∫ sinm(c + dx)(a + b tan(c + dx))n dx

3.83 \(\int \sin ^m(c+d x) (a+b \tan (c+d x))^n \, dx\)

Optimal. Leaf size=24 \[ \text {Int}\left (\sin ^m(c+d x) (a+b \tan (c+d x))^n,x\right ) \]

[Out]

CannotIntegrate(sin(d*x+c)^m*(a+b*tan(d*x+c))^n,x)

________________________________________________________________________________________

Rubi [A] time = 2.46, 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 = {} \[ \int \sin ^m(c+d x) (a+b \tan (c+d x))^n \, dx \]

Veriﬁcation is Not applicable to the result.

[In]

Int[Sin[c + d*x]^m*(a + b*Tan[c + d*x])^n,x]

[Out]

Defer[Int][Sin[c + d*x]^m*(a + b*Tan[c + d*x])^n, x]

Rubi steps

\begin {align*} \int \sin ^m(c+d x) (a+b \tan (c+d x))^n \, dx &=\int \sin ^m(c+d x) (a+b \tan (c+d x))^n \, dx\\ \end {align*}

________________________________________________________________________________________

Mathematica [A] time = 3.50, size = 0, normalized size = 0.00 \[ \int \sin ^m(c+d x) (a+b \tan (c+d x))^n \, dx \]

Veriﬁcation is Not applicable to the result.

[In]

Integrate[Sin[c + d*x]^m*(a + b*Tan[c + d*x])^n,x]

[Out]

Integrate[Sin[c + d*x]^m*(a + b*Tan[c + d*x])^n, x]

________________________________________________________________________________________

fricas [A] time = 0.53, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (b \tan \left (d x + c\right ) + a\right )}^{n} \sin \left (d x + c\right )^{m}, x\right ) \]

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(sin(d*x+c)^m*(a+b*tan(d*x+c))^n,x, algorithm="fricas")

[Out]

integral((b*tan(d*x + c) + a)^n*sin(d*x + c)^m, 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(sin(d*x+c)^m*(a+b*tan(d*x+c))^n,x, algorithm="giac")

[Out]

Exception raised: TypeError >> An error occurred running a Giac command:INPUT:sage2:=int(sage0,x):;OUTPUT:Simp

lification assuming c near 0Unable 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)Simplification a

ssuming a near 0Simplification assuming a near 0Evaluation time: 0.53Unable to divide, perhaps due to rounding

error%%%{-268435456,[0,10,0,10,16,0,4,0]%%%}+%%%{1073741824*i,[0,10,0,10,15,0,5,0]%%%}+%%%{1610612736,[0,10,0

,10,14,0,6,0]%%%}+%%%{-1073741824*i,[0,10,0,10,13,0,7,0]%%%}+%%%{268435456,[0,10,0,10,12,0,8,0]%%%}+%%%{-21474

83648*i,[0,10,0,10,11,0,9,0]%%%}+%%%{-3221225472,[0,10,0,10,10,0,10,0]%%%}+%%%{2147483648*i,[0,10,0,10,9,0,11,

0]%%%}+%%%{268435456,[0,10,0,10,8,0,12,0]%%%}+%%%{1073741824*i,[0,10,0,10,7,0,13,0]%%%}+%%%{1610612736,[0,10,0

,10,6,0,14,0]%%%}+%%%{-1073741824*i,[0,10,0,10,5,0,15,0]%%%}+%%%{-268435456,[0,10,0,10,4,0,16,0]%%%}+%%%{10737

41824,[0,9,0,10,16,1,4,0]%%%}+%%%{-4294967296*i,[0,9,0,10,15,1,5,0]%%%}+%%%{1073741824*i,[0,9,0,10,15,0,5,1]%%

%}+%%%{-6710886400,[0,9,0,10,14,1,6,0]%%%}+%%%{4026531840,[0,9,0,10,14,0,6,1]%%%}+%%%{4831838208*i,[0,9,0,10,1

3,1,7,0]%%%}+%%%{-4831838208*i,[0,9,0,10,13,0,7,1]%%%}+%%%{-1342177280,[0,9,0,10,12,1,8,0]%%%}+%%%{-268435456,

[0,9,0,10,12,0,8,1]%%%}+%%%{9663676416*i,[0,9,0,10,11,1,9,0]%%%}+%%%{-5368709120*i,[0,9,0,10,11,0,9,1]%%%}+%%%

{13421772800,[0,9,0,10,10,1,10,0]%%%}+%%%{-8053063680,[0,9,0,10,10,0,10,1]%%%}+%%%{-8589934592*i,[0,9,0,10,9,1

,11,0]%%%}+%%%{6442450944*i,[0,9,0,10,9,0,11,1]%%%}+%%%{-536870912,[0,9,0,10,8,1,12,0]%%%}+%%%{536870912,[0,9,

0,10,8,0,12,1]%%%}+%%%{-5368709120*i,[0,9,0,10,7,1,13,0]%%%}+%%%{4294967296*i,[0,9,0,10,7,0,13,1]%%%}+%%%{-671

0886400,[0,9,0,10,6,1,14,0]%%%}+%%%{4026531840,[0,9,0,10,6,0,14,1]%%%}+%%%{3758096384*i,[0,9,0,10,5,1,15,0]%%%

