3.1.36 \(\int (a+a \sin (c+d x))^4 \tan (c+d x) \, dx\) [36]
Optimal. Leaf size=88 \[ -\frac {8 a^4 \log (1-\sin (c+d x))}{d}-\frac {8 a^4 \sin (c+d x)}{d}-\frac {7 a^4 \sin ^2(c+d x)}{2 d}-\frac {4 a^4 \sin ^3(c+d x)}{3 d}-\frac {a^4 \sin ^4(c+d x)}{4 d} \]
[Out]
-8*a^4*ln(1-sin(d*x+c))/d-8*a^4*sin(d*x+c)/d-7/2*a^4*sin(d*x+c)^2/d-4/3*a^4*sin(d*x+c)^3/d-1/4*a^4*sin(d*x+c)^
4/d
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Rubi [A]
time = 0.04, antiderivative size = 88, 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^4 \sin ^4(c+d x)}{4 d}-\frac {4 a^4 \sin ^3(c+d x)}{3 d}-\frac {7 a^4 \sin ^2(c+d x)}{2 d}-\frac {8 a^4 \sin (c+d x)}{d}-\frac {8 a^4 \log (1-\sin (c+d x))}{d} \end {gather*}
Antiderivative was successfully verified.
[In]
Int[(a + a*Sin[c + d*x])^4*Tan[c + d*x],x]
[Out]
(-8*a^4*Log[1 - Sin[c + d*x]])/d - (8*a^4*Sin[c + d*x])/d - (7*a^4*Sin[c + d*x]^2)/(2*d) - (4*a^4*Sin[c + d*x]
^3)/(3*d) - (a^4*Sin[c + d*x]^4)/(4*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))^4 \tan (c+d x) \, dx &=\frac {\text {Subst}\left (\int \frac {x (a+x)^3}{a-x} \, dx,x,a \sin (c+d x)\right )}{d}\\ &=\frac {\text {Subst}\left (\int \left (-8 a^3+\frac {8 a^4}{a-x}-7 a^2 x-4 a x^2-x^3\right ) \, dx,x,a \sin (c+d x)\right )}{d}\\ &=-\frac {8 a^4 \log (1-\sin (c+d x))}{d}-\frac {8 a^4 \sin (c+d x)}{d}-\frac {7 a^4 \sin ^2(c+d x)}{2 d}-\frac {4 a^4 \sin ^3(c+d x)}{3 d}-\frac {a^4 \sin ^4(c+d x)}{4 d}\\ \end {align*}
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Mathematica [A]
time = 0.05, size = 62, normalized size = 0.70 \begin {gather*} -\frac {a^4 \left (96 \log (1-\sin (c+d x))+96 \sin (c+d x)+42 \sin ^2(c+d x)+16 \sin ^3(c+d x)+3 \sin ^4(c+d x)\right )}{12 d} \end {gather*}
Antiderivative was successfully verified.
[In]
Integrate[(a + a*Sin[c + d*x])^4*Tan[c + d*x],x]
[Out]
-1/12*(a^4*(96*Log[1 - Sin[c + d*x]] + 96*Sin[c + d*x] + 42*Sin[c + d*x]^2 + 16*Sin[c + d*x]^3 + 3*Sin[c + d*x
]^4))/d
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Maple [A]
time = 0.18, size = 143, normalized size = 1.62
| | |
| method |
result |
size |
| | |
| risch |
\(8 i a^{4} x +\frac {9 i a^{4} {\mathrm e}^{i \left (d x +c \right )}}{2 d}-\frac {9 i a^{4} {\mathrm e}^{-i \left (d x +c \right )}}{2 d}+\frac {16 i a^{4} c}{d}-\frac {16 a^{4} \ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right )}{d}-\frac {a^{4} \cos \left (4 d x +4 c \right )}{32 d}+\frac {a^{4} \sin \left (3 d x +3 c \right )}{3 d}+\frac {15 a^{4} \cos \left (2 d x +2 c \right )}{8 d}\) |
\(127\) |
| derivativedivides |
\(\frac {a^{4} \left (-\frac {\left (\sin ^{4}\left (d x +c \right )\right )}{4}-\frac {\left (\sin ^{2}\left (d x +c \right )\right )}{2}-\ln \left (\cos \left (d x +c \right )\right )\right )+4 a^{4} \left (-\frac {\left (\sin ^{3}\left (d x +c \right )\right )}{3}-\sin \left (d x +c \right )+\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )\right )+6 a^{4} \left (-\frac {\left (\sin ^{2}\left (d x +c \right )\right )}{2}-\ln \left (\cos \left (d x +c \right )\right )\right )+4 a^{4} \left (-\sin \left (d x +c \right )+\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )\right )-a^{4} \ln \left (\cos \left (d x +c \right )\right )}{d}\) |
