rule 5670
Rule 5448
Rule 3308
Rule 2180
Rule 2204
Rule 2205
Rubi steps
\begin{align*} \int (c e+d e x) \left (a+b \cosh ^{-1}(c+d x)\right )^{7/2} \, dx &=\frac{\operatorname{Subst}\left (\int e x \left (a+b \cosh ^{-1}(x)\right )^{7/2} \, dx,x,c+d x\right )}{d}\\ &=\frac{e \operatorname{Subst}\left (\int x \left (a+b \cosh ^{-1}(x)\right )^{7/2} \, dx,x,c+d x\right )}{d}\\ &=\frac{e (c+d x)^2 \left (a+b \cosh ^{-1}(c+d x)\right )^{7/2}}{2 d}-\frac{(7 b e) \operatorname{Subst}\left (\int \frac{x^2 \left (a+b \cosh ^{-1}(x)\right )^{5/2}}{\sqrt{-1+x} \sqrt{1+x}} \, dx,x,c+d x\right )}{4 d}\\ &=-\frac{7 b e \sqrt{-1+c+d x} (c+d x) \sqrt{1+c+d x} \left (a+b \cosh ^{-1}(c+d x)\right )^{5/2}}{8 d}+\frac{e (c+d x)^2 \left (a+b \cosh ^{-1}(c+d x)\right )^{7/2}}{2 d}-\frac{(7 b e) \operatorname{Subst}\left (\int \frac{\left (a+b \cosh ^{-1}(x)\right )^{5/2}}{\sqrt{-1+x} \sqrt{1+x}} \, dx,x,c+d x\right )}{8 d}+\frac{\left (35 b^2 e\right ) \operatorname{Subst}\left (\int x \left (a+b \cosh ^{-1}(x)\right )^{3/2} \, dx,x,c+d x\right )}{16 d}\\ &=\frac{35 b^2 e (c+d x)^2 \left (a+b \cosh ^{-1}(c+d x)\right )^{3/2}}{32 d}-\frac{7 b e \sqrt{-1+c+d x} (c+d x) \sqrt{1+c+d x} \left (a+b \cosh ^{-1}(c+d x)\right )^{5/2}}{8 d}-\frac{e \left (a+b \cosh ^{-1}(c+d x)\right )^{7/2}}{4 d}+\frac{e (c+d x)^2 \left (a+b \cosh ^{-1}(c+d x)\right )^{7/2}}{2 d}-\frac{\left (105 b^3 e\right ) \operatorname{Subst}\left (\int \frac{x^2 \sqrt{a+b \cosh ^{-1}(x)}}{\sqrt{-1+x} \sqrt{1+x}} \, dx,x,c+d x\right )}{64 d}\\ &=-\frac{105 b^3 e \sqrt{-1+c+d x} (c+d x) \sqrt{1+c+d x} \sqrt{a+b \cosh ^{-1}(c+d x)}}{128 d}+\frac{35 b^2 e (c+d x)^2 \left (a+b \cosh ^{-1}(c+d x)\right )^{3/2}}{32 d}-\frac{7 b e \sqrt{-1+c+d x} (c+d x) \sqrt{1+c+d x} \left (a+b \cosh ^{-1}(c+d x)\right )^{5/2}}{8 d}-\frac{e \left (a+b \cosh ^{-1}(c+d x)\right )^{7/2}}{4 d}+\frac{e (c+d x)^2 \left (a+b \cosh ^{-1}(c+d x)\right )^{7/2}}{2 d}-\frac{\left (105 b^3 e\right ) \operatorname{Subst}\left (\int \frac{\sqrt{a+b \cosh ^{-1}(x)}}{\sqrt{-1+x} \sqrt{1+x}} \, dx,x,c+d x\right )}{128 d}+\frac{\left (105 b^4 e\right ) \operatorname{Subst}\left (\int \frac{x}{\sqrt{a+b \cosh ^{-1}(x)}} \, dx,x,c+d x\right )}{256 d}\\ &=-\frac{105 b^3 e \sqrt{-1+c+d x} (c+d x) \sqrt{1+c+d x} \sqrt{a+b \cosh ^{-1}(c+d x)}}{128 d}-\frac{35 b^2 e \left (a+b \cosh ^{-1}(c+d x)\right )^{3/2}}{64 d}+\frac{35 b^2 e (c+d x)^2 \left (a+b \cosh ^{-1}(c+d x)\right )^{3/2}}{32 d}-\frac{7 b e \sqrt{-1+c+d x} (c+d x) \sqrt{1+c+d x} \left (a+b \cosh ^{-1}(c+d x)\right )^{5/2}}{8 d}-\frac{e \left (a+b \cosh ^{-1}(c+d x)\right )^{7/2}}{4 d}+\frac{e (c+d x)^2 \left (a+b \cosh ^{-1}(c+d x)\right )^{7/2}}{2 d}+\frac{\left (105 b^4 e\right ) \operatorname{Subst}\left (\int \frac{\cosh (x) \sinh (x)}{\sqrt{a+b x}} \, dx,x,\cosh ^{-1}(c+d x)\right )}{256 d}\\ &=-\frac{105 b^3 e \sqrt{-1+c+d x} (c+d x) \sqrt{1+c+d x} \sqrt{a+b \cosh ^{-1}(c+d x)}}{128 d}-\frac{35 b^2 e \left (a+b \cosh ^{-1}(c+d x)\right )^{3/2}}{64 d}+\frac{35 b^2 e (c+d x)^2 \left (a+b \cosh ^{-1}(c+d x)\right )^{3/2}}{32 d}-\frac{7 b e \sqrt{-1+c+d x} (c+d x) \sqrt{1+c+d x} \left (a+b \cosh ^{-1}(c+d x)\right )^{5/2}}{8 d}-\frac{e \left (a+b \cosh ^{-1}(c+d x)\right )^{7/2}}{4 d}+\frac{e (c+d x)^2 \left (a+b \cosh ^{-1}(c+d x)\right )^{7/2}}{2 d}+\frac{\left (105 b^4 e\right ) \operatorname{Subst}\left (\int \frac{\sinh (2 x)}{2 \sqrt{a+b x}} \, dx,x,\cosh ^{-1}(c+d x)\right )}{256 d}\\ &=-\frac{105 b^3 e \sqrt{-1+c+d x} (c+d x) \sqrt{1+c+d x} \sqrt{a+b \cosh ^{-1}(c+d x)}}{128 d}-\frac{35 b^2 e \left (a+b \cosh ^{-1}(c+d x)\right )^{3/2}}{64 d}+\frac{35 b^2 e (c+d x)^2 \left (a+b \cosh ^{-1}(c+d x)\right )^{3/2}}{32 d}-\frac{7 b e \sqrt{-1+c+d x} (c+d x) \sqrt{1+c+d x} \left (a+b \cosh ^{-1}(c+d x)\right )^{5/2}}{8 d}-\frac{e \left (a+b \cosh ^{-1}(c+d x)\right )^{7/2}}{4 d}+\frac{e (c+d x)^2 \left (a+b \cosh ^{-1}(c+d x)\right )^{7/2}}{2 d}+\frac{\left (105 b^4 e\right ) \operatorname{Subst}\left (\int \frac{\sinh (2 x)}{\sqrt{a+b x}} \, dx,x,\cosh ^{-1}(c+d x)\right )}{512 d}\\ &=-\frac{105 b^3 e \sqrt{-1+c+d x} (c+d x) \sqrt{1+c+d x} \sqrt{a+b \cosh ^{-1}(c+d x)}}{128 d}-\frac{35 b^2 e \left (a+b \cosh ^{-1}(c+d x)\right )^{3/2}}{64 d}+\frac{35 b^2 e (c+d x)^2 \left (a+b \cosh ^{-1}(c+d x)\right )^{3/2}}{32 d}-\frac{7 b e \sqrt{-1+c+d x} (c+d x) \sqrt{1+c+d x} \left (a+b \cosh ^{-1}(c+d x)\right )^{5/2}}{8 d}-\frac{e \left (a+b \cosh ^{-1}(c+d x)\right )^{7/2}}{4 d}+\frac{e (c+d x)^2 \left (a+b \cosh ^{-1}(c+d x)\right )^{7/2}}{2 d}-\frac{\left (105 b^4 e\right ) \operatorname{Subst}\left (\int \frac{e^{-2 x}}{\sqrt{a+b x}} \, dx,x,\cosh ^{-1}(c+d x)\right )}{1024 d}+\frac{\left (105 b^4 e\right ) \operatorname{Subst}\left (\int \frac{e^{2 x}}{\sqrt{a+b x}} \, dx,x,\cosh ^{-1}(c+d x)\right )}{1024 d}\\ &=-\frac{105 b^3 e \sqrt{-1+c+d x} (c+d x) \sqrt{1+c+d x} \sqrt{a+b \cosh ^{-1}(c+d x)}}{128 d}-\frac{35 b^2 e \left (a+b \cosh ^{-1}(c+d x)\right )^{3/2}}{64 d}+\frac{35 b^2 e (c+d x)^2 \left (a+b \cosh ^{-1}(c+d x)\right )^{3/2}}{32 d}-\frac{7 b e \sqrt{-1+c+d x} (c+d x) \sqrt{1+c+d x} \left (a+b \cosh ^{-1}(c+d x)\right )^{5/2}}{8 