Integrand size = 14, antiderivative size = 182 \[ \int \left (b \tan ^4(c+d x)\right )^{5/2} \, dx=\frac {b^2 \cot (c+d x) \sqrt {b \tan ^4(c+d x)}}{d}-b^2 x \cot ^2(c+d x) \sqrt {b \tan ^4(c+d x)}-\frac {b^2 \tan (c+d x) \sqrt {b \tan ^4(c+d x)}}{3 d}+\frac {b^2 \tan ^3(c+d x) \sqrt {b \tan ^4(c+d x)}}{5 d}-\frac {b^2 \tan ^5(c+d x) \sqrt {b \tan ^4(c+d x)}}{7 d}+\frac {b^2 \tan ^7(c+d x) \sqrt {b \tan ^4(c+d x)}}{9 d} \]
b^2*cot(d*x+c)*(tan(d*x+c)^4*b)^(1/2)/d-b^2*x*cot(d*x+c)^2*(tan(d*x+c)^4*b )^(1/2)-1/3*b^2*(tan(d*x+c)^4*b)^(1/2)*tan(d*x+c)/d+1/5*b^2*(tan(d*x+c)^4* b)^(1/2)*tan(d*x+c)^3/d-1/7*b^2*(tan(d*x+c)^4*b)^(1/2)*tan(d*x+c)^5/d+1/9* b^2*(tan(d*x+c)^4*b)^(1/2)*tan(d*x+c)^7/d
Time = 0.60 (sec) , antiderivative size = 86, normalized size of antiderivative = 0.47 \[ \int \left (b \tan ^4(c+d x)\right )^{5/2} \, dx=\frac {\cot (c+d x) \left (35-45 \cot ^2(c+d x)+63 \cot ^4(c+d x)-105 \cot ^6(c+d x)+315 \cot ^8(c+d x)-315 \arctan (\tan (c+d x)) \cot ^9(c+d x)\right ) \left (b \tan ^4(c+d x)\right )^{5/2}}{315 d} \]
(Cot[c + d*x]*(35 - 45*Cot[c + d*x]^2 + 63*Cot[c + d*x]^4 - 105*Cot[c + d* x]^6 + 315*Cot[c + d*x]^8 - 315*ArcTan[Tan[c + d*x]]*Cot[c + d*x]^9)*(b*Ta n[c + d*x]^4)^(5/2))/(315*d)
Time = 0.57 (sec) , antiderivative size = 100, normalized size of antiderivative = 0.55, number of steps used = 13, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.929, Rules used = {3042, 4141, 3042, 3954, 3042, 3954, 3042, 3954, 3042, 3954, 3042, 3954, 24}
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.
| \(\displaystyle \int \left (b \tan ^4(c+d x)\right )^{5/2} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \left (b \tan (c+d x)^4\right )^{5/2}dx\) |
| \(\Big \downarrow \) 4141 |
| \(\displaystyle b^2 \cot ^2(c+d x) \sqrt {b \tan ^4(c+d x)} \int \tan ^{10}(c+d x)dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle b^2 \cot ^2(c+d x) \sqrt {b \tan ^4(c+d x)} \int \tan (c+d x)^{10}dx\) |
| \(\Big \downarrow \) 3954 |
| \(\displaystyle b^2 \cot ^2(c+d x) \sqrt {b \tan ^4(c+d x)} \left (\frac {\tan ^9(c+d x)}{9 d}-\int \tan ^8(c+d x)dx\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle b^2 \cot ^2(c+d x) \sqrt {b \tan ^4(c+d x)} \left (\frac {\tan ^9(c+d x)}{9 d}-\int \tan (c+d x)^8dx\right )\) |
| \(\Big \downarrow \) 3954 |
| \(\displaystyle b^2 \cot ^2(c+d x) \sqrt {b \tan ^4(c+d x)} \left (\int \tan ^6(c+d x)dx+\frac {\tan ^9(c+d x)}{9 d}-\frac {\tan ^7(c+d x)}{7 d}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle b^2 \cot ^2(c+d x) \sqrt {b \tan ^4(c+d x)} \left (\int \tan (c+d x)^6dx+\frac {\tan ^9(c+d x)}{9 d}-\frac {\tan ^7(c+d x)}{7 d}\right )\) |
| \(\Big \downarrow \) 3954 |
| \(\displaystyle b^2 \cot ^2(c+d x) \sqrt {b \tan ^4(c+d x)} \left (-\int \tan ^4(c+d x)dx+\frac {\tan ^9(c+d x)}{9 d}-\frac {\tan ^7(c+d x)}{7 d}+\frac {\tan ^5(c+d x)}{5 d}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle b^2 \cot ^2(c+d x) \sqrt {b \tan ^4(c+d x)} \left (-\int \tan (c+d x)^4dx+\frac {\tan ^9(c+d x)}{9 d}-\frac {\tan ^7(c+d x)}{7 d}+\frac {\tan ^5(c+d x)}{5 d}\right )\) |
