Integrand size = 24, antiderivative size = 105 \[ \int e^{a+b x} \text {csch}^3(d+b x) \text {sech}^3(d+b x) \, dx=-\frac {8 e^{a+2 d+3 b x}}{b \left (1-e^{4 d+4 b x}\right )^2}+\frac {6 e^{a+2 d+3 b x}}{b \left (1-e^{4 d+4 b x}\right )}-\frac {3 e^{a-d} \arctan \left (e^{d+b x}\right )}{b}+\frac {3 e^{a-d} \text {arctanh}\left (e^{d+b x}\right )}{b} \] Output:
-8*exp(3*b*x+a+2*d)/b/(1-exp(4*b*x+4*d))^2+6*exp(3*b*x+a+2*d)/b/(1-exp(4*b *x+4*d))-3*exp(a-d)*arctan(exp(b*x+d))/b+3*exp(a-d)*arctanh(exp(b*x+d))/b
Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.
Time = 0.14 (sec) , antiderivative size = 64, normalized size of antiderivative = 0.61 \[ \int e^{a+b x} \text {csch}^3(d+b x) \text {sech}^3(d+b x) \, dx=\frac {64 e^{a+2 d+3 b x} \left (-1+\left (-1+e^{4 (d+b x)}\right )^2 \operatorname {Hypergeometric2F1}\left (\frac {3}{4},3,\frac {7}{4},e^{4 (d+b x)}\right )\right )}{5 b \left (-1+e^{4 (d+b x)}\right )^2} \] Input:
Integrate[E^(a + b*x)*Csch[d + b*x]^3*Sech[d + b*x]^3,x]
Output:
(64*E^(a + 2*d + 3*b*x)*(-1 + (-1 + E^(4*(d + b*x)))^2*Hypergeometric2F1[3 /4, 3, 7/4, E^(4*(d + b*x))]))/(5*b*(-1 + E^(4*(d + b*x)))^2)
Time = 0.27 (sec) , antiderivative size = 83, normalized size of antiderivative = 0.79, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.292, Rules used = {2720, 27, 817, 819, 827, 216, 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.
\(\displaystyle \int e^{a+b x} \text {csch}^3(b x+d) \text {sech}^3(b x+d) \, dx\) |
\(\Big \downarrow \) 2720 |
\(\displaystyle \frac {\int -\frac {64 e^{a+6 b x}}{\left (1-e^{4 b x}\right )^3}de^{b x}}{b}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle -\frac {64 e^a \int \frac {e^{6 b x}}{\left (1-e^{4 b x}\right )^3}de^{b x}}{b}\) |
\(\Big \downarrow \) 817 |
\(\displaystyle -\frac {64 e^a \left (\frac {e^{3 b x}}{8 \left (1-e^{4 b x}\right )^2}-\frac {3}{8} \int \frac {e^{2 b x}}{\left (1-e^{4 b x}\right )^2}de^{b x}\right )}{b}\) |
\(\Big \downarrow \) 819 |
\(\displaystyle -\frac {64 e^a \left (\frac {e^{3 b x}}{8 \left (1-e^{4 b x}\right )^2}-\frac {3}{8} \left (\frac {1}{4} \int \frac {e^{2 b x}}{1-e^{4 b x}}de^{b x}+\frac {e^{3 b x}}{4 \left (1-e^{4 b x}\right )}\right )\right )}{b}\) |
\(\Big \downarrow \) 827 |
\(\displaystyle -\frac {64 e^a \left (\frac {e^{3 b x}}{8 \left (1-e^{4 b x}\right )^2}-\frac {3}{8} \left (\frac {1}{4} \left (\frac {1}{2} \int \frac {1}{1-e^{2 b x}}de^{b x}-\frac {1}{2} \int \frac {1}{1+e^{2 b x}}de^{b x}\right )+\frac {e^{3 b x}}{4 \left (1-e^{4 b x}\right )}\right )\right )}{b}\) |
