∫ e−3 tanh−1(a+bx) ________________________

Optimal. Leaf size=257 \[ -\frac {\left (52-51 a+2 a^2\right ) b^3 \sqrt {1-a-b x}}{6 (1-a) (1+a)^4 \sqrt {1+a+b x}}-\frac {(1-a) \sqrt {1-a-b x}}{3 (1+a) x^3 \sqrt {1+a+b x}}+\frac {7 b \sqrt {1-a-b x}}{6 (1+a)^2 x^2 \sqrt {1+a+b x}}-\frac {(19-16 a) b^2 \sqrt {1-a-b x}}{6 (1-a) (1+a)^3 x \sqrt {1+a+b x}}+\frac {\left (11-18 a+6 a^2\right ) b^3 \tanh ^{-1}\left (\frac {\sqrt {1-a} \sqrt {1+a+b x}}{\sqrt {1+a} \sqrt {1-a-b x}}\right )}{(1-a) (1+a)^4 \sqrt {1-a^2}} \]

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Rubi [A]
time = 0.16, antiderivative size = 257, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 7, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {6298, 100, 156, 157, 12, 95, 214} \begin {gather*} -\frac {\left (2 a^2-51 a+52\right ) b^3 \sqrt {-a-b x+1}}{6 (1-a) (a+1)^4 \sqrt {a+b x+1}}+\frac {\left (6 a^2-18 a+11\right ) b^3 \tanh ^{-1}\left (\frac {\sqrt {1-a} \sqrt {a+b x+1}}{\sqrt {a+1} \sqrt {-a-b x+1}}\right )}{(1-a) (a+1)^4 \sqrt {1-a^2}}-\frac {(19-16 a) b^2 \sqrt {-a-b x+1}}{6 (1-a) (a+1)^3 x \sqrt {a+b x+1}}-\frac {(1-a) \sqrt {-a-b x+1}}{3 (a+1) x^3 \sqrt {a+b x+1}}+\frac {7 b \sqrt {-a-b x+1}}{6 (a+1)^2 x^2 \sqrt {a+b x+1}} \end {gather*}

\begin {align*} \int \frac {e^{-3 \tanh ^{-1}(a+b x)}}{x^4} \, dx &=\int \frac {(1-a-b x)^{3/2}}{x^4 (1+a+b x)^{3/2}} \, dx\\ &=-\frac {(1-a) \sqrt {1-a-b x}}{3 (1+a) x^3 \sqrt {1+a+b x}}-\frac {\int \frac {7 (1-a) b-6 b^2 x}{x^3 \sqrt {1-a-b x} (1+a+b x)^{3/2}} \, dx}{3 (1+a)}\\ &=-\frac {(1-a) \sqrt {1-a-b x}}{3 (1+a) x^3 \sqrt {1+a+b x}}+\frac {7 b \sqrt {1-a-b x}}{6 (1+a)^2 x^2 \sqrt {1+a+b x}}+\frac {\int \frac {(19-16 a) (1-a) b^2-14 (1-a) b^3 x}{x^2 \sqrt {1-a-b x} (1+a+b x)^{3/2}} \, dx}{6 (1-a) (1+a)^2}\\ &=-\frac {(1-a) \sqrt {1-a-b x}}{3 (1+a) x^3 \sqrt {1+a+b x}}+\frac {7 b \sqrt {1-a-b x}}{6 (1+a)^2 x^2 \sqrt {1+a+b x}}-\frac {(19-16 a) b^2 \sqrt {1-a-b x}}{6 (1-a) (1+a)^3 x \sqrt {1+a+b x}}-\frac {\int \frac {3 (1-a) \left (11-18 a+6 a^2\right ) b^3-(19-16 a) (1-a) b^4 x}{x \sqrt {1-a-b x} (1+a+b x)^{3/2}} \, dx}{6 (1-a)^2 (1+a)^3}\\ &=-\frac {\left (52-51 a+2 a^2\right ) b^3 \sqrt {1-a-b x}}{6 (1-a) (1+a)^4 \sqrt {1+a+b x}}-\frac {(1-a) \sqrt {1-a-b x}}{3 (1+a) x^3 \sqrt {1+a+b x}}+\frac {7 b \sqrt {1-a-b x}}{6 (1+a)^2 x^2 \sqrt {1+a+b x}}-\frac {(19-16 a) b^2 \sqrt {1-a-b x}}{6 (1-a) (1+a)^3 x \sqrt {1+a+b x}}-\frac {\int \frac {3 (1-a) \left (11-18 a+6 a^2\right ) b^4}{x \sqrt {1-a-b x} \sqrt {1+a+b x}} \, dx}{6 (1-a)^2 (1+a)^4 b}\\ &=-\frac {\left (52-51 a+2 a^2\right ) b^3 \sqrt {1-a-b x}}{6 (1-a) (1+a)^4 \sqrt {1+a+b x}}-\frac {(1-a) \sqrt {1-a-b x}}{3 (1+a) x^3 \sqrt {1+a+b x}}+\frac {7 b \sqrt {1-a-b x}}{6 (1+a)^2 x^2 \sqrt {1+a+b x}}-\frac {(19-16 a) b^2 \sqrt {1-a-b x}}{6 (1-a) (1+a)^3 x \sqrt {1+a+b x}}-\frac {\left (\left (11-18 a+6 a^2\right ) b^3\right ) \int \frac {1}{x \sqrt {1-a-b x} \sqrt {1+a+b x}} \, dx}{2 (1-a) (1+a)^4}\\ &=-\frac {\left (52-51 a+2 a^2\right ) b^3 \sqrt {1-a-b x}}{6 (1-a) (1+a)^4 \sqrt {1+a+b x}}-\frac {(1-a) \sqrt {1-a-b x}}{3 (1+a) x^3 \sqrt {1+a+b x}}+\frac {7 b \sqrt {1-a-b x}}{6 (1+a)^2 x^2 \sqrt {1+a+b x}}-\frac {(19-16 a) b^2 \sqrt {1-a-b x}}{6 (1-a) (1+a)^3 x \sqrt {1+a+b x}}-\frac {\left (\left (11-18 a+6 a^2\right ) b^3\right ) \text {Subst}\left (\int \frac {1}{-1-a-(-1+a) x^2} \, dx,x,\frac {\sqrt {1+a+b x}}{\sqrt {1-a-b x}}\right )}{(1-a) (1+a)^4}\\ &=-\frac {\left (52-51 a+2 a^2\right ) b^3 \sqrt {1-a-b x}}{6 (1-a) (1+a)^4 \sqrt {1+a+b x}}-\frac {(1-a) \sqrt {1-a-b x}}{3 (1+a) x^3 \sqrt {1+a+b x}}+\frac {7 b \sqrt {1-a-b x}}{6 (1+a)^2 x^2 \sqrt {1+a+b x}}-\frac {(19-16 a) b^2 \sqrt {1-a-b x}}{6 (1-a) (1+a)^3 x \sqrt {1+a+b x}}+\frac {\left (11-18 a+6 a^2\right ) b^3 \tanh ^{-1}\left (\frac {\sqrt {1-a} \sqrt {1+a+b x}}{\sqrt {1+a} \sqrt {1-a-b x}}\right )}{(1-a) (1+a)^4 \sqrt {1-a^2}}\\ \end {align*}

