3.22 \(\int (a+b \cos (e+f x))^m (g \tan (e+f x))^p \, dx\)
Optimal. Leaf size=49 \[ (g \tan (e+f x))^p (g \cot (e+f x))^p \text {Int}\left ((g \cot (e+f x))^{-p} (a+b \cos (e+f x))^m,x\right ) \]
[Out]
(g*cot(f*x+e))^p*(g*tan(f*x+e))^p*Unintegrable((a+b*cos(f*x+e))^m/((g*cot(f*x+e))^p),x)
________________________________________________________________________________________
Rubi [A] time = 0.11, antiderivative size = 0, normalized size of antiderivative = 0.00,
number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used =
{} \[ \int (a+b \cos (e+f x))^m (g \tan (e+f x))^p \, dx \]
Verification is Not applicable to the result.
[In]
Int[(a + b*Cos[e + f*x])^m*(g*Tan[e + f*x])^p,x]
[Out]
(g*Cot[e + f*x])^p*(g*Tan[e + f*x])^p*Defer[Int][(a + b*Cos[e + f*x])^m/(g*Cot[e + f*x])^p, x]
Rubi steps
\begin {align*} \int (a+b \cos (e+f x))^m (g \tan (e+f x))^p \, dx &=\left ((g \cot (e+f x))^p (g \tan (e+f x))^p\right ) \int (a+b \cos (e+f x))^m (g \cot (e+f x))^{-p} \, dx\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 2.47, size = 0, normalized size = 0.00 \[ \int (a+b \cos (e+f x))^m (g \tan (e+f x))^p \, dx \]
Verification is Not applicable to the result.
[In]
Integrate[(a + b*Cos[e + f*x])^m*(g*Tan[e + f*x])^p,x]
[Out]
Integrate[(a + b*Cos[e + f*x])^m*(g*Tan[e + f*x])^p, x]
________________________________________________________________________________________
fricas [A] time = 3.21, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (b \cos \left (f x + e\right ) + a\right )}^{m} \left (g \tan \left (f x + e\right )\right )^{p}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((a+b*cos(f*x+e))^m*(g*tan(f*x+e))^p,x, algorithm="fricas")
[Out]
integral((b*cos(f*x + e) + a)^m*(g*tan(f*x + e))^p, x)
________________________________________________________________________________________
giac [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((a+b*cos(f*x+e))^m*(g*tan(f*x+e))^p,x, algorithm="giac")
[Out]
Exception raised: TypeError >> An error occurred running a Giac command:INPUT:sage2:=int(sage0,x):;OUTPUT:Simp
lification assuming g near 0Unable to check sign: (pi/x/2)>(-pi/x/2)Unable to check sign: (pi/x/2)>(-pi/x/2)Si
mplification assuming f near 0Simplification assuming x near 0Simplification assuming a near 0Unable to check
sign: (2*pi/x/2)>(-2*pi/x/2)Unable to check sign: (2*pi/x/2)>(-2*pi/x/2)Unable to check sign: (2*pi/x/2)>(-2*p
