Optimal. Leaf size=107 \[ -\frac {i d \text {Li}_2\left (-e^{2 i (a+b x)}\right )}{2 b^2}+\frac {d \sin (a+b x) \cos (a+b x)}{b^2}+\frac {(c+d x) \log \left (1+e^{2 i (a+b x)}\right )}{b}+\frac {2 (c+d x) \sin ^2(a+b x)}{b}-\frac {d x}{b}-\frac {i (c+d x)^2}{2 d} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.18, antiderivative size = 107, normalized size of antiderivative = 1.00, number of steps used = 13, number of rules used = 9, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.429, Rules used = {4431, 4404, 2635, 8, 4407, 3719, 2190, 2279, 2391} \[ -\frac {i d \text {PolyLog}\left (2,-e^{2 i (a+b x)}\right )}{2 b^2}+\frac {d \sin (a+b x) \cos (a+b x)}{b^2}+\frac {(c+d x) \log \left (1+e^{2 i (a+b x)}\right )}{b}+\frac {2 (c+d x) \sin ^2(a+b x)}{b}-\frac {d x}{b}-\frac {i (c+d x)^2}{2 d} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 8
Rule 2190
Rule 2279
Rule 2391
Rule 2635
Rule 3719
Rule 4404
Rule 4407
Rule 4431
Rubi steps
\begin {align*} \int (c+d x) \sec (a+b x) \sin (3 a+3 b x) \, dx &=\int \left (3 (c+d x) \cos (a+b x) \sin (a+b x)-(c+d x) \sin ^2(a+b x) \tan (a+b x)\right ) \, dx\\ &=3 \int (c+d x) \cos (a+b x) \sin (a+b x) \, dx-\int (c+d x) \sin ^2(a+b x) \tan (a+b x) \, dx\\ &=\frac {3 (c+d x) \sin ^2(a+b x)}{2 b}-\frac {(3 d) \int \sin ^2(a+b x) \, dx}{2 b}+\int (c+d x) \cos (a+b x) \sin (a+b x) \, dx-\int (c+d x) \tan (a+b x) \, dx\\ &=-\frac {i (c+d x)^2}{2 d}+\frac {3 d \cos (a+b x) \sin (a+b x)}{4 b^2}+\frac {2 (c+d x) \sin ^2(a+b x)}{b}+2 i \int \frac {e^{2 i (a+b x)} (c+d x)}{1+e^{2 i (a+b x)}} \, dx-\frac {d \int \sin ^2(a+b x) \, dx}{2 b}-\frac {(3 d) \int 1 \, dx}{4 b}\\ &=-\frac {3 d x}{4 b}-\frac {i (c+d x)^2}{2 d}+\frac {(c+d x) \log \left (1+e^{2 i (a+b x)}\right )}{b}+\frac {d \cos (a+b x) \sin (a+b x)}{b^2}+\frac {2 (c+d x) \sin ^2(a+b x)}{b}-\frac {d \int 1 \, dx}{4 b}-\frac {d \int \log \left (1+e^{2 i (a+b x)}\right ) \, dx}{b}\\ &=-\frac {d x}{b}-\frac {i (c+d x)^2}{2 d}+\frac {(c+d x) \log \left (1+e^{2 i (a+b x)}\right )}{b}+\frac {d \cos (a+b x) \sin (a+b x)}{b^2}+\frac {2 (c+d x) \sin ^2(a+b x)}{b}+\frac {(i d) \operatorname {Subst}\left (\int \frac {\log (1+x)}{x} \, dx,x,e^{2 i (a+b x)}\right )}{2 b^2}\\ &=-\frac {d x}{b}-\frac {i (c+d x)^2}{2 d}+\frac {(c+d x) \log \left (1+e^{2 i (a+b x)}\right )}{b}-\frac {i d \text {Li}_2\left (-e^{2 i (a+b x)}\right )}{2 b^2}+\frac {d \cos (a+b x) \sin (a+b x)}{b^2}+\frac {2 (c+d x) \sin ^2(a+b x)}{b}\\ \end {align*}
________________________________________________________________________________________
Mathematica [B] time = 5.57, size = 254, normalized size = 2.37 \[ \frac {d \csc (a) \sec (a) \left (b^2 x^2 e^{-i \tan ^{-1}(\cot (a))}-\frac {\cot (a) \left (i \text {Li}_2\left (e^{2 i \left (b x-\tan ^{-1}(\cot (a))\right )}\right )+i b x \left (-2 \tan ^{-1}(\cot (a))-\pi \right )-2 \left (b x-\tan ^{-1}(\cot (a))\right ) \log \left (1-e^{2 i \left (b x-\tan ^{-1}(\cot (a))\right )}\right )-2 \tan ^{-1}(\cot (a)) \log \left (\sin \left (b x-\tan ^{-1}(\cot (a))\right )\right )-\pi \log \left (1+e^{-2 i b x}\right )+\pi \log (\cos (b x))\right )}{\sqrt {\cot ^2(a)+1}}\right )}{2 b^2 \sqrt {\csc ^2(a) \left (\sin ^2(a)+\cos ^2(a)\right )}}-\frac {d \cos (2 b x) (2 b x \cos (2 a)-\sin (2 a))}{2 b^2}+\frac {d \sin (2 b x) (2 b x \sin (2 a)+\cos (2 a))}{2 b^2}+\frac {c \left (2 \sin ^2(a+b x)+\log (\cos (a+b x))\right )}{b}-\frac {1}{2} d x^2 \tan (a) \]
Warning: Unable to verify antiderivative.
