Optimal. Leaf size=68 \[ -\frac {2 (-a-b x+1)^{3/2}}{b \sqrt {a+b x+1}}-\frac {3 \sqrt {a+b x+1} \sqrt {-a-b x+1}}{b}-\frac {3 \sin ^{-1}(a+b x)}{b} \]
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Rubi [A] time = 0.03, antiderivative size = 68, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.600, Rules used = {6161, 47, 50, 53, 619, 216} \[ -\frac {2 (-a-b x+1)^{3/2}}{b \sqrt {a+b x+1}}-\frac {3 \sqrt {a+b x+1} \sqrt {-a-b x+1}}{b}-\frac {3 \sin ^{-1}(a+b x)}{b} \]
Antiderivative was successfully verified.
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Rule 47
Rule 50
Rule 53
Rule 216
Rule 619
Rule 6161
Rubi steps
\begin {align*} \int e^{-3 \tanh ^{-1}(a+b x)} \, dx &=\int \frac {(1-a-b x)^{3/2}}{(1+a+b x)^{3/2}} \, dx\\ &=-\frac {2 (1-a-b x)^{3/2}}{b \sqrt {1+a+b x}}-3 \int \frac {\sqrt {1-a-b x}}{\sqrt {1+a+b x}} \, dx\\ &=-\frac {2 (1-a-b x)^{3/2}}{b \sqrt {1+a+b x}}-\frac {3 \sqrt {1-a-b x} \sqrt {1+a+b x}}{b}-3 \int \frac {1}{\sqrt {1-a-b x} \sqrt {1+a+b x}} \, dx\\ &=-\frac {2 (1-a-b x)^{3/2}}{b \sqrt {1+a+b x}}-\frac {3 \sqrt {1-a-b x} \sqrt {1+a+b x}}{b}-3 \int \frac {1}{\sqrt {(1-a) (1+a)-2 a b x-b^2 x^2}} \, dx\\ &=-\frac {2 (1-a-b x)^{3/2}}{b \sqrt {1+a+b x}}-\frac {3 \sqrt {1-a-b x} \sqrt {1+a+b x}}{b}+\frac {3 \operatorname {Subst}\left (\int \frac {1}{\sqrt {1-\frac {x^2}{4 b^2}}} \, dx,x,-2 a b-2 b^2 x\right )}{2 b^2}\\ &=-\frac {2 (1-a-b x)^{3/2}}{b \sqrt {1+a+b x}}-\frac {3 \sqrt {1-a-b x} \sqrt {1+a+b x}}{b}-\frac {3 \sin ^{-1}(a+b x)}{b}\\ \end {align*}
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Mathematica [A] time = 0.04, size = 43, normalized size = 0.63 \[ \frac {\sqrt {1-(a+b x)^2} \left (-\frac {4}{a+b x+1}-1\right )}{b}-\frac {3 \sin ^{-1}(a+b x)}{b} \]
Warning: Unable to verify antiderivative.
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fricas [A] time = 0.63, size = 101, normalized size = 1.49 \[ \frac {3 \, {\left (b x + a + 1\right )} \arctan \left (\frac {\sqrt {-b^{2} x^{2} - 2 \, a b x - a^{2} + 1} {\left (b x + a\right )}}{b^{2} x^{2} + 2 \, a b x + a^{2} - 1}\right ) - \sqrt {-b^{2} x^{2} - 2 \, a b x - a^{2} + 1} {\left (b x + a + 5\right )}}{b^{2} x + {\left (a + 1\right )} b} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.21, size = 94, normalized size = 1.38 \[ \frac {3 \, \arcsin \left (-b x - a\right ) \mathrm {sgn}\relax (b)}{{\left | b \right |}} - \frac {\sqrt {-b^{2} x^{2} - 2 \, a b x - a^{2} + 1}}{b} + \frac {8}{{\left (\frac {\sqrt {-b^{2} x^{2} - 2 \, a b x - a^{2} + 1} {\left | b \right |} + b}{b^{2} x + a b} + 1\right )} {\left | b \right |}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.04, size = 264, normalized size = 3.88 \[ -\frac {\left (-\left (x +\frac {1+a}{b}\right )^{2} b^{2}+2 b \left (x +\frac {1+a}{b}\right )\right )^{\frac {5}{2}}}{b^{4} \left (x +\frac {1}{b}+\frac {a}{b}\right )^{3}}-\frac {2 \left (-\left (x +\frac {1+a}{b}\right )^{2} b^{2}+2 b \left (x +\frac {1+a}{b}\right )\right )^{\frac {5}{2}}}{b^{3} \left (x +\frac {1}{b}+\frac {a}{b}\right )^{2}}-\frac {2 \left (-\left (x +\frac {1+a}{b}\right )^{2} b^{2}+2 b \left (x +\frac {1+a}{b}\right )\right )^{\frac {3}{2}}}{b}-3 \sqrt {-\left (x +\frac {1+a}{b}\right )^{2} b^{2}+2 b \left (x +\frac {1+a}{b}\right )}\, x -\frac {3 \sqrt {-\left (x +\frac {1+a}{b}\right )^{2} b^{2}+2 b \left (x +\frac {1+a}{b}\right )}\, a}{b}-\frac {3 \arctan \left (\frac {\sqrt {b^{2}}\, \left (x +\frac {1+a}{b}-\frac {1}{b}\right )}{\sqrt {-\left (x +\frac {1+a}{b}\right )^{2} b^{2}+2 b \left (x +\frac {1+a}{b}\right )}}\right )}{\sqrt {b^{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.41, size = 104, normalized size = 1.53 \[ \frac {{\left (-b^{2} x^{2} - 2 \, a b x - a^{2} + 1\right )}^{\frac {3}{2}}}{b^{3} x^{2} + 2 \, a b^{2} x + a^{2} b + 2 \, b^{2} x + 2 \, a b + b} - \frac {3 \, \arcsin \left (b x + a\right )}{b} - \frac {6 \, \sqrt {-b^{2} x^{2} - 2 \, a b x - a^{2} + 1}}{b^{2} x + a b + b} \]
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 \[ \int \frac {{\left (1-{\left (a+b\,x\right )}^2\right )}^{3/2}}{{\left (a+b\,x+1\right )}^3} \,d x \]
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 \[ \int \frac {\left (- \left (a + b x - 1\right ) \left (a + b x + 1\right )\right )^{\frac {3}{2}}}{\left (a + b x + 1\right )^{3}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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