Integrand size = 17, antiderivative size = 49 \[ \int \text {csch}^3(a+b x) \text {sech}^2(a+b x) \, dx=\frac {3 \text {arctanh}(\cosh (a+b x))}{2 b}-\frac {3 \text {sech}(a+b x)}{2 b}-\frac {\text {csch}^2(a+b x) \text {sech}(a+b x)}{2 b} \]
Time = 0.12 (sec) , antiderivative size = 86, normalized size of antiderivative = 1.76 \[ \int \text {csch}^3(a+b x) \text {sech}^2(a+b x) \, dx=-\frac {\text {csch}^2\left (\frac {1}{2} (a+b x)\right )}{8 b}+\frac {3 \log \left (\cosh \left (\frac {1}{2} (a+b x)\right )\right )}{2 b}-\frac {3 \log \left (\sinh \left (\frac {1}{2} (a+b x)\right )\right )}{2 b}-\frac {\text {sech}^2\left (\frac {1}{2} (a+b x)\right )}{8 b}-\frac {\text {sech}(a+b x)}{b} \]
-1/8*Csch[(a + b*x)/2]^2/b + (3*Log[Cosh[(a + b*x)/2]])/(2*b) - (3*Log[Sin h[(a + b*x)/2]])/(2*b) - Sech[(a + b*x)/2]^2/(8*b) - Sech[a + b*x]/b
Time = 0.22 (sec) , antiderivative size = 52, normalized size of antiderivative = 1.06, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.353, Rules used = {3042, 26, 3102, 252, 262, 219}
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.
\(\displaystyle \int \text {csch}^3(a+b x) \text {sech}^2(a+b x) \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int -i \csc (i a+i b x)^3 \sec (i a+i b x)^2dx\) |
\(\Big \downarrow \) 26 |
\(\displaystyle -i \int \csc (i a+i b x)^3 \sec (i a+i b x)^2dx\) |
\(\Big \downarrow \) 3102 |
\(\displaystyle -\frac {\int \frac {\text {sech}^4(a+b x)}{\left (1-\text {sech}^2(a+b x)\right )^2}d\text {sech}(a+b x)}{b}\) |
\(\Big \downarrow \) 252 |
\(\displaystyle -\frac {\frac {\text {sech}^3(a+b x)}{2 \left (1-\text {sech}^2(a+b x)\right )}-\frac {3}{2} \int \frac {\text {sech}^2(a+b x)}{1-\text {sech}^2(a+b x)}d\text {sech}(a+b x)}{b}\) |
\(\Big \downarrow \) 262 |
\(\displaystyle -\frac {\frac {\text {sech}^3(a+b x)}{2 \left (1-\text {sech}^2(a+b x)\right )}-\frac {3}{2} \left (\int \frac {1}{1-\text {sech}^2(a+b x)}d\text {sech}(a+b x)-\text {sech}(a+b x)\right )}{b}\) |
\(\Big \downarrow \) 219 |
\(\displaystyle -\frac {\frac {\text {sech}^3(a+b x)}{2 \left (1-\text {sech}^2(a+b x)\right )}-\frac {3}{2} (\text {arctanh}(\text {sech}(a+b x))-\text {sech}(a+b x))}{b}\) |
-(((-3*(ArcTanh[Sech[a + b*x]] - Sech[a + b*x]))/2 + Sech[a + b*x]^3/(2*(1 - Sech[a + b*x]^2)))/b)
3.1.35.3.1 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_)^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] && (Gt Q[a, 0] || LtQ[b, 0])
Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[c*(c*x )^(m - 1)*((a + b*x^2)^(p + 1)/(2*b*(p + 1))), x] - Simp[c^2*((m - 1)/(2*b* (p + 1))) Int[(c*x)^(m - 2)*(a + b*x^2)^(p + 1), x], x] /; FreeQ[{a, b, c }, x] && LtQ[p, -1] && GtQ[m, 1] && !ILtQ[(m + 2*p + 3)/2, 0] && IntBinomi alQ[a, b, c, 2, m, p, x]
Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[c*(c*x) ^(m - 1)*((a + b*x^2)^(p + 1)/(b*(m + 2*p + 1))), x] - Simp[a*c^2*((m - 1)/ (b*(m + 2*p + 1))) Int[(c*x)^(m - 2)*(a + b*x^2)^p, x], x] /; FreeQ[{a, b , c, p}, x] && GtQ[m, 2 - 1] && NeQ[m + 2*p + 1, 0] && IntBinomialQ[a, b, c , 2, m, p, x]
