Integrand size = 21, antiderivative size = 253 \[ \int \cos ^4(e+f x) (d \tan (e+f x))^{5/2} \, dx=-\frac {3 d^{5/2} \arctan \left (1-\frac {\sqrt {2} \sqrt {d \tan (e+f x)}}{\sqrt {d}}\right )}{32 \sqrt {2} f}+\frac {3 d^{5/2} \arctan \left (1+\frac {\sqrt {2} \sqrt {d \tan (e+f x)}}{\sqrt {d}}\right )}{32 \sqrt {2} f}+\frac {3 d^{5/2} \log \left (\sqrt {d}+\sqrt {d} \tan (e+f x)-\sqrt {2} \sqrt {d \tan (e+f x)}\right )}{64 \sqrt {2} f}-\frac {3 d^{5/2} \log \left (\sqrt {d}+\sqrt {d} \tan (e+f x)+\sqrt {2} \sqrt {d \tan (e+f x)}\right )}{64 \sqrt {2} f}+\frac {3 d \cos ^2(e+f x) (d \tan (e+f x))^{3/2}}{16 f}-\frac {d \cos ^4(e+f x) (d \tan (e+f x))^{3/2}}{4 f} \]
-3/64*d^(5/2)*arctan(1-2^(1/2)*(d*tan(f*x+e))^(1/2)/d^(1/2))/f*2^(1/2)+3/6 4*d^(5/2)*arctan(1+2^(1/2)*(d*tan(f*x+e))^(1/2)/d^(1/2))/f*2^(1/2)+3/128*d ^(5/2)*ln(d^(1/2)-2^(1/2)*(d*tan(f*x+e))^(1/2)+d^(1/2)*tan(f*x+e))/f*2^(1/ 2)-3/128*d^(5/2)*ln(d^(1/2)+2^(1/2)*(d*tan(f*x+e))^(1/2)+d^(1/2)*tan(f*x+e ))/f*2^(1/2)+3/16*d*cos(f*x+e)^2*(d*tan(f*x+e))^(3/2)/f-1/4*d*cos(f*x+e)^4 *(d*tan(f*x+e))^(3/2)/f
Time = 0.32 (sec) , antiderivative size = 125, normalized size of antiderivative = 0.49 \[ \int \cos ^4(e+f x) (d \tan (e+f x))^{5/2} \, dx=-\frac {d^2 \left (3 \arcsin (\cos (e+f x)-\sin (e+f x)) \csc (e+f x) \sqrt {\sin (2 (e+f x))}+3 \csc (e+f x) \log \left (\cos (e+f x)+\sin (e+f x)+\sqrt {\sin (2 (e+f x))}\right ) \sqrt {\sin (2 (e+f x))}-2 \sin (2 (e+f x))+2 \sin (4 (e+f x))\right ) \sqrt {d \tan (e+f x)}}{64 f} \]
-1/64*(d^2*(3*ArcSin[Cos[e + f*x] - Sin[e + f*x]]*Csc[e + f*x]*Sqrt[Sin[2* (e + f*x)]] + 3*Csc[e + f*x]*Log[Cos[e + f*x] + Sin[e + f*x] + Sqrt[Sin[2* (e + f*x)]]]*Sqrt[Sin[2*(e + f*x)]] - 2*Sin[2*(e + f*x)] + 2*Sin[4*(e + f* x)])*Sqrt[d*Tan[e + f*x]])/f
Time = 0.45 (sec) , antiderivative size = 262, normalized size of antiderivative = 1.04, number of steps used = 15, number of rules used = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.667, Rules used = {3042, 3087, 252, 253, 266, 27, 826, 1476, 1082, 217, 1479, 25, 27, 1103}
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.
