Integrand size = 21, antiderivative size = 174 \[ \int \sin ^2(a+b x) \sqrt {d \tan (a+b x)} \, dx=-\frac {3 \sqrt {d} \arctan \left (1-\frac {\sqrt {2} \sqrt {d \tan (a+b x)}}{\sqrt {d}}\right )}{4 \sqrt {2} b}+\frac {3 \sqrt {d} \arctan \left (1+\frac {\sqrt {2} \sqrt {d \tan (a+b x)}}{\sqrt {d}}\right )}{4 \sqrt {2} b}-\frac {3 \sqrt {d} \text {arctanh}\left (\frac {\sqrt {2} \sqrt {d \tan (a+b x)}}{\sqrt {d}+\sqrt {d} \tan (a+b x)}\right )}{4 \sqrt {2} b}-\frac {\cos ^2(a+b x) (d \tan (a+b x))^{3/2}}{2 b d} \] Output:
-3/8*d^(1/2)*arctan(1-2^(1/2)*(d*tan(b*x+a))^(1/2)/d^(1/2))*2^(1/2)/b+3/8* d^(1/2)*arctan(1+2^(1/2)*(d*tan(b*x+a))^(1/2)/d^(1/2))*2^(1/2)/b-3/8*d^(1/ 2)*arctanh(2^(1/2)*(d*tan(b*x+a))^(1/2)/(d^(1/2)+d^(1/2)*tan(b*x+a)))*2^(1 /2)/b-1/2*cos(b*x+a)^2*(d*tan(b*x+a))^(3/2)/b/d
Time = 0.27 (sec) , antiderivative size = 104, normalized size of antiderivative = 0.60 \[ \int \sin ^2(a+b x) \sqrt {d \tan (a+b x)} \, dx=-\frac {\left (3 \arcsin (\cos (a+b x)-\sin (a+b x)) \csc (a+b x)+3 \csc (a+b x) \log \left (\cos (a+b x)+\sin (a+b x)+\sqrt {\sin (2 (a+b x))}\right )+2 \sqrt {\sin (2 (a+b x))}\right ) \sqrt {\sin (2 (a+b x))} \sqrt {d \tan (a+b x)}}{8 b} \] Input:
Integrate[Sin[a + b*x]^2*Sqrt[d*Tan[a + b*x]],x]
Output:
-1/8*((3*ArcSin[Cos[a + b*x] - Sin[a + b*x]]*Csc[a + b*x] + 3*Csc[a + b*x] *Log[Cos[a + b*x] + Sin[a + b*x] + Sqrt[Sin[2*(a + b*x)]]] + 2*Sqrt[Sin[2* (a + b*x)]])*Sqrt[Sin[2*(a + b*x)]]*Sqrt[d*Tan[a + b*x]])/b
Time = 0.40 (sec) , antiderivative size = 212, normalized size of antiderivative = 1.22, number of steps used = 13, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.571, Rules used = {3042, 3071, 252, 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.