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Mathematica [A]
time = 0.28, size = 194, normalized size = 0.75 \begin {gather*} \frac {-2 (1-a) (1+a) (1-a-b x)^{5/2}+(3-4 a) b x (1-a-b x)^{5/2}+\frac {\left (11-18 a+6 a^2\right ) b^2 x^2 \left (\sqrt {-1-a} \sqrt {1-a-b x} \left (-1+a^2-5 b x+a b x\right )+6 \sqrt {-1+a} b x \sqrt {1+a+b x} \tanh ^{-1}\left (\frac {\sqrt {-1-a} \sqrt {1-a-b x}}{\sqrt {-1+a} \sqrt {1+a+b x}}\right )\right )}{(-1-a)^{5/2}}}{6 \left (-1+a^2\right )^2 x^3 \sqrt {1+a+b x}} \end {gather*}

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(3699\) vs. \(2(223)=446\).
time = 0.09, size = 3700, normalized size = 14.40


method	result	size

risch	\(-\frac {\left (b^{2} x^{2}+2 a b x +a^{2}-1\right ) \left (2 a^{2} b^{2} x^{2}-2 a^{3} b x -27 a \,b^{2} x^{2}+2 a^{4}+9 a^{2} b x +28 b^{2} x^{2}+2 a b x -4 a^{2}-9 b x +2\right )}{6 \left (1+a \right )^{3} x^{3} \sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}\, \left (a^{2}-1\right )}+\frac {4 b^{2} \sqrt {-b^{2} \left (x +\frac {1+a}{b}\right )^{2}+2 b \left (x +\frac {1+a}{b}\right )}}{\left (a^{2}-1\right ) \left (1+a \right )^{3} \left (x +\frac {1}{b}+\frac {a}{b}\right )}-\frac {4 b^{2} \sqrt {-b^{2} \left (x +\frac {1+a}{b}\right )^{2}+2 b \left (x +\frac {1+a}{b}\right )}\, a}{\left (a^{2}-1\right ) \left (1+a \right )^{3} \left (x +\frac {1}{b}+\frac {a}{b}\right )}-\frac {3 b^{3} \ln \left (\frac {-2 a^{2}+2-2 a b x +2 \sqrt {-a^{2}+1}\, \sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}}{x}\right ) a^{2}}{\left (a^{2}-1\right ) \left (1+a \right )^{3} \sqrt {-a^{2}+1}}+\frac {9 b^{3} \ln \left (\frac {-2 a^{2}+2-2 a b x +2 \sqrt {-a^{2}+1}\, \sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}}{x}\right ) a}{\left (a^{2}-1\right ) \left (1+a \right )^{3} \sqrt {-a^{2}+1}}-\frac {11 b^{3} \ln \left (\frac {-2 a^{2}+2-2 a b x +2 \sqrt {-a^{2}+1}\, \sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}}{x}\right )}{2 \left (a^{2}-1\right ) \left (1+a \right )^{3} \sqrt {-a^{2}+1}}\)	\(474\)
default	\(\text {Expression too large to display}\)	\(3700\)

