Integrand size = 23, antiderivative size = 88 \[ \int \frac {\tan ^3(c+d x)}{(a+b \sec (c+d x))^{3/2}} \, dx=\frac {2 \text {arctanh}\left (\frac {\sqrt {a+b \sec (c+d x)}}{\sqrt {a}}\right )}{a^{3/2} d}+\frac {2 \left (a^2-b^2\right )}{a b^2 d \sqrt {a+b \sec (c+d x)}}+\frac {2 \sqrt {a+b \sec (c+d x)}}{b^2 d} \]
2*arctanh((a+b*sec(d*x+c))^(1/2)/a^(1/2))/a^(3/2)/d+2*(a^2-b^2)/a/b^2/d/(a +b*sec(d*x+c))^(1/2)+2*(a+b*sec(d*x+c))^(1/2)/b^2/d
Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.
Time = 0.26 (sec) , antiderivative size = 66, normalized size of antiderivative = 0.75 \[ \int \frac {\tan ^3(c+d x)}{(a+b \sec (c+d x))^{3/2}} \, dx=\frac {2 \left (-b^2 \operatorname {Hypergeometric2F1}\left (-\frac {1}{2},1,\frac {1}{2},1+\frac {b \sec (c+d x)}{a}\right )+a (2 a+b \sec (c+d x))\right )}{a b^2 d \sqrt {a+b \sec (c+d x)}} \]
(2*(-(b^2*Hypergeometric2F1[-1/2, 1, 1/2, 1 + (b*Sec[c + d*x])/a]) + a*(2* a + b*Sec[c + d*x])))/(a*b^2*d*Sqrt[a + b*Sec[c + d*x]])
Time = 0.33 (sec) , antiderivative size = 79, normalized size of antiderivative = 0.90, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.261, Rules used = {3042, 25, 4373, 517, 1584, 2009}
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.
| \(\displaystyle \int \frac {\tan ^3(c+d x)}{(a+b \sec (c+d x))^{3/2}} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int -\frac {\cot \left (c+d x+\frac {\pi }{2}\right )^3}{\left (a+b \csc \left (c+d x+\frac {\pi }{2}\right )\right )^{3/2}}dx\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\int \frac {\cot \left (\frac {1}{2} (2 c+\pi )+d x\right )^3}{\left (a+b \csc \left (\frac {1}{2} (2 c+\pi )+d x\right )\right )^{3/2}}dx\) |
| \(\Big \downarrow \) 4373 |
| \(\displaystyle -\frac {\int \frac {\cos (c+d x) \left (b^2-b^2 \sec ^2(c+d x)\right )}{b (a+b \sec (c+d x))^{3/2}}d(b \sec (c+d x))}{b^2 d}\) |
| \(\Big \downarrow \) 517 |
| \(\displaystyle -\frac {2 \int \frac {\cos ^2(c+d x) \left (b^4 \sec ^4(c+d x)-2 a b^2 \sec ^2(c+d x)+a^2-b^2\right )}{b^2 \left (a-b^2 \sec ^2(c+d x)\right )}d\sqrt {a+b \sec (c+d x)}}{b^2 d}\) |
| \(\Big \downarrow \) 1584 |
| \(\displaystyle -\frac {2 \int \left (-\frac {b^2}{a \left (a-b^2 \sec ^2(c+d x)\right )}-1+\frac {\left (a^2-b^2\right ) \cos ^2(c+d x)}{a b^2}\right )d\sqrt {a+b \sec (c+d x)}}{b^2 d}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {2 \left (-\frac {b^2 \text {arctanh}\left (\frac {\sqrt {a+b \sec (c+d x)}}{\sqrt {a}}\right )}{a^{3/2}}-\frac {\left (a^2-b^2\right ) \cos (c+d x)}{a b}-\sqrt {a+b \sec (c+d x)}\right )}{b^2 d}\) |
(-2*(-((b^2*ArcTanh[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a]])/a^(3/2)) - ((a^2 - b^2)*Cos[c + d*x])/(a*b) - Sqrt[a + b*Sec[c + d*x]]))/(b^2*d)
