3.170 \(\int \text{csch}^7(c+d x) (a+b \sinh ^3(c+d x))^3 \, dx\)
Optimal. Leaf size=166 \[ -\frac{a^2 b \coth ^3(c+d x)}{d}+\frac{3 a^2 b \coth (c+d x)}{d}+\frac{5 a^3 \tanh ^{-1}(\cosh (c+d x))}{16 d}-\frac{a^3 \coth (c+d x) \text{csch}^5(c+d x)}{6 d}+\frac{5 a^3 \coth (c+d x) \text{csch}^3(c+d x)}{24 d}-\frac{5 a^3 \coth (c+d x) \text{csch}(c+d x)}{16 d}-\frac{3 a b^2 \tanh ^{-1}(\cosh (c+d x))}{d}+\frac{b^3 \sinh (c+d x) \cosh (c+d x)}{2 d}-\frac{b^3 x}{2} \]
[Out]
-(b^3*x)/2 + (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)
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Rubi [A] time = 0.197102, antiderivative size = 166, normalized size of antiderivative = 1.,
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
= {3220, 3770, 3767, 3768, 2635, 8} \[ -\frac{a^2 b \coth ^3(c+d x)}{d}+\frac{3 a^2 b \coth (c+d x)}{d}+\frac{5 a^3 \tanh ^{-1}(\cosh (c+d x))}{16 d}-\frac{a^3 \coth (c+d x) \text{csch}^5(c+d x)}{6 d}+\frac{5 a^3 \coth (c+d x) \text{csch}^3(c+d x)}{24 d}-\frac{5 a^3 \coth (c+d x) \text{csch}(c+d x)}{16 d}-\frac{3 a b^2 \tanh ^{-1}(\cosh (c+d x))}{d}+\frac{b^3 \sinh (c+d x) \cosh (c+d x)}{2 d}-\frac{b^3 x}{2} \]
Antiderivative was successfully verified.
[In]
Int[Csch[c + d*x]^7*(a + b*Sinh[c + d*x]^3)^3,x]
[Out]
-(b^3*x)/2 + (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 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]
Rule 2635
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] && Integer
Q[2*n]
Rule 8
Int[a_, x_Symbol] :> Simp[a*x, x] /; FreeQ[a, 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 ) \operatorname{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 = 1.55176, size = 236, normalized size = 1.42 \[ -\frac{-384 a^2 b \tanh \left (\frac{1}{2} (c+d x)\right )-384 a^2 b \coth \left (\frac{1}{2} (c+d x)\right )-384 a^2 b \sinh ^4\left (\frac{1}{2} (c+d x)\right ) \text{csch}^3(c+d x)-6 a^2 \text{csch}^4\left (\frac{1}{2} (c+d x)\right ) (a-4 b \sinh (c+d x))+a^3 \text{csch}^6\left (\frac{1}{2} (c+d x)\right )+30 a^3 \text{csch}^2\left (\frac{1}{2} (c+d x)\right )+a^3 \text{sech}^6\left (\frac{1}{2} (c+d x)\right )+6 a^3 \text{sech}^4\left (\frac{1}{2} (c+d x)\right )+30 a^3 \text{sech}^2\left (\frac{1}{2} (c+d x)\right )+120 a^3 \log \left (\tanh \left (\frac{1}{2} (c+d x)\right )\right )-1152 a b^2 \log \left (\tanh \left (\frac{1}{2} (c+d x)\right )\right )-96 b^3 \sinh (2 (c+d x))+192 b^3 c+192 b^3 d x}{384 d} \]
Antiderivative was successfully verified.
