3.1.27 \(\int (a+a \sin (c+d x))^3 \tan (c+d x) \, dx\) [27]
Optimal. Leaf size=70 \[ -\frac {4 a^3 \log (1-\sin (c+d x))}{d}-\frac {4 a^3 \sin (c+d x)}{d}-\frac {3 a^3 \sin ^2(c+d x)}{2 d}-\frac {a^3 \sin ^3(c+d x)}{3 d} \]
[Out]
-4*a^3*ln(1-sin(d*x+c))/d-4*a^3*sin(d*x+c)/d-3/2*a^3*sin(d*x+c)^2/d-1/3*a^3*sin(d*x+c)^3/d
________________________________________________________________________________________
Rubi [A]
time = 0.03, antiderivative size = 70, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 2, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.105, Rules used = {2786, 78}
\begin {gather*} -\frac {a^3 \sin ^3(c+d x)}{3 d}-\frac {3 a^3 \sin ^2(c+d x)}{2 d}-\frac {4 a^3 \sin (c+d x)}{d}-\frac {4 a^3 \log (1-\sin (c+d x))}{d} \end {gather*}
Antiderivative was successfully verified.
[In]
Int[(a + a*Sin[c + d*x])^3*Tan[c + d*x],x]
[Out]
(-4*a^3*Log[1 - Sin[c + d*x]])/d - (4*a^3*Sin[c + d*x])/d - (3*a^3*Sin[c + d*x]^2)/(2*d) - (a^3*Sin[c + d*x]^3
)/(3*d)
Rule 78
Int[((a_.) + (b_.)*(x_))*((c_) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Int[ExpandIntegran
d[(a + b*x)*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && NeQ[b*c - a*d, 0] && ((ILtQ[
n, 0] && ILtQ[p, 0]) || EqQ[p, 1] || (IGtQ[p, 0] && ( !IntegerQ[n] || LeQ[9*p + 5*(n + 2), 0] || GeQ[n + p + 1
, 0] || (GeQ[n + p + 2, 0] && RationalQ[a, b, c, d, e, f]))))
Rule 2786
Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*tan[(e_.) + (f_.)*(x_)]^(p_.), x_Symbol] :> Dist[1/f, Subst[I
nt[x^p*((a + x)^(m - (p + 1)/2)/(a - x)^((p + 1)/2)), x], x, b*Sin[e + f*x]], x] /; FreeQ[{a, b, e, f, m}, x]
&& EqQ[a^2 - b^2, 0] && IntegerQ[(p + 1)/2]
Rubi steps
\begin {align*} \int (a+a \sin (c+d x))^3 \tan (c+d x) \, dx &=\frac {\text {Subst}\left (\int \frac {x (a+x)^2}{a-x} \, dx,x,a \sin (c+d x)\right )}{d}\\ &=\frac {\text {Subst}\left (\int \left (-4 a^2+\frac {4 a^3}{a-x}-3 a x-x^2\right ) \, dx,x,a \sin (c+d x)\right )}{d}\\ &=-\frac {4 a^3 \log (1-\sin (c+d x))}{d}-\frac {4 a^3 \sin (c+d x)}{d}-\frac {3 a^3 \sin ^2(c+d x)}{2 d}-\frac {a^3 \sin ^3(c+d x)}{3 d}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 0.03, size = 52, normalized size = 0.74 \begin {gather*} -\frac {a^3 \left (24 \log (1-\sin (c+d x))+24 \sin (c+d x)+9 \sin ^2(c+d x)+2 \sin ^3(c+d x)\right )}{6 d} \end {gather*}
Antiderivative was successfully verified.
