\(\int x \sec ^7(a+b x^2) \, dx\) [15]

Optimal result

Integrand size = 12, antiderivative size = 90 \[ \int x \sec ^7\left (a+b x^2\right ) \, dx=\frac {5 \text {arctanh}\left (\sin \left (a+b x^2\right )\right )}{32 b}+\frac {5 \sec \left (a+b x^2\right ) \tan \left (a+b x^2\right )}{32 b}+\frac {5 \sec ^3\left (a+b x^2\right ) \tan \left (a+b x^2\right )}{48 b}+\frac {\sec ^5\left (a+b x^2\right ) \tan \left (a+b x^2\right )}{12 b} \]

[Out]

5/32*arctanh(sin(b*x^2+a))/b+5/32*sec(b*x^2+a)*tan(b*x^2+a)/b+5/48*sec(b*x^2+a)^3*tan(b*x^2+a)/b+1/12*sec(b*x^
2+a)^5*tan(b*x^2+a)/b

Rubi [A] (verified)

Time = 0.09 (sec) , antiderivative size = 90, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {4289, 3853, 3855} \[ \int x \sec ^7\left (a+b x^2\right ) \, dx=\frac {5 \text {arctanh}\left (\sin \left (a+b x^2\right )\right )}{32 b}+\frac {\tan \left (a+b x^2\right ) \sec ^5\left (a+b x^2\right )}{12 b}+\frac {5 \tan \left (a+b x^2\right ) \sec ^3\left (a+b x^2\right )}{48 b}+\frac {5 \tan \left (a+b x^2\right ) \sec \left (a+b x^2\right )}{32 b} \]

[In]

Int[x*Sec[a + b*x^2]^7,x]

[Out]

(5*ArcTanh[Sin[a + b*x^2]])/(32*b) + (5*Sec[a + b*x^2]*Tan[a + b*x^2])/(32*b) + (5*Sec[a + b*x^2]^3*Tan[a + b*
x^2])/(48*b) + (Sec[a + b*x^2]^5*Tan[a + b*x^2])/(12*b)

Rule 3853

Int[(csc[(c_.) + (d_.)*(x_)]*(b_.))^(n_), x_Symbol] :> Simp[(-b)*Cos[c + d*x]*((b*Csc[c + d*x])^(n - 1)/(d*(n
- 1))), x] + Dist[b^2*((n - 2)/(n - 1)), Int[(b*Csc[c + d*x])^(n - 2), x], x] /; FreeQ[{b, c, d}, x] && GtQ[n,
1] && IntegerQ[2*n]

Rule 3855

Int[csc[(c_.) + (d_.)*(x_)], x_Symbol] :> Simp[-ArcTanh[Cos[c + d*x]]/d, x] /; FreeQ[{c, d}, x]

Rule 4289

Int[(x_)^(m_.)*((a_.) + (b_.)*Sec[(c_.) + (d_.)*(x_)^(n_)])^(p_.), x_Symbol] :> Dist[1/n, Subst[Int[x^(Simplif
y[(m + 1)/n] - 1)*(a + b*Sec[c + d*x])^p, x], x, x^n], x] /; FreeQ[{a, b, c, d, m, n, p}, x] && IGtQ[Simplify[
(m + 1)/n], 0] && IntegerQ[p]

