Optimal. Leaf size=75 \[ \frac {i d \text {Li}_2\left (-i e^{i (a+b x)}\right )}{b^2}-\frac {i d \text {Li}_2\left (i e^{i (a+b x)}\right )}{b^2}-\frac {2 i (c+d x) \tan ^{-1}\left (e^{i (a+b x)}\right )}{b} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.04, antiderivative size = 75, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 3, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {4181, 2279, 2391} \[ \frac {i d \text {Li}_2\left (-i e^{i (a+b x)}\right )}{b^2}-\frac {i d \text {Li}_2\left (i e^{i (a+b x)}\right )}{b^2}-\frac {2 i (c+d x) \tan ^{-1}\left (e^{i (a+b x)}\right )}{b} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 2279
Rule 2391
Rule 4181
Rubi steps
\begin {align*} \int (c+d x) \sec (a+b x) \, dx &=-\frac {2 i (c+d x) \tan ^{-1}\left (e^{i (a+b x)}\right )}{b}-\frac {d \int \log \left (1-i e^{i (a+b x)}\right ) \, dx}{b}+\frac {d \int \log \left (1+i e^{i (a+b x)}\right ) \, dx}{b}\\ &=-\frac {2 i (c+d x) \tan ^{-1}\left (e^{i (a+b x)}\right )}{b}+\frac {(i d) \operatorname {Subst}\left (\int \frac {\log (1-i x)}{x} \, dx,x,e^{i (a+b x)}\right )}{b^2}-\frac {(i d) \operatorname {Subst}\left (\int \frac {\log (1+i x)}{x} \, dx,x,e^{i (a+b x)}\right )}{b^2}\\ &=-\frac {2 i (c+d x) \tan ^{-1}\left (e^{i (a+b x)}\right )}{b}+\frac {i d \text {Li}_2\left (-i e^{i (a+b x)}\right )}{b^2}-\frac {i d \text {Li}_2\left (i e^{i (a+b x)}\right )}{b^2}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.01, size = 87, normalized size = 1.16 \[ \frac {i d \text {Li}_2\left (-i e^{i (a+b x)}\right )}{b^2}-\frac {i d \text {Li}_2\left (i e^{i (a+b x)}\right )}{b^2}+\frac {c \tanh ^{-1}(\sin (a+b x))}{b}-\frac {2 i d x \tan ^{-1}\left (e^{i a+i b x}\right )}{b} \]
Warning: Unable to verify antiderivative.
[In]
[Out]
________________________________________________________________________________________
fricas [B] time = 0.94, size = 306, normalized size = 4.08 \[ \frac {-i \, d {\rm Li}_2\left (i \, \cos \left (b x + a\right ) + \sin \left (b x + a\right )\right ) - i \, d {\rm Li}_2\left (i \, \cos \left (b x + a\right ) - \sin \left (b x + a\right )\right ) + i \, d {\rm Li}_2\left (-i \, \cos \left (b x + a\right ) + \sin \left (b x + a\right )\right ) + i \, d {\rm Li}_2\left (-i \, \cos \left (b x + a\right ) - \sin \left (b x + a\right )\right ) + {\left (b c - a d\right )} \log \left (\cos \left (b x + a\right ) + i \, \sin \left (b x + a\right ) + i\right ) - {\left (b c - a d\right )} \log \left (\cos \left (b x + a\right ) - i \, \sin \left (b x + a\right ) + i\right ) + {\left (b d x + a d\right )} \log \left (i \, \cos \left (b x + a\right ) + \sin \left (b x + a\right ) + 1\right ) - {\left (b d x + a d\right )} \log \left (i \, \cos \left (b x + a\right ) - \sin \left (b x + a\right ) + 1\right ) + {\left (b d x + a d\right )} \log \left (-i \, \cos \left (b x + a\right ) + \sin \left (b x + a\right ) + 1\right ) - {\left (b d x + a d\right )} \log \left (-i \, \cos \left (b x + a\right ) - \sin \left (b x + a\right ) + 1\right ) + {\left (b c - a d\right )} \log \left (-\cos \left (b x + a\right ) + i \, \sin \left (b x + a\right ) + i\right ) - {\left (b c - a d\right )} \log \left (-\cos \left (b x + a\right ) - i \, \sin \left (b x + a\right ) + i\right )}{2 \, b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (d x + c\right )} \sec \left (b x + a\right )\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [B] time = 0.01, size = 172, normalized size = 2.29 \[ -\frac {d \ln \left (1+i {\mathrm e}^{i \left (b x +a \right )}\right ) x}{b}+\frac {d \ln \left (1-i {\mathrm e}^{i \left (b x +a \right )}\right ) x}{b}-\frac {i d \dilog \left (1-i {\mathrm e}^{i \left (b x +a \right )}\right )}{b^{2}}-\frac {d \ln \left (1+i {\mathrm e}^{i \left (b x +a \right )}\right ) a}{b^{2}}+\frac {d \ln \left (1-i {\mathrm e}^{i \left (b x +a \right )}\right ) a}{b^{2}}+\frac {i d \dilog \left (1+i {\mathrm e}^{i \left (b x +a \right )}\right )}{b^{2}}-\frac {d a \ln \left (\sec \left (b x +a \right )+\tan \left (b x +a \right )\right )}{b^{2}}+\frac {c \ln \left (\sec \left (b x +a \right )+\tan \left (b x +a \right )\right )}{b} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {c+d\,x}{\cos \left (a+b\,x\right )} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \left (c + d x\right ) \sec {\left (a + b x \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________