3.56 \(\int \frac {\tan ^9(c+d x)}{a+a \sec (c+d x)} \, dx\)

Optimal. Leaf size=135 \[ \frac {\sec ^7(c+d x)}{7 a d}-\frac {\sec ^6(c+d x)}{6 a d}-\frac {3 \sec ^5(c+d x)}{5 a d}+\frac {3 \sec ^4(c+d x)}{4 a d}+\frac {\sec ^3(c+d x)}{a d}-\frac {3 \sec ^2(c+d x)}{2 a d}-\frac {\sec (c+d x)}{a d}-\frac {\log (\cos (c+d x))}{a d} \]

[Out]

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

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Rubi [A]  time = 0.08, antiderivative size = 135, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.095, Rules used = {3879, 88} \[ \frac {\sec ^7(c+d x)}{7 a d}-\frac {\sec ^6(c+d x)}{6 a d}-\frac {3 \sec ^5(c+d x)}{5 a d}+\frac {3 \sec ^4(c+d x)}{4 a d}+\frac {\sec ^3(c+d x)}{a d}-\frac {3 \sec ^2(c+d x)}{2 a d}-\frac {\sec (c+d x)}{a d}-\frac {\log (\cos (c+d x))}{a d} \]

Antiderivative was successfully verified.

[In]

Int[Tan[c + d*x]^9/(a + a*Sec[c + d*x]),x]

[Out]

-(Log[Cos[c + d*x]]/(a*d)) - Sec[c + d*x]/(a*d) - (3*Sec[c + d*x]^2)/(2*a*d) + Sec[c + d*x]^3/(a*d) + (3*Sec[c
+ d*x]^4)/(4*a*d) - (3*Sec[c + d*x]^5)/(5*a*d) - Sec[c + d*x]^6/(6*a*d) + Sec[c + d*x]^7/(7*a*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 \frac {\tan ^9(c+d x)}{a+a \sec (c+d x)} \, dx &=-\frac {\operatorname {Subst}\left (\int \frac {(a-a x)^4 (a+a x)^3}{x^8} \, dx,x,\cos (c+d x)\right )}{a^8 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^8 d}\\ &=-\frac {\log (\cos (c+d x))}{a d}-\frac {\sec (c+d x)}{a d}-\frac {3 \sec ^2(c+d x)}{2 a d}+\frac {\sec ^3(c+d x)}{a d}+\frac {3 \sec ^4(c+d x)}{4 a d}-\frac {3 \sec ^5(c+d x)}{5 a d}-\frac {\sec ^6(c+d x)}{6 a d}+\frac {\sec ^7(c+d x)}{7 a d}\\ \end {align*}

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Mathematica [A]  time = 0.56, size = 137, normalized size = 1.01 \[ -\frac {\sec ^7(c+d x) (35 \cos (c+d x) (105 \log (\cos (c+d x))+104)+3 (602 \cos (2 (c+d x))+140 \cos (4 (c+d x))+210 \cos (5 (c+d x))+70 \cos (6 (c+d x))+245 \cos (5 (c+d x)) \log (\cos (c+d x))+35 \cos (7 (c+d x)) \log (\cos (c+d x))+105 \cos (3 (c+d x)) (7 \log (\cos (c+d x))+6)+212))}{6720 a d} \]

Antiderivative was successfully verified.

[In]

Integrate[Tan[c + d*x]^9/(a + a*Sec[c + d*x]),x]

[Out]

-1/6720*((35*Cos[c + d*x]*(104 + 105*Log[Cos[c + d*x]]) + 3*(212 + 602*Cos[2*(c + d*x)] + 140*Cos[4*(c + d*x)]
+ 210*Cos[5*(c + d*x)] + 70*Cos[6*(c + d*x)] + 245*Cos[5*(c + d*x)]*Log[Cos[c + d*x]] + 35*Cos[7*(c + d*x)]*L
og[Cos[c + d*x]] + 105*Cos[3*(c + d*x)]*(6 + 7*Log[Cos[c + d*x]])))*Sec[c + d*x]^7)/(a*d)

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fricas [A]  time = 0.50, size = 95, normalized size = 0.70 \[ -\frac {420 \, \cos \left (d x + c\right )^{7} \log \left (-\cos \left (d x + c\right )\right ) + 420 \, \cos \left (d x + c\right )^{6} + 630 \, \cos \left (d x + c\right )^{5} - 420 \, \cos \left (d x + c\right )^{4} - 315 \, \cos \left (d x + c\right )^{3} + 252 \, \cos \left (d x + c\right )^{2} + 70 \, \cos \left (d x + c\right ) - 60}{420 \, a d \cos \left (d x + c\right )^{7}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(tan(d*x+c)^9/(a+a*sec(d*x+c)),x, algorithm="fricas")

[Out]

