Integrand size = 14, antiderivative size = 102 \[ \int (c+d x) \text {sech}^3(a+b x) \, dx=\frac {(c+d x) \arctan \left (e^{a+b x}\right )}{b}-\frac {i d \operatorname {PolyLog}\left (2,-i e^{a+b x}\right )}{2 b^2}+\frac {i d \operatorname {PolyLog}\left (2,i e^{a+b x}\right )}{2 b^2}+\frac {d \text {sech}(a+b x)}{2 b^2}+\frac {(c+d x) \text {sech}(a+b x) \tanh (a+b x)}{2 b} \]
(d*x+c)*arctan(exp(b*x+a))/b-1/2*I*d*polylog(2,-I*exp(b*x+a))/b^2+1/2*I*d* polylog(2,I*exp(b*x+a))/b^2+1/2*d*sech(b*x+a)/b^2+1/2*(d*x+c)*sech(b*x+a)* tanh(b*x+a)/b
Time = 0.61 (sec) , antiderivative size = 156, normalized size of antiderivative = 1.53 \[ \int (c+d x) \text {sech}^3(a+b x) \, dx=\frac {c \arctan (\sinh (a+b x))}{2 b}+\frac {i d \left (b x \left (\log \left (1-i e^{a+b x}\right )-\log \left (1+i e^{a+b x}\right )\right )-\operatorname {PolyLog}\left (2,-i e^{a+b x}\right )+\operatorname {PolyLog}\left (2,i e^{a+b x}\right )\right )}{2 b^2}+\frac {d \text {sech}(a) \text {sech}(a+b x) (\cosh (a)+b x \sinh (a))}{2 b^2}+\frac {d x \text {sech}(a) \text {sech}^2(a+b x) \sinh (b x)}{2 b}+\frac {c \text {sech}(a+b x) \tanh (a+b x)}{2 b} \]
(c*ArcTan[Sinh[a + b*x]])/(2*b) + ((I/2)*d*(b*x*(Log[1 - I*E^(a + b*x)] - Log[1 + I*E^(a + b*x)]) - PolyLog[2, (-I)*E^(a + b*x)] + PolyLog[2, I*E^(a + b*x)]))/b^2 + (d*Sech[a]*Sech[a + b*x]*(Cosh[a] + b*x*Sinh[a]))/(2*b^2) + (d*x*Sech[a]*Sech[a + b*x]^2*Sinh[b*x])/(2*b) + (c*Sech[a + b*x]*Tanh[a + b*x])/(2*b)
Time = 0.41 (sec) , antiderivative size = 104, normalized size of antiderivative = 1.02, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.429, Rules used = {3042, 4673, 3042, 4668, 2715, 2838}
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.
\(\displaystyle \int (c+d x) \text {sech}^3(a+b x) \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int (c+d x) \csc \left (i a+i b x+\frac {\pi }{2}\right )^3dx\) |
\(\Big \downarrow \) 4673 |
\(\displaystyle \frac {1}{2} \int (c+d x) \text {sech}(a+b x)dx+\frac {d \text {sech}(a+b x)}{2 b^2}+\frac {(c+d x) \tanh (a+b x) \text {sech}(a+b x)}{2 b}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {1}{2} \int (c+d x) \csc \left (i a+i b x+\frac {\pi }{2}\right )dx+\frac {d \text {sech}(a+b x)}{2 b^2}+\frac {(c+d x) \tanh (a+b x) \text {sech}(a+b x)}{2 b}\) |
\(\Big \downarrow \) 4668 |
\(\displaystyle \frac {1}{2} \left (-\frac {i d \int \log \left (1-i e^{a+b x}\right )dx}{b}+\frac {i d \int \log \left (1+i e^{a+b x}\right )dx}{b}+\frac {2 (c+d x) \arctan \left (e^{a+b x}\right )}{b}\right )+\frac {d \text {sech}(a+b x)}{2 b^2}+\frac {(c+d x) \tanh (a+b x) \text {sech}(a+b x)}{2 b}\) |
\(\Big \downarrow \) 2715 |
\(\displaystyle \frac {1}{2} \left (-\frac {i d \int e^{-a-b x} \log \left (1-i e^{a+b x}\right )de^{a+b x}}{b^2}+\frac {i d \int e^{-a-b x} \log \left (1+i e^{a+b x}\right )de^{a+b x}}{b^2}+\frac {2 (c+d x) \arctan \left (e^{a+b x}\right )}{b}\right )+\frac {d \text {sech}(a+b x)}{2 b^2}+\frac {(c+d x) \tanh (a+b x) \text {sech}(a+b x)}{2 b}\) |
\(\Big \downarrow \) 2838 |
