Optimal. Leaf size=99 \[ -\frac {2 x^4}{3 b \tanh ^{-1}(\tanh (a+b x))^{3/2}}-\frac {16 x^3}{3 b^2 \sqrt {\tanh ^{-1}(\tanh (a+b x))}}+\frac {32 x^2 \sqrt {\tanh ^{-1}(\tanh (a+b x))}}{b^3}-\frac {128 x \tanh ^{-1}(\tanh (a+b x))^{3/2}}{3 b^4}+\frac {256 \tanh ^{-1}(\tanh (a+b x))^{5/2}}{15 b^5} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.05, antiderivative size = 99, normalized size of antiderivative = 1.00, number of steps
used = 6, number of rules used = 3, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {2199, 2188, 30}
\begin {gather*} \frac {256 \tanh ^{-1}(\tanh (a+b x))^{5/2}}{15 b^5}-\frac {128 x \tanh ^{-1}(\tanh (a+b x))^{3/2}}{3 b^4}+\frac {32 x^2 \sqrt {\tanh ^{-1}(\tanh (a+b x))}}{b^3}-\frac {16 x^3}{3 b^2 \sqrt {\tanh ^{-1}(\tanh (a+b x))}}-\frac {2 x^4}{3 b \tanh ^{-1}(\tanh (a+b x))^{3/2}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 30
Rule 2188
Rule 2199
Rubi steps
\begin {align*} \int \frac {x^4}{\tanh ^{-1}(\tanh (a+b x))^{5/2}} \, dx &=-\frac {2 x^4}{3 b \tanh ^{-1}(\tanh (a+b x))^{3/2}}+\frac {8 \int \frac {x^3}{\tanh ^{-1}(\tanh (a+b x))^{3/2}} \, dx}{3 b}\\ &=-\frac {2 x^4}{3 b \tanh ^{-1}(\tanh (a+b x))^{3/2}}-\frac {16 x^3}{3 b^2 \sqrt {\tanh ^{-1}(\tanh (a+b x))}}+\frac {16 \int \frac {x^2}{\sqrt {\tanh ^{-1}(\tanh (a+b x))}} \, dx}{b^2}\\ &=-\frac {2 x^4}{3 b \tanh ^{-1}(\tanh (a+b x))^{3/2}}-\frac {16 x^3}{3 b^2 \sqrt {\tanh ^{-1}(\tanh (a+b x))}}+\frac {32 x^2 \sqrt {\tanh ^{-1}(\tanh (a+b x))}}{b^3}-\frac {64 \int x \sqrt {\tanh ^{-1}(\tanh (a+b x))} \, dx}{b^3}\\ &=-\frac {2 x^4}{3 b \tanh ^{-1}(\tanh (a+b x))^{3/2}}-\frac {16 x^3}{3 b^2 \sqrt {\tanh ^{-1}(\tanh (a+b x))}}+\frac {32 x^2 \sqrt {\tanh ^{-1}(\tanh (a+b x))}}{b^3}-\frac {128 x \tanh ^{-1}(\tanh (a+b x))^{3/2}}{3 b^4}+\frac {128 \int \tanh ^{-1}(\tanh (a+b x))^{3/2} \, dx}{3 b^4}\\ &=-\frac {2 x^4}{3 b \tanh ^{-1}(\tanh (a+b x))^{3/2}}-\frac {16 x^3}{3 b^2 \sqrt {\tanh ^{-1}(\tanh (a+b x))}}+\frac {32 x^2 \sqrt {\tanh ^{-1}(\tanh (a+b x))}}{b^3}-\frac {128 x \tanh ^{-1}(\tanh (a+b x))^{3/2}}{3 b^4}+\frac {128 \text {Subst}\left (\int x^{3/2} \, dx,x,\tanh ^{-1}(\tanh (a+b x))\right )}{3 b^5}\\ &=-\frac {2 x^4}{3 b \tanh ^{-1}(\tanh (a+b x))^{3/2}}-\frac {16 x^3}{3 b^2 \sqrt {\tanh ^{-1}(\tanh (a+b x))}}+\frac {32 x^2 \sqrt {\tanh ^{-1}(\tanh (a+b x))}}{b^3}-\frac {128 x \tanh ^{-1}(\tanh (a+b x))^{3/2}}{3 b^4}+\frac {256 \tanh ^{-1}(\tanh (a+b x))^{5/2}}{15 b^5}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 0.03, size = 83, normalized size = 0.84 \begin {gather*} -\frac {2 \left (5 b^4 x^4+40 b^3 x^3 \tanh ^{-1}(\tanh (a+b x))-240 b^2 x^2 \tanh ^{-1}(\tanh (a+b x))^2+320 b x \tanh ^{-1}(\tanh (a+b x))^3-128 \tanh ^{-1}(\tanh (a+b x))^4\right )}{15 b^5 \tanh ^{-1}(\tanh (a+b x))^{3/2}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(294\) vs.