i/x/2)Unable to check sign: (2*pi/x/2)>(-2*pi/x/2)Unable to check sign: (2*pi/x/2)>(-2*pi/x/2)Evaluation time:
0.55Unable to divide, perhaps due to rounding error%%%{-268435456,[0,10,0,10,16,0,0,4]%%%}+%%%{1946157056,[0,
10,0,10,14,0,0,6]%%%}+%%%{-5570035712,[0,10,0,10,12,0,0,8]%%%}+%%%{7985954816,[0,10,0,10,10,0,0,10]%%%}+%%%{-5
972688896,[0,10,0,10,8,0,0,12]%%%}+%%%{2147483648,[0,10,0,10,6,0,0,14]%%%}+%%%{-268435456,[0,10,0,10,4,0,0,16]
%%%}+%%%{805306368,[0,9,0,10,16,1,0,4]%%%}+%%%{268435456,[0,9,0,10,16,0,1,4]%%%}+%%%{-5905580032,[0,9,0,10,14,
1,0,6]%%%}+%%%{-2550136832,[0,9,0,10,14,0,1,6]%%%}+%%%{17179869184,[0,9,0,10,12,1,0,8]%%%}+%%%{9529458688,[0,9
,0,10,12,0,1,8]%%%}+%%%{-25232932864,[0,9,0,10,10,1,0,10]%%%}+%%%{-18119393280,[0,9,0,10,10,0,1,10]%%%}+%%%{19
595788288,[0,9,0,10,8,1,0,12]%%%}+%%%{18656264192,[0,9,0,10,8,0,1,12]%%%}+%%%{-7516192768,[0,9,0,10,6,1,0,14]%
%%}+%%%{-9932111872,[0,9,0,10,6,0,1,14]%%%}+%%%{1073741824,[0,9,0,10,4,1,0,16]%%%}+%%%{2147483648,[0,9,0,10,4,
0,1,16]%%%}+%%%{-872415232,[0,8,0,10,16,2,0,4]%%%}+%%%{-939524096,[0,8,0,10,16,1,1,4]%%%}+%%%{738197504,[0,8,0
,10,16,0,2,4]%%%}+%%%{6241124352,[0,8,0,10,14,2,0,6]%%%}+%%%{8455716864,[0,8,0,10,14,1,1,6]%%%}+%%%{-523449139
2,[0,8,0,10,14,0,2,6]%%%}+%%%{-18320719872,[0,8,0,10,12,2,0,8]%%%}+%%%{-30467424256,[0,8,0,10,12,1,1,8]%%%}+%%
%{14763950080,[0,8,0,10,12,0,2,8]%%%}+%%%{27850178560,[0,8,0,10,10,2,0,10]%%%}+%%%{56505663488,[0,8,0,10,10,1,
1,10]%%%}+%%%{-20803747840,[0,8,0,10,10,0,2,10]%%%}+%%%{-22951231488,[0,8,0,10,8,2,0,12]%%%}+%%%{-57176752128,
[0,8,0,10,8,1,1,12]%%%}+%%%{15099494400,[0,8,0,10,8,0,2,12]%%%}+%%%{9663676416,[0,8,0,10,6,2,0,14]%%%}+%%%{300
64771072,[0,8,0,10,6,1,1,14]%%%}+%%%{-5100273664,[0,8,0,10,6,0,2,14]%%%}+%%%{-1610612736,[0,8,0,10,4,2,0,16]%%
%}+%%%{-6442450944,[0,8,0,10,4,1,1,16]%%%}+%%%{536870912,[0,8,0,10,4,0,2,16]%%%}+%%%{402653184,[0,7,0,10,16,3,
0,4]%%%}+%%%{1207959552,[0,7,0,10,16,2,1,4]%%%}+%%%{-2013265920,[0,7,0,10,16,1,2,4]%%%}+%%%{-671088640,[0,7,0,
10,16,0,3,4]%%%}+%%%{-2550136832,[0,7,0,10,14,3,0,6]%%%}+%%%{-10066329600,[0,7,0,10,14,2,1,6]%%%}+%%%{14898167
808,[0,7,0,10,14,1,2,6]%%%}+%%%{6845104128,[0,7,0,10,14,0,3,6]%%%}+%%%{7381975040,[0,7,0,10,12,3,0,8]%%%}+%%%{
34225520640,[0,7,0,10,12,2,1,8]%%%}+%%%{-43352326144,[0,7,0,10,12,1,2,8]%%%}+%%%{-26709327872,[0,7,0,10,12,0,3
