Optimal. Leaf size=125 \[ \frac {\text {ArcTan}\left (\frac {\sqrt [4]{b} \cosh (c+d x)}{\sqrt {\sqrt {a}-\sqrt {b}}}\right )}{2 \sqrt {a} \sqrt {\sqrt {a}-\sqrt {b}} \sqrt [4]{b} d}+\frac {\tanh ^{-1}\left (\frac {\sqrt [4]{b} \cosh (c+d x)}{\sqrt {\sqrt {a}+\sqrt {b}}}\right )}{2 \sqrt {a} \sqrt {\sqrt {a}+\sqrt {b}} \sqrt [4]{b} d} \]
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Rubi [A]
time = 0.08, antiderivative size = 125, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 4, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {3294, 1107,
211, 214} \begin {gather*} \frac {\text {ArcTan}\left (\frac {\sqrt [4]{b} \cosh (c+d x)}{\sqrt {\sqrt {a}-\sqrt {b}}}\right )}{2 \sqrt {a} \sqrt [4]{b} d \sqrt {\sqrt {a}-\sqrt {b}}}+\frac {\tanh ^{-1}\left (\frac {\sqrt [4]{b} \cosh (c+d x)}{\sqrt {\sqrt {a}+\sqrt {b}}}\right )}{2 \sqrt {a} \sqrt [4]{b} d \sqrt {\sqrt {a}+\sqrt {b}}} \end {gather*}
Antiderivative was successfully verified.
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Rule 211
Rule 214
Rule 1107
Rule 3294
Rubi steps
\begin {align*} \int \frac {\sinh (c+d x)}{a-b \sinh ^4(c+d x)} \, dx &=\frac {\text {Subst}\left (\int \frac {1}{a-b+2 b x^2-b x^4} \, dx,x,\cosh (c+d x)\right )}{d}\\ &=-\frac {\sqrt {b} \text {Subst}\left (\int \frac {1}{-\sqrt {a} \sqrt {b}+b-b x^2} \, dx,x,\cosh (c+d x)\right )}{2 \sqrt {a} d}+\frac {\sqrt {b} \text {Subst}\left (\int \frac {1}{\sqrt {a} \sqrt {b}+b-b x^2} \, dx,x,\cosh (c+d x)\right )}{2 \sqrt {a} d}\\ &=\frac {\tan ^{-1}\left (\frac {\sqrt [4]{b} \cosh (c+d x)}{\sqrt {\sqrt {a}-\sqrt {b}}}\right )}{2 \sqrt {a} \sqrt {\sqrt {a}-\sqrt {b}} \sqrt [4]{b} d}+\frac {\tanh ^{-1}\left (\frac {\sqrt [4]{b} \cosh (c+d x)}{\sqrt {\sqrt {a}+\sqrt {b}}}\right )}{2 \sqrt {a} \sqrt {\sqrt {a}+\sqrt {b}} \sqrt [4]{b} d}\\ \end {align*}
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Mathematica [C] Result contains higher order function than in optimal. Order 9 vs. order 3 in
optimal.
time = 0.12, size = 221, normalized size = 1.77 \begin {gather*} -\frac {\text {RootSum}\left [b-4 b \text {$\#$1}^2-16 a \text {$\#$1}^4+6 b \text {$\#$1}^4-4 b \text {$\#$1}^6+b \text {$\#$1}^8\&,\frac {-c \text {$\#$1}-d x \text {$\#$1}-2 \log \left (-\cosh \left (\frac {1}{2} (c+d x)\right )-\sinh \left (\frac {1}{2} (c+d x)\right )+\cosh \left (\frac {1}{2} (c+d x)\right ) \text {$\#$1}-\sinh \left (\frac {1}{2} (c+d x)\right ) \text {$\#$1}\right ) \text {$\#$1}+c \text {$\#$1}^3+d x \text {$\#$1}^3+2 \log \left (-\cosh \left (\frac {1}{2} (c+d x)\right )-\sinh \left (\frac {1}{2} (c+d x)\right )+\cosh \left (\frac {1}{2} (c+d x)\right ) \text {$\#$1}-\sinh \left (\frac {1}{2} (c+d x)\right ) \text {$\#$1}\right ) \text {$\#$1}^3}{-b-8 a \text {$\#$1}^2+3 b \text {$\#$1}^2-3 b \text {$\#$1}^4+b \text {$\#$1}^6}\&\right ]}{2 d} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 1.71, size = 132, normalized size = 1.06
method | result | size |