\(\displaystyle \frac {1}{2} \left (\frac {2 (c+d x) \arctan \left (e^{a+b x}\right )}{b}-\frac {i d \operatorname {PolyLog}\left (2,-i e^{a+b x}\right )}{b^2}+\frac {i d \operatorname {PolyLog}\left (2,i e^{a+b x}\right )}{b^2}\right )+\frac {d \text {sech}(a+b x)}{2 b^2}+\frac {(c+d x) \tanh (a+b x) \text {sech}(a+b x)}{2 b}\) |
((2*(c + d*x)*ArcTan[E^(a + b*x)])/b - (I*d*PolyLog[2, (-I)*E^(a + b*x)])/ b^2 + (I*d*PolyLog[2, I*E^(a + b*x)])/b^2)/2 + (d*Sech[a + b*x])/(2*b^2) + ((c + d*x)*Sech[a + b*x]*Tanh[a + b*x])/(2*b)
3.1.11.3.1 Defintions of rubi rules used
Int[Log[(a_) + (b_.)*((F_)^((e_.)*((c_.) + (d_.)*(x_))))^(n_.)], x_Symbol] :> Simp[1/(d*e*n*Log[F]) Subst[Int[Log[a + b*x]/x, x], x, (F^(e*(c + d*x) ))^n], x] /; FreeQ[{F, a, b, c, d, e, n}, x] && GtQ[a, 0]
Int[Log[(c_.)*((d_) + (e_.)*(x_)^(n_.))]/(x_), x_Symbol] :> Simp[-PolyLog[2 , (-c)*e*x^n]/n, x] /; FreeQ[{c, d, e, n}, x] && EqQ[c*d, 1]
Int[csc[(e_.) + Pi*(k_.) + (Complex[0, fz_])*(f_.)*(x_)]*((c_.) + (d_.)*(x_ ))^(m_.), x_Symbol] :> Simp[-2*(c + d*x)^m*(ArcTanh[E^((-I)*e + f*fz*x)/E^( I*k*Pi)]/(f*fz*I)), x] + (-Simp[d*(m/(f*fz*I)) Int[(c + d*x)^(m - 1)*Log[ 1 - E^((-I)*e + f*fz*x)/E^(I*k*Pi)], x], x] + Simp[d*(m/(f*fz*I)) Int[(c + d*x)^(m - 1)*Log[1 + E^((-I)*e + f*fz*x)/E^(I*k*Pi)], x], x]) /; FreeQ[{c , d, e, f, fz}, x] && IntegerQ[2*k] && IGtQ[m, 0]
Int[(csc[(e_.) + (f_.)*(x_)]*(b_.))^(n_)*((c_.) + (d_.)*(x_)), x_Symbol] :> Simp[(-b^2)*(c + d*x)*Cot[e + f*x]*((b*Csc[e + f*x])^(n - 2)/(f*(n - 1))), x] + (-Simp[b^2*d*((b*Csc[e + f*x])^(n - 2)/(f^2*(n - 1)*(n - 2))), x] + S imp[b^2*((n - 2)/(n - 1)) Int[(c + d*x)*(b*Csc[e + f*x])^(n - 2), x], x]) /; FreeQ[{b, c, d, e, f}, x] && GtQ[n, 1] && NeQ[n, 2]
Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 215 vs. \(2 (87 ) = 174\).
Time = 0.58 (sec) , antiderivative size = 216, normalized size of antiderivative = 2.12
method | result | size |
risch | \(\frac {{\mathrm e}^{b x +a} \left ({\mathrm e}^{2 b x +2 a} b d x +{\mathrm e}^{2 b x +2 a} b c -d x b +{\mathrm e}^{2 b x +2 a} d -c b +d \right )}{b^{2} \left (1+{\mathrm e}^{2 b x +2 a}\right )^{2}}+\frac {c \arctan \left ({\mathrm e}^{b x +a}\right )}{b}-\frac {i d \ln \left (1+i {\mathrm e}^{b x +a}\right ) x}{2 b}-\frac {i d \ln \left (1+i {\mathrm e}^{b x +a}\right ) a}{2 b^{2}}+\frac {i d \ln \left (1-i {\mathrm e}^{b x +a}\right ) x}{2 b}+\frac {i d \ln \left (1-i {\mathrm e}^{b x +a}\right ) a}{2 b^{2}}-\frac {i d \operatorname {dilog}\left (1+i {\mathrm e}^{b x +a}\right )}{2 b^{2}}+\frac {i d \operatorname {dilog}\left (1-i {\mathrm e}^{b x +a}\right )}{2 b^{2}}-\frac {d a \arctan \left ({\mathrm e}^{b x +a}\right )}{b^{2}}\) | \(216\) |
exp(b*x+a)*(exp(2*b*x+2*a)*b*d*x+exp(2*b*x+2*a)*b*c-d*x*b+exp(2*b*x+2*a)*d -c*b+d)/b^2/(1+exp(2*b*x+2*a))^2+1/b*c*arctan(exp(b*x+a))-1/2*I/b*d*ln(1+I *exp(b*x+a))*x-1/2*I/b^2*d*ln(1+I*exp(b*x+a))*a+1/2*I/b*d*ln(1-I*exp(b*x+a ))*x+1/2*I/b^2*d*ln(1-I*exp(b*x+a))*a-1/2*I/b^2*d*dilog(1+I*exp(b*x+a))+1/ 2*I/b^2*d*dilog(1-I*exp(b*x+a))-1/b^2*d*a*arctan(exp(b*x+a))
Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 1267 vs. \(2 (81) = 162\).
