Optimal. Leaf size=151 \[ \frac {5 \sqrt {b} (3 a+7 b) \tan ^{-1}\left (\frac {\sqrt {b} \tanh (c+d x)}{\sqrt {a}}\right )}{8 a^{9/2} d}+\frac {b (7 a+11 b) \tanh (c+d x)}{8 a^4 d \left (a+b \tanh ^2(c+d x)\right )}+\frac {(a+3 b) \coth (c+d x)}{a^4 d}+\frac {b (a+b) \tanh (c+d x)}{4 a^3 d \left (a+b \tanh ^2(c+d x)\right )^2}-\frac {\coth ^3(c+d x)}{3 a^3 d} \]
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Rubi [A] time = 0.20, antiderivative size = 151, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.217, Rules used = {3663, 456, 1259, 1261, 205} \[ \frac {b (7 a+11 b) \tanh (c+d x)}{8 a^4 d \left (a+b \tanh ^2(c+d x)\right )}+\frac {b (a+b) \tanh (c+d x)}{4 a^3 d \left (a+b \tanh ^2(c+d x)\right )^2}+\frac {(a+3 b) \coth (c+d x)}{a^4 d}+\frac {5 \sqrt {b} (3 a+7 b) \tan ^{-1}\left (\frac {\sqrt {b} \tanh (c+d x)}{\sqrt {a}}\right )}{8 a^{9/2} d}-\frac {\coth ^3(c+d x)}{3 a^3 d} \]
Antiderivative was successfully verified.
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Rule 205
Rule 456
Rule 1259
Rule 1261
Rule 3663
Rubi steps
\begin {align*} \int \frac {\text {csch}^4(c+d x)}{\left (a+b \tanh ^2(c+d x)\right )^3} \, dx &=\frac {\operatorname {Subst}\left (\int \frac {1-x^2}{x^4 \left (a+b x^2\right )^3} \, dx,x,\tanh (c+d x)\right )}{d}\\ &=\frac {b (a+b) \tanh (c+d x)}{4 a^3 d \left (a+b \tanh ^2(c+d x)\right )^2}-\frac {b \operatorname {Subst}\left (\int \frac {-\frac {4}{a b}+\frac {4 (a+b) x^2}{a^2 b}-\frac {3 (a+b) x^4}{a^3}}{x^4 \left (a+b x^2\right )^2} \, dx,x,\tanh (c+d x)\right )}{4 d}\\ &=\frac {b (a+b) \tanh (c+d x)}{4 a^3 d \left (a+b \tanh ^2(c+d x)\right )^2}+\frac {b (7 a+11 b) \tanh (c+d x)}{8 a^4 d \left (a+b \tanh ^2(c+d x)\right )}-\frac {\operatorname {Subst}\left (\int \frac {-8 a b+8 b (a+2 b) x^2-\frac {b^2 (7 a+11 b) x^4}{a}}{x^4 \left (a+b x^2\right )} \, dx,x,\tanh (c+d x)\right )}{8 a^3 b d}\\ &=\frac {b (a+b) \tanh (c+d x)}{4 a^3 d \left (a+b \tanh ^2(c+d x)\right )^2}+\frac {b (7 a+11 b) \tanh (c+d x)}{8 a^4 d \left (a+b \tanh ^2(c+d x)\right )}-\frac {\operatorname {Subst}\left (\int \left (-\frac {8 b}{x^4}+\frac {8 b (a+3 b)}{a x^2}-\frac {5 b^2 (3 a+7 b)}{a \left (a+b x^2\right )}\right ) \, dx,x,\tanh (c+d x)\right )}{8 a^3 b d}\\ &=\frac {(a+3 b) \coth (c+d x)}{a^4 d}-\frac {\coth ^3(c+d x)}{3 a^3 d}+\frac {b (a+b) \tanh (c+d x)}{4 a^3 d \left (a+b \tanh ^2(c+d x)\right )^2}+\frac {b (7 a+11 b) \tanh (c+d x)}{8 a^4 d \left (a+b \tanh ^2(c+d x)\right )}+\frac {(5 b (3 a+7 b)) \operatorname {Subst}\left (\int \frac {1}{a+b x^2} \, dx,x,\tanh (c+d x)\right )}{8 a^4 d}\\ &=\frac {5 \sqrt {b} (3 a+7 b) \tan ^{-1}\left (\frac {\sqrt {b} \tanh (c+d x)}{\sqrt {a}}\right )}{8 a^{9/2} d}+\frac {(a+3 b) \coth (c+d x)}{a^4 d}-\frac {\coth ^3(c+d x)}{3 a^3 d}+\frac {b (a+b) \tanh (c+d x)}{4 a^3 d \left (a+b \tanh ^2(c+d x)\right )^2}+\frac {b (7 a+11 b) \tanh (c+d x)}{8 a^4 d \left (a+b \tanh ^2(c+d x)\right )}\\ \end {align*}
