Integrand size = 21, antiderivative size = 277 \[ \int \sin ^4(a+b x) (d \tan (a+b x))^{5/2} \, dx=\frac {77 d^{5/2} \arctan \left (1-\frac {\sqrt {2} \sqrt {d \tan (a+b x)}}{\sqrt {d}}\right )}{32 \sqrt {2} b}-\frac {77 d^{5/2} \arctan \left (1+\frac {\sqrt {2} \sqrt {d \tan (a+b x)}}{\sqrt {d}}\right )}{32 \sqrt {2} b}-\frac {77 d^{5/2} \log \left (\sqrt {d}+\sqrt {d} \tan (a+b x)-\sqrt {2} \sqrt {d \tan (a+b x)}\right )}{64 \sqrt {2} b}+\frac {77 d^{5/2} \log \left (\sqrt {d}+\sqrt {d} \tan (a+b x)+\sqrt {2} \sqrt {d \tan (a+b x)}\right )}{64 \sqrt {2} b}+\frac {77 d (d \tan (a+b x))^{3/2}}{48 b}-\frac {11 \cos ^2(a+b x) (d \tan (a+b x))^{7/2}}{16 b d}-\frac {\cos ^4(a+b x) (d \tan (a+b x))^{11/2}}{4 b d^3} \]
77/64*d^(5/2)*arctan(1-2^(1/2)*(d*tan(b*x+a))^(1/2)/d^(1/2))/b*2^(1/2)-77/ 64*d^(5/2)*arctan(1+2^(1/2)*(d*tan(b*x+a))^(1/2)/d^(1/2))/b*2^(1/2)-77/128 *d^(5/2)*ln(d^(1/2)-2^(1/2)*(d*tan(b*x+a))^(1/2)+d^(1/2)*tan(b*x+a))/b*2^( 1/2)+77/128*d^(5/2)*ln(d^(1/2)+2^(1/2)*(d*tan(b*x+a))^(1/2)+d^(1/2)*tan(b* x+a))/b*2^(1/2)+77/48*d*(d*tan(b*x+a))^(3/2)/b-11/16*cos(b*x+a)^2*(d*tan(b *x+a))^(7/2)/b/d-1/4*cos(b*x+a)^4*(d*tan(b*x+a))^(11/2)/b/d^3
Time = 0.83 (sec) , antiderivative size = 142, normalized size of antiderivative = 0.51 \[ \int \sin ^4(a+b x) (d \tan (a+b x))^{5/2} \, dx=\frac {d \left (128+204 \cos ^2(a+b x)+231 \arcsin (\cos (a+b x)-\sin (a+b x)) \cot (a+b x) \csc (a+b x) \sqrt {\sin (2 (a+b x))}+231 \cot (a+b x) \csc (a+b x) \log \left (\cos (a+b x)+\sin (a+b x)+\sqrt {\sin (2 (a+b x))}\right ) \sqrt {\sin (2 (a+b x))}-6 \cot (a+b x) \sin (4 (a+b x))\right ) (d \tan (a+b x))^{3/2}}{192 b} \]
(d*(128 + 204*Cos[a + b*x]^2 + 231*ArcSin[Cos[a + b*x] - Sin[a + b*x]]*Cot [a + b*x]*Csc[a + b*x]*Sqrt[Sin[2*(a + b*x)]] + 231*Cot[a + b*x]*Csc[a + b *x]*Log[Cos[a + b*x] + Sin[a + b*x] + Sqrt[Sin[2*(a + b*x)]]]*Sqrt[Sin[2*( a + b*x)]] - 6*Cot[a + b*x]*Sin[4*(a + b*x)])*(d*Tan[a + b*x])^(3/2))/(192 *b)
Time = 0.46 (sec) , antiderivative size = 273, normalized size of antiderivative = 0.99, number of steps used = 15, number of rules used = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.667, Rules used = {3042, 3071, 252, 252, 262, 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.
