\(\int \csc ^6(c+d x) (a+b \sec (c+d x))^3 \, dx\) [195]

Optimal result

Integrand size = 21, antiderivative size = 279 \[ \int \csc ^6(c+d x) (a+b \sec (c+d x))^3 \, dx=\frac {3 a^2 b \text {arctanh}(\sin (c+d x))}{d}+\frac {7 b^3 \text {arctanh}(\sin (c+d x))}{2 d}-\frac {a^3 \cot (c+d x)}{d}-\frac {9 a b^2 \cot (c+d x)}{d}-\frac {2 a^3 \cot ^3(c+d x)}{3 d}-\frac {3 a b^2 \cot ^3(c+d x)}{d}-\frac {a^3 \cot ^5(c+d x)}{5 d}-\frac {3 a b^2 \cot ^5(c+d x)}{5 d}-\frac {3 a^2 b \csc (c+d x)}{d}-\frac {7 b^3 \csc (c+d x)}{2 d}-\frac {a^2 b \csc ^3(c+d x)}{d}-\frac {7 b^3 \csc ^3(c+d x)}{6 d}-\frac {3 a^2 b \csc ^5(c+d x)}{5 d}-\frac {7 b^3 \csc ^5(c+d x)}{10 d}+\frac {b^3 \csc ^5(c+d x) \sec ^2(c+d x)}{2 d}+\frac {3 a b^2 \tan (c+d x)}{d} \]

[Out]

3*a^2*b*arctanh(sin(d*x+c))/d+7/2*b^3*arctanh(sin(d*x+c))/d-a^3*cot(d*x+c)/d-9*a*b^2*cot(d*x+c)/d-2/3*a^3*cot(
d*x+c)^3/d-3*a*b^2*cot(d*x+c)^3/d-1/5*a^3*cot(d*x+c)^5/d-3/5*a*b^2*cot(d*x+c)^5/d-3*a^2*b*csc(d*x+c)/d-7/2*b^3
*csc(d*x+c)/d-a^2*b*csc(d*x+c)^3/d-7/6*b^3*csc(d*x+c)^3/d-3/5*a^2*b*csc(d*x+c)^5/d-7/10*b^3*csc(d*x+c)^5/d+1/2
*b^3*csc(d*x+c)^5*sec(d*x+c)^2/d+3*a*b^2*tan(d*x+c)/d

Rubi [A] (verified)

Time = 0.39 (sec) , antiderivative size = 279, normalized size of antiderivative = 1.00, number of steps used = 17, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.429, Rules used = {3957, 2991, 3852, 2701, 308, 213, 2700, 276, 294} \[ \int \csc ^6(c+d x) (a+b \sec (c+d x))^3 \, dx=-\frac {a^3 \cot ^5(c+d x)}{5 d}-\frac {2 a^3 \cot ^3(c+d x)}{3 d}-\frac {a^3 \cot (c+d x)}{d}+\frac {3 a^2 b \text {arctanh}(\sin (c+d x))}{d}-\frac {3 a^2 b \csc ^5(c+d x)}{5 d}-\frac {a^2 b \csc ^3(c+d x)}{d}-\frac {3 a^2 b \csc (c+d x)}{d}+\frac {3 a b^2 \tan (c+d x)}{d}-\frac {3 a b^2 \cot ^5(c+d x)}{5 d}-\frac {3 a b^2 \cot ^3(c+d x)}{d}-\frac {9 a b^2 \cot (c+d x)}{d}+\frac {7 b^3 \text {arctanh}(\sin (c+d x))}{2 d}-\frac {7 b^3 \csc ^5(c+d x)}{10 d}-\frac {7 b^3 \csc ^3(c+d x)}{6 d}-\frac {7 b^3 \csc (c+d x)}{2 d}+\frac {b^3 \csc ^5(c+d x) \sec ^2(c+d x)}{2 d} \]

[In]

Int[Csc[c + d*x]^6*(a + b*Sec[c + d*x])^3,x]

[Out]

(3*a^2*b*ArcTanh[Sin[c + d*x]])/d + (7*b^3*ArcTanh[Sin[c + d*x]])/(2*d) - (a^3*Cot[c + d*x])/d - (9*a*b^2*Cot[
c + d*x])/d - (2*a^3*Cot[c + d*x]^3)/(3*d) - (3*a*b^2*Cot[c + d*x]^3)/d - (a^3*Cot[c + d*x]^5)/(5*d) - (3*a*b^
2*Cot[c + d*x]^5)/(5*d) - (3*a^2*b*Csc[c + d*x])/d - (7*b^3*Csc[c + d*x])/(2*d) - (a^2*b*Csc[c + d*x]^3)/d - (
7*b^3*Csc[c + d*x]^3)/(6*d) - (3*a^2*b*Csc[c + d*x]^5)/(5*d) - (7*b^3*Csc[c + d*x]^5)/(10*d) + (b^3*Csc[c + d*
x]^5*Sec[c + d*x]^2)/(2*d) + (3*a*b^2*Tan[c + d*x])/d

Rule 213

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(-(Rt[-a, 2]*Rt[b, 2])^(-1))*ArcTanh[Rt[b, 2]*(x/Rt[-a, 2])]
, x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (LtQ[a, 0] || GtQ[b, 0])