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Mathematica [A] time = 1.40, size = 149, normalized size = 0.99 \[ \frac {\frac {3 \sqrt {a} b \sinh (2 (c+d x)) \left (\left (9 a^2+20 a b+11 b^2\right ) \cosh (2 (c+d x))+9 a^2+6 a b-11 b^2\right )}{((a+b) \cosh (2 (c+d x))+a-b)^2}+15 \sqrt {b} (3 a+7 b) \tan ^{-1}\left (\frac {\sqrt {b} \tanh (c+d x)}{\sqrt {a}}\right )-8 \sqrt {a} \coth (c+d x) \left (a \text {csch}^2(c+d x)-2 a-9 b\right )}{24 a^{9/2} d} \]
Antiderivative was successfully verified.
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fricas [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.85, size = 407, normalized size = 2.70 \[ \frac {\frac {15 \, {\left (3 \, a b e^{\left (2 \, c\right )} + 7 \, b^{2} e^{\left (2 \, c\right )}\right )} \arctan \left (\frac {a e^{\left (2 \, d x + 2 \, c\right )} + b e^{\left (2 \, d x + 2 \, c\right )} + a - b}{2 \, \sqrt {a b}}\right ) e^{\left (-2 \, c\right )}}{\sqrt {a b} a^{4}} - \frac {6 \, {\left (9 \, a^{3} b e^{\left (6 \, d x + 6 \, c\right )} + 7 \, a^{2} b^{2} e^{\left (6 \, d x + 6 \, c\right )} - 13 \, a b^{3} e^{\left (6 \, d x + 6 \, c\right )} - 11 \, b^{4} e^{\left (6 \, d x + 6 \, c\right )} + 27 \, a^{3} b e^{\left (4 \, d x + 4 \, c\right )} + 15 \, a^{2} b^{2} e^{\left (4 \, d x + 4 \, c\right )} + 5 \, a b^{3} e^{\left (4 \, d x + 4 \, c\right )} + 33 \, b^{4} e^{\left (4 \, d x + 4 \, c\right )} + 27 \, a^{3} b e^{\left (2 \, d x + 2 \, c\right )} + 37 \, a^{2} b^{2} e^{\left (2 \, d x + 2 \, c\right )} - 23 \, a b^{3} e^{\left (2 \, d x + 2 \, c\right )} - 33 \, b^{4} e^{\left (2 \, d x + 2 \, c\right )} + 9 \, a^{3} b + 29 \, a^{2} b^{2} + 31 \, a b^{3} + 11 \, b^{4}\right )}}{{\left (a^{5} + a^{4} b\right )} {\left (a e^{\left (4 \, d x + 4 \, c\right )} + b e^{\left (4 \, d x + 4 \, c\right )} + 2 \, a e^{\left (2 \, d x + 2 \, c\right )} - 2 \, b e^{\left (2 \, d x + 2 \, c\right )} + a + b\right )}^{2}} + \frac {16 \, {\left (9 \, b e^{\left (4 \, d x + 4 \, c\right )} - 6 \, a e^{\left (2 \, d x + 2 \, c\right )} - 18 \, b e^{\left (2 \, d x + 2 \, c\right )} + 2 \, a + 9 \, b\right )}}{a^{4} {\left (e^{\left (2 \, d x + 2 \, c\right )} - 1\right )}^{3}}}{24 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.48, size = 1416, normalized size = 9.38 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.70, size = 615, normalized size = 4.07 \[ \frac {16 \, a^{4} + 147 \, a^{3} b + 351 \, a^{2} b^{2} + 325 \, a b^{3} + 105 \, b^{4} + 2 \, {\left (8 \, a^{4} + 32 \, a^{3} b - 251 \, a^{2} b^{2} - 590 \, a b^{3} - 315 \, b^{4}\right )} e^{\left (-2 \, d x - 2 \, c\right )} - {\left (96 \, a^{4} + 313 \, a^{3} b + 19 \, a^{2} b^{2} - 1725 \, a b^{3} - 1575 \, b^{4}\right )} e^{\left (-4 \, d x - 4 \, c\right )} - 4 \, {\left (56 \, a^{4} + 80 \, a^{3} b - 65 \, a^{2} b^{2} + 400 \, a b^{3} + 525 \, b^{4}\right )} e^{\left (-6 \, d x - 6 \, c\right )} - {\left (176 \, a^{4} + 135 \, a^{3} b + 15 \, a^{2} b^{2} - 1375 \, a b^{3} - 1575 \, b^{4}\right )} e^{\left (-8 \, d x - 8 \, c\right )} - 6 \, {\left (8 \, a^{4} + 45 \, a^{2} b^{2} + 150 \, a b^{3} + 105 \, b^{4}\right )} e^{\left (-10 \, d x - 10 \, c\right )} + 15 \, {\left (3 \, a^{3} b + 13 \, a^{2} b^{2} + 17 \, a b^{3} + 7 \, b^{4}\right )} e^{\left (-12 \, d x - 12 \, c\right )}}{12 \, {\left (a^{7} + 3 \, a^{6} b + 3 \, a^{5} b^{2} + a^{4} b^{3} + {\left (a^{7} - 5 \, a^{6} b - 13 \, a^{5} b^{2} - 7 \, a^{4} b^{3}\right )} e^{\left (-2 \, d x - 2 \, c\right )} - {\left (3 \, a^{7} + a^{6} b - 23 \, a^{5} b^{2} - 21 \, a^{4} b^{3}\right )} e^{\left (-4 \, d x - 4 \, c\right )} - {\left (3 \, a^{7} - 7 \, a^{6} b + 25 \, a^{5} b^{2} + 35 \, a^{4} b^{3}\right )} e^{\left (-6 \, d x - 6 \, c\right )} + {\left (3 \, a^{7} - 7 \, a^{6} b + 25 \, a^{5} b^{2} + 35 \, a^{4} b^{3}\right )} e^{\left (-8 \, d x - 8 \, c\right )} + {\left (3 \, a^{7} + a^{6} b - 23 \, a^{5} b^{2} - 21 \, a^{4} b^{3}\right )} e^{\left (-10 \, d x - 10 \, c\right )} - {\left (a^{7} - 5 \, a^{6} b - 13 \, a^{5} b^{2} - 7 \, a^{4} b^{3}\right )} e^{\left (-12 \, d x - 12 \, c\right )} - {\left (a^{7} + 3 \, a^{6} b + 3 \, a^{5} b^{2} + a^{4} b^{3}\right )} e^{\left (-14 \, d x - 14 \, c\right )}\right )} d} - \frac {5 \, {\left (3 \, a b + 7 \, b^{2}\right )} \arctan \left (\frac {{\left (a + b\right )} e^{\left (-2 \, d x - 2 \, c\right )} + a - b}{2 \, \sqrt {a b}}\right )}{8 \, \sqrt {a b} a^{4} d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {1}{{\mathrm {sinh}\left (c+d\,x\right )}^4\,{\left (b\,{\mathrm {tanh}\left (c+d\,x\right )}^2+a\right )}^3} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\operatorname {csch}^{4}{\left (c + d x \right )}}{\left (a + b \tanh ^{2}{\left (c + d x \right )}\right )^{3}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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