3.160 \(\int \text{csch}^7(c+d x) (a+b \sinh ^3(c+d x))^2 \, dx\)
Optimal. Leaf size=133 \[ \frac{5 a^2 \tanh ^{-1}(\cosh (c+d x))}{16 d}-\frac{a^2 \coth (c+d x) \text{csch}^5(c+d x)}{6 d}+\frac{5 a^2 \coth (c+d x) \text{csch}^3(c+d x)}{24 d}-\frac{5 a^2 \coth (c+d x) \text{csch}(c+d x)}{16 d}-\frac{2 a b \coth ^3(c+d x)}{3 d}+\frac{2 a b \coth (c+d x)}{d}-\frac{b^2 \tanh ^{-1}(\cosh (c+d x))}{d} \]
[Out]
(5*a^2*ArcTanh[Cosh[c + d*x]])/(16*d) - (b^2*ArcTanh[Cosh[c + d*x]])/d + (2*a*b*Coth[c + d*x])/d - (2*a*b*Coth
[c + d*x]^3)/(3*d) - (5*a^2*Coth[c + d*x]*Csch[c + d*x])/(16*d) + (5*a^2*Coth[c + d*x]*Csch[c + d*x]^3)/(24*d)
- (a^2*Coth[c + d*x]*Csch[c + d*x]^5)/(6*d)
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Rubi [A] time = 0.174964, antiderivative size = 133, normalized size of antiderivative = 1.,
number of steps used = 9, number of rules used = 4, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.174, Rules used =
{3220, 3770, 3767, 3768} \[ \frac{5 a^2 \tanh ^{-1}(\cosh (c+d x))}{16 d}-\frac{a^2 \coth (c+d x) \text{csch}^5(c+d x)}{6 d}+\frac{5 a^2 \coth (c+d x) \text{csch}^3(c+d x)}{24 d}-\frac{5 a^2 \coth (c+d x) \text{csch}(c+d x)}{16 d}-\frac{2 a b \coth ^3(c+d x)}{3 d}+\frac{2 a b \coth (c+d x)}{d}-\frac{b^2 \tanh ^{-1}(\cosh (c+d x))}{d} \]
Antiderivative was successfully verified.
[In]
Int[Csch[c + d*x]^7*(a + b*Sinh[c + d*x]^3)^2,x]
[Out]
(5*a^2*ArcTanh[Cosh[c + d*x]])/(16*d) - (b^2*ArcTanh[Cosh[c + d*x]])/d + (2*a*b*Coth[c + d*x])/d - (2*a*b*Coth
[c + d*x]^3)/(3*d) - (5*a^2*Coth[c + d*x]*Csch[c + d*x])/(16*d) + (5*a^2*Coth[c + d*x]*Csch[c + d*x]^3)/(24*d)
- (a^2*Coth[c + d*x]*Csch[c + d*x]^5)/(6*d)
Rule 3220
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 3770
Int[csc[(c_.) + (d_.)*(x_)], x_Symbol] :> -Simp[ArcTanh[Cos[c + d*x]]/d, x] /; FreeQ[{c, d}, x]
Rule 3767
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 3768
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]
Rubi steps
