3.2.67 \(\int \frac {\coth ^3(c+d x)}{(a+b \text {sech}^2(c+d x))^3} \, dx\) [167]
Optimal. Leaf size=152 \[ -\frac {b^4}{4 a^3 (a+b)^2 d \left (b+a \cosh ^2(c+d x)\right )^2}+\frac {b^3 (2 a+b)}{a^3 (a+b)^3 d \left (b+a \cosh ^2(c+d x)\right )}-\frac {\text {csch}^2(c+d x)}{2 (a+b)^3 d}+\frac {b^2 \left (6 a^2+4 a b+b^2\right ) \log \left (b+a \cosh ^2(c+d x)\right )}{2 a^3 (a+b)^4 d}+\frac {(a+4 b) \log (\sinh (c+d x))}{(a+b)^4 d} \]
[Out]
-1/4*b^4/a^3/(a+b)^2/d/(b+a*cosh(d*x+c)^2)^2+b^3*(2*a+b)/a^3/(a+b)^3/d/(b+a*cosh(d*x+c)^2)-1/2*csch(d*x+c)^2/(
a+b)^3/d+1/2*b^2*(6*a^2+4*a*b+b^2)*ln(b+a*cosh(d*x+c)^2)/a^3/(a+b)^4/d+(a+4*b)*ln(sinh(d*x+c))/(a+b)^4/d
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Rubi [A]
time = 0.17, antiderivative size = 152, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 3, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.130, Rules used = {4223, 457, 90}
\begin {gather*} -\frac {b^4}{4 a^3 d (a+b)^2 \left (a \cosh ^2(c+d x)+b\right )^2}+\frac {b^3 (2 a+b)}{a^3 d (a+b)^3 \left (a \cosh ^2(c+d x)+b\right )}+\frac {b^2 \left (6 a^2+4 a b+b^2\right ) \log \left (a \cosh ^2(c+d x)+b\right )}{2 a^3 d (a+b)^4}-\frac {\text {csch}^2(c+d x)}{2 d (a+b)^3}+\frac {(a+4 b) \log (\sinh (c+d x))}{d (a+b)^4} \end {gather*}
Antiderivative was successfully verified.
[In]
Int[Coth[c + d*x]^3/(a + b*Sech[c + d*x]^2)^3,x]
[Out]
-1/4*b^4/(a^3*(a + b)^2*d*(b + a*Cosh[c + d*x]^2)^2) + (b^3*(2*a + b))/(a^3*(a + b)^3*d*(b + a*Cosh[c + d*x]^2
)) - Csch[c + d*x]^2/(2*(a + b)^3*d) + (b^2*(6*a^2 + 4*a*b + b^2)*Log[b + a*Cosh[c + d*x]^2])/(2*a^3*(a + b)^4
*d) + ((a + 4*b)*Log[Sinh[c + d*x]])/((a + b)^4*d)
Rule 90
Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Int[ExpandI
ntegrand[(a + b*x)^m*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, p}, x] && IntegersQ[m, n] &&
(IntegerQ[p] || (GtQ[m, 0] && GeQ[n, -1]))
Rule 457
Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_.), x_Symbol] :> Dist[1/n, Subst[Int
[x^(Simplify[(m + 1)/n] - 1)*(a + b*x)^p*(c + d*x)^q, x], x, x^n], x] /; FreeQ[{a, b, c, d, m, n, p, q}, x] &&
NeQ[b*c - a*d, 0] && IntegerQ[Simplify[(m + 1)/n]]
Rule 4223
Int[((a_) + (b_.)*sec[(e_.) + (f_.)*(x_)]^(n_))^(p_.)*tan[(e_.) + (f_.)*(x_)]^(m_.), x_Symbol] :> Module[{ff =
