Integrand size = 21, antiderivative size = 257 \[ \int \frac {\sin ^4(a+b x)}{(d \tan (a+b x))^{3/2}} \, dx=-\frac {3 \arctan \left (1-\frac {\sqrt {2} \sqrt {d \tan (a+b x)}}{\sqrt {d}}\right )}{32 \sqrt {2} b d^{3/2}}+\frac {3 \arctan \left (1+\frac {\sqrt {2} \sqrt {d \tan (a+b x)}}{\sqrt {d}}\right )}{32 \sqrt {2} b d^{3/2}}+\frac {3 \log \left (\sqrt {d}+\sqrt {d} \tan (a+b x)-\sqrt {2} \sqrt {d \tan (a+b x)}\right )}{64 \sqrt {2} b d^{3/2}}-\frac {3 \log \left (\sqrt {d}+\sqrt {d} \tan (a+b x)+\sqrt {2} \sqrt {d \tan (a+b x)}\right )}{64 \sqrt {2} b d^{3/2}}+\frac {3 \cos ^2(a+b x) (d \tan (a+b x))^{3/2}}{16 b d^3}-\frac {\cos ^4(a+b x) (d \tan (a+b x))^{3/2}}{4 b d^3} \]
-3/64*arctan(1-2^(1/2)*(d*tan(b*x+a))^(1/2)/d^(1/2))/b/d^(3/2)*2^(1/2)+3/6 4*arctan(1+2^(1/2)*(d*tan(b*x+a))^(1/2)/d^(1/2))/b/d^(3/2)*2^(1/2)+3/128*l n(d^(1/2)-2^(1/2)*(d*tan(b*x+a))^(1/2)+d^(1/2)*tan(b*x+a))/b/d^(3/2)*2^(1/ 2)-3/128*ln(d^(1/2)+2^(1/2)*(d*tan(b*x+a))^(1/2)+d^(1/2)*tan(b*x+a))/b/d^( 3/2)*2^(1/2)+3/16*cos(b*x+a)^2*(d*tan(b*x+a))^(3/2)/b/d^3-1/4*cos(b*x+a)^4 *(d*tan(b*x+a))^(3/2)/b/d^3
Time = 0.59 (sec) , antiderivative size = 123, normalized size of antiderivative = 0.48 \[ \int \frac {\sin ^4(a+b x)}{(d \tan (a+b x))^{3/2}} \, dx=\frac {\csc (a+b x) \left (\cos (a+b x)-2 \cos (3 (a+b x))+\cos (5 (a+b x))-3 \arcsin (\cos (a+b x)-\sin (a+b x)) \sqrt {\sin (2 (a+b x))}-3 \log \left (\cos (a+b x)+\sin (a+b x)+\sqrt {\sin (2 (a+b x))}\right ) \sqrt {\sin (2 (a+b x))}\right ) \sqrt {d \tan (a+b x)}}{64 b d^2} \]
(Csc[a + b*x]*(Cos[a + b*x] - 2*Cos[3*(a + b*x)] + Cos[5*(a + b*x)] - 3*Ar cSin[Cos[a + b*x] - Sin[a + b*x]]*Sqrt[Sin[2*(a + b*x)]] - 3*Log[Cos[a + b *x] + Sin[a + b*x] + Sqrt[Sin[2*(a + b*x)]]]*Sqrt[Sin[2*(a + b*x)]])*Sqrt[ d*Tan[a + b*x]])/(64*b*d^2)
Time = 0.46 (sec) , antiderivative size = 257, normalized size of antiderivative = 1.00, number of steps used = 14, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.619, Rules used = {3042, 3071, 252, 253, 266, 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.