\begin{align*} \int \text{csch}^7(c+d x) \left (a+b \sinh ^3(c+d x)\right )^2 \, dx &=-\left (i \int \left (i b^2 \text{csch}(c+d x)+2 i a b \text{csch}^4(c+d x)+i a^2 \text{csch}^7(c+d x)\right ) \, dx\right )\\ &=a^2 \int \text{csch}^7(c+d x) \, dx+(2 a b) \int \text{csch}^4(c+d x) \, dx+b^2 \int \text{csch}(c+d x) \, dx\\ &=-\frac{b^2 \tanh ^{-1}(\cosh (c+d x))}{d}-\frac{a^2 \coth (c+d x) \text{csch}^5(c+d x)}{6 d}-\frac{1}{6} \left (5 a^2\right ) \int \text{csch}^5(c+d x) \, dx+\frac{(2 i a b) \operatorname{Subst}\left (\int \left (1+x^2\right ) \, dx,x,-i \coth (c+d x)\right )}{d}\\ &=-\frac{b^2 \tanh ^{-1}(\cosh (c+d x))}{d}+\frac{2 a b \coth (c+d x)}{d}-\frac{2 a b \coth ^3(c+d x)}{3 d}+\frac{5 a^2 \coth (c+d x) \text{csch}^3(c+d x)}{24 d}-\frac{a^2 \coth (c+d x) \text{csch}^5(c+d x)}{6 d}+\frac{1}{8} \left (5 a^2\right ) \int \text{csch}^3(c+d x) \, dx\\ &=-\frac{b^2 \tanh ^{-1}(\cosh (c+d x))}{d}+\frac{2 a b \coth (c+d x)}{d}-\frac{2 a b \coth ^3(c+d x)}{3 d}-\frac{5 a^2 \coth (c+d x) \text{csch}(c+d x)}{16 d}+\frac{5 a^2 \coth (c+d x) \text{csch}^3(c+d x)}{24 d}-\frac{a^2 \coth (c+d x) \text{csch}^5(c+d x)}{6 d}-\frac{1}{16} \left (5 a^2\right ) \int \text{csch}(c+d x) \, dx\\ &=\frac{5 a^2 \tanh ^{-1}(\cosh (c+d x))}{16 d}-\frac{b^2 \tanh ^{-1}(\cosh (c+d x))}{d}+\frac{2 a b \coth (c+d x)}{d}-\frac{2 a b \coth ^3(c+d x)}{3 d}-\frac{5 a^2 \coth (c+d x) \text{csch}(c+d x)}{16 d}+\frac{5 a^2 \coth (c+d x) \text{csch}^3(c+d x)}{24 d}-\frac{a^2 \coth (c+d x) \text{csch}^5(c+d x)}{6 d}\\ \end{align*}
Mathematica [A] time = 0.0525052, size = 235, normalized size = 1.77 \[ -\frac{a^2 \text{csch}^6\left (\frac{1}{2} (c+d x)\right )}{384 d}+\frac{a^2 \text{csch}^4\left (\frac{1}{2} (c+d x)\right )}{64 d}-\frac{5 a^2 \text{csch}^2\left (\frac{1}{2} (c+d x)\right )}{64 d}-\frac{a^2 \text{sech}^6\left (\frac{1}{2} (c+d x)\right )}{384 d}-\frac{a^2 \text{sech}^4\left (\frac{1}{2} (c+d x)\right )}{64 d}-\frac{5 a^2 \text{sech}^2\left (\frac{1}{2} (c+d x)\right )}{64 d}-\frac{5 a^2 \log \left (\tanh \left (\frac{1}{2} (c+d x)\right )\right )}{16 d}+\frac{4 a b \coth (c+d x)}{3 d}-\frac{2 a b \coth (c+d x) \text{csch}^2(c+d x)}{3 d}+\frac{b^2 \log \left (\sinh \left (\frac{c}{2}+\frac{d x}{2}\right )\right )}{d}-\frac{b^2 \log \left (\cosh \left (\frac{c}{2}+\frac{d x}{2}\right )\right )}{d} \]
Antiderivative was successfully verified.