\(143\) |
| default |
\(\frac {a^{4} \left (-\frac {\left (\sin ^{4}\left (d x +c \right )\right )}{4}-\frac {\left (\sin ^{2}\left (d x +c \right )\right )}{2}-\ln \left (\cos \left (d x +c \right )\right )\right )+4 a^{4} \left (-\frac {\left (\sin ^{3}\left (d x +c \right )\right )}{3}-\sin \left (d x +c \right )+\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )\right )+6 a^{4} \left (-\frac {\left (\sin ^{2}\left (d x +c \right )\right )}{2}-\ln \left (\cos \left (d x +c \right )\right )\right )+4 a^{4} \left (-\sin \left (d x +c \right )+\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )\right )-a^{4} \ln \left (\cos \left (d x +c \right )\right )}{d}\) |
\(143\) |
| | |
|
|
|
|
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((a+a*sin(d*x+c))^4*tan(d*x+c),x,method=_RETURNVERBOSE)
[Out]
1/d*(a^4*(-1/4*sin(d*x+c)^4-1/2*sin(d*x+c)^2-ln(cos(d*x+c)))+4*a^4*(-1/3*sin(d*x+c)^3-sin(d*x+c)+ln(sec(d*x+c)
+tan(d*x+c)))+6*a^4*(-1/2*sin(d*x+c)^2-ln(cos(d*x+c)))+4*a^4*(-sin(d*x+c)+ln(sec(d*x+c)+tan(d*x+c)))-a^4*ln(co
s(d*x+c)))
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Maxima [A]
time = 0.28, size = 70, normalized size = 0.80 \begin {gather*} -\frac {3 \, a^{4} \sin \left (d x + c\right )^{4} + 16 \, a^{4} \sin \left (d x + c\right )^{3} + 42 \, a^{4} \sin \left (d x + c\right )^{2} + 96 \, a^{4} \log \left (\sin \left (d x + c\right ) - 1\right ) + 96 \, a^{4} \sin \left (d x + c\right )}{12 \, d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((a+a*sin(d*x+c))^4*tan(d*x+c),x, algorithm="maxima")
[Out]
-1/12*(3*a^4*sin(d*x + c)^4 + 16*a^4*sin(d*x + c)^3 + 42*a^4*sin(d*x + c)^2 + 96*a^4*log(sin(d*x + c) - 1) + 9
6*a^4*sin(d*x + c))/d
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Fricas [A]
time = 0.35, size = 74, normalized size = 0.84 \begin {gather*} -\frac {3 \, a^{4} \cos \left (d x + c\right )^{4} - 48 \, a^{4} \cos \left (d x + c\right )^{2} + 96 \, a^{4} \log \left (-\sin \left (d x + c\right ) + 1\right ) - 16 \, {\left (a^{4} \cos \left (d x + c\right )^{2} - 7 \, a^{4}\right )} \sin \left (d x + c\right )}{12 \, d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((a+a*sin(d*x+c))^4*tan(d*x+c),x, algorithm="fricas")
[Out]
-1/12*(3*a^4*cos(d*x + c)^4 - 48*a^4*cos(d*x + c)^2 + 96*a^4*log(-sin(d*x + c) + 1) - 16*(a^4*cos(d*x + c)^2 -
7*a^4)*sin(d*x + c))/d
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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} a^{4} \left (\int 4 \sin {\left (c + d x \right )} \tan {\left (c + d x \right )}\, dx + \int 6 \sin ^{2}{\left (c + d x \right )} \tan {\left (c + d x \right )}\, dx + \int 4 \sin ^{3}{\left (c + d x \right )} \tan {\left (c + d x \right )}\, dx + \int \sin ^{4}{\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))**4*tan(d*x+c),x)
[Out]
a**4*(Integral(4*sin(c + d*x)*tan(c + d*x), x) + Integral(6*sin(c + d*x)**2*tan(c + d*x), x) + Integral(4*sin(
c + d*x)**3*tan(c + d*x), x) + Integral(sin(c + d*x)**4*tan(c + d*x), x) + Integral(tan(c + d*x), x))
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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 67058 vs.
