Integrand size = 17, antiderivative size = 89 \[ \int \text {csch}^4(a+b x) \text {sech}^5(a+b x) \, dx=\frac {35 \arctan (\sinh (a+b x))}{8 b}+\frac {35 \text {csch}(a+b x)}{8 b}-\frac {35 \text {csch}^3(a+b x)}{24 b}+\frac {7 \text {csch}^3(a+b x) \text {sech}^2(a+b x)}{8 b}+\frac {\text {csch}^3(a+b x) \text {sech}^4(a+b x)}{4 b} \]
35/8*arctan(sinh(b*x+a))/b+35/8*csch(b*x+a)/b-35/24*csch(b*x+a)^3/b+7/8*cs ch(b*x+a)^3*sech(b*x+a)^2/b+1/4*csch(b*x+a)^3*sech(b*x+a)^4/b
Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.
Time = 0.01 (sec) , antiderivative size = 33, normalized size of antiderivative = 0.37 \[ \int \text {csch}^4(a+b x) \text {sech}^5(a+b x) \, dx=-\frac {\text {csch}^3(a+b x) \operatorname {Hypergeometric2F1}\left (-\frac {3}{2},3,-\frac {1}{2},-\sinh ^2(a+b x)\right )}{3 b} \]
Result contains complex when optimal does not.
Time = 0.26 (sec) , antiderivative size = 105, normalized size of antiderivative = 1.18, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.412, Rules used = {3042, 3101, 25, 252, 252, 254, 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.
\(\displaystyle \int \text {csch}^4(a+b x) \text {sech}^5(a+b x) \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \csc (i a+i b x)^4 \sec (i a+i b x)^5dx\) |
\(\Big \downarrow \) 3101 |
\(\displaystyle \frac {i \int -\frac {\text {csch}^8(a+b x)}{\left (\text {csch}^2(a+b x)+1\right )^3}d(-i \text {csch}(a+b x))}{b}\) |
\(\Big \downarrow \) 25 |
\(\displaystyle -\frac {i \int \frac {\text {csch}^8(a+b x)}{\left (\text {csch}^2(a+b x)+1\right )^3}d(-i \text {csch}(a+b x))}{b}\) |
\(\Big \downarrow \) 252 |
\(\displaystyle \frac {i \left (\frac {7}{4} \int -\frac {\text {csch}^6(a+b x)}{\left (\text {csch}^2(a+b x)+1\right )^2}d(-i \text {csch}(a+b x))-\frac {i \text {csch}^7(a+b x)}{4 \left (\text {csch}^2(a+b x)+1\right )^2}\right )}{b}\) |
\(\Big \downarrow \) 252 |
\(\displaystyle \frac {i \left (\frac {7}{4} \left (-\frac {5}{2} \int \frac {\text {csch}^4(a+b x)}{\text {csch}^2(a+b x)+1}d(-i \text {csch}(a+b x))-\frac {i \text {csch}^5(a+b x)}{2 \left (\text {csch}^2(a+b x)+1\right )}\right )-\frac {i \text {csch}^7(a+b x)}{4 \left (\text {csch}^2(a+b x)+1\right )^2}\right )}{b}\) |
\(\Big \downarrow \) 254 |
\(\displaystyle \frac {i \left (\frac {7}{4} \left (-\frac {5}{2} \int \left (\text {csch}^2(a+b x)+\frac {1}{\text {csch}^2(a+b x)+1}-1\right )d(-i \text {csch}(a+b x))-\frac {i \text {csch}^5(a+b x)}{2 \left (\text {csch}^2(a+b x)+1\right )}\right )-\frac {i \text {csch}^7(a+b x)}{4 \left (\text {csch}^2(a+b x)+1\right )^2}\right )}{b}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {i \left (\frac {7}{4} \left (-\frac {5}{2} \left (-i \arctan (\text {csch}(a+b x))-\frac {1}{3} i \text {csch}^3(a+b x)+i \text {csch}(a+b x)\right )-\frac {i \text {csch}^5(a+b x)}{2 \left (\text {csch}^2(a+b x)+1\right )}\right )-\frac {i \text {csch}^7(a+b x)}{4 \left (\text {csch}^2(a+b x)+1\right )^2}\right )}{b}\) |
