3.167 \(\int \coth ^7(c+d x) (a+b \tanh ^2(c+d x))^3 \, dx\)
Optimal. Leaf size=103 \[ -\frac{a \left (a^2+3 a b+3 b^2\right ) \coth ^2(c+d x)}{2 d}-\frac{a^2 (a+3 b) \coth ^4(c+d x)}{4 d}-\frac{a^3 \coth ^6(c+d x)}{6 d}+\frac{(a+b)^3 \log (\tanh (c+d x))}{d}+\frac{(a+b)^3 \log (\cosh (c+d x))}{d} \]
[Out]
-(a*(a^2 + 3*a*b + 3*b^2)*Coth[c + d*x]^2)/(2*d) - (a^2*(a + 3*b)*Coth[c + d*x]^4)/(4*d) - (a^3*Coth[c + d*x]^
6)/(6*d) + ((a + b)^3*Log[Cosh[c + d*x]])/d + ((a + b)^3*Log[Tanh[c + d*x]])/d
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Rubi [A] time = 0.128969, antiderivative size = 103, normalized size of antiderivative = 1.,
number of steps used = 4, number of rules used = 3, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.13, Rules used =
{3670, 446, 88} \[ -\frac{a \left (a^2+3 a b+3 b^2\right ) \coth ^2(c+d x)}{2 d}-\frac{a^2 (a+3 b) \coth ^4(c+d x)}{4 d}-\frac{a^3 \coth ^6(c+d x)}{6 d}+\frac{(a+b)^3 \log (\tanh (c+d x))}{d}+\frac{(a+b)^3 \log (\cosh (c+d x))}{d} \]
Antiderivative was successfully verified.
[In]
Int[Coth[c + d*x]^7*(a + b*Tanh[c + d*x]^2)^3,x]
[Out]
-(a*(a^2 + 3*a*b + 3*b^2)*Coth[c + d*x]^2)/(2*d) - (a^2*(a + 3*b)*Coth[c + d*x]^4)/(4*d) - (a^3*Coth[c + d*x]^
6)/(6*d) + ((a + b)^3*Log[Cosh[c + d*x]])/d + ((a + b)^3*Log[Tanh[c + d*x]])/d
Rule 3670
Int[((d_.)*tan[(e_.) + (f_.)*(x_)])^(m_.)*((a_) + (b_.)*((c_.)*tan[(e_.) + (f_.)*(x_)])^(n_))^(p_.), x_Symbol]
:> With[{ff = FreeFactors[Tan[e + f*x], x]}, Dist[(c*ff)/f, Subst[Int[(((d*ff*x)/c)^m*(a + b*(ff*x)^n)^p)/(c^
2 + ff^2*x^2), x], x, (c*Tan[e + f*x])/ff], x]] /; FreeQ[{a, b, c, d, e, f, m, n, p}, x] && (IGtQ[p, 0] || EqQ
[n, 2] || EqQ[n, 4] || (IntegerQ[p] && RationalQ[n]))
Rule 446
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 88
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]))
Rubi steps
\begin{align*} \int \coth ^7(c+d x) \left (a+b \tanh ^2(c+d x)\right )^3 \, dx &=\frac{\operatorname{Subst}\left (\int \frac{\left (a+b x^2\right )^3}{x^7 \left (1-x^2\right )} \, dx,x,\tanh (c+d x)\right )}{d}\\ &=\frac{\operatorname{Subst}\left (\int \frac{(a+b x)^3}{(1-x) x^4} \, dx,x,\tanh ^2(c+d x)\right )}{2 d}\\ &=\frac{\operatorname{Subst}\left (\int \left (-\frac{(a+b)^3}{-1+x}+\frac{a^3}{x^4}+\frac{a^2 (a+3 b)}{x^3}+\frac{a \left (a^2+3 a b+3 b^2\right )}{x^2}+\frac{(a+b)^3}{x}\right ) \, dx,x,\tanh ^2(c+d x)\right )}{2 d}\\ &=-\frac{a \left (a^2+3 a b+3 b^2\right ) \coth ^2(c+d x)}{2 d}-\frac{a^2 (a+3 b) \coth ^4(c+d x)}{4 d}-\frac{a^3 \coth ^6(c+d x)}{6 d}+\frac{(a+b)^3 \log (\cosh (c+d x))}{d}+\frac{(a+b)^3 \log (\tanh (c+d x))}{d}\\ \end{align*}
Mathematica [A] time = 0.237595, size = 76, normalized size = 0.74 \[ -\frac{a (a+b)^2 \coth ^2(c+d x)+\frac{1}{2} (a+b) \left (a \coth ^2(c+d x)+b\right )^2+\frac{1}{3} \left (a \coth ^2(c+d x)+b\right )^3-2 (a+b)^3 \log (\sinh (c+d x))}{2 d} \]
Antiderivative was successfully verified.