| \(\displaystyle \int \cos ^4(e+f x) (d \tan (e+f x))^{5/2} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {(d \tan (e+f x))^{5/2}}{\sec (e+f x)^4}dx\) |
| \(\Big \downarrow \) 3087 |
| \(\displaystyle \frac {\int \frac {(d \tan (e+f x))^{5/2}}{\left (\tan ^2(e+f x)+1\right )^3}d\tan (e+f x)}{f}\) |
| \(\Big \downarrow \) 252 |
| \(\displaystyle \frac {\frac {3}{8} d^2 \int \frac {\sqrt {d \tan (e+f x)}}{\left (\tan ^2(e+f x)+1\right )^2}d\tan (e+f x)-\frac {d (d \tan (e+f x))^{3/2}}{4 \left (\tan ^2(e+f x)+1\right )^2}}{f}\) |
| \(\Big \downarrow \) 253 |
| \(\displaystyle \frac {\frac {3}{8} d^2 \left (\frac {1}{4} \int \frac {\sqrt {d \tan (e+f x)}}{\tan ^2(e+f x)+1}d\tan (e+f x)+\frac {(d \tan (e+f x))^{3/2}}{2 d \left (\tan ^2(e+f x)+1\right )}\right )-\frac {d (d \tan (e+f x))^{3/2}}{4 \left (\tan ^2(e+f x)+1\right )^2}}{f}\) |
| \(\Big \downarrow \) 266 |
| \(\displaystyle \frac {\frac {3}{8} d^2 \left (\frac {\int \frac {d^3 \tan (e+f x)}{\tan ^2(e+f x) d^2+d^2}d\sqrt {d \tan (e+f x)}}{2 d}+\frac {(d \tan (e+f x))^{3/2}}{2 d \left (\tan ^2(e+f x)+1\right )}\right )-\frac {d (d \tan (e+f x))^{3/2}}{4 \left (\tan ^2(e+f x)+1\right )^2}}{f}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {3}{8} d^2 \left (\frac {1}{2} d \int \frac {d \tan (e+f x)}{\tan ^2(e+f x) d^2+d^2}d\sqrt {d \tan (e+f x)}+\frac {(d \tan (e+f x))^{3/2}}{2 d \left (\tan ^2(e+f x)+1\right )}\right )-\frac {d (d \tan (e+f x))^{3/2}}{4 \left (\tan ^2(e+f x)+1\right )^2}}{f}\) |
| \(\Big \downarrow \) 826 |
| \(\displaystyle \frac {\frac {3}{8} d^2 \left (\frac {1}{2} d \left (\frac {1}{2} \int \frac {\tan (e+f x) d+d}{\tan ^2(e+f x) d^2+d^2}d\sqrt {d \tan (e+f x)}-\frac {1}{2} \int \frac {d-d \tan (e+f x)}{\tan ^2(e+f x) d^2+d^2}d\sqrt {d \tan (e+f x)}\right )+\frac {(d \tan (e+f x))^{3/2}}{2 d \left (\tan ^2(e+f x)+1\right )}\right )-\frac {d (d \tan (e+f x))^{3/2}}{4 \left (\tan ^2(e+f x)+1\right )^2}}{f}\) |
| \(\Big \downarrow \) 1476 |
| \(\displaystyle \frac {\frac {3}{8} d^2 \left (\frac {1}{2} d \left (\frac {1}{2} \left (\frac {1}{2} \int \frac {1}{\tan (e+f x) d+d-\sqrt {2} \sqrt {d \tan (e+f x)} \sqrt {d}}d\sqrt {d \tan (e+f x)}+\frac {1}{2} \int \frac {1}{\tan (e+f x) d+d+\sqrt {2} \sqrt {d \tan (e+f x)} \sqrt {d}}d\sqrt {d \tan (e+f x)}\right )-\frac {1}{2} \int \frac {d-d \tan (e+f x)}{\tan ^2(e+f x) d^2+d^2}d\sqrt {d \tan (e+f x)}\right )+\frac {(d \tan (e+f x))^{3/2}}{2 d \left (\tan ^2(e+f x)+1\right )}\right )-\frac {d (d \tan (e+f x))^{3/2}}{4 \left (\tan ^2(e+f x)+1\right )^2}}{f}\) |
| \(\Big \downarrow \) 1082 |
