3.31 \(\int (a+a \cos (c+d x))^3 \sec ^5(c+d x) \, dx\)

Optimal. Leaf size=93 \[ \frac {a^3 \tan ^3(c+d x)}{d}+\frac {4 a^3 \tan (c+d x)}{d}+\frac {15 a^3 \tanh ^{-1}(\sin (c+d x))}{8 d}+\frac {a^3 \tan (c+d x) \sec ^3(c+d x)}{4 d}+\frac {15 a^3 \tan (c+d x) \sec (c+d x)}{8 d} \]

[Out]

15/8*a^3*arctanh(sin(d*x+c))/d+4*a^3*tan(d*x+c)/d+15/8*a^3*sec(d*x+c)*tan(d*x+c)/d+1/4*a^3*sec(d*x+c)^3*tan(d*
x+c)/d+a^3*tan(d*x+c)^3/d

________________________________________________________________________________________

Rubi [A]  time = 0.12, antiderivative size = 93, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 5, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.238, Rules used = {2757, 3767, 8, 3768, 3770} \[ \frac {a^3 \tan ^3(c+d x)}{d}+\frac {4 a^3 \tan (c+d x)}{d}+\frac {15 a^3 \tanh ^{-1}(\sin (c+d x))}{8 d}+\frac {a^3 \tan (c+d x) \sec ^3(c+d x)}{4 d}+\frac {15 a^3 \tan (c+d x) \sec (c+d x)}{8 d} \]

Antiderivative was successfully verified.

[In]

Int[(a + a*Cos[c + d*x])^3*Sec[c + d*x]^5,x]

[Out]

(15*a^3*ArcTanh[Sin[c + d*x]])/(8*d) + (4*a^3*Tan[c + d*x])/d + (15*a^3*Sec[c + d*x]*Tan[c + d*x])/(8*d) + (a^
3*Sec[c + d*x]^3*Tan[c + d*x])/(4*d) + (a^3*Tan[c + d*x]^3)/d

Rule 8

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

Rule 2757

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 3767

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 3768

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 3770

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

Rubi steps

\begin {align*} \int (a+a \cos (c+d x))^3 \sec ^5(c+d x) \, dx &=\int \left (a^3 \sec ^2(c+d x)+3 a^3 \sec ^3(c+d x)+3 a^3 \sec ^4(c+d x)+a^3 \sec ^5(c+d x)\right ) \, dx\\ &=a^3 \int \sec ^2(c+d x) \, dx+a^3 \int \sec ^5(c+d x) \, dx+\left (3 a^3\right ) \int \sec ^3(c+d x) \, dx+\left (3 a^3\right ) \int \sec ^4(c+d x) \, dx\\ &=\frac {3 a^3 \sec (c+d x) \tan (c+d x)}{2 d}+\frac {a^3 \sec ^3(c+d x) \tan (c+d x)}{4 d}+\frac {1}{4} \left (3 a^3\right ) \int \sec ^3(c+d x) \, dx+\frac {1}{2} \left (3 a^3\right ) \int \sec (c+d x) \, dx-\frac {a^3 \operatorname {Subst}(\int 1 \, dx,x,-\tan (c+d x))}{d}-\frac {\left (3 a^3\right ) \operatorname {Subst}\left (\int \left (1+x^2\right ) \, dx,x,-\tan (c+d x)\right )}{d}\\ &=\frac {3 a^3 \tanh ^{-1}(\sin (c+d x))}{2 d}+\frac {4 a^3 \tan (c+d x)}{d}+\frac {15 a^3 \sec (c+d x) \tan (c+d x)}{8 d}+\frac {a^3 \sec ^3(c+d x) \tan (c+d x)}{4 d}+\frac {a^3 \tan ^3(c+d x)}{d}+\frac {1}{8} \left (3 a^3\right ) \int \sec (c+d x) \, dx\\ &=\frac {15 a^3 \tanh ^{-1}(\sin (c+d x))}{8 d}+\frac {4 a^3 \tan (c+d x)}{d}+\frac {15 a^3 \sec (c+d x) \tan (c+d x)}{8 d}+\frac {a^3 \sec ^3(c+d x) \tan (c+d x)}{4 d}+\frac {a^3 \tan ^3(c+d x)}{d}\\ \end {align*}