| \(\displaystyle \int \frac {\sin ^4(a+b x)}{(d \tan (a+b x))^{3/2}} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\sin (a+b x)^4}{(d \tan (a+b x))^{3/2}}dx\) |
| \(\Big \downarrow \) 3071 |
| \(\displaystyle \frac {d \int \frac {(d \tan (a+b x))^{5/2}}{\left (\tan ^2(a+b x) d^2+d^2\right )^3}d(d \tan (a+b x))}{b}\) |
| \(\Big \downarrow \) 252 |
| \(\displaystyle \frac {d \left (\frac {3}{8} \int \frac {\sqrt {d \tan (a+b x)}}{\left (\tan ^2(a+b x) d^2+d^2\right )^2}d(d \tan (a+b x))-\frac {(d \tan (a+b x))^{3/2}}{4 \left (d^2 \tan ^2(a+b x)+d^2\right )^2}\right )}{b}\) |
| \(\Big \downarrow \) 253 |
| \(\displaystyle \frac {d \left (\frac {3}{8} \left (\frac {\int \frac {\sqrt {d \tan (a+b x)}}{\tan ^2(a+b x) d^2+d^2}d(d \tan (a+b x))}{4 d^2}+\frac {(d \tan (a+b x))^{3/2}}{2 d^2 \left (d^2 \tan ^2(a+b x)+d^2\right )}\right )-\frac {(d \tan (a+b x))^{3/2}}{4 \left (d^2 \tan ^2(a+b x)+d^2\right )^2}\right )}{b}\) |
| \(\Big \downarrow \) 266 |
| \(\displaystyle \frac {d \left (\frac {3}{8} \left (\frac {\int \frac {d^2 \tan ^2(a+b x)}{d^4 \tan ^4(a+b x)+d^2}d\sqrt {d \tan (a+b x)}}{2 d^2}+\frac {(d \tan (a+b x))^{3/2}}{2 d^2 \left (d^2 \tan ^2(a+b x)+d^2\right )}\right )-\frac {(d \tan (a+b x))^{3/2}}{4 \left (d^2 \tan ^2(a+b x)+d^2\right )^2}\right )}{b}\) |
| \(\Big \downarrow \) 826 |
| \(\displaystyle \frac {d \left (\frac {3}{8} \left (\frac {\frac {1}{2} \int \frac {d^2 \tan ^2(a+b x)+d}{d^4 \tan ^4(a+b x)+d^2}d\sqrt {d \tan (a+b x)}-\frac {1}{2} \int \frac {d-d^2 \tan ^2(a+b x)}{d^4 \tan ^4(a+b x)+d^2}d\sqrt {d \tan (a+b x)}}{2 d^2}+\frac {(d \tan (a+b x))^{3/2}}{2 d^2 \left (d^2 \tan ^2(a+b x)+d^2\right )}\right )-\frac {(d \tan (a+b x))^{3/2}}{4 \left (d^2 \tan ^2(a+b x)+d^2\right )^2}\right )}{b}\) |
| \(\Big \downarrow \) 1476 |
| \(\displaystyle \frac {d \left (\frac {3}{8} \left (\frac {\frac {1}{2} \left (\frac {1}{2} \int \frac {1}{d^2 \tan ^2(a+b x)-\sqrt {2} d^{3/2} \tan (a+b x)+d}d\sqrt {d \tan (a+b x)}+\frac {1}{2} \int \frac {1}{d^2 \tan ^2(a+b x)+\sqrt {2} d^{3/2} \tan (a+b x)+d}d\sqrt {d \tan (a+b x)}\right )-\frac {1}{2} \int \frac {d-d^2 \tan ^2(a+b x)}{d^4 \tan ^4(a+b x)+d^2}d\sqrt {d \tan (a+b x)}}{2 d^2}+\frac {(d \tan (a+b x))^{3/2}}{2 d^2 \left (d^2 \tan ^2(a+b x)+d^2\right )}\right )-\frac {(d \tan (a+b x))^{3/2}}{4 \left (d^2 \tan ^2(a+b x)+d^2\right )^2}\right )}{b}\) |
| \(\Big \downarrow \) 1082 |
