Optimal. Leaf size=210 \[ \frac {\left (4 \sqrt {a}-3 \sqrt {b}\right ) \tanh ^{-1}\left (\frac {\sqrt {\sqrt {a}-\sqrt {b}} \tanh (c+d x)}{\sqrt [4]{a}}\right )}{8 a^{7/4} \left (\sqrt {a}-\sqrt {b}\right )^{3/2} d}+\frac {\left (4 \sqrt {a}+3 \sqrt {b}\right ) \tanh ^{-1}\left (\frac {\sqrt {\sqrt {a}+\sqrt {b}} \tanh (c+d x)}{\sqrt [4]{a}}\right )}{8 a^{7/4} \left (\sqrt {a}+\sqrt {b}\right )^{3/2} d}-\frac {b \tanh (c+d x) \left (1-2 \tanh ^2(c+d x)\right )}{4 a (a-b) d \left (a-2 a \tanh ^2(c+d x)+(a-b) \tanh ^4(c+d x)\right )} \]
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Rubi [A]
time = 0.18, antiderivative size = 210, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 4, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.267, Rules used = {3288, 1219,
1180, 214} \begin {gather*} \frac {\left (4 \sqrt {a}-3 \sqrt {b}\right ) \tanh ^{-1}\left (\frac {\sqrt {\sqrt {a}-\sqrt {b}} \tanh (c+d x)}{\sqrt [4]{a}}\right )}{8 a^{7/4} d \left (\sqrt {a}-\sqrt {b}\right )^{3/2}}+\frac {\left (4 \sqrt {a}+3 \sqrt {b}\right ) \tanh ^{-1}\left (\frac {\sqrt {\sqrt {a}+\sqrt {b}} \tanh (c+d x)}{\sqrt [4]{a}}\right )}{8 a^{7/4} d \left (\sqrt {a}+\sqrt {b}\right )^{3/2}}-\frac {b \tanh (c+d x) \left (1-2 \tanh ^2(c+d x)\right )}{4 a d (a-b) \left ((a-b) \tanh ^4(c+d x)-2 a \tanh ^2(c+d x)+a\right )} \end {gather*}
Antiderivative was successfully verified.
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Rule 214
Rule 1180
Rule 1219
Rule 3288
Rubi steps
\begin {align*} \int \frac {1}{\left (a-b \sinh ^4(c+d x)\right )^2} \, dx &=\frac {\text {Subst}\left (\int \frac {\left (1-x^2\right )^3}{\left (a-2 a x^2+(a-b) x^4\right )^2} \, dx,x,\tanh (c+d x)\right )}{d}\\ &=-\frac {b \tanh (c+d x) \left (1-2 \tanh ^2(c+d x)\right )}{4 a (a-b) d \left (a-2 a \tanh ^2(c+d x)+(a-b) \tanh ^4(c+d x)\right )}-\frac {\text {Subst}\left (\int \frac {-\frac {2 a (4 a-3 b) b}{a-b}+\frac {4 a (2 a-b) b x^2}{a-b}}{a-2 a x^2+(a-b) x^4} \, dx,x,\tanh (c+d x)\right )}{8 a^2 b d}\\ &=-\frac {b \tanh (c+d x) \left (1-2 \tanh ^2(c+d x)\right )}{4 a (a-b) d \left (a-2 a \tanh ^2(c+d x)+(a-b) \tanh ^4(c+d x)\right )}-\frac {\left (4 a-\sqrt {a} \sqrt {b}-3 b\right ) \text {Subst}\left (\int \frac {1}{-a+\sqrt {a} \sqrt {b}+(a-b) x^2} \, dx,x,\tanh (c+d x)\right )}{8 a^{3/2} \left (\sqrt {a}+\sqrt {b}\right ) d}-\frac {\left (4 a+\sqrt {a} \sqrt {b}-3 b\right ) \text {Subst}\left (\int \frac {1}{-a-\sqrt {a} \sqrt {b}+(a-b) x^2} \, dx,x,\tanh (c+d x)\right )}{8 a^{3/2} \left (\sqrt {a}-\sqrt {b}\right ) d}\\ &=\frac {\left (4 \sqrt {a}-3 \sqrt {b}\right ) \tanh ^{-1}\left (\frac {\sqrt {\sqrt {a}-\sqrt {b}} \tanh (c+d x)}{\sqrt [4]{a}}\right )}{8 a^{7/4} \left (\sqrt {a}-\sqrt {b}\right )^{3/2} d}+\frac {\left (4 \sqrt {a}+3 \sqrt {b}\right ) \tanh ^{-1}\left (\frac {\sqrt {\sqrt {a}+\sqrt {b}} \tanh (c+d x)}{\sqrt [4]{a}}\right )}{8 a^{7/4} \left (\sqrt {a}+\sqrt {b}\right )^{3/2} d}-\frac {b \tanh (c+d x) \left (1-2 \tanh ^2(c+d x)\right )}{4 a (a-b) d \left (a-2 a \tanh ^2(c+d x)+(a-b) \tanh ^4(c+d x)\right )}\\ \end {align*}
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Mathematica [A]
time = 2.05, size = 230, normalized size = 1.10 \begin {gather*} \frac {-\frac {\left (4 a+\sqrt {a} \sqrt {b}-3 b\right ) \text {ArcTan}\left (\frac {\left (\sqrt {a}-\sqrt {b}\right ) \tanh (c+d x)}{\sqrt {-a+\sqrt {a} \sqrt {b}}}\right )}{\sqrt {-a+\sqrt {a} \sqrt {b}}}+\frac {\left (4 a-\sqrt {a} \sqrt {b}-3 b\right ) \tanh ^{-1}\left (\frac {\left (\sqrt {a}+\sqrt {b}\right ) \tanh (c+d x)}{\sqrt {a+\sqrt {a} \sqrt {b}}}\right )}{\sqrt {a+\sqrt {a} \sqrt {b}}}+\frac {2 \sqrt {a} b (-6 \sinh (2 (c+d x))+\sinh (4 (c+d x)))}{8 a-3 b+4 b \cosh (2 (c+d x))-b \cosh (4 (c+d x))}}{8 a^{3/2} (a-b) d} \end {gather*}
Antiderivative was successfully verified.
