Integrand size = 21, antiderivative size = 259 \[ \int \frac {\sec ^5(c+d x)}{(a+b \sec (c+d x))^4} \, dx=\frac {\text {arctanh}(\sin (c+d x))}{b^4 d}-\frac {a \left (2 a^6-7 a^4 b^2+8 a^2 b^4-8 b^6\right ) \text {arctanh}\left (\frac {\sqrt {a-b} \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a+b}}\right )}{(a-b)^{7/2} b^4 (a+b)^{7/2} d}-\frac {a^2 \sec ^2(c+d x) \tan (c+d x)}{3 b \left (a^2-b^2\right ) d (a+b \sec (c+d x))^3}+\frac {a^3 \left (3 a^2-8 b^2\right ) \tan (c+d x)}{6 b^3 \left (a^2-b^2\right )^2 d (a+b \sec (c+d x))^2}-\frac {a^2 \left (9 a^4-28 a^2 b^2+34 b^4\right ) \tan (c+d x)}{6 b^3 \left (a^2-b^2\right )^3 d (a+b \sec (c+d x))} \]
arctanh(sin(d*x+c))/b^4/d-a*(2*a^6-7*a^4*b^2+8*a^2*b^4-8*b^6)*arctanh((a-b )^(1/2)*tan(1/2*d*x+1/2*c)/(a+b)^(1/2))/(a-b)^(7/2)/b^4/(a+b)^(7/2)/d-1/3* a^2*sec(d*x+c)^2*tan(d*x+c)/b/(a^2-b^2)/d/(a+b*sec(d*x+c))^3+1/6*a^3*(3*a^ 2-8*b^2)*tan(d*x+c)/b^3/(a^2-b^2)^2/d/(a+b*sec(d*x+c))^2-1/6*a^2*(9*a^4-28 *a^2*b^2+34*b^4)*tan(d*x+c)/b^3/(a^2-b^2)^3/d/(a+b*sec(d*x+c))
Time = 3.99 (sec) , antiderivative size = 250, normalized size of antiderivative = 0.97 \[ \int \frac {\sec ^5(c+d x)}{(a+b \sec (c+d x))^4} \, dx=\frac {\frac {6 a \left (2 a^6-7 a^4 b^2+8 a^2 b^4-8 b^6\right ) \text {arctanh}\left (\frac {(-a+b) \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a^2-b^2}}\right )}{\left (a^2-b^2\right )^{7/2}}-6 \log \left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )+6 \log \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )-\frac {a^2 b \left (11 a^4 b^2-32 a^2 b^4+36 b^6+15 a b \left (a^4-3 a^2 b^2+4 b^4\right ) \cos (c+d x)+a^2 \left (6 a^4-17 a^2 b^2+26 b^4\right ) \cos ^2(c+d x)\right ) \sin (c+d x)}{(a-b)^3 (a+b)^3 (b+a \cos (c+d x))^3}}{6 b^4 d} \]
((6*a*(2*a^6 - 7*a^4*b^2 + 8*a^2*b^4 - 8*b^6)*ArcTanh[((-a + b)*Tan[(c + d *x)/2])/Sqrt[a^2 - b^2]])/(a^2 - b^2)^(7/2) - 6*Log[Cos[(c + d*x)/2] - Sin [(c + d*x)/2]] + 6*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]] - (a^2*b*(11*a ^4*b^2 - 32*a^2*b^4 + 36*b^6 + 15*a*b*(a^4 - 3*a^2*b^2 + 4*b^4)*Cos[c + d* x] + a^2*(6*a^4 - 17*a^2*b^2 + 26*b^4)*Cos[c + d*x]^2)*Sin[c + d*x])/((a - b)^3*(a + b)^3*(b + a*Cos[c + d*x])^3))/(6*b^4*d)
Time = 1.83 (sec) , antiderivative size = 314, normalized size of antiderivative = 1.21, number of steps used = 16, number of rules used = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.714, Rules used = {3042, 4332, 3042, 4578, 3042, 4568, 27, 3042, 4486, 3042, 4257, 4318, 3042, 3138, 221}
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.