[In]
Integrate[Csch[c + d*x]^7*(a + b*Sinh[c + d*x]^3)^2,x]
[Out]
(4*a*b*Coth[c + d*x])/(3*d) - (5*a^2*Csch[(c + d*x)/2]^2)/(64*d) + (a^2*Csch[(c + d*x)/2]^4)/(64*d) - (a^2*Csc
h[(c + d*x)/2]^6)/(384*d) - (2*a*b*Coth[c + d*x]*Csch[c + d*x]^2)/(3*d) - (b^2*Log[Cosh[c/2 + (d*x)/2]])/d + (
b^2*Log[Sinh[c/2 + (d*x)/2]])/d - (5*a^2*Log[Tanh[(c + d*x)/2]])/(16*d) - (5*a^2*Sech[(c + d*x)/2]^2)/(64*d) -
(a^2*Sech[(c + d*x)/2]^4)/(64*d) - (a^2*Sech[(c + d*x)/2]^6)/(384*d)
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Maple [A] time = 0.099, size = 90, normalized size = 0.7 \begin{align*}{\frac{1}{d} \left ({a}^{2} \left ( \left ( -{\frac{ \left ({\rm csch} \left (dx+c\right ) \right ) ^{5}}{6}}+{\frac{5\, \left ({\rm csch} \left (dx+c\right ) \right ) ^{3}}{24}}-{\frac{5\,{\rm csch} \left (dx+c\right )}{16}} \right ){\rm coth} \left (dx+c\right )+{\frac{5\,{\it Artanh} \left ({{\rm e}^{dx+c}} \right ) }{8}} \right ) +2\,ab \left ( 2/3-1/3\, \left ({\rm csch} \left (dx+c\right ) \right ) ^{2} \right ){\rm coth} \left (dx+c\right )-2\,{b}^{2}{\it Artanh} \left ({{\rm e}^{dx+c}} \right ) \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(csch(d*x+c)^7*(a+b*sinh(d*x+c)^3)^2,x)
[Out]
1/d*(a^2*((-1/6*csch(d*x+c)^5+5/24*csch(d*x+c)^3-5/16*csch(d*x+c))*coth(d*x+c)+5/8*arctanh(exp(d*x+c)))+2*a*b*
(2/3-1/3*csch(d*x+c)^2)*coth(d*x+c)-2*b^2*arctanh(exp(d*x+c)))
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Maxima [B] time = 1.19719, size = 427, normalized size = 3.21 \begin{align*} \frac{1}{48} \, a^{2}{\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 )} - 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 )} + \frac{8}{3} \, a 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{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(csch(d*x+c)^7*(a+b*sinh(d*x+c)^3)^2,x, algorithm="maxima")
[Out]
1/48*a^2*(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) - 15*e^(-8*d*x - 8*c) + 6*e^(-10*d*x - 10*c) - e^(-12*d*x - 1
2*c) - 1))) - b^2*(log(e^(-d*x - c) + 1)/d - log(e^(-d*x - c) - 1)/d) + 8/3*a*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)))
