\(\int \csc ^4(c+d x) \sec ^2(c+d x) (a+a \sin (c+d x))^3 \, dx\) [771]

Optimal result

Integrand size = 29, antiderivative size = 98 \[ \int \csc ^4(c+d x) \sec ^2(c+d x) (a+a \sin (c+d x))^3 \, dx=-\frac {11 a^3 \text {arctanh}(\cos (c+d x))}{2 d}-\frac {5 a^3 \cot (c+d x)}{d}-\frac {a^3 \cot ^3(c+d x)}{3 d}-\frac {3 a^3 \cot (c+d x) \csc (c+d x)}{2 d}+\frac {4 a^3 \cos (c+d x)}{d (1-\sin (c+d x))} \]

[Out]

-11/2*a^3*arctanh(cos(d*x+c))/d-5*a^3*cot(d*x+c)/d-1/3*a^3*cot(d*x+c)^3/d-3/2*a^3*cot(d*x+c)*csc(d*x+c)/d+4*a^
3*cos(d*x+c)/d/(1-sin(d*x+c))

Rubi [A] (verified)

Time = 0.16 (sec) , antiderivative size = 98, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.207, Rules used = {2951, 3855, 3852, 8, 3853, 2727} \[ \int \csc ^4(c+d x) \sec ^2(c+d x) (a+a \sin (c+d x))^3 \, dx=-\frac {11 a^3 \text {arctanh}(\cos (c+d x))}{2 d}-\frac {a^3 \cot ^3(c+d x)}{3 d}-\frac {5 a^3 \cot (c+d x)}{d}+\frac {4 a^3 \cos (c+d x)}{d (1-\sin (c+d x))}-\frac {3 a^3 \cot (c+d x) \csc (c+d x)}{2 d} \]

[In]

Int[Csc[c + d*x]^4*Sec[c + d*x]^2*(a + a*Sin[c + d*x])^3,x]

[Out]

(-11*a^3*ArcTanh[Cos[c + d*x]])/(2*d) - (5*a^3*Cot[c + d*x])/d - (a^3*Cot[c + d*x]^3)/(3*d) - (3*a^3*Cot[c + d
*x]*Csc[c + d*x])/(2*d) + (4*a^3*Cos[c + d*x])/(d*(1 - Sin[c + d*x]))

Rule 8

Int[a_, x_Symbol] :> Simp[a*x, x] /; FreeQ[a, x]

Rule 2727

Int[((a_) + (b_.)*sin[(c_.) + (d_.)*(x_)])^(-1), x_Symbol] :> Simp[-Cos[c + d*x]/(d*(b + a*Sin[c + d*x])), x]
/; FreeQ[{a, b, c, d}, x] && EqQ[a^2 - b^2, 0]

Rule 2951

Int[cos[(e_.) + (f_.)*(x_)]^(p_)*((d_.)*sin[(e_.) + (f_.)*(x_)])^(n_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(
m_), x_Symbol] :> Dist[1/a^p, Int[ExpandTrig[(d*sin[e + f*x])^n*(a - b*sin[e + f*x])^(p/2)*(a + b*sin[e + f*x]
)^(m + p/2), x], x], x] /; FreeQ[{a, b, d, e, f}, x] && EqQ[a^2 - b^2, 0] && IntegersQ[m, n, p/2] && ((GtQ[m,
0] && GtQ[p, 0] && LtQ[-m - p, n, -1]) || (GtQ[m, 2] && LtQ[p, 0] && GtQ[m + p/2, 0]))

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 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]