Rubi steps \begin{align*} \text {integral}& = \frac {1}{2} \text {Subst}\left (\int \sec ^7(a+b x) \, dx,x,x^2\right ) \\ & = \frac {\sec ^5\left (a+b x^2\right ) \tan \left (a+b x^2\right )}{12 b}+\frac {5}{12} \text {Subst}\left (\int \sec ^5(a+b x) \, dx,x,x^2\right ) \\ & = \frac {5 \sec ^3\left (a+b x^2\right ) \tan \left (a+b x^2\right )}{48 b}+\frac {\sec ^5\left (a+b x^2\right ) \tan \left (a+b x^2\right )}{12 b}+\frac {5}{16} \text {Subst}\left (\int \sec ^3(a+b x) \, dx,x,x^2\right ) \\ & = \frac {5 \sec \left (a+b x^2\right ) \tan \left (a+b x^2\right )}{32 b}+\frac {5 \sec ^3\left (a+b x^2\right ) \tan \left (a+b x^2\right )}{48 b}+\frac {\sec ^5\left (a+b x^2\right ) \tan \left (a+b x^2\right )}{12 b}+\frac {5}{32} \text {Subst}\left (\int \sec (a+b x) \, dx,x,x^2\right ) \\ & = \frac {5 \text {arctanh}\left (\sin \left (a+b x^2\right )\right )}{32 b}+\frac {5 \sec \left (a+b x^2\right ) \tan \left (a+b x^2\right )}{32 b}+\frac {5 \sec ^3\left (a+b x^2\right ) \tan \left (a+b x^2\right )}{48 b}+\frac {\sec ^5\left (a+b x^2\right ) \tan \left (a+b x^2\right )}{12 b} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.08 (sec) , antiderivative size = 90, normalized size of antiderivative = 1.00 \[ \int x \sec ^7\left (a+b x^2\right ) \, dx=\frac {5 \text {arctanh}\left (\sin \left (a+b x^2\right )\right )}{32 b}+\frac {5 \sec \left (a+b x^2\right ) \tan \left (a+b x^2\right )}{32 b}+\frac {5 \sec ^3\left (a+b x^2\right ) \tan \left (a+b x^2\right )}{48 b}+\frac {\sec ^5\left (a+b x^2\right ) \tan \left (a+b x^2\right )}{12 b} \]

[In]

Integrate[x*Sec[a + b*x^2]^7,x]

[Out]

(5*ArcTanh[Sin[a + b*x^2]])/(32*b) + (5*Sec[a + b*x^2]*Tan[a + b*x^2])/(32*b) + (5*Sec[a + b*x^2]^3*Tan[a + b*
x^2])/(48*b) + (Sec[a + b*x^2]^5*Tan[a + b*x^2])/(12*b)

Maple [A] (verified)

Time = 0.53 (sec) , antiderivative size = 72, normalized size of antiderivative = 0.80

method	result	size
derivativedivides	\(\frac {-\left (-\frac {\sec \left (x^{2} b +a \right )^{5}}{6}-\frac {5 \sec \left (x^{2} b +a \right )^{3}}{24}-\frac {5 \sec \left (x^{2} b +a \right )}{16}\right ) \tan \left (x^{2} b +a \right )+\frac {5 \ln \left (\sec \left (x^{2} b +a \right )+\tan \left (x^{2} b +a \right )\right )}{16}}{2 b}\)	\(72\)
default	\(\frac {-\left (-\frac {\sec \left (x^{2} b +a \right )^{5}}{6}-\frac {5 \sec \left (x^{2} b +a \right )^{3}}{24}-\frac {5 \sec \left (x^{2} b +a \right )}{16}\right ) \tan \left (x^{2} b +a \right )+\frac {5 \ln \left (\sec \left (x^{2} b +a \right )+\tan \left (x^{2} b +a \right )\right )}{16}}{2 b}\)	\(72\)
risch	\(-\frac {i \left (15 \,{\mathrm e}^{11 i \left (x^{2} b +a \right )}+85 \,{\mathrm e}^{9 i \left (x^{2} b +a \right )}+198 \,{\mathrm e}^{7 i \left (x^{2} b +a \right )}-198 \,{\mathrm e}^{5 i \left (x^{2} b +a \right )}-85 \,{\mathrm e}^{3 i \left (x^{2} b +a \right )}-15 \,{\mathrm e}^{i \left (x^{2} b +a \right )}\right )}{48 b \left ({\mathrm e}^{2 i \left (x^{2} b +a \right )}+1\right )^{6}}-\frac {5 \ln \left ({\mathrm e}^{i \left (x^{2} b +a \right )}-i\right )}{32 b}+\frac {5 \ln \left ({\mathrm e}^{i \left (x^{2} b +a \right )}+i\right )}{32 b}\)	\(142\)
parallelrisch	\(\frac {\left (-225 \cos \left (2 x^{2} b +2 a \right )-90 \cos \left (4 x^{2} b +4 a \right )-15 \cos \left (6 x^{2} b +6 a \right )-150\right ) \ln \left (\tan \left (\frac {a}{2}+\frac {x^{2} b}{2}\right )-1\right )+\left (225 \cos \left (2 x^{2} b +2 a \right )+90 \cos \left (4 x^{2} b +4 a \right )+15 \cos \left (6 x^{2} b +6 a \right )+150\right ) \ln \left (\tan \left (\frac {a}{2}+\frac {x^{2} b}{2}\right )+1\right )+396 \sin \left (x^{2} b +a \right )+170 \sin \left (3 x^{2} b +3 a \right )+30 \sin \left (5 x^{2} b +5 a \right )}{96 b \left (10+\cos \left (6 x^{2} b +6 a \right )+6 \cos \left (4 x^{2} b +4 a \right )+15 \cos \left (2 x^{2} b +2 a \right )\right )}\)	\(196\)