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Fricas [B] time = 2.28431, size = 9393, normalized size = 70.62 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(csch(d*x+c)^7*(a+b*sinh(d*x+c)^3)^2,x, algorithm="fricas")
[Out]
-1/48*(30*a^2*cosh(d*x + c)^11 + 330*a^2*cosh(d*x + c)*sinh(d*x + c)^10 + 30*a^2*sinh(d*x + c)^11 - 170*a^2*co
sh(d*x + c)^9 + 384*a*b*cosh(d*x + c)^8 + 10*(165*a^2*cosh(d*x + c)^2 - 17*a^2)*sinh(d*x + c)^9 + 396*a^2*cosh
(d*x + c)^7 + 6*(825*a^2*cosh(d*x + c)^3 - 255*a^2*cosh(d*x + c) + 64*a*b)*sinh(d*x + c)^8 - 1280*a*b*cosh(d*x
+ c)^6 + 12*(825*a^2*cosh(d*x + c)^4 - 510*a^2*cosh(d*x + c)^2 + 256*a*b*cosh(d*x + c) + 33*a^2)*sinh(d*x + c
)^7 + 396*a^2*cosh(d*x + c)^5 + 4*(3465*a^2*cosh(d*x + c)^5 - 3570*a^2*cosh(d*x + c)^3 + 2688*a*b*cosh(d*x + c
)^2 + 693*a^2*cosh(d*x + c) - 320*a*b)*sinh(d*x + c)^6 + 1536*a*b*cosh(d*x + c)^4 + 12*(1155*a^2*cosh(d*x + c)
^6 - 1785*a^2*cosh(d*x + c)^4 + 1792*a*b*cosh(d*x + c)^3 + 693*a^2*cosh(d*x + c)^2 - 640*a*b*cosh(d*x + c) + 3
3*a^2)*sinh(d*x + c)^5 - 170*a^2*cosh(d*x + c)^3 + 12*(825*a^2*cosh(d*x + c)^7 - 1785*a^2*cosh(d*x + c)^5 + 22
40*a*b*cosh(d*x + c)^4 + 1155*a^2*cosh(d*x + c)^3 - 1600*a*b*cosh(d*x + c)^2 + 165*a^2*cosh(d*x + c) + 128*a*b
)*sinh(d*x + c)^4 - 768*a*b*cosh(d*x + c)^2 + 2*(2475*a^2*cosh(d*x + c)^8 - 7140*a^2*cosh(d*x + c)^6 + 10752*a
*b*cosh(d*x + c)^5 + 6930*a^2*cosh(d*x + c)^4 - 12800*a*b*cosh(d*x + c)^3 + 1980*a^2*cosh(d*x + c)^2 + 3072*a*
b*cosh(d*x + c) - 85*a^2)*sinh(d*x + c)^3 + 30*a^2*cosh(d*x + c) + 6*(275*a^2*cosh(d*x + c)^9 - 1020*a^2*cosh(
d*x + c)^7 + 1792*a*b*cosh(d*x + c)^6 + 1386*a^2*cosh(d*x + c)^5 - 3200*a*b*cosh(d*x + c)^4 + 660*a^2*cosh(d*x
+ c)^3 + 1536*a*b*cosh(d*x + c)^2 - 85*a^2*cosh(d*x + c) - 128*a*b)*sinh(d*x + c)^2 + 128*a*b - 3*((5*a^2 - 1
6*b^2)*cosh(d*x + c)^12 + 12*(5*a^2 - 16*b^2)*cosh(d*x + c)*sinh(d*x + c)^11 + (5*a^2 - 16*b^2)*sinh(d*x + c)^
12 - 6*(5*a^2 - 16*b^2)*cosh(d*x + c)^10 + 6*(11*(5*a^2 - 16*b^2)*cosh(d*x + c)^2 - 5*a^2 + 16*b^2)*sinh(d*x +
c)^10 + 20*(11*(5*a^2 - 16*b^2)*cosh(d*x + c)^3 - 3*(5*a^2 - 16*b^2)*cosh(d*x + c))*sinh(d*x + c)^9 + 15*(5*a
^2 - 16*b^2)*cosh(d*x + c)^8 + 15*(33*(5*a^2 - 16*b^2)*cosh(d*x + c)^4 - 18*(5*a^2 - 16*b^2)*cosh(d*x + c)^2 +
5*a^2 - 16*b^2)*sinh(d*x + c)^8 + 24*(33*(5*a^2 - 16*b^2)*cosh(d*x + c)^5 - 30*(5*a^2 - 16*b^2)*cosh(d*x + c)
