3.2.70 \(\int \text {csch}^7(c+d x) (a+b \sinh ^3(c+d x))^3 \, dx\) [170]

Optimal. Leaf size=166 \[ -\frac {b^3 x}{2}+\frac {5 a^3 \tanh ^{-1}(\cosh (c+d x))}{16 d}-\frac {3 a b^2 \tanh ^{-1}(\cosh (c+d x))}{d}+\frac {3 a^2 b \coth (c+d x)}{d}-\frac {a^2 b \coth ^3(c+d x)}{d}-\frac {5 a^3 \coth (c+d x) \text {csch}(c+d x)}{16 d}+\frac {5 a^3 \coth (c+d x) \text {csch}^3(c+d x)}{24 d}-\frac {a^3 \coth (c+d x) \text {csch}^5(c+d x)}{6 d}+\frac {b^3 \cosh (c+d x) \sinh (c+d x)}{2 d} \]

[Out]

-1/2*b^3*x+5/16*a^3*arctanh(cosh(d*x+c))/d-3*a*b^2*arctanh(cosh(d*x+c))/d+3*a^2*b*coth(d*x+c)/d-a^2*b*coth(d*x
+c)^3/d-5/16*a^3*coth(d*x+c)*csch(d*x+c)/d+5/24*a^3*coth(d*x+c)*csch(d*x+c)^3/d-1/6*a^3*coth(d*x+c)*csch(d*x+c
)^5/d+1/2*b^3*cosh(d*x+c)*sinh(d*x+c)/d

________________________________________________________________________________________

Rubi [A]
time = 0.15, antiderivative size = 166, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 6, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.261, Rules used = {3299, 3855, 3852, 3853, 2715, 8} \begin {gather*} \frac {5 a^3 \tanh ^{-1}(\cosh (c+d x))}{16 d}-\frac {a^3 \coth (c+d x) \text {csch}^5(c+d x)}{6 d}+\frac {5 a^3 \coth (c+d x) \text {csch}^3(c+d x)}{24 d}-\frac {5 a^3 \coth (c+d x) \text {csch}(c+d x)}{16 d}-\frac {a^2 b \coth ^3(c+d x)}{d}+\frac {3 a^2 b \coth (c+d x)}{d}-\frac {3 a b^2 \tanh ^{-1}(\cosh (c+d x))}{d}+\frac {b^3 \sinh (c+d x) \cosh (c+d x)}{2 d}-\frac {b^3 x}{2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Csch[c + d*x]^7*(a + b*Sinh[c + d*x]^3)^3,x]

[Out]

-1/2*(b^3*x) + (5*a^3*ArcTanh[Cosh[c + d*x]])/(16*d) - (3*a*b^2*ArcTanh[Cosh[c + d*x]])/d + (3*a^2*b*Coth[c +
d*x])/d - (a^2*b*Coth[c + d*x]^3)/d - (5*a^3*Coth[c + d*x]*Csch[c + d*x])/(16*d) + (5*a^3*Coth[c + d*x]*Csch[c
+ d*x]^3)/(24*d) - (a^3*Coth[c + d*x]*Csch[c + d*x]^5)/(6*d) + (b^3*Cosh[c + d*x]*Sinh[c + d*x])/(2*d)

Rule 8

Int[a_, x_Symbol] :> Simp[a*x, x] /; FreeQ[a, x]

Rule 2715

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

Rule 3299

Int[sin[(e_.) + (f_.)*(x_)]^(m_.)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]^(n_))^(p_.), x_Symbol] :> Int[ExpandTr
ig[sin[e + f*x]^m*(a + b*sin[e + f*x]^n)^p, x], x] /; FreeQ[{a, b, e, f}, x] && IntegersQ[m, p] && (EqQ[n, 4]
|| GtQ[p, 0] || (EqQ[p, -1] && IntegerQ[n]))

