Integrand size = 21, antiderivative size = 247 \[ \int \sin ^2(a+b x) (d \tan (a+b x))^{3/2} \, dx=\frac {5 d^{3/2} \arctan \left (1-\frac {\sqrt {2} \sqrt {d \tan (a+b x)}}{\sqrt {d}}\right )}{4 \sqrt {2} b}-\frac {5 d^{3/2} \arctan \left (1+\frac {\sqrt {2} \sqrt {d \tan (a+b x)}}{\sqrt {d}}\right )}{4 \sqrt {2} b}+\frac {5 d^{3/2} \log \left (\sqrt {d}+\sqrt {d} \tan (a+b x)-\sqrt {2} \sqrt {d \tan (a+b x)}\right )}{8 \sqrt {2} b}-\frac {5 d^{3/2} \log \left (\sqrt {d}+\sqrt {d} \tan (a+b x)+\sqrt {2} \sqrt {d \tan (a+b x)}\right )}{8 \sqrt {2} b}+\frac {5 d \sqrt {d \tan (a+b x)}}{2 b}-\frac {\cos ^2(a+b x) (d \tan (a+b x))^{5/2}}{2 b d} \]
5/8*d^(3/2)*arctan(1-2^(1/2)*(d*tan(b*x+a))^(1/2)/d^(1/2))/b*2^(1/2)-5/8*d ^(3/2)*arctan(1+2^(1/2)*(d*tan(b*x+a))^(1/2)/d^(1/2))/b*2^(1/2)+5/16*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)-5 /16*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)+5/2*d*(d*tan(b*x+a))^(1/2)/b-1/2*cos(b*x+a)^2*(d*tan(b*x+a))^(5/2) /b/d
Time = 0.46 (sec) , antiderivative size = 113, normalized size of antiderivative = 0.46 \[ \int \sin ^2(a+b x) (d \tan (a+b x))^{3/2} \, dx=\frac {d \csc (a+b x) \left (17 \sin (a+b x)+5 \arcsin (\cos (a+b x)-\sin (a+b x)) \sqrt {\sin (2 (a+b x))}-5 \log \left (\cos (a+b x)+\sin (a+b x)+\sqrt {\sin (2 (a+b x))}\right ) \sqrt {\sin (2 (a+b x))}+\sin (3 (a+b x))\right ) \sqrt {d \tan (a+b x)}}{8 b} \]
(d*Csc[a + b*x]*(17*Sin[a + b*x] + 5*ArcSin[Cos[a + b*x] - Sin[a + b*x]]*S qrt[Sin[2*(a + b*x)]] - 5*Log[Cos[a + b*x] + Sin[a + b*x] + Sqrt[Sin[2*(a + b*x)]]]*Sqrt[Sin[2*(a + b*x)]] + Sin[3*(a + b*x)])*Sqrt[d*Tan[a + b*x]]) /(8*b)
Time = 0.45 (sec) , antiderivative size = 238, normalized size of antiderivative = 0.96, 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, 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.
| \(\displaystyle \int \sin ^2(a+b x) (d \tan (a+b x))^{3/2} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \sin (a+b x)^2 (d \tan (a+b x))^{3/2}dx\) |
| \(\Big \downarrow \) 3071 |
| \(\displaystyle \frac {d \int \frac {(d \tan (a+b x))^{7/2}}{\left (\tan ^2(a+b x) d^2+d^2\right )^2}d(d \tan (a+b x))}{b}\) |
| \(\Big \downarrow \) 252 |
| \(\displaystyle \frac {d \left (\frac {5}{4} \int \frac {(d \tan (a+b x))^{3/2}}{\tan ^2(a+b x) d^2+d^2}d(d \tan (a+b x))-\frac {(d \tan (a+b x))^{5/2}}{2 \left (d^2 \tan ^2(a+b x)+d^2\right )}\right )}{b}\) |
| \(\Big \downarrow \) 262 |
| \(\displaystyle \frac {d \left (\frac {5}{4} \left (2 \sqrt {d \tan (a+b x)}-d^2 \int \frac {1}{\sqrt {d \tan (a+b x)} \left (\tan ^2(a+b x) d^2+d^2\right )}d(d \tan (a+b x))\right )-\frac {(d \tan (a+b x))^{5/2}}{2 \left (d^2 \tan ^2(a+b x)+d^2\right )}\right )}{b}\) |
| \(\Big \downarrow \) 266 |