,8]%%%}+%%%{-11945377792,[0,7,0,10,10,3,0,10]%%%}+%%%{-60800630784,[0,7,0,10,10,2,1,10]%%%}+%%%{62948114432,[0
,7,0,10,10,1,2,10]%%%}+%%%{52210696192,[0,7,0,10,10,0,3,10]%%%}+%%%{11005853696,[0,7,0,10,8,3,0,12]%%%}+%%%{59
592671232,[0,7,0,10,8,2,1,12]%%%}+%%%{-47513075712,[0,7,0,10,8,1,2,12]%%%}+%%%{-54760833024,[0,7,0,10,8,0,3,12
]%%%}+%%%{-5368709120,[0,7,0,10,6,3,0,14]%%%}+%%%{-30601641984,[0,7,0,10,6,2,1,14]%%%}+%%%{17179869184,[0,7,0,
10,6,1,2,14]%%%}+%%%{29527900160,[0,7,0,10,6,0,3,14]%%%}+%%%{1073741824,[0,7,0,10,4,3,0,16]%%%}+%%%{6442450944
,[0,7,0,10,4,2,1,16]%%%}+%%%{-2147483648,[0,7,0,10,4,1,2,16]%%%}+%%%{-6442450944,[0,7,0,10,4,0,3,16]%%%}+%%%{-
67108864,[0,6,0,10,16,4,0,4]%%%}+%%%{-671088640,[0,6,0,10,16,3,1,4]%%%}+%%%{1879048192,[0,6,0,10,16,2,2,4]%%%}
+%%%{2281701376,[0,6,0,10,16,1,3,4]%%%}+%%%{-738197504,[0,6,0,10,16,0,4,4]%%%}+%%%{268435456,[0,6,0,10,14,4,0,
6]%%%}+%%%{4966055936,[0,6,0,10,14,3,1,6]%%%}+%%%{-14092861440,[0,6,0,10,14,2,2,6]%%%}+%%%{-22145925120,[0,6,0
,10,14,1,3,6]%%%}+%%%{4831838208,[0,6,0,10,14,0,4,6]%%%}+%%%{-671088640,[0,6,0,10,12,4,0,8]%%%}+%%%{-151666032
64,[0,6,0,10,12,3,1,8]%%%}+%%%{42278584320,[0,6,0,10,12,2,2,8]%%%}+%%%{83886080000,[0,6,0,10,12,1,3,8]%%%}+%%%
{-13019119616,[0,6,0,10,12,0,4,8]%%%}+%%%{1342177280,[0,6,0,10,10,4,0,10]%%%}+%%%{24561844224,[0,6,0,10,10,3,1
,10]%%%}+%%%{-64290291712,[0,6,0,10,10,2,2,10]%%%}+%%%{-160927055872,[0,6,0,10,10,1,3,10]%%%}+%%%{17716740096,
[0,6,0,10,10,0,4,10]%%%}+%%%{-1677721600,[0,6,0,10,8,4,0,12]%%%}+%%%{-22280142848,[0,6,0,10,8,3,1,12]%%%}+%%%{
51942260736,[0,6,0,10,8,2,2,12]%%%}+%%%{166698418176,[0,6,0,10,8,1,3,12]%%%}+%%%{-12280922112,[0,6,0,10,8,0,4,
12]%%%}+%%%{1073741824,[0,6,0,10,6,4,0,14]%%%}+%%%{10737418240,[0,6,0,10,6,3,1,14]%%%}+%%%{-20937965568,[0,6,0
,10,6,2,2,14]%%%}+%%%{-89120571392,[0,6,0,10,6,1,3,14]%%%}+%%%{3758096384,[0,6,0,10,6,0,4,14]%%%}+%%%{-2684354
56,[0,6,0,10,4,4,0,16]%%%}+%%%{-2147483648,[0,6,0,10,4,3,1,16]%%%}+%%%{3221225472,[0,6,0,10,4,2,2,16]%%%}+%%%{
19327352832,[0,6,0,10,4,1,3,16]%%%}+%%%{-268435456,[0,6,0,10,4,0,4,16]%%%}+%%%{134217728,[0,5,0,10,16,4,1,4]%%
%}+%%%{-671088640,[0,5,0,10,16,3,2,4]%%%}+%%%{-2818572288,[0,5,0,10,16,2,3,4]%%%}+%%%{1744830464,[0,5,0,10,16,
1,4,4]%%%}+%%%{536870912,[0,5,0,10,16,0,5,4]%%%}+%%%{-805306368,[0,5,0,10,14,4,1,6]%%%}+%%%{4429185024,[0,5,0,
10,14,3,2,6]%%%}+%%%{25367150592,[0,5,0,10,14,2,3,6]%%%}+%%%{-12482248704,[0,5,0,10,14,1,4,6]%%%}+%%%{-6039797
