Optimal. Leaf size=351 \[ \frac{5 \sqrt{\pi } d^2 e^{a/b} \text{Erf}\left (\frac{\sqrt{a+b \cosh ^{-1}(c x)}}{\sqrt{b}}\right )}{8 b^{3/2} c}-\frac{5 \sqrt{3 \pi } d^2 e^{\frac{3 a}{b}} \text{Erf}\left (\frac{\sqrt{3} \sqrt{a+b \cosh ^{-1}(c x)}}{\sqrt{b}}\right )}{16 b^{3/2} c}+\frac{\sqrt{5 \pi } d^2 e^{\frac{5 a}{b}} \text{Erf}\left (\frac{\sqrt{5} \sqrt{a+b \cosh ^{-1}(c x)}}{\sqrt{b}}\right )}{16 b^{3/2} c}+\frac{5 \sqrt{\pi } d^2 e^{-\frac{a}{b}} \text{Erfi}\left (\frac{\sqrt{a+b \cosh ^{-1}(c x)}}{\sqrt{b}}\right )}{8 b^{3/2} c}-\frac{5 \sqrt{3 \pi } d^2 e^{-\frac{3 a}{b}} \text{Erfi}\left (\frac{\sqrt{3} \sqrt{a+b \cosh ^{-1}(c x)}}{\sqrt{b}}\right )}{16 b^{3/2} c}+\frac{\sqrt{5 \pi } d^2 e^{-\frac{5 a}{b}} \text{Erfi}\left (\frac{\sqrt{5} \sqrt{a+b \cosh ^{-1}(c x)}}{\sqrt{b}}\right )}{16 b^{3/2} c}-\frac{2 d^2 (c x-1)^{5/2} (c x+1)^{5/2}}{b c \sqrt{a+b \cosh ^{-1}(c x)}} \]
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Rubi [A] time = 0.924336, antiderivative size = 351, normalized size of antiderivative = 1., number of steps used = 19, number of rules used = 7, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.269, Rules used = {5695, 5781, 5448, 3307, 2180, 2204, 2205} \[ \frac{5 \sqrt{\pi } d^2 e^{a/b} \text{Erf}\left (\frac{\sqrt{a+b \cosh ^{-1}(c x)}}{\sqrt{b}}\right )}{8 b^{3/2} c}-\frac{5 \sqrt{3 \pi } d^2 e^{\frac{3 a}{b}} \text{Erf}\left (\frac{\sqrt{3} \sqrt{a+b \cosh ^{-1}(c x)}}{\sqrt{b}}\right )}{16 b^{3/2} c}+\frac{\sqrt{5 \pi } d^2 e^{\frac{5 a}{b}} \text{Erf}\left (\frac{\sqrt{5} \sqrt{a+b \cosh ^{-1}(c x)}}{\sqrt{b}}\right )}{16 b^{3/2} c}+\frac{5 \sqrt{\pi } d^2 e^{-\frac{a}{b}} \text{Erfi}\left (\frac{\sqrt{a+b \cosh ^{-1}(c x)}}{\sqrt{b}}\right )}{8 b^{3/2} c}-\frac{5 \sqrt{3 \pi } d^2 e^{-\frac{3 a}{b}} \text{Erfi}\left (\frac{\sqrt{3} \sqrt{a+b \cosh ^{-1}(c x)}}{\sqrt{b}}\right )}{16 b^{3/2} c}+\frac{\sqrt{5 \pi } d^2 e^{-\frac{5 a}{b}} \text{Erfi}\left (\frac{\sqrt{5} \sqrt{a+b \cosh ^{-1}(c x)}}{\sqrt{b}}\right )}{16 b^{3/2} c}-\frac{2 d^2 (c x-1)^{5/2} (c x+1)^{5/2}}{b c \sqrt{a+b \cosh ^{-1}(c x)}} \]
Antiderivative was successfully verified.