| \(\displaystyle \int \sin ^4(a+b x) (d \tan (a+b x))^{5/2} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \sin (a+b x)^4 (d \tan (a+b x))^{5/2}dx\) |
| \(\Big \downarrow \) 3071 |
| \(\displaystyle \frac {d \int \frac {(d \tan (a+b x))^{13/2}}{\left (\tan ^2(a+b x) d^2+d^2\right )^3}d(d \tan (a+b x))}{b}\) |
| \(\Big \downarrow \) 252 |
| \(\displaystyle \frac {d \left (\frac {11}{8} \int \frac {(d \tan (a+b x))^{9/2}}{\left (\tan ^2(a+b x) d^2+d^2\right )^2}d(d \tan (a+b x))-\frac {(d \tan (a+b x))^{11/2}}{4 \left (d^2 \tan ^2(a+b x)+d^2\right )^2}\right )}{b}\) |
| \(\Big \downarrow \) 252 |
| \(\displaystyle \frac {d \left (\frac {11}{8} \left (\frac {7}{4} \int \frac {(d \tan (a+b x))^{5/2}}{\tan ^2(a+b x) d^2+d^2}d(d \tan (a+b x))-\frac {(d \tan (a+b x))^{7/2}}{2 \left (d^2 \tan ^2(a+b x)+d^2\right )}\right )-\frac {(d \tan (a+b x))^{11/2}}{4 \left (d^2 \tan ^2(a+b x)+d^2\right )^2}\right )}{b}\) |
| \(\Big \downarrow \) 262 |
| \(\displaystyle \frac {d \left (\frac {11}{8} \left (\frac {7}{4} \left (\frac {2}{3} (d \tan (a+b x))^{3/2}-d^2 \int \frac {\sqrt {d \tan (a+b x)}}{\tan ^2(a+b x) d^2+d^2}d(d \tan (a+b x))\right )-\frac {(d \tan (a+b x))^{7/2}}{2 \left (d^2 \tan ^2(a+b x)+d^2\right )}\right )-\frac {(d \tan (a+b x))^{11/2}}{4 \left (d^2 \tan ^2(a+b x)+d^2\right )^2}\right )}{b}\) |
| \(\Big \downarrow \) 266 |
| \(\displaystyle \frac {d \left (\frac {11}{8} \left (\frac {7}{4} \left (\frac {2}{3} (d \tan (a+b x))^{3/2}-2 d^2 \int \frac {d^2 \tan ^2(a+b x)}{d^4 \tan ^4(a+b x)+d^2}d\sqrt {d \tan (a+b x)}\right )-\frac {(d \tan (a+b x))^{7/2}}{2 \left (d^2 \tan ^2(a+b x)+d^2\right )}\right )-\frac {(d \tan (a+b x))^{11/2}}{4 \left (d^2 \tan ^2(a+b x)+d^2\right )^2}\right )}{b}\) |
| \(\Big \downarrow \) 826 |
| \(\displaystyle \frac {d \left (\frac {11}{8} \left (\frac {7}{4} \left (\frac {2}{3} (d \tan (a+b x))^{3/2}-2 d^2 \left (\frac {1}{2} \int \frac {d^2 \tan ^2(a+b x)+d}{d^4 \tan ^4(a+b x)+d^2}d\sqrt {d \tan (a+b x)}-\frac {1}{2} \int \frac {d-d^2 \tan ^2(a+b x)}{d^4 \tan ^4(a+b x)+d^2}d\sqrt {d \tan (a+b x)}\right )\right )-\frac {(d \tan (a+b x))^{7/2}}{2 \left (d^2 \tan ^2(a+b x)+d^2\right )}\right )-\frac {(d \tan (a+b x))^{11/2}}{4 \left (d^2 \tan ^2(a+b x)+d^2\right )^2}\right )}{b}\) |
| \(\Big \downarrow \) 1476 |
| \(\displaystyle \frac {d \left (\frac {11}{8} \left (\frac {7}{4} \left (\frac {2}{3} (d \tan (a+b x))^{3/2}-2 d^2 \left (\frac {1}{2} \left (\frac {1}{2} \int \frac {1}{d^2 \tan ^2(a+b x)-\sqrt {2} d^{3/2} \tan (a+b x)+d}d\sqrt {d \tan (a+b x)}+\frac {1}{2} \int \frac {1}{d^2 \tan ^2(a+b x)+\sqrt {2} d^{3/2} \tan (a+b x)+d}d\sqrt {d \tan (a+b x)}\right )-\frac {1}{2} \int \frac {d-d^2 \tan ^2(a+b x)}{d^4 \tan ^4(a+b x)+d^2}d\sqrt {d \tan (a+b x)}\right )\right )-\frac {(d \tan (a+b x))^{7/2}}{2 \left (d^2 \tan ^2(a+b x)+d^2\right )}\right )-\frac {(d \tan (a+b x))^{11/2}}{4 \left (d^2 \tan ^2(a+b x)+d^2\right )^2}\right )}{b}\) |
