Optimal. Leaf size=206 \[ \frac {b^2 \text {sech}(c+d x)}{a^3 d}-\frac {b^2 \tanh ^{-1}(\cosh (c+d x))}{a^3 d}+\frac {b \tanh (c+d x)}{a^2 d}+\frac {b \coth (c+d x)}{a^2 d}+\frac {2 b^5 \tanh ^{-1}\left (\frac {b-a \tanh \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a^2+b^2}}\right )}{a^3 d \left (a^2+b^2\right )^{3/2}}-\frac {b^3 \text {sech}(c+d x) (a \sinh (c+d x)+b)}{a^3 d \left (a^2+b^2\right )}-\frac {3 \text {sech}(c+d x)}{2 a d}+\frac {3 \tanh ^{-1}(\cosh (c+d x))}{2 a d}-\frac {\text {csch}^2(c+d x) \text {sech}(c+d x)}{2 a d} \]
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Rubi [A] time = 0.42, antiderivative size = 206, normalized size of antiderivative = 1.00, number of steps used = 17, number of rules used = 12, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.414, Rules used = {2898, 2622, 321, 207, 2620, 14, 288, 2696, 12, 2660, 618, 204} \[ \frac {2 b^5 \tanh ^{-1}\left (\frac {b-a \tanh \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a^2+b^2}}\right )}{a^3 d \left (a^2+b^2\right )^{3/2}}+\frac {b^2 \text {sech}(c+d x)}{a^3 d}-\frac {b^2 \tanh ^{-1}(\cosh (c+d x))}{a^3 d}-\frac {b^3 \text {sech}(c+d x) (a \sinh (c+d x)+b)}{a^3 d \left (a^2+b^2\right )}+\frac {b \tanh (c+d x)}{a^2 d}+\frac {b \coth (c+d x)}{a^2 d}-\frac {3 \text {sech}(c+d x)}{2 a d}+\frac {3 \tanh ^{-1}(\cosh (c+d x))}{2 a d}-\frac {\text {csch}^2(c+d x) \text {sech}(c+d x)}{2 a d} \]
Antiderivative was successfully verified.
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Rule 12
Rule 14
Rule 204
Rule 207
Rule 288
Rule 321
Rule 618
Rule 2620
Rule 2622
Rule 2660
Rule 2696
Rule 2898
Rubi steps
\begin {align*} \int \frac {\text {csch}^3(c+d x) \text {sech}^2(c+d x)}{a+b \sinh (c+d x)} \, dx &=-\left (i \int \left (\frac {i b^2 \text {csch}(c+d x) \text {sech}^2(c+d x)}{a^3}-\frac {i b \text {csch}^2(c+d x) \text {sech}^2(c+d x)}{a^2}+\frac {i \text {csch}^3(c+d x) \text {sech}^2(c+d x)}{a}-\frac {i b^3 \text {sech}^2(c+d x)}{a^3 (a+b \sinh (c+d x))}\right ) \, dx\right )\\ &=\frac {\int \text {csch}^3(c+d x) \text {sech}^2(c+d x) \, dx}{a}-\frac {b \int \text {csch}^2(c+d x) \text {sech}^2(c+d x) \, dx}{a^2}+\frac {b^2 \int \text {csch}(c+d x) \text {sech}^2(c+d x) \, dx}{a^3}-\frac {b^3 \int \frac {\text {sech}^2(c+d x)}{a+b \sinh (c+d x)} \, dx}{a^3}\\ &=-\frac {b^3 \text {sech}(c+d x) (b+a \sinh (c+d x))}{a^3 \left (a^2+b^2\right ) d}-\frac {b^3 \int \frac {b^2}{a+b \sinh (c+d x)} \, dx}{a^3 \left (a^2+b^2\right )}-\frac {\operatorname {Subst}\left (\int \frac {x^4}{\left (-1+x^2\right )^2} \, dx,x,\text {sech}(c+d x)\right )}{a d}-\frac {(i b) \operatorname {Subst}\left (\int \frac {1+x^2}{x^2} \, dx,x,i \tanh (c+d x)\right )}{a^2 d}+\frac {b^2 \operatorname {Subst}\left (\int \frac {x^2}{-1+x^2} \, dx,x,\text {sech}(c+d x)\right )}{a^3 d}\\ &=\frac {b^2 \text {sech}(c+d x)}{a^3 d}-\frac {\text {csch}^2(c+d x) \text {sech}(c+d x)}{2 a d}-\frac {b^3 \text {sech}(c+d