}+%%%{-1610612736*i,[0,9,0,10,5,0,15,1]%%%}+%%%{805306368,[0,9,0,10,4,1,16,0]%%%}+%%%{-268435456,[0,9,0,10,4,0

,16,1]%%%}+%%%{-1610612736,[0,8,0,10,16,2,4,0]%%%}+%%%{536870912,[0,8,0,10,16,0,4,2]%%%}+%%%{6442450944*i,[0,8

,0,10,15,2,5,0]%%%}+%%%{-4294967296*i,[0,8,0,10,15,1,5,1]%%%}+%%%{-2147483648*i,[0,8,0,10,15,0,5,2]%%%}+%%%{10

468982784,[0,8,0,10,14,2,6,0]%%%}+%%%{-16106127360,[0,8,0,10,14,1,6,1]%%%}+%%%{-1879048192,[0,8,0,10,14,0,6,2]

%%%}+%%%{-8053063680*i,[0,8,0,10,13,2,7,0]%%%}+%%%{19864223744*i,[0,8,0,10,13,1,7,1]%%%}+%%%{-2147483648*i,[0,

8,0,10,13,0,7,2]%%%}+%%%{2348810240,[0,8,0,10,12,2,8,0]%%%}+%%%{2013265920,[0,8,0,10,12,1,8,1]%%%}+%%%{-355676

9792,[0,8,0,10,12,0,8,2]%%%}+%%%{-16106127360*i,[0,8,0,10,11,2,9,0]%%%}+%%%{22548578304*i,[0,8,0,10,11,1,9,1]%

%%}+%%%{-21206401024,[0,8,0,10,10,2,10,0]%%%}+%%%{34896609280,[0,8,0,10,10,1,10,1]%%%}+%%%{-1879048192,[0,8,0,

10,10,0,10,2]%%%}+%%%{12884901888*i,[0,8,0,10,9,2,11,0]%%%}+%%%{-25769803776*i,[0,8,0,10,9,1,11,1]%%%}+%%%{-40

2653184,[0,8,0,10,8,2,12,0]%%%}+%%%{268435456,[0,8,0,10,8,1,12,1]%%%}+%%%{-2550136832,[0,8,0,10,8,0,12,2]%%%}+

%%%{9663676416*i,[0,8,0,10,7,2,13,0]%%%}+%%%{-18253611008*i,[0,8,0,10,7,1,13,1]%%%}+%%%{2147483648*i,[0,8,0,10

,7,0,13,2]%%%}+%%%{10200547328,[0,8,0,10,6,2,14,0]%%%}+%%%{-15569256448,[0,8,0,10,6,1,14,1]%%%}+%%%{-107374182

4,[0,8,0,10,6,0,14,2]%%%}+%%%{-4831838208*i,[0,8,0,10,5,2,15,0]%%%}+%%%{5905580032*i,[0,8,0,10,5,1,15,1]%%%}+%

%%{2147483648*i,[0,8,0,10,5,0,15,2]%%%}+%%%{-872415232,[0,8,0,10,4,2,16,0]%%%}+%%%{939524096,[0,8,0,10,4,1,16,

1]%%%}+%%%{738197504,[0,8,0,10,4,0,16,2]%%%}+%%%{1073741824,[0,7,0,10,16,3,4,0]%%%}+%%%{-2147483648,[0,7,0,10,

16,1,4,2]%%%}+%%%{-4294967296*i,[0,7,0,10,15,3,5,0]%%%}+%%%{6442450944*i,[0,7,0,10,15,2,5,1]%%%}+%%%{858993459

2*i,[0,7,0,10,15,1,5,2]%%%}+%%%{-2147483648*i,[0,7,0,10,15,0,5,3]%%%}+%%%{-7247757312,[0,7,0,10,14,3,6,0]%%%}+

%%%{24159191040,[0,7,0,10,14,2,6,1]%%%}+%%%{8321499136,[0,7,0,10,14,1,6,2]%%%}+%%%{-8053063680,[0,7,0,10,14,0,

6,3]%%%}+%%%{5905580032*i,[0,7,0,10,13,3,7,0]%%%}+%%%{-30601641984*i,[0,7,0,10,13,2,7,1]%%%}+%%%{6979321856*i,

[0,7,0,10,13,1,7,2]%%%}+%%%{9126805504*i,[0,7,0,10,13,0,7,3]%%%}+%%%{-1744830464,[0,7,0,10,12,3,8,0]%%%}+%%%{-

4429185024,[0,7,0,10,12,2,8,1]%%%}+%%%{15166603264,[0,7,0,10,12,1,8,2]%%%}+%%%{-402653184,[0,7,0,10,12,0,8,3]%

%%}+%%%{11811160064*i,[0,7,0,10,11,3,9,0]%%%}+%%%{-35433480192*i,[0,7,0,10,11,2,9,1]%%%}+%%%{-3221225472*i,[0,

7,0,10,11,1,9,2]%%%}+%%%{9663676416*i,[0,7,0,10,11,0,9,3]%%%}+%%%{15032385536,[0,7,0,10,10,3,10,0]%%%}+%%%{-56

371445760,[0,7,0,10,10,2,10,1]%%%}+%%%{6442450944,[0,7,0,10,10,1,10,2]%%%}+%%%{13421772800,[0,7,0,10,10,0,10,3