| \(\displaystyle \frac {d \left (\frac {3}{8} \left (\frac {\frac {1}{2} \left (\frac {\int \frac {1}{-d^2 \tan ^2(a+b x)-1}d\left (1-\sqrt {2} \sqrt {d} \tan (a+b x)\right )}{\sqrt {2} \sqrt {d}}-\frac {\int \frac {1}{-d^2 \tan ^2(a+b x)-1}d\left (\sqrt {2} \sqrt {d} \tan (a+b x)+1\right )}{\sqrt {2} \sqrt {d}}\right )-\frac {1}{2} \int \frac {d-d^2 \tan ^2(a+b x)}{d^4 \tan ^4(a+b x)+d^2}d\sqrt {d \tan (a+b x)}}{2 d^2}+\frac {(d \tan (a+b x))^{3/2}}{2 d^2 \left (d^2 \tan ^2(a+b x)+d^2\right )}\right )-\frac {(d \tan (a+b x))^{3/2}}{4 \left (d^2 \tan ^2(a+b x)+d^2\right )^2}\right )}{b}\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle \frac {d \left (\frac {3}{8} \left (\frac {\frac {1}{2} \left (\frac {\arctan \left (\sqrt {2} \sqrt {d} \tan (a+b x)+1\right )}{\sqrt {2} \sqrt {d}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {d} \tan (a+b x)\right )}{\sqrt {2} \sqrt {d}}\right )-\frac {1}{2} \int \frac {d-d^2 \tan ^2(a+b x)}{d^4 \tan ^4(a+b x)+d^2}d\sqrt {d \tan (a+b x)}}{2 d^2}+\frac {(d \tan (a+b x))^{3/2}}{2 d^2 \left (d^2 \tan ^2(a+b x)+d^2\right )}\right )-\frac {(d \tan (a+b x))^{3/2}}{4 \left (d^2 \tan ^2(a+b x)+d^2\right )^2}\right )}{b}\) |
| \(\Big \downarrow \) 1479 |
| \(\displaystyle \frac {d \left (\frac {3}{8} \left (\frac {\frac {1}{2} \left (\frac {\int -\frac {\sqrt {2} \sqrt {d}-2 \sqrt {d \tan (a+b x)}}{d^2 \tan ^2(a+b x)-\sqrt {2} d^{3/2} \tan (a+b x)+d}d\sqrt {d \tan (a+b x)}}{2 \sqrt {2} \sqrt {d}}+\frac {\int -\frac {\sqrt {2} \left (\sqrt {d}+\sqrt {2} \sqrt {d \tan (a+b x)}\right )}{d^2 \tan ^2(a+b x)+\sqrt {2} d^{3/2} \tan (a+b x)+d}d\sqrt {d \tan (a+b x)}}{2 \sqrt {2} \sqrt {d}}\right )+\frac {1}{2} \left (\frac {\arctan \left (\sqrt {2} \sqrt {d} \tan (a+b x)+1\right )}{\sqrt {2} \sqrt {d}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {d} \tan (a+b x)\right )}{\sqrt {2} \sqrt {d}}\right )}{2 d^2}+\frac {(d \tan (a+b x))^{3/2}}{2 d^2 \left (d^2 \tan ^2(a+b x)+d^2\right )}\right )-\frac {(d \tan (a+b x))^{3/2}}{4 \left (d^2 \tan ^2(a+b x)+d^2\right )^2}\right )}{b}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {d \left (\frac {3}{8} \left (\frac {\frac {1}{2} \left (-\frac {\int \frac {\sqrt {2} \sqrt {d}-2 \sqrt {d \tan (a+b x)}}{d^2 \tan ^2(a+b x)-\sqrt {2} d^{3/2} \tan (a+b x)+d}d\sqrt {d \tan (a+b x)}}{2 \sqrt {2} \sqrt {d}}-\frac {\int \frac {\sqrt {2} \left (\sqrt {d}+\sqrt {2} \sqrt {d \tan (a+b x)}\right )}{d^2 \tan ^2(a+b x)+\sqrt {2} d^{3/2} \tan (a+b x)+d}d\sqrt {d \tan (a+b x)}}{2 \sqrt {2} \sqrt {d}}\right )+\frac {1}{2} \left (\frac {\arctan \left (\sqrt {2} \sqrt {d} \tan (a+b x)+1\right )}{\sqrt {2} \sqrt {d}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {d} \tan (a+b x)\right )}{\sqrt {2} \sqrt {d}}\right )}{2 d^2}+\frac {(d \tan (a+b