Rule 276

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*(a + b*x^n)^p,
x], x] /; FreeQ[{a, b, c, m, n}, x] && IGtQ[p, 0]

Rule 294

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[c^(n - 1)*(c*x)^(m - n + 1)*((a + b*x^
n)^(p + 1)/(b*n*(p + 1))), x] - Dist[c^n*((m - n + 1)/(b*n*(p + 1))), Int[(c*x)^(m - n)*(a + b*x^n)^(p + 1), x
], x] /; FreeQ[{a, b, c}, x] && IGtQ[n, 0] && LtQ[p, -1] && GtQ[m + 1, n] &&  !ILtQ[(m + n*(p + 1) + 1)/n, 0]
&& IntBinomialQ[a, b, c, n, m, p, x]

Rule 308

Int[(x_)^(m_)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> Int[PolynomialDivide[x^m, a + b*x^n, x], x] /; FreeQ[{a,
b}, x] && IGtQ[m, 0] && IGtQ[n, 0] && GtQ[m, 2*n - 1]

Rule 2700

Int[csc[(e_.) + (f_.)*(x_)]^(m_.)*sec[(e_.) + (f_.)*(x_)]^(n_.), x_Symbol] :> Dist[1/f, Subst[Int[(1 + x^2)^((
m + n)/2 - 1)/x^m, x], x, Tan[e + f*x]], x] /; FreeQ[{e, f}, x] && IntegersQ[m, n, (m + n)/2]

Rule 2701

Int[(csc[(e_.) + (f_.)*(x_)]*(a_.))^(m_)*sec[(e_.) + (f_.)*(x_)]^(n_.), x_Symbol] :> Dist[-(f*a^n)^(-1), Subst
[Int[x^(m + n - 1)/(-1 + x^2/a^2)^((n + 1)/2), x], x, a*Csc[e + f*x]], x] /; FreeQ[{a, e, f, m}, x] && Integer
Q[(n + 1)/2] &&  !(IntegerQ[(m + 1)/2] && LtQ[0, m, n])

Rule 2991

Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((d_.)*sin[(e_.) + (f_.)*(x_)])^(n_)*((a_) + (b_.)*sin[(e_.) + (f_.)*
(x_)])^(m_), x_Symbol] :> Int[ExpandTrig[(g*cos[e + f*x])^p, (d*sin[e + f*x])^n*(a + b*sin[e + f*x])^m, x], x]
/; FreeQ[{a, b, d, e, f, g, n, p}, x] && NeQ[a^2 - b^2, 0] && IntegerQ[m] && (GtQ[m, 0] || IntegerQ[n])

Rule 3852

Int[csc[(c_.) + (d_.)*(x_)]^(n_), x_Symbol] :> Dist[-d^(-1), Subst[Int[ExpandIntegrand[(1 + x^2)^(n/2 - 1), x]
, x], x, Cot[c + d*x]], x] /; FreeQ[{c, d}, x] && IGtQ[n/2, 0]

Rule 3957

Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_.)*(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_))^(m_.), x_Symbol] :> Int[(g*Co
s[e + f*x])^p*((b + a*Sin[e + f*x])^m/Sin[e + f*x]^m), x] /; FreeQ[{a, b, e, f, g, p}, x] && IntegerQ[m]