]%%%}+%%%{-8589934592*i,[0,7,0,10,9,3,11,0]%%%}+%%%{38654705664*i,[0,7,0,10,9,2,11,1]%%%}+%%%{-12884901888*i,[

0,7,0,10,9,0,11,3]%%%}+%%%{1342177280,[0,7,0,10,8,3,12,0]%%%}+%%%{-4026531840,[0,7,0,10,8,2,12,1]%%%}+%%%{9395

240960,[0,7,0,10,8,1,12,2]%%%}+%%%{-3489660928,[0,7,0,10,8,0,12,3]%%%}+%%%{-7516192768*i,[0,7,0,10,7,3,13,0]%%

%}+%%%{28991029248*i,[0,7,0,10,7,2,13,1]%%%}+%%%{-5368709120*i,[0,7,0,10,7,1,13,2]%%%}+%%%{-7516192768*i,[0,7,

0,10,7,0,13,3]%%%}+%%%{-6710886400,[0,7,0,10,6,3,14,0]%%%}+%%%{22548578304,[0,7,0,10,6,2,14,1]%%%}+%%%{5637144

576,[0,7,0,10,6,1,14,2]%%%}+%%%{-8589934592,[0,7,0,10,6,0,14,3]%%%}+%%%{2684354560*i,[0,7,0,10,5,3,15,0]%%%}+%

%%{-8053063680*i,[0,7,0,10,5,2,15,1]%%%}+%%%{-6979321856*i,[0,7,0,10,5,1,15,2]%%%}+%%%{3758096384*i,[0,7,0,10,

5,0,15,3]%%%}+%%%{402653184,[0,7,0,10,4,3,16,0]%%%}+%%%{-1207959552,[0,7,0,10,4,2,16,1]%%%}+%%%{-2013265920,[0

,7,0,10,4,1,16,2]%%%}+%%%{671088640,[0,7,0,10,4,0,16,3]%%%}+%%%{-268435456,[0,6,0,10,16,4,4,0]%%%}+%%%{3221225

472,[0,6,0,10,16,2,4,2]%%%}+%%%{-268435456,[0,6,0,10,16,0,4,4]%%%}+%%%{1073741824*i,[0,6,0,10,15,4,5,0]%%%}+%%

%{-4294967296*i,[0,6,0,10,15,3,5,1]%%%}+%%%{-12884901888*i,[0,6,0,10,15,2,5,2]%%%}+%%%{8589934592*i,[0,6,0,10,

15,1,5,3]%%%}+%%%{1073741824*i,[0,6,0,10,15,0,5,4]%%%}+%%%{1879048192,[0,6,0,10,14,4,6,0]%%%}+%%%{-16106127360

,[0,6,0,10,14,3,6,1]%%%}+%%%{-13690208256,[0,6,0,10,14,2,6,2]%%%}+%%%{32212254720,[0,6,0,10,14,1,6,3]%%%}+%%%{

-1073741824,[0,6,0,10,14,0,6,4]%%%}+%%%{-1610612736*i,[0,6,0,10,13,4,7,0]%%%}+%%%{20937965568*i,[0,6,0,10,13,3

,7,1]%%%}+%%%{-8053063680*i,[0,6,0,10,13,2,7,2]%%%}+%%%{-38117834752*i,[0,6,0,10,13,1,7,3]%%%}+%%%{7516192768*

i,[0,6,0,10,13,0,7,4]%%%}+%%%{469762048,[0,6,0,10,12,4,8,0]%%%}+%%%{3892314112,[0,6,0,10,12,3,8,1]%%%}+%%%{-23

890755584,[0,6,0,10,12,2,8,2]%%%}+%%%{-1207959552,[0,6,0,10,12,1,8,3]%%%}+%%%{6241124352,[0,6,0,10,12,0,8,4]%%

%}+%%%{-3221225472*i,[0,6,0,10,11,4,9,0]%%%}+%%%{24696061952*i,[0,6,0,10,11,3,9,1]%%%}+%%%{9663676416*i,[0,6,0

,10,11,2,9,2]%%%}+%%%{-41875931136*i,[0,6,0,10,11,1,9,3]%%%}+%%%{6442450944*i,[0,6,0,10,11,0,9,4]%%%}+%%%{-402

6531840,[0,6,0,10,10,4,10,0]%%%}+%%%{40265318400,[0,6,0,10,10,3,10,1]%%%}+%%%{-6979321856,[0,6,0,10,10,2,10,2]

%%%}+%%%{-61740154880,[0,6,0,10,10,1,10,3]%%%}+%%%{13153337344,[0,6,0,10,10,0,10,4]%%%}+%%%{2147483648*i,[0,6,

0,10,9,4,11,0]%%%}+%%%{-25769803776*i,[0,6,0,10,9,3,11,1]%%%}+%%%{51539607552*i,[0,6,0,10,9,1,11,3]%%%}+%%%{-6

442450944*i,[0,6,0,10,9,0,11,4]%%%}+%%%{-671088640,[0,6,0,10,8,4,12,0]%%%}+%%%{5100273664,[0,6,0,10,8,3,12,1]%