[In]

int(x*sec(b*x^2+a)^7,x,method=_RETURNVERBOSE)

[Out]

1/2/b*(-(-1/6*sec(b*x^2+a)^5-5/24*sec(b*x^2+a)^3-5/16*sec(b*x^2+a))*tan(b*x^2+a)+5/16*ln(sec(b*x^2+a)+tan(b*x^
2+a)))

Fricas [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 100, normalized size of antiderivative = 1.11 \[ \int x \sec ^7\left (a+b x^2\right ) \, dx=\frac {15 \, \cos \left (b x^{2} + a\right )^{6} \log \left (\sin \left (b x^{2} + a\right ) + 1\right ) - 15 \, \cos \left (b x^{2} + a\right )^{6} \log \left (-\sin \left (b x^{2} + a\right ) + 1\right ) + 2 \, {\left (15 \, \cos \left (b x^{2} + a\right )^{4} + 10 \, \cos \left (b x^{2} + a\right )^{2} + 8\right )} \sin \left (b x^{2} + a\right )}{192 \, b \cos \left (b x^{2} + a\right )^{6}} \]

[In]

integrate(x*sec(b*x^2+a)^7,x, algorithm="fricas")

[Out]

1/192*(15*cos(b*x^2 + a)^6*log(sin(b*x^2 + a) + 1) - 15*cos(b*x^2 + a)^6*log(-sin(b*x^2 + a) + 1) + 2*(15*cos(
b*x^2 + a)^4 + 10*cos(b*x^2 + a)^2 + 8)*sin(b*x^2 + a))/(b*cos(b*x^2 + a)^6)

Sympy [F]

\[ \int x \sec ^7\left (a+b x^2\right ) \, dx=\int x \sec ^{7}{\left (a + b x^{2} \right )}\, dx \]

[In]

integrate(x*sec(b*x**2+a)**7,x)

[Out]

Integral(x*sec(a + b*x**2)**7, x)

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 2838 vs. \(2 (82) = 164\).

Time = 0.45 (sec) , antiderivative size = 2838, normalized size of antiderivative = 31.53 \[ \int x \sec ^7\left (a+b x^2\right ) \, dx=\text {Too large to display} \]

[In]

integrate(x*sec(b*x^2+a)^7,x, algorithm="maxima")

[Out]

