3.1.35 \(\int (a+a \sin (c+d x))^4 \tan ^3(c+d x) \, dx\) [35]

3.1.35.1 Optimal result
3.1.35.2 Mathematica [A] (verified)
3.1.35.3 Rubi [A] (verified)
3.1.35.4 Maple [C] (verified)
3.1.35.5 Fricas [A] (verification not implemented)
3.1.35.6 Sympy [F]
3.1.35.7 Maxima [A] (verification not implemented)
3.1.35.8 Giac [F(-1)]
3.1.35.9 Mupad [B] (verification not implemented)

3.1.35.1 Optimal result

Integrand size = 21, antiderivative size = 107 \[ \int (a+a \sin (c+d x))^4 \tan ^3(c+d x) \, dx=\frac {16 a^4 \log (1-\sin (c+d x))}{d}+\frac {12 a^4 \sin (c+d x)}{d}+\frac {4 a^4 \sin ^2(c+d x)}{d}+\frac {4 a^4 \sin ^3(c+d x)}{3 d}+\frac {a^4 \sin ^4(c+d x)}{4 d}+\frac {4 a^5}{d (a-a \sin (c+d x))} \]

output
16*a^4*ln(1-sin(d*x+c))/d+12*a^4*sin(d*x+c)/d+4*a^4*sin(d*x+c)^2/d+4/3*a^4 
*sin(d*x+c)^3/d+1/4*a^4*sin(d*x+c)^4/d+4*a^5/d/(a-a*sin(d*x+c))
 
3.1.35.2 Mathematica [A] (verified)

Time = 0.11 (sec) , antiderivative size = 76, normalized size of antiderivative = 0.71 \[ \int (a+a \sin (c+d x))^4 \tan ^3(c+d x) \, dx=\frac {a^4 \left (192 \log (1-\sin (c+d x))+\frac {48}{1-\sin (c+d x)}+144 \sin (c+d x)+48 \sin ^2(c+d x)+16 \sin ^3(c+d x)+3 \sin ^4(c+d x)\right )}{12 d} \]

input
Integrate[(a + a*Sin[c + d*x])^4*Tan[c + d*x]^3,x]
 
output
(a^4*(192*Log[1 - Sin[c + d*x]] + 48/(1 - Sin[c + d*x]) + 144*Sin[c + d*x] 
 + 48*Sin[c + d*x]^2 + 16*Sin[c + d*x]^3 + 3*Sin[c + d*x]^4))/(12*d)
 
3.1.35.3 Rubi [A] (verified)

Time = 0.26 (sec) , antiderivative size = 94, normalized size of antiderivative = 0.88, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.190, Rules used = {3042, 3186, 99, 2009}

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

\(\displaystyle \int \tan ^3(c+d x) (a \sin (c+d x)+a)^4 \, dx\)

\(\Big \downarrow \) 3042

\(\displaystyle \int \tan (c+d x)^3 (a \sin (c+d x)+a)^4dx\)

\(\Big \downarrow \) 3186

\(\displaystyle \frac {\int \frac {a^3 \sin ^3(c+d x) (\sin (c+d x) a+a)^2}{(a-a \sin (c+d x))^2}d(a \sin (c+d x))}{d}\)

\(\Big \downarrow \) 99

\(\displaystyle \frac {\int \left (\frac {4 a^5}{(a-a \sin (c+d x))^2}-\frac {16 a^4}{a-a \sin (c+d x)}+\sin ^3(c+d x) a^3+4 \sin ^2(c+d x) a^3+8 \sin (c+d x) a^3+12 a^3\right )d(a \sin (c+d x))}{d}\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {\frac {4 a^5}{a-a \sin (c+d x)}+\frac {1}{4} a^4 \sin ^4(c+d x)+\frac {4}{3} a^4 \sin ^3(c+d x)+4 a^4 \sin ^2(c+d x)+12 a^4 \sin (c+d x)+16 a^4 \log (a-a \sin (c+d x))}{d}\)

input
Int[(a + a*Sin[c + d*x])^4*Tan[c + d*x]^3,x]
 
output
(16*a^4*Log[a - a*Sin[c + d*x]] + 12*a^4*Sin[c + d*x] + 4*a^4*Sin[c + d*x] 
^2 + (4*a^4*Sin[c + d*x]^3)/3 + (a^4*Sin[c + d*x]^4)/4 + (4*a^5)/(a - a*Si 
n[c + d*x]))/d
 