\(2(81)=162\).
time = 0.07, size = 295, normalized size = 2.98
method | result | size |
default | \(\frac {\frac {2 \arctanh \left (\tanh \left (b x +a \right )\right )^{\frac {5}{2}}}{5}-\frac {8 \arctanh \left (\tanh \left (b x +a \right )\right )^{\frac {3}{2}} a}{3}-\frac {8 \arctanh \left (\tanh \left (b x +a \right )\right )^{\frac {3}{2}} \left (\arctanh \left (\tanh \left (b x +a \right )\right )-b x -a \right )}{3}+12 \sqrt {\arctanh \left (\tanh \left (b x +a \right )\right )}\, a^{2}+24 a \left (\arctanh \left (\tanh \left (b x +a \right )\right )-b x -a \right ) \sqrt {\arctanh \left (\tanh \left (b x +a \right )\right )}+12 \left (\arctanh \left (\tanh \left (b x +a \right )\right )-b x -a \right )^{2} \sqrt {\arctanh \left (\tanh \left (b x +a \right )\right )}-\frac {2 \left (-4 a^{3}-12 a^{2} \left (\arctanh \left (\tanh \left (b x +a \right )\right )-b x -a \right )-12 a \left (\arctanh \left (\tanh \left (b x +a \right )\right )-b x -a \right )^{2}-4 \left (\arctanh \left (\tanh \left (b x +a \right )\right )-b x -a \right )^{3}\right )}{\sqrt {\arctanh \left (\tanh \left (b x +a \right )\right )}}-\frac {2 \left (a^{4}+4 a^{3} \left (\arctanh \left (\tanh \left (b x +a \right )\right )-b x -a \right )+6 a^{2} \left (\arctanh \left (\tanh \left (b x +a \right )\right )-b x -a \right )^{2}+4 a \left (\arctanh \left (\tanh \left (b x +a \right )\right )-b x -a \right )^{3}+\left (\arctanh \left (\tanh \left (b x +a \right )\right )-b x -a \right )^{4}\right )}{3 \arctanh \left (\tanh \left (b x +a \right )\right )^{\frac {3}{2}}}}{b^{5}}\) | \(295\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A]
time = 0.53, size = 64, normalized size = 0.65 \begin {gather*} \frac {2 \, {\left (3 \, b^{5} x^{5} - 5 \, a b^{4} x^{4} + 40 \, a^{2} b^{3} x^{3} + 240 \, a^{3} b^{2} x^{2} + 320 \, a^{4} b x + 128 \, a^{5}\right )}}{15 \, {\left (b x + a\right )}^{\frac {5}{2}} b^{5}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A]
time = 0.35, size = 74, normalized size = 0.75 \begin {gather*} \frac {2 \, {\left (3 \, b^{4} x^{4} - 8 \, a b^{3} x^{3} + 48 \, a^{2} b^{2} x^{2} + 192 \, a^{3} b x + 128 \, a^{4}\right )} \sqrt {b x + a}}{15 \, {\left (b^{7} x^{2} + 2 \, a b^{6} x + a^{2} b^{5}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^{4}}{\operatorname {atanh}^{\frac {5}{2}}{\left (\tanh {\left (a + b x \right )} \right )}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A]
time = 0.38, size = 75, normalized size = 0.76 \begin {gather*} \frac {2 \, {\left (12 \, {\left (b x + a\right )} a^{3} - a^{4}\right )}}{3 \, {\left (b x + a\right )}^{\frac {3}{2}} b^{5}} + \frac {2 \, {\left (3 \, {\left (b x + a\right )}^{\frac {5}{2}} b^{20} - 20 \, {\left (b x + a\right )}^{\frac {3}{2}} a b^{20} + 90 \, \sqrt {b x + a} a^{2} b^{20}\right )}}{15 \, b^{25}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Mupad [B]