x) (b+a \sinh (c+d x))}{a^3 \left (a^2+b^2\right ) d}-\frac {b^5 \int \frac {1}{a+b \sinh (c+d x)} \, dx}{a^3 \left (a^2+b^2\right )}-\frac {3 \operatorname {Subst}\left (\int \frac {x^2}{-1+x^2} \, dx,x,\text {sech}(c+d x)\right )}{2 a d}-\frac {(i b) \operatorname {Subst}\left (\int \left (1+\frac {1}{x^2}\right ) \, dx,x,i \tanh (c+d x)\right )}{a^2 d}+\frac {b^2 \operatorname {Subst}\left (\int \frac {1}{-1+x^2} \, dx,x,\text {sech}(c+d x)\right )}{a^3 d}\\ &=-\frac {b^2 \tanh ^{-1}(\cosh (c+d x))}{a^3 d}+\frac {b \coth (c+d x)}{a^2 d}-\frac {3 \text {sech}(c+d x)}{2 a d}+\frac {b^2 \text {sech}(c+d x)}{a^3 d}-\frac {\text {csch}^2(c+d x) \text {sech}(c+d x)}{2 a d}-\frac {b^3 \text {sech}(c+d x) (b+a \sinh (c+d x))}{a^3 \left (a^2+b^2\right ) d}+\frac {b \tanh (c+d x)}{a^2 d}-\frac {3 \operatorname {Subst}\left (\int \frac {1}{-1+x^2} \, dx,x,\text {sech}(c+d x)\right )}{2 a d}+\frac {\left (2 i b^5\right ) \operatorname {Subst}\left (\int \frac {1}{a-2 i b x+a x^2} \, dx,x,\tan \left (\frac {1}{2} (i c+i d x)\right )\right )}{a^3 \left (a^2+b^2\right ) d}\\ &=\frac {3 \tanh ^{-1}(\cosh (c+d x))}{2 a d}-\frac {b^2 \tanh ^{-1}(\cosh (c+d x))}{a^3 d}+\frac {b \coth (c+d x)}{a^2 d}-\frac {3 \text {sech}(c+d x)}{2 a d}+\frac {b^2 \text {sech}(c+d x)}{a^3 d}-\frac {\text {csch}^2(c+d x) \text {sech}(c+d x)}{2 a d}-\frac {b^3 \text {sech}(c+d x) (b+a \sinh (c+d x))}{a^3 \left (a^2+b^2\right ) d}+\frac {b \tanh (c+d x)}{a^2 d}-\frac {\left (4 i b^5\right ) \operatorname {Subst}\left (\int \frac {1}{-4 \left (a^2+b^2\right )-x^2} \, dx,x,-2 i b+2 a \tan \left (\frac {1}{2} (i c+i d x)\right )\right )}{a^3 \left (a^2+b^2\right ) d}\\ &=\frac {3 \tanh ^{-1}(\cosh (c+d x))}{2 a d}-\frac {b^2 \tanh ^{-1}(\cosh (c+d x))}{a^3 d}+\frac {2 b^5 \tanh ^{-1}\left (\frac {b-a \tanh \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a^2+b^2}}\right )}{a^3 \left (a^2+b^2\right )^{3/2} d}+\frac {b \coth (c+d x)}{a^2 d}-\frac {3 \text {sech}(c+d x)}{2 a d}+\frac {b^2 \text {sech}(c+d x)}{a^3 d}-\frac {\text {csch}^2(c+d x) \text {sech}(c+d x)}{2 a d}-\frac {b^3 \text {sech}(c+d x) (b+a \sinh (c+d x))}{a^3 \left (a^2+b^2\right ) d}+\frac {b \tanh (c+d x)}{a^2 d}\\ \end {align*}
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Mathematica [A] time = 2.43, size = 185, normalized size = 0.90 \[ \frac {\frac {8 \text {sech}(c+d x) (b \sinh (c+d x)-a)}{a^2+b^2}+\frac {4 b \tanh \left (\frac {1}{2} (c+d x)\right )}{a^2}+\frac {4 b \coth \left (\frac {1}{2} (c+d x)\right )}{a^2}-\frac {4 \left (3 a^2-2 b^2\right ) \log \left (\tanh \left (\frac {1}{2} (c+d x)\right )\right )}{a^3}+\frac {16 b^5 \tan ^{-1}\left (\frac {b-a \tanh \left (\frac {1}{2} (c+d x)\right )}{\sqrt {-a^2-b^2}}\right )}{a^3 \left (-a^2-b^2\right )^{3/2}}-\frac {\text {csch}^2\left (\frac {1}{2} (c+d x)\right )}{a}-\frac {\text {sech}^2\left (\frac {1}{2} (c+d x)\right )}{a}}{8 d} \]
Antiderivative was successfully verified.