| \(\displaystyle \int \frac {\sec ^5(c+d x)}{(a+b \sec (c+d x))^4} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\csc \left (c+d x+\frac {\pi }{2}\right )^5}{\left (a+b \csc \left (c+d x+\frac {\pi }{2}\right )\right )^4}dx\) |
| \(\Big \downarrow \) 4332 |
| \(\displaystyle -\frac {\int \frac {\sec ^2(c+d x) \left (2 a^2-3 b \sec (c+d x) a-3 \left (a^2-b^2\right ) \sec ^2(c+d x)\right )}{(a+b \sec (c+d x))^3}dx}{3 b \left (a^2-b^2\right )}-\frac {a^2 \tan (c+d x) \sec ^2(c+d x)}{3 b d \left (a^2-b^2\right ) (a+b \sec (c+d x))^3}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {\int \frac {\csc \left (c+d x+\frac {\pi }{2}\right )^2 \left (2 a^2-3 b \csc \left (c+d x+\frac {\pi }{2}\right ) a-3 \left (a^2-b^2\right ) \csc \left (c+d x+\frac {\pi }{2}\right )^2\right )}{\left (a+b \csc \left (c+d x+\frac {\pi }{2}\right )\right )^3}dx}{3 b \left (a^2-b^2\right )}-\frac {a^2 \tan (c+d x) \sec ^2(c+d x)}{3 b d \left (a^2-b^2\right ) (a+b \sec (c+d x))^3}\) |
| \(\Big \downarrow \) 4578 |
| \(\displaystyle -\frac {\frac {\int \frac {\sec (c+d x) \left (2 b \left (3 a^2-8 b^2\right ) a^2+\left (3 a^4-10 b^2 a^2+12 b^4\right ) \sec (c+d x) a-6 b \left (a^2-b^2\right )^2 \sec ^2(c+d x)\right )}{(a+b \sec (c+d x))^2}dx}{2 b^2 \left (a^2-b^2\right )}-\frac {a^3 \left (3 a^2-8 b^2\right ) \tan (c+d x)}{2 b^2 d \left (a^2-b^2\right ) (a+b \sec (c+d x))^2}}{3 b \left (a^2-b^2\right )}-\frac {a^2 \tan (c+d x) \sec ^2(c+d x)}{3 b d \left (a^2-b^2\right ) (a+b \sec (c+d x))^3}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {\frac {\int \frac {\csc \left (c+d x+\frac {\pi }{2}\right ) \left (2 b \left (3 a^2-8 b^2\right ) a^2+\left (3 a^4-10 b^2 a^2+12 b^4\right ) \csc \left (c+d x+\frac {\pi }{2}\right ) a-6 b \left (a^2-b^2\right )^2 \csc \left (c+d x+\frac {\pi }{2}\right )^2\right )}{\left (a+b \csc \left (c+d x+\frac {\pi }{2}\right )\right )^2}dx}{2 b^2 \left (a^2-b^2\right )}-\frac {a^3 \left (3 a^2-8 b^2\right ) \tan (c+d x)}{2 b^2 d \left (a^2-b^2\right ) (a+b \sec (c+d x))^2}}{3 b \left (a^2-b^2\right )}-\frac {a^2 \tan (c+d x) \sec ^2(c+d x)}{3 b d \left (a^2-b^2\right ) (a+b \sec (c+d x))^3}\) |
| \(\Big \downarrow \) 4568 |
| \(\displaystyle -\frac {\frac {\frac {a^2 \left (9 a^4-28 a^2 b^2+34 b^4\right ) \tan (c+d x)}{d \left (a^2-b^2\right ) (a+b \sec (c+d x))}-\frac {\int \frac {3 \sec (c+d x) \left (2 b \sec (c+d x) \left (a^2-b^2\right )^3+a b^2 \left (a^4-2 b^2 a^2+6 b^4\right )\right )}{a+b \sec (c+d x)}dx}{b \left (a^2-b^2\right )}}{2 b^2 \left (a^2-b^2\right )}-\frac {a^3 \left (3 a^2-8 b^2\right ) \tan (c+d x)}{2 b^2 d \left (a^2-b^2\right ) (a+b \sec (c+d x))^2}}{3 b \left (a^2-b^2\right )}-\frac {a^2 \tan (c+d x) \sec ^2(c+d x)}{3 b d \left (a^2-b^2\right ) (a+b \sec (c+d x))^3}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {\frac {\frac {a^2 \left (9 a^4-28 a^2 b^2+34 b^4\right ) \tan (c+d x)}{d \left (a^2-b^2\right ) (a+b \sec (c+d x))}-\frac {3 \int \frac {\sec (c+d x) \left (2 b \sec (c+d x) \left (a^2-b^2\right )^3+a b^2 \left (a^4-2 b^2 a^2+6 b^4\right )\right )}{a+b \sec (c+d x)}dx}{b \left (a^2-b^2\right )}}{2 b^2 \left (a^2-b^2\right )}-\frac {a^3 \left (3 a^2-8 b^2\right ) \tan (c+d x)}{2 b^2 d \left (a^2-b^2\right ) (a+b \sec (c+d x))^2}}{3 b \left (a^2-b^2\right )}-\frac {a^2 \tan (c+d x) \sec ^2(c+d x)}{3 b d \left (a^2-b^2\right ) (a+b \sec (c+d x))^3}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {\frac {\frac {a^2 \left (9 a^4-28 a^2 b^2+34 b^4\right ) \tan (c+d x)}{d \left (a^2-b^2\right ) (a+b \sec (c+d x))}-\frac {3 \int \frac {\csc \left (c+d x+\frac {\pi }{2}\right ) \left (2 b \csc \left (c+d x+\frac {\pi }{2}\right ) \left (a^2-b^2\right )^3+a b^2 \left (a^4-2 b^2 a^2+6 b^4\right )\right )}{a+b \csc \left (c+d x+\frac {\pi }{2}\right )}dx}{b \left (a^2-b^2\right )}}{2 b^2 \left (a^2-b^2\right )}-\frac {a^3 \left (3 a^2-8 b^2\right ) \tan (c+d x)}{2 b^2 d \left (a^2-b^2\right ) (a+b \sec (c+d x))^2}}{3 b \left (a^2-b^2\right )}-\frac {a^2 \tan (c+d x) \sec ^2(c+d x)}{3 b d \left (a^2-b^2\right ) (a+b \sec (c+d x))^3}\) |
| \(\Big \downarrow \) 4486 |
| \(\displaystyle -\frac {\frac {\frac {a^2 \left (9 a^4-28 a^2 b^2+34 b^4\right ) \tan (c+d x)}{d \left (a^2-b^2\right ) (a+b \sec (c+d x))}-\frac {3 \left (2 \left (a^2-b^2\right )^3 \int \sec (c+d x)dx-a \left (2 a^6-7 a^4 b^2+8 a^2 b^4-8 b^6\right ) \int \frac {\sec (c+d x)}{a+b \sec (c+d x)}dx\right )}{b \left (a^2-b^2\right )}}{2 b^2 \left (a^2-b^2\right )}-\frac {a^3 \left (3 a^2-8 b^2\right ) \tan (c+d x)}{2 b^2 d \left (a^2-b^2\right ) (a+b \sec (c+d x))^2}}{3 b \left (a^2-b^2\right )}-\frac {a^2 \tan (c+d x) \sec ^2(c+d x)}{3 b d \left (a^2-b^2\right ) (a+b \sec (c+d x))^3}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {\frac {\frac {a^2 \left (9 a^4-28 a^2 b^2+34 b^4\right ) \tan (c+d x)}{d \left (a^2-b^2\right ) (a+b \sec (c+d x))}-\frac {3 \left (2 \left (a^2-b^2\right )^3 \int \csc \left (c+d x+\frac {\pi }{2}\right )dx-a \left (2 a^6-7 a^4 b^2+8 a^2 b^4-8 b^6\right ) \int \frac {\csc \left (c+d x+\frac {\pi }{2}\right )}{a+b \csc \left (c+d x+\frac {\pi }{2}\right )}dx\right )}{b \left (a^2-b^2\right )}}{2 b^2 \left (a^2-b^2\right )}-\frac {a^3 \left (3 a^2-8 b^2\right ) \tan (c+d x)}{2 b^2 d \left (a^2-b^2\right ) (a+b \sec (c+d x))^2}}{3 b \left (a^2-b^2\right )}-\frac {a^2 \tan (c+d x) \sec ^2(c+d x)}{3 b d \left (a^2-b^2\right ) (a+b \sec (c+d x))^3}\) |
| \(\Big \downarrow \) 4257 |
| \(\displaystyle -\frac {\frac {\frac {a^2 \left (9 a^4-28 a^2 b^2+34 b^4\right ) \tan (c+d x)}{d \left (a^2-b^2\right ) (a+b \sec (c+d x))}-\frac {3 \left (\frac {2 \left (a^2-b^2\right )^3 \text {arctanh}(\sin (c+d x))}{d}-a \left (2 a^6-7 a^4 b^2+8 a^2 b^4-8 b^6\right ) \int \frac {\csc \left (c+d x+\frac {\pi }{2}\right )}{a+b \csc \left (c+d x+\frac {\pi }{2}\right )}dx\right )}{b \left (a^2-b^2\right )}}{2 b^2 \left (a^2-b^2\right )}-\frac {a^3 \left (3 a^2-8 b^2\right ) \tan (c+d x)}{2 b^2 d \left (a^2-b^2\right ) (a+b \sec (c+d x))^2}}{3 b \left (a^2-b^2\right )}-\frac {a^2 \tan (c+d x) \sec ^2(c+d x)}{3 b d \left (a^2-b^2\right ) (a+b \sec (c+d x))^3}\) |
| \(\Big \downarrow \) 4318 |
