Optimal. Leaf size=121 \[ \frac {\left (8 a^2-20 a b+15 b^2\right ) x}{8 b^3}-\frac {(a-b)^{5/2} \tanh ^{-1}\left (\frac {\sqrt {a-b} \tanh (c+d x)}{\sqrt {a}}\right )}{\sqrt {a} b^3 d}-\frac {(4 a-7 b) \cosh (c+d x) \sinh (c+d x)}{8 b^2 d}+\frac {\cosh ^3(c+d x) \sinh (c+d x)}{4 b d} \]
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Rubi [A]
time = 0.14, antiderivative size = 121, normalized size of antiderivative = 1.00, number of steps
used = 6, number of rules used = 6, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.261, Rules used = {3270, 425, 541,
536, 212, 214} \begin {gather*} \frac {x \left (8 a^2-20 a b+15 b^2\right )}{8 b^3}-\frac {(a-b)^{5/2} \tanh ^{-1}\left (\frac {\sqrt {a-b} \tanh (c+d x)}{\sqrt {a}}\right )}{\sqrt {a} b^3 d}-\frac {(4 a-7 b) \sinh (c+d x) \cosh (c+d x)}{8 b^2 d}+\frac {\sinh (c+d x) \cosh ^3(c+d x)}{4 b d} \end {gather*}
Antiderivative was successfully verified.
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Rule 212
Rule 214
Rule 425
Rule 536
Rule 541
Rule 3270
Rubi steps
\begin {align*} \int \frac {\cosh ^6(c+d x)}{a+b \sinh ^2(c+d x)} \, dx &=\frac {\text {Subst}\left (\int \frac {1}{\left (1-x^2\right )^3 \left (a-(a-b) x^2\right )} \, dx,x,\tanh (c+d x)\right )}{d}\\ &=\frac {\cosh ^3(c+d x) \sinh (c+d x)}{4 b d}+\frac {\text {Subst}\left (\int \frac {-a+4 b-3 (a-b) x^2}{\left (1-x^2\right )^2 \left (a+(-a+b) x^2\right )} \, dx,x,\tanh (c+d x)\right )}{4 b d}\\ &=-\frac {(4 a-7 b) \cosh (c+d x) \sinh (c+d x)}{8 b^2 d}+\frac {\cosh ^3(c+d x) \sinh (c+d x)}{4 b d}+\frac {\text {Subst}\left (\int \frac {4 a^2-9 a b+8 b^2+(4 a-7 b) (a-b) x^2}{\left (1-x^2\right ) \left (a+(-a+b) x^2\right )} \, dx,x,\tanh (c+d x)\right )}{8 b^2 d}\\ &=-\frac {(4 a-7 b) \cosh (c+d x) \sinh (c+d x)}{8 b^2 d}+\frac {\cosh ^3(c+d x) \sinh (c+d x)}{4 b d}-\frac {(a-b)^3 \text {Subst}\left (\int \frac {1}{a+(-a+b) x^2} \, dx,x,\tanh (c+d x)\right )}{b^3 d}+\frac {\left (8 a^2-20 a b+15 b^2\right ) \text {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\tanh (c+d x)\right )}{8 b^3 d}\\ &=\frac {\left (8 a^2-20 a b+15 b^2\right ) x}{8 b^3}-\frac {(a-b)^{5/2} \tanh ^{-1}\left (\frac {\sqrt {a-b} \tanh (c+d x)}{\sqrt {a}}\right )}{\sqrt {a} b^3 d}-\frac {(4 a-7 b) \cosh (c+d x) \sinh (c+d x)}{8 b^2 d}+\frac {\cosh ^3(c+d x) \sinh (c+d x)}{4 b d}\\ \end {align*}
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Mathematica [A]
time = 0.24, size = 106, normalized size = 0.88 \begin {gather*} \frac {-32 (a-b)^{5/2} \tanh ^{-1}\left (\frac {\sqrt {a-b} \tanh (c+d x)}{\sqrt {a}}\right )+\sqrt {a} \left (4 \left (8 a^2-20 a b+15 b^2\right ) (c+d x)-8 (a-2 b) b \sinh (2 (c+d x))+b^2 \sinh (4 (c+d x))\right )}{32 \sqrt {a} b^3 d} \end {gather*}
Antiderivative was successfully verified.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(437\) vs.
