Optimal. Leaf size=110 \[ -\frac {b^3 \tanh ^{-1}\left (\frac {\sqrt {a-b} \tanh (c+d x)}{\sqrt {a}}\right )}{a^{7/2} \sqrt {a-b} d}-\frac {\left (a^2+a b+b^2\right ) \coth (c+d x)}{a^3 d}+\frac {(2 a+b) \coth ^3(c+d x)}{3 a^2 d}-\frac {\coth ^5(c+d x)}{5 a d} \]
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Rubi [A]
time = 0.10, antiderivative size = 110, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 3, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.130, Rules used = {3266, 472, 214}
\begin {gather*} -\frac {b^3 \tanh ^{-1}\left (\frac {\sqrt {a-b} \tanh (c+d x)}{\sqrt {a}}\right )}{a^{7/2} d \sqrt {a-b}}+\frac {(2 a+b) \coth ^3(c+d x)}{3 a^2 d}-\frac {\left (a^2+a b+b^2\right ) \coth (c+d x)}{a^3 d}-\frac {\coth ^5(c+d x)}{5 a d} \end {gather*}
Antiderivative was successfully verified.
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Rule 214
Rule 472
Rule 3266
Rubi steps
\begin {align*} \int \frac {\text {csch}^6(c+d x)}{a+b \sinh ^2(c+d x)} \, dx &=\frac {\text {Subst}\left (\int \frac {\left (1-x^2\right )^3}{x^6 \left (a-(a-b) x^2\right )} \, dx,x,\tanh (c+d x)\right )}{d}\\ &=\frac {\text {Subst}\left (\int \left (\frac {1}{a x^6}+\frac {-2 a-b}{a^2 x^4}+\frac {a^2+a b+b^2}{a^3 x^2}+\frac {b^3}{a^3 \left (-a+(a-b) x^2\right )}\right ) \, dx,x,\tanh (c+d x)\right )}{d}\\ &=-\frac {\left (a^2+a b+b^2\right ) \coth (c+d x)}{a^3 d}+\frac {(2 a+b) \coth ^3(c+d x)}{3 a^2 d}-\frac {\coth ^5(c+d x)}{5 a d}+\frac {b^3 \text {Subst}\left (\int \frac {1}{-a+(a-b) x^2} \, dx,x,\tanh (c+d x)\right )}{a^3 d}\\ &=-\frac {b^3 \tanh ^{-1}\left (\frac {\sqrt {a-b} \tanh (c+d x)}{\sqrt {a}}\right )}{a^{7/2} \sqrt {a-b} d}-\frac {\left (a^2+a b+b^2\right ) \coth (c+d x)}{a^3 d}+\frac {(2 a+b) \coth ^3(c+d x)}{3 a^2 d}-\frac {\coth ^5(c+d x)}{5 a d}\\ \end {align*}
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Mathematica [A]
time = 0.95, size = 155, normalized size = 1.41 \begin {gather*} -\frac {(2 a-b+b \cosh (2 (c+d x))) \text {csch}^2(c+d x) \left (15 b^3 \tanh ^{-1}\left (\frac {\sqrt {a-b} \tanh (c+d x)}{\sqrt {a}}\right )+\sqrt {a} \sqrt {a-b} \coth (c+d x) \left (8 a^2+10 a b+15 b^2-a (4 a+5 b) \text {csch}^2(c+d x)+3 a^2 \text {csch}^4(c+d x)\right )\right )}{30 a^{7/2} \sqrt {a-b} d \left (b+a \text {csch}^2(c+d x)\right )} \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. \(346\) vs.
