Integrand size = 29, antiderivative size = 158 \[ \int \frac {(e+f x) \text {sech}^2(c+d x)}{a+i a \sinh (c+d x)} \, dx=-\frac {i f \arctan (\sinh (c+d x))}{6 a d^2}-\frac {2 f \log (\cosh (c+d x))}{3 a d^2}+\frac {f \text {sech}^2(c+d x)}{6 a d^2}+\frac {i (e+f x) \text {sech}^3(c+d x)}{3 a d}+\frac {2 (e+f x) \tanh (c+d x)}{3 a d}-\frac {i f \text {sech}(c+d x) \tanh (c+d x)}{6 a d^2}+\frac {(e+f x) \text {sech}^2(c+d x) \tanh (c+d x)}{3 a d} \]
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Time = 0.11 (sec) , antiderivative size = 158, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.241, Rules used = {5690, 4270, 4269, 3556, 5559, 3853, 3855} \[ \int \frac {(e+f x) \text {sech}^2(c+d x)}{a+i a \sinh (c+d x)} \, dx=-\frac {i f \arctan (\sinh (c+d x))}{6 a d^2}+\frac {f \text {sech}^2(c+d x)}{6 a d^2}-\frac {2 f \log (\cosh (c+d x))}{3 a d^2}-\frac {i f \tanh (c+d x) \text {sech}(c+d x)}{6 a d^2}+\frac {2 (e+f x) \tanh (c+d x)}{3 a d}+\frac {i (e+f x) \text {sech}^3(c+d x)}{3 a d}+\frac {(e+f x) \tanh (c+d x) \text {sech}^2(c+d x)}{3 a d} \]
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Rule 3556
Rule 3853
Rule 3855
Rule 4269
Rule 4270
Rule 5559
Rule 5690
Rubi steps \begin{align*} \text {integral}& = -\frac {i \int (e+f x) \text {sech}^3(c+d x) \tanh (c+d x) \, dx}{a}+\frac {\int (e+f x) \text {sech}^4(c+d x) \, dx}{a} \\ & = \frac {f \text {sech}^2(c+d x)}{6 a d^2}+\frac {i (e+f x) \text {sech}^3(c+d x)}{3 a d}+\frac {(e+f x) \text {sech}^2(c+d x) \tanh (c+d x)}{3 a d}+\frac {2 \int (e+f x) \text {sech}^2(c+d x) \, dx}{3 a}-\frac {(i f) \int \text {sech}^3(c+d x) \, dx}{3 a d} \\ & = \frac {f \text {sech}^2(c+d x)}{6 a d^2}+\frac {i (e+f x) \text {sech}^3(c+d x)}{3 a d}+\frac {2 (e+f x) \tanh (c+d x)}{3 a d}-\frac {i f \text {sech}(c+d x) \tanh (c+d x)}{6 a d^2}+\frac {(e+f x) \text {sech}^2(c+d x) \tanh (c+d x)}{3 a d}-\frac {(i f) \int \text {sech}(c+d x) \, dx}{6 a d}-\frac {(2 f) \int \tanh (c+d x) \, dx}{3 a d} \\ & = -\frac {i f \arctan (\sinh (c+d x))}{6 a d^2}-\frac {2 f \log (\cosh (c+d x))}{3 a d^2}+\frac {f \text {sech}^2(c+d x)}{6 a d^2}+\frac {i (e+f x) \text {sech}^3(c+d x)}{3 a d}+\frac {2 (e+f x) \tanh (c+d x)}{3 a d}-\frac {i f \text {sech}(c+d x) \tanh (c+d x)}{6 a d^2}+\frac {(e+f x) \text {sech}^2(c+d x) \tanh (c+d x)}{3 a d} \\ \end{align*}
Time = 2.76 (sec) , antiderivative size = 194, normalized size of antiderivative = 1.23 \[ \int \frac {(e+f x) \text {sech}^2(c+d x)}{a+i a \sinh (c+d x)} \, dx=\frac {2 d (e+f x) (\cosh (2 (c+d x))-2 i \sinh (c+d x))+\cosh (c+d x) \left (-d e-i f+c f-2 f \arctan \left (\tanh \left (\frac {1}{2} (c+d x)\right )\right )+4 i f \log (\cosh (c+d x))-i \left (d e-c f+2 f \arctan \left (\tanh \left (\frac {1}{2} (c+d x)\right )\right )-4 i f \log (\cosh (c+d x))\right ) \sinh (c+d x)\right )}{6 a d^2 \left (\cosh \left (\frac {1}{2} (c+d x)\right )-i \sinh \left (\frac {1}{2} (c+d x)\right )\right ) \left (\cosh \left (\frac {1}{2} (c+d x)\right )+i \sinh \left (\frac {1}{2} (c+d x)\right )\right ) (-i+\sinh (c+d x))} \]
