Integrand size = 23, antiderivative size = 61 \[ \int \coth ^2(c+d x) \left (a+b \text {sech}^2(c+d x)\right )^3 \, dx=a^3 x-\frac {(a+b)^3 \coth (c+d x)}{d}-\frac {b^2 (3 a+2 b) \tanh (c+d x)}{d}+\frac {b^3 \tanh ^3(c+d x)}{3 d} \]
Leaf count is larger than twice the leaf count of optimal. \(162\) vs. \(2(61)=122\).
Time = 6.79 (sec) , antiderivative size = 162, normalized size of antiderivative = 2.66 \[ \int \coth ^2(c+d x) \left (a+b \text {sech}^2(c+d x)\right )^3 \, dx=\frac {8 (a \cosh (c+d x)+b \text {sech}(c+d x))^3 \left (3 a^3 d x \cosh ^3(c+d x)-b^3 \text {sech}(c) \sinh (d x)+\frac {1}{2} \cosh (c+d x) \left (\left (3 a^3+9 a^2 b+18 a b^2+8 b^3\right ) \cosh (d x)+\left (3 a^3+9 a^2 b-2 b^3\right ) \cosh (2 c+d x)\right ) \coth (c+d x) \text {csch}(c) \text {sech}(c) \sinh (d x)-b^3 \cosh (c+d x) \tanh (c)\right )}{3 d (a+2 b+a \cosh (2 (c+d x)))^3} \]
(8*(a*Cosh[c + d*x] + b*Sech[c + d*x])^3*(3*a^3*d*x*Cosh[c + d*x]^3 - b^3* Sech[c]*Sinh[d*x] + (Cosh[c + d*x]*((3*a^3 + 9*a^2*b + 18*a*b^2 + 8*b^3)*C osh[d*x] + (3*a^3 + 9*a^2*b - 2*b^3)*Cosh[2*c + d*x])*Coth[c + d*x]*Csch[c ]*Sech[c]*Sinh[d*x])/2 - b^3*Cosh[c + d*x]*Tanh[c]))/(3*d*(a + 2*b + a*Cos h[2*(c + d*x)])^3)
Time = 0.35 (sec) , antiderivative size = 62, normalized size of antiderivative = 1.02, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.304, Rules used = {3042, 25, 4629, 25, 2075, 364, 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.
\(\displaystyle \int \coth ^2(c+d x) \left (a+b \text {sech}^2(c+d x)\right )^3 \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int -\frac {\left (a+b \sec (i c+i d x)^2\right )^3}{\tan (i c+i d x)^2}dx\) |
\(\Big \downarrow \) 25 |
\(\displaystyle -\int \frac {\left (b \sec (i c+i d x)^2+a\right )^3}{\tan (i c+i d x)^2}dx\) |
\(\Big \downarrow \) 4629 |
\(\displaystyle -\frac {\int -\frac {\coth ^2(c+d x) \left (a+b \left (1-\tanh ^2(c+d x)\right )\right )^3}{1-\tanh ^2(c+d x)}d\tanh (c+d x)}{d}\) |
\(\Big \downarrow \) 25 |
\(\displaystyle \frac {\int \frac {\coth ^2(c+d x) \left (a+b \left (1-\tanh ^2(c+d x)\right )\right )^3}{1-\tanh ^2(c+d x)}d\tanh (c+d x)}{d}\) |
\(\Big \downarrow \) 2075 |
\(\displaystyle \frac {\int \frac {\coth ^2(c+d x) \left (-b \tanh ^2(c+d x)+a+b\right )^3}{1-\tanh ^2(c+d x)}d\tanh (c+d x)}{d}\) |
\(\Big \downarrow \) 364 |
\(\displaystyle \frac {\int \left (-\frac {a^3}{\tanh ^2(c+d x)-1}+(a+b)^3 \coth ^2(c+d x)+b^3 \tanh ^2(c+d x)-b^2 (3 a+2 b)\right )d\tanh (c+d x)}{d}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -\frac {a^3 (-\text {arctanh}(\tanh (c+d x)))+b^2 (3 a+2 b) \tanh (c+d x)+(a+b)^3 \coth (c+d x)-\frac {1}{3} b^3 \tanh ^3(c+d x)}{d}\) |
-((-(a^3*ArcTanh[Tanh[c + d*x]]) + (a + b)^3*Coth[c + d*x] + b^2*(3*a + 2* b)*Tanh[c + d*x] - (b^3*Tanh[c + d*x]^3)/3)/d)