3.1.35.3.1 Defintions of rubi rules used

rule 99
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) 
)^(p_), x_] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d*x)^n*(e + f*x)^p, x], 
 x] /; FreeQ[{a, b, c, d, e, f, p}, x] && IntegersQ[m, n] && (IntegerQ[p] | 
| (GtQ[m, 0] && GeQ[n, -1]))
 

rule 2009
Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

rule 3042
Int[u_, x_Symbol] :> Int[DeactivateTrig[u, x], x] /; FunctionOfTrigOfLinear 
Q[u, x]
 

rule 3186
Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*tan[(e_.) + (f_.)*(x_)]^(p 
_.), x_Symbol] :> Simp[1/f   Subst[Int[x^p*((a + x)^(m - (p + 1)/2)/(a - x) 
^((p + 1)/2)), x], x, b*Sin[e + f*x]], x] /; FreeQ[{a, b, e, f, m}, x] && E 
qQ[a^2 - b^2, 0] && IntegerQ[(p + 1)/2]
 
3.1.35.4 Maple [C] (verified)

Result contains complex when optimal does not.

Time = 14.79 (sec) , antiderivative size = 159, normalized size of antiderivative = 1.49

method result size
risch \(-16 i a^{4} x -\frac {13 i a^{4} {\mathrm e}^{i \left (d x +c \right )}}{2 d}+\frac {13 i a^{4} {\mathrm e}^{-i \left (d x +c \right )}}{2 d}-\frac {32 i a^{4} c}{d}-\frac {8 i a^{4} {\mathrm e}^{i \left (d x +c \right )}}{\left (-i+{\mathrm e}^{i \left (d x +c \right )}\right )^{2} d}+\frac {32 a^{4} \ln \left (-i+{\mathrm e}^{i \left (d x +c \right )}\right )}{d}+\frac {a^{4} \cos \left (4 d x +4 c \right )}{32 d}-\frac {a^{4} \sin \left (3 d x +3 c \right )}{3 d}-\frac {17 a^{4} \cos \left (2 d x +2 c \right )}{8 d}\) \(159\)
derivativedivides \(\frac {a^{4} \left (\frac {\sin ^{8}\left (d x +c \right )}{2 \cos \left (d x +c \right )^{2}}+\frac {\left (\sin ^{6}\left (d x +c \right )\right )}{2}+\frac {3 \left (\sin ^{4}\left (d x +c \right )\right )}{4}+\frac {3 \left (\sin ^{2}\left (d x +c \right )\right )}{2}+3 \ln \left (\cos \left (d x +c \right )\right )\right )+4 a^{4} \left (\frac {\sin ^{7}\left (d x +c \right )}{2 \cos \left (d x +c \right )^{2}}+\frac {\left (\sin ^{5}\left (d x +c \right )\right )}{2}+\frac {5 \left (\sin ^{3}\left (d x +c \right )\right )}{6}+\frac {5 \sin \left (d x +c \right )}{2}-\frac {5 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2}\right )+6 a^{4} \left (\frac {\sin ^{6}\left (d x +c \right )}{2 \cos \left (d x +c \right )^{2}}+\frac {\left (\sin ^{4}\left (d x +c \right )\right )}{2}+\sin ^{2}\left (d x +c \right )+2 \ln \left (\cos \left (d x +c \right )\right )\right )+4 a^{4} \left (\frac {\sin ^{5}\left (d x +c \right )}{2 \cos \left (d x +c \right )^{2}}+\frac {\left (\sin ^{3}\left (d x +c \right )\right )}{2}+\frac {3 \sin \left (d x +c \right )}{2}-\frac {3 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2}\right )+a^{4} \left (\frac {\left (\tan ^{2}\left (d x +c \right )\right )}{2}+\ln \left (\cos \left (d x +c \right )\right )\right )}{d}\) \(267\)
default \(\frac {a^{4} \left (\frac {\sin ^{8}\left (d x +c \right )}{2 \cos \left (d x +c \right )^{2}}+\frac {\left (\sin ^{6}\left (d x +c \right )\right )}{2}+\frac {3 \left (\sin ^{4}\left (d x +c \right )\right )}{4}+\frac {3 \left (\sin ^{2}\left (d x +c \right )\right )}{2}+3 \ln \left (\cos \left (d x +c \right )\right )\right )+4 a^{4} \left (\frac {\sin ^{7}\left (d x +c \right )}{2 \cos \left (d x +c \right )^{2}}+\frac {\left (\sin ^{5}\left (d x +c \right )\right )}{2}+\frac {5 \left (\sin ^{3}\left (d x +c \right )\right )}{6}+\frac {5 \sin \left (d x +c \right )}{2}-\frac {5 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2}\right )+6 a^{4} \left (\frac {\sin ^{6}\left (d x +c \right )}{2 \cos \left (d x +c \right )^{2}}+\frac {\left (\sin ^{4}\left (d x +c \right )\right )}{2}+\sin ^{2}\left (d x +c \right )+2 \ln \left (\cos \left (d x +c \right )\right )\right )+4 a^{4} \left (\frac {\sin ^{5}\left (d x +c \right )}{2 \cos \left (d x +c \right )^{2}}+\frac {\left (\sin ^{3}\left (d x +c \right )\right )}{2}+\frac {3 \sin \left (d x +c \right )}{2}-\frac {3 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2}\right )+a^{4} \left (\frac {\left (\tan ^{2}\left (d x +c \right )\right )}{2}+\ln \left (\cos \left (d x +c \right )\right )\right )}{d}\) \(267\)
parts \(\frac {a^{4} \left (\frac {\left (\tan ^{2}\left (d x +c \right )\right )}{2}-\frac {\ln \left (1+\tan ^{2}\left (d x +c \right )\right )}{2}\right )}{d}+\frac {a^{4} \left (\frac {\sin ^{8}\left (d x +c \right )}{2 \cos \left (d x +c \right )^{2}}+\frac {\left (\sin ^{6}\left (d x +c \right )\right )}{2}+\frac {3 \left (\sin ^{4}\left (d x +c \right )\right )}{4}+\frac {3 \left (\sin ^{2}\left (d x +c \right )\right )}{2}+3 \ln \left (\cos \left (d x +c \right )\right )\right )}{d}+\frac {4 a^{4} \left (\frac {\sin ^{5}\left (d x +c \right )}{2 \cos \left (d x +c \right )^{2}}+\frac {\left (\sin ^{3}\left (d x +c \right )\right )}{2}+\frac {3 \sin \left (d x +c \right )}{2}-\frac {3 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2}\right )}{d}+\frac {6 a^{4} \left (\frac {\sin ^{6}\left (d x +c \right )}{2 \cos \left (d x +c \right )^{2}}+\frac {\left (\sin ^{4}\left (d x +c \right )\right )}{2}+\sin ^{2}\left (d x +c \right )+2 \ln \left (\cos \left (d x +c \right )\right )\right )}{d}+\frac {4 a^{4} \left (\frac {\sin ^{7}\left (d x +c \right )}{2 \cos \left (d x +c \right )^{2}}+\frac {\left (\sin ^{5}\left (d x +c \right )\right )}{2}+\frac {5 \left (\sin ^{3}\left (d x +c \right )\right )}{6}+\frac {5 \sin \left (d x +c \right )}{2}-\frac {5 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2}\right )}{d}\) \(284\)

