3.195 \(\int \text{csch}^7(c+d x) (a+b \sinh ^4(c+d x)) \, dx\)
Optimal. Leaf size=92 \[ \frac{(5 a+8 b) \tanh ^{-1}(\cosh (c+d x))}{16 d}-\frac{(5 a+8 b) \coth (c+d x) \text{csch}(c+d x)}{16 d}-\frac{a \coth (c+d x) \text{csch}^5(c+d x)}{6 d}+\frac{5 a \coth (c+d x) \text{csch}^3(c+d x)}{24 d} \]
[Out]
((5*a + 8*b)*ArcTanh[Cosh[c + d*x]])/(16*d) - ((5*a + 8*b)*Coth[c + d*x]*Csch[c + d*x])/(16*d) + (5*a*Coth[c +
d*x]*Csch[c + d*x]^3)/(24*d) - (a*Coth[c + d*x]*Csch[c + d*x]^5)/(6*d)
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Rubi [A] time = 0.0837145, antiderivative size = 92, normalized size of antiderivative = 1.,
number of steps used = 5, number of rules used = 5, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.238, Rules used =
{3215, 1157, 385, 199, 206} \[ \frac{(5 a+8 b) \tanh ^{-1}(\cosh (c+d x))}{16 d}-\frac{(5 a+8 b) \coth (c+d x) \text{csch}(c+d x)}{16 d}-\frac{a \coth (c+d x) \text{csch}^5(c+d x)}{6 d}+\frac{5 a \coth (c+d x) \text{csch}^3(c+d x)}{24 d} \]
Antiderivative was successfully verified.
[In]
Int[Csch[c + d*x]^7*(a + b*Sinh[c + d*x]^4),x]
[Out]
((5*a + 8*b)*ArcTanh[Cosh[c + d*x]])/(16*d) - ((5*a + 8*b)*Coth[c + d*x]*Csch[c + d*x])/(16*d) + (5*a*Coth[c +
d*x]*Csch[c + d*x]^3)/(24*d) - (a*Coth[c + d*x]*Csch[c + d*x]^5)/(6*d)
Rule 3215
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]
Rule 1157
Int[((d_) + (e_.)*(x_)^2)^(q_)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_.), x_Symbol] :> With[{Qx = PolynomialQ
uotient[(a + b*x^2 + c*x^4)^p, d + e*x^2, x], R = Coeff[PolynomialRemainder[(a + b*x^2 + c*x^4)^p, d + e*x^2,
x], x, 0]}, -Simp[(R*x*(d + e*x^2)^(q + 1))/(2*d*(q + 1)), x] + Dist[1/(2*d*(q + 1)), Int[(d + e*x^2)^(q + 1)*
ExpandToSum[2*d*(q + 1)*Qx + R*(2*q + 3), x], x], x]] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && N
eQ[c*d^2 - b*d*e + a*e^2, 0] && IGtQ[p, 0] && LtQ[q, -1]
Rule 385
Int[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_)), x_Symbol] :> -Simp[((b*c - a*d)*x*(a + b*x^n)^(p +
1))/(a*b*n*(p + 1)), x] - Dist[(a*d - b*c*(n*(p + 1) + 1))/(a*b*n*(p + 1)), Int[(a + b*x^n)^(p + 1), x], x] /
; FreeQ[{a, b, c, d, n, p}, x] && NeQ[b*c - a*d, 0] && (LtQ[p, -1] || ILtQ[1/n + p, 0])
Rule 199
Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> -Simp[(x*(a + b*x^n)^(p + 1))/(a*n*(p + 1)), x] + Dist[(n*(p +
1) + 1)/(a*n*(p + 1)), Int[(a + b*x^n)^(p + 1), x], x] /; FreeQ[{a, b}, x] && IGtQ[n, 0] && LtQ[p, -1] && (In
tegerQ[2*p] || (n == 2 && IntegerQ[4*p]) || (n == 2 && IntegerQ[3*p]) || Denominator[p + 1/n] < Denominator[p]
)
Rule 206
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTanh[(Rt[-b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[-b, 2]), x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])