[In]
Integrate[Csch[c + d*x]^7*(a + b*Sinh[c + d*x]^3)^3,x]
[Out]
-(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*Sec
h[(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])/(384*d)
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Maple [A] time = 0.083, size = 119, normalized size = 0.7 \begin{align*}{\frac{1}{d} \left ({a}^{3} \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 ) +3\,{a}^{2}b \left ( 2/3-1/3\, \left ({\rm csch} \left (dx+c\right ) \right ) ^{2} \right ){\rm coth} \left (dx+c\right )-6\,a{b}^{2}{\it Artanh} \left ({{\rm e}^{dx+c}} \right ) +{b}^{3} \left ({\frac{\cosh \left ( dx+c \right ) \sinh \left ( dx+c \right ) }{2}}-{\frac{dx}{2}}-{\frac{c}{2}} \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)^3,x)
[Out]
1/d*(a^3*((-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)))+3*a^2*
b*(2/3-1/3*csch(d*x+c)^2)*coth(d*x+c)-6*a*b^2*arctanh(exp(d*x+c))+b^3*(1/2*cosh(d*x+c)*sinh(d*x+c)-1/2*d*x-1/2
*c))
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Maxima [B] time = 1.08364, size = 479, normalized size = 2.89 \begin{align*} -\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{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)^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)))
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Fricas [B] time = 2.78234, size = 16188, normalized size = 97.52 \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)^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 + 75*(5*a^3 - 48*a*b^2)*cosh(d*x + c))*sinh(d*x + c)^9 - 20*(5*
a^3 - 48*a*b^2)*cosh(d*x + c)^8 + (3003*(5*a^3 - 48*a*b^2)*cosh(d*x + c)^6 - 2970*(5*a^3 - 48*a*b^2)*cosh(d*x
+ c)^4 - 100*a^3 + 960*a*b^2 + 675*(5*a^3 - 48*a*b^2)*cosh(d*x + c)^2)*sinh(d*x + c)^8 + 8*(429*(5*a^3 - 48*a*
b^2)*cosh(d*x + c)^7 - 594*(5*a^3 - 48*a*b^2)*cosh(d*x + c)^5 + 225*(5*a^3 - 48*a*b^2)*cosh(d*x + c)^3 - 20*(5
*a^3 - 48*a*b^2)*cosh(d*x + c))*sinh(d*x + c)^7 + 15*(5*a^3 - 48*a*b^2)*cosh(d*x + c)^6 + (3003*(5*a^3 - 48*a*
b^2)*cosh(d*x + c)^8 - 5544*(5*a^3 - 48*a*b^2)*cosh(d*x + c)^6 + 3150*(5*a^3 - 48*a*b^2)*cosh(d*x + c)^4 + 75*
a^3 - 720*a*b^2 - 560*(5*a^3 - 48*a*b^2)*cosh(d*x + c)^2)*sinh(d*x + c)^6 + 2*(1001*(5*a^3 - 48*a*b^2)*cosh(d*
x + c)^9 - 2376*(5*a^3 - 48*a*b^2)*cosh(d*x + c)^7 + 1890*(5*a^3 - 48*a*b^2)*cosh(d*x + c)^5 - 560*(5*a^3 - 48