[In]
Integrate[Coth[c + d*x]^7*(a + b*Tanh[c + d*x]^2)^3,x]
[Out]
-(a*(a + b)^2*Coth[c + d*x]^2 + ((a + b)*(b + a*Coth[c + d*x]^2)^2)/2 + (b + a*Coth[c + d*x]^2)^3/3 - 2*(a + b
)^3*Log[Sinh[c + d*x]])/(2*d)
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Maple [A] time = 0.062, size = 161, normalized size = 1.6 \begin{align*}{\frac{{a}^{3}\ln \left ( \sinh \left ( dx+c \right ) \right ) }{d}}-{\frac{{a}^{3} \left ({\rm coth} \left (dx+c\right ) \right ) ^{2}}{2\,d}}-{\frac{{a}^{3} \left ({\rm coth} \left (dx+c\right ) \right ) ^{4}}{4\,d}}-{\frac{{a}^{3} \left ({\rm coth} \left (dx+c\right ) \right ) ^{6}}{6\,d}}+3\,{\frac{{a}^{2}b\ln \left ( \sinh \left ( dx+c \right ) \right ) }{d}}-{\frac{3\,{a}^{2}b \left ({\rm coth} \left (dx+c\right ) \right ) ^{2}}{2\,d}}-{\frac{3\,{a}^{2}b \left ({\rm coth} \left (dx+c\right ) \right ) ^{4}}{4\,d}}+3\,{\frac{a{b}^{2}\ln \left ( \sinh \left ( dx+c \right ) \right ) }{d}}-{\frac{3\,a{b}^{2} \left ({\rm coth} \left (dx+c\right ) \right ) ^{2}}{2\,d}}+{\frac{{b}^{3}\ln \left ( \sinh \left ( dx+c \right ) \right ) }{d}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(coth(d*x+c)^7*(a+b*tanh(d*x+c)^2)^3,x)
[Out]
1/d*a^3*ln(sinh(d*x+c))-1/2*a^3*coth(d*x+c)^2/d-1/4*a^3*coth(d*x+c)^4/d-1/6*a^3*coth(d*x+c)^6/d+3/d*a^2*b*ln(s
inh(d*x+c))-3/2/d*a^2*b*coth(d*x+c)^2-3/4/d*a^2*b*coth(d*x+c)^4+3/d*a*b^2*ln(sinh(d*x+c))-3/2/d*a*b^2*coth(d*x
+c)^2+1/d*b^3*ln(sinh(d*x+c))
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Maxima [B] time = 1.07398, size = 567, normalized size = 5.5 \begin{align*} \frac{1}{3} \, a^{3}{\left (3 \, x + \frac{3 \, c}{d} + \frac{3 \, \log \left (e^{\left (-d x - c\right )} + 1\right )}{d} + \frac{3 \, \log \left (e^{\left (-d x - c\right )} - 1\right )}{d} + \frac{2 \,{\left (9 \, e^{\left (-2 \, d x - 2 \, c\right )} - 18 \, e^{\left (-4 \, d x - 4 \, c\right )} + 34 \, e^{\left (-6 \, d x - 6 \, c\right )} - 18 \, e^{\left (-8 \, d x - 8 \, c\right )} + 9 \, e^{\left (-10 \, d x - 10 \, 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^{2} b{\left (x + \frac{c}{d} + \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{4 \,{\left (e^{\left (-2 \, d x - 2 \, c\right )} - e^{\left (-4 \, d x - 4 \, c\right )} + e^{\left (-6 \, d x - 6 \, c\right )}\right )}}{d{\left (4 \, e^{\left (-2 \, d x - 2 \, c\right )} - 6 \, e^{\left (-4 \, d x - 4 \, c\right )} + 4 \, e^{\left (-6 \, d x - 6 \, c\right )} - e^{\left (-8 \, d x - 8 \, c\right )} - 1\right )}}\right )} + 3 \, a b^{2}{\left (x + \frac{c}{d} + \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 \, e^{\left (-2 \, d x - 2 \, c\right )}}{d{\left (2 \, e^{\left (-2 \, d x - 2 \, c\right )} - e^{\left (-4 \, d x - 4 \, c\right )} - 1\right )}}\right )} + \frac{b^{3} \log \left (e^{\left (d x + c\right )} - e^{\left (-d x - c\right )}\right )}{d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(coth(d*x+c)^7*(a+b*tanh(d*x+c)^2)^3,x, algorithm="maxima")
[Out]
1/3*a^3*(3*x + 3*c/d + 3*log(e^(-d*x - c) + 1)/d + 3*log(e^(-d*x - c) - 1)/d + 2*(9*e^(-2*d*x - 2*c) - 18*e^(-
4*d*x - 4*c) + 34*e^(-6*d*x - 6*c) - 18*e^(-8*d*x - 8*c) + 9*e^(-10*d*x - 10*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)))
+ 3*a^2*b*(x + c/d + log(e^(-d*x - c) + 1)/d + log(e^(-d*x - c) - 1)/d + 4*(e^(-2*d*x - 2*c) - e^(-4*d*x - 4*
c) + e^(-6*d*x - 6*c))/(d*(4*e^(-2*d*x - 2*c) - 6*e^(-4*d*x - 4*c) + 4*e^(-6*d*x - 6*c) - e^(-8*d*x - 8*c) - 1
))) + 3*a*b^2*(x + c/d + log(e^(-d*x - c) + 1)/d + log(e^(-d*x - c) - 1)/d + 2*e^(-2*d*x - 2*c)/(d*(2*e^(-2*d*
x - 2*c) - e^(-4*d*x - 4*c) - 1))) + b^3*log(e^(d*x + c) - e^(-d*x - c))/d
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Fricas [B] time = 2.87791, size = 10689, normalized size = 103.78 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(coth(d*x+c)^7*(a+b*tanh(d*x+c)^2)^3,x, algorithm="fricas")
[Out]
-1/3*(3*(a^3 + 3*a^2*b + 3*a*b^2 + b^3)*d*x*cosh(d*x + c)^12 + 36*(a^3 + 3*a^2*b + 3*a*b^2 + b^3)*d*x*cosh(d*x
+ c)*sinh(d*x + c)^11 + 3*(a^3 + 3*a^2*b + 3*a*b^2 + b^3)*d*x*sinh(d*x + c)^12 + 18*(a^3 + 2*a^2*b + a*b^2 -
