3.36 \(\int \sin ^6(e+f x) (a+b \tan ^2(e+f x)) \, dx\)
Optimal. Leaf size=102 \[ -\frac{(a-b) \sin (e+f x) \cos ^5(e+f x)}{6 f}+\frac{(13 a-19 b) \sin (e+f x) \cos ^3(e+f x)}{24 f}-\frac{(11 a-29 b) \sin (e+f x) \cos (e+f x)}{16 f}+\frac{5}{16} x (a-7 b)+\frac{b \tan (e+f x)}{f} \]
[Out]
(5*(a - 7*b)*x)/16 - ((11*a - 29*b)*Cos[e + f*x]*Sin[e + f*x])/(16*f) + ((13*a - 19*b)*Cos[e + f*x]^3*Sin[e +
f*x])/(24*f) - ((a - b)*Cos[e + f*x]^5*Sin[e + f*x])/(6*f) + (b*Tan[e + f*x])/f
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Rubi [A] time = 0.118024, antiderivative size = 102, normalized size of antiderivative = 1.,
number of steps used = 6, number of rules used = 6, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.286, Rules used =
{3663, 455, 1814, 1157, 388, 203} \[ -\frac{(a-b) \sin (e+f x) \cos ^5(e+f x)}{6 f}+\frac{(13 a-19 b) \sin (e+f x) \cos ^3(e+f x)}{24 f}-\frac{(11 a-29 b) \sin (e+f x) \cos (e+f x)}{16 f}+\frac{5}{16} x (a-7 b)+\frac{b \tan (e+f x)}{f} \]
Antiderivative was successfully verified.
[In]
Int[Sin[e + f*x]^6*(a + b*Tan[e + f*x]^2),x]
[Out]
(5*(a - 7*b)*x)/16 - ((11*a - 29*b)*Cos[e + f*x]*Sin[e + f*x])/(16*f) + ((13*a - 19*b)*Cos[e + f*x]^3*Sin[e +
f*x])/(24*f) - ((a - b)*Cos[e + f*x]^5*Sin[e + f*x])/(6*f) + (b*Tan[e + f*x])/f
Rule 3663
Int[sin[(e_.) + (f_.)*(x_)]^(m_)*((a_) + (b_.)*((c_.)*tan[(e_.) + (f_.)*(x_)])^(n_))^(p_.), x_Symbol] :> With[
{ff = FreeFactors[Tan[e + f*x], x]}, Dist[(c*ff^(m + 1))/f, Subst[Int[(x^m*(a + b*(ff*x)^n)^p)/(c^2 + ff^2*x^2
)^(m/2 + 1), x], x, (c*Tan[e + f*x])/ff], x]] /; FreeQ[{a, b, c, e, f, n, p}, x] && IntegerQ[m/2]
Rule 455
Int[(x_)^(m_)*((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2), x_Symbol] :> Simp[((-a)^(m/2 - 1)*(b*c - a*d)*
x*(a + b*x^2)^(p + 1))/(2*b^(m/2 + 1)*(p + 1)), x] + Dist[1/(2*b^(m/2 + 1)*(p + 1)), Int[(a + b*x^2)^(p + 1)*E
xpandToSum[2*b*(p + 1)*x^2*Together[(b^(m/2)*x^(m - 2)*(c + d*x^2) - (-a)^(m/2 - 1)*(b*c - a*d))/(a + b*x^2)]
- (-a)^(m/2 - 1)*(b*c - a*d), x], x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0] && LtQ[p, -1] && IGtQ[
m/2, 0] && (IntegerQ[p] || EqQ[m + 2*p + 1, 0])
Rule 1814
Int[(Pq_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> With[{Q = PolynomialQuotient[Pq, a + b*x^2, x], f = Coeff[P
olynomialRemainder[Pq, a + b*x^2, x], x, 0], g = Coeff[PolynomialRemainder[Pq, a + b*x^2, x], x, 1]}, Simp[((a
*g - b*f*x)*(a + b*x^2)^(p + 1))/(2*a*b*(p + 1)), x] + Dist[1/(2*a*(p + 1)), Int[(a + b*x^2)^(p + 1)*ExpandToS
um[2*a*(p + 1)*Q + f*(2*p + 3), x], x], x]] /; FreeQ[{a, b}, x] && PolyQ[Pq, x] && LtQ[p, -1]
Rule 1157
Int[((d_) + (e_.)*(x_)^2)^(q_)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_.), x_Symbol] :> With[{Qx = PolynomialQ
uotient[(a + b*x^2 + c*x^4)^p, d + e*x^2, x], R = Coeff[PolynomialRemainder[(a + b*x^2 + c*x^4)^p, d + e*x^2,
x], x, 0]}, -Simp[(R*x*(d + e*x^2)^(q + 1))/(2*d*(q + 1)), x] + Dist[1/(2*d*(q + 1)), Int[(d + e*x^2)^(q + 1)*
ExpandToSum[2*d*(q + 1)*Qx + R*(2*q + 3), x], x], x]] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && N
eQ[c*d^2 - b*d*e + a*e^2, 0] && IGtQ[p, 0] && LtQ[q, -1]
Rule 388