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 3853

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

Rule 3855

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

Rubi steps

\begin {align*} \int \text {csch}^7(c+d x) \left (a+b \sinh ^3(c+d x)\right )^3 \, dx &=-\left (i \int \left (3 i a b^2 \text {csch}(c+d x)+3 i a^2 b \text {csch}^4(c+d x)+i a^3 \text {csch}^7(c+d x)+i b^3 \sinh ^2(c+d x)\right ) \, dx\right )\\ &=a^3 \int \text {csch}^7(c+d x) \, dx+\left (3 a^2 b\right ) \int \text {csch}^4(c+d x) \, dx+\left (3 a b^2\right ) \int \text {csch}(c+d x) \, dx+b^3 \int \sinh ^2(c+d x) \, dx\\ &=-\frac {3 a b^2 \tanh ^{-1}(\cosh (c+d x))}{d}-\frac {a^3 \coth (c+d x) \text {csch}^5(c+d x)}{6 d}+\frac {b^3 \cosh (c+d x) \sinh (c+d x)}{2 d}-\frac {1}{6} \left (5 a^3\right ) \int \text {csch}^5(c+d x) \, dx-\frac {1}{2} b^3 \int 1 \, dx+\frac {\left (3 i a^2 b\right ) \text {Subst}\left (\int \left (1+x^2\right ) \, dx,x,-i \coth (c+d x)\right )}{d}\\ &=-\frac {b^3 x}{2}-\frac {3 a b^2 \tanh ^{-1}(\cosh (c+d x))}{d}+\frac {3 a^2 b \coth (c+d x)}{d}-\frac {a^2 b \coth ^3(c+d x)}{d}+\frac {5 a^3 \coth (c+d x) \text {csch}^3(c+d x)}{24 d}-\frac {a^3 \coth (c+d x) \text {csch}^5(c+d x)}{6 d}+\frac {b^3 \cosh (c+d x) \sinh (c+d x)}{2 d}+\frac {1}{8} \left (5 a^3\right ) \int \text {csch}^3(c+d x) \, dx\\ &=-\frac {b^3 x}{2}-\frac {3 a b^2 \tanh ^{-1}(\cosh (c+d x))}{d}+\frac {3 a^2 b \coth (c+d x)}{d}-\frac {a^2 b \coth ^3(c+d x)}{d}-\frac {5 a^3 \coth (c+d x) \text {csch}(c+d x)}{16 d}+\frac {5 a^3 \coth (c+d x) \text {csch}^3(c+d x)}{24 d}-\frac {a^3 \coth (c+d x) \text {csch}^5(c+d x)}{6 d}+\frac {b^3 \cosh (c+d x) \sinh (c+d x)}{2 d}-\frac {1}{16} \left (5 a^3\right ) \int \text {csch}(c+d x) \, dx\\ &=-\frac {b^3 x}{2}+\frac {5 a^3 \tanh ^{-1}(\cosh (c+d x))}{16 d}-\frac {3 a b^2 \tanh ^{-1}(\cosh (c+d x))}{d}+\frac {3 a^2 b \coth (c+d x)}{d}-\frac {a^2 b \coth ^3(c+d x)}{d}-\frac {5 a^3 \coth (c+d x) \text {csch}(c+d x)}{16 d}+\frac {5 a^3 \coth (c+d x) \text {csch}^3(c+d x)}{24 d}-\frac {a^3 \coth (c+d x) \text {csch}^5(c+d x)}{6 d}+\frac {b^3 \cosh (c+d x) \sinh (c+d x)}{2 d}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]
time = 1.13, size = 236, normalized size = 1.42 \begin {gather*} -\frac {192 b^3 c+192 b^3 d x-384 a^2 b \coth \left (\frac {1}{2} (c+d x)\right )+30 a^3 \text {csch}^2\left (\frac {1}{2} (c+d x)\right )+a^3 \text {csch}^6\left (\frac {1}{2} (c+d x)\right )+120 a^3 \log \left (\tanh \left (\frac {1}{2} (c+d x)\right )\right )-1152 a b^2 \log \left (\tanh \left (\frac {1}{2} (c+d x)\right )\right )+30 a^3 \text {sech}^2\left (\frac {1}{2} (c+d x)\right )+6 a^3 \text {sech}^4\left (\frac {1}{2} (c+d x)\right )+a^3 \text {sech}^6\left (\frac {1}{2} (c+d x)\right )-384 a^2 b \text {csch}^3(c+d x) \sinh ^4\left (\frac {1}{2} (c+d x)\right )-6 a^2 \text {csch}^4\left (\frac {1}{2} (c+d x)\right ) (a-4 b \sinh (c+d x))-96 b^3 \sinh (2 (c+d x))-384 a^2 b \tanh \left (\frac {1}{2} (c+d x)\right )}{384 d} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[Csch[c + d*x]^7*(a + b*Sinh[c + d*x]^3)^3,x]

[Out]

