Optimal. Leaf size=71 \[ -\frac {3 x}{a^3}-\frac {9 \cos (x)}{5 a^3}+\frac {\cos (x) \sin ^3(x)}{5 (a+a \sin (x))^3}+\frac {3 \cos (x) \sin ^2(x)}{5 a (a+a \sin (x))^2}-\frac {3 \cos (x)}{a^3+a^3 \sin (x)} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.16, antiderivative size = 71, normalized size of antiderivative = 1.00, number of steps
used = 7, number of rules used = 7, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.538, Rules used = {2844, 3056,
3047, 3102, 12, 2814, 2727} \begin {gather*} -\frac {3 x}{a^3}-\frac {9 \cos (x)}{5 a^3}-\frac {3 \cos (x)}{a^3 \sin (x)+a^3}+\frac {\sin ^3(x) \cos (x)}{5 (a \sin (x)+a)^3}+\frac {3 \sin ^2(x) \cos (x)}{5 a (a \sin (x)+a)^2} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 12
Rule 2727
Rule 2814
Rule 2844
Rule 3047
Rule 3056
Rule 3102
Rubi steps
\begin {align*} \int \frac {\sin ^4(x)}{(a+a \sin (x))^3} \, dx &=\frac {\cos (x) \sin ^3(x)}{5 (a+a \sin (x))^3}-\frac {\int \frac {\sin ^2(x) (3 a-6 a \sin (x))}{(a+a \sin (x))^2} \, dx}{5 a^2}\\ &=\frac {\cos (x) \sin ^3(x)}{5 (a+a \sin (x))^3}+\frac {3 \cos (x) \sin ^2(x)}{5 a (a+a \sin (x))^2}-\frac {\int \frac {\sin (x) \left (18 a^2-27 a^2 \sin (x)\right )}{a+a \sin (x)} \, dx}{15 a^4}\\ &=\frac {\cos (x) \sin ^3(x)}{5 (a+a \sin (x))^3}+\frac {3 \cos (x) \sin ^2(x)}{5 a (a+a \sin (x))^2}-\frac {\int \frac {18 a^2 \sin (x)-27 a^2 \sin ^2(x)}{a+a \sin (x)} \, dx}{15 a^4}\\ &=-\frac {9 \cos (x)}{5 a^3}+\frac {\cos (x) \sin ^3(x)}{5 (a+a \sin (x))^3}+\frac {3 \cos (x) \sin ^2(x)}{5 a (a+a \sin (x))^2}-\frac {\int \frac {45 a^3 \sin (x)}{a+a \sin (x)} \, dx}{15 a^5}\\ &=-\frac {9 \cos (x)}{5 a^3}+\frac {\cos (x) \sin ^3(x)}{5 (a+a \sin (x))^3}+\frac {3 \cos (x) \sin ^2(x)}{5 a (a+a \sin (x))^2}-\frac {3 \int \frac {\sin (x)}{a+a \sin (x)} \, dx}{a^2}\\ &=-\frac {3 x}{a^3}-\frac {9 \cos (x)}{5 a^3}+\frac {\cos (x) \sin ^3(x)}{5 (a+a \sin (x))^3}+\frac {3 \cos (x) \sin ^2(x)}{5 a (a+a \sin (x))^2}+\frac {3 \int \frac {1}{a+a \sin (x)} \, dx}{a^2}\\ &=-\frac {3 x}{a^3}-\frac {9 \cos (x)}{5 a^3}+\frac {\cos (x) \sin ^3(x)}{5 (a+a \sin (x))^3}+\frac {3 \cos (x) \sin ^2(x)}{5 a (a+a \sin (x))^2}-\frac {3 \cos (x)}{a^3+a^3 \sin (x)}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 0.06, size = 140, normalized size = 1.97 \begin {gather*} \frac {\left (\cos \left (\frac {x}{2}\right )+\sin \left (\frac {x}{2}\right )\right ) \left (-\cos \left (\frac {x}{2}\right )+\sin \left (\frac {x}{2}\right )-12 \sin \left (\frac {x}{2}\right ) \left (\cos \left (\frac {x}{2}\right )+\sin \left (\frac {x}{2}\right )\right )^2+6 \left (\cos \left (\frac {x}{2}\right )+\sin \left (\frac {x}{2}\right )\right )^3+48 \sin \left (\frac {x}{2}\right ) \left (\cos \left (\frac {x}{2}\right )+\sin \left (\frac {x}{2}\right )\right )^4-15 x \left (\cos \left (\frac {x}{2}\right )+\sin \left (\frac {x}{2}\right )\right )^5-5 \cos (x) \left (\cos \left (\frac {x}{2}\right )+\sin \left (\frac {x}{2}\right )\right )^5\right )}{5 (a+a \sin (x))^3} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A]