Int[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_)), x_Symbol] :> Simp[(d*x*(a + b*x^n)^(p + 1))/(b*(n*
(p + 1) + 1)), x] - Dist[(a*d - b*c*(n*(p + 1) + 1))/(b*(n*(p + 1) + 1)), Int[(a + b*x^n)^p, x], x] /; FreeQ[{
a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && NeQ[n*(p + 1) + 1, 0]
Rule 203
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTan[(Rt[b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[b, 2]), x] /;
FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a, 0] || GtQ[b, 0])
Rubi steps
\begin{align*} \int \sin ^6(e+f x) \left (a+b \tan ^2(e+f x)\right ) \, dx &=\frac{\operatorname{Subst}\left (\int \frac{x^6 \left (a+b x^2\right )}{\left (1+x^2\right )^4} \, dx,x,\tan (e+f x)\right )}{f}\\ &=-\frac{(a-b) \cos ^5(e+f x) \sin (e+f x)}{6 f}-\frac{\operatorname{Subst}\left (\int \frac{-a+b+6 (a-b) x^2-6 (a-b) x^4-6 b x^6}{\left (1+x^2\right )^3} \, dx,x,\tan (e+f x)\right )}{6 f}\\ &=\frac{(13 a-19 b) \cos ^3(e+f x) \sin (e+f x)}{24 f}-\frac{(a-b) \cos ^5(e+f x) \sin (e+f x)}{6 f}+\frac{\operatorname{Subst}\left (\int \frac{-3 (3 a-5 b)+24 (a-2 b) x^2+24 b x^4}{\left (1+x^2\right )^2} \, dx,x,\tan (e+f x)\right )}{24 f}\\ &=-\frac{(11 a-29 b) \cos (e+f x) \sin (e+f x)}{16 f}+\frac{(13 a-19 b) \cos ^3(e+f x) \sin (e+f x)}{24 f}-\frac{(a-b) \cos ^5(e+f x) \sin (e+f x)}{6 f}-\frac{\operatorname{Subst}\left (\int \frac{-3 (5 a-19 b)-48 b x^2}{1+x^2} \, dx,x,\tan (e+f x)\right )}{48 f}\\ &=-\frac{(11 a-29 b) \cos (e+f x) \sin (e+f x)}{16 f}+\frac{(13 a-19 b) \cos ^3(e+f x) \sin (e+f x)}{24 f}-\frac{(a-b) \cos ^5(e+f x) \sin (e+f x)}{6 f}+\frac{b \tan (e+f x)}{f}+\frac{(5 (a-7 b)) \operatorname{Subst}\left (\int \frac{1}{1+x^2} \, dx,x,\tan (e+f x)\right )}{16 f}\\ &=\frac{5}{16} (a-7 b) x-\frac{(11 a-29 b) \cos (e+f x) \sin (e+f x)}{16 f}+\frac{(13 a-19 b) \cos ^3(e+f x) \sin (e+f x)}{24 f}-\frac{(a-b) \cos ^5(e+f x) \sin (e+f x)}{6 f}+\frac{b \tan (e+f x)}{f}\\ \end{align*}
Mathematica [A] time = 0.34246, size = 89, normalized size = 0.87 \[ \frac{(141 b-45 a) \sin (2 (e+f x))+3 (3 a-5 b) \sin (4 (e+f x))-a \sin (6 (e+f x))+60 a e+60 a f x+b \sin (6 (e+f x))+192 b \tan (e+f x)-420 b e-420 b f x}{192 f} \]
Antiderivative was successfully verified.
[In]
Integrate[Sin[e + f*x]^6*(a + b*Tan[e + f*x]^2),x]
[Out]
(60*a*e - 420*b*e + 60*a*f*x - 420*b*f*x + (-45*a + 141*b)*Sin[2*(e + f*x)] + 3*(3*a - 5*b)*Sin[4*(e + f*x)] -
a*Sin[6*(e + f*x)] + b*Sin[6*(e + f*x)] + 192*b*Tan[e + f*x])/(192*f)
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Maple [A] time = 0.091, size = 122, normalized size = 1.2 \begin{align*}{\frac{1}{f} \left ( a \left ( -{\frac{\cos \left ( fx+e \right ) }{6} \left ( \left ( \sin \left ( fx+e \right ) \right ) ^{5}+{\frac{5\, \left ( \sin \left ( fx+e \right ) \right ) ^{3}}{4}}+{\frac{15\,\sin \left ( fx+e \right ) }{8}} \right ) }+{\frac{5\,fx}{16}}+{\frac{5\,e}{16}} \right ) +b \left ({\frac{ \left ( \sin \left ( fx+e \right ) \right ) ^{9}}{\cos \left ( fx+e \right ) }}+ \left ( \left ( \sin \left ( fx+e \right ) \right ) ^{7}+{\frac{7\, \left ( \sin \left ( fx+e \right ) \right ) ^{5}}{6}}+{\frac{35\, \left ( \sin \left ( fx+e \right ) \right ) ^{3}}{24}}+{\frac{35\,\sin \left ( fx+e \right ) }{16}} \right ) \cos \left ( fx+e \right ) -{\frac{35\,fx}{16}}-{\frac{35\,e}{16}} \right ) \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(sin(f*x+e)^6*(a+b*tan(f*x+e)^2),x)
