\(\int \csc ^4(c+d x) \sec ^4(c+d x) (a+a \sin (c+d x))^3 \, dx\) [818]

Optimal result

Integrand size = 29, antiderivative size = 128 \[ \int \csc ^4(c+d x) \sec ^4(c+d x) (a+a \sin (c+d x))^3 \, dx=-\frac {17 a^3 \text {arctanh}(\cos (c+d x))}{2 d}-\frac {6 a^3 \cot (c+d x)}{d}-\frac {a^3 \cot ^3(c+d x)}{3 d}-\frac {3 a^3 \cot (c+d x) \csc (c+d x)}{2 d}+\frac {2 a^3 \cos (c+d x)}{3 d (1-\sin (c+d x))^2}+\frac {23 a^3 \cos (c+d x)}{3 d (1-\sin (c+d x))} \]

[Out]

-17/2*a^3*arctanh(cos(d*x+c))/d-6*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+2/3*
a^3*cos(d*x+c)/d/(1-sin(d*x+c))^2+23/3*a^3*cos(d*x+c)/d/(1-sin(d*x+c))

Rubi [A] (verified)

Time = 0.15 (sec) , antiderivative size = 128, normalized size of antiderivative = 1.00, number of steps used = 12, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.241, Rules used = {2951, 3855, 3852, 8, 3853, 2729, 2727} \[ \int \csc ^4(c+d x) \sec ^4(c+d x) (a+a \sin (c+d x))^3 \, dx=-\frac {17 a^3 \text {arctanh}(\cos (c+d x))}{2 d}-\frac {a^3 \cot ^3(c+d x)}{3 d}-\frac {6 a^3 \cot (c+d x)}{d}+\frac {23 a^3 \cos (c+d x)}{3 d (1-\sin (c+d x))}+\frac {2 a^3 \cos (c+d x)}{3 d (1-\sin (c+d x))^2}-\frac {3 a^3 \cot (c+d x) \csc (c+d x)}{2 d} \]

[In]

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

[Out]

(-17*a^3*ArcTanh[Cos[c + d*x]])/(2*d) - (6*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) + (2*a^3*Cos[c + d*x])/(3*d*(1 - Sin[c + d*x])^2) + (23*a^3*Cos[c + d*x])/(3*d*(1 - Si
n[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 2729

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

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^4 \int \left (\frac {7 \csc (c+d x)}{a}+\frac {5 \csc ^2(c+d x)}{a}+\frac {3 \csc ^3(c+d x)}{a}+\frac {\csc ^4(c+d x)}{a}+\frac {2}{a (-1+\sin (c+d x))^2}-\frac {7}{a (-1+\sin (c+d x))}\right ) \, dx \\ & = a^3 \int \csc ^4(c+d x) \, dx+\left (2 a^3\right ) \int \frac {1}{(-1+\sin (c+d x))^2} \, dx+\left (3 a^3\right ) \int \csc ^3(c+d x) \, dx+\left (5 a^3\right ) \int \csc ^2(c+d x) \, dx+\left (7 a^3\right ) \int \csc (c+d x) \, dx-\left (7 a^3\right ) \int \frac {1}{-1+\sin (c+d x)} \, dx \\ & = -\frac {7 a^3 \text {arctanh}(\cos (c+d x))}{d}-\frac {3 a^3 \cot (c+d x) \csc (c+d x)}{2 d}+\frac {2 a^3 \cos (c+d x)}{3 d (1-\sin (c+d x))^2}+\frac {7 a^3 \cos (c+d x)}{d (1-\sin (c+d x))}-\frac {1}{3} \left (2 a^3\right ) \int \frac {1}{-1+\sin (c+d x)} \, dx+\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 (5 a^3\right ) \text {Subst}(\int 1 \, dx,x,\cot (c+d x))}{d} \\ & = -\frac {17 a^3 \text {arctanh}(\cos (c+d x))}{2 d}-\frac {6 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 {2 a^3 \cos (c+d x)}{3 d (1-\sin (c+d x))^2}+\frac {23 a^3 \cos (c+d x)}{3 d (1-\sin (c+d x))} \\ \end{align*}

Mathematica [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(287\) vs. \(2(128)=256\).

