Optimal. Leaf size=158 \[ \frac {(c+d x) \tanh \left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right )}{3 a^2 f}+\frac {(c+d x) \tanh \left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right ) \text {sech}^2\left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right )}{6 a^2 f}+\frac {d \text {sech}^2\left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right )}{6 a^2 f^2}-\frac {2 d \log \left (\cosh \left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right )\right )}{3 a^2 f^2} \]
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Rubi [A] time = 0.11, antiderivative size = 158, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.190, Rules used = {3318, 4185, 4184, 3475} \[ \frac {(c+d x) \tanh \left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right )}{3 a^2 f}+\frac {(c+d x) \tanh \left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right ) \text {sech}^2\left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right )}{6 a^2 f}+\frac {d \text {sech}^2\left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right )}{6 a^2 f^2}-\frac {2 d \log \left (\cosh \left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right )\right )}{3 a^2 f^2} \]
Antiderivative was successfully verified.
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Rule 3318
Rule 3475
Rule 4184
Rule 4185
Rubi steps
\begin {align*} \int \frac {c+d x}{(a+i a \sinh (e+f x))^2} \, dx &=\frac {\int (c+d x) \csc ^4\left (\frac {1}{2} \left (i e+\frac {\pi }{2}\right )+\frac {i f x}{2}\right ) \, dx}{4 a^2}\\ &=\frac {d \text {sech}^2\left (\frac {e}{2}+\frac {i \pi }{4}+\frac {f x}{2}\right )}{6 a^2 f^2}+\frac {(c+d x) \text {sech}^2\left (\frac {e}{2}+\frac {i \pi }{4}+\frac {f x}{2}\right ) \tanh \left (\frac {e}{2}+\frac {i \pi }{4}+\frac {f x}{2}\right )}{6 a^2 f}-\frac {\int (c+d x) \text {csch}^2\left (\frac {e}{2}-\frac {i \pi }{4}+\frac {f x}{2}\right ) \, dx}{6 a^2}\\ &=\frac {d \text {sech}^2\left (\frac {e}{2}+\frac {i \pi }{4}+\frac {f x}{2}\right )}{6 a^2 f^2}+\frac {(c+d x) \tanh \left (\frac {e}{2}+\frac {i \pi }{4}+\frac {f x}{2}\right )}{3 a^2 f}+\frac {(c+d x) \text {sech}^2\left (\frac {e}{2}+\frac {i \pi }{4}+\frac {f x}{2}\right ) \tanh \left (\frac {e}{2}+\frac {i \pi }{4}+\frac {f x}{2}\right )}{6 a^2 f}-\frac {d \int \coth \left (\frac {e}{2}-\frac {i \pi }{4}+\frac {f x}{2}\right ) \, dx}{3 a^2 f}\\ &=-\frac {2 d \log \left (\cosh \left (\frac {e}{2}+\frac {i \pi }{4}+\frac {f x}{2}\right )\right )}{3 a^2 f^2}+\frac {d \text {sech}^2\left (\frac {e}{2}+\frac {i \pi }{4}+\frac {f x}{2}\right )}{6 a^2 f^2}+\frac {(c+d x) \tanh \left (\frac {e}{2}+\frac {i \pi }{4}+\frac {f x}{2}\right )}{3 a^2 f}+\frac {(c+d x) \text {sech}^2\left (\frac {e}{2}+\frac {i \pi }{4}+\frac {f x}{2}\right ) \tanh \left (\frac {e}{2}+\frac {i \pi }{4}+\frac {f x}{2}\right )}{6 a^2 f}\\ \end {align*}
