Integrand size = 14, antiderivative size = 65 \[ \int \left (a+a \tan ^2(c+d x)\right )^4 \, dx=\frac {a^4 \tan (c+d x)}{d}+\frac {a^4 \tan ^3(c+d x)}{d}+\frac {3 a^4 \tan ^5(c+d x)}{5 d}+\frac {a^4 \tan ^7(c+d x)}{7 d} \]
Time = 0.24 (sec) , antiderivative size = 46, normalized size of antiderivative = 0.71 \[ \int \left (a+a \tan ^2(c+d x)\right )^4 \, dx=\frac {a^4 \left (\tan (c+d x)+\tan ^3(c+d x)+\frac {3}{5} \tan ^5(c+d x)+\frac {1}{7} \tan ^7(c+d x)\right )}{d} \]
Time = 0.25 (sec) , antiderivative size = 51, normalized size of antiderivative = 0.78, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.429, Rules used = {3042, 4140, 27, 3042, 4254, 2009}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
\(\displaystyle \int \left (a \tan ^2(c+d x)+a\right )^4 \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \left (a \tan (c+d x)^2+a\right )^4dx\) |
\(\Big \downarrow \) 4140 |
\(\displaystyle \int a^4 \sec ^8(c+d x)dx\) |
\(\Big \downarrow \) 27 |
\(\displaystyle a^4 \int \sec ^8(c+d x)dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle a^4 \int \csc \left (c+d x+\frac {\pi }{2}\right )^8dx\) |
\(\Big \downarrow \) 4254 |
\(\displaystyle -\frac {a^4 \int \left (\tan ^6(c+d x)+3 \tan ^4(c+d x)+3 \tan ^2(c+d x)+1\right )d(-\tan (c+d x))}{d}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -\frac {a^4 \left (-\frac {1}{7} \tan ^7(c+d x)-\frac {3}{5} \tan ^5(c+d x)-\tan ^3(c+d x)-\tan (c+d x)\right )}{d}\) |
3.2.79.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[(u_.)*((a_) + (b_.)*tan[(e_.) + (f_.)*(x_)]^2)^(p_), x_Symbol] :> Int[A ctivateTrig[u*(a*sec[e + f*x]^2)^p], x] /; FreeQ[{a, b, e, f, p}, x] && EqQ [a, b]
Int[csc[(c_.) + (d_.)*(x_)]^(n_), x_Symbol] :> Simp[-d^(-1) Subst[Int[Exp andIntegrand[(1 + x^2)^(n/2 - 1), x], x], x, Cot[c + d*x]], x] /; FreeQ[{c, d}, x] && IGtQ[n/2, 0]
Time = 0.07 (sec) , antiderivative size = 43, normalized size of antiderivative = 0.66
method | result | size |
derivativedivides | \(\frac {a^{4} \left (\frac {\tan \left (d x +c \right )^{7}}{7}+\frac {3 \tan \left (d x +c \right )^{5}}{5}+\tan \left (d x +c \right )^{3}+\tan \left (d x +c \right )\right )}{d}\) | \(43\) |
default | \(\frac {a^{4} \left (\frac {\tan \left (d x +c \right )^{7}}{7}+\frac {3 \tan \left (d x +c \right )^{5}}{5}+\tan \left (d x +c \right )^{3}+\tan \left (d x +c \right )\right )}{d}\) | \(43\) |
parallelrisch | \(\frac {5 a^{4} \tan \left (d x +c \right )^{7}+21 a^{4} \tan \left (d x +c \right )^{5}+35 a^{4} \tan \left (d x +c \right )^{3}+35 a^{4} \tan \left (d x +c \right )}{35 d}\) | \(57\) |
