\(\int (a+a \cos (c+d x))^4 \sec ^4(c+d x) \, dx\) [39]

Optimal result

Integrand size = 21, antiderivative size = 73 \[ \int (a+a \cos (c+d x))^4 \sec ^4(c+d x) \, dx=a^4 x+\frac {6 a^4 \text {arctanh}(\sin (c+d x))}{d}+\frac {7 a^4 \tan (c+d x)}{d}+\frac {2 a^4 \sec (c+d x) \tan (c+d x)}{d}+\frac {a^4 \tan ^3(c+d x)}{3 d} \]

[Out]

a^4*x+6*a^4*arctanh(sin(d*x+c))/d+7*a^4*tan(d*x+c)/d+2*a^4*sec(d*x+c)*tan(d*x+c)/d+1/3*a^4*tan(d*x+c)^3/d

Rubi [A] (verified)

Time = 0.12 (sec) , antiderivative size = 73, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.238, Rules used = {2836, 3855, 3852, 8, 3853} \[ \int (a+a \cos (c+d x))^4 \sec ^4(c+d x) \, dx=\frac {6 a^4 \text {arctanh}(\sin (c+d x))}{d}+\frac {a^4 \tan ^3(c+d x)}{3 d}+\frac {7 a^4 \tan (c+d x)}{d}+\frac {2 a^4 \tan (c+d x) \sec (c+d x)}{d}+a^4 x \]

[In]

Int[(a + a*Cos[c + d*x])^4*Sec[c + d*x]^4,x]

[Out]

a^4*x + (6*a^4*ArcTanh[Sin[c + d*x]])/d + (7*a^4*Tan[c + d*x])/d + (2*a^4*Sec[c + d*x]*Tan[c + d*x])/d + (a^4*
Tan[c + d*x]^3)/(3*d)

Rule 8

Int[a_, x_Symbol] :> Simp[a*x, x] /; FreeQ[a, x]

Rule 2836

Int[((d_.)*sin[(e_.) + (f_.)*(x_)])^(n_.)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.), x_Symbol] :> Int[Expan
dTrig[(a + b*sin[e + f*x])^m*(d*sin[e + f*x])^n, x], x] /; FreeQ[{a, b, d, e, f, n}, x] && EqQ[a^2 - b^2, 0] &
& IGtQ[m, 0] && RationalQ[n]

Rule 3852

Int[csc[(c_.) + (d_.)*(x_)]^(n_), x_Symbol] :> Dist[-d^(-1), Subst[Int[ExpandIntegrand[(1 + x^2)^(n/2 - 1), x]
, x], x, Cot[c + d*x]], x] /; FreeQ[{c, d}, x] && IGtQ[n/2, 0]

Rule 3853

Int[(csc[(c_.) + (d_.)*(x_)]*(b_.))^(n_), x_Symbol] :> Simp[(-b)*Cos[c + d*x]*((b*Csc[c + d*x])^(n - 1)/(d*(n
- 1))), x] + Dist[b^2*((n - 2)/(n - 1)), Int[(b*Csc[c + d*x])^(n - 2), x], x] /; FreeQ[{b, c, d}, x] && GtQ[n,
1] && IntegerQ[2*n]

Rule 3855

Int[csc[(c_.) + (d_.)*(x_)], x_Symbol] :> Simp[-ArcTanh[Cos[c + d*x]]/d, x] /; FreeQ[{c, d}, x]