Time = 6.41 (sec) , antiderivative size = 287, normalized size of antiderivative = 2.24 \[ \int \csc ^4(c+d x) \sec ^4(c+d x) (a+a \sin (c+d x))^3 \, dx=a^3 \left (-\frac {17 \cot \left (\frac {1}{2} (c+d x)\right )}{6 d}-\frac {3 \csc ^2\left (\frac {1}{2} (c+d x)\right )}{8 d}-\frac {\cot \left (\frac {1}{2} (c+d x)\right ) \csc ^2\left (\frac {1}{2} (c+d x)\right )}{24 d}-\frac {17 \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )}{2 d}+\frac {17 \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )}{2 d}+\frac {3 \sec ^2\left (\frac {1}{2} (c+d x)\right )}{8 d}+\frac {2}{3 d \left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )^2}+\frac {4 \sin \left (\frac {1}{2} (c+d x)\right )}{3 d \left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )^3}+\frac {46 \sin \left (\frac {1}{2} (c+d x)\right )}{3 d \left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )}+\frac {17 \tan \left (\frac {1}{2} (c+d x)\right )}{6 d}+\frac {\sec ^2\left (\frac {1}{2} (c+d x)\right ) \tan \left (\frac {1}{2} (c+d x)\right )}{24 d}\right ) \]

[In]

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

[Out]

a^3*((-17*Cot[(c + d*x)/2])/(6*d) - (3*Csc[(c + d*x)/2]^2)/(8*d) - (Cot[(c + d*x)/2]*Csc[(c + d*x)/2]^2)/(24*d
) - (17*Log[Cos[(c + d*x)/2]])/(2*d) + (17*Log[Sin[(c + d*x)/2]])/(2*d) + (3*Sec[(c + d*x)/2]^2)/(8*d) + 2/(3*
d*(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])^2) + (4*Sin[(c + d*x)/2])/(3*d*(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])^3
) + (46*Sin[(c + d*x)/2])/(3*d*(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])) + (17*Tan[(c + d*x)/2])/(6*d) + (Sec[(c
+ d*x)/2]^2*Tan[(c + d*x)/2])/(24*d))

Maple [A] (verified)

Time = 0.40 (sec) , antiderivative size = 145, normalized size of antiderivative = 1.13

method	result	size
parallelrisch	\(\frac {\left (204 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{3} \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )+6 \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+45 \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\cot ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )+6 \left (\cot ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-846 \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+45 \cot \left (\frac {d x}{2}+\frac {c}{2}\right )+1440 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )-762\right ) a^{3}}{24 d \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{3}}\)	\(145\)
risch	\(\frac {-153 i a^{3} {\mathrm e}^{7 i \left (d x +c \right )}+51 a^{3} {\mathrm e}^{8 i \left (d x +c \right )}+459 i a^{3} {\mathrm e}^{5 i \left (d x +c \right )}-289 a^{3} {\mathrm e}^{6 i \left (d x +c \right )}-511 i a^{3} {\mathrm e}^{3 i \left (d x +c \right )}+501 a^{3} {\mathrm e}^{4 i \left (d x +c \right )}+189 i a^{3} {\mathrm e}^{i \left (d x +c \right )}-327 a^{3} {\mathrm e}^{2 i \left (d x +c \right )}+80 a^{3}}{3 \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )^{3} \left ({\mathrm e}^{i \left (d x +c \right )}-i\right )^{3} d}+\frac {17 a^{3} \ln \left ({\mathrm e}^{i \left (d x +c \right )}-1\right )}{2 d}-\frac {17 a^{3} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+1\right )}{2 d}\)	\(196\)
derivativedivides	\(\frac {a^{3} \left (\frac {1}{3 \cos \left (d x +c \right )^{3}}+\frac {1}{\cos \left (d x +c \right )}+\ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )\right )+3 a^{3} \left (\frac {1}{3 \sin \left (d x +c \right ) \cos \left (d x +c \right )^{3}}+\frac {4}{3 \sin \left (d x +c \right ) \cos \left (d x +c \right )}-\frac {8 \cot \left (d x +c \right )}{3}\right )+3 a^{3} \left (\frac {1}{3 \sin \left (d x +c \right )^{2} \cos \left (d x +c \right )^{3}}-\frac {5}{6 \sin \left (d x +c \right )^{2} \cos \left (d x +c \right )}+\frac {5}{2 \cos \left (d x +c \right )}+\frac {5 \ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )}{2}\right )+a^{3} \left (\frac {1}{3 \sin \left (d x +c \right )^{3} \cos \left (d x +c \right )^{3}}-\frac {2}{3 \sin \left (d x +c \right )^{3} \cos \left (d x +c \right )}+\frac {8}{3 \sin \left (d x +c \right ) \cos \left (d x +c \right )}-\frac {16 \cot \left (d x +c \right )}{3}\right )}{d}\)	\(232\)
default	\(\frac {a^{3} \left (\frac {1}{3 \cos \left (d x +c \right )^{3}}+\frac {1}{\cos \left (d x +c \right )}+\ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )\right )+3 a^{3} \left (\frac {1}{3 \sin \left (d x +c \right ) \cos \left (d x +c \right )^{3}}+\frac {4}{3 \sin \left (d x +c \right ) \cos \left (d x +c \right )}-\frac {8 \cot \left (d x +c \right )}{3}\right )+3 a^{3} \left (\frac {1}{3 \sin \left (d x +c \right )^{2} \cos \left (d x +c \right )^{3}}-\frac {5}{6 \sin \left (d x +c \right )^{2} \cos \left (d x +c \right )}+\frac {5}{2 \cos \left (d x +c \right )}+\frac {5 \ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )}{2}\right )+a^{3} \left (\frac {1}{3 \sin \left (d x +c \right )^{3} \cos \left (d x +c \right )^{3}}-\frac {2}{3 \sin \left (d x +c \right )^{3} \cos \left (d x +c \right )}+\frac {8}{3 \sin \left (d x +c \right ) \cos \left (d x +c \right )}-\frac {16 \cot \left (d x +c \right )}{3}\right )}{d}\)	\(232\)

