Integrand size = 12, antiderivative size = 242 \[ \int \frac {1}{(a+b \sec (c+d x))^4} \, dx=\frac {x}{a^4}-\frac {b \left (8 a^6-8 a^4 b^2+7 a^2 b^4-2 b^6\right ) \text {arctanh}\left (\frac {\sqrt {a-b} \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a+b}}\right )}{a^4 (a-b)^{7/2} (a+b)^{7/2} d}+\frac {b^2 \tan (c+d x)}{3 a \left (a^2-b^2\right ) d (a+b \sec (c+d x))^3}+\frac {b^2 \left (8 a^2-3 b^2\right ) \tan (c+d x)}{6 a^2 \left (a^2-b^2\right )^2 d (a+b \sec (c+d x))^2}+\frac {b^2 \left (26 a^4-17 a^2 b^2+6 b^4\right ) \tan (c+d x)}{6 a^3 \left (a^2-b^2\right )^3 d (a+b \sec (c+d x))} \]
x/a^4-b*(8*a^6-8*a^4*b^2+7*a^2*b^4-2*b^6)*arctanh((a-b)^(1/2)*tan(1/2*d*x+ 1/2*c)/(a+b)^(1/2))/a^4/(a-b)^(7/2)/(a+b)^(7/2)/d+1/3*b^2*tan(d*x+c)/a/(a^ 2-b^2)/d/(a+b*sec(d*x+c))^3+1/6*b^2*(8*a^2-3*b^2)*tan(d*x+c)/a^2/(a^2-b^2) ^2/d/(a+b*sec(d*x+c))^2+1/6*b^2*(26*a^4-17*a^2*b^2+6*b^4)*tan(d*x+c)/a^3/( a^2-b^2)^3/d/(a+b*sec(d*x+c))
Time = 1.37 (sec) , antiderivative size = 268, normalized size of antiderivative = 1.11 \[ \int \frac {1}{(a+b \sec (c+d x))^4} \, dx=\frac {(b+a \cos (c+d x)) \sec ^4(c+d x) \left (6 (c+d x) (b+a \cos (c+d x))^3-\frac {6 b \left (-8 a^6+8 a^4 b^2-7 a^2 b^4+2 b^6\right ) \text {arctanh}\left (\frac {(-a+b) \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a^2-b^2}}\right ) (b+a \cos (c+d x))^3}{\left (a^2-b^2\right )^{7/2}}+\frac {2 a b^4 \sin (c+d x)}{(a-b) (a+b)}-\frac {a b^3 \left (12 a^2-7 b^2\right ) (b+a \cos (c+d x)) \sin (c+d x)}{(a-b)^2 (a+b)^2}+\frac {a b^2 \left (36 a^4-32 a^2 b^2+11 b^4\right ) (b+a \cos (c+d x))^2 \sin (c+d x)}{(a-b)^3 (a+b)^3}\right )}{6 a^4 d (a+b \sec (c+d x))^4} \]
((b + a*Cos[c + d*x])*Sec[c + d*x]^4*(6*(c + d*x)*(b + a*Cos[c + d*x])^3 - (6*b*(-8*a^6 + 8*a^4*b^2 - 7*a^2*b^4 + 2*b^6)*ArcTanh[((-a + b)*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]]*(b + a*Cos[c + d*x])^3)/(a^2 - b^2)^(7/2) + (2*a *b^4*Sin[c + d*x])/((a - b)*(a + b)) - (a*b^3*(12*a^2 - 7*b^2)*(b + a*Cos[ c + d*x])*Sin[c + d*x])/((a - b)^2*(a + b)^2) + (a*b^2*(36*a^4 - 32*a^2*b^ 2 + 11*b^4)*(b + a*Cos[c + d*x])^2*Sin[c + d*x])/((a - b)^3*(a + b)^3)))/( 6*a^4*d*(a + b*Sec[c + d*x])^4)
Time = 1.47 (sec) , antiderivative size = 306, normalized size of antiderivative = 1.26, number of steps used = 17, number of rules used = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.333, Rules used = {3042, 4272, 25, 3042, 4548, 25, 3042, 4548, 27, 3042, 4407, 3042, 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.