Rubi steps \begin{align*} \text {integral}& = \int \left (a^4+4 a^4 \sec (c+d x)+6 a^4 \sec ^2(c+d x)+4 a^4 \sec ^3(c+d x)+a^4 \sec ^4(c+d x)\right ) \, dx \\ & = a^4 x+a^4 \int \sec ^4(c+d x) \, dx+\left (4 a^4\right ) \int \sec (c+d x) \, dx+\left (4 a^4\right ) \int \sec ^3(c+d x) \, dx+\left (6 a^4\right ) \int \sec ^2(c+d x) \, dx \\ & = a^4 x+\frac {4 a^4 \text {arctanh}(\sin (c+d x))}{d}+\frac {2 a^4 \sec (c+d x) \tan (c+d x)}{d}+\left (2 a^4\right ) \int \sec (c+d x) \, dx-\frac {a^4 \text {Subst}\left (\int \left (1+x^2\right ) \, dx,x,-\tan (c+d x)\right )}{d}-\frac {\left (6 a^4\right ) \text {Subst}(\int 1 \, dx,x,-\tan (c+d x))}{d} \\ & = a^4 x+\frac {6 a^4 \text {arctanh}(\sin (c+d x))}{d}+\frac {7 a^4 \tan (c+d x)}{d}+\frac {2 a^4 \sec (c+d x) \tan (c+d x)}{d}+\frac {a^4 \tan ^3(c+d x)}{3 d} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.03 (sec) , antiderivative size = 61, normalized size of antiderivative = 0.84 \[ \int (a+a \cos (c+d x))^4 \sec ^4(c+d x) \, dx=a^4 \left (x+\frac {6 \text {arctanh}(\sin (c+d x))}{d}+\frac {7 \tan (c+d x)}{d}+\frac {2 \sec (c+d x) \tan (c+d x)}{d}+\frac {\tan ^3(c+d x)}{3 d}\right ) \]

[In]

Integrate[(a + a*Cos[c + d*x])^4*Sec[c + d*x]^4,x]

[Out]

a^4*(x + (6*ArcTanh[Sin[c + d*x]])/d + (7*Tan[c + d*x])/d + (2*Sec[c + d*x]*Tan[c + d*x])/d + Tan[c + d*x]^3/(
3*d))

Maple [A] (verified)

Time = 3.30 (sec) , antiderivative size = 96, normalized size of antiderivative = 1.32

method	result	size
parts	\(-\frac {a^{4} \left (-\frac {2}{3}-\frac {\left (\sec ^{2}\left (d x +c \right )\right )}{3}\right ) \tan \left (d x +c \right )}{d}+\frac {a^{4} \left (d x +c \right )}{d}+\frac {2 a^{4} \sec \left (d x +c \right ) \tan \left (d x +c \right )}{d}+\frac {6 a^{4} \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{d}+\frac {6 a^{4} \tan \left (d x +c \right )}{d}\)	\(96\)
derivativedivides	\(\frac {a^{4} \left (d x +c \right )+4 a^{4} \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )+6 a^{4} \tan \left (d x +c \right )+4 a^{4} \left (\frac {\sec \left (d x +c \right ) \tan \left (d x +c \right )}{2}+\frac {\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2}\right )-a^{4} \left (-\frac {2}{3}-\frac {\left (\sec ^{2}\left (d x +c \right )\right )}{3}\right ) \tan \left (d x +c \right )}{d}\)	\(104\)
default	\(\frac {a^{4} \left (d x +c \right )+4 a^{4} \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )+6 a^{4} \tan \left (d x +c \right )+4 a^{4} \left (\frac {\sec \left (d x +c \right ) \tan \left (d x +c \right )}{2}+\frac {\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2}\right )-a^{4} \left (-\frac {2}{3}-\frac {\left (\sec ^{2}\left (d x +c \right )\right )}{3}\right ) \tan \left (d x +c \right )}{d}\)	\(104\)
risch	\(a^{4} x -\frac {4 i a^{4} \left (3 \,{\mathrm e}^{5 i \left (d x +c \right )}-9 \,{\mathrm e}^{4 i \left (d x +c \right )}-21 \,{\mathrm e}^{2 i \left (d x +c \right )}-3 \,{\mathrm e}^{i \left (d x +c \right )}-10\right )}{3 d \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )^{3}}+\frac {6 a^{4} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right )}{d}-\frac {6 a^{4} \ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right )}{d}\)	\(117\)
parallelrisch	\(-\frac {18 \left (\left (\cos \left (d x +c \right )+\frac {\cos \left (3 d x +3 c \right )}{3}\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )+\left (-\cos \left (d x +c \right )-\frac {\cos \left (3 d x +3 c \right )}{3}\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )-\frac {d x \cos \left (d x +c \right )}{6}-\frac {d x \cos \left (3 d x +3 c \right )}{18}-\frac {4 \sin \left (d x +c \right )}{9}-\frac {2 \sin \left (2 d x +2 c \right )}{9}-\frac {10 \sin \left (3 d x +3 c \right )}{27}\right ) a^{4}}{d \left (\cos \left (3 d x +3 c \right )+3 \cos \left (d x +c \right )\right )}\)	\(147\)
norman	\(\frac {a^{4} x \left (\tan ^{12}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+a^{4} x \left (\tan ^{14}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-a^{4} x -\frac {18 a^{4} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{d}-\frac {140 a^{4} \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3 d}-\frac {50 a^{4} \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3 d}+\frac {40 a^{4} \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}+\frac {70 a^{4} \left (\tan ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3 d}-\frac {44 a^{4} \left (\tan ^{11}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3 d}-\frac {10 a^{4} \left (\tan ^{13}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}-a^{4} x \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+3 a^{4} x \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+3 a^{4} x \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-3 a^{4} x \left (\tan ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-3 a^{4} x \left (\tan ^{10}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{\left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{4} \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{3}}-\frac {6 a^{4} \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{d}+\frac {6 a^{4} \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{d}\)	\(328\)

