3.1.94 \(\int \frac {1}{(5+3 i \sinh (c+d x))^3} \, dx\) [94]

Optimal. Leaf size=95 \[ \frac {59 x}{2048}-\frac {59 i \text {ArcTan}\left (\frac {\cosh (c+d x)}{3+i \sinh (c+d x)}\right )}{1024 d}-\frac {3 i \cosh (c+d x)}{32 d (5+3 i \sinh (c+d x))^2}-\frac {45 i \cosh (c+d x)}{512 d (5+3 i \sinh (c+d x))} \]

[Out]

59/2048*x-59/1024*I*arctan(cosh(d*x+c)/(3+I*sinh(d*x+c)))/d-3/32*I*cosh(d*x+c)/d/(5+3*I*sinh(d*x+c))^2-45/512*
I*cosh(d*x+c)/d/(5+3*I*sinh(d*x+c))

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Rubi [A]
time = 0.05, antiderivative size = 95, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {2743, 2833, 12, 2736} \begin {gather*} -\frac {59 i \text {ArcTan}\left (\frac {\cosh (c+d x)}{3+i \sinh (c+d x)}\right )}{1024 d}-\frac {45 i \cosh (c+d x)}{512 d (5+3 i \sinh (c+d x))}-\frac {3 i \cosh (c+d x)}{32 d (5+3 i \sinh (c+d x))^2}+\frac {59 x}{2048} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(5 + (3*I)*Sinh[c + d*x])^(-3),x]

[Out]

(59*x)/2048 - (((59*I)/1024)*ArcTan[Cosh[c + d*x]/(3 + I*Sinh[c + d*x])])/d - (((3*I)/32)*Cosh[c + d*x])/(d*(5
+ (3*I)*Sinh[c + d*x])^2) - (((45*I)/512)*Cosh[c + d*x])/(d*(5 + (3*I)*Sinh[c + d*x]))

Rule 12

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

Rule 2736

Int[((a_) + (b_.)*sin[(c_.) + (d_.)*(x_)])^(-1), x_Symbol] :> With[{q = Rt[a^2 - b^2, 2]}, Simp[x/q, x] + Simp
[(2/(d*q))*ArcTan[b*(Cos[c + d*x]/(a + q + b*Sin[c + d*x]))], x]] /; FreeQ[{a, b, c, d}, x] && GtQ[a^2 - b^2,
0] && PosQ[a]

Rule 2743

Int[((a_) + (b_.)*sin[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[(-b)*Cos[c + d*x]*((a + b*Sin[c + d*x])^(n
+ 1)/(d*(n + 1)*(a^2 - b^2))), x] + Dist[1/((n + 1)*(a^2 - b^2)), Int[(a + b*Sin[c + d*x])^(n + 1)*Simp[a*(n +
1) - b*(n + 2)*Sin[c + d*x], x], x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[a^2 - b^2, 0] && LtQ[n, -1] && Integ
erQ[2*n]

Rule 2833

Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[(-(
b*c - a*d))*Cos[e + f*x]*((a + b*Sin[e + f*x])^(m + 1)/(f*(m + 1)*(a^2 - b^2))), x] + Dist[1/((m + 1)*(a^2 - b
^2)), Int[(a + b*Sin[e + f*x])^(m + 1)*Simp[(a*c - b*d)*(m + 1) - (b*c - a*d)*(m + 2)*Sin[e + f*x], x], x], x]
/; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && LtQ[m, -1] && IntegerQ[2*m]