%%}+%%%{-11274289152,[0,6,0,10,8,2,12,2]%%%}+%%%{6710886400,[0,6,0,10,8,1,12,3]%%%}+%%%{3892314112,[0,6,0,10,8

,0,12,4]%%%}+%%%{2147483648*i,[0,6,0,10,7,4,13,0]%%%}+%%%{-20401094656*i,[0,6,0,10,7,3,13,1]%%%}+%%%{322122547

2*i,[0,6,0,10,7,2,13,2]%%%}+%%%{33285996544*i,[0,6,0,10,7,1,13,3]%%%}+%%%{-7516192768*i,[0,6,0,10,7,0,13,4]%%%

}+%%%{1610612736,[0,6,0,10,6,4,14,0]%%%}+%%%{-14495514624,[0,6,0,10,6,3,14,1]%%%}+%%%{-9395240960,[0,6,0,10,6,

2,14,2]%%%}+%%%{32749125632,[0,6,0,10,6,1,14,3]%%%}+%%%{-2952790016,[0,6,0,10,6,0,14,4]%%%}+%%%{-536870912*i,[

0,6,0,10,5,4,15,0]%%%}+%%%{4831838208*i,[0,6,0,10,5,3,15,1]%%%}+%%%{8053063680*i,[0,6,0,10,5,2,15,2]%%%}+%%%{-

13421772800*i,[0,6,0,10,5,1,15,3]%%%}+%%%{-1073741824*i,[0,6,0,10,5,0,15,4]%%%}+%%%{-67108864,[0,6,0,10,4,4,16

,0]%%%}+%%%{671088640,[0,6,0,10,4,3,16,1]%%%}+%%%{1879048192,[0,6,0,10,4,2,16,2]%%%}+%%%{-2281701376,[0,6,0,10

,4,1,16,3]%%%}+%%%{-738197504,[0,6,0,10,4,0,16,4]%%%}+%%%{-2147483648,[0,5,0,10,16,3,4,2]%%%}+%%%{1073741824,[

0,5,0,10,16,1,4,4]%%%}+%%%{1073741824*i,[0,5,0,10,15,4,5,1]%%%}+%%%{8589934592*i,[0,5,0,10,15,3,5,2]%%%}+%%%{-

12884901888*i,[0,5,0,10,15,2,5,3]%%%}+%%%{-4294967296*i,[0,5,0,10,15,1,5,4]%%%}+%%%{1073741824*i,[0,5,0,10,15,

0,5,5]%%%}+%%%{4026531840,[0,5,0,10,14,4,6,1]%%%}+%%%{9932111872,[0,5,0,10,14,3,6,2]%%%}+%%%{-48318382080,[0,5

,0,10,14,2,6,3]%%%}+%%%{3489660928,[0,5,0,10,14,1,6,4]%%%}+%%%{4026531840,[0,5,0,10,14,0,6,5]%%%}+%%%{-5368709

120*i,[0,5,0,10,13,4,7,1]%%%}+%%%{3758096384*i,[0,5,0,10,13,3,7,2]%%%}+%%%{59592671232*i,[0,5,0,10,13,2,7,3]%%

%}+%%%{-28454158336*i,[0,5,0,10,13,1,7,4]%%%}+%%%{-3758096384*i,[0,5,0,10,13,0,7,5]%%%}+%%%{-1207959552,[0,5,0

,10,12,4,8,1]%%%}+%%%{16508780544,[0,5,0,10,12,3,8,2]%%%}+%%%{6039797760,[0,5,0,10,12,2,8,3]%%%}+%%%{-26172456

960,[0,5,0,10,12,1,8,4]%%%}+%%%{1610612736,[0,5,0,10,12,0,8,5]%%%}+%%%{-6442450944*i,[0,5,0,10,11,4,9,1]%%%}+%

%%{-9663676416*i,[0,5,0,10,11,3,9,2]%%%}+%%%{67645734912*i,[0,5,0,10,11,2,9,3]%%%}+%%%{-22548578304*i,[0,5,0,1

0,11,1,9,4]%%%}+%%%{-3221225472*i,[0,5,0,10,11,0,9,5]%%%}+%%%{-10737418240,[0,5,0,10,10,4,10,1]%%%}+%%%{214748

3648,[0,5,0,10,10,3,10,2]%%%}+%%%{104689827840,[0,5,0,10,10,2,10,3]%%%}+%%%{-52613349376,[0,5,0,10,10,1,10,4]%

%%}+%%%{-2684354560,[0,5,0,10,10,0,10,5]%%%}+%%%{6442450944*i,[0,5,0,10,9,4,11,1]%%%}+%%%{-77309411328*i,[0,5,

0,10,9,2,11,3]%%%}+%%%{25769803776*i,[0,5,0,10,9,1,11,4]%%%}+%%%{6442450944*i,[0,5,0,10,9,0,11,5]%%%}+%%%{-187

9048192,[0,5,0,10,8,4,12,1]%%%}+%%%{4563402752,[0,5,0,10,8,3,12,2]%%%}+%%%{805306368,[0,5,0,10,8,2,12,3]%%%}+%

%%{-16374562816,[0,5,0,10,8,1,12,4]%%%}+%%%{5368709120,[0,5,0,10,8,0,12,5]%%%}+%%%{5368709120*i,[0,5,0,10,7,4,