[Out]
1/f*(a*(-1/6*(sin(f*x+e)^5+5/4*sin(f*x+e)^3+15/8*sin(f*x+e))*cos(f*x+e)+5/16*f*x+5/16*e)+b*(sin(f*x+e)^9/cos(f
*x+e)+(sin(f*x+e)^7+7/6*sin(f*x+e)^5+35/24*sin(f*x+e)^3+35/16*sin(f*x+e))*cos(f*x+e)-35/16*f*x-35/16*e))
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Maxima [A] time = 1.44308, size = 150, normalized size = 1.47 \begin{align*} \frac{15 \,{\left (f x + e\right )}{\left (a - 7 \, b\right )} + 48 \, b \tan \left (f x + e\right ) - \frac{3 \,{\left (11 \, a - 29 \, b\right )} \tan \left (f x + e\right )^{5} + 8 \,{\left (5 \, a - 17 \, b\right )} \tan \left (f x + e\right )^{3} + 3 \,{\left (5 \, a - 19 \, b\right )} \tan \left (f x + e\right )}{\tan \left (f x + e\right )^{6} + 3 \, \tan \left (f x + e\right )^{4} + 3 \, \tan \left (f x + e\right )^{2} + 1}}{48 \, f} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(sin(f*x+e)^6*(a+b*tan(f*x+e)^2),x, algorithm="maxima")
[Out]
1/48*(15*(f*x + e)*(a - 7*b) + 48*b*tan(f*x + e) - (3*(11*a - 29*b)*tan(f*x + e)^5 + 8*(5*a - 17*b)*tan(f*x +
e)^3 + 3*(5*a - 19*b)*tan(f*x + e))/(tan(f*x + e)^6 + 3*tan(f*x + e)^4 + 3*tan(f*x + e)^2 + 1))/f
________________________________________________________________________________________
Fricas [A] time = 1.9969, size = 230, normalized size = 2.25 \begin{align*} \frac{15 \,{\left (a - 7 \, b\right )} f x \cos \left (f x + e\right ) -{\left (8 \,{\left (a - b\right )} \cos \left (f x + e\right )^{6} - 2 \,{\left (13 \, a - 19 \, b\right )} \cos \left (f x + e\right )^{4} + 3 \,{\left (11 \, a - 29 \, b\right )} \cos \left (f x + e\right )^{2} - 48 \, b\right )} \sin \left (f x + e\right )}{48 \, f \cos \left (f x + e\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(sin(f*x+e)^6*(a+b*tan(f*x+e)^2),x, algorithm="fricas")
[Out]
1/48*(15*(a - 7*b)*f*x*cos(f*x + e) - (8*(a - b)*cos(f*x + e)^6 - 2*(13*a - 19*b)*cos(f*x + e)^4 + 3*(11*a - 2
9*b)*cos(f*x + e)^2 - 48*b)*sin(f*x + e))/(f*cos(f*x + e))
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(sin(f*x+e)**6*(a+b*tan(f*x+e)**2),x)
[Out]
Timed out
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Giac [B] time = 6.02839, size = 10661, normalized size = 104.52 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(sin(f*x+e)^6*(a+b*tan(f*x+e)^2),x, algorithm="giac")
[Out]
1/192*(21*pi*b*sgn(2*tan(f*x)^2*tan(e)^2 - 2)*sgn(-2*tan(f*x)^2*tan(e) + 2*tan(f*x)*tan(e)^2 + 2*tan(f*x) - 2*
tan(e))*tan(f*x)^7*tan(e)^7 + 60*a*f*x*tan(f*x)^7*tan(e)^7 - 420*b*f*x*tan(f*x)^7*tan(e)^7 + 21*pi*b*sgn(-2*ta
n(f*x)^2*tan(e) + 2*tan(f*x)*tan(e)^2 + 2*tan(f*x) - 2*tan(e))*tan(f*x)^7*tan(e)^7 + 63*pi*b*sgn(2*tan(f*x)^2*
tan(e)^2 - 2)*sgn(-2*tan(f*x)^2*tan(e) + 2*tan(f*x)*tan(e)^2 + 2*tan(f*x) - 2*tan(e))*tan(f*x)^7*tan(e)^5 - 21
*pi*b*sgn(2*tan(f*x)^2*tan(e)^2 - 2)*sgn(-2*tan(f*x)^2*tan(e) + 2*tan(f*x)*tan(e)^2 + 2*tan(f*x) - 2*tan(e))*t
an(f*x)^6*tan(e)^6 + 63*pi*b*sgn(2*tan(f*x)^2*tan(e)^2 - 2)*sgn(-2*tan(f*x)^2*tan(e) + 2*tan(f*x)*tan(e)^2 + 2