| \(\Big \downarrow \) 1082 |
| \(\displaystyle \frac {d \left (\frac {11}{8} \left (\frac {7}{4} \left (\frac {2}{3} (d \tan (a+b x))^{3/2}-2 d^2 \left (\frac {1}{2} \left (\frac {\int \frac {1}{-d^2 \tan ^2(a+b x)-1}d\left (1-\sqrt {2} \sqrt {d} \tan (a+b x)\right )}{\sqrt {2} \sqrt {d}}-\frac {\int \frac {1}{-d^2 \tan ^2(a+b x)-1}d\left (\sqrt {2} \sqrt {d} \tan (a+b x)+1\right )}{\sqrt {2} \sqrt {d}}\right )-\frac {1}{2} \int \frac {d-d^2 \tan ^2(a+b x)}{d^4 \tan ^4(a+b x)+d^2}d\sqrt {d \tan (a+b x)}\right )\right )-\frac {(d \tan (a+b x))^{7/2}}{2 \left (d^2 \tan ^2(a+b x)+d^2\right )}\right )-\frac {(d \tan (a+b x))^{11/2}}{4 \left (d^2 \tan ^2(a+b x)+d^2\right )^2}\right )}{b}\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle \frac {d \left (\frac {11}{8} \left (\frac {7}{4} \left (\frac {2}{3} (d \tan (a+b x))^{3/2}-2 d^2 \left (\frac {1}{2} \left (\frac {\arctan \left (\sqrt {2} \sqrt {d} \tan (a+b x)+1\right )}{\sqrt {2} \sqrt {d}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {d} \tan (a+b x)\right )}{\sqrt {2} \sqrt {d}}\right )-\frac {1}{2} \int \frac {d-d^2 \tan ^2(a+b x)}{d^4 \tan ^4(a+b x)+d^2}d\sqrt {d \tan (a+b x)}\right )\right )-\frac {(d \tan (a+b x))^{7/2}}{2 \left (d^2 \tan ^2(a+b x)+d^2\right )}\right )-\frac {(d \tan (a+b x))^{11/2}}{4 \left (d^2 \tan ^2(a+b x)+d^2\right )^2}\right )}{b}\) |
| \(\Big \downarrow \) 1479 |
| \(\displaystyle \frac {d \left (\frac {11}{8} \left (\frac {7}{4} \left (\frac {2}{3} (d \tan (a+b x))^{3/2}-2 d^2 \left (\frac {1}{2} \left (\frac {\int -\frac {\sqrt {2} \sqrt {d}-2 \sqrt {d \tan (a+b x)}}{d^2 \tan ^2(a+b x)-\sqrt {2} d^{3/2} \tan (a+b x)+d}d\sqrt {d \tan (a+b x)}}{2 \sqrt {2} \sqrt {d}}+\frac {\int -\frac {\sqrt {2} \left (\sqrt {d}+\sqrt {2} \sqrt {d \tan (a+b x)}\right )}{d^2 \tan ^2(a+b x)+\sqrt {2} d^{3/2} \tan (a+b x)+d}d\sqrt {d \tan (a+b x)}}{2 \sqrt {2} \sqrt {d}}\right )+\frac {1}{2} \left (\frac {\arctan \left (\sqrt {2} \sqrt {d} \tan (a+b x)+1\right )}{\sqrt {2} \sqrt {d}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {d} \tan (a+b x)\right )}{\sqrt {2} \sqrt {d}}\right )\right )\right )-\frac {(d \tan (a+b x))^{7/2}}{2 \left (d^2 \tan ^2(a+b x)+d^2\right )}\right )-\frac {(d \tan (a+b x))^{11/2}}{4 \left (d^2 \tan ^2(a+b x)+d^2\right )^2}\right )}{b}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {d \left (\frac {11}{8} \left (\frac {7}{4} \left (\frac {2}{3} (d \tan (a+b x))^{3/2}-2 d^2 \left (\frac {1}{2} \left (-\frac {\int \frac {\sqrt {2} \sqrt {d}-2 \sqrt {d \tan (a+b x)}}{d^2 \tan ^2(a+b x)-\sqrt {2} d^{3/2} \tan (a+b x)+d}d\sqrt {d \tan (a+b x)}}{2 \sqrt {2} \sqrt {d}}-\frac {\int \frac {\sqrt {2} \left (\sqrt {d}+\sqrt {2} \sqrt {d \tan (a+b x)}\right )}{d^2 \tan ^2(a+b x)+\sqrt {2} d^{3/2} \tan (a+b x)+d}d\sqrt {d \tan (a+b x)}}{2 \sqrt {2} \sqrt {d}}\right )+\frac {1}{2} \left (\frac {\arctan \left (\sqrt {2} \sqrt {d} \tan (a+b x)+1\right )}{\sqrt {2} \sqrt {d}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {d} \tan (a+b x)\right )}{\sqrt {2} \sqrt {d}}\right )\right )\right )-\frac {(d \tan (a+b x))^{7/2}}{2 \left (d^2 \tan ^2(a+b x)+d^2\right )}\right )-\frac {(d \tan (a+b x))^{11/2}}{4 \left (d^2 \tan ^2(a+b x)+d^2\right )^2}\right )}{b}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {d \left (\frac {11}{8} \left (\frac {7}{4} \left (\frac {2}{3} (d \tan (a+b x))^{3/2}-2 d^2 \left (\frac {1}{2} \left (-\frac {\int \frac {\sqrt {2} \sqrt {d}-2 \sqrt {d \tan (a+b x)}}{d^2 \tan ^2(a+b x)-\sqrt {2} d^{3/2} \tan (a+b x)+d}d\sqrt {d \tan (a+b x)}}{2 \sqrt {2} \sqrt {d}}-\frac {\int \frac {\sqrt {d}+\sqrt {2} \sqrt {d \tan (a+b x)}}{d^2 \tan ^2(a+b x)+\sqrt {2} d^{3/2} \tan (a+b x)+d}d\sqrt {d \tan (a+b x)}}{2 \sqrt {d}}\right )+\frac {1}{2} \left (\frac {\arctan \left (\sqrt {2} \sqrt {d} \tan (a+b x)+1\right )}{\sqrt {2} \sqrt {d}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {d} \tan (a+b x)\right )}{\sqrt {2} \sqrt {d}}\right )\right )\right )-\frac {(d \tan (a+b x))^{7/2}}{2 \left (d^2 \tan ^2(a+b x)+d^2\right )}\right )-\frac {(d \tan (a+b x))^{11/2}}{4 \left (d^2 \tan ^2(a+b x)+d^2\right )^2}\right )}{b}\) |
| \(\Big \downarrow \) 1103 |
| \(\displaystyle \frac {d \left (\frac {11}{8} \left (\frac {7}{4} \left (\frac {2}{3} (d \tan (a+b x))^{3/2}-2 d^2 \left (\frac {1}{2} \left (\frac {\arctan \left (\sqrt {2} \sqrt {d} \tan (a+b x)+1\right )}{\sqrt {2} \sqrt {d}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {d} \tan (a+b x)\right )}{\sqrt {2} \sqrt {d}}\right )+\frac {1}{2} \left (\frac {\log \left (-\sqrt {2} d^{3/2} \tan (a+b x)+d^2 \tan ^2(a+b x)+d\right )}{2 \sqrt {2} \sqrt {d}}-\frac {\log \left (\sqrt {2} d^{3/2} \tan (a+b x)+d^2 \tan ^2(a+b x)+d\right )}{2 \sqrt {2} \sqrt {d}}\right )\right )\right )-\frac {(d \tan (a+b x))^{7/2}}{2 \left (d^2 \tan ^2(a+b x)+d^2\right )}\right )-\frac {(d \tan (a+b x))^{11/2}}{4 \left (d^2 \tan ^2(a+b x)+d^2\right )^2}\right )}{b}\) |
(d*(-1/4*(d*Tan[a + b*x])^(11/2)/(d^2 + d^2*Tan[a + b*x]^2)^2 + (11*(-1/2* (d*Tan[a + b*x])^(7/2)/(d^2 + d^2*Tan[a + b*x]^2) + (7*(-2*d^2*((-(ArcTan[ 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*Tan[ a + b*x])^(3/2))/3))/4))/8))/b
3.1.73.3.1 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*(c*x) ^(m - 1)*((a + b*x^2)^(p + 1)/(b*(m + 2*p + 1))), x] - Simp[a*c^2*((m - 1)/ (b*(m + 2*p + 1))) Int[(c*x)^(m - 2)*(a + b*x^2)^p, x], x] /; FreeQ[{a, b , c, p}, x] && GtQ[m, 2 - 1] && NeQ[m + 2*p + 1, 0] && 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. \(525\) vs. \(2(213)=426\).