Rubi steps
\begin{align*} \int \text{csch}^7(c+d x) \left (a+b \sinh ^4(c+d x)\right ) \, dx &=\frac{\operatorname{Subst}\left (\int \frac{a+b-2 b x^2+b x^4}{\left (1-x^2\right )^4} \, dx,x,\cosh (c+d x)\right )}{d}\\ &=-\frac{a \coth (c+d x) \text{csch}^5(c+d x)}{6 d}-\frac{\operatorname{Subst}\left (\int \frac{-5 a-6 b+6 b x^2}{\left (1-x^2\right )^3} \, dx,x,\cosh (c+d x)\right )}{6 d}\\ &=\frac{5 a \coth (c+d x) \text{csch}^3(c+d x)}{24 d}-\frac{a \coth (c+d x) \text{csch}^5(c+d x)}{6 d}+\frac{(5 a+8 b) \operatorname{Subst}\left (\int \frac{1}{\left (1-x^2\right )^2} \, dx,x,\cosh (c+d x)\right )}{8 d}\\ &=-\frac{(5 a+8 b) \coth (c+d x) \text{csch}(c+d x)}{16 d}+\frac{5 a \coth (c+d x) \text{csch}^3(c+d x)}{24 d}-\frac{a \coth (c+d x) \text{csch}^5(c+d x)}{6 d}+\frac{(5 a+8 b) \operatorname{Subst}\left (\int \frac{1}{1-x^2} \, dx,x,\cosh (c+d x)\right )}{16 d}\\ &=\frac{(5 a+8 b) \tanh ^{-1}(\cosh (c+d x))}{16 d}-\frac{(5 a+8 b) \coth (c+d x) \text{csch}(c+d x)}{16 d}+\frac{5 a \coth (c+d x) \text{csch}^3(c+d x)}{24 d}-\frac{a \coth (c+d x) \text{csch}^5(c+d x)}{6 d}\\ \end{align*}
Mathematica [B] time = 0.035123, size = 199, normalized size = 2.16 \[ -\frac{a \text{csch}^6\left (\frac{1}{2} (c+d x)\right )}{384 d}+\frac{a \text{csch}^4\left (\frac{1}{2} (c+d x)\right )}{64 d}-\frac{5 a \text{csch}^2\left (\frac{1}{2} (c+d x)\right )}{64 d}-\frac{a \text{sech}^6\left (\frac{1}{2} (c+d x)\right )}{384 d}-\frac{a \text{sech}^4\left (\frac{1}{2} (c+d x)\right )}{64 d}-\frac{5 a \text{sech}^2\left (\frac{1}{2} (c+d x)\right )}{64 d}-\frac{5 a \log \left (\tanh \left (\frac{1}{2} (c+d x)\right )\right )}{16 d}-\frac{b \text{csch}^2\left (\frac{1}{2} (c+d x)\right )}{8 d}-\frac{b \text{sech}^2\left (\frac{1}{2} (c+d x)\right )}{8 d}-\frac{b \log \left (\tanh \left (\frac{1}{2} (c+d x)\right )\right )}{2 d} \]
Antiderivative was successfully verified.
[In]
Integrate[Csch[c + d*x]^7*(a + b*Sinh[c + d*x]^4),x]
[Out]
(-5*a*Csch[(c + d*x)/2]^2)/(64*d) - (b*Csch[(c + d*x)/2]^2)/(8*d) + (a*Csch[(c + d*x)/2]^4)/(64*d) - (a*Csch[(
c + d*x)/2]^6)/(384*d) - (5*a*Log[Tanh[(c + d*x)/2]])/(16*d) - (b*Log[Tanh[(c + d*x)/2]])/(2*d) - (5*a*Sech[(c
+ d*x)/2]^2)/(64*d) - (b*Sech[(c + d*x)/2]^2)/(8*d) - (a*Sech[(c + d*x)/2]^4)/(64*d) - (a*Sech[(c + d*x)/2]^6
)/(384*d)
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Maple [A] time = 0.041, size = 78, normalized size = 0.9 \begin{align*}{\frac{1}{d} \left ( a \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 ) +b \left ( -{\frac{{\rm csch} \left (dx+c\right ){\rm coth} \left (dx+c\right )}{2}}+{\it Artanh} \left ({{\rm e}^{dx+c}} \right ) \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)^4),x)
