3.3.10 \(\int \text {csch}(c+d x) (a+b \sinh ^4(c+d x))^3 \, dx\) [210]

Optimal. Leaf size=158 \[ -\frac {a^3 \tanh ^{-1}(\cosh (c+d x))}{d}-\frac {b \left (3 a^2+3 a b+b^2\right ) \cosh (c+d x)}{d}+\frac {b \left (3 a^2+9 a b+5 b^2\right ) \cosh ^3(c+d x)}{3 d}-\frac {b^2 (9 a+10 b) \cosh ^5(c+d x)}{5 d}+\frac {b^2 (3 a+10 b) \cosh ^7(c+d x)}{7 d}-\frac {5 b^3 \cosh ^9(c+d x)}{9 d}+\frac {b^3 \cosh ^{11}(c+d x)}{11 d} \]

[Out]

-a^3*arctanh(cosh(d*x+c))/d-b*(3*a^2+3*a*b+b^2)*cosh(d*x+c)/d+1/3*b*(3*a^2+9*a*b+5*b^2)*cosh(d*x+c)^3/d-1/5*b^
2*(9*a+10*b)*cosh(d*x+c)^5/d+1/7*b^2*(3*a+10*b)*cosh(d*x+c)^7/d-5/9*b^3*cosh(d*x+c)^9/d+1/11*b^3*cosh(d*x+c)^1
1/d

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Rubi [A]
time = 0.10, antiderivative size = 158, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {3294, 1167, 212} \begin {gather*} -\frac {a^3 \tanh ^{-1}(\cosh (c+d x))}{d}+\frac {b \left (3 a^2+9 a b+5 b^2\right ) \cosh ^3(c+d x)}{3 d}-\frac {b \left (3 a^2+3 a b+b^2\right ) \cosh (c+d x)}{d}+\frac {b^2 (3 a+10 b) \cosh ^7(c+d x)}{7 d}-\frac {b^2 (9 a+10 b) \cosh ^5(c+d x)}{5 d}+\frac {b^3 \cosh ^{11}(c+d x)}{11 d}-\frac {5 b^3 \cosh ^9(c+d x)}{9 d} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Csch[c + d*x]*(a + b*Sinh[c + d*x]^4)^3,x]

[Out]

-((a^3*ArcTanh[Cosh[c + d*x]])/d) - (b*(3*a^2 + 3*a*b + b^2)*Cosh[c + d*x])/d + (b*(3*a^2 + 9*a*b + 5*b^2)*Cos
h[c + d*x]^3)/(3*d) - (b^2*(9*a + 10*b)*Cosh[c + d*x]^5)/(5*d) + (b^2*(3*a + 10*b)*Cosh[c + d*x]^7)/(7*d) - (5
*b^3*Cosh[c + d*x]^9)/(9*d) + (b^3*Cosh[c + d*x]^11)/(11*d)

Rule 212

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))*ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rule 1167

Int[((d_) + (e_.)*(x_)^2)^(q_.)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_.), x_Symbol] :> Int[ExpandIntegrand[(
d + e*x^2)^q*(a + b*x^2 + c*x^4)^p, x], x] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 -
b*d*e + a*e^2, 0] && IGtQ[p, 0] && IGtQ[q, -2]

Rule 3294

Int[sin[(e_.) + (f_.)*(x_)]^(m_.)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]^4)^(p_.), x_Symbol] :> With[{ff = Free
Factors[Cos[e + f*x], x]}, Dist[-ff/f, Subst[Int[(1 - ff^2*x^2)^((m - 1)/2)*(a + b - 2*b*ff^2*x^2 + b*ff^4*x^4
)^p, x], x, Cos[e + f*x]/ff], x]] /; FreeQ[{a, b, e, f, p}, x] && IntegerQ[(m - 1)/2]

