Optimal. Leaf size=73 \[ -\frac {b c}{15 x^{5/2}}-\frac {b c^3}{9 x^{3/2}}-\frac {b c^5}{3 \sqrt {x}}+\frac {1}{3} b c^6 \tanh ^{-1}\left (c \sqrt {x}\right )-\frac {a+b \tanh ^{-1}\left (c \sqrt {x}\right )}{3 x^3} \]
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Rubi [A]
time = 0.02, antiderivative size = 73, normalized size of antiderivative = 1.00, number of steps
used = 6, number of rules used = 4, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {6037, 53, 65,
212} \begin {gather*} -\frac {a+b \tanh ^{-1}\left (c \sqrt {x}\right )}{3 x^3}+\frac {1}{3} b c^6 \tanh ^{-1}\left (c \sqrt {x}\right )-\frac {b c^5}{3 \sqrt {x}}-\frac {b c^3}{9 x^{3/2}}-\frac {b c}{15 x^{5/2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 53
Rule 65
Rule 212
Rule 6037
Rubi steps
\begin {align*} \int \frac {a+b \tanh ^{-1}\left (c \sqrt {x}\right )}{x^4} \, dx &=-\frac {a+b \tanh ^{-1}\left (c \sqrt {x}\right )}{3 x^3}+\frac {1}{6} (b c) \int \frac {1}{x^{7/2} \left (1-c^2 x\right )} \, dx\\ &=-\frac {b c}{15 x^{5/2}}-\frac {a+b \tanh ^{-1}\left (c \sqrt {x}\right )}{3 x^3}+\frac {1}{6} \left (b c^3\right ) \int \frac {1}{x^{5/2} \left (1-c^2 x\right )} \, dx\\ &=-\frac {b c}{15 x^{5/2}}-\frac {b c^3}{9 x^{3/2}}-\frac {a+b \tanh ^{-1}\left (c \sqrt {x}\right )}{3 x^3}+\frac {1}{6} \left (b c^5\right ) \int \frac {1}{x^{3/2} \left (1-c^2 x\right )} \, dx\\ &=-\frac {b c}{15 x^{5/2}}-\frac {b c^3}{9 x^{3/2}}-\frac {b c^5}{3 \sqrt {x}}-\frac {a+b \tanh ^{-1}\left (c \sqrt {x}\right )}{3 x^3}+\frac {1}{6} \left (b c^7\right ) \int \frac {1}{\sqrt {x} \left (1-c^2 x\right )} \, dx\\ &=-\frac {b c}{15 x^{5/2}}-\frac {b c^3}{9 x^{3/2}}-\frac {b c^5}{3 \sqrt {x}}-\frac {a+b \tanh ^{-1}\left (c \sqrt {x}\right )}{3 x^3}+\frac {1}{3} \left (b c^7\right ) \text {Subst}\left (\int \frac {1}{1-c^2 x^2} \, dx,x,\sqrt {x}\right )\\ &=-\frac {b c}{15 x^{5/2}}-\frac {b c^3}{9 x^{3/2}}-\frac {b c^5}{3 \sqrt {x}}+\frac {1}{3} b c^6 \tanh ^{-1}\left (c \sqrt {x}\right )-\frac {a+b \tanh ^{-1}\left (c \sqrt {x}\right )}{3 x^3}\\ \end {align*}
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Mathematica [A]
time = 0.02, size = 99, normalized size = 1.36 \begin {gather*} -\frac {a}{3 x^3}-\frac {b c}{15 x^{5/2}}-\frac {b c^3}{9 x^{3/2}}-\frac {b c^5}{3 \sqrt {x}}-\frac {b \tanh ^{-1}\left (c \sqrt {x}\right )}{3 x^3}-\frac {1}{6} b c^6 \log \left (1-c \sqrt {x}\right )+\frac {1}{6} b c^6 \log \left (1+c \sqrt {x}\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.07, size = 80, normalized size = 1.10
method | result | size |
derivativedivides | \(2 c^{6} \left (-\frac {a}{6 c^{6} x^{3}}-\frac {b \arctanh \left (c \sqrt {x}\right )}{6 c^{6} x^{3}}-\frac {b}{30 c^{5} x^{\frac {5}{2}}}-\frac {b}{18 c^{3} x^{\frac {3}{2}}}-\frac {b}{6 c \sqrt {x}}-\frac {b \ln \left (c \sqrt {x}-1\right )}{12}+\frac {b \ln \left (1+c \sqrt {x}\right )}{12}\right )\) | \(80\) |