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fricas [B] time = 1.23, size = 2653, normalized size = 12.88 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.57, size = 224, normalized size = 1.09 \[ -\frac {\frac {2 \, b^{5} \log \left (\frac {{\left | 2 \, b e^{\left (d x + c\right )} + 2 \, a - 2 \, \sqrt {a^{2} + b^{2}} \right |}}{{\left | 2 \, b e^{\left (d x + c\right )} + 2 \, a + 2 \, \sqrt {a^{2} + b^{2}} \right |}}\right )}{{\left (a^{5} + a^{3} b^{2}\right )} \sqrt {a^{2} + b^{2}}} + \frac {4 \, {\left (a e^{\left (d x + c\right )} + b\right )}}{{\left (a^{2} + b^{2}\right )} {\left (e^{\left (2 \, d x + 2 \, c\right )} + 1\right )}} - \frac {{\left (3 \, a^{2} - 2 \, b^{2}\right )} \log \left (e^{\left (d x + c\right )} + 1\right )}{a^{3}} + \frac {{\left (3 \, a^{2} - 2 \, b^{2}\right )} \log \left ({\left | e^{\left (d x + c\right )} - 1 \right |}\right )}{a^{3}} + \frac {2 \, {\left (a e^{\left (3 \, d x + 3 \, c\right )} - 2 \, b e^{\left (2 \, d x + 2 \, c\right )} + a e^{\left (d x + c\right )} + 2 \, b\right )}}{a^{2} {\left (e^{\left (2 \, d x + 2 \, c\right )} - 1\right )}^{2}}}{2 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 233, normalized size = 1.13 \[ \frac {\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )}{8 d a}+\frac {\tanh \left (\frac {d x}{2}+\frac {c}{2}\right ) b}{2 d \,a^{2}}-\frac {2 b^{5} \arctanh \left (\frac {2 a \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-2 b}{2 \sqrt {a^{2}+b^{2}}}\right )}{d \,a^{3} \left (a^{2}+b^{2}\right )^{\frac {3}{2}}}-\frac {1}{8 d a \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}-\frac {3 \ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2 d a}+\frac {\ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )\right ) b^{2}}{d \,a^{3}}+\frac {b}{2 d \,a^{2} \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}+\frac {2 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right ) b}{d \left (a^{2}+b^{2}\right ) \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}-\frac {2 a}{d \left (a^{2}+b^{2}\right ) \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.41, size = 334, normalized size = 1.62 \[ -\frac {b^{5} \log \left (\frac {b e^{\left (-d x - c\right )} - a - \sqrt {a^{2} + b^{2}}}{b e^{\left (-d x - c\right )} - a + \sqrt {a^{2} + b^{2}}}\right )}{{\left (a^{5} + a^{3} b^{2}\right )} \sqrt {a^{2} + b^{2}} d} - \frac {4 \, a^{2} b e^{\left (-2 \, d x - 2 \, c\right )} + 2 \, b^{3} e^{\left (-4 \, d