\(\displaystyle \int \sin ^2(a+b x) \sqrt {d \tan (a+b x)} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \sin (a+b x)^2 \sqrt {d \tan (a+b x)}dx\) |
\(\Big \downarrow \) 3071 |
\(\displaystyle \frac {d \int \frac {(d \tan (a+b x))^{5/2}}{\left (\tan ^2(a+b x) d^2+d^2\right )^2}d(d \tan (a+b x))}{b}\) |
\(\Big \downarrow \) 252 |
\(\displaystyle \frac {d \left (\frac {3}{4} \int \frac {\sqrt {d \tan (a+b x)}}{\tan ^2(a+b x) d^2+d^2}d(d \tan (a+b x))-\frac {(d \tan (a+b x))^{3/2}}{2 \left (d^2 \tan ^2(a+b x)+d^2\right )}\right )}{b}\) |
\(\Big \downarrow \) 266 |
\(\displaystyle \frac {d \left (\frac {3}{2} \int \frac {d^2 \tan ^2(a+b x)}{d^4 \tan ^4(a+b x)+d^2}d\sqrt {d \tan (a+b x)}-\frac {(d \tan (a+b x))^{3/2}}{2 \left (d^2 \tan ^2(a+b x)+d^2\right )}\right )}{b}\) |
\(\Big \downarrow \) 826 |
\(\displaystyle \frac {d \left (\frac {3}{2} \left (\frac {1}{2} \int \frac {d^2 \tan ^2(a+b x)+d}{d^4 \tan ^4(a+b x)+d^2}d\sqrt {d \tan (a+b x)}-\frac {1}{2} \int \frac {d-d^2 \tan ^2(a+b x)}{d^4 \tan ^4(a+b x)+d^2}d\sqrt {d \tan (a+b x)}\right )-\frac {(d \tan (a+b x))^{3/2}}{2 \left (d^2 \tan ^2(a+b x)+d^2\right )}\right )}{b}\) |
\(\Big \downarrow \) 1476 |
\(\displaystyle \frac {d \left (\frac {3}{2} \left (\frac {1}{2} \left (\frac {1}{2} \int \frac {1}{d^2 \tan ^2(a+b x)-\sqrt {2} d^{3/2} \tan (a+b x)+d}d\sqrt {d \tan (a+b x)}+\frac {1}{2} \int \frac {1}{d^2 \tan ^2(a+b x)+\sqrt {2} d^{3/2} \tan (a+b x)+d}d\sqrt {d \tan (a+b x)}\right )-\frac {1}{2} \int \frac {d-d^2 \tan ^2(a+b x)}{d^4 \tan ^4(a+b x)+d^2}d\sqrt {d \tan (a+b x)}\right )-\frac {(d \tan (a+b x))^{3/2}}{2 \left (d^2 \tan ^2(a+b x)+d^2\right )}\right )}{b}\) |
\(\Big \downarrow \) 1082 |
\(\displaystyle \frac {d \left (\frac {3}{2} \left (\frac {1}{2} \left (\frac {\int \frac {1}{-d^2 \tan ^2(a+b x)-1}d\left (1-\sqrt {2} \sqrt {d} \tan (a+b x)\right )}{\sqrt {2} \sqrt {d}}-\frac {\int \frac {1}{-d^2 \tan ^2(a+b x)-1}d\left (\sqrt {2} \sqrt {d} \tan (a+b x)+1\right )}{\sqrt {2} \sqrt {d}}\right )-\frac {1}{2} \int \frac {d-d^2 \tan ^2(a+b x)}{d^4 \tan ^4(a+b x)+d^2}d\sqrt {d \tan (a+b x)}\right )-\frac {(d \tan (a+b x))^{3/2}}{2 \left (d^2 \tan ^2(a+b x)+d^2\right )}\right )}{b}\) |
\(\Big \downarrow \) 217 |
\(\displaystyle \frac {d \left (\frac {3}{2} \left (\frac {1}{2} \left (\frac {\arctan \left (\sqrt {2} \sqrt {d} \tan (a+b x)+1\right )}{\sqrt {2} \sqrt {d}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {d} \tan (a+b x)\right )}{\sqrt {2} \sqrt {d}}\right )-\frac {1}{2} \int \frac {d-d^2 \tan ^2(a+b x)}{d^4 \tan ^4(a+b x)+d^2}d\sqrt {d \tan (a+b x)}\right )-\frac {(d \tan (a+b x))^{3/2}}{2 \left (d^2 \tan ^2(a+b x)+d^2\right )}\right )}{b}\) |
