Integrand size = 21, antiderivative size = 277 \[ \int \sin ^4(a+b x) (d \tan (a+b x))^{3/2} \, dx=\frac {45 d^{3/2} \arctan \left (1-\frac {\sqrt {2} \sqrt {d \tan (a+b x)}}{\sqrt {d}}\right )}{32 \sqrt {2} b}-\frac {45 d^{3/2} \arctan \left (1+\frac {\sqrt {2} \sqrt {d \tan (a+b x)}}{\sqrt {d}}\right )}{32 \sqrt {2} b}+\frac {45 d^{3/2} \log \left (\sqrt {d}+\sqrt {d} \tan (a+b x)-\sqrt {2} \sqrt {d \tan (a+b x)}\right )}{64 \sqrt {2} b}-\frac {45 d^{3/2} \log \left (\sqrt {d}+\sqrt {d} \tan (a+b x)+\sqrt {2} \sqrt {d \tan (a+b x)}\right )}{64 \sqrt {2} b}+\frac {45 d \sqrt {d \tan (a+b x)}}{16 b}-\frac {9 \cos ^2(a+b x) (d \tan (a+b x))^{5/2}}{16 b d}-\frac {\cos ^4(a+b x) (d \tan (a+b x))^{9/2}}{4 b d^3} \]
45/64*d^(3/2)*arctan(1-2^(1/2)*(d*tan(b*x+a))^(1/2)/d^(1/2))/b*2^(1/2)-45/ 64*d^(3/2)*arctan(1+2^(1/2)*(d*tan(b*x+a))^(1/2)/d^(1/2))/b*2^(1/2)+45/128 *d^(3/2)*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)-45/128*d^(3/2)*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)+45/16*d*(d*tan(b*x+a))^(1/2)/b-9/16*cos(b*x+a)^2*(d*tan(b* x+a))^(5/2)/b/d-1/4*cos(b*x+a)^4*(d*tan(b*x+a))^(9/2)/b/d^3
Time = 0.77 (sec) , antiderivative size = 123, normalized size of antiderivative = 0.44 \[ \int \sin ^4(a+b x) (d \tan (a+b x))^{3/2} \, dx=-\frac {d \csc (a+b x) \left (-143 \sin (a+b x)-45 \arcsin (\cos (a+b x)-\sin (a+b x)) \sqrt {\sin (2 (a+b x))}+45 \log \left (\cos (a+b x)+\sin (a+b x)+\sqrt {\sin (2 (a+b x))}\right ) \sqrt {\sin (2 (a+b x))}-14 \sin (3 (a+b x))+\sin (5 (a+b x))\right ) \sqrt {d \tan (a+b x)}}{64 b} \]
-1/64*(d*Csc[a + b*x]*(-143*Sin[a + b*x] - 45*ArcSin[Cos[a + b*x] - Sin[a + b*x]]*Sqrt[Sin[2*(a + b*x)]] + 45*Log[Cos[a + b*x] + Sin[a + b*x] + Sqrt [Sin[2*(a + b*x)]]]*Sqrt[Sin[2*(a + b*x)]] - 14*Sin[3*(a + b*x)] + Sin[5*( a + b*x)])*Sqrt[d*Tan[a + b*x]])/b
Time = 0.47 (sec) , antiderivative size = 277, normalized size of antiderivative = 1.00, 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, 755, 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.