^3 + 5*(5*a^2 - 16*b^2)*cosh(d*x + c))*sinh(d*x + c)^7 - 20*(5*a^2 - 16*b^2)*cosh(d*x + c)^6 + 4*(231*(5*a^2 -
16*b^2)*cosh(d*x + c)^6 - 315*(5*a^2 - 16*b^2)*cosh(d*x + c)^4 + 105*(5*a^2 - 16*b^2)*cosh(d*x + c)^2 - 25*a^
2 + 80*b^2)*sinh(d*x + c)^6 + 24*(33*(5*a^2 - 16*b^2)*cosh(d*x + c)^7 - 63*(5*a^2 - 16*b^2)*cosh(d*x + c)^5 +
35*(5*a^2 - 16*b^2)*cosh(d*x + c)^3 - 5*(5*a^2 - 16*b^2)*cosh(d*x + c))*sinh(d*x + c)^5 + 15*(5*a^2 - 16*b^2)*
cosh(d*x + c)^4 + 15*(33*(5*a^2 - 16*b^2)*cosh(d*x + c)^8 - 84*(5*a^2 - 16*b^2)*cosh(d*x + c)^6 + 70*(5*a^2 -
16*b^2)*cosh(d*x + c)^4 - 20*(5*a^2 - 16*b^2)*cosh(d*x + c)^2 + 5*a^2 - 16*b^2)*sinh(d*x + c)^4 + 20*(11*(5*a^
2 - 16*b^2)*cosh(d*x + c)^9 - 36*(5*a^2 - 16*b^2)*cosh(d*x + c)^7 + 42*(5*a^2 - 16*b^2)*cosh(d*x + c)^5 - 20*(
5*a^2 - 16*b^2)*cosh(d*x + c)^3 + 3*(5*a^2 - 16*b^2)*cosh(d*x + c))*sinh(d*x + c)^3 - 6*(5*a^2 - 16*b^2)*cosh(
d*x + c)^2 + 6*(11*(5*a^2 - 16*b^2)*cosh(d*x + c)^10 - 45*(5*a^2 - 16*b^2)*cosh(d*x + c)^8 + 70*(5*a^2 - 16*b^
2)*cosh(d*x + c)^6 - 50*(5*a^2 - 16*b^2)*cosh(d*x + c)^4 + 15*(5*a^2 - 16*b^2)*cosh(d*x + c)^2 - 5*a^2 + 16*b^
2)*sinh(d*x + c)^2 + 5*a^2 - 16*b^2 + 12*((5*a^2 - 16*b^2)*cosh(d*x + c)^11 - 5*(5*a^2 - 16*b^2)*cosh(d*x + c)
^9 + 10*(5*a^2 - 16*b^2)*cosh(d*x + c)^7 - 10*(5*a^2 - 16*b^2)*cosh(d*x + c)^5 + 5*(5*a^2 - 16*b^2)*cosh(d*x +
c)^3 - (5*a^2 - 16*b^2)*cosh(d*x + c))*sinh(d*x + c))*log(cosh(d*x + c) + sinh(d*x + c) + 1) + 3*((5*a^2 - 16
*b^2)*cosh(d*x + c)^12 + 12*(5*a^2 - 16*b^2)*cosh(d*x + c)*sinh(d*x + c)^11 + (5*a^2 - 16*b^2)*sinh(d*x + c)^1
2 - 6*(5*a^2 - 16*b^2)*cosh(d*x + c)^10 + 6*(11*(5*a^2 - 16*b^2)*cosh(d*x + c)^2 - 5*a^2 + 16*b^2)*sinh(d*x +
c)^10 + 20*(11*(5*a^2 - 16*b^2)*cosh(d*x + c)^3 - 3*(5*a^2 - 16*b^2)*cosh(d*x + c))*sinh(d*x + c)^9 + 15*(5*a^
2 - 16*b^2)*cosh(d*x + c)^8 + 15*(33*(5*a^2 - 16*b^2)*cosh(d*x + c)^4 - 18*(5*a^2 - 16*b^2)*cosh(d*x + c)^2 +
5*a^2 - 16*b^2)*sinh(d*x + c)^8 + 24*(33*(5*a^2 - 16*b^2)*cosh(d*x + c)^5 - 30*(5*a^2 - 16*b^2)*cosh(d*x + c)^
3 + 5*(5*a^2 - 16*b^2)*cosh(d*x + c))*sinh(d*x + c)^7 - 20*(5*a^2 - 16*b^2)*cosh(d*x + c)^6 + 4*(231*(5*a^2 -
16*b^2)*cosh(d*x + c)^6 - 315*(5*a^2 - 16*b^2)*cosh(d*x + c)^4 + 105*(5*a^2 - 16*b^2)*cosh(d*x + c)^2 - 25*a^2