FreeFactors[Cos[e + f*x], x]}, Dist[-(f*ff^(m + n*p - 1))^(-1), Subst[Int[(1 - ff^2*x^2)^((m - 1)/2)*((b + a*
(ff*x)^n)^p/x^(m + n*p)), x], x, Cos[e + f*x]/ff], x]] /; FreeQ[{a, b, e, f, n}, x] && IntegerQ[(m - 1)/2] &&
IntegerQ[n] && IntegerQ[p]
Rubi steps
\begin {align*} \int \frac {\coth ^3(c+d x)}{\left (a+b \text {sech}^2(c+d x)\right )^3} \, dx &=\frac {\text {Subst}\left (\int \frac {x^9}{\left (1-x^2\right )^2 \left (b+a x^2\right )^3} \, dx,x,\cosh (c+d x)\right )}{d}\\ &=\frac {\text {Subst}\left (\int \frac {x^4}{(1-x)^2 (b+a x)^3} \, dx,x,\cosh ^2(c+d x)\right )}{2 d}\\ &=\frac {\text {Subst}\left (\int \left (\frac {1}{(a+b)^3 (-1+x)^2}+\frac {a+4 b}{(a+b)^4 (-1+x)}+\frac {b^4}{a^2 (a+b)^2 (b+a x)^3}-\frac {2 b^3 (2 a+b)}{a^2 (a+b)^3 (b+a x)^2}+\frac {b^2 \left (6 a^2+4 a b+b^2\right )}{a^2 (a+b)^4 (b+a x)}\right ) \, dx,x,\cosh ^2(c+d x)\right )}{2 d}\\ &=-\frac {b^4}{4 a^3 (a+b)^2 d \left (b+a \cosh ^2(c+d x)\right )^2}+\frac {b^3 (2 a+b)}{a^3 (a+b)^3 d \left (b+a \cosh ^2(c+d x)\right )}-\frac {\text {csch}^2(c+d x)}{2 (a+b)^3 d}+\frac {b^2 \left (6 a^2+4 a b+b^2\right ) \log \left (b+a \cosh ^2(c+d x)\right )}{2 a^3 (a+b)^4 d}+\frac {(a+4 b) \log (\sinh (c+d x))}{(a+b)^4 d}\\ \end {align*}
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Mathematica [A]
time = 1.24, size = 172, normalized size = 1.13 \begin {gather*} -\frac {(a+2 b+a \cosh (2 (c+d x)))^3 \text {sech}^6(c+d x) \left (2 (a+b) \text {csch}^2(c+d x)-4 (a+4 b) \log (\sinh (c+d x))-\frac {2 b^2 \left (6 a^2+4 a b+b^2\right ) \log \left (a+b+a \sinh ^2(c+d x)\right )}{a^3}+\frac {b^4 (a+b)^2}{a^3 \left (a+b+a \sinh ^2(c+d x)\right )^2}-\frac {4 b^3 (a+b) (2 a+b)}{a^3 \left (a+b+a \sinh ^2(c+d x)\right )}\right )}{32 (a+b)^4 d \left (a+b \text {sech}^2(c+d x)\right )^3} \end {gather*}
Antiderivative was successfully verified.
[In]
Integrate[Coth[c + d*x]^3/(a + b*Sech[c + d*x]^2)^3,x]
[Out]
-1/32*((a + 2*b + a*Cosh[2*(c + d*x)])^3*Sech[c + d*x]^6*(2*(a + b)*Csch[c + d*x]^2 - 4*(a + 4*b)*Log[Sinh[c +
d*x]] - (2*b^2*(6*a^2 + 4*a*b + b^2)*Log[a + b + a*Sinh[c + d*x]^2])/a^3 + (b^4*(a + b)^2)/(a^3*(a + b + a*Si
nh[c + d*x]^2)^2) - (4*b^3*(a + b)*(2*a + b))/(a^3*(a + b + a*Sinh[c + d*x]^2))))/((a + b)^4*d*(a + b*Sech[c +
d*x]^2)^3)
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(353\) vs.
