3.1.47 \(\int (a+a \sec (c+d x))^3 \tan ^6(c+d x) \, dx\) [47]

Optimal. Leaf size=237 \[ -a^3 x-\frac {125 a^3 \tanh ^{-1}(\sin (c+d x))}{128 d}+\frac {a^3 \tan (c+d x)}{d}+\frac {115 a^3 \sec (c+d x) \tan (c+d x)}{128 d}+\frac {5 a^3 \sec ^3(c+d x) \tan (c+d x)}{64 d}-\frac {a^3 \tan ^3(c+d x)}{3 d}-\frac {5 a^3 \sec (c+d x) \tan ^3(c+d x)}{8 d}-\frac {5 a^3 \sec ^3(c+d x) \tan ^3(c+d x)}{48 d}+\frac {a^3 \tan ^5(c+d x)}{5 d}+\frac {a^3 \sec (c+d x) \tan ^5(c+d x)}{2 d}+\frac {a^3 \sec ^3(c+d x) \tan ^5(c+d x)}{8 d}+\frac {3 a^3 \tan ^7(c+d x)}{7 d} \]

[Out]

-a^3*x-125/128*a^3*arctanh(sin(d*x+c))/d+a^3*tan(d*x+c)/d+115/128*a^3*sec(d*x+c)*tan(d*x+c)/d+5/64*a^3*sec(d*x
+c)^3*tan(d*x+c)/d-1/3*a^3*tan(d*x+c)^3/d-5/8*a^3*sec(d*x+c)*tan(d*x+c)^3/d-5/48*a^3*sec(d*x+c)^3*tan(d*x+c)^3
/d+1/5*a^3*tan(d*x+c)^5/d+1/2*a^3*sec(d*x+c)*tan(d*x+c)^5/d+1/8*a^3*sec(d*x+c)^3*tan(d*x+c)^5/d+3/7*a^3*tan(d*
x+c)^7/d

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Rubi [A]
time = 0.23, antiderivative size = 237, normalized size of antiderivative = 1.00, number of steps used = 17, number of rules used = 8, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.381, Rules used = {3971, 3554, 8, 2691, 3855, 2687, 30, 3853} \begin {gather*} \frac {3 a^3 \tan ^7(c+d x)}{7 d}+\frac {a^3 \tan ^5(c+d x)}{5 d}-\frac {a^3 \tan ^3(c+d x)}{3 d}+\frac {a^3 \tan (c+d x)}{d}-\frac {125 a^3 \tanh ^{-1}(\sin (c+d x))}{128 d}+\frac {a^3 \tan ^5(c+d x) \sec ^3(c+d x)}{8 d}-\frac {5 a^3 \tan ^3(c+d x) \sec ^3(c+d x)}{48 d}+\frac {5 a^3 \tan (c+d x) \sec ^3(c+d x)}{64 d}+\frac {a^3 \tan ^5(c+d x) \sec (c+d x)}{2 d}-\frac {5 a^3 \tan ^3(c+d x) \sec (c+d x)}{8 d}+\frac {115 a^3 \tan (c+d x) \sec (c+d x)}{128 d}-a^3 x \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a + a*Sec[c + d*x])^3*Tan[c + d*x]^6,x]

[Out]

-(a^3*x) - (125*a^3*ArcTanh[Sin[c + d*x]])/(128*d) + (a^3*Tan[c + d*x])/d + (115*a^3*Sec[c + d*x]*Tan[c + d*x]
)/(128*d) + (5*a^3*Sec[c + d*x]^3*Tan[c + d*x])/(64*d) - (a^3*Tan[c + d*x]^3)/(3*d) - (5*a^3*Sec[c + d*x]*Tan[
c + d*x]^3)/(8*d) - (5*a^3*Sec[c + d*x]^3*Tan[c + d*x]^3)/(48*d) + (a^3*Tan[c + d*x]^5)/(5*d) + (a^3*Sec[c + d
*x]*Tan[c + d*x]^5)/(2*d) + (a^3*Sec[c + d*x]^3*Tan[c + d*x]^5)/(8*d) + (3*a^3*Tan[c + d*x]^7)/(7*d)

Rule 8

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

Rule 30

Int[(x_)^(m_.), x_Symbol] :> Simp[x^(m + 1)/(m + 1), x] /; FreeQ[m, x] && NeQ[m, -1]