\(2(98)=196\).
time = 1.58, size = 347, normalized size = 3.15
method | result | size |
risch | \(-\frac {2 \left (15 b^{2} {\mathrm e}^{8 d x +8 c}-30 a b \,{\mathrm e}^{6 d x +6 c}-60 b^{2} {\mathrm e}^{6 d x +6 c}+80 a^{2} {\mathrm e}^{4 d x +4 c}+70 a b \,{\mathrm e}^{4 d x +4 c}+90 b^{2} {\mathrm e}^{4 d x +4 c}-40 a^{2} {\mathrm e}^{2 d x +2 c}-50 a b \,{\mathrm e}^{2 d x +2 c}-60 b^{2} {\mathrm e}^{2 d x +2 c}+8 a^{2}+10 a b +15 b^{2}\right )}{15 d \,a^{3} \left ({\mathrm e}^{2 d x +2 c}-1\right )^{5}}+\frac {b^{3} \ln \left ({\mathrm e}^{2 d x +2 c}+\frac {2 a \sqrt {a^{2}-a b}-b \sqrt {a^{2}-a b}+2 a^{2}-2 a b}{b \sqrt {a^{2}-a b}}\right )}{2 \sqrt {a^{2}-a b}\, d \,a^{3}}-\frac {b^{3} \ln \left ({\mathrm e}^{2 d x +2 c}+\frac {2 a \sqrt {a^{2}-a b}-b \sqrt {a^{2}-a b}-2 a^{2}+2 a b}{b \sqrt {a^{2}-a b}}\right )}{2 \sqrt {a^{2}-a b}\, d \,a^{3}}\) | \(325\) |
derivativedivides | \(\frac {-\frac {\frac {a^{2} \left (\tanh ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{5}-\frac {5 a^{2} \left (\tanh ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3}-\frac {4 a \left (\tanh ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) b}{3}+10 a^{2} \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+12 a \tanh \left (\frac {d x}{2}+\frac {c}{2}\right ) b +16 b^{2} \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{32 a^{3}}+\frac {2 b^{3} \left (\frac {\left (\sqrt {-b \left (a -b \right )}+b \right ) \arctan \left (\frac {a \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{\sqrt {\left (2 \sqrt {-b \left (a -b \right )}-a +2 b \right ) a}}\right )}{2 a \sqrt {-b \left (a -b \right )}\, \sqrt {\left (2 \sqrt {-b \left (a -b \right )}-a +2 b \right ) a}}-\frac {\left (\sqrt {-b \left (a -b \right )}-b \right ) \arctanh \left (\frac {a \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{\sqrt {\left (2 \sqrt {-b \left (a -b \right )}+a -2 b \right ) a}}\right )}{2 a \sqrt {-b \left (a -b \right )}\, \sqrt {\left (2 \sqrt {-b \left (a -b \right )}+a -2 b \right ) a}}\right )}{a^{2}}-\frac {1}{160 a \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}-\frac {-5 a -4 b}{96 a^{2} \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}-\frac {10 a^{2}+12 a b +16 b^{2}}{32 a^{3} \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}}{d}\) | \(347\) |
default | \(\frac {-\frac {\frac {a^{2} \left (\tanh ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{5}-\frac {5 a^{2} \left (\tanh ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3}-\frac {4 a \left (\tanh ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) b}{3}+10 a^{2} \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+12 a \tanh \left (\frac {d x}{2}+\frac {c}{2}\right ) b +16 b^{2} \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{32 a^{3}}+\frac {2 b^{3} \left (\frac {\left (\sqrt {-b \left (a -b \right )}+b \right ) \arctan \left (\frac {a \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{\sqrt {\left (2 \sqrt {-b \left (a -b \right )}-a +2 b \right ) a}}\right )}{2 a \sqrt {-b \left (a -b \right )}\, \sqrt {\left (2 \sqrt {-b \left (a -b \right )}-a +2 b \right ) a}}-\frac {\left (\sqrt {-b \left (a -b \right )}-b \right ) \arctanh \left (\frac {a \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{\sqrt {\left (2 \sqrt {-b \left (a -b \right )}+a -2 b \right ) a}}\right )}{2 a \sqrt {-b \left (a -b \right )}\, \sqrt {\left (2 \sqrt {-b \left (a -b \right )}+a -2 b \right ) a}}\right )}{a^{2}}-\frac {1}{160 a \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}-\frac {-5 a -4 b}{96 a^{2} \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}-\frac {10 a^{2}+12 a b +16 b^{2}}{32 a^{3} \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}}{d}\) | \(347\) |
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. 2142 vs.
\(2 (98) = 196\).
time = 0.48, size = 4540, normalized size = 41.27 \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. 213 vs.
