Optimal. Leaf size=79 \[ \frac {\text {ArcTan}\left (\frac {\sqrt {b \tanh (c+d x)}}{\sqrt {b}}\right )}{b^{5/2} d}+\frac {\tanh ^{-1}\left (\frac {\sqrt {b \tanh (c+d x)}}{\sqrt {b}}\right )}{b^{5/2} d}-\frac {2}{3 b d (b \tanh (c+d x))^{3/2}} \]
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Rubi [A]
time = 0.04, antiderivative size = 79, normalized size of antiderivative = 1.00, number of steps
used = 6, number of rules used = 6, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {3555, 3557,
335, 218, 212, 209} \begin {gather*} \frac {\text {ArcTan}\left (\frac {\sqrt {b \tanh (c+d x)}}{\sqrt {b}}\right )}{b^{5/2} d}+\frac {\tanh ^{-1}\left (\frac {\sqrt {b \tanh (c+d x)}}{\sqrt {b}}\right )}{b^{5/2} d}-\frac {2}{3 b d (b \tanh (c+d x))^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 209
Rule 212
Rule 218
Rule 335
Rule 3555
Rule 3557
Rubi steps
\begin {align*} \int \frac {1}{(b \tanh (c+d x))^{5/2}} \, dx &=-\frac {2}{3 b d (b \tanh (c+d x))^{3/2}}+\frac {\int \frac {1}{\sqrt {b \tanh (c+d x)}} \, dx}{b^2}\\ &=-\frac {2}{3 b d (b \tanh (c+d x))^{3/2}}-\frac {\text {Subst}\left (\int \frac {1}{\sqrt {x} \left (-b^2+x^2\right )} \, dx,x,b \tanh (c+d x)\right )}{b d}\\ &=-\frac {2}{3 b d (b \tanh (c+d x))^{3/2}}-\frac {2 \text {Subst}\left (\int \frac {1}{-b^2+x^4} \, dx,x,\sqrt {b \tanh (c+d x)}\right )}{b d}\\ &=-\frac {2}{3 b d (b \tanh (c+d x))^{3/2}}+\frac {\text {Subst}\left (\int \frac {1}{b-x^2} \, dx,x,\sqrt {b \tanh (c+d x)}\right )}{b^2 d}+\frac {\text {Subst}\left (\int \frac {1}{b+x^2} \, dx,x,\sqrt {b \tanh (c+d x)}\right )}{b^2 d}\\ &=\frac {\tan ^{-1}\left (\frac {\sqrt {b \tanh (c+d x)}}{\sqrt {b}}\right )}{b^{5/2} d}+\frac {\tanh ^{-1}\left (\frac {\sqrt {b \tanh (c+d x)}}{\sqrt {b}}\right )}{b^{5/2} d}-\frac {2}{3 b d (b \tanh (c+d x))^{3/2}}\\ \end {align*}
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Mathematica [C] Result contains higher order function than in optimal. Order 5 vs. order 3 in
optimal.
time = 0.05, size = 38, normalized size = 0.48 \begin {gather*} -\frac {2 \, _2F_1\left (-\frac {3}{4},1;\frac {1}{4};\tanh ^2(c+d x)\right )}{3 b d (b \tanh (c+d x))^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 2.01, size = 63, normalized size = 0.80
method | result | size |
derivativedivides | \(-\frac {2 b \left (-\frac {\arctan \left (\frac {\sqrt {b \tanh \left (d x +c \right )}}{\sqrt {b}}\right )}{2 b^{\frac {7}{2}}}+\frac {1}{3 b^{2} \left (b \tanh \left (d x +c \right )\right )^{\frac {3}{2}}}-\frac {\arctanh \left (\frac {\sqrt {b \tanh \left (d x +c \right )}}{\sqrt {b}}\right )}{2 b^{\frac {7}{2}}}\right )}{d}\) | \(63\) |
default | \(-\frac {2 b \left (-\frac {\arctan \left (\frac {\sqrt {b \tanh \left (d x +c \right )}}{\sqrt {b}}\right )}{2 b^{\frac {7}{2}}}+\frac {1}{3 b^{2} \left (b \tanh \left (d x +c \right )\right )^{\frac {3}{2}}}-\frac {\arctanh \left (\frac {\sqrt {b \tanh \left (d x +c \right )}}{\sqrt {b}}\right )}{2 b^{\frac {7}{2}}}\right )}{d}\) | \(63\) |
Verification of antiderivative is not currently implemented for this CAS.
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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.
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 691 vs.
\(2 (63) = 126\).
time = 0.43, size = 1436, normalized size = 18.18 \begin {gather*} \text {Too large to display} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{\left (b \tanh {\left (c + d x \right )}\right )^{\frac {5}{2}}}\, dx \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. 173 vs.
\(2 (63) = 126\).
time = 0.57, size = 173, normalized size = 2.19 \begin {gather*} \frac {\frac {6 \, \arctan \left (-\frac {\sqrt {b} e^{\left (2 \, d x + 2 \, c\right )} - \sqrt {b e^{\left (4 \, d x + 4 \, c\right )} - b}}{\sqrt {b}}\right )}{b^{\frac {5}{2}}} - \frac {3 \, \log \left ({\left | -\sqrt {b} e^{\left (2 \, d x + 2 \, c\right )} + \sqrt {b e^{\left (4 \, d x + 4 \, c\right )} - b} \right |}\right )}{b^{\frac {5}{2}}} + \frac {8 \, {\left (3 \, {\left (\sqrt {b} e^{\left (2 \, d x + 2 \, c\right )} - \sqrt {b e^{\left (4 \, d x + 4 \, c\right )} - b}\right )}^{2} + b\right )}}{{\left (\sqrt {b} e^{\left (2 \, d x + 2 \, c\right )} - \sqrt {b e^{\left (4 \, d x + 4 \, c\right )} - b} - \sqrt {b}\right )}^{3} b^{2}}}{6 \, d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 1.36, size = 63, normalized size = 0.80 \begin {gather*} \frac {\mathrm {atan}\left (\frac {\sqrt {b\,\mathrm {tanh}\left (c+d\,x\right )}}{\sqrt {b}}\right )}{b^{5/2}\,d}-\frac {2}{3\,b\,d\,{\left (b\,\mathrm {tanh}\left (c+d\,x\right )\right )}^{3/2}}+\frac {\mathrm {atanh}\left (\frac {\sqrt {b\,\mathrm {tanh}\left (c+d\,x\right )}}{\sqrt {b}}\right )}{b^{5/2}\,d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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