| \(\displaystyle \int \sin ^4(a+b x) (d \tan (a+b x))^{3/2} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \sin (a+b x)^4 (d \tan (a+b x))^{3/2}dx\) |
| \(\Big \downarrow \) 3071 |
| \(\displaystyle \frac {d \int \frac {(d \tan (a+b x))^{11/2}}{\left (\tan ^2(a+b x) d^2+d^2\right )^3}d(d \tan (a+b x))}{b}\) |
| \(\Big \downarrow \) 252 |
| \(\displaystyle \frac {d \left (\frac {9}{8} \int \frac {(d \tan (a+b x))^{7/2}}{\left (\tan ^2(a+b x) d^2+d^2\right )^2}d(d \tan (a+b x))-\frac {(d \tan (a+b x))^{9/2}}{4 \left (d^2 \tan ^2(a+b x)+d^2\right )^2}\right )}{b}\) |
| \(\Big \downarrow \) 252 |
| \(\displaystyle \frac {d \left (\frac {9}{8} \left (\frac {5}{4} \int \frac {(d \tan (a+b x))^{3/2}}{\tan ^2(a+b x) d^2+d^2}d(d \tan (a+b x))-\frac {(d \tan (a+b x))^{5/2}}{2 \left (d^2 \tan ^2(a+b x)+d^2\right )}\right )-\frac {(d \tan (a+b x))^{9/2}}{4 \left (d^2 \tan ^2(a+b x)+d^2\right )^2}\right )}{b}\) |
| \(\Big \downarrow \) 262 |
| \(\displaystyle \frac {d \left (\frac {9}{8} \left (\frac {5}{4} \left (2 \sqrt {d \tan (a+b x)}-d^2 \int \frac {1}{\sqrt {d \tan (a+b x)} \left (\tan ^2(a+b x) d^2+d^2\right )}d(d \tan (a+b x))\right )-\frac {(d \tan (a+b x))^{5/2}}{2 \left (d^2 \tan ^2(a+b x)+d^2\right )}\right )-\frac {(d \tan (a+b x))^{9/2}}{4 \left (d^2 \tan ^2(a+b x)+d^2\right )^2}\right )}{b}\) |
| \(\Big \downarrow \) 266 |
| \(\displaystyle \frac {d \left (\frac {9}{8} \left (\frac {5}{4} \left (2 \sqrt {d \tan (a+b x)}-2 d^2 \int \frac {1}{d^4 \tan ^4(a+b x)+d^2}d\sqrt {d \tan (a+b x)}\right )-\frac {(d \tan (a+b x))^{5/2}}{2 \left (d^2 \tan ^2(a+b x)+d^2\right )}\right )-\frac {(d \tan (a+b x))^{9/2}}{4 \left (d^2 \tan ^2(a+b x)+d^2\right )^2}\right )}{b}\) |
| \(\Big \downarrow \) 755 |
| \(\displaystyle \frac {d \left (\frac {9}{8} \left (\frac {5}{4} \left (2 \sqrt {d \tan (a+b x)}-2 d^2 \left (\frac {\int \frac {d-d^2 \tan ^2(a+b x)}{d^4 \tan ^4(a+b x)+d^2}d\sqrt {d \tan (a+b x)}}{2 d}+\frac {\int \frac {d^2 \tan ^2(a+b x)+d}{d^4 \tan ^4(a+b x)+d^2}d\sqrt {d \tan (a+b x)}}{2 d}\right )\right )-\frac {(d \tan (a+b x))^{5/2}}{2 \left (d^2 \tan ^2(a+b x)+d^2\right )}\right )-\frac {(d \tan (a+b x))^{9/2}}{4 \left (d^2 \tan ^2(a+b x)+d^2\right )^2}\right )}{b}\) |
| \(\Big \downarrow \) 1476 |