time = 0.20, size = 66, normalized size = 0.93
method | result | size |
default | \(\frac {-\frac {8}{5 \left (\tan \left (\frac {x}{2}\right )+1\right )^{5}}+\frac {4}{\left (\tan \left (\frac {x}{2}\right )+1\right )^{4}}-\frac {4}{\left (\tan \left (\frac {x}{2}\right )+1\right )^{2}}-\frac {6}{\tan \left (\frac {x}{2}\right )+1}-\frac {2}{\tan ^{2}\left (\frac {x}{2}\right )+1}-6 \arctan \left (\tan \left (\frac {x}{2}\right )\right )}{a^{3}}\) | \(66\) |
risch | \(-\frac {3 x}{a^{3}}-\frac {{\mathrm e}^{i x}}{2 a^{3}}-\frac {{\mathrm e}^{-i x}}{2 a^{3}}-\frac {4 \left (-70 \,{\mathrm e}^{2 i x}+50 i {\mathrm e}^{3 i x}+15 \,{\mathrm e}^{4 i x}-45 i {\mathrm e}^{i x}+12\right )}{5 \left ({\mathrm e}^{i x}+i\right )^{5} a^{3}}\) | \(75\) |
norman | \(\frac {-\frac {2172 \left (\tan ^{6}\left (\frac {x}{2}\right )\right )}{5 a}-\frac {90 x \left (\tan ^{3}\left (\frac {x}{2}\right )\right )}{a}-\frac {170 \left (\tan ^{9}\left (\frac {x}{2}\right )\right )}{a}-\frac {252 x \left (\tan ^{7}\left (\frac {x}{2}\right )\right )}{a}-\frac {252 x \left (\tan ^{6}\left (\frac {x}{2}\right )\right )}{a}-\frac {213 x \left (\tan ^{8}\left (\frac {x}{2}\right )\right )}{a}-\frac {213 x \left (\tan ^{5}\left (\frac {x}{2}\right )\right )}{a}-\frac {153 x \left (\tan ^{4}\left (\frac {x}{2}\right )\right )}{a}-\frac {522 \left (\tan ^{2}\left (\frac {x}{2}\right )\right )}{5 a}-\frac {206 \left (\tan ^{3}\left (\frac {x}{2}\right )\right )}{a}-\frac {1598 \left (\tan ^{4}\left (\frac {x}{2}\right )\right )}{5 a}-\frac {1428 \left (\tan ^{8}\left (\frac {x}{2}\right )\right )}{5 a}-\frac {396 \left (\tan ^{5}\left (\frac {x}{2}\right )\right )}{a}-\frac {42 x \left (\tan ^{2}\left (\frac {x}{2}\right )\right )}{a}-\frac {82 \left (\tan ^{10}\left (\frac {x}{2}\right )\right )}{a}-\frac {42 \tan \left (\frac {x}{2}\right )}{a}-\frac {372 \left (\tan ^{7}\left (\frac {x}{2}\right )\right )}{a}-\frac {42 x \left (\tan ^{11}\left (\frac {x}{2}\right )\right )}{a}-\frac {6 \left (\tan ^{12}\left (\frac {x}{2}\right )\right )}{a}-\frac {15 x \tan \left (\frac {x}{2}\right )}{a}-\frac {90 x \left (\tan ^{10}\left (\frac {x}{2}\right )\right )}{a}-\frac {3 x}{a}-\frac {15 x \left (\tan ^{12}\left (\frac {x}{2}\right )\right )}{a}-\frac {153 x \left (\tan ^{9}\left (\frac {x}{2}\right )\right )}{a}-\frac {3 x \left (\tan ^{13}\left (\frac {x}{2}\right )\right )}{a}-\frac {30 \left (\tan ^{11}\left (\frac {x}{2}\right )\right )}{a}-\frac {48}{5 a}}{\left (\tan ^{2}\left (\frac {x}{2}\right )+1\right )^{4} a^{2} \left (\tan \left (\frac {x}{2}\right )+1\right )^{5}}\) | \(319\) |
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. 198 vs.