Rubi steps \begin{align*} \text {integral}& = -\int (-b-a \cos (c+d x))^3 \csc ^6(c+d x) \sec ^3(c+d x) \, dx \\ & = \int \left (a^3 \csc ^6(c+d x)+3 a^2 b \csc ^6(c+d x) \sec (c+d x)+3 a b^2 \csc ^6(c+d x) \sec ^2(c+d x)+b^3 \csc ^6(c+d x) \sec ^3(c+d x)\right ) \, dx \\ & = a^3 \int \csc ^6(c+d x) \, dx+\left (3 a^2 b\right ) \int \csc ^6(c+d x) \sec (c+d x) \, dx+\left (3 a b^2\right ) \int \csc ^6(c+d x) \sec ^2(c+d x) \, dx+b^3 \int \csc ^6(c+d x) \sec ^3(c+d x) \, dx \\ & = -\frac {a^3 \text {Subst}\left (\int \left (1+2 x^2+x^4\right ) \, dx,x,\cot (c+d x)\right )}{d}-\frac {\left (3 a^2 b\right ) \text {Subst}\left (\int \frac {x^6}{-1+x^2} \, dx,x,\csc (c+d x)\right )}{d}+\frac {\left (3 a b^2\right ) \text {Subst}\left (\int \frac {\left (1+x^2\right )^3}{x^6} \, dx,x,\tan (c+d x)\right )}{d}-\frac {b^3 \text {Subst}\left (\int \frac {x^8}{\left (-1+x^2\right )^2} \, dx,x,\csc (c+d x)\right )}{d} \\ & = -\frac {a^3 \cot (c+d x)}{d}-\frac {2 a^3 \cot ^3(c+d x)}{3 d}-\frac {a^3 \cot ^5(c+d x)}{5 d}+\frac {b^3 \csc ^5(c+d x) \sec ^2(c+d x)}{2 d}-\frac {\left (3 a^2 b\right ) \text {Subst}\left (\int \left (1+x^2+x^4+\frac {1}{-1+x^2}\right ) \, dx,x,\csc (c+d x)\right )}{d}+\frac {\left (3 a b^2\right ) \text {Subst}\left (\int \left (1+\frac {1}{x^6}+\frac {3}{x^4}+\frac {3}{x^2}\right ) \, dx,x,\tan (c+d x)\right )}{d}-\frac {\left (7 b^3\right ) \text {Subst}\left (\int \frac {x^6}{-1+x^2} \, dx,x,\csc (c+d x)\right )}{2 d} \\ & = -\frac {a^3 \cot (c+d x)}{d}-\frac {9 a b^2 \cot (c+d x)}{d}-\frac {2 a^3 \cot ^3(c+d x)}{3 d}-\frac {3 a b^2 \cot ^3(c+d x)}{d}-\frac {a^3 \cot ^5(c+d x)}{5 d}-\frac {3 a b^2 \cot ^5(c+d x)}{5 d}-\frac {3 a^2 b \csc (c+d x)}{d}-\frac {a^2 b \csc ^3(c+d x)}{d}-\frac {3 a^2 b \csc ^5(c+d x)}{5 d}+\frac {b^3 \csc ^5(c+d x) \sec ^2(c+d x)}{2 d}+\frac {3 a b^2 \tan (c+d x)}{d}-\frac {\left (3 a^2 b\right ) \text {Subst}\left (\int \frac {1}{-1+x^2} \, dx,x,\csc (c+d x)\right )}{d}-\frac {\left (7 b^3\right ) \text {Subst}\left (\int \left (1+x^2+x^4+\frac {1}{-1+x^2}\right ) \, dx,x,\csc (c+d x)\right )}{2 d} \\ & = \frac {3 a^2 b \text {arctanh}(\sin (c+d x))}{d}-\frac {a^3 \cot (c+d x)}{d}-\frac {9 a b^2 \cot (c+d x)}{d}-\frac {2 a^3 \cot ^3(c+d x)}{3 d}-\frac {3 a b^2 \cot ^3(c+d x)}{d}-\frac {a^3 \cot ^5(c+d x)}{5 d}-\frac {3 a b^2 \cot ^5(c+d x)}{5 d}-\frac {3 a^2 b \csc (c+d x)}{d}-\frac {7 b^3 \csc (c+d x)}{2 d}-\frac {a^2 b \csc ^3(c+d x)}{d}-\frac {7 b^3 \csc ^3(c+d x)}{6 d}-\frac {3 a^2 b \csc ^5(c+d x)}{5 d}-\frac {7 b^3 \csc ^5(c+d x)}{10 d}+\frac {b^3 \csc ^5(c+d x) \sec ^2(c+d x)}{2 d}+\frac {3 a b^2 \tan (c+d x)}{d}-\frac {\left (7 b^3\right ) \text {Subst}\left (\int \frac {1}{-1+x^2} \, dx,x,\csc (c+d x)\right )}{2 d} \\ & = \frac {3 a^2 b \text {arctanh}(\sin (c+d x))}{d}+\frac {7 b^3 \text {arctanh}(\sin (c+d x))}{2 d}-\frac {a^3 \cot (c+d x)}{d}-\frac {9 a b^2 \cot (c+d x)}{d}-\frac {2 a^3 \cot ^3(c+d x)}{3 d}-\frac {3 a b^2 \cot ^3(c+d x)}{d}-\frac {a^3 \cot ^5(c+d x)}{5 d}-\frac {3 a b^2 \cot ^5(c+d x)}{5 d}-\frac {3 a^2 b \csc (c+d x)}{d}-\frac {7 b^3 \csc (c+d x)}{2 d}-\frac {a^2 b \csc ^3(c+d x)}{d}-\frac {7 b^3 \csc ^3(c+d x)}{6 d}-\frac {3 a^2 b \csc ^5(c+d x)}{5 d}-\frac {7 b^3 \csc ^5(c+d x)}{10 d}+\frac {b^3 \csc ^5(c+d x) \sec ^2(c+d x)}{2 d}+\frac {3 a b^2 \tan (c+d x)}{d} \\ \end{align*}

Mathematica [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(812\) vs. \(2(279)=558\).

Time = 2.16 (sec) , antiderivative size = 812, normalized size of antiderivative = 2.91 \[ \int \csc ^6(c+d x) (a+b \sec (c+d x))^3 \, dx=-\frac {\csc ^9\left (\frac {1}{2} (c+d x)\right ) \sec ^5\left (\frac {1}{2} (c+d x)\right ) \left (1176 a^2 b+412 b^3+80 a \left (5 a^2+18 b^2\right ) \cos (c+d x)+66 \left (6 a^2 b+7 b^3\right ) \cos (2 (c+d x))+16 a^3 \cos (3 (c+d x))+288 a b^2 \cos (3 (c+d x))-600 a^2 b \cos (4 (c+d x))-700 b^3 \cos (4 (c+d x))-48 a^3 \cos (5 (c+d x))-864 a b^2 \cos (5 (c+d x))+180 a^2 b \cos (6 (c+d x))+210 b^3 \cos (6 (c+d x))+16 a^3 \cos (7 (c+d x))+288 a b^2 \cos (7 (c+d x))+450 a^2 b \log \left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right ) \sin (c+d x)+525 b^3 \log \left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right ) \sin (c+d x)-450 a^2 b \log \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right ) \sin (c+d x)-525 b^3 \log \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right ) \sin (c+d x)+90 a^2 b \log \left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right ) \sin (3 (c+d x))+105 b^3 \log \left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right ) \sin (3 (c+d x))-90 a^2 b \log \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right ) \sin (3 (c+d x))-105 b^3 \log \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right ) \sin (3 (c+d x))-270 a^2 b \log \left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right ) \sin (5 (c+d x))-315 b^3 \log \left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right ) \sin (5 (c+d x))+270 a^2 b \log \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right ) \sin (5 (c+d x))+315 b^3 \log \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right ) \sin (5 (c+d x))+90 a^2 b \log \left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right ) \sin (7 (c+d x))+105 b^3 \log \left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right ) \sin (7 (c+d x))-90 a^2 b \log \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right ) \sin (7 (c+d x))-105 b^3 \log \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right ) \sin (7 (c+d x))\right )}{61440 d \left (-1+\cot ^2\left (\frac {1}{2} (c+d x)\right )\right )^2} \]