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Mathematica [A] time = 1.05, size = 241, normalized size = 1.53 \[ \frac {\left (\sinh \left (\frac {1}{2} (e+f x)\right )-i \cosh \left (\frac {1}{2} (e+f x)\right )\right ) \left (\cosh \left (\frac {3}{2} (e+f x)\right ) \left (2 c f+2 d \tan ^{-1}\left (\tanh \left (\frac {1}{2} (e+f x)\right )\right )-i d \log (\cosh (e+f x))-d e+d f x\right )+2 i \sinh \left (\frac {1}{2} (e+f x)\right ) \left (-3 c f-4 d \tan ^{-1}\left (\tanh \left (\frac {1}{2} (e+f x)\right )\right )+2 i d \log (\cosh (e+f x))+d \cosh (e+f x) \left (-2 \tan ^{-1}\left (\tanh \left (\frac {1}{2} (e+f x)\right )\right )+i \log (\cosh (e+f x))+e+f x\right )+2 d e-d f x-i d\right )+d \cosh \left (\frac {1}{2} (e+f x)\right ) \left (-6 \tan ^{-1}\left (\tanh \left (\frac {1}{2} (e+f x)\right )\right )+3 i \log (\cosh (e+f x))+3 e+3 f x-2 i\right )\right )}{6 a^2 f^2 (\sinh (e+f x)-i)^2} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.44, size = 162, normalized size = 1.03 \[ \frac {2 \, d f x e^{\left (3 \, f x + 3 \, e\right )} - 2 i \, c f + {\left (-6 i \, d f x - 2 i \, d\right )} e^{\left (2 \, f x + 2 \, e\right )} + 2 \, {\left (3 \, c f - d\right )} e^{\left (f x + e\right )} - {\left (2 \, d e^{\left (3 \, f x + 3 \, e\right )} - 6 i \, d e^{\left (2 \, f x + 2 \, e\right )} - 6 \, d e^{\left (f x + e\right )} + 2 i \, d\right )} \log \left (e^{\left (f x + e\right )} - i\right )}{3 \, a^{2} f^{2} e^{\left (3 \, f x + 3 \, e\right )} - 9 i \, a^{2} f^{2} e^{\left (2 \, f x + 2 \, e\right )} - 9 \, a^{2} f^{2} e^{\left (f x + e\right )} + 3 i \, a^{2} f^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.22, size = 211, normalized size = 1.34 \[ \frac {2 \, d f x e^{\left (3 \, f x + 3 \, e\right )} - 6 i \, d f x e^{\left (2 \, f x + 2 \, e\right )} + 6 \, c f e^{\left (f x + e\right )} - 2 \, d e^{\left (3 \, f x + 3 \, e\right )} \log \left (e^{\left (f x + e\right )} - i\right ) + 6 i \, d e^{\left (2 \, f x + 2 \, e\right )} \log \left (e^{\left (f x + e\right )} - i\right ) + 6 \, d e^{\left (f x + e\right )} \log \left (e^{\left (f x + e\right )} - i\right ) - 2 i \, c f - 2 i \, d e^{\left (2 \, f x + 2 \, e\right )} - 2 \, d e^{\left (f x + e\right )} - 2 i \, d \log \left (e^{\left (f x + e\right )} - i\right )}{3 \, a^{2} f^{2} e^{\left (3 \, f x + 3 \, e\right )} - 9 i \, a^{2} f^{2} e^{\left (2 \, f x + 2 \, e\right )} - 9 \, a^{2} f^{2} e^{\left (f x + e\right )} + 3 i \, a^{2} f^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.23, size = 113, normalized size = 0.72 \[ \frac {2 d x}{3 f \,a^{2}}+\frac {2 d e}{3 f^{2} a^{2}}-\frac {2 i \left (3 i f d x \,{\mathrm e}^{f x +e}+3 i f c \,{\mathrm e}^{f x +e}-i d \,{\mathrm e}^{f x +e}+d f x +{\mathrm e}^{2 f x +2 e} d +c f \right )}{3 \left ({\mathrm e}^{f x +e}-i\right )^{3} f^{2} a^{2}}-\frac {2 d \ln \left ({\mathrm e}^{f x +e}-i\right )}{3 f^{2} a^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.35, size = 257, normalized size = 1.63 \[ \frac {1}{3} \, d {\left (\frac {3 \, {\left (2 \, f x e^{\left (3 \, f x + 3 \, e\right )} + {\left (-6 i \, f x e^{\left (2 \, e\right )} - 2 i \, e^{\left (2 \, e\right )}\right )} e^{\left (2 \, f x\right )} - 2 \, e^{\left (f x + e\right )}\right )}}{3 \, a^{2} f^{2} e^{\left (3 \, f x + 3 \, e\right )} - 9 i \, a^{2} f^{2} e^{\left (2 \, f x + 2 \, e\right )} - 9 \, a^{2} f^{2} e^{\left (f x + e\right )} + 3 i \, a^{2} f^{2}} - \frac {2 \, \log \left (-i \, {\left (i \, e^{\left (f x + e\right )} + 1\right )} e^{\left (-e\right )}\right )}{a^{2} f^{2}}\right )} + 2 \, c {\left (\frac {3 \, e^{\left (-f x - e\right )}}{{\left (9 \, a^{2} e^{\left (-f x - e\right )} - 9 i \, a^{2} e^{\left (-2 \, f x - 2 \, e\right )} - 3 \, a^{2} e^{\left (-3 \, f x - 3 \, e\right )} + 3 i \, a^{2}\right )} f} + \frac {i}{{\left (9 \, a^{2} e^{\left (-f x - e\right )} - 9 i \, a^{2} e^{\left (-2 \, f x - 2 \, e\right )} - 3 \, a^{2} e^{\left (-3 \, f x - 3 \, e\right )} + 3 i \, a^{2}\right )} f}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.54, size = 160, normalized size = 1.01 \[ -\frac {\frac {2\,d\,\ln \left ({\mathrm {e}}^{f\,x}\,{\mathrm {e}}^e-\mathrm {i}\right )}{3}+{\mathrm {e}}^{e+f\,x}\,\left (-\frac {d\,2{}\mathrm {i}}{3}+c\,f\,2{}\mathrm {i}+d\,\ln \left ({\mathrm {e}}^{f\,x}\,{\mathrm {e}}^e-\mathrm {i}\right )\,2{}\mathrm {i}\right )+\frac {2\,d\,{\mathrm {e}}^{2\,e+2\,f\,x}}{3}+f\,\left (\frac {2\,c}{3}+2\,d\,x\,{\mathrm {e}}^{2\,e+2\,f\,x}+\frac {d\,x\,{\mathrm {e}}^{3\,e+3\,f\,x}\,2{}\mathrm {i}}{3}\right )-2\,d\,{\mathrm {e}}^{2\,e+2\,f\,x}\,\ln \left ({\mathrm {e}}^{f\,x}\,{\mathrm {e}}^e-\mathrm {i}\right )-\frac {d\,{\mathrm {e}}^{3\,e+3\,f\,x}\,\ln \left ({\mathrm {e}}^{f\,x}\,{\mathrm {e}}^e-\mathrm {i}\right )\,2{}\mathrm {i}}{3}}{a^2\,f^2\,{\left (1+{\mathrm {e}}^{e+f\,x}\,1{}\mathrm {i}\right )}^3} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.52, size = 184, normalized size = 1.16 \[ \frac {2 c f e^{3 e} + 2 d f x e^{3 e} - 2 d e^{e} e^{- 2 f x} + \left (- 6 i c f e^{2 e} - 6 i d f x e^{2 e} - 2 i d e^{2 e}\right ) e^{- f x}}{3 a^{2} f^{2} e^{3 e} - 9 i a^{2} f^{2} e^{2 e} e^{- f x} - 9 a^{2} f^{2} e^{e} e^{- 2 f x} + 3 i a^{2} f^{2} e^{- 3 f x}} - \frac {2 d x}{3 a^{2} f} - \frac {2 d \log {\left (i e^{e} + e^{- f x} \right )}}{3 a^{2} f^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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