1/192*(4*(15*sin(11*b*x^2 + 11*a) + 85*sin(9*b*x^2 + 9*a) + 198*sin(7*b*x^2 + 7*a) - 198*sin(5*b*x^2 + 5*a) -
85*sin(3*b*x^2 + 3*a) - 15*sin(b*x^2 + a))*cos(12*b*x^2 + 12*a) - 60*(6*sin(10*b*x^2 + 10*a) + 15*sin(8*b*x^2
+ 8*a) + 20*sin(6*b*x^2 + 6*a) + 15*sin(4*b*x^2 + 4*a) + 6*sin(2*b*x^2 + 2*a))*cos(11*b*x^2 + 11*a) + 24*(85*s
in(9*b*x^2 + 9*a) + 198*sin(7*b*x^2 + 7*a) - 198*sin(5*b*x^2 + 5*a) - 85*sin(3*b*x^2 + 3*a) - 15*sin(b*x^2 + a
))*cos(10*b*x^2 + 10*a) - 340*(15*sin(8*b*x^2 + 8*a) + 20*sin(6*b*x^2 + 6*a) + 15*sin(4*b*x^2 + 4*a) + 6*sin(2
*b*x^2 + 2*a))*cos(9*b*x^2 + 9*a) + 60*(198*sin(7*b*x^2 + 7*a) - 198*sin(5*b*x^2 + 5*a) - 85*sin(3*b*x^2 + 3*a
) - 15*sin(b*x^2 + a))*cos(8*b*x^2 + 8*a) - 792*(20*sin(6*b*x^2 + 6*a) + 15*sin(4*b*x^2 + 4*a) + 6*sin(2*b*x^2
+ 2*a))*cos(7*b*x^2 + 7*a) - 80*(198*sin(5*b*x^2 + 5*a) + 85*sin(3*b*x^2 + 3*a) + 15*sin(b*x^2 + a))*cos(6*b*
x^2 + 6*a) + 2376*(5*sin(4*b*x^2 + 4*a) + 2*sin(2*b*x^2 + 2*a))*cos(5*b*x^2 + 5*a) - 300*(17*sin(3*b*x^2 + 3*a
) + 3*sin(b*x^2 + a))*cos(4*b*x^2 + 4*a) - 15*(2*(6*cos(10*b*x^2 + 10*a) + 15*cos(8*b*x^2 + 8*a) + 20*cos(6*b*
x^2 + 6*a) + 15*cos(4*b*x^2 + 4*a) + 6*cos(2*b*x^2 + 2*a) + 1)*cos(12*b*x^2 + 12*a) + cos(12*b*x^2 + 12*a)^2 +
12*(15*cos(8*b*x^2 + 8*a) + 20*cos(6*b*x^2 + 6*a) + 15*cos(4*b*x^2 + 4*a) + 6*cos(2*b*x^2 + 2*a) + 1)*cos(10*
b*x^2 + 10*a) + 36*cos(10*b*x^2 + 10*a)^2 + 30*(20*cos(6*b*x^2 + 6*a) + 15*cos(4*b*x^2 + 4*a) + 6*cos(2*b*x^2
+ 2*a) + 1)*cos(8*b*x^2 + 8*a) + 225*cos(8*b*x^2 + 8*a)^2 + 40*(15*cos(4*b*x^2 + 4*a) + 6*cos(2*b*x^2 + 2*a) +
1)*cos(6*b*x^2 + 6*a) + 400*cos(6*b*x^2 + 6*a)^2 + 30*(6*cos(2*b*x^2 + 2*a) + 1)*cos(4*b*x^2 + 4*a) + 225*cos
(4*b*x^2 + 4*a)^2 + 36*cos(2*b*x^2 + 2*a)^2 + 2*(6*sin(10*b*x^2 + 10*a) + 15*sin(8*b*x^2 + 8*a) + 20*sin(6*b*x
^2 + 6*a) + 15*sin(4*b*x^2 + 4*a) + 6*sin(2*b*x^2 + 2*a))*sin(12*b*x^2 + 12*a) + sin(12*b*x^2 + 12*a)^2 + 12*(
15*sin(8*b*x^2 + 8*a) + 20*sin(6*b*x^2 + 6*a) + 15*sin(4*b*x^2 + 4*a) + 6*sin(2*b*x^2 + 2*a))*sin(10*b*x^2 + 1
0*a) + 36*sin(10*b*x^2 + 10*a)^2 + 30*(20*sin(6*b*x^2 + 6*a) + 15*sin(4*b*x^2 + 4*a) + 6*sin(2*b*x^2 + 2*a))*s
in(8*b*x^2 + 8*a) + 225*sin(8*b*x^2 + 8*a)^2 + 120*(5*sin(4*b*x^2 + 4*a) + 2*sin(2*b*x^2 + 2*a))*sin(6*b*x^2 +
6*a) + 400*sin(6*b*x^2 + 6*a)^2 + 225*sin(4*b*x^2 + 4*a)^2 + 180*sin(4*b*x^2 + 4*a)*sin(2*b*x^2 + 2*a) + 36*s
in(2*b*x^2 + 2*a)^2 + 12*cos(2*b*x^2 + 2*a) + 1)*log((cos(b*x^2 + 2*a)^2 + cos(a)^2 - 2*cos(a)*sin(b*x^2 + 2*a
) + sin(b*x^2 + 2*a)^2 + 2*cos(b*x^2 + 2*a)*sin(a) + sin(a)^2)/(cos(b*x^2 + 2*a)^2 + cos(a)^2 + 2*cos(a)*sin(b
*x^2 + 2*a) + sin(b*x^2 + 2*a)^2 - 2*cos(b*x^2 + 2*a)*sin(a) + sin(a)^2)) - 4*(15*cos(11*b*x^2 + 11*a) + 85*co