[Out]
1/d*(a*((-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)))+b*(-1/2*
csch(d*x+c)*coth(d*x+c)+arctanh(exp(d*x+c))))
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Maxima [B] time = 1.05621, size = 362, normalized size = 3.93 \begin{align*} \frac{1}{48} \, a{\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 )} + \frac{1}{2} \, b{\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} + \frac{2 \,{\left (e^{\left (-d x - c\right )} + e^{\left (-3 \, d x - 3 \, c\right )}\right )}}{d{\left (2 \, e^{\left (-2 \, d x - 2 \, c\right )} - e^{\left (-4 \, d x - 4 \, 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)^4),x, algorithm="maxima")
[Out]
1/48*a*(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) + 1
98*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 - 12*
c) - 1))) + 1/2*b*(log(e^(-d*x - c) + 1)/d - log(e^(-d*x - c) - 1)/d + 2*(e^(-d*x - c) + e^(-3*d*x - 3*c))/(d*
(2*e^(-2*d*x - 2*c) - e^(-4*d*x - 4*c) - 1)))
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Fricas [B] time = 1.88138, size = 8357, normalized size = 90.84 \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)^4),x, algorithm="fricas")
[Out]
-1/48*(6*(5*a + 8*b)*cosh(d*x + c)^11 + 66*(5*a + 8*b)*cosh(d*x + c)*sinh(d*x + c)^10 + 6*(5*a + 8*b)*sinh(d*x
+ c)^11 - 2*(85*a + 72*b)*cosh(d*x + c)^9 + 2*(165*(5*a + 8*b)*cosh(d*x + c)^2 - 85*a - 72*b)*sinh(d*x + c)^9
+ 18*(55*(5*a + 8*b)*cosh(d*x + c)^3 - (85*a + 72*b)*cosh(d*x + c))*sinh(d*x + c)^8 + 12*(33*a + 8*b)*cosh(d*
x + c)^7 + 12*(165*(5*a + 8*b)*cosh(d*x + c)^4 - 6*(85*a + 72*b)*cosh(d*x + c)^2 + 33*a + 8*b)*sinh(d*x + c)^7
+ 84*(33*(5*a + 8*b)*cosh(d*x + c)^5 - 2*(85*a + 72*b)*cosh(d*x + c)^3 + (33*a + 8*b)*cosh(d*x + c))*sinh(d*x
+ c)^6 + 12*(33*a + 8*b)*cosh(d*x + c)^5 + 12*(231*(5*a + 8*b)*cosh(d*x + c)^6 - 21*(85*a + 72*b)*cosh(d*x +
c)^4 + 21*(33*a + 8*b)*cosh(d*x + c)^2 + 33*a + 8*b)*sinh(d*x + c)^5 + 12*(165*(5*a + 8*b)*cosh(d*x + c)^7 - 2
1*(85*a + 72*b)*cosh(d*x + c)^5 + 35*(33*a + 8*b)*cosh(d*x + c)^3 + 5*(33*a + 8*b)*cosh(d*x + c))*sinh(d*x + c
)^4 - 2*(85*a + 72*b)*cosh(d*x + c)^3 + 2*(495*(5*a + 8*b)*cosh(d*x + c)^8 - 84*(85*a + 72*b)*cosh(d*x + c)^6
+ 210*(33*a + 8*b)*cosh(d*x + c)^4 + 60*(33*a + 8*b)*cosh(d*x + c)^2 - 85*a - 72*b)*sinh(d*x + c)^3 + 6*(55*(5
*a + 8*b)*cosh(d*x + c)^9 - 12*(85*a + 72*b)*cosh(d*x + c)^7 + 42*(33*a + 8*b)*cosh(d*x + c)^5 + 20*(33*a + 8*
b)*cosh(d*x + c)^3 - (85*a + 72*b)*cosh(d*x + c))*sinh(d*x + c)^2 + 6*(5*a + 8*b)*cosh(d*x + c) - 3*((5*a + 8*
b)*cosh(d*x + c)^12 + 12*(5*a + 8*b)*cosh(d*x + c)*sinh(d*x + c)^11 + (5*a + 8*b)*sinh(d*x + c)^12 - 6*(5*a +