| \(\displaystyle \frac {d \left (\frac {9}{8} \left (\frac {5}{4} \left (2 \sqrt {d \tan (a+b x)}-2 d^2 \left (\frac {\frac {1}{2} \int \frac {1}{d^2 \tan ^2(a+b x)-\sqrt {2} d^{3/2} \tan (a+b x)+d}d\sqrt {d \tan (a+b x)}+\frac {1}{2} \int \frac {1}{d^2 \tan ^2(a+b x)+\sqrt {2} d^{3/2} \tan (a+b x)+d}d\sqrt {d \tan (a+b x)}}{2 d}+\frac {\int \frac {d-d^2 \tan ^2(a+b x)}{d^4 \tan ^4(a+b x)+d^2}d\sqrt {d \tan (a+b x)}}{2 d}\right )\right )-\frac {(d \tan (a+b x))^{5/2}}{2 \left (d^2 \tan ^2(a+b x)+d^2\right )}\right )-\frac {(d \tan (a+b x))^{9/2}}{4 \left (d^2 \tan ^2(a+b x)+d^2\right )^2}\right )}{b}\) |
| \(\Big \downarrow \) 1082 |
| \(\displaystyle \frac {d \left (\frac {9}{8} \left (\frac {5}{4} \left (2 \sqrt {d \tan (a+b x)}-2 d^2 \left (\frac {\frac {\int \frac {1}{-d^2 \tan ^2(a+b x)-1}d\left (1-\sqrt {2} \sqrt {d} \tan (a+b x)\right )}{\sqrt {2} \sqrt {d}}-\frac {\int \frac {1}{-d^2 \tan ^2(a+b x)-1}d\left (\sqrt {2} \sqrt {d} \tan (a+b x)+1\right )}{\sqrt {2} \sqrt {d}}}{2 d}+\frac {\int \frac {d-d^2 \tan ^2(a+b x)}{d^4 \tan ^4(a+b x)+d^2}d\sqrt {d \tan (a+b x)}}{2 d}\right )\right )-\frac {(d \tan (a+b x))^{5/2}}{2 \left (d^2 \tan ^2(a+b x)+d^2\right )}\right )-\frac {(d \tan (a+b x))^{9/2}}{4 \left (d^2 \tan ^2(a+b x)+d^2\right )^2}\right )}{b}\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle \frac {d \left (\frac {9}{8} \left (\frac {5}{4} \left (2 \sqrt {d \tan (a+b x)}-2 d^2 \left (\frac {\int \frac {d-d^2 \tan ^2(a+b x)}{d^4 \tan ^4(a+b x)+d^2}d\sqrt {d \tan (a+b x)}}{2 d}+\frac {\frac {\arctan \left (\sqrt {2} \sqrt {d} \tan (a+b x)+1\right )}{\sqrt {2} \sqrt {d}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {d} \tan (a+b x)\right )}{\sqrt {2} \sqrt {d}}}{2 d}\right )\right )-\frac {(d \tan (a+b x))^{5/2}}{2 \left (d^2 \tan ^2(a+b x)+d^2\right )}\right )-\frac {(d \tan (a+b x))^{9/2}}{4 \left (d^2 \tan ^2(a+b x)+d^2\right )^2}\right )}{b}\) |
| \(\Big \downarrow \) 1479 |
| \(\displaystyle \frac {d \left (\frac {9}{8} \left (\frac {5}{4} \left (2 \sqrt {d \tan (a+b x)}-2 d^2 \left (\frac {-\frac {\int -\frac {\sqrt {2} \sqrt {d}-2 \sqrt {d \tan (a+b x)}}{d^2 \tan ^2(a+b x)-\sqrt {2} d^{3/2} \tan (a+b x)+d}d\sqrt {d \tan (a+b x)}}{2 \sqrt {2} \sqrt {d}}-\frac {\int -\frac {\sqrt {2} \left (\sqrt {d}+\sqrt {2} \sqrt {d \tan (a+b x)}\right )}{d^2 \tan ^2(a+b x)+\sqrt {2} d^{3/2} \tan (a+b x)+d}d\sqrt {d \tan (a+b x)}}{2 \sqrt {2} \sqrt {d}}}{2 d}+\frac {\frac {\arctan \left (\sqrt {2} \sqrt {d} \tan (a+b x)+1\right )}{\sqrt {2} \sqrt {d}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {d} \tan (a+b x)\right )}{\sqrt {2} \sqrt {d}}}{2 d}\right )\right )-\frac {(d \tan (a+b