Optimal. Leaf size=245 \[ \frac {d \left (420 a^6 c^3+378 a^4 c^2 d+180 a^2 c d^2+35 d^3\right ) x^2}{630 a^7}+\frac {d^2 \left (378 a^4 c^2+180 a^2 c d+35 d^2\right ) x^4}{1260 a^5}+\frac {d^3 \left (36 a^2 c+7 d\right ) x^6}{378 a^3}+\frac {d^4 x^8}{72 a}+c^4 x \tanh ^{-1}(a x)+\frac {4}{3} c^3 d x^3 \tanh ^{-1}(a x)+\frac {6}{5} c^2 d^2 x^5 \tanh ^{-1}(a x)+\frac {4}{7} c d^3 x^7 \tanh ^{-1}(a x)+\frac {1}{9} d^4 x^9 \tanh ^{-1}(a x)+\frac {\left (315 a^8 c^4+420 a^6 c^3 d+378 a^4 c^2 d^2+180 a^2 c d^3+35 d^4\right ) \log \left (1-a^2 x^2\right )}{630 a^9} \]
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Rubi [A]
time = 0.14, antiderivative size = 245, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 4, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {200, 6123,
1824, 266} \begin {gather*} \frac {d^3 x^6 \left (36 a^2 c+7 d\right )}{378 a^3}+\frac {d^2 x^4 \left (378 a^4 c^2+180 a^2 c d+35 d^2\right )}{1260 a^5}+\frac {d x^2 \left (420 a^6 c^3+378 a^4 c^2 d+180 a^2 c d^2+35 d^3\right )}{630 a^7}+\frac {\left (315 a^8 c^4+420 a^6 c^3 d+378 a^4 c^2 d^2+180 a^2 c d^3+35 d^4\right ) \log \left (1-a^2 x^2\right )}{630 a^9}+c^4 x \tanh ^{-1}(a x)+\frac {4}{3} c^3 d x^3 \tanh ^{-1}(a x)+\frac {6}{5} c^2 d^2 x^5 \tanh ^{-1}(a x)+\frac {4}{7} c d^3 x^7 \tanh ^{-1}(a x)+\frac {1}{9} d^4 x^9 \tanh ^{-1}(a x)+\frac {d^4 x^8}{72 a} \end {gather*}
Antiderivative was successfully verified.
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Rule 200
Rule 266
Rule 1824
Rule 6123
Rubi steps
\begin {align*} \int \left (c+d x^2\right )^4 \tanh ^{-1}(a x) \, dx &=c^4 x \tanh ^{-1}(a x)+\frac {4}{3} c^3 d x^3 \tanh ^{-1}(a x)+\frac {6}{5} c^2 d^2 x^5 \tanh ^{-1}(a x)+\frac {4}{7} c d^3 x^7 \tanh ^{-1}(a x)+\frac {1}{9} d^4 x^9 \tanh ^{-1}(a x)-a \int \frac {c^4 x+\frac {4}{3} c^3 d x^3+\frac {6}{5} c^2 d^2 x^5+\frac {4}{7} c d^3 x^7+\frac {d^4 x^9}{9}}{1-a^2 x^2} \, dx\\ &=c^4 x \tanh ^{-1}(a x)+\frac {4}{3} c^3 d x^3 \tanh ^{-1}(a x)+\frac {6}{5} c^2 d^2 x^5 \tanh ^{-1}(a x)+\frac {4}{7} c d^3 x^7 \tanh ^{-1}(a x)+\frac {1}{9} d^4 x^9 \tanh ^{-1}(a x)-a \int \left (-\frac {d \left (420 a^6 c^3+378 a^4 c^2 d+180 a^2 c d^2+35 d^3\right ) x}{315 a^8}-\frac {d^2 \left (378 a^4 c^2+180 a^2 c d+35 d^2\right ) x^3}{315 a^6}-\frac {d^3 \left (36 a^2 c+7 d\right ) x^5}{63 a^4}-\frac {d^4 x^7}{9 a^2}+\frac {\left (315 a^8 c^4+420 a^6 c^3 d+378 a^4 c^2 d^2+180 a^2 c d^3+35 d^4\right ) x}{315 a^8 \left (1-a^2 x^2\right )}\right ) \, dx\\ &=\frac {d \left (420 a^6 c^3+378 a^4 c^2 d+180 a^2 c d^2+35 d^3\right ) x^2}{630 a^7}+\frac {d^2 \left (378 a^4 c^2+180 a^2 c d+35 d^2\right ) x^4}{1260 a^5}+\frac {d^3 \left (36 a^2 c+7 d\right ) x^6}{378 a^3}+\frac {d^4 x^8}{72 a}+c^4 x \tanh ^{-1}(a x)+\frac {4}{3} c^3 d x^3 \tanh ^{-1}(a x)+\frac {6}{5} c^2 d^2 x^5 \tanh ^{-1}(a x)+\frac {4}{7} c d^3 x^7 \tanh ^{-1}(a x)+\frac {1}{9} d^4 x^9 \tanh ^{-1}(a x)-\frac {\left (315 a^8 c^4+420 a^6 c^3 d+378 a^4 c^2 d^2+180 a^2 c d^3+35 d^4\right ) \int \frac {x}{1-a^2 x^2} \, dx}{315 a^7}\\ &=\frac {d \left (420 a^6 c^3+378 a^4 c^2 d+180 a^2 c d^2+35 d^3\right ) x^2}{630 a^7}+\frac {d^2 \left (378 a^4 c^2+180 a^2 c d+35 d^2\right ) x^4}{1260 a^5}+\frac {d^3 \left (36 a^2 c+7 d\right ) x^6}{378 a^3}+\frac {d^4 x^8}{72 a}+c^4 x \tanh ^{-1}(a x)+\frac {4}{3} c^3 d x^3 \tanh ^{-1}(a x)+\frac {6}{5} c^2 d^2 x^5 \tanh ^{-1}(a x)+\frac {4}{7} c d^3 x^7 \tanh ^{-1}(a x)+\frac {1}{9} d^4 x^9 \tanh ^{-1}(a x)+\frac {\left (315 a^8 c^4+420 a^6 c^3 d+378 a^4 c^2 d^2+180 a^2 c d^3+35 d^4\right ) \log \left (1-a^2 x^2\right )}{630 a^9}\\ \end {align*}
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Mathematica [A]
time = 0.06, size = 213, normalized size = 0.87 \begin {gather*} \frac {a^2 d x^2 \left (420 d^3+30 a^2 d^2 \left (72 c+7 d x^2\right )+4 a^4 d \left (1134 c^2+270 c d x^2+35 d^2 x^4\right )+3 a^6 \left (1680 c^3+756 c^2 d x^2+240 c d^2 x^4+35 d^3 x^6\right )\right )+24 a^9 x \left (315 c^4+420 c^3 d x^2+378 c^2 d^2 x^4+180 c d^3 x^6+35 d^4 x^8\right ) \tanh ^{-1}(a x)+12 \left (315 a^8 c^4+420 a^6 c^3 d+378 a^4 c^2 d^2+180 a^2 c d^3+35 d^4\right ) \log \left (1-a^2 x^2\right )}{7560 a^9} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.90, size = 301, normalized size = 1.23
method | result | size |
derivativedivides | \(\frac {\arctanh \left (a x \right ) c^{4} a x +\frac {4 a \arctanh \left (a x \right ) c^{3} d \,x^{3}}{3}+\frac {6 a \arctanh \left (a x \right ) c^{2} d^{2} x^{5}}{5}+\frac {4 a \arctanh \left (a x \right ) c \,d^{3} x^{7}}{7}+\frac {a \arctanh \left (a x \right ) d^{4} x^{9}}{9}-\frac {\frac {\left (-315 a^{8} c^{4}-420 a^{6} c^{3} d -378 a^{4} c^{2} d^{2}-180 a^{2} c \,d^{3}-35 d^{4}\right ) \ln \left (a x +1\right )}{2}-\frac {\left (315 a^{8} c^{4}+420 a^{6} c^{3} d +378 a^{4} c^{2} d^{2}+180 a^{2} c \,d^{3}+35 d^{4}\right ) \ln \left (a x -1\right )}{2}-\frac {35 d^{4} a^{6} x^{6}}{6}-\frac {35 d^{4} a^{2} x^{2}}{2}-\frac {35 d^{4} a^{4} x^{4}}{4}-\frac {35 d^{4} a^{8} x^{8}}{8}-45 a^{6} c \,d^{3} x^{4}-90 a^{4} c \,d^{3} x^{2}-30 c \,a^{8} d^{3} x^{6}-\frac {189 c^{2} a^{8} d^{2} x^{4}}{2}-210 c^{3} a^{8} d \,x^{2}-189 c^{2} a^{6} d^{2} x^{2}}{315 a^{8}}}{a}\) | \(301\) |