(a^3 + 3*a^2*b + 3*a*b^2 + b^3)*d*x)*cosh(d*x + c)^10 + 18*(11*(a^3 + 3*a^2*b + 3*a*b^2 + b^3)*d*x*cosh(d*x +
c)^2 + a^3 + 2*a^2*b + a*b^2 - (a^3 + 3*a^2*b + 3*a*b^2 + b^3)*d*x)*sinh(d*x + c)^10 + 60*(11*(a^3 + 3*a^2*b +
3*a*b^2 + b^3)*d*x*cosh(d*x + c)^3 + 3*(a^3 + 2*a^2*b + a*b^2 - (a^3 + 3*a^2*b + 3*a*b^2 + b^3)*d*x)*cosh(d*x
+ c))*sinh(d*x + c)^9 - 9*(4*a^3 + 12*a^2*b + 8*a*b^2 - 5*(a^3 + 3*a^2*b + 3*a*b^2 + b^3)*d*x)*cosh(d*x + c)^
8 + 9*(165*(a^3 + 3*a^2*b + 3*a*b^2 + b^3)*d*x*cosh(d*x + c)^4 - 4*a^3 - 12*a^2*b - 8*a*b^2 + 5*(a^3 + 3*a^2*b
+ 3*a*b^2 + b^3)*d*x + 90*(a^3 + 2*a^2*b + a*b^2 - (a^3 + 3*a^2*b + 3*a*b^2 + b^3)*d*x)*cosh(d*x + c)^2)*sinh
(d*x + c)^8 + 72*(33*(a^3 + 3*a^2*b + 3*a*b^2 + b^3)*d*x*cosh(d*x + c)^5 + 30*(a^3 + 2*a^2*b + a*b^2 - (a^3 +
3*a^2*b + 3*a*b^2 + b^3)*d*x)*cosh(d*x + c)^3 - (4*a^3 + 12*a^2*b + 8*a*b^2 - 5*(a^3 + 3*a^2*b + 3*a*b^2 + b^3
)*d*x)*cosh(d*x + c))*sinh(d*x + c)^7 + 4*(17*a^3 + 36*a^2*b + 27*a*b^2 - 15*(a^3 + 3*a^2*b + 3*a*b^2 + b^3)*d
*x)*cosh(d*x + c)^6 + 4*(693*(a^3 + 3*a^2*b + 3*a*b^2 + b^3)*d*x*cosh(d*x + c)^6 + 945*(a^3 + 2*a^2*b + a*b^2
- (a^3 + 3*a^2*b + 3*a*b^2 + b^3)*d*x)*cosh(d*x + c)^4 + 17*a^3 + 36*a^2*b + 27*a*b^2 - 15*(a^3 + 3*a^2*b + 3*
a*b^2 + b^3)*d*x - 63*(4*a^3 + 12*a^2*b + 8*a*b^2 - 5*(a^3 + 3*a^2*b + 3*a*b^2 + b^3)*d*x)*cosh(d*x + c)^2)*si
nh(d*x + c)^6 + 24*(99*(a^3 + 3*a^2*b + 3*a*b^2 + b^3)*d*x*cosh(d*x + c)^7 + 189*(a^3 + 2*a^2*b + a*b^2 - (a^3
+ 3*a^2*b + 3*a*b^2 + b^3)*d*x)*cosh(d*x + c)^5 - 21*(4*a^3 + 12*a^2*b + 8*a*b^2 - 5*(a^3 + 3*a^2*b + 3*a*b^2
+ b^3)*d*x)*cosh(d*x + c)^3 + (17*a^3 + 36*a^2*b + 27*a*b^2 - 15*(a^3 + 3*a^2*b + 3*a*b^2 + b^3)*d*x)*cosh(d*
x + c))*sinh(d*x + c)^5 - 9*(4*a^3 + 12*a^2*b + 8*a*b^2 - 5*(a^3 + 3*a^2*b + 3*a*b^2 + b^3)*d*x)*cosh(d*x + c)
^4 + 3*(495*(a^3 + 3*a^2*b + 3*a*b^2 + b^3)*d*x*cosh(d*x + c)^8 + 1260*(a^3 + 2*a^2*b + a*b^2 - (a^3 + 3*a^2*b
+ 3*a*b^2 + b^3)*d*x)*cosh(d*x + c)^6 - 210*(4*a^3 + 12*a^2*b + 8*a*b^2 - 5*(a^3 + 3*a^2*b + 3*a*b^2 + b^3)*d