\(\Big \downarrow \) 216 |
\(\displaystyle -\frac {64 e^a \left (\frac {e^{3 b x}}{8 \left (1-e^{4 b x}\right )^2}-\frac {3}{8} \left (\frac {1}{4} \left (\frac {1}{2} \int \frac {1}{1-e^{2 b x}}de^{b x}-\frac {1}{2} \arctan \left (e^{b x}\right )\right )+\frac {e^{3 b x}}{4 \left (1-e^{4 b x}\right )}\right )\right )}{b}\) |
\(\Big \downarrow \) 219 |
\(\displaystyle -\frac {64 e^a \left (\frac {e^{3 b x}}{8 \left (1-e^{4 b x}\right )^2}-\frac {3}{8} \left (\frac {1}{4} \left (\frac {1}{2} \text {arctanh}\left (e^{b x}\right )-\frac {1}{2} \arctan \left (e^{b x}\right )\right )+\frac {e^{3 b x}}{4 \left (1-e^{4 b x}\right )}\right )\right )}{b}\) |
Input:
Int[E^(a + b*x)*Csch[d + b*x]^3*Sech[d + b*x]^3,x]
Output:
(-64*E^a*(E^(3*b*x)/(8*(1 - E^(4*b*x))^2) - (3*(E^(3*b*x)/(4*(1 - E^(4*b*x ))) + (-1/2*ArcTan[E^(b*x)] + ArcTanh[E^(b*x)]/2)/4))/8))/b
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[b, 2]))*A rcTan[Rt[b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a , 0] || GtQ[b, 0])
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_)^(n_))^(p_), x_Symbol] :> Simp[c^( n - 1)*(c*x)^(m - n + 1)*((a + b*x^n)^(p + 1)/(b*n*(p + 1))), x] - Simp[c^n *((m - n + 1)/(b*n*(p + 1))) Int[(c*x)^(m - n)*(a + b*x^n)^(p + 1), x], x ] /; FreeQ[{a, b, c}, x] && IGtQ[n, 0] && LtQ[p, -1] && GtQ[m + 1, n] && ! ILtQ[(m + n*(p + 1) + 1)/n, 0] && IntBinomialQ[a, b, c, n, m, p, x]
Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(-( c*x)^(m + 1))*((a + b*x^n)^(p + 1)/(a*c*n*(p + 1))), x] + Simp[(m + n*(p + 1) + 1)/(a*n*(p + 1)) Int[(c*x)^m*(a + b*x^n)^(p + 1), x], x] /; FreeQ[{a , b, c, m}, x] && IGtQ[n, 0] && LtQ[p, -1] && IntBinomialQ[a, b, c, n, m, p , x]
Int[(x_)^2/((a_) + (b_.)*(x_)^4), x_Symbol] :> With[{r = Numerator[Rt[-a/b, 2]], s = Denominator[Rt[-a/b, 2]]}, Simp[s/(2*b) Int[1/(r + s*x^2), x], x] - Simp[s/(2*b) Int[1/(r - s*x^2), x], x]] /; FreeQ[{a, b}, x] && !GtQ [a/b, 0]
Int[u_, x_Symbol] :> With[{v = FunctionOfExponential[u, x]}, Simp[v/D[v, x] Subst[Int[FunctionOfExponentialFunction[u, x]/x, x], x, v], x]] /; Funct ionOfExponentialQ[u, x] && !MatchQ[u, (w_)*((a_.)*(v_)^(n_))^(m_) /; FreeQ [{a, m, n}, x] && IntegerQ[m*n]] && !MatchQ[u, E^((c_.)*((a_.) + (b_.)*x)) *(F_)[v_] /; FreeQ[{a, b, c}, x] && InverseFunctionQ[F[x]]]
Result contains complex when optimal does not.
