3.2 \(\int (a+a \sec (c+d x)) \tan ^7(c+d x) \, dx\)

Optimal. Leaf size=118 \[ \frac {a \sec ^7(c+d x)}{7 d}+\frac {a \sec ^6(c+d x)}{6 d}-\frac {3 a \sec ^5(c+d x)}{5 d}-\frac {3 a \sec ^4(c+d x)}{4 d}+\frac {a \sec ^3(c+d x)}{d}+\frac {3 a \sec ^2(c+d x)}{2 d}-\frac {a \sec (c+d x)}{d}+\frac {a \log (\cos (c+d x))}{d} \]

[Out]

a*ln(cos(d*x+c))/d-a*sec(d*x+c)/d+3/2*a*sec(d*x+c)^2/d+a*sec(d*x+c)^3/d-3/4*a*sec(d*x+c)^4/d-3/5*a*sec(d*x+c)^
5/d+1/6*a*sec(d*x+c)^6/d+1/7*a*sec(d*x+c)^7/d

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Rubi [A]  time = 0.06, antiderivative size = 118, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.105, Rules used = {3879, 88} \[ \frac {a \sec ^7(c+d x)}{7 d}+\frac {a \sec ^6(c+d x)}{6 d}-\frac {3 a \sec ^5(c+d x)}{5 d}-\frac {3 a \sec ^4(c+d x)}{4 d}+\frac {a \sec ^3(c+d x)}{d}+\frac {3 a \sec ^2(c+d x)}{2 d}-\frac {a \sec (c+d x)}{d}+\frac {a \log (\cos (c+d x))}{d} \]

Antiderivative was successfully verified.

[In]

Int[(a + a*Sec[c + d*x])*Tan[c + d*x]^7,x]

[Out]

(a*Log[Cos[c + d*x]])/d - (a*Sec[c + d*x])/d + (3*a*Sec[c + d*x]^2)/(2*d) + (a*Sec[c + d*x]^3)/d - (3*a*Sec[c
+ d*x]^4)/(4*d) - (3*a*Sec[c + d*x]^5)/(5*d) + (a*Sec[c + d*x]^6)/(6*d) + (a*Sec[c + d*x]^7)/(7*d)

Rule 88

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Int[ExpandI
ntegrand[(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 3879

Int[cot[(c_.) + (d_.)*(x_)]^(m_.)*(csc[(c_.) + (d_.)*(x_)]*(b_.) + (a_))^(n_.), x_Symbol] :> Dist[1/(a^(m - n
- 1)*b^n*d), Subst[Int[((a - b*x)^((m - 1)/2)*(a + b*x)^((m - 1)/2 + n))/x^(m + n), x], x, Sin[c + d*x]], x] /
; FreeQ[{a, b, c, d}, x] && IntegerQ[(m - 1)/2] && EqQ[a^2 - b^2, 0] && IntegerQ[n]

Rubi steps

\begin {align*} \int (a+a \sec (c+d x)) \tan ^7(c+d x) \, dx &=-\frac {\operatorname {Subst}\left (\int \frac {(a-a x)^3 (a+a x)^4}{x^8} \, dx,x,\cos (c+d x)\right )}{a^6 d}\\ &=-\frac {\operatorname {Subst}\left (\int \left (\frac {a^7}{x^8}+\frac {a^7}{x^7}-\frac {3 a^7}{x^6}-\frac {3 a^7}{x^5}+\frac {3 a^7}{x^4}+\frac {3 a^7}{x^3}-\frac {a^7}{x^2}-\frac {a^7}{x}\right ) \, dx,x,\cos (c+d x)\right )}{a^6 d}\\ &=\frac {a \log (\cos (c+d x))}{d}-\frac {a \sec (c+d x)}{d}+\frac {3 a \sec ^2(c+d x)}{2 d}+\frac {a \sec ^3(c+d x)}{d}-\frac {3 a \sec ^4(c+d x)}{4 d}-\frac {3 a \sec ^5(c+d x)}{5 d}+\frac {a \sec ^6(c+d x)}{6 d}+\frac {a \sec ^7(c+d x)}{7 d}\\ \end {align*}