default | \(\frac {\arctanh \left (a x \right ) c^{4} a x +\frac {4 a \arctanh \left (a x \right ) c^{3} d \,x^{3}}{3}+\frac {6 a \arctanh \left (a x \right ) c^{2} d^{2} x^{5}}{5}+\frac {4 a \arctanh \left (a x \right ) c \,d^{3} x^{7}}{7}+\frac {a \arctanh \left (a x \right ) d^{4} x^{9}}{9}-\frac {\frac {\left (-315 a^{8} c^{4}-420 a^{6} c^{3} d -378 a^{4} c^{2} d^{2}-180 a^{2} c \,d^{3}-35 d^{4}\right ) \ln \left (a x +1\right )}{2}-\frac {\left (315 a^{8} c^{4}+420 a^{6} c^{3} d +378 a^{4} c^{2} d^{2}+180 a^{2} c \,d^{3}+35 d^{4}\right ) \ln \left (a x -1\right )}{2}-\frac {35 d^{4} a^{6} x^{6}}{6}-\frac {35 d^{4} a^{2} x^{2}}{2}-\frac {35 d^{4} a^{4} x^{4}}{4}-\frac {35 d^{4} a^{8} x^{8}}{8}-45 a^{6} c \,d^{3} x^{4}-90 a^{4} c \,d^{3} x^{2}-30 c \,a^{8} d^{3} x^{6}-\frac {189 c^{2} a^{8} d^{2} x^{4}}{2}-210 c^{3} a^{8} d \,x^{2}-189 c^{2} a^{6} d^{2} x^{2}}{315 a^{8}}}{a}\) | \(301\) |
risch | \(\left (\frac {1}{18} d^{4} x^{9}+\frac {2}{7} d^{3} c \,x^{7}+\frac {3}{5} c^{2} d^{2} x^{5}+\frac {2}{3} d \,c^{3} x^{3}+\frac {1}{2} c^{4} x \right ) \ln \left (a x +1\right )-\frac {d^{4} x^{9} \ln \left (-a x +1\right )}{18}-\frac {2 c \,d^{3} x^{7} \ln \left (-a x +1\right )}{7}+\frac {d^{4} x^{8}}{72 a}-\frac {3 c^{2} d^{2} x^{5} \ln \left (-a x +1\right )}{5}+\frac {2 c \,d^{3} x^{6}}{21 a}-\frac {2 c^{3} d \,x^{3} \ln \left (-a x +1\right )}{3}+\frac {3 c^{2} d^{2} x^{4}}{10 a}+\frac {d^{4} x^{6}}{54 a^{3}}-\frac {c^{4} x \ln \left (-a x +1\right )}{2}+\frac {2 c^{3} d \,x^{2}}{3 a}+\frac {c \,d^{3} x^{4}}{7 a^{3}}+\frac {\ln \left (a^{2} x^{2}-1\right ) c^{4}}{2 a}+\frac {3 c^{2} d^{2} x^{2}}{5 a^{3}}+\frac {d^{4} x^{4}}{36 a^{5}}+\frac {2 \ln \left (a^{2} x^{2}-1\right ) c^{3} d}{3 a^{3}}+\frac {2 c \,d^{3} x^{2}}{7 a^{5}}+\frac {3 \ln \left (a^{2} x^{2}-1\right ) c^{2} d^{2}}{5 a^{5}}+\frac {d^{4} x^{2}}{18 a^{7}}+\frac {2 \ln \left (a^{2} x^{2}-1\right ) c \,d^{3}}{7 a^{7}}+\frac {\ln \left (a^{2} x^{2}-1\right ) d^{4}}{18 a^{9}}\) | \(346\) |
meijerg | \(-\frac {d^{4} \left (-\frac {a^{2} x^{2} \left (15 a^{6} x^{6}+20 a^{4} x^{4}+30 a^{2} x^{2}+60\right )}{270}+\frac {2 a^{10} x^{10} \left (\ln \left (1-\sqrt {a^{2} x^{2}}\right )-\ln \left (1+\sqrt {a^{2} x^{2}}\right )\right )}{9 \sqrt {a^{2} x^{2}}}-\frac {2 \ln \left (-a^{2} x^{2}+1\right )}{9}\right )}{4 a^{9}}+\frac {d^{3} c \left (\frac {a^{2} x^{2} \left (4 a^{4} x^{4}+6 a^{2} x^{2}+12\right )}{42}-\frac {2 a^{8} x^{8} \left (\ln \left (1-\sqrt {a^{2} x^{2}}\right )-\ln \left (1+\sqrt {a^{2} x^{2}}\right )\right )}{7 \sqrt {a^{2} x^{2}}}+\frac {2 \ln \left (-a^{2} x^{2}+1\right )}{7}\right )}{a^{7}}-\frac {3 d^{2} c^{2} \left (-\frac {a^{2} x^{2} \left (3 