risch | \(\frac {32 i a^{4} \left (35 \,{\mathrm e}^{6 i \left (d x +c \right )}+21 \,{\mathrm e}^{4 i \left (d x +c \right )}+7 \,{\mathrm e}^{2 i \left (d x +c \right )}+1\right )}{35 d \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )^{7}}\) | \(58\) |
norman | \(\frac {a^{4} \tan \left (d x +c \right )}{d}+\frac {a^{4} \tan \left (d x +c \right )^{3}}{d}+\frac {3 a^{4} \tan \left (d x +c \right )^{5}}{5 d}+\frac {a^{4} \tan \left (d x +c \right )^{7}}{7 d}\) | \(62\) |
parts | \(x \,a^{4}+\frac {a^{4} \left (\frac {\tan \left (d x +c \right )^{7}}{7}-\frac {\tan \left (d x +c \right )^{5}}{5}+\frac {\tan \left (d x +c \right )^{3}}{3}-\tan \left (d x +c \right )+\arctan \left (\tan \left (d x +c \right )\right )\right )}{d}+\frac {4 a^{4} \left (\tan \left (d x +c \right )-\arctan \left (\tan \left (d x +c \right )\right )\right )}{d}+\frac {6 a^{4} \left (\frac {\tan \left (d x +c \right )^{3}}{3}-\tan \left (d x +c \right )+\arctan \left (\tan \left (d x +c \right )\right )\right )}{d}+\frac {4 a^{4} \left (\frac {\tan \left (d x +c \right )^{5}}{5}-\frac {\tan \left (d x +c \right )^{3}}{3}+\tan \left (d x +c \right )-\arctan \left (\tan \left (d x +c \right )\right )\right )}{d}\) | \(162\) |
Time = 0.25 (sec) , antiderivative size = 56, normalized size of antiderivative = 0.86 \[ \int \left (a+a \tan ^2(c+d x)\right )^4 \, dx=\frac {5 \, a^{4} \tan \left (d x + c\right )^{7} + 21 \, a^{4} \tan \left (d x + c\right )^{5} + 35 \, a^{4} \tan \left (d x + c\right )^{3} + 35 \, a^{4} \tan \left (d x + c\right )}{35 \, d} \]
1/35*(5*a^4*tan(d*x + c)^7 + 21*a^4*tan(d*x + c)^5 + 35*a^4*tan(d*x + c)^3 + 35*a^4*tan(d*x + c))/d
Time = 0.20 (sec) , antiderivative size = 68, normalized size of antiderivative = 1.05 \[ \int \left (a+a \tan ^2(c+d x)\right )^4 \, dx=\begin {cases} \frac {a^{4} \tan ^{7}{\left (c + d x \right )}}{7 d} + \frac {3 a^{4} \tan ^{5}{\left (c + d x \right )}}{5 d} + \frac {a^{4} \tan ^{3}{\left (c + d x \right )}}{d} + \frac {a^{4} \tan {\left (c + d x \right )}}{d} & \text {for}\: d \neq 0 \\x \left (a \tan ^{2}{\left (c \right )} + a\right )^{4} & \text {otherwise} \end {cases} \]
Piecewise((a**4*tan(c + d*x)**7/(7*d) + 3*a**4*tan(c + d*x)**5/(5*d) + a** 4*tan(c + d*x)**3/d + a**4*tan(c + d*x)/d, Ne(d, 0)), (x*(a*tan(c)**2 + a) **4, True))
Leaf count of result is larger than twice the leaf count of optimal. 157 vs. \(2 (61) = 122\).
