Optimal. Leaf size=37 \[ -\frac {\cos ^3(a+b x)}{3 b}+\frac {2 \cos (a+b x)}{b}+\frac {\sec (a+b x)}{b} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.03, antiderivative size = 37, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.118, Rules used = {2590, 270} \[ -\frac {\cos ^3(a+b x)}{3 b}+\frac {2 \cos (a+b x)}{b}+\frac {\sec (a+b x)}{b} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 270
Rule 2590
Rubi steps
\begin {align*} \int \sin ^3(a+b x) \tan ^2(a+b x) \, dx &=-\frac {\operatorname {Subst}\left (\int \frac {\left (1-x^2\right )^2}{x^2} \, dx,x,\cos (a+b x)\right )}{b}\\ &=-\frac {\operatorname {Subst}\left (\int \left (-2+\frac {1}{x^2}+x^2\right ) \, dx,x,\cos (a+b x)\right )}{b}\\ &=\frac {2 \cos (a+b x)}{b}-\frac {\cos ^3(a+b x)}{3 b}+\frac {\sec (a+b x)}{b}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.03, size = 39, normalized size = 1.05 \[ \frac {7 \cos (a+b x)}{4 b}-\frac {\cos (3 (a+b x))}{12 b}+\frac {\sec (a+b x)}{b} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.49, size = 33, normalized size = 0.89 \[ -\frac {\cos \left (b x + a\right )^{4} - 6 \, \cos \left (b x + a\right )^{2} - 3}{3 \, b \cos \left (b x + a\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [B] time = 0.31, size = 99, normalized size = 2.68 \[ \frac {2 \, {\left (\frac {3}{\frac {\cos \left (b x + a\right ) - 1}{\cos \left (b x + a\right ) + 1} + 1} + \frac {\frac {12 \, {\left (\cos \left (b x + a\right ) - 1\right )}}{\cos \left (b x + a\right ) + 1} - \frac {3 \, {\left (\cos \left (b x + a\right ) - 1\right )}^{2}}{{\left (\cos \left (b x + a\right ) + 1\right )}^{2}} - 5}{{\left (\frac {\cos \left (b x + a\right ) - 1}{\cos \left (b x + a\right ) + 1} - 1\right )}^{3}}\right )}}{3 \, b} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.03, size = 50, normalized size = 1.35 \[ \frac {\frac {\sin ^{6}\left (b x +a \right )}{\cos \left (b x +a \right )}+\left (\frac {8}{3}+\sin ^{4}\left (b x +a \right )+\frac {4 \left (\sin ^{2}\left (b x +a \right )\right )}{3}\right ) \cos \left (b x +a \right )}{b} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 0.32, size = 32, normalized size = 0.86 \[ -\frac {\cos \left (b x + a\right )^{3} - \frac {3}{\cos \left (b x + a\right )} - 6 \, \cos \left (b x + a\right )}{3 \, b} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 0.50, size = 31, normalized size = 0.84 \[ -\frac {{\left (\cos \left (a+b\,x\right )+1\right )}^3\,\left (\cos \left (a+b\,x\right )-3\right )}{3\,b\,\cos \left (a+b\,x\right )} \]
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]
________________________________________________________________________________________