*a*b^2)*cosh(d*x + c)^3 + 45*(5*a^3 - 48*a*b^2)*cosh(d*x + c))*sinh(d*x + c)^5 - 6*(5*a^3 - 48*a*b^2)*cosh(d*x
+ c)^4 + (1001*(5*a^3 - 48*a*b^2)*cosh(d*x + c)^10 - 2970*(5*a^3 - 48*a*b^2)*cosh(d*x + c)^8 + 3150*(5*a^3 -
48*a*b^2)*cosh(d*x + c)^6 - 1400*(5*a^3 - 48*a*b^2)*cosh(d*x + c)^4 - 30*a^3 + 288*a*b^2 + 225*(5*a^3 - 48*a*b
^2)*cosh(d*x + c)^2)*sinh(d*x + c)^4 + 4*(91*(5*a^3 - 48*a*b^2)*cosh(d*x + c)^11 - 330*(5*a^3 - 48*a*b^2)*cosh
(d*x + c)^9 + 450*(5*a^3 - 48*a*b^2)*cosh(d*x + c)^7 - 280*(5*a^3 - 48*a*b^2)*cosh(d*x + c)^5 + 75*(5*a^3 - 48
*a*b^2)*cosh(d*x + c)^3 - 6*(5*a^3 - 48*a*b^2)*cosh(d*x + c))*sinh(d*x + c)^3 + (5*a^3 - 48*a*b^2)*cosh(d*x +
c)^2 + (91*(5*a^3 - 48*a*b^2)*cosh(d*x + c)^12 - 396*(5*a^3 - 48*a*b^2)*cosh(d*x + c)^10 + 675*(5*a^3 - 48*a*b
^2)*cosh(d*x + c)^8 - 560*(5*a^3 - 48*a*b^2)*cosh(d*x + c)^6 + 225*(5*a^3 - 48*a*b^2)*cosh(d*x + c)^4 + 5*a^3
- 48*a*b^2 - 36*(5*a^3 - 48*a*b^2)*cosh(d*x + c)^2)*sinh(d*x + c)^2 + 2*(7*(5*a^3 - 48*a*b^2)*cosh(d*x + c)^13
- 36*(5*a^3 - 48*a*b^2)*cosh(d*x + c)^11 + 75*(5*a^3 - 48*a*b^2)*cosh(d*x + c)^9 - 80*(5*a^3 - 48*a*b^2)*cosh
(d*x + c)^7 + 45*(5*a^3 - 48*a*b^2)*cosh(d*x + c)^5 - 12*(5*a^3 - 48*a*b^2)*cosh(d*x + c)^3 + (5*a^3 - 48*a*b^
2)*cosh(d*x + c))*sinh(d*x + c))*log(cosh(d*x + c) + sinh(d*x + c) + 1) - 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 - 4
8*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 - 4
8*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)*c
osh(d*x + c)^3 + 75*(5*a^3 - 48*a*b^2)*cosh(d*x + c))*sinh(d*x + c)^9 - 20*(5*a^3 - 48*a*b^2)*cosh(d*x + c)^8
+ (3003*(5*a^3 - 48*a*b^2)*cosh(d*x + c)^6 - 2970*(5*a^3 - 48*a*b^2)*cosh(d*x + c)^4 - 100*a^3 + 960*a*b^2 + 6
75*(5*a^3 - 48*a*b^2)*cosh(d*x + c)^2)*sinh(d*x + c)^8 + 8*(429*(5*a^3 - 48*a*b^2)*cosh(d*x + c)^7 - 594*(5*a^
3 - 48*a*b^2)*cosh(d*x + c)^5 + 225*(5*a^3 - 48*a*b^2)*cosh(d*x + c)^3 - 20*(5*a^3 - 48*a*b^2)*cosh(d*x + c))*
sinh(d*x + c)^7 + 15*(5*a^3 - 48*a*b^2)*cosh(d*x + c)^6 + (3003*(5*a^3 - 48*a*b^2)*cosh(d*x + c)^8 - 5544*(5*a
^3 - 48*a*b^2)*cosh(d*x + c)^6 + 3150*(5*a^3 - 48*a*b^2)*cosh(d*x + c)^4 + 75*a^3 - 720*a*b^2 - 560*(5*a^3 - 4
8*a*b^2)*cosh(d*x + c)^2)*sinh(d*x + c)^6 + 2*(1001*(5*a^3 - 48*a*b^2)*cosh(d*x + c)^9 - 2376*(5*a^3 - 48*a*b^
2)*cosh(d*x + c)^7 + 1890*(5*a^3 - 48*a*b^2)*cosh(d*x + c)^5 - 560*(5*a^3 - 48*a*b^2)*cosh(d*x + c)^3 + 45*(5*