a^{2} x^{2}+6\right )}{15}+\frac {2 a^{6} x^{6} \left (\ln \left (1-\sqrt {a^{2} x^{2}}\right )-\ln \left (1+\sqrt {a^{2} x^{2}}\right )\right )}{5 \sqrt {a^{2} x^{2}}}-\frac {2 \ln \left (-a^{2} x^{2}+1\right )}{5}\right )}{2 a^{5}}+\frac {d \,c^{3} \left (\frac {2 a^{2} x^{2}}{3}-\frac {2 a^{4} x^{4} \left (\ln \left (1-\sqrt {a^{2} x^{2}}\right )-\ln \left (1+\sqrt {a^{2} x^{2}}\right )\right )}{3 \sqrt {a^{2} x^{2}}}+\frac {2 \ln \left (-a^{2} x^{2}+1\right )}{3}\right )}{a^{3}}-\frac {c^{4} \left (\frac {2 a^{2} x^{2} \left (\ln \left (1-\sqrt {a^{2} x^{2}}\right )-\ln \left (1+\sqrt {a^{2} x^{2}}\right )\right )}{\sqrt {a^{2} x^{2}}}-2 \ln \left (-a^{2} x^{2}+1\right )\right )}{4 a}\) | \(431\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.26, size = 276, normalized size = 1.13 \begin {gather*} \frac {1}{7560} \, a {\left (\frac {105 \, a^{6} d^{4} x^{8} + 20 \, {\left (36 \, a^{6} c d^{3} + 7 \, a^{4} d^{4}\right )} x^{6} + 6 \, {\left (378 \, a^{6} c^{2} d^{2} + 180 \, a^{4} c d^{3} + 35 \, a^{2} d^{4}\right )} x^{4} + 12 \, {\left (420 \, a^{6} c^{3} d + 378 \, a^{4} c^{2} d^{2} + 180 \, a^{2} c d^{3} + 35 \, d^{4}\right )} x^{2}}{a^{8}} + \frac {12 \, {\left (315 \, a^{8} c^{4} + 420 \, a^{6} c^{3} d + 378 \, a^{4} c^{2} d^{2} + 180 \, a^{2} c d^{3} + 35 \, d^{4}\right )} \log \left (a x + 1\right )}{a^{10}} + \frac {12 \, {\left (315 \, a^{8} c^{4} + 420 \, a^{6} c^{3} d + 378 \, a^{4} c^{2} d^{2} + 180 \, a^{2} c d^{3} + 35 \, d^{4}\right )} \log \left (a x - 1\right )}{a^{10}}\right )} + \frac {1}{315} \, {\left (35 \, d^{4} x^{9} + 180 \, c d^{3} x^{7} + 378 \, c^{2} d^{2} x^{5} + 420 \, c^{3} d x^{3} + 315 \, c^{4} x\right )} \operatorname {artanh}\left (a x\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.35, size = 248, normalized size = 1.01 \begin {gather*} \frac {105 \, a^{8} d^{4} x^{8} + 20 \, {\left (36 \, a^{8} c d^{3} + 7 \, a^{6} d^{4}\right )} x^{6} + 6 \, {\left (378 \, a^{8} c^{2} d^{2} + 180 \, a^{6} c d^{3} + 35 \, a^{4} d^{4}\right )} x^{4} + 12 \, {\left (420 \, a^{8} c^{3} d + 378 \, a^{6} c^{2} d^{2} + 180 \, a^{4} c d^{3} + 35 \, a^{2} d^{4}\right )} x^{2} + 12 \, {\left (315 \, a^{8} c^{4} + 420 \, a^{6} c^{3} d + 378 \, a^{4} c^{2} d^{2} + 180 \, a^{2} c d^{3} + 35 \, d^{4}\right )} \log \left (a^{2} x^{2} - 1\right ) + 12 \, {\left (35 \, a^{9} d^{4} x^{9} + 180 \, a^{9} c d^{3} x^{7} + 378 \, a^{9} c^{2} d^{2} x^{5} + 420 \, a^{9} c^{3} d x^{3} + 315 \, a^{9} c^{4} x\right )} \log \left (-\frac {a x + 1}{a x - 1}\right )}{7560 \, a^{9}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.85, size = 372, normalized size = 1.52 \begin {gather*} \begin {cases} c^{4} x \operatorname {atanh}{\left (a x \right )} + \frac {4 c^{3} d x^{3} \operatorname {atanh}{\left (a x \right )}}{3} + \frac {6 c^{2} d^{2} x^{5} \operatorname {atanh}{\left (a x \right )}}{5} + \frac {4 c d^{3} x^{7} \operatorname {atanh}{\left (a x \right )}}{7} + \frac {d^{4} x^{9} \operatorname {atanh}{\left (a x \right )}}{9} + \frac {c^{4} \log {\left (x - \frac {1}{a} \right )}}{a} + \frac {c^{4} \operatorname {atanh}{\left (a x \right )}}{a} + \frac {2 c^{3} d x^{2}}{3 a} + \frac {3 c^{2} d^{2} x^{4}}{10 a} + \frac {2 c d^{3} x^{6}}{21 a} + \frac {d^{4} x^{8}}{72 a} + \frac {4 c^{3} d \log {\left (x - \frac {1}{a} \right )}}{3 a^{3}} + \frac {4 c^{3} d \operatorname {atanh}{\left (a x \right )}}{3 a^{3}} + \frac {3 c^{2} d^{2} x^{2}}{5 a^{3}} + \frac {c d^{3} x^{4}}{7 a^{3}} + \frac {d^{4} x^{6}}{54 a^{3}} + \frac {6 c^{2} d^{2} \log {\left (x - \frac {1}{a} \right )}}{5 a^{5}} + \frac {6 c^{2} d^{2} \operatorname {atanh}{\left (a x \right )}}{5 a^{5}} + \frac {2 c d^{3} x^{2}}{7 a^{5}} + \frac {d^{4} x^{4}}{36 a^{5}} + \frac {4 c d^{3} \log {\left (x - \frac {1}{a} \right )}}{7 a^{7}} + \frac {4 c d^{3} \operatorname {atanh}{\left (a x \right )}}{7 a^{7}} + \frac {d^{4} x^{2}}{18 a^{7}} + \frac {d^{4} \log {\left (x - \frac {1}{a} \right )}}{9 a^{9}} + \frac {d^{4} \operatorname {atanh}{\left (a x \right )}}{9 a^{9}} & \text {for}\: a \neq 0 \\0 & \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. 1471 vs.
\(2 (227) = 454\).
time = 0.46, size = 1471, normalized size = 6.00 \begin {gather*} \text {Too large to display} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 1.28, size = 288, normalized size = 1.18 \begin {gather*} x^2\,\left (\frac {\frac {\frac {d^4}{9\,a^3}+\frac {4\,c\,d^3}{7\,a}}{a^2}+\frac {6\,c^2\,d^2}{5\,a}}{2\,a^2}+\frac {2\,c^3\,d}{3\,a}\right )+x^6\,\left (\frac {d^4}{54\,a^3}+\frac {2\,c\,d^3}{21\,a}\right )+\ln \left (a\,x+1\right )\,\left (\frac {c^4\,x}{2}+\frac {2\,c^3\,d\,x^3}{3}+\frac {3\,c^2\,d^2\,x^5}{5}+\frac {2\,c\,d^3\,x^7}{7}+\frac {d^4\,x^9}{18}\right )-\ln \left (1-a\,x\right )\,\left (\frac {c^4\,x}{2}+\frac {2\,c^3\,d\,x^3}{3}+\frac {3\,c^2\,d^2\,x^5}{5}+\frac {2\,c\,d^3\,x^7}{7}+\frac {d^4\,x^9}{18}\right )+x^4\,\left (\frac {\frac {d^4}{9\,a^3}+\frac {4\,c\,d^3}{7\,a}}{4\,a^2}+\frac {3\,c^2\,d^2}{10\,a}\right )+\frac {\ln \left (a^2\,x^2-1\right )\,\left (315\,a^8\,c^4+420\,a^6\,c^3\,d+378\,a^4\,c^2\,d^2+180\,a^2\,c\,d^3+35\,d^4\right )}{630\,a^9}+\frac {d^4\,x^8}{72\,a} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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