| \(\displaystyle \int \frac {1}{(a+b \sec (c+d x))^4} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {1}{\left (a+b \csc \left (c+d x+\frac {\pi }{2}\right )\right )^4}dx\) |
| \(\Big \downarrow \) 4272 |
| \(\displaystyle \frac {b^2 \tan (c+d x)}{3 a d \left (a^2-b^2\right ) (a+b \sec (c+d x))^3}-\frac {\int -\frac {2 b^2 \sec ^2(c+d x)-3 a b \sec (c+d x)+3 \left (a^2-b^2\right )}{(a+b \sec (c+d x))^3}dx}{3 a \left (a^2-b^2\right )}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\int \frac {2 b^2 \sec ^2(c+d x)-3 a b \sec (c+d x)+3 \left (a^2-b^2\right )}{(a+b \sec (c+d x))^3}dx}{3 a \left (a^2-b^2\right )}+\frac {b^2 \tan (c+d x)}{3 a d \left (a^2-b^2\right ) (a+b \sec (c+d x))^3}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\int \frac {2 b^2 \csc \left (c+d x+\frac {\pi }{2}\right )^2-3 a b \csc \left (c+d x+\frac {\pi }{2}\right )+3 \left (a^2-b^2\right )}{\left (a+b \csc \left (c+d x+\frac {\pi }{2}\right )\right )^3}dx}{3 a \left (a^2-b^2\right )}+\frac {b^2 \tan (c+d x)}{3 a d \left (a^2-b^2\right ) (a+b \sec (c+d x))^3}\) |
| \(\Big \downarrow \) 4548 |
| \(\displaystyle \frac {\frac {b^2 \left (8 a^2-3 b^2\right ) \tan (c+d x)}{2 a d \left (a^2-b^2\right ) (a+b \sec (c+d x))^2}-\frac {\int -\frac {6 \left (a^2-b^2\right )^2+b^2 \left (8 a^2-3 b^2\right ) \sec ^2(c+d x)-2 a b \left (6 a^2-b^2\right ) \sec (c+d x)}{(a+b \sec (c+d x))^2}dx}{2 a \left (a^2-b^2\right )}}{3 a \left (a^2-b^2\right )}+\frac {b^2 \tan (c+d x)}{3 a d \left (a^2-b^2\right ) (a+b \sec (c+d x))^3}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\frac {\int \frac {6 \left (a^2-b^2\right )^2+b^2 \left (8 a^2-3 b^2\right ) \sec ^2(c+d x)-2 a b \left (6 a^2-b^2\right ) \sec (c+d x)}{(a+b \sec (c+d x))^2}dx}{2 a \left (a^2-b^2\right )}+\frac {b^2 \left (8 a^2-3 b^2\right ) \tan (c+d x)}{2 a d \left (a^2-b^2\right ) (a+b \sec (c+d x))^2}}{3 a \left (a^2-b^2\right )}+\frac {b^2 \tan (c+d x)}{3 a d \left (a^2-b^2\right ) (a+b \sec (c+d x))^3}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {\int \frac {6 \left (a^2-b^2\right )^2+b^2 \left (8 a^2-3 b^2\right ) \csc \left (c+d x+\frac {\pi }{2}\right )^2-2 a b \left (6 a^2-b^2\right ) \csc \left (c+d x+\frac {\pi }{2}\right )}{\left (a+b \csc \left (c+d x+\frac {\pi }{2}\right )\right )^2}dx}{2 a \left (a^2-b^2\right )}+\frac {b^2 \left (8 a^2-3 b^2\right ) \tan (c+d x)}{2 a d \left (a^2-b^2\right ) (a+b \sec (c+d x))^2}}{3 a \left (a^2-b^2\right )}+\frac {b^2 \tan (c+d x)}{3 a d \left (a^2-b^2\right ) (a+b \sec (c+d x))^3}\) |
| \(\Big \downarrow \) 4548 |
| \(\displaystyle \frac {\frac {\frac {b^2 \left (26 a^4-17 a^2 b^2+6 b^4\right ) \tan (c+d x)}{a d \left (a^2-b^2\right ) (a+b \sec (c+d x))}-\frac {\int -\frac {3 \left (2 \left (a^2-b^2\right )^3-a b \left (6 a^4-2 b^2 a^2+b^4\right ) \sec (c+d x)\right )}{a+b \sec (c+d x)}dx}{a \left (a^2-b^2\right )}}{2 a \left (a^2-b^2\right )}+\frac {b^2 \left (8 a^2-3 b^2\right ) \tan (c+d x)}{2 a d \left (a^2-b^2\right ) (a+b \sec (c+d x))^2}}{3 a \left (a^2-b^2\right )}+\frac {b^2 \tan (c+d x)}{3 a d \left (a^2-b^2\right ) (a+b \sec (c+d x))^3}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {\frac {3 \int \frac {2 \left (a^2-b^2\right )^3-a b \left (6 a^4-2 b^2 a^2+b^4\right ) \sec (c+d x)}{a+b \sec (c+d x)}dx}{a \left (a^2-b^2\right )}+\frac {b^2 \left (26 a^4-17 a^2 b^2+6 b^4\right ) \tan (c+d x)}{a d \left (a^2-b^2\right ) (a+b \sec (c+d x))}}{2 a \left (a^2-b^2\right )}+\frac {b^2 \left (8 a^2-3 b^2\right ) \tan (c+d x)}{2 a d \left (a^2-b^2\right ) (a+b \sec (c+d x))^2}}{3 a \left (a^2-b^2\right )}+\frac {b^2 \tan (c+d x)}{3 a d \left (a^2-b^2\right ) (a+b \sec (c+d x))^3}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {\frac {3 \int \frac {2 \left (a^2-b^2\right )^3-a b \left (6 a^4-2 b^2 a^2+b^4\right ) \csc \left (c+d x+\frac {\pi }{2}\right )}{a+b \csc \left (c+d x+\frac {\pi }{2}\right )}dx}{a \left (a^2-b^2\right )}+\frac {b^2 \left (26 a^4-17 a^2 b^2+6 b^4\right ) \tan (c+d x)}{a d \left (a^2-b^2\right ) (a+b \sec (c+d x))}}{2 a \left (a^2-b^2\right )}+\frac {b^2 \left (8 a^2-3 b^2\right ) \tan (c+d x)}{2 a d \left (a^2-b^2\right ) (a+b \sec (c+d x))^2}}{3 a \left (a^2-b^2\right )}+\frac {b^2 \tan (c+d x)}{3 a d \left (a^2-b^2\right ) (a+b \sec (c+d x))^3}\) |
| \(\Big \downarrow \) 4407 |
| \(\displaystyle \frac {\frac {\frac {3 \left (\frac {2 x \left (a^2-b^2\right )^3}{a}-\frac {b \left (8 a^6-8 a^4 b^2+7 a^2 b^4-2 b^6\right ) \int \frac {\sec (c+d x)}{a+b \sec (c+d x)}dx}{a}\right )}{a \left (a^2-b^2\right )}+\frac {b^2 \left (26 a^4-17 a^2 b^2+6 b^4\right ) \tan (c+d x)}{a d \left (a^2-b^2\right ) (a+b \sec (c+d x))}}{2 a \left (a^2-b^2\right )}+\frac {b^2 \left (8 a^2-3 b^2\right ) \tan (c+d x)}{2 a d \left (a^2-b^2\right ) (a+b \sec (c+d x))^2}}{3 a \left (a^2-b^2\right )}+\frac {b^2 \tan (c+d x)}{3 a d \left (a^2-b^2\right ) (a+b \sec (c+d x))^3}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {\frac {3 \left (\frac {2 x \left (a^2-b^2\right )^3}{a}-\frac {b \left (8 a^6-8 a^4 b^2+7 a^2 b^4-2 b^6\right ) \int \frac {\csc \left (c+d x+\frac {\pi }{2}\right )}{a+b \csc \left (c+d x+\frac {\pi }{2}\right )}dx}{a}\right )}{a \left (a^2-b^2\right )}+\frac {b^2 \left (26 a^4-17 a^2 b^2+6 b^4\right ) \tan (c+d x)}{a d \left (a^2-b^2\right ) (a+b \sec (c+d x))}}{2 a \left (a^2-b^2\right )}+\frac {b^2 \left (8 a^2-3 b^2\right ) \tan (c+d x)}{2 a d \left (a^2-b^2\right ) (a+b \sec (c+d x))^2}}{3 a \left (a^2-b^2\right )}+\frac {b^2 \tan (c+d x)}{3 a d \left (a^2-b^2\right ) (a+b \sec (c+d x))^3}\) |
| \(\Big \downarrow \) 4318 |
| \(\displaystyle \frac {\frac {\frac {3 \left (\frac {2 x \left (a^2-b^2\right )^3}{a}-\frac {\left (8 a^6-8 a^4 b^2+7 a^2 b^4-2 b^6\right ) \int \frac {1}{\frac {a \cos (c+d x)}{b}+1}dx}{a}\right )}{a \left (a^2-b^2\right )}+\frac {b^2 \left (26 a^4-17 a^2 b^2+6 b^4\right ) \tan (c+d x)}{a d \left (a^2-b^2\right ) (a+b \sec (c+d x))}}{2 a \left (a^2-b^2\right )}+\frac {b^2 \left (8 a^2-3 b^2\right ) \tan (c+d x)}{2 a d \left (a^2-b^2\right ) (a+b \sec (c+d x))^2}}{3 a \left (a^2-b^2\right )}+\frac {b^2 \tan (c+d x)}{3 a d \left (a^2-b^2\right ) (a+b \sec (c+d x))^3}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {\frac {3 \left (\frac {2 x \left (a^2-b^2\right )^3}{a}-\frac {\left (8 a^6-8 a^4 b^2+7 a^2 b^4-2 b^6\right ) \int \frac {1}{\frac {a \sin \left (c+d x+\frac {\pi }{2}\right )}{b}+1}dx}{a}\right )}{a \left (a^2-b^2\right )}+\frac {b^2 \left (26 a^4-17 a^2 b^2+6 b^4\right ) \tan (c+d x)}{a d \left (a^2-b^2\right ) (a+b \sec (c+d x))}}{2 a \left (a^2-b^2\right )}+\frac {b^2 \left (8 a^2-3 b^2\right ) \tan (c+d x)}{2 a d \left (a^2-b^2\right ) (a+b \sec (c+d x))^2}}{3 a \left (a^2-b^2\right )}+\frac {b^2 \tan (c+d x)}{3 a d \left (a^2-b^2\right ) (a+b \sec (c+d x))^3}\) |
