3.3.85 \(\int \frac {(e+f x) \text {sech}^3(c+d x)}{a+i a \sinh (c+d x)} \, dx\) [285]

Optimal. Leaf size=233 \[ \frac {3 (e+f x) \text {ArcTan}\left (e^{c+d x}\right )}{4 a d}-\frac {3 i f \text {PolyLog}\left (2,-i e^{c+d x}\right )}{8 a d^2}+\frac {3 i f \text {PolyLog}\left (2,i e^{c+d x}\right )}{8 a d^2}+\frac {3 f \text {sech}(c+d x)}{8 a d^2}+\frac {f \text {sech}^3(c+d x)}{12 a d^2}+\frac {i (e+f x) \text {sech}^4(c+d x)}{4 a d}-\frac {i f \tanh (c+d x)}{4 a d^2}+\frac {3 (e+f x) \text {sech}(c+d x) \tanh (c+d x)}{8 a d}+\frac {(e+f x) \text {sech}^3(c+d x) \tanh (c+d x)}{4 a d}+\frac {i f \tanh ^3(c+d x)}{12 a d^2} \]

[Out]

3/4*(f*x+e)*arctan(exp(d*x+c))/a/d-3/8*I*f*polylog(2,-I*exp(d*x+c))/a/d^2+3/8*I*f*polylog(2,I*exp(d*x+c))/a/d^
2+3/8*f*sech(d*x+c)/a/d^2+1/12*f*sech(d*x+c)^3/a/d^2+1/4*I*(f*x+e)*sech(d*x+c)^4/a/d-1/4*I*f*tanh(d*x+c)/a/d^2
+3/8*(f*x+e)*sech(d*x+c)*tanh(d*x+c)/a/d+1/4*(f*x+e)*sech(d*x+c)^3*tanh(d*x+c)/a/d+1/12*I*f*tanh(d*x+c)^3/a/d^
2

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Rubi [A]
time = 0.14, antiderivative size = 233, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 7, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.241, Rules used = {5690, 4270, 4265, 2317, 2438, 5559, 3852} \begin {gather*} \frac {3 (e+f x) \text {ArcTan}\left (e^{c+d x}\right )}{4 a d}-\frac {3 i f \text {Li}_2\left (-i e^{c+d x}\right )}{8 a d^2}+\frac {3 i f \text {Li}_2\left (i e^{c+d x}\right )}{8 a d^2}+\frac {i f \tanh ^3(c+d x)}{12 a d^2}-\frac {i f \tanh (c+d x)}{4 a d^2}+\frac {f \text {sech}^3(c+d x)}{12 a d^2}+\frac {3 f \text {sech}(c+d x)}{8 a d^2}+\frac {i (e+f x) \text {sech}^4(c+d x)}{4 a d}+\frac {(e+f x) \tanh (c+d x) \text {sech}^3(c+d x)}{4 a d}+\frac {3 (e+f x) \tanh (c+d x) \text {sech}(c+d x)}{8 a d} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[((e + f*x)*Sech[c + d*x]^3)/(a + I*a*Sinh[c + d*x]),x]

[Out]

(3*(e + f*x)*ArcTan[E^(c + d*x)])/(4*a*d) - (((3*I)/8)*f*PolyLog[2, (-I)*E^(c + d*x)])/(a*d^2) + (((3*I)/8)*f*
PolyLog[2, I*E^(c + d*x)])/(a*d^2) + (3*f*Sech[c + d*x])/(8*a*d^2) + (f*Sech[c + d*x]^3)/(12*a*d^2) + ((I/4)*(
e + f*x)*Sech[c + d*x]^4)/(a*d) - ((I/4)*f*Tanh[c + d*x])/(a*d^2) + (3*(e + f*x)*Sech[c + d*x]*Tanh[c + d*x])/
(8*a*d) + ((e + f*x)*Sech[c + d*x]^3*Tanh[c + d*x])/(4*a*d) + ((I/12)*f*Tanh[c + d*x]^3)/(a*d^2)

Rule 2317

Int[Log[(a_) + (b_.)*((F_)^((e_.)*((c_.) + (d_.)*(x_))))^(n_.)], x_Symbol] :> Dist[1/(d*e*n*Log[F]), Subst[Int
[Log[a + b*x]/x, x], x, (F^(e*(c + d*x)))^n], x] /; FreeQ[{F, a, b, c, d, e, n}, x] && GtQ[a, 0]