Time = 0.25 (sec) , antiderivative size = 188, normalized size of antiderivative = 1.79
\[-\frac {2 \left (3 \,{\mathrm e}^{4 b x +4 a +4 d}+{\mathrm e}^{4 a}\right ) {\mathrm e}^{3 b x +5 a +2 d}}{\left ({\mathrm e}^{2 b x +2 a +2 d}+{\mathrm e}^{2 a}\right )^{2} \left (-{\mathrm e}^{2 b x +2 a +2 d}+{\mathrm e}^{2 a}\right )^{2} b}+\frac {3 \ln \left ({\mathrm e}^{b x +a}+{\mathrm e}^{a -d}\right ) {\mathrm e}^{a -d}}{2 b}-\frac {3 \ln \left ({\mathrm e}^{b x +a}-{\mathrm e}^{a -d}\right ) {\mathrm e}^{a -d}}{2 b}+\frac {3 i \ln \left ({\mathrm e}^{b x +a}-i {\mathrm e}^{a -d}\right ) {\mathrm e}^{a -d}}{2 b}-\frac {3 i \ln \left ({\mathrm e}^{b x +a}+i {\mathrm e}^{a -d}\right ) {\mathrm e}^{a -d}}{2 b}\]
Input:
int(exp(b*x+a)*csch(b*x+d)^3*sech(b*x+d)^3,x)
Output:
-2/(exp(2*b*x+2*a+2*d)+exp(2*a))^2/(-exp(2*b*x+2*a+2*d)+exp(2*a))^2/b*(3*e xp(4*b*x+4*a+4*d)+exp(4*a))*exp(3*b*x+5*a+2*d)+3/2*ln(exp(b*x+a)+exp(a-d)) /b*exp(a-d)-3/2*ln(exp(b*x+a)-exp(a-d))/b*exp(a-d)+3/2*I*ln(exp(b*x+a)-I*e xp(a-d))/b*exp(a-d)-3/2*I*ln(exp(b*x+a)+I*exp(a-d))/b*exp(a-d)
Leaf count of result is larger than twice the leaf count of optimal. 2075 vs. \(2 (93) = 186\).
Time = 0.10 (sec) , antiderivative size = 2075, normalized size of antiderivative = 19.76 \[ \int e^{a+b x} \text {csch}^3(d+b x) \text {sech}^3(d+b x) \, dx=\text {Too large to display} \] Input:
integrate(exp(b*x+a)*csch(b*x+d)^3*sech(b*x+d)^3,x, algorithm="fricas")
Output:
-1/2*(12*cosh(b*x + d)^7*cosh(-a + d) + 12*(cosh(-a + d) - sinh(-a + d))*s inh(b*x + d)^7 + 84*(cosh(b*x + d)*cosh(-a + d) - cosh(b*x + d)*sinh(-a + d))*sinh(b*x + d)^6 + 252*(cosh(b*x + d)^2*cosh(-a + d) - cosh(b*x + d)^2* sinh(-a + d))*sinh(b*x + d)^5 + 420*(cosh(b*x + d)^3*cosh(-a + d) - cosh(b *x + d)^3*sinh(-a + d))*sinh(b*x + d)^4 + 4*cosh(b*x + d)^3*cosh(-a + d) + 4*(105*cosh(b*x + d)^4*cosh(-a + d) - (105*cosh(b*x + d)^4 + 1)*sinh(-a + d) + cosh(-a + d))*sinh(b*x + d)^3 + 12*(21*cosh(b*x + d)^5*cosh(-a + d) + cosh(b*x + d)*cosh(-a + d) - (21*cosh(b*x + d)^5 + cosh(b*x + d))*sinh(- a + d))*sinh(b*x + d)^2 + 6*(cosh(b*x + d)^8*cosh(-a + d) + (cosh(-a + d) - sinh(-a + d))*sinh(b*x + d)^8 + 8*(cosh(b*x + d)*cosh(-a + d) - cosh(b*x + d)*sinh(-a + d))*sinh(b*x + d)^7 + 28*(cosh(b*x + d)^2*cosh(-a + d) - c osh(b*x + d)^2*sinh(-a + d))*sinh(b*x + d)^6 + 56*(cosh(b*x + d)^3*cosh(-a + d) - cosh(b*x + d)^3*sinh(-a + d))*sinh(b*x + d)^5 - 2*cosh(b*x + d)^4* cosh(-a + d) + 2*(35*cosh(b*x + d)^4*cosh(-a + d) - (35*cosh(b*x + d)^4 - 1)*sinh(-a + d) - cosh(-a + d))*sinh(b*x + d)^4 + 8*(7*cosh(b*x + d)^5*cos h(-a + d) - cosh(b*x + d)*cosh(-a + d) - (7*cosh(b*x + d)^5 - cosh(b*x + d ))*sinh(-a + d))*sinh(b*x + d)^3 + 4*(7*cosh(b*x + d)^6*cosh(-a + d) - 3*c osh(b*x + d)^2*cosh(-a + d) - (7*cosh(b*x + d)^6 - 3*cosh(b*x + d)^2)*sinh (-a + d))*sinh(b*x + d)^2 + 8*(cosh(b*x + d)^7*cosh(-a + d) - cosh(b*x + d )^3*cosh(-a + d) - (cosh(b*x + d)^7 - cosh(b*x + d)^3)*sinh(-a + d))*si...
\[ \int e^{a+b x} \text {csch}^3(d+b x) \text {sech}^3(d+b x) \, dx=e^{a} \int e^{b x} \operatorname {csch}^{3}{\left (b x + d \right )} \operatorname {sech}^{3}{\left (b x + d \right )}\, dx \] Input:
integrate(exp(b*x+a)*csch(b*x+d)**3*sech(b*x+d)**3,x)
Output:
exp(a)*Integral(exp(b*x)*csch(b*x + d)**3*sech(b*x + d)**3, x)
Time = 0.12 (sec) , antiderivative size = 130, normalized size of antiderivative = 1.24 \[ \int e^{a+b x} \text {csch}^3(d+b x) \text {sech}^3(d+b x) \, dx=-\frac {3 \, \arctan \left (e^{\left (b x + d\right )}\right ) e^{\left (a - d\right )}}{b} + \frac {3 \, e^{\left (a - d\right )} \log \left (e^{\left (b x + a + d\right )} + e^{a}\right )}{2 \, b} - \frac {3 \, e^{\left (a - d\right )} \log \left (e^{\left (b x + a + d\right )} - e^{a}\right )}{2 \, b} - \frac {2 \, {\left (3 \, e^{\left (7 \, b x + 9 \, a + 6 \, d\right )} + e^{\left (3 \, b x + 9 \, a + 2 \, d\right )}\right )}}{b {\left (e^{\left (8 \, b x + 8 \, a + 8 \, d\right )} - 2 \, e^{\left (4 \, b x + 8 \, a + 4 \, d\right )} + e^{\left (8 \, a\right )}\right )}} \] Input:
integrate(exp(b*x+a)*csch(b*x+d)^3*sech(b*x+d)^3,x, algorithm="maxima")
Output:
-3*arctan(e^(b*x + d))*e^(a - d)/b + 3/2*e^(a - d)*log(e^(b*x + a + d) + e ^a)/b - 3/2*e^(a - d)*log(e^(b*x + a + d) - e^a)/b - 2*(3*e^(7*b*x + 9*a + 6*d) + e^(3*b*x + 9*a + 2*d))/(b*(e^(8*b*x + 8*a + 8*d) - 2*e^(4*b*x + 8* a + 4*d) + e^(8*a)))
