∫ csch(x)_ (i+sinh(x))2 dx [52]

Optimal. Leaf size=34 \[ \tanh ^{-1}(\cosh (x))+\frac {\cosh (x)}{3 (i+\sinh (x))^2}-\frac {4 i \cosh (x)}{3 (i+\sinh (x))} \]

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Rubi [A]
time = 0.06, antiderivative size = 34, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.364, Rules used = {2845, 3057, 12, 3855} \begin {gather*} -\frac {4 i \cosh (x)}{3 (\sinh (x)+i)}+\frac {\cosh (x)}{3 (\sinh (x)+i)^2}+\tanh ^{-1}(\cosh (x)) \end {gather*}

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] && !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Sim

p[b^2*Cos[e + f*x]*(a + b*Sin[e + f*x])^m*((c + d*Sin[e + f*x])^(n + 1)/(a*f*(2*m + 1)*(b*c - a*d))), x] + Dis

t[1/(a*(2*m + 1)*(b*c - a*d)), Int[(a + b*Sin[e + f*x])^(m + 1)*(c + d*Sin[e + f*x])^n*Simp[b*c*(m + 1) - a*d*

(2*m + n + 2) + b*d*(m + n + 2)*Sin[e + f*x], x], x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && NeQ[b*c - a*d,

0] && EqQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && LtQ[m, -1] && !GtQ[n, 0] && (IntegersQ[2*m, 2*n] || (IntegerQ

Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)])*((c_.) + (d_.)*sin[(e_

.) + (f_.)*(x_)])^(n_), x_Symbol] :> Simp[b*(A*b - a*B)*Cos[e + f*x]*(a + b*Sin[e + f*x])^m*((c + d*Sin[e + f*

x])^(n + 1)/(a*f*(2*m + 1)*(b*c - a*d))), x] + Dist[1/(a*(2*m + 1)*(b*c - a*d)), Int[(a + b*Sin[e + f*x])^(m +

1)*(c + d*Sin[e + f*x])^n*Simp[B*(a*c*m + b*d*(n + 1)) + A*(b*c*(m + 1) - a*d*(2*m + n + 2)) + d*(A*b - a*B)*

(m + n + 2)*Sin[e + f*x], x], x], x] /; FreeQ[{a, b, c, d, e, f, A, B, n}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2

- b^2, 0] && NeQ[c^2 - d^2, 0] && LtQ[m, -2^(-1)] && !GtQ[n, 0] && IntegerQ[2*m] && (IntegerQ[2*n] || EqQ[c,

Int[csc[(c_.) + (d_.)*(x_)], x_Symbol] :> Simp[-ArcTanh[Cos[c + d*x]]/d, x] /; FreeQ[{c, d}, x]

\begin {align*} \int \frac {\text {csch}(x)}{(i+\sinh (x))^2} \, dx &=\frac {\cosh (x)}{3 (i+\sinh (x))^2}-\frac {1}{3} \int \frac {\text {csch}(x) (3 i-\sinh (x))}{i+\sinh (x)} \, dx\\ &=\frac {\cosh (x)}{3 (i+\sinh (x))^2}-\frac {4 i \cosh (x)}{3 (i+\sinh (x))}+\frac {1}{3} i \int 3 i \text {csch}(x) \, dx\\ &=\frac {\cosh (x)}{3 (i+\sinh (x))^2}-\frac {4 i \cosh (x)}{3 (i+\sinh (x))}-\int \text {csch}(x) \, dx\\ &=\tanh ^{-1}(\cosh (x))+\frac {\cosh (x)}{3 (i+\sinh (x))^2}-\frac {4 i \cosh (x)}{3 (i+\sinh (x))}\\ \end {align*}

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Mathematica [B] Both result and optimal contain complex but leaf count is larger than twice the leaf count of optimal. \(91\) vs. \(2(34)=68\).
time = 0.06, size = 91, normalized size = 2.68 \begin {gather*} \frac {\cosh \left (\frac {x}{2}\right ) \left (6-9 \log \left (\tanh \left (\frac {x}{2}\right )\right )\right )+\cosh \left (\frac {3 x}{2}\right ) \left (-8+3 \log \left (\tanh \left (\frac {x}{2}\right )\right )\right )+6 i \left (-3+2 \log \left (\tanh \left (\frac {x}{2}\right )\right )+\cosh (x) \log \left (\tanh \left (\frac {x}{2}\right )\right )\right ) \sinh \left (\frac {x}{2}\right )}{6 \left (\cosh \left (\frac {x}{2}\right )-i \sinh \left (\frac {x}{2}\right )\right )^3} \end {gather*}