Int[csc[(e_.) + (f_.)*(x_)]^(n_.)*((a_.)*sec[(e_.) + (f_.)*(x_)])^(m_), x_S ymbol] :> Simp[1/(f*a^n) Subst[Int[x^(m + n - 1)/(-1 + x^2/a^2)^((n + 1)/ 2), x], x, a*Sec[e + f*x]], x] /; FreeQ[{a, e, f, m}, x] && IntegerQ[(n + 1 )/2] && !(IntegerQ[(m + 1)/2] && LtQ[0, m, n])
Time = 4.12 (sec) , antiderivative size = 43, normalized size of antiderivative = 0.88
method | result | size |
derivativedivides | \(\frac {-\frac {1}{2 \sinh \left (b x +a \right )^{2} \cosh \left (b x +a \right )}-\frac {3}{2 \cosh \left (b x +a \right )}+3 \,\operatorname {arctanh}\left ({\mathrm e}^{b x +a}\right )}{b}\) | \(43\) |
default | \(\frac {-\frac {1}{2 \sinh \left (b x +a \right )^{2} \cosh \left (b x +a \right )}-\frac {3}{2 \cosh \left (b x +a \right )}+3 \,\operatorname {arctanh}\left ({\mathrm e}^{b x +a}\right )}{b}\) | \(43\) |
risch | \(-\frac {{\mathrm e}^{b x +a} \left (3 \,{\mathrm e}^{4 b x +4 a}-2 \,{\mathrm e}^{2 b x +2 a}+3\right )}{b \left ({\mathrm e}^{2 b x +2 a}-1\right )^{2} \left (1+{\mathrm e}^{2 b x +2 a}\right )}-\frac {3 \ln \left ({\mathrm e}^{b x +a}-1\right )}{2 b}+\frac {3 \ln \left ({\mathrm e}^{b x +a}+1\right )}{2 b}\) | \(91\) |
Leaf count of result is larger than twice the leaf count of optimal. 709 vs. \(2 (43) = 86\).
Time = 0.26 (sec) , antiderivative size = 709, normalized size of antiderivative = 14.47 \[ \int \text {csch}^3(a+b x) \text {sech}^2(a+b x) \, dx=-\frac {6 \, \cosh \left (b x + a\right )^{5} + 30 \, \cosh \left (b x + a\right ) \sinh \left (b x + a\right )^{4} + 6 \, \sinh \left (b x + a\right )^{5} + 4 \, {\left (15 \, \cosh \left (b x + a\right )^{2} - 1\right )} \sinh \left (b x + a\right )^{3} - 4 \, \cosh \left (b x + a\right )^{3} + 12 \, {\left (5 \, \cosh \left (b x + a\right )^{3} - \cosh \left (b x + a\right )\right )} \sinh \left (b x + a\right )^{2} - 3 \, {\left (\cosh \left (b x + a\right )^{6} + 6 \, \cosh \left (b x + a\right ) \sinh \left (b x + a\right )^{5} + \sinh \left (b x + a\right )^{6} + {\left (15 \, \cosh \left (b x + a\right )^{2} - 1\right )} \sinh \left (b x + a\right )^{4} - \cosh \left (b x + a\right )^{4} + 4 \, {\left (5 \, \cosh \left (b x + a\right )^{3} - \cosh \left (b x + a\right )\right )} \sinh \left (b x + a\right )^{3} + {\left (15 \, \cosh \left (b x + a\right )^{4} - 6 \, \cosh \left (b x + a\right )^{2} - 1\right )} \sinh \left (b x + a\right )^{2} - \cosh \left (b x + a\right )^{2} + 2 \, {\left (3 \, \cosh \left (b x + a\right )^{5} - 2 \, \cosh \left (b x + a\right )^{3} - \cosh \left (b x + a\right )\right )} \sinh \left (b x + a\right ) + 1\right )} \log \left (\cosh \left (b x + a\right ) + \sinh \left (b x + a\right ) + 1\right ) + 3 \, {\left (\cosh \left (b x + a\right )^{6} + 6 \, \cosh \left (b x + a\right ) \sinh \left (b x + a\right )^{5} + \sinh \left (b x + a\right )^{6} + {\left (15 \, \cosh \left (b x + a\right )^{2} - 1\right )} \sinh \left (b x + a\right )^{4} - \cosh \left (b x + a\right )^{4} + 4 \, {\left (5 \, \cosh \left (b x + a\right )^{3} - \cosh \left (b x + a\right )\right )} \sinh \left (b x + a\right )^{3} + {\left (15 \, \cosh \left (b x + a\right )^{4} - 6 \, \cosh \left (b x + a\right )^{2} - 1\right )} \sinh \left (b x + a\right )^{2} - \cosh \left (b x + a\right )^{2} + 2 \, {\left (3 \, \cosh \left (b x + a\right )^{5} - 2 \, \cosh \left (b x + a\right )^{3} - \cosh \left (b x + a\right )\right )} \sinh \left (b x + a\right ) + 1\right )} \log \left (\cosh \left (b x + a\right ) + \sinh \left (b x + a\right ) - 1\right ) + 6 \, {\left (5 \, \cosh \left (b x + a\right )^{4} - 2 \, \cosh \left (b x + a\right )^{2} + 1\right )} \sinh \left (b x + a\right ) + 6 \, \cosh \left (b x + a\right )}{2 \, {\left (b \cosh \left (b x + a\right )^{6} + 6 \, b \cosh \left (b x + a\right ) \sinh \left (b x + a\right )^{5} + b \sinh \left (b x + a\right )^{6} - b \cosh \left (b x + a\right )^{4} + {\left (15 \, b \cosh \left (b x + a\right )^{2} - b\right )} \sinh \left (b x + a\right )^{4} + 4 \, {\left (5 \, b \cosh \left (b x + a\right )^{3} - b \cosh \left (b x + a\right )\right )} \sinh \left (b x + a\right )^{3} - b \cosh \left (b x + a\right )^{2} + {\left (15 \, b \cosh \left (b x + a\right )^{4} - 6 \, b \cosh \left (b x + a\right )^{2} - b\right )} \sinh \left (b x + a\right )^{2} + 2 \, {\left (3 \, b \cosh \left (b x + a\right )^{5} - 2 \, b \cosh \left (b x + a\right )^{3} - b \cosh \left (b x + a\right )\right )} \sinh \left (b x + a\right ) + b\right )}} \]
-1/2*(6*cosh(b*x + a)^5 + 30*cosh(b*x + a)*sinh(b*x + a)^4 + 6*sinh(b*x + a)^5 + 4*(15*cosh(b*x + a)^2 - 1)*sinh(b*x + a)^3 - 4*cosh(b*x + a)^3 + 12 *(5*cosh(b*x + a)^3 - cosh(b*x + a))*sinh(b*x + a)^2 - 3*(cosh(b*x + a)^6 + 6*cosh(b*x + a)*sinh(b*x + a)^5 + sinh(b*x + a)^6 + (15*cosh(b*x + a)^2 - 1)*sinh(b*x + a)^4 - cosh(b*x + a)^4 + 4*(5*cosh(b*x + a)^3 - cosh(b*x + a))*sinh(b*x + a)^3 + (15*cosh(b*x + a)^4 - 6*cosh(b*x + a)^2 - 1)*sinh(b *x + a)^2 - cosh(b*x + a)^2 + 2*(3*cosh(b*x + a)^5 - 2*cosh(b*x + a)^3 - c osh(b*x + a))*sinh(b*x + a) + 1)*log(cosh(b*x + a) + sinh(b*x + a) + 1) + 3*(cosh(b*x + a)^6 + 6*cosh(b*x + a)*sinh(b*x + a)^5 + sinh(b*x + a)^6 + ( 15*cosh(b*x + a)^2 - 1)*sinh(b*x + a)^4 - cosh(b*x + a)^4 + 4*(5*cosh(b*x + a)^3 - cosh(b*x + a))*sinh(b*x + a)^3 + (15*cosh(b*x + a)^4 - 6*cosh(b*x + a)^2 - 1)*sinh(b*x + a)^2 - cosh(b*x + a)^2 + 2*(3*cosh(b*x + a)^5 - 2* cosh(b*x + a)^3 - cosh(b*x + a))*sinh(b*x + a) + 1)*log(cosh(b*x + a) + si nh(b*x + a) - 1) + 6*(5*cosh(b*x + a)^4 - 2*cosh(b*x + a)^2 + 1)*sinh(b*x + a) + 6*cosh(b*x + a))/(b*cosh(b*x + a)^6 + 6*b*cosh(b*x + a)*sinh(b*x + a)^5 + b*sinh(b*x + a)^6 - b*cosh(b*x + a)^4 + (15*b*cosh(b*x + a)^2 - b)* sinh(b*x + a)^4 + 4*(5*b*cosh(b*x + a)^3 - b*cosh(b*x + a))*sinh(b*x + a)^ 3 - b*cosh(b*x + a)^2 + (15*b*cosh(b*x + a)^4 - 6*b*cosh(b*x + a)^2 - b)*s inh(b*x + a)^2 + 2*(3*b*cosh(b*x + a)^5 - 2*b*cosh(b*x + a)^3 - b*cosh(b*x + a))*sinh(b*x + a) + b)
\[ \int \text {csch}^3(a+b x) \text {sech}^2(a+b x) \, dx=\int \operatorname {csch}^{3}{\left (a + b x \right )} \operatorname {sech}^{2}{\left (a + b x \right )}\, dx \]
Leaf count of result is larger than twice the leaf count of optimal. 106 vs. \(2 (43) = 86\).
