Optimal. Leaf size=189 \[ -\frac {a}{3 x^3}-\frac {b}{2 x^2}+\frac {a b^2 \sqrt {-1+a+b x} \sqrt {1+a+b x}}{2 \left (1-a^2\right )^2 x}-\frac {a b \sqrt {-1+a+b x} (1+a+b x)^{3/2}}{2 (1-a) (1+a)^2 x^2}+\frac {(-1+a+b x)^{3/2} (1+a+b x)^{3/2}}{3 \left (1-a^2\right ) x^3}-\frac {a b^3 \text {ArcTan}\left (\frac {\sqrt {1-a} \sqrt {1+a+b x}}{\sqrt {1+a} \sqrt {-1+a+b x}}\right )}{\left (1-a^2\right )^{5/2}} \]
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Rubi [A]
time = 0.12, antiderivative size = 189, normalized size of antiderivative = 1.00, number of steps
used = 8, number of rules used = 6, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {6020, 14, 98,
96, 95, 211} \begin {gather*} -\frac {a b^3 \text {ArcTan}\left (\frac {\sqrt {1-a} \sqrt {a+b x+1}}{\sqrt {a+1} \sqrt {a+b x-1}}\right )}{\left (1-a^2\right )^{5/2}}+\frac {a b^2 \sqrt {a+b x-1} \sqrt {a+b x+1}}{2 \left (1-a^2\right )^2 x}+\frac {(a+b x-1)^{3/2} (a+b x+1)^{3/2}}{3 \left (1-a^2\right ) x^3}-\frac {a b \sqrt {a+b x-1} (a+b x+1)^{3/2}}{2 (1-a) (a+1)^2 x^2}-\frac {a}{3 x^3}-\frac {b}{2 x^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 14
Rule 95
Rule 96
Rule 98
Rule 211
Rule 6020
Rubi steps
\begin {align*} \int \frac {e^{\cosh ^{-1}(a+b x)}}{x^4} \, dx &=\int \frac {a+b x+\sqrt {-1+a+b x} \sqrt {1+a+b x}}{x^4} \, dx\\ &=\int \left (\frac {a}{x^4}+\frac {b}{x^3}+\frac {\sqrt {-1+a+b x} \sqrt {1+a+b x}}{x^4}\right ) \, dx\\ &=-\frac {a}{3 x^3}-\frac {b}{2 x^2}+\int \frac {\sqrt {-1+a+b x} \sqrt {1+a+b x}}{x^4} \, dx\\ &=-\frac {a}{3 x^3}-\frac {b}{2 x^2}+\frac {(-1+a+b x)^{3/2} (1+a+b x)^{3/2}}{3 \left (1-a^2\right ) x^3}+\frac {(a b) \int \frac {\sqrt {-1+a+b x} \sqrt {1+a+b x}}{x^3} \, dx}{1-a^2}\\ &=-\frac {a}{3 x^3}-\frac {b}{2 x^2}-\frac {a b \sqrt {-1+a+b x} (1+a+b x)^{3/2}}{2 (1-a) (1+a)^2 x^2}+\frac {(-1+a+b x)^{3/2} (1+a+b x)^{3/2}}{3 \left (1-a^2\right ) x^3}+\frac {\left (a b^2\right ) \int \frac {\sqrt {1+a+b x}}{x^2 \sqrt {-1+a+b x}} \, dx}{2 (1-a) (1+a)^2}\\ &=-\frac {a}{3 x^3}-\frac {b}{2 x^2}+\frac {a b^2 \sqrt {-1+a+b x} \sqrt {1+a+b x}}{2 \left (1-a^2\right )^2 x}-\frac {a b \sqrt {-1+a+b x} (1+a+b x)^{3/2}}{2 (1-a) (1+a)^2 x^2}+\frac {(-1+a+b x)^{3/2} (1+a+b x)^{3/2}}{3 \left (1-a^2\right ) x^3}+\frac {\left (a b^3\right ) \int \frac {1}{x \sqrt {-1+a+b x} \sqrt {1+a+b x}} \, dx}{2 \left (1-a^2\right )^2}\\ &=-\frac {a}{3 x^3}-\frac {b}{2 x^2}+\frac {a b^2 \sqrt {-1+a+b x} \sqrt {1+a+b x}}{2 \left (1-a^2\right )^2 x}-\frac {a b \sqrt {-1+a+b x} (1+a+b x)^{3/2}}{2 (1-a) (1+a)^2 x^2}+\frac {(-1+a+b x)^{3/2} (1+a+b x)^{3/2}}{3 \left (1-a^2\right ) x^3}+\frac {\left (a 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 )}{\left (1-a^2\right )^2}\\ &=-\frac {a}{3 x^3}-\frac {b}{2 x^2}+\frac {a b^2 \sqrt {-1+a+b x} \sqrt {1+a+b x}}{2 \left (1-a^2\right )^2 x}-\frac {a b \sqrt {-1+a+b x} (1+a+b x)^{3/2}}{2 (1-a) (1+a)^2 x^2}+\frac {(-1+a+b x)^{3/2} (1+a+b x)^{3/2}}{3 \left (1-a^2\right ) x^3}-\frac {a b^3 \tan ^{-1}\left (\frac {\sqrt {1-a} \sqrt {1+a+b x}}{\sqrt {1+a} \sqrt {-1+a+b x}}\right )}{\left (1-a^2\right )^{5/2}}\\ \end {align*}
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Mathematica [C] Result contains complex when optimal does not.