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

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Fricas [A]
time = 0.44, size = 697, normalized size = 2.71 \begin {gather*} \left [-\frac {3 \, {\left ({\left (6 \, a^{2} - 18 \, a + 11\right )} b^{4} x^{4} + {\left (6 \, a^{3} - 12 \, a^{2} - 7 \, a + 11\right )} b^{3} x^{3}\right )} \sqrt {-a^{2} + 1} \log \left (\frac {{\left (2 \, a^{2} - 1\right )} b^{2} x^{2} + 2 \, a^{4} + 4 \, {\left (a^{3} - a\right )} b x + 2 \, \sqrt {-b^{2} x^{2} - 2 \, a b x - a^{2} + 1} {\left (a b x + a^{2} - 1\right )} \sqrt {-a^{2} + 1} - 4 \, a^{2} + 2}{x^{2}}\right ) - 2 \, {\left (2 \, a^{7} + {\left (2 \, a^{4} - 51 \, a^{3} + 50 \, a^{2} + 51 \, a - 52\right )} b^{3} x^{3} + 2 \, a^{6} - 6 \, a^{5} - {\left (16 \, a^{4} - 3 \, a^{3} - 35 \, a^{2} + 3 \, a + 19\right )} b^{2} x^{2} - 6 \, a^{4} + 6 \, a^{3} + 7 \, {\left (a^{5} + a^{4} - 2 \, a^{3} - 2 \, a^{2} + a + 1\right )} b x + 6 \, a^{2} - 2 \, a - 2\right )} \sqrt {-b^{2} x^{2} - 2 \, a b x - a^{2} + 1}}{12 \, {\left ({\left (a^{7} + 3 \, a^{6} + a^{5} - 5 \, a^{4} - 5 \, a^{3} + a^{2} + 3 \, a + 1\right )} b x^{4} + {\left (a^{8} + 4 \, a^{7} + 4 \, a^{6} - 4 \, a^{5} - 10 \, a^{4} - 4 \, a^{3} + 4 \, a^{2} + 4 \, a + 1\right )} x^{3}\right )}}, \frac {3 \, {\left ({\left (6 \, a^{2} - 18 \, a + 11\right )} b^{4} x^{4} + {\left (6 \, a^{3} - 12 \, a^{2} - 7 \, a + 11\right )} b^{3} x^{3}\right )} \sqrt {a^{2} - 1} \arctan \left (\frac {\sqrt {-b^{2} x^{2} - 2 \, a b x - a^{2} + 1} {\left (a b x + a^{2} - 1\right )} \sqrt {a^{2} - 1}}{{\left (a^{2} - 1\right )} b^{2} x^{2} + a^{4} + 2 \, {\left (a^{3} - a\right )} b x - 2 \, a^{2} + 1}\right ) + {\left (2 \, a^{7} + {\left (2 \, a^{4} - 51 \, a^{3} + 50 \, a^{2} + 51 \, a - 52\right )} b^{3} x^{3} + 2 \, a^{6} - 6 \, a^{5} - {\left (16 \, a^{4} - 3 \, a^{3} - 35 \, a^{2} + 3 \, a + 19\right )} b^{2} x^{2} - 6 \, a^{4} + 6 \, a^{3} + 7 \, {\left (a^{5} + a^{4} - 2 \, a^{3} - 2 \, a^{2} + a + 1\right )} b x + 6 \, a^{2} - 2 \, a - 2\right )} \sqrt {-b^{2} x^{2} - 2 \, a b x - a^{2} + 1}}{6 \, {\left ({\left (a^{7} + 3 \, a^{6} + a^{5} - 5 \, a^{4} - 5 \, a^{3} + a^{2} + 3 \, a + 1\right )} b x^{4} + {\left (a^{8} + 4 \, a^{7} + 4 \, a^{6} - 4 \, a^{5} - 10 \, a^{4} - 4 \, a^{3} + 4 \, a^{2} + 4 \, a + 1\right )} x^{3}\right )}}\right ] \end {gather*}

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (- \left (a + b x - 1\right ) \left (a + b x + 1\right )\right )^{\frac {3}{2}}}{x^{4} \left (a + b x + 1\right )^{3}}\, dx \end {gather*}

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 1839 vs. \(2 (216) = 432\).
time = 0.45, size = 1839, normalized size = 7.16 \begin {gather*} \text {Too large to display} \end {gather*}

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {{\left (1-{\left (a+b\,x\right )}^2\right )}^{3/2}}{x^4\,{\left (a+b\,x+1\right )}^3} \,d x \end {gather*}

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3.9.67 \(\int \frac {e^{-3 \tanh ^{-1}(a+b x)}}{x^4} \, dx\) [867]