| \(\displaystyle -\frac {\frac {\frac {a^2 \left (9 a^4-28 a^2 b^2+34 b^4\right ) \tan (c+d x)}{d \left (a^2-b^2\right ) (a+b \sec (c+d x))}-\frac {3 \left (\frac {2 \left (a^2-b^2\right )^3 \text {arctanh}(\sin (c+d x))}{d}-\frac {a \left (2 a^6-7 a^4 b^2+8 a^2 b^4-8 b^6\right ) \int \frac {1}{\frac {a \cos (c+d x)}{b}+1}dx}{b}\right )}{b \left (a^2-b^2\right )}}{2 b^2 \left (a^2-b^2\right )}-\frac {a^3 \left (3 a^2-8 b^2\right ) \tan (c+d x)}{2 b^2 d \left (a^2-b^2\right ) (a+b \sec (c+d x))^2}}{3 b \left (a^2-b^2\right )}-\frac {a^2 \tan (c+d x) \sec ^2(c+d x)}{3 b d \left (a^2-b^2\right ) (a+b \sec (c+d x))^3}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {\frac {\frac {a^2 \left (9 a^4-28 a^2 b^2+34 b^4\right ) \tan (c+d x)}{d \left (a^2-b^2\right ) (a+b \sec (c+d x))}-\frac {3 \left (\frac {2 \left (a^2-b^2\right )^3 \text {arctanh}(\sin (c+d x))}{d}-\frac {a \left (2 a^6-7 a^4 b^2+8 a^2 b^4-8 b^6\right ) \int \frac {1}{\frac {a \sin \left (c+d x+\frac {\pi }{2}\right )}{b}+1}dx}{b}\right )}{b \left (a^2-b^2\right )}}{2 b^2 \left (a^2-b^2\right )}-\frac {a^3 \left (3 a^2-8 b^2\right ) \tan (c+d x)}{2 b^2 d \left (a^2-b^2\right ) (a+b \sec (c+d x))^2}}{3 b \left (a^2-b^2\right )}-\frac {a^2 \tan (c+d x) \sec ^2(c+d x)}{3 b d \left (a^2-b^2\right ) (a+b \sec (c+d x))^3}\) |
| \(\Big \downarrow \) 3138 |
| \(\displaystyle -\frac {\frac {\frac {a^2 \left (9 a^4-28 a^2 b^2+34 b^4\right ) \tan (c+d x)}{d \left (a^2-b^2\right ) (a+b \sec (c+d x))}-\frac {3 \left (\frac {2 \left (a^2-b^2\right )^3 \text {arctanh}(\sin (c+d x))}{d}-\frac {2 a \left (2 a^6-7 a^4 b^2+8 a^2 b^4-8 b^6\right ) \int \frac {1}{\left (1-\frac {a}{b}\right ) \tan ^2\left (\frac {1}{2} (c+d x)\right )+\frac {a+b}{b}}d\tan \left (\frac {1}{2} (c+d x)\right )}{b d}\right )}{b \left (a^2-b^2\right )}}{2 b^2 \left (a^2-b^2\right )}-\frac {a^3 \left (3 a^2-8 b^2\right ) \tan (c+d x)}{2 b^2 d \left (a^2-b^2\right ) (a+b \sec (c+d x))^2}}{3 b \left (a^2-b^2\right )}-\frac {a^2 \tan (c+d x) \sec ^2(c+d x)}{3 b d \left (a^2-b^2\right ) (a+b \sec (c+d x))^3}\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle -\frac {a^2 \tan (c+d x) \sec ^2(c+d x)}{3 b d \left (a^2-b^2\right ) (a+b \sec (c+d x))^3}-\frac {\frac {\frac {a^2 \left (9 a^4-28 a^2 b^2+34 b^4\right ) \tan (c+d x)}{d \left (a^2-b^2\right ) (a+b \sec (c+d x))}-\frac {3 \left (\frac {2 \left (a^2-b^2\right )^3 \text {arctanh}(\sin (c+d x))}{d}-\frac {2 a \left (2 a^6-7 a^4 b^2+8 a^2 b^4-8 b^6\right ) \text {arctanh}\left (\frac {\sqrt {a-b} \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a+b}}\right )}{d \sqrt {a-b} \sqrt {a+b}}\right )}{b \left (a^2-b^2\right )}}{2 b^2 \left (a^2-b^2\right )}-\frac {a^3 \left (3 a^2-8 b^2\right ) \tan (c+d x)}{2 b^2 d \left (a^2-b^2\right ) (a+b \sec (c+d x))^2}}{3 b \left (a^2-b^2\right )}\) |
-1/3*(a^2*Sec[c + d*x]^2*Tan[c + d*x])/(b*(a^2 - b^2)*d*(a + b*Sec[c + d*x ])^3) - (-1/2*(a^3*(3*a^2 - 8*b^2)*Tan[c + d*x])/(b^2*(a^2 - b^2)*d*(a + b *Sec[c + d*x])^2) + ((-3*((2*(a^2 - b^2)^3*ArcTanh[Sin[c + d*x]])/d - (2*a *(2*a^6 - 7*a^4*b^2 + 8*a^2*b^4 - 8*b^6)*ArcTanh[(Sqrt[a - b]*Tan[(c + d*x )/2])/Sqrt[a + b]])/(Sqrt[a - b]*Sqrt[a + b]*d)))/(b*(a^2 - b^2)) + (a^2*( 9*a^4 - 28*a^2*b^2 + 34*b^4)*Tan[c + d*x])/((a^2 - b^2)*d*(a + b*Sec[c + d *x])))/(2*b^2*(a^2 - b^2)))/(3*b*(a^2 - b^2))
3.6.15.3.1 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/b, 2]/a)*ArcTanh[x /Rt[-a/b, 2]], x] /; FreeQ[{a, b}, x] && NegQ[a/b]
Int[((a_) + (b_.)*sin[Pi/2 + (c_.) + (d_.)*(x_)])^(-1), x_Symbol] :> With[{ e = FreeFactors[Tan[(c + d*x)/2], x]}, Simp[2*(e/d) Subst[Int[1/(a + b + (a - b)*e^2*x^2), x], x, Tan[(c + d*x)/2]/e], x]] /; FreeQ[{a, b, c, d}, x] && NeQ[a^2 - b^2, 0]