time = 1.42, size = 817, normalized size = 8.25 \begin {gather*} \frac {\sqrt {\frac {\ln \left (\frac {2\,{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}+1}\right )}{2}-\frac {\ln \left (\frac {2}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}+1}\right )}{2}}\,\left (\frac {3\,{\left (\ln \left (\frac {2}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}+1}\right )-\ln \left (\frac {2\,{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}+1}\right )+2\,b\,x\right )}^2}{2\,b^4}+\frac {2\,\left (\frac {2\,\left (\ln \left (\frac {2}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}+1}\right )-\ln \left (\frac {2\,{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}+1}\right )+2\,b\,x\right )}{b^3}+\frac {8\,\left (\frac {\ln \left (\frac {2}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}+1}\right )}{2}-\frac {\ln \left (\frac {2\,{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}+1}\right )}{2}+b\,x\right )}{5\,b^3}\right )\,\left (\frac {\ln \left (\frac {2}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}+1}\right )}{2}-\frac {\ln \left (\frac {2\,{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}+1}\right )}{2}+b\,x\right )}{3\,b}\right )}{b}+\frac {2\,x^2\,\sqrt {\frac {\ln \left (\frac {2\,{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}+1}\right )}{2}-\frac {\ln \left (\frac {2}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}+1}\right )}{2}}}{5\,b^3}+\frac {x\,\left (\frac {2\,\left (\ln \left (\frac {2}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}+1}\right )-\ln \left (\frac {2\,{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}+1}\right )+2\,b\,x\right )}{b^3}+\frac {8\,\left (\frac {\ln \left (\frac {2}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}+1}\right )}{2}-\frac {\ln \left (\frac {2\,{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}+1}\right )}{2}+b\,x\right )}{5\,b^3}\right )\,\sqrt {\frac {\ln \left (\frac {2\,{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}+1}\right )}{2}-\frac {\ln \left (\frac {2}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}+1}\right )}{2}}}{3\,b}-\frac {2\,\sqrt {\frac {\ln \left (\frac {2\,{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}+1}\right )}{2}-\frac {\ln \left (\frac {2}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}+1}\right )}{2}}\,{\left (\ln \left (\frac {2}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}+1}\right )-\ln \left (\frac {2\,{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}+1}\right )+2\,b\,x\right )}^3}{b^5\,\left (\ln \left (\frac {2\,{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}+1}\right )-\ln \left (\frac {2}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}+1}\right )\right )}-\frac {\sqrt {\frac {\ln \left (\frac {2\,{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}+1}\right )}{2}-\frac {\ln \left (\frac {2}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}+1}\right )}{2}}\,{\left (\ln \left (\frac {2}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}+1}\right )-\ln \left (\frac {2\,{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}+1}\right )+2\,b\,x\right )}^4}{6\,b^5\,{\left (\ln \left (\frac {2\,{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}+1}\right )-\ln \left (\frac {2}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}+1}\right )\right )}^2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________