[In]
Integrate[(a + a*Sin[c + d*x])^3*Tan[c + d*x],x]
[Out]
-1/6*(a^3*(24*Log[1 - Sin[c + d*x]] + 24*Sin[c + d*x] + 9*Sin[c + d*x]^2 + 2*Sin[c + d*x]^3))/d
________________________________________________________________________________________
Maple [A]
time = 0.16, size = 108, normalized size = 1.54
| | |
| method |
result |
size |
| | |
| derivativedivides |
\(\frac {a^{3} \left (-\frac {\left (\sin ^{3}\left (d x +c \right )\right )}{3}-\sin \left (d x +c \right )+\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )\right )+3 a^{3} \left (-\frac {\left (\sin ^{2}\left (d x +c \right )\right )}{2}-\ln \left (\cos \left (d x +c \right )\right )\right )+3 a^{3} \left (-\sin \left (d x +c \right )+\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )\right )-a^{3} \ln \left (\cos \left (d x +c \right )\right )}{d}\) |
\(108\) |
| default |
\(\frac {a^{3} \left (-\frac {\left (\sin ^{3}\left (d x +c \right )\right )}{3}-\sin \left (d x +c \right )+\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )\right )+3 a^{3} \left (-\frac {\left (\sin ^{2}\left (d x +c \right )\right )}{2}-\ln \left (\cos \left (d x +c \right )\right )\right )+3 a^{3} \left (-\sin \left (d x +c \right )+\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )\right )-a^{3} \ln \left (\cos \left (d x +c \right )\right )}{d}\) |
\(108\) |
| risch |
\(4 i a^{3} x +\frac {17 i a^{3} {\mathrm e}^{i \left (d x +c \right )}}{8 d}-\frac {17 i a^{3} {\mathrm e}^{-i \left (d x +c \right )}}{8 d}+\frac {8 i a^{3} c}{d}-\frac {8 a^{3} \ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right )}{d}+\frac {a^{3} \sin \left (3 d x +3 c \right )}{12 d}+\frac {3 a^{3} \cos \left (2 d x +2 c \right )}{4 d}\) |
\(110\) |
| | |
|
|
|
|
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((a+a*sin(d*x+c))^3*tan(d*x+c),x,method=_RETURNVERBOSE)
[Out]
1/d*(a^3*(-1/3*sin(d*x+c)^3-sin(d*x+c)+ln(sec(d*x+c)+tan(d*x+c)))+3*a^3*(-1/2*sin(d*x+c)^2-ln(cos(d*x+c)))+3*a
^3*(-sin(d*x+c)+ln(sec(d*x+c)+tan(d*x+c)))-a^3*ln(cos(d*x+c)))
________________________________________________________________________________________
Maxima [A]
time = 0.29, size = 57, normalized size = 0.81 \begin {gather*} -\frac {2 \, a^{3} \sin \left (d x + c\right )^{3} + 9 \, a^{3} \sin \left (d x + c\right )^{2} + 24 \, a^{3} \log \left (\sin \left (d x + c\right ) - 1\right ) + 24 \, a^{3} \sin \left (d x + c\right )}{6 \, d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((a+a*sin(d*x+c))^3*tan(d*x+c),x, algorithm="maxima")
[Out]
-1/6*(2*a^3*sin(d*x + c)^3 + 9*a^3*sin(d*x + c)^2 + 24*a^3*log(sin(d*x + c) - 1) + 24*a^3*sin(d*x + c))/d
________________________________________________________________________________________
Fricas [A]
time = 0.36, size = 61, normalized size = 0.87 \begin {gather*} \frac {9 \, a^{3} \cos \left (d x + c\right )^{2} - 24 \, a^{3} \log \left (-\sin \left (d x + c\right ) + 1\right ) + 2 \, {\left (a^{3} \cos \left (d x + c\right )^{2} - 13 \, a^{3}\right )} \sin \left (d x + c\right )}{6 \, d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((a+a*sin(d*x+c))^3*tan(d*x+c),x, algorithm="fricas")
[Out]
1/6*(9*a^3*cos(d*x + c)^2 - 24*a^3*log(-sin(d*x + c) + 1) + 2*(a^3*cos(d*x + c)^2 - 13*a^3)*sin(d*x + c))/d
________________________________________________________________________________________
Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} a^{3} \left (\int 3 \sin {\left (c + d x \right )} \tan {\left (c + d x \right )}\, dx + \int 3 \sin ^{2}{\left (c + d x \right )} \tan {\left (c + d x \right )}\, dx + \int \sin ^{3}{\left (c + d x \right )} \tan {\left (c + d x \right )}\, dx + \int \tan {\left (c + d x \right )}\, dx\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((a+a*sin(d*x+c))**3*tan(d*x+c),x)
[Out]
a**3*(Integral(3*sin(c + d*x)*tan(c + d*x), x) + Integral(3*sin(c + d*x)**2*tan(c + d*x), x) + Integral(sin(c
+ d*x)**3*tan(c + d*x), x) + Integral(tan(c + d*x), x))
________________________________________________________________________________________
Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 28789 vs.