d}-\frac{e \left (a+b \cosh ^{-1}(c+d x)\right )^{7/2}}{4 d}+\frac{e (c+d x)^2 \left (a+b \cosh ^{-1}(c+d x)\right )^{7/2}}{2 d}-\frac{\left (105 b^3 e\right ) \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 )}{512 d}+\frac{\left (105 b^3 e\right ) \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 )}{512 d}\\ &=-\frac{105 b^3 e \sqrt{-1+c+d x} (c+d x) \sqrt{1+c+d x} \sqrt{a+b \cosh ^{-1}(c+d x)}}{128 d}-\frac{35 b^2 e \left (a+b \cosh ^{-1}(c+d x)\right )^{3/2}}{64 d}+\frac{35 b^2 e (c+d x)^2 \left (a+b \cosh ^{-1}(c+d x)\right )^{3/2}}{32 d}-\frac{7 b e \sqrt{-1+c+d x} (c+d x) \sqrt{1+c+d x} \left (a+b \cosh ^{-1}(c+d x)\right )^{5/2}}{8 d}-\frac{e \left (a+b \cosh ^{-1}(c+d x)\right )^{7/2}}{4 d}+\frac{e (c+d x)^2 \left (a+b \cosh ^{-1}(c+d x)\right )^{7/2}}{2 d}-\frac{105 b^{7/2} 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 )}{1024 d}+\frac{105 b^{7/2} 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 )}{1024 d}\\ \end{align*}
Mathematica [A] time = 5.43985, size = 288, normalized size = 0.9 \[ \frac{e \left (8 \sqrt{a+b \cosh ^{-1}(c+d x)} \left (4 a \left (16 a^2+35 b^2\right ) \cosh \left (2 \cosh ^{-1}(c+d x)\right )+4 b \cosh ^{-1}(c+d x) \left (\left (48 a^2+35 b^2\right ) \cosh \left (2 \cosh ^{-1}(c+d x)\right )-56 a b \sinh \left (2 \cosh ^{-1}(c+d x)\right )\right )-7 b \left (16 a^2+15 b^2\right ) \sinh \left (2 \cosh ^{-1}(c+d x)\right )+16 b^2 \cosh ^{-1}(c+d x)^2 \left (12 a \cosh \left (2 \cosh ^{-1}(c+d x)\right )-7 b \sinh \left (2 \cosh ^{-1}(c+d x)\right )\right )+64 b^3 \cosh \left (2 \cosh ^{-1}(c+d x)\right ) \cosh ^{-1}(c+d x)^3\right )-105 \sqrt{2 \pi } b^{7/2} \left (\sinh \left (\frac{2 a}{b}\right )+\cosh \left (\frac{2 a}{b}\right )\right ) \text{Erf}\left (\frac{\sqrt{2} \sqrt{a+b \cosh ^{-1}(c+d x)}}{\sqrt{b}}\right )+105 \sqrt{2 \pi } b^{7/2} \left (\cosh \left (\frac{2 a}{b}\right )-\sinh \left (\frac{2 a}{b}\right )\right ) \text{Erfi}\left (\frac{\sqrt{2} \sqrt{a+b \cosh ^{-1}(c+d x)}}{\sqrt{b}}\right )\right )}{2048 d} \]
Warning: Unable to verify antiderivative.
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Maple [F] time = 0.132, size = 0, normalized size = 0. \begin{align*} \int \left ( dex+ce \right ) \left ( a+b{\rm arccosh} \left (dx+c\right ) \right ) ^{{\frac{7}{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{\left (d e x + c e\right )}{\left (b \operatorname{arcosh}\left (d x + c\right ) + a\right )}^{\frac{7}{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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