Optimal. Leaf size=266 \[ \frac{8 \sqrt{2 \pi } e e^{\frac{2 a}{b}} \text{Erf}\left (\frac{\sqrt{2} \sqrt{a+b \cosh ^{-1}(c+d x)}}{\sqrt{b}}\right )}{15 b^{7/2} d}+\frac{8 \sqrt{2 \pi } e e^{-\frac{2 a}{b}} \text{Erfi}\left (\frac{\sqrt{2} \sqrt{a+b \cosh ^{-1}(c+d x)}}{\sqrt{b}}\right )}{15 b^{7/2} d}-\frac{8 e (c+d x)^2}{15 b^2 d \left (a+b \cosh ^{-1}(c+d x)\right )^{3/2}}-\frac{32 e \sqrt{c+d x-1} \sqrt{c+d x+1} (c+d x)}{15 b^3 d \sqrt{a+b \cosh ^{-1}(c+d x)}}+\frac{4 e}{15 b^2 d \left (a+b \cosh ^{-1}(c+d x)\right )^{3/2}}-\frac{2 e \sqrt{c+d x-1} \sqrt{c+d x+1} (c+d x)}{5 b d \left (a+b \cosh ^{-1}(c+d x)\right )^{5/2}} \]
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Rubi [A] time = 0.784347, antiderivative size = 266, normalized size of antiderivative = 1., number of steps used = 11, number of rules used = 10, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.435, Rules used = {5866, 12, 5668, 5775, 5666, 3307, 2180, 2204, 2205, 5676} \[ \frac{8 \sqrt{2 \pi } e e^{\frac{2 a}{b}} \text{Erf}\left (\frac{\sqrt{2} \sqrt{a+b \cosh ^{-1}(c+d x)}}{\sqrt{b}}\right )}{15 b^{7/2} d}+\frac{8 \sqrt{2 \pi } e e^{-\frac{2 a}{b}} \text{Erfi}\left (\frac{\sqrt{2} \sqrt{a+b \cosh ^{-1}(c+d x)}}{\sqrt{b}}\right )}{15 b^{7/2} d}-\frac{8 e (c+d x)^2}{15 b^2 d \left (a+b \cosh ^{-1}(c+d x)\right )^{3/2}}-\frac{32 e \sqrt{c+d x-1} \sqrt{c+d x+1} (c+d x)}{15 b^3 d \sqrt{a+b \cosh ^{-1}(c+d x)}}+\frac{4 e}{15 b^2 d \left (a+b \cosh ^{-1}(c+d x)\right )^{3/2}}-\frac{2 e \sqrt{c+d x-1} \sqrt{c+d x+1} (c+d x)}{5 b d \left (a+b \cosh ^{-1}(c+d x)\right )^{5/2}} \]
Antiderivative was successfully verified.
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Rule 5866
Rule 12
Rule 5668
Rule 5775
Rule 5666
Rule 3307
Rule 2180
Rule 2204
Rule 2205
Rule 5676
Rubi steps
\begin{align*} \int \frac{c e+d e x}{\left (a+b \cosh ^{-1}(c+d x)\right )^{7/2}} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{e x}{\left (a+b \cosh ^{-1}(x)\right )^{7/2}} \, dx,x,c+d x\right )}{d}\\ &=\frac{e \operatorname{Subst}\left (\int \frac{x}{\left (a+b \cosh ^{-1}(x)\right )^{7/2}} \, dx,x,c+d x\right )}{d}\\ &=-\frac{2 e \sqrt{-1+c+d x} (c+d x) \sqrt{1+c+d x}}{5 b d \left (a+b \cosh ^{-1}(c+d x)\right )^{5/2}}-\frac{(2 e) \operatorname{Subst}\left (\int \frac{1}{\sqrt{-1+x} \sqrt{1+x} \left (a+b \cosh ^{-1}(x)\right )^{5/2}} \, dx,x,c+d x\right )}{5 b d}+\frac{(4 e) \operatorname{Subst}\left (\int \frac{x^2}{\sqrt{-1+x} \sqrt{1+x} \left (a+b \cosh ^{-1}(x)\right )^{5/2}} \, dx,x,c+d x\right )}{5 b d}\\ &=-\frac{2 e \sqrt{-1+c+d x} (c+d x) \sqrt{1+c+d x}}{5 b d \left (a+b \cosh ^{-1}(c+d x)\right )^{5/2}}+\frac{4 e}{15 b^2 d \left (a+b \cosh ^{-1}(c+d x)\right )^{3/2}}-\frac{8 