3.4.36.3.1 Defintions of rubi rules used
Int[((e_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_)*((a_) + (b_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[2*(e^m/d^(m + 2*p + 1)) Subst[Int[x^(2*n + 1)*(-c + x^ 2)^m*(b*c^2 + a*d^2 - 2*b*c*x^2 + b*x^4)^p, x], x, Sqrt[c + d*x]], x] /; Fr eeQ[{a, b, c, d, e}, x] && IGtQ[p, 0] && ILtQ[m, 0] && IntegerQ[n + 1/2]
Int[((f_.)*(x_))^(m_.)*((d_) + (e_.)*(x_)^2)^(q_.)*((a_) + (b_.)*(x_)^2 + ( c_.)*(x_)^4)^(p_.), x_Symbol] :> Int[ExpandIntegrand[(f*x)^m*(d + e*x^2)^q* (a + b*x^2 + c*x^4)^p, x], x] /; FreeQ[{a, b, c, d, e, f, m, q}, x] && NeQ[ b^2 - 4*a*c, 0] && IGtQ[p, 0] && IGtQ[q, -2]
Int[cot[(c_.) + (d_.)*(x_)]^(m_.)*(csc[(c_.) + (d_.)*(x_)]*(b_.) + (a_))^(n _), x_Symbol] :> Simp[-(-1)^((m - 1)/2)/(d*b^(m - 1)) Subst[Int[(b^2 - x^ 2)^((m - 1)/2)*((a + x)^n/x), x], x, b*Csc[c + d*x]], x] /; FreeQ[{a, b, c, d, n}, x] && IntegerQ[(m - 1)/2] && NeQ[a^2 - b^2, 0]
Leaf count of result is larger than twice the leaf count of optimal. \(730\) vs. \(2(78)=156\).
Time = 9.56 (sec) , antiderivative size = 731, normalized size of antiderivative = 8.31
| method | result | size |
| default | \(\frac {\left (\cos \left (d x +c \right )^{3} a^{\frac {5}{2}} \ln \left (4 \cos \left (d x +c \right ) \sqrt {\frac {\left (b +a \cos \left (d x +c \right )\right ) \cos \left (d x +c \right )}{\left (\cos \left (d x +c \right )+1\right )^{2}}}\, \sqrt {a}+4 a \cos \left (d x +c \right )+4 \sqrt {a}\, \sqrt {\frac {\left (b +a \cos \left (d x +c \right )\right ) \cos \left (d x +c \right )}{\left (\cos \left (d x +c \right )+1\right )^{2}}}+2 b \right ) b^{2}+2 \cos \left (d x +c \right )^{2} a^{\frac {3}{2}} \ln \left (4 \cos \left (d x +c \right ) \sqrt {\frac {\left (b +a \cos \left (d x +c \right )\right ) \cos \left (d x +c \right )}{\left (\cos \left (d x +c \right )+1\right )^{2}}}\, \sqrt {a}+4 a \cos \left (d x +c \right )+4 \sqrt {a}\, \sqrt {\frac {\left (b +a \cos \left (d x +c \right )\right ) \cos \left (d x +c \right )}{\left (\cos \left (d x +c \right )+1\right )^{2}}}+2 b \right ) b^{3}+2 \sin \left (d x +c \right )^{2} \cos \left (d x +c \right ) \sqrt {\frac {\left (b +a \cos \left (d x +c \right )\right ) \cos \left (d x +c \right )}{\left (\cos \left (d x +c \right )+1\right )^{2}}}\, a^{2} b^{2}+4 \cos \left (d x +c \right )^{3} \sqrt {\frac {\left (b +a \cos \left (d x +c \right )\right ) \cos \left (d x +c \right )}{\left (\cos \left (d x +c \right )+1\right )^{2}}}\, a^{4}+2 \sqrt {\frac {\left (b +a \cos \left (d x +c \right )\right ) \cos \left (d x +c \right )}{\left (\cos \left (d x +c \right )+1\right )^{2}}}\, a^{2} b^{2} \sin \left (d x +c \right )^{2}+4 \cos \left (d x +c \right )^{2} \sqrt {\frac {\left (b +a \cos \left (d x +c \right )\right ) \cos \left (d x +c \right )}{\left (\cos \left (d x +c \right )+1\right )^{2}}}\, a^{4}+6 \cos \left (d x +c \right )^{2} \sqrt {\frac {\left (b +a \cos \left (d x +c \right )\right ) \cos \left (d x +c \right )}{\left (\cos \left (d x +c \right )+1\right )^{2}}}\, a^{3} b -2 \cos \left (d x +c \right )^{2} \sqrt {\frac {\left (b +a \cos \left (d x +c \right )\right ) \cos \left (d x +c \right )}{\left (\cos \left (d x +c \right )+1\right )^{2}}}\, a \,b^{3}+\cos \left (d x +c \right ) \ln \left (4 \cos \left (d x +c \right ) \sqrt {\frac {\left (b +a \cos \left (d x +c \right )\right ) \cos \left (d x +c \right )}{\left (\cos \left (d x +c \right )+1\right )^{2}}}\, \sqrt {a}+4 a \cos \left (d x +c \right )+4 \sqrt {a}\, \sqrt {\frac {\left (b +a \cos \left (d x +c \right )\right ) \cos \left (d x +c \right )}{\left (\cos \left (d x +c \right )+1\right )^{2}}}+2 b \right ) \sqrt {a}\, b^{4}+6 \cos \left (d x +c \right ) \sqrt {\frac {\left (b +a \cos \left (d x +c \right )\right ) \cos \left (d x +c \right )}{\left (\cos \left (d x +c \right )+1\right )^{2}}}\, a^{3} b -2 \cos \left (d x +c \right ) \sqrt {\frac {\left (b +a \cos \left (d x +c \right )\right ) \cos \left (d x +c \right )}{\left (\cos \left (d x +c \right )+1\right )^{2}}}\, a \,b^{3}\right ) \sqrt {a +b \sec \left (d x +c \right )}}{d \,a^{2} b^{2} \left (b +a \cos \left (d x +c \right )\right )^{2} \sqrt {\frac {\left (b +a \cos \left (d x +c \right )\right ) \cos \left (d x +c \right )}{\left (\cos \left (d x +c \right )+1\right )^{2}}}\, \left (\cos \left (d x +c \right )+1\right )}\) | \(731\) |
1/d/a^2/b^2*(cos(d*x+c)^3*a^(5/2)*ln(4*cos(d*x+c)*((b+a*cos(d*x+c))*cos(d* x+c)/(cos(d*x+c)+1)^2)^(1/2)*a^(1/2)+4*a*cos(d*x+c)+4*a^(1/2)*((b+a*cos(d* x+c))*cos(d*x+c)/(cos(d*x+c)+1)^2)^(1/2)+2*b)*b^2+2*cos(d*x+c)^2*a^(3/2)*l n(4*cos(d*x+c)*((b+a*cos(d*x+c))*cos(d*x+c)/(cos(d*x+c)+1)^2)^(1/2)*a^(1/2 )+4*a*cos(d*x+c)+4*a^(1/2)*((b+a*cos(d*x+c))*cos(d*x+c)/(cos(d*x+c)+1)^2)^ (1/2)+2*b)*b^3+2*sin(d*x+c)^2*cos(d*x+c)*((b+a*cos(d*x+c))*cos(d*x+c)/(cos (d*x+c)+1)^2)^(1/2)*a^2*b^2+4*cos(d*x+c)^3*((b+a*cos(d*x+c))*cos(d*x+c)/(c os(d*x+c)+1)^2)^(1/2)*a^4+2*((b+a*cos(d*x+c))*cos(d*x+c)/(cos(d*x+c)+1)^2) ^(1/2)*a^2*b^2*sin(d*x+c)^2+4*cos(d*x+c)^2*((b+a*cos(d*x+c))*cos(d*x+c)/(c os(d*x+c)+1)^2)^(1/2)*a^4+6*cos(d*x+c)^2*((b+a*cos(d*x+c))*cos(d*x+c)/(cos (d*x+c)+1)^2)^(1/2)*a^3*b-2*cos(d*x+c)^2*((b+a*cos(d*x+c))*cos(d*x+c)/(cos (d*x+c)+1)^2)^(1/2)*a*b^3+cos(d*x+c)*ln(4*cos(d*x+c)*((b+a*cos(d*x+c))*cos (d*x+c)/(cos(d*x+c)+1)^2)^(1/2)*a^(1/2)+4*a*cos(d*x+c)+4*a^(1/2)*((b+a*cos (d*x+c))*cos(d*x+c)/(cos(d*x+c)+1)^2)^(1/2)+2*b)*a^(1/2)*b^4+6*cos(d*x+c)* ((b+a*cos(d*x+c))*cos(d*x+c)/(cos(d*x+c)+1)^2)^(1/2)*a^3*b-2*cos(d*x+c)*(( b+a*cos(d*x+c))*cos(d*x+c)/(cos(d*x+c)+1)^2)^(1/2)*a*b^3)*(a+b*sec(d*x+c)) ^(1/2)/(b+a*cos(d*x+c))^2/((b+a*cos(d*x+c))*cos(d*x+c)/(cos(d*x+c)+1)^2)^( 1/2)/(cos(d*x+c)+1)
Time = 0.44 (sec) , antiderivative size = 317, normalized size of antiderivative = 3.60 \[ \int \frac {\tan ^3(c+d x)}{(a+b \sec (c+d x))^{3/2}} \, dx=\left [\frac {{\left (a b^{2} \cos \left (d x + c\right ) + b^{3}\right )} \sqrt {a} \log \left (-8 \, a^{2} \cos \left (d x + c\right )^{2} - 8 \, a b \cos \left (d x + c\right ) - b^{2} - 4 \, {\left (2 \, a \cos \left (d x + c\right )^{2} + b \cos \left (d x + c\right )\right )} \sqrt {a} \sqrt {\frac {a \cos \left (d x + c\right ) + b}{\cos \left (d x + c\right )}}\right ) + 4 \, {\left (a^{2} b + {\left (2 \, a^{3} - a b^{2}\right )} \cos \left (d x + c\right )\right )} \sqrt {\frac {a \cos \left (d x + c\right ) + b}{\cos \left (d x + c\right )}}}{2 \, {\left (a^{3} b^{2} d \cos \left (d x + c\right ) + a^{2} b^{3} d\right )}}, -\frac {{\left (a b^{2} \cos \left (d x + c\right ) + b^{3}\right )} \sqrt {-a} \arctan \left (\frac {2 \, \sqrt {-a} \sqrt {\frac {a \cos \left (d x + c\right ) + b}{\cos \left (d x + c\right )}} \cos \left (d x + c\right )}{2 \, a \cos \left (d x + c\right ) + b}\right ) - 2 \, {\left (a^{2} b + {\left (2 \, a^{3} - a b^{2}\right )} \cos \left (d x + c\right )\right )} \sqrt {\frac {a \cos \left (d x + c\right ) + b}{\cos \left (d x + c\right )}}}{a^{3} b^{2} d \cos \left (d x + c\right ) + a^{2} b^{3} d}\right ] \]
[1/2*((a*b^2*cos(d*x + c) + b^3)*sqrt(a)*log(-8*a^2*cos(d*x + c)^2 - 8*a*b *cos(d*x + c) - b^2 - 4*(2*a*cos(d*x + c)^2 + b*cos(d*x + c))*sqrt(a)*sqrt ((a*cos(d*x + c) + b)/cos(d*x + c))) + 4*(a^2*b + (2*a^3 - a*b^2)*cos(d*x + c))*sqrt((a*cos(d*x + c) + b)/cos(d*x + c)))/(a^3*b^2*d*cos(d*x + c) + a ^2*b^3*d), -((a*b^2*cos(d*x + c) + b^3)*sqrt(-a)*arctan(2*sqrt(-a)*sqrt((a *cos(d*x + c) + b)/cos(d*x + c))*cos(d*x + c)/(2*a*cos(d*x + c) + b)) - 2* (a^2*b + (2*a^3 - a*b^2)*cos(d*x + c))*sqrt((a*cos(d*x + c) + b)/cos(d*x + c)))/(a^3*b^2*d*cos(d*x + c) + a^2*b^3*d)]
\[ \int \frac {\tan ^3(c+d x)}{(a+b \sec (c+d x))^{3/2}} \, dx=\int \frac {\tan ^{3}{\left (c + d x \right )}}{\left (a + b \sec {\left (c + d x \right )}\right )^{\frac {3}{2}}}\, dx \]