[In]

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

[Out]

1/24*(204*(tan(1/2*d*x+1/2*c)-1)^3*ln(tan(1/2*d*x+1/2*c))+tan(1/2*d*x+1/2*c)^6+6*tan(1/2*d*x+1/2*c)^5+45*tan(1
/2*d*x+1/2*c)^4+cot(1/2*d*x+1/2*c)^3+6*cot(1/2*d*x+1/2*c)^2-846*tan(1/2*d*x+1/2*c)^2+45*cot(1/2*d*x+1/2*c)+144
0*tan(1/2*d*x+1/2*c)-762)*a^3/d/(tan(1/2*d*x+1/2*c)-1)^3

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 528 vs. \(2 (114) = 228\).

Time = 0.29 (sec) , antiderivative size = 528, normalized size of antiderivative = 4.12 \[ \int \csc ^4(c+d x) \sec ^4(c+d x) (a+a \sin (c+d x))^3 \, dx=\frac {160 \, a^{3} \cos \left (d x + c\right )^{5} - 58 \, a^{3} \cos \left (d x + c\right )^{4} - 356 \, a^{3} \cos \left (d x + c\right )^{3} + 70 \, a^{3} \cos \left (d x + c\right )^{2} + 200 \, a^{3} \cos \left (d x + c\right ) - 8 \, a^{3} - 51 \, {\left (a^{3} \cos \left (d x + c\right )^{5} + 2 \, a^{3} \cos \left (d x + c\right )^{4} - 2 \, a^{3} \cos \left (d x + c\right )^{3} - 4 \, a^{3} \cos \left (d x + c\right )^{2} + a^{3} \cos \left (d x + c\right ) + 2 \, a^{3} - {\left (a^{3} \cos \left (d x + c\right )^{4} - a^{3} \cos \left (d x + c\right )^{3} - 3 \, a^{3} \cos \left (d x + c\right )^{2} + a^{3} \cos \left (d x + c\right ) + 2 \, 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 ) + 51 \, {\left (a^{3} \cos \left (d x + c\right )^{5} + 2 \, a^{3} \cos \left (d x + c\right )^{4} - 2 \, a^{3} \cos \left (d x + c\right )^{3} - 4 \, a^{3} \cos \left (d x + c\right )^{2} + a^{3} \cos \left (d x + c\right ) + 2 \, a^{3} - {\left (a^{3} \cos \left (d x + c\right )^{4} - a^{3} \cos \left (d x + c\right )^{3} - 3 \, a^{3} \cos \left (d x + c\right )^{2} + a^{3} \cos \left (d x + c\right ) + 2 \, 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 (80 \, a^{3} \cos \left (d x + c\right )^{4} + 109 \, a^{3} \cos \left (d x + c\right )^{3} - 69 \, a^{3} \cos \left (d x + c\right )^{2} - 104 \, a^{3} \cos \left (d x + c\right ) - 4 \, a^{3}\right )} \sin \left (d x + c\right )}{12 \, {\left (d \cos \left (d x + c\right )^{5} + 2 \, d \cos \left (d x + c\right )^{4} - 2 \, d \cos \left (d x + c\right )^{3} - 4 \, d \cos \left (d x + c\right )^{2} + d \cos \left (d x + c\right ) - {\left (d \cos \left (d x + c\right )^{4} - d \cos \left (d x + c\right )^{3} - 3 \, d \cos \left (d x + c\right )^{2} + d \cos \left (d x + c\right ) + 2 \, d\right )} \sin \left (d x + c\right ) + 2 \, d\right )}} \]