(Cosh[x/2]*(6 - 9*Log[Tanh[x/2]]) + Cosh[(3*x)/2]*(-8 + 3*Log[Tanh[x/2]]) + (6*I)*(-3 + 2*Log[Tanh[x/2]] + Cos

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method	result	size

risch	\(-\frac {2 \left (9 i {\mathrm e}^{x}+3 \,{\mathrm e}^{2 x}-4\right )}{3 \left ({\mathrm e}^{x}+i\right )^{3}}+\ln \left ({\mathrm e}^{x}+1\right )-\ln \left ({\mathrm e}^{x}-1\right )\)	\(36\)
default	\(-\ln \left (\tanh \left (\frac {x}{2}\right )\right )+\frac {4 i}{3 \left (\tanh \left (\frac {x}{2}\right )+i\right )^{3}}-\frac {4 i}{\tanh \left (\frac {x}{2}\right )+i}-\frac {2}{\left (\tanh \left (\frac {x}{2}\right )+i\right )^{2}}\)	\(44\)

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Maxima [B] Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 55 vs. \(2 (24) = 48\).
time = 0.30, size = 55, normalized size = 1.62 \begin {gather*} \frac {2 \, {\left (-9 i \, e^{\left (-x\right )} + 3 \, e^{\left (-2 \, x\right )} - 4\right )}}{3 \, {\left (3 \, e^{\left (-x\right )} + 3 i \, e^{\left (-2 \, x\right )} - e^{\left (-3 \, x\right )} - i\right )}} + \log \left (e^{\left (-x\right )} + 1\right ) - \log \left (e^{\left (-x\right )} - 1\right ) \end {gather*}

2/3*(-9*I*e^(-x) + 3*e^(-2*x) - 4)/(3*e^(-x) + 3*I*e^(-2*x) - e^(-3*x) - I) + log(e^(-x) + 1) - log(e^(-x) - 1

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Fricas [B] Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 78 vs. \(2 (24) = 48\).
time = 0.34, size = 78, normalized size = 2.29 \begin {gather*} \frac {3 \, {\left (e^{\left (3 \, x\right )} + 3 i \, e^{\left (2 \, x\right )} - 3 \, e^{x} - i\right )} \log \left (e^{x} + 1\right ) - 3 \, {\left (e^{\left (3 \, x\right )} + 3 i \, e^{\left (2 \, x\right )} - 3 \, e^{x} - i\right )} \log \left (e^{x} - 1\right ) - 6 \, e^{\left (2 \, x\right )} - 18 i \, e^{x} + 8}{3 \, {\left (e^{\left (3 \, x\right )} + 3 i \, e^{\left (2 \, x\right )} - 3 \, e^{x} - i\right )}} \end {gather*}

1/3*(3*(e^(3*x) + 3*I*e^(2*x) - 3*e^x - I)*log(e^x + 1) - 3*(e^(3*x) + 3*I*e^(2*x) - 3*e^x - I)*log(e^x - 1) -

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\operatorname {csch}{\left (x \right )}}{\left (\sinh {\left (x \right )} + i\right )^{2}}\, dx \end {gather*}

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Giac [A]
time = 0.45, size = 34, normalized size = 1.00 \begin {gather*} -\frac {2 \, {\left (3 \, e^{\left (2 \, x\right )} + 9 i \, e^{x} - 4\right )}}{3 \, {\left (e^{x} + i\right )}^{3}} + \log \left (e^{x} + 1\right ) - \log \left ({\left | e^{x} - 1 \right |}\right ) \end {gather*}

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Mupad [B]
time = 0.28, size = 41, normalized size = 1.21 \begin {gather*} \ln \left ({\mathrm {e}}^x+1\right )-\ln \left ({\mathrm {e}}^x-1\right )-\frac {2}{{\mathrm {e}}^x+1{}\mathrm {i}}-\frac {2{}\mathrm {i}}{{\left ({\mathrm {e}}^x+1{}\mathrm {i}\right )}^2}-\frac {4}{3\,{\left ({\mathrm {e}}^x+1{}\mathrm {i}\right )}^3} \end {gather*}

log(exp(x) + 1) - log(exp(x) - 1) - 2/(exp(x) + 1i) - 2i/(exp(x) + 1i)^2 - 4/(3*(exp(x) + 1i)^3)

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3.1.52 \(\int \frac {\text {csch}(x)}{(i+\sinh (x))^2} \, dx\) [52]