\(2(146)=292\).
time = 3.37, size = 354, normalized size = 2.33 Too large to display
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(coth(d*x+c)^3/(a+b*sech(d*x+c)^2)^3,x,method=_RETURNVERBOSE)
[Out]
1/d*(-1/8*tanh(1/2*d*x+1/2*c)^2/(a^3+3*a^2*b+3*a*b^2+b^3)+b^2/(a+b)^4/a^3*(((-8*a^3*b-10*a^2*b^2-2*a*b^3)*tanh
(1/2*d*x+1/2*c)^6-4*(4*a^2-2*a*b-b^2)*a*b*tanh(1/2*d*x+1/2*c)^4-2*(4*a^2+5*a*b+b^2)*a*b*tanh(1/2*d*x+1/2*c)^2)
/(a*tanh(1/2*d*x+1/2*c)^4+b*tanh(1/2*d*x+1/2*c)^4+2*a*tanh(1/2*d*x+1/2*c)^2-2*b*tanh(1/2*d*x+1/2*c)^2+a+b)^2+1
/2*(6*a^2+4*a*b+b^2)*ln(a*tanh(1/2*d*x+1/2*c)^4+b*tanh(1/2*d*x+1/2*c)^4+2*a*tanh(1/2*d*x+1/2*c)^2-2*b*tanh(1/2
*d*x+1/2*c)^2+a+b))-1/a^3*ln(tanh(1/2*d*x+1/2*c)-1)-1/8/(a+b)^3/tanh(1/2*d*x+1/2*c)^2+1/4/(a+b)^4*(4*a+16*b)*l
n(tanh(1/2*d*x+1/2*c))-1/a^3*ln(tanh(1/2*d*x+1/2*c)+1))
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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 692 vs.
\(2 (146) = 292\).
time = 0.31, size = 692, normalized size = 4.55 \begin {gather*} \frac {{\left (6 \, a^{2} b^{2} + 4 \, a b^{3} + b^{4}\right )} \log \left (2 \, {\left (a + 2 \, b\right )} e^{\left (-2 \, d x - 2 \, c\right )} + a e^{\left (-4 \, d x - 4 \, c\right )} + a\right )}{2 \, {\left (a^{7} + 4 \, a^{6} b + 6 \, a^{5} b^{2} + 4 \, a^{4} b^{3} + a^{3} b^{4}\right )} d} + \frac {{\left (a + 4 \, b\right )} \log \left (e^{\left (-d x - c\right )} + 1\right )}{{\left (a^{4} + 4 \, a^{3} b + 6 \, a^{2} b^{2} + 4 \, a b^{3} + b^{4}\right )} d} + \frac {{\left (a + 4 \, b\right )} \log \left (e^{\left (-d x - c\right )} - 1\right )}{{\left (a^{4} + 4 \, a^{3} b + 6 \, a^{2} b^{2} + 4 \, a b^{3} + b^{4}\right )} d} - \frac {2 \, {\left ({\left (a^{5} - 4 \, a^{2} b^{3} - 2 \, a b^{4}\right )} e^{\left (-2 \, d x - 2 \, c\right )} + 2 \, {\left (2 \, a^{5} + 4 \, a^{4} b - 7 \, a b^{4} - 3 \, b^{5}\right )} e^{\left (-4 \, d x - 4 \, c\right )} + 2 \, {\left (3 \, a^{5} + 8 \, a^{4} b + 8 \, a^{3} b^{2} + 4 \, a^{2} b^{3} + 16 \, a b^{4} + 6 \, b^{5}\right )} e^{\left (-6 \, d x - 6 \, c\right )} + 2 \, {\left (2 \, a^{5} + 4 \, a^{4} b - 7 \, a b^{4} - 3 \, b^{5}\right )} e^{\left (-8 \, d x - 8 \, c\right )} + {\left (a^{5} - 4 \, a^{2} b^{3} - 2 \, a b^{4}\right )} e^{\left (-10 \, d x - 10 \, c\right )}\right )}}{{\left (a^{8} + 