________________________________________________________________________________________

Mathematica [B]  time = 6.39, size = 797, normalized size = 8.57 \[ -\frac {15 (\cos (c+d x) a+a)^3 \log \left (\cos \left (\frac {c}{2}+\frac {d x}{2}\right )-\sin \left (\frac {c}{2}+\frac {d x}{2}\right )\right ) \sec ^6\left (\frac {c}{2}+\frac {d x}{2}\right )}{64 d}+\frac {15 (\cos (c+d x) a+a)^3 \log \left (\cos \left (\frac {c}{2}+\frac {d x}{2}\right )+\sin \left (\frac {c}{2}+\frac {d x}{2}\right )\right ) \sec ^6\left (\frac {c}{2}+\frac {d x}{2}\right )}{64 d}+\frac {3 (\cos (c+d x) a+a)^3 \sin \left (\frac {d x}{2}\right ) \sec ^6\left (\frac {c}{2}+\frac {d x}{2}\right )}{8 d \left (\cos \left (\frac {c}{2}\right )-\sin \left (\frac {c}{2}\right )\right ) \left (\cos \left (\frac {c}{2}+\frac {d x}{2}\right )-\sin \left (\frac {c}{2}+\frac {d x}{2}\right )\right )}+\frac {3 (\cos (c+d x) a+a)^3 \sin \left (\frac {d x}{2}\right ) \sec ^6\left (\frac {c}{2}+\frac {d x}{2}\right )}{8 d \left (\cos \left (\frac {c}{2}\right )+\sin \left (\frac {c}{2}\right )\right ) \left (\cos \left (\frac {c}{2}+\frac {d x}{2}\right )+\sin \left (\frac {c}{2}+\frac {d x}{2}\right )\right )}+\frac {(\cos (c+d x) a+a)^3 \left (19 \cos \left (\frac {c}{2}\right )-11 \sin \left (\frac {c}{2}\right )\right ) \sec ^6\left (\frac {c}{2}+\frac {d x}{2}\right )}{128 d \left (\cos \left (\frac {c}{2}\right )-\sin \left (\frac {c}{2}\right )\right ) \left (\cos \left (\frac {c}{2}+\frac {d x}{2}\right )-\sin \left (\frac {c}{2}+\frac {d x}{2}\right )\right )^2}+\frac {(\cos (c+d x) a+a)^3 \left (-19 \cos \left (\frac {c}{2}\right )-11 \sin \left (\frac {c}{2}\right )\right ) \sec ^6\left (\frac {c}{2}+\frac {d x}{2}\right )}{128 d \left (\cos \left (\frac {c}{2}\right )+\sin \left (\frac {c}{2}\right )\right ) \left (\cos \left (\frac {c}{2}+\frac {d x}{2}\right )+\sin \left (\frac {c}{2}+\frac {d x}{2}\right )\right )^2}+\frac {(\cos (c+d x) a+a)^3 \sin \left (\frac {d x}{2}\right ) \sec ^6\left (\frac {c}{2}+\frac {d x}{2}\right )}{16 d \left (\cos \left (\frac {c}{2}\right )-\sin \left (\frac {c}{2}\right )\right ) \left (\cos \left (\frac {c}{2}+\frac {d x}{2}\right )-\sin \left (\frac {c}{2}+\frac {d x}{2}\right )\right )^3}+\frac {(\cos (c+d x) a+a)^3 \sin \left (\frac {d x}{2}\right ) \sec ^6\left (\frac {c}{2}+\frac {d x}{2}\right )}{16 d \left (\cos \left (\frac {c}{2}\right )+\sin \left (\frac {c}{2}\right )\right ) \left (\cos \left (\frac {c}{2}+\frac {d x}{2}\right )+\sin \left (\frac {c}{2}+\frac {d x}{2}\right )\right )^3}+\frac {(\cos (c+d x) a+a)^3 \sec ^6\left (\frac {c}{2}+\frac {d x}{2}\right )}{128 d \left (\cos \left (\frac {c}{2}+\frac {d x}{2}\right )-\sin \left (\frac {c}{2}+\frac {d x}{2}\right )\right )^4}-\frac {(\cos (c+d x) a+a)^3 \sec ^6\left (\frac {c}{2}+\frac {d x}{2}\right )}{128 d \left (\cos \left (\frac {c}{2}+\frac {d x}{2}\right )+\sin \left (\frac {c}{2}+\frac {d x}{2}\right )\right )^4} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[(a + a*Cos[c + d*x])^3*Sec[c + d*x]^5,x]

[Out]