-1/384*(192*b^3*c + 192*b^3*d*x - 384*a^2*b*Coth[(c + d*x)/2] + 30*a^3*Csch[(c + d*x)/2]^2 + a^3*Csch[(c + d*x
)/2]^6 + 120*a^3*Log[Tanh[(c + d*x)/2]] - 1152*a*b^2*Log[Tanh[(c + d*x)/2]] + 30*a^3*Sech[(c + d*x)/2]^2 + 6*a
^3*Sech[(c + d*x)/2]^4 + a^3*Sech[(c + d*x)/2]^6 - 384*a^2*b*Csch[c + d*x]^3*Sinh[(c + d*x)/2]^4 - 6*a^2*Csch[
(c + d*x)/2]^4*(a - 4*b*Sinh[c + d*x]) - 96*b^3*Sinh[2*(c + d*x)] - 384*a^2*b*Tanh[(c + d*x)/2])/d

________________________________________________________________________________________

Maple [A]
time = 2.13, size = 254, normalized size = 1.53

method	result	size
risch	\(-\frac {b^{3} x}{2}+\frac {b^{3} {\mathrm e}^{2 d x +2 c}}{8 d}-\frac {b^{3} {\mathrm e}^{-2 d x -2 c}}{8 d}-\frac {a^{2} \left (15 a \,{\mathrm e}^{11 d x +11 c}-85 a \,{\mathrm e}^{9 d x +9 c}+288 b \,{\mathrm e}^{8 d x +8 c}+198 a \,{\mathrm e}^{7 d x +7 c}-960 b \,{\mathrm e}^{6 d x +6 c}+198 a \,{\mathrm e}^{5 d x +5 c}+1152 b \,{\mathrm e}^{4 d x +4 c}-85 a \,{\mathrm e}^{3 d x +3 c}-576 b \,{\mathrm e}^{2 d x +2 c}+15 a \,{\mathrm e}^{d x +c}+96 b \right )}{24 d \left ({\mathrm e}^{2 d x +2 c}-1\right )^{6}}+\frac {5 a^{3} \ln \left ({\mathrm e}^{d x +c}+1\right )}{16 d}-\frac {3 a \ln \left ({\mathrm e}^{d x +c}+1\right ) b^{2}}{d}-\frac {5 a^{3} \ln \left ({\mathrm e}^{d x +c}-1\right )}{16 d}+\frac {3 a \ln \left ({\mathrm e}^{d x +c}-1\right ) b^{2}}{d}\)	\(254\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(csch(d*x+c)^7*(a+b*sinh(d*x+c)^3)^3,x,method=_RETURNVERBOSE)

[Out]

-1/2*b^3*x+1/8*b^3/d*exp(2*d*x+2*c)-1/8*b^3/d*exp(-2*d*x-2*c)-1/24*a^2*(15*a*exp(11*d*x+11*c)-85*a*exp(9*d*x+9
*c)+288*b*exp(8*d*x+8*c)+198*a*exp(7*d*x+7*c)-960*b*exp(6*d*x+6*c)+198*a*exp(5*d*x+5*c)+1152*b*exp(4*d*x+4*c)-
85*a*exp(3*d*x+3*c)-576*b*exp(2*d*x+2*c)+15*a*exp(d*x+c)+96*b)/d/(exp(2*d*x+2*c)-1)^6+5/16*a^3/d*ln(exp(d*x+c)
+1)-3*a/d*ln(exp(d*x+c)+1)*b^2-5/16*a^3/d*ln(exp(d*x+c)-1)+3*a/d*ln(exp(d*x+c)-1)*b^2