3 \, a^{7} b + 3 \, a^{6} b^{2} + a^{5} b^{3} + 2 \, {\left (a^{8} + 7 \, a^{7} b + 15 \, a^{6} b^{2} + 13 \, a^{5} b^{3} + 4 \, a^{4} b^{4}\right )} e^{\left (-2 \, d x - 2 \, c\right )} - {\left (a^{8} + 3 \, a^{7} b - 13 \, a^{6} b^{2} - 47 \, a^{5} b^{3} - 48 \, a^{4} b^{4} - 16 \, a^{3} b^{5}\right )} e^{\left (-4 \, d x - 4 \, c\right )} - 4 \, {\left (a^{8} + 7 \, a^{7} b + 23 \, a^{6} b^{2} + 37 \, a^{5} b^{3} + 28 \, a^{4} b^{4} + 8 \, a^{3} b^{5}\right )} e^{\left (-6 \, d x - 6 \, c\right )} - {\left (a^{8} + 3 \, a^{7} b - 13 \, a^{6} b^{2} - 47 \, a^{5} b^{3} - 48 \, a^{4} b^{4} - 16 \, a^{3} b^{5}\right )} e^{\left (-8 \, d x - 8 \, c\right )} + 2 \, {\left (a^{8} + 7 \, a^{7} b + 15 \, a^{6} b^{2} + 13 \, a^{5} b^{3} + 4 \, a^{4} b^{4}\right )} e^{\left (-10 \, d x - 10 \, c\right )} + {\left (a^{8} + 3 \, a^{7} b + 3 \, a^{6} b^{2} + a^{5} b^{3}\right )} e^{\left (-12 \, d x - 12 \, c\right )}\right )} d} + \frac {d x + c}{a^{3} d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(coth(d*x+c)^3/(a+b*sech(d*x+c)^2)^3,x, algorithm="maxima")
[Out]
1/2*(6*a^2*b^2 + 4*a*b^3 + b^4)*log(2*(a + 2*b)*e^(-2*d*x - 2*c) + a*e^(-4*d*x - 4*c) + a)/((a^7 + 4*a^6*b + 6
*a^5*b^2 + 4*a^4*b^3 + a^3*b^4)*d) + (a + 4*b)*log(e^(-d*x - c) + 1)/((a^4 + 4*a^3*b + 6*a^2*b^2 + 4*a*b^3 + b
^4)*d) + (a + 4*b)*log(e^(-d*x - c) - 1)/((a^4 + 4*a^3*b + 6*a^2*b^2 + 4*a*b^3 + b^4)*d) - 2*((a^5 - 4*a^2*b^3
- 2*a*b^4)*e^(-2*d*x - 2*c) + 2*(2*a^5 + 4*a^4*b - 7*a*b^4 - 3*b^5)*e^(-4*d*x - 4*c) + 2*(3*a^5 + 8*a^4*b + 8
*a^3*b^2 + 4*a^2*b^3 + 16*a*b^4 + 6*b^5)*e^(-6*d*x - 6*c) + 2*(2*a^5 + 4*a^4*b - 7*a*b^4 - 3*b^5)*e^(-8*d*x -
8*c) + (a^5 - 4*a^2*b^3 - 2*a*b^4)*e^(-10*d*x - 10*c))/((a^8 + 3*a^7*b + 3*a^6*b^2 + a^5*b^3 + 2*(a^8 + 7*a^7*
b + 15*a^6*b^2 + 13*a^5*b^3 + 4*a^4*b^4)*e^(-2*d*x - 2*c) - (a^8 + 3*a^7*b - 13*a^6*b^2 - 47*a^5*b^3 - 48*a^4*
b^4 - 16*a^3*b^5)*e^(-4*d*x - 4*c) - 4*(a^8 + 7*a^7*b + 23*a^6*b^2 + 37*a^5*b^3 + 28*a^4*b^4 + 8*a^3*b^5)*e^(-
6*d*x - 6*c) - (a^8 + 3*a^7*b - 13*a^6*b^2 - 47*a^5*b^3 - 48*a^4*b^4 - 16*a^3*b^5)*e^(-8*d*x - 8*c) + 2*(a^8 +
7*a^7*b + 15*a^6*b^2 + 13*a^5*b^3 + 4*a^4*b^4)*e^(-10*d*x - 10*c) + (a^8 + 3*a^7*b + 3*a^6*b^2 + a^5*b^3)*e^(
-12*d*x - 12*c))*d) + (d*x + c)/(a^3*d)
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 10255 vs.
