Optimal. Leaf size=72 \[ \frac {\text {ArcTan}\left (\sqrt {\tanh \left (a+b \log \left (c x^n\right )\right )}\right )}{b n}+\frac {\tanh ^{-1}\left (\sqrt {\tanh \left (a+b \log \left (c x^n\right )\right )}\right )}{b n}-\frac {2}{3 b n \tanh ^{\frac {3}{2}}\left (a+b \log \left (c x^n\right )\right )} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.04, antiderivative size = 72, normalized size of antiderivative = 1.00, number of steps
used = 7, number of rules used = 6, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.316, Rules used = {3555, 3557,
335, 218, 212, 209} \begin {gather*} \frac {\text {ArcTan}\left (\sqrt {\tanh \left (a+b \log \left (c x^n\right )\right )}\right )}{b n}+\frac {\tanh ^{-1}\left (\sqrt {\tanh \left (a+b \log \left (c x^n\right )\right )}\right )}{b n}-\frac {2}{3 b n \tanh ^{\frac {3}{2}}\left (a+b \log \left (c x^n\right )\right )} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 209
Rule 212
Rule 218
Rule 335
Rule 3555
Rule 3557
Rubi steps
\begin {align*} \int \frac {1}{x \tanh ^{\frac {5}{2}}\left (a+b \log \left (c x^n\right )\right )} \, dx &=\frac {\text {Subst}\left (\int \frac {1}{\tanh ^{\frac {5}{2}}(a+b x)} \, dx,x,\log \left (c x^n\right )\right )}{n}\\ &=-\frac {2}{3 b n \tanh ^{\frac {3}{2}}\left (a+b \log \left (c x^n\right )\right )}+\frac {\text {Subst}\left (\int \frac {1}{\sqrt {\tanh (a+b x)}} \, dx,x,\log \left (c x^n\right )\right )}{n}\\ &=-\frac {2}{3 b n \tanh ^{\frac {3}{2}}\left (a+b \log \left (c x^n\right )\right )}-\frac {\text {Subst}\left (\int \frac {1}{\sqrt {x} \left (-1+x^2\right )} \, dx,x,\tanh \left (a+b \log \left (c x^n\right )\right )\right )}{b n}\\ &=-\frac {2}{3 b n \tanh ^{\frac {3}{2}}\left (a+b \log \left (c x^n\right )\right )}-\frac {2 \text {Subst}\left (\int \frac {1}{-1+x^4} \, dx,x,\sqrt {\tanh \left (a+b \log \left (c x^n\right )\right )}\right )}{b n}\\ &=-\frac {2}{3 b n \tanh ^{\frac {3}{2}}\left (a+b \log \left (c x^n\right )\right )}+\frac {\text {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\sqrt {\tanh \left (a+b \log \left (c x^n\right )\right )}\right )}{b n}+\frac {\text {Subst}\left (\int \frac {1}{1+x^2} \, dx,x,\sqrt {\tanh \left (a+b \log \left (c x^n\right )\right )}\right )}{b n}\\ &=\frac {\tan ^{-1}\left (\sqrt {\tanh \left (a+b \log \left (c x^n\right )\right )}\right )}{b n}+\frac {\tanh ^{-1}\left (\sqrt {\tanh \left (a+b \log \left (c x^n\right )\right )}\right )}{b n}-\frac {2}{3 b n \tanh ^{\frac {3}{2}}\left (a+b \log \left (c x^n\right )\right )}\\ \end {align*}
________________________________________________________________________________________
Mathematica [C] Result contains higher order function than in optimal. Order 5 vs. order 3 in
optimal.
time = 0.15, size = 46, normalized size = 0.64 \begin {gather*} -\frac {2 \, _2F_1\left (-\frac {3}{4},1;\frac {1}{4};\tanh ^2\left (a+b \log \left (c x^n\right )\right )\right )}{3 b n \tanh ^{\frac {3}{2}}\left (a+b \log \left (c x^n\right )\right )} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A]
time = 4.20, size = 74, normalized size = 1.03
method | result | size |
derivativedivides | \(\frac {-\frac {2}{3 \tanh \left (a +b \ln \left (c \,x^{n}\right )\right )^{\frac {3}{2}}}+\frac {\ln \left (\sqrt {\tanh }\left (a +b \ln \left (c \,x^{n}\right )\right )+1\right )}{2}-\frac {\ln \left (\sqrt {\tanh }\left (a +b \ln \left (c \,x^{n}\right )\right )-1\right )}{2}+\arctan \left (\sqrt {\tanh }\left (a +b \ln \left (c \,x^{n}\right )\right )\right )}{n b}\) | \(74\) |
default | \(\frac {-\frac {2}{3 \tanh \left (a +b \ln \left (c \,x^{n}\right )\right )^{\frac {3}{2}}}+\frac {\ln \left (\sqrt {\tanh }\left (a +b \ln \left (c \,x^{n}\right )\right )+1\right )}{2}-\frac {\ln \left (\sqrt {\tanh }\left (a +b \ln \left (c \,x^{n}\right )\right )-1\right )}{2}+\arctan \left (\sqrt {\tanh }\left (a +b \ln \left (c \,x^{n}\right )\right )\right )}{n b}\) | \(74\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 1110 vs.
\(2 (64) = 128\).
time = 0.37, size = 1110, normalized size = 15.42 \begin {gather*} \text {Too large to display} \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 {1}{x \tanh ^{\frac {5}{2}}{\left (a + b \log {\left (c x^{n} \right )} \right )}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [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.
[In]
[Out]
________________________________________________________________________________________
Mupad [B]
time = 2.48, size = 64, normalized size = 0.89 \begin {gather*} \frac {\mathrm {atan}\left (\sqrt {\mathrm {tanh}\left (a+b\,\ln \left (c\,x^n\right )\right )}\right )}{b\,n}+\frac {\mathrm {atanh}\left (\sqrt {\mathrm {tanh}\left (a+b\,\ln \left (c\,x^n\right )\right )}\right )}{b\,n}-\frac {2}{3\,b\,n\,{\mathrm {tanh}\left (a+b\,\ln \left (c\,x^n\right )\right )}^{3/2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________