[In]

Integrate[Csc[c + d*x]^6*(a + b*Sec[c + d*x])^3,x]

[Out]

-1/61440*(Csc[(c + d*x)/2]^9*Sec[(c + d*x)/2]^5*(1176*a^2*b + 412*b^3 + 80*a*(5*a^2 + 18*b^2)*Cos[c + d*x] + 6
6*(6*a^2*b + 7*b^3)*Cos[2*(c + d*x)] + 16*a^3*Cos[3*(c + d*x)] + 288*a*b^2*Cos[3*(c + d*x)] - 600*a^2*b*Cos[4*
(c + d*x)] - 700*b^3*Cos[4*(c + d*x)] - 48*a^3*Cos[5*(c + d*x)] - 864*a*b^2*Cos[5*(c + d*x)] + 180*a^2*b*Cos[6
*(c + d*x)] + 210*b^3*Cos[6*(c + d*x)] + 16*a^3*Cos[7*(c + d*x)] + 288*a*b^2*Cos[7*(c + d*x)] + 450*a^2*b*Log[
Cos[(c + d*x)/2] - Sin[(c + d*x)/2]]*Sin[c + d*x] + 525*b^3*Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]]*Sin[c + d
*x] - 450*a^2*b*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]]*Sin[c + d*x] - 525*b^3*Log[Cos[(c + d*x)/2] + Sin[(c
+ d*x)/2]]*Sin[c + d*x] + 90*a^2*b*Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]]*Sin[3*(c + d*x)] + 105*b^3*Log[Cos
[(c + d*x)/2] - Sin[(c + d*x)/2]]*Sin[3*(c + d*x)] - 90*a^2*b*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]]*Sin[3*(
c + d*x)] - 105*b^3*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]]*Sin[3*(c + d*x)] - 270*a^2*b*Log[Cos[(c + d*x)/2]
- Sin[(c + d*x)/2]]*Sin[5*(c + d*x)] - 315*b^3*Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]]*Sin[5*(c + d*x)] + 27
0*a^2*b*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]]*Sin[5*(c + d*x)] + 315*b^3*Log[Cos[(c + d*x)/2] + Sin[(c + d*
x)/2]]*Sin[5*(c + d*x)] + 90*a^2*b*Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]]*Sin[7*(c + d*x)] + 105*b^3*Log[Cos
[(c + d*x)/2] - Sin[(c + d*x)/2]]*Sin[7*(c + d*x)] - 90*a^2*b*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]]*Sin[7*(
c + d*x)] - 105*b^3*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]]*Sin[7*(c + d*x)]))/(d*(-1 + Cot[(c + d*x)/2]^2)^2
)

Maple [A] (verified)

Time = 1.87 (sec) , antiderivative size = 243, normalized size of antiderivative = 0.87