| \(\Big \downarrow \) 3138 |
| \(\displaystyle \frac {\frac {\frac {3 \left (\frac {2 x \left (a^2-b^2\right )^3}{a}-\frac {2 \left (8 a^6-8 a^4 b^2+7 a^2 b^4-2 b^6\right ) \int \frac {1}{\left (1-\frac {a}{b}\right ) \tan ^2\left (\frac {1}{2} (c+d x)\right )+\frac {a+b}{b}}d\tan \left (\frac {1}{2} (c+d x)\right )}{a d}\right )}{a \left (a^2-b^2\right )}+\frac {b^2 \left (26 a^4-17 a^2 b^2+6 b^4\right ) \tan (c+d x)}{a d \left (a^2-b^2\right ) (a+b \sec (c+d x))}}{2 a \left (a^2-b^2\right )}+\frac {b^2 \left (8 a^2-3 b^2\right ) \tan (c+d x)}{2 a d \left (a^2-b^2\right ) (a+b \sec (c+d x))^2}}{3 a \left (a^2-b^2\right )}+\frac {b^2 \tan (c+d x)}{3 a d \left (a^2-b^2\right ) (a+b \sec (c+d x))^3}\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle \frac {b^2 \tan (c+d x)}{3 a d \left (a^2-b^2\right ) (a+b \sec (c+d x))^3}+\frac {\frac {b^2 \left (8 a^2-3 b^2\right ) \tan (c+d x)}{2 a d \left (a^2-b^2\right ) (a+b \sec (c+d x))^2}+\frac {\frac {b^2 \left (26 a^4-17 a^2 b^2+6 b^4\right ) \tan (c+d x)}{a d \left (a^2-b^2\right ) (a+b \sec (c+d x))}+\frac {3 \left (\frac {2 x \left (a^2-b^2\right )^3}{a}-\frac {2 b \left (8 a^6-8 a^4 b^2+7 a^2 b^4-2 b^6\right ) \text {arctanh}\left (\frac {\sqrt {a-b} \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a+b}}\right )}{a d \sqrt {a-b} \sqrt {a+b}}\right )}{a \left (a^2-b^2\right )}}{2 a \left (a^2-b^2\right )}}{3 a \left (a^2-b^2\right )}\) |
(b^2*Tan[c + d*x])/(3*a*(a^2 - b^2)*d*(a + b*Sec[c + d*x])^3) + ((b^2*(8*a ^2 - 3*b^2)*Tan[c + d*x])/(2*a*(a^2 - b^2)*d*(a + b*Sec[c + d*x])^2) + ((3 *((2*(a^2 - b^2)^3*x)/a - (2*b*(8*a^6 - 8*a^4*b^2 + 7*a^2*b^4 - 2*b^6)*Arc Tanh[(Sqrt[a - b]*Tan[(c + d*x)/2])/Sqrt[a + b]])/(a*Sqrt[a - b]*Sqrt[a + b]*d)))/(a*(a^2 - b^2)) + (b^2*(26*a^4 - 17*a^2*b^2 + 6*b^4)*Tan[c + d*x]) /(a*(a^2 - b^2)*d*(a + b*Sec[c + d*x])))/(2*a*(a^2 - b^2)))/(3*a*(a^2 - b^ 2))
3.6.20.3.1 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_)]*(b_.) + (a_))^(n_), x_Symbol] :> Simp[b^2*Cot[ c + d*x]*((a + b*Csc[c + d*x])^(n + 1)/(a*d*(n + 1)*(a^2 - b^2))), x] + Sim p[1/(a*(n + 1)*(a^2 - b^2)) Int[(a + b*Csc[c + d*x])^(n + 1)*Simp[(a^2 - b^2)*(n + 1) - a*b*(n + 1)*Csc[c + d*x] + b^2*(n + 2)*Csc[c + d*x]^2, x], x ], x] /; FreeQ[{a, b, c, d}, x] && NeQ[a^2 - b^2, 0] && LtQ[n, -1] && Integ erQ[2*n]
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_.) + (c_))/(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_)), x_Symbol] :> Simp[c*(x/a), x] - Simp[(b*c - a*d)/a Int[Csc[e + f* x]/(a + b*Csc[e + f*x]), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0]