[In]

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

[Out]

1/12*(160*a^3*cos(d*x + c)^5 - 58*a^3*cos(d*x + c)^4 - 356*a^3*cos(d*x + c)^3 + 70*a^3*cos(d*x + c)^2 + 200*a^
3*cos(d*x + c) - 8*a^3 - 51*(a^3*cos(d*x + c)^5 + 2*a^3*cos(d*x + c)^4 - 2*a^3*cos(d*x + c)^3 - 4*a^3*cos(d*x
+ c)^2 + a^3*cos(d*x + c) + 2*a^3 - (a^3*cos(d*x + c)^4 - a^3*cos(d*x + c)^3 - 3*a^3*cos(d*x + c)^2 + a^3*cos(
d*x + c) + 2*a^3)*sin(d*x + c))*log(1/2*cos(d*x + c) + 1/2) + 51*(a^3*cos(d*x + c)^5 + 2*a^3*cos(d*x + c)^4 -
2*a^3*cos(d*x + c)^3 - 4*a^3*cos(d*x + c)^2 + a^3*cos(d*x + c) + 2*a^3 - (a^3*cos(d*x + c)^4 - a^3*cos(d*x + c
)^3 - 3*a^3*cos(d*x + c)^2 + a^3*cos(d*x + c) + 2*a^3)*sin(d*x + c))*log(-1/2*cos(d*x + c) + 1/2) + 2*(80*a^3*
cos(d*x + c)^4 + 109*a^3*cos(d*x + c)^3 - 69*a^3*cos(d*x + c)^2 - 104*a^3*cos(d*x + c) - 4*a^3)*sin(d*x + c))/
(d*cos(d*x + c)^5 + 2*d*cos(d*x + c)^4 - 2*d*cos(d*x + c)^3 - 4*d*cos(d*x + c)^2 + d*cos(d*x + c) - (d*cos(d*x
+ c)^4 - d*cos(d*x + c)^3 - 3*d*cos(d*x + c)^2 + d*cos(d*x + c) + 2*d)*sin(d*x + c) + 2*d)

Sympy [F(-1)]

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

[In]

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

[Out]

Timed out

Maxima [A] (verification not implemented)

none

Time = 0.22 (sec) , antiderivative size = 205, normalized size of antiderivative = 1.60 \[ \int \csc ^4(c+d x) \sec ^4(c+d x) (a+a \sin (c+d x))^3 \, dx=\frac {12 \, {\left (\tan \left (d x + c\right )^{3} - \frac {3}{\tan \left (d x + c\right )} + 6 \, \tan \left (d x + c\right )\right )} a^{3} + 4 \, {\left (\tan \left (d x + c\right )^{3} - \frac {9 \, \tan \left (d x + c\right )^{2} + 1}{\tan \left (d x + c\right )^{3}} + 9 \, \tan \left (d x + c\right )\right )} a^{3} + 3 \, a^{3} {\left (\frac {2 \, {\left (15 \, \cos \left (d x + c\right )^{4} - 10 \, \cos \left (d x + c\right )^{2} - 2\right )}}{\cos \left (d x + c\right )^{5} - \cos \left (d x + c\right )^{3}} - 15 \, \log \left (\cos \left (d x + c\right ) + 1\right ) + 15 \, \log \left (\cos \left (d x + c\right ) - 1\right )\right )} + 2 \, a^{3} {\left (\frac {2 \, {\left (3 \, \cos \left (d x + c\right )^{2} + 1\right )}}{\cos \left (d x + c\right )^{3}} - 3 \, \log \left (\cos \left (d x + c\right ) + 1\right ) + 3 \, \log \left (\cos \left (d x + c\right ) - 1\right )\right )}}{12 \, d} \]