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Mathematica [A]  time = 0.47, size = 106, normalized size = 0.90 \[ \frac {a \sec ^7(c+d x)}{7 d}-\frac {3 a \sec ^5(c+d x)}{5 d}+\frac {a \sec ^3(c+d x)}{d}-\frac {a \sec (c+d x)}{d}+\frac {a \left (2 \tan ^6(c+d x)-3 \tan ^4(c+d x)+6 \tan ^2(c+d x)+12 \log (\cos (c+d x))\right )}{12 d} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + a*Sec[c + d*x])*Tan[c + d*x]^7,x]

[Out]

-((a*Sec[c + d*x])/d) + (a*Sec[c + d*x]^3)/d - (3*a*Sec[c + d*x]^5)/(5*d) + (a*Sec[c + d*x]^7)/(7*d) + (a*(12*
Log[Cos[c + d*x]] + 6*Tan[c + d*x]^2 - 3*Tan[c + d*x]^4 + 2*Tan[c + d*x]^6))/(12*d)

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fricas [A]  time = 0.80, size = 101, normalized size = 0.86 \[ \frac {420 \, a \cos \left (d x + c\right )^{7} \log \left (-\cos \left (d x + c\right )\right ) - 420 \, a \cos \left (d x + c\right )^{6} + 630 \, a \cos \left (d x + c\right )^{5} + 420 \, a \cos \left (d x + c\right )^{4} - 315 \, a \cos \left (d x + c\right )^{3} - 252 \, a \cos \left (d x + c\right )^{2} + 70 \, a \cos \left (d x + c\right ) + 60 \, a}{420 \, d \cos \left (d x + c\right )^{7}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+a*sec(d*x+c))*tan(d*x+c)^7,x, algorithm="fricas")

[Out]

1/420*(420*a*cos(d*x + c)^7*log(-cos(d*x + c)) - 420*a*cos(d*x + c)^6 + 630*a*cos(d*x + c)^5 + 420*a*cos(d*x +
c)^4 - 315*a*cos(d*x + c)^3 - 252*a*cos(d*x + c)^2 + 70*a*cos(d*x + c) + 60*a)/(d*cos(d*x + c)^7)

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giac [B]  time = 7.98, size = 247, normalized size = 2.09 \[ -\frac {420 \, a \log \left ({\left | -\frac {\cos \left (d x + c\right ) - 1}{\cos \left (d x + c\right ) + 1} + 1 \right |}\right ) - 420 \, a \log \left ({\left | -\frac {\cos \left (d x + c\right ) - 1}{\cos \left (d x + c\right ) + 1} - 1 \right |}\right ) + \frac {1473 \, a + \frac {11151 \, a {\left (\cos \left (d x + c\right ) - 1\right )}}{\cos \left (d x + c\right ) + 1} + \frac {36813 \, a {\left (\cos \left (d x + c\right ) - 1\right )}^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {69475 \, a {\left (\cos \left (d x + c\right ) - 1\right )}^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac {56035 \, a {\left (\cos \left (d x + c\right ) - 1\right )}^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} + \frac {28749 \, a {\left (\cos \left (d x + c\right ) - 1\right )}^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} + \frac {8463 \, a {\left (\cos \left (d x + c\right ) - 1\right )}^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}} + \frac {1089 \, a {\left (\cos \left (d x + c\right ) - 1\right )}^{7}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{7}}}{{\left (\frac {\cos \left (d x + c\right ) - 1}{\cos \left (d x + c\right ) + 1} + 1\right )}^{7}}}{420 \, d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+a*sec(d*x+c))*tan(d*x+c)^7,x, algorithm="giac")