13,1]%%%}+%%%{1073741824*i,[0,5,0,10,7,3,13,2]%%%}+%%%{-54760833024*i,[0,5,0,10,7,2,13,3]%%%}+%%%{26843545600*

i,[0,5,0,10,7,1,13,4]%%%}+%%%{2147483648*i,[0,5,0,10,7,0,13,5]%%%}+%%%{3489660928,[0,5,0,10,6,4,14,1]%%%}+%%%{

6174015488,[0,5,0,10,6,3,14,2]%%%}+%%%{-46707769344,[0,5,0,10,6,2,14,3]%%%}+%%%{9395240960,[0,5,0,10,6,1,14,4]

%%%}+%%%{5100273664,[0,5,0,10,6,0,14,5]%%%}+%%%{-1073741824*i,[0,5,0,10,5,4,15,1]%%%}+%%%{-3758096384*i,[0,5,0

,10,5,3,15,2]%%%}+%%%{17716740096*i,[0,5,0,10,5,2,15,3]%%%}+%%%{2684354560*i,[0,5,0,10,5,1,15,4]%%%}+%%%{-2684

354560*i,[0,5,0,10,5,0,15,5]%%%}+%%%{-134217728,[0,5,0,10,4,4,16,1]%%%}+%%%{-671088640,[0,5,0,10,4,3,16,2]%%%}

+%%%{2818572288,[0,5,0,10,4,2,16,3]%%%}+%%%{1744830464,[0,5,0,10,4,1,16,4]%%%}+%%%{-536870912,[0,5,0,10,4,0,16

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67296*i,[0,4,0,10,15,1,5,5]%%%}+%%%{-2684354560,[0,4,0,10,14,4,6,2]%%%}+%%%{32212254720,[0,4,0,10,14,3,6,3]%%%

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0*i,[0,4,0,10,13,2,7,4]%%%}+%%%{16642998272*i,[0,4,0,10,13,1,7,5]%%%}+%%%{-4294967296*i,[0,4,0,10,13,0,7,6]%%%

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0,4,0,10,11,4,9,2]%%%}+%%%{-48318382080*i,[0,4,0,10,11,3,9,3]%%%}+%%%{28991029248*i,[0,4,0,10,11,2,9,4]%%%}+%%

%{16106127360*i,[0,4,0,10,11,1,9,5]%%%}+%%%{-4294967296*i,[0,4,0,10,11,0,9,6]%%%}+%%%{268435456,[0,4,0,10,10,4

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[0,4,0,10,10,1,10,5]%%%}+%%%{-7784628224,[0,4,0,10,10,0,10,6]%%%}+%%%{51539607552*i,[0,4,0,10,9,3,11,3]%%%}+%%

%{-38654705664*i,[0,4,0,10,9,2,11,4]%%%}+%%%{-25769803776*i,[0,4,0,10,9,1,11,5]%%%}+%%%{4294967296*i,[0,4,0,10

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[0,4,0,10,8,2,12,4]%%%}+%%%{-14227079168,[0,4,0,10,8,1,12,5]%%%}+%%%{-1207959552,[0,4,0,10,8,0,12,6]%%%}+%%%{-

1073741824*i,[0,4,0,10,7,4,13,2]%%%}+%%%{39728447488*i,[0,4,0,10,7,3,13,3]%%%}+%%%{-35433480192*i,[0,4,0,10,7,

2,13,4]%%%}+%%%{-11811160064*i,[0,4,0,10,7,1,13,5]%%%}+%%%{4294967296*i,[0,4,0,10,7,0,13,6]%%%}+%%%{-134217728

0,[0,4,0,10,6,4,14,2]%%%}+%%%{29527900160,[0,4,0,10,6,3,14,3]%%%}+%%%{-12079595520,[0,4,0,10,6,2,14,4]%%%}+%%%

{-18790481920,[0,4,0,10,6,1,14,5]%%%}+%%%{2684354560,[0,4,0,10,6,0,14,6]%%%}+%%%{536870912*i,[0,4,0,10,5,4,15,

2]%%%}+%%%{-10200547328*i,[0,4,0,10,5,3,15,3]%%%}+%%%{-1610612736*i,[0,4,0,10,5,2,15,4]%%%}+%%%{9126805504*i,[

0,4,0,10,5,1,15,5]%%%}+%%%{67108864,[0,4,0,10,4,4,16,2]%%%}+%%%{-1476395008,[0,4,0,10,4,3,16,3]%%%}+%%%{-12079

59552,[0,4,0,10,4,2,16,4]%%%}+%%%{1744830464,[0,4,0,10,4,1,16,5]%%%}+%%%{335544320,[0,4,0,10,4,0,16,6]%%%}+%%%

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13248,[0,3,0,10,12,1,8,6]%%%}+%%%{-939524096,[0,3,0,10,12,0,8,7]%%%}+%%%{12884901888*i,[0,3,0,10,11,4,9,3]%%%}

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0,10,11,1,9,6]%%%}+%%%{-1073741824*i,[0,3,0,10,11,0,9,7]%%%}+%%%{21474836480,[0,3,0,10,10,4,10,3]%%%}+%%%{-493

92123904,[0,3,0,10,10,3,10,4]%%%}+%%%{-40265318400,[0,3,0,10,10,2,10,5]%%%}+%%%{32212254720,[0,3,0,10,10,1,10,