x))^{3/2}}{2 d^2 \left (d^2 \tan ^2(a+b x)+d^2\right )}\right )-\frac {(d \tan (a+b x))^{3/2}}{4 \left (d^2 \tan ^2(a+b x)+d^2\right )^2}\right )}{b}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {d \left (\frac {3}{8} \left (\frac {\frac {1}{2} \left (-\frac {\int \frac {\sqrt {2} \sqrt {d}-2 \sqrt {d \tan (a+b x)}}{d^2 \tan ^2(a+b x)-\sqrt {2} d^{3/2} \tan (a+b x)+d}d\sqrt {d \tan (a+b x)}}{2 \sqrt {2} \sqrt {d}}-\frac {\int \frac {\sqrt {d}+\sqrt {2} \sqrt {d \tan (a+b x)}}{d^2 \tan ^2(a+b x)+\sqrt {2} d^{3/2} \tan (a+b x)+d}d\sqrt {d \tan (a+b x)}}{2 \sqrt {d}}\right )+\frac {1}{2} \left (\frac {\arctan \left (\sqrt {2} \sqrt {d} \tan (a+b x)+1\right )}{\sqrt {2} \sqrt {d}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {d} \tan (a+b x)\right )}{\sqrt {2} \sqrt {d}}\right )}{2 d^2}+\frac {(d \tan (a+b x))^{3/2}}{2 d^2 \left (d^2 \tan ^2(a+b x)+d^2\right )}\right )-\frac {(d \tan (a+b x))^{3/2}}{4 \left (d^2 \tan ^2(a+b x)+d^2\right )^2}\right )}{b}\) |
| \(\Big \downarrow \) 1103 |
| \(\displaystyle \frac {d \left (\frac {3}{8} \left (\frac {\frac {1}{2} \left (\frac {\arctan \left (\sqrt {2} \sqrt {d} \tan (a+b x)+1\right )}{\sqrt {2} \sqrt {d}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {d} \tan (a+b x)\right )}{\sqrt {2} \sqrt {d}}\right )+\frac {1}{2} \left (\frac {\log \left (-\sqrt {2} d^{3/2} \tan (a+b x)+d^2 \tan ^2(a+b x)+d\right )}{2 \sqrt {2} \sqrt {d}}-\frac {\log \left (\sqrt {2} d^{3/2} \tan (a+b x)+d^2 \tan ^2(a+b x)+d\right )}{2 \sqrt {2} \sqrt {d}}\right )}{2 d^2}+\frac {(d \tan (a+b x))^{3/2}}{2 d^2 \left (d^2 \tan ^2(a+b x)+d^2\right )}\right )-\frac {(d \tan (a+b x))^{3/2}}{4 \left (d^2 \tan ^2(a+b x)+d^2\right )^2}\right )}{b}\) |
(d*(-1/4*(d*Tan[a + b*x])^(3/2)/(d^2 + d^2*Tan[a + b*x]^2)^2 + (3*(((-(Arc Tan[1 - Sqrt[2]*Sqrt[d]*Tan[a + b*x]]/(Sqrt[2]*Sqrt[d])) + ArcTan[1 + Sqrt [2]*Sqrt[d]*Tan[a + b*x]]/(Sqrt[2]*Sqrt[d]))/2 + (Log[d - Sqrt[2]*d^(3/2)* Tan[a + b*x] + d^2*Tan[a + b*x]^2]/(2*Sqrt[2]*Sqrt[d]) - Log[d + Sqrt[2]*d ^(3/2)*Tan[a + b*x] + d^2*Tan[a + b*x]^2]/(2*Sqrt[2]*Sqrt[d]))/2)/(2*d^2) + (d*Tan[a + b*x])^(3/2)/(2*d^2*(d^2 + d^2*Tan[a + b*x]^2))))/8))/b
3.1.94.3.1 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[sin[(e_.) + (f_.)*(x_)]^(m_)*((b_.)*tan[(e_.) + (f_.)*(x_)])^(n_.), x_S ymbol] :> With[{ff = FreeFactors[Tan[e + f*x], x]}, Simp[b*(ff/f) Subst[I nt[(ff*x)^(m + n)/(b^2 + ff^2*x^2)^(m/2 + 1), x], x, b*(Tan[e + f*x]/ff)], x]] /; FreeQ[{b, e, f, n}, x] && IntegerQ[m/2]
Leaf count of result is larger than twice the leaf count of optimal. \(620\) vs. \(2(197)=394\).