Time = 0.28 (sec) , antiderivative size = 1267, normalized size of antiderivative = 12.42 \[ \int (c+d x) \text {sech}^3(a+b x) \, dx=\text {Too large to display} \]
1/2*(2*(b*d*x + b*c + d)*cosh(b*x + a)^3 + 6*(b*d*x + b*c + d)*cosh(b*x + a)*sinh(b*x + a)^2 + 2*(b*d*x + b*c + d)*sinh(b*x + a)^3 - 2*(b*d*x + b*c - d)*cosh(b*x + a) + (I*d*cosh(b*x + a)^4 + 4*I*d*cosh(b*x + a)*sinh(b*x + a)^3 + I*d*sinh(b*x + a)^4 + 2*I*d*cosh(b*x + a)^2 - 2*(-3*I*d*cosh(b*x + a)^2 - I*d)*sinh(b*x + a)^2 - 4*(-I*d*cosh(b*x + a)^3 - I*d*cosh(b*x + a) )*sinh(b*x + a) + I*d)*dilog(I*cosh(b*x + a) + I*sinh(b*x + a)) + (-I*d*co sh(b*x + a)^4 - 4*I*d*cosh(b*x + a)*sinh(b*x + a)^3 - I*d*sinh(b*x + a)^4 - 2*I*d*cosh(b*x + a)^2 - 2*(3*I*d*cosh(b*x + a)^2 + I*d)*sinh(b*x + a)^2 - 4*(I*d*cosh(b*x + a)^3 + I*d*cosh(b*x + a))*sinh(b*x + a) - I*d)*dilog(- I*cosh(b*x + a) - I*sinh(b*x + a)) + ((I*b*c - I*a*d)*cosh(b*x + a)^4 - 4* (-I*b*c + I*a*d)*cosh(b*x + a)*sinh(b*x + a)^3 + (I*b*c - I*a*d)*sinh(b*x + a)^4 - 2*(-I*b*c + I*a*d)*cosh(b*x + a)^2 - 2*(3*(-I*b*c + I*a*d)*cosh(b *x + a)^2 - I*b*c + I*a*d)*sinh(b*x + a)^2 + I*b*c - I*a*d - 4*((-I*b*c + I*a*d)*cosh(b*x + a)^3 + (-I*b*c + I*a*d)*cosh(b*x + a))*sinh(b*x + a))*lo g(cosh(b*x + a) + sinh(b*x + a) + I) + ((-I*b*c + I*a*d)*cosh(b*x + a)^4 - 4*(I*b*c - I*a*d)*cosh(b*x + a)*sinh(b*x + a)^3 + (-I*b*c + I*a*d)*sinh(b *x + a)^4 - 2*(I*b*c - I*a*d)*cosh(b*x + a)^2 - 2*(3*(I*b*c - I*a*d)*cosh( b*x + a)^2 + I*b*c - I*a*d)*sinh(b*x + a)^2 - I*b*c + I*a*d - 4*((I*b*c - I*a*d)*cosh(b*x + a)^3 + (I*b*c - I*a*d)*cosh(b*x + a))*sinh(b*x + a))*log (cosh(b*x + a) + sinh(b*x + a) - I) + ((-I*b*d*x - I*a*d)*cosh(b*x + a)...
\[ \int (c+d x) \text {sech}^3(a+b x) \, dx=\int \left (c + d x\right ) \operatorname {sech}^{3}{\left (a + b x \right )}\, dx \]
\[ \int (c+d x) \text {sech}^3(a+b x) \, dx=\int { {\left (d x + c\right )} \operatorname {sech}\left (b x + a\right )^{3} \,d x } \]
d*(((b*x*e^(3*a) + e^(3*a))*e^(3*b*x) - (b*x*e^a - e^a)*e^(b*x))/(b^2*e^(4 *b*x + 4*a) + 2*b^2*e^(2*b*x + 2*a) + b^2) + 8*integrate(1/8*x*e^(b*x + a) /(e^(2*b*x + 2*a) + 1), x)) - c*(arctan(e^(-b*x - a))/b - (e^(-b*x - a) - e^(-3*b*x - 3*a))/(b*(2*e^(-2*b*x - 2*a) + e^(-4*b*x - 4*a) + 1)))
\[ \int (c+d x) \text {sech}^3(a+b x) \, dx=\int { {\left (d x + c\right )} \operatorname {sech}\left (b x + a\right )^{3} \,d x } \]
Timed out. \[ \int (c+d x) \text {sech}^3(a+b x) \, dx=\int \frac {c+d\,x}{{\mathrm {cosh}\left (a+b\,x\right )}^3} \,d x \]