6]%%%}+%%%{-2684354560,[0,3,0,10,10,0,10,7]%%%}+%%%{-12884901888*i,[0,3,0,10,9,4,11,3]%%%}+%%%{25769803776*i,[

0,3,0,10,9,3,11,4]%%%}+%%%{38654705664*i,[0,3,0,10,9,2,11,5]%%%}+%%%{-17179869184*i,[0,3,0,10,9,1,11,6]%%%}+%%

%{3758096384,[0,3,0,10,8,4,12,3]%%%}+%%%{-13153337344,[0,3,0,10,8,3,12,4]%%%}+%%%{10468982784,[0,3,0,10,8,2,12

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6106127360*i,[0,3,0,10,7,1,13,6]%%%}+%%%{1073741824*i,[0,3,0,10,7,0,13,7]%%%}+%%%{-6979321856,[0,3,0,10,6,4,14

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10,6,1,14,6]%%%}+%%%{-536870912,[0,3,0,10,6,0,14,7]%%%}+%%%{2147483648*i,[0,3,0,10,5,4,15,3]%%%}+%%%{-53687091

2*i,[0,3,0,10,5,3,15,4]%%%}+%%%{-11274289152*i,[0,3,0,10,5,2,15,5]%%%}+%%%{536870912*i,[0,3,0,10,5,1,15,6]%%%}

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4]%%%}+%%%{-2013265920,[0,3,0,10,4,2,16,5]%%%}+%%%{-671088640,[0,3,0,10,4,1,16,6]%%%}+%%%{134217728,[0,3,0,10,

4,0,16,7]%%%}+%%%{-268435456,[0,2,0,10,16,4,4,4]%%%}+%%%{1073741824*i,[0,2,0,10,15,4,5,4]%%%}+%%%{-4294967296*

i,[0,2,0,10,15,3,5,5]%%%}+%%%{-268435456,[0,2,0,10,14,4,6,4]%%%}+%%%{-16106127360,[0,2,0,10,14,3,6,5]%%%}+%%%{

7247757312,[0,2,0,10,14,2,6,6]%%%}+%%%{5905580032*i,[0,2,0,10,13,4,7,4]%%%}+%%%{19864223744*i,[0,2,0,10,13,3,7

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2,0,10,12,4,8,4]%%%}+%%%{2013265920,[0,2,0,10,12,3,8,5]%%%}+%%%{-19058917376,[0,2,0,10,12,2,8,6]%%%}+%%%{28185

72288,[0,2,0,10,12,1,8,7]%%%}+%%%{-67108864,[0,2,0,10,12,0,8,8]%%%}+%%%{3221225472*i,[0,2,0,10,11,4,9,4]%%%}+%

%%{22548578304*i,[0,2,0,10,11,3,9,5]%%%}+%%%{-22548578304*i,[0,2,0,10,11,2,9,6]%%%}+%%%{3221225472*i,[0,2,0,10

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992,[0,2,0,10,10,2,10,6]%%%}+%%%{8053063680,[0,2,0,10,10,1,10,7]%%%}+%%%{-268435456,[0,2,0,10,10,0,10,8]%%%}+%

%%{-6442450944*i,[0,2,0,10,9,4,11,4]%%%}+%%%{-25769803776*i,[0,2,0,10,9,3,11,5]%%%}+%%%{25769803776*i,[0,2,0,1

0,9,2,11,6]%%%}+%%%{2281701376,[0,2,0,10,8,4,12,4]%%%}+%%%{268435456,[0,2,0,10,8,3,12,5]%%%}+%%%{-11274289152,

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4967296*i,[0,2,0,10,7,4,13,4]%%%}+%%%{-18253611008*i,[0,2,0,10,7,3,13,5]%%%}+%%%{22548578304*i,[0,2,0,10,7,2,1

3,6]%%%}+%%%{-3221225472*i,[0,2,0,10,7,1,13,7]%%%}+%%%{-2147483648,[0,2,0,10,6,4,14,4]%%%}+%%%{-15569256448,[0

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4,16,4]%%%}+%%%{939524096,[0,2,0,10,4,3,16,5]%%%}+%%%{268435456,[0,2,0,10,4,2,16,6]%%%}+%%%{-402653184,[0,2,0,

10,4,1,16,7]%%%}+%%%{-67108864,[0,2,0,10,4,0,16,8]%%%}+%%%{1073741824*i,[0,1,0,10,15,4,5,5]%%%}+%%%{4026531840

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{15569256448*i,[0,1,0,10,13,3,7,6]%%%}+%%%{-1610612736*i,[0,1,0,10,13,2,7,7]%%%}+%%%{-1207959552,[0,1,0,10,12,

4,8,5]%%%}+%%%{13019119616,[0,1,0,10,12,3,8,6]%%%}+%%%{-2818572288,[0,1,0,10,12,2,8,7]%%%}+%%%{134217728,[0,1,

0,10,12,1,8,8]%%%}+%%%{-6442450944*i,[0,1,0,10,11,4,9,5]%%%}+%%%{13958643712*i,[0,1,0,10,11,3,9,6]%%%}+%%%{-32