Int[((A_.) + csc[(e_.) + (f_.)*(x_)]*(B_.) + csc[(e_.) + (f_.)*(x_)]^2*(C_. ))*(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_))^(m_), x_Symbol] :> Simp[(A*b^2 - a*b*B + a^2*C)*Cot[e + f*x]*((a + b*Csc[e + f*x])^(m + 1)/(a*f*(m + 1)*(a^2 - b^2))), x] + Simp[1/(a*(m + 1)*(a^2 - b^2)) Int[(a + b*Csc[e + f*x])^( m + 1)*Simp[A*(a^2 - b^2)*(m + 1) - a*(A*b - a*B + b*C)*(m + 1)*Csc[e + f*x ] + (A*b^2 - a*b*B + a^2*C)*(m + 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 = 0.63 (sec) , antiderivative size = 365, normalized size of antiderivative = 1.51
| method | result | size |
| derivativedivides | \(\frac {\frac {2 \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a^{4}}+\frac {2 b \left (\frac {-\frac {\left (12 a^{4}+4 a^{3} b -6 a^{2} b^{2}-a \,b^{3}+2 b^{4}\right ) a b \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}{2 \left (a -b \right ) \left (a^{3}+3 a^{2} b +3 a \,b^{2}+b^{3}\right )}+\frac {2 \left (18 a^{4}-11 a^{2} b^{2}+3 b^{4}\right ) a b \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{3 \left (a^{2}+2 a b +b^{2}\right ) \left (a^{2}-2 a b +b^{2}\right )}-\frac {\left (12 a^{4}-4 a^{3} b -6 a^{2} b^{2}+a \,b^{3}+2 b^{4}\right ) a b \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{2 \left (a +b \right ) \left (a^{3}-3 a^{2} b +3 a \,b^{2}-b^{3}\right )}}{\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} a -\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} b -a -b \right )^{3}}-\frac {\left (8 a^{6}-8 a^{4} b^{2}+7 a^{2} b^{4}-2 b^{6}\right ) \operatorname {arctanh}\left (\frac {\left (a -b \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{\sqrt {\left (a -b \right ) \left (a +b \right )}}\right )}{2 \left (a^{6}-3 a^{4} b^{2}+3 a^{2} b^{4}-b^{6}\right ) \sqrt {\left (a -b \right ) \left (a +b \right )}}\right )}{a^{4}}}{d}\) | \(365\) |
| default | \(\frac {\frac {2 \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a^{4}}+\frac {2 b \left (\frac {-\frac {\left (12 a^{4}+4 a^{3} b -6 a^{2} b^{2}-a \,b^{3}+2 b^{4}\right ) a b \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}{2 \left (a -b \right ) \left (a^{3}+3 a^{2} b +3 a \,b^{2}+b^{3}\right )}+\frac {2 \left (18 a^{4}-11 a^{2} b^{2}+3 b^{4}\right ) a b \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{3 \left (a^{2}+2 a b +b^{2}\right ) \left (a^{2}-2 a b +b^{2}\right )}-\frac {\left (12 a^{4}-4 a^{3} b -6 a^{2} b^{2}+a \,b^{3}+2 b^{4}\right ) a b \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{2 \left (a +b \right ) \left (a^{3}-3 a^{2} b +3 a \,b^{2}-b^{3}\right )}}{\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} a -\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} b -a -b \right )^{3}}-\frac {\left (8 a^{6}-8 a^{4} b^{2}+7 a^{2} b^{4}-2 b^{6}\right ) \operatorname {arctanh}\left (\frac {\left (a -b \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{\sqrt {\left (a -b \right ) \left (a +b \right )}}\right )}{2 \left (a^{6}-3 a^{4} b^{2}+3 a^{2} b^{4}-b^{6}\right ) \sqrt {\left (a -b \right ) \left (a +b \right )}}\right )}{a^{4}}}{d}\) | \(365\) |