Int[csc[(c_.) + (d_.)*(x_)], x_Symbol] :> Simp[-ArcTanh[Cos[c + d*x]]/d, x] /; FreeQ[{c, d}, x]
Int[csc[(e_.) + (f_.)*(x_)]/(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_)), x_Symbo l] :> Simp[1/b Int[1/(1 + (a/b)*Sin[e + f*x]), x], x] /; FreeQ[{a, b, e, f}, x] && NeQ[a^2 - b^2, 0]
Int[(csc[(e_.) + (f_.)*(x_)]*(d_.))^(n_)*(csc[(e_.) + (f_.)*(x_)]*(b_.) + ( a_))^(m_), x_Symbol] :> Simp[(-a^2)*d^3*Cot[e + f*x]*(a + b*Csc[e + f*x])^( m + 1)*((d*Csc[e + f*x])^(n - 3)/(b*f*(m + 1)*(a^2 - b^2))), x] + Simp[d^3/ (b*(m + 1)*(a^2 - b^2)) Int[(a + b*Csc[e + f*x])^(m + 1)*(d*Csc[e + f*x]) ^(n - 3)*Simp[a^2*(n - 3) + a*b*(m + 1)*Csc[e + f*x] - (a^2*(n - 2) + b^2*( m + 1))*Csc[e + f*x]^2, x], x], x] /; FreeQ[{a, b, d, e, f}, x] && NeQ[a^2 - b^2, 0] && LtQ[m, -1] && (IGtQ[n, 3] || (IntegersQ[n + 1/2, 2*m] && GtQ[n , 2]))
Int[(csc[(e_.) + (f_.)*(x_)]*(csc[(e_.) + (f_.)*(x_)]*(B_.) + (A_)))/(csc[( e_.) + (f_.)*(x_)]*(b_.) + (a_)), x_Symbol] :> Simp[B/b Int[Csc[e + f*x], x], x] + Simp[(A*b - a*B)/b Int[Csc[e + f*x]/(a + b*Csc[e + f*x]), x], x ] /; FreeQ[{a, b, e, f, A, B}, x] && NeQ[A*b - a*B, 0]
Int[csc[(e_.) + (f_.)*(x_)]*((A_.) + csc[(e_.) + (f_.)*(x_)]*(B_.) + csc[(e _.) + (f_.)*(x_)]^2*(C_.))*(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_))^(m_), x_S ymbol] :> Simp[(-(A*b^2 - a*b*B + a^2*C))*Cot[e + f*x]*((a + b*Csc[e + f*x] )^(m + 1)/(b*f*(m + 1)*(a^2 - b^2))), x] + Simp[1/(b*(m + 1)*(a^2 - b^2)) Int[Csc[e + f*x]*(a + b*Csc[e + f*x])^(m + 1)*Simp[b*(a*A - b*B + a*C)*(m + 1) - (A*b^2 - a*b*B + a^2*C + b*(A*b - a*B + b*C)*(m + 1))*Csc[e + f*x], x], x], x] /; FreeQ[{a, b, e, f, A, B, C}, x] && LtQ[m, -1] && NeQ[a^2 - b^ 2, 0]
Int[csc[(e_.) + (f_.)*(x_)]^2*((A_.) + csc[(e_.) + (f_.)*(x_)]*(B_.) + csc[ (e_.) + (f_.)*(x_)]^2*(C_.))*(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_))^(m_), x _Symbol] :> Simp[a*(A*b^2 - a*b*B + a^2*C)*Cot[e + f*x]*((a + b*Csc[e + f*x ])^(m + 1)/(b^2*f*(m + 1)*(a^2 - b^2))), x] - Simp[1/(b^2*(m + 1)*(a^2 - b^ 2)) Int[Csc[e + f*x]*(a + b*Csc[e + f*x])^(m + 1)*Simp[b*(m + 1)*((-a)*(b *B - a*C) + A*b^2) + (b*B*(a^2 + b^2*(m + 1)) - a*(A*b^2*(m + 2) + C*(a^2 + b^2*(m + 1))))*Csc[e + f*x] - b*C*(m + 1)*(a^2 - b^2)*Csc[e + f*x]^2, x], x], x] /; FreeQ[{a, b, e, f, A, B, C}, x] && NeQ[a^2 - b^2, 0] && LtQ[m, -1 ]
Time = 1.50 (sec) , antiderivative size = 383, normalized size of antiderivative = 1.48
| method | result | size |
| derivativedivides | \(\frac {\frac {2 a \left (\frac {\frac {\left (2 a^{4}-a^{3} b -6 a^{2} b^{2}+4 a \,b^{3}+12 b^{4}\right ) a b \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}{2 \left (a -b \right ) \left (a^{3}+3 a^{2} b +3 a \,b^{2}+b^{3}\right )}-\frac {2 \left (3 a^{4}-11 a^{2} b^{2}+18 b^{4}\right ) a b \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{3 \left (a^{2}+2 a b +b^{2}\right ) \left (a^{2}-2 a b +b^{2}\right )}+\frac {\left (2 a^{4}+a^{3} b -6 a^{2} b^{2}-4 a \,b^{3}+12 b^{4}\right ) a b \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{2 \left (a +b \right ) \left (a^{3}-3 a^{2} b +3 a \,b^{2}-b^{3}\right )}}{\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} a -\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} b -a -b \right )^{3}}-\frac {\left (2 a^{6}-7 a^{4} b^{2}+8 a^{2} b^{4}-8 b^{6}\right ) \operatorname {arctanh}\left (\frac {\left (a -b \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{\sqrt {\left (a -b \right ) \left (a +b \right )}}\right )}{2 \left (a^{6}-3 a^{4} b^{2}+3 a^{2} b^{4}-b^{6}\right ) \sqrt {\left (a -b \right ) \left (a +b \right )}}\right )}{b^{4}}+\frac {\ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{b^{4}}-\frac {\ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{b^{4}}}{d}\) | \(383\) |
| default | \(\frac {\frac {2 a \left (\frac {\frac {\left (2 a^{4}-a^{3} b -6 a^{2} b^{2}+4 a \,b^{3}+12 b^{4}\right ) a b \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}{2 \left (a -b \right ) \left (a^{3}+3 a^{2} b +3 a \,b^{2}+b^{3}\right )}-\frac {2 \left (3 a^{4}-11 a^{2} b^{2}+18 b^{4}\right ) a b \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{3 \left (a^{2}+2 a b +b^{2}\right ) \left (a^{2}-2 a b +b^{2}\right )}+\frac {\left (2 a^{4}+a^{3} b -6 a^{2} b^{2}-4 a \,b^{3}+12 b^{4}\right ) a b \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{2 \left (a +b \right ) \left (a^{3}-3 a^{2} b +3 a \,b^{2}-b^{3}\right )}}{\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} a -\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} b -a -b \right )^{3}}-\frac {\left (2 a^{6}-7 a^{4} b^{2}+8 a^{2} b^{4}-8 b^{6}\right ) \operatorname {arctanh}\left (\frac {\left (a -b \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{\sqrt {\left (a -b \right ) \left (a +b \right )}}\right )}{2 \left (a^{6}-3 a^{4} b^{2}+3 a^{2} b^{4}-b^{6}\right ) \sqrt {\left (a -b \right ) \left (a +b \right )}}\right )}{b^{4}}+\frac {\ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{b^{4}}-\frac {\ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{b^{4}}}{d}\) | \(383\) |
| risch | \(\text {Expression too large to display}\) | \(1085\) |
1/d*(2*a/b^4*((1/2*(2*a^4-a^3*b-6*a^2*b^2+4*a*b^3+12*b^4)*a*b/(a-b)/(a^3+3 *a^2*b+3*a*b^2+b^3)*tan(1/2*d*x+1/2*c)^5-2/3*(3*a^4-11*a^2*b^2+18*b^4)*a*b /(a^2+2*a*b+b^2)/(a^2-2*a*b+b^2)*tan(1/2*d*x+1/2*c)^3+1/2*(2*a^4+a^3*b-6*a ^2*b^2-4*a*b^3+12*b^4)*a*b/(a+b)/(a^3-3*a^2*b+3*a*b^2-b^3)*tan(1/2*d*x+1/2 *c))/(tan(1/2*d*x+1/2*c)^2*a-tan(1/2*d*x+1/2*c)^2*b-a-b)^3-1/2*(2*a^6-7*a^ 4*b^2+8*a^2*b^4-8*b^6)/(a^6-3*a^4*b^2+3*a^2*b^4-b^6)/((a-b)*(a+b))^(1/2)*a rctanh((a-b)*tan(1/2*d*x+1/2*c)/((a-b)*(a+b))^(1/2)))+1/b^4*ln(tan(1/2*d*x +1/2*c)+1)-1/b^4*ln(tan(1/2*d*x+1/2*c)-1))
Leaf count of result is larger than twice the leaf count of optimal. 882 vs. \(2 (244) = 488\).
Time = 1.84 (sec) , antiderivative size = 1822, normalized size of antiderivative = 7.03 \[ \int \frac {\sec ^5(c+d x)}{(a+b \sec (c+d x))^4} \, dx=\text {Too large to display} \]
[1/12*(3*(2*a^7*b^3 - 7*a^5*b^5 + 8*a^3*b^7 - 8*a*b^9 + (2*a^10 - 7*a^8*b^ 2 + 8*a^6*b^4 - 8*a^4*b^6)*cos(d*x + c)^3 + 3*(2*a^9*b - 7*a^7*b^3 + 8*a^5 *b^5 - 8*a^3*b^7)*cos(d*x + c)^2 + 3*(2*a^8*b^2 - 7*a^6*b^4 + 8*a^4*b^6 - 8*a^2*b^8)*cos(d*x + c))*sqrt(a^2 - b^2)*log((2*a*b*cos(d*x + c) - (a^2 - 2*b^2)*cos(d*x + c)^2 - 2*sqrt(a^2 - b^2)*(b*cos(d*x + c) + a)*sin(d*x + c ) + 2*a^2 - b^2)/(a^2*cos(d*x + c)^2 + 2*a*b*cos(d*x + c) + b^2)) + 6*(a^8 *b^3 - 4*a^6*b^5 + 6*a^4*b^7 - 4*a^2*b^9 + b^11 + (a^11 - 4*a^9*b^2 + 6*a^ 7*b^4 - 4*a^5*b^6 + a^3*b^8)*cos(d*x + c)^3 + 3*(a^10*b - 4*a^8*b^3 + 6*a^ 6*b^5 - 4*a^4*b^7 + a^2*b^9)*cos(d*x + c)^2 + 3*(a^9*b^2 - 4*a^7*b^4 + 6*a ^5*b^6 - 4*a^3*b^8 + a*b^10)*cos(d*x + c))*log(sin(d*x + c) + 1) - 6*(a^8* b^3 - 4*a^6*b^5 + 6*a^4*b^7 - 4*a^2*b^9 + b^11 + (a^11 - 4*a^9*b^2 + 6*a^7 *b^4 - 4*a^5*b^6 + a^3*b^8)*cos(d*x + c)^3 + 3*(a^10*b - 4*a^8*b^3 + 6*a^6 *b^5 - 4*a^4*b^7 + a^2*b^9)*cos(d*x + c)^2 + 3*(a^9*b^2 - 4*a^7*b^4 + 6*a^ 5*b^6 - 4*a^3*b^8 + a*b^10)*cos(d*x + c))*log(-sin(d*x + c) + 1) - 2*(11*a ^8*b^3 - 43*a^6*b^5 + 68*a^4*b^7 - 36*a^2*b^9 + (6*a^10*b - 23*a^8*b^3 + 4 3*a^6*b^5 - 26*a^4*b^7)*cos(d*x + c)^2 + 15*(a^9*b^2 - 4*a^7*b^4 + 7*a^5*b ^6 - 4*a^3*b^8)*cos(d*x + c))*sin(d*x + c))/((a^11*b^4 - 4*a^9*b^6 + 6*a^7 *b^8 - 4*a^5*b^10 + a^3*b^12)*d*cos(d*x + c)^3 + 3*(a^10*b^5 - 4*a^8*b^7 + 6*a^6*b^9 - 4*a^4*b^11 + a^2*b^13)*d*cos(d*x + c)^2 + 3*(a^9*b^6 - 4*a^7* b^8 + 6*a^5*b^10 - 4*a^3*b^12 + a*b^14)*d*cos(d*x + c) + (a^8*b^7 - 4*a...
\[ \int \frac {\sec ^5(c+d x)}{(a+b \sec (c+d x))^4} \, dx=\int \frac {\sec ^{5}{\left (c + d x \right )}}{\left (a + b \sec {\left (c + d x \right )}\right )^{4}}\, dx \]
Exception generated. \[ \int \frac {\sec ^5(c+d x)}{(a+b \sec (c+d x))^4} \, dx=\text {Exception raised: ValueError} \]
Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'assume' command before evaluation *may* help (example of legal syntax is 'assume(4*a^2-4*b^2>0)', see `assume?` f or more de
Leaf count of result is larger than twice the leaf count of optimal. 559 vs. \(2 (244) = 488\).
Time = 0.39 (sec) , antiderivative size = 559, normalized size of antiderivative = 2.16 \[ \int \frac {\sec ^5(c+d x)}{(a+b \sec (c+d x))^4} \, dx=\frac {\frac {3 \, {\left (2 \, a^{7} - 7 \, a^{5} b^{2} + 8 \, a^{3} b^{4} - 8 \, a b^{6}\right )} {\left (\pi \left \lfloor \frac {d x + c}{2 \, \pi } + \frac {1}{2} \right \rfloor \mathrm {sgn}\left (2 \, a - 2 \, b\right ) + \arctan \left (\frac {a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{\sqrt {-a^{2} + b^{2}}}\right )\right )}}{{\left (a^{6} b^{4} - 3 \, a^{4} b^{6} + 3 \, a^{2} b^{8} - b^{10}\right )} \sqrt {-a^{2} + b^{2}}} + \frac {6 \, a^{8} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 15 \, a^{7} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 6 \, a^{6} b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 45 \, a^{5} b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 6 \, a^{4} b^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 60 \, a^{3} b^{5} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 36 \, a^{2} b^{6} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 12 \, a^{8} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 56 \, a^{6} b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 116 \, a^{4} b^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 72 \, a^{2} b^{6} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 6 \, a^{8} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 15 \, a^{7} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 6 \, a^{6} b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 45 \, a^{5} b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 6 \, a^{4} b^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 