Optimal. Leaf size=85 \[ \frac {(a+2 b) \tanh ^{-1}(\cosh (c+d x))}{2 a^2 d}-\frac {\sqrt {b} \sqrt {a+b} \tanh ^{-1}\left (\frac {\sqrt {b} \text {sech}(c+d x)}{\sqrt {a+b}}\right )}{a^2 d}-\frac {\coth (c+d x) \text {csch}(c+d x)}{2 a d} \]
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Rubi [A] time = 0.12, antiderivative size = 85, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.217, Rules used = {3664, 471, 522, 207, 208} \[ \frac {(a+2 b) \tanh ^{-1}(\cosh (c+d x))}{2 a^2 d}-\frac {\sqrt {b} \sqrt {a+b} \tanh ^{-1}\left (\frac {\sqrt {b} \text {sech}(c+d x)}{\sqrt {a+b}}\right )}{a^2 d}-\frac {\coth (c+d x) \text {csch}(c+d x)}{2 a d} \]
Antiderivative was successfully verified.
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Rule 207
Rule 208
Rule 471
Rule 522
Rule 3664
Rubi steps
\begin {align*} \int \frac {\text {csch}^3(c+d x)}{a+b \tanh ^2(c+d x)} \, dx &=-\frac {\operatorname {Subst}\left (\int \frac {x^2}{\left (-1+x^2\right )^2 \left (a+b-b x^2\right )} \, dx,x,\text {sech}(c+d x)\right )}{d}\\ &=-\frac {\coth (c+d x) \text {csch}(c+d x)}{2 a d}-\frac {\operatorname {Subst}\left (\int \frac {a+b+b x^2}{\left (-1+x^2\right ) \left (a+b-b x^2\right )} \, dx,x,\text {sech}(c+d x)\right )}{2 a d}\\ &=-\frac {\coth (c+d x) \text {csch}(c+d x)}{2 a d}-\frac {(b (a+b)) \operatorname {Subst}\left (\int \frac {1}{a+b-b x^2} \, dx,x,\text {sech}(c+d x)\right )}{a^2 d}-\frac {(a+2 b) \operatorname {Subst}\left (\int \frac {1}{-1+x^2} \, dx,x,\text {sech}(c+d x)\right )}{2 a^2 d}\\ &=\frac {(a+2 b) \tanh ^{-1}(\cosh (c+d x))}{2 a^2 d}-\frac {\sqrt {b} \sqrt {a+b} \tanh ^{-1}\left (\frac {\sqrt {b} \text {sech}(c+d x)}{\sqrt {a+b}}\right )}{a^2 d}-\frac {\coth (c+d x) \text {csch}(c+d x)}{2 a d}\\ \end {align*}
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Mathematica [C] time = 0.70, size = 170, normalized size = 2.00 \[ -\frac {8 i \sqrt {b} \sqrt {a+b} \tan ^{-1}\left (\frac {-\sqrt {a} \tanh \left (\frac {1}{2} (c+d x)\right )-i \sqrt {a+b}}{\sqrt {b}}\right )+8 i \sqrt {b} \sqrt {a+b} \tan ^{-1}\left (\frac {\sqrt {a} \tanh \left (\frac {1}{2} (c+d x)\right )-i \sqrt {a+b}}{\sqrt {b}}\right )+a \text {csch}^2\left (\frac {1}{2} (c+d x)\right )+a \text {sech}^2\left (\frac {1}{2} (c+d x)\right )+4 a \log \left (\tanh \left (\frac {1}{2} (c+d x)\right )\right )+8 b \log \left (\tanh \left (\frac {1}{2} (c+d x)\right )\right )}{8 a^2 d} \]
Antiderivative was successfully verified.