3.2.30.3.1 Defintions of rubi rules used
Int[(((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_))/((c_) + (d_.)*(x_)^2), x_Symbol] :> Int[ExpandIntegrand[(e*x)^m*((a + b*x^2)^p/(c + d*x^2)), x], x ] /; FreeQ[{a, b, c, d, e, m}, x] && NeQ[b*c - a*d, 0] && IGtQ[p, 0] && (In tegerQ[m] || IGtQ[2*(m + 1), 0] || !RationalQ[m])
Int[(u_)^(p_.)*(v_)^(q_.)*((e_.)*(x_))^(m_.), x_Symbol] :> Int[(e*x)^m*Expa ndToSum[u, x]^p*ExpandToSum[v, x]^q, x] /; FreeQ[{e, m, p, q}, x] && Binomi alQ[{u, v}, x] && EqQ[BinomialDegree[u, x] - BinomialDegree[v, x], 0] && ! BinomialMatchQ[{u, v}, x]
Int[((a_) + (b_.)*sec[(e_.) + (f_.)*(x_)]^(n_))^(p_.)*((d_.)*tan[(e_.) + (f _.)*(x_)])^(m_), x_Symbol] :> With[{ff = FreeFactors[Tan[e + f*x], x]}, Sim p[ff/f Subst[Int[(d*ff*x)^m*((a + b*(1 + ff^2*x^2)^(n/2))^p/(1 + ff^2*x^2 )), x], x, Tan[e + f*x]/ff], x]] /; FreeQ[{a, b, d, e, f, m, p}, x] && Inte gerQ[n/2] && (IntegerQ[m/2] || EqQ[n, 2])
Time = 47.32 (sec) , antiderivative size = 111, normalized size of antiderivative = 1.82
method | result | size |
derivativedivides | \(\frac {a^{3} \left (d x +c -\coth \left (d x +c \right )\right )-3 a^{2} b \coth \left (d x +c \right )+3 a \,b^{2} \left (-\frac {1}{\sinh \left (d x +c \right ) \cosh \left (d x +c \right )}-2 \tanh \left (d x +c \right )\right )+b^{3} \left (-\frac {1}{\sinh \left (d x +c \right ) \cosh \left (d x +c \right )^{3}}-4 \left (\frac {2}{3}+\frac {\operatorname {sech}\left (d x +c \right )^{2}}{3}\right ) \tanh \left (d x +c \right )\right )}{d}\) | \(111\) |
default | \(\frac {a^{3} \left (d x +c -\coth \left (d x +c \right )\right )-3 a^{2} b \coth \left (d x +c \right )+3 a \,b^{2} \left (-\frac {1}{\sinh \left (d x +c \right ) \cosh \left (d x +c \right )}-2 \tanh \left (d x +c \right )\right )+b^{3} \left (-\frac {1}{\sinh \left (d x +c \right ) \cosh \left (d x +c \right )^{3}}-4 \left (\frac {2}{3}+\frac {\operatorname {sech}\left (d x +c \right )^{2}}{3}\right ) \tanh \left (d x +c \right )\right )}{d}\) | \(111\) |
risch | \(a^{3} x -\frac {2 \left (3 a^{3} {\mathrm e}^{6 d x +6 c}+9 a^{2} b \,{\mathrm e}^{6 d x +6 c}+9 a^{3} {\mathrm e}^{4 d x +4 c}+27 a^{2} b \,{\mathrm e}^{4 d x +4 c}+18 a \,b^{2} {\mathrm e}^{4 d x +4 c}+9 a^{3} {\mathrm e}^{2 d x +2 c}+27 a^{2} b \,{\mathrm e}^{2 d x +2 c}+36 \,{\mathrm e}^{2 d x +2 c} a \,b^{2}+16 \,{\mathrm e}^{2 d x +2 c} b^{3}+3 a^{3}+9 a^{2} b +18 a \,b^{2}+8 b^{3}\right )}{3 d \left ({\mathrm e}^{2 d x +2 c}+1\right )^{3} \left ({\mathrm e}^{2 d x +2 c}-1\right )}\) | \(192\) |
1/d*(a^3*(d*x+c-coth(d*x+c))-3*a^2*b*coth(d*x+c)+3*a*b^2*(-1/sinh(d*x+c)/c osh(d*x+c)-2*tanh(d*x+c))+b^3*(-1/sinh(d*x+c)/cosh(d*x+c)^3-4*(2/3+1/3*sec h(d*x+c)^2)*tanh(d*x+c)))
Leaf count of result is larger than twice the leaf count of optimal. 359 vs. \(2 (59) = 118\).