Rule 2438

Int[Log[(c_.)*((d_) + (e_.)*(x_)^(n_.))]/(x_), x_Symbol] :> Simp[-PolyLog[2, (-c)*e*x^n]/n, x] /; FreeQ[{c, d,
e, n}, x] && EqQ[c*d, 1]

Rule 3852

Int[csc[(c_.) + (d_.)*(x_)]^(n_), x_Symbol] :> Dist[-d^(-1), Subst[Int[ExpandIntegrand[(1 + x^2)^(n/2 - 1), x]
, x], x, Cot[c + d*x]], x] /; FreeQ[{c, d}, x] && IGtQ[n/2, 0]

Rule 4265

Int[csc[(e_.) + Pi*(k_.) + (Complex[0, fz_])*(f_.)*(x_)]*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> Simp[-2*(c +
d*x)^m*(ArcTanh[E^((-I)*e + f*fz*x)/E^(I*k*Pi)]/(f*fz*I)), x] + (-Dist[d*(m/(f*fz*I)), Int[(c + d*x)^(m - 1)*
Log[1 - E^((-I)*e + f*fz*x)/E^(I*k*Pi)], x], x] + Dist[d*(m/(f*fz*I)), Int[(c + d*x)^(m - 1)*Log[1 + E^((-I)*e
+ f*fz*x)/E^(I*k*Pi)], x], x]) /; FreeQ[{c, d, e, f, fz}, x] && IntegerQ[2*k] && IGtQ[m, 0]

Rule 4270

Int[(csc[(e_.) + (f_.)*(x_)]*(b_.))^(n_)*((c_.) + (d_.)*(x_)), x_Symbol] :> Simp[(-b^2)*(c + d*x)*Cot[e + f*x]
*((b*Csc[e + f*x])^(n - 2)/(f*(n - 1))), x] + (Dist[b^2*((n - 2)/(n - 1)), Int[(c + d*x)*(b*Csc[e + f*x])^(n -
2), x], x] - Simp[b^2*d*((b*Csc[e + f*x])^(n - 2)/(f^2*(n - 1)*(n - 2))), x]) /; FreeQ[{b, c, d, e, f}, x] &&
GtQ[n, 1] && NeQ[n, 2]

Rule 5559

Int[((c_.) + (d_.)*(x_))^(m_.)*Sech[(a_.) + (b_.)*(x_)]^(n_.)*Tanh[(a_.) + (b_.)*(x_)]^(p_.), x_Symbol] :> Sim
p[(-(c + d*x)^m)*(Sech[a + b*x]^n/(b*n)), x] + Dist[d*(m/(b*n)), Int[(c + d*x)^(m - 1)*Sech[a + b*x]^n, x], x]
/; FreeQ[{a, b, c, d, n}, x] && EqQ[p, 1] && GtQ[m, 0]

Rule 5690

Int[(((e_.) + (f_.)*(x_))^(m_.)*Sech[(c_.) + (d_.)*(x_)]^(n_.))/((a_) + (b_.)*Sinh[(c_.) + (d_.)*(x_)]), x_Sym
bol] :> Dist[1/a, Int[(e + f*x)^m*Sech[c + d*x]^(n + 2), x], x] + Dist[1/b, Int[(e + f*x)^m*Sech[c + d*x]^(n +
1)*Tanh[c + d*x], x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && IGtQ[m, 0] && EqQ[a^2 + b^2, 0]