Rule 2687

Int[sec[(e_.) + (f_.)*(x_)]^(m_)*((b_.)*tan[(e_.) + (f_.)*(x_)])^(n_.), x_Symbol] :> Dist[1/f, Subst[Int[(b*x)
^n*(1 + x^2)^(m/2 - 1), x], x, Tan[e + f*x]], x] /; FreeQ[{b, e, f, n}, x] && IntegerQ[m/2] &&  !(IntegerQ[(n
- 1)/2] && LtQ[0, n, m - 1])

Rule 2691

Int[((a_.)*sec[(e_.) + (f_.)*(x_)])^(m_.)*((b_.)*tan[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Simp[b*(a*Sec[e +
f*x])^m*((b*Tan[e + f*x])^(n - 1)/(f*(m + n - 1))), x] - Dist[b^2*((n - 1)/(m + n - 1)), Int[(a*Sec[e + f*x])
^m*(b*Tan[e + f*x])^(n - 2), x], x] /; FreeQ[{a, b, e, f, m}, x] && GtQ[n, 1] && NeQ[m + n - 1, 0] && Integers
Q[2*m, 2*n]

Rule 3554

Int[((b_.)*tan[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[b*((b*Tan[c + d*x])^(n - 1)/(d*(n - 1))), x] - Dis
t[b^2, Int[(b*Tan[c + d*x])^(n - 2), x], x] /; FreeQ[{b, c, d}, x] && GtQ[n, 1]

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]

Rule 3971

Int[(cot[(c_.) + (d_.)*(x_)]*(e_.))^(m_)*(csc[(c_.) + (d_.)*(x_)]*(b_.) + (a_))^(n_), x_Symbol] :> Int[ExpandI
ntegrand[(e*Cot[c + d*x])^m, (a + b*Csc[c + d*x])^n, x], x] /; FreeQ[{a, b, c, d, e, m}, x] && IGtQ[n, 0]