760,[0,5,0,10,14,0,5,6]%%%}+%%%{1879048192,[0,5,0,10,12,4,1,8]%%%}+%%%{-13555990528,[0,5,0,10,12,3,2,8]%%%}+%%
%{-91402272768,[0,5,0,10,12,2,3,8]%%%}+%%%{35567697920,[0,5,0,10,12,1,4,8]%%%}+%%%{24830279680,[0,5,0,10,12,0,
5,8]%%%}+%%%{-2147483648,[0,5,0,10,10,4,1,10]%%%}+%%%{22951231488,[0,5,0,10,10,3,2,10]%%%}+%%%{169516990464,[0
,5,0,10,10,2,3,10]%%%}+%%%{-50331648000,[0,5,0,10,10,1,4,10]%%%}+%%%{-50063212544,[0,5,0,10,10,0,5,10]%%%}+%%%
{1207959552,[0,5,0,10,8,4,1,12]%%%}+%%%{-21743271936,[0,5,0,10,8,3,2,12]%%%}+%%%{-171530256384,[0,5,0,10,8,2,3
,12]%%%}+%%%{36238786560,[0,5,0,10,8,1,4,12]%%%}+%%%{53552873472,[0,5,0,10,8,0,5,12]%%%}+%%%{-268435456,[0,5,0
,10,6,4,1,14]%%%}+%%%{10737418240,[0,5,0,10,6,3,2,14]%%%}+%%%{90194313216,[0,5,0,10,6,2,3,14]%%%}+%%%{-1181116
0064,[0,5,0,10,6,1,4,14]%%%}+%%%{-29259464704,[0,5,0,10,6,0,5,14]%%%}+%%%{-2147483648,[0,5,0,10,4,3,2,16]%%%}+
%%%{-19327352832,[0,5,0,10,4,2,3,16]%%%}+%%%{1073741824,[0,5,0,10,4,1,4,16]%%%}+%%%{6442450944,[0,5,0,10,4,0,5
,16]%%%}+%%%{67108864,[0,4,0,10,16,4,2,4]%%%}+%%%{1476395008,[0,4,0,10,16,3,3,4]%%%}+%%%{-1207959552,[0,4,0,10
,16,2,4,4]%%%}+%%%{-1744830464,[0,4,0,10,16,1,5,4]%%%}+%%%{335544320,[0,4,0,10,16,0,6,4]%%%}+%%%{-11676942336,
[0,4,0,10,14,3,3,6]%%%}+%%%{9663676416,[0,4,0,10,14,2,4,6]%%%}+%%%{18924699648,[0,4,0,10,14,1,5,6]%%%}+%%%{-17
44830464,[0,4,0,10,14,0,6,6]%%%}+%%%{-134217728,[0,4,0,10,12,4,2,8]%%%}+%%%{37983617024,[0,4,0,10,12,3,3,8]%%%
}+%%%{-29796335616,[0,4,0,10,12,2,4,8]%%%}+%%%{-76369887232,[0,4,0,10,12,1,5,8]%%%}+%%%{4026531840,[0,4,0,10,1
2,0,6,8]%%%}+%%%{-805306368,[0,4,0,10,10,4,2,10]%%%}+%%%{-65095598080,[0,4,0,10,10,3,3,10]%%%}+%%%{45097156608
,[0,4,0,10,10,2,4,10]%%%}+%%%{152337121280,[0,4,0,10,10,1,5,10]%%%}+%%%{-4966055936,[0,4,0,10,10,0,6,10]%%%}+%
%%{2214592512,[0,4,0,10,8,4,2,12]%%%}+%%%{62008590336,[0,4,0,10,8,3,3,12]%%%}+%%%{-35030827008,[0,4,0,10,8,2,4
,12]%%%}+%%%{-161866579968,[0,4,0,10,8,1,5,12]%%%}+%%%{3154116608,[0,4,0,10,8,0,6,12]%%%}+%%%{-1879048192,[0,4
,0,10,6,4,2,14]%%%}+%%%{-31138512896,[0,4,0,10,6,3,3,14]%%%}+%%%{12884901888,[0,4,0,10,6,2,4,14]%%%}+%%%{88046
829568,[0,4,0,10,6,1,5,14]%%%}+%%%{-805306368,[0,4,0,10,6,0,6,14]%%%}+%%%{536870912,[0,4,0,10,4,4,2,16]%%%}+%%
%{6442450944,[0,4,0,10,4,3,3,16]%%%}+%%%{-1610612736,[0,4,0,10,4,2,4,16]%%%}+%%%{-19327352832,[0,4,0,10,4,1,5,