default | \(2 c^{6} \left (-\frac {a}{6 c^{6} x^{3}}-\frac {b \arctanh \left (c \sqrt {x}\right )}{6 c^{6} x^{3}}-\frac {b}{30 c^{5} x^{\frac {5}{2}}}-\frac {b}{18 c^{3} x^{\frac {3}{2}}}-\frac {b}{6 c \sqrt {x}}-\frac {b \ln \left (c \sqrt {x}-1\right )}{12}+\frac {b \ln \left (1+c \sqrt {x}\right )}{12}\right )\) | \(80\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.26, size = 72, normalized size = 0.99 \begin {gather*} \frac {1}{90} \, {\left ({\left (15 \, c^{5} \log \left (c \sqrt {x} + 1\right ) - 15 \, c^{5} \log \left (c \sqrt {x} - 1\right ) - \frac {2 \, {\left (15 \, c^{4} x^{2} + 5 \, c^{2} x + 3\right )}}{x^{\frac {5}{2}}}\right )} c - \frac {30 \, \operatorname {artanh}\left (c \sqrt {x}\right )}{x^{3}}\right )} b - \frac {a}{3 \, x^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.35, size = 74, normalized size = 1.01 \begin {gather*} \frac {15 \, {\left (b c^{6} x^{3} - b\right )} \log \left (-\frac {c^{2} x + 2 \, c \sqrt {x} + 1}{c^{2} x - 1}\right ) - 2 \, {\left (15 \, b c^{5} x^{2} + 5 \, b c^{3} x + 3 \, b c\right )} \sqrt {x} - 30 \, a}{90 \, x^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 371 vs.
\(2 (68) = 136\).
time = 25.83, size = 371, normalized size = 5.08 \begin {gather*} \begin {cases} - \frac {a}{3 x^{3}} + \frac {b \operatorname {atanh}{\left (\sqrt {x} \sqrt {\frac {1}{x}} \right )}}{3 x^{3}} & \text {for}\: c = - \sqrt {\frac {1}{x}} \\- \frac {a}{3 x^{3}} - \frac {b \operatorname {atanh}{\left (\sqrt {x} \sqrt {\frac {1}{x}} \right )}}{3 x^{3}} & \text {for}\: c = \sqrt {\frac {1}{x}} \\- \frac {15 a c^{2} x^{\frac {3}{2}}}{45 c^{2} x^{\frac {9}{2}} - 45 x^{\frac {7}{2}}} + \frac {15 a \sqrt {x}}{45 c^{2} x^{\frac {9}{2}} - 45 x^{\frac {7}{2}}} + \frac {15 b c^{8} x^{\frac {9}{2}} \operatorname {atanh}{\left (c \sqrt {x} \right )}}{45 c^{2} x^{\frac {9}{2}} - 45 x^{\frac {7}{2}}} - \frac {15 b c^{7} x^{4}}{45 c^{2} x^{\frac {9}{2}} - 45 x^{\frac {7}{2}}} - \frac {15 b c^{6} x^{\frac {7}{2}} \operatorname {atanh}{\left (c \sqrt {x} \right )}}{45 c^{2} x^{\frac {9}{2}} - 45 x^{\frac {7}{2}}} + \frac {10 b c^{5} x^{3}}{45 c^{2} x^{\frac {9}{2}} - 45 x^{\frac {7}{2}}} + \frac {2 b c^{3} x^{2}}{45 c^{2} x^{\frac {9}{2}} - 45 x^{\frac {7}{2}}} - \frac {15 b c^{2} x^{\frac {3}{2}} \operatorname {atanh}{\left (c \sqrt {x} \right )}}{45 c^{2} x^{\frac {9}{2}} - 45 x^{\frac {7}{2}}} + \frac {3 b c x}{45 c^{2} x^{\frac {9}{2}} - 45 x^{\frac {7}{2}}} + \frac {15 b \sqrt {x} \operatorname {atanh}{\left (c \sqrt {x} \right )}}{45 c^{2} x^{\frac {9}{2}} - 45 x^{\frac {7}{2}}} & \text {otherwise} \end {cases} \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. 534 vs.