x - 4 \, c\right )} - 4 \, a^{2} b - 2 \, b^{3} + {\left (3 \, a^{3} + a b^{2}\right )} e^{\left (-d x - c\right )} - 2 \, {\left (a^{3} - a b^{2}\right )} e^{\left (-3 \, d x - 3 \, c\right )} + {\left (3 \, a^{3} + a b^{2}\right )} e^{\left (-5 \, d x - 5 \, c\right )}}{{\left (a^{4} + a^{2} b^{2} - {\left (a^{4} + a^{2} b^{2}\right )} e^{\left (-2 \, d x - 2 \, c\right )} - {\left (a^{4} + a^{2} b^{2}\right )} e^{\left (-4 \, d x - 4 \, c\right )} + {\left (a^{4} + a^{2} b^{2}\right )} e^{\left (-6 \, d x - 6 \, c\right )}\right )} d} + \frac {{\left (3 \, a^{2} - 2 \, b^{2}\right )} \log \left (e^{\left (-d x - c\right )} + 1\right )}{2 \, a^{3} d} - \frac {{\left (3 \, a^{2} - 2 \, b^{2}\right )} \log \left (e^{\left (-d x - c\right )} - 1\right )}{2 \, a^{3} d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 3.47, size = 531, normalized size = 2.58 \[ \frac {b^5\,\ln \left (2\,a^4\,b-4\,a^5\,{\mathrm {e}}^{c+d\,x}+b^5+3\,a^2\,b^3+4\,a^2\,{\mathrm {e}}^{c+d\,x}\,\sqrt {{\left (a^2+b^2\right )}^3}+b^2\,{\mathrm {e}}^{c+d\,x}\,\sqrt {{\left (a^2+b^2\right )}^3}-7\,a^3\,b^2\,{\mathrm {e}}^{c+d\,x}-2\,a\,b\,\sqrt {{\left (a^2+b^2\right )}^3}-3\,a\,b^4\,{\mathrm {e}}^{c+d\,x}\right )\,\sqrt {{\left (a^2+b^2\right )}^3}}{d\,a^9+3\,d\,a^7\,b^2+3\,d\,a^5\,b^4+d\,a^3\,b^6}-\frac {\frac {{\mathrm {e}}^{c+d\,x}}{a\,d}-\frac {2\,\left (a^2\,b+b^3\right )}{a^2\,d\,\left (a^2+b^2\right )}}{{\mathrm {e}}^{2\,c+2\,d\,x}-1}-\frac {\frac {2\,b}{d\,\left (a^2+b^2\right )}+\frac {2\,a\,{\mathrm {e}}^{c+d\,x}}{d\,\left (a^2+b^2\right )}}{{\mathrm {e}}^{2\,c+2\,d\,x}+1}-\frac {\ln \left ({\mathrm {e}}^{c+d\,x}-1\right )\,\left (3\,a^2-2\,b^2\right )}{2\,a^3\,d}+\frac {\ln \left ({\mathrm {e}}^{c+d\,x}+1\right )\,\left (3\,a^2-2\,b^2\right )}{2\,a^3\,d}-\frac {b^5\,\ln \left (4\,a^5\,{\mathrm {e}}^{c+d\,x}-2\,a^4\,b-b^5-3\,a^2\,b^3+4\,a^2\,{\mathrm {e}}^{c+d\,x}\,\sqrt {{\left (a^2+b^2\right )}^3}+b^2\,{\mathrm {e}}^{c+d\,x}\,\sqrt {{\left (a^2+b^2\right )}^3}+7\,a^3\,b^2\,{\mathrm {e}}^{c+d\,x}-2\,a\,b\,\sqrt {{\left (a^2+b^2\right )}^3}+3\,a\,b^4\,{\mathrm {e}}^{c+d\,x}\right )\,\sqrt {{\left (a^2+b^2\right )}^3}}{d\,a^9+3\,d\,a^7\,b^2+3\,d\,a^5\,b^4+d\,a^3\,b^6}-\frac {2\,{\mathrm {e}}^{c+d\,x}}{a\,d\,\left ({\mathrm {e}}^{4\,c+4\,d\,x}-2\,{\mathrm {e}}^{2\,c+2\,d\,x}+1\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-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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