[In]

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

[Out]

1/12*(12*(tan(d*x + c)^3 - 3/tan(d*x + c) + 6*tan(d*x + c))*a^3 + 4*(tan(d*x + c)^3 - (9*tan(d*x + c)^2 + 1)/t
an(d*x + c)^3 + 9*tan(d*x + c))*a^3 + 3*a^3*(2*(15*cos(d*x + c)^4 - 10*cos(d*x + c)^2 - 2)/(cos(d*x + c)^5 - c
os(d*x + c)^3) - 15*log(cos(d*x + c) + 1) + 15*log(cos(d*x + c) - 1)) + 2*a^3*(2*(3*cos(d*x + c)^2 + 1)/cos(d*
x + c)^3 - 3*log(cos(d*x + c) + 1) + 3*log(cos(d*x + c) - 1)))/d

Giac [A] (verification not implemented)

none

Time = 0.36 (sec) , antiderivative size = 194, normalized size of antiderivative = 1.52 \[ \int \csc ^4(c+d x) \sec ^4(c+d x) (a+a \sin (c+d x))^3 \, dx=\frac {a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 9 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 204 \, a^{3} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) \right |}\right ) + 69 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - \frac {187 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} - 60 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 405 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 394 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 45 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 6 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - a^{3}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}^{3}}}{24 \, d} \]

[In]

integrate(csc(d*x+c)^4*sec(d*x+c)^4*(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 + 204*a^3*log(abs(tan(1/2*d*x + 1/2*c))) + 69*
a^3*tan(1/2*d*x + 1/2*c) - (187*a^3*tan(1/2*d*x + 1/2*c)^6 - 60*a^3*tan(1/2*d*x + 1/2*c)^5 - 405*a^3*tan(1/2*d
*x + 1/2*c)^4 + 394*a^3*tan(1/2*d*x + 1/2*c)^3 - 45*a^3*tan(1/2*d*x + 1/2*c)^2 - 6*a^3*tan(1/2*d*x + 1/2*c) -
a^3)/(tan(1/2*d*x + 1/2*c)^2 - tan(1/2*d*x + 1/2*c))^3)/d

Mupad [B] (verification not implemented)

Time = 14.33 (sec) , antiderivative size = 239, normalized size of antiderivative = 1.87 \[ \int \csc ^4(c+d x) \sec ^4(c+d x) (a+a \sin (c+d x))^3 \, dx=\frac {a^3\,\left (6\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )+45\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2-581\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3+897\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4-303\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5-181\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6+45\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7+6\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9-204\,\ln \left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3+612\,\ln \left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4-612\,\ln \left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5+204\,\ln \left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6+1\right )}{24\,d\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3\,{\left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )-1\right )}^3} \]

[In]

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

[Out]

(a^3*(6*tan(c/2 + (d*x)/2) + 45*tan(c/2 + (d*x)/2)^2 - 581*tan(c/2 + (d*x)/2)^3 + 897*tan(c/2 + (d*x)/2)^4 - 3
03*tan(c/2 + (d*x)/2)^5 - 181*tan(c/2 + (d*x)/2)^6 + 45*tan(c/2 + (d*x)/2)^7 + 6*tan(c/2 + (d*x)/2)^8 + tan(c/
2 + (d*x)/2)^9 - 204*log(tan(c/2 + (d*x)/2))*tan(c/2 + (d*x)/2)^3 + 612*log(tan(c/2 + (d*x)/2))*tan(c/2 + (d*x
)/2)^4 - 612*log(tan(c/2 + (d*x)/2))*tan(c/2 + (d*x)/2)^5 + 204*log(tan(c/2 + (d*x)/2))*tan(c/2 + (d*x)/2)^6 +
1))/(24*d*tan(c/2 + (d*x)/2)^3*(tan(c/2 + (d*x)/2) - 1)^3)