\(\Big \downarrow \) 1479 |
\(\displaystyle \frac {d \left (\frac {3}{2} \left (\frac {1}{2} \left (\frac {\int -\frac {\sqrt {2} \sqrt {d}-2 \sqrt {d \tan (a+b x)}}{d^2 \tan ^2(a+b x)-\sqrt {2} d^{3/2} \tan (a+b x)+d}d\sqrt {d \tan (a+b x)}}{2 \sqrt {2} \sqrt {d}}+\frac {\int -\frac {\sqrt {2} \left (\sqrt {d}+\sqrt {2} \sqrt {d \tan (a+b x)}\right )}{d^2 \tan ^2(a+b x)+\sqrt {2} d^{3/2} \tan (a+b x)+d}d\sqrt {d \tan (a+b x)}}{2 \sqrt {2} \sqrt {d}}\right )+\frac {1}{2} \left (\frac {\arctan \left (\sqrt {2} \sqrt {d} \tan (a+b x)+1\right )}{\sqrt {2} \sqrt {d}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {d} \tan (a+b x)\right )}{\sqrt {2} \sqrt {d}}\right )\right )-\frac {(d \tan (a+b x))^{3/2}}{2 \left (d^2 \tan ^2(a+b x)+d^2\right )}\right )}{b}\) |
\(\Big \downarrow \) 25 |
\(\displaystyle \frac {d \left (\frac {3}{2} \left (\frac {1}{2} \left (-\frac {\int \frac {\sqrt {2} \sqrt {d}-2 \sqrt {d \tan (a+b x)}}{d^2 \tan ^2(a+b x)-\sqrt {2} d^{3/2} \tan (a+b x)+d}d\sqrt {d \tan (a+b x)}}{2 \sqrt {2} \sqrt {d}}-\frac {\int \frac {\sqrt {2} \left (\sqrt {d}+\sqrt {2} \sqrt {d \tan (a+b x)}\right )}{d^2 \tan ^2(a+b x)+\sqrt {2} d^{3/2} \tan (a+b x)+d}d\sqrt {d \tan (a+b x)}}{2 \sqrt {2} \sqrt {d}}\right )+\frac {1}{2} \left (\frac {\arctan \left (\sqrt {2} \sqrt {d} \tan (a+b x)+1\right )}{\sqrt {2} \sqrt {d}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {d} \tan (a+b x)\right )}{\sqrt {2} \sqrt {d}}\right )\right )-\frac {(d \tan (a+b x))^{3/2}}{2 \left (d^2 \tan ^2(a+b x)+d^2\right )}\right )}{b}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {d \left (\frac {3}{2} \left (\frac {1}{2} \left (-\frac {\int \frac {\sqrt {2} \sqrt {d}-2 \sqrt {d \tan (a+b x)}}{d^2 \tan ^2(a+b x)-\sqrt {2} d^{3/2} \tan (a+b x)+d}d\sqrt {d \tan (a+b x)}}{2 \sqrt {2} \sqrt {d}}-\frac {\int \frac {\sqrt {d}+\sqrt {2} \sqrt {d \tan (a+b x)}}{d^2 \tan ^2(a+b x)+\sqrt {2} d^{3/2} \tan (a+b x)+d}d\sqrt {d \tan (a+b x)}}{2 \sqrt {d}}\right )+\frac {1}{2} \left (\frac {\arctan \left (\sqrt {2} \sqrt {d} \tan (a+b x)+1\right )}{\sqrt {2} \sqrt {d}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {d} \tan (a+b x)\right )}{\sqrt {2} \sqrt {d}}\right )\right )-\frac {(d \tan (a+b x))^{3/2}}{2 \left (d^2 \tan ^2(a+b x)+d^2\right )}\right )}{b}\) |
\(\Big \downarrow \) 1103 |