*tan(f*x) - 2*tan(e))*tan(f*x)^5*tan(e)^7 + 42*b*arctan((tan(f*x) + tan(e))/(tan(f*x)*tan(e) - 1))*tan(f*x)^7*
tan(e)^7 - 42*b*arctan(-(tan(f*x) - tan(e))/(tan(f*x)*tan(e) + 1))*tan(f*x)^7*tan(e)^7 + 180*a*f*x*tan(f*x)^7*
tan(e)^5 - 1260*b*f*x*tan(f*x)^7*tan(e)^5 + 63*pi*b*sgn(-2*tan(f*x)^2*tan(e) + 2*tan(f*x)*tan(e)^2 + 2*tan(f*x
) - 2*tan(e))*tan(f*x)^7*tan(e)^5 - 60*a*f*x*tan(f*x)^6*tan(e)^6 + 420*b*f*x*tan(f*x)^6*tan(e)^6 - 21*pi*b*sgn
(-2*tan(f*x)^2*tan(e) + 2*tan(f*x)*tan(e)^2 + 2*tan(f*x) - 2*tan(e))*tan(f*x)^6*tan(e)^6 + 180*a*f*x*tan(f*x)^
5*tan(e)^7 - 1260*b*f*x*tan(f*x)^5*tan(e)^7 + 63*pi*b*sgn(-2*tan(f*x)^2*tan(e) + 2*tan(f*x)*tan(e)^2 + 2*tan(f
*x) - 2*tan(e))*tan(f*x)^5*tan(e)^7 + 63*pi*b*sgn(2*tan(f*x)^2*tan(e)^2 - 2)*sgn(-2*tan(f*x)^2*tan(e) + 2*tan(
f*x)*tan(e)^2 + 2*tan(f*x) - 2*tan(e))*tan(f*x)^7*tan(e)^3 - 63*pi*b*sgn(2*tan(f*x)^2*tan(e)^2 - 2)*sgn(-2*tan
(f*x)^2*tan(e) + 2*tan(f*x)*tan(e)^2 + 2*tan(f*x) - 2*tan(e))*tan(f*x)^6*tan(e)^4 + 189*pi*b*sgn(2*tan(f*x)^2*
tan(e)^2 - 2)*sgn(-2*tan(f*x)^2*tan(e) + 2*tan(f*x)*tan(e)^2 + 2*tan(f*x) - 2*tan(e))*tan(f*x)^5*tan(e)^5 + 12
6*b*arctan((tan(f*x) + tan(e))/(tan(f*x)*tan(e) - 1))*tan(f*x)^7*tan(e)^5 - 126*b*arctan(-(tan(f*x) - tan(e))/
(tan(f*x)*tan(e) + 1))*tan(f*x)^7*tan(e)^5 - 63*pi*b*sgn(2*tan(f*x)^2*tan(e)^2 - 2)*sgn(-2*tan(f*x)^2*tan(e) +
2*tan(f*x)*tan(e)^2 + 2*tan(f*x) - 2*tan(e))*tan(f*x)^4*tan(e)^6 - 42*b*arctan((tan(f*x) + tan(e))/(tan(f*x)*
tan(e) - 1))*tan(f*x)^6*tan(e)^6 + 42*b*arctan(-(tan(f*x) - tan(e))/(tan(f*x)*tan(e) + 1))*tan(f*x)^6*tan(e)^6
+ 60*a*tan(f*x)^7*tan(e)^6 - 420*b*tan(f*x)^7*tan(e)^6 + 63*pi*b*sgn(2*tan(f*x)^2*tan(e)^2 - 2)*sgn(-2*tan(f*
x)^2*tan(e) + 2*tan(f*x)*tan(e)^2 + 2*tan(f*x) - 2*tan(e))*tan(f*x)^3*tan(e)^7 + 126*b*arctan((tan(f*x) + tan(
e))/(tan(f*x)*tan(e) - 1))*tan(f*x)^5*tan(e)^7 - 126*b*arctan(-(tan(f*x) - tan(e))/(tan(f*x)*tan(e) + 1))*tan(
f*x)^5*tan(e)^7 + 60*a*tan(f*x)^6*tan(e)^7 - 420*b*tan(f*x)^6*tan(e)^7 + 180*a*f*x*tan(f*x)^7*tan(e)^3 - 1260*
b*f*x*tan(f*x)^7*tan(e)^3 + 63*pi*b*sgn(-2*tan(f*x)^2*tan(e) + 2*tan(f*x)*tan(e)^2 + 2*tan(f*x) - 2*tan(e))*ta
n(f*x)^7*tan(e)^3 - 180*a*f*x*tan(f*x)^6*tan(e)^4 + 1260*b*f*x*tan(f*x)^6*tan(e)^4 - 63*pi*b*sgn(-2*tan(f*x)^2
*tan(e) + 2*tan(f*x)*tan(e)^2 + 2*tan(f*x) - 2*tan(e))*tan(f*x)^6*tan(e)^4 + 540*a*f*x*tan(f*x)^5*tan(e)^5 - 3
780*b*f*x*tan(f*x)^5*tan(e)^5 + 189*pi*b*sgn(-2*tan(f*x)^2*tan(e) + 2*tan(f*x)*tan(e)^2 + 2*tan(f*x) - 2*tan(e
))*tan(f*x)^5*tan(e)^5 - 180*a*f*x*tan(f*x)^4*tan(e)^6 + 1260*b*f*x*tan(f*x)^4*tan(e)^6 - 63*pi*b*sgn(-2*tan(f
*x)^2*tan(e) + 2*tan(f*x)*tan(e)^2 + 2*tan(f*x) - 2*tan(e))*tan(f*x)^4*tan(e)^6 + 180*a*f*x*tan(f*x)^3*tan(e)^
7 - 1260*b*f*x*tan(f*x)^3*tan(e)^7 + 63*pi*b*sgn(-2*tan(f*x)^2*tan(e) + 2*tan(f*x)*tan(e)^2 + 2*tan(f*x) - 2*t
an(e))*tan(f*x)^3*tan(e)^7 + 21*pi*b*sgn(2*tan(f*x)^2*tan(e)^2 - 2)*sgn(-2*tan(f*x)^2*tan(e) + 2*tan(f*x)*tan(
e)^2 + 2*tan(f*x) - 2*tan(e))*tan(f*x)^7*tan(e) - 63*pi*b*sgn(2*tan(f*x)^2*tan(e)^2 - 2)*sgn(-2*tan(f*x)^2*tan
(e) + 2*tan(f*x)*tan(e)^2 + 2*tan(f*x) - 2*tan(e))*tan(f*x)^6*tan(e)^2 + 189*pi*b*sgn(2*tan(f*x)^2*tan(e)^2 -