21225472*i,[0,1,0,10,11,2,9,7]%%%}+%%%{-10737418240,[0,1,0,10,10,4,10,5]%%%}+%%%{32212254720,[0,1,0,10,10,3,10

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0,10,9,4,11,5]%%%}+%%%{-17179869184*i,[0,1,0,10,9,3,11,6]%%%}+%%%{-1879048192,[0,1,0,10,8,4,12,5]%%%}+%%%{7247

757312,[0,1,0,10,8,3,12,6]%%%}+%%%{-7247757312,[0,1,0,10,8,2,12,7]%%%}+%%%{805306368,[0,1,0,10,8,1,12,8]%%%}+%

%%{5368709120*i,[0,1,0,10,7,4,13,5]%%%}+%%%{-13958643712*i,[0,1,0,10,7,3,13,6]%%%}+%%%{3221225472*i,[0,1,0,10,

7,2,13,7]%%%}+%%%{3489660928,[0,1,0,10,6,4,14,5]%%%}+%%%{-7247757312,[0,1,0,10,6,3,14,6]%%%}+%%%{-1610612736,[

0,1,0,10,6,2,14,7]%%%}+%%%{536870912,[0,1,0,10,6,1,14,8]%%%}+%%%{-1073741824*i,[0,1,0,10,5,4,15,5]%%%}+%%%{161

0612736*i,[0,1,0,10,5,3,15,6]%%%}+%%%{1610612736*i,[0,1,0,10,5,2,15,7]%%%}+%%%{-134217728,[0,1,0,10,4,4,16,5]%

%%}+%%%{134217728,[0,1,0,10,4,3,16,6]%%%}+%%%{402653184,[0,1,0,10,4,2,16,7]%%%}+%%%{134217728,[0,1,0,10,4,1,16

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0,10,13,3,7,7]%%%}+%%%{-3288334336,[0,0,0,10,12,4,8,6]%%%}+%%%{939524096,[0,0,0,10,12,3,8,7]%%%}+%%%{-67108864

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%{-7784628224,[0,0,0,10,10,4,10,6]%%%}+%%%{2684354560,[0,0,0,10,10,3,10,7]%%%}+%%%{-268435456,[0,0,0,10,10,2,1

0,8]%%%}+%%%{4294967296*i,[0,0,0,10,9,4,11,6]%%%}+%%%{-1476395008,[0,0,0,10,8,4,12,6]%%%}+%%%{2415919104,[0,0,

0,10,8,3,12,7]%%%}+%%%{-402653184,[0,0,0,10,8,2,12,8]%%%}+%%%{3221225472*i,[0,0,0,10,7,4,13,6]%%%}+%%%{-107374

1824*i,[0,0,0,10,7,3,13,7]%%%}+%%%{1879048192,[0,0,0,10,6,4,14,6]%%%}+%%%{536870912,[0,0,0,10,6,3,14,7]%%%}+%%

%{-268435456,[0,0,0,10,6,2,14,8]%%%}+%%%{-536870912*i,[0,0,0,10,5,4,15,6]%%%}+%%%{-536870912*i,[0,0,0,10,5,3,1

5,7]%%%}+%%%{-67108864,[0,0,0,10,4,4,16,6]%%%}+%%%{-134217728,[0,0,0,10,4,3,16,7]%%%}+%%%{-67108864,[0,0,0,10,

4,2,16,8]%%%} / %%%{1024,[0,4,0,4,8,0,0,0]%%%}+%%%{-4096*i,[0,4,0,4,7,0,1,0]%%%}+%%%{-8192,[0,4,0,4,6,0,2,0]%%

%}+%%%{12288*i,[0,4,0,4,5,0,3,0]%%%}+%%%{14336,[0,4,0,4,4,0,4,0]%%%}+%%%{-12288*i,[0,4,0,4,3,0,5,0]%%%}+%%%{-8

192,[0,4,0,4,2,0,6,0]%%%}+%%%{4096*i,[0,4,0,4,1,0,7,0]%%%}+%%%{1024,[0,4,0,4,0,0,8,0]%%%}+%%%{-2048,[0,3,0,4,8

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696,[0,3,0,4,4,1,4,0]%%%}+%%%{31744,[0,3,0,4,4,0,4,1]%%%}+%%%{24576*i,[0,3,0,4,3,1,5,0]%%%}+%%%{-28672*i,[0,3,

0,4,3,0,5,1]%%%}+%%%{15360,[0,3,0,4,2,1,6,0]%%%}+%%%{-17408,[0,3,0,4,2,0,6,1]%%%}+%%%{-6144*i,[0,3,0,4,1,1,7,0

]%%%}+%%%{6144*i,[0,3,0,4,1,0,7,1]%%%}+%%%{-1024,[0,3,0,4,0,1,8,0]%%%}+%%%{1024,[0,3,0,4,0,0,8,1]%%%}+%%%{1024

,[0,2,0,4,8,2,0,0]%%%}+%%%{-4096*i,[0,2,0,4,7,2,1,0]%%%}+%%%{8192*i,[0,2,0,4,7,1,1,1]%%%}+%%%{-9216,[0,2,0,4,6

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%%%{-55296*i,[0,2,0,4,5,1,3,1]%%%}+%%%{16384*i,[0,2,0,4,5,0,3,2]%%%}+%%%{15616,[0,2,0,4,4,2,4,0]%%%}+%%%{-6707