[In]
[Out]
________________________________________________________________________________________
fricas [B] time = 0.53, size = 340, normalized size = 3.18 \[ \frac {2 \, b d x - 4 \, {\left (b d x + b c\right )} \cos \left (b x + a\right )^{2} + 2 \, d \cos \left (b x + a\right ) \sin \left (b x + a\right ) + i \, d {\rm Li}_2\left (i \, \cos \left (b x + a\right ) + \sin \left (b x + a\right )\right ) - i \, d {\rm Li}_2\left (i \, \cos \left (b x + a\right ) - \sin \left (b x + a\right )\right ) - i \, d {\rm Li}_2\left (-i \, \cos \left (b x + a\right ) + \sin \left (b x + a\right )\right ) + i \, d {\rm Li}_2\left (-i \, \cos \left (b x + a\right ) - \sin \left (b x + a\right )\right ) + {\left (b c - a d\right )} \log \left (\cos \left (b x + a\right ) + i \, \sin \left (b x + a\right ) + i\right ) + {\left (b c - a d\right )} \log \left (\cos \left (b x + a\right ) - i \, \sin \left (b x + a\right ) + i\right ) + {\left (b d x + a d\right )} \log \left (i \, \cos \left (b x + a\right ) + \sin \left (b x + a\right ) + 1\right ) + {\left (b d x + a d\right )} \log \left (i \, \cos \left (b x + a\right ) - \sin \left (b x + a\right ) + 1\right ) + {\left (b d x + a d\right )} \log \left (-i \, \cos \left (b x + a\right ) + \sin \left (b x + a\right ) + 1\right ) + {\left (b d x + a d\right )} \log \left (-i \, \cos \left (b x + a\right ) - \sin \left (b x + a\right ) + 1\right ) + {\left (b c - a d\right )} \log \left (-\cos \left (b x + a\right ) + i \, \sin \left (b x + a\right ) + i\right ) + {\left (b c - a d\right )} \log \left (-\cos \left (b x + a\right ) - i \, \sin \left (b x + a\right ) + i\right )}{2 \, b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (d x + c\right )} \sec \left (b x + a\right ) \sin \left (3 \, b x + 3 \, a\right )\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.31, size = 177, normalized size = 1.65 \[ -\frac {i d \,x^{2}}{2}+i c x -\frac {\left (2 b d x +2 c b +i d \right ) {\mathrm e}^{2 i \left (b x +a \right )}}{4 b^{2}}-\frac {\left (2 b d x +2 c b -i d \right ) {\mathrm e}^{-2 i \left (b x +a \right )}}{4 b^{2}}+\frac {c \ln \left (1+{\mathrm e}^{2 i \left (b x +a \right )}\right )}{b}-\frac {2 c \ln \left ({\mathrm e}^{i \left (b x +a \right )}\right )}{b}-\frac {2 i d a x}{b}-\frac {i d \,a^{2}}{b^{2}}+\frac {d \ln \left (1+{\mathrm e}^{2 i \left (b x +a \right )}\right ) x}{b}-\frac {i d \polylog \left (2, -{\mathrm e}^{2 i \left (b x +a \right )}\right )}{2 b^{2}}+\frac {2 d a \ln \left ({\mathrm e}^{i \left (b x +a \right )}\right )}{b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ -\frac {c {\left (2 \, \cos \left (2 \, b x + 2 \, a\right ) - \log \left (\cos \left (2 \, b x\right )^{2} + 2 \, \cos \left (2 \, b x\right ) \cos \left (2 \, a\right ) + \cos \left (2 \, a\right )^{2} + \sin \left (2 \, b x\right )^{2} - 2 \, \sin \left (2 \, b x\right ) \sin \left (2 \, a\right ) + \sin \left (2 \, a\right )^{2}\right )\right )}}{2 \, b} - \frac {{\left (i \, b^{2} x^{2} - 2 i \, b x \arctan \left (\sin \left (2 \, b x + 2 \, a\right ), \cos \left (2 \, b x + 2 \, a\right ) + 1\right ) + 2 \, b x \cos \left (2 \, b x + 2 \, a\right ) - b x \log \left (\cos \left (2 \, b x + 2 \, a\right )^{2} + \sin \left (2 \, b x + 2 \, a\right )^{2} + 2 \, \cos \left (2 \, b x + 2 \, a\right ) + 1\right ) + i \, {\rm Li}_2\left (-e^{\left (2 i \, b x + 2 i \, a\right )}\right ) - \sin \left (2 \, b x + 2 \, a\right )\right )} d}{2 \, b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {\sin \left (3\,a+3\,b\,x\right )\,\left (c+d\,x\right )}{\cos \left (a+b\,x\right )} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [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]
________________________________________________________________________________________