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

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giac [A]  time = 17.82, size = 245, normalized size = 1.81 \[ \frac {\frac {420 \, \log \left ({\left | -\frac {\cos \left (d x + c\right ) - 1}{\cos \left (d x + c\right ) + 1} + 1 \right |}\right )}{a} - \frac {420 \, \log \left ({\left | -\frac {\cos \left (d x + c\right ) - 1}{\cos \left (d x + c\right ) + 1} - 1 \right |}\right )}{a} + \frac {\frac {5775 \, {\left (\cos \left (d x + c\right ) - 1\right )}}{\cos \left (d x + c\right ) + 1} + \frac {20685 \, {\left (\cos \left (d x + c\right ) - 1\right )}^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {42595 \, {\left (\cos \left (d x + c\right ) - 1\right )}^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac {56035 \, {\left (\cos \left (d x + c\right ) - 1\right )}^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} + \frac {28749 \, {\left (\cos \left (d x + c\right ) - 1\right )}^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} + \frac {8463 \, {\left (\cos \left (d x + c\right ) - 1\right )}^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}} + \frac {1089 \, {\left (\cos \left (d x + c\right ) - 1\right )}^{7}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{7}} + 705}{a {\left (\frac {\cos \left (d x + c\right ) - 1}{\cos \left (d x + c\right ) + 1} + 1\right )}^{7}}}{420 \, d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(tan(d*x+c)^9/(a+a*sec(d*x+c)),x, algorithm="giac")

[Out]

1/420*(420*log(abs(-(cos(d*x + c) - 1)/(cos(d*x + c) + 1) + 1))/a - 420*log(abs(-(cos(d*x + c) - 1)/(cos(d*x +
c) + 1) - 1))/a + (5775*(cos(d*x + c) - 1)/(cos(d*x + c) + 1) + 20685*(cos(d*x + c) - 1)^2/(cos(d*x + c) + 1)
^2 + 42595*(cos(d*x + c) - 1)^3/(cos(d*x + c) + 1)^3 + 56035*(cos(d*x + c) - 1)^4/(cos(d*x + c) + 1)^4 + 28749
*(cos(d*x + c) - 1)^5/(cos(d*x + c) + 1)^5 + 8463*(cos(d*x + c) - 1)^6/(cos(d*x + c) + 1)^6 + 1089*(cos(d*x +
c) - 1)^7/(cos(d*x + c) + 1)^7 + 705)/(a*((cos(d*x + c) - 1)/(cos(d*x + c) + 1) + 1)^7))/d

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maple [A]  time = 0.63, size = 125, normalized size = 0.93 \[ \frac {\sec ^{7}\left (d x +c \right )}{7 d a}-\frac {\sec ^{6}\left (d x +c \right )}{6 d a}-\frac {3 \left (\sec ^{5}\left (d x +c \right )\right )}{5 d a}+\frac {3 \left (\sec ^{4}\left (d x +c \right )\right )}{4 d a}+\frac {\sec ^{3}\left (d x +c \right )}{d a}-\frac {3 \left (\sec ^{2}\left (d x +c \right )\right )}{2 d a}-\frac {\sec \left (d x +c \right )}{d a}+\frac {\ln \left (\sec \left (d x +c \right )\right )}{a d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(tan(d*x+c)^9/(a+a*sec(d*x+c)),x)

[Out]

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

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maxima [A]  time = 0.32, size = 90, normalized size = 0.67 \[ -\frac {\frac {420 \, \log \left (\cos \left (d x + c\right )\right )}{a} + \frac {420 \, \cos \left (d x + c\right )^{6} + 630 \, \cos \left (d x + c\right )^{5} - 420 \, \cos \left (d x + c\right )^{4} - 315 \, \cos \left (d x + c\right )^{3} + 252 \, \cos \left (d x + c\right )^{2} + 70 \, \cos \left (d x + c\right ) - 60}{a \cos \left (d x + c\right )^{7}}}{420 \, d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(tan(d*x+c)^9/(a+a*sec(d*x+c)),x, algorithm="maxima")

[Out]

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

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mupad [B]  time = 5.14, size = 208, normalized size = 1.54 \[ \frac {2\,\mathrm {atanh}\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2\right )}{a\,d}-\frac {-2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{12}+14\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}-\frac {128\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8}{3}+\frac {32\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6}{3}+\frac {26\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4}{5}-\frac {22\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{5}+\frac {32}{35}}{d\,\left (-a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{14}+7\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{12}-21\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}+35\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8-35\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6+21\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4-7\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+a\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(tan(c + d*x)^9/(a + a/cos(c + d*x)),x)

[Out]

(2*atanh(tan(c/2 + (d*x)/2)^2))/(a*d) - ((26*tan(c/2 + (d*x)/2)^4)/5 - (22*tan(c/2 + (d*x)/2)^2)/5 + (32*tan(c
/2 + (d*x)/2)^6)/3 - (128*tan(c/2 + (d*x)/2)^8)/3 + 14*tan(c/2 + (d*x)/2)^10 - 2*tan(c/2 + (d*x)/2)^12 + 32/35
)/(d*(a - 7*a*tan(c/2 + (d*x)/2)^2 + 21*a*tan(c/2 + (d*x)/2)^4 - 35*a*tan(c/2 + (d*x)/2)^6 + 35*a*tan(c/2 + (d
*x)/2)^8 - 21*a*tan(c/2 + (d*x)/2)^10 + 7*a*tan(c/2 + (d*x)/2)^12 - a*tan(c/2 + (d*x)/2)^14))

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \frac {\int \frac {\tan ^{9}{\left (c + d x \right )}}{\sec {\left (c + d x \right )} + 1}\, dx}{a} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(tan(d*x+c)**9/(a+a*sec(d*x+c)),x)

[Out]

Integral(tan(c + d*x)**9/(sec(c + d*x) + 1), x)/a

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