2,[0,2,0,4,4,1,4,1]%%%}+%%%{21760,[0,2,0,4,4,0,4,2]%%%}+%%%{-12288*i,[0,2,0,4,3,2,5,0]%%%}+%%%{57344*i,[0,2,0,

4,3,1,5,1]%%%}+%%%{-16384*i,[0,2,0,4,3,0,5,2]%%%}+%%%{-6656,[0,2,0,4,2,2,6,0]%%%}+%%%{31744,[0,2,0,4,2,1,6,1]%

%%}+%%%{-6656,[0,2,0,4,2,0,6,2]%%%}+%%%{2048*i,[0,2,0,4,1,2,7,0]%%%}+%%%{-10240*i,[0,2,0,4,1,1,7,1]%%%}+%%%{25

6,[0,2,0,4,0,2,8,0]%%%}+%%%{-1536,[0,2,0,4,0,1,8,1]%%%}+%%%{-768,[0,2,0,4,0,0,8,2]%%%}+%%%{-4096*i,[0,1,0,4,7,

2,1,1]%%%}+%%%{-15360,[0,1,0,4,6,2,2,1]%%%}+%%%{9216,[0,1,0,4,6,1,2,2]%%%}+%%%{28672*i,[0,1,0,4,5,2,3,1]%%%}+%

%%{-30720*i,[0,1,0,4,5,1,3,2]%%%}+%%%{2048*i,[0,1,0,4,5,0,3,3]%%%}+%%%{35328,[0,1,0,4,4,2,4,1]%%%}+%%%{-43008,

[0,1,0,4,4,1,4,2]%%%}+%%%{3584,[0,1,0,4,4,0,4,3]%%%}+%%%{-28672*i,[0,1,0,4,3,2,5,1]%%%}+%%%{32768*i,[0,1,0,4,3

,1,5,2]%%%}+%%%{-14336,[0,1,0,4,2,2,6,1]%%%}+%%%{13312,[0,1,0,4,2,1,6,2]%%%}+%%%{3072,[0,1,0,4,2,0,6,3]%%%}+%%

%{4096*i,[0,1,0,4,1,2,7,1]%%%}+%%%{-2048*i,[0,1,0,4,1,1,7,2]%%%}+%%%{-2048*i,[0,1,0,4,1,0,7,3]%%%}+%%%{512,[0,

1,0,4,0,2,8,1]%%%}+%%%{-512,[0,1,0,4,0,0,8,3]%%%}+%%%{-4096,[0,0,0,4,6,2,2,2]%%%}+%%%{14336*i,[0,0,0,4,5,2,3,2

]%%%}+%%%{-2048*i,[0,0,0,4,5,1,3,3]%%%}+%%%{20736,[0,0,0,4,4,2,4,2]%%%}+%%%{-3584,[0,0,0,4,4,1,4,3]%%%}+%%%{25

6,[0,0,0,4,4,0,4,4]%%%}+%%%{-16384*i,[0,0,0,4,3,2,5,2]%%%}+%%%{-7680,[0,0,0,4,2,2,6,2]%%%}+%%%{-3072,[0,0,0,4,

2,1,6,3]%%%}+%%%{512,[0,0,0,4,2,0,6,4]%%%}+%%%{2048*i,[0,0,0,4,1,2,7,2]%%%}+%%%{2048*i,[0,0,0,4,1,1,7,3]%%%}+%

%%{256,[0,0,0,4,0,2,8,2]%%%}+%%%{512,[0,0,0,4,0,1,8,3]%%%}+%%%{256,[0,0,0,4,0,0,8,4]%%%} Error: Bad Argument V

alue

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maple [A] time = 1.55, size = 0, normalized size = 0.00 \[ \int \left (\sin ^{m}\left (d x +c \right )\right ) \left (a +b \tan \left (d x +c \right )\right )^{n}\, dx \]

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

int(sin(d*x+c)^m*(a+b*tan(d*x+c))^n,x)

[Out]

int(sin(d*x+c)^m*(a+b*tan(d*x+c))^n,x)

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maxima [A] time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (b \tan \left (d x + c\right ) + a\right )}^{n} \sin \left (d x + c\right )^{m}\,{d x} \]

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(sin(d*x+c)^m*(a+b*tan(d*x+c))^n,x, algorithm="maxima")

[Out]

integrate((b*tan(d*x + c) + a)^n*sin(d*x + c)^m, x)

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mupad [A] time = 0.00, size = -1, normalized size = -0.04 \[ \int {\sin \left (c+d\,x\right )}^m\,{\left (a+b\,\mathrm {tan}\left (c+d\,x\right )\right )}^n \,d x \]

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

int(sin(c + d*x)^m*(a + b*tan(c + d*x))^n,x)

[Out]

int(sin(c + d*x)^m*(a + b*tan(c + d*x))^n, x)

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sympy [A] time = 0.00, size = 0, normalized size = 0.00 \[ \int \left (a + b \tan {\left (c + d x \right )}\right )^{n} \sin ^{m}{\left (c + d x \right )}\, dx \]

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(sin(d*x+c)**m*(a+b*tan(d*x+c))**n,x)

[Out]

Integral((a + b*tan(c + d*x))**n*sin(c + d*x)**m, x)

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