*x)*cosh(d*x + c)^4 - 12*a^3 - 36*a^2*b - 24*a*b^2 + 15*(a^3 + 3*a^2*b + 3*a*b^2 + b^3)*d*x + 20*(17*a^3 + 36*
a^2*b + 27*a*b^2 - 15*(a^3 + 3*a^2*b + 3*a*b^2 + b^3)*d*x)*cosh(d*x + c)^2)*sinh(d*x + c)^4 + 4*(165*(a^3 + 3*
a^2*b + 3*a*b^2 + b^3)*d*x*cosh(d*x + c)^9 + 540*(a^3 + 2*a^2*b + a*b^2 - (a^3 + 3*a^2*b + 3*a*b^2 + b^3)*d*x)
*cosh(d*x + c)^7 - 126*(4*a^3 + 12*a^2*b + 8*a*b^2 - 5*(a^3 + 3*a^2*b + 3*a*b^2 + b^3)*d*x)*cosh(d*x + c)^5 +
20*(17*a^3 + 36*a^2*b + 27*a*b^2 - 15*(a^3 + 3*a^2*b + 3*a*b^2 + b^3)*d*x)*cosh(d*x + c)^3 - 9*(4*a^3 + 12*a^2
*b + 8*a*b^2 - 5*(a^3 + 3*a^2*b + 3*a*b^2 + b^3)*d*x)*cosh(d*x + c))*sinh(d*x + c)^3 + 3*(a^3 + 3*a^2*b + 3*a*
b^2 + b^3)*d*x + 18*(a^3 + 2*a^2*b + a*b^2 - (a^3 + 3*a^2*b + 3*a*b^2 + b^3)*d*x)*cosh(d*x + c)^2 + 6*(33*(a^3
+ 3*a^2*b + 3*a*b^2 + b^3)*d*x*cosh(d*x + c)^10 + 135*(a^3 + 2*a^2*b + a*b^2 - (a^3 + 3*a^2*b + 3*a*b^2 + b^3
)*d*x)*cosh(d*x + c)^8 - 42*(4*a^3 + 12*a^2*b + 8*a*b^2 - 5*(a^3 + 3*a^2*b + 3*a*b^2 + b^3)*d*x)*cosh(d*x + c)
^6 + 10*(17*a^3 + 36*a^2*b + 27*a*b^2 - 15*(a^3 + 3*a^2*b + 3*a*b^2 + b^3)*d*x)*cosh(d*x + c)^4 + 3*a^3 + 6*a^
2*b + 3*a*b^2 - 3*(a^3 + 3*a^2*b + 3*a*b^2 + b^3)*d*x - 9*(4*a^3 + 12*a^2*b + 8*a*b^2 - 5*(a^3 + 3*a^2*b + 3*a
*b^2 + b^3)*d*x)*cosh(d*x + c)^2)*sinh(d*x + c)^2 - 3*((a^3 + 3*a^2*b + 3*a*b^2 + b^3)*cosh(d*x + c)^12 + 12*(
a^3 + 3*a^2*b + 3*a*b^2 + b^3)*cosh(d*x + c)*sinh(d*x + c)^11 + (a^3 + 3*a^2*b + 3*a*b^2 + b^3)*sinh(d*x + c)^
12 - 6*(a^3 + 3*a^2*b + 3*a*b^2 + b^3)*cosh(d*x + c)^10 - 6*(a^3 + 3*a^2*b + 3*a*b^2 + b^3 - 11*(a^3 + 3*a^2*b
+ 3*a*b^2 + b^3)*cosh(d*x + c)^2)*sinh(d*x + c)^10 + 20*(11*(a^3 + 3*a^2*b + 3*a*b^2 + b^3)*cosh(d*x + c)^3 -
3*(a^3 + 3*a^2*b + 3*a*b^2 + b^3)*cosh(d*x + c))*sinh(d*x + c)^9 + 15*(a^3 + 3*a^2*b + 3*a*b^2 + b^3)*cosh(d*
x + c)^8 + 15*(33*(a^3 + 3*a^2*b + 3*a*b^2 + b^3)*cosh(d*x + c)^4 + a^3 + 3*a^2*b + 3*a*b^2 + b^3 - 18*(a^3 +
3*a^2*b + 3*a*b^2 + b^3)*cosh(d*x + c)^2)*sinh(d*x + c)^8 + 24*(33*(a^3 + 3*a^2*b + 3*a*b^2 + b^3)*cosh(d*x +
c)^5 - 30*(a^3 + 3*a^2*b + 3*a*b^2 + b^3)*cosh(d*x + c)^3 + 5*(a^3 + 3*a^2*b + 3*a*b^2 + b^3)*cosh(d*x + c))*s