\(\displaystyle \frac {d \left (\frac {3}{2} \left (\frac {1}{2} \left (\frac {\arctan \left (\sqrt {2} \sqrt {d} \tan (a+b x)+1\right )}{\sqrt {2} \sqrt {d}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {d} \tan (a+b x)\right )}{\sqrt {2} \sqrt {d}}\right )+\frac {1}{2} \left (\frac {\log \left (-\sqrt {2} d^{3/2} \tan (a+b x)+d^2 \tan ^2(a+b x)+d\right )}{2 \sqrt {2} \sqrt {d}}-\frac {\log \left (\sqrt {2} d^{3/2} \tan (a+b x)+d^2 \tan ^2(a+b x)+d\right )}{2 \sqrt {2} \sqrt {d}}\right )\right )-\frac {(d \tan (a+b x))^{3/2}}{2 \left (d^2 \tan ^2(a+b x)+d^2\right )}\right )}{b}\) |
Input:
Int[Sin[a + b*x]^2*Sqrt[d*Tan[a + b*x]],x]
Output:
(d*((3*((-(ArcTan[1 - Sqrt[2]*Sqrt[d]*Tan[a + b*x]]/(Sqrt[2]*Sqrt[d])) + A rcTan[1 + Sqrt[2]*Sqrt[d]*Tan[a + b*x]]/(Sqrt[2]*Sqrt[d]))/2 + (Log[d - Sq rt[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)/(2*(d^2 + d^2*Tan[a + b*x]^2))))/b
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] :> 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. \(403\) vs. \(2(132)=264\).
Time = 0.54 (sec) , antiderivative size = 404, normalized size of antiderivative = 2.32
method | result | size |
default | \(-\frac {\left (-3 \ln \left (-\frac {\cot \left (b x +a \right ) \cos \left (b x +a \right )-2 \cot \left (b x +a \right )-2 \sin \left (b x +a \right ) \sqrt {-\frac {2 \sin \left (b x +a \right ) \cos \left (b x +a \right )}{\left (\cos \left (b x +a \right )+1\right )^{2}}}-2 \cos \left (b x +a \right )+\csc \left (b x +a \right )-\sin \left (b x +a \right )+2}{-1+\cos \left (b x +a \right )}\right )+3 \ln \left (-\frac {\cot \left (b x +a \right ) \cos \left (b x +a \right )-2 \cot \left (b x +a \right )+2 \sin \left (b x +a \right ) \sqrt {-\frac {2 \sin \left (b x +a \right ) \cos \left (b x +a \right )}{\left (\cos \left (b x +a \right )+1\right )^{2}}}-2 \cos \left (b x +a \right )+\csc \left (b x +a \right )-\sin \left (b x +a \right )+2}{-1+\cos \left (b x +a \right )}\right )+6 \arctan \left (\frac {-\sin \left (b x +a \right ) \sqrt {-\frac {2 \sin \left (b x +a \right ) \cos \left (b x +a \right )}{\left (\cos \left (b x +a \right )+1\right )^{2}}}+\cos \left (b x +a \right )-1}{-1+\cos \left (b x +a \right )}\right )-6 \arctan \left (\frac {\sin \left (b x +a \right ) \sqrt {-\frac {2 \sin \left (b x +a \right ) \cos \left (b x +a \right )}{\left (\cos \left (b x +a \right )+1\right )^{2}}}+\cos \left (b x +a \right )-1}{-1+\cos \left (b x +a \right )}\right )+\left (4 \cos \left (b x +a \right )+4\right ) \sin \left (b x +a \right ) \sqrt {-\frac {2 \sin \left (b x +a \right ) \cos \left (b x +a \right )}{\left (\cos \left (b x +a \right )+1\right )^{2}}}\right ) \cos \left (b x +a \right ) \sqrt {d \tan \left (b x +a \right )}\, \sqrt {2}}{16 b \left (\cos \left (b x +a \right )+1\right ) \sqrt {-\frac {\sin \left (b x +a \right ) \cos \left (b x +a \right )}{\left (\cos \left (b x +a \right )+1\right )^{2}}}}\) | \(404\) |