| risch | \(\frac {x}{a^{4}}-\frac {i b^{2} \left (48 a^{6} b \,{\mathrm e}^{5 i \left (d x +c \right )}-51 a^{4} b^{3} {\mathrm e}^{5 i \left (d x +c \right )}+18 a^{2} b^{5} {\mathrm e}^{5 i \left (d x +c \right )}+36 a^{7} {\mathrm e}^{4 i \left (d x +c \right )}+132 a^{5} b^{2} {\mathrm e}^{4 i \left (d x +c \right )}-147 a^{3} b^{4} {\mathrm e}^{4 i \left (d x +c \right )}+54 a \,b^{6} {\mathrm e}^{4 i \left (d x +c \right )}+216 a^{6} b \,{\mathrm e}^{3 i \left (d x +c \right )}-48 a^{4} b^{3} {\mathrm e}^{3 i \left (d x +c \right )}-62 a^{2} b^{5} {\mathrm e}^{3 i \left (d x +c \right )}+44 b^{7} {\mathrm e}^{3 i \left (d x +c \right )}+72 a^{7} {\mathrm e}^{2 i \left (d x +c \right )}+204 a^{5} b^{2} {\mathrm e}^{2 i \left (d x +c \right )}-204 a^{3} b^{4} {\mathrm e}^{2 i \left (d x +c \right )}+78 a \,b^{6} {\mathrm e}^{2 i \left (d x +c \right )}+168 a^{6} b \,{\mathrm e}^{i \left (d x +c \right )}-141 a^{4} b^{3} {\mathrm e}^{i \left (d x +c \right )}+48 a^{2} b^{5} {\mathrm e}^{i \left (d x +c \right )}+36 a^{7}-32 a^{5} b^{2}+11 b^{4} a^{3}\right )}{3 a^{4} \left (-a^{2}+b^{2}\right )^{3} d \left (a \,{\mathrm e}^{2 i \left (d x +c \right )}+2 b \,{\mathrm e}^{i \left (d x +c \right )}+a \right )^{3}}+\frac {4 b \,a^{2} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+\frac {-i a^{2}+i b^{2}+b \sqrt {a^{2}-b^{2}}}{a \sqrt {a^{2}-b^{2}}}\right )}{\sqrt {a^{2}-b^{2}}\, \left (a +b \right )^{3} \left (a -b \right )^{3} d}-\frac {4 b^{3} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+\frac {-i a^{2}+i b^{2}+b \sqrt {a^{2}-b^{2}}}{a \sqrt {a^{2}-b^{2}}}\right )}{\sqrt {a^{2}-b^{2}}\, \left (a +b \right )^{3} \left (a -b \right )^{3} d}+\frac {7 b^{5} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+\frac {-i a^{2}+i b^{2}+b \sqrt {a^{2}-b^{2}}}{a \sqrt {a^{2}-b^{2}}}\right )}{2 \sqrt {a^{2}-b^{2}}\, \left (a +b \right )^{3} \left (a -b \right )^{3} d \,a^{2}}-\frac {b^{7} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+\frac {-i a^{2}+i b^{2}+b \sqrt {a^{2}-b^{2}}}{a \sqrt {a^{2}-b^{2}}}\right )}{\sqrt {a^{2}-b^{2}}\, \left (a +b \right )^{3} \left (a -b \right )^{3} d \,a^{4}}-\frac {4 a^{2} b \ln \left ({\mathrm e}^{i \left (d x +c \right )}+\frac {i a^{2}-i b^{2}+b \sqrt {a^{2}-b^{2}}}{\sqrt {a^{2}-b^{2}}\, a}\right )}{\sqrt {a^{2}-b^{2}}\, \left (a +b \right )^{3} \left (a -b \right )^{3} d}+\frac {4 b^{3} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+\frac {i a^{2}-i b^{2}+b \sqrt {a^{2}-b^{2}}}{\sqrt {a^{2}-b^{2}}\, a}\right )}{\sqrt {a^{2}-b^{2}}\, \left (a +b \right )^{3} \left (a -b \right )^{3} d}-\frac {7 b^{5} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+\frac {i a^{2}-i b^{2}+b \sqrt {a^{2}-b^{2}}}{\sqrt {a^{2}-b^{2}}\, a}\right )}{2 \sqrt {a^{2}-b^{2}}\, \left (a +b \right )^{3} \left (a -b \right )^{3} d \,a^{2}}+\frac {b^{7} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+\frac {i a^{2}-i b^{2}+b \sqrt {a^{2}-b^{2}}}{\sqrt {a^{2}-b^{2}}\, a}\right )}{\sqrt {a^{2}-b^{2}}\, \left (a +b \right )^{3} \left (a -b \right )^{3} d \,a^{4}}\) | \(1043\) |
1/d*(2/a^4*arctan(tan(1/2*d*x+1/2*c))+2*b/a^4*((-1/2*(12*a^4+4*a^3*b-6*a^2 *b^2-a*b^3+2*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*(18*a^4-11*a^2*b^2+3*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*(12*a^4-4*a^3*b-6*a^2*b^2+a*b^3+2*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*(8*a^6-8*a^4*b^2+7*a^2*b^4-2*b^6)/(a^6-3*a^4*b^2+3* a^2*b^4-b^6)/((a-b)*(a+b))^(1/2)*arctanh((a-b)*tan(1/2*d*x+1/2*c)/((a-b)*( a+b))^(1/2))))
Leaf count of result is larger than twice the leaf count of optimal. 699 vs. \(2 (227) = 454\).
