Optimal. Leaf size=28 \[ \frac {\tan ^3(a+b x)}{3 b}-\frac {\tan (a+b x)}{b}+x \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.02, antiderivative size = 28, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {3473, 8} \[ \frac {\tan ^3(a+b x)}{3 b}-\frac {\tan (a+b x)}{b}+x \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 8
Rule 3473
Rubi steps
\begin {align*} \int \tan ^4(a+b x) \, dx &=\frac {\tan ^3(a+b x)}{3 b}-\int \tan ^2(a+b x) \, dx\\ &=-\frac {\tan (a+b x)}{b}+\frac {\tan ^3(a+b x)}{3 b}+\int 1 \, dx\\ &=x-\frac {\tan (a+b x)}{b}+\frac {\tan ^3(a+b x)}{3 b}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.01, size = 38, normalized size = 1.36 \[ \frac {\tan ^{-1}(\tan (a+b x))}{b}+\frac {\tan ^3(a+b x)}{3 b}-\frac {\tan (a+b x)}{b} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.45, size = 46, normalized size = 1.64 \[ \frac {3 \, b x \cos \left (b x + a\right )^{3} - {\left (4 \, \cos \left (b x + a\right )^{2} - 1\right )} \sin \left (b x + a\right )}{3 \, b \cos \left (b x + a\right )^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.25, size = 29, normalized size = 1.04 \[ \frac {\tan \left (b x + a\right )^{3} + 3 \, b x + 3 \, a - 3 \, \tan \left (b x + a\right )}{3 \, b} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.03, size = 28, normalized size = 1.00 \[ \frac {\frac {\left (\tan ^{3}\left (b x +a \right )\right )}{3}-\tan \left (b x +a \right )+b x +a}{b} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 0.42, size = 29, normalized size = 1.04 \[ \frac {\tan \left (b x + a\right )^{3} + 3 \, b x + 3 \, a - 3 \, \tan \left (b x + a\right )}{3 \, b} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 0.40, size = 24, normalized size = 0.86 \[ x-\frac {\mathrm {tan}\left (a+b\,x\right )-\frac {{\mathrm {tan}\left (a+b\,x\right )}^3}{3}}{b} \]
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]
________________________________________________________________________________________