\(2 (98) = 196\).
time = 0.69, size = 213, normalized size = 1.94 \begin {gather*} -\frac {\frac {15 \, b^{3} \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} a^{3}} + \frac {2 \, {\left (15 \, b^{2} e^{\left (8 \, d x + 8 \, c\right )} - 30 \, a b e^{\left (6 \, d x + 6 \, c\right )} - 60 \, b^{2} e^{\left (6 \, d x + 6 \, c\right )} + 80 \, a^{2} e^{\left (4 \, d x + 4 \, c\right )} + 70 \, a b e^{\left (4 \, d x + 4 \, c\right )} + 90 \, b^{2} e^{\left (4 \, d x + 4 \, c\right )} - 40 \, a^{2} e^{\left (2 \, d x + 2 \, c\right )} - 50 \, a b e^{\left (2 \, d x + 2 \, c\right )} - 60 \, b^{2} e^{\left (2 \, d x + 2 \, c\right )} + 8 \, a^{2} + 10 \, a b + 15 \, b^{2}\right )}}{a^{3} {\left (e^{\left (2 \, d x + 2 \, c\right )} - 1\right )}^{5}}}{15 \, d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 1.21, size = 479, normalized size = 4.35 \begin {gather*} \frac {4\,b}{a^2\,d\,\left ({\mathrm {e}}^{4\,c+4\,d\,x}-2\,{\mathrm {e}}^{2\,c+2\,d\,x}+1\right )}-\frac {32}{5\,a\,d\,\left (5\,{\mathrm {e}}^{2\,c+2\,d\,x}-10\,{\mathrm {e}}^{4\,c+4\,d\,x}+10\,{\mathrm {e}}^{6\,c+6\,d\,x}-5\,{\mathrm {e}}^{8\,c+8\,d\,x}+{\mathrm {e}}^{10\,c+10\,d\,x}-1\right )}-\frac {2\,b^2}{a^3\,d\,\left ({\mathrm {e}}^{2\,c+2\,d\,x}-1\right )}-\frac {8\,\left (4\,a-b\right )}{3\,a^2\,d\,\left (3\,{\mathrm {e}}^{2\,c+2\,d\,x}-3\,{\mathrm {e}}^{4\,c+4\,d\,x}+{\mathrm {e}}^{6\,c+6\,d\,x}-1\right )}-\frac {16}{a\,d\,\left (6\,{\mathrm {e}}^{4\,c+4\,d\,x}-4\,{\mathrm {e}}^{2\,c+2\,d\,x}-4\,{\mathrm {e}}^{6\,c+6\,d\,x}+{\mathrm {e}}^{8\,c+8\,d\,x}+1\right )}+\frac {b^3\,\ln \left (\frac {4\,b^4\,\left (2\,a\,b-b^2+8\,a^2\,{\mathrm {e}}^{2\,c+2\,d\,x}+b^2\,{\mathrm {e}}^{2\,c+2\,d\,x}-8\,a\,b\,{\mathrm {e}}^{2\,c+2\,d\,x}\right )}{a^7\,\left (a-b\right )}-\frac {8\,b^4\,\left (b+4\,a\,{\mathrm {e}}^{2\,c+2\,d\,x}-2\,b\,{\mathrm {e}}^{2\,c+2\,d\,x}\right )}{a^{13/2}\,\sqrt {a-b}}\right )}{2\,a^{7/2}\,d\,\sqrt {a-b}}-\frac {b^3\,\ln \left (\frac {4\,b^4\,\left (2\,a\,b-b^2+8\,a^2\,{\mathrm {e}}^{2\,c+2\,d\,x}+b^2\,{\mathrm {e}}^{2\,c+2\,d\,x}-8\,a\,b\,{\mathrm {e}}^{2\,c+2\,d\,x}\right )}{a^7\,\left (a-b\right )}+\frac {8\,b^4\,\left (b+4\,a\,{\mathrm {e}}^{2\,c+2\,d\,x}-2\,b\,{\mathrm {e}}^{2\,c+2\,d\,x}\right )}{a^{13/2}\,\sqrt {a-b}}\right )}{2\,a^{7/2}\,d\,\sqrt {a-b}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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