Optimal. Leaf size=38 \[ \frac {4 a^2 \sin (c+d x) \sqrt [4]{\sec (c+d x)}}{d \sqrt {a \cos (c+d x)+a}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.12, antiderivative size = 38, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.120, Rules used = {4222, 2762, 8} \[ \frac {4 a^2 \sin (c+d x) \sqrt [4]{\sec (c+d x)}}{d \sqrt {a \cos (c+d x)+a}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 8
Rule 2762
Rule 4222
Rubi steps
\begin {align*} \int (a+a \cos (c+d x))^{3/2} \sec ^{\frac {5}{4}}(c+d x) \, dx &=\left (\sqrt [4]{\cos (c+d x)} \sqrt [4]{\sec (c+d x)}\right ) \int \frac {(a+a \cos (c+d x))^{3/2}}{\cos ^{\frac {5}{4}}(c+d x)} \, dx\\ &=\frac {4 a^2 \sqrt [4]{\sec (c+d x)} \sin (c+d x)}{d \sqrt {a+a \cos (c+d x)}}-\left (4 a \sqrt [4]{\cos (c+d x)} \sqrt [4]{\sec (c+d x)}\right ) \int 0 \, dx\\ &=\frac {4 a^2 \sqrt [4]{\sec (c+d x)} \sin (c+d x)}{d \sqrt {a+a \cos (c+d x)}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.12, size = 51, normalized size = 1.34 \[ \frac {2 \tan \left (\frac {1}{2} (c+d x)\right ) \sec ^2\left (\frac {1}{2} (c+d x)\right ) \sqrt [4]{\sec (c+d x)} (a (\cos (c+d x)+1))^{3/2}}{d} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 1.09, size = 41, normalized size = 1.08 \[ \frac {4 \, \sqrt {a \cos \left (d x + c\right ) + a} a \sin \left (d x + c\right )}{{\left (d \cos \left (d x + c\right ) + d\right )} \cos \left (d x + c\right )^{\frac {1}{4}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [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]
________________________________________________________________________________________
maple [F] time = 0.23, size = 0, normalized size = 0.00 \[ \int \left (a +a \cos \left (d x +c \right )\right )^{\frac {3}{2}} \left (\sec ^{\frac {5}{4}}\left (d x +c \right )\right )\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [B] time = 1.01, size = 121, normalized size = 3.18 \[ \frac {4 \, {\left (\frac {\sqrt {2} a^{\frac {3}{2}} \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac {\sqrt {2} a^{\frac {3}{2}} \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}}\right )}}{d {\left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + 1\right )}^{\frac {5}{4}} {\left (-\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + 1\right )}^{\frac {5}{4}} {\left (\frac {\sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + 1\right )}^{\frac {1}{4}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 0.74, size = 44, normalized size = 1.16 \[ \frac {4\,a\,\sin \left (c+d\,x\right )\,\sqrt {a\,\left (\cos \left (c+d\,x\right )+1\right )}\,{\left (\frac {1}{\cos \left (c+d\,x\right )}\right )}^{1/4}}{d\,\left (\cos \left (c+d\,x\right )+1\right )} \]
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]
________________________________________________________________________________________