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Rule 5695
Rule 5781
Rule 5448
Rule 3307
Rule 2180
Rule 2204
Rule 2205
Rubi steps
\begin{align*} \int \frac{\left (d-c^2 d x^2\right )^2}{\left (a+b \cosh ^{-1}(c x)\right )^{3/2}} \, dx &=-\frac{2 d^2 (-1+c x)^{5/2} (1+c x)^{5/2}}{b c \sqrt{a+b \cosh ^{-1}(c x)}}+\frac{\left (10 c d^2\right ) \int \frac{x (-1+c x)^{3/2} (1+c x)^{3/2}}{\sqrt{a+b \cosh ^{-1}(c x)}} \, dx}{b}\\ &=-\frac{2 d^2 (-1+c x)^{5/2} (1+c x)^{5/2}}{b c \sqrt{a+b \cosh ^{-1}(c x)}}+\frac{\left (10 d^2\right ) \operatorname{Subst}\left (\int \frac{\cosh (x) \sinh ^4(x)}{\sqrt{a+b x}} \, dx,x,\cosh ^{-1}(c x)\right )}{b c}\\ &=-\frac{2 d^2 (-1+c x)^{5/2} (1+c x)^{5/2}}{b c \sqrt{a+b \cosh ^{-1}(c x)}}+\frac{\left (10 d^2\right ) \operatorname{Subst}\left (\int \left (\frac{\cosh (x)}{8 \sqrt{a+b x}}-\frac{3 \cosh (3 x)}{16 \sqrt{a+b x}}+\frac{\cosh (5 x)}{16 \sqrt{a+b x}}\right ) \, dx,x,\cosh ^{-1}(c x)\right )}{b c}\\ &=-\frac{2 d^2 (-1+c x)^{5/2} (1+c x)^{5/2}}{b c \sqrt{a+b \cosh ^{-1}(c x)}}+\frac{\left (5 d^2\right ) \operatorname{Subst}\left (\int \frac{\cosh (5 x)}{\sqrt{a+b x}} \, dx,x,\cosh ^{-1}(c x)\right )}{8 b c}+\frac{\left (5 d^2\right ) \operatorname{Subst}\left (\int \frac{\cosh (x)}{\sqrt{a+b x}} \, dx,x,\cosh ^{-1}(c x)\right )}{4 b c}-\frac{\left (15 d^2\right ) \operatorname{Subst}\left (\int \frac{\cosh (3 x)}{\sqrt{a+b x}} \, dx,x,\cosh ^{-1}(c x)\right )}{8 b c}\\ &=-\frac{2 d^2 (-1+c x)^{5/2} (1+c x)^{5/2}}{b c \sqrt{a+b \cosh ^{-1}(c x)}}+\frac{\left (5 d^2\right ) \operatorname{Subst}\left (\int \frac{e^{-5 x}}{\sqrt{a+b x}} \, dx,x,\cosh ^{-1}(c x)\right )}{16 b c}+\frac{\left (5 d^2\right ) \operatorname{Subst}\left (\int \frac{e^{5 x}}{\sqrt{a+b x}} \, dx,x,\cosh ^{-1}(c x)\right )}{16 b c}+\frac{\left (5 d^2\right ) \operatorname{Subst}\left (\int \frac{e^{-x}}{\sqrt{a+b x}} \, dx,x,\cosh ^{-1}(c x)\right )}{8 b c}+\frac{\left (5 d^2\right ) \operatorname{Subst}\left (\int \frac{e^x}{\sqrt{a+b x}} \, dx,x,\cosh ^{-1}(c x)\right )}{8 b c}-\frac{\left (15 d^2\right ) \operatorname{Subst}\left (\int \frac{e^{-3 x}}{\sqrt{a+b x}} \, dx,x,\cosh ^{-1}(c x)\right )}{16 b c}-\frac{\left (15 d^2\right ) \operatorname{Subst}\left (\int \frac{e^{3 x}}{\sqrt{a+b x}} \, dx,x,\cosh ^{-1}(c x)\right )}{16 b c}\\ &=-\frac{2 d^2 (-1+c x)^{5/2} (1+c x)^{5/2}}{b c \sqrt{a+b \cosh ^{-1}(c x)}}+\frac{\left (5 d^2\right ) \operatorname{Subst}\left (\int e^{\frac{5 a}{b}-\frac{5 x^2}{b}} \, dx,x,\sqrt{a+b \cosh ^{-1}(c x)}\right )}{8 b^2 c}+\frac{\left (5 d^2\right ) \operatorname{Subst}\left (\int e^{-\frac{5 a}{b}+\frac{5 x^2}{b}} \, dx,x,\sqrt{a+b \cosh ^{-1}(c x)}\right )}{8 b^2 c}+\frac{\left (5 d^2\right ) \operatorname{Subst}\left (\int e^{\frac{a}{b}-\frac{x^2}{b}} \, dx,x,\sqrt{a+b \cosh ^{-1}(c x)}\right )}{4 b^2 c}+\frac{\left (5 d^2\right ) \operatorname{Subst}\left (\int e^{-\frac{a}{b}+\frac{x^2}{b}} \, dx,x,\sqrt{a+b \cosh ^{-1}(c x)}\right )}{4 b^2 c}-\frac{\left (15 d^2\right ) \operatorname{Subst}\left (\int e^{\frac{3 a}{b}-\frac{3 x^2}{b}} \, dx,x,\sqrt{a+b \cosh ^{-1}(c x)}\right )}{8 b^2 c}-\frac{\left (15 d^2\right ) \operatorname{Subst}\left (\int e^{-\frac{3 a}{b}+\frac{3 x^2}{b}} \, dx,x,\sqrt{a+b \cosh ^{-1}(c x)}\right )}{8 b^2 c}\\ &=-\frac{2 d^2 (-1+c x)^{5/2} (1+c