Input:
int(sin(b*x+a)^2*(d*tan(b*x+a))^(1/2),x,method=_RETURNVERBOSE)
Output:
-1/16/b*(-3*ln(-(cot(b*x+a)*cos(b*x+a)-2*cot(b*x+a)-2*sin(b*x+a)*(-2*sin(b *x+a)*cos(b*x+a)/(cos(b*x+a)+1)^2)^(1/2)-2*cos(b*x+a)+csc(b*x+a)-sin(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)*(-2*sin(b*x+a)*cos(b*x+a)/(cos(b*x+a)+1)^2)^(1/2)-2*cos(b*x+a)+csc(b*x+ a)-sin(b*x+a)+2)/(-1+cos(b*x+a)))+6*arctan((-sin(b*x+a)*(-2*sin(b*x+a)*cos (b*x+a)/(cos(b*x+a)+1)^2)^(1/2)+cos(b*x+a)-1)/(-1+cos(b*x+a)))-6*arctan((s in(b*x+a)*(-2*sin(b*x+a)*cos(b*x+a)/(cos(b*x+a)+1)^2)^(1/2)+cos(b*x+a)-1)/ (-1+cos(b*x+a)))+(4*cos(b*x+a)+4)*sin(b*x+a)*(-2*sin(b*x+a)*cos(b*x+a)/(co s(b*x+a)+1)^2)^(1/2))*cos(b*x+a)*(d*tan(b*x+a))^(1/2)/(cos(b*x+a)+1)/(-sin (b*x+a)*cos(b*x+a)/(cos(b*x+a)+1)^2)^(1/2)*2^(1/2)
Leaf count of result is larger than twice the leaf count of optimal. 440 vs. \(2 (132) = 264\).
Time = 0.13 (sec) , antiderivative size = 440, normalized size of antiderivative = 2.53 \[ \int \sin ^2(a+b x) \sqrt {d \tan (a+b x)} \, dx=-\frac {16 \, \sqrt {\frac {d \sin \left (b x + a\right )}{\cos \left (b x + a\right )}} \cos \left (b x + a\right ) \sin \left (b x + a\right ) + 6 \, \sqrt {2} \sqrt {d} \arctan \left (-\frac {\sqrt {2} \sqrt {d} \sqrt {\frac {d \sin \left (b x + a\right )}{\cos \left (b x + a\right )}} \cos \left (b x + a\right )}{d \cos \left (b x + a\right ) - d \sin \left (b x + a\right )}\right ) + 3 \, \sqrt {2} \sqrt {d} \arctan \left (\frac {2 \, d \cos \left (b x + a\right )^{2} - 2 \, d \cos \left (b x + a\right ) \sin \left (b x + a\right ) + \sqrt {2} \sqrt {d} \sqrt {\frac {d \sin \left (b x + a\right )}{\cos \left (b x + a\right )}} - 2 \, d}{2 \, {\left (d \cos \left (b x + a\right )^{2} + d \cos \left (b x + a\right ) \sin \left (b x + a\right ) - d\right )}}\right ) + 3 \, \sqrt {2} \sqrt {d} \arctan \left (-\frac {2 \, d \cos \left (b x + a\right )^{2} - 2 \, d \cos \left (b x + a\right ) \sin \left (b x + a\right ) - \sqrt {2} \sqrt {d} \sqrt {\frac {d \sin \left (b x + a\right )}{\cos \left (b x + a\right )}} - 2 \, d}{2 \, {\left (d \cos \left (b x + a\right )^{2} + d \cos \left (b x + a\right ) \sin \left (b x + a\right ) - d\right )}}\right ) + 3 \, \sqrt {2} \sqrt {d} \log \left (4 \, d \cos \left (b x + a\right ) \sin \left (b x + a\right ) + 2 \, \sqrt {2} {\left (\cos \left (b x + a\right )^{2} + \cos \left (b x + a\right ) \sin \left (b x + a\right )\right )} \sqrt {d} \sqrt {\frac {d \sin \left (b x + a\right )}{\cos \left (b x + a\right )}} + d\right ) - 3 \, \sqrt {2} \sqrt {d} \log \left (4 \, d \cos \left (b x + a\right ) \sin \left (b x + a\right ) - 2 \, \sqrt {2} {\left (\cos \left (b x + a\right )^{2} + \cos \left (b x + a\right ) \sin \left (b x + a\right )\right )} \sqrt {d} \sqrt {\frac {d \sin \left (b x + a\right )}{\cos \left (b x + a\right )}} + d\right )}{32 \, b} \] Input:
integrate(sin(b*x+a)^2*(d*tan(b*x+a))^(1/2),x, algorithm="fricas")
Output:
-1/32*(16*sqrt(d*sin(b*x + a)/cos(b*x + a))*cos(b*x + a)*sin(b*x + a) + 6* sqrt(2)*sqrt(d)*arctan(-sqrt(2)*sqrt(d)*sqrt(d*sin(b*x + a)/cos(b*x + a))* cos(b*x + a)/(d*cos(b*x + a) - d*sin(b*x + a))) + 3*sqrt(2)*sqrt(d)*arctan (1/2*(2*d*cos(b*x + a)^2 - 2*d*cos(b*x + a)*sin(b*x + a) + sqrt(2)*sqrt(d) *sqrt(d*sin(b*x + a)/cos(b*x + a)) - 2*d)/(d*cos(b*x + a)^2 + d*cos(b*x + a)*sin(b*x + a) - d)) + 3*sqrt(2)*sqrt(d)*arctan(-1/2*(2*d*cos(b*x + a)^2 - 2*d*cos(b*x + a)*sin(b*x + a) - sqrt(2)*sqrt(d)*sqrt(d*sin(b*x + a)/cos( b*x + a)) - 2*d)/(d*cos(b*x + a)^2 + d*cos(b*x + a)*sin(b*x + a) - d)) + 3 *sqrt(2)*sqrt(d)*log(4*d*cos(b*x + a)*sin(b*x + a) + 2*sqrt(2)*(cos(b*x + a)^2 + cos(b*x + a)*sin(b*x + a))*sqrt(d)*sqrt(d*sin(b*x + a)/cos(b*x + a) ) + d) - 3*sqrt(2)*sqrt(d)*log(4*d*cos(b*x + a)*sin(b*x + a) - 2*sqrt(2)*( cos(b*x + a)^2 + cos(b*x + a)*sin(b*x + a))*sqrt(d)*sqrt(d*sin(b*x + a)/co s(b*x + a)) + d))/b
\[ \int \sin ^2(a+b x) \sqrt {d \tan (a+b x)} \, dx=\int \sqrt {d \tan {\left (a + b x \right )}} \sin ^{2}{\left (a + b x \right )}\, dx \] Input:
integrate(sin(b*x+a)**2*(d*tan(b*x+a))**(1/2),x)
Output:
Integral(sqrt(d*tan(a + b*x))*sin(a + b*x)**2, x)
Time = 0.11 (sec) , antiderivative size = 194, normalized size of antiderivative = 1.11 \[ \int \sin ^2(a+b x) \sqrt {d \tan (a+b x)} \, dx=\frac {3 \, d^{4} {\left (\frac {2 \, \sqrt {2} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} \sqrt {d} + 2 \, \sqrt {d \tan \left (b x + a\right )}\right )}}{2 \, \sqrt {d}}\right )}{\sqrt {d}} + \frac {2 \, \sqrt {2} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} \sqrt {d} - 2 \, \sqrt {d \tan \left (b x + a\right )}\right )}}{2 \, \sqrt {d}}\right )}{\sqrt {d}} - \frac {\sqrt {2} \log \left (d \tan \left (b x + a\right ) + \sqrt {2} \sqrt {d \tan \left (b x + a\right )} \sqrt {d} + d\right )}{\sqrt {d}} + \frac {\sqrt {2} \log \left (d \tan \left (b x + a\right ) - \sqrt {2} \sqrt {d \tan \left (b x + a\right )} \sqrt {d} + d\right )}{\sqrt {d}}\right )} - \frac {8 \, \left (d \tan \left (b x + a\right )\right )^{\frac {3}{2}} d^{4}}{d^{2} \tan \left (b x + a\right )^{2} + d^{2}}}{16 \, b d^{3}} \] Input:
