3.95 \(\int \frac {1}{(5+3 i \sinh (c+d x))^4} \, dx\)

Optimal. Leaf size=124 \[ -\frac {311 i \cosh (c+d x)}{8192 d (5+3 i \sinh (c+d x))}-\frac {25 i \cosh (c+d x)}{512 d (5+3 i \sinh (c+d x))^2}-\frac {i \cosh (c+d x)}{16 d (5+3 i \sinh (c+d x))^3}-\frac {385 i \tan ^{-1}\left (\frac {\cosh (c+d x)}{3+i \sinh (c+d x)}\right )}{16384 d}+\frac {385 x}{32768} \]

[Out]

385/32768*x-385/16384*I*arctan(cosh(d*x+c)/(3+I*sinh(d*x+c)))/d-1/16*I*cosh(d*x+c)/d/(5+3*I*sinh(d*x+c))^3-25/
512*I*cosh(d*x+c)/d/(5+3*I*sinh(d*x+c))^2-311/8192*I*cosh(d*x+c)/d/(5+3*I*sinh(d*x+c))

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Rubi [A]  time = 0.10, antiderivative size = 124, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {2664, 2754, 12, 2657} \[ -\frac {311 i \cosh (c+d x)}{8192 d (5+3 i \sinh (c+d x))}-\frac {25 i \cosh (c+d x)}{512 d (5+3 i \sinh (c+d x))^2}-\frac {i \cosh (c+d x)}{16 d (5+3 i \sinh (c+d x))^3}-\frac {385 i \tan ^{-1}\left (\frac {\cosh (c+d x)}{3+i \sinh (c+d x)}\right )}{16384 d}+\frac {385 x}{32768} \]

Antiderivative was successfully verified.

[In]

Int[(5 + (3*I)*Sinh[c + d*x])^(-4),x]

[Out]

(385*x)/32768 - (((385*I)/16384)*ArcTan[Cosh[c + d*x]/(3 + I*Sinh[c + d*x])])/d - ((I/16)*Cosh[c + d*x])/(d*(5
+ (3*I)*Sinh[c + d*x])^3) - (((25*I)/512)*Cosh[c + d*x])/(d*(5 + (3*I)*Sinh[c + d*x])^2) - (((311*I)/8192)*Co
sh[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 2657

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

Rule 2664

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] && Integer
Q[2*n]

Rule 2754

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))^4} \, dx &=-\frac {i \cosh (c+d x)}{16 d (5+3 i \sinh (c+d x))^3}-\frac {1}{48} \int \frac {-15+6 i \sinh (c+d x)}{(5+3 i \sinh (c+d x))^3} \, dx\\ &=-\frac {i \cosh (c+d x)}{16 d (5+3 i \sinh (c+d x))^3}-\frac {25 i \cosh (c+d x)}{512 d (5+3 i \sinh (c+d x))^2}+\frac {\int \frac {186-75 i \sinh (c+d x)}{(5+3 i \sinh (c+d x))^2} \, dx}{1536}\\ &=-\frac {i \cosh (c+d x)}{16 d (5+3 i \sinh (c+d x))^3}-\frac {25 i \cosh (c+d x)}{512 d (5+3 i \sinh (c+d x))^2}-\frac {311 i \cosh (c+d x)}{8192 d (5+3 i \sinh (c+d x))}-\frac {\int -\frac {1155}{5+3 i \sinh (c+d x)} \, dx}{24576}\\ &=-\frac {i \cosh (c+d x)}{16 d (5+3 i \sinh (c+d x))^3}-\frac {25 i \cosh (c+d x)}{512 d (5+3 i \sinh (c+d x))^2}-\frac {311 i \cosh (c+d x)}{8192 d (5+3 i \sinh (c+d x))}+\frac {385 \int \frac {1}{5+3 i \sinh (c+d x)} \, dx}{8192}\\ &=\frac {385 x}{32768}-\frac {385 i \tan ^{-1}\left (\frac {\cosh (c+d x)}{3+i \sinh (c+d x)}\right )}{16384 d}-\frac {i \cosh (c+d x)}{16 d (5+3 i \sinh (c+d x))^3}-\frac {25 i \cosh (c+d x)}{512 d (5+3 i \sinh (c+d x))^2}-\frac {311 i \cosh (c+d x)}{8192 d (5+3 i \sinh (c+d x))}\\ \end {align*}