________________________________________________________________________________________

Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 355 vs. \(2 (154) = 308\).
time = 0.28, size = 355, normalized size = 2.14 \begin {gather*} -\frac {1}{8} \, b^{3} {\left (4 \, x - \frac {e^{\left (2 \, d x + 2 \, c\right )}}{d} + \frac {e^{\left (-2 \, d x - 2 \, c\right )}}{d}\right )} + \frac {1}{48} \, a^{3} {\left (\frac {15 \, \log \left (e^{\left (-d x - c\right )} + 1\right )}{d} - \frac {15 \, \log \left (e^{\left (-d x - c\right )} - 1\right )}{d} + \frac {2 \, {\left (15 \, e^{\left (-d x - c\right )} - 85 \, e^{\left (-3 \, d x - 3 \, c\right )} + 198 \, e^{\left (-5 \, d x - 5 \, c\right )} + 198 \, e^{\left (-7 \, d x - 7 \, c\right )} - 85 \, e^{\left (-9 \, d x - 9 \, c\right )} + 15 \, e^{\left (-11 \, d x - 11 \, c\right )}\right )}}{d {\left (6 \, e^{\left (-2 \, d x - 2 \, c\right )} - 15 \, e^{\left (-4 \, d x - 4 \, c\right )} + 20 \, e^{\left (-6 \, d x - 6 \, c\right )} - 15 \, e^{\left (-8 \, d x - 8 \, c\right )} + 6 \, e^{\left (-10 \, d x - 10 \, c\right )} - e^{\left (-12 \, d x - 12 \, c\right )} - 1\right )}}\right )} - 3 \, a b^{2} {\left (\frac {\log \left (e^{\left (-d x - c\right )} + 1\right )}{d} - \frac {\log \left (e^{\left (-d x - c\right )} - 1\right )}{d}\right )} + 4 \, a^{2} b {\left (\frac {3 \, e^{\left (-2 \, d x - 2 \, c\right )}}{d {\left (3 \, e^{\left (-2 \, d x - 2 \, c\right )} - 3 \, e^{\left (-4 \, d x - 4 \, c\right )} + e^{\left (-6 \, d x - 6 \, c\right )} - 1\right )}} - \frac {1}{d {\left (3 \, e^{\left (-2 \, d x - 2 \, c\right )} - 3 \, e^{\left (-4 \, d x - 4 \, c\right )} + e^{\left (-6 \, d x - 6 \, c\right )} - 1\right )}}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csch(d*x+c)^7*(a+b*sinh(d*x+c)^3)^3,x, algorithm="maxima")

[Out]

-1/8*b^3*(4*x - e^(2*d*x + 2*c)/d + e^(-2*d*x - 2*c)/d) + 1/48*a^3*(15*log(e^(-d*x - c) + 1)/d - 15*log(e^(-d*
x - c) - 1)/d + 2*(15*e^(-d*x - c) - 85*e^(-3*d*x - 3*c) + 198*e^(-5*d*x - 5*c) + 198*e^(-7*d*x - 7*c) - 85*e^
(-9*d*x - 9*c) + 15*e^(-11*d*x - 11*c))/(d*(6*e^(-2*d*x - 2*c) - 15*e^(-4*d*x - 4*c) + 20*e^(-6*d*x - 6*c) - 1
5*e^(-8*d*x - 8*c) + 6*e^(-10*d*x - 10*c) - e^(-12*d*x - 12*c) - 1))) - 3*a*b^2*(log(e^(-d*x - c) + 1)/d - log
(e^(-d*x - c) - 1)/d) + 4*a^2*b*(3*e^(-2*d*x - 2*c)/(d*(3*e^(-2*d*x - 2*c) - 3*e^(-4*d*x - 4*c) + e^(-6*d*x -
6*c) - 1)) - 1/(d*(3*e^(-2*d*x - 2*c) - 3*e^(-4*d*x - 4*c) + e^(-6*d*x - 6*c) - 1)))

________________________________________________________________________________________

Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 6210 vs. \(2 (154) = 308\).
time = 0.51, size = 6210, normalized size = 37.41 \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)^7*(a+b*sinh(d*x+c)^3)^3,x, algorithm="fricas")

[Out]

