Integrand size = 23, antiderivative size = 81 \[ \int \frac {\tanh ^3(c+d x)}{\left (a+b \text {sech}^2(c+d x)\right )^3} \, dx=-\frac {b (a+b)}{4 a^3 d \left (b+a \cosh ^2(c+d x)\right )^2}+\frac {a+2 b}{2 a^3 d \left (b+a \cosh ^2(c+d x)\right )}+\frac {\log \left (b+a \cosh ^2(c+d x)\right )}{2 a^3 d} \]
-1/4*b*(a+b)/a^3/d/(b+a*cosh(d*x+c)^2)^2+1/2*(a+2*b)/a^3/d/(b+a*cosh(d*x+c )^2)+1/2*ln(b+a*cosh(d*x+c)^2)/a^3/d
Time = 1.45 (sec) , antiderivative size = 131, normalized size of antiderivative = 1.62 \[ \int \frac {\tanh ^3(c+d x)}{\left (a+b \text {sech}^2(c+d x)\right )^3} \, dx=\frac {2 \left (a^2+3 a b+3 b^2\right )+(a+2 b)^2 \log (a+2 b+a \cosh (2 (c+d x)))+a^2 \cosh ^2(2 (c+d x)) \log (a+2 b+a \cosh (2 (c+d x)))+2 a (a+2 b) \cosh (2 (c+d x)) (1+\log (a+2 b+a \cosh (2 (c+d x))))}{2 a^3 d (a+2 b+a \cosh (2 (c+d x)))^2} \]
(2*(a^2 + 3*a*b + 3*b^2) + (a + 2*b)^2*Log[a + 2*b + a*Cosh[2*(c + d*x)]] + a^2*Cosh[2*(c + d*x)]^2*Log[a + 2*b + a*Cosh[2*(c + d*x)]] + 2*a*(a + 2* b)*Cosh[2*(c + d*x)]*(1 + Log[a + 2*b + a*Cosh[2*(c + d*x)]]))/(2*a^3*d*(a + 2*b + a*Cosh[2*(c + d*x)])^2)
Time = 0.32 (sec) , antiderivative size = 75, normalized size of antiderivative = 0.93, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.261, Rules used = {3042, 26, 4626, 354, 86, 2009}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
\(\displaystyle \int \frac {\tanh ^3(c+d x)}{\left (a+b \text {sech}^2(c+d x)\right )^3} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {i \tan (i c+i d x)^3}{\left (a+b \sec (i c+i d x)^2\right )^3}dx\) |
\(\Big \downarrow \) 26 |
\(\displaystyle i \int \frac {\tan (i c+i d x)^3}{\left (b \sec (i c+i d x)^2+a\right )^3}dx\) |
\(\Big \downarrow \) 4626 |
\(\displaystyle -\frac {\int \frac {\cosh ^3(c+d x) \left (1-\cosh ^2(c+d x)\right )}{\left (a \cosh ^2(c+d x)+b\right )^3}d\cosh (c+d x)}{d}\) |
\(\Big \downarrow \) 354 |
\(\displaystyle -\frac {\int \frac {\cosh ^2(c+d x) \left (1-\cosh ^2(c+d x)\right )}{\left (a \cosh ^2(c+d x)+b\right )^3}d\cosh ^2(c+d x)}{2 d}\) |
\(\Big \downarrow \) 86 |
\(\displaystyle -\frac {\int \left (-\frac {b (a+b)}{a^2 \left (a \cosh ^2(c+d x)+b\right )^3}-\frac {1}{a^2 \left (a \cosh ^2(c+d x)+b\right )}+\frac {a+2 b}{a^2 \left (a \cosh ^2(c+d x)+b\right )^2}\right )d\cosh ^2(c+d x)}{2 d}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -\frac {\frac {b (a+b)}{2 a^3 \left (a \cosh ^2(c+d x)+b\right )^2}-\frac {a+2 b}{a^3 \left (a \cosh ^2(c+d x)+b\right )}-\frac {\log \left (a \cosh ^2(c+d x)+b\right )}{a^3}}{2 d}\) |
-1/2*((b*(a + b))/(2*a^3*(b + a*Cosh[c + d*x]^2)^2) - (a + 2*b)/(a^3*(b + a*Cosh[c + d*x]^2)) - Log[b + a*Cosh[c + d*x]^2]/a^3)/d
3.2.61.3.1 Defintions of rubi rules used
Int[(Complex[0, a_])*(Fx_), x_Symbol] :> Simp[(Complex[Identity[0], a]) I nt[Fx, x], x] /; FreeQ[a, x] && EqQ[a^2, 1]
Int[((a_.) + (b_.)*(x_))*((c_) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_ .), x_] :> Int[ExpandIntegrand[(a + b*x)*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && ((ILtQ[n, 0] && ILtQ[p, 0]) || EqQ[p, 1 ] || (IGtQ[p, 0] && ( !IntegerQ[n] || LeQ[9*p + 5*(n + 2), 0] || GeQ[n + p + 1, 0] || (GeQ[n + p + 2, 0] && RationalQ[a, b, c, d, e, f]))))
Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^2)^(p_.)*((c_) + (d_.)*(x_)^2)^(q_.), x_S ymbol] :> Simp[1/2 Subst[Int[x^((m - 1)/2)*(a + b*x)^p*(c + d*x)^q, x], x , x^2], x] /; FreeQ[{a, b, c, d, p, q}, x] && NeQ[b*c - a*d, 0] && IntegerQ [(m - 1)/2]