Time = 48.45 (sec) , antiderivative size = 526, normalized size of antiderivative = 1.90
| method | result | size |
| default | \(-\frac {\tan \left (b x +a \right ) \sqrt {d \tan \left (b x +a \right )}\, \left (48 \left (\cos ^{5}\left (b x +a \right )\right ) \sqrt {2}-48 \sqrt {2}\, \left (\cos ^{4}\left (b x +a \right )\right )-228 \left (\cos ^{3}\left (b x +a \right )\right ) \sqrt {2}+228 \left (\cos ^{2}\left (b x +a \right )\right ) \sqrt {2}-462 \sqrt {-\frac {\cos \left (b x +a \right ) \sin \left (b x +a \right )}{\left (\cos \left (b x +a \right )+1\right )^{2}}}\, \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 ) \cos \left (b x +a \right )+462 \sqrt {-\frac {\cos \left (b x +a \right ) \sin \left (b x +a \right )}{\left (\cos \left (b x +a \right )+1\right )^{2}}}\, \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 ) \cos \left (b x +a \right )-231 \cos \left (b x +a \right ) \sqrt {-\frac {\cos \left (b x +a \right ) \sin \left (b x +a \right )}{\left (\cos \left (b x +a \right )+1\right )^{2}}}\, \ln \left (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}}}\, \sqrt {2}\, \csc \left (b x +a \right )+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}}}\, \sqrt {2}\, \cot \left (b x +a \right )-2 \cot \left (b x +a \right )+2\right )+231 \cos \left (b x +a \right ) \sqrt {-\frac {\cos \left (b x +a \right ) \sin \left (b x +a \right )}{\left (\cos \left (b x +a \right )+1\right )^{2}}}\, \ln \left (-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}}}\, \sqrt {2}\, \csc \left (b x +a \right )-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}}}\, \sqrt {2}\, \cot \left (b x +a \right )-2 \cot \left (b x +a \right )+2\right )-128 \sqrt {2}\, \cos \left (b x +a \right )+128 \sqrt {2}\right ) d^{2} \sqrt {2}}{384 b \left (-1+\cos \left (b x +a \right )\right )}\) | \(526\) |
-1/384/b*tan(b*x+a)*(d*tan(b*x+a))^(1/2)*(48*cos(b*x+a)^5*2^(1/2)-48*2^(1/ 2)*cos(b*x+a)^4-228*cos(b*x+a)^3*2^(1/2)+228*cos(b*x+a)^2*2^(1/2)-462*(-co s(b*x+a)*sin(b*x+a)/(cos(b*x+a)+1)^2)^(1/2)*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) ))*cos(b*x+a)+462*(-cos(b*x+a)*sin(b*x+a)/(cos(b*x+a)+1)^2)^(1/2)*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)))*cos(b*x+a)-231*cos(b*x+a)*(-cos(b*x+a)*sin(b*x+a)/ (cos(b*x+a)+1)^2)^(1/2)*ln(2*(-cos(b*x+a)*sin(b*x+a)/(cos(b*x+a)+1)^2)^(1/ 2)*2^(1/2)*csc(b*x+a)+2*(-cos(b*x+a)*sin(b*x+a)/(cos(b*x+a)+1)^2)^(1/2)*2^ (1/2)*cot(b*x+a)-2*cot(b*x+a)+2)+231*cos(b*x+a)*(-cos(b*x+a)*sin(b*x+a)/(c os(b*x+a)+1)^2)^(1/2)*ln(-2*(-cos(b*x+a)*sin(b*x+a)/(cos(b*x+a)+1)^2)^(1/2 )*2^(1/2)*csc(b*x+a)-2*(-cos(b*x+a)*sin(b*x+a)/(cos(b*x+a)+1)^2)^(1/2)*2^( 1/2)*cot(b*x+a)-2*cot(b*x+a)+2)-128*2^(1/2)*cos(b*x+a)+128*2^(1/2))/(-1+co s(b*x+a))*d^2*2^(1/2)
Result contains complex when optimal does not.
