Optimal. Leaf size=65 \[ -\frac {8}{11} \cot ^{11}(x)-\frac {16 \cot ^9(x)}{9}-\frac {9 \cot ^7(x)}{7}-\frac {\cot ^5(x)}{5}+\frac {8 \csc ^{11}(x)}{11}-\frac {20 \csc ^9(x)}{9}+\frac {16 \csc ^7(x)}{7}-\frac {4 \csc ^5(x)}{5} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.21, antiderivative size = 65, normalized size of antiderivative = 1.00, number of steps used = 18, number of rules used = 6, integrand size = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.857, Rules used = {4397, 2711, 2607, 14, 2606, 270} \[ -\frac {8}{11} \cot ^{11}(x)-\frac {16 \cot ^9(x)}{9}-\frac {9 \cot ^7(x)}{7}-\frac {\cot ^5(x)}{5}+\frac {8 \csc ^{11}(x)}{11}-\frac {20 \csc ^9(x)}{9}+\frac {16 \csc ^7(x)}{7}-\frac {4 \csc ^5(x)}{5} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 14
Rule 270
Rule 2606
Rule 2607
Rule 2711
Rule 4397
Rubi steps
\begin {align*} \int \frac {1}{(\sin (x)+\tan (x))^4} \, dx &=\int \frac {\cot ^4(x)}{(1+\cos (x))^4} \, dx\\ &=\int \left (\cot ^8(x) \csc ^4(x)-4 \cot ^7(x) \csc ^5(x)+6 \cot ^6(x) \csc ^6(x)-4 \cot ^5(x) \csc ^7(x)+\cot ^4(x) \csc ^8(x)\right ) \, dx\\ &=-\left (4 \int \cot ^7(x) \csc ^5(x) \, dx\right )-4 \int \cot ^5(x) \csc ^7(x) \, dx+6 \int \cot ^6(x) \csc ^6(x) \, dx+\int \cot ^8(x) \csc ^4(x) \, dx+\int \cot ^4(x) \csc ^8(x) \, dx\\ &=4 \operatorname {Subst}\left (\int x^6 \left (-1+x^2\right )^2 \, dx,x,\csc (x)\right )+4 \operatorname {Subst}\left (\int x^4 \left (-1+x^2\right )^3 \, dx,x,\csc (x)\right )+6 \operatorname {Subst}\left (\int x^6 \left (1+x^2\right )^2 \, dx,x,-\cot (x)\right )+\operatorname {Subst}\left (\int x^8 \left (1+x^2\right ) \, dx,x,-\cot (x)\right )+\operatorname {Subst}\left (\int x^4 \left (1+x^2\right )^3 \, dx,x,-\cot (x)\right )\\ &=4 \operatorname {Subst}\left (\int \left (-x^4+3 x^6-3 x^8+x^{10}\right ) \, dx,x,\csc (x)\right )+4 \operatorname {Subst}\left (\int \left (x^6-2 x^8+x^{10}\right ) \, dx,x,\csc (x)\right )+6 \operatorname {Subst}\left (\int \left (x^6+2 x^8+x^{10}\right ) \, dx,x,-\cot (x)\right )+\operatorname {Subst}\left (\int \left (x^8+x^{10}\right ) \, dx,x,-\cot (x)\right )+\operatorname {Subst}\left (\int \left (x^4+3 x^6+3 x^8+x^{10}\right ) \, dx,x,-\cot (x)\right )\\ &=-\frac {1}{5} \cot ^5(x)-\frac {9 \cot ^7(x)}{7}-\frac {16 \cot ^9(x)}{9}-\frac {8 \cot ^{11}(x)}{11}-\frac {4 \csc ^5(x)}{5}+\frac {16 \csc ^7(x)}{7}-\frac {20 \csc ^9(x)}{9}+\frac {8 \csc ^{11}(x)}{11}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.02, size = 129, normalized size = 1.98 \[ -\frac {2749 \tan \left (\frac {x}{2}\right )}{110880}+\frac {1}{96} \cot \left (\frac {x}{2}\right )-\frac {1}{384} \cot \left (\frac {x}{2}\right ) \csc ^2\left (\frac {x}{2}\right )+\frac {\tan \left (\frac {x}{2}\right ) \sec ^{10}\left (\frac {x}{2}\right )}{1408}-\frac {7 \tan \left (\frac {x}{2}\right ) \sec ^8\left (\frac {x}{2}\right )}{1584}+\frac {641 \tan \left (\frac {x}{2}\right ) \sec ^6\left (\frac {x}{2}\right )}{88704}+\frac {179 \tan \left (\frac {x}{2}\right ) \sec ^4\left (\frac {x}{2}\right )}{73920}-\frac {2033 \tan \left (\frac {x}{2}\right ) \sec ^2\left (\frac {x}{2}\right )}{443520} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 1.06, size = 78, normalized size = 1.20 \[ \frac {122 \, \cos \relax (x)^{7} + 488 \, \cos \relax (x)^{6} + 549 \, \cos \relax (x)^{5} - 244 \, \cos \relax (x)^{4} - 64 \, \cos \relax (x)^{3} + 144 \, \cos \relax (x)^{2} + 128 \, \cos \relax (x) + 32}{3465 \, {\left (\cos \relax (x)^{6} + 4 \, \cos \relax (x)^{5} + 5 \, \cos \relax (x)^{4} - 5 \, \cos \relax (x)^{2} - 4 \, \cos \relax (x) - 1\right )} \sin \relax (x)} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.15, size = 65, normalized size = 1.00 \[ \frac {1}{1408} \, \tan \left (\frac {1}{2} \, x\right )^{11} - \frac {1}{1152} \, \tan \left (\frac {1}{2} \, x\right )^{9} - \frac {3}{896} \, \tan \left (\frac {1}{2} \, x\right )^{7} + \frac {3}{640} \, \tan \left (\frac {1}{2} \, x\right )^{5} + \frac {1}{128} \, \tan \left (\frac {1}{2} \, x\right )^{3} + \frac {3 \, \tan \left (\frac {1}{2} \, x\right )^{2} - 1}{384 \, \tan \left (\frac {1}{2} \, x\right )^{3}} - \frac {3}{128} \, \tan \left (\frac {1}{2} \, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.13, size = 64, normalized size = 0.98 \[ \frac {\left (\tan ^{11}\left (\frac {x}{2}\right )\right )}{1408}-\frac {\left (\tan ^{9}\left (\frac {x}{2}\right )\right )}{1152}-\frac {3 \left (\tan ^{7}\left (\frac {x}{2}\right )\right )}{896}+\frac {3 \left (\tan ^{5}\left (\frac {x}{2}\right )\right )}{640}+\frac {\left (\tan ^{3}\left (\frac {x}{2}\right )\right )}{128}-\frac {3 \tan \left (\frac {x}{2}\right )}{128}+\frac {1}{128 \tan \left (\frac {x}{2}\right )}-\frac {1}{384 \tan \left (\frac {x}{2}\right )^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 0.32, size = 97, normalized size = 1.49 \[ \frac {{\left (\frac {3 \, \sin \relax (x)^{2}}{{\left (\cos \relax (x) + 1\right )}^{2}} - 1\right )} {\left (\cos \relax (x) + 1\right )}^{3}}{384 \, \sin \relax (x)^{3}} - \frac {3 \, \sin \relax (x)}{128 \, {\left (\cos \relax (x) + 1\right )}} + \frac {\sin \relax (x)^{3}}{128 \, {\left (\cos \relax (x) + 1\right )}^{3}} + \frac {3 \, \sin \relax (x)^{5}}{640 \, {\left (\cos \relax (x) + 1\right )}^{5}} - \frac {3 \, \sin \relax (x)^{7}}{896 \, {\left (\cos \relax (x) + 1\right )}^{7}} - \frac {\sin \relax (x)^{9}}{1152 \, {\left (\cos \relax (x) + 1\right )}^{9}} + \frac {\sin \relax (x)^{11}}{1408 \, {\left (\cos \relax (x) + 1\right )}^{11}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 2.45, size = 87, normalized size = 1.34 \[ -\frac {15616\,{\cos \left (\frac {x}{2}\right )}^{14}-23424\,{\cos \left (\frac {x}{2}\right )}^{12}+5856\,{\cos \left (\frac {x}{2}\right )}^{10}+976\,{\cos \left (\frac {x}{2}\right )}^8+7296\,{\cos \left (\frac {x}{2}\right )}^6-7440\,{\cos \left (\frac {x}{2}\right )}^4+2590\,{\cos \left (\frac {x}{2}\right )}^2-315}{443520\,\left ({\cos \left (\frac {x}{2}\right )}^{11}\,\sin \left (\frac {x}{2}\right )-{\cos \left (\frac {x}{2}\right )}^{13}\,\sin \left (\frac {x}{2}\right )\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\left (\sin {\relax (x )} + \tan {\relax (x )}\right )^{4}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________