Rubi steps \begin{align*} \text {integral}& = a^2 \int \left (4 a \csc (c+d x)+4 a \csc ^2(c+d x)+3 a \csc ^3(c+d x)+a \csc ^4(c+d x)-\frac {4 a}{-1+\sin (c+d x)}\right ) \, dx \\ & = a^3 \int \csc ^4(c+d x) \, dx+\left (3 a^3\right ) \int \csc ^3(c+d x) \, dx+\left (4 a^3\right ) \int \csc (c+d x) \, dx+\left (4 a^3\right ) \int \csc ^2(c+d x) \, dx-\left (4 a^3\right ) \int \frac {1}{-1+\sin (c+d x)} \, dx \\ & = -\frac {4 a^3 \text {arctanh}(\cos (c+d x))}{d}-\frac {3 a^3 \cot (c+d x) \csc (c+d x)}{2 d}+\frac {4 a^3 \cos (c+d x)}{d (1-\sin (c+d x))}+\frac {1}{2} \left (3 a^3\right ) \int \csc (c+d x) \, dx-\frac {a^3 \text {Subst}\left (\int \left (1+x^2\right ) \, dx,x,\cot (c+d x)\right )}{d}-\frac {\left (4 a^3\right ) \text {Subst}(\int 1 \, dx,x,\cot (c+d x))}{d} \\ & = -\frac {11 a^3 \text {arctanh}(\cos (c+d x))}{2 d}-\frac {5 a^3 \cot (c+d x)}{d}-\frac {a^3 \cot ^3(c+d x)}{3 d}-\frac {3 a^3 \cot (c+d x) \csc (c+d x)}{2 d}+\frac {4 a^3 \cos (c+d x)}{d (1-\sin (c+d x))} \\ \end{align*}

Mathematica [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(211\) vs. \(2(98)=196\).

Time = 6.48 (sec) , antiderivative size = 211, normalized size of antiderivative = 2.15 \[ \int \csc ^4(c+d x) \sec ^2(c+d x) (a+a \sin (c+d x))^3 \, dx=a^3 \left (-\frac {7 \cot \left (\frac {1}{2} (c+d x)\right )}{3 d}-\frac {3 \csc ^2\left (\frac {1}{2} (c+d x)\right )}{8 d}-\frac {\cot \left (\frac {1}{2} (c+d x)\right ) \csc ^2\left (\frac {1}{2} (c+d x)\right )}{24 d}-\frac {11 \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )}{2 d}+\frac {11 \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )}{2 d}+\frac {3 \sec ^2\left (\frac {1}{2} (c+d x)\right )}{8 d}+\frac {8 \sin \left (\frac {1}{2} (c+d x)\right )}{d \left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )}+\frac {7 \tan \left (\frac {1}{2} (c+d x)\right )}{3 d}+\frac {\sec ^2\left (\frac {1}{2} (c+d x)\right ) \tan \left (\frac {1}{2} (c+d x)\right )}{24 d}\right ) \]

[In]

Integrate[Csc[c + d*x]^4*Sec[c + d*x]^2*(a + a*Sin[c + d*x])^3,x]

[Out]

a^3*((-7*Cot[(c + d*x)/2])/(3*d) - (3*Csc[(c + d*x)/2]^2)/(8*d) - (Cot[(c + d*x)/2]*Csc[(c + d*x)/2]^2)/(24*d)
- (11*Log[Cos[(c + d*x)/2]])/(2*d) + (11*Log[Sin[(c + d*x)/2]])/(2*d) + (3*Sec[(c + d*x)/2]^2)/(8*d) + (8*Sin
[(c + d*x)/2])/(d*(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])) + (7*Tan[(c + d*x)/2])/(3*d) + (Sec[(c + d*x)/2]^2*Ta
n[(c + d*x)/2])/(24*d))

Maple [A] (verified)

Time = 0.32 (sec) , antiderivative size = 119, normalized size of antiderivative = 1.21