Rubi steps

\begin {align*} \int \text {csch}(c+d x) \left (a+b \sinh ^4(c+d x)\right )^3 \, dx &=-\frac {\text {Subst}\left (\int \frac {\left (a+b-2 b x^2+b x^4\right )^3}{1-x^2} \, dx,x,\cosh (c+d x)\right )}{d}\\ &=-\frac {\text {Subst}\left (\int \left (b \left (3 a^2+3 a b+b^2\right )-b \left (3 a^2+9 a b+5 b^2\right ) x^2+b^2 (9 a+10 b) x^4-b^2 (3 a+10 b) x^6+5 b^3 x^8-b^3 x^{10}+\frac {a^3}{1-x^2}\right ) \, dx,x,\cosh (c+d x)\right )}{d}\\ &=-\frac {b \left (3 a^2+3 a b+b^2\right ) \cosh (c+d x)}{d}+\frac {b \left (3 a^2+9 a b+5 b^2\right ) \cosh ^3(c+d x)}{3 d}-\frac {b^2 (9 a+10 b) \cosh ^5(c+d x)}{5 d}+\frac {b^2 (3 a+10 b) \cosh ^7(c+d x)}{7 d}-\frac {5 b^3 \cosh ^9(c+d x)}{9 d}+\frac {b^3 \cosh ^{11}(c+d x)}{11 d}-\frac {a^3 \text {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\cosh (c+d x)\right )}{d}\\ &=-\frac {a^3 \tanh ^{-1}(\cosh (c+d x))}{d}-\frac {b \left (3 a^2+3 a b+b^2\right ) \cosh (c+d x)}{d}+\frac {b \left (3 a^2+9 a b+5 b^2\right ) \cosh ^3(c+d x)}{3 d}-\frac {b^2 (9 a+10 b) \cosh ^5(c+d x)}{5 d}+\frac {b^2 (3 a+10 b) \cosh ^7(c+d x)}{7 d}-\frac {5 b^3 \cosh ^9(c+d x)}{9 d}+\frac {b^3 \cosh ^{11}(c+d x)}{11 d}\\ \end {align*}

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Mathematica [A]
time = 0.28, size = 139, normalized size = 0.88 \begin {gather*} \frac {-20790 b \left (384 a^2+280 a b+77 b^2\right ) \cosh (c+d x)+6930 b (8 a+5 b) (16 a+11 b) \cosh (3 (c+d x))-2079 b^2 (112 a+55 b) \cosh (5 (c+d x))+495 b^2 (48 a+55 b) \cosh (7 (c+d x))-4235 b^3 \cosh (9 (c+d x))+315 b^3 \cosh (11 (c+d x))+3548160 a^3 \log \left (\tanh \left (\frac {1}{2} (c+d x)\right )\right )}{3548160 d} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[Csch[c + d*x]*(a + b*Sinh[c + d*x]^4)^3,x]

[Out]

(-20790*b*(384*a^2 + 280*a*b + 77*b^2)*Cosh[c + d*x] + 6930*b*(8*a + 5*b)*(16*a + 11*b)*Cosh[3*(c + d*x)] - 20
79*b^2*(112*a + 55*b)*Cosh[5*(c + d*x)] + 495*b^2*(48*a + 55*b)*Cosh[7*(c + d*x)] - 4235*b^3*Cosh[9*(c + d*x)]
+ 315*b^3*Cosh[11*(c + d*x)] + 3548160*a^3*Log[Tanh[(c + d*x)/2]])/(3548160*d)

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(532\) vs. \(2(148)=296\).
time = 1.35, size = 533, normalized size = 3.37