2)*sgn(-2*tan(f*x)^2*tan(e) + 2*tan(f*x)*tan(e)^2 + 2*tan(f*x) - 2*tan(e))*tan(f*x)^5*tan(e)^3 + 126*b*arctan(
(tan(f*x) + tan(e))/(tan(f*x)*tan(e) - 1))*tan(f*x)^7*tan(e)^3 - 126*b*arctan(-(tan(f*x) - tan(e))/(tan(f*x)*t
an(e) + 1))*tan(f*x)^7*tan(e)^3 - 189*pi*b*sgn(2*tan(f*x)^2*tan(e)^2 - 2)*sgn(-2*tan(f*x)^2*tan(e) + 2*tan(f*x
)*tan(e)^2 + 2*tan(f*x) - 2*tan(e))*tan(f*x)^4*tan(e)^4 - 126*b*arctan((tan(f*x) + tan(e))/(tan(f*x)*tan(e) -
1))*tan(f*x)^6*tan(e)^4 + 126*b*arctan(-(tan(f*x) - tan(e))/(tan(f*x)*tan(e) + 1))*tan(f*x)^6*tan(e)^4 + 160*a
*tan(f*x)^7*tan(e)^4 - 1120*b*tan(f*x)^7*tan(e)^4 + 189*pi*b*sgn(2*tan(f*x)^2*tan(e)^2 - 2)*sgn(-2*tan(f*x)^2*
tan(e) + 2*tan(f*x)*tan(e)^2 + 2*tan(f*x) - 2*tan(e))*tan(f*x)^3*tan(e)^5 + 378*b*arctan((tan(f*x) + tan(e))/(
tan(f*x)*tan(e) - 1))*tan(f*x)^5*tan(e)^5 - 378*b*arctan(-(tan(f*x) - tan(e))/(tan(f*x)*tan(e) + 1))*tan(f*x)^
5*tan(e)^5 + 120*a*tan(f*x)^6*tan(e)^5 - 840*b*tan(f*x)^6*tan(e)^5 - 63*pi*b*sgn(2*tan(f*x)^2*tan(e)^2 - 2)*sg
n(-2*tan(f*x)^2*tan(e) + 2*tan(f*x)*tan(e)^2 + 2*tan(f*x) - 2*tan(e))*tan(f*x)^2*tan(e)^6 - 126*b*arctan((tan(
f*x) + tan(e))/(tan(f*x)*tan(e) - 1))*tan(f*x)^4*tan(e)^6 + 126*b*arctan(-(tan(f*x) - tan(e))/(tan(f*x)*tan(e)
+ 1))*tan(f*x)^4*tan(e)^6 + 120*a*tan(f*x)^5*tan(e)^6 - 840*b*tan(f*x)^5*tan(e)^6 + 21*pi*b*sgn(2*tan(f*x)^2*
tan(e)^2 - 2)*sgn(-2*tan(f*x)^2*tan(e) + 2*tan(f*x)*tan(e)^2 + 2*tan(f*x) - 2*tan(e))*tan(f*x)*tan(e)^7 + 126*
b*arctan((tan(f*x) + tan(e))/(tan(f*x)*tan(e) - 1))*tan(f*x)^3*tan(e)^7 - 126*b*arctan(-(tan(f*x) - tan(e))/(t
an(f*x)*tan(e) + 1))*tan(f*x)^3*tan(e)^7 + 160*a*tan(f*x)^4*tan(e)^7 - 1120*b*tan(f*x)^4*tan(e)^7 + 60*a*f*x*t
an(f*x)^7*tan(e) - 420*b*f*x*tan(f*x)^7*tan(e) + 21*pi*b*sgn(-2*tan(f*x)^2*tan(e) + 2*tan(f*x)*tan(e)^2 + 2*ta
n(f*x) - 2*tan(e))*tan(f*x)^7*tan(e) - 180*a*f*x*tan(f*x)^6*tan(e)^2 + 1260*b*f*x*tan(f*x)^6*tan(e)^2 - 63*pi*
b*sgn(-2*tan(f*x)^2*tan(e) + 2*tan(f*x)*tan(e)^2 + 2*tan(f*x) - 2*tan(e))*tan(f*x)^6*tan(e)^2 + 540*a*f*x*tan(
f*x)^5*tan(e)^3 - 3780*b*f*x*tan(f*x)^5*tan(e)^3 + 189*pi*b*sgn(-2*tan(f*x)^2*tan(e) + 2*tan(f*x)*tan(e)^2 + 2
*tan(f*x) - 2*tan(e))*tan(f*x)^5*tan(e)^3 - 540*a*f*x*tan(f*x)^4*tan(e)^4 + 3780*b*f*x*tan(f*x)^4*tan(e)^4 - 1
89*pi*b*sgn(-2*tan(f*x)^2*tan(e) + 2*tan(f*x)*tan(e)^2 + 2*tan(f*x) - 2*tan(e))*tan(f*x)^4*tan(e)^4 + 540*a*f*
x*tan(f*x)^3*tan(e)^5 - 3780*b*f*x*tan(f*x)^3*tan(e)^5 + 189*pi*b*sgn(-2*tan(f*x)^2*tan(e) + 2*tan(f*x)*tan(e)
^2 + 2*tan(f*x) - 2*tan(e))*tan(f*x)^3*tan(e)^5 - 180*a*f*x*tan(f*x)^2*tan(e)^6 + 1260*b*f*x*tan(f*x)^2*tan(e)
^6 - 63*pi*b*sgn(-2*tan(f*x)^2*tan(e) + 2*tan(f*x)*tan(e)^2 + 2*tan(f*x) - 2*tan(e))*tan(f*x)^2*tan(e)^6 + 60*
a*f*x*tan(f*x)*tan(e)^7 - 420*b*f*x*tan(f*x)*tan(e)^7 + 21*pi*b*sgn(-2*tan(f*x)^2*tan(e) + 2*tan(f*x)*tan(e)^2
+ 2*tan(f*x) - 2*tan(e))*tan(f*x)*tan(e)^7 - 21*pi*b*sgn(2*tan(f*x)^2*tan(e)^2 - 2)*sgn(-2*tan(f*x)^2*tan(e)
+ 2*tan(f*x)*tan(e)^2 + 2*tan(f*x) - 2*tan(e))*tan(f*x)^6 + 63*pi*b*sgn(2*tan(f*x)^2*tan(e)^2 - 2)*sgn(-2*tan(
f*x)^2*tan(e) + 2*tan(f*x)*tan(e)^2 + 2*tan(f*x) - 2*tan(e))*tan(f*x)^5*tan(e) + 42*b*arctan((tan(f*x) + tan(e
))/(tan(f*x)*tan(e) - 1))*tan(f*x)^7*tan(e) - 42*b*arctan(-(tan(f*x) - tan(e))/(tan(f*x)*tan(e) + 1))*tan(f*x)
