Integrand size = 21, antiderivative size = 167 \[ \int \cos ^2(a+b x) (d \tan (a+b x))^{3/2} \, dx=-\frac {d^{3/2} \arctan \left (1-\frac {\sqrt {2} \sqrt {d \tan (a+b x)}}{\sqrt {d}}\right )}{4 \sqrt {2} b}+\frac {d^{3/2} \arctan \left (1+\frac {\sqrt {2} \sqrt {d \tan (a+b x)}}{\sqrt {d}}\right )}{4 \sqrt {2} b}+\frac {d^{3/2} \text {arctanh}\left (\frac {\sqrt {2} \sqrt {d \tan (a+b x)}}{\sqrt {d} (1+\tan (a+b x))}\right )}{4 \sqrt {2} b}-\frac {d \cos ^2(a+b x) \sqrt {d \tan (a+b x)}}{2 b} \] Output:
-1/8*d^(3/2)*arctan(1-2^(1/2)*(d*tan(b*x+a))^(1/2)/d^(1/2))*2^(1/2)/b+1/8* d^(3/2)*arctan(1+2^(1/2)*(d*tan(b*x+a))^(1/2)/d^(1/2))*2^(1/2)/b+1/8*d^(3/ 2)*arctanh(2^(1/2)*(d*tan(b*x+a))^(1/2)/d^(1/2)/(1+tan(b*x+a)))*2^(1/2)/b- 1/2*d*cos(b*x+a)^2*(d*tan(b*x+a))^(1/2)/b
Time = 0.35 (sec) , antiderivative size = 110, normalized size of antiderivative = 0.66 \[ \int \cos ^2(a+b x) (d \tan (a+b x))^{3/2} \, dx=-\frac {d \csc (a+b x) \left (\sin (a+b x)+\arcsin (\cos (a+b x)-\sin (a+b x)) \sqrt {\sin (2 (a+b x))}-\log \left (\cos (a+b x)+\sin (a+b x)+\sqrt {\sin (2 (a+b x))}\right ) \sqrt {\sin (2 (a+b x))}+\sin (3 (a+b x))\right ) \sqrt {d \tan (a+b x)}}{8 b} \] Input:
Integrate[Cos[a + b*x]^2*(d*Tan[a + b*x])^(3/2),x]
Output:
-1/8*(d*Csc[a + b*x]*(Sin[a + b*x] + ArcSin[Cos[a + b*x] - Sin[a + b*x]]*S qrt[Sin[2*(a + b*x)]] - 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]])/b
Time = 0.40 (sec) , antiderivative size = 225, normalized size of antiderivative = 1.35, number of steps used = 14, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.619, Rules used = {3042, 3087, 252, 266, 755, 27, 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.
| \(\displaystyle \int \cos ^2(a+b x) (d \tan (a+b x))^{3/2} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {(d \tan (a+b x))^{3/2}}{\sec (a+b x)^2}dx\) |
| \(\Big \downarrow \) 3087 |
| \(\displaystyle \frac {\int \frac {(d \tan (a+b x))^{3/2}}{\left (\tan ^2(a+b x)+1\right )^2}d\tan (a+b x)}{b}\) |
| \(\Big \downarrow \) 252 |
| \(\displaystyle \frac {\frac {1}{4} d^2 \int \frac {1}{\sqrt {d \tan (a+b x)} \left (\tan ^2(a+b x)+1\right )}d\tan (a+b x)-\frac {d \sqrt {d \tan (a+b x)}}{2 \left (\tan ^2(a+b x)+1\right )}}{b}\) |
| \(\Big \downarrow \) 266 |
| \(\displaystyle \frac {\frac {1}{2} d \int \frac {1}{\tan ^2(a+b x)+1}d\sqrt {d \tan (a+b x)}-\frac {d \sqrt {d \tan (a+b x)}}{2 \left (\tan ^2(a+b x)+1\right )}}{b}\) |
| \(\Big \downarrow \) 755 |
| \(\displaystyle \frac {\frac {1}{2} d \left (\frac {\int \frac {d^2 (d-d \tan (a+b x))}{\tan ^2(a+b x) d^2+d^2}d\sqrt {d \tan (a+b x)}}{2 d}+\frac {\int \frac {d^2 (\tan (a+b x) d+d)}{\tan ^2(a+b x) d^2+d^2}d\sqrt {d \tan (a+b x)}}{2 d}\right )-\frac {d \sqrt {d \tan (a+b x)}}{2 \left (\tan ^2(a+b x)+1\right )}}{b}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {1}{2} d \left (\frac {1}{2} d \int \frac {d-d \tan (a+b x)}{\tan ^2(a+b x) d^2+d^2}d\sqrt {d \tan (a+b x)}+\frac {1}{2} d \int \frac {\tan (a+b x) d+d}{\tan ^2(a+b x) d^2+d^2}d\sqrt {d \tan (a+b x)}\right )-\frac {d \sqrt {d \tan (a+b x)}}{2 \left (\tan ^2(a+b x)+1\right )}}{b}\) |