e (c+d x)^2}{15 b^2 d \left (a+b \cosh ^{-1}(c+d x)\right )^{3/2}}+\frac{(16 e) \operatorname{Subst}\left (\int \frac{x}{\left (a+b \cosh ^{-1}(x)\right )^{3/2}} \, dx,x,c+d x\right )}{15 b^2 d}\\ &=-\frac{2 e \sqrt{-1+c+d x} (c+d x) \sqrt{1+c+d x}}{5 b d \left (a+b \cosh ^{-1}(c+d x)\right )^{5/2}}+\frac{4 e}{15 b^2 d \left (a+b \cosh ^{-1}(c+d x)\right )^{3/2}}-\frac{8 e (c+d x)^2}{15 b^2 d \left (a+b \cosh ^{-1}(c+d x)\right )^{3/2}}-\frac{32 e \sqrt{-1+c+d x} (c+d x) \sqrt{1+c+d x}}{15 b^3 d \sqrt{a+b \cosh ^{-1}(c+d x)}}+\frac{(32 e) \operatorname{Subst}\left (\int \frac{\cosh (2 x)}{\sqrt{a+b x}} \, dx,x,\cosh ^{-1}(c+d x)\right )}{15 b^3 d}\\ &=-\frac{2 e \sqrt{-1+c+d x} (c+d x) \sqrt{1+c+d x}}{5 b d \left (a+b \cosh ^{-1}(c+d x)\right )^{5/2}}+\frac{4 e}{15 b^2 d \left (a+b \cosh ^{-1}(c+d x)\right )^{3/2}}-\frac{8 e (c+d x)^2}{15 b^2 d \left (a+b \cosh ^{-1}(c+d x)\right )^{3/2}}-\frac{32 e \sqrt{-1+c+d x} (c+d x) \sqrt{1+c+d x}}{15 b^3 d \sqrt{a+b \cosh ^{-1}(c+d x)}}+\frac{(16 e) \operatorname{Subst}\left (\int \frac{e^{-2 x}}{\sqrt{a+b x}} \, dx,x,\cosh ^{-1}(c+d x)\right )}{15 b^3 d}+\frac{(16 e) \operatorname{Subst}\left (\int \frac{e^{2 x}}{\sqrt{a+b x}} \, dx,x,\cosh ^{-1}(c+d x)\right )}{15 b^3 d}\\ &=-\frac{2 e \sqrt{-1+c+d x} (c+d x) \sqrt{1+c+d x}}{5 b d \left (a+b \cosh ^{-1}(c+d x)\right )^{5/2}}+\frac{4 e}{15 b^2 d \left (a+b \cosh ^{-1}(c+d x)\right )^{3/2}}-\frac{8 e (c+d x)^2}{15 b^2 d \left (a+b \cosh ^{-1}(c+d x)\right )^{3/2}}-\frac{32 e \sqrt{-1+c+d x} (c+d x) \sqrt{1+c+d x}}{15 b^3 d \sqrt{a+b \cosh ^{-1}(c+d x)}}+\frac{(32 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 )}{15 b^4 d}+\frac{(32 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 )}{15 b^4 d}\\ &=-\frac{2 e \sqrt{-1+c+d x} (c+d x) \sqrt{1+c+d x}}{5 b d \left (a+b \cosh ^{-1}(c+d x)\right )^{5/2}}+\frac{4 e}{15 b^2 d \left (a+b \cosh ^{-1}(c+d x)\right )^{3/2}}-\frac{8 e (c+d x)^2}{15 b^2 d \left (a+b \cosh ^{-1}(c+d x)\right )^{3/2}}-\frac{32 e \sqrt{-1+c+d x} (c+d x) \sqrt{1+c+d x}}{15 b^3 d \sqrt{a+b \cosh ^{-1}(c+d x)}}+\frac{8 e e^{\frac{2 a}{b}} \sqrt{2 \pi } \text{erf}\left (\frac{\sqrt{2} \sqrt{a+b \cosh ^{-1}(c+d x)}}{\sqrt{b}}\right )}{15 b^{7/2} d}+\frac{8 e e^{-\frac{2 a}{b}} \sqrt{2 \pi } \text{erfi}\left (\frac{\sqrt{2} \sqrt{a+b \cosh ^{-1}(c+d x)}}{\sqrt{b}}\right )}{15 b^{7/2} d}\\ \end{align*}