input
int((a+a*sin(d*x+c))^4*tan(d*x+c)^3,x,method=_RETURNVERBOSE)
 
output
-16*I*a^4*x-13/2*I/d*a^4*exp(I*(d*x+c))+13/2*I/d*a^4*exp(-I*(d*x+c))-32*I/ 
d*a^4*c-8*I*a^4*exp(I*(d*x+c))/(-I+exp(I*(d*x+c)))^2/d+32/d*a^4*ln(-I+exp( 
I*(d*x+c)))+1/32/d*a^4*cos(4*d*x+4*c)-1/3*a^4/d*sin(3*d*x+3*c)-17/8/d*a^4* 
cos(2*d*x+2*c)
 
3.1.35.5 Fricas [A] (verification not implemented)

Time = 0.33 (sec) , antiderivative size = 116, normalized size of antiderivative = 1.08 \[ \int (a+a \sin (c+d x))^4 \tan ^3(c+d x) \, dx=\frac {104 \, a^{4} \cos \left (d x + c\right )^{4} - 976 \, a^{4} \cos \left (d x + c\right )^{2} + 689 \, a^{4} + 1536 \, {\left (a^{4} \sin \left (d x + c\right ) - a^{4}\right )} \log \left (-\sin \left (d x + c\right ) + 1\right ) + {\left (24 \, a^{4} \cos \left (d x + c\right )^{4} - 304 \, a^{4} \cos \left (d x + c\right )^{2} - 1073 \, a^{4}\right )} \sin \left (d x + c\right )}{96 \, {\left (d \sin \left (d x + c\right ) - d\right )}} \]

input
integrate((a+a*sin(d*x+c))^4*tan(d*x+c)^3,x, algorithm="fricas")
 
output
1/96*(104*a^4*cos(d*x + c)^4 - 976*a^4*cos(d*x + c)^2 + 689*a^4 + 1536*(a^ 
4*sin(d*x + c) - a^4)*log(-sin(d*x + c) + 1) + (24*a^4*cos(d*x + c)^4 - 30 
4*a^4*cos(d*x + c)^2 - 1073*a^4)*sin(d*x + c))/(d*sin(d*x + c) - d)
 
