Optimal. Leaf size=115 \[ \frac {\tanh ^{-1}\left (\frac {\sqrt {\sqrt {a}-\sqrt {b}} \tanh (c+d x)}{\sqrt [4]{a}}\right )}{2 a^{3/4} \sqrt {\sqrt {a}-\sqrt {b}} d}+\frac {\tanh ^{-1}\left (\frac {\sqrt {\sqrt {a}+\sqrt {b}} \tanh (c+d x)}{\sqrt [4]{a}}\right )}{2 a^{3/4} \sqrt {\sqrt {a}+\sqrt {b}} d} \]
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Rubi [A]
time = 0.07, antiderivative size = 115, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 3, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {3288, 1180,
214} \begin {gather*} \frac {\tanh ^{-1}\left (\frac {\sqrt {\sqrt {a}-\sqrt {b}} \tanh (c+d x)}{\sqrt [4]{a}}\right )}{2 a^{3/4} d \sqrt {\sqrt {a}-\sqrt {b}}}+\frac {\tanh ^{-1}\left (\frac {\sqrt {\sqrt {a}+\sqrt {b}} \tanh (c+d x)}{\sqrt [4]{a}}\right )}{2 a^{3/4} d \sqrt {\sqrt {a}+\sqrt {b}}} \end {gather*}
Antiderivative was successfully verified.
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Rule 214
Rule 1180
Rule 3288
Rubi steps
\begin {align*} \int \frac {1}{a-b \sinh ^4(c+d x)} \, dx &=\frac {\text {Subst}\left (\int \frac {1-x^2}{a-2 a x^2+(a-b) x^4} \, dx,x,\tanh (c+d x)\right )}{d}\\ &=-\frac {\left (1-\frac {\sqrt {b}}{\sqrt {a}}\right ) \text {Subst}\left (\int \frac {1}{-a+\sqrt {a} \sqrt {b}+(a-b) x^2} \, dx,x,\tanh (c+d x)\right )}{2 d}-\frac {\left (1+\frac {\sqrt {b}}{\sqrt {a}}\right ) \text {Subst}\left (\int \frac {1}{-a-\sqrt {a} \sqrt {b}+(a-b) x^2} \, dx,x,\tanh (c+d x)\right )}{2 d}\\ &=\frac {\tanh ^{-1}\left (\frac {\sqrt {\sqrt {a}-\sqrt {b}} \tanh (c+d x)}{\sqrt [4]{a}}\right )}{2 a^{3/4} \sqrt {\sqrt {a}-\sqrt {b}} d}+\frac {\tanh ^{-1}\left (\frac {\sqrt {\sqrt {a}+\sqrt {b}} \tanh (c+d x)}{\sqrt [4]{a}}\right )}{2 a^{3/4} \sqrt {\sqrt {a}+\sqrt {b}} d}\\ \end {align*}
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Mathematica [A]
time = 0.17, size = 128, normalized size = 1.11 \begin {gather*} \frac {-\frac {\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 {\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}}}}{2 \sqrt {a} 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 = 1.67, size = 102, normalized size = 0.89
method | result | size |
derivativedivides | \(\frac {\munderset {\textit {\_R} =\RootOf \left (a \,\textit {\_Z}^{8}-4 a \,\textit {\_Z}^{6}+\left (6 a -16 b \right ) \textit {\_Z}^{4}-4 a \,\textit {\_Z}^{2}+a \right )}{\sum }\frac {\left (-\textit {\_R}^{6}+3 \textit {\_R}^{4}-3 \textit {\_R}^{2}+1\right ) \ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-\textit {\_R} \right )}{\textit {\_R}^{7} a -3 \textit {\_R}^{5} a +3 \textit {\_R}^{3} a -8 \textit {\_R}^{3} b -\textit {\_R} a}}{4 d}\) | \(102\) |