| \(\Big \downarrow \) 1476 |
| \(\displaystyle \frac {\frac {1}{2} d \left (\frac {1}{2} d \int \frac {d-d \tan (a+b x)}{\tan ^2(a+b x) d^2+d^2}d\sqrt {d \tan (a+b x)}+\frac {1}{2} d \left (\frac {1}{2} \int \frac {1}{\tan (a+b x) d+d-\sqrt {2} \sqrt {d \tan (a+b x)} \sqrt {d}}d\sqrt {d \tan (a+b x)}+\frac {1}{2} \int \frac {1}{\tan (a+b x) d+d+\sqrt {2} \sqrt {d \tan (a+b x)} \sqrt {d}}d\sqrt {d \tan (a+b x)}\right )\right )-\frac {d \sqrt {d \tan (a+b x)}}{2 \left (\tan ^2(a+b x)+1\right )}}{b}\) |
| \(\Big \downarrow \) 1082 |
| \(\displaystyle \frac {\frac {1}{2} d \left (\frac {1}{2} d \int \frac {d-d \tan (a+b x)}{\tan ^2(a+b x) d^2+d^2}d\sqrt {d \tan (a+b x)}+\frac {1}{2} d \left (\frac {\int \frac {1}{-d \tan (a+b x)-1}d\left (1-\frac {\sqrt {2} \sqrt {d \tan (a+b x)}}{\sqrt {d}}\right )}{\sqrt {2} \sqrt {d}}-\frac {\int \frac {1}{-d \tan (a+b x)-1}d\left (\frac {\sqrt {2} \sqrt {d \tan (a+b x)}}{\sqrt {d}}+1\right )}{\sqrt {2} \sqrt {d}}\right )\right )-\frac {d \sqrt {d \tan (a+b x)}}{2 \left (\tan ^2(a+b x)+1\right )}}{b}\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle \frac {\frac {1}{2} d \left (\frac {1}{2} d \int \frac {d-d \tan (a+b x)}{\tan ^2(a+b x) d^2+d^2}d\sqrt {d \tan (a+b x)}+\frac {1}{2} d \left (\frac {\arctan \left (\frac {\sqrt {2} \sqrt {d \tan (a+b x)}}{\sqrt {d}}+1\right )}{\sqrt {2} \sqrt {d}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt {d \tan (a+b x)}}{\sqrt {d}}\right )}{\sqrt {2} \sqrt {d}}\right )\right )-\frac {d \sqrt {d \tan (a+b x)}}{2 \left (\tan ^2(a+b x)+1\right )}}{b}\) |
| \(\Big \downarrow \) 1479 |
| \(\displaystyle \frac {\frac {1}{2} d \left (\frac {1}{2} d \left (-\frac {\int -\frac {\sqrt {2} \sqrt {d}-2 \sqrt {d \tan (a+b x)}}{\tan (a+b x) d+d-\sqrt {2} \sqrt {d \tan (a+b x)} \sqrt {d}}d\sqrt {d \tan (a+b x)}}{2 \sqrt {2} \sqrt {d}}-\frac {\int -\frac {\sqrt {2} \left (\sqrt {d}+\sqrt {2} \sqrt {d \tan (a+b x)}\right )}{\tan (a+b x) d+d+\sqrt {2} \sqrt {d \tan (a+b x)} \sqrt {d}}d\sqrt {d \tan (a+b x)}}{2 \sqrt {2} \sqrt {d}}\right )+\frac {1}{2} d \left (\frac {\arctan \left (\frac {\sqrt {2} \sqrt {d \tan (a+b x)}}{\sqrt {d}}+1\right )}{\sqrt {2} \sqrt {d}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt {d \tan (a+b x)}}{\sqrt {d}}\right )}{\sqrt {2} \sqrt {d}}\right )\right )-\frac {d \sqrt {d \tan (a+b x)}}{2 \left (\tan ^2(a+b x)+1\right )}}{b}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\frac {1}{2} d \left (\frac {1}{2} d \left (\frac {\int \frac {\sqrt {2} \sqrt {d}-2 \sqrt {d \tan (a+b x)}}{\tan (a+b x) d+d-\sqrt {2} \sqrt {d \tan (a+b x)} \sqrt {d}}d\sqrt {d \tan (a+b x)}}{2 \sqrt {2} \sqrt {d}}+\frac {\int \frac {\sqrt {2} \left (\sqrt {d}+\sqrt {2} \sqrt {d \tan (a+b x)}\right )}{\tan (a+b x) d+d+\sqrt {2} \sqrt {d \tan (a+b x)} \sqrt {d}}d\sqrt {d \tan (a+b x)}}{2 \sqrt {2} \sqrt {d}}\right )+\frac {1}{2} d \left (\frac {\arctan \left (\frac {\sqrt {2} \sqrt {d \tan (a+b x)}}{\sqrt {d}}+1\right )}{\sqrt {2} \sqrt {d}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt {d \tan (a+b x)}}{\sqrt {d}}\right )}{\sqrt {2} \sqrt {d}}\right )\right )-\frac {d \sqrt {d \tan (a+b x)}}{2 \left (\tan ^2(a+b x)+1\right )}}{b}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {1}{2} d \left (\frac {1}{2} d \left (\frac {\int \frac {\sqrt {2} \sqrt {d}-2 \sqrt {d \tan (a+b x)}}{\tan (a+b x) d+d-\sqrt {2} \sqrt {d \tan (a+b x)} \sqrt {d}}d\sqrt {d \tan (a+b x)}}{2 \sqrt {2} \sqrt {d}}+\frac {\int \frac {\sqrt {d}+\sqrt {2} \sqrt {d \tan (a+b x)}}{\tan (a+b x) d+d+\sqrt {2} \sqrt {d \tan (a+b x)} \sqrt {d}}d\sqrt {d \tan (a+b x)}}{2 \sqrt {d}}\right )+\frac {1}{2} d \left (\frac {\arctan \left (\frac {\sqrt {2} \sqrt {d \tan (a+b x)}}{\sqrt {d}}+1\right )}{\sqrt {2} \sqrt {d}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt {d \tan (a+b x)}}{\sqrt {d}}\right )}{\sqrt {2} \sqrt {d}}\right )\right )-\frac {d \sqrt {d \tan (a+b x)}}{2 \left (\tan ^2(a+b x)+1\right )}}{b}\) |
| \(\Big \downarrow \) 1103 |
| \(\displaystyle \frac {\frac {1}{2} d \left (\frac {1}{2} d \left (\frac {\arctan \left (\frac {\sqrt {2} \sqrt {d \tan (a+b x)}}{\sqrt {d}}+1\right )}{\sqrt {2} \sqrt {d}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt {d \tan (a+b x)}}{\sqrt {d}}\right )}{\sqrt {2} \sqrt {d}}\right )+\frac {1}{2} d \left (\frac {\log \left (d \tan (a+b x)+\sqrt {2} \sqrt {d} \sqrt {d \tan (a+b x)}+d\right )}{2 \sqrt {2} \sqrt {d}}-\frac {\log \left (d \tan (a+b x)-\sqrt {2} \sqrt {d} \sqrt {d \tan (a+b x)}+d\right )}{2 \sqrt {2} \sqrt {d}}\right )\right )-\frac {d \sqrt {d \tan (a+b x)}}{2 \left (\tan ^2(a+b x)+1\right )}}{b}\) |
Input:
Int[Cos[a + b*x]^2*(d*Tan[a + b*x])^(3/2),x]
Output:
((d*((d*(-(ArcTan[1 - (Sqrt[2]*Sqrt[d*Tan[a + b*x]])/Sqrt[d]]/(Sqrt[2]*Sqr t[d])) + ArcTan[1 + (Sqrt[2]*Sqrt[d*Tan[a + b*x]])/Sqrt[d]]/(Sqrt[2]*Sqrt[ d])))/2 + (d*(-1/2*Log[d + d*Tan[a + b*x] - Sqrt[2]*Sqrt[d]*Sqrt[d*Tan[a + b*x]]]/(Sqrt[2]*Sqrt[d]) + Log[d + d*Tan[a + b*x] + Sqrt[2]*Sqrt[d]*Sqrt[ d*Tan[a + b*x]]]/(2*Sqrt[2]*Sqrt[d])))/2))/2 - (d*Sqrt[d*Tan[a + b*x]])/(2 *(1 + 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[((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[sec[(e_.) + (f_.)*(x_)]^(m_)*((b_.)*tan[(e_.) + (f_.)*(x_)])^(n_.), x_S ymbol] :> Simp[1/f Subst[Int[(b*x)^n*(1 + x^2)^(m/2 - 1), x], x, Tan[e + f*x]], x] /; FreeQ[{b, e, f, n}, x] && IntegerQ[m/2] && !(IntegerQ[(n - 1) /2] && LtQ[0, n, m - 1])
Leaf count of result is larger than twice the leaf count of optimal. \(403\) vs. \(2(127)=254\).
