Optimal. Leaf size=165 \[ -\frac {5 \sqrt {b} (3 a-4 b) \tanh ^{-1}\left (\frac {\sqrt {b} \tanh (c+d x)}{\sqrt {a+b}}\right )}{8 d (a+b)^{9/2}}-\frac {b (7 a-4 b) \tanh (c+d x)}{8 d (a+b)^4 \left (a-b \tanh ^2(c+d x)+b\right )}-\frac {a b \tanh (c+d x)}{4 d (a+b)^3 \left (a-b \tanh ^2(c+d x)+b\right )^2}-\frac {\coth ^3(c+d x)}{3 d (a+b)^3}+\frac {(a-2 b) \coth (c+d x)}{d (a+b)^4} \]
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Rubi [A] time = 0.27, antiderivative size = 165, 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 = {4132, 456, 1259, 1261, 208} \[ -\frac {5 \sqrt {b} (3 a-4 b) \tanh ^{-1}\left (\frac {\sqrt {b} \tanh (c+d x)}{\sqrt {a+b}}\right )}{8 d (a+b)^{9/2}}-\frac {b (7 a-4 b) \tanh (c+d x)}{8 d (a+b)^4 \left (a-b \tanh ^2(c+d x)+b\right )}-\frac {a b \tanh (c+d x)}{4 d (a+b)^3 \left (a-b \tanh ^2(c+d x)+b\right )^2}-\frac {\coth ^3(c+d x)}{3 d (a+b)^3}+\frac {(a-2 b) \coth (c+d x)}{d (a+b)^4} \]
Antiderivative was successfully verified.
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Rule 208
Rule 456
Rule 1259
Rule 1261
Rule 4132
Rubi steps
\begin {align*} \int \frac {\text {csch}^4(c+d x)}{\left (a+b \text {sech}^2(c+d x)\right )^3} \, dx &=\frac {\operatorname {Subst}\left (\int \frac {1-x^2}{x^4 \left (a+b-b x^2\right )^3} \, dx,x,\tanh (c+d x)\right )}{d}\\ &=-\frac {a b \tanh (c+d x)}{4 (a+b)^3 d \left (a+b-b \tanh ^2(c+d x)\right )^2}+\frac {b \operatorname {Subst}\left (\int \frac {\frac {4}{b (a+b)}-\frac {4 a x^2}{b (a+b)^2}-\frac {3 a x^4}{(a+b)^3}}{x^4 \left (a+b-b x^2\right )^2} \, dx,x,\tanh (c+d x)\right )}{4 d}\\ &=-\frac {a b \tanh (c+d x)}{4 (a+b)^3 d \left (a+b-b \tanh ^2(c+d x)\right )^2}-\frac {(7 a-4 b) b \tanh (c+d x)}{8 (a+b)^4 d \left (a+b-b \tanh ^2(c+d x)\right )}+\frac {\operatorname {Subst}\left (\int \frac {8 b (a+b)-8 (a-b) b x^2-\frac {(7 a-4 b) b^2 x^4}{a+b}}{x^4 \left (a+b-b x^2\right )} \, dx,x,\tanh (c+d x)\right )}{8 b (a+b)^3 d}\\ &=-\frac {a b \tanh (c+d x)}{4 (a+b)^3 d \left (a+b-b \tanh ^2(c+d x)\right )^2}-\frac {(7 a-4 b) b \tanh (c+d x)}{8 (a+b)^4 d \left (a+b-b \tanh ^2(c+d x)\right )}+\frac {\operatorname {Subst}\left (\int \left (\frac {8 b}{x^4}+\frac {8 b (-a+2 b)}{(a+b) x^2}-\frac {5 b^2 (-3 a+4 b)}{(a+b) \left (-a-b+b x^2\right )}\right ) \, dx,x,\tanh (c+d x)\right )}{8 b (a+b)^3 d}\\ &=\frac {(a-2 b) \coth (c+d x)}{(a+b)^4 d}-\frac {\coth ^3(c+d x)}{3 (a+b)^3 d}-\frac {a b \tanh (c+d x)}{4 (a+b)^3 d \left (a+b-b \tanh ^2(c+d x)\right )^2}-\frac {(7 a-4 b) b \tanh (c+d x)}{8 (a+b)^4 d \left (a+b-b \tanh ^2(c+d x)\right )}+\frac {(5 (3 a-4 b) b) \operatorname {Subst}\left (\int \frac {1}{-a-b+b x^2} \, dx,x,\tanh (c+d x)\right )}{8 (a+b)^4 d}\\ &=-\frac {5 (3 a-4 b) \sqrt {b} \tanh ^{-1}\left (\frac {\sqrt {b} \tanh (c+d x)}{\sqrt {a+b}}\right )}{8 (a+b)^{9/2} d}+\frac {(a-2 b) \coth (c+d x)}{(a+b)^4 d}-\frac {\coth ^3(c+d x)}{3 (a+b)^3 d}-\frac {a b \tanh (c+d x)}{4 (a+b)^3 d \left (a+b-b \tanh ^2(c+d x)\right )^2}-\frac {(7 a-4 b) b \tanh (c+d x)}{8 (a+b)^4 d \left (a+b-b \tanh ^2(c+d x)\right )}\\ \end {align*}
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Mathematica [B] time = 5.53, size = 985, normalized size = 5.97 \[ -\frac {(\cosh (2 (c+d x)) a+a+2 b) \text {sech}^6(c+d x) \left (\frac {\text {csch}(c) \text {sech}(2 c) \left (224 \sinh (2 c-d x) a^4-224 \sinh (2 c+d x) a^4+176 \sinh (4 c+d x) a^4+48 \sinh (2 c+3 d x) a^4-96 \sinh (4 c+3 d x) a^4+48 \sinh (6 c+3 d x) a^4+16 \sinh (2 c+5 d x) a^4+16 \sinh (6 c+5 d x) a^4+16 \sinh (4 c+7 d x) a^4+16 \sinh (8 c+7 d x) a^4+576 b \sinh (2 c-d x) a^3-657 b \sinh (2 c+d x) a^3+569 b \sinh (4 c+d x) a^3+111 b \sinh (2 c+3 d x) a^3-152 b \sinh (4 c+3 d x) a^3+192 b \sinh (6 c+3 d x) a^3+72 b \sinh (4 c+5 d x) a^3+27 b \sinh (6 c+5 d x) a^3+45 b \sinh (8 c+5 d x) a^3-83 b \sinh (4 c+7 d x) a^3+27 b \sinh (6 c+7 d x) a^3-56 b \sinh (8 c+7 d x) a^3+124 b^2 \sinh (2 c-d x) a^2-538 b^2 \sinh (2 c+d x) a^2+666 b^2 \sinh (4 c+d x) a^2+360 b^2 \sinh (2 c+3 d x) a^2+146 b^2 \sinh (4 c+3 d x) a^2+558 b^2 \sinh (6 c+3 d x) a^2-598 b^2 \sinh (2 c+5 d x) a^2+150 b^2 \sinh (4 c+5 d x) a^2-388 b^2 \sinh (6 c+5 d x) a^2-60 b^2 \sinh (8 c+5 d x) a^2+6 b^2 \sinh (4 c+7 d x) a^2-6 b^2 \sinh (6 c+7 d x) a^2-2184 b^3 \sinh (2 c-d x) a+984 b^3 \sinh (2 c+d x) a+1704 b^3 \sinh (4 c+d x) a+312 b^3 \sinh (2 c+3 d x) a-728 b^3 \sinh (4 c+3 d x) a-168 b^3 \sinh (6 c+3 d x) a+48 b^3 \sinh (2 c+5 d x) a-48 b^3 \sinh (4 c+5 d x) a+4 \left (44 a^4+122 b a^3+63 b^2 a^2+126 b^3 a+36 b^4\right ) \sinh (d x)+\left (-96 a^4-71 b a^3+344 b^2 a^2-1208 b^3 a+48 b^4\right ) \sinh (3 d x)+144 b^4 \sinh (2 c-d x)+144 b^4 \sinh (2 c+d x)-144 b^4 \sinh (4 c+d x)-48 b^4 \sinh (2 c+3 d x)-48 b^4 \sinh (4 c+3 d x)+48 b^4 \sinh (6 c+3 d x)\right ) \text {csch}^3(c+d x)}{a}+\frac {480 (3 a-4 b) b \tanh ^{-1}\left (\frac {\text {sech}(d x) (\cosh (2 c)-\sinh (2 c)) ((a+2 b) \sinh (d x)-a \sinh (2 c+d x))}{2 \sqrt {a+b} \sqrt {b (\cosh (c)-\sinh (c))^4}}\right ) (\cosh (2 (c+d x)) a+a+2 b)^2 (\cosh (2 c)-\sinh (2 c))}{\sqrt {a+b} \sqrt {b (\cosh (c)-\sinh (c))^4}}\right )}{6144 (a+b)^4 d \left (b \text {sech}^2(c+d x)+a\right )^3} \]
Warning: Unable to verify antiderivative.