\(2 (146) = 292\).
time = 0.92, size = 10255, normalized size = 67.47 \begin {gather*} \text {Too large to display} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(coth(d*x+c)^3/(a+b*sech(d*x+c)^2)^3,x, algorithm="fricas")
[Out]
-1/2*(2*(a^6 + 4*a^5*b + 6*a^4*b^2 + 4*a^3*b^3 + a^2*b^4)*d*x*cosh(d*x + c)^12 + 24*(a^6 + 4*a^5*b + 6*a^4*b^2
+ 4*a^3*b^3 + a^2*b^4)*d*x*cosh(d*x + c)*sinh(d*x + c)^11 + 2*(a^6 + 4*a^5*b + 6*a^4*b^2 + 4*a^3*b^3 + a^2*b^
4)*d*x*sinh(d*x + c)^12 + 4*(a^6 + a^5*b - 4*a^3*b^3 - 6*a^2*b^4 - 2*a*b^5 + (a^6 + 8*a^5*b + 22*a^4*b^2 + 28*
a^3*b^3 + 17*a^2*b^4 + 4*a*b^5)*d*x)*cosh(d*x + c)^10 + 4*(a^6 + a^5*b - 4*a^3*b^3 - 6*a^2*b^4 - 2*a*b^5 + 33*
(a^6 + 4*a^5*b + 6*a^4*b^2 + 4*a^3*b^3 + a^2*b^4)*d*x*cosh(d*x + c)^2 + (a^6 + 8*a^5*b + 22*a^4*b^2 + 28*a^3*b
^3 + 17*a^2*b^4 + 4*a*b^5)*d*x)*sinh(d*x + c)^10 + 40*(11*(a^6 + 4*a^5*b + 6*a^4*b^2 + 4*a^3*b^3 + a^2*b^4)*d*
x*cosh(d*x + c)^3 + (a^6 + a^5*b - 4*a^3*b^3 - 6*a^2*b^4 - 2*a*b^5 + (a^6 + 8*a^5*b + 22*a^4*b^2 + 28*a^3*b^3
+ 17*a^2*b^4 + 4*a*b^5)*d*x)*cosh(d*x + c))*sinh(d*x + c)^9 + 2*(8*a^6 + 24*a^5*b + 16*a^4*b^2 - 28*a^2*b^4 -
40*a*b^5 - 12*b^6 - (a^6 + 4*a^5*b - 10*a^4*b^2 - 60*a^3*b^3 - 95*a^2*b^4 - 64*a*b^5 - 16*b^6)*d*x)*cosh(d*x +
c)^8 + 2*(495*(a^6 + 4*a^5*b + 6*a^4*b^2 + 4*a^3*b^3 + a^2*b^4)*d*x*cosh(d*x + c)^4 + 8*a^6 + 24*a^5*b + 16*a
^4*b^2 - 28*a^2*b^4 - 40*a*b^5 - 12*b^6 - (a^6 + 4*a^5*b - 10*a^4*b^2 - 60*a^3*b^3 - 95*a^2*b^4 - 64*a*b^5 - 1
6*b^6)*d*x + 90*(a^6 + a^5*b - 4*a^3*b^3 - 6*a^2*b^4 - 2*a*b^5 + (a^6 + 8*a^5*b + 22*a^4*b^2 + 28*a^3*b^3 + 17
*a^2*b^4 + 4*a*b^5)*d*x)*cosh(d*x + c)^2)*sinh(d*x + c)^8 + 16*(99*(a^6 + 4*a^5*b + 6*a^4*b^2 + 4*a^3*b^3 + a^
2*b^4)*d*x*cosh(d*x + c)^5 + 30*(a^6 + a^5*b - 4*a^3*b^3 - 6*a^2*b^4 - 2*a*b^5 + (a^6 + 8*a^5*b + 22*a^4*b^2 +
28*a^3*b^3 + 17*a^2*b^4 + 4*a*b^5)*d*x)*cosh(d*x + c)^3 + (8*a^6 + 24*a^5*b + 16*a^4*b^2 - 28*a^2*b^4 - 40*a*
b^5 - 12*b^6 - (a^6 + 4*a^5*b - 10*a^4*b^2 - 60*a^3*b^3 - 95*a^2*b^4 - 64*a*b^5 - 16*b^6)*d*x)*cosh(d*x + c))*
sinh(d*x + c)^7 + 8*(3*a^6 + 11*a^5*b + 16*a^4*b^2 + 12*a^3*b^3 + 20*a^2*b^4 + 22*a*b^5 + 6*b^6 - (a^6 + 8*a^5
*b + 30*a^4*b^2 + 60*a^3*b^3 + 65*a^2*b^4 + 36*a*b^5 + 8*b^6)*d*x)*cosh(d*x + c)^6 + 8*(231*(a^6 + 4*a^5*b + 6