\(2 (82) = 164\).
time = 19.83, size = 67058, normalized size = 762.02 \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))^4*tan(d*x+c),x, algorithm="giac")
[Out]
-1/96*(384*a^4*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)^4*tan(1/2*d*x)^6*tan
(1/2*c)^6*tan(c)^4 - 384*a^4*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)^4*tan(
1/2*d*x)^6*tan(1/2*c)^6*tan(c)^4 + 384*a^4*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)^4*tan(1/2*d*x)^6*tan(1/2*c)^6*tan(c)^4 - 17
7*a^4*tan(d*x)^4*tan(1/2*d*x)^6*tan(1/2*c)^6*tan(c)^4 + 768*a^4*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)^4*tan(1/2*d*x)^6*tan(1/2*c)^6*tan(c)^2 - 768*a^4*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*ta
n(1/2*c) + 1)/(tan(1/2*c)^2 + 1))*tan(d*x)^4*tan(1/2*d*x)^6*tan(1/2*c)^6*tan(c)^2 + 768*a^4*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)^4*tan(1/2*d*x)^6*tan(1/2*c)^6*tan(c)^2 + 1152*a^4*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)^4*tan(1/2*d*x)^6*tan(1/2*c)^4*tan(c)^4 - 1152*a^4*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)^4*tan(1/2*d*x)^6*tan(1/2*c)^4*tan(c)^4 + 1152*a^4*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)^
4*tan(1/2*d*x)^6*tan(1/2*c)^4*tan(c)^4 - 1536*a^4*tan(d*x)^4*tan(1/2*d*x)^6*tan(1/2*c)^5*tan(c)^4 + 1152*a^4*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(d*x)^4*tan(1/2*d*x)^4*tan(1/2*c)^6*tan(c)
^4 - 1152*a^4*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)^4*tan(1/2*d*x)^4*tan(
1/2*c)^6*tan(c)^4 + 1152*a^4*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)^4*tan(1/2*d*x)^4*tan(1/2*c)^6*tan(c)^4 - 1536*a^4*tan(d*x
)^4*tan(1/2*d*x)^5*tan(1/2*c)^6*tan(c)^4 + 768*a^4*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)^4 - 768*a^4*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)^4 + 768*a^4*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)^4 - 18*a^4*tan(d*x)^4*tan(1/2*d*x)^6*tan(1/2*c)^6*tan(c)^2 + 672*a^4*tan(d*x)^3*
tan(1/2*d*x)^6*tan(1/2*c)^6*tan(c)^3 - 531*a^4*tan(d*x)^4*tan(1/2*d*x)^6*tan(1/2*c)^4*tan(c)^4 - 531*a^4*tan(d
*x)^4*tan(1/2*d*x)^4*tan(1/2*c)^6*tan(c)^4 - 18*a^4*tan(d*x)^2*tan(1/2*d*x)^6*tan(1/2*c)^6*tan(c)^4 + 384*a^4*
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...
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Mupad [B]
time = 6.63, size = 131, normalized size = 1.49 \begin {gather*} \frac {8\,a^4\,\ln \left (\frac {1}{{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}\right )}{d}-\frac {28\,a^4\,\sin \left (c+d\,x\right )}{3\,d}-\frac {16\,a^4\,\ln \left (\frac {\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )-\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )}{d}+\frac {4\,a^4\,{\cos \left (c+d\,x\right )}^2}{d}-\frac {a^4\,{\cos \left (c+d\,x\right )}^4}{4\,d}+\frac {4\,a^4\,{\cos \left (c+d\,x\right )}^2\,\sin \left (c+d\,x\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))^4,x)
[Out]
(8*a^4*log(1/cos(c/2 + (d*x)/2)^2))/d - (28*a^4*sin(c + d*x))/(3*d) - (16*a^4*log((cos(c/2 + (d*x)/2) - sin(c/
2 + (d*x)/2))/cos(c/2 + (d*x)/2)))/d + (4*a^4*cos(c + d*x)^2)/d - (a^4*cos(c + d*x)^4)/(4*d) + (4*a^4*cos(c +
d*x)^2*sin(c + d*x))/(3*d)
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