(I*(((-1/4*I)*Csch[a + b*x]^7)/(1 + Csch[a + b*x]^2)^2 + (7*(((-1/2*I)*Csc h[a + b*x]^5)/(1 + Csch[a + b*x]^2) - (5*((-I)*ArcTan[Csch[a + b*x]] + I*C sch[a + b*x] - (I/3)*Csch[a + b*x]^3))/2))/4))/b
3.1.43.3.1 Defintions of rubi rules used
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[(x_)^(m_)/((a_) + (b_.)*(x_)^2), x_Symbol] :> Int[PolynomialDivide[x^m, a + b*x^2, x], x] /; FreeQ[{a, b}, x] && IGtQ[m, 3]
Int[(csc[(e_.) + (f_.)*(x_)]*(a_.))^(m_)*sec[(e_.) + (f_.)*(x_)]^(n_.), x_S ymbol] :> Simp[-(f*a^n)^(-1) Subst[Int[x^(m + n - 1)/(-1 + x^2/a^2)^((n + 1)/2), x], x, a*Csc[e + f*x]], x] /; FreeQ[{a, e, f, m}, x] && IntegerQ[(n + 1)/2] && !(IntegerQ[(m + 1)/2] && LtQ[0, m, n])
Time = 151.89 (sec) , antiderivative size = 78, normalized size of antiderivative = 0.88
method | result | size |
derivativedivides | \(\frac {-\frac {1}{3 \sinh \left (b x +a \right )^{3} \cosh \left (b x +a \right )^{4}}+\frac {7}{3 \sinh \left (b x +a \right ) \cosh \left (b x +a \right )^{4}}+\frac {35 \left (\frac {\operatorname {sech}\left (b x +a \right )^{3}}{4}+\frac {3 \,\operatorname {sech}\left (b x +a \right )}{8}\right ) \tanh \left (b x +a \right )}{3}+\frac {35 \arctan \left ({\mathrm e}^{b x +a}\right )}{4}}{b}\) | \(78\) |
default | \(\frac {-\frac {1}{3 \sinh \left (b x +a \right )^{3} \cosh \left (b x +a \right )^{4}}+\frac {7}{3 \sinh \left (b x +a \right ) \cosh \left (b x +a \right )^{4}}+\frac {35 \left (\frac {\operatorname {sech}\left (b x +a \right )^{3}}{4}+\frac {3 \,\operatorname {sech}\left (b x +a \right )}{8}\right ) \tanh \left (b x +a \right )}{3}+\frac {35 \arctan \left ({\mathrm e}^{b x +a}\right )}{4}}{b}\) | \(78\) |
risch | \(\frac {{\mathrm e}^{b x +a} \left (105 \,{\mathrm e}^{12 b x +12 a}+70 \,{\mathrm e}^{10 b x +10 a}-329 \,{\mathrm e}^{8 b x +8 a}-204 \,{\mathrm e}^{6 b x +6 a}-329 \,{\mathrm e}^{4 b x +4 a}+70 \,{\mathrm e}^{2 b x +2 a}+105\right )}{12 b \left (1+{\mathrm e}^{2 b x +2 a}\right )^{4} \left ({\mathrm e}^{2 b x +2 a}-1\right )^{3}}+\frac {35 i \ln \left ({\mathrm e}^{b x +a}+i\right )}{8 b}-\frac {35 i \ln \left ({\mathrm e}^{b x +a}-i\right )}{8 b}\) | \(139\) |
1/b*(-1/3/sinh(b*x+a)^3/cosh(b*x+a)^4+7/3/sinh(b*x+a)/cosh(b*x+a)^4+35/3*( 1/4*sech(b*x+a)^3+3/8*sech(b*x+a))*tanh(b*x+a)+35/4*arctan(exp(b*x+a)))
Leaf count of result is larger than twice the leaf count of optimal. 2092 vs. \(2 (79) = 158\).