Int[((a_) + (b_.)*sec[(e_.) + (f_.)*(x_)]^(n_))^(p_.)*tan[(e_.) + (f_.)*(x_ )]^(m_.), x_Symbol] :> Module[{ff = FreeFactors[Cos[e + f*x], x]}, Simp[-(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]
Leaf count of result is larger than twice the leaf count of optimal. \(186\) vs. \(2(75)=150\).
Time = 46.04 (sec) , antiderivative size = 187, normalized size of antiderivative = 2.31
method | result | size |
risch | \(-\frac {x}{a^{3}}-\frac {2 c}{a^{3} d}+\frac {2 \,{\mathrm e}^{2 d x +2 c} \left (a^{2} {\mathrm e}^{4 d x +4 c}+2 a b \,{\mathrm e}^{4 d x +4 c}+2 a^{2} {\mathrm e}^{2 d x +2 c}+6 a b \,{\mathrm e}^{2 d x +2 c}+6 \,{\mathrm e}^{2 d x +2 c} b^{2}+a^{2}+2 a b \right )}{a^{3} d \left (a \,{\mathrm e}^{4 d x +4 c}+2 \,{\mathrm e}^{2 d x +2 c} a +4 b \,{\mathrm e}^{2 d x +2 c}+a \right )^{2}}+\frac {\ln \left ({\mathrm e}^{4 d x +4 c}+\frac {2 \left (a +2 b \right ) {\mathrm e}^{2 d x +2 c}}{a}+1\right )}{2 a^{3} d}\) | \(187\) |
derivativedivides | \(\frac {-\frac {\ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{a^{3}}+\frac {\frac {\left (-2 a^{2}-2 a b \right ) \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{6}-\frac {4 a \left (a^{2}+a b -b^{2}\right ) \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}}{a +b}-2 a \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} \left (a +b \right )}{\left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{4} a +\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{4} b +2 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} a -2 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} b +a +b \right )^{2}}+\frac {\ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{4} a +\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{4} b +2 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} a -2 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} b +a +b \right )}{2}}{a^{3}}-\frac {\ln \left (1+\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a^{3}}}{d}\) | \(236\) |
default | \(\frac {-\frac {\ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{a^{3}}+\frac {\frac {\left (-2 a^{2}-2 a b \right ) \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{6}-\frac {4 a \left (a^{2}+a b -b^{2}\right ) \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}}{a +b}-2 a \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} \left (a +b \right )}{\left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{4} a +\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{4} b +2 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} a -2 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} b +a +b \right )^{2}}+\frac {\ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{4} a +\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{4} b +2 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} a -2 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} b +a +b \right )}{2}}{a^{3}}-\frac {\ln \left (1+\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a^{3}}}{d}\) | \(236\) |
-x/a^3-2/a^3/d*c+2/a^3*exp(2*d*x+2*c)*(a^2*exp(4*d*x+4*c)+2*a*b*exp(4*d*x+ 4*c)+2*a^2*exp(2*d*x+2*c)+6*a*b*exp(2*d*x+2*c)+6*exp(2*d*x+2*c)*b^2+a^2+2* a*b)/d/(a*exp(4*d*x+4*c)+2*exp(2*d*x+2*c)*a+4*b*exp(2*d*x+2*c)+a)^2+1/2/a^ 3/d*ln(exp(4*d*x+4*c)+2*(a+2*b)/a*exp(2*d*x+2*c)+1)
Leaf count of result is larger than twice the leaf count of optimal. 1753 vs. \(2 (75) = 150\).
