3.73 \(\int \frac {\tan ^5(c+d x)}{(a+a \sec (c+d x))^2} \, dx\)

Optimal. Leaf size=48 \[ \frac {\sec ^2(c+d x)}{2 a^2 d}-\frac {2 \sec (c+d x)}{a^2 d}-\frac {\log (\cos (c+d x))}{a^2 d} \]

[Out]

-ln(cos(d*x+c))/a^2/d-2*sec(d*x+c)/a^2/d+1/2*sec(d*x+c)^2/a^2/d

________________________________________________________________________________________

Rubi [A]  time = 0.05, antiderivative size = 48, 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, 43} \[ \frac {\sec ^2(c+d x)}{2 a^2 d}-\frac {2 \sec (c+d x)}{a^2 d}-\frac {\log (\cos (c+d x))}{a^2 d} \]

Antiderivative was successfully verified.

[In]

Int[Tan[c + d*x]^5/(a + a*Sec[c + d*x])^2,x]

[Out]

-(Log[Cos[c + d*x]]/(a^2*d)) - (2*Sec[c + d*x])/(a^2*d) + Sec[c + d*x]^2/(2*a^2*d)

Rule 43

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

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

________________________________________________________________________________________

Mathematica [A]  time = 0.12, size = 51, normalized size = 1.06 \[ -\frac {\sec ^2(c+d x) (4 \cos (c+d x)+\cos (2 (c+d x)) \log (\cos (c+d x))+\log (\cos (c+d x))-1)}{2 a^2 d} \]

Antiderivative was successfully verified.

[In]

Integrate[Tan[c + d*x]^5/(a + a*Sec[c + d*x])^2,x]

[Out]

-1/2*((-1 + 4*Cos[c + d*x] + Log[Cos[c + d*x]] + Cos[2*(c + d*x)]*Log[Cos[c + d*x]])*Sec[c + d*x]^2)/(a^2*d)

________________________________________________________________________________________

fricas [A]  time = 0.48, size = 45, normalized size = 0.94 \[ -\frac {2 \, \cos \left (d x + c\right )^{2} \log \left (-\cos \left (d x + c\right )\right ) + 4 \, \cos \left (d x + c\right ) - 1}{2 \, a^{2} d \cos \left (d x + c\right )^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(tan(d*x+c)^5/(a+a*sec(d*x+c))^2,x, algorithm="fricas")

[Out]

-1/2*(2*cos(d*x + c)^2*log(-cos(d*x + c)) + 4*cos(d*x + c) - 1)/(a^2*d*cos(d*x + c)^2)

________________________________________________________________________________________

giac [B]  time = 8.11, size = 136, normalized size = 2.83 \[ \frac {\frac {2 \, \log \left ({\left | -\frac {\cos \left (d x + c\right ) - 1}{\cos \left (d x + c\right ) + 1} + 1 \right |}\right )}{a^{2}} - \frac {2 \, \log \left ({\left | -\frac {\cos \left (d x + c\right ) - 1}{\cos \left (d x + c\right ) + 1} - 1 \right |}\right )}{a^{2}} - \frac {\frac {6 \, {\left (\cos \left (d x + c\right ) - 1\right )}}{\cos \left (d x + c\right ) + 1} - \frac {3 \, {\left (\cos \left (d x + c\right ) - 1\right )}^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + 5}{a^{2} {\left (\frac {\cos \left (d x + c\right ) - 1}{\cos \left (d x + c\right ) + 1} + 1\right )}^{2}}}{2 \, d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(tan(d*x+c)^5/(a+a*sec(d*x+c))^2,x, algorithm="giac")

[Out]

1/2*(2*log(abs(-(cos(d*x + c) - 1)/(cos(d*x + c) + 1) + 1))/a^2 - 2*log(abs(-(cos(d*x + c) - 1)/(cos(d*x + c)
+ 1) - 1))/a^2 - (6*(cos(d*x + c) - 1)/(cos(d*x + c) + 1) - 3*(cos(d*x + c) - 1)^2/(cos(d*x + c) + 1)^2 + 5)/(
a^2*((cos(d*x + c) - 1)/(cos(d*x + c) + 1) + 1)^2))/d

________________________________________________________________________________________

maple [A]  time = 0.54, size = 46, normalized size = 0.96 \[ \frac {\sec ^{2}\left (d x +c \right )}{2 a^{2} d}-\frac {2 \sec \left (d x +c \right )}{a^{2} d}+\frac {\ln \left (\sec \left (d x +c \right )\right )}{a^{2} d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(tan(d*x+c)^5/(a+a*sec(d*x+c))^2,x)

[Out]

1/2*sec(d*x+c)^2/a^2/d-2*sec(d*x+c)/a^2/d+1/a^2/d*ln(sec(d*x+c))

________________________________________________________________________________________

maxima [A]  time = 0.87, size = 40, normalized size = 0.83 \[ -\frac {\frac {2 \, \log \left (\cos \left (d x + c\right )\right )}{a^{2}} + \frac {4 \, \cos \left (d x + c\right ) - 1}{a^{2} \cos \left (d x + c\right )^{2}}}{2 \, d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(tan(d*x+c)^5/(a+a*sec(d*x+c))^2,x, algorithm="maxima")

[Out]

-1/2*(2*log(cos(d*x + c))/a^2 + (4*cos(d*x + c) - 1)/(a^2*cos(d*x + c)^2))/d

________________________________________________________________________________________

mupad [B]  time = 1.35, size = 77, normalized size = 1.60 \[ \frac {6\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2-4}{d\,\left (a^2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4-2\,a^2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+a^2\right )}+\frac {2\,\mathrm {atanh}\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2\right )}{a^2\,d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(tan(c + d*x)^5/(a + a/cos(c + d*x))^2,x)

[Out]

(6*tan(c/2 + (d*x)/2)^2 - 4)/(d*(a^2*tan(c/2 + (d*x)/2)^4 - 2*a^2*tan(c/2 + (d*x)/2)^2 + a^2)) + (2*atanh(tan(
c/2 + (d*x)/2)^2))/(a^2*d)

________________________________________________________________________________________

sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \frac {\int \frac {\tan ^{5}{\left (c + d x \right )}}{\sec ^{2}{\left (c + d x \right )} + 2 \sec {\left (c + d x \right )} + 1}\, dx}{a^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(tan(d*x+c)**5/(a+a*sec(d*x+c))**2,x)

[Out]

Integral(tan(c + d*x)**5/(sec(c + d*x)**2 + 2*sec(c + d*x) + 1), x)/a**2

________________________________________________________________________________________