\(2(107)=214\).
time = 1.82, size = 438, normalized size = 3.62
method | result | size |
risch | \(\frac {x \,a^{2}}{b^{3}}-\frac {5 a x}{2 b^{2}}+\frac {15 x}{8 b}+\frac {{\mathrm e}^{4 d x +4 c}}{64 b d}+\frac {{\mathrm e}^{2 d x +2 c}}{4 b d}-\frac {a \,{\mathrm e}^{2 d x +2 c}}{8 b^{2} d}-\frac {{\mathrm e}^{-2 d x -2 c}}{4 b d}+\frac {a \,{\mathrm e}^{-2 d x -2 c}}{8 b^{2} d}-\frac {{\mathrm e}^{-4 d x -4 c}}{64 b d}+\frac {a \sqrt {a \left (a -b \right )}\, \ln \left ({\mathrm e}^{2 d x +2 c}+\frac {2 \sqrt {a \left (a -b \right )}+2 a -b}{b}\right )}{2 d \,b^{3}}-\frac {\sqrt {a \left (a -b \right )}\, \ln \left ({\mathrm e}^{2 d x +2 c}+\frac {2 \sqrt {a \left (a -b \right )}+2 a -b}{b}\right )}{d \,b^{2}}+\frac {\sqrt {a \left (a -b \right )}\, \ln \left ({\mathrm e}^{2 d x +2 c}+\frac {2 \sqrt {a \left (a -b \right )}+2 a -b}{b}\right )}{2 a d b}-\frac {a \sqrt {a \left (a -b \right )}\, \ln \left ({\mathrm e}^{2 d x +2 c}-\frac {2 \sqrt {a \left (a -b \right )}-2 a +b}{b}\right )}{2 d \,b^{3}}+\frac {\sqrt {a \left (a -b \right )}\, \ln \left ({\mathrm e}^{2 d x +2 c}-\frac {2 \sqrt {a \left (a -b \right )}-2 a +b}{b}\right )}{d \,b^{2}}-\frac {\sqrt {a \left (a -b \right )}\, \ln \left ({\mathrm e}^{2 d x +2 c}-\frac {2 \sqrt {a \left (a -b \right )}-2 a +b}{b}\right )}{2 a d b}\) | \(431\) |
derivativedivides | \(\frac {\frac {2 a \left (a^{3}-3 a^{2} b +3 a \,b^{2}-b^{3}\right ) \left (\frac {\left (\sqrt {-b \left (a -b \right )}+b \right ) \arctan \left (\frac {a \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{\sqrt {\left (2 \sqrt {-b \left (a -b \right )}-a +2 b \right ) a}}\right )}{2 a \sqrt {-b \left (a -b \right )}\, \sqrt {\left (2 \sqrt {-b \left (a -b \right )}-a +2 b \right ) a}}-\frac {\left (\sqrt {-b \left (a -b \right )}-b \right ) \arctanh \left (\frac {a \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{\sqrt {\left (2 \sqrt {-b \left (a -b \right )}+a -2 b \right ) a}}\right )}{2 a \sqrt {-b \left (a -b \right )}\, \sqrt {\left (2 \sqrt {-b \left (a -b \right )}+a -2 b \right ) a}}\right )}{b^{3}}-\frac {1}{4 b \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{4}}+\frac {1}{2 b \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{3}}-\frac {-9 b +4 a}{8 b^{2} \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}-\frac {11 b -4 a}{8 b^{2} \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{2}}+\frac {\left (8 a^{2}-20 a b +15 b^{2}\right ) \ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{8 b^{3}}+\frac {1}{4 b \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{4}}+\frac {1}{2 b \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{3}}-\frac {-11 b +4 a}{8 b^{2} \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{2}}-\frac {-9 b +4 a}{8 b^{2} \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}+\frac {\left (-8 a^{2}+20 a b -15 b^{2}\right ) \ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{8 b^{3}}}{d}\) | \(438\) |
default | \(\frac {\frac {2 a \left (a^{3}-3 a^{2} b +3 a \,b^{2}-b^{3}\right ) \left (\frac {\left (\sqrt {-b \left (a -b \right )}+b \right ) \arctan \left (\frac {a \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{\sqrt {\left (2 \sqrt {-b \left (a -b \right )}-a +2 b \right ) a}}\right )}{2 a \sqrt {-b \left (a -b \right )}\, \sqrt {\left (2 \sqrt {-b \left (a -b \right )}-a +2 b \right ) a}}-\frac {\left (\sqrt {-b \left (a -b \right )}-b \right ) \arctanh \left (\frac {a \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{\sqrt {\left (2 \sqrt {-b \left (a -b \right )}+a -2 b \right ) a}}\right )}{2 a \sqrt {-b \left (a -b \right )}\, \sqrt {\left (2 \sqrt {-b \left (a -b \right )}+a -2 b \right ) a}}\right )}{b^{3}}-\frac {1}{4 b \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{4}}+\frac {1}{2 b \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{3}}-\frac {-9 b +4 a}{8 b^{2} \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}-\frac {11 b -4 a}{8 b^{2} \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{2}}+\frac {\left (8 a^{2}-20 a b +15 b^{2}\right ) \ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{8 b^{3}}+\frac {1}{4 b \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{4}}+\frac {1}{2 b \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{3}}-\frac {-11 b +4 a}{8 b^{2} \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{2}}-\frac {-9 b +4 a}{8 b^{2} \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}+\frac {\left (-8 a^{2}+20 a b -15 b^{2}\right ) \ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{8 b^{3}}}{d}\) | \(438\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: ValueError} \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. 776 vs.