Rubi steps

\begin {align*} \int (a+a \sec (c+d x))^3 \tan ^6(c+d x) \, dx &=\int \left (a^3 \tan ^6(c+d x)+3 a^3 \sec (c+d x) \tan ^6(c+d x)+3 a^3 \sec ^2(c+d x) \tan ^6(c+d x)+a^3 \sec ^3(c+d x) \tan ^6(c+d x)\right ) \, dx\\ &=a^3 \int \tan ^6(c+d x) \, dx+a^3 \int \sec ^3(c+d x) \tan ^6(c+d x) \, dx+\left (3 a^3\right ) \int \sec (c+d x) \tan ^6(c+d x) \, dx+\left (3 a^3\right ) \int \sec ^2(c+d x) \tan ^6(c+d x) \, dx\\ &=\frac {a^3 \tan ^5(c+d x)}{5 d}+\frac {a^3 \sec (c+d x) \tan ^5(c+d x)}{2 d}+\frac {a^3 \sec ^3(c+d x) \tan ^5(c+d x)}{8 d}-\frac {1}{8} \left (5 a^3\right ) \int \sec ^3(c+d x) \tan ^4(c+d x) \, dx-a^3 \int \tan ^4(c+d x) \, dx-\frac {1}{2} \left (5 a^3\right ) \int \sec (c+d x) \tan ^4(c+d x) \, dx+\frac {\left (3 a^3\right ) \text {Subst}\left (\int x^6 \, dx,x,\tan (c+d x)\right )}{d}\\ &=-\frac {a^3 \tan ^3(c+d x)}{3 d}-\frac {5 a^3 \sec (c+d x) \tan ^3(c+d x)}{8 d}-\frac {5 a^3 \sec ^3(c+d x) \tan ^3(c+d x)}{48 d}+\frac {a^3 \tan ^5(c+d x)}{5 d}+\frac {a^3 \sec (c+d x) \tan ^5(c+d x)}{2 d}+\frac {a^3 \sec ^3(c+d x) \tan ^5(c+d x)}{8 d}+\frac {3 a^3 \tan ^7(c+d x)}{7 d}+\frac {1}{16} \left (5 a^3\right ) \int \sec ^3(c+d x) \tan ^2(c+d x) \, dx+a^3 \int \tan ^2(c+d x) \, dx+\frac {1}{8} \left (15 a^3\right ) \int \sec (c+d x) \tan ^2(c+d x) \, dx\\ &=\frac {a^3 \tan (c+d x)}{d}+\frac {15 a^3 \sec (c+d x) \tan (c+d x)}{16 d}+\frac {5 a^3 \sec ^3(c+d x) \tan (c+d x)}{64 d}-\frac {a^3 \tan ^3(c+d x)}{3 d}-\frac {5 a^3 \sec (c+d x) \tan ^3(c+d x)}{8 d}-\frac {5 a^3 \sec ^3(c+d x) \tan ^3(c+d x)}{48 d}+\frac {a^3 \tan ^5(c+d x)}{5 d}+\frac {a^3 \sec (c+d x) \tan ^5(c+d x)}{2 d}+\frac {a^3 \sec ^3(c+d x) \tan ^5(c+d x)}{8 d}+\frac {3 a^3 \tan ^7(c+d x)}{7 d}-\frac {1}{64} \left (5 a^3\right ) \int \sec ^3(c+d x) \, dx-\frac {1}{16} \left (15 a^3\right ) \int \sec (c+d x) \, dx-a^3 \int 1 \, dx\\ &=-a^3 x-\frac {15 a^3 \tanh ^{-1}(\sin (c+d x))}{16 d}+\frac {a^3 \tan (c+d x)}{d}+\frac {115 a^3 \sec (c+d x) \tan (c+d x)}{128 d}+\frac {5 a^3 \sec ^3(c+d x) \tan (c+d x)}{64 d}-\frac {a^3 \tan ^3(c+d x)}{3 d}-\frac {5 a^3 \sec (c+d x) \tan ^3(c+d x)}{8 d}-\frac {5 a^3 \sec ^3(c+d x) \tan ^3(c+d x)}{48 d}+\frac {a^3 \tan ^5(c+d x)}{5 d}+\frac {a^3 \sec (c+d x) \tan ^5(c+d x)}{2 d}+\frac {a^3 \sec ^3(c+d x) \tan ^5(c+d x)}{8 d}+\frac {3 a^3 \tan ^7(c+d x)}{7 d}-\frac {1}{128} \left (5 a^3\right ) \int \sec (c+d x) \, dx\\ &=-a^3 x-\frac {125 a^3 \tanh ^{-1}(\sin (c+d x))}{128 d}+\frac {a^3 \tan (c+d x)}{d}+\frac {115 a^3 \sec (c+d x) \tan (c+d x)}{128 d}+\frac {5 a^3 \sec ^3(c+d x) \tan (c+d x)}{64 d}-\frac {a^3 \tan ^3(c+d x)}{3 d}-\frac {5 a^3 \sec (c+d x) \tan ^3(c+d x)}{8 d}-\frac {5 a^3 \sec ^3(c+d x) \tan ^3(c+d x)}{48 d}+\frac {a^3 \tan ^5(c+d x)}{5 d}+\frac {a^3 \sec (c+d x) \tan ^5(c+d x)}{2 d}+\frac {a^3 \sec ^3(c+d x) \tan ^5(c+d x)}{8 d}+\frac {3 a^3 \tan ^7(c+d x)}{7 d}\\ \end {align*}

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Mathematica [A]
time = 2.13, size = 363, normalized size = 1.53 \begin {gather*} \frac {a^3 (1+\cos (c+d x))^3 \sec ^6\left (\frac {1}{2} (c+d x)\right ) \sec ^8(c+d x) \left (1680000 \cos ^8(c+d x) \left (\log \left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )-\log \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )\right )-\sec (c) (470400 d x \cos (c)+376320 d x \cos (c+2 d x)+376320 d x \cos (3 c+2 d x)+188160 d x \cos (3 c+4 d x)+188160 d x \cos (5 c+4 d x)+53760 d x \cos (5 c+6 d x)+53760 d x \cos (7 c+6 d x)+6720 d x \cos (7 c+8 d x)+6720 d x \cos (9 c+8 d x)+519680 \sin (c)-133175 \sin (d x)-133175 \sin (2 c+d x)-544768 \sin (c+2 d x)+286720 \sin (3 c+2 d x)-63595 \sin (2 c+3 d x)-63595 \sin (4 c+3 d x)-254464 \sin (3 c+4 d x)+161280 \sin (5 c+4 d x)-65135 \sin (4 c+5 d x)-65135 \sin (6 c+5 d x)-118784 \sin (5 c+6 d x)-27195 \sin (6 c+7 d x)-27195 \sin (8 c+7 d x)-14848 \sin (7 c+8 d x))\right )}{13762560 d} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(a + a*Sec[c + d*x])^3*Tan[c + d*x]^6,x]