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Mathematica [B]  time = 1.93, size = 308, normalized size = 2.48 \[ \frac {\frac {2 (-298563 i \sinh (c+d x)+89364 i \sinh (2 (c+d x))+8397 i \sinh (3 (c+d x))+166615 \cosh (c+d x)+82530 \cosh (2 (c+d x))-13995 \cosh (3 (c+d x))-235150)}{(3 \sinh (c+d x)-5 i)^3}+\frac {2656-192 i}{\left ((1+2 i) \cosh \left (\frac {1}{2} (c+d x)\right )-(2+i) \sinh \left (\frac {1}{2} (c+d x)\right )\right )^2}+\frac {2656+192 i}{\left ((1+2 i) \sinh \left (\frac {1}{2} (c+d x)\right )+(2+i) \cosh \left (\frac {1}{2} (c+d x)\right )\right )^2}-1925 \log (5 \cosh (c+d x)-4 \sinh (c+d x))+1925 \log (4 \sinh (c+d x)+5 \cosh (c+d x))-3850 i \tan ^{-1}\left (\frac {2 \cosh \left (\frac {1}{2} (c+d x)\right )-\sinh \left (\frac {1}{2} (c+d x)\right )}{\cosh \left (\frac {1}{2} (c+d x)\right )-2 \sinh \left (\frac {1}{2} (c+d x)\right )}\right )+3850 i \tan ^{-1}\left (\frac {2 \sinh \left (\frac {1}{2} (c+d x)\right )+\cosh \left (\frac {1}{2} (c+d x)\right )}{\sinh \left (\frac {1}{2} (c+d x)\right )+2 \cosh \left (\frac {1}{2} (c+d x)\right )}\right )}{327680 d} \]

Antiderivative was successfully verified.

[In]

Integrate[(5 + (3*I)*Sinh[c + d*x])^(-4),x]

[Out]

((-3850*I)*ArcTan[(2*Cosh[(c + d*x)/2] - Sinh[(c + d*x)/2])/(Cosh[(c + d*x)/2] - 2*Sinh[(c + d*x)/2])] + (3850
*I)*ArcTan[(Cosh[(c + d*x)/2] + 2*Sinh[(c + d*x)/2])/(2*Cosh[(c + d*x)/2] + Sinh[(c + d*x)/2])] - 1925*Log[5*C
osh[c + d*x] - 4*Sinh[c + d*x]] + 1925*Log[5*Cosh[c + d*x] + 4*Sinh[c + d*x]] + (2656 - 192*I)/((1 + 2*I)*Cosh
[(c + d*x)/2] - (2 + I)*Sinh[(c + d*x)/2])^2 + (2656 + 192*I)/((2 + I)*Cosh[(c + d*x)/2] + (1 + 2*I)*Sinh[(c +
d*x)/2])^2 + (2*(-235150 + 166615*Cosh[c + d*x] + 82530*Cosh[2*(c + d*x)] - 13995*Cosh[3*(c + d*x)] - (298563
*I)*Sinh[c + d*x] + (89364*I)*Sinh[2*(c + d*x)] + (8397*I)*Sinh[3*(c + d*x)]))/(-5*I + 3*Sinh[c + d*x])^3)/(32
7680*d)