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fricas [B] time = 1.27, size = 1790, normalized size = 21.06 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.38, size = 475, normalized size = 5.59 \[ \frac {\frac {2 \, {\left (3 \, a b - b^{2} - \sqrt {-a b} {\left (a - 3 \, b\right )}\right )} {\left | a e^{\left (2 \, c\right )} + b e^{\left (2 \, c\right )} \right |} \arctan \left (\frac {e^{\left (d x\right )}}{\sqrt {\frac {a^{3} e^{\left (2 \, c\right )} - a^{2} b e^{\left (2 \, c\right )} + \sqrt {{\left (a^{3} e^{\left (2 \, c\right )} - a^{2} b e^{\left (2 \, c\right )}\right )}^{2} - {\left (a^{3} e^{\left (4 \, c\right )} + a^{2} b e^{\left (4 \, c\right )}\right )} {\left (a^{3} + a^{2} b\right )}}}{a^{3} e^{\left (4 \, c\right )} + a^{2} b e^{\left (4 \, c\right )}}}}\right ) e^{\left (-2 \, c\right )}}{{\left (a^{3} - a^{2} b + 2 \, \sqrt {-a b} a^{2}\right )} \sqrt {a^{2} - b^{2} + 2 \, \sqrt {-a b} {\left (a + b\right )}}} + \frac {2 \, {\left (3 \, a b - b^{2} + \sqrt {-a b} {\left (a - 3 \, b\right )}\right )} {\left | a e^{\left (2 \, c\right )} + b e^{\left (2 \, c\right )} \right |} \arctan \left (\frac {e^{\left (d x\right )}}{\sqrt {\frac {a^{3} e^{\left (2 \, c\right )} - a^{2} b e^{\left (2 \, c\right )} - \sqrt {{\left (a^{3} e^{\left (2 \, c\right )} - a^{2} b e^{\left (2 \, c\right )}\right )}^{2} - {\left (a^{3} e^{\left (4 \, c\right )} + a^{2} b e^{\left (4 \, c\right )}\right )} {\left (a^{3} + a^{2} b\right )}}}{a^{3} e^{\left (4 \, c\right )} + a^{2} b e^{\left (4 \, c\right )}}}}\right ) e^{\left (-2 \, c\right )}}{{\left (a^{3} - a^{2} b - 2 \, \sqrt {-a b} a^{2}\right )} \sqrt {a^{2} - b^{2} - 2 \, \sqrt {-a b} {\left (a + b\right )}}} + \frac {{\left (a e^{c} + 2 \, b e^{c}\right )} e^{\left (-c\right )} \log \left (e^{\left (d x + c\right )} + 1\right )}{a^{2}} - \frac {{\left (a e^{c} + 2 \, b e^{c}\right )} e^{\left (-c\right )} \log \left ({\left | e^{\left (d x + c\right )} - 1 \right |}\right )}{a^{2}} - \frac {2 \, {\left (e^{\left (3 \, d x + 3 \, c\right )} + e^{\left (d x + c\right )}\right )}}{a {\left (e^{\left (2 \, d x + 2 \, c\right )} - 1\right )}^{2}}}{2 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.39, size = 181, normalized size = 2.13 \[ \frac {\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )}{8 d a}-\frac {b \arctanh \left (\frac {2 \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a +2 a +4 b}{4 \sqrt {a b +b^{2}}}\right )}{d a \sqrt {a b +b^{2}}}-\frac {b^{2} \arctanh \left (\frac {2 \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a +2 a +4 b}{4 \sqrt {a b +b^{2}}}\right )}{d \,a^{2} \sqrt {a b +b^{2}}}-\frac {1}{8 d a \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}-\frac {\ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2 d a}-\frac {b \ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d \,a^{2}} \]