1/48*(6*b^3*cosh(d*x + c)^16 + 96*b^3*cosh(d*x + c)*sinh(d*x + c)^15 + 6*b^3*sinh(d*x + c)^16 - 30*a^3*cosh(d*
x + c)^13 - 12*(2*b^3*d*x + 3*b^3)*cosh(d*x + c)^14 - 12*(2*b^3*d*x - 60*b^3*cosh(d*x + c)^2 + 3*b^3)*sinh(d*x
+ c)^14 + 170*a^3*cosh(d*x + c)^11 + 6*(560*b^3*cosh(d*x + c)^3 - 5*a^3 - 28*(2*b^3*d*x + 3*b^3)*cosh(d*x + c
))*sinh(d*x + c)^13 + 12*(12*b^3*d*x + 7*b^3)*cosh(d*x + c)^12 + 6*(1820*b^3*cosh(d*x + c)^4 + 24*b^3*d*x - 65
*a^3*cosh(d*x + c) + 14*b^3 - 182*(2*b^3*d*x + 3*b^3)*cosh(d*x + c)^2)*sinh(d*x + c)^12 - 396*a^3*cosh(d*x + c
)^9 + 2*(13104*b^3*cosh(d*x + c)^5 - 1170*a^3*cosh(d*x + c)^2 - 2184*(2*b^3*d*x + 3*b^3)*cosh(d*x + c)^3 + 85*
a^3 + 72*(12*b^3*d*x + 7*b^3)*cosh(d*x + c))*sinh(d*x + c)^11 - 12*(30*b^3*d*x + 48*a^2*b + 7*b^3)*cosh(d*x +
c)^10 + 2*(24024*b^3*cosh(d*x + c)^6 - 4290*a^3*cosh(d*x + c)^3 - 180*b^3*d*x - 6006*(2*b^3*d*x + 3*b^3)*cosh(
d*x + c)^4 + 935*a^3*cosh(d*x + c) - 288*a^2*b - 42*b^3 + 396*(12*b^3*d*x + 7*b^3)*cosh(d*x + c)^2)*sinh(d*x +
c)^10 - 396*a^3*cosh(d*x + c)^7 + 2*(34320*b^3*cosh(d*x + c)^7 - 10725*a^3*cosh(d*x + c)^4 - 12012*(2*b^3*d*x
+ 3*b^3)*cosh(d*x + c)^5 + 4675*a^3*cosh(d*x + c)^2 + 1320*(12*b^3*d*x + 7*b^3)*cosh(d*x + c)^3 - 198*a^3 - 6
0*(30*b^3*d*x + 48*a^2*b + 7*b^3)*cosh(d*x + c))*sinh(d*x + c)^9 + 480*(b^3*d*x + 4*a^2*b)*cosh(d*x + c)^8 + 6
*(12870*b^3*cosh(d*x + c)^8 - 6435*a^3*cosh(d*x + c)^5 - 6006*(2*b^3*d*x + 3*b^3)*cosh(d*x + c)^6 + 4675*a^3*c
osh(d*x + c)^3 + 80*b^3*d*x + 990*(12*b^3*d*x + 7*b^3)*cosh(d*x + c)^4 - 594*a^3*cosh(d*x + c) + 320*a^2*b - 9
0*(30*b^3*d*x + 48*a^2*b + 7*b^3)*cosh(d*x + c)^2)*sinh(d*x + c)^8 + 170*a^3*cosh(d*x + c)^5 + 12*(5720*b^3*co
sh(d*x + c)^9 - 4290*a^3*cosh(d*x + c)^6 - 3432*(2*b^3*d*x + 3*b^3)*cosh(d*x + c)^7 + 4675*a^3*cosh(d*x + c)^4
+ 792*(12*b^3*d*x + 7*b^3)*cosh(d*x + c)^5 - 1188*a^3*cosh(d*x + c)^2 - 120*(30*b^3*d*x + 48*a^2*b + 7*b^3)*c
osh(d*x + c)^3 - 33*a^3 + 320*(b^3*d*x + 4*a^2*b)*cosh(d*x + c))*sinh(d*x + c)^7 - 12*(30*b^3*d*x + 192*a^2*b
- 7*b^3)*cosh(d*x + c)^6 + 12*(4004*b^3*cosh(d*x + c)^10 - 4290*a^3*cosh(d*x + c)^7 - 3003*(2*b^3*d*x + 3*b^3)
*cosh(d*x + c)^8 + 6545*a^3*cosh(d*x + c)^5 + 924*(12*b^3*d*x + 7*b^3)*cosh(d*x + c)^6 - 2772*a^3*cosh(d*x + c
)^3 - 30*b^3*d*x - 210*(30*b^3*d*x + 48*a^2*b + 7*b^3)*cosh(d*x + c)^4 - 231*a^3*cosh(d*x + c) - 192*a^2*b + 7
*b^3 + 1120*(b^3*d*x + 4*a^2*b)*cosh(d*x + c)^2)*sinh(d*x + c)^6 - 30*a^3*cosh(d*x + c)^3 + 2*(13104*b^3*cosh(
d*x + c)^11 - 19305*a^3*cosh(d*x + c)^8 - 12012*(2*b^3*d*x + 3*b^3)*cosh(d*x + c)^9 + 39270*a^3*cosh(d*x + c)^