method	result	size
risch	\(-\frac {21 a \,b^{2} {\mathrm e}^{5 d x +5 c}}{640 d}+\frac {21 a \,b^{2} {\mathrm e}^{3 d x +3 c}}{128 d}+\frac {a^{3} \ln \left ({\mathrm e}^{d x +c}-1\right )}{d}-\frac {a^{3} \ln \left ({\mathrm e}^{d x +c}+1\right )}{d}-\frac {9 b \,{\mathrm e}^{d x +c} a^{2}}{8 d}-\frac {11 b^{3} {\mathrm e}^{9 d x +9 c}}{18432 d}-\frac {33 \,{\mathrm e}^{5 d x +5 c} b^{3}}{2048 d}+\frac {55 \,{\mathrm e}^{3 d x +3 c} b^{3}}{1024 d}+\frac {b^{3} {\mathrm e}^{11 d x +11 c}}{22528 d}-\frac {11 b^{3} {\mathrm e}^{-9 d x -9 c}}{18432 d}+\frac {b^{3} {\mathrm e}^{-11 d x -11 c}}{22528 d}-\frac {105 a \,{\mathrm e}^{-d x -c} b^{2}}{128 d}+\frac {21 a \,b^{2} {\mathrm e}^{-3 d x -3 c}}{128 d}-\frac {231 b^{3} {\mathrm e}^{d x +c}}{1024 d}-\frac {231 b^{3} {\mathrm e}^{-d x -c}}{1024 d}+\frac {55 b^{3} {\mathrm e}^{-3 d x -3 c}}{1024 d}+\frac {55 b^{3} {\mathrm e}^{7 d x +7 c}}{14336 d}-\frac {21 a \,b^{2} {\mathrm e}^{-5 d x -5 c}}{640 d}-\frac {33 \,{\mathrm e}^{-5 d x -5 c} b^{3}}{2048 d}+\frac {55 b^{3} {\mathrm e}^{-7 d x -7 c}}{14336 d}-\frac {105 a \,{\mathrm e}^{d x +c} b^{2}}{128 d}-\frac {9 \,{\mathrm e}^{-d x -c} a^{2} b}{8 d}+\frac {{\mathrm e}^{-3 d x -3 c} a^{2} b}{8 d}+\frac {3 b^{2} {\mathrm e}^{7 d x +7 c} a}{896 d}+\frac {3 b^{2} {\mathrm e}^{-7 d x -7 c} a}{896 d}+\frac {{\mathrm e}^{3 d x +3 c} a^{2} b}{8 d}\)	\(446\)
default	\(\frac {-6 a \,b^{2} \arctanh \left ({\mathrm e}^{d x +c}\right )-6 a^{2} b \arctanh \left ({\mathrm e}^{d x +c}\right )-2 a^{3} \arctanh \left ({\mathrm e}^{d x +c}\right )-2 b^{3} \arctanh \left ({\mathrm e}^{d x +c}\right )+3 a \,b^{2} \left (\frac {\left (\cosh ^{7}\left (d x +c \right )\right )}{7}+\frac {\left (\cosh ^{5}\left (d x +c \right )\right )}{5}+\frac {\left (\cosh ^{3}\left (d x +c \right )\right )}{3}+\cosh \left (d x +c \right )-2 \arctanh \left ({\mathrm e}^{d x +c}\right )\right )-12 a \,b^{2} \left (\frac {\left (\cosh ^{5}\left (d x +c \right )\right )}{5}+\frac {\left (\cosh ^{3}\left (d x +c \right )\right )}{3}+\cosh \left (d x +c \right )-2 \arctanh \left ({\mathrm e}^{d x +c}\right )\right )+3 a^{2} b \left (\frac {\left (\cosh ^{3}\left (d x +c \right )\right )}{3}+\cosh \left (d x +c \right )-2 \arctanh \left ({\mathrm e}^{d x +c}\right )\right )+18 a \,b^{2} \left (\frac {\left (\cosh ^{3}\left (d x +c \right )\right )}{3}+\cosh \left (d x +c \right )-2 \arctanh \left ({\mathrm e}^{d x +c}\right )\right )-6 a^{2} b \left (\cosh \left (d x +c \right )-2 \arctanh \left ({\mathrm e}^{d x +c}\right )\right )-12 a \,b^{2} \left (\cosh \left (d x +c \right )-2 \arctanh \left ({\mathrm e}^{d x +c}\right )\right )-6 b^{3} \left (\cosh \left (d x +c \right )-2 \arctanh \left ({\mathrm e}^{d x +c}\right )\right )+b^{3} \left (\frac {\left (\cosh ^{11}\left (d x +c \right )\right )}{11}+\frac {\left (\cosh ^{9}\left (d x +c \right )\right )}{9}+\frac {\left (\cosh ^{7}\left (d x +c \right )\right )}{7}+\frac {\left (\cosh ^{5}\left (d x +c \right )\right )}{5}+\frac {\left (\cosh ^{3}\left (d x +c \right )\right )}{3}+\cosh \left (d x +c \right )-2 \arctanh \left ({\mathrm e}^{d x +c}\right )\right )-6 b^{3} \left (\frac {\left (\cosh ^{9}\left (d x +c \right )\right )}{9}+\frac {\left (\cosh ^{7}\left (d x +c \right )\right )}{7}+\frac {\left (\cosh ^{5}\left (d x +c \right )\right )}{5}+\frac {\left (\cosh ^{3}\left (d x +c \right )\right )}{3}+\cosh \left (d x +c \right )-2 \arctanh \left ({\mathrm e}^{d x +c}\right )\right )+15 b^{3} \left (\frac {\left (\cosh ^{7}\left (d x +c \right )\right )}{7}+\frac {\left (\cosh ^{5}\left (d x +c \right )\right )}{5}+\frac {\left (\cosh ^{3}\left (d x +c \right )\right )}{3}+\cosh \left (d x +c \right )-2 \arctanh \left ({\mathrm e}^{d x +c}\right )\right )-20 b^{3} \left (\frac {\left (\cosh ^{5}\left (d x +c \right )\right )}{5}+\frac {\left (\cosh ^{3}\left (d x +c \right )\right )}{3}+\cosh \left (d x +c \right )-2 \arctanh \left ({\mathrm e}^{d x +c}\right )\right )+15 b^{3} \left (\frac {\left (\cosh ^{3}\left (d x +c \right )\right )}{3}+\cosh \left (d x +c \right )-2 \arctanh \left ({\mathrm e}^{d x +c}\right )\right )}{d}\)	\(533\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(csch(d*x+c)*(a+b*sinh(d*x+c)^4)^3,x,method=_RETURNVERBOSE)

[Out]

