3.2.95 \(\int \text {csch}^7(c+d x) (a+b \sinh ^4(c+d x)) \, dx\) [195]
Optimal. Leaf size=92 \[ \frac {(5 a+8 b) \tanh ^{-1}(\cosh (c+d x))}{16 d}-\frac {(5 a+8 b) \coth (c+d x) \text {csch}(c+d x)}{16 d}+\frac {5 a \coth (c+d x) \text {csch}^3(c+d x)}{24 d}-\frac {a \coth (c+d x) \text {csch}^5(c+d x)}{6 d} \]
[Out]
1/16*(5*a+8*b)*arctanh(cosh(d*x+c))/d-1/16*(5*a+8*b)*coth(d*x+c)*csch(d*x+c)/d+5/24*a*coth(d*x+c)*csch(d*x+c)^
3/d-1/6*a*coth(d*x+c)*csch(d*x+c)^5/d
________________________________________________________________________________________
Rubi [A]
time = 0.06, antiderivative size = 92, normalized size of antiderivative = 1.00, 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 = {3294, 1171,
393, 205, 212} \begin {gather*} \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} \end {gather*}
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 205
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] && (
IntegerQ[2*p] || (n == 2 && IntegerQ[4*p]) || (n == 2 && IntegerQ[3*p]) || Denominator[p + 1/n] < Denominator[
p])
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 393
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 1171
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] &&
NeQ[c*d^2 - b*d*e + a*e^2, 0] && IGtQ[p, 0] && LtQ[q, -1]
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}^7(c+d x) \left (a+b \sinh ^4(c+d x)\right ) \, dx &=\frac {\text {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 {\text {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) \text {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) \text {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*}
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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(199\) vs. \(2(92)=184\).
time = 0.03, size = 199, normalized size = 2.16 \begin {gather*} -\frac {5 a \text {csch}^2\left (\frac {1}{2} (c+d x)\right )}{64 d}-\frac {b \text {csch}^2\left (\frac {1}{2} (c+d x)\right )}{8 d}+\frac {a \text {csch}^4\left (\frac {1}{2} (c+d x)\right )}{64 d}-\frac {a \text {csch}^6\left (\frac {1}{2} (c+d x)\right )}{384 d}-\frac {5 a \log \left (\tanh \left (\frac {1}{2} (c+d x)\right )\right )}{16 d}-\frac {b \log \left (\tanh \left (\frac {1}{2} (c+d x)\right )\right )}{2 d}-\frac {5 a \text {sech}^2\left (\frac {1}{2} (c+d x)\right )}{64 d}-\frac {b \text {sech}^2\left (\frac {1}{2} (c+d x)\right )}{8 d}-\frac {a \text {sech}^4\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} \end {gather*}
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 [B] Leaf count of result is larger than twice the leaf count of optimal. \(212\) vs.
\(2(84)=168\).
time = 1.29, size = 213, normalized size = 2.32
| | |
| method |
result |
size |
| | |
| risch |
\(-\frac {{\mathrm e}^{d x +c} \left (15 a \,{\mathrm e}^{10 d x +10 c}+24 b \,{\mathrm e}^{10 d x +10 c}-85 a \,{\mathrm e}^{8 d x +8 c}-72 b \,{\mathrm e}^{8 d x +8 c}+198 a \,{\mathrm e}^{6 d x +6 c}+48 b \,{\mathrm e}^{6 d x +6 c}+198 a \,{\mathrm e}^{4 d x +4 c}+48 b \,{\mathrm e}^{4 d x +4 c}-85 a \,{\mathrm e}^{2 d x +2 c}-72 b \,{\mathrm e}^{2 d x +2 c}+15 a +24 b \right )}{24 d \left ({\mathrm e}^{2 d x +2 c}-1\right )^{6}}-\frac {5 a \ln \left ({\mathrm e}^{d x +c}-1\right )}{16 d}-\frac {\ln \left ({\mathrm e}^{d x +c}-1\right ) b}{2 d}+\frac {5 a \ln \left ({\mathrm e}^{d x +c}+1\right )}{16 d}+\frac {\ln \left ({\mathrm e}^{d x +c}+1\right ) b}{2 d}\) |
\(213\) |
| | |
|
|
|
|
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,method=_RETURNVERBOSE)
[Out]
-1/24*exp(d*x+c)*(15*a*exp(10*d*x+10*c)+24*b*exp(10*d*x+10*c)-85*a*exp(8*d*x+8*c)-72*b*exp(8*d*x+8*c)+198*a*ex
p(6*d*x+6*c)+48*b*exp(6*d*x+6*c)+198*a*exp(4*d*x+4*c)+48*b*exp(4*d*x+4*c)-85*a*exp(2*d*x+2*c)-72*b*exp(2*d*x+2
*c)+15*a+24*b)/d/(exp(2*d*x+2*c)-1)^6-5/16*a/d*ln(exp(d*x+c)-1)-1/2/d*ln(exp(d*x+c)-1)*b+5/16*a/d*ln(exp(d*x+c
)+1)+1/2/d*ln(exp(d*x+c)+1)*b
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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 268 vs.