method	result	size
derivativedivides	\(\frac {a^{3} \left (-\frac {8}{15}-\frac {\csc \left (d x +c \right )^{4}}{5}-\frac {4 \csc \left (d x +c \right )^{2}}{15}\right ) \cot \left (d x +c \right )+3 a^{2} b \left (-\frac {1}{5 \sin \left (d x +c \right )^{5}}-\frac {1}{3 \sin \left (d x +c \right )^{3}}-\frac {1}{\sin \left (d x +c \right )}+\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )\right )+3 a \,b^{2} \left (-\frac {1}{5 \sin \left (d x +c \right )^{5} \cos \left (d x +c \right )}-\frac {2}{5 \sin \left (d x +c \right )^{3} \cos \left (d x +c \right )}+\frac {8}{5 \sin \left (d x +c \right ) \cos \left (d x +c \right )}-\frac {16 \cot \left (d x +c \right )}{5}\right )+b^{3} \left (-\frac {1}{5 \sin \left (d x +c \right )^{5} \cos \left (d x +c \right )^{2}}-\frac {7}{15 \sin \left (d x +c \right )^{3} \cos \left (d x +c \right )^{2}}+\frac {7}{6 \sin \left (d x +c \right ) \cos \left (d x +c \right )^{2}}-\frac {7}{2 \sin \left (d x +c \right )}+\frac {7 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2}\right )}{d}\)	\(243\)
default	\(\frac {a^{3} \left (-\frac {8}{15}-\frac {\csc \left (d x +c \right )^{4}}{5}-\frac {4 \csc \left (d x +c \right )^{2}}{15}\right ) \cot \left (d x +c \right )+3 a^{2} b \left (-\frac {1}{5 \sin \left (d x +c \right )^{5}}-\frac {1}{3 \sin \left (d x +c \right )^{3}}-\frac {1}{\sin \left (d x +c \right )}+\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )\right )+3 a \,b^{2} \left (-\frac {1}{5 \sin \left (d x +c \right )^{5} \cos \left (d x +c \right )}-\frac {2}{5 \sin \left (d x +c \right )^{3} \cos \left (d x +c \right )}+\frac {8}{5 \sin \left (d x +c \right ) \cos \left (d x +c \right )}-\frac {16 \cot \left (d x +c \right )}{5}\right )+b^{3} \left (-\frac {1}{5 \sin \left (d x +c \right )^{5} \cos \left (d x +c \right )^{2}}-\frac {7}{15 \sin \left (d x +c \right )^{3} \cos \left (d x +c \right )^{2}}+\frac {7}{6 \sin \left (d x +c \right ) \cos \left (d x +c \right )^{2}}-\frac {7}{2 \sin \left (d x +c \right )}+\frac {7 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2}\right )}{d}\)	\(243\)
parallelrisch	\(\frac {-46080 \left (1+\cos \left (2 d x +2 c \right )\right ) b \left (a^{2}+\frac {7 b^{2}}{6}\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )+46080 \left (1+\cos \left (2 d x +2 c \right )\right ) b \left (a^{2}+\frac {7 b^{2}}{6}\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )+24 \csc \left (\frac {d x}{2}+\frac {c}{2}\right )^{5} \sec \left (\frac {d x}{2}+\frac {c}{2}\right )^{5} \left (\left (-\frac {33}{4} a^{2} b -\frac {77}{8} b^{3}\right ) \cos \left (2 d x +2 c \right )+\left (-\frac {1}{3} a^{3}-6 a \,b^{2}\right ) \cos \left (3 d x +3 c \right )+\left (\frac {25}{2} a^{2} b +\frac {175}{12} b^{3}\right ) \cos \left (4 d x +4 c \right )+\left (a^{3}+18 a \,b^{2}\right ) \cos \left (5 d x +5 c \right )+\left (-\frac {15}{4} a^{2} b -\frac {35}{8} b^{3}\right ) \cos \left (6 d x +6 c \right )+\left (-\frac {1}{3} a^{3}-6 a \,b^{2}\right ) \cos \left (7 d x +7 c \right )+\left (-\frac {25}{3} a^{3}-30 a \,b^{2}\right ) \cos \left (d x +c \right )-\frac {49 a^{2} b}{2}-\frac {103 b^{3}}{12}\right )}{15360 d \left (1+\cos \left (2 d x +2 c \right )\right )}\)	\(275\)
norman	\(\frac {-\frac {a^{3}+3 a^{2} b +3 a \,b^{2}+b^{3}}{160 d}+\frac {\left (a^{3}-3 a^{2} b +3 a \,b^{2}-b^{3}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{14}}{160 d}+\frac {\left (19 a^{3}-87 a^{2} b +117 a \,b^{2}-49 b^{3}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{12}}{480 d}-\frac {\left (19 a^{3}+87 a^{2} b +117 a \,b^{2}+49 b^{3}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}{480 d}-\frac {\left (85 a^{3}-177 a^{2} b +1575 a \,b^{2}-259 b^{3}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{8}}{96 d}+\frac {\left (85 a^{3}+177 a^{2} b +1575 a \,b^{2}+259 b^{3}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{6}}{96 d}+\frac {\left (103 a^{3}-789 a^{2} b +1449 a \,b^{2}-763 b^{3}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{10}}{480 d}-\frac {\left (103 a^{3}+789 a^{2} b +1449 a \,b^{2}+763 b^{3}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}}{480 d}}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5} \left (-1+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right )^{2}}-\frac {b \left (6 a^{2}+7 b^{2}\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{2 d}+\frac {b \left (6 a^{2}+7 b^{2}\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{2 d}\)	\(383\)
risch	\(-\frac {i \left (90 a^{2} b \,{\mathrm e}^{13 i \left (d x +c \right )}+105 b^{3} {\mathrm e}^{13 i \left (d x +c \right )}-300 a^{2} b \,{\mathrm e}^{11 i \left (d x +c \right )}-350 b^{3} {\mathrm e}^{11 i \left (d x +c \right )}+198 a^{2} b \,{\mathrm e}^{9 i \left (d x +c \right )}+231 b^{3} {\mathrm e}^{9 i \left (d x +c \right )}+160 a^{3} {\mathrm e}^{8 i \left (d x +c \right )}+1176 a^{2} b \,{\mathrm e}^{7 i \left (d x +c \right )}+412 b^{3} {\mathrm e}^{7 i \left (d x +c \right )}+240 a^{3} {\mathrm e}^{6 i \left (d x +c \right )}+1440 a \,b^{2} {\mathrm e}^{6 i \left (d x +c \right )}+198 a^{2} b \,{\mathrm e}^{5 i \left (d x +c \right )}+231 b^{3} {\mathrm e}^{5 i \left (d x +c \right )}+16 a^{3} {\mathrm e}^{4 i \left (d x +c \right )}+288 a \,b^{2} {\mathrm e}^{4 i \left (d x +c \right )}-300 a^{2} b \,{\mathrm e}^{3 i \left (d x +c \right )}-350 b^{3} {\mathrm e}^{3 i \left (d x +c \right )}-48 a^{3} {\mathrm e}^{2 i \left (d x +c \right )}-864 a \,b^{2} {\mathrm e}^{2 i \left (d x +c \right )}+90 a^{2} b \,{\mathrm e}^{i \left (d x +c \right )}+105 b^{3} {\mathrm e}^{i \left (d x +c \right )}+16 a^{3}+288 a \,b^{2}\right )}{15 d \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )^{5} \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )^{2}}+\frac {3 b \ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right ) a^{2}}{d}+\frac {7 b^{3} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right )}{2 d}-\frac {3 b \ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right ) a^{2}}{d}-\frac {7 b^{3} \ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right )}{2 d}\)	\(436\)