Time = 0.29 (sec) , antiderivative size = 1753, normalized size of antiderivative = 21.64 \[ \int \frac {\tanh ^3(c+d x)}{\left (a+b \text {sech}^2(c+d x)\right )^3} \, dx=\text {Too large to display} \]
-1/2*(2*a^2*d*x*cosh(d*x + c)^8 + 16*a^2*d*x*cosh(d*x + c)*sinh(d*x + c)^7 + 2*a^2*d*x*sinh(d*x + c)^8 + 4*(2*(a^2 + 2*a*b)*d*x - a^2 - 2*a*b)*cosh( d*x + c)^6 + 4*(14*a^2*d*x*cosh(d*x + c)^2 + 2*(a^2 + 2*a*b)*d*x - a^2 - 2 *a*b)*sinh(d*x + c)^6 + 8*(14*a^2*d*x*cosh(d*x + c)^3 + 3*(2*(a^2 + 2*a*b) *d*x - a^2 - 2*a*b)*cosh(d*x + c))*sinh(d*x + c)^5 + 4*((3*a^2 + 8*a*b + 8 *b^2)*d*x - 2*a^2 - 6*a*b - 6*b^2)*cosh(d*x + c)^4 + 4*(35*a^2*d*x*cosh(d* x + c)^4 + (3*a^2 + 8*a*b + 8*b^2)*d*x + 15*(2*(a^2 + 2*a*b)*d*x - a^2 - 2 *a*b)*cosh(d*x + c)^2 - 2*a^2 - 6*a*b - 6*b^2)*sinh(d*x + c)^4 + 2*a^2*d*x + 16*(7*a^2*d*x*cosh(d*x + c)^5 + 5*(2*(a^2 + 2*a*b)*d*x - a^2 - 2*a*b)*c osh(d*x + c)^3 + ((3*a^2 + 8*a*b + 8*b^2)*d*x - 2*a^2 - 6*a*b - 6*b^2)*cos h(d*x + c))*sinh(d*x + c)^3 + 4*(2*(a^2 + 2*a*b)*d*x - a^2 - 2*a*b)*cosh(d *x + c)^2 + 4*(14*a^2*d*x*cosh(d*x + c)^6 + 15*(2*(a^2 + 2*a*b)*d*x - a^2 - 2*a*b)*cosh(d*x + c)^4 + 2*(a^2 + 2*a*b)*d*x + 6*((3*a^2 + 8*a*b + 8*b^2 )*d*x - 2*a^2 - 6*a*b - 6*b^2)*cosh(d*x + c)^2 - a^2 - 2*a*b)*sinh(d*x + c )^2 - (a^2*cosh(d*x + c)^8 + 8*a^2*cosh(d*x + c)*sinh(d*x + c)^7 + a^2*sin h(d*x + c)^8 + 4*(a^2 + 2*a*b)*cosh(d*x + c)^6 + 4*(7*a^2*cosh(d*x + c)^2 + a^2 + 2*a*b)*sinh(d*x + c)^6 + 8*(7*a^2*cosh(d*x + c)^3 + 3*(a^2 + 2*a*b )*cosh(d*x + c))*sinh(d*x + c)^5 + 2*(3*a^2 + 8*a*b + 8*b^2)*cosh(d*x + c) ^4 + 2*(35*a^2*cosh(d*x + c)^4 + 30*(a^2 + 2*a*b)*cosh(d*x + c)^2 + 3*a^2 + 8*a*b + 8*b^2)*sinh(d*x + c)^4 + 8*(7*a^2*cosh(d*x + c)^5 + 10*(a^2 +...