Time = 2.03 (sec) , antiderivative size = 404, normalized size of antiderivative = 2.42
| method | result | size |
| default | \(-\frac {\left (\cos \left (b x +a \right ) \left (4 \cos \left (b x +a \right )+4\right ) \sqrt {-\frac {2 \sin \left (b x +a \right ) \cos \left (b x +a \right )}{\left (\cos \left (b x +a \right )+1\right )^{2}}}+2 \arctan \left (\frac {\sin \left (b x +a \right ) \sqrt {-\frac {2 \sin \left (b x +a \right ) \cos \left (b x +a \right )}{\left (\cos \left (b x +a \right )+1\right )^{2}}}-\cos \left (b x +a \right )+1}{-1+\cos \left (b x +a \right )}\right )+\ln \left (-\frac {\cot \left (b x +a \right ) \cos \left (b x +a \right )-2 \cot \left (b x +a \right )+2 \sin \left (b x +a \right ) \sqrt {-\frac {2 \sin \left (b x +a \right ) \cos \left (b x +a \right )}{\left (\cos \left (b x +a \right )+1\right )^{2}}}-2 \cos \left (b x +a \right )+\csc \left (b x +a \right )-\sin \left (b x +a \right )+2}{-1+\cos \left (b x +a \right )}\right )-\ln \left (\frac {2 \sin \left (b x +a \right ) \sqrt {-\frac {2 \sin \left (b x +a \right ) \cos \left (b x +a \right )}{\left (\cos \left (b x +a \right )+1\right )^{2}}}-\cot \left (b x +a \right ) \cos \left (b x +a \right )+\sin \left (b x +a \right )+2 \cos \left (b x +a \right )-\csc \left (b x +a \right )+2 \cot \left (b x +a \right )-2}{-1+\cos \left (b x +a \right )}\right )+2 \arctan \left (\frac {\sin \left (b x +a \right ) \sqrt {-\frac {2 \sin \left (b x +a \right ) \cos \left (b x +a \right )}{\left (\cos \left (b x +a \right )+1\right )^{2}}}+\cos \left (b x +a \right )-1}{-1+\cos \left (b x +a \right )}\right )\right ) \cos \left (b x +a \right ) \sqrt {d \tan \left (b x +a \right )}\, d \sqrt {2}}{16 b \left (\cos \left (b x +a \right )+1\right ) \sqrt {-\frac {\sin \left (b x +a \right ) \cos \left (b x +a \right )}{\left (\cos \left (b x +a \right )+1\right )^{2}}}}\) | \(404\) |
Input:
int(cos(b*x+a)^2*(d*tan(b*x+a))^(3/2),x,method=_RETURNVERBOSE)
Output:
-1/16/b*(cos(b*x+a)*(4*cos(b*x+a)+4)*(-2*sin(b*x+a)*cos(b*x+a)/(cos(b*x+a) +1)^2)^(1/2)+2*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)))+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*co s(b*x+a)+csc(b*x+a)-sin(b*x+a)+2)/(-1+cos(b*x+a)))-ln((2*sin(b*x+a)*(-2*si n(b*x+a)*cos(b*x+a)/(cos(b*x+a)+1)^2)^(1/2)-cot(b*x+a)*cos(b*x+a)+sin(b*x+ a)+2*cos(b*x+a)-csc(b*x+a)+2*cot(b*x+a)-2)/(-1+cos(b*x+a)))+2*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))))*cos(b*x+a)*(d*tan(b*x+a))^(1/2)*d/(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. 431 vs. \(2 (127) = 254\).