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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order
3.
time = 2.05, size = 307, normalized size = 1.46
method | result | size |
derivativedivides | \(\frac {-\frac {2 \left (\frac {b \left (\tanh ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4 a \left (a -b \right )}-\frac {5 b \left (\tanh ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4 a \left (a -b \right )}-\frac {5 b \left (\tanh ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4 a \left (a -b \right )}+\frac {b \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{4 a \left (a -b \right )}\right )}{a \left (\tanh ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-4 a \left (\tanh ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+6 a \left (\tanh ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-16 b \left (\tanh ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-4 a \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+a}-\frac {\munderset {\textit {\_R} =\RootOf \left (a \,\textit {\_Z}^{8}-4 a \,\textit {\_Z}^{6}+\left (6 a -16 b \right ) \textit {\_Z}^{4}-4 a \,\textit {\_Z}^{2}+a \right )}{\sum }\frac {\left (\left (4 a -3 b \right ) \textit {\_R}^{6}+\left (-12 a +5 b \right ) \textit {\_R}^{4}+\left (12 a -5 b \right ) \textit {\_R}^{2}-4 a +3 b \right ) \ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-\textit {\_R} \right )}{\textit {\_R}^{7} a -3 \textit {\_R}^{5} a +3 \textit {\_R}^{3} a -8 \textit {\_R}^{3} b -\textit {\_R} a}}{16 a \left (a -b \right )}}{d}\) | \(307\) |
default | \(\frac {-\frac {2 \left (\frac {b \left (\tanh ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4 a \left (a -b \right )}-\frac {5 b \left (\tanh ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4 a \left (a -b \right )}-\frac {5 b \left (\tanh ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4 a \left (a -b \right )}+\frac {b \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{4 a \left (a -b \right )}\right )}{a \left (\tanh ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-4 a \left (\tanh ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+6 a \left (\tanh ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-16 b \left (\tanh ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-4 a \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+a}-\frac {\munderset {\textit {\_R} =\RootOf \left (a \,\textit {\_Z}^{8}-4 a \,\textit {\_Z}^{6}+\left (6 a -16 b \right ) \textit {\_Z}^{4}-4 a \,\textit {\_Z}^{2}+a \right )}{\sum }\frac {\left (\left (4 a -3 b \right ) \textit {\_R}^{6}+\left (-12 a +5 b \right ) \textit {\_R}^{4}+\left (12 a -5 b \right ) \textit {\_R}^{2}-4 a +3 b \right ) \ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-\textit {\_R} \right )}{\textit {\_R}^{7} a -3 \textit {\_R}^{5} a +3 \textit {\_R}^{3} a -8 \textit {\_R}^{3} b -\textit {\_R} a}}{16 a \left (a -b \right )}}{d}\) | \(307\) |