a^3 - 48*a*b^2)*cosh(d*x + c))*sinh(d*x + c)^5 - 6*(5*a^3 - 48*a*b^2)*cosh(d*x + c)^4 + (1001*(5*a^3 - 48*a*b^
2)*cosh(d*x + c)^10 - 2970*(5*a^3 - 48*a*b^2)*cosh(d*x + c)^8 + 3150*(5*a^3 - 48*a*b^2)*cosh(d*x + c)^6 - 1400
*(5*a^3 - 48*a*b^2)*cosh(d*x + c)^4 - 30*a^3 + 288*a*b^2 + 225*(5*a^3 - 48*a*b^2)*cosh(d*x + c)^2)*sinh(d*x +
c)^4 + 4*(91*(5*a^3 - 48*a*b^2)*cosh(d*x + c)^11 - 330*(5*a^3 - 48*a*b^2)*cosh(d*x + c)^9 + 450*(5*a^3 - 48*a*
b^2)*cosh(d*x + c)^7 - 280*(5*a^3 - 48*a*b^2)*cosh(d*x + c)^5 + 75*(5*a^3 - 48*a*b^2)*cosh(d*x + c)^3 - 6*(5*a
^3 - 48*a*b^2)*cosh(d*x + c))*sinh(d*x + c)^3 + (5*a^3 - 48*a*b^2)*cosh(d*x + c)^2 + (91*(5*a^3 - 48*a*b^2)*co
sh(d*x + c)^12 - 396*(5*a^3 - 48*a*b^2)*cosh(d*x + c)^10 + 675*(5*a^3 - 48*a*b^2)*cosh(d*x + c)^8 - 560*(5*a^3
- 48*a*b^2)*cosh(d*x + c)^6 + 225*(5*a^3 - 48*a*b^2)*cosh(d*x + c)^4 + 5*a^3 - 48*a*b^2 - 36*(5*a^3 - 48*a*b^
2)*cosh(d*x + c)^2)*sinh(d*x + c)^2 + 2*(7*(5*a^3 - 48*a*b^2)*cosh(d*x + c)^13 - 36*(5*a^3 - 48*a*b^2)*cosh(d*
x + c)^11 + 75*(5*a^3 - 48*a*b^2)*cosh(d*x + c)^9 - 80*(5*a^3 - 48*a*b^2)*cosh(d*x + c)^7 + 45*(5*a^3 - 48*a*b
^2)*cosh(d*x + c)^5 - 12*(5*a^3 - 48*a*b^2)*cosh(d*x + c)^3 + (5*a^3 - 48*a*b^2)*cosh(d*x + c))*sinh(d*x + c))
*log(cosh(d*x + c) + sinh(d*x + c) - 1) + 2*(48*b^3*cosh(d*x + c)^15 - 195*a^3*cosh(d*x + c)^12 - 84*(2*b^3*d*
x + 3*b^3)*cosh(d*x + c)^13 + 935*a^3*cosh(d*x + c)^10 + 72*(12*b^3*d*x + 7*b^3)*cosh(d*x + c)^11 - 1782*a^3*c
osh(d*x + c)^8 - 60*(30*b^3*d*x + 48*a^2*b + 7*b^3)*cosh(d*x + c)^9 - 1386*a^3*cosh(d*x + c)^6 + 1920*(b^3*d*x
+ 4*a^2*b)*cosh(d*x + c)^7 + 425*a^3*cosh(d*x + c)^4 - 36*(30*b^3*d*x + 192*a^2*b - 7*b^3)*cosh(d*x + c)^5 -
45*a^3*cosh(d*x + c)^2 + 24*(12*b^3*d*x + 96*a^2*b - 7*b^3)*cosh(d*x + c)^3 - 12*(2*b^3*d*x + 16*a^2*b - 3*b^3
)*cosh(d*x + c))*sinh(d*x + c))/(d*cosh(d*x + c)^14 + 14*d*cosh(d*x + c)*sinh(d*x + c)^13 + d*sinh(d*x + c)^14
- 6*d*cosh(d*x + c)^12 + (91*d*cosh(d*x + c)^2 - 6*d)*sinh(d*x + c)^12 + 4*(91*d*cosh(d*x + c)^3 - 18*d*cosh(
d*x + c))*sinh(d*x + c)^11 + 15*d*cosh(d*x + c)^10 + (1001*d*cosh(d*x + c)^4 - 396*d*cosh(d*x + c)^2 + 15*d)*s
inh(d*x + c)^10 + 2*(1001*d*cosh(d*x + c)^5 - 660*d*cosh(d*x + c)^3 + 75*d*cosh(d*x + c))*sinh(d*x + c)^9 - 20
*d*cosh(d*x + c)^8 + (3003*d*cosh(d*x + c)^6 - 2970*d*cosh(d*x + c)^4 + 675*d*cosh(d*x + c)^2 - 20*d)*sinh(d*x