Time = 0.12 (sec) , antiderivative size = 101, normalized size of antiderivative = 0.96 \[ \int e^{a+b x} \text {csch}^3(d+b x) \text {sech}^3(d+b x) \, dx=-\frac {1}{2} \, {\left (\frac {6 \, \arctan \left (e^{\left (b x + d\right )}\right ) e^{\left (-7 \, d\right )}}{b} - \frac {3 \, e^{\left (-7 \, d\right )} \log \left (e^{\left (b x + d\right )} + 1\right )}{b} + \frac {3 \, e^{\left (-7 \, d\right )} \log \left ({\left | e^{\left (b x + d\right )} - 1 \right |}\right )}{b} + \frac {4 \, {\left (e^{\left (3 \, b x\right )} + 3 \, e^{\left (7 \, b x + 4 \, d\right )}\right )} e^{\left (-4 \, d\right )}}{b {\left (e^{\left (4 \, b x + 4 \, d\right )} - 1\right )}^{2}}\right )} e^{\left (a + 6 \, d\right )} \] Input:
integrate(exp(b*x+a)*csch(b*x+d)^3*sech(b*x+d)^3,x, algorithm="giac")
Output:
-1/2*(6*arctan(e^(b*x + d))*e^(-7*d)/b - 3*e^(-7*d)*log(e^(b*x + d) + 1)/b + 3*e^(-7*d)*log(abs(e^(b*x + d) - 1))/b + 4*(e^(3*b*x) + 3*e^(7*b*x + 4* d))*e^(-4*d)/(b*(e^(4*b*x + 4*d) - 1)^2))*e^(a + 6*d)
Timed out. \[ \int e^{a+b x} \text {csch}^3(d+b x) \text {sech}^3(d+b x) \, dx=\int \frac {{\mathrm {e}}^{a+b\,x}}{{\mathrm {cosh}\left (d+b\,x\right )}^3\,{\mathrm {sinh}\left (d+b\,x\right )}^3} \,d x \] Input:
int(exp(a + b*x)/(cosh(d + b*x)^3*sinh(d + b*x)^3),x)
Output:
int(exp(a + b*x)/(cosh(d + b*x)^3*sinh(d + b*x)^3), x)
Time = 0.24 (sec) , antiderivative size = 226, normalized size of antiderivative = 2.15 \[ \int e^{a+b x} \text {csch}^3(d+b x) \text {sech}^3(d+b x) \, dx=\frac {e^{a} \left (-6 e^{8 b x +8 d} \mathit {atan} \left (e^{b x +d}\right )+12 e^{4 b x +4 d} \mathit {atan} \left (e^{b x +d}\right )-6 \mathit {atan} \left (e^{b x +d}\right )-3 e^{8 b x +8 d} \mathrm {log}\left (e^{b x +d}-1\right )+3 e^{8 b x +8 d} \mathrm {log}\left (e^{b x +d}+1\right )-12 e^{7 b x +7 d}+6 e^{4 b x +4 d} \mathrm {log}\left (e^{b x +d}-1\right )-6 e^{4 b x +4 d} \mathrm {log}\left (e^{b x +d}+1\right )-4 e^{3 b x +3 d}-3 \,\mathrm {log}\left (e^{b x +d}-1\right )+3 \,\mathrm {log}\left (e^{b x +d}+1\right )\right )}{2 e^{d} b \left (e^{8 b x +8 d}-2 e^{4 b x +4 d}+1\right )} \] Input:
int(exp(b*x+a)*csch(b*x+d)^3*sech(b*x+d)^3,x)
Output:
(e**a*( - 6*e**(8*b*x + 8*d)*atan(e**(b*x + d)) + 12*e**(4*b*x + 4*d)*atan (e**(b*x + d)) - 6*atan(e**(b*x + d)) - 3*e**(8*b*x + 8*d)*log(e**(b*x + d ) - 1) + 3*e**(8*b*x + 8*d)*log(e**(b*x + d) + 1) - 12*e**(7*b*x + 7*d) + 6*e**(4*b*x + 4*d)*log(e**(b*x + d) - 1) - 6*e**(4*b*x + 4*d)*log(e**(b*x + d) + 1) - 4*e**(3*b*x + 3*d) - 3*log(e**(b*x + d) - 1) + 3*log(e**(b*x + d) + 1)))/(2*e**d*b*(e**(8*b*x + 8*d) - 2*e**(4*b*x + 4*d) + 1))