x))^{5/2}}{2 \left (d^2 \tan ^2(a+b x)+d^2\right )}\right )-\frac {(d \tan (a+b x))^{9/2}}{4 \left (d^2 \tan ^2(a+b x)+d^2\right )^2}\right )}{b}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {d \left (\frac {9}{8} \left (\frac {5}{4} \left (2 \sqrt {d \tan (a+b x)}-2 d^2 \left (\frac {\frac {\int \frac {\sqrt {2} \sqrt {d}-2 \sqrt {d \tan (a+b x)}}{d^2 \tan ^2(a+b x)-\sqrt {2} d^{3/2} \tan (a+b x)+d}d\sqrt {d \tan (a+b x)}}{2 \sqrt {2} \sqrt {d}}+\frac {\int \frac {\sqrt {2} \left (\sqrt {d}+\sqrt {2} \sqrt {d \tan (a+b x)}\right )}{d^2 \tan ^2(a+b x)+\sqrt {2} d^{3/2} \tan (a+b x)+d}d\sqrt {d \tan (a+b x)}}{2 \sqrt {2} \sqrt {d}}}{2 d}+\frac {\frac {\arctan \left (\sqrt {2} \sqrt {d} \tan (a+b x)+1\right )}{\sqrt {2} \sqrt {d}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {d} \tan (a+b x)\right )}{\sqrt {2} \sqrt {d}}}{2 d}\right )\right )-\frac {(d \tan (a+b x))^{5/2}}{2 \left (d^2 \tan ^2(a+b x)+d^2\right )}\right )-\frac {(d \tan (a+b x))^{9/2}}{4 \left (d^2 \tan ^2(a+b x)+d^2\right )^2}\right )}{b}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {d \left (\frac {9}{8} \left (\frac {5}{4} \left (2 \sqrt {d \tan (a+b x)}-2 d^2 \left (\frac {\frac {\int \frac {\sqrt {2} \sqrt {d}-2 \sqrt {d \tan (a+b x)}}{d^2 \tan ^2(a+b x)-\sqrt {2} d^{3/2} \tan (a+b x)+d}d\sqrt {d \tan (a+b x)}}{2 \sqrt {2} \sqrt {d}}+\frac {\int \frac {\sqrt {d}+\sqrt {2} \sqrt {d \tan (a+b x)}}{d^2 \tan ^2(a+b x)+\sqrt {2} d^{3/2} \tan (a+b x)+d}d\sqrt {d \tan (a+b x)}}{2 \sqrt {d}}}{2 d}+\frac {\frac {\arctan \left (\sqrt {2} \sqrt {d} \tan (a+b x)+1\right )}{\sqrt {2} \sqrt {d}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {d} \tan (a+b x)\right )}{\sqrt {2} \sqrt {d}}}{2 d}\right )\right )-\frac {(d \tan (a+b x))^{5/2}}{2 \left (d^2 \tan ^2(a+b x)+d^2\right )}\right )-\frac {(d \tan (a+b x))^{9/2}}{4 \left (d^2 \tan ^2(a+b x)+d^2\right )^2}\right )}{b}\) |
| \(\Big \downarrow \) 1103 |
| \(\displaystyle \frac {d \left (\frac {9}{8} \left (\frac {5}{4} \left (2 \sqrt {d \tan (a+b x)}-2 d^2 \left (\frac {\frac {\arctan \left (\sqrt {2} \sqrt {d} \tan (a+b x)+1\right )}{\sqrt {2} \sqrt {d}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {d} \tan (a+b x)\right )}{\sqrt {2} \sqrt {d}}}{2 d}+\frac {\frac {\log \left (\sqrt {2} d^{3/2} \tan (a+b x)+d^2 \tan ^2(a+b x)+d\right )}{2 \sqrt {2} \sqrt {d}}-\frac {\log \left (-\sqrt {2} d^{3/2} \tan (a+b x)+d^2 \tan ^2(a+b x)+d\right )}{2 \sqrt {2} \sqrt {d}}}{2 d}\right )\right )-\frac {(d \tan (a+b x))^{5/2}}{2 \left (d^2 \tan ^2(a+b x)+d^2\right )}\right )-\frac {(d \tan (a+b x))^{9/2}}{4 \left (d^2 \tan ^2(a+b x)+d^2\right )^2}\right )}{b}\) |