Time = 0.26 (sec) , antiderivative size = 2092, normalized size of antiderivative = 23.51 \[ \int \text {csch}^4(a+b x) \text {sech}^5(a+b x) \, dx=\text {Too large to display} \]
1/12*(105*cosh(b*x + a)^13 + 1365*cosh(b*x + a)*sinh(b*x + a)^12 + 105*sin h(b*x + a)^13 + 70*(117*cosh(b*x + a)^2 + 1)*sinh(b*x + a)^11 + 70*cosh(b* x + a)^11 + 770*(39*cosh(b*x + a)^3 + cosh(b*x + a))*sinh(b*x + a)^10 + 7* (10725*cosh(b*x + a)^4 + 550*cosh(b*x + a)^2 - 47)*sinh(b*x + a)^9 - 329*c osh(b*x + a)^9 + 21*(6435*cosh(b*x + a)^5 + 550*cosh(b*x + a)^3 - 141*cosh (b*x + a))*sinh(b*x + a)^8 + 12*(15015*cosh(b*x + a)^6 + 1925*cosh(b*x + a )^4 - 987*cosh(b*x + a)^2 - 17)*sinh(b*x + a)^7 - 204*cosh(b*x + a)^7 + 84 *(2145*cosh(b*x + a)^7 + 385*cosh(b*x + a)^5 - 329*cosh(b*x + a)^3 - 17*co sh(b*x + a))*sinh(b*x + a)^6 + 7*(19305*cosh(b*x + a)^8 + 4620*cosh(b*x + a)^6 - 5922*cosh(b*x + a)^4 - 612*cosh(b*x + a)^2 - 47)*sinh(b*x + a)^5 - 329*cosh(b*x + a)^5 + 7*(10725*cosh(b*x + a)^9 + 3300*cosh(b*x + a)^7 - 59 22*cosh(b*x + a)^5 - 1020*cosh(b*x + a)^3 - 235*cosh(b*x + a))*sinh(b*x + a)^4 + 14*(2145*cosh(b*x + a)^10 + 825*cosh(b*x + a)^8 - 1974*cosh(b*x + a )^6 - 510*cosh(b*x + a)^4 - 235*cosh(b*x + a)^2 + 5)*sinh(b*x + a)^3 + 70* cosh(b*x + a)^3 + 14*(585*cosh(b*x + a)^11 + 275*cosh(b*x + a)^9 - 846*cos h(b*x + a)^7 - 306*cosh(b*x + a)^5 - 235*cosh(b*x + a)^3 + 15*cosh(b*x + a ))*sinh(b*x + a)^2 + 105*(cosh(b*x + a)^14 + 14*cosh(b*x + a)*sinh(b*x + a )^13 + sinh(b*x + a)^14 + (91*cosh(b*x + a)^2 + 1)*sinh(b*x + a)^12 + cosh (b*x + a)^12 + 4*(91*cosh(b*x + a)^3 + 3*cosh(b*x + a))*sinh(b*x + a)^11 + (1001*cosh(b*x + a)^4 + 66*cosh(b*x + a)^2 - 3)*sinh(b*x + a)^10 - 3*c...
\[ \int \text {csch}^4(a+b x) \text {sech}^5(a+b x) \, dx=\int \operatorname {csch}^{4}{\left (a + b x \right )} \operatorname {sech}^{5}{\left (a + b x \right )}\, dx \]
Leaf count of result is larger than twice the leaf count of optimal. 178 vs. \(2 (79) = 158\).
Time = 0.27 (sec) , antiderivative size = 178, normalized size of antiderivative = 2.00 \[ \int \text {csch}^4(a+b x) \text {sech}^5(a+b x) \, dx=-\frac {35 \, \arctan \left (e^{\left (-b x - a\right )}\right )}{4 \, b} + \frac {105 \, e^{\left (-b x - a\right )} + 70 \, e^{\left (-3 \, b x - 3 \, a\right )} - 329 \, e^{\left (-5 \, b x - 5 \, a\right )} - 204 \, e^{\left (-7 \, b x - 7 \, a\right )} - 329 \, e^{\left (-9 \, b x - 9 \, a\right )} + 70 \, e^{\left (-11 \, b x - 11 \, a\right )} + 105 \, e^{\left (-13 \, b x - 13 \, a\right )}}{12 \, b {\left (e^{\left (-2 \, b x - 2 \, a\right )} - 3 \, e^{\left (-4 \, b x - 4 \, a\right )} - 3 \, e^{\left (-6 \, b x - 6 \, a\right )} + 3 \, e^{\left (-8 \, b x - 8 \, a\right )} + 3 \, e^{\left (-10 \, b x - 10 \, a\right )} - e^{\left (-12 \, b x - 12 \, a\right )} - e^{\left (-14 \, b x - 14 \, a\right )} + 1\right )}} \]