inh(d*x + c)^7 - 20*(a^3 + 3*a^2*b + 3*a*b^2 + b^3)*cosh(d*x + c)^6 + 4*(231*(a^3 + 3*a^2*b + 3*a*b^2 + b^3)*c
osh(d*x + c)^6 - 315*(a^3 + 3*a^2*b + 3*a*b^2 + b^3)*cosh(d*x + c)^4 - 5*a^3 - 15*a^2*b - 15*a*b^2 - 5*b^3 + 1
05*(a^3 + 3*a^2*b + 3*a*b^2 + b^3)*cosh(d*x + c)^2)*sinh(d*x + c)^6 + 24*(33*(a^3 + 3*a^2*b + 3*a*b^2 + b^3)*c
osh(d*x + c)^7 - 63*(a^3 + 3*a^2*b + 3*a*b^2 + b^3)*cosh(d*x + c)^5 + 35*(a^3 + 3*a^2*b + 3*a*b^2 + b^3)*cosh(
d*x + c)^3 - 5*(a^3 + 3*a^2*b + 3*a*b^2 + b^3)*cosh(d*x + c))*sinh(d*x + c)^5 + 15*(a^3 + 3*a^2*b + 3*a*b^2 +
b^3)*cosh(d*x + c)^4 + 15*(33*(a^3 + 3*a^2*b + 3*a*b^2 + b^3)*cosh(d*x + c)^8 - 84*(a^3 + 3*a^2*b + 3*a*b^2 +
b^3)*cosh(d*x + c)^6 + 70*(a^3 + 3*a^2*b + 3*a*b^2 + b^3)*cosh(d*x + c)^4 + a^3 + 3*a^2*b + 3*a*b^2 + b^3 - 20
*(a^3 + 3*a^2*b + 3*a*b^2 + b^3)*cosh(d*x + c)^2)*sinh(d*x + c)^4 + 20*(11*(a^3 + 3*a^2*b + 3*a*b^2 + b^3)*cos
h(d*x + c)^9 - 36*(a^3 + 3*a^2*b + 3*a*b^2 + b^3)*cosh(d*x + c)^7 + 42*(a^3 + 3*a^2*b + 3*a*b^2 + b^3)*cosh(d*
x + c)^5 - 20*(a^3 + 3*a^2*b + 3*a*b^2 + b^3)*cosh(d*x + c)^3 + 3*(a^3 + 3*a^2*b + 3*a*b^2 + b^3)*cosh(d*x + c
))*sinh(d*x + c)^3 + a^3 + 3*a^2*b + 3*a*b^2 + b^3 - 6*(a^3 + 3*a^2*b + 3*a*b^2 + b^3)*cosh(d*x + c)^2 + 6*(11
*(a^3 + 3*a^2*b + 3*a*b^2 + b^3)*cosh(d*x + c)^10 - 45*(a^3 + 3*a^2*b + 3*a*b^2 + b^3)*cosh(d*x + c)^8 + 70*(a
^3 + 3*a^2*b + 3*a*b^2 + b^3)*cosh(d*x + c)^6 - 50*(a^3 + 3*a^2*b + 3*a*b^2 + b^3)*cosh(d*x + c)^4 - a^3 - 3*a
^2*b - 3*a*b^2 - b^3 + 15*(a^3 + 3*a^2*b + 3*a*b^2 + b^3)*cosh(d*x + c)^2)*sinh(d*x + c)^2 + 12*((a^3 + 3*a^2*
b + 3*a*b^2 + b^3)*cosh(d*x + c)^11 - 5*(a^3 + 3*a^2*b + 3*a*b^2 + b^3)*cosh(d*x + c)^9 + 10*(a^3 + 3*a^2*b +
3*a*b^2 + b^3)*cosh(d*x + c)^7 - 10*(a^3 + 3*a^2*b + 3*a*b^2 + b^3)*cosh(d*x + c)^5 + 5*(a^3 + 3*a^2*b + 3*a*b
^2 + b^3)*cosh(d*x + c)^3 - (a^3 + 3*a^2*b + 3*a*b^2 + b^3)*cosh(d*x + c))*sinh(d*x + c))*log(2*sinh(d*x + c)/
(cosh(d*x + c) - sinh(d*x + c))) + 12*(3*(a^3 + 3*a^2*b + 3*a*b^2 + b^3)*d*x*cosh(d*x + c)^11 + 15*(a^3 + 2*a^