s(9*b*x^2 + 9*a) + 198*cos(7*b*x^2 + 7*a) - 198*cos(5*b*x^2 + 5*a) - 85*cos(3*b*x^2 + 3*a) - 15*cos(b*x^2 + a)
)*sin(12*b*x^2 + 12*a) + 60*(6*cos(10*b*x^2 + 10*a) + 15*cos(8*b*x^2 + 8*a) + 20*cos(6*b*x^2 + 6*a) + 15*cos(4
*b*x^2 + 4*a) + 6*cos(2*b*x^2 + 2*a) + 1)*sin(11*b*x^2 + 11*a) - 24*(85*cos(9*b*x^2 + 9*a) + 198*cos(7*b*x^2 +
7*a) - 198*cos(5*b*x^2 + 5*a) - 85*cos(3*b*x^2 + 3*a) - 15*cos(b*x^2 + a))*sin(10*b*x^2 + 10*a) + 340*(15*cos
(8*b*x^2 + 8*a) + 20*cos(6*b*x^2 + 6*a) + 15*cos(4*b*x^2 + 4*a) + 6*cos(2*b*x^2 + 2*a) + 1)*sin(9*b*x^2 + 9*a)
- 60*(198*cos(7*b*x^2 + 7*a) - 198*cos(5*b*x^2 + 5*a) - 85*cos(3*b*x^2 + 3*a) - 15*cos(b*x^2 + a))*sin(8*b*x^
2 + 8*a) + 792*(20*cos(6*b*x^2 + 6*a) + 15*cos(4*b*x^2 + 4*a) + 6*cos(2*b*x^2 + 2*a) + 1)*sin(7*b*x^2 + 7*a) +
80*(198*cos(5*b*x^2 + 5*a) + 85*cos(3*b*x^2 + 3*a) + 15*cos(b*x^2 + a))*sin(6*b*x^2 + 6*a) - 792*(15*cos(4*b*
x^2 + 4*a) + 6*cos(2*b*x^2 + 2*a) + 1)*sin(5*b*x^2 + 5*a) + 300*(17*cos(3*b*x^2 + 3*a) + 3*cos(b*x^2 + a))*sin
(4*b*x^2 + 4*a) - 340*(6*cos(2*b*x^2 + 2*a) + 1)*sin(3*b*x^2 + 3*a) + 2040*cos(3*b*x^2 + 3*a)*sin(2*b*x^2 + 2*
a) + 360*cos(b*x^2 + a)*sin(2*b*x^2 + 2*a) - 360*cos(2*b*x^2 + 2*a)*sin(b*x^2 + a) - 60*sin(b*x^2 + a))/(b*cos
(12*b*x^2 + 12*a)^2 + 36*b*cos(10*b*x^2 + 10*a)^2 + 225*b*cos(8*b*x^2 + 8*a)^2 + 400*b*cos(6*b*x^2 + 6*a)^2 +
225*b*cos(4*b*x^2 + 4*a)^2 + 36*b*cos(2*b*x^2 + 2*a)^2 + b*sin(12*b*x^2 + 12*a)^2 + 36*b*sin(10*b*x^2 + 10*a)^
2 + 225*b*sin(8*b*x^2 + 8*a)^2 + 400*b*sin(6*b*x^2 + 6*a)^2 + 225*b*sin(4*b*x^2 + 4*a)^2 + 180*b*sin(4*b*x^2 +
4*a)*sin(2*b*x^2 + 2*a) + 36*b*sin(2*b*x^2 + 2*a)^2 + 2*(6*b*cos(10*b*x^2 + 10*a) + 15*b*cos(8*b*x^2 + 8*a) +
20*b*cos(6*b*x^2 + 6*a) + 15*b*cos(4*b*x^2 + 4*a) + 6*b*cos(2*b*x^2 + 2*a) + b)*cos(12*b*x^2 + 12*a) + 12*(15
*b*cos(8*b*x^2 + 8*a) + 20*b*cos(6*b*x^2 + 6*a) + 15*b*cos(4*b*x^2 + 4*a) + 6*b*cos(2*b*x^2 + 2*a) + b)*cos(10
*b*x^2 + 10*a) + 30*(20*b*cos(6*b*x^2 + 6*a) + 15*b*cos(4*b*x^2 + 4*a) + 6*b*cos(2*b*x^2 + 2*a) + b)*cos(8*b*x
^2 + 8*a) + 40*(15*b*cos(4*b*x^2 + 4*a) + 6*b*cos(2*b*x^2 + 2*a) + b)*cos(6*b*x^2 + 6*a) + 30*(6*b*cos(2*b*x^2
+ 2*a) + b)*cos(4*b*x^2 + 4*a) + 12*b*cos(2*b*x^2 + 2*a) + 2*(6*b*sin(10*b*x^2 + 10*a) + 15*b*sin(8*b*x^2 + 8
*a) + 20*b*sin(6*b*x^2 + 6*a) + 15*b*sin(4*b*x^2 + 4*a) + 6*b*sin(2*b*x^2 + 2*a))*sin(12*b*x^2 + 12*a) + 12*(1
5*b*sin(8*b*x^2 + 8*a) + 20*b*sin(6*b*x^2 + 6*a) + 15*b*sin(4*b*x^2 + 4*a) + 6*b*sin(2*b*x^2 + 2*a))*sin(10*b*
x^2 + 10*a) + 30*(20*b*sin(6*b*x^2 + 6*a) + 15*b*sin(4*b*x^2 + 4*a) + 6*b*sin(2*b*x^2 + 2*a))*sin(8*b*x^2 + 8*
a) + 120*(5*b*sin(4*b*x^2 + 4*a) + 2*b*sin(2*b*x^2 + 2*a))*sin(6*b*x^2 + 6*a) + b)