| \(\displaystyle \frac {\frac {3}{8} d^2 \left (\frac {1}{2} d \left (\frac {1}{2} \left (\frac {\int \frac {1}{-d \tan (e+f x)-1}d\left (1-\frac {\sqrt {2} \sqrt {d \tan (e+f x)}}{\sqrt {d}}\right )}{\sqrt {2} \sqrt {d}}-\frac {\int \frac {1}{-d \tan (e+f x)-1}d\left (\frac {\sqrt {2} \sqrt {d \tan (e+f x)}}{\sqrt {d}}+1\right )}{\sqrt {2} \sqrt {d}}\right )-\frac {1}{2} \int \frac {d-d \tan (e+f x)}{\tan ^2(e+f x) d^2+d^2}d\sqrt {d \tan (e+f x)}\right )+\frac {(d \tan (e+f x))^{3/2}}{2 d \left (\tan ^2(e+f x)+1\right )}\right )-\frac {d (d \tan (e+f x))^{3/2}}{4 \left (\tan ^2(e+f x)+1\right )^2}}{f}\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle \frac {\frac {3}{8} d^2 \left (\frac {1}{2} d \left (\frac {1}{2} \left (\frac {\arctan \left (\frac {\sqrt {2} \sqrt {d \tan (e+f x)}}{\sqrt {d}}+1\right )}{\sqrt {2} \sqrt {d}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt {d \tan (e+f x)}}{\sqrt {d}}\right )}{\sqrt {2} \sqrt {d}}\right )-\frac {1}{2} \int \frac {d-d \tan (e+f x)}{\tan ^2(e+f x) d^2+d^2}d\sqrt {d \tan (e+f x)}\right )+\frac {(d \tan (e+f x))^{3/2}}{2 d \left (\tan ^2(e+f x)+1\right )}\right )-\frac {d (d \tan (e+f x))^{3/2}}{4 \left (\tan ^2(e+f x)+1\right )^2}}{f}\) |
| \(\Big \downarrow \) 1479 |
| \(\displaystyle \frac {\frac {3}{8} d^2 \left (\frac {1}{2} d \left (\frac {1}{2} \left (\frac {\int -\frac {\sqrt {2} \sqrt {d}-2 \sqrt {d \tan (e+f x)}}{\tan (e+f x) d+d-\sqrt {2} \sqrt {d \tan (e+f x)} \sqrt {d}}d\sqrt {d \tan (e+f x)}}{2 \sqrt {2} \sqrt {d}}+\frac {\int -\frac {\sqrt {2} \left (\sqrt {d}+\sqrt {2} \sqrt {d \tan (e+f x)}\right )}{\tan (e+f x) d+d+\sqrt {2} \sqrt {d \tan (e+f x)} \sqrt {d}}d\sqrt {d \tan (e+f x)}}{2 \sqrt {2} \sqrt {d}}\right )+\frac {1}{2} \left (\frac {\arctan \left (\frac {\sqrt {2} \sqrt {d \tan (e+f x)}}{\sqrt {d}}+1\right )}{\sqrt {2} \sqrt {d}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt {d \tan (e+f x)}}{\sqrt {d}}\right )}{\sqrt {2} \sqrt {d}}\right )\right )+\frac {(d \tan (e+f x))^{3/2}}{2 d \left (\tan ^2(e+f x)+1\right )}\right )-\frac {d (d \tan (e+f x))^{3/2}}{4 \left (\tan ^2(e+f x)+1\right )^2}}{f}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\frac {3}{8} d^2 \left (\frac {1}{2} d \left (\frac {1}{2} \left (-\frac {\int \frac {\sqrt {2} \sqrt {d}-2 \sqrt {d \tan (e+f x)}}{\tan (e+f x) d+d-\sqrt {2} \sqrt {d \tan (e+f x)} \sqrt {d}}d\sqrt {d \tan (e+f x)}}{2 \sqrt {2} \sqrt {d}}-\frac {\int \frac {\sqrt {2} \left (\sqrt {d}+\sqrt {2} \sqrt {d \tan (e+f x)}\right )}{\tan (e+f x) d+d+\sqrt {2} \sqrt {d \tan (e+f x)} \sqrt {d}}d\sqrt {d \tan (e+f x)}}{2 \sqrt {2} \sqrt {d}}\right )+\frac {1}{2} \left (\frac {\arctan \left (\frac {\sqrt {2} \sqrt {d \tan (e+f x)}}{\sqrt {d}}+1\right )}{\sqrt {2} \sqrt {d}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt {d \tan (e+f x)}}{\sqrt {d}}\right )}{\sqrt {2} \sqrt {d}}\right )\right )+\frac {(d \tan (e+f x))^{3/2}}{2 d \left (\tan ^2(e+f x)+1\right )}\right )-\frac {d (d \tan (e+f x))^{3/2}}{4 \left (\tan ^2(e+f x)+1\right )^2}}{f}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {3}{8} d^2 \left (\frac {1}{2} d \left (\frac {1}{2} \left (-\frac {\int \frac {\sqrt {2} \sqrt {d}-2 \sqrt {d \tan (e+f x)}}{\tan (e+f x) d+d-\sqrt {2} \sqrt {d \tan (e+f x)} \sqrt {d}}d\sqrt {d \tan (e+f x)}}{2 \sqrt {2} \sqrt {d}}-\frac {\int \frac {\sqrt {d}+\sqrt {2} \sqrt {d \tan (e+f x)}}{\tan (e+f x) d+d+\sqrt {2} \sqrt {d \tan (e+f x)} \sqrt {d}}d\sqrt {d \tan (e+f x)}}{2 \sqrt {d}}\right )+\frac {1}{2} \left (\frac {\arctan \left (\frac {\sqrt {2} \sqrt {d \tan (e+f x)}}{\sqrt {d}}+1\right )}{\sqrt {2} \sqrt {d}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt {d \tan (e+f x)}}{\sqrt {d}}\right )}{\sqrt {2} \sqrt {d}}\right )\right )+\frac {(d \tan (e+f x))^{3/2}}{2 d \left (\tan ^2(e+f x)+1\right )}\right )-\frac {d (d \tan (e+f x))^{3/2}}{4 \left (\tan ^2(e+f x)+1\right )^2}}{f}\) |
| \(\Big \downarrow \) 1103 |
| \(\displaystyle \frac {\frac {3}{8} d^2 \left (\frac {1}{2} d \left (\frac {1}{2} \left (\frac {\arctan \left (\frac {\sqrt {2} \sqrt {d \tan (e+f x)}}{\sqrt {d}}+1\right )}{\sqrt {2} \sqrt {d}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt {d \tan (e+f x)}}{\sqrt {d}}\right )}{\sqrt {2} \sqrt {d}}\right )+\frac {1}{2} \left (\frac {\log \left (d \tan (e+f x)-\sqrt {2} \sqrt {d} \sqrt {d \tan (e+f x)}+d\right )}{2 \sqrt {2} \sqrt {d}}-\frac {\log \left (d \tan (e+f x)+\sqrt {2} \sqrt {d} \sqrt {d \tan (e+f x)}+d\right )}{2 \sqrt {2} \sqrt {d}}\right )\right )+\frac {(d \tan (e+f x))^{3/2}}{2 d \left (\tan ^2(e+f x)+1\right )}\right )-\frac {d (d \tan (e+f x))^{3/2}}{4 \left (\tan ^2(e+f x)+1\right )^2}}{f}\) |
(-1/4*(d*(d*Tan[e + f*x])^(3/2))/(1 + Tan[e + f*x]^2)^2 + (3*d^2*((d*((-(A rcTan[1 - (Sqrt[2]*Sqrt[d*Tan[e + f*x]])/Sqrt[d]]/(Sqrt[2]*Sqrt[d])) + Arc Tan[1 + (Sqrt[2]*Sqrt[d*Tan[e + f*x]])/Sqrt[d]]/(Sqrt[2]*Sqrt[d]))/2 + (Lo g[d + d*Tan[e + f*x] - Sqrt[2]*Sqrt[d]*Sqrt[d*Tan[e + f*x]]]/(2*Sqrt[2]*Sq rt[d]) - Log[d + d*Tan[e + f*x] + Sqrt[2]*Sqrt[d]*Sqrt[d*Tan[e + f*x]]]/(2 *Sqrt[2]*Sqrt[d]))/2))/2 + (d*Tan[e + f*x])^(3/2)/(2*d*(1 + Tan[e + f*x]^2 ))))/8)/f
3.3.52.3.1 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[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(-(Rt[-a, 2]*Rt[-b, 2])^( -1))*ArcTan[Rt[-b, 2]*(x/Rt[-a, 2])], x] /; FreeQ[{a, b}, x] && PosQ[a/b] & & (LtQ[a, 0] || LtQ[b, 0])
Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[c*(c*x )^(m - 1)*((a + b*x^2)^(p + 1)/(2*b*(p + 1))), x] - Simp[c^2*((m - 1)/(2*b* (p + 1))) Int[(c*x)^(m - 2)*(a + b*x^2)^(p + 1), x], x] /; FreeQ[{a, b, c }, x] && LtQ[p, -1] && GtQ[m, 1] && !ILtQ[(m + 2*p + 3)/2, 0] && IntBinomi alQ[a, b, c, 2, m, p, x]
Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(-(c*x )^(m + 1))*((a + b*x^2)^(p + 1)/(2*a*c*(p + 1))), x] + Simp[(m + 2*p + 3)/( 2*a*(p + 1)) Int[(c*x)^m*(a + b*x^2)^(p + 1), x], x] /; FreeQ[{a, b, c, m }, x] && LtQ[p, -1] && IntBinomialQ[a, b, c, 2, m, p, x]
Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> With[{k = De nominator[m]}, Simp[k/c Subst[Int[x^(k*(m + 1) - 1)*(a + b*(x^(2*k)/c^2)) ^p, x], x, (c*x)^(1/k)], x]] /; FreeQ[{a, b, c, p}, x] && FractionQ[m] && I ntBinomialQ[a, b, c, 2, m, p, x]