1/d*(-6*a*b^2*arctanh(exp(d*x+c))-6*a^2*b*arctanh(exp(d*x+c))-2*a^3*arctanh(exp(d*x+c))-2*b^3*arctanh(exp(d*x+
c))+3*a*b^2*(1/7*cosh(d*x+c)^7+1/5*cosh(d*x+c)^5+1/3*cosh(d*x+c)^3+cosh(d*x+c)-2*arctanh(exp(d*x+c)))-12*a*b^2
*(1/5*cosh(d*x+c)^5+1/3*cosh(d*x+c)^3+cosh(d*x+c)-2*arctanh(exp(d*x+c)))+3*a^2*b*(1/3*cosh(d*x+c)^3+cosh(d*x+c
)-2*arctanh(exp(d*x+c)))+18*a*b^2*(1/3*cosh(d*x+c)^3+cosh(d*x+c)-2*arctanh(exp(d*x+c)))-6*a^2*b*(cosh(d*x+c)-2
*arctanh(exp(d*x+c)))-12*a*b^2*(cosh(d*x+c)-2*arctanh(exp(d*x+c)))-6*b^3*(cosh(d*x+c)-2*arctanh(exp(d*x+c)))+b
^3*(1/11*cosh(d*x+c)^11+1/9*cosh(d*x+c)^9+1/7*cosh(d*x+c)^7+1/5*cosh(d*x+c)^5+1/3*cosh(d*x+c)^3+cosh(d*x+c)-2*
arctanh(exp(d*x+c)))-6*b^3*(1/9*cosh(d*x+c)^9+1/7*cosh(d*x+c)^7+1/5*cosh(d*x+c)^5+1/3*cosh(d*x+c)^3+cosh(d*x+c
)-2*arctanh(exp(d*x+c)))+15*b^3*(1/7*cosh(d*x+c)^7+1/5*cosh(d*x+c)^5+1/3*cosh(d*x+c)^3+cosh(d*x+c)-2*arctanh(e
xp(d*x+c)))-20*b^3*(1/5*cosh(d*x+c)^5+1/3*cosh(d*x+c)^3+cosh(d*x+c)-2*arctanh(exp(d*x+c)))+15*b^3*(1/3*cosh(d*
x+c)^3+cosh(d*x+c)-2*arctanh(exp(d*x+c))))

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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 327 vs. \(2 (148) = 296\).
time = 0.28, size = 327, normalized size = 2.07 \begin {gather*} -\frac {1}{1419264} \, b^{3} {\left (\frac {{\left (847 \, e^{\left (-2 \, d x - 2 \, c\right )} - 5445 \, e^{\left (-4 \, d x - 4 \, c\right )} + 22869 \, e^{\left (-6 \, d x - 6 \, c\right )} - 76230 \, e^{\left (-8 \, d x - 8 \, c\right )} + 320166 \, e^{\left (-10 \, d x - 10 \, c\right )} - 63\right )} e^{\left (11 \, d x + 11 \, c\right )}}{d} + \frac {320166 \, e^{\left (-d x - c\right )} - 76230 \, e^{\left (-3 \, d x - 3 \, c\right )} + 22869 \, e^{\left (-5 \, d x - 5 \, c\right )} - 5445 \, e^{\left (-7 \, d x - 7 \, c\right )} + 847 \, e^{\left (-9 \, d x - 9 \, c\right )} - 63 \, e^{\left (-11 \, d x - 11 \, c\right )}}{d}\right )} - \frac {3}{4480} \, a b^{2} {\left (\frac {{\left (49 \, e^{\left (-2 \, d x - 2 \, c\right )} - 245 \, e^{\left (-4 \, d x - 4 \, c\right )} + 1225 \, e^{\left (-6 \, d x - 6 \, c\right )} - 5\right )} e^{\left (7 \, d x + 7 \, c\right )}}{d} + \frac {1225 \, e^{\left (-d x - c\right )} - 245 \, e^{\left (-3 \, d x - 3 \, c\right )} + 49 \, e^{\left (-5 \, d x - 5 \, c\right )} - 5 \, e^{\left (-7 \, d x - 7 \, c\right )}}{d}\right )} + \frac {1}{8} \, a^{2} b {\left (\frac {e^{\left (3 \, d x + 3 \, c\right )}}{d} - \frac {9 \, e^{\left (d x + c\right )}}{d} - \frac {9 \, e^{\left (-d x - c\right )}}{d} + \frac {e^{\left (-3 \, d x - 3 \, c\right )}}{d}\right )} + \frac {a^{3} \log \left (\tanh \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}{d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csch(d*x+c)*(a+b*sinh(d*x+c)^4)^3,x, algorithm="maxima")

[Out]