\(2 (65) = 130\).
time = 0.53, size = 198, normalized size = 2.79 \begin {gather*} -\frac {2 \, {\left (\frac {105 \, \sin \left (x\right )}{\cos \left (x\right ) + 1} + \frac {189 \, \sin \left (x\right )^{2}}{{\left (\cos \left (x\right ) + 1\right )}^{2}} + \frac {200 \, \sin \left (x\right )^{3}}{{\left (\cos \left (x\right ) + 1\right )}^{3}} + \frac {160 \, \sin \left (x\right )^{4}}{{\left (\cos \left (x\right ) + 1\right )}^{4}} + \frac {75 \, \sin \left (x\right )^{5}}{{\left (\cos \left (x\right ) + 1\right )}^{5}} + \frac {15 \, \sin \left (x\right )^{6}}{{\left (\cos \left (x\right ) + 1\right )}^{6}} + 24\right )}}{5 \, {\left (a^{3} + \frac {5 \, a^{3} \sin \left (x\right )}{\cos \left (x\right ) + 1} + \frac {11 \, a^{3} \sin \left (x\right )^{2}}{{\left (\cos \left (x\right ) + 1\right )}^{2}} + \frac {15 \, a^{3} \sin \left (x\right )^{3}}{{\left (\cos \left (x\right ) + 1\right )}^{3}} + \frac {15 \, a^{3} \sin \left (x\right )^{4}}{{\left (\cos \left (x\right ) + 1\right )}^{4}} + \frac {11 \, a^{3} \sin \left (x\right )^{5}}{{\left (\cos \left (x\right ) + 1\right )}^{5}} + \frac {5 \, a^{3} \sin \left (x\right )^{6}}{{\left (\cos \left (x\right ) + 1\right )}^{6}} + \frac {a^{3} \sin \left (x\right )^{7}}{{\left (\cos \left (x\right ) + 1\right )}^{7}}\right )}} - \frac {6 \, \arctan \left (\frac {\sin \left (x\right )}{\cos \left (x\right ) + 1}\right )}{a^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 132 vs.
\(2 (65) = 130\).
time = 0.34, size = 132, normalized size = 1.86 \begin {gather*} -\frac {3 \, {\left (5 \, x + 13\right )} \cos \left (x\right )^{3} + 5 \, \cos \left (x\right )^{4} + {\left (45 \, x - 28\right )} \cos \left (x\right )^{2} - 3 \, {\left (10 \, x + 21\right )} \cos \left (x\right ) + {\left ({\left (15 \, x - 34\right )} \cos \left (x\right )^{2} + 5 \, \cos \left (x\right )^{3} - 2 \, {\left (15 \, x + 31\right )} \cos \left (x\right ) - 60 \, x + 1\right )} \sin \left (x\right ) - 60 \, x - 1}{5 \, {\left (a^{3} \cos \left (x\right )^{3} + 3 \, a^{3} \cos \left (x\right )^{2} - 2 \, a^{3} \cos \left (x\right ) - 4 \, a^{3} + {\left (a^{3} \cos \left (x\right )^{2} - 2 \, a^{3} \cos \left (x\right ) - 4 \, a^{3}\right )} \sin \left (x\right )\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. 1425 vs.
\(2 (71) = 142\).
time = 8.84, size = 1425, normalized size = 20.07 \begin {gather*} \text {Too large to display} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A]
time = 0.49, size = 67, normalized size = 0.94 \begin {gather*} -\frac {3 \, x}{a^{3}} - \frac {2}{{\left (\tan \left (\frac {1}{2} \, x\right )^{2} + 1\right )} a^{3}} - \frac {2 \, {\left (15 \, \tan \left (\frac {1}{2} \, x\right )^{4} + 70 \, \tan \left (\frac {1}{2} \, x\right )^{3} + 120 \, \tan \left (\frac {1}{2} \, x\right )^{2} + 80 \, \tan \left (\frac {1}{2} \, x\right ) + 19\right )}}{5 \, a^{3} {\left (\tan \left (\frac {1}{2} \, x\right ) + 1\right )}^{5}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Mupad [B]
time = 6.89, size = 78, normalized size = 1.10 \begin {gather*} -\frac {3\,x}{a^3}-\frac {6\,{\mathrm {tan}\left (\frac {x}{2}\right )}^6+30\,{\mathrm {tan}\left (\frac {x}{2}\right )}^5+64\,{\mathrm {tan}\left (\frac {x}{2}\right )}^4+80\,{\mathrm {tan}\left (\frac {x}{2}\right )}^3+\frac {378\,{\mathrm {tan}\left (\frac {x}{2}\right )}^2}{5}+42\,\mathrm {tan}\left (\frac {x}{2}\right )+\frac {48}{5}}{a^3\,\left ({\mathrm {tan}\left (\frac {x}{2}\right )}^2+1\right )\,{\left (\mathrm {tan}\left (\frac {x}{2}\right )+1\right )}^5} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________