x)^{5/2}}{b c \sqrt{a+b \cosh ^{-1}(c x)}}+\frac{5 d^2 e^{a/b} \sqrt{\pi } \text{erf}\left (\frac{\sqrt{a+b \cosh ^{-1}(c x)}}{\sqrt{b}}\right )}{8 b^{3/2} c}-\frac{5 d^2 e^{\frac{3 a}{b}} \sqrt{3 \pi } \text{erf}\left (\frac{\sqrt{3} \sqrt{a+b \cosh ^{-1}(c x)}}{\sqrt{b}}\right )}{16 b^{3/2} c}+\frac{d^2 e^{\frac{5 a}{b}} \sqrt{5 \pi } \text{erf}\left (\frac{\sqrt{5} \sqrt{a+b \cosh ^{-1}(c x)}}{\sqrt{b}}\right )}{16 b^{3/2} c}+\frac{5 d^2 e^{-\frac{a}{b}} \sqrt{\pi } \text{erfi}\left (\frac{\sqrt{a+b \cosh ^{-1}(c x)}}{\sqrt{b}}\right )}{8 b^{3/2} c}-\frac{5 d^2 e^{-\frac{3 a}{b}} \sqrt{3 \pi } \text{erfi}\left (\frac{\sqrt{3} \sqrt{a+b \cosh ^{-1}(c x)}}{\sqrt{b}}\right )}{16 b^{3/2} c}+\frac{d^2 e^{-\frac{5 a}{b}} \sqrt{5 \pi } \text{erfi}\left (\frac{\sqrt{5} \sqrt{a+b \cosh ^{-1}(c x)}}{\sqrt{b}}\right )}{16 b^{3/2} c}\\ \end{align*}
Mathematica [A] time = 1.93068, size = 387, normalized size = 1.1 \[ -\frac{d^2 e^{-\frac{5 a}{b}} \left (10 e^{\frac{6 a}{b}} \sqrt{\frac{a}{b}+\cosh ^{-1}(c x)} \text{Gamma}\left (\frac{1}{2},\frac{a}{b}+\cosh ^{-1}(c x)\right )-\sqrt{5} \sqrt{-\frac{a+b \cosh ^{-1}(c x)}{b}} \text{Gamma}\left (\frac{1}{2},-\frac{5 \left (a+b \cosh ^{-1}(c x)\right )}{b}\right )+5 \sqrt{3} e^{\frac{2 a}{b}} \sqrt{-\frac{a+b \cosh ^{-1}(c x)}{b}} \text{Gamma}\left (\frac{1}{2},-\frac{3 \left (a+b \cosh ^{-1}(c x)\right )}{b}\right )-10 e^{\frac{4 a}{b}} \sqrt{-\frac{a+b \cosh ^{-1}(c x)}{b}} \text{Gamma}\left (\frac{1}{2},-\frac{a+b \cosh ^{-1}(c x)}{b}\right )-5 \sqrt{3} e^{\frac{8 a}{b}} \sqrt{\frac{a}{b}+\cosh ^{-1}(c x)} \text{Gamma}\left (\frac{1}{2},\frac{3 \left (a+b \cosh ^{-1}(c x)\right )}{b}\right )+\sqrt{5} e^{\frac{10 a}{b}} \sqrt{\frac{a}{b}+\cosh ^{-1}(c x)} \text{Gamma}\left (\frac{1}{2},\frac{5 \left (a+b \cosh ^{-1}(c x)\right )}{b}\right )+20 c x e^{\frac{5 a}{b}} \sqrt{\frac{c x-1}{c x+1}}+20 e^{\frac{5 a}{b}} \sqrt{\frac{c x-1}{c x+1}}-10 e^{\frac{5 a}{b}} \sinh \left (3 \cosh ^{-1}(c x)\right )+2 e^{\frac{5 a}{b}} \sinh \left (5 \cosh ^{-1}(c x)\right )\right )}{16 b c \sqrt{a+b \cosh ^{-1}(c x)}} \]
Warning: Unable to verify antiderivative.
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Maple [F] time = 0.28, size = 0, normalized size = 0. \begin{align*} \int{ \left ( -{c}^{2}d{x}^{2}+d \right ) ^{2} \left ( a+b{\rm arccosh} \left (cx\right ) \right ) ^{-{\frac{3}{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 (c^{2} d x^{2} - d\right )}^{2}}{{\left (b \operatorname{arcosh}\left (c x\right ) + a\right )}^{\frac{3}{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] time = 0., size = 0, normalized size = 0. \begin{align*} d^{2} \left (\int - \frac{2 c^{2} x^{2}}{a \sqrt{a + b \operatorname{acosh}{\left (c x \right )}} + b \sqrt{a + b \operatorname{acosh}{\left (c x \right )}} \operatorname{acosh}{\left (c x \right )}}\, dx + \int \frac{c^{4} x^{4}}{a \sqrt{a + b \operatorname{acosh}{\left (c x \right )}} + b \sqrt{a + b \operatorname{acosh}{\left (c x \right )}} \operatorname{acosh}{\left (c x \right )}}\, dx + \int \frac{1}{a \sqrt{a + b \operatorname{acosh}{\left (c x \right )}} + b \sqrt{a + b \operatorname{acosh}{\left (c x \right )}} \operatorname{acosh}{\left (c 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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