(-15*(a + a*Cos[c + d*x])^3*Log[Cos[c/2 + (d*x)/2] - Sin[c/2 + (d*x)/2]]*Sec[c/2 + (d*x)/2]^6)/(64*d) + (15*(a
+ a*Cos[c + d*x])^3*Log[Cos[c/2 + (d*x)/2] + Sin[c/2 + (d*x)/2]]*Sec[c/2 + (d*x)/2]^6)/(64*d) + ((a + a*Cos[c
+ d*x])^3*Sec[c/2 + (d*x)/2]^6)/(128*d*(Cos[c/2 + (d*x)/2] - Sin[c/2 + (d*x)/2])^4) + ((a + a*Cos[c + d*x])^3
*Sec[c/2 + (d*x)/2]^6*Sin[(d*x)/2])/(16*d*(Cos[c/2] - Sin[c/2])*(Cos[c/2 + (d*x)/2] - Sin[c/2 + (d*x)/2])^3) +
((a + a*Cos[c + d*x])^3*Sec[c/2 + (d*x)/2]^6*(19*Cos[c/2] - 11*Sin[c/2]))/(128*d*(Cos[c/2] - Sin[c/2])*(Cos[c
/2 + (d*x)/2] - Sin[c/2 + (d*x)/2])^2) + (3*(a + a*Cos[c + d*x])^3*Sec[c/2 + (d*x)/2]^6*Sin[(d*x)/2])/(8*d*(Co
s[c/2] - Sin[c/2])*(Cos[c/2 + (d*x)/2] - Sin[c/2 + (d*x)/2])) - ((a + a*Cos[c + d*x])^3*Sec[c/2 + (d*x)/2]^6)/
(128*d*(Cos[c/2 + (d*x)/2] + Sin[c/2 + (d*x)/2])^4) + ((a + a*Cos[c + d*x])^3*Sec[c/2 + (d*x)/2]^6*Sin[(d*x)/2
])/(16*d*(Cos[c/2] + Sin[c/2])*(Cos[c/2 + (d*x)/2] + Sin[c/2 + (d*x)/2])^3) + ((a + a*Cos[c + d*x])^3*Sec[c/2
+ (d*x)/2]^6*(-19*Cos[c/2] - 11*Sin[c/2]))/(128*d*(Cos[c/2] + Sin[c/2])*(Cos[c/2 + (d*x)/2] + Sin[c/2 + (d*x)/
2])^2) + (3*(a + a*Cos[c + d*x])^3*Sec[c/2 + (d*x)/2]^6*Sin[(d*x)/2])/(8*d*(Cos[c/2] + Sin[c/2])*(Cos[c/2 + (d
*x)/2] + Sin[c/2 + (d*x)/2]))

________________________________________________________________________________________

fricas [A]  time = 1.03, size = 111, normalized size = 1.19 \[ \frac {15 \, a^{3} \cos \left (d x + c\right )^{4} \log \left (\sin \left (d x + c\right ) + 1\right ) - 15 \, a^{3} \cos \left (d x + c\right )^{4} \log \left (-\sin \left (d x + c\right ) + 1\right ) + 2 \, {\left (24 \, a^{3} \cos \left (d x + c\right )^{3} + 15 \, a^{3} \cos \left (d x + c\right )^{2} + 8 \, a^{3} \cos \left (d x + c\right ) + 2 \, a^{3}\right )} \sin \left (d x + c\right )}{16 \, d \cos \left (d x + c\right )^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+a*cos(d*x+c))^3*sec(d*x+c)^5,x, algorithm="fricas")

[Out]

1/16*(15*a^3*cos(d*x + c)^4*log(sin(d*x + c) + 1) - 15*a^3*cos(d*x + c)^4*log(-sin(d*x + c) + 1) + 2*(24*a^3*c
os(d*x + c)^3 + 15*a^3*cos(d*x + c)^2 + 8*a^3*cos(d*x + c) + 2*a^3)*sin(d*x + c))/(d*cos(d*x + c)^4)

________________________________________________________________________________________

giac [A]  time = 0.83, size = 122, normalized size = 1.31 \[ \frac {15 \, a^{3} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1 \right |}\right ) - 15 \, a^{3} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1 \right |}\right ) - \frac {2 \, {\left (15 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 55 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 73 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 49 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 1\right )}^{4}}}{8 \, d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+a*cos(d*x+c))^3*sec(d*x+c)^5,x, algorithm="giac")

[Out]