Giac [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 85, normalized size of antiderivative = 0.94 \[ \int x \sec ^7\left (a+b x^2\right ) \, dx=-\frac {\frac {2 \, {\left (15 \, \sin \left (b x^{2} + a\right )^{5} - 40 \, \sin \left (b x^{2} + a\right )^{3} + 33 \, \sin \left (b x^{2} + a\right )\right )}}{{\left (\sin \left (b x^{2} + a\right )^{2} - 1\right )}^{3}} - 15 \, \log \left (\sin \left (b x^{2} + a\right ) + 1\right ) + 15 \, \log \left (-\sin \left (b x^{2} + a\right ) + 1\right )}{192 \, b} \]

[In]

integrate(x*sec(b*x^2+a)^7,x, algorithm="giac")

[Out]

-1/192*(2*(15*sin(b*x^2 + a)^5 - 40*sin(b*x^2 + a)^3 + 33*sin(b*x^2 + a))/(sin(b*x^2 + a)^2 - 1)^3 - 15*log(si
n(b*x^2 + a) + 1) + 15*log(-sin(b*x^2 + a) + 1))/b

Mupad [B] (verification not implemented)

Time = 25.42 (sec) , antiderivative size = 496, normalized size of antiderivative = 5.51 \[ \int x \sec ^7\left (a+b x^2\right ) \, dx=\frac {5\,\ln \left (-\frac {x\,5{}\mathrm {i}}{8}-\frac {5\,x\,{\mathrm {e}}^{a\,1{}\mathrm {i}}\,{\mathrm {e}}^{b\,x^2\,1{}\mathrm {i}}}{8}\right )}{32\,b}-\frac {5\,\ln \left (\frac {x\,5{}\mathrm {i}}{8}-\frac {5\,x\,{\mathrm {e}}^{a\,1{}\mathrm {i}}\,{\mathrm {e}}^{b\,x^2\,1{}\mathrm {i}}}{8}\right )}{32\,b}+\frac {{\mathrm {e}}^{3{}\mathrm {i}\,b\,x^2+a\,3{}\mathrm {i}}\,8{}\mathrm {i}}{3\,b\,\left (5\,{\mathrm {e}}^{2{}\mathrm {i}\,b\,x^2+a\,2{}\mathrm {i}}+10\,{\mathrm {e}}^{4{}\mathrm {i}\,b\,x^2+a\,4{}\mathrm {i}}+10\,{\mathrm {e}}^{6{}\mathrm {i}\,b\,x^2+a\,6{}\mathrm {i}}+5\,{\mathrm {e}}^{8{}\mathrm {i}\,b\,x^2+a\,8{}\mathrm {i}}+{\mathrm {e}}^{10{}\mathrm {i}\,b\,x^2+a\,10{}\mathrm {i}}+1\right )}-\frac {{\mathrm {e}}^{1{}\mathrm {i}\,b\,x^2+a\,1{}\mathrm {i}}\,1{}\mathrm {i}}{6\,b\,\left (3\,{\mathrm {e}}^{2{}\mathrm {i}\,b\,x^2+a\,2{}\mathrm {i}}+3\,{\mathrm {e}}^{4{}\mathrm {i}\,b\,x^2+a\,4{}\mathrm {i}}+{\mathrm {e}}^{6{}\mathrm {i}\,b\,x^2+a\,6{}\mathrm {i}}+1\right )}-\frac {{\mathrm {e}}^{1{}\mathrm {i}\,b\,x^2+a\,1{}\mathrm {i}}\,5{}\mathrm {i}}{16\,b\,\left ({\mathrm {e}}^{2{}\mathrm {i}\,b\,x^2+a\,2{}\mathrm {i}}+1\right )}+\frac {{\mathrm {e}}^{5{}\mathrm {i}\,b\,x^2+a\,5{}\mathrm {i}}\,16{}\mathrm {i}}{3\,b\,\left (6\,{\mathrm {e}}^{2{}\mathrm {i}\,b\,x^2+a\,2{}\mathrm {i}}+15\,{\mathrm {e}}^{4{}\mathrm {i}\,b\,x^2+a\,4{}\mathrm {i}}+20\,{\mathrm {e}}^{6{}\mathrm {i}\,b\,x^2+a\,6{}\mathrm {i}}+15\,{\mathrm {e}}^{8{}\mathrm {i}\,b\,x^2+a\,8{}\mathrm {i}}+6\,{\mathrm {e}}^{10{}\mathrm {i}\,b\,x^2+a\,10{}\mathrm {i}}+{\mathrm {e}}^{12{}\mathrm {i}\,b\,x^2+a\,12{}\mathrm {i}}+1\right )}+\frac {{\mathrm {e}}^{1{}\mathrm {i}\,b\,x^2+a\,1{}\mathrm {i}}\,1{}\mathrm {i}}{b\,\left (4\,{\mathrm {e}}^{2{}\mathrm {i}\,b\,x^2+a\,2{}\mathrm {i}}+6\,{\mathrm {e}}^{4{}\mathrm {i}\,b\,x^2+a\,4{}\mathrm {i}}+4\,{\mathrm {e}}^{6{}\mathrm {i}\,b\,x^2+a\,6{}\mathrm {i}}+{\mathrm {e}}^{8{}\mathrm {i}\,b\,x^2+a\,8{}\mathrm {i}}+1\right )}-\frac {{\mathrm {e}}^{1{}\mathrm {i}\,b\,x^2+a\,1{}\mathrm {i}}\,5{}\mathrm {i}}{24\,b\,\left (2\,{\mathrm {e}}^{2{}\mathrm {i}\,b\,x^2+a\,2{}\mathrm {i}}+{\mathrm {e}}^{4{}\mathrm {i}\,b\,x^2+a\,4{}\mathrm {i}}+1\right )} \]

[In]

int(x/cos(a + b*x^2)^7,x)

[Out]

(5*log(- (x*5i)/8 - (5*x*exp(a*1i)*exp(b*x^2*1i))/8))/(32*b) - (5*log((x*5i)/8 - (5*x*exp(a*1i)*exp(b*x^2*1i))
/8))/(32*b) + (exp(a*3i + b*x^2*3i)*8i)/(3*b*(5*exp(a*2i + b*x^2*2i) + 10*exp(a*4i + b*x^2*4i) + 10*exp(a*6i +
b*x^2*6i) + 5*exp(a*8i + b*x^2*8i) + exp(a*10i + b*x^2*10i) + 1)) - (exp(a*1i + b*x^2*1i)*1i)/(6*b*(3*exp(a*2
i + b*x^2*2i) + 3*exp(a*4i + b*x^2*4i) + exp(a*6i + b*x^2*6i) + 1)) - (exp(a*1i + b*x^2*1i)*5i)/(16*b*(exp(a*2
i + b*x^2*2i) + 1)) + (exp(a*5i + b*x^2*5i)*16i)/(3*b*(6*exp(a*2i + b*x^2*2i) + 15*exp(a*4i + b*x^2*4i) + 20*e
xp(a*6i + b*x^2*6i) + 15*exp(a*8i + b*x^2*8i) + 6*exp(a*10i + b*x^2*10i) + exp(a*12i + b*x^2*12i) + 1)) + (exp
(a*1i + b*x^2*1i)*1i)/(b*(4*exp(a*2i + b*x^2*2i) + 6*exp(a*4i + b*x^2*4i) + 4*exp(a*6i + b*x^2*6i) + exp(a*8i
+ b*x^2*8i) + 1)) - (exp(a*1i + b*x^2*1i)*5i)/(24*b*(2*exp(a*2i + b*x^2*2i) + exp(a*4i + b*x^2*4i) + 1))