Time = 0.15 (sec) , antiderivative size = 431, normalized size of antiderivative = 2.58 \[ \int \cos ^2(a+b x) (d \tan (a+b x))^{3/2} \, dx=-\frac {16 \, d \sqrt {\frac {d \sin \left (b x + a\right )}{\cos \left (b x + a\right )}} \cos \left (b x + a\right )^{2} - 2 \, \sqrt {2} d^{\frac {3}{2}} \arctan \left (-\frac {\sqrt {2} \sqrt {\frac {d \sin \left (b x + a\right )}{\cos \left (b x + a\right )}} {\left (\cos \left (b x + a\right ) - \sin \left (b x + a\right )\right )}}{2 \, \sqrt {d} \sin \left (b x + a\right )}\right ) + \sqrt {2} d^{\frac {3}{2}} \arctan \left (\frac {2 \, d \cos \left (b x + a\right )^{2} - 2 \, d \cos \left (b x + a\right ) \sin \left (b x + a\right ) + \sqrt {2} \sqrt {d} \sqrt {\frac {d \sin \left (b x + a\right )}{\cos \left (b x + a\right )}} - 2 \, d}{2 \, {\left (d \cos \left (b x + a\right )^{2} + d \cos \left (b x + a\right ) \sin \left (b x + a\right ) - d\right )}}\right ) + \sqrt {2} d^{\frac {3}{2}} \arctan \left (-\frac {2 \, d \cos \left (b x + a\right )^{2} - 2 \, d \cos \left (b x + a\right ) \sin \left (b x + a\right ) - \sqrt {2} \sqrt {d} \sqrt {\frac {d \sin \left (b x + a\right )}{\cos \left (b x + a\right )}} - 2 \, d}{2 \, {\left (d \cos \left (b x + a\right )^{2} + d \cos \left (b x + a\right ) \sin \left (b x + a\right ) - d\right )}}\right ) - \sqrt {2} d^{\frac {3}{2}} \log \left (4 \, d \cos \left (b x + a\right ) \sin \left (b x + a\right ) + 2 \, \sqrt {2} {\left (\cos \left (b x + a\right )^{2} + \cos \left (b x + a\right ) \sin \left (b x + a\right )\right )} \sqrt {d} \sqrt {\frac {d \sin \left (b x + a\right )}{\cos \left (b x + a\right )}} + d\right ) + \sqrt {2} d^{\frac {3}{2}} \log \left (4 \, d \cos \left (b x + a\right ) \sin \left (b x + a\right ) - 2 \, \sqrt {2} {\left (\cos \left (b x + a\right )^{2} + \cos \left (b x + a\right ) \sin \left (b x + a\right )\right )} \sqrt {d} \sqrt {\frac {d \sin \left (b x + a\right )}{\cos \left (b x + a\right )}} + d\right )}{32 \, b} \] Input:
integrate(cos(b*x+a)^2*(d*tan(b*x+a))^(3/2),x, algorithm="fricas")
Output:
-1/32*(16*d*sqrt(d*sin(b*x + a)/cos(b*x + a))*cos(b*x + a)^2 - 2*sqrt(2)*d ^(3/2)*arctan(-1/2*sqrt(2)*sqrt(d*sin(b*x + a)/cos(b*x + a))*(cos(b*x + a) - sin(b*x + a))/(sqrt(d)*sin(b*x + a))) + sqrt(2)*d^(3/2)*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*s in(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)) + sqrt(2)*d^(3/2)*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)) - sqrt(2)*d^(3 /2)*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) + sqrt (2)*d^(3/2)*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 ))/b
Timed out. \[ \int \cos ^2(a+b x) (d \tan (a+b x))^{3/2} \, dx=\text {Timed out} \] Input:
integrate(cos(b*x+a)**2*(d*tan(b*x+a))**(3/2),x)
Output:
Timed out
