Optimal. Leaf size=96 \[ \frac {x \cosh ^{-1}(a x)}{c \sqrt {c+d x^2}}-\frac {\sqrt {a^2 x^2-1} \tanh ^{-1}\left (\frac {\sqrt {d} \sqrt {a^2 x^2-1}}{a \sqrt {c+d x^2}}\right )}{c \sqrt {d} \sqrt {a x-1} \sqrt {a x+1}} \]
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Rubi [A] time = 0.19, antiderivative size = 96, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 8, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {191, 5705, 12, 519, 444, 63, 217, 206} \[ \frac {x \cosh ^{-1}(a x)}{c \sqrt {c+d x^2}}-\frac {\sqrt {a^2 x^2-1} \tanh ^{-1}\left (\frac {\sqrt {d} \sqrt {a^2 x^2-1}}{a \sqrt {c+d x^2}}\right )}{c \sqrt {d} \sqrt {a x-1} \sqrt {a x+1}} \]
Antiderivative was successfully verified.
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Rule 12
Rule 63
Rule 191
Rule 206
Rule 217
Rule 444
Rule 519
Rule 5705
Rubi steps
\begin {align*} \int \frac {\cosh ^{-1}(a x)}{\left (c+d x^2\right )^{3/2}} \, dx &=\frac {x \cosh ^{-1}(a x)}{c \sqrt {c+d x^2}}-a \int \frac {x}{c \sqrt {-1+a x} \sqrt {1+a x} \sqrt {c+d x^2}} \, dx\\ &=\frac {x \cosh ^{-1}(a x)}{c \sqrt {c+d x^2}}-\frac {a \int \frac {x}{\sqrt {-1+a x} \sqrt {1+a x} \sqrt {c+d x^2}} \, dx}{c}\\ &=\frac {x \cosh ^{-1}(a x)}{c \sqrt {c+d x^2}}-\frac {\left (a \sqrt {-1+a^2 x^2}\right ) \int \frac {x}{\sqrt {-1+a^2 x^2} \sqrt {c+d x^2}} \, dx}{c \sqrt {-1+a x} \sqrt {1+a x}}\\ &=\frac {x \cosh ^{-1}(a x)}{c \sqrt {c+d x^2}}-\frac {\left (a \sqrt {-1+a^2 x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {-1+a^2 x} \sqrt {c+d x}} \, dx,x,x^2\right )}{2 c \sqrt {-1+a x} \sqrt {1+a x}}\\ &=\frac {x \cosh ^{-1}(a x)}{c \sqrt {c+d x^2}}-\frac {\sqrt {-1+a^2 x^2} \operatorname {Subst}\left (\int \frac {1}{\sqrt {c+\frac {d}{a^2}+\frac {d x^2}{a^2}}} \, dx,x,\sqrt {-1+a^2 x^2}\right )}{a c \sqrt {-1+a x} \sqrt {1+a x}}\\ &=\frac {x \cosh ^{-1}(a x)}{c \sqrt {c+d x^2}}-\frac {\sqrt {-1+a^2 x^2} \operatorname {Subst}\left (\int \frac {1}{1-\frac {d x^2}{a^2}} \, dx,x,\frac {\sqrt {-1+a^2 x^2}}{\sqrt {c+d x^2}}\right )}{a c \sqrt {-1+a x} \sqrt {1+a x}}\\ &=\frac {x \cosh ^{-1}(a x)}{c \sqrt {c+d x^2}}-\frac {\sqrt {-1+a^2 x^2} \tanh ^{-1}\left (\frac {\sqrt {d} \sqrt {-1+a^2 x^2}}{a \sqrt {c+d x^2}}\right )}{c \sqrt {d} \sqrt {-1+a x} \sqrt {1+a x}}\\ \end {align*}
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Mathematica [C] time = 3.23, size = 551, normalized size = 5.74 \[ \frac {x \cosh ^{-1}(a x)+\frac {2 (a x-1)^{3/2} \sqrt {\frac {(a x+1) \left (a \sqrt {c}-i \sqrt {d}\right )}{(a x-1) \left (a \sqrt {c}+i \sqrt {d}\right )}} \left (a \sqrt {c} \left (-a \sqrt {c}+i \sqrt {d}\right ) \sqrt {\frac {\left (a^2 c+d\right ) \left (c+d x^2\right )}{c d (a x-1)^2}} \sqrt {-\frac {a \left (x+\frac {i \sqrt {c}}{\sqrt {d}}\right )+\frac {i \sqrt {d} x}{\sqrt {c}}-1}{1-a