risch | \(\munderset {\textit {\_R} =\RootOf \left (-1+\left (256 b \,d^{4} a^{3}-256 a^{2} b^{2} d^{4}\right ) \textit {\_Z}^{4}+32 a \,d^{2} \textit {\_Z}^{2} b \right )}{\sum }\textit {\_R} \ln \left ({\mathrm e}^{2 d x +2 c}+\left (\left (128 a^{2} b \,d^{3}-128 a \,b^{2} d^{3}\right ) \textit {\_R}^{3}+\left (8 a d +8 b d \right ) \textit {\_R} \right ) {\mathrm e}^{d x +c}+1\right )\) | \(98\) |
derivativedivides | \(\frac {2 a \left (\frac {\arctan \left (\frac {2 a \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+4 \sqrt {a b}-2 a}{4 \sqrt {\sqrt {a b}\, a -a b}}\right )}{4 a \sqrt {\sqrt {a b}\, a -a b}}-\frac {\arctan \left (\frac {-2 a \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+4 \sqrt {a b}+2 a}{4 \sqrt {-\sqrt {a b}\, a -a b}}\right )}{4 a \sqrt {-\sqrt {a b}\, a -a b}}\right )}{d}\) | \(132\) |
default | \(\frac {2 a \left (\frac {\arctan \left (\frac {2 a \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+4 \sqrt {a b}-2 a}{4 \sqrt {\sqrt {a b}\, a -a b}}\right )}{4 a \sqrt {\sqrt {a b}\, a -a b}}-\frac {\arctan \left (\frac {-2 a \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+4 \sqrt {a b}+2 a}{4 \sqrt {-\sqrt {a b}\, a -a b}}\right )}{4 a \sqrt {-\sqrt {a b}\, a -a b}}\right )}{d}\) | \(132\) |
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. 979 vs.
\(2 (85) = 170\).
time = 0.40, size = 979, normalized size = 7.83 \begin {gather*} \frac {1}{4} \, \sqrt {-\frac {{\left (a^{2} - a b\right )} d^{2} \sqrt {\frac {1}{{\left (a^{3} b - 2 \, a^{2} b^{2} + a b^{3}\right )} d^{4}}} + 1}{{\left (a^{2} - a b\right )} d^{2}}} \log \left (\cosh \left (d x + c\right )^{2} + 2 \, \cosh \left (d x + c\right ) \sinh \left (d x + c\right ) + \sinh \left (d x + c\right )^{2} + 2 \, {\left (a d \cosh \left (d x + c\right ) + a d \sinh \left (d x + c\right ) - {\left ({\left (a^{2} b - a b^{2}\right )} d^{3} \cosh \left (d x + c\right ) + {\left (a^{2} b - a b^{2}\right )} d^{3} \sinh \left (d x + c\right )\right )} \sqrt {\frac {1}{{\left (a^{3} b - 2 \, a^{2} b^{2} + a b^{3}\right )} d^{4}}}\right )} \sqrt {-\frac {{\left (a^{2} - a b\right )} d^{2} \sqrt {\frac {1}{{\left (a^{3} b - 2 \, a^{2} b^{2} + a b^{3}\right )} d^{4}}} + 1}{{\left (a^{2} - a b\right )} d^{2}}} + 1\right ) - \frac {1}{4} \, \sqrt {-\frac {{\left (a^{2} - a b\right )} d^{2} \sqrt {\frac {1}{{\left (a^{3} b - 2 \, a^{2} b^{2} + a b^{3}\right )} d^{4}}} + 1}{{\left (a^{2} - a b\right )} d^{2}}} \log \left (\cosh \left (d x + c\right )^{2} + 2 \, \cosh \left (d x + c\right ) \sinh \left (d x + c\right ) + \sinh \left (d x + c\right )^{2} - 2 \, {\left (a d \cosh \left (d x + c\right ) + a d \sinh \left (d x + c\right ) - {\left ({\left (a^{2} b - a b^{2}\right )} d^{3} \cosh \left (d x + c\right ) + {\left (a^{2} b - a b^{2}\right )} d^{3} \sinh \left (d x + c\right )\right )} \sqrt {\frac {1}{{\left (a^{3} b - 2 \, a^{2} b^{2} + a b^{3}\right )} d^{4}}}\right )} \sqrt {-\frac {{\left (a^{2} - a b\right )} d^{2} \sqrt {\frac {1}{{\left (a^{3} b - 2 \, a^{2} b^{2} + a b^{3}\right )} d^{4}}} + 1}{{\left (a^{2} - a b\right )} d^{2}}} + 1\right ) + \frac {1}{4} \, \sqrt {\frac {{\left (a^{2} - a b\right )} d^{2} \sqrt {\frac {1}{{\left (a^{3} b - 2 \, a^{2} b^{2} + a b^{3}\right )} d^{4}}} - 1}{{\left (a^{2} - a b\right )} d^{2}}} \log \left (\cosh \left (d x + c\right )^{2} + 2 \, \cosh \left (d x + c\right ) \sinh \left (d x + c\right ) + \sinh \left (d x + c\right )^{2} + 2 \, {\left (a d \cosh \left (d x + c\right ) + a d \sinh \left (d x + c\right ) + {\left ({\left (a^{2} b - a b^{2}\right )} d^{3} \cosh \left (d x + c\right ) + {\left (a^{2} b - a b^{2}\right )} d^{3} \sinh \left (d x + c\right )\right )} \sqrt {\frac {1}{{\left (a^{3} b - 2 \, a^{2} b^{2} + a b^{3}\right )} d^{4}}}\right )} \sqrt {\frac {{\left (a^{2} - a b\right )} d^{2} \sqrt {\frac {1}{{\left (a^{3} b - 2 \, a^{2} b^{2} + a b^{3}\right )} d^{4}}} - 1}{{\left (a^{2} - a b\right )} d^{2}}} + 1\right ) - \frac {1}{4} \, \sqrt {\frac {{\left (a^{2} - a b\right )} d^{2} \sqrt {\frac {1}{{\left (a^{3} b - 2 \, a^{2} b^{2} + a b^{3}\right )} d^{4}}} - 1}{{\left (a^{2} - a b\right )} d^{2}}} \log \left (\cosh \left (d x + c\right )^{2} + 2 \, \cosh \left (d x + c\right ) \sinh \left (d x + c\right ) + \sinh \left (d x + c\right )^{2} - 2 \, {\left (a d \cosh \left (d x + c\right ) + a d \sinh \left (d x + c\right ) + {\left ({\left (a^{2} b - a b^{2}\right )} d^{3} \cosh \left (d x + c\right ) + {\left (a^{2} b - a b^{2}\right )} d^{3} \sinh \left (d x + c\right )\right )} \sqrt {\frac {1}{{\left (a^{3} b - 2 \, a^{2} b^{2} + a b^{3}\right )} d^{4}}}\right )} \sqrt {\frac {{\left (a^{2} - a b\right )} d^{2} \sqrt {\frac {1}{{\left (a^{3} b - 2 \, a^{2} b^{2} + a b^{3}\right )} d^{4}}} - 1}{{\left (a^{2} - a b\right )} d^{2}}} + 1\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 [B] Leaf count of result is larger than twice the leaf count of optimal. 332 vs.
\(2 (85) = 170\).
time = 0.54, size = 332, normalized size = 2.66 \begin {gather*} \frac {\frac {{\left (4 \, \sqrt {-b^{2} - \sqrt {a b} b} a^{2} b + 5 \, \sqrt {-b^{2} - \sqrt {a b} b} a b^{2} + 4 \, \sqrt {a b} \sqrt {-b^{2} - \sqrt {a b} b} a b + 5 \, \sqrt {a b} \sqrt {-b^{2} - \sqrt {a b} b} b^{2}\right )} {\left | b \right |} \arctan \left (\frac {e^{\left (d x + c\right )} + e^{\left (-d x - c\right )}}{2 \, \sqrt {-\frac {b + \sqrt {{\left (a - b\right )} b + b^{2}}}{b}}}\right )}{4 \, a^{3} b^{3} + a^{2} b^{4} - 5 \, a b^{5}} + \frac {{\left (4 \, \sqrt {-b^{2} + \sqrt {a b} b} a^{2} b + 5 \, \sqrt {-b^{2} + \sqrt {a b} b} a b^{2} + 4 \, \sqrt {a b} \sqrt {-b^{2} + \sqrt {a b} b} a b + 5 \, \sqrt {a b} \sqrt {-b^{2} + \sqrt {a b} b} b^{2}\right )} {\left | b \right |} \arctan \left (\frac {e^{\left (d x + c\right )} + e^{\left (-d x - c\right )}}{2 \, \sqrt {-\frac {b - \sqrt {{\left (a - b\right )} b + b^{2}}}{b}}}\right )}{4 \, a^{3} b^{3} + a^{2} b^{4} - 5 \, a b^{5}}}{2 \, d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 8.15, size = 1007, normalized size = 8.06 \begin {gather*} \ln \left (\frac {\left (\frac {\left (\frac {4194304\,a^2\,d^2\,\left ({\mathrm {e}}^{2\,c+2\,d\,x}+1\right )\,\left (3\,a+b\right )}{b^5\,{\left (a-b\right )}^2}+\frac {16777216\,a^3\,d^3\,{\mathrm {e}}^{c+d\,x}\,\sqrt {-\frac {a\,b-\sqrt {a^3\,b}}{a^2\,b\,d^2\,\left (a-b\right )}}}{b^5\,\left (a-b\right )}\right )\,\sqrt {-\frac {a\,b-\sqrt {a^3\,b}}{a^2\,b\,d^2\,\left (a-b\right )}}}{4}-\frac {2097152\,a^2\,d\,{\mathrm {e}}^{c+d\,x}}{b^6\,\left (a-b\right )}\right )\,\sqrt {-\frac {a\,b-\sqrt {a^3\,b}}{a^2\,b\,d^2\,\left (a-b\right )}}}{4}-\frac {262144\,a\,\left ({\mathrm {e}}^{2\,c+2\,d\,x}+1\right )\,\left (a+b\right )}{b^6\,{\left (a-b\right )}^2}\right )\,\sqrt {-\frac {a\,b-\sqrt {a^3\,b}}{16\,\left (a^3\,b\,d^2-a^2\,b^2\,d^2\right )}}-\ln \left (\frac {\left (\frac {\left (\frac {4194304\,a^2\,d^2\,\left ({\mathrm {e}}^{2\,c+2\,d\,x}+1\right )\,\left (3\,a+b\right )}{b^5\,{\left (a-b\right )}^2}-\frac {16777216\,a^3\,d^3\,{\mathrm {e}}^{c+d\,x}\,\sqrt {-\frac {a\,b-\sqrt {a^3\,b}}{a^2\,b\,d^2\,\left (a-b\right )}}}{b^5\,\left (a-b\right )}\right )\,\sqrt {-\frac {a\,b-\sqrt {a^3\,b}}{a^2\,b\,d^2\,\left (a-b\right )}}}{4}+\frac {2097152\,a^2\,d\,{\mathrm {e}}^{c+d\,x}}{b^6\,\left (a-b\right )}\right )\,\sqrt {-\frac {a\,b-\sqrt {a^3\,b}}{a^2\,b\,d^2\,\left (a-b\right )}}}{4}-\frac {262144\,a\,\left ({\mathrm {e}}^{2\,c+2\,d\,x}+1\right )\,\left (a+b\right )}{b^6\,{\left (a-b\right )}^2}\right )\,\sqrt {-\frac {a\,b-\sqrt {a^3\,b}}{16\,\left (a^3\,b\,d^2-a^2\,b^2\,d^2\right )}}-\ln \left (\frac {\left (\frac {\left (\frac {4194304\,a^2\,d^2\,\left ({\mathrm {e}}^{2\,c+2\,d\,x}+1\right )\,\left (3\,a+b\right )}{b^5\,{\left (a-b\right )}^2}-\frac {16777216\,a^3\,d^3\,{\mathrm {e}}^{c+d\,x}\,\sqrt {-\frac {a\,b+\sqrt {a^3\,b}}{a^2\,b\,d^2\,\left (a-b\right )}}}{b^5\,\left (a-b\right )}\right )\,\sqrt {-\frac {a\,b+\sqrt {a^3\,b}}{a^2\,b\,d^2\,\left (a-b\right )}}}{4}+\frac {2097152\,a^2\,d\,{\mathrm {e}}^{c+d\,x}}{b^6\,\left (a-b\right )}\right )\,\sqrt {-\frac {a\,b+\sqrt {a^3\,b}}{a^2\,b\,d^2\,\left (a-b\right )}}}{4}-\frac {262144\,a\,\left ({\mathrm {e}}^{2\,c+2\,d\,x}+1\right )\,\left (a+b\right )}{b^6\,{\left (a-b\right )}^2}\right )\,\sqrt {-\frac {a\,b+\sqrt {a^3\,b}}{16\,\left (a^3\,b\,d^2-a^2\,b^2\,d^2\right )}}+\ln \left (\frac {\left (\frac {\left (\frac {4194304\,a^2\,d^2\,\left ({\mathrm {e}}^{2\,c+2\,d\,x}+1\right )\,\left (3\,a+b\right )}{b^5\,{\left (a-b\right )}^2}+\frac {16777216\,a^3\,d^3\,{\mathrm {e}}^{c+d\,x}\,\sqrt {-\frac {a\,b+\sqrt {a^3\,b}}{a^2\,b\,d^2\,\left (a-b\right )}}}{b^5\,\left (a-b\right )}\right )\,\sqrt {-\frac {a\,b+\sqrt {a^3\,b}}{a^2\,b\,d^2\,\left (a-b\right )}}}{4}-\frac {2097152\,a^2\,d\,{\mathrm {e}}^{c+d\,x}}{b^6\,\left (a-b\right )}\right )\,\sqrt {-\frac {a\,b+\sqrt {a^3\,b}}{a^2\,b\,d^2\,\left (a-b\right )}}}{4}-\frac {262144\,a\,\left ({\mathrm {e}}^{2\,c+2\,d\,x}+1\right )\,\left (a+b\right )}{b^6\,{\left (a-b\right )}^2}\right )\,\sqrt {-\frac {a\,b+\sqrt {a^3\,b}}{16\,\left (a^3\,b\,d^2-a^2\,b^2\,d^2\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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