6 + 4752*(12*b^3*d*x + 7*b^3)*cosh(d*x + c)^7 - 24948*a^3*cosh(d*x + c)^4 - 1512*(30*b^3*d*x + 48*a^2*b + 7*b^
3)*cosh(d*x + c)^5 - 4158*a^3*cosh(d*x + c)^2 + 13440*(b^3*d*x + 4*a^2*b)*cosh(d*x + c)^3 + 85*a^3 - 36*(30*b^
3*d*x + 192*a^2*b - 7*b^3)*cosh(d*x + c))*sinh(d*x + c)^5 + 12*(12*b^3*d*x + 96*a^2*b - 7*b^3)*cosh(d*x + c)^4
+ 2*(5460*b^3*cosh(d*x + c)^12 - 10725*a^3*cosh(d*x + c)^9 - 6006*(2*b^3*d*x + 3*b^3)*cosh(d*x + c)^10 + 2805
0*a^3*cosh(d*x + c)^7 + 2970*(12*b^3*d*x + 7*b^3)*cosh(d*x + c)^8 - 24948*a^3*cosh(d*x + c)^5 - 1260*(30*b^3*d
*x + 48*a^2*b + 7*b^3)*cosh(d*x + c)^6 - 6930*a^3*cosh(d*x + c)^3 + 72*b^3*d*x + 16800*(b^3*d*x + 4*a^2*b)*cos
h(d*x + c)^4 + 425*a^3*cosh(d*x + c) + 576*a^2*b - 42*b^3 - 90*(30*b^3*d*x + 192*a^2*b - 7*b^3)*cosh(d*x + c)^
2)*sinh(d*x + c)^4 + 2*(1680*b^3*cosh(d*x + c)^13 - 4290*a^3*cosh(d*x + c)^10 - 2184*(2*b^3*d*x + 3*b^3)*cosh(
d*x + c)^11 + 14025*a^3*cosh(d*x + c)^8 + 1320*(12*b^3*d*x + 7*b^3)*cosh(d*x + c)^9 - 16632*a^3*cosh(d*x + c)^
6 - 720*(30*b^3*d*x + 48*a^2*b + 7*b^3)*cosh(d*x + c)^7 - 6930*a^3*cosh(d*x + c)^4 + 13440*(b^3*d*x + 4*a^2*b)
*cosh(d*x + c)^5 + 850*a^3*cosh(d*x + c)^2 - 120*(30*b^3*d*x + 192*a^2*b - 7*b^3)*cosh(d*x + c)^3 - 15*a^3 + 2
4*(12*b^3*d*x + 96*a^2*b - 7*b^3)*cosh(d*x + c))*sinh(d*x + c)^3 - 6*b^3 - 12*(2*b^3*d*x + 16*a^2*b - 3*b^3)*c
osh(d*x + c)^2 + 2*(360*b^3*cosh(d*x + c)^14 - 1170*a^3*cosh(d*x + c)^11 - 546*(2*b^3*d*x + 3*b^3)*cosh(d*x +
c)^12 + 4675*a^3*cosh(d*x + c)^9 + 396*(12*b^3*d*x + 7*b^3)*cosh(d*x + c)^10 - 7128*a^3*cosh(d*x + c)^7 - 270*
(30*b^3*d*x + 48*a^2*b + 7*b^3)*cosh(d*x + c)^8 - 4158*a^3*cosh(d*x + c)^5 + 6720*(b^3*d*x + 4*a^2*b)*cosh(d*x
+ c)^6 + 850*a^3*cosh(d*x + c)^3 - 12*b^3*d*x - 90*(30*b^3*d*x + 192*a^2*b - 7*b^3)*cosh(d*x + c)^4 - 45*a^3*
cosh(d*x + c) - 96*a^2*b + 18*b^3 + 36*(12*b^3*d*x + 96*a^2*b - 7*b^3)*cosh(d*x + c)^2)*sinh(d*x + c)^2 + 3*((
5*a^3 - 48*a*b^2)*cosh(d*x + c)^14 + 14*(5*a^3 - 48*a*b^2)*cosh(d*x + c)*sinh(d*x + c)^13 + (5*a^3 - 48*a*b^2)
*sinh(d*x + c)^14 - 6*(5*a^3 - 48*a*b^2)*cosh(d*x + c)^12 - (30*a^3 - 288*a*b^2 - 91*(5*a^3 - 48*a*b^2)*cosh(d
*x + c)^2)*sinh(d*x + c)^12 + 4*(91*(5*a^3 - 48*a*b^2)*cosh(d*x + c)^3 - 18*(5*a^3 - 48*a*b^2)*cosh(d*x + c))*
sinh(d*x + c)^11 + 15*(5*a^3 - 48*a*b^2)*cosh(d*x + c)^10 + (1001*(5*a^3 - 48*a*b^2)*cosh(d*x + c)^4 + 75*a^3
- 720*a*b^2 - 396*(5*a^3 - 48*a*b^2)*cosh(d*x + c)^2)*sinh(d*x + c)^10 + 2*(1001*(5*a^3 - 48*a*b^2)*cosh(d*x +
c)^5 - 660*(5*a^3 - 48*a*b^2)*cosh(d*x + c)^3 ...