Time = 0.37 (sec) , antiderivative size = 1456, normalized size of antiderivative = 6.02 \[ \int \frac {1}{(a+b \sec (c+d x))^4} \, dx=\text {Too large to display} \]
[1/12*(12*(a^11 - 4*a^9*b^2 + 6*a^7*b^4 - 4*a^5*b^6 + a^3*b^8)*d*x*cos(d*x + c)^3 + 36*(a^10*b - 4*a^8*b^3 + 6*a^6*b^5 - 4*a^4*b^7 + a^2*b^9)*d*x*co s(d*x + c)^2 + 36*(a^9*b^2 - 4*a^7*b^4 + 6*a^5*b^6 - 4*a^3*b^8 + a*b^10)*d *x*cos(d*x + c) + 12*(a^8*b^3 - 4*a^6*b^5 + 6*a^4*b^7 - 4*a^2*b^9 + b^11)* d*x + 3*(8*a^6*b^4 - 8*a^4*b^6 + 7*a^2*b^8 - 2*b^10 + (8*a^9*b - 8*a^7*b^3 + 7*a^5*b^5 - 2*a^3*b^7)*cos(d*x + c)^3 + 3*(8*a^8*b^2 - 8*a^6*b^4 + 7*a^ 4*b^6 - 2*a^2*b^8)*cos(d*x + c)^2 + 3*(8*a^7*b^3 - 8*a^5*b^5 + 7*a^3*b^7 - 2*a*b^9)*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)) + 2*(26*a ^7*b^4 - 43*a^5*b^6 + 23*a^3*b^8 - 6*a*b^10 + (36*a^9*b^2 - 68*a^7*b^4 + 4 3*a^5*b^6 - 11*a^3*b^8)*cos(d*x + c)^2 + 15*(4*a^8*b^3 - 7*a^6*b^5 + 4*a^4 *b^7 - a^2*b^9)*cos(d*x + c))*sin(d*x + c))/((a^15 - 4*a^13*b^2 + 6*a^11*b ^4 - 4*a^9*b^6 + a^7*b^8)*d*cos(d*x + c)^3 + 3*(a^14*b - 4*a^12*b^3 + 6*a^ 10*b^5 - 4*a^8*b^7 + a^6*b^9)*d*cos(d*x + c)^2 + 3*(a^13*b^2 - 4*a^11*b^4 + 6*a^9*b^6 - 4*a^7*b^8 + a^5*b^10)*d*cos(d*x + c) + (a^12*b^3 - 4*a^10*b^ 5 + 6*a^8*b^7 - 4*a^6*b^9 + a^4*b^11)*d), 1/6*(6*(a^11 - 4*a^9*b^2 + 6*a^7 *b^4 - 4*a^5*b^6 + a^3*b^8)*d*x*cos(d*x + c)^3 + 18*(a^10*b - 4*a^8*b^3 + 6*a^6*b^5 - 4*a^4*b^7 + a^2*b^9)*d*x*cos(d*x + c)^2 + 18*(a^9*b^2 - 4*a^7* b^4 + 6*a^5*b^6 - 4*a^3*b^8 + a*b^10)*d*x*cos(d*x + c) + 6*(a^8*b^3 - 4...
\[ \int \frac {1}{(a+b \sec (c+d x))^4} \, dx=\int \frac {1}{\left (a + b \sec {\left (c + d x \right )}\right )^{4}}\, dx \]
Exception generated. \[ \int \frac {1}{(a+b \sec (c+d x))^4} \, dx=\text {Exception raised: ValueError} \]
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. 532 vs. \(2 (227) = 454\).