[Out]

-1/420*(420*a*log(abs(-(cos(d*x + c) - 1)/(cos(d*x + c) + 1) + 1)) - 420*a*log(abs(-(cos(d*x + c) - 1)/(cos(d*
x + c) + 1) - 1)) + (1473*a + 11151*a*(cos(d*x + c) - 1)/(cos(d*x + c) + 1) + 36813*a*(cos(d*x + c) - 1)^2/(co
s(d*x + c) + 1)^2 + 69475*a*(cos(d*x + c) - 1)^3/(cos(d*x + c) + 1)^3 + 56035*a*(cos(d*x + c) - 1)^4/(cos(d*x
+ c) + 1)^4 + 28749*a*(cos(d*x + c) - 1)^5/(cos(d*x + c) + 1)^5 + 8463*a*(cos(d*x + c) - 1)^6/(cos(d*x + c) +
1)^6 + 1089*a*(cos(d*x + c) - 1)^7/(cos(d*x + c) + 1)^7)/((cos(d*x + c) - 1)/(cos(d*x + c) + 1) + 1)^7)/d

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maple [A]  time = 0.86, size = 216, normalized size = 1.83 \[ \frac {a \left (\tan ^{6}\left (d x +c \right )\right )}{6 d}-\frac {a \left (\tan ^{4}\left (d x +c \right )\right )}{4 d}+\frac {a \left (\tan ^{2}\left (d x +c \right )\right )}{2 d}+\frac {a \ln \left (\cos \left (d x +c \right )\right )}{d}+\frac {a \left (\sin ^{8}\left (d x +c \right )\right )}{7 d \cos \left (d x +c \right )^{7}}-\frac {a \left (\sin ^{8}\left (d x +c \right )\right )}{35 d \cos \left (d x +c \right )^{5}}+\frac {a \left (\sin ^{8}\left (d x +c \right )\right )}{35 d \cos \left (d x +c \right )^{3}}-\frac {a \left (\sin ^{8}\left (d x +c \right )\right )}{7 d \cos \left (d x +c \right )}-\frac {16 a \cos \left (d x +c \right )}{35 d}-\frac {a \cos \left (d x +c \right ) \left (\sin ^{6}\left (d x +c \right )\right )}{7 d}-\frac {6 a \cos \left (d x +c \right ) \left (\sin ^{4}\left (d x +c \right )\right )}{35 d}-\frac {8 a \cos \left (d x +c \right ) \left (\sin ^{2}\left (d x +c \right )\right )}{35 d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+a*sec(d*x+c))*tan(d*x+c)^7,x)

[Out]

1/6/d*a*tan(d*x+c)^6-1/4*a*tan(d*x+c)^4/d+1/2*a*tan(d*x+c)^2/d+a*ln(cos(d*x+c))/d+1/7/d*a*sin(d*x+c)^8/cos(d*x
+c)^7-1/35/d*a*sin(d*x+c)^8/cos(d*x+c)^5+1/35/d*a*sin(d*x+c)^8/cos(d*x+c)^3-1/7/d*a*sin(d*x+c)^8/cos(d*x+c)-16
/35*a*cos(d*x+c)/d-1/7/d*a*cos(d*x+c)*sin(d*x+c)^6-6/35/d*a*cos(d*x+c)*sin(d*x+c)^4-8/35/d*a*cos(d*x+c)*sin(d*
x+c)^2

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maxima [A]  time = 0.35, size = 94, normalized size = 0.80 \[ \frac {420 \, a \log \left (\cos \left (d x + c\right )\right ) - \frac {420 \, a \cos \left (d x + c\right )^{6} - 630 \, a \cos \left (d x + c\right )^{5} - 420 \, a \cos \left (d x + c\right )^{4} + 315 \, a \cos \left (d x + c\right )^{3} + 252 \, a \cos \left (d x + c\right )^{2} - 70 \, a \cos \left (d x + c\right ) - 60 \, a}{\cos \left (d x + c\right )^{7}}}{420 \, d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+a*sec(d*x+c))*tan(d*x+c)^7,x, algorithm="maxima")