Rubi steps

\begin {align*} \int \frac {(e+f x) \text {sech}^3(c+d x)}{a+i a \sinh (c+d x)} \, dx &=-\frac {i \int (e+f x) \text {sech}^4(c+d x) \tanh (c+d x) \, dx}{a}+\frac {\int (e+f x) \text {sech}^5(c+d x) \, dx}{a}\\ &=\frac {f \text {sech}^3(c+d x)}{12 a d^2}+\frac {i (e+f x) \text {sech}^4(c+d x)}{4 a d}+\frac {(e+f x) \text {sech}^3(c+d x) \tanh (c+d x)}{4 a d}+\frac {3 \int (e+f x) \text {sech}^3(c+d x) \, dx}{4 a}-\frac {(i f) \int \text {sech}^4(c+d x) \, dx}{4 a d}\\ &=\frac {3 f \text {sech}(c+d x)}{8 a d^2}+\frac {f \text {sech}^3(c+d x)}{12 a d^2}+\frac {i (e+f x) \text {sech}^4(c+d x)}{4 a d}+\frac {3 (e+f x) \text {sech}(c+d x) \tanh (c+d x)}{8 a d}+\frac {(e+f x) \text {sech}^3(c+d x) \tanh (c+d x)}{4 a d}+\frac {3 \int (e+f x) \text {sech}(c+d x) \, dx}{8 a}+\frac {f \text {Subst}\left (\int \left (1+x^2\right ) \, dx,x,-i \tanh (c+d x)\right )}{4 a d^2}\\ &=\frac {3 (e+f x) \tan ^{-1}\left (e^{c+d x}\right )}{4 a d}+\frac {3 f \text {sech}(c+d x)}{8 a d^2}+\frac {f \text {sech}^3(c+d x)}{12 a d^2}+\frac {i (e+f x) \text {sech}^4(c+d x)}{4 a d}-\frac {i f \tanh (c+d x)}{4 a d^2}+\frac {3 (e+f x) \text {sech}(c+d x) \tanh (c+d x)}{8 a d}+\frac {(e+f x) \text {sech}^3(c+d x) \tanh (c+d x)}{4 a d}+\frac {i f \tanh ^3(c+d x)}{12 a d^2}-\frac {(3 i f) \int \log \left (1-i e^{c+d x}\right ) \, dx}{8 a d}+\frac {(3 i f) \int \log \left (1+i e^{c+d x}\right ) \, dx}{8 a d}\\ &=\frac {3 (e+f x) \tan ^{-1}\left (e^{c+d x}\right )}{4 a d}+\frac {3 f \text {sech}(c+d x)}{8 a d^2}+\frac {f \text {sech}^3(c+d x)}{12 a d^2}+\frac {i (e+f x) \text {sech}^4(c+d x)}{4 a d}-\frac {i f \tanh (c+d x)}{4 a d^2}+\frac {3 (e+f x) \text {sech}(c+d x) \tanh (c+d x)}{8 a d}+\frac {(e+f x) \text {sech}^3(c+d x) \tanh (c+d x)}{4 a d}+\frac {i f \tanh ^3(c+d x)}{12 a d^2}-\frac {(3 i f) \text {Subst}\left (\int \frac {\log (1-i x)}{x} \, dx,x,e^{c+d x}\right )}{8 a d^2}+\frac {(3 i f) \text {Subst}\left (\int \frac {\log (1+i x)}{x} \, dx,x,e^{c+d x}\right )}{8 a d^2}\\ &=\frac {3 (e+f x) \tan ^{-1}\left (e^{c+d x}\right )}{4 a d}-\frac {3 i f \text {Li}_2\left (-i e^{c+d x}\right )}{8 a d^2}+\frac {3 i f \text {Li}_2\left (i e^{c+d x}\right )}{8 a d^2}+\frac {3 f \text {sech}(c+d x)}{8 a d^2}+\frac {f \text {sech}^3(c+d x)}{12 a d^2}+\frac {i (e+f x) \text {sech}^4(c+d x)}{4 a d}-\frac {i f \tanh (c+d x)}{4 a d^2}+\frac {3 (e+f x) \text {sech}(c+d x) \tanh (c+d x)}{8 a d}+\frac {(e+f x) \text {sech}^3(c+d x) \tanh (c+d x)}{4 a d}+\frac {i f \tanh ^3(c+d x)}{12 a d^2}\\ \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. \(929\) vs. \(2(233)=466\).
