Optimal. Leaf size=25 \[ \frac {b x \sqrt {b \sec (c+d x)}}{\sqrt {\sec (c+d x)}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.00, antiderivative size = 25, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.087, Rules used = {17, 8} \[ \frac {b x \sqrt {b \sec (c+d x)}}{\sqrt {\sec (c+d x)}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 8
Rule 17
Rubi steps
\begin {align*} \int \frac {(b \sec (c+d x))^{3/2}}{\sec ^{\frac {3}{2}}(c+d x)} \, dx &=\frac {\left (b \sqrt {b \sec (c+d x)}\right ) \int 1 \, dx}{\sqrt {\sec (c+d x)}}\\ &=\frac {b x \sqrt {b \sec (c+d x)}}{\sqrt {\sec (c+d x)}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.03, size = 24, normalized size = 0.96 \[ \frac {x (b \sec (c+d x))^{3/2}}{\sec ^{\frac {3}{2}}(c+d x)} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.79, size = 99, normalized size = 3.96 \[ \left [\frac {\sqrt {-b} b \log \left (-2 \, \sqrt {-b} \sqrt {\frac {b}{\cos \left (d x + c\right )}} \cos \left (d x + c\right )^{\frac {3}{2}} \sin \left (d x + c\right ) + 2 \, b \cos \left (d x + c\right )^{2} - b\right )}{2 \, d}, \frac {b^{\frac {3}{2}} \arctan \left (\frac {\sqrt {\frac {b}{\cos \left (d x + c\right )}} \sin \left (d x + c\right )}{\sqrt {b} \sqrt {\cos \left (d x + c\right )}}\right )}{d}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\left (b \sec \left (d x + c\right )\right )^{\frac {3}{2}}}{\sec \left (d x + c\right )^{\frac {3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.98, size = 32, normalized size = 1.28 \[ \frac {\left (\frac {b}{\cos \left (d x +c \right )}\right )^{\frac {3}{2}} \left (d x +c \right )}{d \left (\frac {1}{\cos \left (d x +c \right )}\right )^{\frac {3}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 0.77, size = 26, normalized size = 1.04 \[ \frac {2 \, b^{\frac {3}{2}} \arctan \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right )}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 0.20, size = 25, normalized size = 1.00 \[ \frac {b\,x\,\sqrt {\frac {b}{\cos \left (c+d\,x\right )}}}{\sqrt {\frac {1}{\cos \left (c+d\,x\right )}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [A] time = 18.08, size = 5, normalized size = 0.20 \[ b^{\frac {3}{2}} x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________