Optimal. Leaf size=101 \[ \frac {x \left (a+b \cosh ^{-1}(c x)\right )}{d \sqrt {d+e x^2}}-\frac {b \sqrt {c^2 x^2-1} \tanh ^{-1}\left (\frac {\sqrt {e} \sqrt {c^2 x^2-1}}{c \sqrt {d+e x^2}}\right )}{d \sqrt {e} \sqrt {c x-1} \sqrt {c x+1}} \]
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Rubi [A] time = 0.19, antiderivative size = 101, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 8, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {191, 5705, 12, 519, 444, 63, 217, 206} \[ \frac {x \left (a+b \cosh ^{-1}(c x)\right )}{d \sqrt {d+e x^2}}-\frac {b \sqrt {c^2 x^2-1} \tanh ^{-1}\left (\frac {\sqrt {e} \sqrt {c^2 x^2-1}}{c \sqrt {d+e x^2}}\right )}{d \sqrt {e} \sqrt {c x-1} \sqrt {c x+1}} \]
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 {a+b \cosh ^{-1}(c x)}{\left (d+e x^2\right )^{3/2}} \, dx &=\frac {x \left (a+b \cosh ^{-1}(c x)\right )}{d \sqrt {d+e x^2}}-(b c) \int \frac {x}{d \sqrt {-1+c x} \sqrt {1+c x} \sqrt {d+e x^2}} \, dx\\ &=\frac {x \left (a+b \cosh ^{-1}(c x)\right )}{d \sqrt {d+e x^2}}-\frac {(b c) \int \frac {x}{\sqrt {-1+c x} \sqrt {1+c x} \sqrt {d+e x^2}} \, dx}{d}\\ &=\frac {x \left (a+b \cosh ^{-1}(c x)\right )}{d \sqrt {d+e x^2}}-\frac {\left (b c \sqrt {-1+c^2 x^2}\right ) \int \frac {x}{\sqrt {-1+c^2 x^2} \sqrt {d+e x^2}} \, dx}{d \sqrt {-1+c x} \sqrt {1+c x}}\\ &=\frac {x \left (a+b \cosh ^{-1}(c x)\right )}{d \sqrt {d+e x^2}}-\frac {\left (b c \sqrt {-1+c^2 x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {-1+c^2 x} \sqrt {d+e x}} \, dx,x,x^2\right )}{2 d \sqrt {-1+c x} \sqrt {1+c x}}\\ &=\frac {x \left (a+b \cosh ^{-1}(c x)\right )}{d \sqrt {d+e x^2}}-\frac {\left (b \sqrt {-1+c^2 x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {d+\frac {e}{c^2}+\frac {e x^2}{c^2}}} \, dx,x,\sqrt {-1+c^2 x^2}\right )}{c d \sqrt {-1+c x} \sqrt {1+c x}}\\ &=\frac {x \left (a+b \cosh ^{-1}(c x)\right )}{d \sqrt {d+e x^2}}-\frac {\left (b \sqrt {-1+c^2 x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{1-\frac {e x^2}{c^2}} \, dx,x,\frac {\sqrt {-1+c^2 x^2}}{\sqrt {d+e x^2}}\right )}{c d \sqrt {-1+c x} \sqrt {1+c x}}\\ &=\frac {x \left (a+b \cosh ^{-1}(c x)\right )}{d \sqrt {d+e x^2}}-\frac {b \sqrt {-1+c^2 x^2} \tanh ^{-1}\left (\frac {\sqrt {e} \sqrt {-1+c^2 x^2}}{c \sqrt {d+e x^2}}\right )}{d \sqrt {e} \sqrt {-1+c x} \sqrt {1+c x}}\\ \end {align*}
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Mathematica [C] time = 3.36, size = 556, normalized size = 5.50 \[ \frac {a x+\frac {2 b (c x-1)^{3/2} \sqrt {\frac {(c x+1) \left (c \sqrt {d}-i \sqrt {e}\right )}{(c x-1) \left (c \sqrt {d}+i \sqrt {e}\right )}} \left (c \sqrt {d} \left (-c \sqrt {d}+i \sqrt {e}\right ) \sqrt {\frac {\left (c^2 d+e\right ) \left (d+e x^2\right )}{d e (c x-1)^2}} \sqrt {-\frac {c \left (x+\frac {i \sqrt {d}}{\sqrt {e}}\right )+\frac {i \sqrt {e} x}{\sqrt {d}}-1}{1-c x}} \Pi \left (\frac {2 c \sqrt {d}}{\sqrt {d} c+i \sqrt {e}};\sin ^{-1}\left (\sqrt {-\frac {\frac {i \sqrt {e} x}{\sqrt {d}}+c \left (x+\frac {i \sqrt {d}}{\sqrt {e}}\right )-1}{2-2 c x}}\right )|\frac {4 i c \sqrt {d} \sqrt {e}}{\left (\sqrt {d} c+i \sqrt {e}\right )^2}\right )+\frac {c \left (\sqrt {e}-i c \sqrt {d}\right ) \left (\sqrt {e} x+i \sqrt {d}\right ) \sqrt {\frac {\frac {i c \sqrt {d}}{\sqrt {e}}+c (-x)+\frac {i \sqrt {e} x}{\sqrt {d}}+1}{1-c x}} F\left (\sin ^{-1}\left (\sqrt {-\frac {\frac {i \sqrt {e} x}{\sqrt {d}}+c \left (x+\frac {i \sqrt {d}}{\sqrt {e}}\right )-1}{2-2 c x}}\right )|\frac {4 i c \sqrt {d} \sqrt {e}}{\left (\sqrt {d} c+i \sqrt {e}\right )^2}\right )}{c x-1}\right )}{c \sqrt {c x+1} \left (c^2 d+e\right ) \sqrt {-\frac {c \left (x+\frac {i \sqrt {d}}{\sqrt {e}}\right )+\frac {i \sqrt {e} x}{\sqrt {d}}-1}{1-c x}}}+b x \cosh ^{-1}(c x)}{d \sqrt {d+e x^2}} \]
Antiderivative was successfully verified.
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fricas [A] time = 1.43, size = 332, normalized size = 3.29 \[ \left [\frac {4 \, \sqrt {e x^{2} + d} b e x \log \left (c x + \sqrt {c^{2} x^{2} - 1}\right ) + 4 \, \sqrt {e x^{2} + d} a e x + {\left (b e x^{2} + b d\right )} \sqrt {e} \log \left (8 \, c^{4} e^{2} x^{4} + c^{4} d^{2} - 6 \, c^{2} d e + 8 \, {\left (c^{4} d e - c^{2} e^{2}\right )} x^{2} - 4 \, {\left (2 \, c^{3} e x^{2} + c^{3} d - c e\right )} \sqrt {c^{2} x^{2} - 1} \sqrt {e x^{2} + d} \sqrt {e} + e^{2}\right )}{4 \, {\left (d e^{2} x^{2} + d^{2} e\right )}}, \frac {2 \, \sqrt {e x^{2} + d} b e x \log \left (c x + \sqrt {c^{2} x^{2} - 1}\right ) + 2 \, \sqrt {e x^{2} + d} a e x + {\left (b e x^{2} + b d\right )} \sqrt {-e} \arctan \left (\frac {{\left (2 \, c^{2} e x^{2} + c^{2} d - e\right )} \sqrt {c^{2} x^{2} - 1} \sqrt {e x^{2} + d} \sqrt {-e}}{2 \, {\left (c^{3} e^{2} x^{4} - c d e + {\left (c^{3} d e - c e^{2}\right )} x^{2}\right )}}\right )}{2 \, {\left (d e^{2} x^{2} + d^{2} e\right )}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {b \operatorname {arcosh}\left (c x\right ) + a}{{\left (e x^{2} + d\right )}^{\frac {3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.64, size = 0, normalized size = 0.00 \[ \int \frac {a +b \,\mathrm {arccosh}\left (c x \right )}{\left (e \,x^{2}+d \right )^{\frac {3}{2}}}\, dx \]
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 \[ \int \frac {a+b\,\mathrm {acosh}\left (c\,x\right )}{{\left (e\,x^2+d\right )}^{3/2}} \,d x \]
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 \[ \int \frac {a + b \operatorname {acosh}{\left (c x \right )}}{\left (d + e x^{2}\right )^{\frac {3}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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