| \(\displaystyle \frac {d \left (\frac {5}{4} \left (2 \sqrt {d \tan (a+b x)}-2 d^2 \int \frac {1}{d^4 \tan ^4(a+b x)+d^2}d\sqrt {d \tan (a+b x)}\right )-\frac {(d \tan (a+b x))^{5/2}}{2 \left (d^2 \tan ^2(a+b x)+d^2\right )}\right )}{b}\) |
| \(\Big \downarrow \) 755 |
| \(\displaystyle \frac {d \left (\frac {5}{4} \left (2 \sqrt {d \tan (a+b x)}-2 d^2 \left (\frac {\int \frac {d-d^2 \tan ^2(a+b x)}{d^4 \tan ^4(a+b x)+d^2}d\sqrt {d \tan (a+b x)}}{2 d}+\frac {\int \frac {d^2 \tan ^2(a+b x)+d}{d^4 \tan ^4(a+b x)+d^2}d\sqrt {d \tan (a+b x)}}{2 d}\right )\right )-\frac {(d \tan (a+b x))^{5/2}}{2 \left (d^2 \tan ^2(a+b x)+d^2\right )}\right )}{b}\) |
| \(\Big \downarrow \) 1476 |
| \(\displaystyle \frac {d \left (\frac {5}{4} \left (2 \sqrt {d \tan (a+b x)}-2 d^2 \left (\frac {\frac {1}{2} \int \frac {1}{d^2 \tan ^2(a+b x)-\sqrt {2} d^{3/2} \tan (a+b x)+d}d\sqrt {d \tan (a+b x)}+\frac {1}{2} \int \frac {1}{d^2 \tan ^2(a+b x)+\sqrt {2} d^{3/2} \tan (a+b x)+d}d\sqrt {d \tan (a+b x)}}{2 d}+\frac {\int \frac {d-d^2 \tan ^2(a+b x)}{d^4 \tan ^4(a+b x)+d^2}d\sqrt {d \tan (a+b x)}}{2 d}\right )\right )-\frac {(d \tan (a+b x))^{5/2}}{2 \left (d^2 \tan ^2(a+b x)+d^2\right )}\right )}{b}\) |
| \(\Big \downarrow \) 1082 |
| \(\displaystyle \frac {d \left (\frac {5}{4} \left (2 \sqrt {d \tan (a+b x)}-2 d^2 \left (\frac {\frac {\int \frac {1}{-d^2 \tan ^2(a+b x)-1}d\left (1-\sqrt {2} \sqrt {d} \tan (a+b x)\right )}{\sqrt {2} \sqrt {d}}-\frac {\int \frac {1}{-d^2 \tan ^2(a+b x)-1}d\left (\sqrt {2} \sqrt {d} \tan (a+b x)+1\right )}{\sqrt {2} \sqrt {d}}}{2 d}+\frac {\int \frac {d-d^2 \tan ^2(a+b x)}{d^4 \tan ^4(a+b x)+d^2}d\sqrt {d \tan (a+b x)}}{2 d}\right )\right )-\frac {(d \tan (a+b x))^{5/2}}{2 \left (d^2 \tan ^2(a+b x)+d^2\right )}\right )}{b}\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle \frac {d \left (\frac {5}{4} \left (2 \sqrt {d \tan (a+b x)}-2 d^2 \left (\frac {\int \frac {d-d^2 \tan ^2(a+b x)}{d^4 \tan ^4(a+b x)+d^2}d\sqrt {d \tan (a+b x)}}{2 d}+\frac {\frac {\arctan \left (\sqrt {2} \sqrt {d} \tan (a+b x)+1\right )}{\sqrt {2} \sqrt {d}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {d} \tan (a+b x)\right )}{\sqrt {2} \sqrt {d}}}{2 d}\right )\right )-\frac {(d \tan (a+b x))^{5/2}}{2 \left (d^2 \tan ^2(a+b x)+d^2\right )}\right )}{b}\) |
| \(\Big \downarrow \) 1479 |
| \(\displaystyle \frac {d \left (\frac {5}{4} \left (2 \sqrt {d \tan (a+b x)}-2 d^2 \left (\frac {-\frac {\int -\frac {\sqrt {2} \sqrt {d}-2 \sqrt {d \tan (a+b x)}}{d^2 \tan ^2(a+b x)-\sqrt {2} d^{3/2} \tan (a+b x)+d}d\sqrt {d \tan (a+b x)}}{2 \sqrt {2} \sqrt {d}}-\frac {\int -\frac {\sqrt {2} \left (\sqrt {d}+\sqrt {2} \sqrt {d \tan (a+b x)}\right )}{d^2 \tan ^2(a+b x)+\sqrt {2} d^{3/2} \tan (a+b x)+d}d\sqrt {d \tan (a+b x)}}{2 \sqrt {2} \sqrt {d}}}{2 d}+\frac {\frac {\arctan \left (\sqrt {2} \sqrt {d} \tan (a+b x)+1\right )}{\sqrt {2} \sqrt {d}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {d} \tan (a+b x)\right )}{\sqrt {2} \sqrt {d}}}{2 d}\right )\right )-\frac {(d \tan (a+b x))^{5/2}}{2 \left (d^2 \tan ^2(a+b x)+d^2\right )}\right )}{b}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {d \left (\frac {5}{4} \left (2 \sqrt {d \tan (a+b x)}-2 d^2 \left (\frac {\frac {\int \frac {\sqrt {2} \sqrt {d}-2 \sqrt {d \tan (a+b x)}}{d^2 \tan ^2(a+b x)-\sqrt {2} d^{3/2} \tan (a+b x)+d}d\sqrt {d \tan (a+b x)}}{2 \sqrt {2} \sqrt {d}}+\frac {\int \frac {\sqrt {2} \left (\sqrt {d}+\sqrt {2} \sqrt {d \tan (a+b x)}\right )}{d^2 \tan ^2(a+b x)+\sqrt {2} d^{3/2} \tan (a+b x)+d}d\sqrt {d \tan (a+b x)}}{2 \sqrt {2} \sqrt {d}}}{2 d}+\frac {\frac {\arctan \left (\sqrt {2} \sqrt {d} \tan (a+b x)+1\right )}{\sqrt {2} \sqrt {d}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {d} \tan (a+b x)\right )}{\sqrt {2} \sqrt {d}}}{2 d}\right )\right )-\frac {(d \tan (a+b x))^{5/2}}{2 \left (d^2 \tan ^2(a+b x)+d^2\right )}\right )}{b}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {d \left (\frac {5}{4} \left (2 \sqrt {d \tan (a+b x)}-2 d^2 \left (\frac {\frac {\int \frac {\sqrt {2} \sqrt {d}-2 \sqrt {d \tan (a+b x)}}{d^2 \tan ^2(a+b x)-\sqrt {2} d^{3/2} \tan (a+b x)+d}d\sqrt {d \tan (a+b x)}}{2 \sqrt {2} \sqrt {d}}+\frac {\int \frac {\sqrt {d}+\sqrt {2} \sqrt {d \tan (a+b x)}}{d^2 \tan ^2(a+b x)+\sqrt {2} d^{3/2} \tan (a+b x)+d}d\sqrt {d \tan (a+b x)}}{2 \sqrt {d}}}{2 d}+\frac {\frac {\arctan \left (\sqrt {2} \sqrt {d} \tan (a+b x)+1\right )}{\sqrt {2} \sqrt {d}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {d} \tan (a+b x)\right )}{\sqrt {2} \sqrt {d}}}{2 d}\right )\right )-\frac {(d \tan (a+b x))^{5/2}}{2 \left (d^2 \tan ^2(a+b x)+d^2\right )}\right )}{b}\) |
| \(\Big \downarrow \) 1103 |
| \(\displaystyle \frac {d \left (\frac {5}{4} \left (2 \sqrt {d \tan (a+b x)}-2 d^2 \left (\frac {\frac {\arctan \left (\sqrt {2} \sqrt {d} \tan (a+b x)+1\right )}{\sqrt {2} \sqrt {d}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {d} \tan (a+b x)\right )}{\sqrt {2} \sqrt {d}}}{2 d}+\frac {\frac {\log \left (\sqrt {2} d^{3/2} \tan (a+b x)+d^2 \tan ^2(a+b x)+d\right )}{2 \sqrt {2} \sqrt {d}}-\frac {\log \left (-\sqrt {2} d^{3/2} \tan (a+b x)+d^2 \tan ^2(a+b x)+d\right )}{2 \sqrt {2} \sqrt {d}}}{2 d}\right )\right )-\frac {(d \tan (a+b x))^{5/2}}{2 \left (d^2 \tan ^2(a+b x)+d^2\right )}\right )}{b}\) |
(d*(-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]*Sqrt[d]*Tan[a + b*x]]/(Sqrt[2]*Sqrt[d]))/(2*d) + (-1/2*Log[d - Sq rt[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))/b
3.1.65.3.1 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. \(1082\) vs. \(2(187)=374\).