Time = 0.26 (sec) , antiderivative size = 359, normalized size of antiderivative = 5.89 \[ \int \coth ^2(c+d x) \left (a+b \text {sech}^2(c+d x)\right )^3 \, dx=-\frac {{\left (3 \, a^{3} + 9 \, a^{2} b + 18 \, a b^{2} + 8 \, b^{3}\right )} \cosh \left (d x + c\right )^{4} - 4 \, {\left (3 \, a^{3} d x + 3 \, a^{3} + 9 \, a^{2} b + 18 \, a b^{2} + 8 \, b^{3}\right )} \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{3} + {\left (3 \, a^{3} + 9 \, a^{2} b + 18 \, a b^{2} + 8 \, b^{3}\right )} \sinh \left (d x + c\right )^{4} + 9 \, a^{3} + 27 \, a^{2} b + 18 \, a b^{2} + 4 \, {\left (3 \, a^{3} + 9 \, a^{2} b + 9 \, a b^{2} + 4 \, b^{3}\right )} \cosh \left (d x + c\right )^{2} + 2 \, {\left (6 \, a^{3} + 18 \, a^{2} b + 18 \, a b^{2} + 8 \, b^{3} + 3 \, {\left (3 \, a^{3} + 9 \, a^{2} b + 18 \, a b^{2} + 8 \, b^{3}\right )} \cosh \left (d x + c\right )^{2}\right )} \sinh \left (d x + c\right )^{2} - 4 \, {\left ({\left (3 \, a^{3} d x + 3 \, a^{3} + 9 \, a^{2} b + 18 \, a b^{2} + 8 \, b^{3}\right )} \cosh \left (d x + c\right )^{3} + {\left (3 \, a^{3} d x + 3 \, a^{3} + 9 \, a^{2} b + 18 \, a b^{2} + 8 \, b^{3}\right )} \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )}{12 \, {\left (d \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{3} + {\left (d \cosh \left (d x + c\right )^{3} + d \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )\right )}} \]
-1/12*((3*a^3 + 9*a^2*b + 18*a*b^2 + 8*b^3)*cosh(d*x + c)^4 - 4*(3*a^3*d*x + 3*a^3 + 9*a^2*b + 18*a*b^2 + 8*b^3)*cosh(d*x + c)*sinh(d*x + c)^3 + (3* a^3 + 9*a^2*b + 18*a*b^2 + 8*b^3)*sinh(d*x + c)^4 + 9*a^3 + 27*a^2*b + 18* a*b^2 + 4*(3*a^3 + 9*a^2*b + 9*a*b^2 + 4*b^3)*cosh(d*x + c)^2 + 2*(6*a^3 + 18*a^2*b + 18*a*b^2 + 8*b^3 + 3*(3*a^3 + 9*a^2*b + 18*a*b^2 + 8*b^3)*cosh (d*x + c)^2)*sinh(d*x + c)^2 - 4*((3*a^3*d*x + 3*a^3 + 9*a^2*b + 18*a*b^2 + 8*b^3)*cosh(d*x + c)^3 + (3*a^3*d*x + 3*a^3 + 9*a^2*b + 18*a*b^2 + 8*b^3 )*cosh(d*x + c))*sinh(d*x + c))/(d*cosh(d*x + c)*sinh(d*x + c)^3 + (d*cosh (d*x + c)^3 + d*cosh(d*x + c))*sinh(d*x + c))
Timed out. \[ \int \coth ^2(c+d x) \left (a+b \text {sech}^2(c+d x)\right )^3 \, dx=\text {Timed out} \]
Leaf count of result is larger than twice the leaf count of optimal. 172 vs. \(2 (59) = 118\).
Time = 0.20 (sec) , antiderivative size = 172, normalized size of antiderivative = 2.82 \[ \int \coth ^2(c+d x) \left (a+b \text {sech}^2(c+d x)\right )^3 \, dx=a^{3} {\left (x + \frac {c}{d} + \frac {2}{d {\left (e^{\left (-2 \, d x - 2 \, c\right )} - 1\right )}}\right )} - \frac {16}{3} \, b^{3} {\left (\frac {2 \, e^{\left (-2 \, d x - 2 \, c\right )}}{d {\left (2 \, e^{\left (-2 \, d x - 2 \, c\right )} - 2 \, e^{\left (-6 \, d x - 6 \, c\right )} - e^{\left (-8 \, d x - 8 \, c\right )} + 1\right )}} + \frac {1}{d {\left (2 \, e^{\left (-2 \, d x - 2 \, c\right )} - 2 \, e^{\left (-6 \, d x - 6 \, c\right )} - e^{\left (-8 \, d x - 8 \, c\right )} + 1\right )}}\right )} + \frac {6 \, a^{2} b}{d {\left (e^{\left (-2 \, d x - 2 \, c\right )} - 1\right )}} + \frac {12 \, a b^{2}}{d {\left (e^{\left (-4 \, d x - 4 \, c\right )} - 1\right )}} \]
a^3*(x + c/d + 2/(d*(e^(-2*d*x - 2*c) - 1))) - 16/3*b^3*(2*e^(-2*d*x - 2*c )/(d*(2*e^(-2*d*x - 2*c) - 2*e^(-6*d*x - 6*c) - e^(-8*d*x - 8*c) + 1)) + 1 /(d*(2*e^(-2*d*x - 2*c) - 2*e^(-6*d*x - 6*c) - e^(-8*d*x - 8*c) + 1))) + 6 *a^2*b/(d*(e^(-2*d*x - 2*c) - 1)) + 12*a*b^2/(d*(e^(-4*d*x - 4*c) - 1))
Leaf count of result is larger than twice the leaf count of optimal. 135 vs. \(2 (59) = 118\).