*a^4*b^2 + 4*a^3*b^3 + a^2*b^4)*d*x*cosh(d*x + c)^6 + 3*a^6 + 11*a^5*b + 16*a^4*b^2 + 12*a^3*b^3 + 20*a^2*b^4
+ 22*a*b^5 + 6*b^6 + 105*(a^6 + a^5*b - 4*a^3*b^3 - 6*a^2*b^4 - 2*a*b^5 + (a^6 + 8*a^5*b + 22*a^4*b^2 + 28*a^3
*b^3 + 17*a^2*b^4 + 4*a*b^5)*d*x)*cosh(d*x + c)^4 - (a^6 + 8*a^5*b + 30*a^4*b^2 + 60*a^3*b^3 + 65*a^2*b^4 + 36
*a*b^5 + 8*b^6)*d*x + 7*(8*a^6 + 24*a^5*b + 16*a^4*b^2 - 28*a^2*b^4 - 40*a*b^5 - 12*b^6 - (a^6 + 4*a^5*b - 10*
a^4*b^2 - 60*a^3*b^3 - 95*a^2*b^4 - 64*a*b^5 - 16*b^6)*d*x)*cosh(d*x + c)^2)*sinh(d*x + c)^6 + 16*(99*(a^6 + 4
*a^5*b + 6*a^4*b^2 + 4*a^3*b^3 + a^2*b^4)*d*x*cosh(d*x + c)^7 + 63*(a^6 + a^5*b - 4*a^3*b^3 - 6*a^2*b^4 - 2*a*
b^5 + (a^6 + 8*a^5*b + 22*a^4*b^2 + 28*a^3*b^3 + 17*a^2*b^4 + 4*a*b^5)*d*x)*cosh(d*x + c)^5 + 7*(8*a^6 + 24*a^
5*b + 16*a^4*b^2 - 28*a^2*b^4 - 40*a*b^5 - 12*b^6 - (a^6 + 4*a^5*b - 10*a^4*b^2 - 60*a^3*b^3 - 95*a^2*b^4 - 64
*a*b^5 - 16*b^6)*d*x)*cosh(d*x + c)^3 + 3*(3*a^6 + 11*a^5*b + 16*a^4*b^2 + 12*a^3*b^3 + 20*a^2*b^4 + 22*a*b^5
+ 6*b^6 - (a^6 + 8*a^5*b + 30*a^4*b^2 + 60*a^3*b^3 + 65*a^2*b^4 + 36*a*b^5 + 8*b^6)*d*x)*cosh(d*x + c))*sinh(d
*x + c)^5 + 2*(8*a^6 + 24*a^5*b + 16*a^4*b^2 - 28*a^2*b^4 - 40*a*b^5 - 12*b^6 - (a^6 + 4*a^5*b - 10*a^4*b^2 -
60*a^3*b^3 - 95*a^2*b^4 - 64*a*b^5 - 16*b^6)*d*x)*cosh(d*x + c)^4 + 2*(495*(a^6 + 4*a^5*b + 6*a^4*b^2 + 4*a^3*
b^3 + a^2*b^4)*d*x*cosh(d*x + c)^8 + 420*(a^6 + a^5*b - 4*a^3*b^3 - 6*a^2*b^4 - 2*a*b^5 + (a^6 + 8*a^5*b + 22*
a^4*b^2 + 28*a^3*b^3 + 17*a^2*b^4 + 4*a*b^5)*d*x)*cosh(d*x + c)^6 + 8*a^6 + 24*a^5*b + 16*a^4*b^2 - 28*a^2*b^4
- 40*a*b^5 - 12*b^6 + 70*(8*a^6 + 24*a^5*b + 16*a^4*b^2 - 28*a^2*b^4 - 40*a*b^5 - 12*b^6 - (a^6 + 4*a^5*b - 1
0*a^4*b^2 - 60*a^3*b^3 - 95*a^2*b^4 - 64*a*b^5 - 16*b^6)*d*x)*cosh(d*x + c)^4 - (a^6 + 4*a^5*b - 10*a^4*b^2 -
60*a^3*b^3 - 95*a^2*b^4 - 64*a*b^5 - 16*b^6)*d*x + 60*(3*a^6 + 11*a^5*b + 16*a^4*b^2 + 12*a^3*b^3 + 20*a^2*b^4
+ 22*a*b^5 + 6*b^6 - (a^6 + 8*a^5*b + 30*a^4*b^2 + 60*a^3*b^3 + 65*a^2*b^4 + 36*a*b^5 + 8*b^6)*d*x)*cosh(d*x
+ c)^2)*sinh(d*x + c)^4 + 8*(55*(a^6 + 4*a^5*b + 6*a^4*b^2 + 4*a^3*b^3 + a^2*b^4)*d*x*cosh(d*x + c)^9 + 60*(a^