time = 0.18, size = 179, normalized size = 0.95 \begin {gather*} \frac {1}{6} \left (-\frac {2 a}{x^3}-\frac {3 b}{x^2}+\frac {\sqrt {-1+a+b x} \sqrt {1+a+b x} \left (-2-2 a^4+a b x-a^3 b x+2 b^2 x^2+a^2 \left (4+b^2 x^2\right )\right )}{\left (-1+a^2\right )^2 x^3}-\frac {3 i a b^3 \log \left (\frac {4 \left (1-a^2\right )^{3/2} \left (-i+i a^2+i a b x+\sqrt {1-a^2} \sqrt {-1+a+b x} \sqrt {1+a+b x}\right )}{a b^3 x}\right )}{\left (1-a^2\right )^{5/2}}\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(373\) vs.
\(2(157)=314\).
time = 0.07, size = 374, normalized size = 1.98
method | result | size |
default | \(-\frac {\sqrt {b x +a -1}\, \sqrt {b x +a +1}\, \left (3 \sqrt {a^{2}-1}\, \ln \left (\frac {2 a^{2}-2+2 a b x +2 \sqrt {a^{2}-1}\, \sqrt {b^{2} x^{2}+2 a b x +a^{2}-1}}{x}\right ) a \,b^{3} x^{3}-a^{4} b^{2} x^{2} \sqrt {b^{2} x^{2}+2 a b x +a^{2}-1}+a^{5} b x \sqrt {b^{2} x^{2}+2 a b x +a^{2}-1}+2 a^{6} \sqrt {b^{2} x^{2}+2 a b x +a^{2}-1}-\sqrt {b^{2} x^{2}+2 a b x +a^{2}-1}\, a^{2} b^{2} x^{2}-2 a^{3} b x \sqrt {b^{2} x^{2}+2 a b x +a^{2}-1}-6 a^{4} \sqrt {b^{2} x^{2}+2 a b x +a^{2}-1}+2 \sqrt {b^{2} x^{2}+2 a b x +a^{2}-1}\, b^{2} x^{2}+\sqrt {b^{2} x^{2}+2 a b x +a^{2}-1}\, a b x +6 \sqrt {b^{2} x^{2}+2 a b x +a^{2}-1}\, a^{2}-2 \sqrt {b^{2} x^{2}+2 a b x +a^{2}-1}\right )}{6 \sqrt {b^{2} x^{2}+2 a b x +a^{2}-1}\, \left (a^{2}-1\right )^{3} x^{3}}-\frac {a}{3 x^{3}}-\frac {b}{2 x^{2}}\) | \(374\) |
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 \begin {gather*} \text {Exception raised: ValueError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.40, size = 431, normalized size = 2.28 \begin {gather*} \left [\frac {3 \, \sqrt {a^{2} - 1} a b^{3} x^{3} \log \left (\frac {a^{2} b x + a^{3} + {\left (a^{2} - \sqrt {a^{2} - 1} a - 1\right )} \sqrt {b x + a + 1} \sqrt {b x + a - 1} - {\left (a b x + a^{2} - 1\right )} \sqrt {a^{2} - 1} - a}{x}\right ) - 2 \, a^{7} + {\left (a^{4} + a^{2} - 2\right )} b^{3} x^{3} + 6 \, a^{5} - 6 \, a^{3} - 3 \, {\left (a^{6} - 3 \, a^{4} + 3 \, a^{2} - 1\right )} b x - {\left (2 \, a^{6} - {\left (a^{4} + a^{2} - 2\right )} b^{2} x^{2} - 6 \, a^{4} + {\left (a^{5} - 2 \, a^{3} + a\right )} b x + 6 \, a^{2} - 2\right )} \sqrt {b x + a + 1} \sqrt {b x + a - 1} + 2 \, a}{6 \, {\left (a^{6} - 3 \, a^{4} + 3 \, a^{2} - 1\right )} x^{3}}, \frac {6 \, \sqrt {-a^{2} + 1} a b^{3} x^{3} \arctan \left (-\frac {\sqrt {-a^{2} + 1} b x - \sqrt {-a^{2} + 1} \sqrt {b x + a + 1} \sqrt {b x + a - 1}}{a^{2} - 1}\right ) - 2 \, a^{7} + {\left (a^{4} + a^{2} - 2\right )} b^{3} x^{3} + 6 \, a^{5} - 6 \, a^{3} - 3 \, {\left (a^{6} - 3 \, a^{4} + 3 \, a^{2} - 1\right )} b x - {\left (2 \, a^{6} - {\left (a^{4} + a^{2} - 2\right )} b^{2} x^{2} - 6 \, a^{4} + {\left (a^{5} - 2 \, a^{3} + a\right )} b x + 6 \, a^{2} - 2\right )} \sqrt {b x + a + 1} \sqrt {b x + a - 1} + 2 \, a}{6 \, {\left (a^{6} - 3 \, a^{4} + 3 \, a^{2} - 1\right )} x^{3}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 487 vs.
\(2 (151) = 302\).
time = 0.54, size = 487, normalized size = 2.58 \begin {gather*} -\frac {\frac {6 \, a b^{4} \arctan \left (\frac {{\left (\sqrt {b x + a + 1} - \sqrt {b x + a - 1}\right )}^{2} - 2 \, a}{2 \, \sqrt {-a^{2} + 1}}\right )}{{\left (a^{4} - 2 \, a^{2} + 1\right )} \sqrt {-a^{2} + 1}} - \frac {4 \, {\left (12 \, a^{4} b^{4} {\left (\sqrt {b x + a + 1} - \sqrt {b x + a - 1}\right )}^{8} - 16 \, a^{5} b^{4} {\left (\sqrt {b x + a + 1} - \sqrt {b x + a - 1}\right )}^{6} - 3 \, a b^{4} {\left (\sqrt {b x + a + 1} - \sqrt {b x + a - 1}\right )}^{10} + 6 \, a^{2} b^{4} {\left (\sqrt {b x + a + 1} - \sqrt {b x + a - 1}\right )}^{8} - 56 \, a^{3} b^{4} {\left (\sqrt {b x + a + 1} - \sqrt {b x + a - 1}\right )}^{6} + 48 \, a^{4} b^{4} {\left (\sqrt {b x + a + 1} - \sqrt {b x + a - 1}\right )}^{4} + 12 \, b^{4} {\left (\sqrt {b x + a + 1} - \sqrt {b x + a - 1}\right )}^{8} - 48 \, a b^{4} {\left (\sqrt {b x + a + 1} - \sqrt {b x + a - 1}\right )}^{6} + 192 \, a^{2} b^{4} {\left (\sqrt {b x + a + 1} - \sqrt {b x + a - 1}\right )}^{4} - 96 \, a^{3} b^{4} {\left (\sqrt {b x + a + 1} - \sqrt {b x + a - 1}\right )}^{2} - 144 \, a b^{4} {\left (\sqrt {b x + a + 1} - \sqrt {b x + a - 1}\right )}^{2} + 32 \, a^{2} b^{4} + 64 \, b^{4}\right )}}{{\left (a^{4} - 2 \, a^{2} + 1\right )} {\left ({\left (\sqrt {b x + a + 1} - \sqrt {b x + a - 1}\right )}^{4} - 4 \, a {\left (\sqrt {b x + a + 1} - \sqrt {b x + a - 1}\right )}^{2} + 4\right )}^{3}} + \frac {3 \, {\left (b x + a + 1\right )} b^{4} - a b^{4} - 3 \, b^{4}}{b^{3} x^{3}}}{6 \, b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 18.89, size = 