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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 = 1.66, size = 406, normalized size = 2.46 \[ -\frac {\frac {15 \, {\left (3 \, a b - 4 \, b^{2}\right )} \arctan \left (\frac {a e^{\left (2 \, d x + 2 \, c\right )} + a + 2 \, b}{2 \, \sqrt {-a b - b^{2}}}\right )}{{\left (a^{4} + 4 \, a^{3} b + 6 \, a^{2} b^{2} + 4 \, a b^{3} + b^{4}\right )} \sqrt {-a b - b^{2}}} - \frac {6 \, {\left (9 \, a^{3} b e^{\left (6 \, d x + 6 \, c\right )} + 20 \, a^{2} b^{2} e^{\left (6 \, d x + 6 \, c\right )} + 27 \, a^{3} b e^{\left (4 \, d x + 4 \, c\right )} + 66 \, a^{2} b^{2} e^{\left (4 \, d x + 4 \, c\right )} + 56 \, a b^{3} e^{\left (4 \, d x + 4 \, c\right )} - 16 \, b^{4} e^{\left (4 \, d x + 4 \, c\right )} + 27 \, a^{3} b e^{\left (2 \, d x + 2 \, c\right )} + 44 \, a^{2} b^{2} e^{\left (2 \, d x + 2 \, c\right )} - 16 \, a b^{3} e^{\left (2 \, d x + 2 \, c\right )} + 9 \, a^{3} b - 2 \, a^{2} b^{2}\right )}}{{\left (a^{5} + 4 \, a^{4} b + 6 \, a^{3} b^{2} + 4 \, a^{2} b^{3} + a b^{4}\right )} {\left (a e^{\left (4 \, d x + 4 \, c\right )} + 2 \, a e^{\left (2 \, d x + 2 \, c\right )} + 4 \, b e^{\left (2 \, d x + 2 \, c\right )} + a\right )}^{2}} + \frac {16 \, {\left (9 \, b e^{\left (4 \, d x + 4 \, c\right )} + 6 \, a e^{\left (2 \, d x + 2 \, c\right )} - 12 \, b e^{\left (2 \, d x + 2 \, c\right )} - 2 \, a + 7 \, b\right )}}{{\left (a^{4} + 4 \, a^{3} b + 6 \, a^{2} b^{2} + 4 \, a b^{3} + b^{4}\right )} {\left (e^{\left (2 \, d x + 2 \, c\right )} - 1\right )}^{3}}}{24 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.48, size = 1443, normalized size = 8.75 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.60, size = 782, normalized size = 4.74 \[ \frac {5 \, {\left (3 \, a b - 4 \, b^{2}\right )} \log \left (\frac {a e^{\left (-2 \, d x - 2 \, c\right )} + a + 2 \, b - 2 \, \sqrt {{\left (a + b\right )} b}}{a e^{\left (-2 \, d x - 2 \, c\right )} + a + 2 \, b + 2 \, \sqrt {{\left (a + b\right )} b}}\right )}{16 \, {\left (a^{4} + 4 \, a^{3} b + 6 \, a^{2} b^{2} + 4 \, a b^{3} + b^{4}\right )} \sqrt {{\left (a + b\right )} b} d} + \frac {16 \, a^{4} - 83 \, a^{3} b + 6 \, a^{2} b^{2} + 2 \, {\left (8 \, a^{4} - 299 \, a^{2} b^{2} + 24 \, a b^{3}\right )} e^{\left (-2 \, d x - 2 \, c\right )} - {\left (96 \, a^{4} + 71 \, a^{3} b - 344 \, a^{2} b^{2} + 1208 \, a b^{3} - 