| \(\Big \downarrow \) 3954 |
| \(\displaystyle b^2 \cot ^2(c+d x) \sqrt {b \tan ^4(c+d x)} \left (\int \tan ^2(c+d x)dx+\frac {\tan ^9(c+d x)}{9 d}-\frac {\tan ^7(c+d x)}{7 d}+\frac {\tan ^5(c+d x)}{5 d}-\frac {\tan ^3(c+d x)}{3 d}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle b^2 \cot ^2(c+d x) \sqrt {b \tan ^4(c+d x)} \left (\int \tan (c+d x)^2dx+\frac {\tan ^9(c+d x)}{9 d}-\frac {\tan ^7(c+d x)}{7 d}+\frac {\tan ^5(c+d x)}{5 d}-\frac {\tan ^3(c+d x)}{3 d}\right )\) |
| \(\Big \downarrow \) 3954 |
| \(\displaystyle b^2 \cot ^2(c+d x) \sqrt {b \tan ^4(c+d x)} \left (-\int 1dx+\frac {\tan ^9(c+d x)}{9 d}-\frac {\tan ^7(c+d x)}{7 d}+\frac {\tan ^5(c+d x)}{5 d}-\frac {\tan ^3(c+d x)}{3 d}+\frac {\tan (c+d x)}{d}\right )\) |
| \(\Big \downarrow \) 24 |
| \(\displaystyle b^2 \left (\frac {\tan ^9(c+d x)}{9 d}-\frac {\tan ^7(c+d x)}{7 d}+\frac {\tan ^5(c+d x)}{5 d}-\frac {\tan ^3(c+d x)}{3 d}+\frac {\tan (c+d x)}{d}-x\right ) \cot ^2(c+d x) \sqrt {b \tan ^4(c+d x)}\) |
b^2*Cot[c + d*x]^2*Sqrt[b*Tan[c + d*x]^4]*(-x + Tan[c + d*x]/d - Tan[c + d *x]^3/(3*d) + Tan[c + d*x]^5/(5*d) - Tan[c + d*x]^7/(7*d) + Tan[c + d*x]^9 /(9*d))
3.1.36.3.1 Defintions of rubi rules used
Int[((b_.)*tan[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[b*((b*Tan[c + d *x])^(n - 1)/(d*(n - 1))), x] - Simp[b^2 Int[(b*Tan[c + d*x])^(n - 2), x] , x] /; FreeQ[{b, c, d}, x] && GtQ[n, 1]
Int[(u_.)*((b_.)*tan[(e_.) + (f_.)*(x_)]^(n_))^(p_), x_Symbol] :> With[{ff = FreeFactors[Tan[e + f*x], x]}, Simp[(b*ff^n)^IntPart[p]*((b*Tan[e + f*x]^ n)^FracPart[p]/(Tan[e + f*x]/ff)^(n*FracPart[p])) Int[ActivateTrig[u]*(Ta n[e + f*x]/ff)^(n*p), x], x]] /; FreeQ[{b, e, f, n, p}, x] && !IntegerQ[p] && IntegerQ[n] && (EqQ[u, 1] || MatchQ[u, ((d_.)*(trig_)[e + f*x])^(m_.) / ; FreeQ[{d, m}, x] && MemberQ[{sin, cos, tan, cot, sec, csc}, trig]])
Time = 0.23 (sec) , antiderivative size = 84, normalized size of antiderivative = 0.46
| method | result | size |
| derivativedivides | \(-\frac {{\left (\left (\tan ^{4}\left (d x +c \right )\right ) b \right )}^{\frac {5}{2}} \left (-35 \left (\tan ^{9}\left (d x +c \right )\right )+45 \left (\tan ^{7}\left (d x +c \right )\right )-63 \left (\tan ^{5}\left (d x +c \right )\right )+105 \left (\tan ^{3}\left (d x +c \right )\right )+315 \arctan \left (\tan \left (d x +c \right )\right )-315 \tan \left (d x +c \right )\right )}{315 d \tan \left (d x +c \right )^{10}}\) | \(84\) |
| default | \(-\frac {{\left (\left (\tan ^{4}\left (d x +c \right )\right ) b \right )}^{\frac {5}{2}} \left (-35 \left (\tan ^{9}\left (d x +c \right )\right )+45 \left (\tan ^{7}\left (d x +c \right )\right )-63 \left (\tan ^{5}\left (d x +c \right )\right )+105 \left (\tan ^{3}\left (d x +c \right )\right )+315 \arctan \left (\tan \left (d x +c \right )\right )-315 \tan \left (d x +c \right )\right )}{315 d \tan \left (d x +c \right )^{10}}\) | \(84\) |