Time = 0.28 (sec) , antiderivative size = 110, normalized size of antiderivative = 1.25 \[ \int \frac {\tan ^3(c+d x)}{(a+b \sec (c+d x))^{3/2}} \, dx=-\frac {\frac {\log \left (\frac {\sqrt {a + \frac {b}{\cos \left (d x + c\right )}} - \sqrt {a}}{\sqrt {a + \frac {b}{\cos \left (d x + c\right )}} + \sqrt {a}}\right )}{a^{\frac {3}{2}}} + \frac {2}{\sqrt {a + \frac {b}{\cos \left (d x + c\right )}} a} - \frac {2 \, \sqrt {a + \frac {b}{\cos \left (d x + c\right )}}}{b^{2}} - \frac {2 \, a}{\sqrt {a + \frac {b}{\cos \left (d x + c\right )}} b^{2}}}{d} \]
-(log((sqrt(a + b/cos(d*x + c)) - sqrt(a))/(sqrt(a + b/cos(d*x + c)) + sqr t(a)))/a^(3/2) + 2/(sqrt(a + b/cos(d*x + c))*a) - 2*sqrt(a + b/cos(d*x + c ))/b^2 - 2*a/(sqrt(a + b/cos(d*x + c))*b^2))/d
Leaf count of result is larger than twice the leaf count of optimal. 258 vs. \(2 (78) = 156\).
Time = 1.10 (sec) , antiderivative size = 258, normalized size of antiderivative = 2.93 \[ \int \frac {\tan ^3(c+d x)}{(a+b \sec (c+d x))^{3/2}} \, dx=-\frac {2 \, {\left (\frac {\frac {{\left (2 \, a^{3} \mathrm {sgn}\left (\cos \left (d x + c\right )\right ) - a^{2} b \mathrm {sgn}\left (\cos \left (d x + c\right )\right ) - a b^{2} \mathrm {sgn}\left (\cos \left (d x + c\right )\right )\right )} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2}}{a^{2} b^{2}} - \frac {2 \, a^{3} \mathrm {sgn}\left (\cos \left (d x + c\right )\right ) + a^{2} b \mathrm {sgn}\left (\cos \left (d x + c\right )\right ) - a b^{2} \mathrm {sgn}\left (\cos \left (d x + c\right )\right )}{a^{2} b^{2}}}{\sqrt {a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 2 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + a + b}} + \frac {\arctan \left (-\frac {\sqrt {a - b} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - \sqrt {a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 2 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + a + b} + \sqrt {a - b}}{2 \, \sqrt {-a}}\right )}{\sqrt {-a} a \mathrm {sgn}\left (\cos \left (d x + c\right )\right )}\right )}}{d} \]
-2*(((2*a^3*sgn(cos(d*x + c)) - a^2*b*sgn(cos(d*x + c)) - a*b^2*sgn(cos(d* x + c)))*tan(1/2*d*x + 1/2*c)^2/(a^2*b^2) - (2*a^3*sgn(cos(d*x + c)) + a^2 *b*sgn(cos(d*x + c)) - a*b^2*sgn(cos(d*x + c)))/(a^2*b^2))/sqrt(a*tan(1/2* d*x + 1/2*c)^4 - b*tan(1/2*d*x + 1/2*c)^4 - 2*a*tan(1/2*d*x + 1/2*c)^2 + a + b) + arctan(-1/2*(sqrt(a - b)*tan(1/2*d*x + 1/2*c)^2 - sqrt(a*tan(1/2*d *x + 1/2*c)^4 - b*tan(1/2*d*x + 1/2*c)^4 - 2*a*tan(1/2*d*x + 1/2*c)^2 + a + b) + sqrt(a - b))/sqrt(-a))/(sqrt(-a)*a*sgn(cos(d*x + c))))/d
Timed out. \[ \int \frac {\tan ^3(c+d x)}{(a+b \sec (c+d x))^{3/2}} \, dx=\int \frac {{\mathrm {tan}\left (c+d\,x\right )}^3}{{\left (a+\frac {b}{\cos \left (c+d\,x\right )}\right )}^{3/2}} \,d x \]