default | \(\frac {\munderset {\textit {\_R} =\RootOf \left (a \,\textit {\_Z}^{8}-4 a \,\textit {\_Z}^{6}+\left (6 a -16 b \right ) \textit {\_Z}^{4}-4 a \,\textit {\_Z}^{2}+a \right )}{\sum }\frac {\left (-\textit {\_R}^{6}+3 \textit {\_R}^{4}-3 \textit {\_R}^{2}+1\right ) \ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-\textit {\_R} \right )}{\textit {\_R}^{7} a -3 \textit {\_R}^{5} a +3 \textit {\_R}^{3} a -8 \textit {\_R}^{3} b -\textit {\_R} a}}{4 d}\) | \(102\) |
risch | \(\munderset {\textit {\_R} =\RootOf \left (1+\left (256 a^{4} d^{4}-256 b \,d^{4} a^{3}\right ) \textit {\_Z}^{4}-32 a^{2} d^{2} \textit {\_Z}^{2}\right )}{\sum }\textit {\_R} \ln \left ({\mathrm e}^{2 d x +2 c}+\left (-\frac {128 d^{3} a^{4}}{b}+128 a^{3} d^{3}\right ) \textit {\_R}^{3}+\left (\frac {32 a^{3} d^{2}}{b}-32 a^{2} d^{2}\right ) \textit {\_R}^{2}+\left (\frac {8 a^{2} d}{b}+8 a d \right ) \textit {\_R} -\frac {2 a}{b}-1\right )\) | \(124\) |
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. 975 vs.
\(2 (79) = 158\).
time = 0.42, size = 975, normalized size = 8.48 \begin {gather*} -\frac {1}{4} \, \sqrt {\frac {{\left (a^{2} - a b\right )} d^{2} \sqrt {\frac {b}{{\left (a^{5} - 2 \, a^{4} b + a^{3} b^{2}\right )} d^{4}}} + 1}{{\left (a^{2} - a b\right )} d^{2}}} \log \left (2 \, {\left (a^{3} - a^{2} b\right )} d^{2} \sqrt {\frac {b}{{\left (a^{5} - 2 \, a^{4} b + a^{3} b^{2}\right )} d^{4}}} + b \cosh \left (d x + c\right )^{2} + 2 \, b \cosh \left (d x + c\right ) \sinh \left (d x + c\right ) + b \sinh \left (d x + c\right )^{2} + 2 \, {\left ({\left (a^{4} - a^{3} b\right )} d^{3} \sqrt {\frac {b}{{\left (a^{5} - 2 \, a^{4} b + a^{3} b^{2}\right )} d^{4}}} - a b d\right )} \sqrt {\frac {{\left (a^{2} - a b\right )} d^{2} \sqrt {\frac {b}{{\left (a^{5} - 2 \, a^{4} b + a^{3} b^{2}\right )} d^{4}}} + 1}{{\left (a^{2} - a b\right )} d^{2}}} - b\right ) + \frac {1}{4} \, \sqrt {\frac {{\left (a^{2} - a b\right )} d^{2} \sqrt {\frac {b}{{\left (a^{5} - 2 \, a^{4} b + a^{3} b^{2}\right )} d^{4}}} + 1}{{\left (a^{2} - a b\right )} d^{2}}} \log \left (2 \, {\left (a^{3} - a^{2} b\right )} d^{2} \sqrt {\frac {b}{{\left (a^{5} - 2 \, a^{4} b + a^{3} b^{2}\right )} d^{4}}} + b \cosh \left (d x + c\right )^{2} + 2 \, b \cosh \left (d x + c\right ) \sinh \left (d x + c\right ) + b \sinh \left (d x + c\right )^{2} - 2 \, {\left ({\left (a^{4} - a^{3} b\right )} d^{3} \sqrt {\frac {b}{{\left (a^{5} - 2 \, a^{4} b + a^{3} b^{2}\right )} d^{4}}} - a b d\right )} \sqrt {\frac {{\left (a^{2} - a b\right )} d^{2} \sqrt {\frac {b}{{\left (a^{5} - 2 \, a^{4} b + a^{3} b^{2}\right )} d^{4}}} + 1}{{\left (a^{2} - a b\right )} d^{2}}} - b\right ) + \frac {1}{4} \, \sqrt {-\frac {{\left (a^{2} - a b\right )} d^{2} \sqrt {\frac {b}{{\left (a^{5} - 2 \, a^{4} b + a^{3} b^{2}\right )} d^{4}}} - 1}{{\left (a^{2} - a b\right )} d^{2}}} \log \left (-2 \, {\left (a^{3} - a^{2} b\right )} d^{2} \sqrt {\frac {b}{{\left (a^{5} - 2 \, a^{4} b + a^{3} b^{2}\right )} d^{4}}} + b \cosh \left (d x + c\right )^{2} + 2 \, b \cosh \left (d x + c\right ) \sinh \left (d x + c\right ) + b \sinh \left (d x + c\right )^{2} + 2 \, {\left ({\left (a^{4} - a^{3} b\right )} d^{3} \sqrt {\frac {b}{{\left (a^{5} - 2 \, a^{4} b + a^{3} b^{2}\right )} d^{4}}} + a b d\right )} \sqrt {-\frac {{\left (a^{2} - a b\right )} d^{2} \sqrt {\frac {b}{{\left (a^{5} - 2 \, a^{4} b + a^{3} b^{2}\right )} d^{4}}} - 1}{{\left (a^{2} - a b\right )} d^{2}}} - b\right ) - \frac {1}{4} \, \sqrt {-\frac {{\left (a^{2} - a b\right )} d^{2} \sqrt {\frac {b}{{\left (a^{5} - 2 \, a^{4} b + a^{3} b^{2}\right )} d^{4}}} - 1}{{\left (a^{2} - a b\right )} d^{2}}} \log \left (-2 \, {\left (a^{3} - a^{2} b\right )} d^{2} \sqrt {\frac {b}{{\left (a^{5} - 2 \, a^{4} b + a^{3} b^{2}\right )} d^{4}}} + b \cosh \left (d x + c\right )^{2} + 2 \, b \cosh \left (d x + c\right ) \sinh \left (d x + c\right ) + b \sinh \left (d x + c\right )^{2} - 2 \, {\left ({\left (a^{4} - a^{3} b\right )} d^{3} \sqrt {\frac {b}{{\left (a^{5} - 2 \, a^{4} b + a^{3} b^{2}\right )} d^{4}}} + a b d\right )} \sqrt {-\frac {{\left (a^{2} - a b\right )} d^{2} \sqrt {\frac {b}{{\left (a^{5} - 2 \, a^{4} b + a^{3} b^{2}\right )} d^{4}}} - 1}{{\left (a^{2} - a b\right )} d^{2}}} - b\right ) \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 = 0.43, size = 1, normalized size = 0.01 \begin {gather*} 0 \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 10.10, size = 1787, normalized size = 15.54 \begin {gather*} \ln \left (\frac {\left (\frac {\left (\frac {524288\,d^2\,\left (31\,a\,b^2-128\,a^2\,b+128\,a^3-b^3+256\,a^3\,{\mathrm {e}}^{2\,c+2\,d\,x}+b^3\,{\mathrm {e}}^{2\,c+2\,d\,x}+21\,a\,b^2\,{\mathrm {e}}^{2\,c+2\,d\,x}-240\,a^2\,b\,{\mathrm {e}}^{2\,c+2\,d\,x}\right )}{b^5\,\left (a-b\right )}+\frac {1048576\,a\,d^3\,\sqrt {\frac {a^2-\sqrt {a^3\,b}}{a^3\,d^2\,\left (a-b\right )}}\,\left (45\,a\,b^2-104\,a^2\,b+64\,a^3-3\,b^3+4\,b^3\,{\mathrm {e}}^{2\,c+2\,d\,x}-50\,a\,b^2\,{\mathrm {e}}^{2\,c+2\,d\,x}+48\,a^2\,b\,{\mathrm {e}}^{2\,c+2\,d\,x}\right )}{b^5\,\left (a-b\right )}\right )\,\sqrt {\frac {a^2-\sqrt {a^3\,b}}{a^3\,d^2\,\left (a-b\right )}}}{4}+\frac {262144\,d\,\left (72\,a\,b-64\,a^2-9\,b^2+256\,a^2\,{\mathrm {e}}^{2\,c+2\,d\,x}+31\,b^2\,{\mathrm {e}}^{2\,c+2\,d\,x}-288\,a\,b\,{\mathrm {e}}^{2\,c+2\,d\,x}\right )}{b^5\,\left (a-b\right )}\right )\,\sqrt {\frac {a^2-\sqrt {a^3\,b}}{a^3\,d^2\,\left (a-b\right )}}}{4}+\frac {32768\,\left (128\,a\,b-128\,a^2-15\,b^2+256\,a^2\,{\mathrm {e}}^{2\,c+2\,d\,x}+29\,b^2\,{\mathrm {e}}^{2\,c+2\,d\,x}-304\,a\,b\,{\mathrm {e}}^{2\,c+2\,d\,x}\right )}{a\,b^5\,\left (a-b\right )}\right )\,\sqrt {\frac {a^2-\sqrt {a^3\,b}}{16\,\left (a^4\,d^2-a^3\,b\,d^2\right )}}-\ln \left (\frac {\left (\frac {\left (\frac {524288\,d^2\,\left (31\,a\,b^2-128\,a^2\,b+128\,a^3-b^3+256\,a^3\,{\mathrm {e}}^{2\,c+2\,d\,x}+b^3\,{\mathrm {e}}^{2\,c+2\,d\,x}+21\,a\,b^2\,{\mathrm {e}}^{2\,c+2\,d\,x}-240\,a^2\,b\,{\mathrm {e}}^{2\,c+2\,d\,x}\right )}{b^5\,\left (a-b\right )}-\frac {1048576\,a\,d^3\,\sqrt {\frac {a^2-\sqrt {a^3\,b}}{a^3\,d^2\,\left (a-b\right )}}\,\left (45\,a\,b^2-104\,a^2\,b+64\,a^3-3\,b^3+4\,b^3\,{\mathrm {e}}^{2\,c+2\,d\,x}-50\,a\,b^2\,{\mathrm {e}}^{2\,c+2\,d\,x}+48\,a^2\,b\,{\mathrm {e}}^{2\,c+2\,d\,x}\right )}{b^5\,\left (a-b\right )}\right )\,\sqrt {\frac {a^2-\sqrt {a^3\,b}}{a^3\,d^2\,\left (a-b\right )}}}{4}-\frac {262144\,d\,\left (72\,a\,b-64\,a^2-9\,b^2+256\,a^2\,{\mathrm {e}}^{2\,c+2\,d\,x}+31\,b^2\,{\mathrm {e}}^{2\,c+2\,d\,x}-288\,a\,b\,{\mathrm {e}}^{2\,c+2\,d\,x}\right )}{b^5\,\left (a-b\right )}\right )\,\sqrt {\frac {a^2-\sqrt {a^3\,b}}{a^3\,d^2\,\left (a-b\right )}}}{4}+\frac {32768\,\left (128\,a\,b-128\,a^2-15\,b^2+256\,a^2\,{\mathrm {e}}^{2\,c+2\,d\,x}+29\,b^2\,{\mathrm {e}}^{2\,c+2\,d\,x}-304\,a\,b\,{\mathrm {e}}^{2\,c+2\,d\,x}\right )}{a\,b^5\,\left (a-b\right )}\right )\,\sqrt {\frac {a^2-\sqrt {a^3\,b}}{16\,\left (a^4\,d^2-a^3\,b\,d^2\right )}}-\ln \left (\frac {\left (\frac {\left (\frac {524288\,d^2\,\left (31\,a\,b^2-128\,a^2\,b+128\,a^3-b^3+256\,a^3\,{\mathrm {e}}^{2\,c+2\,d\,x}+b^3\,{\mathrm {e}}^{2\,c+2\,d\,x}+21\,a\,b^2\,{\mathrm {e}}^{2\,c+2\,d\,x}-240\,a^2\,b\,{\mathrm {e}}^{2\,c+2\,d\,x}\right )}{b^5\,\left (a-b\right )}-\frac {1048576\,a\,d^3\,\sqrt {\frac {a^2+\sqrt {a^3\,b}}{a^3\,d^2\,\left (a-b\right )}}\,\left (45\,a\,b^2-104\,a^2\,b+64\,a^3-3\,b^3+4\,b^3\,{\mathrm {e}}^{2\,c+2\,d\,x}-50\,a\,b^2\,{\mathrm {e}}^{2\,c+2\,d\,x}+48\,a^2\,b\,{\mathrm {e}}^{2\,c+2\,d\,x}\right )}{b^5\,\left (a-b\right )}\right )\,\sqrt {\frac {a^2+\sqrt {a^3\,b}}{a^3\,d^2\,\left (a-b\right )}}}{4}-\frac {262144\,d\,\left (72\,a\,b-64\,a^2-9\,b^2+256\,a^2\,{\mathrm {e}}^{2\,c+2\,d\,x}+31\,b^2\,{\mathrm {e}}^{2\,c+2\,d\,x}-288\,a\,b\,{\mathrm {e}}^{2\,c+2\,d\,x}\right )}{b^5\,\left (a-b\right )}\right )\,\sqrt {\frac {a^2+\sqrt {a^3\,b}}{a^3\,d^2\,\left (a-b\right )}}}{4}+\frac {32768\,\left (128\,a\,b-128\,a^2-15\,b^2+256\,a^2\,{\mathrm {e}}^{2\,c+2\,d\,x}+29\,b^2\,{\mathrm {e}}^{2\,c+2\,d\,x}-304\,a\,b\,{\mathrm {e}}^{2\,c+2\,d\,x}\right )}{a\,b^5\,\left (a-b\right )}\right )\,\sqrt {\frac {a^2+\sqrt {a^3\,b}}{16\,\left (a^4\,d^2-a^3\,b\,d^2\right )}}+\ln \left (\frac {\left (\frac {\left (\frac {524288\,d^2\,\left (31\,a\,b^2-128\,a^2\,b+128\,a^3-b^3+256\,a^3\,{\mathrm {e}}^{2\,c+2\,d\,x}+b^3\,{\mathrm {e}}^{2\,c+2\,d\,x}+21\,a\,b^2\,{\mathrm {e}}^{2\,c+2\,d\,x}-240\,a^2\,b\,{\mathrm {e}}^{2\,c+2\,d\,x}\right )}{b^5\,\left (a-b\right )}+\frac {1048576\,a\,d^3\,\sqrt {\frac {a^2+\sqrt {a^3\,b}}{a^3\,d^2\,\left (a-b\right )}}\,\left (45\,a\,b^2-104\,a^2\,b+64\,a^3-3\,b^3+4\,b^3\,{\mathrm {e}}^{2\,c+2\,d\,x}-50\,a\,b^2\,{\mathrm {e}}^{2\,c+2\,d\,x}+48\,a^2\,b\,{\mathrm {e}}^{2\,c+2\,d\,x}\right )}{b^5\,\left (a-b\right )}\right )\,\sqrt {\frac {a^2+\sqrt {a^3\,b}}{a^3\,d^2\,\left (a-b\right )}}}{4}+\frac {262144\,d\,\left (72\,a\,b-64\,a^2-9\,b^2+256\,a^2\,{\mathrm {e}}^{2\,c+2\,d\,x}+31\,b^2\,{\mathrm {e}}^{2\,c+2\,d\,x}-288\,a\,b\,{\mathrm {e}}^{2\,c+2\,d\,x}\right )}{b^5\,\left (a-b\right )}\right )\,\sqrt {\frac {a^2+\sqrt {a^3\,b}}{a^3\,d^2\,\left (a-b\right )}}}{4}+\frac {32768\,\left (128\,a\,b-128\,a^2-15\,b^2+256\,a^2\,{\mathrm {e}}^{2\,c+2\,d\,x}+29\,b^2\,{\mathrm {e}}^{2\,c+2\,d\,x}-304\,a\,b\,{\mathrm {e}}^{2\,c+2\,d\,x}\right )}{a\,b^5\,\left (a-b\right )}\right )\,\sqrt {\frac {a^2+\sqrt {a^3\,b}}{16\,\left (a^4\,d^2-a^3\,b\,d^2\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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