Optimal. Leaf size=102 \[ -\frac {i d \text {Li}_2\left (-i e^{a+b x}\right )}{2 b^2}+\frac {i d \text {Li}_2\left (i e^{a+b x}\right )}{2 b^2}+\frac {d \text {sech}(a+b x)}{2 b^2}+\frac {(c+d x) \tan ^{-1}\left (e^{a+b x}\right )}{b}+\frac {(c+d x) \tanh (a+b x) \text {sech}(a+b x)}{2 b} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.06, antiderivative size = 102, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 4, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {4185, 4180, 2279, 2391} \[ -\frac {i d \text {PolyLog}\left (2,-i e^{a+b x}\right )}{2 b^2}+\frac {i d \text {PolyLog}\left (2,i e^{a+b x}\right )}{2 b^2}+\frac {d \text {sech}(a+b x)}{2 b^2}+\frac {(c+d x) \tan ^{-1}\left (e^{a+b x}\right )}{b}+\frac {(c+d x) \tanh (a+b x) \text {sech}(a+b x)}{2 b} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 2279
Rule 2391
Rule 4180
Rule 4185
Rubi steps
\begin {align*} \int (c+d x) \text {sech}^3(a+b x) \, dx &=\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}+\frac {1}{2} \int (c+d x) \text {sech}(a+b x) \, dx\\ &=\frac {(c+d x) \tan ^{-1}\left (e^{a+b x}\right )}{b}+\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}-\frac {(i d) \int \log \left (1-i e^{a+b x}\right ) \, dx}{2 b}+\frac {(i d) \int \log \left (1+i e^{a+b x}\right ) \, dx}{2 b}\\ &=\frac {(c+d x) \tan ^{-1}\left (e^{a+b x}\right )}{b}+\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}-\frac {(i d) \operatorname {Subst}\left (\int \frac {\log (1-i x)}{x} \, dx,x,e^{a+b x}\right )}{2 b^2}+\frac {(i d) \operatorname {Subst}\left (\int \frac {\log (1+i x)}{x} \, dx,x,e^{a+b x}\right )}{2 b^2}\\ &=\frac {(c+d x) \tan ^{-1}\left (e^{a+b x}\right )}{b}-\frac {i d \text {Li}_2\left (-i e^{a+b x}\right )}{2 b^2}+\frac {i d \text {Li}_2\left (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}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 2.96, size = 178, normalized size = 1.75 \[ \frac {b c \tan ^{-1}(\sinh (a+b x))+b c \tanh (a+b x) \text {sech}(a+b x)+\frac {1}{2} d \left (-2 i \left (\text {Li}_2\left (-i e^{a+b x}\right )-\text {Li}_2\left (i e^{a+b x}\right )\right )-\left ((-2 i a-2 i b x+\pi ) \left (\log \left (1-i e^{a+b x}\right )-\log \left (1+i e^{a+b x}\right )\right )\right )+(\pi -2 i a) \log \left (\cot \left (\frac {1}{4} (2 i a+2 i b x+\pi )\right )\right )\right )+b d x \text {sech}(a) \sinh (b x) \text {sech}^2(a+b x)+d (b x \tanh (a)+1) \text {sech}(a+b x)}{2 b^2} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [B] time = 0.45, size = 1243, normalized size = 12.19 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (d x + c\right )} \operatorname {sech}\left (b x + a\right )^{3}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [B] time = 0.33, size = 216, normalized size = 2.12 \[ \frac {{\mathrm e}^{b x +a} \left (b d x \,{\mathrm e}^{2 b x +2 a}+b c \,{\mathrm e}^{2 b x +2 a}-b d x +d \,{\mathrm e}^{2 b x +2 a}-c b +d \right )}{b^{2} \left (1+{\mathrm e}^{2 b x +2 a}\right )^{2}}+\frac {c \arctan \left ({\mathrm e}^{b x +a}\right )}{b}-\frac {i d \ln \left (1+i {\mathrm e}^{b x +a}\right ) x}{2 b}-\frac {i d \ln \left (1+i {\mathrm e}^{b x +a}\right ) a}{2 b^{2}}+\frac {i d \ln \left (1-i {\mathrm e}^{b x +a}\right ) x}{2 b}+\frac {i d \ln \left (1-i {\mathrm e}^{b x +a}\right ) a}{2 b^{2}}-\frac {i d \dilog \left (1+i {\mathrm e}^{b x +a}\right )}{2 b^{2}}+\frac {i d \dilog \left (1-i {\mathrm e}^{b x +a}\right )}{2 b^{2}}-\frac {d a \arctan \left ({\mathrm e}^{b x +a}\right )}{b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ d {\left (\frac {{\left (b x e^{\left (3 \, a\right )} + e^{\left (3 \, a\right )}\right )} e^{\left (3 \, b x\right )} - {\left (b x e^{a} - e^{a}\right )} e^{\left (b x\right )}}{b^{2} e^{\left (4 \, b x + 4 \, a\right )} + 2 \, b^{2} e^{\left (2 \, b x + 2 \, a\right )} + b^{2}} + 8 \, \int \frac {x e^{\left (b x + a\right )}}{8 \, {\left (e^{\left (2 \, b x + 2 \, a\right )} + 1\right )}}\,{d x}\right )} - c {\left (\frac {\arctan \left (e^{\left (-b x - a\right )}\right )}{b} - \frac {e^{\left (-b x - a\right )} - e^{\left (-3 \, b x - 3 \, a\right )}}{b {\left (2 \, e^{\left (-2 \, b x - 2 \, a\right )} + e^{\left (-4 \, b x - 4 \, a\right )} + 1\right )}}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {c+d\,x}{{\mathrm {cosh}\left (a+b\,x\right )}^3} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \left (c + d x\right ) \operatorname {sech}^{3}{\left (a + b x \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________