time = 5.02, size = 929, normalized size = 3.99 \begin {gather*} \frac {2 (f+6 i d (e+f x))+\frac {6 i d (e+f x)}{\left (\cosh \left (\frac {1}{2} (c+d x)\right )+i \sinh \left (\frac {1}{2} (c+d x)\right )\right )^2}-9 (c+d x) (c f-d (2 e+f x)) \left (\cosh \left (\frac {1}{2} (c+d x)\right )+i \sinh \left (\frac {1}{2} (c+d x)\right )\right )^2-9 d e \left (c+d x-2 i \log \left (\cosh \left (\frac {1}{2} (c+d x)\right )-i \sinh \left (\frac {1}{2} (c+d x)\right )\right )\right ) \left (\cosh \left (\frac {1}{2} (c+d x)\right )+i \sinh \left (\frac {1}{2} (c+d x)\right )\right )^2+9 c f \left (c+d x-2 i \log \left (\cosh \left (\frac {1}{2} (c+d x)\right )-i \sinh \left (\frac {1}{2} (c+d x)\right )\right )\right ) \left (\cosh \left (\frac {1}{2} (c+d x)\right )+i \sinh \left (\frac {1}{2} (c+d x)\right )\right )^2-9 d e \left (c+d x+2 i \log \left (\cosh \left (\frac {1}{2} (c+d x)\right )+i \sinh \left (\frac {1}{2} (c+d x)\right )\right )\right ) \left (\cosh \left (\frac {1}{2} (c+d x)\right )+i \sinh \left (\frac {1}{2} (c+d x)\right )\right )^2+9 c f \left (c+d x+2 i \log \left (\cosh \left (\frac {1}{2} (c+d x)\right )+i \sinh \left (\frac {1}{2} (c+d x)\right )\right )\right ) \left (\cosh \left (\frac {1}{2} (c+d x)\right )+i \sinh \left (\frac {1}{2} (c+d x)\right )\right )^2-\frac {9 f \left (-2 (-1)^{3/4} (c+d x)^2+\sqrt {2} \left (2 (-2 i c+\pi -2 i d x) \log \left (1+i e^{-c-d x}\right )+\pi \left (3 c+3 d x-4 \log \left (1+e^{c+d x}\right )+4 \log \left (\cosh \left (\frac {1}{2} (c+d x)\right )\right )-2 \log \left (-\sin \left (\frac {1}{4} (\pi -2 i (c+d x))\right )\right )\right )+4 i \text {PolyLog}\left (2,-i e^{-c-d x}\right )\right )\right ) \left (\cosh \left (\frac {1}{2} (c+d x)\right )+i \sinh \left (\frac {1}{2} (c+d x)\right )\right )^2}{2 \sqrt {2}}-\frac {9 f \left (2 \sqrt [4]{-1} (c+d x)^2+\sqrt {2} \left (2 (2 i c+\pi +2 i d x) \log \left (1-i e^{-c-d x}\right )-\pi \left (c+d x-4 \log \left (1+e^{c+d x}\right )+4 \log \left (\cosh \left (\frac {1}{2} (c+d x)\right )\right )+2 \log \left (\sin \left (\frac {1}{4} (\pi +2 i (c+d x))\right )\right )\right )-4 i \text {PolyLog}\left (2,i e^{-c-d x}\right )\right )\right ) \left (\cosh \left (\frac {1}{2} (c+d x)\right )+i \sinh \left (\frac {1}{2} (c+d x)\right )\right )^2}{2 \sqrt {2}}-\frac {6 i d (e+f x) \left (\cosh \left (\frac {1}{2} (c+d x)\right )+i \sinh \left (\frac {1}{2} (c+d x)\right )\right )^2}{\left (\cosh \left (\frac {1}{2} (c+d x)\right )-i \sinh \left (\frac {1}{2} (c+d x)\right )\right )^2}-\frac {4 i f \sinh \left (\frac {1}{2} (c+d x)\right )}{\cosh \left (\frac {1}{2} (c+d x)\right )+i \sinh \left (\frac {1}{2} (c+d x)\right )}+\frac {12 i f \left (\cosh \left (\frac {1}{2} (c+d x)\right )+i \sinh \left (\frac {1}{2} (c+d x)\right )\right )^2 \sinh \left (\frac {1}{2} (c+d x)\right )}{\cosh \left (\frac {1}{2} (c+d x)\right )-i \sinh \left (\frac {1}{2} (c+d x)\right )}+28 f \sinh \left (\frac {1}{2} (c+d x)\right ) \left (-i \cosh \left (\frac {1}{2} (c+d x)\right )+\sinh \left (\frac {1}{2} (c+d x)\right )\right )}{48 d^2 (a+i a \sinh (c+d x))} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[((e + f*x)*Sech[c + d*x]^3)/(a + I*a*Sinh[c + d*x]),x]