Time = 0.17 (sec) , antiderivative size = 106, normalized size of antiderivative = 2.16 \[ \int \text {csch}^3(a+b x) \text {sech}^2(a+b x) \, dx=\frac {3 \, \log \left (e^{\left (-b x - a\right )} + 1\right )}{2 \, b} - \frac {3 \, \log \left (e^{\left (-b x - a\right )} - 1\right )}{2 \, b} + \frac {3 \, e^{\left (-b x - a\right )} - 2 \, e^{\left (-3 \, b x - 3 \, a\right )} + 3 \, e^{\left (-5 \, b x - 5 \, a\right )}}{b {\left (e^{\left (-2 \, b x - 2 \, a\right )} + e^{\left (-4 \, b x - 4 \, a\right )} - e^{\left (-6 \, b x - 6 \, a\right )} - 1\right )}} \]
3/2*log(e^(-b*x - a) + 1)/b - 3/2*log(e^(-b*x - a) - 1)/b + (3*e^(-b*x - a ) - 2*e^(-3*b*x - 3*a) + 3*e^(-5*b*x - 5*a))/(b*(e^(-2*b*x - 2*a) + e^(-4* b*x - 4*a) - e^(-6*b*x - 6*a) - 1))
Leaf count of result is larger than twice the leaf count of optimal. 110 vs. \(2 (43) = 86\).
Time = 0.28 (sec) , antiderivative size = 110, normalized size of antiderivative = 2.24 \[ \int \text {csch}^3(a+b x) \text {sech}^2(a+b x) \, dx=-\frac {\frac {4 \, {\left (3 \, {\left (e^{\left (b x + a\right )} + e^{\left (-b x - a\right )}\right )}^{2} - 8\right )}}{{\left (e^{\left (b x + a\right )} + e^{\left (-b x - a\right )}\right )}^{3} - 4 \, e^{\left (b x + a\right )} - 4 \, e^{\left (-b x - a\right )}} - 3 \, \log \left (e^{\left (b x + a\right )} + e^{\left (-b x - a\right )} + 2\right ) + 3 \, \log \left (e^{\left (b x + a\right )} + e^{\left (-b x - a\right )} - 2\right )}{4 \, b} \]
-1/4*(4*(3*(e^(b*x + a) + e^(-b*x - a))^2 - 8)/((e^(b*x + a) + e^(-b*x - a ))^3 - 4*e^(b*x + a) - 4*e^(-b*x - a)) - 3*log(e^(b*x + a) + e^(-b*x - a) + 2) + 3*log(e^(b*x + a) + e^(-b*x - a) - 2))/b
Time = 2.21 (sec) , antiderivative size = 111, normalized size of antiderivative = 2.27 \[ \int \text {csch}^3(a+b x) \text {sech}^2(a+b x) \, dx=\frac {3\,\mathrm {atan}\left (\frac {{\mathrm {e}}^{b\,x}\,{\mathrm {e}}^a\,\sqrt {-b^2}}{b}\right )}{\sqrt {-b^2}}-\frac {2\,{\mathrm {e}}^{a+b\,x}}{b\,\left ({\mathrm {e}}^{4\,a+4\,b\,x}-2\,{\mathrm {e}}^{2\,a+2\,b\,x}+1\right )}-\frac {{\mathrm {e}}^{a+b\,x}}{b\,\left ({\mathrm {e}}^{2\,a+2\,b\,x}-1\right )}-\frac {2\,{\mathrm {e}}^{a+b\,x}}{b\,\left ({\mathrm {e}}^{2\,a+2\,b\,x}+1\right )} \]