Mathematica [B] time = 4.52608, size = 916, normalized size = 3.44 \[ \frac{e \left (8 c \sqrt{\frac{c+d x-1}{c+d x+1}} (c+d x+1) \cosh ^{-1}(c+d x)^2 b^{5/2}+4 c (c+d x) \cosh ^{-1}(c+d x) b^{5/2}-4 \cosh ^{-1}(c+d x) \cosh \left (2 \cosh ^{-1}(c+d x)\right ) b^{5/2}-16 \cosh ^{-1}(c+d x)^2 \sinh \left (2 \cosh ^{-1}(c+d x)\right ) b^{5/2}-3 \sinh \left (2 \cosh ^{-1}(c+d x)\right ) b^{5/2}+4 a c (c+d x) b^{3/2}+16 a c \sqrt{\frac{c+d x-1}{c+d x+1}} (c+d x+1) \cosh ^{-1}(c+d x) b^{3/2}-4 a \cosh \left (2 \cosh ^{-1}(c+d x)\right ) b^{3/2}-32 a \cosh ^{-1}(c+d x) \sinh \left (2 \cosh ^{-1}(c+d x)\right ) b^{3/2}+8 a^2 c \sqrt{\frac{c+d x-1}{c+d x+1}} (c+d x+1) \sqrt{b}-2 c e^{-\cosh ^{-1}(c+d x)} \left (a+b \cosh ^{-1}(c+d x)\right ) \left (-2 a+b-2 b \cosh ^{-1}(c+d x)+2 e^{\frac{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 )\right ) \sqrt{b}-2 c e^{-\frac{a}{b}} \left (a+b \cosh ^{-1}(c+d x)\right ) \left (2 b \text{Gamma}\left (\frac{1}{2},-\frac{a+b \cosh ^{-1}(c+d x)}{b}\right ) \left (-\frac{a+b \cosh ^{-1}(c+d x)}{b}\right )^{3/2}+e^{\frac{a}{b}+\cosh ^{-1}(c+d x)} \left (2 a+b+2 b \cosh ^{-1}(c+d x)\right )\right ) \sqrt{b}-16 a^2 \sinh \left (2 \cosh ^{-1}(c+d x)\right ) \sqrt{b}-4 c \sqrt{\pi } \left (a+b \cosh ^{-1}(c+d x)\right )^{5/2} \cosh \left (\frac{a}{b}\right ) \text{Erf}\left (\frac{\sqrt{a+b \cosh ^{-1}(c+d x)}}{\sqrt{b}}\right )+8 \sqrt{2 \pi } \left (a+b \cosh ^{-1}(c+d x)\right )^{5/2} \cosh \left (\frac{2 a}{b}\right ) \text{Erf}\left (\frac{\sqrt{2} \sqrt{a+b \cosh ^{-1}(c+d x)}}{\sqrt{b}}\right )-4 c \sqrt{\pi } \left (a+b \cosh ^{-1}(c+d x)\right )^{5/2} \cosh \left (\frac{a}{b}\right ) \text{Erfi}\left (\frac{\sqrt{a+b \cosh ^{-1}(c+d x)}}{\sqrt{b}}\right )+8 \sqrt{2 \pi } \left (a+b \cosh ^{-1}(c+d x)\right )^{5/2} \cosh \left (\frac{2 a}{b}\right ) \text{Erfi}\left (\frac{\sqrt{2} \sqrt{a+b \cosh ^{-1}(c+d x)}}{\sqrt{b}}\right )-4 c \sqrt{\pi } \left (a+b \cosh ^{-1}(c+d x)\right )^{5/2} \text{Erf}\left (\frac{\sqrt{a+b \cosh ^{-1}(c+d x)}}{\sqrt{b}}\right ) \sinh \left (\frac{a}{b}\right )+4 c \sqrt{\pi } \left (a+b \cosh ^{-1}(c+d x)\right )^{5/2} \text{Erfi}\left (\frac{\sqrt{a+b \cosh ^{-1}(c+d x)}}{\sqrt{b}}\right ) \sinh \left (\frac{a}{b}\right )+8 \sqrt{2 \pi } \left (a+b \cosh ^{-1}(c+d x)\right )^{5/2} \text{Erf}\left (\frac{\sqrt{2} \sqrt{a+b \cosh ^{-1}(c+d x)}}{\sqrt{b}}\right ) \sinh \left (\frac{2 a}{b}\right )-8 \sqrt{2 \pi } \left (a+b \cosh ^{-1}(c+d x)\right )^{5/2} \text{Erfi}\left (\frac{\sqrt{2} \sqrt{a+b \cosh ^{-1}(c+d x)}}{\sqrt{b}}\right ) \sinh \left (\frac{2 a}{b}\right )\right )}{15 b^{7/2} d \left (a+b \cosh ^{-1}(c+d x)\right )^{5/2}} \]
Warning: Unable to verify antiderivative.
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Maple [F] time = 0.128, size = 0, normalized size = 0. \begin{align*} \int{(dex+ce) \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 \frac{d e x + c e}{{\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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