-1/1419264*b^3*((847*e^(-2*d*x - 2*c) - 5445*e^(-4*d*x - 4*c) + 22869*e^(-6*d*x - 6*c) - 76230*e^(-8*d*x - 8*c
) + 320166*e^(-10*d*x - 10*c) - 63)*e^(11*d*x + 11*c)/d + (320166*e^(-d*x - c) - 76230*e^(-3*d*x - 3*c) + 2286
9*e^(-5*d*x - 5*c) - 5445*e^(-7*d*x - 7*c) + 847*e^(-9*d*x - 9*c) - 63*e^(-11*d*x - 11*c))/d) - 3/4480*a*b^2*(
(49*e^(-2*d*x - 2*c) - 245*e^(-4*d*x - 4*c) + 1225*e^(-6*d*x - 6*c) - 5)*e^(7*d*x + 7*c)/d + (1225*e^(-d*x - c
) - 245*e^(-3*d*x - 3*c) + 49*e^(-5*d*x - 5*c) - 5*e^(-7*d*x - 7*c))/d) + 1/8*a^2*b*(e^(3*d*x + 3*c)/d - 9*e^(
d*x + c)/d - 9*e^(-d*x - c)/d + e^(-3*d*x - 3*c)/d) + a^3*log(tanh(1/2*d*x + 1/2*c))/d

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 3824 vs. \(2 (148) = 296\).
time = 0.43, size = 3824, normalized size = 24.20 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csch(d*x+c)*(a+b*sinh(d*x+c)^4)^3,x, algorithm="fricas")

[Out]