Time = 0.42 (sec) , antiderivative size = 1048, normalized size of antiderivative = 3.78 \[ \int \sin ^4(a+b x) (d \tan (a+b x))^{5/2} \, dx=\text {Too large to display} \]
-1/768*(231*(-d^10/b^4)^(1/4)*b*cos(b*x + a)*log(-456533/2*d^8*cos(b*x + a )*sin(b*x + a) + 456533/4*(2*b^2*d^3*cos(b*x + a)^2 - b^2*d^3)*sqrt(-d^10/ b^4) + 456533/2*((-d^10/b^4)^(1/4)*b*d^5*cos(b*x + a)*sin(b*x + a) - (-d^1 0/b^4)^(3/4)*b^3*cos(b*x + a)^2)*sqrt(d*sin(b*x + a)/cos(b*x + a))) - 231* (-d^10/b^4)^(1/4)*b*cos(b*x + a)*log(-456533/2*d^8*cos(b*x + a)*sin(b*x + a) + 456533/4*(2*b^2*d^3*cos(b*x + a)^2 - b^2*d^3)*sqrt(-d^10/b^4) - 45653 3/2*((-d^10/b^4)^(1/4)*b*d^5*cos(b*x + a)*sin(b*x + a) - (-d^10/b^4)^(3/4) *b^3*cos(b*x + a)^2)*sqrt(d*sin(b*x + a)/cos(b*x + a))) - 231*I*(-d^10/b^4 )^(1/4)*b*cos(b*x + a)*log(-456533/2*d^8*cos(b*x + a)*sin(b*x + a) - 45653 3/4*(2*b^2*d^3*cos(b*x + a)^2 - b^2*d^3)*sqrt(-d^10/b^4) - 456533/2*(I*(-d ^10/b^4)^(1/4)*b*d^5*cos(b*x + a)*sin(b*x + a) + I*(-d^10/b^4)^(3/4)*b^3*c os(b*x + a)^2)*sqrt(d*sin(b*x + a)/cos(b*x + a))) + 231*I*(-d^10/b^4)^(1/4 )*b*cos(b*x + a)*log(-456533/2*d^8*cos(b*x + a)*sin(b*x + a) - 456533/4*(2 *b^2*d^3*cos(b*x + a)^2 - b^2*d^3)*sqrt(-d^10/b^4) - 456533/2*(-I*(-d^10/b ^4)^(1/4)*b*d^5*cos(b*x + a)*sin(b*x + a) - I*(-d^10/b^4)^(3/4)*b^3*cos(b* x + a)^2)*sqrt(d*sin(b*x + a)/cos(b*x + a))) - 231*(-d^10/b^4)^(1/4)*b*cos (b*x + a)*log(456533*d^8 + 913066*((-d^10/b^4)^(1/4)*b*d^5*cos(b*x + a)^2 - (-d^10/b^4)^(3/4)*b^3*cos(b*x + a)*sin(b*x + a))*sqrt(d*sin(b*x + a)/cos (b*x + a))) + 231*(-d^10/b^4)^(1/4)*b*cos(b*x + a)*log(456533*d^8 - 913066 *((-d^10/b^4)^(1/4)*b*d^5*cos(b*x + a)^2 - (-d^10/b^4)^(3/4)*b^3*cos(b*...
