Optimal. Leaf size=144 \[ \frac {b^2 \text {sech}(c+d x) (a \sinh (c+d x)+b)}{a^2 d \left (a^2+b^2\right )}-\frac {2 b^4 \tanh ^{-1}\left (\frac {b-a \tanh \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a^2+b^2}}\right )}{a^2 d \left (a^2+b^2\right )^{3/2}}-\frac {b \text {sech}(c+d x)}{a^2 d}+\frac {b \tanh ^{-1}(\cosh (c+d x))}{a^2 d}-\frac {\tanh (c+d x)}{a d}-\frac {\coth (c+d x)}{a d} \]
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Rubi [A] time = 0.31, antiderivative size = 144, normalized size of antiderivative = 1.00, number of steps used = 13, number of rules used = 11, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.379, Rules used = {2898, 2622, 321, 207, 2620, 14, 2696, 12, 2660, 618, 204} \[ -\frac {2 b^4 \tanh ^{-1}\left (\frac {b-a \tanh \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a^2+b^2}}\right )}{a^2 d \left (a^2+b^2\right )^{3/2}}+\frac {b^2 \text {sech}(c+d x) (a \sinh (c+d x)+b)}{a^2 d \left (a^2+b^2\right )}-\frac {b \text {sech}(c+d x)}{a^2 d}+\frac {b \tanh ^{-1}(\cosh (c+d x))}{a^2 d}-\frac {\tanh (c+d x)}{a d}-\frac {\coth (c+d x)}{a d} \]
Antiderivative was successfully verified.
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Rule 12
Rule 14
Rule 204
Rule 207
Rule 321
Rule 618
Rule 2620
Rule 2622
Rule 2660
Rule 2696
Rule 2898
Rubi steps
\begin {align*} \int \frac {\text {csch}^2(c+d x) \text {sech}^2(c+d x)}{a+b \sinh (c+d x)} \, dx &=-\int \left (\frac {b \text {csch}(c+d x) \text {sech}^2(c+d x)}{a^2}-\frac {\text {csch}^2(c+d x) \text {sech}^2(c+d x)}{a}-\frac {b^2 \text {sech}^2(c+d x)}{a^2 (a+b \sinh (c+d x))}\right ) \, dx\\ &=\frac {\int \text {csch}^2(c+d x) \text {sech}^2(c+d x) \, dx}{a}-\frac {b \int \text {csch}(c+d x) \text {sech}^2(c+d x) \, dx}{a^2}+\frac {b^2 \int \frac {\text {sech}^2(c+d x)}{a+b \sinh (c+d x)} \, dx}{a^2}\\ &=\frac {b^2 \text {sech}(c+d x) (b+a \sinh (c+d x))}{a^2 \left (a^2+b^2\right ) d}+\frac {b^2 \int \frac {b^2}{a+b \sinh (c+d x)} \, dx}{a^2 \left (a^2+b^2\right )}+\frac {i \operatorname {Subst}\left (\int \frac {1+x^2}{x^2} \, dx,x,i \tanh (c+d x)\right )}{a d}-\frac {b \operatorname {Subst}\left (\int \frac {x^2}{-1+x^2} \, dx,x,\text {sech}(c+d x)\right )}{a^2 d}\\ &=-\frac {b \text {sech}(c+d x)}{a^2 d}+\frac {b^2 \text {sech}(c+d x) (b+a \sinh (c+d x))}{a^2 \left (a^2+b^2\right ) d}+\frac {b^4 \int \frac {1}{a+b \sinh (c+d x)} \, dx}{a^2 \left (a^2+b^2\right )}+\frac {i \operatorname {Subst}\left (\int \left (1+\frac {1}{x^2}\right ) \, dx,x,i \tanh (c+d x)\right )}{a d}-\frac {b \operatorname {Subst}\left (\int \frac {1}{-1+x^2} \, dx,x,\text {sech}(c+d x)\right )}{a^2 d}\\ &=\frac {b \tanh ^{-1}(\cosh (c+d x))}{a^2 d}-\frac {\coth (c+d x)}{a d}-\frac {b \text {sech}(c+d x)}{a^2 d}+\frac {b^2 \text {sech}(c+d x) (b+a \sinh (c+d x))}{a^2 \left (a^2+b^2\right ) d}-\frac {\tanh (c+d x)}{a d}-\frac {\left (2 i b^4\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^2 \left (a^2+b^2\right ) d}\\ &=\frac {b \tanh ^{-1}(\cosh (c+d x))}{a^2 d}-\frac {\coth (c+d x)}{a d}-\frac {b \text {sech}(c+d x)}{a^2 d}+\frac {b^2 \text {sech}(c+d x) (b+a \sinh (c+d x))}{a^2 \left (a^2+b^2\right ) d}-\frac {\tanh (c+d x)}{a d}+\frac {\left (4 i b^4\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^2 \left (a^2+b^2\right ) d}\\ &=\frac {b \tanh ^{-1}(\cosh (c+d x))}{a^2 d}-\frac {2 b^4 \tanh ^{-1}\left (\frac {b-a \tanh \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a^2+b^2}}\right )}{a^2 \left (a^2+b^2\right )^{3/2} d}-\frac {\coth (c+d x)}{a d}-\frac {b \text {sech}(c+d x)}{a^2 d}+\frac {b^2 \text {sech}(c+d x) (b+a \sinh (c+d x))}{a^2 \left (a^2+b^2\right ) d}-\frac {\tanh (c+d x)}{a d}\\ \end {align*}
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Mathematica [A] time = 2.57, size = 135, normalized size = 0.94 \[ -\frac {\frac {2 \text {sech}(c+d x) (a \sinh (c+d x)+b)}{a^2+b^2}+\frac {4 b^4 \tan ^{-1}\left (\frac {b-a \tanh \left (\frac {1}{2} (c+d x)\right )}{\sqrt {-a^2-b^2}}\right )}{a^2 \left (-a^2-b^2\right )^{3/2}}+\frac {2 b \log \left (\tanh \left (\frac {1}{2} (c+d x)\right )\right )}{a^2}+\frac {\tanh \left (\frac {1}{2} (c+d x)\right )}{a}+\frac {\coth \left (\frac {1}{2} (c+d x)\right )}{a}}{2 d} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.61, size = 1040, normalized size = 7.22 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.18, size = 185, normalized size = 1.28 \[ \frac {\frac {b^{4} \log \left (\frac {{\left | 2 \, b e^{\left (d x + c\right )} + 2 \, a - 2 \, \sqrt {a^{2} + b^{2}} \right |}}{{\left | 2 \, b e^{\left (d x + c\right )} + 2 \, a + 2 \, \sqrt {a^{2} + b^{2}} \right |}}\right )}{{\left (a^{4} + a^{2} b^{2}\right )} \sqrt {a^{2} + b^{2}}} + \frac {b \log \left (e^{\left (d x + c\right )} + 1\right )}{a^{2}} - \frac {b \log \left ({\left | e^{\left (d x + c\right )} - 1 \right |}\right )}{a^{2}} - \frac {2 \, {\left (a b e^{\left (3 \, d x + 3 \, c\right )} + b^{2} e^{\left (2 \, d x + 2 \, c\right )} - a b e^{\left (d x + c\right )} + 2 \, a^{2} + b^{2}\right )}}{{\left (a^{3} + a b^{2}\right )} {\left (e^{\left (4 \, d x + 4 \, c\right )} - 1\right )}}}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 174, normalized size = 1.21 \[ -\frac {\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{2 d a}+\frac {2 b^{4} \arctanh \left (\frac {2 a \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-2 b}{2 \sqrt {a^{2}+b^{2}}}\right )}{d \,a^{2} \left (a^{2}+b^{2}\right )^{\frac {3}{2}}}-\frac {1}{2 d a \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}-\frac {b \ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d \,a^{2}}-\frac {2 a \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{d \left (a^{2}+b^{2}\right ) \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}-\frac {2 b}{d \left (a^{2}+b^{2}\right ) \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.41, size = 208, normalized size = 