Time = 10.06 (sec) , antiderivative size = 621, normalized size of antiderivative = 2.42
| method | result | size |
| default | \(\frac {\csc \left (b x +a \right ) \left (-1+\cos \left (b x +a \right )\right ) \left (16 \sqrt {2}\, \sqrt {-\frac {\cos \left (b x +a \right ) \sin \left (b x +a \right )}{\left (\cos \left (b x +a \right )+1\right )^{2}}}\, \left (\cos ^{3}\left (b x +a \right )\right ) \sin \left (b x +a \right )+16 \left (\cos ^{2}\left (b x +a \right )\right ) \sin \left (b x +a \right ) \sqrt {2}\, \sqrt {-\frac {\cos \left (b x +a \right ) \sin \left (b x +a \right )}{\left (\cos \left (b x +a \right )+1\right )^{2}}}-12 \cos \left (b x +a \right ) \sin \left (b x +a \right ) \sqrt {2}\, \sqrt {-\frac {\cos \left (b x +a \right ) \sin \left (b x +a \right )}{\left (\cos \left (b x +a \right )+1\right )^{2}}}-12 \sin \left (b x +a \right ) \sqrt {2}\, \sqrt {-\frac {\cos \left (b x +a \right ) \sin \left (b x +a \right )}{\left (\cos \left (b x +a \right )+1\right )^{2}}}-6 \arctan \left (\frac {\sin \left (b x +a \right ) \sqrt {2}\, \sqrt {-\frac {\cos \left (b x +a \right ) \sin \left (b x +a \right )}{\left (\cos \left (b x +a \right )+1\right )^{2}}}+\cos \left (b x +a \right )-1}{-1+\cos \left (b x +a \right )}\right )-6 \arctan \left (\frac {\sin \left (b x +a \right ) \sqrt {2}\, \sqrt {-\frac {\cos \left (b x +a \right ) \sin \left (b x +a \right )}{\left (\cos \left (b x +a \right )+1\right )^{2}}}-\cos \left (b x +a \right )+1}{-1+\cos \left (b x +a \right )}\right )-3 \ln \left (-\frac {\cot \left (b x +a \right ) \cos \left (b x +a \right )-2 \cot \left (b x +a \right )-2 \sin \left (b x +a \right ) \sqrt {-\left (\cot ^{3}\left (b x +a \right )\right )+3 \left (\cot ^{2}\left (b x +a \right )\right ) \csc \left (b x +a \right )-3 \cot \left (b x +a \right ) \left (\csc ^{2}\left (b x +a \right )\right )+\csc ^{3}\left (b x +a \right )+\cot \left (b x +a \right )-\csc \left (b x +a \right )}-2 \cos \left (b x +a \right )-\sin \left (b x +a \right )+\csc \left (b x +a \right )+2}{-1+\cos \left (b x +a \right )}\right )+3 \ln \left (-\frac {\cot \left (b x +a \right ) \cos \left (b x +a \right )-2 \cot \left (b x +a \right )+2 \sin \left (b x +a \right ) \sqrt {-\left (\cot ^{3}\left (b x +a \right )\right )+3 \left (\cot ^{2}\left (b x +a \right )\right ) \csc \left (b x +a \right )-3 \cot \left (b x +a \right ) \left (\csc ^{2}\left (b x +a \right )\right )+\csc ^{3}\left (b x +a \right )+\cot \left (b x +a \right )-\csc \left (b x +a \right )}-2 \cos \left (b x +a \right )-\sin \left (b x +a \right )+\csc \left (b x +a \right )+2}{-1+\cos \left (b x +a \right )}\right )\right ) \sqrt {2}}{128 b \sqrt {-\frac {\cos \left (b x +a \right ) \sin \left (b x +a \right )}{\left (\cos \left (b x +a \right )+1\right )^{2}}}\, \sqrt {d \tan \left (b x +a \right )}\, d}\) | \(621\) |
1/128/b*csc(b*x+a)*(-1+cos(b*x+a))*(16*2^(1/2)*(-cos(b*x+a)*sin(b*x+a)/(co s(b*x+a)+1)^2)^(1/2)*cos(b*x+a)^3*sin(b*x+a)+16*cos(b*x+a)^2*sin(b*x+a)*2^ (1/2)*(-cos(b*x+a)*sin(b*x+a)/(cos(b*x+a)+1)^2)^(1/2)-12*cos(b*x+a)*sin(b* x+a)*2^(1/2)*(-cos(b*x+a)*sin(b*x+a)/(cos(b*x+a)+1)^2)^(1/2)-12*sin(b*x+a) *2^(1/2)*(-cos(b*x+a)*sin(b*x+a)/(cos(b*x+a)+1)^2)^(1/2)-6*arctan((sin(b*x +a)*2^(1/2)*(-cos(b*x+a)*sin(b*x+a)/(cos(b*x+a)+1)^2)^(1/2)+cos(b*x+a)-1)/ (-1+cos(b*x+a)))-6*arctan((sin(b*x+a)*2^(1/2)*(-cos(b*x+a)*sin(b*x+a)/(cos (b*x+a)+1)^2)^(1/2)-cos(b*x+a)+1)/(-1+cos(b*x+a)))-3*ln(-(cot(b*x+a)*cos(b *x+a)-2*cot(b*x+a)-2*sin(b*x+a)*(-cot(b*x+a)^3+3*cot(b*x+a)^2*csc(b*x+a)-3 *cot(b*x+a)*csc(b*x+a)^2+csc(b*x+a)^3+cot(b*x+a)-csc(b*x+a))^(1/2)-2*cos(b *x+a)-sin(b*x+a)+csc(b*x+a)+2)/(-1+cos(b*x+a)))+3*ln(-(cot(b*x+a)*cos(b*x+ a)-2*cot(b*x+a)+2*sin(b*x+a)*(-cot(b*x+a)^3+3*cot(b*x+a)^2*csc(b*x+a)-3*co t(b*x+a)*csc(b*x+a)^2+csc(b*x+a)^3+cot(b*x+a)-csc(b*x+a))^(1/2)-2*cos(b*x+ a)-sin(b*x+a)+csc(b*x+a)+2)/(-1+cos(b*x+a))))/(-cos(b*x+a)*sin(b*x+a)/(cos (b*x+a)+1)^2)^(1/2)/(d*tan(b*x+a))^(1/2)/d*2^(1/2)
Result contains complex when optimal does not.
Time = 0.40 (sec) , antiderivative size = 986, normalized size of antiderivative = 3.84 \[ \int \frac {\sin ^4(a+b x)}{(d \tan (a+b x))^{3/2}} \, dx=\text {Too large to display} \]
1/256*(3*b*d^2*(-1/(b^4*d^6))^(1/4)*log(1/2*cos(b*x + a)*sin(b*x + a) + 1/ 2*(b^3*d^4*(-1/(b^4*d^6))^(3/4)*cos(b*x + a)^2 - b*d*(-1/(b^4*d^6))^(1/4)* cos(b*x + a)*sin(b*x + a))*sqrt(d*sin(b*x + a)/cos(b*x + a)) - 1/4*(2*b^2* d^3*cos(b*x + a)^2 - b^2*d^3)*sqrt(-1/(b^4*d^6))) - 3*b*d^2*(-1/(b^4*d^6)) ^(1/4)*log(1/2*cos(b*x + a)*sin(b*x + a) - 1/2*(b^3*d^4*(-1/(b^4*d^6))^(3/ 4)*cos(b*x + a)^2 - b*d*(-1/(b^4*d^6))^(1/4)*cos(b*x + a)*sin(b*x + a))*sq rt(d*sin(b*x + a)/cos(b*x + a)) - 1/4*(2*b^2*d^3*cos(b*x + a)^2 - b^2*d^3) *sqrt(-1/(b^4*d^6))) - 3*I*b*d^2*(-1/(b^4*d^6))^(1/4)*log(1/2*cos(b*x + a) *sin(b*x + a) + 