Time = 0.31 (sec) , antiderivative size = 157, normalized size of antiderivative = 2.42 \[ \int \left (a+a \tan ^2(c+d x)\right )^4 \, dx=a^{4} x + \frac {{\left (15 \, \tan \left (d x + c\right )^{7} - 21 \, \tan \left (d x + c\right )^{5} + 35 \, \tan \left (d x + c\right )^{3} + 105 \, d x + 105 \, c - 105 \, \tan \left (d x + c\right )\right )} a^{4}}{105 \, d} + \frac {4 \, {\left (3 \, \tan \left (d x + c\right )^{5} - 5 \, \tan \left (d x + c\right )^{3} - 15 \, d x - 15 \, c + 15 \, \tan \left (d x + c\right )\right )} a^{4}}{15 \, d} + \frac {2 \, {\left (\tan \left (d x + c\right )^{3} + 3 \, d x + 3 \, c - 3 \, \tan \left (d x + c\right )\right )} a^{4}}{d} - \frac {4 \, {\left (d x + c - \tan \left (d x + c\right )\right )} a^{4}}{d} \]
a^4*x + 1/105*(15*tan(d*x + c)^7 - 21*tan(d*x + c)^5 + 35*tan(d*x + c)^3 + 105*d*x + 105*c - 105*tan(d*x + c))*a^4/d + 4/15*(3*tan(d*x + c)^5 - 5*ta n(d*x + c)^3 - 15*d*x - 15*c + 15*tan(d*x + c))*a^4/d + 2*(tan(d*x + c)^3 + 3*d*x + 3*c - 3*tan(d*x + c))*a^4/d - 4*(d*x + c - tan(d*x + c))*a^4/d
Leaf count of result is larger than twice the leaf count of optimal. 519 vs. \(2 (61) = 122\).
Time = 1.03 (sec) , antiderivative size = 519, normalized size of antiderivative = 7.98 \[ \int \left (a+a \tan ^2(c+d x)\right )^4 \, dx=-\frac {35 \, a^{4} \tan \left (d x\right )^{7} \tan \left (c\right )^{6} + 35 \, a^{4} \tan \left (d x\right )^{6} \tan \left (c\right )^{7} + 35 \, a^{4} \tan \left (d x\right )^{7} \tan \left (c\right )^{4} - 105 \, a^{4} \tan \left (d x\right )^{6} \tan \left (c\right )^{5} - 105 \, a^{4} \tan \left (d x\right )^{5} \tan \left (c\right )^{6} + 35 \, a^{4} \tan \left (d x\right )^{4} \tan \left (c\right )^{7} + 21 \, a^{4} \tan \left (d x\right )^{7} \tan \left (c\right )^{2} - 35 \, a^{4} \tan \left (d x\right )^{6} \tan \left (c\right )^{3} + 315 \, a^{4} \tan \left (d x\right )^{5} \tan \left (c\right )^{4} + 315 \, a^{4} \tan \left (d x\right )^{4} \tan \left (c\right )^{5} - 35 \, a^{4} \tan \left (d x\right )^{3} \tan \left (c\right )^{6} + 21 \, a^{4} \tan \left (d x\right )^{2} \tan \left (c\right )^{7} + 5 \, a^{4} \tan \left (d x\right )^{7} - 7 \, a^{4} \tan \left (d x\right )^{6} \tan \left (c\right ) + 105 \, a^{4} \tan \left (d x\right )^{5} \tan \left (c\right )^{2} - 315 \, a^{4} \tan \left (d x\right )^{4} \tan \left (c\right )^{3} - 315 \, a^{4} \tan \left (d x\right )^{3} \tan \left (c\right )^{4} + 105 \, a^{4} \tan \left (d x\right )^{2} \tan \left (c\right )^{5} - 7 \, a^{4} \tan \left (d x\right ) \tan \left (c\right )^{6} + 5 \, a^{4} \tan \left (c\right )^{7} + 21 \, a^{4} \tan \left (d x\right )^{5} - 35 \, a^{4} \tan \left (d x\right )^{4} \tan \left (c\right ) + 315 \, a^{4} \tan \left (d x\right )^{3} \tan \left (c\right )^{2} + 315 \, a^{4} \tan \left (d x\right )^{2} \tan \left (c\right )^{3} - 35 \, a^{4} \tan \left (d x\right ) \tan \left (c\right )^{4} + 21 \, a^{4} \tan \left (c\right )^{5} + 35 \, a^{4} \tan \left (d x\right )^{3} - 105 \, a^{4} \tan \left (d x\right )^{2} \tan \left (c\right ) - 105 \, a^{4} \tan \left (d x\right ) \tan \left (c\right )^{2} + 35 \, a^{4} \tan \left (c\right )^{3} + 35 \, a^{4} \tan \left (d x\right ) + 35 \, a^{4} \tan \left (c\right )}{35 \, {\left (d \tan \left (d x\right )^{7} \tan \left (c\right )^{7} - 7 \, d \tan \left (d x\right )^{6} \tan \left (c\right )^{6} + 21 \, d \tan \left (d x\right )^{5} \tan \left (c\right )^{5} - 35 \, d \tan \left (d x\right )^{4} \tan \left (c\right )^{4} + 35 \, d \tan \left (d x\right )^{3} \tan \left (c\right )^{3} - 21 \, d \tan \left (d x\right )^{2} \tan \left (c\right )^{2} + 7 \, d \tan \left (d x\right ) \tan \left (c\right ) - d\right )}} \]
-1/35*(35*a^4*tan(d*x)^7*tan(c)^6 + 35*a^4*tan(d*x)^6*tan(c)^7 + 35*a^4*ta n(d*x)^7*tan(c)^4 - 105*a^4*tan(d*x)^6*tan(c)^5 - 105*a^4*tan(d*x)^5*tan(c )^6 + 35*a^4*tan(d*x)^4*tan(c)^7 + 21*a^4*tan(d*x)^7*tan(c)^2 - 35*a^4*tan (d*x)^6*tan(c)^3 + 315*a^4*tan(d*x)^5*tan(c)^4 + 315*a^4*tan(d*x)^4*tan(c) ^5 - 35*a^4*tan(d*x)^3*tan(c)^6 + 21*a^4*tan(d*x)^2*tan(c)^7 + 5*a^4*tan(d *x)^7 - 7*a^4*tan(d*x)^6*tan(c) + 105*a^4*tan(d*x)^5*tan(c)^2 - 315*a^4*ta n(d*x)^4*tan(c)^3 - 315*a^4*tan(d*x)^3*tan(c)^4 + 105*a^4*tan(d*x)^2*tan(c )^5 - 7*a^4*tan(d*x)*tan(c)^6 + 5*a^4*tan(c)^7 + 21*a^4*tan(d*x)^5 - 35*a^ 4*tan(d*x)^4*tan(c) + 315*a^4*tan(d*x)^3*tan(c)^2 + 315*a^4*tan(d*x)^2*tan (c)^3 - 35*a^4*tan(d*x)*tan(c)^4 + 21*a^4*tan(c)^5 + 35*a^4*tan(d*x)^3 - 1 05*a^4*tan(d*x)^2*tan(c) - 105*a^4*tan(d*x)*tan(c)^2 + 35*a^4*tan(c)^3 + 3 5*a^4*tan(d*x) + 35*a^4*tan(c))/(d*tan(d*x)^7*tan(c)^7 - 7*d*tan(d*x)^6*ta n(c)^6 + 21*d*tan(d*x)^5*tan(c)^5 - 35*d*tan(d*x)^4*tan(c)^4 + 35*d*tan(d* x)^3*tan(c)^3 - 21*d*tan(d*x)^2*tan(c)^2 + 7*d*tan(d*x)*tan(c) - d)
Time = 11.24 (sec) , antiderivative size = 53, normalized size of antiderivative = 0.82 \[ \int \left (a+a \tan ^2(c+d x)\right )^4 \, dx=\frac {\frac {a^4\,{\mathrm {tan}\left (c+d\,x\right )}^7}{7}+\frac {3\,a^4\,{\mathrm {tan}\left (c+d\,x\right )}^5}{5}+a^4\,{\mathrm {tan}\left (c+d\,x\right )}^3+a^4\,\mathrm {tan}\left (c+d\,x\right )}{d} \]