[In]

int((a+cos(d*x+c)*a)^4*sec(d*x+c)^4,x,method=_RETURNVERBOSE)

[Out]

-a^4/d*(-2/3-1/3*sec(d*x+c)^2)*tan(d*x+c)+a^4/d*(d*x+c)+2*a^4*sec(d*x+c)*tan(d*x+c)/d+6*a^4/d*ln(sec(d*x+c)+ta
n(d*x+c))+6*a^4*tan(d*x+c)/d

Fricas [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 110, normalized size of antiderivative = 1.51 \[ \int (a+a \cos (c+d x))^4 \sec ^4(c+d x) \, dx=\frac {3 \, a^{4} d x \cos \left (d x + c\right )^{3} + 9 \, a^{4} \cos \left (d x + c\right )^{3} \log \left (\sin \left (d x + c\right ) + 1\right ) - 9 \, a^{4} \cos \left (d x + c\right )^{3} \log \left (-\sin \left (d x + c\right ) + 1\right ) + {\left (20 \, a^{4} \cos \left (d x + c\right )^{2} + 6 \, a^{4} \cos \left (d x + c\right ) + a^{4}\right )} \sin \left (d x + c\right )}{3 \, d \cos \left (d x + c\right )^{3}} \]

[In]

integrate((a+a*cos(d*x+c))^4*sec(d*x+c)^4,x, algorithm="fricas")

[Out]

1/3*(3*a^4*d*x*cos(d*x + c)^3 + 9*a^4*cos(d*x + c)^3*log(sin(d*x + c) + 1) - 9*a^4*cos(d*x + c)^3*log(-sin(d*x
+ c) + 1) + (20*a^4*cos(d*x + c)^2 + 6*a^4*cos(d*x + c) + a^4)*sin(d*x + c))/(d*cos(d*x + c)^3)

Sympy [F]

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

[In]

integrate((a+a*cos(d*x+c))**4*sec(d*x+c)**4,x)

[Out]

a**4*(Integral(4*cos(c + d*x)*sec(c + d*x)**4, x) + Integral(6*cos(c + d*x)**2*sec(c + d*x)**4, x) + Integral(
4*cos(c + d*x)**3*sec(c + d*x)**4, x) + Integral(cos(c + d*x)**4*sec(c + d*x)**4, x) + Integral(sec(c + d*x)**
4, x))