1/7096320*(315*b^3*cosh(d*x + c)^22 + 6930*b^3*cosh(d*x + c)*sinh(d*x + c)^21 + 315*b^3*sinh(d*x + c)^22 - 423
5*b^3*cosh(d*x + c)^20 + 385*(189*b^3*cosh(d*x + c)^2 - 11*b^3)*sinh(d*x + c)^20 + 7700*(63*b^3*cosh(d*x + c)^
3 - 11*b^3*cosh(d*x + c))*sinh(d*x + c)^19 + 495*(48*a*b^2 + 55*b^3)*cosh(d*x + c)^18 + 55*(41895*b^3*cosh(d*x
+ c)^4 - 14630*b^3*cosh(d*x + c)^2 + 432*a*b^2 + 495*b^3)*sinh(d*x + c)^18 + 330*(25137*b^3*cosh(d*x + c)^5 -
14630*b^3*cosh(d*x + c)^3 + 27*(48*a*b^2 + 55*b^3)*cosh(d*x + c))*sinh(d*x + c)^17 - 2079*(112*a*b^2 + 55*b^3
)*cosh(d*x + c)^16 + 33*(712215*b^3*cosh(d*x + c)^6 - 621775*b^3*cosh(d*x + c)^4 - 7056*a*b^2 - 3465*b^3 + 229
5*(48*a*b^2 + 55*b^3)*cosh(d*x + c)^2)*sinh(d*x + c)^16 + 528*(101745*b^3*cosh(d*x + c)^7 - 124355*b^3*cosh(d*
x + c)^5 + 765*(48*a*b^2 + 55*b^3)*cosh(d*x + c)^3 - 63*(112*a*b^2 + 55*b^3)*cosh(d*x + c))*sinh(d*x + c)^15 +
6930*(128*a^2*b + 168*a*b^2 + 55*b^3)*cosh(d*x + c)^14 + 330*(305235*b^3*cosh(d*x + c)^8 - 497420*b^3*cosh(d*
x + c)^6 + 4590*(48*a*b^2 + 55*b^3)*cosh(d*x + c)^4 + 2688*a^2*b + 3528*a*b^2 + 1155*b^3 - 756*(112*a*b^2 + 55
*b^3)*cosh(d*x + c)^2)*sinh(d*x + c)^14 + 4620*(33915*b^3*cosh(d*x + c)^9 - 71060*b^3*cosh(d*x + c)^7 + 918*(4
8*a*b^2 + 55*b^3)*cosh(d*x + c)^5 - 252*(112*a*b^2 + 55*b^3)*cosh(d*x + c)^3 + 21*(128*a^2*b + 168*a*b^2 + 55*
b^3)*cosh(d*x + c))*sinh(d*x + c)^13 - 20790*(384*a^2*b + 280*a*b^2 + 77*b^3)*cosh(d*x + c)^12 + 2310*(88179*b
^3*cosh(d*x + c)^10 - 230945*b^3*cosh(d*x + c)^8 + 3978*(48*a*b^2 + 55*b^3)*cosh(d*x + c)^6 - 1638*(112*a*b^2
+ 55*b^3)*cosh(d*x + c)^4 - 3456*a^2*b - 2520*a*b^2 - 693*b^3 + 273*(128*a^2*b + 168*a*b^2 + 55*b^3)*cosh(d*x
+ c)^2)*sinh(d*x + c)^12 + 8*(27776385*b^3*cosh(d*x + c)^11 - 88913825*b^3*cosh(d*x + c)^9 + 1969110*(48*a*b^2
+ 55*b^3)*cosh(d*x + c)^7 - 1135134*(112*a*b^2 + 55*b^3)*cosh(d*x + c)^5 + 315315*(128*a^2*b + 168*a*b^2 + 55
*b^3)*cosh(d*x + c)^3 - 31185*(384*a^2*b + 280*a*b^2 + 77*b^3)*cosh(d*x + c))*sinh(d*x + c)^11 - 20790*(384*a^
2*b + 280*a*b^2 + 77*b^3)*cosh(d*x + c)^10 + 22*(9258795*b^3*cosh(d*x + c)^12 - 35565530*b^3*cosh(d*x + c)^10
+ 984555*(48*a*b^2 + 55*b^3)*cosh(d*x + c)^8 - 756756*(112*a*b^2 + 55*b^3)*cosh(d*x + c)^6 + 315315*(128*a^2*b
+ 168*a*b^2 + 55*b^3)*cosh(d*x + c)^4 - 362880*a^2*b - 264600*a*b^2 - 72765*b^3 - 62370*(384*a^2*b + 280*a*b^
2 + 77*b^3)*cosh(d*x + c)^2)*sinh(d*x + c)^10 + 220*(712215*b^3*cosh(d*x + c)^13 - 3233230*b^3*cosh(d*x + c)^1
1 + 109395*(48*a*b^2 + 55*b^3)*cosh(d*x + c)^9 - 108108*(112*a*b^2 + 55*b^3)*cosh(d*x + c)^7 + 63063*(128*a^2*
b + 168*a*b^2 + 55*b^3)*cosh(d*x + c)^5 - 20790*(384*a^2*b + 280*a*b^2 + 77*b^3)*cosh(d*x + c)^3 - 945*(384*a^
2*b + 280*a*b^2 + 77*b^3)*cosh(d*x + c))*sinh(d*x + c)^9 + 6930*(128*a^2*b + 168*a*b^2 + 55*b^3)*cosh(d*x + c)
^8 + 330*(305235*b^3*cosh(d*x + c)^14 - 1616615*b^3*cosh(d*x + c)^12 + 65637*(48*a*b^2 + 55*b^3)*cosh(d*x + c)
^10 - 81081*(112*a*b^2 + 55*b^3)*cosh(d*x + c)^8 + 63063*(128*a^2*b + 168*a*b^2 + 55*b^3)*cosh(d*x + c)^6 - 31
185*(384*a^2*b + 280*a*b^2 + 77*b^3)*cosh(d*x + c)^4 + 2688*a^2*b + 3528*a*b^2 + 1155*b^3 - 2835*(384*a^2*b +
280*a*b^2 + 77*b^3)*cosh(d*x + c)^2)*sinh(d*x + c)^8 + 2640*(20349*b^3*cosh(d*x + c)^15 - 124355*b^3*cosh(d*x
+ c)^13 + 5967*(48*a*b^2 + 55*b^3)*cosh(d*x + c)^11 - 9009*(112*a*b^2 + 55*b^3)*cosh(d*x + c)^9 + 9009*(128*a^
2*b + 168*a*b^2 + 55*b^3)*cosh(d*x + c)^7 - 6237*(384*a^2*b + 280*a*b^2 + 77*b^3)*cosh(d*x + c)^5 - 945*(384*a
^2*b + 280*a*b^2 + 77*b^3)*cosh(d*x + c)^3 + 21*(128*a^2*b + 168*a*b^2 + 55*b^3)*cosh(d*x + c))*sinh(d*x + c)^
7 - 2079*(112*a*b^2 + 55*b^3)*cosh(d*x + c)^6 + 231*(101745*b^3*cosh(d*x + c)^16 - 710600*b^3*cosh(d*x + c)^14
+ 39780*(48*a*b^2 + 55*b^3)*cosh(d*x + c)^12 - 72072*(112*a*b^2 + 55*b^3)*cosh(d*x + c)^10 + 90090*(128*a^2*b
+ 168*a*b^2 + 55*b^3)*cosh(d*x + c)^8 - 83160*(384*a^2*b + 280*a*b^2 + 77*b^3)*cosh(d*x + c)^6 - 18900*(384*a
^2*b + 280*a*b^2 + 77*b^3)*cosh(d*x + c)^4 - 1008*a*b^2 - 495*b^3 + 840*(128*a^2*b + 168*a*b^2 + 55*b^3)*cosh(
d*x + c)^2)*sinh(d*x + c)^6 + 462*(17955*b^3*cosh(d*x + c)^17 - 142120*b^3*cosh(d*x + c)^15 + 9180*(48*a*b^2 +
55*b^3)*cosh(d*x + c)^13 - 19656*(112*a*b^2 + 55*b^3)*cosh(d*x + c)^11 + 30030*(128*a^2*b + 168*a*b^2 + 55*b^
3)*cosh(d*x + c)^9 - 35640*(384*a^2*b + 280*a*b^2 + 77*b^3)*cosh(d*x + c)^7 - 11340*(384*a^2*b + 280*a*b^2 + 7
7*b^3)*cosh(d*x + c)^5 + 840*(128*a^2*b + 168*a*b^2 + 55*b^3)*cosh(d*x + c)^3 - 27*(112*a*b^2 + 55*b^3)*cosh(d
*x + c))*sinh(d*x + c)^5 - 4235*b^3*cosh(d*x + c)^2 + 495*(48*a*b^2 + 55*b^3)*cosh(d*x + c)^4 + 165*(13965*b^3
*cosh(d*x + c)^18 - 124355*b^3*cosh(d*x + c)^16 + 9180*(48*a*b^2 + 55*b^3)*cosh(d*x + c)^14 - 22932*(112*a*b^2
+ 55*b^3)*cosh(d*x + c)^12 + 42042*(128*a^2*b + 168*a*b^2 + 55*b^3)*cosh(d*x + c)^10 - 62370*(384*a^2*b + 280
*a*b^2 + 77*b^3)*cosh(d*x + c)^8 - 26460*(384*a^2*b + 280*a*b^2 + 77*b^3)*cosh(d*x + c)^6 + 2940*(128*a^2*b +
168*a*b^2 + 55*b^3)*cosh(d*x + c)^4 + 144*a*b^2 + 165*b^3 - 189*(112*a*b^2 + 55*b^3)*cosh(d*x + c)^2)*sinh(d*x
+ c)^4 + 660*(735*b^3*cosh(d*x + c)^19 - 7315*...