\(2 (84) = 168\).
time = 0.29, size = 268, normalized size = 2.91 \begin {gather*} \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 {gather*}
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] Leaf count of result is larger than twice the leaf count of optimal. 3115 vs.
\(2 (84) = 168\).
time = 0.42, size = 3115, normalized size = 33.86 \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)^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 + ...
________________________________________________________________________________________
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)**4),x)
[Out]
Timed out
________________________________________________________________________________________
Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 207 vs.
\(2 (84) = 168\).
time = 0.45, size = 207, normalized size = 2.25 \begin {gather*} \frac {3 \, {\left (5 \, a + 8 \, b\right )} \log \left (e^{\left (d x + c\right )} + e^{\left (-d x - c\right )} + 2\right ) - 3 \, {\left (5 \, a + 8 \, b\right )} \log \left (e^{\left (d x + c\right )} + e^{\left (-d x - c\right )} - 2\right ) - \frac {4 \, {\left (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 )}\right )}}{{\left ({\left (e^{\left (d x + c\right )} + e^{\left (-d x - c\right )}\right )}^{2} - 4\right )}^{3}}}{96 \, 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)^4),x, algorithm="giac")
[Out]
1/96*(3*(5*a + 8*b)*log(e^(d*x + c) + e^(-d*x - c) + 2) - 3*(5*a + 8*b)*log(e^(d*x + c) + e^(-d*x - c) - 2) -
4*(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
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Mupad [B]
time = 0.77, size = 472, normalized size = 5.13 \begin {gather*} \frac {\mathrm {atan}\left (\frac {{\mathrm {e}}^{d\,x}\,{\mathrm {e}}^c\,\left (5\,a\,\sqrt {-d^2}+8\,b\,\sqrt {-d^2}\right )}{d\,\sqrt {25\,a^2+80\,a\,b+64\,b^2}}\right )\,\sqrt {25\,a^2+80\,a\,b+64\,b^2}}{8\,\sqrt {-d^2}}-\frac {\frac {2\,b\,{\mathrm {e}}^{9\,c+9\,d\,x}}{3\,d}-\frac {8\,b\,{\mathrm {e}}^{7\,c+7\,d\,x}}{3\,d}-\frac {8\,b\,{\mathrm {e}}^{3\,c+3\,d\,x}}{3\,d}+\frac {4\,{\mathrm {e}}^{5\,c+5\,d\,x}\,\left (8\,a+3\,b\right )}{3\,d}+\frac {2\,b\,{\mathrm {e}}^{c+d\,x}}{3\,d}}{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}+\frac {{\mathrm {e}}^{c+d\,x}\,\left (5\,a-16\,b\right )}{12\,d\,\left ({\mathrm {e}}^{4\,c+4\,d\,x}-2\,{\mathrm {e}}^{2\,c+2\,d\,x}+1\right )}-\frac {a\,{\mathrm {e}}^{c+d\,x}}{3\,d\,\left (3\,{\mathrm {e}}^{2\,c+2\,d\,x}-3\,{\mathrm {e}}^{4\,c+4\,d\,x}+{\mathrm {e}}^{6\,c+6\,d\,x}-1\right )}-\frac {22\,a\,{\mathrm {e}}^{c+d\,x}}{3\,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 {{\mathrm {e}}^{c+d\,x}\,\left (5\,a+8\,b\right )}{8\,d\,\left ({\mathrm {e}}^{2\,c+2\,d\,x}-1\right )}-\frac {16\,a\,{\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 )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((a + b*sinh(c + d*x)^4)/sinh(c + d*x)^7,x)
[Out]
(atan((exp(d*x)*exp(c)*(5*a*(-d^2)^(1/2) + 8*b*(-d^2)^(1/2)))/(d*(80*a*b + 25*a^2 + 64*b^2)^(1/2)))*(80*a*b +
25*a^2 + 64*b^2)^(1/2))/(8*(-d^2)^(1/2)) - ((2*b*exp(9*c + 9*d*x))/(3*d) - (8*b*exp(7*c + 7*d*x))/(3*d) - (8*b
*exp(3*c + 3*d*x))/(3*d) + (4*exp(5*c + 5*d*x)*(8*a + 3*b))/(3*d) + (2*b*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) + (exp(c + d*x)*(5*a - 16*b))/(12*d*(exp(4*c + 4*d*x) - 2*exp(2*c + 2*d*x) + 1)) - (a*exp(c + d*x))/(
3*d*(3*exp(2*c + 2*d*x) - 3*exp(4*c + 4*d*x) + exp(6*c + 6*d*x) - 1)) - (22*a*exp(c + d*x))/(3*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)) - (exp(c + d*x)*(5*a + 8*b))/(8*d*(e
xp(2*c + 2*d*x) - 1)) - (16*a*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))
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