+ 80*b^2)*sinh(d*x + c)^6 + 24*(33*(5*a^2 - 16*b^2)*cosh(d*x + c)^7 - 63*(5*a^2 - 16*b^2)*cosh(d*x + c)^5 + 3
5*(5*a^2 - 16*b^2)*cosh(d*x + c)^3 - 5*(5*a^2 - 16*b^2)*cosh(d*x + c))*sinh(d*x + c)^5 + 15*(5*a^2 - 16*b^2)*c
osh(d*x + c)^4 + 15*(33*(5*a^2 - 16*b^2)*cosh(d*x + c)^8 - 84*(5*a^2 - 16*b^2)*cosh(d*x + c)^6 + 70*(5*a^2 - 1
6*b^2)*cosh(d*x + c)^4 - 20*(5*a^2 - 16*b^2)*cosh(d*x + c)^2 + 5*a^2 - 16*b^2)*sinh(d*x + c)^4 + 20*(11*(5*a^2
- 16*b^2)*cosh(d*x + c)^9 - 36*(5*a^2 - 16*b^2)*cosh(d*x + c)^7 + 42*(5*a^2 - 16*b^2)*cosh(d*x + c)^5 - 20*(5
*a^2 - 16*b^2)*cosh(d*x + c)^3 + 3*(5*a^2 - 16*b^2)*cosh(d*x + c))*sinh(d*x + c)^3 - 6*(5*a^2 - 16*b^2)*cosh(d
*x + c)^2 + 6*(11*(5*a^2 - 16*b^2)*cosh(d*x + c)^10 - 45*(5*a^2 - 16*b^2)*cosh(d*x + c)^8 + 70*(5*a^2 - 16*b^2
)*cosh(d*x + c)^6 - 50*(5*a^2 - 16*b^2)*cosh(d*x + c)^4 + 15*(5*a^2 - 16*b^2)*cosh(d*x + c)^2 - 5*a^2 + 16*b^2
)*sinh(d*x + c)^2 + 5*a^2 - 16*b^2 + 12*((5*a^2 - 16*b^2)*cosh(d*x + c)^11 - 5*(5*a^2 - 16*b^2)*cosh(d*x + c)^
9 + 10*(5*a^2 - 16*b^2)*cosh(d*x + c)^7 - 10*(5*a^2 - 16*b^2)*cosh(d*x + c)^5 + 5*(5*a^2 - 16*b^2)*cosh(d*x +
c)^3 - (5*a^2 - 16*b^2)*cosh(d*x + c))*sinh(d*x + c))*log(cosh(d*x + c) + sinh(d*x + c) - 1) + 6*(55*a^2*cosh(
d*x + c)^10 - 255*a^2*cosh(d*x + c)^8 + 512*a*b*cosh(d*x + c)^7 + 462*a^2*cosh(d*x + c)^6 - 1280*a*b*cosh(d*x
+ c)^5 + 330*a^2*cosh(d*x + c)^4 + 1024*a*b*cosh(d*x + c)^3 - 85*a^2*cosh(d*x + c)^2 - 256*a*b*cosh(d*x + c) +
5*a^2)*sinh(d*x + c))/(d*cosh(d*x + c)^12 + 12*d*cosh(d*x + c)*sinh(d*x + c)^11 + d*sinh(d*x + c)^12 - 6*d*co
sh(d*x + c)^10 + 6*(11*d*cosh(d*x + c)^2 - d)*sinh(d*x + c)^10 + 20*(11*d*cosh(d*x + c)^3 - 3*d*cosh(d*x + c))
*sinh(d*x + c)^9 + 15*d*cosh(d*x + c)^8 + 15*(33*d*cosh(d*x + c)^4 - 18*d*cosh(d*x + c)^2 + d)*sinh(d*x + c)^8
+ 24*(33*d*cosh(d*x + c)^5 - 30*d*cosh(d*x + c)^3 + 5*d*cosh(d*x + c))*sinh(d*x + c)^7 - 20*d*cosh(d*x + c)^6
+ 4*(231*d*cosh(d*x + c)^6 - 315*d*cosh(d*x + c)^4 + 105*d*cosh(d*x + c)^2 - 5*d)*sinh(d*x + c)^6 + 24*(33*d*
cosh(d*x + c)^7 - 63*d*cosh(d*x + c)^5 + 35*d*cosh(d*x + c)^3 - 5*d*cosh(d*x + c))*sinh(d*x + c)^5 + 15*d*cosh