________________________________________________________________________________________

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)**7*(a+b*sinh(d*x+c)**3)**3,x)

[Out]

Timed out

________________________________________________________________________________________

Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 327 vs. \(2 (154) = 308\).
time = 0.52, size = 327, normalized size = 1.97 \begin {gather*} -\frac {24 \, {\left (d x + c\right )} b^{3} - 6 \, b^{3} e^{\left (2 \, d x + 2 \, c\right )} - 3 \, {\left (5 \, a^{3} - 48 \, a b^{2}\right )} \log \left (e^{\left (d x + c\right )} + 1\right ) + 3 \, {\left (5 \, a^{3} - 48 \, a b^{2}\right )} \log \left ({\left | e^{\left (d x + c\right )} - 1 \right |}\right ) + \frac {2 \, {\left (15 \, a^{3} e^{\left (13 \, d x + 13 \, c\right )} + 3 \, b^{3} e^{\left (12 \, d x + 12 \, c\right )} - 85 \, a^{3} e^{\left (11 \, d x + 11 \, c\right )} + 198 \, a^{3} e^{\left (9 \, d x + 9 \, c\right )} + 198 \, a^{3} e^{\left (7 \, d x + 7 \, c\right )} - 85 \, a^{3} e^{\left (5 \, d x + 5 \, c\right )} + 15 \, a^{3} e^{\left (3 \, d x + 3 \, c\right )} + 3 \, b^{3} + 18 \, {\left (16 \, a^{2} b - b^{3}\right )} e^{\left (10 \, d x + 10 \, c\right )} - 15 \, {\left (64 \, a^{2} b - 3 \, b^{3}\right )} e^{\left (8 \, d x + 8 \, c\right )} + 12 \, {\left (96 \, a^{2} b - 5 \, b^{3}\right )} e^{\left (6 \, d x + 6 \, c\right )} - 9 \, {\left (64 \, a^{2} b - 5 \, b^{3}\right )} e^{\left (4 \, d x + 4 \, c\right )} + 6 \, {\left (16 \, a^{2} b - 3 \, b^{3}\right )} e^{\left (2 \, d x + 2 \, c\right )}\right )} e^{\left (-2 \, d x - 2 \, c\right )}}{{\left (e^{\left (d x + c\right )} + 1\right )}^{6} {\left (e^{\left (d x + c\right )} - 1\right )}^{6}}}{48 \, d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csch(d*x+c)^7*(a+b*sinh(d*x+c)^3)^3,x, algorithm="giac")

[Out]

-1/48*(24*(d*x + c)*b^3 - 6*b^3*e^(2*d*x + 2*c) - 3*(5*a^3 - 48*a*b^2)*log(e^(d*x + c) + 1) + 3*(5*a^3 - 48*a*
b^2)*log(abs(e^(d*x + c) - 1)) + 2*(15*a^3*e^(13*d*x + 13*c) + 3*b^3*e^(12*d*x + 12*c) - 85*a^3*e^(11*d*x + 11
*c) + 198*a^3*e^(9*d*x + 9*c) + 198*a^3*e^(7*d*x + 7*c) - 85*a^3*e^(5*d*x + 5*c) + 15*a^3*e^(3*d*x + 3*c) + 3*
b^3 + 18*(16*a^2*b - b^3)*e^(10*d*x + 10*c) - 15*(64*a^2*b - 3*b^3)*e^(8*d*x + 8*c) + 12*(96*a^2*b - 5*b^3)*e^
(6*d*x + 6*c) - 9*(64*a^2*b - 5*b^3)*e^(4*d*x + 4*c) + 6*(16*a^2*b - 3*b^3)*e^(2*d*x + 2*c))*e^(-2*d*x - 2*c)/
((e^(d*x + c) + 1)^6*(e^(d*x + c) - 1)^6))/d