\(2 (66) = 132\).
time = 32.14, size = 28789, normalized size = 411.27 \begin {gather*} \text {Too large to display} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((a+a*sin(d*x+c))^3*tan(d*x+c),x, algorithm="giac")
[Out]
-1/12*(24*a^3*log(2*(tan(1/2*d*x)^4*tan(1/2*c)^2 + 2*tan(1/2*d*x)^4*tan(1/2*c) + 2*tan(1/2*d*x)^3*tan(1/2*c)^2
+ tan(1/2*d*x)^4 + 2*tan(1/2*d*x)^2*tan(1/2*c)^2 - 2*tan(1/2*d*x)^3 + 2*tan(1/2*d*x)*tan(1/2*c)^2 + 2*tan(1/2
*d*x)^2 + tan(1/2*c)^2 - 2*tan(1/2*d*x) - 2*tan(1/2*c) + 1)/(tan(1/2*c)^2 + 1))*tan(d*x)^2*tan(1/2*d*x)^6*tan(
1/2*c)^6*tan(c)^2 - 24*a^3*log(2*(tan(1/2*d*x)^4*tan(1/2*c)^2 - 2*tan(1/2*d*x)^4*tan(1/2*c) - 2*tan(1/2*d*x)^3
*tan(1/2*c)^2 + tan(1/2*d*x)^4 + 2*tan(1/2*d*x)^2*tan(1/2*c)^2 + 2*tan(1/2*d*x)^3 - 2*tan(1/2*d*x)*tan(1/2*c)^
2 + 2*tan(1/2*d*x)^2 + tan(1/2*c)^2 + 2*tan(1/2*d*x) + 2*tan(1/2*c) + 1)/(tan(1/2*c)^2 + 1))*tan(d*x)^2*tan(1/
2*d*x)^6*tan(1/2*c)^6*tan(c)^2 + 24*a^3*log(4*(tan(d*x)^4*tan(c)^2 - 2*tan(d*x)^3*tan(c) + tan(d*x)^2*tan(c)^2
+ tan(d*x)^2 - 2*tan(d*x)*tan(c) + 1)/(tan(c)^2 + 1))*tan(d*x)^2*tan(1/2*d*x)^6*tan(1/2*c)^6*tan(c)^2 - 9*a^3
*tan(d*x)^2*tan(1/2*d*x)^6*tan(1/2*c)^6*tan(c)^2 + 24*a^3*log(2*(tan(1/2*d*x)^4*tan(1/2*c)^2 + 2*tan(1/2*d*x)^
4*tan(1/2*c) + 2*tan(1/2*d*x)^3*tan(1/2*c)^2 + tan(1/2*d*x)^4 + 2*tan(1/2*d*x)^2*tan(1/2*c)^2 - 2*tan(1/2*d*x)
^3 + 2*tan(1/2*d*x)*tan(1/2*c)^2 + 2*tan(1/2*d*x)^2 + tan(1/2*c)^2 - 2*tan(1/2*d*x) - 2*tan(1/2*c) + 1)/(tan(1
/2*c)^2 + 1))*tan(d*x)^2*tan(1/2*d*x)^6*tan(1/2*c)^6 - 24*a^3*log(2*(tan(1/2*d*x)^4*tan(1/2*c)^2 - 2*tan(1/2*d
*x)^4*tan(1/2*c) - 2*tan(1/2*d*x)^3*tan(1/2*c)^2 + tan(1/2*d*x)^4 + 2*tan(1/2*d*x)^2*tan(1/2*c)^2 + 2*tan(1/2*
d*x)^3 - 2*tan(1/2*d*x)*tan(1/2*c)^2 + 2*tan(1/2*d*x)^2 + tan(1/2*c)^2 + 2*tan(1/2*d*x) + 2*tan(1/2*c) + 1)/(t
an(1/2*c)^2 + 1))*tan(d*x)^2*tan(1/2*d*x)^6*tan(1/2*c)^6 + 24*a^3*log(4*(tan(d*x)^4*tan(c)^2 - 2*tan(d*x)^3*ta
n(c) + tan(d*x)^2*tan(c)^2 + tan(d*x)^2 - 2*tan(d*x)*tan(c) + 1)/(tan(c)^2 + 1))*tan(d*x)^2*tan(1/2*d*x)^6*tan
(1/2*c)^6 + 72*a^3*log(2*(tan(1/2*d*x)^4*tan(1/2*c)^2 + 2*tan(1/2*d*x)^4*tan(1/2*c) + 2*tan(1/2*d*x)^3*tan(1/2
*c)^2 + tan(1/2*d*x)^4 + 2*tan(1/2*d*x)^2*tan(1/2*c)^2 - 2*tan(1/2*d*x)^3 + 2*tan(1/2*d*x)*tan(1/2*c)^2 + 2*ta
n(1/2*d*x)^2 + tan(1/2*c)^2 - 2*tan(1/2*d*x) - 2*tan(1/2*c) + 1)/(tan(1/2*c)^2 + 1))*tan(d*x)^2*tan(1/2*d*x)^6
*tan(1/2*c)^4*tan(c)^2 - 72*a^3*log(2*(tan(1/2*d*x)^4*tan(1/2*c)^2 - 2*tan(1/2*d*x)^4*tan(1/2*c) - 2*tan(1/2*d
*x)^3*tan(1/2*c)^2 + tan(1/2*d*x)^4 + 2*tan(1/2*d*x)^2*tan(1/2*c)^2 + 2*tan(1/2*d*x)^3 - 2*tan(1/2*d*x)*tan(1/