1/2*(I*b^3*d^4*(-1/(b^4*d^6))^(3/4)*cos(b*x + a)^2 + I*b*d *(-1/(b^4*d^6))^(1/4)*cos(b*x + a)*sin(b*x + a))*sqrt(d*sin(b*x + a)/cos(b *x + a)) + 1/4*(2*b^2*d^3*cos(b*x + a)^2 - b^2*d^3)*sqrt(-1/(b^4*d^6))) + 3*I*b*d^2*(-1/(b^4*d^6))^(1/4)*log(1/2*cos(b*x + a)*sin(b*x + a) + 1/2*(-I *b^3*d^4*(-1/(b^4*d^6))^(3/4)*cos(b*x + a)^2 - I*b*d*(-1/(b^4*d^6))^(1/4)* cos(b*x + a)*sin(b*x + a))*sqrt(d*sin(b*x + a)/cos(b*x + a)) + 1/4*(2*b^2* d^3*cos(b*x + a)^2 - b^2*d^3)*sqrt(-1/(b^4*d^6))) + 3*b*d^2*(-1/(b^4*d^6)) ^(1/4)*log(2*(b^3*d^4*(-1/(b^4*d^6))^(3/4)*cos(b*x + a)*sin(b*x + a) - b*d *(-1/(b^4*d^6))^(1/4)*cos(b*x + a)^2)*sqrt(d*sin(b*x + a)/cos(b*x + a)) + 1) - 3*b*d^2*(-1/(b^4*d^6))^(1/4)*log(-2*(b^3*d^4*(-1/(b^4*d^6))^(3/4)*cos (b*x + a)*sin(b*x + a) - b*d*(-1/(b^4*d^6))^(1/4)*cos(b*x + a)^2)*sqrt(d*s in(b*x + a)/cos(b*x + a)) + 1) + 3*I*b*d^2*(-1/(b^4*d^6))^(1/4)*log(-2*...
\[ \int \frac {\sin ^4(a+b x)}{(d \tan (a+b x))^{3/2}} \, dx=\int \frac {\sin ^{4}{\left (a + b x \right )}}{\left (d \tan {\left (a + b x \right )}\right )^{\frac {3}{2}}}\, dx \]
Time = 0.37 (sec) , antiderivative size = 225, normalized size of antiderivative = 0.88 \[ \int \frac {\sin ^4(a+b x)}{(d \tan (a+b x))^{3/2}} \, dx=\frac {3 \, d^{4} {\left (\frac {2 \, \sqrt {2} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} \sqrt {d} + 2 \, \sqrt {d \tan \left (b x + a\right )}\right )}}{2 \, \sqrt {d}}\right )}{\sqrt {d}} + \frac {2 \, \sqrt {2} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} \sqrt {d} - 2 \, \sqrt {d \tan \left (b x + a\right )}\right )}}{2 \, \sqrt {d}}\right )}{\sqrt {d}} - \frac {\sqrt {2} \log \left (d \tan \left (b x + a\right ) + \sqrt {2} \sqrt {d \tan \left (b x + a\right )} \sqrt {d} + d\right )}{\sqrt {d}} + \frac {\sqrt {2} \log \left (d \tan \left (b x + a\right ) - \sqrt {2} \sqrt {d \tan \left (b x + a\right )} \sqrt {d} + d\right )}{\sqrt {d}}\right )} + \frac {8 \, {\left (3 \, \left (d \tan \left (b x + a\right )\right )^{\frac {7}{2}} d^{4} - \left (d \tan \left (b x + a\right )\right )^{\frac {3}{2}} d^{6}\right )}}{d^{4} \tan \left (b x + a\right )^{4} + 2 \, d^{4} \tan \left (b x + a\right )^{2} + d^{4}}}{128 \, b d^{5}} \]