Rubi steps

\begin {align*} \int \frac {1}{(5+3 i \sinh (c+d x))^3} \, dx &=-\frac {3 i \cosh (c+d x)}{32 d (5+3 i \sinh (c+d x))^2}-\frac {1}{32} \int \frac {-10+3 i \sinh (c+d x)}{(5+3 i \sinh (c+d x))^2} \, dx\\ &=-\frac {3 i \cosh (c+d x)}{32 d (5+3 i \sinh (c+d x))^2}-\frac {45 i \cosh (c+d x)}{512 d (5+3 i \sinh (c+d x))}+\frac {1}{512} \int \frac {59}{5+3 i \sinh (c+d x)} \, dx\\ &=-\frac {3 i \cosh (c+d x)}{32 d (5+3 i \sinh (c+d x))^2}-\frac {45 i \cosh (c+d x)}{512 d (5+3 i \sinh (c+d x))}+\frac {59}{512} \int \frac {1}{5+3 i \sinh (c+d x)} \, dx\\ &=\frac {59 x}{2048}-\frac {59 i \tan ^{-1}\left (\frac {\cosh (c+d x)}{3+i \sinh (c+d x)}\right )}{1024 d}-\frac {3 i \cosh (c+d x)}{32 d (5+3 i \sinh (c+d x))^2}-\frac {45 i \cosh (c+d x)}{512 d (5+3 i \sinh (c+d 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. \(277\) vs. \(2(95)=190\).
time = 0.51, size = 277, normalized size = 2.92 \begin {gather*} \frac {-118 i \text {ArcTan}\left (\frac {2 \cosh \left (\frac {1}{2} (c+d x)\right )-\sinh \left (\frac {1}{2} (c+d x)\right )}{\cosh \left (\frac {1}{2} (c+d x)\right )-2 \sinh \left (\frac {1}{2} (c+d x)\right )}\right )+118 i \text {ArcTan}\left (\frac {\cosh \left (\frac {1}{2} (c+d x)\right )+2 \sinh \left (\frac {1}{2} (c+d x)\right )}{2 \cosh \left (\frac {1}{2} (c+d x)\right )+\sinh \left (\frac {1}{2} (c+d x)\right )}\right )-59 \log (5 \cosh (c+d x)-4 \sinh (c+d x))+59 \log (5 \cosh (c+d x)+4 \sinh (c+d x))+\frac {48}{\left ((1+2 i) \cosh \left (\frac {1}{2} (c+d x)\right )-(2+i) \sinh \left (\frac {1}{2} (c+d x)\right )\right )^2}+\frac {48}{\left ((2+i) \cosh \left (\frac {1}{2} (c+d x)\right )+(1+2 i) \sinh \left (\frac {1}{2} (c+d x)\right )\right )^2}-\frac {144 \sinh \left (\frac {1}{2} (c+d x)\right ) \left (-3 i \cosh \left (\frac {1}{2} (c+d x)\right )+5 \sinh \left (\frac {1}{2} (c+d x)\right )\right )}{-5 i+3 \sinh (c+d x)}}{4096 d} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(5 + (3*I)*Sinh[c + d*x])^(-3),x]

[Out]

((-118*I)*ArcTan[(2*Cosh[(c + d*x)/2] - Sinh[(c + d*x)/2])/(Cosh[(c + d*x)/2] - 2*Sinh[(c + d*x)/2])] + (118*I
)*ArcTan[(Cosh[(c + d*x)/2] + 2*Sinh[(c + d*x)/2])/(2*Cosh[(c + d*x)/2] + Sinh[(c + d*x)/2])] - 59*Log[5*Cosh[
c + d*x] - 4*Sinh[c + d*x]] + 59*Log[5*Cosh[c + d*x] + 4*Sinh[c + d*x]] + 48/((1 + 2*I)*Cosh[(c + d*x)/2] - (2
+ I)*Sinh[(c + d*x)/2])^2 + 48/((2 + I)*Cosh[(c + d*x)/2] + (1 + 2*I)*Sinh[(c + d*x)/2])^2 - (144*Sinh[(c + d
*x)/2]*((-3*I)*Cosh[(c + d*x)/2] + 5*Sinh[(c + d*x)/2]))/(-5*I + 3*Sinh[c + d*x]))/(4096*d)

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Maple [A]
time = 1.30, size = 126, normalized size = 1.33

method	result	size
risch	\(-\frac {3 i \left (-295 i {\mathrm e}^{2 d x +2 c}+59 \,{\mathrm e}^{3 d x +3 c}+45 i-241 \,{\mathrm e}^{d x +c}\right )}{256 d \left (3 \,{\mathrm e}^{2 d x +2 c}-3-10 i {\mathrm e}^{d x +c}\right )^{2}}-\frac {59 \ln \left (-3 i+{\mathrm e}^{d x +c}\right )}{2048 d}+\frac {59 \ln \left ({\mathrm e}^{d x +c}-\frac {i}{3}\right )}{2048 d}\)	\(96\)
derivativedivides	\(\frac {\frac {-\frac {63}{3200}-\frac {27 i}{400}}{\left (5 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-4-3 i\right )^{2}}+\frac {-\frac {963}{12800}+\frac {123 i}{1600}}{5 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-4-3 i}-\frac {59 \ln \left (5 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-4-3 i\right )}{2048}+\frac {\frac {63}{3200}-\frac {27 i}{400}}{\left (5 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+4-3 i\right )^{2}}+\frac {-\frac {963}{12800}-\frac {123 i}{1600}}{5 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+4-3 i}+\frac {59 \ln \left (5 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+4-3 i\right )}{2048}}{d}\)	\(126\)
default	\(\frac {\frac {-\frac {63}{3200}-\frac {27 i}{400}}{\left (5 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-4-3 i\right )^{2}}+\frac {-\frac {963}{12800}+\frac {123 i}{1600}}{5 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-4-3 i}-\frac {59 \ln \left (5 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-4-3 i\right )}{2048}+\frac {\frac {63}{3200}-\frac {27 i}{400}}{\left (5 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+4-3 i\right )^{2}}+\frac {-\frac {963}{12800}-\frac {123 i}{1600}}{5 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+4-3 i}+\frac {59 \ln \left (5 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+4-3 i\right )}{2048}}{d}\)	\(126\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(5+3*I*sinh(d*x+c))^3,x,method=_RETURNVERBOSE)