2*c)^2 + 2*tan(1/2*d*x)^2 + tan(1/2*c)^2 + 2*tan(1/2*d*x) + 2*tan(1/2*c) + 1)/(tan(1/2*c)^2 + 1))*tan(d*x)^2*t
an(1/2*d*x)^6*tan(1/2*c)^4*tan(c)^2 + 72*a^3*log(4*(tan(d*x)^4*tan(c)^2 - 2*tan(d*x)^3*tan(c) + tan(d*x)^2*tan
(c)^2 + tan(d*x)^2 - 2*tan(d*x)*tan(c) + 1)/(tan(c)^2 + 1))*tan(d*x)^2*tan(1/2*d*x)^6*tan(1/2*c)^4*tan(c)^2 -
96*a^3*tan(d*x)^2*tan(1/2*d*x)^6*tan(1/2*c)^5*tan(c)^2 + 72*a^3*log(2*(tan(1/2*d*x)^4*tan(1/2*c)^2 + 2*tan(1/2
*d*x)^4*tan(1/2*c) + 2*tan(1/2*d*x)^3*tan(1/2*c)^2 + tan(1/2*d*x)^4 + 2*tan(1/2*d*x)^2*tan(1/2*c)^2 - 2*tan(1/
2*d*x)^3 + 2*tan(1/2*d*x)*tan(1/2*c)^2 + 2*tan(1/2*d*x)^2 + tan(1/2*c)^2 - 2*tan(1/2*d*x) - 2*tan(1/2*c) + 1)/
(tan(1/2*c)^2 + 1))*tan(d*x)^2*tan(1/2*d*x)^4*tan(1/2*c)^6*tan(c)^2 - 72*a^3*log(2*(tan(1/2*d*x)^4*tan(1/2*c)^
2 - 2*tan(1/2*d*x)^4*tan(1/2*c) - 2*tan(1/2*d*x)^3*tan(1/2*c)^2 + tan(1/2*d*x)^4 + 2*tan(1/2*d*x)^2*tan(1/2*c)
^2 + 2*tan(1/2*d*x)^3 - 2*tan(1/2*d*x)*tan(1/2*c)^2 + 2*tan(1/2*d*x)^2 + tan(1/2*c)^2 + 2*tan(1/2*d*x) + 2*tan
(1/2*c) + 1)/(tan(1/2*c)^2 + 1))*tan(d*x)^2*tan(1/2*d*x)^4*tan(1/2*c)^6*tan(c)^2 + 72*a^3*log(4*(tan(d*x)^4*ta
n(c)^2 - 2*tan(d*x)^3*tan(c) + tan(d*x)^2*tan(c)^2 + tan(d*x)^2 - 2*tan(d*x)*tan(c) + 1)/(tan(c)^2 + 1))*tan(d
*x)^2*tan(1/2*d*x)^4*tan(1/2*c)^6*tan(c)^2 - 96*a^3*tan(d*x)^2*tan(1/2*d*x)^5*tan(1/2*c)^6*tan(c)^2 + 24*a^3*l
og(2*(tan(1/2*d*x)^4*tan(1/2*c)^2 + 2*tan(1/2*d*x)^4*tan(1/2*c) + 2*tan(1/2*d*x)^3*tan(1/2*c)^2 + tan(1/2*d*x)
^4 + 2*tan(1/2*d*x)^2*tan(1/2*c)^2 - 2*tan(1/2*d*x)^3 + 2*tan(1/2*d*x)*tan(1/2*c)^2 + 2*tan(1/2*d*x)^2 + tan(1
/2*c)^2 - 2*tan(1/2*d*x) - 2*tan(1/2*c) + 1)/(tan(1/2*c)^2 + 1))*tan(1/2*d*x)^6*tan(1/2*c)^6*tan(c)^2 - 24*a^3
*log(2*(tan(1/2*d*x)^4*tan(1/2*c)^2 - 2*tan(1/2*d*x)^4*tan(1/2*c) - 2*tan(1/2*d*x)^3*tan(1/2*c)^2 + tan(1/2*d*
x)^4 + 2*tan(1/2*d*x)^2*tan(1/2*c)^2 + 2*tan(1/2*d*x)^3 - 2*tan(1/2*d*x)*tan(1/2*c)^2 + 2*tan(1/2*d*x)^2 + tan
(1/2*c)^2 + 2*tan(1/2*d*x) + 2*tan(1/2*c) + 1)/(tan(1/2*c)^2 + 1))*tan(1/2*d*x)^6*tan(1/2*c)^6*tan(c)^2 + 24*a
^3*log(4*(tan(d*x)^4*tan(c)^2 - 2*tan(d*x)^3*tan(c) + tan(d*x)^2*tan(c)^2 + tan(d*x)^2 - 2*tan(d*x)*tan(c) + 1
)/(tan(c)^2 + 1))*tan(1/2*d*x)^6*tan(1/2*c)^6*tan(c)^2 + 9*a^3*tan(d*x)^2*tan(1/2*d*x)^6*tan(1/2*c)^6 + 36*a^3
*tan(d*x)*tan(1/2*d*x)^6*tan(1/2*c)^6*tan(c) - 27*a^3*tan(d*x)^2*tan(1/2*d*x)^6*tan(1/2*c)^4*tan(c)^2 - 27*a^3
*tan(d*x)^2*tan(1/2*d*x)^4*tan(1/2*c)^6*tan(c)^2 + 9*a^3*tan(1/2*d*x)^6*tan(1/2*c)^6*tan(c)^2 + 72*a^3*log(2*(