[In]

int(csc(d*x+c)^6*(a+b*sec(d*x+c))^3,x,method=_RETURNVERBOSE)

[Out]

1/d*(a^3*(-8/15-1/5*csc(d*x+c)^4-4/15*csc(d*x+c)^2)*cot(d*x+c)+3*a^2*b*(-1/5/sin(d*x+c)^5-1/3/sin(d*x+c)^3-1/s
in(d*x+c)+ln(sec(d*x+c)+tan(d*x+c)))+3*a*b^2*(-1/5/sin(d*x+c)^5/cos(d*x+c)-2/5/sin(d*x+c)^3/cos(d*x+c)+8/5/sin
(d*x+c)/cos(d*x+c)-16/5*cot(d*x+c))+b^3*(-1/5/sin(d*x+c)^5/cos(d*x+c)^2-7/15/sin(d*x+c)^3/cos(d*x+c)^2+7/6/sin
(d*x+c)/cos(d*x+c)^2-7/2/sin(d*x+c)+7/2*ln(sec(d*x+c)+tan(d*x+c))))

Fricas [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 354, normalized size of antiderivative = 1.27 \[ \int \csc ^6(c+d x) (a+b \sec (c+d x))^3 \, dx=-\frac {32 \, {\left (a^{3} + 18 \, a b^{2}\right )} \cos \left (d x + c\right )^{7} + 30 \, {\left (6 \, a^{2} b + 7 \, b^{3}\right )} \cos \left (d x + c\right )^{6} - 80 \, {\left (a^{3} + 18 \, a b^{2}\right )} \cos \left (d x + c\right )^{5} - 70 \, {\left (6 \, a^{2} b + 7 \, b^{3}\right )} \cos \left (d x + c\right )^{4} - 180 \, a b^{2} \cos \left (d x + c\right ) + 60 \, {\left (a^{3} + 18 \, a b^{2}\right )} \cos \left (d x + c\right )^{3} - 30 \, b^{3} + 46 \, {\left (6 \, a^{2} b + 7 \, b^{3}\right )} \cos \left (d x + c\right )^{2} - 15 \, {\left ({\left (6 \, a^{2} b + 7 \, b^{3}\right )} \cos \left (d x + c\right )^{6} - 2 \, {\left (6 \, a^{2} b + 7 \, b^{3}\right )} \cos \left (d x + c\right )^{4} + {\left (6 \, a^{2} b + 7 \, b^{3}\right )} \cos \left (d x + c\right )^{2}\right )} \log \left (\sin \left (d x + c\right ) + 1\right ) \sin \left (d x + c\right ) + 15 \, {\left ({\left (6 \, a^{2} b + 7 \, b^{3}\right )} \cos \left (d x + c\right )^{6} - 2 \, {\left (6 \, a^{2} b + 7 \, b^{3}\right )} \cos \left (d x + c\right )^{4} + {\left (6 \, a^{2} b + 7 \, b^{3}\right )} \cos \left (d x + c\right )^{2}\right )} \log \left (-\sin \left (d x + c\right ) + 1\right ) \sin \left (d x + c\right )}{60 \, {\left (d \cos \left (d x + c\right )^{6} - 2 \, d \cos \left (d x + c\right )^{4} + d \cos \left (d x + c\right )^{2}\right )} \sin \left (d x + c\right )} \]

[In]

integrate(csc(d*x+c)^6*(a+b*sec(d*x+c))^3,x, algorithm="fricas")

[Out]

-1/60*(32*(a^3 + 18*a*b^2)*cos(d*x + c)^7 + 30*(6*a^2*b + 7*b^3)*cos(d*x + c)^6 - 80*(a^3 + 18*a*b^2)*cos(d*x
+ c)^5 - 70*(6*a^2*b + 7*b^3)*cos(d*x + c)^4 - 180*a*b^2*cos(d*x + c) + 60*(a^3 + 18*a*b^2)*cos(d*x + c)^3 - 3
0*b^3 + 46*(6*a^2*b + 7*b^3)*cos(d*x + c)^2 - 15*((6*a^2*b + 7*b^3)*cos(d*x + c)^6 - 2*(6*a^2*b + 7*b^3)*cos(d
*x + c)^4 + (6*a^2*b + 7*b^3)*cos(d*x + c)^2)*log(sin(d*x + c) + 1)*sin(d*x + c) + 15*((6*a^2*b + 7*b^3)*cos(d
*x + c)^6 - 2*(6*a^2*b + 7*b^3)*cos(d*x + c)^4 + (6*a^2*b + 7*b^3)*cos(d*x + c)^2)*log(-sin(d*x + c) + 1)*sin(
d*x + c))/((d*cos(d*x + c)^6 - 2*d*cos(d*x + c)^4 + d*cos(d*x + c)^2)*sin(d*x + c))

