Integrand size = 25, antiderivative size = 239 \[ \int \frac {\left (1-c^2 x^2\right )^{3/2}}{a+b \text {arccosh}(c x)} \, dx=\frac {\sqrt {1-c x} \cosh \left (\frac {2 a}{b}\right ) \text {Chi}\left (\frac {2 (a+b \text {arccosh}(c x))}{b}\right )}{2 b c \sqrt {-1+c x}}-\frac {\sqrt {1-c x} \cosh \left (\frac {4 a}{b}\right ) \text {Chi}\left (\frac {4 (a+b \text {arccosh}(c x))}{b}\right )}{8 b c \sqrt {-1+c x}}-\frac {3 \sqrt {1-c x} \log (a+b \text {arccosh}(c x))}{8 b c \sqrt {-1+c x}}-\frac {\sqrt {1-c x} \sinh \left (\frac {2 a}{b}\right ) \text {Shi}\left (\frac {2 (a+b \text {arccosh}(c x))}{b}\right )}{2 b c \sqrt {-1+c x}}+\frac {\sqrt {1-c x} \sinh \left (\frac {4 a}{b}\right ) \text {Shi}\left (\frac {4 (a+b \text {arccosh}(c x))}{b}\right )}{8 b c \sqrt {-1+c x}} \]
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Time = 0.17 (sec) , antiderivative size = 239, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {5906, 3393, 3384, 3379, 3382} \[ \int \frac {\left (1-c^2 x^2\right )^{3/2}}{a+b \text {arccosh}(c x)} \, dx=\frac {\sqrt {1-c x} \cosh \left (\frac {2 a}{b}\right ) \text {Chi}\left (\frac {2 (a+b \text {arccosh}(c x))}{b}\right )}{2 b c \sqrt {c x-1}}-\frac {\sqrt {1-c x} \cosh \left (\frac {4 a}{b}\right ) \text {Chi}\left (\frac {4 (a+b \text {arccosh}(c x))}{b}\right )}{8 b c \sqrt {c x-1}}-\frac {\sqrt {1-c x} \sinh \left (\frac {2 a}{b}\right ) \text {Shi}\left (\frac {2 (a+b \text {arccosh}(c x))}{b}\right )}{2 b c \sqrt {c x-1}}+\frac {\sqrt {1-c x} \sinh \left (\frac {4 a}{b}\right ) \text {Shi}\left (\frac {4 (a+b \text {arccosh}(c x))}{b}\right )}{8 b c \sqrt {c x-1}}-\frac {3 \sqrt {1-c x} \log (a+b \text {arccosh}(c x))}{8 b c \sqrt {c x-1}} \]
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Rule 3379
Rule 3382
Rule 3384
Rule 3393
Rule 5906
Rubi steps \begin{align*} \text {integral}& = -\frac {\sqrt {1-c x} \text {Subst}\left (\int \frac {\sinh ^4\left (\frac {a}{b}-\frac {x}{b}\right )}{x} \, dx,x,a+b \text {arccosh}(c x)\right )}{b c \sqrt {-1+c x}} \\ & = -\frac {\sqrt {1-c x} \text {Subst}\left (\int \left (\frac {3}{8 x}+\frac {\cosh \left (\frac {4 a}{b}-\frac {4 x}{b}\right )}{8 x}-\frac {\cosh \left (\frac {2 a}{b}-\frac {2 x}{b}\right )}{2 x}\right ) \, dx,x,a+b \text {arccosh}(c x)\right )}{b c \sqrt {-1+c x}} \\ & = -\frac {3 \sqrt {1-c x} \log (a+b \text {arccosh}(c x))}{8 b c \sqrt {-1+c x}}-\frac {\sqrt {1-c x} \text {Subst}\left (\int \frac {\cosh \left (\frac {4 a}{b}-\frac {4 x}{b}\right )}{x} \, dx,x,a+b \text {arccosh}(c x)\right )}{8 b c \sqrt {-1+c x}}+\frac {\sqrt {1-c x} \text {Subst}\left (\int \frac {\cosh \left (\frac {2 a}{b}-\frac {2 x}{b}\right )}{x} \, dx,x,a+b \text {arccosh}(c x)\right )}{2 b c \sqrt {-1+c x}} \\ & = -\frac {3 \sqrt {1-c x} \log (a+b \text {arccosh}(c x))}{8 b c \sqrt {-1+c x}}+\frac {\left (\sqrt {1-c x} \cosh \left (\frac {2 a}{b}\right )\right ) \text {Subst}\left (\int \frac {\cosh \left (\frac {2 x}{b}\right )}{x} \, dx,x,a+b \text {arccosh}(c x)\right )}{2 b c \sqrt {-1+c x}}-\frac {\left (\sqrt {1-c x} \cosh \left (\frac {4 a}{b}\right )\right ) \text {Subst}\left (\int \frac {\cosh \left (\frac {4 x}{b}\right )}{x} \, dx,x,a+b \text {arccosh}(c x)\right )}{8 b c \sqrt {-1+c x}}-\frac {\left (\sqrt {1-c x} \sinh \left (\frac {2 a}{b}\right )\right ) \text {Subst}\left (\int \frac {\sinh \left (\frac {2 x}{b}\right )}{x} \, dx,x,a+b \text {arccosh}(c x)\right )}{2 b c \sqrt {-1+c x}}+\frac {\left (\sqrt {1-c x} \sinh \left (\frac {4 a}{b}\right )\right ) \text {Subst}\left (\int \frac {\sinh \left (\frac {4 x}{b}\right )}{x} \, dx,x,a+b \text {arccosh}(c x)\right )}{8 b c \sqrt {-1+c x}} \\ & = \frac {\sqrt {1-c x} \cosh \left (\frac {2 a}{b}\right ) \text {Chi}\left (\frac {2 (a+b \text {arccosh}(c x))}{b}\right )}{2 b c \sqrt {-1+c x}}-\frac {\sqrt {1-c x} \cosh \left (\frac {4 a}{b}\right ) \text {Chi}\left (\frac {4 (a+b \text {arccosh}(c x))}{b}\right )}{8 b c \sqrt {-1+c x}}-\frac {3 \sqrt {1-c x} \log (a+b \text {arccosh}(c x))}{8 b c \sqrt {-1+c x}}-\frac {\sqrt {1-c x} \sinh \left (\frac {2 a}{b}\right ) \text {Shi}\left (\frac {2 (a+b \text {arccosh}(c x))}{b}\right )}{2 b c \sqrt {-1+c x}}+\frac {\sqrt {1-c x} \sinh \left (\frac {4 a}{b}\right ) \text {Shi}\left (\frac {4 (a+b \text {arccosh}(c x))}{b}\right )}{8 b c \sqrt {-1+c x}} \\ \end{align*}