Time = 0.34 (sec) , antiderivative size = 135, normalized size of antiderivative = 2.21 \[ \int \coth ^2(c+d x) \left (a+b \text {sech}^2(c+d x)\right )^3 \, dx=\frac {3 \, {\left (d x + c\right )} a^{3} - \frac {6 \, {\left (a^{3} + 3 \, a^{2} b + 3 \, a b^{2} + b^{3}\right )}}{e^{\left (2 \, d x + 2 \, c\right )} - 1} + \frac {2 \, {\left (9 \, a b^{2} e^{\left (4 \, d x + 4 \, c\right )} + 3 \, b^{3} e^{\left (4 \, d x + 4 \, c\right )} + 18 \, a b^{2} e^{\left (2 \, d x + 2 \, c\right )} + 12 \, b^{3} e^{\left (2 \, d x + 2 \, c\right )} + 9 \, a b^{2} + 5 \, b^{3}\right )}}{{\left (e^{\left (2 \, d x + 2 \, c\right )} + 1\right )}^{3}}}{3 \, d} \]
1/3*(3*(d*x + c)*a^3 - 6*(a^3 + 3*a^2*b + 3*a*b^2 + b^3)/(e^(2*d*x + 2*c) - 1) + 2*(9*a*b^2*e^(4*d*x + 4*c) + 3*b^3*e^(4*d*x + 4*c) + 18*a*b^2*e^(2* d*x + 2*c) + 12*b^3*e^(2*d*x + 2*c) + 9*a*b^2 + 5*b^3)/(e^(2*d*x + 2*c) + 1)^3)/d
Time = 0.13 (sec) , antiderivative size = 234, normalized size of antiderivative = 3.84 \[ \int \coth ^2(c+d x) \left (a+b \text {sech}^2(c+d x)\right )^3 \, dx=\frac {\frac {2\,\left (b^3+3\,a\,b^2\right )}{3\,d}+\frac {4\,{\mathrm {e}}^{2\,c+2\,d\,x}\,\left (b^3+a\,b^2\right )}{d}+\frac {2\,{\mathrm {e}}^{4\,c+4\,d\,x}\,\left (b^3+3\,a\,b^2\right )}{3\,d}}{3\,{\mathrm {e}}^{2\,c+2\,d\,x}+3\,{\mathrm {e}}^{4\,c+4\,d\,x}+{\mathrm {e}}^{6\,c+6\,d\,x}+1}+a^3\,x+\frac {\frac {2\,\left (b^3+a\,b^2\right )}{d}+\frac {2\,{\mathrm {e}}^{2\,c+2\,d\,x}\,\left (b^3+3\,a\,b^2\right )}{3\,d}}{2\,{\mathrm {e}}^{2\,c+2\,d\,x}+{\mathrm {e}}^{4\,c+4\,d\,x}+1}+\frac {2\,\left (b^3+3\,a\,b^2\right )}{3\,d\,\left ({\mathrm {e}}^{2\,c+2\,d\,x}+1\right )}-\frac {2\,\left (a^3+3\,a^2\,b+3\,a\,b^2+b^3\right )}{d\,\left ({\mathrm {e}}^{2\,c+2\,d\,x}-1\right )} \]
((2*(3*a*b^2 + b^3))/(3*d) + (4*exp(2*c + 2*d*x)*(a*b^2 + b^3))/d + (2*exp (4*c + 4*d*x)*(3*a*b^2 + b^3))/(3*d))/(3*exp(2*c + 2*d*x) + 3*exp(4*c + 4* d*x) + exp(6*c + 6*d*x) + 1) + a^3*x + ((2*(a*b^2 + b^3))/d + (2*exp(2*c + 2*d*x)*(3*a*b^2 + b^3))/(3*d))/(2*exp(2*c + 2*d*x) + exp(4*c + 4*d*x) + 1 ) + (2*(3*a*b^2 + b^3))/(3*d*(exp(2*c + 2*d*x) + 1)) - (2*(3*a*b^2 + 3*a^2 *b + a^3 + b^3))/(d*(exp(2*c + 2*d*x) - 1))