risch | \(\frac {-b \,{\mathrm e}^{6 d x +6 c}+8 a \,{\mathrm e}^{4 d x +4 c}-3 b \,{\mathrm e}^{4 d x +4 c}+5 b \,{\mathrm e}^{2 d x +2 c}-b}{2 a d \left (a -b \right ) \left (-b \,{\mathrm e}^{8 d x +8 c}+4 b \,{\mathrm e}^{6 d x +6 c}+16 a \,{\mathrm e}^{4 d x +4 c}-6 b \,{\mathrm e}^{4 d x +4 c}+4 b \,{\mathrm e}^{2 d x +2 c}-b \right )}+\left (\munderset {\textit {\_R} =\RootOf \left (\left (65536 a^{10} d^{4}-196608 a^{9} b \,d^{4}+196608 a^{8} b^{2} d^{4}-65536 a^{7} b^{3} d^{4}\right ) \textit {\_Z}^{4}+\left (-8192 a^{6} d^{2}+7680 a^{5} b \,d^{2}-1536 b^{2} d^{2} a^{4}\right ) \textit {\_Z}^{2}+256 a^{2}-288 a b +81 b^{2}\right )}{\sum }\textit {\_R} \ln \left ({\mathrm e}^{2 d x +2 c}+\left (-\frac {32768 d^{3} a^{10}}{384 a^{3} b -680 a^{2} b^{2}+405 a \,b^{3}-81 b^{4}}+\frac {114688 a^{9} b \,d^{3}}{384 a^{3} b -680 a^{2} b^{2}+405 a \,b^{3}-81 b^{4}}-\frac {147456 a^{8} b^{2} d^{3}}{384 a^{3} b -680 a^{2} b^{2}+405 a \,b^{3}-81 b^{4}}+\frac {81920 a^{7} b^{3} d^{3}}{384 a^{3} b -680 a^{2} b^{2}+405 a \,b^{3}-81 b^{4}}-\frac {16384 a^{6} b^{4} d^{3}}{384 a^{3} b -680 a^{2} b^{2}+405 a \,b^{3}-81 b^{4}}\right ) \textit {\_R}^{3}+\left (\frac {8192 d^{2} a^{8}}{384 a^{3} b -680 a^{2} b^{2}+405 a \,b^{3}-81 b^{4}}-\frac {29184 a^{7} b \,d^{2}}{384 a^{3} b -680 a^{2} b^{2}+405 a \,b^{3}-81 b^{4}}+\frac {38400 a^{6} b^{2} d^{2}}{384 a^{3} b -680 a^{2} b^{2}+405 a \,b^{3}-81 b^{4}}-\frac {22016 a^{5} b^{3} d^{2}}{384 a^{3} b -680 a^{2} b^{2}+405 a \,b^{3}-81 b^{4}}+\frac {4608 a^{4} b^{4} d^{2}}{384 a^{3} b -680 a^{2} b^{2}+405 a \,b^{3}-81 b^{4}}\right ) \textit {\_R}^{2}+\left (\frac {2048 d \,a^{6}}{384 a^{3} b -680 a^{2} b^{2}+405 a \,b^{3}-81 b^{4}}+\frac {896 a^{5} b d}{384 a^{3} b -680 a^{2} b^{2}+405 a \,b^{3}-81 b^{4}}-\frac {5600 a^{4} b^{2} d}{384 a^{3} b -680 a^{2} b^{2}+405 a \,b^{3}-81 b^{4}}+\frac {4032 a^{3} b^{3} d}{384 a^{3} b -680 a^{2} b^{2}+405 a \,b^{3}-81 b^{4}}-\frac {864 d \,b^{4} a^{2}}{384 a^{3} b -680 a^{2} b^{2}+405 a \,b^{3}-81 b^{4}}\right ) \textit {\_R} -\frac {512 a^{4}}{384 a^{3} b -680 a^{2} b^{2}+405 a \,b^{3}-81 b^{4}}+\frac {384 a^{3} b}{384 a^{3} b -680 a^{2} b^{2}+405 a \,b^{3}-81 b^{4}}+\frac {314 a^{2} b^{2}}{384 a^{3} b -680 a^{2} b^{2}+405 a \,b^{3}-81 b^{4}}-\frac {351 a \,b^{3}}{384 a^{3} b -680 a^{2} b^{2}+405 a \,b^{3}-81 b^{4}}+\frac {81 b^{4}}{384 a^{3} b -680 a^{2} b^{2}+405 a \,b^{3}-81 b^{4}}\right )\right )\) | \(987\) |
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. 6522 vs.
\(2 (164) = 328\).
time = 0.64, size = 6522, normalized size = 31.06 \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(-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.
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Giac [A]
time = 0.46, size = 127, normalized size = 0.60 \begin {gather*} \frac {b e^{\left (6 \, d x + 6 \, c\right )} - 8 \, a e^{\left (4 \, d x + 4 \, c\right )} + 3 \, b e^{\left (4 \, d x + 4 \, c\right )} - 5 \, b e^{\left (2 \, d x + 2 \, c\right )} + b}{2 \, {\left (a^{2} - a b\right )} {\left (b e^{\left (8 \, d x + 8 \, c\right )} - 4 \, b e^{\left (6 \, d x + 6 \, c\right )} - 16 \, a e^{\left (4 \, d x + 4 \, c\right )} + 6 \, b e^{\left (4 \, d x + 4 \, c\right )} - 4 \, b e^{\left (2 \, d x + 2 \, c\right )} + b\right )} d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {1}{{\left (a-b\,{\mathrm {sinh}\left (c+d\,x\right )}^4\right )}^2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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