1/128*(3*d^4*(2*sqrt(2)*arctan(1/2*sqrt(2)*(sqrt(2)*sqrt(d) + 2*sqrt(d*tan (b*x + a)))/sqrt(d))/sqrt(d) + 2*sqrt(2)*arctan(-1/2*sqrt(2)*(sqrt(2)*sqrt (d) - 2*sqrt(d*tan(b*x + a)))/sqrt(d))/sqrt(d) - sqrt(2)*log(d*tan(b*x + a ) + sqrt(2)*sqrt(d*tan(b*x + a))*sqrt(d) + d)/sqrt(d) + sqrt(2)*log(d*tan( b*x + a) - sqrt(2)*sqrt(d*tan(b*x + a))*sqrt(d) + d)/sqrt(d)) + 8*(3*(d*ta n(b*x + a))^(7/2)*d^4 - (d*tan(b*x + a))^(3/2)*d^6)/(d^4*tan(b*x + a)^4 + 2*d^4*tan(b*x + a)^2 + d^4))/(b*d^5)
Time = 0.43 (sec) , antiderivative size = 257, normalized size of antiderivative = 1.00 \[ \int \frac {\sin ^4(a+b x)}{(d \tan (a+b x))^{3/2}} \, dx=\frac {\frac {6 \, \sqrt {2} {\left | d \right |}^{\frac {3}{2}} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} \sqrt {{\left | d \right |}} + 2 \, \sqrt {d \tan \left (b x + a\right )}\right )}}{2 \, \sqrt {{\left | d \right |}}}\right )}{b d^{2}} + \frac {6 \, \sqrt {2} {\left | d \right |}^{\frac {3}{2}} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} \sqrt {{\left | d \right |}} - 2 \, \sqrt {d \tan \left (b x + a\right )}\right )}}{2 \, \sqrt {{\left | d \right |}}}\right )}{b d^{2}} - \frac {3 \, \sqrt {2} {\left | d \right |}^{\frac {3}{2}} \log \left (d \tan \left (b x + a\right ) + \sqrt {2} \sqrt {d \tan \left (b x + a\right )} \sqrt {{\left | d \right |}} + {\left | d \right |}\right )}{b d^{2}} + \frac {3 \, \sqrt {2} {\left | d \right |}^{\frac {3}{2}} \log \left (d \tan \left (b x + a\right ) - \sqrt {2} \sqrt {d \tan \left (b x + a\right )} \sqrt {{\left | d \right |}} + {\left | d \right |}\right )}{b d^{2}} + \frac {8 \, {\left (3 \, \sqrt {d \tan \left (b x + a\right )} d^{3} \tan \left (b x + a\right )^{3} - \sqrt {d \tan \left (b x + a\right )} d^{3} \tan \left (b x + a\right )\right )}}{{\left (d^{2} \tan \left (b x + a\right )^{2} + d^{2}\right )}^{2} b}}{128 \, d} \]
1/128*(6*sqrt(2)*abs(d)^(3/2)*arctan(1/2*sqrt(2)*(sqrt(2)*sqrt(abs(d)) + 2 *sqrt(d*tan(b*x + a)))/sqrt(abs(d)))/(b*d^2) + 6*sqrt(2)*abs(d)^(3/2)*arct an(-1/2*sqrt(2)*(sqrt(2)*sqrt(abs(d)) - 2*sqrt(d*tan(b*x + a)))/sqrt(abs(d )))/(b*d^2) - 3*sqrt(2)*abs(d)^(3/2)*log(d*tan(b*x + a) + sqrt(2)*sqrt(d*t an(b*x + a))*sqrt(abs(d)) + abs(d))/(b*d^2) + 3*sqrt(2)*abs(d)^(3/2)*log(d *tan(b*x + a) - sqrt(2)*sqrt(d*tan(b*x + a))*sqrt(abs(d)) + abs(d))/(b*d^2 ) + 8*(3*sqrt(d*tan(b*x + a))*d^3*tan(b*x + a)^3 - sqrt(d*tan(b*x + a))*d^ 3*tan(b*x + a))/((d^2*tan(b*x + a)^2 + d^2)^2*b))/d
Timed out. \[ \int \frac {\sin ^4(a+b x)}{(d \tan (a+b x))^{3/2}} \, dx=\int \frac {{\sin \left (a+b\,x\right )}^4}{{\left (d\,\mathrm {tan}\left (a+b\,x\right )\right )}^{3/2}} \,d x \]