(d*(-1/4*(d*Tan[a + b*x])^(9/2)/(d^2 + d^2*Tan[a + b*x]^2)^2 + (9*(-1/2*(d *Tan[a + b*x])^(5/2)/(d^2 + d^2*Tan[a + b*x]^2) + (5*(-2*d^2*((-(ArcTan[1 - Sqrt[2]*Sqrt[d]*Tan[a + b*x]]/(Sqrt[2]*Sqrt[d])) + ArcTan[1 + Sqrt[2]*Sq rt[d]*Tan[a + b*x]]/(Sqrt[2]*Sqrt[d]))/(2*d) + (-1/2*Log[d - Sqrt[2]*d^(3/ 2)*Tan[a + b*x] + d^2*Tan[a + b*x]^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*d)) + 2 *Sqrt[d*Tan[a + b*x]]))/4))/8))/b
3.1.64.3.1 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[((a_) + (b_.)*(x_)^4)^(-1), x_Symbol] :> With[{r = Numerator[Rt[a/b, 2] ], s = Denominator[Rt[a/b, 2]]}, Simp[1/(2*r) Int[(r - s*x^2)/(a + b*x^4) , x], x] + Simp[1/(2*r) 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. \(889\) vs. \(2(213)=426\).
Time = 3.46 (sec) , antiderivative size = 890, normalized size of antiderivative = 3.21
1/128/b*(d*tan(b*x+a))^(1/2)*(-16*2^(1/2)*cos(b*x+a)^4+68*cos(b*x+a)^2*2^( 1/2)+45*cot(b*x+a)*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)*(-cos(b*x+a)*sin(b*x+a)/(cos(b*x+a)+1)^2)^(1/ 2)-45*cot(b*x+a)*(-cos(b*x+a)*sin(b*x+a)/(cos(b*x+a)+1)^2)^(1/2)*ln(2*(-co s(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)- 90*cot(b*x+a)*(-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)))+90*cot(b*x+a)*(-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)))+45*csc(b*x+a)*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)*(-cos(b*x +a)*sin(b*x+a)/(cos(b*x+a)+1)^2)^(1/2)-45*csc(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)+128*2^(1/2)-90*csc(b*x+a)*(-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)))+90*...
Result contains complex when optimal does not.
Time = 0.39 (sec) , antiderivative size = 954, normalized size of antiderivative = 3.44 \[ \int \sin ^4(a+b x) (d \tan (a+b x))^{3/2} \, dx=\text {Too large to display} \]
-1/256*(45*I*(-d^6/b^4)^(1/4)*b*log(182250*d^5*cos(b*x + a)^2 + 182250*sqr t(-d^6/b^4)*b^2*d^2*cos(b*x + a)*sin(b*x + a) - 91125*d^5 - 182250*(I*(-d^ 6/b^4)^(1/4)*b*d^3*cos(b*x + a)*sin(b*x + a) - I*(-d^6/b^4)^(3/4)*b^3*cos( b*x + a)^2)*sqrt(d*sin(b*x + a)/cos(b*x + a))) - 45*I*(-d^6/b^4)^(1/4)*b*l og(182250*d^5*cos(b*x + a)^2 + 182250*sqrt(-d^6/b^4)*b^2*d^2*cos(b*x + a)* sin(b*x + a) - 91125*d^5 - 182250*(-I*(-d^6/b^4)^(1/4)*b*d^3*cos(b*x + a)* sin(b*x + a) + I*(-d^6/b^4)^(3/4)*b^3*cos(b*x + a)^2)*sqrt(d*sin(b*x + a)/ cos(b*x + a))) - 45*(-d^6/b^4)^(1/4)*b*log(182250*d^5*cos(b*x + a)^2 - 182 250*sqrt(-d^6/b^4)*b^2*d^2*cos(b*x + a)*sin(b*x + a) - 91125*d^5 + 182250* ((-d^6/b^4)^(1/4)*b*d^3*cos(b*x + a)*sin(b*x + a) + (-d^6/b^4)^(3/4)*b^3*c os(b*x + a)^2)*sqrt(d*sin(b*x + a)/cos(b*x + a))) + 45*(-d^6/b^4)^(1/4)*b* log(182250*d^5*cos(b*x + a)^2 - 182250*sqrt(-d^6/b^4)*b^2*d^2*cos(b*x + a) *sin(b*x + a) - 91125*d^5 - 182250*((-d^6/b^4)^(1/4)*b*d^3*cos(b*x + a)*si n(b*x + a) + (-d^6/b^4)^(3/4)*b^3*cos(b*x + a)^2)*sqrt(d*sin(b*x + a)/cos( b*x + a))) - 45*(-d^6/b^4)^(1/4)*b*log(-91125*d^5 + 182250*((-d^6/b^4)^(1/ 4)*b*d^3*cos(b*x + a)*sin(b*x + a) - (-d^6/b^4)^(3/4)*b^3*cos(b*x + a)^2)* sqrt(d*sin(b*x + a)/cos(b*x + a))) + 45*(-d^6/b^4)^(1/4)*b*log(-91125*d^5 - 182250*((-d^6/b^4)^(1/4)*b*d^3*cos(b*x + a)*sin(b*x + a) - (-d^6/b^4)^(3 /4)*b^3*cos(b*x + a)^2)*sqrt(d*sin(b*x + a)/cos(b*x + a))) + 45*I*(-d^6/b^ 4)^(1/4)*b*log(-91125*d^5 - 182250*(I*(-d^6/b^4)^(1/4)*b*d^3*cos(b*x + ...
Timed out. \[ \int \sin ^4(a+b x) (d \tan (a+b x))^{3/2} \, dx=\text {Timed out} \]
Time = 0.37 (sec) , antiderivative size = 235, normalized size of antiderivative = 0.85 \[ \int \sin ^4(a+b x) (d \tan (a+b x))^{3/2} \, dx=-\frac {90 \, \sqrt {2} d^{\frac {13}{2}} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} \sqrt {d} + 2 \, \sqrt {d \tan \left (b x + a\right )}\right )}}{2 \, \sqrt {d}}\right ) + 90 \, \sqrt {2} d^{\frac {13}{2}} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} \sqrt {d} - 2 \, \sqrt {d \tan \left (b x + a\right )}\right )}}{2 \, \sqrt {d}}\right ) + 45 \, \sqrt {2} d^{\frac {13}{2}} \log \left (d \tan \left (b x + a\right ) + \sqrt {2} \sqrt {d \tan \left (b x + a\right )} \sqrt {d} + d\right ) - 45 \, \sqrt {2} d^{\frac {13}{2}} \log \left (d \tan \left (b x + a\right ) - \sqrt {2} \sqrt {d \tan \left (b x + a\right )} \sqrt {d} + d\right ) - 256 \, \sqrt {d \tan \left (b x + a\right )} d^{6} - \frac {8 \, {\left (17 \, \left (d \tan \left (b x + a\right )\right )^{\frac {5}{2}} d^{8} + 13 \, \sqrt {d \tan \left (b x + a\right )} d^{10}\right )}}{d^{4} \tan \left (b x + a\right )^{4} + 2 \, d^{4} \tan \left (b x + a\right )^{2} + d^{4}}}{128 \, b d^{5}} \]