8*b)*cosh(d*x + c)^10 + 6*(11*(5*a + 8*b)*cosh(d*x + c)^2 - 5*a - 8*b)*sinh(d*x + c)^10 + 20*(11*(5*a + 8*b)*c
osh(d*x + c)^3 - 3*(5*a + 8*b)*cosh(d*x + c))*sinh(d*x + c)^9 + 15*(5*a + 8*b)*cosh(d*x + c)^8 + 15*(33*(5*a +
8*b)*cosh(d*x + c)^4 - 18*(5*a + 8*b)*cosh(d*x + c)^2 + 5*a + 8*b)*sinh(d*x + c)^8 + 24*(33*(5*a + 8*b)*cosh(
d*x + c)^5 - 30*(5*a + 8*b)*cosh(d*x + c)^3 + 5*(5*a + 8*b)*cosh(d*x + c))*sinh(d*x + c)^7 - 20*(5*a + 8*b)*co
sh(d*x + c)^6 + 4*(231*(5*a + 8*b)*cosh(d*x + c)^6 - 315*(5*a + 8*b)*cosh(d*x + c)^4 + 105*(5*a + 8*b)*cosh(d*
x + c)^2 - 25*a - 40*b)*sinh(d*x + c)^6 + 24*(33*(5*a + 8*b)*cosh(d*x + c)^7 - 63*(5*a + 8*b)*cosh(d*x + c)^5
+ 35*(5*a + 8*b)*cosh(d*x + c)^3 - 5*(5*a + 8*b)*cosh(d*x + c))*sinh(d*x + c)^5 + 15*(5*a + 8*b)*cosh(d*x + c)
^4 + 15*(33*(5*a + 8*b)*cosh(d*x + c)^8 - 84*(5*a + 8*b)*cosh(d*x + c)^6 + 70*(5*a + 8*b)*cosh(d*x + c)^4 - 20
*(5*a + 8*b)*cosh(d*x + c)^2 + 5*a + 8*b)*sinh(d*x + c)^4 + 20*(11*(5*a + 8*b)*cosh(d*x + c)^9 - 36*(5*a + 8*b
)*cosh(d*x + c)^7 + 42*(5*a + 8*b)*cosh(d*x + c)^5 - 20*(5*a + 8*b)*cosh(d*x + c)^3 + 3*(5*a + 8*b)*cosh(d*x +
c))*sinh(d*x + c)^3 - 6*(5*a + 8*b)*cosh(d*x + c)^2 + 6*(11*(5*a + 8*b)*cosh(d*x + c)^10 - 45*(5*a + 8*b)*cos
h(d*x + c)^8 + 70*(5*a + 8*b)*cosh(d*x + c)^6 - 50*(5*a + 8*b)*cosh(d*x + c)^4 + 15*(5*a + 8*b)*cosh(d*x + c)^
2 - 5*a - 8*b)*sinh(d*x + c)^2 + 12*((5*a + 8*b)*cosh(d*x + c)^11 - 5*(5*a + 8*b)*cosh(d*x + c)^9 + 10*(5*a +
8*b)*cosh(d*x + c)^7 - 10*(5*a + 8*b)*cosh(d*x + c)^5 + 5*(5*a + 8*b)*cosh(d*x + c)^3 - (5*a + 8*b)*cosh(d*x +
c))*sinh(d*x + c) + 5*a + 8*b)*log(cosh(d*x + c) + sinh(d*x + c) + 1) + 3*((5*a + 8*b)*cosh(d*x + c)^12 + 12*
(5*a + 8*b)*cosh(d*x + c)*sinh(d*x + c)^11 + (5*a + 8*b)*sinh(d*x + c)^12 - 6*(5*a + 8*b)*cosh(d*x + c)^10 + 6
*(11*(5*a + 8*b)*cosh(d*x + c)^2 - 5*a - 8*b)*sinh(d*x + c)^10 + 20*(11*(5*a + 8*b)*cosh(d*x + c)^3 - 3*(5*a +
8*b)*cosh(d*x + c))*sinh(d*x + c)^9 + 15*(5*a + 8*b)*cosh(d*x + c)^8 + 15*(33*(5*a + 8*b)*cosh(d*x + c)^4 - 1
8*(5*a + 8*b)*cosh(d*x + c)^2 + 5*a + 8*b)*sinh(d*x + c)^8 + 24*(33*(5*a + 8*b)*cosh(d*x + c)^5 - 30*(5*a + 8*
b)*cosh(d*x + c)^3 + 5*(5*a + 8*b)*cosh(d*x + c))*sinh(d*x + c)^7 - 20*(5*a + 8*b)*cosh(d*x + c)^6 + 4*(231*(5
*a + 8*b)*cosh(d*x + c)^6 - 315*(5*a + 8*b)*cosh(d*x + c)^4 + 105*(5*a + 8*b)*cosh(d*x + c)^2 - 25*a - 40*b)*s
inh(d*x + c)^6 + 24*(33*(5*a + 8*b)*cosh(d*x + c)^7 - 63*(5*a + 8*b)*cosh(d*x + c)^5 + 35*(5*a + 8*b)*cosh(d*x
+ c)^3 - 5*(5*a + 8*b)*cosh(d*x + c))*sinh(d*x + c)^5 + 15*(5*a + 8*b)*cosh(d*x + c)^4 + 15*(33*(5*a + 8*b)*c
osh(d*x + c)^8 - 84*(5*a + 8*b)*cosh(d*x + c)^6 + 70*(5*a + 8*b)*cosh(d*x + c)^4 - 20*(5*a + 8*b)*cosh(d*x + c