^7*tan(e) - 189*pi*b*sgn(2*tan(f*x)^2*tan(e)^2 - 2)*sgn(-2*tan(f*x)^2*tan(e) + 2*tan(f*x)*tan(e)^2 + 2*tan(f*x
) - 2*tan(e))*tan(f*x)^4*tan(e)^2 - 126*b*arctan((tan(f*x) + tan(e))/(tan(f*x)*tan(e) - 1))*tan(f*x)^6*tan(e)^
2 + 126*b*arctan(-(tan(f*x) - tan(e))/(tan(f*x)*tan(e) + 1))*tan(f*x)^6*tan(e)^2 + 132*a*tan(f*x)^7*tan(e)^2 -
924*b*tan(f*x)^7*tan(e)^2 + 189*pi*b*sgn(2*tan(f*x)^2*tan(e)^2 - 2)*sgn(-2*tan(f*x)^2*tan(e) + 2*tan(f*x)*tan
(e)^2 + 2*tan(f*x) - 2*tan(e))*tan(f*x)^3*tan(e)^3 + 378*b*arctan((tan(f*x) + tan(e))/(tan(f*x)*tan(e) - 1))*t
an(f*x)^5*tan(e)^3 - 378*b*arctan(-(tan(f*x) - tan(e))/(tan(f*x)*tan(e) + 1))*tan(f*x)^5*tan(e)^3 + 20*a*tan(f
*x)^6*tan(e)^3 - 140*b*tan(f*x)^6*tan(e)^3 - 189*pi*b*sgn(2*tan(f*x)^2*tan(e)^2 - 2)*sgn(-2*tan(f*x)^2*tan(e)
+ 2*tan(f*x)*tan(e)^2 + 2*tan(f*x) - 2*tan(e))*tan(f*x)^2*tan(e)^4 - 378*b*arctan((tan(f*x) + tan(e))/(tan(f*x
)*tan(e) - 1))*tan(f*x)^4*tan(e)^4 + 378*b*arctan(-(tan(f*x) - tan(e))/(tan(f*x)*tan(e) + 1))*tan(f*x)^4*tan(e
)^4 + 300*a*tan(f*x)^5*tan(e)^4 - 2100*b*tan(f*x)^5*tan(e)^4 + 63*pi*b*sgn(2*tan(f*x)^2*tan(e)^2 - 2)*sgn(-2*t
an(f*x)^2*tan(e) + 2*tan(f*x)*tan(e)^2 + 2*tan(f*x) - 2*tan(e))*tan(f*x)*tan(e)^5 + 378*b*arctan((tan(f*x) + t
an(e))/(tan(f*x)*tan(e) - 1))*tan(f*x)^3*tan(e)^5 - 378*b*arctan(-(tan(f*x) - tan(e))/(tan(f*x)*tan(e) + 1))*t
an(f*x)^3*tan(e)^5 + 300*a*tan(f*x)^4*tan(e)^5 - 2100*b*tan(f*x)^4*tan(e)^5 - 21*pi*b*sgn(2*tan(f*x)^2*tan(e)^
2 - 2)*sgn(-2*tan(f*x)^2*tan(e) + 2*tan(f*x)*tan(e)^2 + 2*tan(f*x) - 2*tan(e))*tan(e)^6 - 126*b*arctan((tan(f*
x) + tan(e))/(tan(f*x)*tan(e) - 1))*tan(f*x)^2*tan(e)^6 + 126*b*arctan(-(tan(f*x) - tan(e))/(tan(f*x)*tan(e) +
1))*tan(f*x)^2*tan(e)^6 + 20*a*tan(f*x)^3*tan(e)^6 - 140*b*tan(f*x)^3*tan(e)^6 + 42*b*arctan((tan(f*x) + tan(
e))/(tan(f*x)*tan(e) - 1))*tan(f*x)*tan(e)^7 - 42*b*arctan(-(tan(f*x) - tan(e))/(tan(f*x)*tan(e) + 1))*tan(f*x
)*tan(e)^7 + 132*a*tan(f*x)^2*tan(e)^7 - 924*b*tan(f*x)^2*tan(e)^7 - 60*a*f*x*tan(f*x)^6 + 420*b*f*x*tan(f*x)^
6 - 21*pi*b*sgn(-2*tan(f*x)^2*tan(e) + 2*tan(f*x)*tan(e)^2 + 2*tan(f*x) - 2*tan(e))*tan(f*x)^6 + 180*a*f*x*tan
(f*x)^5*tan(e) - 1260*b*f*x*tan(f*x)^5*tan(e) + 63*pi*b*sgn(-2*tan(f*x)^2*tan(e) + 2*tan(f*x)*tan(e)^2 + 2*tan
(f*x) - 2*tan(e))*tan(f*x)^5*tan(e) - 540*a*f*x*tan(f*x)^4*tan(e)^2 + 3780*b*f*x*tan(f*x)^4*tan(e)^2 - 189*pi*
b*sgn(-2*tan(f*x)^2*tan(e) + 2*tan(f*x)*tan(e)^2 + 2*tan(f*x) - 2*tan(e))*tan(f*x)^4*tan(e)^2 + 540*a*f*x*tan(
f*x)^3*tan(e)^3 - 3780*b*f*x*tan(f*x)^3*tan(e)^3 + 189*pi*b*sgn(-2*tan(f*x)^2*tan(e) + 2*tan(f*x)*tan(e)^2 + 2
*tan(f*x) - 2*tan(e))*tan(f*x)^3*tan(e)^3 - 540*a*f*x*tan(f*x)^2*tan(e)^4 + 3780*b*f*x*tan(f*x)^2*tan(e)^4 - 1
89*pi*b*sgn(-2*tan(f*x)^2*tan(e) + 2*tan(f*x)*tan(e)^2 + 2*tan(f*x) - 2*tan(e))*tan(f*x)^2*tan(e)^4 + 180*a*f*
x*tan(f*x)*tan(e)^5 - 1260*b*f*x*tan(f*x)*tan(e)^5 + 63*pi*b*sgn(-2*tan(f*x)^2*tan(e) + 2*tan(f*x)*tan(e)^2 +
2*tan(f*x) - 2*tan(e))*tan(f*x)*tan(e)^5 - 60*a*f*x*tan(e)^6 + 420*b*f*x*tan(e)^6 - 21*pi*b*sgn(-2*tan(f*x)^2*
tan(e) + 2*tan(f*x)*tan(e)^2 + 2*tan(f*x) - 2*tan(e))*tan(e)^6 - 63*pi*b*sgn(2*tan(f*x)^2*tan(e)^2 - 2)*sgn(-2