[Out]

1/d*((-63/3200-27/400*I)/(5*tanh(1/2*d*x+1/2*c)-4-3*I)^2+(-963/12800+123/1600*I)/(5*tanh(1/2*d*x+1/2*c)-4-3*I)
-59/2048*ln(5*tanh(1/2*d*x+1/2*c)-4-3*I)+(63/3200-27/400*I)/(5*tanh(1/2*d*x+1/2*c)+4-3*I)^2-(963/12800+123/160
0*I)/(5*tanh(1/2*d*x+1/2*c)+4-3*I)+59/2048*ln(5*tanh(1/2*d*x+1/2*c)+4-3*I))

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Maxima [A]
time = 0.60, size = 108, normalized size = 1.14 \begin {gather*} -\frac {59 i \, \arctan \left (\frac {3}{4} \, e^{\left (-d x - c\right )} + \frac {5}{4} i\right )}{1024 \, d} + \frac {3 \, {\left (241 i \, e^{\left (-d x - c\right )} + 295 \, e^{\left (-2 \, d x - 2 \, c\right )} - 59 i \, e^{\left (-3 \, d x - 3 \, c\right )} - 45\right )}}{-256 \, d {\left (60 i \, e^{\left (-d x - c\right )} + 118 \, e^{\left (-2 \, d x - 2 \, c\right )} - 60 i \, e^{\left (-3 \, d x - 3 \, c\right )} - 9 \, e^{\left (-4 \, d x - 4 \, c\right )} - 9\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(5+3*I*sinh(d*x+c))^3,x, algorithm="maxima")

[Out]

-59/1024*I*arctan(3/4*e^(-d*x - c) + 5/4*I)/d + 3*(241*I*e^(-d*x - c) + 295*e^(-2*d*x - 2*c) - 59*I*e^(-3*d*x
- 3*c) - 45)/(d*(-15360*I*e^(-d*x - c) - 30208*e^(-2*d*x - 2*c) + 15360*I*e^(-3*d*x - 3*c) + 2304*e^(-4*d*x -
4*c) + 2304))

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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. 193 vs. \(2 (75) = 150\).
time = 0.39, size = 193, normalized size = 2.03 \begin {gather*} \frac {59 \, {\left (9 \, e^{\left (4 \, d x + 4 \, c\right )} - 60 i \, e^{\left (3 \, d x + 3 \, c\right )} - 118 \, e^{\left (2 \, d x + 2 \, c\right )} + 60 i \, e^{\left (d x + c\right )} + 9\right )} \log \left (e^{\left (d x + c\right )} - \frac {1}{3} i\right ) - 59 \, {\left (9 \, e^{\left (4 \, d x + 4 \, c\right )} - 60 i \, e^{\left (3 \, d x + 3 \, c\right )} - 118 \, e^{\left (2 \, d x + 2 \, c\right )} + 60 i \, e^{\left (d x + c\right )} + 9\right )} \log \left (e^{\left (d x + c\right )} - 3 i\right ) - 1416 i \, e^{\left (3 \, d x + 3 \, c\right )} - 7080 \, e^{\left (2 \, d x + 2 \, c\right )} + 5784 i \, e^{\left (d x + c\right )} + 1080}{2048 \, {\left (9 \, d e^{\left (4 \, d x + 4 \, c\right )} - 60 i \, d e^{\left (3 \, d x + 3 \, c\right )} - 118 \, d e^{\left (2 \, d x + 2 \, c\right )} + 60 i \, d e^{\left (d x + c\right )} + 9 \, d\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(5+3*I*sinh(d*x+c))^3,x, algorithm="fricas")

[Out]

1/2048*(59*(9*e^(4*d*x + 4*c) - 60*I*e^(3*d*x + 3*c) - 118*e^(2*d*x + 2*c) + 60*I*e^(d*x + c) + 9)*log(e^(d*x
+ c) - 1/3*I) - 59*(9*e^(4*d*x + 4*c) - 60*I*e^(3*d*x + 3*c) - 118*e^(2*d*x + 2*c) + 60*I*e^(d*x + c) + 9)*log
(e^(d*x + c) - 3*I) - 1416*I*e^(3*d*x + 3*c) - 7080*e^(2*d*x + 2*c) + 5784*I*e^(d*x + c) + 1080)/(9*d*e^(4*d*x
+ 4*c) - 60*I*d*e^(3*d*x + 3*c) - 118*d*e^(2*d*x + 2*c) + 60*I*d*e^(d*x + c) + 9*d)