-35/4*arctan(e^(-b*x - a))/b + 1/12*(105*e^(-b*x - a) + 70*e^(-3*b*x - 3*a ) - 329*e^(-5*b*x - 5*a) - 204*e^(-7*b*x - 7*a) - 329*e^(-9*b*x - 9*a) + 7 0*e^(-11*b*x - 11*a) + 105*e^(-13*b*x - 13*a))/(b*(e^(-2*b*x - 2*a) - 3*e^ (-4*b*x - 4*a) - 3*e^(-6*b*x - 6*a) + 3*e^(-8*b*x - 8*a) + 3*e^(-10*b*x - 10*a) - e^(-12*b*x - 12*a) - e^(-14*b*x - 14*a) + 1))
Time = 0.28 (sec) , antiderivative size = 148, normalized size of antiderivative = 1.66 \[ \int \text {csch}^4(a+b x) \text {sech}^5(a+b x) \, dx=\frac {105 \, \pi + \frac {12 \, {\left (11 \, {\left (e^{\left (b x + a\right )} - e^{\left (-b x - a\right )}\right )}^{3} + 52 \, e^{\left (b x + a\right )} - 52 \, e^{\left (-b x - a\right )}\right )}}{{\left ({\left (e^{\left (b x + a\right )} - e^{\left (-b x - a\right )}\right )}^{2} + 4\right )}^{2}} + \frac {32 \, {\left (9 \, {\left (e^{\left (b x + a\right )} - e^{\left (-b x - a\right )}\right )}^{2} - 4\right )}}{{\left (e^{\left (b x + a\right )} - e^{\left (-b x - a\right )}\right )}^{3}} + 210 \, \arctan \left (\frac {1}{2} \, {\left (e^{\left (2 \, b x + 2 \, a\right )} - 1\right )} e^{\left (-b x - a\right )}\right )}{48 \, b} \]
1/48*(105*pi + 12*(11*(e^(b*x + a) - e^(-b*x - a))^3 + 52*e^(b*x + a) - 52 *e^(-b*x - a))/((e^(b*x + a) - e^(-b*x - a))^2 + 4)^2 + 32*(9*(e^(b*x + a) - e^(-b*x - a))^2 - 4)/(e^(b*x + a) - e^(-b*x - a))^3 + 210*arctan(1/2*(e ^(2*b*x + 2*a) - 1)*e^(-b*x - a)))/b
Time = 2.16 (sec) , antiderivative size = 291, normalized size of antiderivative = 3.27 \[ \int \text {csch}^4(a+b x) \text {sech}^5(a+b x) \, dx=\frac {35\,\mathrm {atan}\left (\frac {{\mathrm {e}}^{b\,x}\,{\mathrm {e}}^a\,\sqrt {b^2}}{b}\right )}{4\,\sqrt {b^2}}-\frac {8\,{\mathrm {e}}^{a+b\,x}}{3\,b\,\left ({\mathrm {e}}^{4\,a+4\,b\,x}-2\,{\mathrm {e}}^{2\,a+2\,b\,x}+1\right )}-\frac {7\,{\mathrm {e}}^{a+b\,x}}{2\,b\,\left (2\,{\mathrm {e}}^{2\,a+2\,b\,x}+{\mathrm {e}}^{4\,a+4\,b\,x}+1\right )}-\frac {8\,{\mathrm {e}}^{a+b\,x}}{3\,b\,\left (3\,{\mathrm {e}}^{2\,a+2\,b\,x}-3\,{\mathrm {e}}^{4\,a+4\,b\,x}+{\mathrm {e}}^{6\,a+6\,b\,x}-1\right )}-\frac {6\,{\mathrm {e}}^{a+b\,x}}{b\,\left (3\,{\mathrm {e}}^{2\,a+2\,b\,x}+3\,{\mathrm {e}}^{4\,a+4\,b\,x}+{\mathrm {e}}^{6\,a+6\,b\,x}+1\right )}+\frac {4\,{\mathrm {e}}^{a+b\,x}}{b\,\left (4\,{\mathrm {e}}^{2\,a+2\,b\,x}+6\,{\mathrm {e}}^{4\,a+4\,b\,x}+4\,{\mathrm {e}}^{6\,a+6\,b\,x}+{\mathrm {e}}^{8\,a+8\,b\,x}+1\right )}+\frac {6\,{\mathrm {e}}^{a+b\,x}}{b\,\left ({\mathrm {e}}^{2\,a+2\,b\,x}-1\right )}+\frac {11\,{\mathrm {e}}^{a+b\,x}}{4\,b\,\left ({\mathrm {e}}^{2\,a+2\,b\,x}+1\right )} \]
(35*atan((exp(b*x)*exp(a)*(b^2)^(1/2))/b))/(4*(b^2)^(1/2)) - (8*exp(a + b* x))/(3*b*(exp(4*a + 4*b*x) - 2*exp(2*a + 2*b*x) + 1)) - (7*exp(a + b*x))/( 2*b*(2*exp(2*a + 2*b*x) + exp(4*a + 4*b*x) + 1)) - (8*exp(a + b*x))/(3*b*( 3*exp(2*a + 2*b*x) - 3*exp(4*a + 4*b*x) + exp(6*a + 6*b*x) - 1)) - (6*exp( a + b*x))/(b*(3*exp(2*a + 2*b*x) + 3*exp(4*a + 4*b*x) + exp(6*a + 6*b*x) + 1)) + (4*exp(a + b*x))/(b*(4*exp(2*a + 2*b*x) + 6*exp(4*a + 4*b*x) + 4*ex p(6*a + 6*b*x) + exp(8*a + 8*b*x) + 1)) + (6*exp(a + b*x))/(b*(exp(2*a + 2 *b*x) - 1)) + (11*exp(a + b*x))/(4*b*(exp(2*a + 2*b*x) + 1))