Timed out. \[ \int \frac {\tanh ^3(c+d x)}{\left (a+b \text {sech}^2(c+d x)\right )^3} \, dx=\text {Timed out} \]
Leaf count of result is larger than twice the leaf count of optimal. 209 vs. \(2 (75) = 150\).
Time = 0.21 (sec) , antiderivative size = 209, normalized size of antiderivative = 2.58 \[ \int \frac {\tanh ^3(c+d x)}{\left (a+b \text {sech}^2(c+d x)\right )^3} \, dx=\frac {2 \, {\left ({\left (a^{2} + 2 \, a b\right )} e^{\left (-2 \, d x - 2 \, c\right )} + 2 \, {\left (a^{2} + 3 \, a b + 3 \, b^{2}\right )} e^{\left (-4 \, d x - 4 \, c\right )} + {\left (a^{2} + 2 \, a b\right )} e^{\left (-6 \, d x - 6 \, c\right )}\right )}}{{\left (a^{5} e^{\left (-8 \, d x - 8 \, c\right )} + a^{5} + 4 \, {\left (a^{5} + 2 \, a^{4} b\right )} e^{\left (-2 \, d x - 2 \, c\right )} + 2 \, {\left (3 \, a^{5} + 8 \, a^{4} b + 8 \, a^{3} b^{2}\right )} e^{\left (-4 \, d x - 4 \, c\right )} + 4 \, {\left (a^{5} + 2 \, a^{4} b\right )} e^{\left (-6 \, d x - 6 \, c\right )}\right )} d} + \frac {d x + c}{a^{3} d} + \frac {\log \left (2 \, {\left (a + 2 \, b\right )} e^{\left (-2 \, d x - 2 \, c\right )} + a e^{\left (-4 \, d x - 4 \, c\right )} + a\right )}{2 \, a^{3} d} \]
2*((a^2 + 2*a*b)*e^(-2*d*x - 2*c) + 2*(a^2 + 3*a*b + 3*b^2)*e^(-4*d*x - 4* c) + (a^2 + 2*a*b)*e^(-6*d*x - 6*c))/((a^5*e^(-8*d*x - 8*c) + a^5 + 4*(a^5 + 2*a^4*b)*e^(-2*d*x - 2*c) + 2*(3*a^5 + 8*a^4*b + 8*a^3*b^2)*e^(-4*d*x - 4*c) + 4*(a^5 + 2*a^4*b)*e^(-6*d*x - 6*c))*d) + (d*x + c)/(a^3*d) + 1/2*l og(2*(a + 2*b)*e^(-2*d*x - 2*c) + a*e^(-4*d*x - 4*c) + a)/(a^3*d)
\[ \int \frac {\tanh ^3(c+d x)}{\left (a+b \text {sech}^2(c+d x)\right )^3} \, dx=\int { \frac {\tanh \left (d x + c\right )^{3}}{{\left (b \operatorname {sech}\left (d x + c\right )^{2} + a\right )}^{3}} \,d x } \]
Timed out. \[ \int \frac {\tanh ^3(c+d x)}{\left (a+b \text {sech}^2(c+d x)\right )^3} \, dx=\int \frac {{\mathrm {cosh}\left (c+d\,x\right )}^6\,{\mathrm {tanh}\left (c+d\,x\right )}^3}{{\left (a\,{\mathrm {cosh}\left (c+d\,x\right )}^2+b\right )}^3} \,d x \]