Time = 0.55 (sec) , antiderivative size = 147, normalized size of antiderivative = 0.62 \[ \int \frac {\left (1-c^2 x^2\right )^{3/2}}{a+b \text {arccosh}(c x)} \, dx=-\frac {\sqrt {1-c^2 x^2} \left (-4 \cosh \left (\frac {2 a}{b}\right ) \text {Chi}\left (2 \left (\frac {a}{b}+\text {arccosh}(c x)\right )\right )+\cosh \left (\frac {4 a}{b}\right ) \text {Chi}\left (4 \left (\frac {a}{b}+\text {arccosh}(c x)\right )\right )+3 \log (a+b \text {arccosh}(c x))+4 \sinh \left (\frac {2 a}{b}\right ) \text {Shi}\left (2 \left (\frac {a}{b}+\text {arccosh}(c x)\right )\right )-\sinh \left (\frac {4 a}{b}\right ) \text {Shi}\left (4 \left (\frac {a}{b}+\text {arccosh}(c x)\right )\right )\right )}{8 b c \sqrt {\frac {-1+c x}{1+c x}} (1+c x)} \]
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Time = 0.86 (sec) , antiderivative size = 231, normalized size of antiderivative = 0.97
method | result | size |
default | \(-\frac {\sqrt {-c^{2} x^{2}+1}\, \left (-\sqrt {c x -1}\, \sqrt {c x +1}\, c x +c^{2} x^{2}-1\right ) \left (-6 \sqrt {c x -1}\, \sqrt {c x +1}\, \ln \left (a +b \,\operatorname {arccosh}\left (c x \right )\right )-6 \ln \left (a +b \,\operatorname {arccosh}\left (c x \right )\right ) c x +\operatorname {Ei}_{1}\left (4 \,\operatorname {arccosh}\left (c x \right )+\frac {4 a}{b}\right ) {\mathrm e}^{\frac {b \,\operatorname {arccosh}\left (c x \right )+4 a}{b}}+\operatorname {Ei}_{1}\left (-4 \,\operatorname {arccosh}\left (c x \right )-\frac {4 a}{b}\right ) {\mathrm e}^{-\frac {-b \,\operatorname {arccosh}\left (c x \right )+4 a}{b}}-4 \,\operatorname {Ei}_{1}\left (2 \,\operatorname {arccosh}\left (c x \right )+\frac {2 a}{b}\right ) {\mathrm e}^{\frac {b \,\operatorname {arccosh}\left (c x \right )+2 a}{b}}-4 \,\operatorname {Ei}_{1}\left (-2 \,\operatorname {arccosh}\left (c x \right )-\frac {2 a}{b}\right ) {\mathrm e}^{-\frac {-b \,\operatorname {arccosh}\left (c x \right )+2 a}{b}}\right )}{16 \left (c x -1\right ) \left (c x +1\right ) c b}\) | \(231\) |
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\[ \int \frac {\left (1-c^2 x^2\right )^{3/2}}{a+b \text {arccosh}(c x)} \, dx=\int { \frac {{\left (-c^{2} x^{2} + 1\right )}^{\frac {3}{2}}}{b \operatorname {arcosh}\left (c x\right ) + a} \,d x } \]
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\[ \int \frac {\left (1-c^2 x^2\right )^{3/2}}{a+b \text {arccosh}(c x)} \, dx=\int \frac {\left (- \left (c x - 1\right ) \left (c x + 1\right )\right )^{\frac {3}{2}}}{a + b \operatorname {acosh}{\left (c x \right )}}\, dx \]
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\[ \int \frac {\left (1-c^2 x^2\right )^{3/2}}{a+b \text {arccosh}(c x)} \, dx=\int { \frac {{\left (-c^{2} x^{2} + 1\right )}^{\frac {3}{2}}}{b \operatorname {arcosh}\left (c x\right ) + a} \,d x } \]
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\[ \int \frac {\left (1-c^2 x^2\right )^{3/2}}{a+b \text {arccosh}(c x)} \, dx=\int { \frac {{\left (-c^{2} x^{2} + 1\right )}^{\frac {3}{2}}}{b \operatorname {arcosh}\left (c x\right ) + a} \,d x } \]
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Timed out. \[ \int \frac {\left (1-c^2 x^2\right )^{3/2}}{a+b \text {arccosh}(c x)} \, dx=\int \frac {{\left (1-c^2\,x^2\right )}^{3/2}}{a+b\,\mathrm {acosh}\left (c\,x\right )} \,d x \]
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