[Out]

1/420*(420*a*log(cos(d*x + c)) - (420*a*cos(d*x + c)^6 - 630*a*cos(d*x + c)^5 - 420*a*cos(d*x + c)^4 + 315*a*c
os(d*x + c)^3 + 252*a*cos(d*x + c)^2 - 70*a*cos(d*x + c) - 60*a)/cos(d*x + c)^7)/d

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mupad [B]  time = 5.69, size = 204, normalized size = 1.73 \[ \frac {2\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{12}-14\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}+\frac {128\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8}{3}-\frac {224\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6}{3}+\frac {166\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4}{5}-\frac {42\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{5}+\frac {32\,a}{35}}{d\,\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{14}-7\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{12}+21\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}-35\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8+35\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6-21\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4+7\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2-1\right )}-\frac {2\,a\,\mathrm {atanh}\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2\right )}{d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(tan(c + d*x)^7*(a + a/cos(c + d*x)),x)

[Out]

((32*a)/35 - (42*a*tan(c/2 + (d*x)/2)^2)/5 + (166*a*tan(c/2 + (d*x)/2)^4)/5 - (224*a*tan(c/2 + (d*x)/2)^6)/3 +
(128*a*tan(c/2 + (d*x)/2)^8)/3 - 14*a*tan(c/2 + (d*x)/2)^10 + 2*a*tan(c/2 + (d*x)/2)^12)/(d*(7*tan(c/2 + (d*x
)/2)^2 - 21*tan(c/2 + (d*x)/2)^4 + 35*tan(c/2 + (d*x)/2)^6 - 35*tan(c/2 + (d*x)/2)^8 + 21*tan(c/2 + (d*x)/2)^1
0 - 7*tan(c/2 + (d*x)/2)^12 + tan(c/2 + (d*x)/2)^14 - 1)) - (2*a*atanh(tan(c/2 + (d*x)/2)^2))/d

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sympy [A]  time = 8.46, size = 148, normalized size = 1.25 \[ \begin {cases} - \frac {a \log {\left (\tan ^{2}{\left (c + d x \right )} + 1 \right )}}{2 d} + \frac {a \tan ^{6}{\left (c + d x \right )} \sec {\left (c + d x \right )}}{7 d} + \frac {a \tan ^{6}{\left (c + d x \right )}}{6 d} - \frac {6 a \tan ^{4}{\left (c + d x \right )} \sec {\left (c + d x \right )}}{35 d} - \frac {a \tan ^{4}{\left (c + d x \right )}}{4 d} + \frac {8 a \tan ^{2}{\left (c + d x \right )} \sec {\left (c + d x \right )}}{35 d} + \frac {a \tan ^{2}{\left (c + d x \right )}}{2 d} - \frac {16 a \sec {\left (c + d x \right )}}{35 d} & \text {for}\: d \neq 0 \\x \left (a \sec {\relax (c )} + a\right ) \tan ^{7}{\relax (c )} & \text {otherwise} \end {cases} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+a*sec(d*x+c))*tan(d*x+c)**7,x)

[Out]

Piecewise((-a*log(tan(c + d*x)**2 + 1)/(2*d) + a*tan(c + d*x)**6*sec(c + d*x)/(7*d) + a*tan(c + d*x)**6/(6*d)
- 6*a*tan(c + d*x)**4*sec(c + d*x)/(35*d) - a*tan(c + d*x)**4/(4*d) + 8*a*tan(c + d*x)**2*sec(c + d*x)/(35*d)
+ a*tan(c + d*x)**2/(2*d) - 16*a*sec(c + d*x)/(35*d), Ne(d, 0)), (x*(a*sec(c) + a)*tan(c)**7, True))

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