Optimal. Leaf size=78 \[ \frac {\tanh ^{-1}(\sin (a+b x))}{16 b}+\frac {\tan ^3(a+b x) \sec ^3(a+b x)}{6 b}-\frac {\tan (a+b x) \sec ^3(a+b x)}{8 b}+\frac {\tan (a+b x) \sec (a+b x)}{16 b} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.07, antiderivative size = 78, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.176, Rules used = {2611, 3768, 3770} \[ \frac {\tanh ^{-1}(\sin (a+b x))}{16 b}+\frac {\tan ^3(a+b x) \sec ^3(a+b x)}{6 b}-\frac {\tan (a+b x) \sec ^3(a+b x)}{8 b}+\frac {\tan (a+b x) \sec (a+b x)}{16 b} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 2611
Rule 3768
Rule 3770
Rubi steps
\begin {align*} \int \sec ^3(a+b x) \tan ^4(a+b x) \, dx &=\frac {\sec ^3(a+b x) \tan ^3(a+b x)}{6 b}-\frac {1}{2} \int \sec ^3(a+b x) \tan ^2(a+b x) \, dx\\ &=-\frac {\sec ^3(a+b x) \tan (a+b x)}{8 b}+\frac {\sec ^3(a+b x) \tan ^3(a+b x)}{6 b}+\frac {1}{8} \int \sec ^3(a+b x) \, dx\\ &=\frac {\sec (a+b x) \tan (a+b x)}{16 b}-\frac {\sec ^3(a+b x) \tan (a+b x)}{8 b}+\frac {\sec ^3(a+b x) \tan ^3(a+b x)}{6 b}+\frac {1}{16} \int \sec (a+b x) \, dx\\ &=\frac {\tanh ^{-1}(\sin (a+b x))}{16 b}+\frac {\sec (a+b x) \tan (a+b x)}{16 b}-\frac {\sec ^3(a+b x) \tan (a+b x)}{8 b}+\frac {\sec ^3(a+b x) \tan ^3(a+b x)}{6 b}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.03, size = 99, normalized size = 1.27 \[ \frac {\tanh ^{-1}(\sin (a+b x))}{16 b}-\frac {\tan (a+b x) \sec ^5(a+b x)}{6 b}+\frac {\tan ^3(a+b x) \sec ^3(a+b x)}{3 b}+\frac {\tan (a+b x) \sec ^3(a+b x)}{24 b}+\frac {\tan (a+b x) \sec (a+b x)}{16 b} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.44, size = 84, normalized size = 1.08 \[ \frac {3 \, \cos \left (b x + a\right )^{6} \log \left (\sin \left (b x + a\right ) + 1\right ) - 3 \, \cos \left (b x + a\right )^{6} \log \left (-\sin \left (b x + a\right ) + 1\right ) + 2 \, {\left (3 \, \cos \left (b x + a\right )^{4} - 14 \, \cos \left (b x + a\right )^{2} + 8\right )} \sin \left (b x + a\right )}{96 \, b \cos \left (b x + a\right )^{6}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.53, size = 73, normalized size = 0.94 \[ -\frac {\frac {2 \, {\left (3 \, \sin \left (b x + a\right )^{5} + 8 \, \sin \left (b x + a\right )^{3} - 3 \, \sin \left (b x + a\right )\right )}}{{\left (\sin \left (b x + a\right )^{2} - 1\right )}^{3}} - 3 \, \log \left ({\left | \sin \left (b x + a\right ) + 1 \right |}\right ) + 3 \, \log \left ({\left | \sin \left (b x + a\right ) - 1 \right |}\right )}{96 \, b} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.04, size = 108, normalized size = 1.38 \[ \frac {\sin ^{5}\left (b x +a \right )}{6 b \cos \left (b x +a \right )^{6}}+\frac {\sin ^{5}\left (b x +a \right )}{24 b \cos \left (b x +a \right )^{4}}-\frac {\sin ^{5}\left (b x +a \right )}{48 b \cos \left (b x +a \right )^{2}}-\frac {\sin ^{3}\left (b x +a \right )}{48 b}-\frac {\sin \left (b x +a \right )}{16 b}+\frac {\ln \left (\sec \left (b x +a \right )+\tan \left (b x +a \right )\right )}{16 b} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 0.53, size = 91, normalized size = 1.17 \[ -\frac {\frac {2 \, {\left (3 \, \sin \left (b x + a\right )^{5} + 8 \, \sin \left (b x + a\right )^{3} - 3 \, \sin \left (b x + a\right )\right )}}{\sin \left (b x + a\right )^{6} - 3 \, \sin \left (b x + a\right )^{4} + 3 \, \sin \left (b x + a\right )^{2} - 1} - 3 \, \log \left (\sin \left (b x + a\right ) + 1\right ) + 3 \, \log \left (\sin \left (b x + a\right ) - 1\right )}{96 \, b} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 7.38, size = 177, normalized size = 2.27 \[ \frac {\mathrm {atanh}\left (\mathrm {tan}\left (\frac {a}{2}+\frac {b\,x}{2}\right )\right )}{8\,b}+\frac {-\frac {{\mathrm {tan}\left (\frac {a}{2}+\frac {b\,x}{2}\right )}^{11}}{8}+\frac {17\,{\mathrm {tan}\left (\frac {a}{2}+\frac {b\,x}{2}\right )}^9}{24}+\frac {19\,{\mathrm {tan}\left (\frac {a}{2}+\frac {b\,x}{2}\right )}^7}{4}+\frac {19\,{\mathrm {tan}\left (\frac {a}{2}+\frac {b\,x}{2}\right )}^5}{4}+\frac {17\,{\mathrm {tan}\left (\frac {a}{2}+\frac {b\,x}{2}\right )}^3}{24}-\frac {\mathrm {tan}\left (\frac {a}{2}+\frac {b\,x}{2}\right )}{8}}{b\,\left ({\mathrm {tan}\left (\frac {a}{2}+\frac {b\,x}{2}\right )}^{12}-6\,{\mathrm {tan}\left (\frac {a}{2}+\frac {b\,x}{2}\right )}^{10}+15\,{\mathrm {tan}\left (\frac {a}{2}+\frac {b\,x}{2}\right )}^8-20\,{\mathrm {tan}\left (\frac {a}{2}+\frac {b\,x}{2}\right )}^6+15\,{\mathrm {tan}\left (\frac {a}{2}+\frac {b\,x}{2}\right )}^4-6\,{\mathrm {tan}\left (\frac {a}{2}+\frac {b\,x}{2}\right )}^2+1\right )} \]
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]
________________________________________________________________________________________