Optimal. Leaf size=55 \[ \frac {3}{4 \sqrt {1+2 \sin (x)}}-\frac {4}{\sqrt [4]{1+2 \sin (x)}}-\frac {1}{2} \sqrt {1+2 \sin (x)}+\frac {1}{12} (1+2 \sin (x))^{3/2} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.10, antiderivative size = 55, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 2, integrand size = 32, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.062, Rules used = {4441, 14}
\begin {gather*} \frac {1}{12} (2 \sin (x)+1)^{3/2}-\frac {1}{2} \sqrt {2 \sin (x)+1}-\frac {4}{\sqrt [4]{2 \sin (x)+1}}+\frac {3}{4 \sqrt {2 \sin (x)+1}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 14
Rule 4441
Rubi steps
\begin {align*} \int \frac {\cos (x) \left (-\cos ^2(x)+2 \sqrt [4]{1+2 \sin (x)}\right )}{(1+2 \sin (x))^{3/2}} \, dx &=\text {Subst}\left (\int \frac {-1+x^2+2 \sqrt [4]{1+2 x}}{(1+2 x)^{3/2}} \, dx,x,\sin (x)\right )\\ &=\frac {1}{2} \text {Subst}\left (\int \frac {-3+8 x-2 x^4+x^8}{x^3} \, dx,x,\sqrt [4]{1+2 \sin (x)}\right )\\ &=\frac {1}{2} \text {Subst}\left (\int \left (-\frac {3}{x^3}+\frac {8}{x^2}-2 x+x^5\right ) \, dx,x,\sqrt [4]{1+2 \sin (x)}\right )\\ &=\frac {3}{4 \sqrt {1+2 \sin (x)}}-\frac {4}{\sqrt [4]{1+2 \sin (x)}}-\frac {1}{2} \sqrt {1+2 \sin (x)}+\frac {1}{12} (1+2 \sin (x))^{3/2}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 0.06, size = 36, normalized size = 0.65 \begin {gather*} -\frac {-3+\cos (2 x)+4 \sin (x)+24 \sqrt [4]{1+2 \sin (x)}}{6 \sqrt {1+2 \sin (x)}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A]
time = 0.69, size = 31, normalized size = 0.56
method | result | size |
default | \(\frac {\sin ^{2}\left (x \right )-2 \sin \left (x \right )-12 \left (1+2 \sin \left (x \right )\right )^{\frac {1}{4}}+1}{3 \sqrt {1+2 \sin \left (x \right )}}\) | \(31\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A]
time = 2.27, size = 43, normalized size = 0.78 \begin {gather*} \frac {1}{12} \, {\left (2 \, \sin \left (x\right ) + 1\right )}^{\frac {3}{2}} - \frac {16 \, {\left (2 \, \sin \left (x\right ) + 1\right )}^{\frac {1}{4}} - 3}{4 \, \sqrt {2 \, \sin \left (x\right ) + 1}} - \frac {1}{2} \, \sqrt {2 \, \sin \left (x\right ) + 1} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A]
time = 1.07, size = 40, normalized size = 0.73 \begin {gather*} -\frac {{\left (\cos \left (x\right )^{2} + 2 \, \sin \left (x\right ) - 2\right )} \sqrt {2 \, \sin \left (x\right ) + 1} + 12 \, {\left (2 \, \sin \left (x\right ) + 1\right )}^{\frac {3}{4}}}{3 \, {\left (2 \, \sin \left (x\right ) + 1\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 230 vs.
\(2 (48) = 96\).
time = 47.11, size = 230, normalized size = 4.18 \begin {gather*} \frac {4 \left (2 \sin {\left (x \right )} + 1\right )^{\frac {3}{4}} \sin ^{2}{\left (x \right )}}{6 \sqrt [4]{2 \sin {\left (x \right )} + 1} \sin {\left (x \right )} + 3 \sqrt [4]{2 \sin {\left (x \right )} + 1}} - \frac {2 \left (2 \sin {\left (x \right )} + 1\right )^{\frac {3}{4}} \sin {\left (x \right )}}{6 \sqrt [4]{2 \sin {\left (x \right )} + 1} \sin {\left (x \right )} + 3 \sqrt [4]{2 \sin {\left (x \right )} + 1}} + \frac {3 \left (2 \sin {\left (x \right )} + 1\right )^{\frac {3}{4}} \cos ^{2}{\left (x \right )}}{6 \sqrt [4]{2 \sin {\left (x \right )} + 1} \sin {\left (x \right )} + 3 \sqrt [4]{2 \sin {\left (x \right )} + 1}} - \frac {2 \left (2 \sin {\left (x \right )} + 1\right )^{\frac {3}{4}}}{6 \sqrt [4]{2 \sin {\left (x \right )} + 1} \sin {\left (x \right )} + 3 \sqrt [4]{2 \sin {\left (x \right )} + 1}} - \frac {24 \sin {\left (x \right )}}{6 \sqrt [4]{2 \sin {\left (x \right )} + 1} \sin {\left (x \right )} + 3 \sqrt [4]{2 \sin {\left (x \right )} + 1}} - \frac {12}{6 \sqrt [4]{2 \sin {\left (x \right )} + 1} \sin {\left (x \right )} + 3 \sqrt [4]{2 \sin {\left (x \right )} + 1}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A]
time = 1.19, size = 43, normalized size = 0.78 \begin {gather*} \frac {1}{12} \, {\left (2 \, \sin \left (x\right ) + 1\right )}^{\frac {3}{2}} - \frac {16 \, {\left (2 \, \sin \left (x\right ) + 1\right )}^{\frac {1}{4}} - 3}{4 \, \sqrt {2 \, \sin \left (x\right ) + 1}} - \frac {1}{2} \, \sqrt {2 \, \sin \left (x\right ) + 1} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Mupad [F]
time = 0.00, size = -1, normalized size = -0.02 \begin {gather*} -\int -\frac {\cos \left (x\right )\,\left (2\,{\left (2\,\sin \left (x\right )+1\right )}^{1/4}-{\cos \left (x\right )}^2\right )}{{\left (2\,\sin \left (x\right )+1\right )}^{3/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________