16]%%%}+%%%{-268435456,[0,3,0,10,16,4,3,4]%%%}+%%%{134217728,[0,3,0,10,16,3,4,4]%%%}+%%%{2013265920,[0,3,0,10,
16,2,5,4]%%%}+%%%{-671088640,[0,3,0,10,16,1,6,4]%%%}+%%%{-134217728,[0,3,0,10,16,0,7,4]%%%}+%%%{1610612736,[0,
3,0,10,14,4,3,6]%%%}+%%%{-1207959552,[0,3,0,10,14,3,4,6]%%%}+%%%{-20535312384,[0,3,0,10,14,2,5,6]%%%}+%%%{3892
314112,[0,3,0,10,14,1,6,6]%%%}+%%%{1744830464,[0,3,0,10,14,0,7,6]%%%}+%%%{-3758096384,[0,3,0,10,12,4,3,8]%%%}+
%%%{4966055936,[0,3,0,10,12,3,4,8]%%%}+%%%{80127983616,[0,3,0,10,12,2,5,8]%%%}+%%%{-9797894144,[0,3,0,10,12,1,
6,8]%%%}+%%%{-7650410496,[0,3,0,10,12,0,7,8]%%%}+%%%{4294967296,[0,3,0,10,10,4,3,10]%%%}+%%%{-10066329600,[0,3
,0,10,10,3,4,10]%%%}+%%%{-156632088576,[0,3,0,10,10,2,5,10]%%%}+%%%{12750684160,[0,3,0,10,10,1,6,10]%%%}+%%%{1
5971909632,[0,3,0,10,10,0,7,10]%%%}+%%%{-2415919104,[0,3,0,10,8,4,3,12]%%%}+%%%{10468982784,[0,3,0,10,8,3,4,12
]%%%}+%%%{164282499072,[0,3,0,10,8,2,5,12]%%%}+%%%{-8321499136,[0,3,0,10,8,1,6,12]%%%}+%%%{-17448304640,[0,3,0
,10,8,0,7,12]%%%}+%%%{536870912,[0,3,0,10,6,4,3,14]%%%}+%%%{-5368709120,[0,3,0,10,6,3,4,14]%%%}+%%%{-885837004
80,[0,3,0,10,6,2,5,14]%%%}+%%%{2147483648,[0,3,0,10,6,1,6,14]%%%}+%%%{9663676416,[0,3,0,10,6,0,7,14]%%%}+%%%{1
073741824,[0,3,0,10,4,3,4,16]%%%}+%%%{19327352832,[0,3,0,10,4,2,5,16]%%%}+%%%{-2147483648,[0,3,0,10,4,0,7,16]%
%%}+%%%{67108864,[0,2,0,10,16,4,4,4]%%%}+%%%{-939524096,[0,2,0,10,16,3,5,4]%%%}+%%%{268435456,[0,2,0,10,16,2,6
,4]%%%}+%%%{402653184,[0,2,0,10,16,1,7,4]%%%}+%%%{-67108864,[0,2,0,10,16,0,8,4]%%%}+%%%{-805306368,[0,2,0,10,1
4,4,4,6]%%%}+%%%{8455716864,[0,2,0,10,14,3,5,6]%%%}+%%%{-2013265920,[0,2,0,10,14,2,6,6]%%%}+%%%{-5234491392,[0
,2,0,10,14,1,7,6]%%%}+%%%{201326592,[0,2,0,10,14,0,8,6]%%%}+%%%{2281701376,[0,2,0,10,12,4,4,8]%%%}+%%%{-304674
24256,[0,2,0,10,12,3,5,8]%%%}+%%%{6039797760,[0,2,0,10,12,2,6,8]%%%}+%%%{22951231488,[0,2,0,10,12,1,7,8]%%%}+%
%%{-201326592,[0,2,0,10,12,0,8,8]%%%}+%%%{-2415919104,[0,2,0,10,10,4,4,10]%%%}+%%%{56505663488,[0,2,0,10,10,3,
5,10]%%%}+%%%{-8724152320,[0,2,0,10,10,2,6,10]%%%}+%%%{-47915728896,[0,2,0,10,10,1,7,10]%%%}+%%%{67108864,[0,2
,0,10,10,0,8,10]%%%}+%%%{603979776,[0,2,0,10,8,4,4,12]%%%}+%%%{-57176752128,[0,2,0,10,8,3,5,12]%%%}+%%%{603979
7760,[0,2,0,10,8,2,6,12]%%%}+%%%{52344913920,[0,2,0,10,8,1,7,12]%%%}+%%%{536870912,[0,2,0,10,6,4,4,14]%%%}+%%%
{30064771072,[0,2,0,10,6,3,5,14]%%%}+%%%{-1610612736,[0,2,0,10,6,2,6,14]%%%}+%%%{-28991029248,[0,2,0,10,6,1,7,