[Out]

(a^3*(1 + Cos[c + d*x])^3*Sec[(c + d*x)/2]^6*Sec[c + d*x]^8*(1680000*Cos[c + d*x]^8*(Log[Cos[(c + d*x)/2] - Si
n[(c + d*x)/2]] - Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]]) - Sec[c]*(470400*d*x*Cos[c] + 376320*d*x*Cos[c + 2
*d*x] + 376320*d*x*Cos[3*c + 2*d*x] + 188160*d*x*Cos[3*c + 4*d*x] + 188160*d*x*Cos[5*c + 4*d*x] + 53760*d*x*Co
s[5*c + 6*d*x] + 53760*d*x*Cos[7*c + 6*d*x] + 6720*d*x*Cos[7*c + 8*d*x] + 6720*d*x*Cos[9*c + 8*d*x] + 519680*S
in[c] - 133175*Sin[d*x] - 133175*Sin[2*c + d*x] - 544768*Sin[c + 2*d*x] + 286720*Sin[3*c + 2*d*x] - 63595*Sin[
2*c + 3*d*x] - 63595*Sin[4*c + 3*d*x] - 254464*Sin[3*c + 4*d*x] + 161280*Sin[5*c + 4*d*x] - 65135*Sin[4*c + 5*
d*x] - 65135*Sin[6*c + 5*d*x] - 118784*Sin[5*c + 6*d*x] - 27195*Sin[6*c + 7*d*x] - 27195*Sin[8*c + 7*d*x] - 14
848*Sin[7*c + 8*d*x])))/(13762560*d)

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Maple [A]
time = 0.14, size = 290, normalized size = 1.22