method	result	size
parallelrisch	\(\frac {\left (-306+132 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )+\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )+\cot ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )+8 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+8 \left (\cot ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+48 \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+48 \cot \left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a^{3}}{24 d \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}\)	\(119\)
derivativedivides	\(\frac {a^{3} \left (\frac {1}{\cos \left (d x +c \right )}+\ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )\right )+3 a^{3} \left (\frac {1}{\sin \left (d x +c \right ) \cos \left (d x +c \right )}-2 \cot \left (d x +c \right )\right )+3 a^{3} \left (-\frac {1}{2 \sin \left (d x +c \right )^{2} \cos \left (d x +c \right )}+\frac {3}{2 \cos \left (d x +c \right )}+\frac {3 \ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )}{2}\right )+a^{3} \left (-\frac {1}{3 \sin \left (d x +c \right )^{3} \cos \left (d x +c \right )}+\frac {4}{3 \sin \left (d x +c \right ) \cos \left (d x +c \right )}-\frac {8 \cot \left (d x +c \right )}{3}\right )}{d}\)	\(167\)
default	\(\frac {a^{3} \left (\frac {1}{\cos \left (d x +c \right )}+\ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )\right )+3 a^{3} \left (\frac {1}{\sin \left (d x +c \right ) \cos \left (d x +c \right )}-2 \cot \left (d x +c \right )\right )+3 a^{3} \left (-\frac {1}{2 \sin \left (d x +c \right )^{2} \cos \left (d x +c \right )}+\frac {3}{2 \cos \left (d x +c \right )}+\frac {3 \ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )}{2}\right )+a^{3} \left (-\frac {1}{3 \sin \left (d x +c \right )^{3} \cos \left (d x +c \right )}+\frac {4}{3 \sin \left (d x +c \right ) \cos \left (d x +c \right )}-\frac {8 \cot \left (d x +c \right )}{3}\right )}{d}\)	\(167\)
risch	\(\frac {-33 i a^{3} {\mathrm e}^{5 i \left (d x +c \right )}+33 a^{3} {\mathrm e}^{6 i \left (d x +c \right )}+60 i a^{3} {\mathrm e}^{3 i \left (d x +c \right )}-96 a^{3} {\mathrm e}^{4 i \left (d x +c \right )}-19 i a^{3} {\mathrm e}^{i \left (d x +c \right )}+123 a^{3} {\mathrm e}^{2 i \left (d x +c \right )}-52 a^{3}}{3 \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )^{3} \left ({\mathrm e}^{i \left (d x +c \right )}-i\right ) d}+\frac {11 a^{3} \ln \left ({\mathrm e}^{i \left (d x +c \right )}-1\right )}{2 d}-\frac {11 a^{3} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+1\right )}{2 d}\)	\(167\)
norman	\(\frac {\frac {a^{3}}{24 d}+\frac {3 a^{3} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{8 d}+\frac {59 a^{3} \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{24 d}-\frac {45 a^{3} \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8 d}-\frac {231 a^{3} \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8 d}-\frac {51 a^{3} \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2 d}-\frac {231 a^{3} \left (\tan ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8 d}-\frac {45 a^{3} \left (\tan ^{10}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8 d}+\frac {59 a^{3} \left (\tan ^{12}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{24 d}+\frac {3 a^{3} \left (\tan ^{13}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8 d}+\frac {a^{3} \left (\tan ^{14}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{24 d}-\frac {13 a^{3} \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2 d}-\frac {79 a^{3} \left (\tan ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8 d}-\frac {183 a^{3} \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8 d}}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3} \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )-1\right ) \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{3}}+\frac {11 a^{3} \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2 d}\)	\(316\)

[In]

int(csc(d*x+c)^4*sec(d*x+c)^2*(a+a*sin(d*x+c))^3,x,method=_RETURNVERBOSE)

[Out]

1/24*(-306+132*ln(tan(1/2*d*x+1/2*c))*(tan(1/2*d*x+1/2*c)-1)+tan(1/2*d*x+1/2*c)^4+cot(1/2*d*x+1/2*c)^3+8*tan(1
/2*d*x+1/2*c)^3+8*cot(1/2*d*x+1/2*c)^2+48*tan(1/2*d*x+1/2*c)^2+48*cot(1/2*d*x+1/2*c))*a^3/d/(tan(1/2*d*x+1/2*c
)-1)

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 354 vs. \(2 (90) = 180\).