\(2 (107) = 214\).
time = 0.45, size = 1817, normalized size = 15.02 \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 [B] Leaf count of result is larger than twice the leaf count of optimal. 226 vs.
\(2 (107) = 214\).
time = 2.42, size = 226, normalized size = 1.87 \begin {gather*} \frac {\frac {8 \, {\left (8 \, a^{2} - 20 \, a b + 15 \, b^{2}\right )} {\left (d x + c\right )}}{b^{3}} + \frac {b e^{\left (4 \, d x + 4 \, c\right )} - 8 \, a e^{\left (2 \, d x + 2 \, c\right )} + 16 \, b e^{\left (2 \, d x + 2 \, c\right )}}{b^{2}} - \frac {{\left (48 \, a^{2} e^{\left (4 \, d x + 4 \, c\right )} - 120 \, a b e^{\left (4 \, d x + 4 \, c\right )} + 90 \, b^{2} e^{\left (4 \, d x + 4 \, c\right )} - 8 \, a b e^{\left (2 \, d x + 2 \, c\right )} + 16 \, b^{2} e^{\left (2 \, d x + 2 \, c\right )} + b^{2}\right )} e^{\left (-4 \, d x - 4 \, c\right )}}{b^{3}} - \frac {64 \, {\left (a^{3} - 3 \, a^{2} b + 3 \, a b^{2} - b^{3}\right )} \arctan \left (\frac {b e^{\left (2 \, d x + 2 \, c\right )} + 2 \, a - b}{2 \, \sqrt {-a^{2} + a b}}\right )}{\sqrt {-a^{2} + a b} b^{3}}}{64 \, d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 1.37, size = 264, normalized size = 2.18 \begin {gather*} \frac {x\,\left (8\,a^2-20\,a\,b+15\,b^2\right )}{8\,b^3}-\frac {{\mathrm {e}}^{-4\,c-4\,d\,x}}{64\,b\,d}+\frac {{\mathrm {e}}^{4\,c+4\,d\,x}}{64\,b\,d}+\frac {{\mathrm {e}}^{-2\,c-2\,d\,x}\,\left (a-2\,b\right )}{8\,b^2\,d}-\frac {{\mathrm {e}}^{2\,c+2\,d\,x}\,\left (a-2\,b\right )}{8\,b^2\,d}+\frac {\ln \left (\frac {4\,{\mathrm {e}}^{2\,c+2\,d\,x}\,{\left (a-b\right )}^3}{b^4}-\frac {2\,{\left (a-b\right )}^{5/2}\,\left (b+2\,a\,{\mathrm {e}}^{2\,c+2\,d\,x}-b\,{\mathrm {e}}^{2\,c+2\,d\,x}\right )}{\sqrt {a}\,b^4}\right )\,{\left (a-b\right )}^{5/2}}{2\,\sqrt {a}\,b^3\,d}-\frac {\ln \left (\frac {4\,{\mathrm {e}}^{2\,c+2\,d\,x}\,{\left (a-b\right )}^3}{b^4}+\frac {2\,{\left (a-b\right )}^{5/2}\,\left (b+2\,a\,{\mathrm {e}}^{2\,c+2\,d\,x}-b\,{\mathrm {e}}^{2\,c+2\,d\,x}\right )}{\sqrt {a}\,b^4}\right )\,{\left (a-b\right )}^{5/2}}{2\,\sqrt {a}\,b^3\,d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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