[Out]

(2*(f + (6*I)*d*(e + f*x)) + ((6*I)*d*(e + f*x))/(Cosh[(c + d*x)/2] + I*Sinh[(c + d*x)/2])^2 - 9*(c + d*x)*(c*
f - d*(2*e + f*x))*(Cosh[(c + d*x)/2] + I*Sinh[(c + d*x)/2])^2 - 9*d*e*(c + d*x - (2*I)*Log[Cosh[(c + d*x)/2]
- I*Sinh[(c + d*x)/2]])*(Cosh[(c + d*x)/2] + I*Sinh[(c + d*x)/2])^2 + 9*c*f*(c + d*x - (2*I)*Log[Cosh[(c + d*x
)/2] - I*Sinh[(c + d*x)/2]])*(Cosh[(c + d*x)/2] + I*Sinh[(c + d*x)/2])^2 - 9*d*e*(c + d*x + (2*I)*Log[Cosh[(c
+ d*x)/2] + I*Sinh[(c + d*x)/2]])*(Cosh[(c + d*x)/2] + I*Sinh[(c + d*x)/2])^2 + 9*c*f*(c + d*x + (2*I)*Log[Cos
h[(c + d*x)/2] + I*Sinh[(c + d*x)/2]])*(Cosh[(c + d*x)/2] + I*Sinh[(c + d*x)/2])^2 - (9*f*(-2*(-1)^(3/4)*(c +
d*x)^2 + Sqrt[2]*(2*((-2*I)*c + Pi - (2*I)*d*x)*Log[1 + I*E^(-c - d*x)] + Pi*(3*c + 3*d*x - 4*Log[1 + E^(c + d
*x)] + 4*Log[Cosh[(c + d*x)/2]] - 2*Log[-Sin[(Pi - (2*I)*(c + d*x))/4]]) + (4*I)*PolyLog[2, (-I)*E^(-c - d*x)]
))*(Cosh[(c + d*x)/2] + I*Sinh[(c + d*x)/2])^2)/(2*Sqrt[2]) - (9*f*(2*(-1)^(1/4)*(c + d*x)^2 + Sqrt[2]*(2*((2*
I)*c + Pi + (2*I)*d*x)*Log[1 - I*E^(-c - d*x)] - Pi*(c + d*x - 4*Log[1 + E^(c + d*x)] + 4*Log[Cosh[(c + d*x)/2
]] + 2*Log[Sin[(Pi + (2*I)*(c + d*x))/4]]) - (4*I)*PolyLog[2, I*E^(-c - d*x)]))*(Cosh[(c + d*x)/2] + I*Sinh[(c
+ d*x)/2])^2)/(2*Sqrt[2]) - ((6*I)*d*(e + f*x)*(Cosh[(c + d*x)/2] + I*Sinh[(c + d*x)/2])^2)/(Cosh[(c + d*x)/2
] - I*Sinh[(c + d*x)/2])^2 - ((4*I)*f*Sinh[(c + d*x)/2])/(Cosh[(c + d*x)/2] + I*Sinh[(c + d*x)/2]) + ((12*I)*f
*(Cosh[(c + d*x)/2] + I*Sinh[(c + d*x)/2])^2*Sinh[(c + d*x)/2])/(Cosh[(c + d*x)/2] - I*Sinh[(c + d*x)/2]) + 28
*f*Sinh[(c + d*x)/2]*((-I)*Cosh[(c + d*x)/2] + Sinh[(c + d*x)/2]))/(48*d^2*(a + I*a*Sinh[c + d*x]))

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Maple [B] Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 444 vs. \(2 (203 ) = 406\).
time = 4.92, size = 445, normalized size = 1.91

method	result	size
risch	\(\frac {9 d f x \,{\mathrm e}^{5 d x +5 c}+9 d f x \,{\mathrm e}^{d x +c}-18 i d e \,{\mathrm e}^{4 d x +4 c}-4 i f -22 i f \,{\mathrm e}^{2 d x +2 c}+9 f \,{\mathrm e}^{5 d x +5 c}+8 f \,{\mathrm e}^{3 d x +3 c}-f \,{\mathrm e}^{d x +c}+9 d e \,{\mathrm e}^{5 d x +5 c}+9 d e \,{\mathrm e}^{d x +c}-18 i d f x \,{\mathrm e}^{4 d x +4 c}+18 i d e \,{\mathrm e}^{2 d x +2 c}+6 d e \,{\mathrm e}^{3 d x +3 c}+6 d f x \,{\mathrm e}^{3 d x +3 c}-18 i f \,{\mathrm e}^{4 d x +4 c}+18 i d f x \,{\mathrm e}^{2 d x +2 c}}{12 \left ({\mathrm e}^{d x +c}+i\right )^{2} \left ({\mathrm e}^{d x +c}-i\right )^{4} d^{2} a}+\frac {3 i e \ln \left ({\mathrm e}^{d x +c}+i\right )}{8 d a}-\frac {3 i e \ln \left ({\mathrm e}^{d x +c}-i\right )}{8 d a}+\frac {3 i f \ln \left (1-i {\mathrm e}^{d x +c}\right ) x}{8 d a}+\frac {3 i f \ln \left (1-i {\mathrm e}^{d x +c}\right ) c}{8 d^{2} a}+\frac {3 i f \polylog \left (2, i {\mathrm e}^{d x +c}\right )}{8 a \,d^{2}}-\frac {3 i f \ln \left (1+i {\mathrm e}^{d x +c}\right ) x}{8 d a}-\frac {3 i f \ln \left (1+i {\mathrm e}^{d x +c}\right ) c}{8 d^{2} a}-\frac {3 i f \polylog \left (2, -i {\mathrm e}^{d x +c}\right )}{8 a \,d^{2}}-\frac {3 i f c \ln \left ({\mathrm e}^{d x +c}+i\right )}{8 d^{2} a}+\frac {3 i f c \ln \left ({\mathrm e}^{d x +c}-i\right )}{8 d^{2} a}\)	\(445\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((f*x+e)*sech(d*x+c)^3/(a+I*a*sinh(d*x+c)),x,method=_RETURNVERBOSE)

