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]

-(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)

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Rubi [A]  time = 0.0784222, antiderivative size = 135, normalized size of antiderivative = 1., 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 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]

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]))

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*}

Mathematica [A]  time = 0.560865, 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]

-((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)]*Log[Cos[
c + d*x]] + 105*Cos[3*(c + d*x)]*(6 + 7*Log[Cos[c + d*x]])))*Sec[c + d*x]^7)/(6720*a*d)

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Maple [A]  time = 0.089, size = 125, normalized size = 0.9 \begin{align*}{\frac{ \left ( \sec \left ( dx+c \right ) \right ) ^{7}}{7\,da}}-{\frac{ \left ( \sec \left ( dx+c \right ) \right ) ^{6}}{6\,da}}-{\frac{3\, \left ( \sec \left ( dx+c \right ) \right ) ^{5}}{5\,da}}+{\frac{3\, \left ( \sec \left ( dx+c \right ) \right ) ^{4}}{4\,da}}+{\frac{ \left ( \sec \left ( dx+c \right ) \right ) ^{3}}{da}}-{\frac{3\, \left ( \sec \left ( dx+c \right ) \right ) ^{2}}{2\,da}}-{\frac{\sec \left ( dx+c \right ) }{da}}+{\frac{\ln \left ( \sec \left ( dx+c \right ) \right ) }{da}} \end{align*}

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 = 1.12235, size = 122, normalized size = 0.9 \begin{align*} -\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} \end{align*}

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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Fricas [A]  time = 1.23928, size = 266, normalized size = 1.97 \begin{align*} -\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}} \end{align*}

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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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

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]

Timed out

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Giac [A]  time = 17.6367, size = 331, normalized size = 2.45 \begin{align*} \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} \end{align*}

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