Optimal. Leaf size=76 \[ \frac {i e^{c (a+b x)}}{b c}-\frac {2 i e^{c (a+b x)} \, _2F_1\left (1,-\frac {i b c}{2 e};1-\frac {i b c}{2 e};e^{2 i (d+e x)}\right )}{b c} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.06, antiderivative size = 76, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 3, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.188, Rules used = {4528, 2225,
2283} \begin {gather*} \frac {i e^{c (a+b x)}}{b c}-\frac {2 i e^{c (a+b x)} \, _2F_1\left (1,-\frac {i b c}{2 e};1-\frac {i b c}{2 e};e^{2 i (d+e x)}\right )}{b c} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 2225
Rule 2283
Rule 4528
Rubi steps
\begin {align*} \int e^{c (a+b x)} \cot (d+e x) \, dx &=-\left (i \int \left (-e^{c (a+b x)}-\frac {2 e^{c (a+b x)}}{-1+e^{2 i (d+e x)}}\right ) \, dx\right )\\ &=i \int e^{c (a+b x)} \, dx+2 i \int \frac {e^{c (a+b x)}}{-1+e^{2 i (d+e x)}} \, dx\\ &=\frac {i e^{c (a+b x)}}{b c}-\frac {2 i e^{c (a+b x)} \, _2F_1\left (1,-\frac {i b c}{2 e};1-\frac {i b c}{2 e};e^{2 i (d+e x)}\right )}{b c}\\ \end {align*}
________________________________________________________________________________________
Mathematica [B] Both result and optimal contain complex but leaf count is larger than twice
the leaf count of optimal. \(163\) vs. \(2(76)=152\).
time = 1.05, size = 163, normalized size = 2.14 \begin {gather*} \frac {e^{c (a+b x)} \left (2 i b c e^{2 i (d+e x)} \, _2F_1\left (1,1-\frac {i b c}{2 e};2-\frac {i b c}{2 e};e^{2 i (d+e x)}\right )+i (b c+2 i e) \left (1+e^{2 i d}-2 e^{2 i d} \, _2F_1\left (1,-\frac {i b c}{2 e};1-\frac {i b c}{2 e};e^{2 i (d+e x)}\right )\right )\right )}{b c (b c+2 i e) \left (-1+e^{2 i d}\right )} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [F]
time = 0.04, size = 0, normalized size = 0.00 \[\int {\mathrm e}^{c \left (b x +a \right )} \cot \left (e x +d \right )\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} e^{a c} \int e^{b c x} \cot {\left (d + e x \right )}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \mathrm {cot}\left (d+e\,x\right )\,{\mathrm {e}}^{c\,\left (a+b\,x\right )} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________