Time = 0.31 (sec) , antiderivative size = 532, normalized size of antiderivative = 2.20 \[ \int \frac {1}{(a+b \sec (c+d x))^4} \, dx=\frac {\frac {3 \, {\left (8 \, a^{6} b - 8 \, a^{4} b^{3} + 7 \, a^{2} b^{5} - 2 \, b^{7}\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^{10} - 3 \, a^{8} b^{2} + 3 \, a^{6} b^{4} - a^{4} b^{6}\right )} \sqrt {-a^{2} + b^{2}}} - \frac {36 \, a^{6} b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 60 \, 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} + 45 \, a^{3} b^{5} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 6 \, a^{2} b^{6} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 15 \, a b^{7} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 6 \, b^{8} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 72 \, 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} - 56 \, a^{2} b^{6} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 12 \, b^{8} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 36 \, a^{6} b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 60 \, 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 ) - 45 \, a^{3} b^{5} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 6 \, a^{2} b^{6} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 15 \, a b^{7} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 6 \, b^{8} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{{\left (a^{9} - 3 \, a^{7} b^{2} + 3 \, a^{5} b^{4} - a^{3} b^{6}\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 \, {\left (d x + c\right )}}{a^{4}}}{3 \, d} \]
1/3*(3*(8*a^6*b - 8*a^4*b^3 + 7*a^2*b^5 - 2*b^7)*(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^10 - 3*a^8*b^2 + 3*a^6*b^4 - a^4*b^6)*sqrt (-a^2 + b^2)) - (36*a^6*b^2*tan(1/2*d*x + 1/2*c)^5 - 60*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 + 45*a^3*b^5*tan(1/2*d*x + 1/2*c)^5 - 6*a^2*b^6*tan(1/2*d*x + 1/2*c)^5 - 15*a*b^7*tan(1/2*d*x + 1/2* c)^5 + 6*b^8*tan(1/2*d*x + 1/2*c)^5 - 72*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 - 56*a^2*b^6*tan(1/2*d*x + 1/2*c)^3 + 1 2*b^8*tan(1/2*d*x + 1/2*c)^3 + 36*a^6*b^2*tan(1/2*d*x + 1/2*c) + 60*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) - 45*a^3*b^5*tan(1 /2*d*x + 1/2*c) - 6*a^2*b^6*tan(1/2*d*x + 1/2*c) + 15*a*b^7*tan(1/2*d*x + 1/2*c) + 6*b^8*tan(1/2*d*x + 1/2*c))/((a^9 - 3*a^7*b^2 + 3*a^5*b^4 - a^3*b ^6)*(a*tan(1/2*d*x + 1/2*c)^2 - b*tan(1/2*d*x + 1/2*c)^2 - a - b)^3) + 3*( d*x + c)/a^4)/d
Time = 27.00 (sec) , antiderivative size = 7234, normalized size of antiderivative = 29.89 \[ \int \frac {1}{(a+b \sec (c+d x))^4} \, dx=\text {Too large to display} \]
(2*atan((((((8*(16*a^20*b - 4*a^21 + 4*a^8*b^13 - 2*a^9*b^12 - 26*a^10*b^1 1 + 14*a^11*b^10 + 70*a^12*b^9 - 30*a^13*b^8 - 110*a^14*b^7 + 30*a^15*b^6 + 110*a^16*b^5 - 20*a^17*b^4 - 64*a^18*b^3 + 12*a^19*b^2))/(a^19*b + a^20 - a^9*b^11 - a^10*b^10 + 5*a^11*b^9 + 5*a^12*b^8 - 10*a^13*b^7 - 10*a^14*b ^6 + 10*a^15*b^5 + 10*a^16*b^4 - 5*a^17*b^3 - 5*a^18*b^2) - (tan(c/2 + (d* x)/2)*(8*a^21*b - 8*a^8*b^14 + 8*a^9*b^13 + 48*a^10*b^12 - 48*a^11*b^11 - 120*a^12*b^10 + 120*a^13*b^9 + 160*a^14*b^8 - 160*a^15*b^7 - 120*a^16*b^6 + 120*a^17*b^5 + 48*a^18*b^4 - 48*a^19*b^3 - 8*a^20*b^2)*8i)/(a^4*(a^16*b + a^17 - a^6*b^11 - a^7*b^10 + 5*a^8*b^9 + 5*a^9*b^8 - 10*a^10*b^7 - 10*a^ 11*b^6 + 10*a^12*b^5 + 10*a^13*b^4 - 5*a^14*b^3 - 5*a^15*b^2)))*1i)/a^4 + (8*tan(c/2 + (d*x)/2)*(4*a^14 - 8*a^13*b - 8*a*b^13 + 8*b^14 - 48*a^2*b^12 + 48*a^3*b^11 + 117*a^4*b^10 - 120*a^5*b^9 - 164*a^6*b^8 + 160*a^7*b^7 + 156*a^8*b^6 - 120*a^9*b^5 - 92*a^10*b^4 + 48*a^11*b^3 + 44*a^12*b^2))/(a^1 6*b + a^17 - a^6*b^11 - a^7*b^10 + 5*a^8*b^9 + 5*a^9*b^8 - 10*a^10*b^7 - 1 0*a^11*b^6 + 10*a^12*b^5 + 10*a^13*b^4 - 5*a^14*b^3 - 5*a^15*b^2))/a^4 - ( (((8*(16*a^20*b - 4*a^21 + 4*a^8*b^13 - 2*a^9*b^12 - 26*a^10*b^11 + 14*a^1 1*b^10 + 70*a^12*b^9 - 30*a^13*b^8 - 110*a^14*b^7 + 30*a^15*b^6 + 110*a^16 *b^5 - 20*a^17*b^4 - 64*a^18*b^3 + 12*a^19*b^2))/(a^19*b + a^20 - a^9*b^11 - a^10*b^10 + 5*a^11*b^9 + 5*a^12*b^8 - 10*a^13*b^7 - 10*a^14*b^6 + 10*a^ 15*b^5 + 10*a^16*b^4 - 5*a^17*b^3 - 5*a^18*b^2) + (tan(c/2 + (d*x)/2)*(...