Time = 2.33 (sec) , antiderivative size = 1083, normalized size of antiderivative = 4.38
-1/16/b*sin(b*x+a)*(4*cos(b*x+a)^2*sin(b*x+a)*2^(1/2)+5*(-cos(b*x+a)*sin(b *x+a)/(cos(b*x+a)+1)^2)^(1/2)*ln(-(cot(b*x+a)*cos(b*x+a)-2*cot(b*x+a)-2*si n(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)))*cos(b*x+a)-5*(-cos(b*x+a)*sin(b*x+a)/(cos(b*x+a)+ 1)^2)^(1/2)*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)))*cos(b*x+a)+10*(-cos(b*x+a)*sin(b*x+a)/(cos(b*x+a)+1)^2)^(1/2)*arcta n((-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)-10*(-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)+16*sin(b*x+a)*2 ^(1/2)+5*(-cos(b*x+a)*sin(b*x+a)/(cos(b*x+a)+1)^2)^(1/2)*ln(-(cot(b*x+a)*c os(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*c os(b*x+a)-sin(b*x+a)+csc(b*x+a)+2)/(-1+cos(b*x+a)))-5*(-cos(b*x+a)*sin(b*x +a)/(cos(b*x+a)+1)^2)^(1/2)*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...
Result contains complex when optimal does not.
Time = 0.38 (sec) , antiderivative size = 942, normalized size of antiderivative = 3.81 \[ \int \sin ^2(a+b x) (d \tan (a+b x))^{3/2} \, dx=\text {Too large to display} \]
-1/32*(5*I*(-d^6/b^4)^(1/4)*b*log(250*d^5*cos(b*x + a)^2 + 250*sqrt(-d^6/b ^4)*b^2*d^2*cos(b*x + a)*sin(b*x + a) - 125*d^5 - 250*(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)*s qrt(d*sin(b*x + a)/cos(b*x + a))) - 5*I*(-d^6/b^4)^(1/4)*b*log(250*d^5*cos (b*x + a)^2 + 250*sqrt(-d^6/b^4)*b^2*d^2*cos(b*x + a)*sin(b*x + a) - 125*d ^5 - 250*(-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))) - 5*(-d^6/ b^4)^(1/4)*b*log(250*d^5*cos(b*x + a)^2 - 250*sqrt(-d^6/b^4)*b^2*d^2*cos(b *x + a)*sin(b*x + a) - 125*d^5 + 250*((-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)/co s(b*x + a))) + 5*(-d^6/b^4)^(1/4)*b*log(250*d^5*cos(b*x + a)^2 - 250*sqrt( -d^6/b^4)*b^2*d^2*cos(b*x + a)*sin(b*x + a) - 125*d^5 - 250*((-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))) - 5*(-d^6/b^4)^(1/4)*b*log(-125*d^5 + 250*((-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))) + 5*(-d^6/b^4)^(1/4) *b*log(-125*d^5 - 250*((-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))) + 5*I*(-d^6/b^4)^(1/4)*b*log(-125*d^5 - 250*(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...
