Optimal. Leaf size=69 \[ -\frac {i x^2}{2}+i x^2 \, _2F_1\left (1,-\frac {i}{b d n};1-\frac {i}{b d n};-e^{2 i a d} \left (c x^n\right )^{2 i b d}\right ) \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.04, antiderivative size = 69, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 4, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.267, Rules used = {4593, 4591,
470, 371} \begin {gather*} i x^2 \, _2F_1\left (1,-\frac {i}{b d n};1-\frac {i}{b d n};-e^{2 i a d} \left (c x^n\right )^{2 i b d}\right )-\frac {i x^2}{2} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 371
Rule 470
Rule 4591
Rule 4593
Rubi steps
\begin {align*} \int x \tan \left (d \left (a+b \log \left (c x^n\right )\right )\right ) \, dx &=\int x \tan \left (d \left (a+b \log \left (c x^n\right )\right )\right ) \, dx\\ \end {align*}
________________________________________________________________________________________
Mathematica [B] Both result and optimal contain complex but leaf count is larger than twice
the leaf count of optimal. \(146\) vs. \(2(69)=138\).
time = 6.66, size = 146, normalized size = 2.12 \begin {gather*} \frac {x^2 \left (i e^{2 i d \left (a+b \log \left (c x^n\right )\right )} \, _2F_1\left (1,1-\frac {i}{b d n};2-\frac {i}{b d n};-e^{2 i d \left (a+b \log \left (c x^n\right )\right )}\right )+(-i+b d n) \, _2F_1\left (1,-\frac {i}{b d n};1-\frac {i}{b d n};-e^{2 i d \left (a+b \log \left (c x^n\right )\right )}\right )\right )}{-2-2 i b d n} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [F]
time = 0.01, size = 0, normalized size = 0.00 \[\int x \tan \left (d \left (a +b \ln \left (c \,x^{n}\right )\right )\right )\, dx\]
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*} \int x \tan {\left (a d + b d \log {\left (c x^{n} \right )} \right )}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \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 x\,\mathrm {tan}\left (d\,\left (a+b\,\ln \left (c\,x^n\right )\right )\right ) \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________