[Out]

1/12*(9*d*f*x*exp(5*d*x+5*c)+9*d*f*x*exp(d*x+c)-18*I*d*e*exp(4*d*x+4*c)-4*I*f-22*I*f*exp(2*d*x+2*c)+9*f*exp(5*
d*x+5*c)+8*f*exp(3*d*x+3*c)-f*exp(d*x+c)+9*d*e*exp(5*d*x+5*c)+9*d*e*exp(d*x+c)-18*I*d*f*x*exp(4*d*x+4*c)+18*I*
d*e*exp(2*d*x+2*c)+6*d*e*exp(3*d*x+3*c)+6*d*f*x*exp(3*d*x+3*c)-18*I*f*exp(4*d*x+4*c)+18*I*d*f*x*exp(2*d*x+2*c)
)/(exp(d*x+c)+I)^2/(exp(d*x+c)-I)^4/d^2/a+3/8*I/d/a*e*ln(exp(d*x+c)+I)-3/8*I/d/a*e*ln(exp(d*x+c)-I)+3/8*I/d/a*
f*ln(1-I*exp(d*x+c))*x+3/8*I/d^2/a*f*ln(1-I*exp(d*x+c))*c+3/8*I*f*polylog(2,I*exp(d*x+c))/a/d^2-3/8*I/d/a*f*ln
(1+I*exp(d*x+c))*x-3/8*I/d^2/a*f*ln(1+I*exp(d*x+c))*c-3/8*I*f*polylog(2,-I*exp(d*x+c))/a/d^2-3/8*I/d^2/a*f*c*l
n(exp(d*x+c)+I)+3/8*I/d^2/a*f*c*ln(exp(d*x+c)-I)

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Maxima [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: RuntimeError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((f*x+e)*sech(d*x+c)^3/(a+I*a*sinh(d*x+c)),x, algorithm="maxima")

[Out]

Exception raised: RuntimeError >> ECL says: THROW: The catch RAT-ERR is undefined.