(d*x + c)^4 + 15*(33*d*cosh(d*x + c)^8 - 84*d*cosh(d*x + c)^6 + 70*d*cosh(d*x + c)^4 - 20*d*cosh(d*x + c)^2 +
d)*sinh(d*x + c)^4 + 20*(11*d*cosh(d*x + c)^9 - 36*d*cosh(d*x + c)^7 + 42*d*cosh(d*x + c)^5 - 20*d*cosh(d*x +
c)^3 + 3*d*cosh(d*x + c))*sinh(d*x + c)^3 - 6*d*cosh(d*x + c)^2 + 6*(11*d*cosh(d*x + c)^10 - 45*d*cosh(d*x + c
)^8 + 70*d*cosh(d*x + c)^6 - 50*d*cosh(d*x + c)^4 + 15*d*cosh(d*x + c)^2 - d)*sinh(d*x + c)^2 + 12*(d*cosh(d*x
+ c)^11 - 5*d*cosh(d*x + c)^9 + 10*d*cosh(d*x + c)^7 - 10*d*cosh(d*x + c)^5 + 5*d*cosh(d*x + c)^3 - d*cosh(d*
x + c))*sinh(d*x + c) + d)
________________________________________________________________________________________
Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(csch(d*x+c)**7*(a+b*sinh(d*x+c)**3)**2,x)
[Out]
Timed out
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Giac [A] time = 1.21541, size = 281, normalized size = 2.11 \begin{align*} \frac{{\left (5 \, a^{2} - 16 \, b^{2}\right )} \log \left (e^{\left (d x + c\right )} + 1\right )}{16 \, d} - \frac{{\left (5 \, a^{2} - 16 \, b^{2}\right )} \log \left ({\left | e^{\left (d x + c\right )} - 1 \right |}\right )}{16 \, d} - \frac{15 \, a^{2} e^{\left (11 \, d x + 11 \, c\right )} - 85 \, a^{2} e^{\left (9 \, d x + 9 \, c\right )} + 192 \, a b e^{\left (8 \, d x + 8 \, c\right )} + 198 \, a^{2} e^{\left (7 \, d x + 7 \, c\right )} - 640 \, a b e^{\left (6 \, d x + 6 \, c\right )} + 198 \, a^{2} e^{\left (5 \, d x + 5 \, c\right )} + 768 \, a b e^{\left (4 \, d x + 4 \, c\right )} - 85 \, a^{2} e^{\left (3 \, d x + 3 \, c\right )} - 384 \, a b e^{\left (2 \, d x + 2 \, c\right )} + 15 \, a^{2} e^{\left (d x + c\right )} + 64 \, a b}{24 \, d{\left (e^{\left (2 \, d x + 2 \, c\right )} - 1\right )}^{6}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(csch(d*x+c)^7*(a+b*sinh(d*x+c)^3)^2,x, algorithm="giac")
[Out]
1/16*(5*a^2 - 16*b^2)*log(e^(d*x + c) + 1)/d - 1/16*(5*a^2 - 16*b^2)*log(abs(e^(d*x + c) - 1))/d - 1/24*(15*a^
2*e^(11*d*x + 11*c) - 85*a^2*e^(9*d*x + 9*c) + 192*a*b*e^(8*d*x + 8*c) + 198*a^2*e^(7*d*x + 7*c) - 640*a*b*e^(
6*d*x + 6*c) + 198*a^2*e^(5*d*x + 5*c) + 768*a*b*e^(4*d*x + 4*c) - 85*a^2*e^(3*d*x + 3*c) - 384*a*b*e^(2*d*x +
2*c) + 15*a^2*e^(d*x + c) + 64*a*b)/(d*(e^(2*d*x + 2*c) - 1)^6)