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fricas [B]  time = 0.55, size = 281, normalized size = 2.27 \[ \frac {{\left (31185 \, e^{\left (6 \, d x + 6 \, c\right )} - 311850 i \, e^{\left (5 \, d x + 5 \, c\right )} - 1133055 \, e^{\left (4 \, d x + 4 \, c\right )} + 1778700 i \, e^{\left (3 \, d x + 3 \, c\right )} + 1133055 \, e^{\left (2 \, d x + 2 \, c\right )} - 311850 i \, e^{\left (d x + c\right )} - 31185\right )} \log \left (e^{\left (d x + c\right )} - \frac {1}{3} i\right ) - {\left (31185 \, e^{\left (6 \, d x + 6 \, c\right )} - 311850 i \, e^{\left (5 \, d x + 5 \, c\right )} - 1133055 \, e^{\left (4 \, d x + 4 \, c\right )} + 1778700 i \, e^{\left (3 \, d x + 3 \, c\right )} + 1133055 \, e^{\left (2 \, d x + 2 \, c\right )} - 311850 i \, e^{\left (d x + c\right )} - 31185\right )} \log \left (e^{\left (d x + c\right )} - 3 i\right ) - 83160 i \, e^{\left (5 \, d x + 5 \, c\right )} - 693000 \, e^{\left (4 \, d x + 4 \, c\right )} + 1915760 i \, e^{\left (3 \, d x + 3 \, c\right )} + 1747728 \, e^{\left (2 \, d x + 2 \, c\right )} - 588600 i \, e^{\left (d x + c\right )} - 67176}{2654208 \, d e^{\left (6 \, d x + 6 \, c\right )} - 26542080 i \, d e^{\left (5 \, d x + 5 \, c\right )} - 96436224 \, d e^{\left (4 \, d x + 4 \, c\right )} + 151388160 i \, d e^{\left (3 \, d x + 3 \, c\right )} + 96436224 \, d e^{\left (2 \, d x + 2 \, c\right )} - 26542080 i \, d e^{\left (d x + c\right )} - 2654208 \, d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(5+3*I*sinh(d*x+c))^4,x, algorithm="fricas")

[Out]

((31185*e^(6*d*x + 6*c) - 311850*I*e^(5*d*x + 5*c) - 1133055*e^(4*d*x + 4*c) + 1778700*I*e^(3*d*x + 3*c) + 113
3055*e^(2*d*x + 2*c) - 311850*I*e^(d*x + c) - 31185)*log(e^(d*x + c) - 1/3*I) - (31185*e^(6*d*x + 6*c) - 31185
0*I*e^(5*d*x + 5*c) - 1133055*e^(4*d*x + 4*c) + 1778700*I*e^(3*d*x + 3*c) + 1133055*e^(2*d*x + 2*c) - 311850*I
*e^(d*x + c) - 31185)*log(e^(d*x + c) - 3*I) - 83160*I*e^(5*d*x + 5*c) - 693000*e^(4*d*x + 4*c) + 1915760*I*e^
(3*d*x + 3*c) + 1747728*e^(2*d*x + 2*c) - 588600*I*e^(d*x + c) - 67176)/(2654208*d*e^(6*d*x + 6*c) - 26542080*
I*d*e^(5*d*x + 5*c) - 96436224*d*e^(4*d*x + 4*c) + 151388160*I*d*e^(3*d*x + 3*c) + 96436224*d*e^(2*d*x + 2*c)
- 26542080*I*d*e^(d*x + c) - 2654208*d)

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giac [A]  time = 0.17, size = 109, normalized size = 0.88 \[ -\frac {\frac {8 \, {\left (10395 i \, e^{\left (5 \, d x + 5 \, c\right )} + 86625 \, e^{\left (4 \, d x + 4 \, c\right )} - 239470 i \, e^{\left (3 \, d x + 3 \, c\right )} - 218466 \, e^{\left (2 \, d x + 2 \, c\right )} + 73575 i \, e^{\left (d x + c\right )} + 8397\right )}}{{\left (3 \, e^{\left (2 \, d x + 2 \, c\right )} - 10 i \, e^{\left (d x + c\right )} - 3\right )}^{3}} - 1155 \, \log \left (3 \, e^{\left (d x + c\right )} - i\right ) + 1155 \, \log \left (e^{\left (d x + c\right )} - 3 i\right )}{98304 \, d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(5+3*I*sinh(d*x+c))^4,x, algorithm="giac")

[Out]