Int[(x_)^2/((a_) + (b_.)*(x_)^4), x_Symbol] :> With[{r = Numerator[Rt[a/b, 2]], s = Denominator[Rt[a/b, 2]]}, Simp[1/(2*s) Int[(r + s*x^2)/(a + b*x^ 4), x], x] - Simp[1/(2*s) Int[(r - s*x^2)/(a + b*x^4), x], x]] /; FreeQ[{ a, b}, x] && (GtQ[a/b, 0] || (PosQ[a/b] && AtomQ[SplitProduct[SumBaseQ, a]] && AtomQ[SplitProduct[SumBaseQ, b]]))
Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> With[{q = 1 - 4*S implify[a*(c/b^2)]}, Simp[-2/b Subst[Int[1/(q - x^2), x], x, 1 + 2*c*(x/b )], x] /; RationalQ[q] && (EqQ[q^2, 1] || !RationalQ[b^2 - 4*a*c])] /; Fre eQ[{a, b, c}, x]
Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> S imp[d*(Log[RemoveContent[a + b*x + c*x^2, x]]/b), x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[2*c*d - b*e, 0]
Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[ 2*(d/e), 2]}, Simp[e/(2*c) Int[1/Simp[d/e + q*x + x^2, x], x], x] + Simp[ e/(2*c) Int[1/Simp[d/e - q*x + x^2, x], x], x]] /; FreeQ[{a, c, d, e}, x] && EqQ[c*d^2 - a*e^2, 0] && PosQ[d*e]
Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[ -2*(d/e), 2]}, Simp[e/(2*c*q) Int[(q - 2*x)/Simp[d/e + q*x - x^2, x], x], x] + Simp[e/(2*c*q) Int[(q + 2*x)/Simp[d/e - q*x - x^2, x], x], x]] /; F reeQ[{a, c, d, e}, x] && EqQ[c*d^2 - a*e^2, 0] && NegQ[d*e]
Int[sec[(e_.) + (f_.)*(x_)]^(m_)*((b_.)*tan[(e_.) + (f_.)*(x_)])^(n_.), x_S ymbol] :> Simp[1/f Subst[Int[(b*x)^n*(1 + x^2)^(m/2 - 1), x], x, Tan[e + f*x]], x] /; FreeQ[{b, e, f, n}, x] && IntegerQ[m/2] && !(IntegerQ[(n - 1) /2] && LtQ[0, n, m - 1])
Leaf count of result is larger than twice the leaf count of optimal. \(640\) vs. \(2(193)=386\).
Time = 59.81 (sec) , antiderivative size = 641, normalized size of antiderivative = 2.53
| method | result | size |
| default | \(\frac {\cos \left (f x +e \right ) \left (\sin ^{2}\left (f x +e \right )\right ) \left (16 \sqrt {2}\, \left (\cos ^{3}\left (f x +e \right )\right ) \sqrt {-\frac {\sin \left (f x +e \right ) \cos \left (f x +e \right )}{\left (\cos \left (f x +e \right )+1\right )^{2}}}\, \sin \left (f x +e \right )+16 \sqrt {2}\, \left (\cos ^{2}\left (f x +e \right )\right ) \sqrt {-\frac {\sin \left (f x +e \right ) \cos \left (f x +e \right )}{\left (\cos \left (f x +e \right )+1\right )^{2}}}\, \sin \left (f x +e \right )-12 \sqrt {2}\, \cos \left (f x +e \right ) \sqrt {-\frac {\sin \left (f x +e \right ) \cos \left (f x +e \right )}{\left (\cos \left (f x +e \right )+1\right )^{2}}}\, \sin \left (f x +e \right )-12 \sqrt {2}\, \sqrt {-\frac {\sin \left (f x +e \right ) \cos \left (f x +e \right )}{\left (\cos \left (f x +e \right )+1\right )^{2}}}\, \sin \left (f x +e \right )+3 \ln \left (-\frac {\cot \left (f x +e \right ) \cos \left (f x +e \right )-2 \cot \left (f x +e \right )+2 \sin \left (f x +e \right ) \sqrt {-\left (\cot ^{3}\left (f x +e \right )\right )+3 \left (\cot ^{2}\left (f x +e \right )\right ) \csc \left (f x +e \right )-3 \left (\csc ^{2}\left (f x +e \right )\right ) \cot \left (f x +e \right )+\csc ^{3}\left (f x +e \right )+\cot \left (f x +e \right )-\csc \left (f x +e \right )}-2 \cos \left (f x +e \right )-\sin \left (f x +e \right )+\csc \left (f x +e \right )+2}{\cos \left (f x +e \right )-1}\right )-3 \ln \left (\frac {2 \sin \left (f x +e \right ) \sqrt {-\left (\cot ^{3}\left (f x +e \right )\right )+3 \left (\cot ^{2}\left (f x +e \right )\right ) \csc \left (f x +e \right )-3 \left (\csc ^{2}\left (f x +e \right )\right ) \cot \left (f x +e \right )+\csc ^{3}\left (f x +e \right )+\cot \left (f x +e \right )-\csc \left (f x +e \right )}-\cot \left (f x +e \right ) \cos \left (f x +e \right )+\sin \left (f x +e \right )+2 \cos \left (f x +e \right )+2 \cot \left (f x +e \right )-\csc \left (f x +e \right )-2}{\cos \left (f x +e \right )-1}\right )-6 \arctan \left (\frac {\sqrt {2}\, \sqrt {-\frac {\sin \left (f x +e \right ) \cos \left (f x +e \right )}{\left (\cos \left (f x +e \right )+1\right )^{2}}}\, \sin \left (f x +e \right )-\cos \left (f x +e \right )+1}{\cos \left (f x +e \right )-1}\right )-6 \arctan \left (\frac {\sqrt {2}\, \sqrt {-\frac {\sin \left (f x +e \right ) \cos \left (f x +e \right )}{\left (\cos \left (f x +e \right )+1\right )^{2}}}\, \sin \left (f x +e \right )+\cos \left (f x +e \right )-1}{\cos \left (f x +e \right )-1}\right )\right ) \sqrt {d \tan \left (f x +e \right )}\, d^{2} \sqrt {2}}{128 f \left (\cos \left (f x +e \right )-1\right ) \left (\cos \left (f x +e \right )+1\right )^{2} \sqrt {-\frac {\sin \left (f x +e \right ) \cos \left (f x +e \right )}{\left (\cos \left (f x +e \right )+1\right )^{2}}}}\) | \(641\) |
1/128/f*cos(f*x+e)*sin(f*x+e)^2*(16*2^(1/2)*cos(f*x+e)^3*(-sin(f*x+e)*cos( f*x+e)/(cos(f*x+e)+1)^2)^(1/2)*sin(f*x+e)+16*2^(1/2)*cos(f*x+e)^2*(-sin(f* x+e)*cos(f*x+e)/(cos(f*x+e)+1)^2)^(1/2)*sin(f*x+e)-12*2^(1/2)*cos(f*x+e)*( -sin(f*x+e)*cos(f*x+e)/(cos(f*x+e)+1)^2)^(1/2)*sin(f*x+e)-12*2^(1/2)*(-sin (f*x+e)*cos(f*x+e)/(cos(f*x+e)+1)^2)^(1/2)*sin(f*x+e)+3*ln(-(cot(f*x+e)*co s(f*x+e)-2*cot(f*x+e)+2*sin(f*x+e)*(-cot(f*x+e)^3+3*cot(f*x+e)^2*csc(f*x+e )-3*csc(f*x+e)^2*cot(f*x+e)+csc(f*x+e)^3+cot(f*x+e)-csc(f*x+e))^(1/2)-2*co s(f*x+e)-sin(f*x+e)+csc(f*x+e)+2)/(cos(f*x+e)-1))-3*ln((2*sin(f*x+e)*(-cot (f*x+e)^3+3*cot(f*x+e)^2*csc(f*x+e)-3*csc(f*x+e)^2*cot(f*x+e)+csc(f*x+e)^3 +cot(f*x+e)-csc(f*x+e))^(1/2)-cot(f*x+e)*cos(f*x+e)+sin(f*x+e)+2*cos(f*x+e )+2*cot(f*x+e)-csc(f*x+e)-2)/(cos(f*x+e)-1))-6*arctan((2^(1/2)*(-sin(f*x+e )*cos(f*x+e)/(cos(f*x+e)+1)^2)^(1/2)*sin(f*x+e)-cos(f*x+e)+1)/(cos(f*x+e)- 1))-6*arctan((2^(1/2)*(-sin(f*x+e)*cos(f*x+e)/(cos(f*x+e)+1)^2)^(1/2)*sin( f*x+e)+cos(f*x+e)-1)/(cos(f*x+e)-1)))*(d*tan(f*x+e))^(1/2)*d^2/(cos(f*x+e) -1)/(cos(f*x+e)+1)^2/(-sin(f*x+e)*cos(f*x+e)/(cos(f*x+e)+1)^2)^(1/2)*2^(1/ 2)
Result contains complex when optimal does not.