Timed out. \[ \int \sin ^4(a+b x) (d \tan (a+b x))^{5/2} \, dx=\text {Timed out} \]
Time = 0.31 (sec) , antiderivative size = 240, normalized size of antiderivative = 0.87 \[ \int \sin ^4(a+b x) (d \tan (a+b x))^{5/2} \, dx=-\frac {231 \, d^{8} {\left (\frac {2 \, \sqrt {2} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} \sqrt {d} + 2 \, \sqrt {d \tan \left (b x + a\right )}\right )}}{2 \, \sqrt {d}}\right )}{\sqrt {d}} + \frac {2 \, \sqrt {2} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} \sqrt {d} - 2 \, \sqrt {d \tan \left (b x + a\right )}\right )}}{2 \, \sqrt {d}}\right )}{\sqrt {d}} - \frac {\sqrt {2} \log \left (d \tan \left (b x + a\right ) + \sqrt {2} \sqrt {d \tan \left (b x + a\right )} \sqrt {d} + d\right )}{\sqrt {d}} + \frac {\sqrt {2} \log \left (d \tan \left (b x + a\right ) - \sqrt {2} \sqrt {d \tan \left (b x + a\right )} \sqrt {d} + d\right )}{\sqrt {d}}\right )} - 256 \, \left (d \tan \left (b x + a\right )\right )^{\frac {3}{2}} d^{6} - \frac {24 \, {\left (19 \, \left (d \tan \left (b x + a\right )\right )^{\frac {7}{2}} d^{8} + 15 \, \left (d \tan \left (b x + a\right )\right )^{\frac {3}{2}} d^{10}\right )}}{d^{4} \tan \left (b x + a\right )^{4} + 2 \, d^{4} \tan \left (b x + a\right )^{2} + d^{4}}}{384 \, b d^{5}} \]
-1/384*(231*d^8*(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)*s qrt(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*t an(b*x + a) - sqrt(2)*sqrt(d*tan(b*x + a))*sqrt(d) + d)/sqrt(d)) - 256*(d* tan(b*x + a))^(3/2)*d^6 - 24*(19*(d*tan(b*x + a))^(7/2)*d^8 + 15*(d*tan(b* x + a))^(3/2)*d^10)/(d^4*tan(b*x + a)^4 + 2*d^4*tan(b*x + a)^2 + d^4))/(b* d^5)
Time = 0.35 (sec) , antiderivative size = 278, normalized size of antiderivative = 1.00 \[ \int \sin ^4(a+b x) (d \tan (a+b x))^{5/2} \, dx=-\frac {1}{384} \, d^{2} {\left (\frac {462 \, \sqrt {2} {\left | d \right |}^{\frac {3}{2}} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} \sqrt {{\left | d \right |}} + 2 \, \sqrt {d \tan \left (b x + a\right )}\right )}}{2 \, \sqrt {{\left | d \right |}}}\right )}{b d} + \frac {462 \, \sqrt {2} {\left | d \right |}^{\frac {3}{2}} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} \sqrt {{\left | d \right |}} - 2 \, \sqrt {d \tan \left (b x + a\right )}\right )}}{2 \, \sqrt {{\left | d \right |}}}\right )}{b d} - \frac {231 \, \sqrt {2} {\left | d \right |}^{\frac {3}{2}} \log \left (d \tan \left (b x + a\right ) + \sqrt {2} \sqrt {d \tan \left (b x + a\right )} \sqrt {{\left | d \right |}} + {\left | d \right |}\right )}{b d} + \frac {231 \, \sqrt {2} {\left | d \right |}^{\frac {3}{2}} \log \left (d \tan \left (b x + a\right ) - \sqrt {2} \sqrt {d \tan \left (b x + a\right )} \sqrt {{\left | d \right |}} + {\left | d \right |}\right )}{b d} - \frac {256 \, \sqrt {d \tan \left (b x + a\right )} \tan \left (b x + a\right )}{b} - \frac {24 \, {\left (19 \, \sqrt {d \tan \left (b x + a\right )} d^{4} \tan \left (b x + a\right )^{3} + 15 \, \sqrt {d \tan \left (b x + a\right )} d^{4} \tan \left (b x + a\right )\right )}}{{\left (d^{2} \tan \left (b x + a\right )^{2} + d^{2}\right )}^{2} b}\right )} \]
-1/384*d^2*(462*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) + 462*sqrt(2)*abs(d)^(3/ 2)*arctan(-1/2*sqrt(2)*(sqrt(2)*sqrt(abs(d)) - 2*sqrt(d*tan(b*x + a)))/sqr t(abs(d)))/(b*d) - 231*sqrt(2)*abs(d)^(3/2)*log(d*tan(b*x + a) + sqrt(2)*s qrt(d*tan(b*x + a))*sqrt(abs(d)) + abs(d))/(b*d) + 231*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) - 256*sqrt(d*tan(b*x + a))*tan(b*x + a)/b - 24*(19*sqrt(d*tan(b*x + a))*d^4*tan(b*x + a)^3 + 15*sqrt(d*tan(b*x + a))*d^4*tan(b*x + a))/((d^2* tan(b*x + a)^2 + d^2)^2*b))
Timed out. \[ \int \sin ^4(a+b x) (d \tan (a+b x))^{5/2} \, dx=\int {\sin \left (a+b\,x\right )}^4\,{\left (d\,\mathrm {tan}\left (a+b\,x\right )\right )}^{5/2} \,d x \]