2*b + a*b^2 - (a^3 + 3*a^2*b + 3*a*b^2 + b^3)*d*x)*cosh(d*x + c)^9 - 6*(4*a^3 + 12*a^2*b + 8*a*b^2 - 5*(a^3 +
3*a^2*b + 3*a*b^2 + b^3)*d*x)*cosh(d*x + c)^7 + 2*(17*a^3 + 36*a^2*b + 27*a*b^2 - 15*(a^3 + 3*a^2*b + 3*a*b^2
+ b^3)*d*x)*cosh(d*x + c)^5 - 3*(4*a^3 + 12*a^2*b + 8*a*b^2 - 5*(a^3 + 3*a^2*b + 3*a*b^2 + b^3)*d*x)*cosh(d*x
+ c)^3 + 3*(a^3 + 2*a^2*b + a*b^2 - (a^3 + 3*a^2*b + 3*a*b^2 + b^3)*d*x)*cosh(d*x + c))*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*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 - 5
0*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(coth(d*x+c)**7*(a+b*tanh(d*x+c)**2)**3,x)
[Out]
Timed out
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Giac [B] time = 1.53015, size = 297, normalized size = 2.88 \begin{align*} -\frac{{\left (a^{3} + 3 \, a^{2} b + 3 \, a b^{2} + b^{3}\right )}{\left (d x + c\right )}}{d} + \frac{{\left (a^{3} + 3 \, a^{2} b + 3 \, a b^{2} + b^{3}\right )} \log \left ({\left | e^{\left (2 \, d x + 2 \, c\right )} - 1 \right |}\right )}{d} - \frac{2 \,{\left (9 \,{\left (a^{3} + 2 \, a^{2} b + a b^{2}\right )} e^{\left (10 \, d x + 10 \, c\right )} - 18 \,{\left (a^{3} + 3 \, a^{2} b + 2 \, a b^{2}\right )} e^{\left (8 \, d x + 8 \, c\right )} + 2 \,{\left (17 \, a^{3} + 36 \, a^{2} b + 27 \, a b^{2}\right )} e^{\left (6 \, d x + 6 \, c\right )} - 18 \,{\left (a^{3} + 3 \, a^{2} b + 2 \, a b^{2}\right )} e^{\left (4 \, d x + 4 \, c\right )} + 9 \,{\left (a^{3} + 2 \, a^{2} b + a b^{2}\right )} e^{\left (2 \, d x + 2 \, c\right )}\right )}}{3 \, 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(coth(d*x+c)^7*(a+b*tanh(d*x+c)^2)^3,x, algorithm="giac")
[Out]
-(a^3 + 3*a^2*b + 3*a*b^2 + b^3)*(d*x + c)/d + (a^3 + 3*a^2*b + 3*a*b^2 + b^3)*log(abs(e^(2*d*x + 2*c) - 1))/d
- 2/3*(9*(a^3 + 2*a^2*b + a*b^2)*e^(10*d*x + 10*c) - 18*(a^3 + 3*a^2*b + 2*a*b^2)*e^(8*d*x + 8*c) + 2*(17*a^3
+ 36*a^2*b + 27*a*b^2)*e^(6*d*x + 6*c) - 18*(a^3 + 3*a^2*b + 2*a*b^2)*e^(4*d*x + 4*c) + 9*(a^3 + 2*a^2*b + a*
b^2)*e^(2*d*x + 2*c))/(d*(e^(2*d*x + 2*c) - 1)^6)