Time = 0.43 (sec) , antiderivative size = 985, normalized size of antiderivative = 3.89 \[ \int \cos ^4(e+f x) (d \tan (e+f x))^{5/2} \, dx=\text {Too large to display} \]
-1/256*(16*(4*d^2*cos(f*x + e)^3 - 3*d^2*cos(f*x + e))*sqrt(d*sin(f*x + e) /cos(f*x + e))*sin(f*x + e) - 3*I*(-d^10/f^4)^(1/4)*f*log(27/2*d^8*cos(f*x + e)*sin(f*x + e) + 27/4*(2*d^3*f^2*cos(f*x + e)^2 - d^3*f^2)*sqrt(-d^10/ f^4) - 27/2*(I*(-d^10/f^4)^(1/4)*d^5*f*cos(f*x + e)*sin(f*x + e) + I*(-d^1 0/f^4)^(3/4)*f^3*cos(f*x + e)^2)*sqrt(d*sin(f*x + e)/cos(f*x + e))) + 3*I* (-d^10/f^4)^(1/4)*f*log(27/2*d^8*cos(f*x + e)*sin(f*x + e) + 27/4*(2*d^3*f ^2*cos(f*x + e)^2 - d^3*f^2)*sqrt(-d^10/f^4) - 27/2*(-I*(-d^10/f^4)^(1/4)* d^5*f*cos(f*x + e)*sin(f*x + e) - I*(-d^10/f^4)^(3/4)*f^3*cos(f*x + e)^2)* sqrt(d*sin(f*x + e)/cos(f*x + e))) + 3*(-d^10/f^4)^(1/4)*f*log(27/2*d^8*co s(f*x + e)*sin(f*x + e) - 27/4*(2*d^3*f^2*cos(f*x + e)^2 - d^3*f^2)*sqrt(- d^10/f^4) + 27/2*((-d^10/f^4)^(1/4)*d^5*f*cos(f*x + e)*sin(f*x + e) - (-d^ 10/f^4)^(3/4)*f^3*cos(f*x + e)^2)*sqrt(d*sin(f*x + e)/cos(f*x + e))) - 3*( -d^10/f^4)^(1/4)*f*log(27/2*d^8*cos(f*x + e)*sin(f*x + e) - 27/4*(2*d^3*f^ 2*cos(f*x + e)^2 - d^3*f^2)*sqrt(-d^10/f^4) - 27/2*((-d^10/f^4)^(1/4)*d^5* f*cos(f*x + e)*sin(f*x + e) - (-d^10/f^4)^(3/4)*f^3*cos(f*x + e)^2)*sqrt(d *sin(f*x + e)/cos(f*x + e))) + 3*(-d^10/f^4)^(1/4)*f*log(27*d^8 + 54*((-d^ 10/f^4)^(1/4)*d^5*f*cos(f*x + e)^2 - (-d^10/f^4)^(3/4)*f^3*cos(f*x + e)*si n(f*x + e))*sqrt(d*sin(f*x + e)/cos(f*x + e))) - 3*(-d^10/f^4)^(1/4)*f*log (27*d^8 - 54*((-d^10/f^4)^(1/4)*d^5*f*cos(f*x + e)^2 - (-d^10/f^4)^(3/4)*f ^3*cos(f*x + e)*sin(f*x + e))*sqrt(d*sin(f*x + e)/cos(f*x + e))) - 3*I*...
Timed out. \[ \int \cos ^4(e+f x) (d \tan (e+f x))^{5/2} \, dx=\text {Timed out} \]
Time = 0.59 (sec) , antiderivative size = 225, normalized size of antiderivative = 0.89 \[ \int \cos ^4(e+f x) (d \tan (e+f x))^{5/2} \, dx=\frac {3 \, d^{4} {\left (\frac {2 \, \sqrt {2} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} \sqrt {d} + 2 \, \sqrt {d \tan \left (f x + e\right )}\right )}}{2 \, \sqrt {d}}\right )}{\sqrt {d}} + \frac {2 \, \sqrt {2} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} \sqrt {d} - 2 \, \sqrt {d \tan \left (f x + e\right )}\right )}}{2 \, \sqrt {d}}\right )}{\sqrt {d}} - \frac {\sqrt {2} \log \left (d \tan \left (f x + e\right ) + \sqrt {2} \sqrt {d \tan \left (f x + e\right )} \sqrt {d} + d\right )}{\sqrt {d}} + \frac {\sqrt {2} \log \left (d \tan \left (f x + e\right ) - \sqrt {2} \sqrt {d \tan \left (f x + e\right )} \sqrt {d} + d\right )}{\sqrt {d}}\right )} + \frac {8 \, {\left (3 \, \left (d \tan \left (f x + e\right )\right )^{\frac {7}{2}} d^{4} - \left (d \tan \left (f x + e\right )\right )^{\frac {3}{2}} d^{6}\right )}}{d^{4} \tan \left (f x + e\right )^{4} + 2 \, d^{4} \tan \left (f x + e\right )^{2} + d^{4}}}{128 \, d f} \]