+ c)^8 + 8*(429*d*cosh(d*x + c)^7 - 594*d*cosh(d*x + c)^5 + 225*d*cosh(d*x + c)^3 - 20*d*cosh(d*x + c))*sinh(
d*x + c)^7 + 15*d*cosh(d*x + c)^6 + (3003*d*cosh(d*x + c)^8 - 5544*d*cosh(d*x + c)^6 + 3150*d*cosh(d*x + c)^4
- 560*d*cosh(d*x + c)^2 + 15*d)*sinh(d*x + c)^6 + 2*(1001*d*cosh(d*x + c)^9 - 2376*d*cosh(d*x + c)^7 + 1890*d*
cosh(d*x + c)^5 - 560*d*cosh(d*x + c)^3 + 45*d*cosh(d*x + c))*sinh(d*x + c)^5 - 6*d*cosh(d*x + c)^4 + (1001*d*
cosh(d*x + c)^10 - 2970*d*cosh(d*x + c)^8 + 3150*d*cosh(d*x + c)^6 - 1400*d*cosh(d*x + c)^4 + 225*d*cosh(d*x +
c)^2 - 6*d)*sinh(d*x + c)^4 + 4*(91*d*cosh(d*x + c)^11 - 330*d*cosh(d*x + c)^9 + 450*d*cosh(d*x + c)^7 - 280*
d*cosh(d*x + c)^5 + 75*d*cosh(d*x + c)^3 - 6*d*cosh(d*x + c))*sinh(d*x + c)^3 + d*cosh(d*x + c)^2 + (91*d*cosh
(d*x + c)^12 - 396*d*cosh(d*x + c)^10 + 675*d*cosh(d*x + c)^8 - 560*d*cosh(d*x + c)^6 + 225*d*cosh(d*x + c)^4
- 36*d*cosh(d*x + c)^2 + d)*sinh(d*x + c)^2 + 2*(7*d*cosh(d*x + c)^13 - 36*d*cosh(d*x + c)^11 + 75*d*cosh(d*x
+ c)^9 - 80*d*cosh(d*x + c)^7 + 45*d*cosh(d*x + c)^5 - 12*d*cosh(d*x + c)^3 + d*cosh(d*x + c))*sinh(d*x + c))
________________________________________________________________________________________
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)**3,x)
[Out]
Timed out
________________________________________________________________________________________
Giac [B] time = 1.46106, size = 455, normalized size = 2.74 \begin{align*} -\frac{{\left (d x + c\right )} b^{3}}{2 \, d} + \frac{b^{3} e^{\left (2 \, d x + 2 \, c\right )}}{8 \, d} + \frac{{\left (5 \, a^{3} - 48 \, a b^{2}\right )} \log \left (e^{\left (d x + c\right )} + 1\right )}{16 \, d} - \frac{{\left (5 \, a^{3} - 48 \, a b^{2}\right )} \log \left ({\left | e^{\left (d x + c\right )} - 1 \right |}\right )}{16 \, d} - \frac{{\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 )}}{24 \, d{\left (e^{\left (d x + c\right )} + 1\right )}^{6}{\left (e^{\left (d x + 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)^3,x, algorithm="giac")
[Out]
-1/2*(d*x + c)*b^3/d + 1/8*b^3*e^(2*d*x + 2*c)/d + 1/16*(5*a^3 - 48*a*b^2)*log(e^(d*x + c) + 1)/d - 1/16*(5*a^
3 - 48*a*b^2)*log(abs(e^(d*x + c) - 1))/d - 1/24*(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)/(d*(e^(d*x + c) + 1)^6*(e^(d*x + c) - 1)^6)