tan(1/2*d*x)^4*tan(1/2*c)^2 + 2*tan(1/2*d*x)^4*tan(1/2*c) + 2*tan(1/2*d*x)^3*tan(1/2*c)^2 + tan(1/2*d*x)^4 + 2
*tan(1/2*d*x)^2*tan(1/2*c)^2 - 2*tan(1/2*d*x)^3 + 2*tan(1/2*d*x)*tan(1/2*c)^2 + 2*tan(1/2*d*x)^2 + tan(1/2*c)^
2 - 2*tan(1/2*d*x) - 2*tan(1/2*c) + 1)/(tan(1/2...
________________________________________________________________________________________
Mupad [B]
time = 7.26, size = 281, normalized size = 4.01 \begin {gather*} -\frac {\frac {56\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{3}+8\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5-{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2\,\left (2\,a^3\,\left (12\,\ln \left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )-1\right )-6\,\ln \left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+1\right )\right )-\frac {2\,a^3\,\left (36\,\ln \left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )-1\right )-18\,\ln \left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+1\right )+9\right )}{3}\right )-{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4\,\left (2\,a^3\,\left (12\,\ln \left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )-1\right )-6\,\ln \left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+1\right )\right )-\frac {2\,a^3\,\left (36\,\ln \left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )-1\right )-18\,\ln \left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+1\right )+9\right )}{3}\right )+8\,a^3\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{d\,{\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+1\right )}^3}-\frac {2\,a^3\,\left (12\,\ln \left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )-1\right )-6\,\ln \left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+1\right )\right )}{3\,d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(tan(c + d*x)*(a + a*sin(c + d*x))^3,x)
[Out]
- ((56*a^3*tan(c/2 + (d*x)/2)^3)/3 + 8*a^3*tan(c/2 + (d*x)/2)^5 - tan(c/2 + (d*x)/2)^2*(2*a^3*(12*log(tan(c/2
+ (d*x)/2) - 1) - 6*log(tan(c/2 + (d*x)/2)^2 + 1)) - (2*a^3*(36*log(tan(c/2 + (d*x)/2) - 1) - 18*log(tan(c/2 +
(d*x)/2)^2 + 1) + 9))/3) - tan(c/2 + (d*x)/2)^4*(2*a^3*(12*log(tan(c/2 + (d*x)/2) - 1) - 6*log(tan(c/2 + (d*x
)/2)^2 + 1)) - (2*a^3*(36*log(tan(c/2 + (d*x)/2) - 1) - 18*log(tan(c/2 + (d*x)/2)^2 + 1) + 9))/3) + 8*a^3*tan(
c/2 + (d*x)/2))/(d*(tan(c/2 + (d*x)/2)^2 + 1)^3) - (2*a^3*(12*log(tan(c/2 + (d*x)/2) - 1) - 6*log(tan(c/2 + (d
*x)/2)^2 + 1)))/(3*d)
________________________________________________________________________________________