1/128*(3*d^4*(2*sqrt(2)*arctan(1/2*sqrt(2)*(sqrt(2)*sqrt(d) + 2*sqrt(d*tan (f*x + e)))/sqrt(d))/sqrt(d) + 2*sqrt(2)*arctan(-1/2*sqrt(2)*(sqrt(2)*sqrt (d) - 2*sqrt(d*tan(f*x + e)))/sqrt(d))/sqrt(d) - sqrt(2)*log(d*tan(f*x + e ) + sqrt(2)*sqrt(d*tan(f*x + e))*sqrt(d) + d)/sqrt(d) + sqrt(2)*log(d*tan( f*x + e) - sqrt(2)*sqrt(d*tan(f*x + e))*sqrt(d) + d)/sqrt(d)) + 8*(3*(d*ta n(f*x + e))^(7/2)*d^4 - (d*tan(f*x + e))^(3/2)*d^6)/(d^4*tan(f*x + e)^4 + 2*d^4*tan(f*x + e)^2 + d^4))/(d*f)
Time = 0.43 (sec) , antiderivative size = 257, normalized size of antiderivative = 1.02 \[ \int \cos ^4(e+f x) (d \tan (e+f x))^{5/2} \, dx=\frac {1}{128} \, d^{2} {\left (\frac {6 \, \sqrt {2} {\left | d \right |}^{\frac {3}{2}} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} \sqrt {{\left | d \right |}} + 2 \, \sqrt {d \tan \left (f x + e\right )}\right )}}{2 \, \sqrt {{\left | d \right |}}}\right )}{d f} + \frac {6 \, \sqrt {2} {\left | d \right |}^{\frac {3}{2}} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} \sqrt {{\left | d \right |}} - 2 \, \sqrt {d \tan \left (f x + e\right )}\right )}}{2 \, \sqrt {{\left | d \right |}}}\right )}{d f} - \frac {3 \, \sqrt {2} {\left | d \right |}^{\frac {3}{2}} \log \left (d \tan \left (f x + e\right ) + \sqrt {2} \sqrt {d \tan \left (f x + e\right )} \sqrt {{\left | d \right |}} + {\left | d \right |}\right )}{d f} + \frac {3 \, \sqrt {2} {\left | d \right |}^{\frac {3}{2}} \log \left (d \tan \left (f x + e\right ) - \sqrt {2} \sqrt {d \tan \left (f x + e\right )} \sqrt {{\left | d \right |}} + {\left | d \right |}\right )}{d f} + \frac {8 \, {\left (3 \, \sqrt {d \tan \left (f x + e\right )} d^{4} \tan \left (f x + e\right )^{3} - \sqrt {d \tan \left (f x + e\right )} d^{4} \tan \left (f x + e\right )\right )}}{{\left (d^{2} \tan \left (f x + e\right )^{2} + d^{2}\right )}^{2} f}\right )} \]
1/128*d^2*(6*sqrt(2)*abs(d)^(3/2)*arctan(1/2*sqrt(2)*(sqrt(2)*sqrt(abs(d)) + 2*sqrt(d*tan(f*x + e)))/sqrt(abs(d)))/(d*f) + 6*sqrt(2)*abs(d)^(3/2)*ar ctan(-1/2*sqrt(2)*(sqrt(2)*sqrt(abs(d)) - 2*sqrt(d*tan(f*x + e)))/sqrt(abs (d)))/(d*f) - 3*sqrt(2)*abs(d)^(3/2)*log(d*tan(f*x + e) + sqrt(2)*sqrt(d*t an(f*x + e))*sqrt(abs(d)) + abs(d))/(d*f) + 3*sqrt(2)*abs(d)^(3/2)*log(d*t an(f*x + e) - sqrt(2)*sqrt(d*tan(f*x + e))*sqrt(abs(d)) + abs(d))/(d*f) + 8*(3*sqrt(d*tan(f*x + e))*d^4*tan(f*x + e)^3 - sqrt(d*tan(f*x + e))*d^4*ta n(f*x + e))/((d^2*tan(f*x + e)^2 + d^2)^2*f))
Timed out. \[ \int \cos ^4(e+f x) (d \tan (e+f x))^{5/2} \, dx=\int {\cos \left (e+f\,x\right )}^4\,{\left (d\,\mathrm {tan}\left (e+f\,x\right )\right )}^{5/2} \,d x \]