48 \, b^{4}\right )} e^{\left (-4 \, d x - 4 \, c\right )} - 4 \, {\left (56 \, a^{4} + 144 \, a^{3} b + 31 \, a^{2} b^{2} - 546 \, a b^{3} + 36 \, b^{4}\right )} e^{\left (-6 \, d x - 6 \, c\right )} - {\left (176 \, a^{4} + 569 \, a^{3} b + 666 \, a^{2} b^{2} + 1704 \, a b^{3} - 144 \, b^{4}\right )} e^{\left (-8 \, d x - 8 \, c\right )} - 6 \, {\left (8 \, a^{4} + 32 \, a^{3} b + 93 \, a^{2} b^{2} - 28 \, a b^{3} + 8 \, b^{4}\right )} e^{\left (-10 \, d x - 10 \, c\right )} - 15 \, {\left (3 \, a^{3} b - 4 \, a^{2} b^{2}\right )} e^{\left (-12 \, d x - 12 \, c\right )}}{12 \, {\left (a^{7} + 4 \, a^{6} b + 6 \, a^{5} b^{2} + 4 \, a^{4} b^{3} + a^{3} b^{4} + {\left (a^{7} + 12 \, a^{6} b + 38 \, a^{5} b^{2} + 52 \, a^{4} b^{3} + 33 \, a^{3} b^{4} + 8 \, a^{2} b^{5}\right )} e^{\left (-2 \, d x - 2 \, c\right )} - {\left (3 \, a^{7} + 20 \, a^{6} b + 34 \, a^{5} b^{2} - 4 \, a^{4} b^{3} - 61 \, a^{3} b^{4} - 56 \, a^{2} b^{5} - 16 \, a b^{6}\right )} e^{\left (-4 \, d x - 4 \, c\right )} - {\left (3 \, a^{7} + 28 \, a^{6} b + 130 \, a^{5} b^{2} + 300 \, a^{4} b^{3} + 355 \, a^{3} b^{4} + 208 \, a^{2} b^{5} + 48 \, a b^{6}\right )} e^{\left (-6 \, d x - 6 \, c\right )} + {\left (3 \, a^{7} + 28 \, a^{6} b + 130 \, a^{5} b^{2} + 300 \, a^{4} b^{3} + 355 \, a^{3} b^{4} + 208 \, a^{2} b^{5} + 48 \, a b^{6}\right )} e^{\left (-8 \, d x - 8 \, c\right )} + {\left (3 \, a^{7} + 20 \, a^{6} b + 34 \, a^{5} b^{2} - 4 \, a^{4} b^{3} - 61 \, a^{3} b^{4} - 56 \, a^{2} b^{5} - 16 \, a b^{6}\right )} e^{\left (-10 \, d x - 10 \, c\right )} - {\left (a^{7} + 12 \, a^{6} b + 38 \, a^{5} b^{2} + 52 \, a^{4} b^{3} + 33 \, a^{3} b^{4} + 8 \, a^{2} b^{5}\right )} e^{\left (-12 \, d x - 12 \, c\right )} - {\left (a^{7} + 4 \, a^{6} b + 6 \, a^{5} b^{2} + 4 \, a^{4} b^{3} + a^{3} b^{4}\right )} e^{\left (-14 \, d x - 14 \, c\right )}\right )} 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 \[ \int \frac {{\mathrm {cosh}\left (c+d\,x\right )}^6}{{\mathrm {sinh}\left (c+d\,x\right )}^4\,{\left (a\,{\mathrm {cosh}\left (c+d\,x\right )}^2+b\right )}^3} \,d x \]
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 \[ \int \frac {\operatorname {csch}^{4}{\left (c + d x \right )}}{\left (a + b \operatorname {sech}^{2}{\left (c + d x \right )}\right )^{3}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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