-1/98304*(8*(10395*I*e^(5*d*x + 5*c) + 86625*e^(4*d*x + 4*c) - 239470*I*e^(3*d*x + 3*c) - 218466*e^(2*d*x + 2*
c) + 73575*I*e^(d*x + c) + 8397)/(3*e^(2*d*x + 2*c) - 10*I*e^(d*x + c) - 3)^3 - 1155*log(3*e^(d*x + c) - I) +
1155*log(e^(d*x + c) - 3*I))/d

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maple [B]  time = 0.09, size = 314, normalized size = 2.53 \[ \frac {1053}{32000 d \left (5 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-4-3 i\right )^{3}}+\frac {99 i}{8000 d \left (5 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-4-3 i\right )^{3}}-\frac {783}{128000 d \left (5 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-4-3 i\right )^{2}}-\frac {3753 i}{64000 d \left (5 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-4-3 i\right )^{2}}-\frac {39933}{1024000 d \left (5 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-4-3 i\right )}+\frac {8361 i}{256000 d \left (5 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-4-3 i\right )}-\frac {385 \ln \left (5 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-4-3 i\right )}{32768 d}+\frac {1053}{32000 d \left (5 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+4-3 i\right )^{3}}-\frac {99 i}{8000 d \left (5 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+4-3 i\right )^{3}}+\frac {783}{128000 d \left (5 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+4-3 i\right )^{2}}-\frac {3753 i}{64000 d \left (5 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+4-3 i\right )^{2}}-\frac {39933}{1024000 d \left (5 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+4-3 i\right )}-\frac {8361 i}{256000 d \left (5 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+4-3 i\right )}+\frac {385 \ln \left (5 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+4-3 i\right )}{32768 d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(5+3*I*sinh(d*x+c))^4,x)

[Out]

1053/32000/d/(5*tanh(1/2*d*x+1/2*c)-4-3*I)^3+99/8000*I/d/(5*tanh(1/2*d*x+1/2*c)-4-3*I)^3-783/128000/d/(5*tanh(
1/2*d*x+1/2*c)-4-3*I)^2-3753/64000*I/d/(5*tanh(1/2*d*x+1/2*c)-4-3*I)^2-39933/1024000/d/(5*tanh(1/2*d*x+1/2*c)-
4-3*I)+8361/256000*I/d/(5*tanh(1/2*d*x+1/2*c)-4-3*I)-385/32768/d*ln(5*tanh(1/2*d*x+1/2*c)-4-3*I)+1053/32000/d/
(5*tanh(1/2*d*x+1/2*c)+4-3*I)^3-99/8000*I/d/(5*tanh(1/2*d*x+1/2*c)+4-3*I)^3+783/128000/d/(5*tanh(1/2*d*x+1/2*c
)+4-3*I)^2-3753/64000*I/d/(5*tanh(1/2*d*x+1/2*c)+4-3*I)^2-39933/1024000/d/(5*tanh(1/2*d*x+1/2*c)+4-3*I)-8361/2
56000*I/d/(5*tanh(1/2*d*x+1/2*c)+4-3*I)+385/32768/d*ln(5*tanh(1/2*d*x+1/2*c)+4-3*I)

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maxima [A]  time = 0.42, size = 152, normalized size = 1.23 \[ -\frac {385 i \, \arctan \left (\frac {3}{4} \, e^{\left (-d x - c\right )} + \frac {5}{4} i\right )}{16384 \, d} - \frac {73575 i \, e^{\left (-d x - c\right )} + 218466 \, e^{\left (-2 \, d x - 2 \, c\right )} - 239470 i \, e^{\left (-3 \, d x - 3 \, c\right )} - 86625 \, e^{\left (-4 \, d x - 4 \, c\right )} + 10395 i \, e^{\left (-5 \, d x - 5 \, c\right )} - 8397}{d {\left (3317760 i \, e^{\left (-d x - c\right )} + 12054528 \, e^{\left (-2 \, d x - 2 \, c\right )} - 18923520 i \, e^{\left (-3 \, d x - 3 \, c\right )} - 12054528 \, e^{\left (-4 \, d x - 4 \, c\right )} + 3317760 i \, e^{\left (-5 \, d x - 5 \, c\right )} + 331776 \, e^{\left (-6 \, d x - 6 \, c\right )} - 331776\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(5+3*I*sinh(d*x+c))^4,x, algorithm="maxima")