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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csch(d*x+c)*(a+b*sinh(d*x+c)**4)**3,x)

[Out]

Timed out

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 377 vs. \(2 (148) = 296\).
time = 0.54, size = 377, normalized size = 2.39 \begin {gather*} \frac {315 \, b^{3} e^{\left (11 \, d x + 11 \, c\right )} - 4235 \, b^{3} e^{\left (9 \, d x + 9 \, c\right )} + 23760 \, a b^{2} e^{\left (7 \, d x + 7 \, c\right )} + 27225 \, b^{3} e^{\left (7 \, d x + 7 \, c\right )} - 232848 \, a b^{2} e^{\left (5 \, d x + 5 \, c\right )} - 114345 \, b^{3} e^{\left (5 \, d x + 5 \, c\right )} + 887040 \, a^{2} b e^{\left (3 \, d x + 3 \, c\right )} + 1164240 \, a b^{2} e^{\left (3 \, d x + 3 \, c\right )} + 381150 \, b^{3} e^{\left (3 \, d x + 3 \, c\right )} - 7983360 \, a^{2} b e^{\left (d x + c\right )} - 5821200 \, a b^{2} e^{\left (d x + c\right )} - 1600830 \, b^{3} e^{\left (d x + c\right )} - 7096320 \, a^{3} \log \left (e^{\left (d x + c\right )} + 1\right ) + 7096320 \, a^{3} \log \left ({\left | e^{\left (d x + c\right )} - 1 \right |}\right ) - {\left (7983360 \, a^{2} b e^{\left (10 \, d x + 10 \, c\right )} + 5821200 \, a b^{2} e^{\left (10 \, d x + 10 \, c\right )} + 1600830 \, b^{3} e^{\left (10 \, d x + 10 \, c\right )} - 887040 \, a^{2} b e^{\left (8 \, d x + 8 \, c\right )} - 1164240 \, a b^{2} e^{\left (8 \, d x + 8 \, c\right )} - 381150 \, b^{3} e^{\left (8 \, d x + 8 \, c\right )} + 232848 \, a b^{2} e^{\left (6 \, d x + 6 \, c\right )} + 114345 \, b^{3} e^{\left (6 \, d x + 6 \, c\right )} - 23760 \, a b^{2} e^{\left (4 \, d x + 4 \, c\right )} - 27225 \, b^{3} e^{\left (4 \, d x + 4 \, c\right )} + 4235 \, b^{3} e^{\left (2 \, d x + 2 \, c\right )} - 315 \, b^{3}\right )} e^{\left (-11 \, d x - 11 \, c\right )}}{7096320 \, d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csch(d*x+c)*(a+b*sinh(d*x+c)^4)^3,x, algorithm="giac")

[Out]