*tan(f*x)^2*tan(e) + 2*tan(f*x)*tan(e)^2 + 2*tan(f*x) - 2*tan(e))*tan(f*x)^4 - 42*b*arctan((tan(f*x) + tan(e))
/(tan(f*x)*tan(e) - 1))*tan(f*x)^6 + 42*b*arctan(-(tan(f*x) - tan(e))/(tan(f*x)*tan(e) + 1))*tan(f*x)^6 - 192*
b*tan(f*x)^7 + 63*pi*b*sgn(2*tan(f*x)^2*tan(e)^2 - 2)*sgn(-2*tan(f*x)^2*tan(e) + 2*tan(f*x)*tan(e)^2 + 2*tan(f
*x) - 2*tan(e))*tan(f*x)^3*tan(e) + 126*b*arctan((tan(f*x) + tan(e))/(tan(f*x)*tan(e) - 1))*tan(f*x)^5*tan(e)
- 126*b*arctan(-(tan(f*x) - tan(e))/(tan(f*x)*tan(e) + 1))*tan(f*x)^5*tan(e) - 264*a*tan(f*x)^6*tan(e) + 504*b
*tan(f*x)^6*tan(e) - 189*pi*b*sgn(2*tan(f*x)^2*tan(e)^2 - 2)*sgn(-2*tan(f*x)^2*tan(e) + 2*tan(f*x)*tan(e)^2 +
2*tan(f*x) - 2*tan(e))*tan(f*x)^2*tan(e)^2 - 378*b*arctan((tan(f*x) + tan(e))/(tan(f*x)*tan(e) - 1))*tan(f*x)^
4*tan(e)^2 + 378*b*arctan(-(tan(f*x) - tan(e))/(tan(f*x)*tan(e) + 1))*tan(f*x)^4*tan(e)^2 - 360*a*tan(f*x)^5*t
an(e)^2 - 1512*b*tan(f*x)^5*tan(e)^2 + 63*pi*b*sgn(2*tan(f*x)^2*tan(e)^2 - 2)*sgn(-2*tan(f*x)^2*tan(e) + 2*tan
(f*x)*tan(e)^2 + 2*tan(f*x) - 2*tan(e))*tan(f*x)*tan(e)^3 + 378*b*arctan((tan(f*x) + tan(e))/(tan(f*x)*tan(e)
- 1))*tan(f*x)^3*tan(e)^3 - 378*b*arctan(-(tan(f*x) - tan(e))/(tan(f*x)*tan(e) + 1))*tan(f*x)^3*tan(e)^3 - 960
*a*tan(f*x)^4*tan(e)^3 - 63*pi*b*sgn(2*tan(f*x)^2*tan(e)^2 - 2)*sgn(-2*tan(f*x)^2*tan(e) + 2*tan(f*x)*tan(e)^2
+ 2*tan(f*x) - 2*tan(e))*tan(e)^4 - 378*b*arctan((tan(f*x) + tan(e))/(tan(f*x)*tan(e) - 1))*tan(f*x)^2*tan(e)
^4 + 378*b*arctan(-(tan(f*x) - tan(e))/(tan(f*x)*tan(e) + 1))*tan(f*x)^2*tan(e)^4 - 960*a*tan(f*x)^3*tan(e)^4
+ 126*b*arctan((tan(f*x) + tan(e))/(tan(f*x)*tan(e) - 1))*tan(f*x)*tan(e)^5 - 126*b*arctan(-(tan(f*x) - tan(e)
)/(tan(f*x)*tan(e) + 1))*tan(f*x)*tan(e)^5 - 360*a*tan(f*x)^2*tan(e)^5 - 1512*b*tan(f*x)^2*tan(e)^5 - 42*b*arc
tan((tan(f*x) + tan(e))/(tan(f*x)*tan(e) - 1))*tan(e)^6 + 42*b*arctan(-(tan(f*x) - tan(e))/(tan(f*x)*tan(e) +
1))*tan(e)^6 - 264*a*tan(f*x)*tan(e)^6 + 504*b*tan(f*x)*tan(e)^6 - 192*b*tan(e)^7 - 180*a*f*x*tan(f*x)^4 + 126
0*b*f*x*tan(f*x)^4 - 63*pi*b*sgn(-2*tan(f*x)^2*tan(e) + 2*tan(f*x)*tan(e)^2 + 2*tan(f*x) - 2*tan(e))*tan(f*x)^
4 + 180*a*f*x*tan(f*x)^3*tan(e) - 1260*b*f*x*tan(f*x)^3*tan(e) + 63*pi*b*sgn(-2*tan(f*x)^2*tan(e) + 2*tan(f*x)
*tan(e)^2 + 2*tan(f*x) - 2*tan(e))*tan(f*x)^3*tan(e) - 540*a*f*x*tan(f*x)^2*tan(e)^2 + 3780*b*f*x*tan(f*x)^2*t
an(e)^2 - 189*pi*b*sgn(-2*tan(f*x)^2*tan(e) + 2*tan(f*x)*tan(e)^2 + 2*tan(f*x) - 2*tan(e))*tan(f*x)^2*tan(e)^2
+ 180*a*f*x*tan(f*x)*tan(e)^3 - 1260*b*f*x*tan(f*x)*tan(e)^3 + 63*pi*b*sgn(-2*tan(f*x)^2*tan(e) + 2*tan(f*x)*
tan(e)^2 + 2*tan(f*x) - 2*tan(e))*tan(f*x)*tan(e)^3 - 180*a*f*x*tan(e)^4 + 1260*b*f*x*tan(e)^4 - 63*pi*b*sgn(-
2*tan(f*x)^2*tan(e) + 2*tan(f*x)*tan(e)^2 + 2*tan(f*x) - 2*tan(e))*tan(e)^4 - 63*pi*b*sgn(2*tan(f*x)^2*tan(e)^
2 - 2)*sgn(-2*tan(f*x)^2*tan(e) + 2*tan(f*x)*tan(e)^2 + 2*tan(f*x) - 2*tan(e))*tan(f*x)^2 - 126*b*arctan((tan(
f*x) + tan(e))/(tan(f*x)*tan(e) - 1))*tan(f*x)^4 + 126*b*arctan(-(tan(f*x) - tan(e))/(tan(f*x)*tan(e) + 1))*ta
n(f*x)^4 + 132*a*tan(f*x)^5 - 924*b*tan(f*x)^5 + 21*pi*b*sgn(2*tan(f*x)^2*tan(e)^2 - 2)*sgn(-2*tan(f*x)^2*tan(