[Out]

-385/16384*I*arctan(3/4*e^(-d*x - c) + 5/4*I)/d - (73575*I*e^(-d*x - c) + 218466*e^(-2*d*x - 2*c) - 239470*I*e
^(-3*d*x - 3*c) - 86625*e^(-4*d*x - 4*c) + 10395*I*e^(-5*d*x - 5*c) - 8397)/(d*(3317760*I*e^(-d*x - c) + 12054
528*e^(-2*d*x - 2*c) - 18923520*I*e^(-3*d*x - 3*c) - 12054528*e^(-4*d*x - 4*c) + 3317760*I*e^(-5*d*x - 5*c) +
331776*e^(-6*d*x - 6*c) - 331776))

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mupad [B]  time = 2.11, size = 232, normalized size = 1.87 \[ \frac {\frac {1925}{36864\,d}+\frac {{\mathrm {e}}^{c+d\,x}\,385{}\mathrm {i}}{12288\,d}}{1-{\mathrm {e}}^{2\,c+2\,d\,x}+\frac {{\mathrm {e}}^{c+d\,x}\,10{}\mathrm {i}}{3}}+\frac {\frac {41}{486\,d}+\frac {{\mathrm {e}}^{c+d\,x}\,365{}\mathrm {i}}{1458\,d}}{\frac {109\,{\mathrm {e}}^{4\,c+4\,d\,x}}{3}-\frac {109\,{\mathrm {e}}^{2\,c+2\,d\,x}}{3}-{\mathrm {e}}^{6\,c+6\,d\,x}+1+{\mathrm {e}}^{c+d\,x}\,10{}\mathrm {i}-\frac {{\mathrm {e}}^{3\,c+3\,d\,x}\,1540{}\mathrm {i}}{27}+{\mathrm {e}}^{5\,c+5\,d\,x}\,10{}\mathrm {i}}-\frac {385\,\ln \left (-\frac {385\,{\mathrm {e}}^{c+d\,x}}{4}+\frac {1155}{4}{}\mathrm {i}\right )}{32768\,d}+\frac {385\,\ln \left (\frac {3465\,{\mathrm {e}}^{c+d\,x}}{4}-\frac {1155}{4}{}\mathrm {i}\right )}{32768\,d}-\frac {\frac {3461}{31104\,d}+\frac {{\mathrm {e}}^{c+d\,x}\,385{}\mathrm {i}}{10368\,d}}{{\mathrm {e}}^{4\,c+4\,d\,x}-\frac {118\,{\mathrm {e}}^{2\,c+2\,d\,x}}{9}+1+\frac {{\mathrm {e}}^{c+d\,x}\,20{}\mathrm {i}}{3}-\frac {{\mathrm {e}}^{3\,c+3\,d\,x}\,20{}\mathrm {i}}{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(sinh(c + d*x)*3i + 5)^4,x)

[Out]

((exp(c + d*x)*385i)/(12288*d) + 1925/(36864*d))/((exp(c + d*x)*10i)/3 - exp(2*c + 2*d*x) + 1) + ((exp(c + d*x
)*365i)/(1458*d) + 41/(486*d))/(exp(c + d*x)*10i - (109*exp(2*c + 2*d*x))/3 - (exp(3*c + 3*d*x)*1540i)/27 + (1
09*exp(4*c + 4*d*x))/3 + exp(5*c + 5*d*x)*10i - exp(6*c + 6*d*x) + 1) - (385*log(1155i/4 - (385*exp(c + d*x))/
4))/(32768*d) + (385*log((3465*exp(c + d*x))/4 - 1155i/4))/(32768*d) - ((exp(c + d*x)*385i)/(10368*d) + 3461/(
31104*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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sympy [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: NotInvertible} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(5+3*I*sinh(d*x+c))**4,x)

[Out]

Exception raised: NotInvertible

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