method	result	size
risch	\(-a^{3} x -\frac {i a^{3} \left (27195 \,{\mathrm e}^{15 i \left (d x +c \right )}+65135 \,{\mathrm e}^{13 i \left (d x +c \right )}-161280 \,{\mathrm e}^{12 i \left (d x +c \right )}+63595 \,{\mathrm e}^{11 i \left (d x +c \right )}-286720 \,{\mathrm e}^{10 i \left (d x +c \right )}+133175 \,{\mathrm e}^{9 i \left (d x +c \right )}-519680 \,{\mathrm e}^{8 i \left (d x +c \right )}-133175 \,{\mathrm e}^{7 i \left (d x +c \right )}-544768 \,{\mathrm e}^{6 i \left (d x +c \right )}-63595 \,{\mathrm e}^{5 i \left (d x +c \right )}-254464 \,{\mathrm e}^{4 i \left (d x +c \right )}-65135 \,{\mathrm e}^{3 i \left (d x +c \right )}-118784 \,{\mathrm e}^{2 i \left (d x +c \right )}-27195 \,{\mathrm e}^{i \left (d x +c \right )}-14848\right )}{6720 d \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )^{8}}-\frac {125 a^{3} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right )}{128 d}+\frac {125 a^{3} \ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right )}{128 d}\)	\(228\)
derivativedivides	\(\frac {a^{3} \left (\frac {\sin ^{7}\left (d x +c \right )}{8 \cos \left (d x +c \right )^{8}}+\frac {\sin ^{7}\left (d x +c \right )}{48 \cos \left (d x +c \right )^{6}}-\frac {\sin ^{7}\left (d x +c \right )}{192 \cos \left (d x +c \right )^{4}}+\frac {\sin ^{7}\left (d x +c \right )}{128 \cos \left (d x +c \right )^{2}}+\frac {\left (\sin ^{5}\left (d x +c \right )\right )}{128}+\frac {5 \left (\sin ^{3}\left (d x +c \right )\right )}{384}+\frac {5 \sin \left (d x +c \right )}{128}-\frac {5 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{128}\right )+\frac {3 a^{3} \left (\sin ^{7}\left (d x +c \right )\right )}{7 \cos \left (d x +c \right )^{7}}+3 a^{3} \left (\frac {\sin ^{7}\left (d x +c \right )}{6 \cos \left (d x +c \right )^{6}}-\frac {\sin ^{7}\left (d x +c \right )}{24 \cos \left (d x +c \right )^{4}}+\frac {\sin ^{7}\left (d x +c \right )}{16 \cos \left (d x +c \right )^{2}}+\frac {\left (\sin ^{5}\left (d x +c \right )\right )}{16}+\frac {5 \left (\sin ^{3}\left (d x +c \right )\right )}{48}+\frac {5 \sin \left (d x +c \right )}{16}-\frac {5 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{16}\right )+a^{3} \left (\frac {\left (\tan ^{5}\left (d x +c \right )\right )}{5}-\frac {\left (\tan ^{3}\left (d x +c \right )\right )}{3}+\tan \left (d x +c \right )-d x -c \right )}{d}\)	\(290\)
default	\(\frac {a^{3} \left (\frac {\sin ^{7}\left (d x +c \right )}{8 \cos \left (d x +c \right )^{8}}+\frac {\sin ^{7}\left (d x +c \right )}{48 \cos \left (d x +c \right )^{6}}-\frac {\sin ^{7}\left (d x +c \right )}{192 \cos \left (d x +c \right )^{4}}+\frac {\sin ^{7}\left (d x +c \right )}{128 \cos \left (d x +c \right )^{2}}+\frac {\left (\sin ^{5}\left (d x +c \right )\right )}{128}+\frac {5 \left (\sin ^{3}\left (d x +c \right )\right )}{384}+\frac {5 \sin \left (d x +c \right )}{128}-\frac {5 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{128}\right )+\frac {3 a^{3} \left (\sin ^{7}\left (d x +c \right )\right )}{7 \cos \left (d x +c \right )^{7}}+3 a^{3} \left (\frac {\sin ^{7}\left (d x +c \right )}{6 \cos \left (d x +c \right )^{6}}-\frac {\sin ^{7}\left (d x +c \right )}{24 \cos \left (d x +c \right )^{4}}+\frac {\sin ^{7}\left (d x +c \right )}{16 \cos \left (d x +c \right )^{2}}+\frac {\left (\sin ^{5}\left (d x +c \right )\right )}{16}+\frac {5 \left (\sin ^{3}\left (d x +c \right )\right )}{48}+\frac {5 \sin \left (d x +c \right )}{16}-\frac {5 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{16}\right )+a^{3} \left (\frac {\left (\tan ^{5}\left (d x +c \right )\right )}{5}-\frac {\left (\tan ^{3}\left (d x +c \right )\right )}{3}+\tan \left (d x +c \right )-d x -c \right )}{d}\)	\(290\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+a*sec(d*x+c))^3*tan(d*x+c)^6,x,method=_RETURNVERBOSE)

[Out]

1/d*(a^3*(1/8*sin(d*x+c)^7/cos(d*x+c)^8+1/48*sin(d*x+c)^7/cos(d*x+c)^6-1/192*sin(d*x+c)^7/cos(d*x+c)^4+1/128*s
in(d*x+c)^7/cos(d*x+c)^2+1/128*sin(d*x+c)^5+5/384*sin(d*x+c)^3+5/128*sin(d*x+c)-5/128*ln(sec(d*x+c)+tan(d*x+c)
))+3/7*a^3*sin(d*x+c)^7/cos(d*x+c)^7+3*a^3*(1/6*sin(d*x+c)^7/cos(d*x+c)^6-1/24*sin(d*x+c)^7/cos(d*x+c)^4+1/16*
sin(d*x+c)^7/cos(d*x+c)^2+1/16*sin(d*x+c)^5+5/48*sin(d*x+c)^3+5/16*sin(d*x+c)-5/16*ln(sec(d*x+c)+tan(d*x+c)))+
a^3*(1/5*tan(d*x+c)^5-1/3*tan(d*x+c)^3+tan(d*x+c)-d*x-c))