1537, normalized size = 8.13 \begin {gather*} \frac {\frac {{\left (\sqrt {a-1}-\sqrt {a+b\,x-1}\right )}^2\,\left (\frac {3\,b^3}{32}-\frac {a^2\,b^3}{32}\right )}{{\left (\sqrt {a+1}-\sqrt {a+b\,x+1}\right )}^2\,\left (a^4-2\,a^2+1\right )}-\frac {b^3}{192\,\left (a^2-1\right )}+\frac {{\left (\sqrt {a-1}-\sqrt {a+b\,x-1}\right )}^4\,\left (\frac {5\,a^4\,b^3}{8}+\frac {9\,a^2\,b^3}{8}-\frac {b^3}{2}\right )}{{\left (\sqrt {a+1}-\sqrt {a+b\,x+1}\right )}^4\,\left (a^6-3\,a^4+3\,a^2-1\right )}+\frac {{\left (\sqrt {a-1}-\sqrt {a+b\,x-1}\right )}^8\,\left (\frac {3\,a^4\,b^3}{64}+\frac {a^2\,b^3}{32}-\frac {21\,b^3}{64}\right )}{{\left (\sqrt {a+1}-\sqrt {a+b\,x+1}\right )}^8\,\left (a^6-3\,a^4+3\,a^2-1\right )}+\frac {{\left (\sqrt {a-1}-\sqrt {a+b\,x-1}\right )}^6\,\left (\frac {67\,a^6\,b^3}{96}+\frac {11\,a^4\,b^3}{32}-\frac {121\,a^2\,b^3}{32}+\frac {103\,b^3}{96}\right )}{{\left (\sqrt {a+1}-\sqrt {a+b\,x+1}\right )}^6\,\left (a^8-4\,a^6+6\,a^4-4\,a^2+1\right )}-\frac {{\left (\sqrt {a-1}-\sqrt {a+b\,x-1}\right )}^3\,\sqrt {a-1}\,\sqrt {a+1}\,\left (\frac {17\,a^3\,b^3}{96}+\frac {17\,a\,b^3}{32}\right )}{{\left (\sqrt {a+1}-\sqrt {a+b\,x+1}\right )}^3\,\left (a^6-3\,a^4+3\,a^2-1\right )}-\frac {{\left (\sqrt {a-1}-\sqrt {a+b\,x-1}\right )}^7\,\sqrt {a-1}\,\sqrt {a+1}\,\left (\frac {9\,a^5\,b^3}{32}+\frac {3\,a^3\,b^3}{16}-\frac {63\,a\,b^3}{32}\right )}{{\left (\sqrt {a+1}-\sqrt {a+b\,x+1}\right )}^7\,\left (a^8-4\,a^6+6\,a^4-4\,a^2+1\right )}-\frac {{\left (\sqrt {a-1}-\sqrt {a+b\,x-1}\right )}^5\,\sqrt {a-1}\,\sqrt {a+1}\,\left (\frac {29\,a^5\,b^3}{32}+\frac {17\,a^3\,b^3}{16}-\frac {79\,a\,b^3}{32}\right )}{{\left (\sqrt {a+1}-\sqrt {a+b\,x+1}\right )}^5\,\left (a^8-4\,a^6+6\,a^4-4\,a^2+1\right )}+\frac {a\,b^3\,\left (\sqrt {a-1}-\sqrt {a+b\,x-1}\right )\,\sqrt {a-1}\,\sqrt {a+1}}{32\,\left (\sqrt {a+1}-\sqrt {a+b\,x+1}\right )\,\left (a^4-2\,a^2+1\right )}}{\frac {{\left (\sqrt {a-1}-\sqrt {a+b\,x-1}\right )}^3}{{\left (\sqrt {a+1}-\sqrt {a+b\,x+1}\right )}^3}+\frac {{\left (\sqrt {a-1}-\sqrt {a+b\,x-1}\right )}^9}{{\left (\sqrt {a+1}-\sqrt {a+b\,x+1}\right )}^9}+\frac {{\left (\sqrt {a-1}-\sqrt {a+b\,x-1}\right )}^5\,\left (15\,a^2-3\right )}{\left (a^2-1\right )\,{\left (\sqrt {a+1}-\sqrt {a+b\,x+1}\right )}^5}+\frac {{\left (\sqrt {a-1}-\sqrt {a+b\,x-1}\right )}^7\,\left (15\,a^2-3\right )}{\left (a^2-1\right )\,{\left (\sqrt {a+1}-\sqrt {a+b\,x+1}\right )}^7}+\frac {{\left (\sqrt {a-1}-\sqrt {a+b\,x-1}\right )}^6\,\left (12\,a-20\,a^3\right )\,\sqrt {a-1}\,\sqrt {a+1}}{{\left (\sqrt {a+1}-\sqrt {a+b\,x+1}\right )}^6\,\left (a^4-2\,a^2+1\right )}-\frac {6\,a\,{\left (\sqrt {a-1}-\sqrt {a+b\,x-1}\right )}^4\,\sqrt {a-1}\,\sqrt {a+1}}{\left (a^2-1\right )\,{\left (\sqrt {a+1}-\sqrt {a+b\,x+1}\right )}^4}-\frac {6\,a\,{\left (\sqrt {a-1}-\sqrt {a+b\,x-1}\right )}^8\,\sqrt {a-1}\,\sqrt {a+1}}{\left (a^2-1\right )\,{\left (\sqrt {a+1}-\sqrt {a+b\,x+1}\right )}^8}}-\frac {\frac {a}{3}+\frac {b\,x}{2}}{x^3}+\frac {\left (\sqrt {a-1}-\sqrt {a+b\,x-1}\right )\,\left (\frac {b^3\,\left (1792\,a^4-2048\,a^2+256\right )}{4096\,{\left (a^2-1\right )}^3}-\frac {b^3\,\left (25\,a^2-9\right )}{64\,{\left (a^2-1\right )}^2}+\frac {8\,a\,{\left (a-1\right )}^{3/2}\,{\left (a+1\right )}^{3/2}\,\left (\frac {a\,b^3\,\sqrt {a-1}\,\sqrt {a+1}}{8\,{\left (a^2-1\right )}^2}-\frac {a\,b^3\,{\left (a-1\right )}^{3/2}\,{\left (a+1\right )}^{3/2}}{8\,{\left (a^2-1\right )}^3}\right )}{{\left (a^2-1\right )}^2}\right )}{\sqrt {a+1}-\sqrt {a+b\,x+1}}+\frac {{\left (\sqrt {a-1}-\sqrt {a+b\,x-1}\right )}^2\,\left (\frac {a\,b^3\,\sqrt {a-1}\,\sqrt {a+1}}{16\,{\left (a^2-1\right )}^2}-\frac {a\,b^3\,{\left (a-1\right )}^{3/2}\,{\left (a+1\right )}^{3/2}}{16\,{\left (a^2-1\right )}^3}\right )}{{\left (\sqrt {a+1}-\sqrt {a+b\,x+1}\right )}^2}-\frac {b^3\,{\left (\sqrt {a-1}-\sqrt {a+b\,x-1}\right )}^3}{192\,\left (a^2-1\right )\,{\left (\sqrt {a+1}-\sqrt {a+b\,x+1}\right )}^3}+\frac {a\,b^3\,\ln \left (\frac {{\left (\sqrt {a-1}-\sqrt {a+b\,x-1}\right )}^2}{{\left (\sqrt {a+1}-\sqrt {a+b\,x+1}\right )}^2}-a^2-\frac {a^2\,{\left (\sqrt {a-1}-\sqrt {a+b\,x-1}\right )}^2}{{\left (\sqrt {a+1}-\sqrt {a+b\,x+1}\right )}^2}+\frac {2\,a\,\left (\sqrt {a-1}-\sqrt {a+b\,x-1}\right )\,\sqrt {a-1}\,\sqrt {a+1}}{\sqrt {a+1}-\sqrt {a+b\,x+1}}+1\right )\,\sqrt {a-1}\,\sqrt {a+1}}{2\,a^6-6\,a^4+6\,a^2-2}-\frac {a\,b^3\,\ln \left (\frac {\sqrt {a-1}-\sqrt {a+b\,x-1}}{\sqrt {a+1}-\sqrt {a+b\,x+1}}\right )\,\sqrt {a-1}\,\sqrt {a+1}}{2\,a^6-6\,a^4+6\,a^2-2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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