________________________________________________________________________________________

Mupad [B]
time = 0.29, size = 486, normalized size = 2.93 \begin {gather*} \frac {\mathrm {atan}\left (\frac {{\mathrm {e}}^{d\,x}\,{\mathrm {e}}^c\,\left (5\,a^3\,\sqrt {-d^2}-48\,a\,b^2\,\sqrt {-d^2}\right )}{d\,\sqrt {25\,a^6-480\,a^4\,b^2+2304\,a^2\,b^4}}\right )\,\sqrt {25\,a^6-480\,a^4\,b^2+2304\,a^2\,b^4}}{8\,\sqrt {-d^2}}-\frac {b^3\,x}{2}-\frac {\frac {12\,a^2\,b}{d}-\frac {5\,a^3\,{\mathrm {e}}^{c+d\,x}}{12\,d}}{{\mathrm {e}}^{4\,c+4\,d\,x}-2\,{\mathrm {e}}^{2\,c+2\,d\,x}+1}-\frac {\frac {8\,a^2\,b}{d}+\frac {a^3\,{\mathrm {e}}^{c+d\,x}}{3\,d}}{3\,{\mathrm {e}}^{2\,c+2\,d\,x}-3\,{\mathrm {e}}^{4\,c+4\,d\,x}+{\mathrm {e}}^{6\,c+6\,d\,x}-1}-\frac {b^3\,{\mathrm {e}}^{-2\,c-2\,d\,x}}{8\,d}+\frac {b^3\,{\mathrm {e}}^{2\,c+2\,d\,x}}{8\,d}-\frac {18\,a^3\,{\mathrm {e}}^{c+d\,x}}{d\,\left (6\,{\mathrm {e}}^{4\,c+4\,d\,x}-4\,{\mathrm {e}}^{2\,c+2\,d\,x}-4\,{\mathrm {e}}^{6\,c+6\,d\,x}+{\mathrm {e}}^{8\,c+8\,d\,x}+1\right )}-\frac {80\,a^3\,{\mathrm {e}}^{c+d\,x}}{3\,d\,\left (5\,{\mathrm {e}}^{2\,c+2\,d\,x}-10\,{\mathrm {e}}^{4\,c+4\,d\,x}+10\,{\mathrm {e}}^{6\,c+6\,d\,x}-5\,{\mathrm {e}}^{8\,c+8\,d\,x}+{\mathrm {e}}^{10\,c+10\,d\,x}-1\right )}-\frac {32\,a^3\,{\mathrm {e}}^{c+d\,x}}{3\,d\,\left (15\,{\mathrm {e}}^{4\,c+4\,d\,x}-6\,{\mathrm {e}}^{2\,c+2\,d\,x}-20\,{\mathrm {e}}^{6\,c+6\,d\,x}+15\,{\mathrm {e}}^{8\,c+8\,d\,x}-6\,{\mathrm {e}}^{10\,c+10\,d\,x}+{\mathrm {e}}^{12\,c+12\,d\,x}+1\right )}-\frac {5\,a^3\,{\mathrm {e}}^{c+d\,x}}{8\,d\,\left ({\mathrm {e}}^{2\,c+2\,d\,x}-1\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*sinh(c + d*x)^3)^3/sinh(c + d*x)^7,x)

[Out]

(atan((exp(d*x)*exp(c)*(5*a^3*(-d^2)^(1/2) - 48*a*b^2*(-d^2)^(1/2)))/(d*(25*a^6 + 2304*a^2*b^4 - 480*a^4*b^2)^
(1/2)))*(25*a^6 + 2304*a^2*b^4 - 480*a^4*b^2)^(1/2))/(8*(-d^2)^(1/2)) - (b^3*x)/2 - ((12*a^2*b)/d - (5*a^3*exp
(c + d*x))/(12*d))/(exp(4*c + 4*d*x) - 2*exp(2*c + 2*d*x) + 1) - ((8*a^2*b)/d + (a^3*exp(c + d*x))/(3*d))/(3*e
xp(2*c + 2*d*x) - 3*exp(4*c + 4*d*x) + exp(6*c + 6*d*x) - 1) - (b^3*exp(- 2*c - 2*d*x))/(8*d) + (b^3*exp(2*c +
2*d*x))/(8*d) - (18*a^3*exp(c + d*x))/(d*(6*exp(4*c + 4*d*x) - 4*exp(2*c + 2*d*x) - 4*exp(6*c + 6*d*x) + exp(
8*c + 8*d*x) + 1)) - (80*a^3*exp(c + d*x))/(3*d*(5*exp(2*c + 2*d*x) - 10*exp(4*c + 4*d*x) + 10*exp(6*c + 6*d*x
) - 5*exp(8*c + 8*d*x) + exp(10*c + 10*d*x) - 1)) - (32*a^3*exp(c + d*x))/(3*d*(15*exp(4*c + 4*d*x) - 6*exp(2*
c + 2*d*x) - 20*exp(6*c + 6*d*x) + 15*exp(8*c + 8*d*x) - 6*exp(10*c + 10*d*x) + exp(12*c + 12*d*x) + 1)) - (5*
a^3*exp(c + d*x))/(8*d*(exp(2*c + 2*d*x) - 1))

________________________________________________________________________________________