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 \[ -\frac {e^{\left (3 \, d x + 3 \, c\right )} + e^{\left (d x + c\right )}}{a d e^{\left (4 \, d x + 4 \, c\right )} - 2 \, a d e^{\left (2 \, d x + 2 \, c\right )} + a d} + \frac {{\left (a + 2 \, b\right )} \log \left ({\left (e^{\left (d x + c\right )} + 1\right )} e^{\left (-c\right )}\right )}{2 \, a^{2} d} - \frac {{\left (a + 2 \, b\right )} \log \left ({\left (e^{\left (d x + c\right )} - 1\right )} e^{\left (-c\right )}\right )}{2 \, a^{2} d} + 8 \, \int \frac {{\left (a b e^{\left (3 \, c\right )} + b^{2} e^{\left (3 \, c\right )}\right )} e^{\left (3 \, d x\right )} - {\left (a b e^{c} + b^{2} e^{c}\right )} e^{\left (d x\right )}}{4 \, {\left (a^{3} + a^{2} b + {\left (a^{3} e^{\left (4 \, c\right )} + a^{2} b e^{\left (4 \, c\right )}\right )} e^{\left (4 \, d x\right )} + 2 \, {\left (a^{3} e^{\left (2 \, c\right )} - a^{2} b e^{\left (2 \, c\right )}\right )} e^{\left (2 \, d x\right )}\right )}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.80, size = 787, normalized size = 9.26 \[ \frac {\mathrm {atan}\left (\frac {{\mathrm {e}}^{d\,x}\,{\mathrm {e}}^c\,\left (18\,b^7\,\sqrt {-a^4\,d^2}+48\,a^2\,b^5\,\sqrt {-a^4\,d^2}+27\,a^3\,b^4\,\sqrt {-a^4\,d^2}+8\,a^4\,b^3\,\sqrt {-a^4\,d^2}+a^5\,b^2\,\sqrt {-a^4\,d^2}+45\,a\,b^6\,\sqrt {-a^4\,d^2}\right )}{9\,a^2\,b^6\,d\,\sqrt {a^2+4\,a\,b+4\,b^2}+18\,a^3\,b^5\,d\,\sqrt {a^2+4\,a\,b+4\,b^2}+15\,a^4\,b^4\,d\,\sqrt {a^2+4\,a\,b+4\,b^2}+6\,a^5\,b^3\,d\,\sqrt {a^2+4\,a\,b+4\,b^2}+a^6\,b^2\,d\,\sqrt {a^2+4\,a\,b+4\,b^2}}\right )\,\sqrt {a^2+4\,a\,b+4\,b^2}}{\sqrt {-a^4\,d^2}}-\frac {\left (2\,\mathrm {atan}\left (\frac {{\mathrm {e}}^{d\,x}\,{\mathrm {e}}^c\,\left (a+b\right )\,\sqrt {-a^4\,d^2}}{2\,a^2\,d\,\sqrt {b\,\left (a+b\right )}}\right )+2\,\mathrm {atan}\left (\frac {\left ({\mathrm {e}}^{d\,x}\,{\mathrm {e}}^c\,\left (\frac {64\,\left (2\,a^4\,b\,d\,\sqrt {b^2+a\,b}+6\,a^2\,b^3\,d\,\sqrt {b^2+a\,b}+6\,a^3\,b^2\,d\,\sqrt {b^2+a\,b}\right )}{a^9\,d^2\,{\left (a+b\right )}^2\,\left (a^2+2\,a\,b+b^2\right )}-\frac {32\,\left (3\,b^4\,\sqrt {-a^4\,d^2}+4\,a^2\,b^2\,\sqrt {-a^4\,d^2}+6\,a\,b^3\,\sqrt {-a^4\,d^2}+a^3\,b\,\sqrt {-a^4\,d^2}\right )}{a^7\,d\,\left (a+b\right )\,\sqrt {-a^4\,d^2}\,\sqrt {b\,\left (a+b\right )}\,\left (a^2+2\,a\,b+b^2\right )}\right )-\frac {32\,{\mathrm {e}}^{3\,c}\,{\mathrm {e}}^{3\,d\,x}\,\left (3\,b^4\,\sqrt {-a^4\,d^2}+4\,a^2\,b^2\,\sqrt {-a^4\,d^2}+6\,a\,b^3\,\sqrt {-a^4\,d^2}+a^3\,b\,\sqrt {-a^4\,d^2}\right )}{a^7\,d\,\left (a+b\right )\,\sqrt {-a^4\,d^2}\,\sqrt {b\,\left (a+b\right )}\,\left (a^2+2\,a\,b+b^2\right )}\right )\,\left (a^8\,\sqrt {-a^4\,d^2}+a^5\,b^3\,\sqrt {-a^4\,d^2}+3\,a^6\,b^2\,\sqrt {-a^4\,d^2}+3\,a^7\,b\,\sqrt {-a^4\,d^2}\right )}{64\,a^2\,b+192\,a\,b^2+192\,b^3}\right )\right )\,\sqrt {b^2+a\,b}}{2\,\sqrt {-a^4\,d^2}}-\frac {{\mathrm {e}}^{c+d\,x}}{a\,d\,\left ({\mathrm {e}}^{2\,c+2\,d\,x}-1\right )}-\frac {2\,{\mathrm {e}}^{c+d\,x}}{a\,d\,\left ({\mathrm {e}}^{4\,c+4\,d\,x}-2\,{\mathrm {e}}^{2\,c+2\,d\,x}+1\right )} \]
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 \[ \int \frac {\operatorname {csch}^{3}{\left (c + d x \right )}}{a + b \tanh ^{2}{\left (c + d x \right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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