integrate(sin(b*x+a)^2*(d*tan(b*x+a))^(1/2),x, algorithm="maxima")
Output:
1/16*(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*(d*tan(b* x + a))^(3/2)*d^4/(d^2*tan(b*x + a)^2 + d^2))/(b*d^3)
Time = 0.14 (sec) , antiderivative size = 219, normalized size of antiderivative = 1.26 \[ \int \sin ^2(a+b x) \sqrt {d \tan (a+b x)} \, dx=-\frac {\frac {8 \, \sqrt {d \tan \left (b x + a\right )} d^{3} \tan \left (b x + a\right )}{{\left (d^{2} \tan \left (b x + a\right )^{2} + d^{2}\right )} b} - \frac {6 \, \sqrt {2} {\left | d \right |}^{\frac {3}{2}} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} \sqrt {{\left | d \right |}} + 2 \, \sqrt {d \tan \left (b x + a\right )}\right )}}{2 \, \sqrt {{\left | d \right |}}}\right )}{b} - \frac {6 \, \sqrt {2} {\left | d \right |}^{\frac {3}{2}} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} \sqrt {{\left | d \right |}} - 2 \, \sqrt {d \tan \left (b x + a\right )}\right )}}{2 \, \sqrt {{\left | d \right |}}}\right )}{b} + \frac {3 \, \sqrt {2} {\left | d \right |}^{\frac {3}{2}} \log \left (d \tan \left (b x + a\right ) + \sqrt {2} \sqrt {d \tan \left (b x + a\right )} \sqrt {{\left | d \right |}} + {\left | d \right |}\right )}{b} - \frac {3 \, \sqrt {2} {\left | d \right |}^{\frac {3}{2}} \log \left (d \tan \left (b x + a\right ) - \sqrt {2} \sqrt {d \tan \left (b x + a\right )} \sqrt {{\left | d \right |}} + {\left | d \right |}\right )}{b}}{16 \, d} \] Input:
integrate(sin(b*x+a)^2*(d*tan(b*x+a))^(1/2),x, algorithm="giac")
Output:
-1/16*(8*sqrt(d*tan(b*x + a))*d^3*tan(b*x + a)/((d^2*tan(b*x + a)^2 + d^2) *b) - 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 - 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 + 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 - 3*sqrt(2)*abs(d)^(3/2)*log(d*tan(b*x + a) - sqr t(2)*sqrt(d*tan(b*x + a))*sqrt(abs(d)) + abs(d))/b)/d
Timed out. \[ \int \sin ^2(a+b x) \sqrt {d \tan (a+b x)} \, dx=\int {\sin \left (a+b\,x\right )}^2\,\sqrt {d\,\mathrm {tan}\left (a+b\,x\right )} \,d x \] Input:
int(sin(a + b*x)^2*(d*tan(a + b*x))^(1/2),x)
Output:
int(sin(a + b*x)^2*(d*tan(a + b*x))^(1/2), x)
\[ \int \sin ^2(a+b x) \sqrt {d \tan (a+b x)} \, dx=\sqrt {d}\, \left (\int \sqrt {\tan \left (b x +a \right )}\, \sin \left (b x +a \right )^{2}d x \right ) \] Input:
int(sin(b*x+a)^2*(d*tan(b*x+a))^(1/2),x)
Output:
sqrt(d)*int(sqrt(tan(a + b*x))*sin(a + b*x)**2,x)