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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. 936 vs. \(2 (201) = 402\).
time = 0.37, size = 936, normalized size = 4.02 \begin {gather*} -\frac {9 \, {\left (-i \, f e^{\left (6 \, d x + 6 \, c\right )} - 2 \, f e^{\left (5 \, d x + 5 \, c\right )} - i \, f e^{\left (4 \, d x + 4 \, c\right )} - 4 \, f e^{\left (3 \, d x + 3 \, c\right )} + i \, f e^{\left (2 \, d x + 2 \, c\right )} - 2 \, f e^{\left (d x + c\right )} + i \, f\right )} {\rm Li}_2\left (i \, e^{\left (d x + c\right )}\right ) + 9 \, {\left (i \, f e^{\left (6 \, d x + 6 \, c\right )} + 2 \, f e^{\left (5 \, d x + 5 \, c\right )} + i \, f e^{\left (4 \, d x + 4 \, c\right )} + 4 \, f e^{\left (3 \, d x + 3 \, c\right )} - i \, f e^{\left (2 \, d x + 2 \, c\right )} + 2 \, f e^{\left (d x + c\right )} - i \, f\right )} {\rm Li}_2\left (-i \, e^{\left (d x + c\right )}\right ) - 18 \, {\left (d f x + d e + f\right )} e^{\left (5 \, d x + 5 \, c\right )} + 36 \, {\left (i \, d f x + i \, d e + i \, f\right )} e^{\left (4 \, d x + 4 \, c\right )} - 4 \, {\left (3 \, d f x + 3 \, d e + 4 \, f\right )} e^{\left (3 \, d x + 3 \, c\right )} + 4 \, {\left (-9 i \, d f x - 9 i \, d e + 11 i \, f\right )} e^{\left (2 \, d x + 2 \, c\right )} - 2 \, {\left (9 \, d f x + 9 \, d e - f\right )} e^{\left (d x + c\right )} + 9 \, {\left (-i \, c f + i \, d e + {\left (i \, c f - i \, d e\right )} e^{\left (6 \, d x + 6 \, c\right )} + 2 \, {\left (c f - d e\right )} e^{\left (5 \, d x + 5 \, c\right )} + {\left (i \, c f - i \, d e\right )} e^{\left (4 \, d x + 4 \, c\right )} + 4 \, {\left (c f - d e\right )} e^{\left (3 \, d x + 3 \, c\right )} + {\left (-i \, c f + i \, d e\right )} e^{\left (2 \, d x + 2 \, c\right )} + 2 \, {\left (c f - d e\right )} e^{\left (d x + c\right )}\right )} \log \left (e^{\left (d x + c\right )} + i\right ) + 9 \, {\left (i \, c f - i \, d e + {\left (-i \, c f + i \, d e\right )} e^{\left (6 \, d x + 6 \, c\right )} - 2 \, {\left (c f - d e\right )} e^{\left (5 \, d x + 5 \, c\right )} + {\left (-i \, c f + i \, d e\right )} e^{\left (4 \, d x + 4 \, c\right )} - 4 \, {\left (c f - d e\right )} e^{\left (3 \, d x + 3 \, c\right )} + {\left (i \, c f - i \, d e\right )} e^{\left (2 \, d x + 2 \, c\right )} - 2 \, {\left (c f - d e\right )} e^{\left (d x + c\right )}\right )} \log \left (e^{\left (d x + c\right )} - i\right ) + 9 \, {\left (-i \, d f x - i \, c f + {\left (i \, d f x + i \, c f\right )} e^{\left (6 \, d x + 6 \, c\right )} + 2 \, {\left (d f x + c f\right )} e^{\left (5 \, d x + 5 \, c\right )} + {\left (i \, d f x + i \, c f\right )} e^{\left (4 \, d x + 4 \, c\right )} + 4 \, {\left (d f x + c f\right )} e^{\left (3 \, d x + 3 \, c\right )} + {\left (-i \, d f x - i \, c f\right )} e^{\left (2 \, d x + 2 \, c\right )} + 2 \, {\left (d f x + c f\right )} e^{\left (d x + c\right )}\right )} \log \left (i \, e^{\left (d x + c\right )} + 1\right ) + 9 \, {\left (i \, d f x + i \, c f + {\left (-i \, d f x - i \, c f\right )} e^{\left (6 \, d x + 6 \, c\right )} - 2 \, {\left (d f x + c f\right )} e^{\left (5 \, d x + 5 \, c\right )} + {\left (-i \, d f x - i \, c f\right )} e^{\left (4 \, d x + 4 \, c\right )} - 4 \, {\left (d f x + c f\right )} e^{\left (3 \, d x + 3 \, c\right )} + {\left (i \, d f x + i \, c f\right )} e^{\left (2 \, d x + 2 \, c\right )} - 2 \, {\left (d f x + c f\right )} e^{\left (d x + c\right )}\right )} \log \left (-i \, e^{\left (d x + c\right )} + 1\right ) + 8 i \, f}{24 \, {\left (a d^{2} e^{\left (6 \, d x + 6 \, c\right )} - 2 i \, a d^{2} e^{\left (5 \, d x + 5 \, c\right )} + a d^{2} e^{\left (4 \, d x + 4 \, c\right )} - 4 i \, a d^{2} e^{\left (3 \, d x + 3 \, c\right )} - a d^{2} e^{\left (2 \, d x + 2 \, c\right )} - 2 i \, a d^{2} e^{\left (d x + c\right )} - a d^{2}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((f*x+e)*sech(d*x+c)^3/(a+I*a*sinh(d*x+c)),x, algorithm="fricas")

[Out]