6 + a^5*b - 4*a^3*b^3 - 6*a^2*b^4 - 2*a*b^5 + (a^6 + 8*a^5*b + 22*a^4*b^2 + 28*a^3*b^3 + 17*a^2*b^4 + 4*a*b^5)
*d*x)*cosh(d*x + c)^7 + 14*(8*a^6 + 24*a^5*b + 16*a^4*b^2 - 28*a^2*b^4 - 40*a*b^5 - 12*b^6 - (a^6 + 4*a^5*b -
10*a^4*b^2 - 60*a^3*b^3 - 95*a^2*b^4 - 64*a*b^5 - 16*b^6)*d*x)*cosh(d*x + c)^5 + 20*(3*a^6 + 11*a^5*b + 16*a^4
*b^2 + 12*a^3*b^3 + 20*a^2*b^4 + 22*a*b^5 + 6*b^6 - (a^6 + 8*a^5*b + 30*a^4*b^2 + 60*a^3*b^3 + 65*a^2*b^4 + 36
*a*b^5 + 8*b^6)*d*x)*cosh(d*x + c)^3 + (8*a^6 + 24*a^5*b + 16*a^4*b^2 - 28*a^2*b^4 - 40*a*b^5 - 12*b^6 - (a^6
+ 4*a^5*b - 10*a^4*b^2 - 60*a^3*b^3 - 95*a^2*b^4 - 64*a*b^5 - 16*b^6)*d*x)*cosh(d*x + c))*sinh(d*x + c)^3 + 2*
(a^6 + 4*a^5*b + 6*a^4*b^2 + 4*a^3*b^3 + a^2*b^4)*d*x + 4*(a^6 + a^5*b - 4*a^3*b^3 - 6*a^2*b^4 - 2*a*b^5 + (a^
6 + 8*a^5*b + 22*a^4*b^2 + 28*a^3*b^3 + 17*a^2*b^4 + 4*a*b^5)*d*x)*cosh(d*x + c)^2 + 4*(33*(a^6 + 4*a^5*b + 6*
a^4*b^2 + 4*a^3*b^3 + a^2*b^4)*d*x*cosh(d*x + c...
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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\coth ^{3}{\left (c + d x \right )}}{\left (a + b \operatorname {sech}^{2}{\left (c + d x \right )}\right )^{3}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(coth(d*x+c)**3/(a+b*sech(d*x+c)**2)**3,x)
[Out]
Integral(coth(c + d*x)**3/(a + b*sech(c + d*x)**2)**3, x)
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Giac [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(coth(d*x+c)^3/(a+b*sech(d*x+c)^2)^3,x, algorithm="giac")
[Out]
Exception raised: TypeError >> An error occurred running a Giac command:INPUT:sage2:=int(sage0,sageVARx):;OUTP
UT:Warning, need to choose a branch for the root of a polynomial with parameters. This might be wrong.The choi
ce was done
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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {{\mathrm {cosh}\left (c+d\,x\right )}^6\,{\mathrm {coth}\left (c+d\,x\right )}^3}{{\left (a\,{\mathrm {cosh}\left (c+d\,x\right )}^2+b\right )}^3} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(coth(c + d*x)^3/(a + b/cosh(c + d*x)^2)^3,x)
[Out]
int((cosh(c + d*x)^6*coth(c + d*x)^3)/(b + a*cosh(c + d*x)^2)^3, x)
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