1/8*(15*a^3*log(abs(tan(1/2*d*x + 1/2*c) + 1)) - 15*a^3*log(abs(tan(1/2*d*x + 1/2*c) - 1)) - 2*(15*a^3*tan(1/2
*d*x + 1/2*c)^7 - 55*a^3*tan(1/2*d*x + 1/2*c)^5 + 73*a^3*tan(1/2*d*x + 1/2*c)^3 - 49*a^3*tan(1/2*d*x + 1/2*c))
/(tan(1/2*d*x + 1/2*c)^2 - 1)^4)/d

________________________________________________________________________________________

maple [A]  time = 0.13, size = 101, normalized size = 1.09 \[ \frac {3 a^{3} \tan \left (d x +c \right )}{d}+\frac {15 a^{3} \sec \left (d x +c \right ) \tan \left (d x +c \right )}{8 d}+\frac {15 a^{3} \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{8 d}+\frac {a^{3} \tan \left (d x +c \right ) \left (\sec ^{2}\left (d x +c \right )\right )}{d}+\frac {a^{3} \left (\sec ^{3}\left (d x +c \right )\right ) \tan \left (d x +c \right )}{4 d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+a*cos(d*x+c))^3*sec(d*x+c)^5,x)

[Out]

3*a^3*tan(d*x+c)/d+15/8*a^3*sec(d*x+c)*tan(d*x+c)/d+15/8/d*a^3*ln(sec(d*x+c)+tan(d*x+c))+1/d*a^3*tan(d*x+c)*se
c(d*x+c)^2+1/4*a^3*sec(d*x+c)^3*tan(d*x+c)/d

________________________________________________________________________________________

maxima [A]  time = 1.10, size = 156, normalized size = 1.68 \[ \frac {16 \, {\left (\tan \left (d x + c\right )^{3} + 3 \, \tan \left (d x + c\right )\right )} a^{3} - a^{3} {\left (\frac {2 \, {\left (3 \, \sin \left (d x + c\right )^{3} - 5 \, \sin \left (d x + c\right )\right )}}{\sin \left (d x + c\right )^{4} - 2 \, \sin \left (d x + c\right )^{2} + 1} - 3 \, \log \left (\sin \left (d x + c\right ) + 1\right ) + 3 \, \log \left (\sin \left (d x + c\right ) - 1\right )\right )} - 12 \, a^{3} {\left (\frac {2 \, \sin \left (d x + c\right )}{\sin \left (d x + c\right )^{2} - 1} - \log \left (\sin \left (d x + c\right ) + 1\right ) + \log \left (\sin \left (d x + c\right ) - 1\right )\right )} + 16 \, a^{3} \tan \left (d x + c\right )}{16 \, d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+a*cos(d*x+c))^3*sec(d*x+c)^5,x, algorithm="maxima")

[Out]

1/16*(16*(tan(d*x + c)^3 + 3*tan(d*x + c))*a^3 - a^3*(2*(3*sin(d*x + c)^3 - 5*sin(d*x + c))/(sin(d*x + c)^4 -
2*sin(d*x + c)^2 + 1) - 3*log(sin(d*x + c) + 1) + 3*log(sin(d*x + c) - 1)) - 12*a^3*(2*sin(d*x + c)/(sin(d*x +
c)^2 - 1) - log(sin(d*x + c) + 1) + log(sin(d*x + c) - 1)) + 16*a^3*tan(d*x + c))/d

________________________________________________________________________________________

mupad [B]  time = 3.29, size = 141, normalized size = 1.52 \[ \frac {15\,a^3\,\mathrm {atanh}\left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )}{4\,d}-\frac {\frac {15\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7}{4}-\frac {55\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5}{4}+\frac {73\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{4}-\frac {49\,a^3\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{4}}{d\,\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8-4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6+6\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4-4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + a*cos(c + d*x))^3/cos(c + d*x)^5,x)

[Out]

(15*a^3*atanh(tan(c/2 + (d*x)/2)))/(4*d) - ((73*a^3*tan(c/2 + (d*x)/2)^3)/4 - (55*a^3*tan(c/2 + (d*x)/2)^5)/4
+ (15*a^3*tan(c/2 + (d*x)/2)^7)/4 - (49*a^3*tan(c/2 + (d*x)/2))/4)/(d*(6*tan(c/2 + (d*x)/2)^4 - 4*tan(c/2 + (d
*x)/2)^2 - 4*tan(c/2 + (d*x)/2)^6 + tan(c/2 + (d*x)/2)^8 + 1))

________________________________________________________________________________________

sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+a*cos(d*x+c))**3*sec(d*x+c)**5,x)

[Out]

Timed out

________________________________________________________________________________________