-1/24*(9*(-I*f*e^(6*d*x + 6*c) - 2*f*e^(5*d*x + 5*c) - I*f*e^(4*d*x + 4*c) - 4*f*e^(3*d*x + 3*c) + I*f*e^(2*d*
x + 2*c) - 2*f*e^(d*x + c) + I*f)*dilog(I*e^(d*x + c)) + 9*(I*f*e^(6*d*x + 6*c) + 2*f*e^(5*d*x + 5*c) + I*f*e^
(4*d*x + 4*c) + 4*f*e^(3*d*x + 3*c) - I*f*e^(2*d*x + 2*c) + 2*f*e^(d*x + c) - I*f)*dilog(-I*e^(d*x + c)) - 18*
(d*f*x + d*e + f)*e^(5*d*x + 5*c) + 36*(I*d*f*x + I*d*e + I*f)*e^(4*d*x + 4*c) - 4*(3*d*f*x + 3*d*e + 4*f)*e^(
3*d*x + 3*c) + 4*(-9*I*d*f*x - 9*I*d*e + 11*I*f)*e^(2*d*x + 2*c) - 2*(9*d*f*x + 9*d*e - f)*e^(d*x + c) + 9*(-I
*c*f + I*d*e + (I*c*f - I*d*e)*e^(6*d*x + 6*c) + 2*(c*f - d*e)*e^(5*d*x + 5*c) + (I*c*f - I*d*e)*e^(4*d*x + 4*
c) + 4*(c*f - d*e)*e^(3*d*x + 3*c) + (-I*c*f + I*d*e)*e^(2*d*x + 2*c) + 2*(c*f - d*e)*e^(d*x + c))*log(e^(d*x
+ c) + I) + 9*(I*c*f - I*d*e + (-I*c*f + I*d*e)*e^(6*d*x + 6*c) - 2*(c*f - d*e)*e^(5*d*x + 5*c) + (-I*c*f + I*
d*e)*e^(4*d*x + 4*c) - 4*(c*f - d*e)*e^(3*d*x + 3*c) + (I*c*f - I*d*e)*e^(2*d*x + 2*c) - 2*(c*f - d*e)*e^(d*x
+ c))*log(e^(d*x + c) - I) + 9*(-I*d*f*x - I*c*f + (I*d*f*x + I*c*f)*e^(6*d*x + 6*c) + 2*(d*f*x + c*f)*e^(5*d*
x + 5*c) + (I*d*f*x + I*c*f)*e^(4*d*x + 4*c) + 4*(d*f*x + c*f)*e^(3*d*x + 3*c) + (-I*d*f*x - I*c*f)*e^(2*d*x +
2*c) + 2*(d*f*x + c*f)*e^(d*x + c))*log(I*e^(d*x + c) + 1) + 9*(I*d*f*x + I*c*f + (-I*d*f*x - I*c*f)*e^(6*d*x
+ 6*c) - 2*(d*f*x + c*f)*e^(5*d*x + 5*c) + (-I*d*f*x - I*c*f)*e^(4*d*x + 4*c) - 4*(d*f*x + c*f)*e^(3*d*x + 3*
c) + (I*d*f*x + I*c*f)*e^(2*d*x + 2*c) - 2*(d*f*x + c*f)*e^(d*x + c))*log(-I*e^(d*x + c) + 1) + 8*I*f)/(a*d^2*
e^(6*d*x + 6*c) - 2*I*a*d^2*e^(5*d*x + 5*c) + a*d^2*e^(4*d*x + 4*c) - 4*I*a*d^2*e^(3*d*x + 3*c) - a*d^2*e^(2*d
*x + 2*c) - 2*I*a*d^2*e^(d*x + c) - a*d^2)

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} - \frac {i \left (\int \frac {e \operatorname {sech}^{3}{\left (c + d x \right )}}{\sinh {\left (c + d x \right )} - i}\, dx + \int \frac {f x \operatorname {sech}^{3}{\left (c + d x \right )}}{\sinh {\left (c + d x \right )} - i}\, dx\right )}{a} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((f*x+e)*sech(d*x+c)**3/(a+I*a*sinh(d*x+c)),x)

[Out]

-I*(Integral(e*sech(c + d*x)**3/(sinh(c + d*x) - I), x) + Integral(f*x*sech(c + d*x)**3/(sinh(c + d*x) - I), x
))/a

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((f*x+e)*sech(d*x+c)^3/(a+I*a*sinh(d*x+c)),x, algorithm="giac")

[Out]

integrate((f*x + e)*sech(d*x + c)^3/(I*a*sinh(d*x + c) + a), x)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {e+f\,x}{{\mathrm {cosh}\left (c+d\,x\right )}^3\,\left (a+a\,\mathrm {sinh}\left (c+d\,x\right )\,1{}\mathrm {i}\right )} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((e + f*x)/(cosh(c + d*x)^3*(a + a*sinh(c + d*x)*1i)),x)

[Out]

int((e + f*x)/(cosh(c + d*x)^3*(a + a*sinh(c + d*x)*1i)), x)

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