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Maxima [A]
time = 0.50, size = 262, normalized size = 1.11 \begin {gather*} \frac {11520 \, a^{3} \tan \left (d x + c\right )^{7} + 1792 \, {\left (3 \, \tan \left (d x + c\right )^{5} - 5 \, \tan \left (d x + c\right )^{3} - 15 \, d x - 15 \, c + 15 \, \tan \left (d x + c\right )\right )} a^{3} + 35 \, a^{3} {\left (\frac {2 \, {\left (15 \, \sin \left (d x + c\right )^{7} + 73 \, \sin \left (d x + c\right )^{5} - 55 \, \sin \left (d x + c\right )^{3} + 15 \, \sin \left (d x + c\right )\right )}}{\sin \left (d x + c\right )^{8} - 4 \, \sin \left (d x + c\right )^{6} + 6 \, \sin \left (d x + c\right )^{4} - 4 \, \sin \left (d x + c\right )^{2} + 1} - 15 \, \log \left (\sin \left (d x + c\right ) + 1\right ) + 15 \, \log \left (\sin \left (d x + c\right ) - 1\right )\right )} - 840 \, a^{3} {\left (\frac {2 \, {\left (33 \, \sin \left (d x + c\right )^{5} - 40 \, \sin \left (d x + c\right )^{3} + 15 \, \sin \left (d x + c\right )\right )}}{\sin \left (d x + c\right )^{6} - 3 \, \sin \left (d x + c\right )^{4} + 3 \, \sin \left (d x + c\right )^{2} - 1} + 15 \, \log \left (\sin \left (d x + c\right ) + 1\right ) - 15 \, \log \left (\sin \left (d x + c\right ) - 1\right )\right )}}{26880 \, d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+a*sec(d*x+c))^3*tan(d*x+c)^6,x, algorithm="maxima")

[Out]

1/26880*(11520*a^3*tan(d*x + c)^7 + 1792*(3*tan(d*x + c)^5 - 5*tan(d*x + c)^3 - 15*d*x - 15*c + 15*tan(d*x + c
))*a^3 + 35*a^3*(2*(15*sin(d*x + c)^7 + 73*sin(d*x + c)^5 - 55*sin(d*x + c)^3 + 15*sin(d*x + c))/(sin(d*x + c)
^8 - 4*sin(d*x + c)^6 + 6*sin(d*x + c)^4 - 4*sin(d*x + c)^2 + 1) - 15*log(sin(d*x + c) + 1) + 15*log(sin(d*x +
c) - 1)) - 840*a^3*(2*(33*sin(d*x + c)^5 - 40*sin(d*x + c)^3 + 15*sin(d*x + c))/(sin(d*x + c)^6 - 3*sin(d*x +
c)^4 + 3*sin(d*x + c)^2 - 1) + 15*log(sin(d*x + c) + 1) - 15*log(sin(d*x + c) - 1)))/d

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Fricas [A]
time = 2.61, size = 178, normalized size = 0.75 \begin {gather*} -\frac {26880 \, a^{3} d x \cos \left (d x + c\right )^{8} + 13125 \, a^{3} \cos \left (d x + c\right )^{8} \log \left (\sin \left (d x + c\right ) + 1\right ) - 13125 \, a^{3} \cos \left (d x + c\right )^{8} \log \left (-\sin \left (d x + c\right ) + 1\right ) - 2 \, {\left (14848 \, a^{3} \cos \left (d x + c\right )^{7} + 27195 \, a^{3} \cos \left (d x + c\right )^{6} + 7424 \, a^{3} \cos \left (d x + c\right )^{5} - 17710 \, a^{3} \cos \left (d x + c\right )^{4} - 14592 \, a^{3} \cos \left (d x + c\right )^{3} + 1960 \, a^{3} \cos \left (d x + c\right )^{2} + 5760 \, a^{3} \cos \left (d x + c\right ) + 1680 \, a^{3}\right )} \sin \left (d x + c\right )}{26880 \, d \cos \left (d x + c\right )^{8}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+a*sec(d*x+c))^3*tan(d*x+c)^6,x, algorithm="fricas")

[Out]

-1/26880*(26880*a^3*d*x*cos(d*x + c)^8 + 13125*a^3*cos(d*x + c)^8*log(sin(d*x + c) + 1) - 13125*a^3*cos(d*x +
c)^8*log(-sin(d*x + c) + 1) - 2*(14848*a^3*cos(d*x + c)^7 + 27195*a^3*cos(d*x + c)^6 + 7424*a^3*cos(d*x + c)^5
- 17710*a^3*cos(d*x + c)^4 - 14592*a^3*cos(d*x + c)^3 + 1960*a^3*cos(d*x + c)^2 + 5760*a^3*cos(d*x + c) + 168
0*a^3)*sin(d*x + c))/(d*cos(d*x + c)^8)