60 \, a^{3} b^{5} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 36 \, a^{2} b^{6} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{{\left (a^{6} b^{3} - 3 \, a^{4} b^{5} + 3 \, a^{2} b^{7} - b^{9}\right )} {\left (a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - a - b\right )}^{3}} + \frac {3 \, \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1 \right |}\right )}{b^{4}} - \frac {3 \, \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1 \right |}\right )}{b^{4}}}{3 \, d} \]
1/3*(3*(2*a^7 - 7*a^5*b^2 + 8*a^3*b^4 - 8*a*b^6)*(pi*floor(1/2*(d*x + c)/p i + 1/2)*sgn(2*a - 2*b) + arctan((a*tan(1/2*d*x + 1/2*c) - b*tan(1/2*d*x + 1/2*c))/sqrt(-a^2 + b^2)))/((a^6*b^4 - 3*a^4*b^6 + 3*a^2*b^8 - b^10)*sqrt (-a^2 + b^2)) + (6*a^8*tan(1/2*d*x + 1/2*c)^5 - 15*a^7*b*tan(1/2*d*x + 1/2 *c)^5 - 6*a^6*b^2*tan(1/2*d*x + 1/2*c)^5 + 45*a^5*b^3*tan(1/2*d*x + 1/2*c) ^5 - 6*a^4*b^4*tan(1/2*d*x + 1/2*c)^5 - 60*a^3*b^5*tan(1/2*d*x + 1/2*c)^5 + 36*a^2*b^6*tan(1/2*d*x + 1/2*c)^5 - 12*a^8*tan(1/2*d*x + 1/2*c)^3 + 56*a ^6*b^2*tan(1/2*d*x + 1/2*c)^3 - 116*a^4*b^4*tan(1/2*d*x + 1/2*c)^3 + 72*a^ 2*b^6*tan(1/2*d*x + 1/2*c)^3 + 6*a^8*tan(1/2*d*x + 1/2*c) + 15*a^7*b*tan(1 /2*d*x + 1/2*c) - 6*a^6*b^2*tan(1/2*d*x + 1/2*c) - 45*a^5*b^3*tan(1/2*d*x + 1/2*c) - 6*a^4*b^4*tan(1/2*d*x + 1/2*c) + 60*a^3*b^5*tan(1/2*d*x + 1/2*c ) + 36*a^2*b^6*tan(1/2*d*x + 1/2*c))/((a^6*b^3 - 3*a^4*b^5 + 3*a^2*b^7 - b ^9)*(a*tan(1/2*d*x + 1/2*c)^2 - b*tan(1/2*d*x + 1/2*c)^2 - a - b)^3) + 3*l og(abs(tan(1/2*d*x + 1/2*c) + 1))/b^4 - 3*log(abs(tan(1/2*d*x + 1/2*c) - 1 ))/b^4)/d
Time = 27.13 (sec) , antiderivative size = 7222, normalized size of antiderivative = 27.88 \[ \int \frac {\sec ^5(c+d x)}{(a+b \sec (c+d x))^4} \, dx=\text {Too large to display} \]
- ((tan(c/2 + (d*x)/2)^5*(2*a^6 - a^5*b + 12*a^2*b^4 + 4*a^3*b^3 - 6*a^4*b ^2))/((a*b^3 - b^4)*(a + b)^3) - (4*tan(c/2 + (d*x)/2)^3*(3*a^6 + 18*a^2*b ^4 - 11*a^4*b^2))/(3*(a + b)^2*(b^5 - 2*a*b^4 + a^2*b^3)) + (tan(c/2 + (d* x)/2)*(a^5*b + 2*a^6 + 12*a^2*b^4 - 4*a^3*b^3 - 6*a^4*b^2))/((a + b)*(3*a* b^5 - b^6 - 3*a^2*b^4 + a^3*b^3)))/(d*(tan(c/2 + (d*x)/2)^2*(3*a*b^2 - 3*a ^2*b - 3*a^3 + 3*b^3) - tan(c/2 + (d*x)/2)^4*(3*a*b^2 + 3*a^2*b - 3*a^3 - 3*b^3) + 3*a*b^2 + 3*a^2*b + a^3 + b^3 - tan(c/2 + (d*x)/2)^6*(3*a*b^2 - 3 *a^2*b + a^3 - b^3))) - (atan((((((8*(16*a*b^20 - 4*b^21 + 12*a^2*b^19 - 6 4*a^3*b^18 - 20*a^4*b^17 + 110*a^5*b^16 + 30*a^6*b^15 - 110*a^7*b^14 - 30* a^8*b^13 + 70*a^9*b^12 + 14*a^10*b^11 - 26*a^11*b^10 - 2*a^12*b^9 + 4*a^13 *b^8))/(a*b^19 + b^20 - 5*a^2*b^18 - 5*a^3*b^17 + 10*a^4*b^16 + 10*a^5*b^1 5 - 10*a^6*b^14 - 10*a^7*b^13 + 5*a^8*b^12 + 5*a^9*b^11 - a^10*b^10 - a^11 *b^9) - (8*tan(c/2 + (d*x)/2)*(8*a*b^21 - 8*a^2*b^20 - 48*a^3*b^19 + 48*a^ 4*b^18 + 120*a^5*b^17 - 120*a^6*b^16 - 160*a^7*b^15 + 160*a^8*b^14 + 120*a ^9*b^13 - 120*a^10*b^12 - 48*a^11*b^11 + 48*a^12*b^10 + 8*a^13*b^9 - 8*a^1 4*b^8))/(b^4*(a*b^16 + b^17 - 5*a^2*b^15 - 5*a^3*b^14 + 10*a^4*b^13 + 10*a ^5*b^12 - 10*a^6*b^11 - 10*a^7*b^10 + 5*a^8*b^9 + 5*a^9*b^8 - a^10*b^7 - a ^11*b^6)))/b^4 - (8*tan(c/2 + (d*x)/2)*(8*a^14 - 8*a^13*b - 8*a*b^13 + 4*b ^14 + 44*a^2*b^12 + 48*a^3*b^11 - 92*a^4*b^10 - 120*a^5*b^9 + 156*a^6*b^8 + 160*a^7*b^7 - 164*a^8*b^6 - 120*a^9*b^5 + 117*a^10*b^4 + 48*a^11*b^3 ...