Time = 0.27 (sec) , antiderivative size = 354, normalized size of antiderivative = 3.61 \[ \int \csc ^4(c+d x) \sec ^2(c+d x) (a+a \sin (c+d x))^3 \, dx=-\frac {104 \, a^{3} \cos \left (d x + c\right )^{4} + 38 \, a^{3} \cos \left (d x + c\right )^{3} - 156 \, a^{3} \cos \left (d x + c\right )^{2} - 42 \, a^{3} \cos \left (d x + c\right ) + 48 \, a^{3} + 33 \, {\left (a^{3} \cos \left (d x + c\right )^{4} - 2 \, a^{3} \cos \left (d x + c\right )^{2} + a^{3} + {\left (a^{3} \cos \left (d x + c\right )^{3} + a^{3} \cos \left (d x + c\right )^{2} - a^{3} \cos \left (d x + c\right ) - a^{3}\right )} \sin \left (d x + c\right )\right )} \log \left (\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) - 33 \, {\left (a^{3} \cos \left (d x + c\right )^{4} - 2 \, a^{3} \cos \left (d x + c\right )^{2} + a^{3} + {\left (a^{3} \cos \left (d x + c\right )^{3} + a^{3} \cos \left (d x + c\right )^{2} - a^{3} \cos \left (d x + c\right ) - a^{3}\right )} \sin \left (d x + c\right )\right )} \log \left (-\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) - 2 \, {\left (52 \, a^{3} \cos \left (d x + c\right )^{3} + 33 \, a^{3} \cos \left (d x + c\right )^{2} - 45 \, a^{3} \cos \left (d x + c\right ) - 24 \, a^{3}\right )} \sin \left (d x + c\right )}{12 \, {\left (d \cos \left (d x + c\right )^{4} - 2 \, d \cos \left (d x + c\right )^{2} + {\left (d \cos \left (d x + c\right )^{3} + d \cos \left (d x + c\right )^{2} - d \cos \left (d x + c\right ) - d\right )} \sin \left (d x + c\right ) + d\right )}} \]

[In]

integrate(csc(d*x+c)^4*sec(d*x+c)^2*(a+a*sin(d*x+c))^3,x, algorithm="fricas")

[Out]

-1/12*(104*a^3*cos(d*x + c)^4 + 38*a^3*cos(d*x + c)^3 - 156*a^3*cos(d*x + c)^2 - 42*a^3*cos(d*x + c) + 48*a^3
+ 33*(a^3*cos(d*x + c)^4 - 2*a^3*cos(d*x + c)^2 + a^3 + (a^3*cos(d*x + c)^3 + a^3*cos(d*x + c)^2 - a^3*cos(d*x
+ c) - a^3)*sin(d*x + c))*log(1/2*cos(d*x + c) + 1/2) - 33*(a^3*cos(d*x + c)^4 - 2*a^3*cos(d*x + c)^2 + a^3 +
(a^3*cos(d*x + c)^3 + a^3*cos(d*x + c)^2 - a^3*cos(d*x + c) - a^3)*sin(d*x + c))*log(-1/2*cos(d*x + c) + 1/2)
- 2*(52*a^3*cos(d*x + c)^3 + 33*a^3*cos(d*x + c)^2 - 45*a^3*cos(d*x + c) - 24*a^3)*sin(d*x + c))/(d*cos(d*x +
c)^4 - 2*d*cos(d*x + c)^2 + (d*cos(d*x + c)^3 + d*cos(d*x + c)^2 - d*cos(d*x + c) - d)*sin(d*x + c) + d)

Sympy [F(-1)]

Timed out. \[ \int \csc ^4(c+d x) \sec ^2(c+d x) (a+a \sin (c+d x))^3 \, dx=\text {Timed out} \]

[In]

integrate(csc(d*x+c)**4*sec(d*x+c)**2*(a+a*sin(d*x+c))**3,x)

[Out]

Timed out

Maxima [A] (verification not implemented)

none

Time = 0.21 (sec) , antiderivative size = 160, normalized size of antiderivative = 1.63 \[ \int \csc ^4(c+d x) \sec ^2(c+d x) (a+a \sin (c+d x))^3 \, dx=\frac {9 \, a^{3} {\left (\frac {2 \, {\left (3 \, \cos \left (d x + c\right )^{2} - 2\right )}}{\cos \left (d x + c\right )^{3} - \cos \left (d x + c\right )} - 3 \, \log \left (\cos \left (d x + c\right ) + 1\right ) + 3 \, \log \left (\cos \left (d x + c\right ) - 1\right )\right )} + 6 \, a^{3} {\left (\frac {2}{\cos \left (d x + c\right )} - \log \left (\cos \left (d x + c\right ) + 1\right ) + \log \left (\cos \left (d x + c\right ) - 1\right )\right )} - 36 \, a^{3} {\left (\frac {1}{\tan \left (d x + c\right )} - \tan \left (d x + c\right )\right )} - 4 \, a^{3} {\left (\frac {6 \, \tan \left (d x + c\right )^{2} + 1}{\tan \left (d x + c\right )^{3}} - 3 \, \tan \left (d x + c\right )\right )}}{12 \, d} \]