Maxima [A] (verification not implemented)

none

Time = 0.30 (sec) , antiderivative size = 120, normalized size of antiderivative = 1.64 \[ \int (a+a \cos (c+d x))^4 \sec ^4(c+d x) \, dx=\frac {{\left (\tan \left (d x + c\right )^{3} + 3 \, \tan \left (d x + c\right )\right )} a^{4} + 3 \, {\left (d x + c\right )} a^{4} - 3 \, a^{4} {\left (\frac {2 \, \sin \left (d x + c\right )}{\sin \left (d x + c\right )^{2} - 1} - \log \left (\sin \left (d x + c\right ) + 1\right ) + \log \left (\sin \left (d x + c\right ) - 1\right )\right )} + 6 \, a^{4} {\left (\log \left (\sin \left (d x + c\right ) + 1\right ) - \log \left (\sin \left (d x + c\right ) - 1\right )\right )} + 18 \, a^{4} \tan \left (d x + c\right )}{3 \, d} \]

[In]

integrate((a+a*cos(d*x+c))^4*sec(d*x+c)^4,x, algorithm="maxima")

[Out]

1/3*((tan(d*x + c)^3 + 3*tan(d*x + c))*a^4 + 3*(d*x + c)*a^4 - 3*a^4*(2*sin(d*x + c)/(sin(d*x + c)^2 - 1) - lo
g(sin(d*x + c) + 1) + log(sin(d*x + c) - 1)) + 6*a^4*(log(sin(d*x + c) + 1) - log(sin(d*x + c) - 1)) + 18*a^4*
tan(d*x + c))/d

Giac [A] (verification not implemented)

none

Time = 0.35 (sec) , antiderivative size = 116, normalized size of antiderivative = 1.59 \[ \int (a+a \cos (c+d x))^4 \sec ^4(c+d x) \, dx=\frac {3 \, {\left (d x + c\right )} a^{4} + 18 \, a^{4} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1 \right |}\right ) - 18 \, a^{4} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1 \right |}\right ) - \frac {2 \, {\left (15 \, a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 38 \, a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 27 \, a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 1\right )}^{3}}}{3 \, d} \]

[In]

integrate((a+a*cos(d*x+c))^4*sec(d*x+c)^4,x, algorithm="giac")

[Out]

1/3*(3*(d*x + c)*a^4 + 18*a^4*log(abs(tan(1/2*d*x + 1/2*c) + 1)) - 18*a^4*log(abs(tan(1/2*d*x + 1/2*c) - 1)) -
2*(15*a^4*tan(1/2*d*x + 1/2*c)^5 - 38*a^4*tan(1/2*d*x + 1/2*c)^3 + 27*a^4*tan(1/2*d*x + 1/2*c))/(tan(1/2*d*x
+ 1/2*c)^2 - 1)^3)/d

Mupad [B] (verification not implemented)

Time = 14.64 (sec) , antiderivative size = 117, normalized size of antiderivative = 1.60 \[ \int (a+a \cos (c+d x))^4 \sec ^4(c+d x) \, dx=a^4\,x+\frac {12\,a^4\,\mathrm {atanh}\left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )}{d}-\frac {10\,a^4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5-\frac {76\,a^4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{3}+18\,a^4\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{d\,\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6-3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4+3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2-1\right )} \]

[In]

int((a + a*cos(c + d*x))^4/cos(c + d*x)^4,x)

[Out]

a^4*x + (12*a^4*atanh(tan(c/2 + (d*x)/2)))/d - (10*a^4*tan(c/2 + (d*x)/2)^5 - (76*a^4*tan(c/2 + (d*x)/2)^3)/3
+ 18*a^4*tan(c/2 + (d*x)/2))/(d*(3*tan(c/2 + (d*x)/2)^2 - 3*tan(c/2 + (d*x)/2)^4 + tan(c/2 + (d*x)/2)^6 - 1))