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} a^{3} \left (\int 3 \tan ^{6}{\left (c + d x \right )} \sec {\left (c + d x \right )}\, dx + \int 3 \tan ^{6}{\left (c + d x \right )} \sec ^{2}{\left (c + d x \right )}\, dx + \int \tan ^{6}{\left (c + d x \right )} \sec ^{3}{\left (c + d x \right )}\, dx + \int \tan ^{6}{\left (c + d x \right )}\, dx\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+a*sec(d*x+c))**3*tan(d*x+c)**6,x)

[Out]

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

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Giac [A]
time = 2.68, size = 196, normalized size = 0.83 \begin {gather*} -\frac {13440 \, {\left (d x + c\right )} a^{3} + 13125 \, a^{3} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1 \right |}\right ) - 13125 \, a^{3} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1 \right |}\right ) + \frac {2 \, {\left (315 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{15} - 11375 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{13} + 79723 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{11} - 269879 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} + 550089 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 749973 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 212625 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 26565 \, 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 )}^{8}}}{13440 \, d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+a*sec(d*x+c))^3*tan(d*x+c)^6,x, algorithm="giac")

[Out]

-1/13440*(13440*(d*x + c)*a^3 + 13125*a^3*log(abs(tan(1/2*d*x + 1/2*c) + 1)) - 13125*a^3*log(abs(tan(1/2*d*x +
1/2*c) - 1)) + 2*(315*a^3*tan(1/2*d*x + 1/2*c)^15 - 11375*a^3*tan(1/2*d*x + 1/2*c)^13 + 79723*a^3*tan(1/2*d*x
+ 1/2*c)^11 - 269879*a^3*tan(1/2*d*x + 1/2*c)^9 + 550089*a^3*tan(1/2*d*x + 1/2*c)^7 - 749973*a^3*tan(1/2*d*x
+ 1/2*c)^5 + 212625*a^3*tan(1/2*d*x + 1/2*c)^3 - 26565*a^3*tan(1/2*d*x + 1/2*c))/(tan(1/2*d*x + 1/2*c)^2 - 1)^
8)/d

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Mupad [B]
time = 2.43, size = 263, normalized size = 1.11 \begin {gather*} -a^3\,x-\frac {125\,a^3\,\mathrm {atanh}\left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )}{64\,d}-\frac {\frac {3\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{15}}{64}-\frac {325\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{13}}{192}+\frac {11389\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{11}}{960}-\frac {269879\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9}{6720}+\frac {183363\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7}{2240}-\frac {35713\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5}{320}+\frac {2025\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{64}-\frac {253\,a^3\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{64}}{d\,\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{16}-8\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{14}+28\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{12}-56\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}+70\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8-56\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6+28\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4-8\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+1\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(tan(c + d*x)^6*(a + a/cos(c + d*x))^3,x)

[Out]

- a^3*x - (125*a^3*atanh(tan(c/2 + (d*x)/2)))/(64*d) - ((2025*a^3*tan(c/2 + (d*x)/2)^3)/64 - (35713*a^3*tan(c/
2 + (d*x)/2)^5)/320 + (183363*a^3*tan(c/2 + (d*x)/2)^7)/2240 - (269879*a^3*tan(c/2 + (d*x)/2)^9)/6720 + (11389
*a^3*tan(c/2 + (d*x)/2)^11)/960 - (325*a^3*tan(c/2 + (d*x)/2)^13)/192 + (3*a^3*tan(c/2 + (d*x)/2)^15)/64 - (25
3*a^3*tan(c/2 + (d*x)/2))/64)/(d*(28*tan(c/2 + (d*x)/2)^4 - 8*tan(c/2 + (d*x)/2)^2 - 56*tan(c/2 + (d*x)/2)^6 +
70*tan(c/2 + (d*x)/2)^8 - 56*tan(c/2 + (d*x)/2)^10 + 28*tan(c/2 + (d*x)/2)^12 - 8*tan(c/2 + (d*x)/2)^14 + tan
(c/2 + (d*x)/2)^16 + 1))

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