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Time = 11.56 (sec) , antiderivative size = 143, normalized size of antiderivative = 0.91
method | result | size |
risch | \(\frac {4 f x}{3 a d}+\frac {4 f c}{3 a \,d^{2}}-\frac {i \left (-8 d f x \,{\mathrm e}^{d x +c}+f \,{\mathrm e}^{3 d x +3 c}-8 d e \,{\mathrm e}^{d x +c}+f \,{\mathrm e}^{d x +c}+4 i d f x +4 i d e \right )}{3 \left ({\mathrm e}^{d x +c}+i\right ) \left ({\mathrm e}^{d x +c}-i\right )^{3} d^{2} a}-\frac {f \ln \left ({\mathrm e}^{d x +c}+i\right )}{2 a \,d^{2}}-\frac {5 f \ln \left ({\mathrm e}^{d x +c}-i\right )}{6 a \,d^{2}}\) | \(143\) |
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Time = 0.26 (sec) , antiderivative size = 201, normalized size of antiderivative = 1.27 \[ \int \frac {(e+f x) \text {sech}^2(c+d x)}{a+i a \sinh (c+d x)} \, dx=\frac {8 \, d f x e^{\left (4 \, d x + 4 \, c\right )} + 8 \, d e - 2 \, {\left (8 i \, d f x + i \, f\right )} e^{\left (3 \, d x + 3 \, c\right )} - 2 \, {\left (-8 i \, d e + i \, f\right )} e^{\left (d x + c\right )} - 3 \, {\left (f e^{\left (4 \, d x + 4 \, c\right )} - 2 i \, f e^{\left (3 \, d x + 3 \, c\right )} - 2 i \, f e^{\left (d x + c\right )} - f\right )} \log \left (e^{\left (d x + c\right )} + i\right ) - 5 \, {\left (f e^{\left (4 \, d x + 4 \, c\right )} - 2 i \, f e^{\left (3 \, d x + 3 \, c\right )} - 2 i \, f e^{\left (d x + c\right )} - f\right )} \log \left (e^{\left (d x + c\right )} - i\right )}{6 \, {\left (a d^{2} e^{\left (4 \, d x + 4 \, c\right )} - 2 i \, a d^{2} e^{\left (3 \, d x + 3 \, c\right )} - 2 i \, a d^{2} e^{\left (d x + c\right )} - a d^{2}\right )}} \]
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\[ \int \frac {(e+f x) \text {sech}^2(c+d x)}{a+i a \sinh (c+d x)} \, dx=- \frac {i \left (\int \frac {e \operatorname {sech}^{2}{\left (c + d x \right )}}{\sinh {\left (c + d x \right )} - i}\, dx + \int \frac {f x \operatorname {sech}^{2}{\left (c + d x \right )}}{\sinh {\left (c + d x \right )} - i}\, dx\right )}{a} \]
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Time = 0.26 (sec) , antiderivative size = 251, normalized size of antiderivative = 1.59 \[ \int \frac {(e+f x) \text {sech}^2(c+d x)}{a+i a \sinh (c+d x)} \, dx=\frac {1}{6} \, f {\left (\frac {24 \, {\left (4 i \, d x e^{\left (4 \, d x + 4 \, c\right )} + {\left (8 \, d x e^{\left (3 \, c\right )} + e^{\left (3 \, c\right )}\right )} e^{\left (3 \, d x\right )} + e^{\left (d x + c\right )}\right )}}{12 i \, a d^{2} e^{\left (4 \, d x + 4 \, c\right )} + 24 \, a d^{2} e^{\left (3 \, d x + 3 \, c\right )} + 24 \, a d^{2} e^{\left (d x + c\right )} - 12 i \, a d^{2}} - \frac {3 \, \log \left ({\left (e^{\left (d x + c\right )} + i\right )} e^{\left (-c\right )}\right )}{a d^{2}} - \frac {5 \, \log \left (-i \, {\left (i \, e^{\left (d x + c\right )} + 1\right )} e^{\left (-c\right )}\right )}{a d^{2}}\right )} + \frac {4}{3} \, e {\left (\frac {2 \, e^{\left (-d x - c\right )}}{{\left (2 \, a e^{\left (-d x - c\right )} + 2 \, a e^{\left (-3 \, d x - 3 \, c\right )} - i \, a e^{\left (-4 \, d x - 4 \, c\right )} + i \, a\right )} d} + \frac {i}{{\left (2 \, a e^{\left (-d x - c\right )} + 2 \, a e^{\left (-3 \, d x - 3 \, c\right )} - i \, a e^{\left (-4 \, d x - 4 \, c\right )} + i \, a\right )} d}\right )} \]
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Time = 0.28 (sec) , antiderivative size = 260, normalized size of antiderivative = 1.65 \[ \int \frac {(e+f x) \text {sech}^2(c+d x)}{a+i a \sinh (c+d x)} \, dx=\frac {8 \, d f x e^{\left (4 \, d x + 4 \, c\right )} - 16 i \, d f x e^{\left (3 \, d x + 3 \, c\right )} + 16 i \, d e e^{\left (d x + c\right )} - 3 \, f e^{\left (4 \, d x + 4 \, c\right )} \log \left (e^{\left (d x + c\right )} + i\right ) + 6 i \, f e^{\left (3 \, d x + 3 \, c\right )} \log \left (e^{\left (d x + c\right )} + i\right ) + 6 i \, f e^{\left (d x + c\right )} \log \left (e^{\left (d x + c\right )} + i\right ) - 5 \, f e^{\left (4 \, d x + 4 \, c\right )} \log \left (e^{\left (d x + c\right )} - i\right ) + 10 i \, f e^{\left (3 \, d x + 3 \, c\right )} \log \left (e^{\left (d x + c\right )} - i\right ) + 10 i \, f e^{\left (d x + c\right )} \log \left (e^{\left (d x + c\right )} - i\right ) + 8 \, d e - 2 i \, f e^{\left (3 \, d x + 3 \, c\right )} - 2 i \, f e^{\left (d x + c\right )} + 3 \, f \log \left (e^{\left (d x + c\right )} + i\right ) + 5 \, f \log \left (e^{\left (d x + c\right )} - i\right )}{6 \, {\left (a d^{2} e^{\left (4 \, d x + 4 \, c\right )} - 2 i \, a d^{2} e^{\left (3 \, d x + 3 \, c\right )} - 2 i \, a d^{2} e^{\left (d x + c\right )} - a d^{2}\right )}} \]
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Time = 3.46 (sec) , antiderivative size = 205, normalized size of antiderivative = 1.30 \[ \int \frac {(e+f x) \text {sech}^2(c+d x)}{a+i a \sinh (c+d x)} \, dx=\frac {4\,f\,x}{3\,a\,d}-\frac {f+3\,d\,e+3\,d\,f\,x}{3\,a\,d^2\,\left (1-{\mathrm {e}}^{2\,c+2\,d\,x}+{\mathrm {e}}^{c+d\,x}\,2{}\mathrm {i}\right )}-\frac {5\,f\,\ln \left (f+f\,{\mathrm {e}}^{c+d\,x}\,1{}\mathrm {i}\right )}{6\,a\,d^2}-\frac {\left (e+f\,x\right )\,2{}\mathrm {i}}{3\,a\,d\,\left (3\,{\mathrm {e}}^{c+d\,x}+{\mathrm {e}}^{2\,c+2\,d\,x}\,3{}\mathrm {i}-{\mathrm {e}}^{3\,c+3\,d\,x}-\mathrm {i}\right )}-\frac {\left (e+f\,x\right )\,1{}\mathrm {i}}{2\,a\,d\,\left ({\mathrm {e}}^{c+d\,x}+1{}\mathrm {i}\right )}-\frac {f\,\ln \left (-1+{\mathrm {e}}^{c+d\,x}\,1{}\mathrm {i}\right )}{2\,a\,d^2}+\frac {\left (3\,d\,e-2\,f+3\,d\,f\,x\right )\,1{}\mathrm {i}}{6\,a\,d^2\,\left ({\mathrm {e}}^{c+d\,x}-\mathrm {i}\right )} \]
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