3.1.35.6 Sympy [F]

\[ \int (a+a \sin (c+d x))^4 \tan ^3(c+d x) \, dx=a^{4} \left (\int 4 \sin {\left (c + d x \right )} \tan ^{3}{\left (c + d x \right )}\, dx + \int 6 \sin ^{2}{\left (c + d x \right )} \tan ^{3}{\left (c + d x \right )}\, dx + \int 4 \sin ^{3}{\left (c + d x \right )} \tan ^{3}{\left (c + d x \right )}\, dx + \int \sin ^{4}{\left (c + d x \right )} \tan ^{3}{\left (c + d x \right )}\, dx + \int \tan ^{3}{\left (c + d x \right )}\, dx\right ) \]

input
integrate((a+a*sin(d*x+c))**4*tan(d*x+c)**3,x)
 
output
a**4*(Integral(4*sin(c + d*x)*tan(c + d*x)**3, x) + Integral(6*sin(c + d*x 
)**2*tan(c + d*x)**3, x) + Integral(4*sin(c + d*x)**3*tan(c + d*x)**3, x) 
+ Integral(sin(c + d*x)**4*tan(c + d*x)**3, x) + Integral(tan(c + d*x)**3, 
 x))
 
3.1.35.7 Maxima [A] (verification not implemented)

Time = 0.19 (sec) , antiderivative size = 85, normalized size of antiderivative = 0.79 \[ \int (a+a \sin (c+d x))^4 \tan ^3(c+d x) \, dx=\frac {3 \, a^{4} \sin \left (d x + c\right )^{4} + 16 \, a^{4} \sin \left (d x + c\right )^{3} + 48 \, a^{4} \sin \left (d x + c\right )^{2} + 192 \, a^{4} \log \left (\sin \left (d x + c\right ) - 1\right ) + 144 \, a^{4} \sin \left (d x + c\right ) - \frac {48 \, a^{4}}{\sin \left (d x + c\right ) - 1}}{12 \, d} \]

input
integrate((a+a*sin(d*x+c))^4*tan(d*x+c)^3,x, algorithm="maxima")
 
output
1/12*(3*a^4*sin(d*x + c)^4 + 16*a^4*sin(d*x + c)^3 + 48*a^4*sin(d*x + c)^2 
 + 192*a^4*log(sin(d*x + c) - 1) + 144*a^4*sin(d*x + c) - 48*a^4/(sin(d*x 
+ c) - 1))/d
 
3.1.35.8 Giac [F(-1)]

Timed out. \[ \int (a+a \sin (c+d x))^4 \tan ^3(c+d x) \, dx=\text {Timed out} \]

input
integrate((a+a*sin(d*x+c))^4*tan(d*x+c)^3,x, algorithm="giac")
 
output
Timed out
 
3.1.35.9 Mupad [B] (verification not implemented)

Time = 7.07 (sec) , antiderivative size = 320, normalized size of antiderivative = 2.99 \[ \int (a+a \sin (c+d x))^4 \tan ^3(c+d x) \, dx=\frac {32\,a^4\,\ln \left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )-1\right )}{d}+\frac {32\,a^4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9-32\,a^4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8+\frac {320\,a^4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7}{3}-\frac {340\,a^4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6}{3}+\frac {424\,a^4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5}{3}-\frac {340\,a^4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4}{3}+\frac {320\,a^4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{3}-32\,a^4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+32\,a^4\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{d\,\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}-2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9+5\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8-8\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7+10\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6-12\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5+10\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4-8\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3+5\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2-2\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )+1\right )}-\frac {16\,a^4\,\ln \left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+1\right )}{d} \]

input
int(tan(c + d*x)^3*(a + a*sin(c + d*x))^4,x)
 
output
(32*a^4*log(tan(c/2 + (d*x)/2) - 1))/d + ((320*a^4*tan(c/2 + (d*x)/2)^3)/3 
 - 32*a^4*tan(c/2 + (d*x)/2)^2 - (340*a^4*tan(c/2 + (d*x)/2)^4)/3 + (424*a 
^4*tan(c/2 + (d*x)/2)^5)/3 - (340*a^4*tan(c/2 + (d*x)/2)^6)/3 + (320*a^4*t 
an(c/2 + (d*x)/2)^7)/3 - 32*a^4*tan(c/2 + (d*x)/2)^8 + 32*a^4*tan(c/2 + (d 
*x)/2)^9 + 32*a^4*tan(c/2 + (d*x)/2))/(d*(5*tan(c/2 + (d*x)/2)^2 - 2*tan(c 
/2 + (d*x)/2) - 8*tan(c/2 + (d*x)/2)^3 + 10*tan(c/2 + (d*x)/2)^4 - 12*tan( 
c/2 + (d*x)/2)^5 + 10*tan(c/2 + (d*x)/2)^6 - 8*tan(c/2 + (d*x)/2)^7 + 5*ta 
n(c/2 + (d*x)/2)^8 - 2*tan(c/2 + (d*x)/2)^9 + tan(c/2 + (d*x)/2)^10 + 1)) 
- (16*a^4*log(tan(c/2 + (d*x)/2)^2 + 1))/d