)^2 + 5*a + 8*b)*sinh(d*x + c)^4 + 20*(11*(5*a + 8*b)*cosh(d*x + c)^9 - 36*(5*a + 8*b)*cosh(d*x + c)^7 + 42*(5
*a + 8*b)*cosh(d*x + c)^5 - 20*(5*a + 8*b)*cosh(d*x + c)^3 + 3*(5*a + 8*b)*cosh(d*x + c))*sinh(d*x + c)^3 - 6*
(5*a + 8*b)*cosh(d*x + c)^2 + 6*(11*(5*a + 8*b)*cosh(d*x + c)^10 - 45*(5*a + 8*b)*cosh(d*x + c)^8 + 70*(5*a +
8*b)*cosh(d*x + c)^6 - 50*(5*a + 8*b)*cosh(d*x + c)^4 + 15*(5*a + 8*b)*cosh(d*x + c)^2 - 5*a - 8*b)*sinh(d*x +
c)^2 + 12*((5*a + 8*b)*cosh(d*x + c)^11 - 5*(5*a + 8*b)*cosh(d*x + c)^9 + 10*(5*a + 8*b)*cosh(d*x + c)^7 - 10
*(5*a + 8*b)*cosh(d*x + c)^5 + 5*(5*a + 8*b)*cosh(d*x + c)^3 - (5*a + 8*b)*cosh(d*x + c))*sinh(d*x + c) + 5*a
+ 8*b)*log(cosh(d*x + c) + sinh(d*x + c) - 1) + 6*(11*(5*a + 8*b)*cosh(d*x + c)^10 - 3*(85*a + 72*b)*cosh(d*x
+ c)^8 + 14*(33*a + 8*b)*cosh(d*x + c)^6 + 10*(33*a + 8*b)*cosh(d*x + c)^4 - (85*a + 72*b)*cosh(d*x + c)^2 + 5
*a + 8*b)*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*
cosh(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*co
sh(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)
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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)**4),x)
[Out]
Timed out
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Giac [B] time = 1.17426, size = 285, normalized size = 3.1 \begin{align*} \frac{{\left (5 \, a + 8 \, b\right )} \log \left (e^{\left (d x + c\right )} + e^{\left (-d x - c\right )} + 2\right )}{32 \, d} - \frac{{\left (5 \, a + 8 \, b\right )} \log \left (e^{\left (d x + c\right )} + e^{\left (-d x - c\right )} - 2\right )}{32 \, d} - \frac{15 \, a{\left (e^{\left (d x + c\right )} + e^{\left (-d x - c\right )}\right )}^{5} + 24 \, b{\left (e^{\left (d x + c\right )} + e^{\left (-d x - c\right )}\right )}^{5} - 160 \, a{\left (e^{\left (d x + c\right )} + e^{\left (-d x - c\right )}\right )}^{3} - 192 \, b{\left (e^{\left (d x + c\right )} + e^{\left (-d x - c\right )}\right )}^{3} + 528 \, a{\left (e^{\left (d x + c\right )} + e^{\left (-d x - c\right )}\right )} + 384 \, b{\left (e^{\left (d x + c\right )} + e^{\left (-d x - c\right )}\right )}}{24 \,{\left ({\left (e^{\left (d x + c\right )} + e^{\left (-d x - c\right )}\right )}^{2} - 4\right )}^{3} d} \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)^4),x, algorithm="giac")
[Out]
1/32*(5*a + 8*b)*log(e^(d*x + c) + e^(-d*x - c) + 2)/d - 1/32*(5*a + 8*b)*log(e^(d*x + c) + e^(-d*x - c) - 2)/
d - 1/24*(15*a*(e^(d*x + c) + e^(-d*x - c))^5 + 24*b*(e^(d*x + c) + e^(-d*x - c))^5 - 160*a*(e^(d*x + c) + e^(
-d*x - c))^3 - 192*b*(e^(d*x + c) + e^(-d*x - c))^3 + 528*a*(e^(d*x + c) + e^(-d*x - c)) + 384*b*(e^(d*x + c)
+ e^(-d*x - c)))/(((e^(d*x + c) + e^(-d*x - c))^2 - 4)^3*d)