\(2 (53) = 106\).
time = 0.44, size = 534, normalized size = 7.32 \begin {gather*} \frac {2}{45} \, c {\left (\frac {15 \, {\left (\frac {3 \, {\left (c \sqrt {x} + 1\right )}^{5} b c^{5}}{{\left (c \sqrt {x} - 1\right )}^{5}} + \frac {10 \, {\left (c \sqrt {x} + 1\right )}^{3} b c^{5}}{{\left (c \sqrt {x} - 1\right )}^{3}} + \frac {3 \, {\left (c \sqrt {x} + 1\right )} b c^{5}}{c \sqrt {x} - 1}\right )} \log \left (-\frac {c \sqrt {x} + 1}{c \sqrt {x} - 1}\right )}{\frac {{\left (c \sqrt {x} + 1\right )}^{6}}{{\left (c \sqrt {x} - 1\right )}^{6}} + \frac {6 \, {\left (c \sqrt {x} + 1\right )}^{5}}{{\left (c \sqrt {x} - 1\right )}^{5}} + \frac {15 \, {\left (c \sqrt {x} + 1\right )}^{4}}{{\left (c \sqrt {x} - 1\right )}^{4}} + \frac {20 \, {\left (c \sqrt {x} + 1\right )}^{3}}{{\left (c \sqrt {x} - 1\right )}^{3}} + \frac {15 \, {\left (c \sqrt {x} + 1\right )}^{2}}{{\left (c \sqrt {x} - 1\right )}^{2}} + \frac {6 \, {\left (c \sqrt {x} + 1\right )}}{c \sqrt {x} - 1} + 1} + \frac {\frac {90 \, {\left (c \sqrt {x} + 1\right )}^{5} a c^{5}}{{\left (c \sqrt {x} - 1\right )}^{5}} + \frac {300 \, {\left (c \sqrt {x} + 1\right )}^{3} a c^{5}}{{\left (c \sqrt {x} - 1\right )}^{3}} + \frac {90 \, {\left (c \sqrt {x} + 1\right )} a c^{5}}{c \sqrt {x} - 1} + \frac {45 \, {\left (c \sqrt {x} + 1\right )}^{5} b c^{5}}{{\left (c \sqrt {x} - 1\right )}^{5}} + \frac {135 \, {\left (c \sqrt {x} + 1\right )}^{4} b c^{5}}{{\left (c \sqrt {x} - 1\right )}^{4}} + \frac {230 \, {\left (c \sqrt {x} + 1\right )}^{3} b c^{5}}{{\left (c \sqrt {x} - 1\right )}^{3}} + \frac {210 \, {\left (c \sqrt {x} + 1\right )}^{2} b c^{5}}{{\left (c \sqrt {x} - 1\right )}^{2}} + \frac {93 \, {\left (c \sqrt {x} + 1\right )} b c^{5}}{c \sqrt {x} - 1} + 23 \, b c^{5}}{\frac {{\left (c \sqrt {x} + 1\right )}^{6}}{{\left (c \sqrt {x} - 1\right )}^{6}} + \frac {6 \, {\left (c \sqrt {x} + 1\right )}^{5}}{{\left (c \sqrt {x} - 1\right )}^{5}} + \frac {15 \, {\left (c \sqrt {x} + 1\right )}^{4}}{{\left (c \sqrt {x} - 1\right )}^{4}} + \frac {20 \, {\left (c \sqrt {x} + 1\right )}^{3}}{{\left (c \sqrt {x} - 1\right )}^{3}} + \frac {15 \, {\left (c \sqrt {x} + 1\right )}^{2}}{{\left (c \sqrt {x} - 1\right )}^{2}} + \frac {6 \, {\left (c \sqrt {x} + 1\right )}}{c \sqrt {x} - 1} + 1}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 1.39, size = 69, normalized size = 0.95 \begin {gather*} \frac {b\,c^6\,\mathrm {atanh}\left (c\,\sqrt {x}\right )}{3}-\frac {b\,\left (15\,\ln \left (c\,\sqrt {x}+1\right )-15\,\ln \left (1-c\,\sqrt {x}\right )+6\,c\,\sqrt {x}+10\,c^3\,x^{3/2}+30\,c^5\,x^{5/2}\right )}{90\,x^3}-\frac {a}{3\,x^3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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