| risch | \(\frac {b^{2} \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )^{2} \sqrt {\frac {\left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )^{4} b}{\left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )^{4}}}\, x}{\left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )^{2}}-\frac {2 i b^{2} \sqrt {\frac {\left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )^{4} b}{\left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )^{4}}}\, \left (1575 \,{\mathrm e}^{16 i \left (d x +c \right )}+6300 \,{\mathrm e}^{14 i \left (d x +c \right )}+21000 \,{\mathrm e}^{12 i \left (d x +c \right )}+31500 \,{\mathrm e}^{10 i \left (d x +c \right )}+39438 \,{\mathrm e}^{8 i \left (d x +c \right )}+26292 \,{\mathrm e}^{6 i \left (d x +c \right )}+13968 \,{\mathrm e}^{4 i \left (d x +c \right )}+3492 \,{\mathrm e}^{2 i \left (d x +c \right )}+563\right )}{315 \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )^{2} \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )^{7} d}\) | \(218\) |
-1/315/d*(tan(d*x+c)^4*b)^(5/2)*(-35*tan(d*x+c)^9+45*tan(d*x+c)^7-63*tan(d *x+c)^5+105*tan(d*x+c)^3+315*arctan(tan(d*x+c))-315*tan(d*x+c))/tan(d*x+c) ^10
Time = 0.24 (sec) , antiderivative size = 96, normalized size of antiderivative = 0.53 \[ \int \left (b \tan ^4(c+d x)\right )^{5/2} \, dx=\frac {{\left (35 \, b^{2} \tan \left (d x + c\right )^{9} - 45 \, b^{2} \tan \left (d x + c\right )^{7} + 63 \, b^{2} \tan \left (d x + c\right )^{5} - 105 \, b^{2} \tan \left (d x + c\right )^{3} - 315 \, b^{2} d x + 315 \, b^{2} \tan \left (d x + c\right )\right )} \sqrt {b \tan \left (d x + c\right )^{4}}}{315 \, d \tan \left (d x + c\right )^{2}} \]
1/315*(35*b^2*tan(d*x + c)^9 - 45*b^2*tan(d*x + c)^7 + 63*b^2*tan(d*x + c) ^5 - 105*b^2*tan(d*x + c)^3 - 315*b^2*d*x + 315*b^2*tan(d*x + c))*sqrt(b*t an(d*x + c)^4)/(d*tan(d*x + c)^2)
\[ \int \left (b \tan ^4(c+d x)\right )^{5/2} \, dx=\int \left (b \tan ^{4}{\left (c + d x \right )}\right )^{\frac {5}{2}}\, dx \]
Time = 0.38 (sec) , antiderivative size = 79, normalized size of antiderivative = 0.43 \[ \int \left (b \tan ^4(c+d x)\right )^{5/2} \, dx=\frac {35 \, b^{\frac {5}{2}} \tan \left (d x + c\right )^{9} - 45 \, b^{\frac {5}{2}} \tan \left (d x + c\right )^{7} + 63 \, b^{\frac {5}{2}} \tan \left (d x + c\right )^{5} - 105 \, b^{\frac {5}{2}} \tan \left (d x + c\right )^{3} - 315 \, {\left (d x + c\right )} b^{\frac {5}{2}} + 315 \, b^{\frac {5}{2}} \tan \left (d x + c\right )}{315 \, d} \]
1/315*(35*b^(5/2)*tan(d*x + c)^9 - 45*b^(5/2)*tan(d*x + c)^7 + 63*b^(5/2)* tan(d*x + c)^5 - 105*b^(5/2)*tan(d*x + c)^3 - 315*(d*x + c)*b^(5/2) + 315* b^(5/2)*tan(d*x + c))/d
Leaf count of result is larger than twice the leaf count of optimal. 960 vs. \(2 (162) = 324\).