14]%%%}+%%%{-268435456,[0,2,0,10,4,4,4,16]%%%}+%%%{-6442450944,[0,2,0,10,4,3,5,16]%%%}+%%%{6442450944,[0,2,0,1
0,4,1,7,16]%%%}+%%%{134217728,[0,1,0,10,16,4,5,4]%%%}+%%%{134217728,[0,1,0,10,16,3,6,4]%%%}+%%%{-402653184,[0,
1,0,10,16,2,7,4]%%%}+%%%{134217728,[0,1,0,10,16,1,8,4]%%%}+%%%{-805306368,[0,1,0,10,14,4,5,6]%%%}+%%%{-6710886
40,[0,1,0,10,14,3,6,6]%%%}+%%%{5234491392,[0,1,0,10,14,2,7,6]%%%}+%%%{-402653184,[0,1,0,10,14,1,8,6]%%%}+%%%{1
879048192,[0,1,0,10,12,4,5,8]%%%}+%%%{1207959552,[0,1,0,10,12,3,6,8]%%%}+%%%{-22951231488,[0,1,0,10,12,2,7,8]%
%%}+%%%{402653184,[0,1,0,10,12,1,8,8]%%%}+%%%{-2147483648,[0,1,0,10,10,4,5,10]%%%}+%%%{-939524096,[0,1,0,10,10
,3,6,10]%%%}+%%%{47915728896,[0,1,0,10,10,2,7,10]%%%}+%%%{-134217728,[0,1,0,10,10,1,8,10]%%%}+%%%{1207959552,[
0,1,0,10,8,4,5,12]%%%}+%%%{268435456,[0,1,0,10,8,3,6,12]%%%}+%%%{-52344913920,[0,1,0,10,8,2,7,12]%%%}+%%%{-268
435456,[0,1,0,10,6,4,5,14]%%%}+%%%{28991029248,[0,1,0,10,6,2,7,14]%%%}+%%%{-6442450944,[0,1,0,10,4,2,7,16]%%%}
+%%%{-67108864,[0,0,0,10,16,4,6,4]%%%}+%%%{134217728,[0,0,0,10,16,3,7,4]%%%}+%%%{-67108864,[0,0,0,10,16,2,8,4]
%%%}+%%%{536870912,[0,0,0,10,14,4,6,6]%%%}+%%%{-1744830464,[0,0,0,10,14,3,7,6]%%%}+%%%{201326592,[0,0,0,10,14,
2,8,6]%%%}+%%%{-1476395008,[0,0,0,10,12,4,6,8]%%%}+%%%{7650410496,[0,0,0,10,12,3,7,8]%%%}+%%%{-201326592,[0,0,
0,10,12,2,8,8]%%%}+%%%{1879048192,[0,0,0,10,10,4,6,10]%%%}+%%%{-15971909632,[0,0,0,10,10,3,7,10]%%%}+%%%{67108
864,[0,0,0,10,10,2,8,10]%%%}+%%%{-1140850688,[0,0,0,10,8,4,6,12]%%%}+%%%{17448304640,[0,0,0,10,8,3,7,12]%%%}+%
%%{268435456,[0,0,0,10,6,4,6,14]%%%}+%%%{-9663676416,[0,0,0,10,6,3,7,14]%%%}+%%%{2147483648,[0,0,0,10,4,3,7,16
]%%%} / %%%{1024,[0,4,0,4,8,0,0,0]%%%}+%%%{-4352,[0,4,0,4,6,0,0,2]%%%}+%%%{5120,[0,4,0,4,4,0,0,4]%%%}+%%%{-102
4,[0,4,0,4,2,0,0,6]%%%}+%%%{-1024,[0,3,0,4,8,1,0,0]%%%}+%%%{-1024,[0,3,0,4,8,0,1,0]%%%}+%%%{4608,[0,3,0,4,6,1,
0,2]%%%}+%%%{6656,[0,3,0,4,6,0,1,2]%%%}+%%%{-6144,[0,3,0,4,4,1,0,4]%%%}+%%%{-13312,[0,3,0,4,4,0,1,4]%%%}+%%%{2
048,[0,3,0,4,2,1,0,6]%%%}+%%%{8192,[0,3,0,4,2,0,1,6]%%%}+%%%{256,[0,2,0,4,8,2,0,0]%%%}+%%%{1536,[0,2,0,4,8,1,1
,0]%%%}+%%%{-768,[0,2,0,4,8,0,2,0]%%%}+%%%{-256,[0,2,0,4,6,2,0,2]%%%}+%%%{-8192,[0,2,0,4,6,1,1,2]%%%}+%%%{2816
,[0,2,0,4,6,0,2,2]%%%}+%%%{1024,[0,2,0,4,4,2,0,4]%%%}+%%%{14336,[0,2,0,4,4,1,1,4]%%%}+%%%{-3072,[0,2,0,4,4,0,2