Sympy [F(-1)]

Timed out. \[ \int \csc ^6(c+d x) (a+b \sec (c+d x))^3 \, dx=\text {Timed out} \]

[In]

integrate(csc(d*x+c)**6*(a+b*sec(d*x+c))**3,x)

[Out]

Timed out

Maxima [A] (verification not implemented)

none

Time = 0.20 (sec) , antiderivative size = 230, normalized size of antiderivative = 0.82 \[ \int \csc ^6(c+d x) (a+b \sec (c+d x))^3 \, dx=-\frac {b^{3} {\left (\frac {2 \, {\left (105 \, \sin \left (d x + c\right )^{6} - 70 \, \sin \left (d x + c\right )^{4} - 14 \, \sin \left (d x + c\right )^{2} - 6\right )}}{\sin \left (d x + c\right )^{7} - \sin \left (d x + c\right )^{5}} - 105 \, \log \left (\sin \left (d x + c\right ) + 1\right ) + 105 \, \log \left (\sin \left (d x + c\right ) - 1\right )\right )} + 6 \, a^{2} b {\left (\frac {2 \, {\left (15 \, \sin \left (d x + c\right )^{4} + 5 \, \sin \left (d x + c\right )^{2} + 3\right )}}{\sin \left (d x + c\right )^{5}} - 15 \, \log \left (\sin \left (d x + c\right ) + 1\right ) + 15 \, \log \left (\sin \left (d x + c\right ) - 1\right )\right )} + 36 \, a b^{2} {\left (\frac {15 \, \tan \left (d x + c\right )^{4} + 5 \, \tan \left (d x + c\right )^{2} + 1}{\tan \left (d x + c\right )^{5}} - 5 \, \tan \left (d x + c\right )\right )} + \frac {4 \, {\left (15 \, \tan \left (d x + c\right )^{4} + 10 \, \tan \left (d x + c\right )^{2} + 3\right )} a^{3}}{\tan \left (d x + c\right )^{5}}}{60 \, d} \]

[In]

integrate(csc(d*x+c)^6*(a+b*sec(d*x+c))^3,x, algorithm="maxima")

[Out]

-1/60*(b^3*(2*(105*sin(d*x + c)^6 - 70*sin(d*x + c)^4 - 14*sin(d*x + c)^2 - 6)/(sin(d*x + c)^7 - sin(d*x + c)^
5) - 105*log(sin(d*x + c) + 1) + 105*log(sin(d*x + c) - 1)) + 6*a^2*b*(2*(15*sin(d*x + c)^4 + 5*sin(d*x + c)^2
+ 3)/sin(d*x + c)^5 - 15*log(sin(d*x + c) + 1) + 15*log(sin(d*x + c) - 1)) + 36*a*b^2*((15*tan(d*x + c)^4 + 5
*tan(d*x + c)^2 + 1)/tan(d*x + c)^5 - 5*tan(d*x + c)) + 4*(15*tan(d*x + c)^4 + 10*tan(d*x + c)^2 + 3)*a^3/tan(
d*x + c)^5)/d

Giac [A] (verification not implemented)

none

Time = 0.42 (sec) , antiderivative size = 498, normalized size of antiderivative = 1.78 \[ \int \csc ^6(c+d x) (a+b \sec (c+d x))^3 \, dx=\frac {3 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 9 \, a^{2} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 9 \, a b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 3 \, b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 25 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 105 \, a^{2} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 135 \, a b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 55 \, b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 150 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 990 \, a^{2} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1710 \, a b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 870 \, b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 240 \, {\left (6 \, a^{2} b + 7 \, b^{3}\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1 \right |}\right ) - 240 \, {\left (6 \, a^{2} b + 7 \, b^{3}\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1 \right |}\right ) - \frac {480 \, {\left (6 \, a b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 6 \, a b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 1\right )}^{2}} - \frac {150 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 990 \, a^{2} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 1710 \, a b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 870 \, b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 25 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 105 \, a^{2} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 135 \, a b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 55 \, b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 3 \, a^{3} + 9 \, a^{2} b + 9 \, a b^{2} + 3 \, b^{3}}{\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5}}}{480 \, d} \]

[In]

integrate(csc(d*x+c)^6*(a+b*sec(d*x+c))^3,x, algorithm="giac")

[Out]