[In]

integrate(csc(d*x+c)^4*sec(d*x+c)^2*(a+a*sin(d*x+c))^3,x, algorithm="maxima")

[Out]

1/12*(9*a^3*(2*(3*cos(d*x + c)^2 - 2)/(cos(d*x + c)^3 - cos(d*x + c)) - 3*log(cos(d*x + c) + 1) + 3*log(cos(d*
x + c) - 1)) + 6*a^3*(2/cos(d*x + c) - log(cos(d*x + c) + 1) + log(cos(d*x + c) - 1)) - 36*a^3*(1/tan(d*x + c)
- tan(d*x + c)) - 4*a^3*((6*tan(d*x + c)^2 + 1)/tan(d*x + c)^3 - 3*tan(d*x + c)))/d

Giac [A] (verification not implemented)

none

Time = 0.39 (sec) , antiderivative size = 148, normalized size of antiderivative = 1.51 \[ \int \csc ^4(c+d x) \sec ^2(c+d x) (a+a \sin (c+d x))^3 \, dx=\frac {a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 9 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 132 \, a^{3} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) \right |}\right ) + 57 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - \frac {192 \, a^{3}}{\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1} - \frac {242 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 57 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 9 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + a^{3}}{\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3}}}{24 \, d} \]

[In]

integrate(csc(d*x+c)^4*sec(d*x+c)^2*(a+a*sin(d*x+c))^3,x, algorithm="giac")

[Out]

1/24*(a^3*tan(1/2*d*x + 1/2*c)^3 + 9*a^3*tan(1/2*d*x + 1/2*c)^2 + 132*a^3*log(abs(tan(1/2*d*x + 1/2*c))) + 57*
a^3*tan(1/2*d*x + 1/2*c) - 192*a^3/(tan(1/2*d*x + 1/2*c) - 1) - (242*a^3*tan(1/2*d*x + 1/2*c)^3 + 57*a^3*tan(1
/2*d*x + 1/2*c)^2 + 9*a^3*tan(1/2*d*x + 1/2*c) + a^3)/tan(1/2*d*x + 1/2*c)^3)/d

Mupad [B] (verification not implemented)

Time = 10.89 (sec) , antiderivative size = 160, normalized size of antiderivative = 1.63 \[ \int \csc ^4(c+d x) \sec ^2(c+d x) (a+a \sin (c+d x))^3 \, dx=\frac {3\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{8\,d}+\frac {a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{24\,d}-\frac {-83\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3+16\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+\frac {8\,a^3\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{3}+\frac {a^3}{3}}{d\,\left (8\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3-8\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4\right )}+\frac {11\,a^3\,\ln \left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )}{2\,d}+\frac {19\,a^3\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{8\,d} \]

[In]

int((a + a*sin(c + d*x))^3/(cos(c + d*x)^2*sin(c + d*x)^4),x)

[Out]

(3*a^3*tan(c/2 + (d*x)/2)^2)/(8*d) + (a^3*tan(c/2 + (d*x)/2)^3)/(24*d) - (16*a^3*tan(c/2 + (d*x)/2)^2 - 83*a^3
*tan(c/2 + (d*x)/2)^3 + a^3/3 + (8*a^3*tan(c/2 + (d*x)/2))/3)/(d*(8*tan(c/2 + (d*x)/2)^3 - 8*tan(c/2 + (d*x)/2
)^4)) + (11*a^3*log(tan(c/2 + (d*x)/2)))/(2*d) + (19*a^3*tan(c/2 + (d*x)/2))/(8*d)