1/7096320*(315*b^3*e^(11*d*x + 11*c) - 4235*b^3*e^(9*d*x + 9*c) + 23760*a*b^2*e^(7*d*x + 7*c) + 27225*b^3*e^(7
*d*x + 7*c) - 232848*a*b^2*e^(5*d*x + 5*c) - 114345*b^3*e^(5*d*x + 5*c) + 887040*a^2*b*e^(3*d*x + 3*c) + 11642
40*a*b^2*e^(3*d*x + 3*c) + 381150*b^3*e^(3*d*x + 3*c) - 7983360*a^2*b*e^(d*x + c) - 5821200*a*b^2*e^(d*x + c)
- 1600830*b^3*e^(d*x + c) - 7096320*a^3*log(e^(d*x + c) + 1) + 7096320*a^3*log(abs(e^(d*x + c) - 1)) - (798336
0*a^2*b*e^(10*d*x + 10*c) + 5821200*a*b^2*e^(10*d*x + 10*c) + 1600830*b^3*e^(10*d*x + 10*c) - 887040*a^2*b*e^(
8*d*x + 8*c) - 1164240*a*b^2*e^(8*d*x + 8*c) - 381150*b^3*e^(8*d*x + 8*c) + 232848*a*b^2*e^(6*d*x + 6*c) + 114
345*b^3*e^(6*d*x + 6*c) - 23760*a*b^2*e^(4*d*x + 4*c) - 27225*b^3*e^(4*d*x + 4*c) + 4235*b^3*e^(2*d*x + 2*c) -
315*b^3)*e^(-11*d*x - 11*c))/d

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Mupad [B]
time = 0.65, size = 326, normalized size = 2.06 \begin {gather*} \frac {{\mathrm {e}}^{-3\,c-3\,d\,x}\,\left (128\,a^2\,b+168\,a\,b^2+55\,b^3\right )}{1024\,d}-\frac {2\,\mathrm {atan}\left (\frac {a^3\,{\mathrm {e}}^{d\,x}\,{\mathrm {e}}^c\,\sqrt {-d^2}}{d\,\sqrt {a^6}}\right )\,\sqrt {a^6}}{\sqrt {-d^2}}+\frac {{\mathrm {e}}^{3\,c+3\,d\,x}\,\left (128\,a^2\,b+168\,a\,b^2+55\,b^3\right )}{1024\,d}-\frac {11\,b^3\,{\mathrm {e}}^{-9\,c-9\,d\,x}}{18432\,d}-\frac {11\,b^3\,{\mathrm {e}}^{9\,c+9\,d\,x}}{18432\,d}+\frac {b^3\,{\mathrm {e}}^{-11\,c-11\,d\,x}}{22528\,d}+\frac {b^3\,{\mathrm {e}}^{11\,c+11\,d\,x}}{22528\,d}-\frac {3\,b\,{\mathrm {e}}^{-c-d\,x}\,\left (384\,a^2+280\,a\,b+77\,b^2\right )}{1024\,d}+\frac {b^2\,{\mathrm {e}}^{-7\,c-7\,d\,x}\,\left (48\,a+55\,b\right )}{14336\,d}+\frac {b^2\,{\mathrm {e}}^{7\,c+7\,d\,x}\,\left (48\,a+55\,b\right )}{14336\,d}-\frac {3\,b^2\,{\mathrm {e}}^{-5\,c-5\,d\,x}\,\left (112\,a+55\,b\right )}{10240\,d}-\frac {3\,b^2\,{\mathrm {e}}^{5\,c+5\,d\,x}\,\left (112\,a+55\,b\right )}{10240\,d}-\frac {3\,b\,{\mathrm {e}}^{c+d\,x}\,\left (384\,a^2+280\,a\,b+77\,b^2\right )}{1024\,d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*sinh(c + d*x)^4)^3/sinh(c + d*x),x)

[Out]

(exp(- 3*c - 3*d*x)*(168*a*b^2 + 128*a^2*b + 55*b^3))/(1024*d) - (2*atan((a^3*exp(d*x)*exp(c)*(-d^2)^(1/2))/(d
*(a^6)^(1/2)))*(a^6)^(1/2))/(-d^2)^(1/2) + (exp(3*c + 3*d*x)*(168*a*b^2 + 128*a^2*b + 55*b^3))/(1024*d) - (11*
b^3*exp(- 9*c - 9*d*x))/(18432*d) - (11*b^3*exp(9*c + 9*d*x))/(18432*d) + (b^3*exp(- 11*c - 11*d*x))/(22528*d)
+ (b^3*exp(11*c + 11*d*x))/(22528*d) - (3*b*exp(- c - d*x)*(280*a*b + 384*a^2 + 77*b^2))/(1024*d) + (b^2*exp(
- 7*c - 7*d*x)*(48*a + 55*b))/(14336*d) + (b^2*exp(7*c + 7*d*x)*(48*a + 55*b))/(14336*d) - (3*b^2*exp(- 5*c -
5*d*x)*(112*a + 55*b))/(10240*d) - (3*b^2*exp(5*c + 5*d*x)*(112*a + 55*b))/(10240*d) - (3*b*exp(c + d*x)*(280*
a*b + 384*a^2 + 77*b^2))/(1024*d)

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