1.44 \[ \frac {b^{4} \log \left (\frac {b e^{\left (-d x - c\right )} - a - \sqrt {a^{2} + b^{2}}}{b e^{\left (-d x - c\right )} - a + \sqrt {a^{2} + b^{2}}}\right )}{{\left (a^{4} + a^{2} b^{2}\right )} \sqrt {a^{2} + b^{2}} d} - \frac {2 \, {\left (a b e^{\left (-d x - c\right )} + b^{2} e^{\left (-2 \, d x - 2 \, c\right )} - a b e^{\left (-3 \, d x - 3 \, c\right )} + 2 \, a^{2} + b^{2}\right )}}{{\left (a^{3} + a b^{2} - {\left (a^{3} + a b^{2}\right )} e^{\left (-4 \, d x - 4 \, c\right )}\right )} d} + \frac {b \log \left (e^{\left (-d x - c\right )} + 1\right )}{a^{2} d} - \frac {b \log \left (e^{\left (-d x - c\right )} - 1\right )}{a^{2} d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 5.23, size = 768, normalized size = 5.33 \[ \frac {b^4\,\ln \left (\frac {64\,b^8\,\sqrt {{\left (a^2+b^2\right )}^3}-96\,a\,b^{10}-384\,a^3\,b^8-512\,a^5\,b^6-288\,a^7\,b^4-64\,a^9\,b^2+288\,a^2\,b^9\,{\mathrm {e}}^{c+d\,x}+960\,a^4\,b^7\,{\mathrm {e}}^{c+d\,x}+1152\,a^6\,b^5\,{\mathrm {e}}^{c+d\,x}+608\,a^8\,b^3\,{\mathrm {e}}^{c+d\,x}+128\,a^{10}\,b\,{\mathrm {e}}^{c+d\,x}-64\,a\,b^7\,{\mathrm {e}}^{c+d\,x}\,\sqrt {{\left (a^2+b^2\right )}^3}+32\,a^3\,b^5\,{\mathrm {e}}^{c+d\,x}\,\sqrt {{\left (a^2+b^2\right )}^3}}{a^3\,{\left ({\left (a^2+b^2\right )}^3\right )}^{3/2}\,\left (a^2+b^2\right )}-\frac {32\,b\,\left (-4\,{\mathrm {e}}^{c+d\,x}\,a^3+2\,a^2\,b-5\,{\mathrm {e}}^{c+d\,x}\,a\,b^2+2\,b^3\right )}{a^3\,{\left (a^2+b^2\right )}^2}\right )\,\sqrt {{\left (a^2+b^2\right )}^3}}{d\,a^8+3\,d\,a^6\,b^2+3\,d\,a^4\,b^4+d\,a^2\,b^6}-\frac {\frac {2\,b^4\,{\mathrm {e}}^{3\,c+3\,d\,x}}{d\,\left (a^2\,b^3+b^5\right )}-\frac {2\,b^4\,{\mathrm {e}}^{c+d\,x}}{d\,\left (a^2\,b^3+b^5\right )}+\frac {2\,b^3\,\left (2\,a^2+b^2\right )}{a\,d\,\left (a^2\,b^3+b^5\right )}+\frac {2\,b^5\,{\mathrm {e}}^{2\,c+2\,d\,x}}{a\,d\,\left (a^2\,b^3+b^5\right )}}{{\mathrm {e}}^{4\,c+4\,d\,x}-1}-\frac {b^4\,\ln \left (\frac {96\,a\,b^{10}+64\,b^8\,\sqrt {{\left (a^2+b^2\right )}^3}+384\,a^3\,b^8+512\,a^5\,b^6+288\,a^7\,b^4+64\,a^9\,b^2-288\,a^2\,b^9\,{\mathrm {e}}^{c+d\,x}-960\,a^4\,b^7\,{\mathrm {e}}^{c+d\,x}-1152\,a^6\,b^5\,{\mathrm {e}}^{c+d\,x}-608\,a^8\,b^3\,{\mathrm {e}}^{c+d\,x}-128\,a^{10}\,b\,{\mathrm {e}}^{c+d\,x}-64\,a\,b^7\,{\mathrm {e}}^{c+d\,x}\,\sqrt {{\left (a^2+b^2\right )}^3}+32\,a^3\,b^5\,{\mathrm {e}}^{c+d\,x}\,\sqrt {{\left (a^2+b^2\right )}^3}}{a^3\,{\left ({\left (a^2+b^2\right )}^3\right )}^{3/2}\,\left (a^2+b^2\right )}-\frac {32\,b\,\left (-4\,{\mathrm {e}}^{c+d\,x}\,a^3+2\,a^2\,b-5\,{\mathrm {e}}^{c+d\,x}\,a\,b^2+2\,b^3\right )}{a^3\,{\left (a^2+b^2\right )}^2}\right )\,\sqrt {{\left (a^2+b^2\right )}^3}}{d\,a^8+3\,d\,a^6\,b^2+3\,d\,a^4\,b^4+d\,a^2\,b^6}-\frac {b\,\ln \left ({\mathrm {e}}^{c+d\,x}-1\right )}{a^2\,d}+\frac {b\,\ln \left ({\mathrm {e}}^{c+d\,x}+1\right )}{a^2\,d} \]
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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