x}} \Pi \left (\frac {2 a \sqrt {c}}{\sqrt {c} a+i \sqrt {d}};\sin ^{-1}\left (\sqrt {-\frac {\frac {i \sqrt {d} x}{\sqrt {c}}+a \left (x+\frac {i \sqrt {c}}{\sqrt {d}}\right )-1}{2-2 a x}}\right )|\frac {4 i a \sqrt {c} \sqrt {d}}{\left (\sqrt {c} a+i \sqrt {d}\right )^2}\right )+\frac {a \left (\sqrt {d}-i a \sqrt {c}\right ) \left (\sqrt {d} x+i \sqrt {c}\right ) \sqrt {\frac {\frac {i a \sqrt {c}}{\sqrt {d}}+a (-x)+\frac {i \sqrt {d} x}{\sqrt {c}}+1}{1-a x}} F\left (\sin ^{-1}\left (\sqrt {-\frac {\frac {i \sqrt {d} x}{\sqrt {c}}+a \left (x+\frac {i \sqrt {c}}{\sqrt {d}}\right )-1}{2-2 a x}}\right )|\frac {4 i a \sqrt {c} \sqrt {d}}{\left (\sqrt {c} a+i \sqrt {d}\right )^2}\right )}{a x-1}\right )}{a \sqrt {a x+1} \left (a^2 c+d\right ) \sqrt {-\frac {a \left (x+\frac {i \sqrt {c}}{\sqrt {d}}\right )+\frac {i \sqrt {d} x}{\sqrt {c}}-1}{1-a x}}}}{c \sqrt {c+d x^2}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.59, size = 296, normalized size = 3.08 \[ \left [\frac {4 \, \sqrt {d x^{2} + c} d x \log \left (a x + \sqrt {a^{2} x^{2} - 1}\right ) + {\left (d x^{2} + c\right )} \sqrt {d} \log \left (8 \, a^{4} d^{2} x^{4} + a^{4} c^{2} - 6 \, a^{2} c d + 8 \, {\left (a^{4} c d - a^{2} d^{2}\right )} x^{2} - 4 \, {\left (2 \, a^{3} d x^{2} + a^{3} c - a d\right )} \sqrt {a^{2} x^{2} - 1} \sqrt {d x^{2} + c} \sqrt {d} + d^{2}\right )}{4 \, {\left (c d^{2} x^{2} + c^{2} d\right )}}, \frac {2 \, \sqrt {d x^{2} + c} d x \log \left (a x + \sqrt {a^{2} x^{2} - 1}\right ) + {\left (d x^{2} + c\right )} \sqrt {-d} \arctan \left (\frac {{\left (2 \, a^{2} d x^{2} + a^{2} c - d\right )} \sqrt {a^{2} x^{2} - 1} \sqrt {d x^{2} + c} \sqrt {-d}}{2 \, {\left (a^{3} d^{2} x^{4} - a c d + {\left (a^{3} c d - a d^{2}\right )} x^{2}\right )}}\right )}{2 \, {\left (c d^{2} x^{2} + c^{2} d\right )}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.62, size = 82, normalized size = 0.85 \[ \frac {x \log \left (a x + \sqrt {a^{2} x^{2} - 1}\right )}{\sqrt {d x^{2} + c} c} + \frac {a \log \left ({\left | -\sqrt {a^{2} x^{2} - 1} \sqrt {d} + \sqrt {a^{2} c + {\left (a^{2} x^{2} - 1\right )} d + d} \right |}\right )}{c \sqrt {d} {\left | a \right |}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.33, size = 0, normalized size = 0.00 \[ \int \frac {\mathrm {arccosh}\left (a x \right )}{\left (d \,x^{2}+c \right )^{\frac {3}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: ValueError} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {\mathrm {acosh}\left (a\,x\right )}{{\left (d\,x^2+c\right )}^{3/2}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\operatorname {acosh}{\left (a x \right )}}{\left (c + d x^{2}\right )^{\frac {3}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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