Optimal. Leaf size=112 \[ \frac {2 \sin (c+d x)}{5 d \sqrt {1-\cos (c+d x)} \cos ^{\frac {5}{2}}(c+d x)}-\frac {8 \sin (c+d x)}{15 d \sqrt {1-\cos (c+d x)} \cos ^{\frac {3}{2}}(c+d x)}+\frac {16 \sin (c+d x)}{15 d \sqrt {1-\cos (c+d x)} \sqrt {\cos (c+d x)}} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.09, antiderivative size = 112, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 2, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.080, Rules used = {2851, 2850}
\begin {gather*} -\frac {8 \sin (c+d x)}{15 d \sqrt {1-\cos (c+d x)} \cos ^{\frac {3}{2}}(c+d x)}+\frac {2 \sin (c+d x)}{5 d \sqrt {1-\cos (c+d x)} \cos ^{\frac {5}{2}}(c+d x)}+\frac {16 \sin (c+d x)}{15 d \sqrt {1-\cos (c+d x)} \sqrt {\cos (c+d x)}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 2850
Rule 2851
Rubi steps
\begin {align*} \int \frac {\sqrt {1-\cos (c+d x)}}{\cos ^{\frac {7}{2}}(c+d x)} \, dx &=\frac {2 \sin (c+d x)}{5 d \sqrt {1-\cos (c+d x)} \cos ^{\frac {5}{2}}(c+d x)}-\frac {4}{5} \int \frac {\sqrt {1-\cos (c+d x)}}{\cos ^{\frac {5}{2}}(c+d x)} \, dx\\ &=\frac {2 \sin (c+d x)}{5 d \sqrt {1-\cos (c+d x)} \cos ^{\frac {5}{2}}(c+d x)}-\frac {8 \sin (c+d x)}{15 d \sqrt {1-\cos (c+d x)} \cos ^{\frac {3}{2}}(c+d x)}+\frac {8}{15} \int \frac {\sqrt {1-\cos (c+d x)}}{\cos ^{\frac {3}{2}}(c+d x)} \, dx\\ &=\frac {2 \sin (c+d x)}{5 d \sqrt {1-\cos (c+d x)} \cos ^{\frac {5}{2}}(c+d x)}-\frac {8 \sin (c+d x)}{15 d \sqrt {1-\cos (c+d x)} \cos ^{\frac {3}{2}}(c+d x)}+\frac {16 \sin (c+d x)}{15 d \sqrt {1-\cos (c+d x)} \sqrt {\cos (c+d x)}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 0.12, size = 61, normalized size = 0.54 \begin {gather*} \frac {2 \sqrt {1-\cos (c+d x)} \left (3-4 \cos (c+d x)+8 \cos ^2(c+d x)\right ) \cot \left (\frac {1}{2} (c+d x)\right )}{15 d \cos ^{\frac {5}{2}}(c+d x)} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A]
time = 0.14, size = 65, normalized size = 0.58
method | result | size |
default | \(-\frac {\left (8 \left (\cos ^{2}\left (d x +c \right )\right )-4 \cos \left (d x +c \right )+3\right ) \sqrt {2-2 \cos \left (d x +c \right )}\, \sin \left (d x +c \right ) \sqrt {2}}{15 d \left (-1+\cos \left (d x +c \right )\right ) \cos \left (d x +c \right )^{\frac {5}{2}}}\) | \(65\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 209 vs.
\(2 (94) = 188\).
time = 0.53, size = 209, normalized size = 1.87 \begin {gather*} \frac {2 \, {\left (7 \, \sqrt {2} - \frac {17 \, \sqrt {2} \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {25 \, \sqrt {2} \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} - \frac {15 \, \sqrt {2} \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}}\right )} {\left (\frac {\sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + 1\right )}^{3}}{15 \, d {\left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + 1\right )}^{\frac {7}{2}} {\left (-\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + 1\right )}^{\frac {7}{2}} {\left (\frac {3 \, \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {3 \, \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} + \frac {\sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}} + 1\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A]
time = 0.41, size = 63, normalized size = 0.56 \begin {gather*} \frac {2 \, {\left (8 \, \cos \left (d x + c\right )^{3} + 4 \, \cos \left (d x + c\right )^{2} - \cos \left (d x + c\right ) + 3\right )} \sqrt {-\cos \left (d x + c\right ) + 1}}{15 \, d \cos \left (d x + c\right )^{\frac {5}{2}} \sin \left (d x + c\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: SystemError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A]
time = 0.61, size = 117, normalized size = 1.04 \begin {gather*} -\frac {2 \, \sqrt {2} {\left ({\left ({\left ({\left ({\left (7 \, \tan \left (\frac {1}{4} \, d x + \frac {1}{4} \, c\right )^{2} - 75\right )} \tan \left (\frac {1}{4} \, d x + \frac {1}{4} \, c\right )^{2} + 430\right )} \tan \left (\frac {1}{4} \, d x + \frac {1}{4} \, c\right )^{2} - 430\right )} \tan \left (\frac {1}{4} \, d x + \frac {1}{4} \, c\right )^{2} + 75\right )} \tan \left (\frac {1}{4} \, d x + \frac {1}{4} \, c\right )^{2} - 7\right )} \mathrm {sgn}\left (\sin \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}{15 \, {\left (\tan \left (\frac {1}{4} \, d x + \frac {1}{4} \, c\right )^{4} - 6 \, \tan \left (\frac {1}{4} \, d x + \frac {1}{4} \, c\right )^{2} + 1\right )}^{\frac {5}{2}} d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Mupad [B]
time = 2.00, size = 156, normalized size = 1.39 \begin {gather*} \frac {8\,\sqrt {2\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}\,\left (7\,\sin \left (c+d\,x\right )-4\,\sin \left (2\,c+2\,d\,x\right )+9\,\sin \left (3\,c+3\,d\,x\right )-2\,\sin \left (4\,c+4\,d\,x\right )+2\,\sin \left (5\,c+5\,d\,x\right )\right )}{15\,d\,\sqrt {1-2\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}\,\left (-16\,{\sin \left (c+d\,x\right )}^2-4\,{\sin \left (2\,c+2\,d\,x\right )}^2+20\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+10\,{\sin \left (\frac {3\,c}{2}+\frac {3\,d\,x}{2}\right )}^2+2\,{\sin \left (\frac {5\,c}{2}+\frac {5\,d\,x}{2}\right )}^2\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________