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Sympy [A]
time = 0.23, size = 141, normalized size = 1.48 \begin {gather*} \frac {- 177 i e^{3 c} e^{3 d x} - 885 e^{2 c} e^{2 d x} + 723 i e^{c} e^{d x} + 135}{2304 d e^{4 c} e^{4 d x} - 15360 i d e^{3 c} e^{3 d x} - 30208 d e^{2 c} e^{2 d x} + 15360 i d e^{c} e^{d x} + 2304 d} + \frac {- \frac {59 \log {\left (e^{d x} - 3 i e^{- c} \right )}}{2048} + \frac {59 \log {\left (e^{d x} - \frac {i e^{- c}}{3} \right )}}{2048}}{d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(5+3*I*sinh(d*x+c))**3,x)

[Out]

(-177*I*exp(3*c)*exp(3*d*x) - 885*exp(2*c)*exp(2*d*x) + 723*I*exp(c)*exp(d*x) + 135)/(2304*d*exp(4*c)*exp(4*d*
x) - 15360*I*d*exp(3*c)*exp(3*d*x) - 30208*d*exp(2*c)*exp(2*d*x) + 15360*I*d*exp(c)*exp(d*x) + 2304*d) + (-59*
log(exp(d*x) - 3*I*exp(-c))/2048 + 59*log(exp(d*x) - I*exp(-c)/3)/2048)/d

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Giac [A]
time = 0.42, size = 87, normalized size = 0.92 \begin {gather*} -\frac {\frac {24 \, {\left (-59 i \, e^{\left (3 \, d x + 3 \, c\right )} - 295 \, e^{\left (2 \, d x + 2 \, c\right )} + 241 i \, e^{\left (d x + c\right )} + 45\right )}}{{\left (-3 i \, e^{\left (2 \, d x + 2 \, c\right )} - 10 \, e^{\left (d x + c\right )} + 3 i\right )}^{2}} - 59 \, \log \left (3 \, e^{\left (d x + c\right )} - i\right ) + 59 \, \log \left (e^{\left (d x + c\right )} - 3 i\right )}{2048 \, d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(5+3*I*sinh(d*x+c))^3,x, algorithm="giac")

[Out]

-1/2048*(24*(-59*I*e^(3*d*x + 3*c) - 295*e^(2*d*x + 2*c) + 241*I*e^(d*x + c) + 45)/(-3*I*e^(2*d*x + 2*c) - 10*
e^(d*x + c) + 3*I)^2 - 59*log(3*e^(d*x + c) - I) + 59*log(e^(d*x + c) - 3*I))/d

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Mupad [B]
time = 1.62, size = 143, normalized size = 1.51 \begin {gather*} \frac {\frac {295}{2304\,d}+\frac {{\mathrm {e}}^{c+d\,x}\,59{}\mathrm {i}}{768\,d}}{1-{\mathrm {e}}^{2\,c+2\,d\,x}+\frac {{\mathrm {e}}^{c+d\,x}\,10{}\mathrm {i}}{3}}-\frac {59\,\ln \left (-\frac {59\,{\mathrm {e}}^{c+d\,x}}{4}+\frac {177}{4}{}\mathrm {i}\right )}{2048\,d}+\frac {59\,\ln \left (\frac {531\,{\mathrm {e}}^{c+d\,x}}{4}-\frac {177}{4}{}\mathrm {i}\right )}{2048\,d}-\frac {\frac {5}{72\,d}+\frac {{\mathrm {e}}^{c+d\,x}\,41{}\mathrm {i}}{216\,d}}{{\mathrm {e}}^{4\,c+4\,d\,x}-\frac {118\,{\mathrm {e}}^{2\,c+2\,d\,x}}{9}+1+\frac {{\mathrm {e}}^{c+d\,x}\,20{}\mathrm {i}}{3}-\frac {{\mathrm {e}}^{3\,c+3\,d\,x}\,20{}\mathrm {i}}{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(sinh(c + d*x)*3i + 5)^3,x)

[Out]

((exp(c + d*x)*59i)/(768*d) + 295/(2304*d))/((exp(c + d*x)*10i)/3 - exp(2*c + 2*d*x) + 1) - (59*log(177i/4 - (
59*exp(c + d*x))/4))/(2048*d) + (59*log((531*exp(c + d*x))/4 - 177i/4))/(2048*d) - ((exp(c + d*x)*41i)/(216*d)
+ 5/(72*d))/((exp(c + d*x)*20i)/3 - (118*exp(2*c + 2*d*x))/9 - (exp(3*c + 3*d*x)*20i)/3 + exp(4*c + 4*d*x) +
1)

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