e) + 2*tan(f*x)*tan(e)^2 + 2*tan(f*x) - 2*tan(e))*tan(f*x)*tan(e) + 126*b*arctan((tan(f*x) + tan(e))/(tan(f*x)
*tan(e) - 1))*tan(f*x)^3*tan(e) - 126*b*arctan(-(tan(f*x) - tan(e))/(tan(f*x)*tan(e) + 1))*tan(f*x)^3*tan(e) +
20*a*tan(f*x)^4*tan(e) - 140*b*tan(f*x)^4*tan(e) - 63*pi*b*sgn(2*tan(f*x)^2*tan(e)^2 - 2)*sgn(-2*tan(f*x)^2*t
an(e) + 2*tan(f*x)*tan(e)^2 + 2*tan(f*x) - 2*tan(e))*tan(e)^2 - 378*b*arctan((tan(f*x) + tan(e))/(tan(f*x)*tan
(e) - 1))*tan(f*x)^2*tan(e)^2 + 378*b*arctan(-(tan(f*x) - tan(e))/(tan(f*x)*tan(e) + 1))*tan(f*x)^2*tan(e)^2 +
300*a*tan(f*x)^3*tan(e)^2 - 2100*b*tan(f*x)^3*tan(e)^2 + 126*b*arctan((tan(f*x) + tan(e))/(tan(f*x)*tan(e) -
1))*tan(f*x)*tan(e)^3 - 126*b*arctan(-(tan(f*x) - tan(e))/(tan(f*x)*tan(e) + 1))*tan(f*x)*tan(e)^3 + 300*a*tan
(f*x)^2*tan(e)^3 - 2100*b*tan(f*x)^2*tan(e)^3 - 126*b*arctan((tan(f*x) + tan(e))/(tan(f*x)*tan(e) - 1))*tan(e)
^4 + 126*b*arctan(-(tan(f*x) - tan(e))/(tan(f*x)*tan(e) + 1))*tan(e)^4 + 20*a*tan(f*x)*tan(e)^4 - 140*b*tan(f*
x)*tan(e)^4 + 132*a*tan(e)^5 - 924*b*tan(e)^5 - 180*a*f*x*tan(f*x)^2 + 1260*b*f*x*tan(f*x)^2 - 63*pi*b*sgn(-2*
tan(f*x)^2*tan(e) + 2*tan(f*x)*tan(e)^2 + 2*tan(f*x) - 2*tan(e))*tan(f*x)^2 + 60*a*f*x*tan(f*x)*tan(e) - 420*b
*f*x*tan(f*x)*tan(e) + 21*pi*b*sgn(-2*tan(f*x)^2*tan(e) + 2*tan(f*x)*tan(e)^2 + 2*tan(f*x) - 2*tan(e))*tan(f*x
)*tan(e) - 180*a*f*x*tan(e)^2 + 1260*b*f*x*tan(e)^2 - 63*pi*b*sgn(-2*tan(f*x)^2*tan(e) + 2*tan(f*x)*tan(e)^2 +
2*tan(f*x) - 2*tan(e))*tan(e)^2 - 21*pi*b*sgn(2*tan(f*x)^2*tan(e)^2 - 2)*sgn(-2*tan(f*x)^2*tan(e) + 2*tan(f*x
)*tan(e)^2 + 2*tan(f*x) - 2*tan(e)) - 126*b*arctan((tan(f*x) + tan(e))/(tan(f*x)*tan(e) - 1))*tan(f*x)^2 + 126
*b*arctan(-(tan(f*x) - tan(e))/(tan(f*x)*tan(e) + 1))*tan(f*x)^2 + 160*a*tan(f*x)^3 - 1120*b*tan(f*x)^3 + 42*b
*arctan((tan(f*x) + tan(e))/(tan(f*x)*tan(e) - 1))*tan(f*x)*tan(e) - 42*b*arctan(-(tan(f*x) - tan(e))/(tan(f*x
)*tan(e) + 1))*tan(f*x)*tan(e) + 120*a*tan(f*x)^2*tan(e) - 840*b*tan(f*x)^2*tan(e) - 126*b*arctan((tan(f*x) +
tan(e))/(tan(f*x)*tan(e) - 1))*tan(e)^2 + 126*b*arctan(-(tan(f*x) - tan(e))/(tan(f*x)*tan(e) + 1))*tan(e)^2 +
120*a*tan(f*x)*tan(e)^2 - 840*b*tan(f*x)*tan(e)^2 + 160*a*tan(e)^3 - 1120*b*tan(e)^3 - 60*a*f*x + 420*b*f*x -
21*pi*b*sgn(-2*tan(f*x)^2*tan(e) + 2*tan(f*x)*tan(e)^2 + 2*tan(f*x) - 2*tan(e)) - 42*b*arctan((tan(f*x) + tan(
e))/(tan(f*x)*tan(e) - 1)) + 42*b*arctan(-(tan(f*x) - tan(e))/(tan(f*x)*tan(e) + 1)) + 60*a*tan(f*x) - 420*b*t
an(f*x) + 60*a*tan(e) - 420*b*tan(e))/(f*tan(f*x)^7*tan(e)^7 + 3*f*tan(f*x)^7*tan(e)^5 - f*tan(f*x)^6*tan(e)^6
+ 3*f*tan(f*x)^5*tan(e)^7 + 3*f*tan(f*x)^7*tan(e)^3 - 3*f*tan(f*x)^6*tan(e)^4 + 9*f*tan(f*x)^5*tan(e)^5 - 3*f
*tan(f*x)^4*tan(e)^6 + 3*f*tan(f*x)^3*tan(e)^7 + f*tan(f*x)^7*tan(e) - 3*f*tan(f*x)^6*tan(e)^2 + 9*f*tan(f*x)^
5*tan(e)^3 - 9*f*tan(f*x)^4*tan(e)^4 + 9*f*tan(f*x)^3*tan(e)^5 - 3*f*tan(f*x)^2*tan(e)^6 + f*tan(f*x)*tan(e)^7
- f*tan(f*x)^6 + 3*f*tan(f*x)^5*tan(e) - 9*f*tan(f*x)^4*tan(e)^2 + 9*f*tan(f*x)^3*tan(e)^3 - 9*f*tan(f*x)^2*t
an(e)^4 + 3*f*tan(f*x)*tan(e)^5 - f*tan(e)^6 - 3*f*tan(f*x)^4 + 3*f*tan(f*x)^3*tan(e) - 9*f*tan(f*x)^2*tan(e)^
2 + 3*f*tan(f*x)*tan(e)^3 - 3*f*tan(e)^4 - 3*f*tan(f*x)^2 + f*tan(f*x)*tan(e) - 3*f*tan(e)^2 - f)