,4]%%%}+%%%{-1024,[0,2,0,4,2,2,0,6]%%%}+%%%{-8192,[0,2,0,4,2,1,1,6]%%%}+%%%{-512,[0,1,0,4,8,2,1,0]%%%}+%%%{512
,[0,1,0,4,8,0,3,0]%%%}+%%%{1536,[0,1,0,4,6,2,1,2]%%%}+%%%{-1536,[0,1,0,4,6,1,2,2]%%%}+%%%{-5120,[0,1,0,4,6,0,3
,2]%%%}+%%%{-1024,[0,1,0,4,4,2,1,4]%%%}+%%%{2048,[0,1,0,4,4,1,2,4]%%%}+%%%{12288,[0,1,0,4,4,0,3,4]%%%}+%%%{-81
92,[0,1,0,4,2,0,3,6]%%%}+%%%{256,[0,0,0,4,8,2,2,0]%%%}+%%%{-512,[0,0,0,4,8,1,3,0]%%%}+%%%{256,[0,0,0,4,8,0,4,0
]%%%}+%%%{-1280,[0,0,0,4,6,2,2,2]%%%}+%%%{5120,[0,0,0,4,6,1,3,2]%%%}+%%%{1024,[0,0,0,4,4,2,2,4]%%%}+%%%{-12288
,[0,0,0,4,4,1,3,4]%%%}+%%%{8192,[0,0,0,4,2,1,3,6]%%%} Error: Bad Argument Value
________________________________________________________________________________________
maple [A] time = 1.33, size = 0, normalized size = 0.00 \[ \int \left (a +b \cos \left (f x +e \right )\right )^{m} \left (g \tan \left (f x +e \right )\right )^{p}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((a+b*cos(f*x+e))^m*(g*tan(f*x+e))^p,x)
[Out]
int((a+b*cos(f*x+e))^m*(g*tan(f*x+e))^p,x)
________________________________________________________________________________________
maxima [A] time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (b \cos \left (f x + e\right ) + a\right )}^{m} \left (g \tan \left (f x + e\right )\right )^{p}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((a+b*cos(f*x+e))^m*(g*tan(f*x+e))^p,x, algorithm="maxima")
[Out]
integrate((b*cos(f*x + e) + a)^m*(g*tan(f*x + e))^p, x)
________________________________________________________________________________________
mupad [A] time = 0.00, size = -1, normalized size = -0.02 \[ \int {\left (g\,\mathrm {tan}\left (e+f\,x\right )\right )}^p\,{\left (a+b\,\cos \left (e+f\,x\right )\right )}^m \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((g*tan(e + f*x))^p*(a + b*cos(e + f*x))^m,x)
[Out]
int((g*tan(e + f*x))^p*(a + b*cos(e + f*x))^m, x)
________________________________________________________________________________________
sympy [A] time = 0.00, size = 0, normalized size = 0.00 \[ \int \left (g \tan {\left (e + f x \right )}\right )^{p} \left (a + b \cos {\left (e + f x \right )}\right )^{m}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((a+b*cos(f*x+e))**m*(g*tan(f*x+e))**p,x)
[Out]
Integral((g*tan(e + f*x))**p*(a + b*cos(e + f*x))**m, x)
________________________________________________________________________________________