Time = 6.98 (sec) , antiderivative size = 960, normalized size of antiderivative = 5.27 \[ \int \left (b \tan ^4(c+d x)\right )^{5/2} \, dx=\text {Too large to display} \]
-1/315*(315*b^2*d*x*tan(d*x)^9*tan(c)^9 - 2835*b^2*d*x*tan(d*x)^8*tan(c)^8 + 315*b^2*tan(d*x)^9*tan(c)^8 + 315*b^2*tan(d*x)^8*tan(c)^9 + 11340*b^2*d *x*tan(d*x)^7*tan(c)^7 - 105*b^2*tan(d*x)^9*tan(c)^6 - 2835*b^2*tan(d*x)^8 *tan(c)^7 - 2835*b^2*tan(d*x)^7*tan(c)^8 - 105*b^2*tan(d*x)^6*tan(c)^9 - 2 6460*b^2*d*x*tan(d*x)^6*tan(c)^6 + 63*b^2*tan(d*x)^9*tan(c)^4 + 945*b^2*ta n(d*x)^8*tan(c)^5 + 11340*b^2*tan(d*x)^7*tan(c)^6 + 11340*b^2*tan(d*x)^6*t an(c)^7 + 945*b^2*tan(d*x)^5*tan(c)^8 + 63*b^2*tan(d*x)^4*tan(c)^9 + 39690 *b^2*d*x*tan(d*x)^5*tan(c)^5 - 45*b^2*tan(d*x)^9*tan(c)^2 - 567*b^2*tan(d* x)^8*tan(c)^3 - 3780*b^2*tan(d*x)^7*tan(c)^4 - 26460*b^2*tan(d*x)^6*tan(c) ^5 - 26460*b^2*tan(d*x)^5*tan(c)^6 - 3780*b^2*tan(d*x)^4*tan(c)^7 - 567*b^ 2*tan(d*x)^3*tan(c)^8 - 45*b^2*tan(d*x)^2*tan(c)^9 - 39690*b^2*d*x*tan(d*x )^4*tan(c)^4 + 35*b^2*tan(d*x)^9 + 405*b^2*tan(d*x)^8*tan(c) + 2268*b^2*ta n(d*x)^7*tan(c)^2 + 8820*b^2*tan(d*x)^6*tan(c)^3 + 39690*b^2*tan(d*x)^5*ta n(c)^4 + 39690*b^2*tan(d*x)^4*tan(c)^5 + 8820*b^2*tan(d*x)^3*tan(c)^6 + 22 68*b^2*tan(d*x)^2*tan(c)^7 + 405*b^2*tan(d*x)*tan(c)^8 + 35*b^2*tan(c)^9 + 26460*b^2*d*x*tan(d*x)^3*tan(c)^3 - 45*b^2*tan(d*x)^7 - 567*b^2*tan(d*x)^ 6*tan(c) - 3780*b^2*tan(d*x)^5*tan(c)^2 - 26460*b^2*tan(d*x)^4*tan(c)^3 - 26460*b^2*tan(d*x)^3*tan(c)^4 - 3780*b^2*tan(d*x)^2*tan(c)^5 - 567*b^2*tan (d*x)*tan(c)^6 - 45*b^2*tan(c)^7 - 11340*b^2*d*x*tan(d*x)^2*tan(c)^2 + 63* b^2*tan(d*x)^5 + 945*b^2*tan(d*x)^4*tan(c) + 11340*b^2*tan(d*x)^3*tan(c...
Timed out. \[ \int \left (b \tan ^4(c+d x)\right )^{5/2} \, dx=\int {\left (b\,{\mathrm {tan}\left (c+d\,x\right )}^4\right )}^{5/2} \,d x \]