\[ \int \sin ^2(a+b x) (d \tan (a+b x))^{3/2} \, dx=\int \left (d \tan {\left (a + b x \right )}\right )^{\frac {3}{2}} \sin ^{2}{\left (a + b x \right )}\, dx \]
Time = 0.38 (sec) , antiderivative size = 204, normalized size of antiderivative = 0.83 \[ \int \sin ^2(a+b x) (d \tan (a+b x))^{3/2} \, dx=-\frac {10 \, \sqrt {2} d^{\frac {9}{2}} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} \sqrt {d} + 2 \, \sqrt {d \tan \left (b x + a\right )}\right )}}{2 \, \sqrt {d}}\right ) + 10 \, \sqrt {2} d^{\frac {9}{2}} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} \sqrt {d} - 2 \, \sqrt {d \tan \left (b x + a\right )}\right )}}{2 \, \sqrt {d}}\right ) + 5 \, \sqrt {2} d^{\frac {9}{2}} \log \left (d \tan \left (b x + a\right ) + \sqrt {2} \sqrt {d \tan \left (b x + a\right )} \sqrt {d} + d\right ) - 5 \, \sqrt {2} d^{\frac {9}{2}} \log \left (d \tan \left (b x + a\right ) - \sqrt {2} \sqrt {d \tan \left (b x + a\right )} \sqrt {d} + d\right ) - \frac {8 \, \sqrt {d \tan \left (b x + a\right )} d^{6}}{d^{2} \tan \left (b x + a\right )^{2} + d^{2}} - 32 \, \sqrt {d \tan \left (b x + a\right )} d^{4}}{16 \, b d^{3}} \]
-1/16*(10*sqrt(2)*d^(9/2)*arctan(1/2*sqrt(2)*(sqrt(2)*sqrt(d) + 2*sqrt(d*t an(b*x + a)))/sqrt(d)) + 10*sqrt(2)*d^(9/2)*arctan(-1/2*sqrt(2)*(sqrt(2)*s qrt(d) - 2*sqrt(d*tan(b*x + a)))/sqrt(d)) + 5*sqrt(2)*d^(9/2)*log(d*tan(b* x + a) + sqrt(2)*sqrt(d*tan(b*x + a))*sqrt(d) + d) - 5*sqrt(2)*d^(9/2)*log (d*tan(b*x + a) - sqrt(2)*sqrt(d*tan(b*x + a))*sqrt(d) + d) - 8*sqrt(d*tan (b*x + a))*d^6/(d^2*tan(b*x + a)^2 + d^2) - 32*sqrt(d*tan(b*x + a))*d^4)/( b*d^3)
Time = 0.33 (sec) , antiderivative size = 226, normalized size of antiderivative = 0.91 \[ \int \sin ^2(a+b x) (d \tan (a+b x))^{3/2} \, dx=-\frac {1}{16} \, d {\left (\frac {10 \, \sqrt {2} \sqrt {{\left | d \right |}} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} \sqrt {{\left | d \right |}} + 2 \, \sqrt {d \tan \left (b x + a\right )}\right )}}{2 \, \sqrt {{\left | d \right |}}}\right )}{b} + \frac {10 \, \sqrt {2} \sqrt {{\left | d \right |}} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} \sqrt {{\left | d \right |}} - 2 \, \sqrt {d \tan \left (b x + a\right )}\right )}}{2 \, \sqrt {{\left | d \right |}}}\right )}{b} + \frac {5 \, \sqrt {2} \sqrt {{\left | d \right |}} \log \left (d \tan \left (b x + a\right ) + \sqrt {2} \sqrt {d \tan \left (b x + a\right )} \sqrt {{\left | d \right |}} + {\left | d \right |}\right )}{b} - \frac {5 \, \sqrt {2} \sqrt {{\left | d \right |}} \log \left (d \tan \left (b x + a\right ) - \sqrt {2} \sqrt {d \tan \left (b x + a\right )} \sqrt {{\left | d \right |}} + {\left | d \right |}\right )}{b} - \frac {8 \, \sqrt {d \tan \left (b x + a\right )} d^{2}}{{\left (d^{2} \tan \left (b x + a\right )^{2} + d^{2}\right )} b} - \frac {32 \, \sqrt {d \tan \left (b x + a\right )}}{b}\right )} \]
-1/16*d*(10*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 + 10*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 + 5*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 - 5*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 - 8*sqrt(d*tan(b*x + a))*d^2/((d^2*tan(b*x + a)^2 + d^2)*b) - 32*sqrt(d*tan(b*x + a))/b)
Timed out. \[ \int \sin ^2(a+b x) (d \tan (a+b x))^{3/2} \, dx=\int {\sin \left (a+b\,x\right )}^2\,{\left (d\,\mathrm {tan}\left (a+b\,x\right )\right )}^{3/2} \,d x \]