-1/128*(90*sqrt(2)*d^(13/2)*arctan(1/2*sqrt(2)*(sqrt(2)*sqrt(d) + 2*sqrt(d *tan(b*x + a)))/sqrt(d)) + 90*sqrt(2)*d^(13/2)*arctan(-1/2*sqrt(2)*(sqrt(2 )*sqrt(d) - 2*sqrt(d*tan(b*x + a)))/sqrt(d)) + 45*sqrt(2)*d^(13/2)*log(d*t an(b*x + a) + sqrt(2)*sqrt(d*tan(b*x + a))*sqrt(d) + d) - 45*sqrt(2)*d^(13 /2)*log(d*tan(b*x + a) - sqrt(2)*sqrt(d*tan(b*x + a))*sqrt(d) + d) - 256*s qrt(d*tan(b*x + a))*d^6 - 8*(17*(d*tan(b*x + a))^(5/2)*d^8 + 13*sqrt(d*tan (b*x + a))*d^10)/(d^4*tan(b*x + a)^4 + 2*d^4*tan(b*x + a)^2 + d^4))/(b*d^5 )
Time = 0.38 (sec) , antiderivative size = 252, normalized size of antiderivative = 0.91 \[ \int \sin ^4(a+b x) (d \tan (a+b x))^{3/2} \, dx=-\frac {1}{128} \, d {\left (\frac {90 \, \sqrt {2} \sqrt {{\left | d \right |}} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} \sqrt {{\left | d \right |}} + 2 \, \sqrt {d \tan \left (b x + a\right )}\right )}}{2 \, \sqrt {{\left | d \right |}}}\right )}{b} + \frac {90 \, \sqrt {2} \sqrt {{\left | d \right |}} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} \sqrt {{\left | d \right |}} - 2 \, \sqrt {d \tan \left (b x + a\right )}\right )}}{2 \, \sqrt {{\left | d \right |}}}\right )}{b} + \frac {45 \, \sqrt {2} \sqrt {{\left | d \right |}} \log \left (d \tan \left (b x + a\right ) + \sqrt {2} \sqrt {d \tan \left (b x + a\right )} \sqrt {{\left | d \right |}} + {\left | d \right |}\right )}{b} - \frac {45 \, \sqrt {2} \sqrt {{\left | d \right |}} \log \left (d \tan \left (b x + a\right ) - \sqrt {2} \sqrt {d \tan \left (b x + a\right )} \sqrt {{\left | d \right |}} + {\left | d \right |}\right )}{b} - \frac {256 \, \sqrt {d \tan \left (b x + a\right )}}{b} - \frac {8 \, {\left (17 \, \sqrt {d \tan \left (b x + a\right )} d^{4} \tan \left (b x + a\right )^{2} + 13 \, \sqrt {d \tan \left (b x + a\right )} d^{4}\right )}}{{\left (d^{2} \tan \left (b x + a\right )^{2} + d^{2}\right )}^{2} b}\right )} \]
-1/128*d*(90*sqrt(2)*sqrt(abs(d))*arctan(1/2*sqrt(2)*(sqrt(2)*sqrt(abs(d)) + 2*sqrt(d*tan(b*x + a)))/sqrt(abs(d)))/b + 90*sqrt(2)*sqrt(abs(d))*arcta n(-1/2*sqrt(2)*(sqrt(2)*sqrt(abs(d)) - 2*sqrt(d*tan(b*x + a)))/sqrt(abs(d) ))/b + 45*sqrt(2)*sqrt(abs(d))*log(d*tan(b*x + a) + sqrt(2)*sqrt(d*tan(b*x + a))*sqrt(abs(d)) + abs(d))/b - 45*sqrt(2)*sqrt(abs(d))*log(d*tan(b*x + a) - sqrt(2)*sqrt(d*tan(b*x + a))*sqrt(abs(d)) + abs(d))/b - 256*sqrt(d*ta n(b*x + a))/b - 8*(17*sqrt(d*tan(b*x + a))*d^4*tan(b*x + a)^2 + 13*sqrt(d* tan(b*x + a))*d^4)/((d^2*tan(b*x + a)^2 + d^2)^2*b))
Timed out. \[ \int \sin ^4(a+b x) (d \tan (a+b x))^{3/2} \, dx=\int {\sin \left (a+b\,x\right )}^4\,{\left (d\,\mathrm {tan}\left (a+b\,x\right )\right )}^{3/2} \,d x \]