1/480*(3*a^3*tan(1/2*d*x + 1/2*c)^5 - 9*a^2*b*tan(1/2*d*x + 1/2*c)^5 + 9*a*b^2*tan(1/2*d*x + 1/2*c)^5 - 3*b^3*
tan(1/2*d*x + 1/2*c)^5 + 25*a^3*tan(1/2*d*x + 1/2*c)^3 - 105*a^2*b*tan(1/2*d*x + 1/2*c)^3 + 135*a*b^2*tan(1/2*
d*x + 1/2*c)^3 - 55*b^3*tan(1/2*d*x + 1/2*c)^3 + 150*a^3*tan(1/2*d*x + 1/2*c) - 990*a^2*b*tan(1/2*d*x + 1/2*c)
+ 1710*a*b^2*tan(1/2*d*x + 1/2*c) - 870*b^3*tan(1/2*d*x + 1/2*c) + 240*(6*a^2*b + 7*b^3)*log(abs(tan(1/2*d*x
+ 1/2*c) + 1)) - 240*(6*a^2*b + 7*b^3)*log(abs(tan(1/2*d*x + 1/2*c) - 1)) - 480*(6*a*b^2*tan(1/2*d*x + 1/2*c)^
3 - b^3*tan(1/2*d*x + 1/2*c)^3 - 6*a*b^2*tan(1/2*d*x + 1/2*c) - b^3*tan(1/2*d*x + 1/2*c))/(tan(1/2*d*x + 1/2*c
)^2 - 1)^2 - (150*a^3*tan(1/2*d*x + 1/2*c)^4 + 990*a^2*b*tan(1/2*d*x + 1/2*c)^4 + 1710*a*b^2*tan(1/2*d*x + 1/2
*c)^4 + 870*b^3*tan(1/2*d*x + 1/2*c)^4 + 25*a^3*tan(1/2*d*x + 1/2*c)^2 + 105*a^2*b*tan(1/2*d*x + 1/2*c)^2 + 13
5*a*b^2*tan(1/2*d*x + 1/2*c)^2 + 55*b^3*tan(1/2*d*x + 1/2*c)^2 + 3*a^3 + 9*a^2*b + 9*a*b^2 + 3*b^3)/tan(1/2*d*
x + 1/2*c)^5)/d

Mupad [B] (verification not implemented)

Time = 13.68 (sec) , antiderivative size = 363, normalized size of antiderivative = 1.30 \[ \int \csc ^6(c+d x) (a+b \sec (c+d x))^3 \, dx=\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5\,{\left (a-b\right )}^3}{160\,d}-\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2\,\left (\frac {19\,a^3}{15}+\frac {29\,a^2\,b}{5}+\frac {39\,a\,b^2}{5}+\frac {49\,b^3}{15}\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8\,\left (10\,a^3+66\,a^2\,b+306\,a\,b^2+26\,b^3\right )-{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6\,\left (\frac {55\,a^3}{3}+125\,a^2\,b+411\,a\,b^2+\frac {433\,b^3}{3}\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4\,\left (\frac {103\,a^3}{15}+\frac {263\,a^2\,b}{5}+\frac {483\,a\,b^2}{5}+\frac {763\,b^3}{15}\right )+\frac {3\,a\,b^2}{5}+\frac {3\,a^2\,b}{5}+\frac {a^3}{5}+\frac {b^3}{5}}{d\,\left (32\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9-64\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7+32\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5\right )}+\frac {\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (\frac {21\,a\,b^2}{16}-\frac {3\,a^2\,b}{8}-\frac {a^3}{16}-\frac {7\,b^3}{8}+\frac {3\,{\left (a-b\right )}^2\,\left (a-4\,b\right )}{16}+\frac {3\,{\left (a-b\right )}^3}{16}\right )}{d}+\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3\,\left (\frac {{\left (a-b\right )}^2\,\left (a-4\,b\right )}{48}+\frac {{\left (a-b\right )}^3}{32}\right )}{d}-\frac {\mathrm {atanh}\left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )\,\left (a^2\,b\,6{}\mathrm {i}+b^3\,7{}\mathrm {i}\right )\,1{}\mathrm {i}}{d} \]

[In]

int((a + b/cos(c + d*x))^3/sin(c + d*x)^6,x)

[Out]

(tan(c/2 + (d*x)/2)^5*(a - b)^3)/(160*d) - (tan(c/2 + (d*x)/2)^2*((39*a*b^2)/5 + (29*a^2*b)/5 + (19*a^3)/15 +
(49*b^3)/15) + tan(c/2 + (d*x)/2)^8*(306*a*b^2 + 66*a^2*b + 10*a^3 + 26*b^3) - tan(c/2 + (d*x)/2)^6*(411*a*b^2
+ 125*a^2*b + (55*a^3)/3 + (433*b^3)/3) + tan(c/2 + (d*x)/2)^4*((483*a*b^2)/5 + (263*a^2*b)/5 + (103*a^3)/15
+ (763*b^3)/15) + (3*a*b^2)/5 + (3*a^2*b)/5 + a^3/5 + b^3/5)/(d*(32*tan(c/2 + (d*x)/2)^5 - 64*tan(c/2 + (d*x)/
2)^7 + 32*tan(c/2 + (d*x)/2)^9)) - (atanh(tan(c/2 + (d*x)/2))*(a^2*b*6i + b^3*7i)*1i)/d + (tan(c/2 + (d*x)/2)*
((21*a*b^2)/16 - (3*a^2*b)/8 - a^3/16 - (7*b^3)/8 + (3*(a - b)^2*(a - 4*b))/16 + (3*(a - b)^3)/16))/d + (tan(c
/2 + (d*x)/2)^3*(((a - b)^2*(a - 4*b))/48 + (a - b)^3/32))/d