Time = 0.13 (sec) , antiderivative size = 188, normalized size of antiderivative = 1.13 \[ \int \cos ^2(a+b x) (d \tan (a+b x))^{3/2} \, dx=\frac {2 \, \sqrt {2} d^{\frac {5}{2}} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} \sqrt {d} + 2 \, \sqrt {d \tan \left (b x + a\right )}\right )}}{2 \, \sqrt {d}}\right ) + 2 \, \sqrt {2} d^{\frac {5}{2}} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} \sqrt {d} - 2 \, \sqrt {d \tan \left (b x + a\right )}\right )}}{2 \, \sqrt {d}}\right ) + \sqrt {2} d^{\frac {5}{2}} \log \left (d \tan \left (b x + a\right ) + \sqrt {2} \sqrt {d \tan \left (b x + a\right )} \sqrt {d} + d\right ) - \sqrt {2} d^{\frac {5}{2}} \log \left (d \tan \left (b x + a\right ) - \sqrt {2} \sqrt {d \tan \left (b x + a\right )} \sqrt {d} + d\right ) - \frac {8 \, \sqrt {d \tan \left (b x + a\right )} d^{4}}{d^{2} \tan \left (b x + a\right )^{2} + d^{2}}}{16 \, b d} \] Input:
integrate(cos(b*x+a)^2*(d*tan(b*x+a))^(3/2),x, algorithm="maxima")
Output:
1/16*(2*sqrt(2)*d^(5/2)*arctan(1/2*sqrt(2)*(sqrt(2)*sqrt(d) + 2*sqrt(d*tan (b*x + a)))/sqrt(d)) + 2*sqrt(2)*d^(5/2)*arctan(-1/2*sqrt(2)*(sqrt(2)*sqrt (d) - 2*sqrt(d*tan(b*x + a)))/sqrt(d)) + sqrt(2)*d^(5/2)*log(d*tan(b*x + a ) + sqrt(2)*sqrt(d*tan(b*x + a))*sqrt(d) + d) - sqrt(2)*d^(5/2)*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^4/(d^2*tan(b*x + a)^2 + d^2))/(b*d)
Time = 0.15 (sec) , antiderivative size = 224, normalized size of antiderivative = 1.34 \[ \int \cos ^2(a+b x) (d \tan (a+b x))^{3/2} \, dx=\frac {\frac {2 \, \sqrt {2} d^{2} \sqrt {{\left | d \right |}} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} \sqrt {{\left | d \right |}} + 2 \, \sqrt {d \tan \left (b x + a\right )}\right )}}{2 \, \sqrt {{\left | d \right |}}}\right )}{b} + \frac {2 \, \sqrt {2} d^{2} \sqrt {{\left | d \right |}} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} \sqrt {{\left | d \right |}} - 2 \, \sqrt {d \tan \left (b x + a\right )}\right )}}{2 \, \sqrt {{\left | d \right |}}}\right )}{b} + \frac {\sqrt {2} d^{2} \sqrt {{\left | d \right |}} \log \left (d \tan \left (b x + a\right ) + \sqrt {2} \sqrt {d \tan \left (b x + a\right )} \sqrt {{\left | d \right |}} + {\left | d \right |}\right )}{b} - \frac {\sqrt {2} d^{2} \sqrt {{\left | d \right |}} \log \left (d \tan \left (b x + a\right ) - \sqrt {2} \sqrt {d \tan \left (b x + a\right )} \sqrt {{\left | d \right |}} + {\left | d \right |}\right )}{b} - \frac {8 \, \sqrt {d \tan \left (b x + a\right )} d^{4}}{{\left (d^{2} \tan \left (b x + a\right )^{2} + d^{2}\right )} b}}{16 \, d} \] Input:
integrate(cos(b*x+a)^2*(d*tan(b*x+a))^(3/2),x, algorithm="giac")
Output:
1/16*(2*sqrt(2)*d^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 + 2*sqrt(2)*d^2*sqrt(abs(d))*arc tan(-1/2*sqrt(2)*(sqrt(2)*sqrt(abs(d)) - 2*sqrt(d*tan(b*x + a)))/sqrt(abs( d)))/b + sqrt(2)*d^2*sqrt(abs(d))*log(d*tan(b*x + a) + sqrt(2)*sqrt(d*tan( b*x + a))*sqrt(abs(d)) + abs(d))/b - sqrt(2)*d^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^4/((d^2*tan(b*x + a)^2 + d^2)*b))/d
Timed out. \[ \int \cos ^2(a+b x) (d \tan (a+b x))^{3/2} \, dx=\int {\cos \left (a+b\,x\right )}^2\,{\left (d\,\mathrm {tan}\left (a+b\,x\right )\right )}^{3/2} \,d x \] Input:
int(cos(a + b*x)^2*(d*tan(a + b*x))^(3/2),x)
Output:
int(cos(a + b*x)^2*(d*tan(a + b*x))^(3/2), x)
\[ \int \cos ^2(a+b x) (d \tan (a+b x))^{3/2} \, dx=\sqrt {d}\, \left (\int \sqrt {\tan \left (b x +a \right )}\, \cos \left (b x +a \right )^{2} \tan \left (b x +a \right )d x \right ) d \] Input:
int(cos(b*x+a)^2*(d*tan(b*x+a))^(3/2),x)
Output:
sqrt(d)*int(sqrt(tan(a + b*x))*cos(a + b*x)**2*tan(a + b*x),x)*d