\(\int \csc ^3(c+d x) \sec ^2(c+d x) (a+a \sin (c+d x))^3 \, dx\) [770]

Optimal result

Integrand size = 29, antiderivative size = 80 \[ \int \csc ^3(c+d x) \sec ^2(c+d x) (a+a \sin (c+d x))^3 \, dx=-\frac {9 a^3 \text {arctanh}(\cos (c+d x))}{2 d}-\frac {3 a^3 \cot (c+d x)}{d}-\frac {a^3 \cot (c+d x) \csc (c+d x)}{2 d}+\frac {4 a^3 \cos (c+d x)}{d (1-\sin (c+d x))} \] Output:

-9/2*a^3*arctanh(cos(d*x+c))/d-3*a^3*cot(d*x+c)/d-1/2*a^3*cot(d*x+c)*csc(d 
*x+c)/d+4*a^3*cos(d*x+c)/d/(1-sin(d*x+c))
 

Mathematica [A] (verified)

Time = 1.66 (sec) , antiderivative size = 124, normalized size of antiderivative = 1.55 \[ \int \csc ^3(c+d x) \sec ^2(c+d x) (a+a \sin (c+d x))^3 \, dx=\frac {a^3 \left (-12 \cot \left (\frac {1}{2} (c+d x)\right )-\csc ^2\left (\frac {1}{2} (c+d x)\right )-36 \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )+36 \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )+\sec ^2\left (\frac {1}{2} (c+d x)\right )+\frac {64 \sin \left (\frac {1}{2} (c+d x)\right )}{\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )}+12 \tan \left (\frac {1}{2} (c+d x)\right )\right )}{8 d} \] Input:

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

Output:

(a^3*(-12*Cot[(c + d*x)/2] - Csc[(c + d*x)/2]^2 - 36*Log[Cos[(c + d*x)/2]] 
 + 36*Log[Sin[(c + d*x)/2]] + Sec[(c + d*x)/2]^2 + (64*Sin[(c + d*x)/2])/( 
Cos[(c + d*x)/2] - Sin[(c + d*x)/2]) + 12*Tan[(c + d*x)/2]))/(8*d)
 

Rubi [A] (verified)

Time = 0.37 (sec) , antiderivative size = 76, normalized size of antiderivative = 0.95, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.103, Rules used = {3042, 3351, 2009}

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

\(\displaystyle \int \csc ^3(c+d x) \sec ^2(c+d x) (a \sin (c+d x)+a)^3 \, dx\)

\(\Big \downarrow \) 3042

\(\displaystyle \int \frac {(a \sin (c+d x)+a)^3}{\sin (c+d x)^3 \cos (c+d x)^2}dx\)

\(\Big \downarrow \) 3351

\(\displaystyle a^2 \int \left (a \csc ^3(c+d x)+3 a \csc ^2(c+d x)+4 a \csc (c+d x)+\frac {4 a}{1-\sin (c+d x)}\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle a^2 \left (-\frac {9 a \text {arctanh}(\cos (c+d x))}{2 d}-\frac {3 a \cot (c+d x)}{d}+\frac {4 a \cos (c+d x)}{d (1-\sin (c+d x))}-\frac {a \cot (c+d x) \csc (c+d x)}{2 d}\right )\)

Input:

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

Output:

a^2*((-9*a*ArcTanh[Cos[c + d*x]])/(2*d) - (3*a*Cot[c + d*x])/d - (a*Cot[c 
+ d*x]*Csc[c + d*x])/(2*d) + (4*a*Cos[c + d*x])/(d*(1 - Sin[c + d*x])))
 

Defintions of rubi rules used

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

rule 3042

Int[u_, x_Symbol] :> Int[DeactivateTrig[u, x], x] /; FunctionOfTrigOfLinear 
Q[u, x]
 

rule 3351

Int[cos[(e_.) + (f_.)*(x_)]^(p_)*((d_.)*sin[(e_.) + (f_.)*(x_)])^(n_)*((a_) 
 + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_), x_Symbol] :> Simp[1/a^p   Int[Expan 
dTrig[(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] && In 
tegersQ[m, n, p/2] && ((GtQ[m, 0] && GtQ[p, 0] && LtQ[-m - p, n, -1]) || (G 
tQ[m, 2] && LtQ[p, 0] && GtQ[m + p/2, 0]))
 

Maple [A] (verified)

Time = 1.04 (sec) , antiderivative size = 93, normalized size of antiderivative = 1.16

method	result	size
parallelrisch	\(\frac {\left (-88+36 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}+\cot \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}+11 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}+11 \cot \left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a^{3}}{8 d \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}\)	\(93\)
risch	\(\frac {a^{3} \left (-7 i {\mathrm e}^{3 i \left (d x +c \right )}+9 \,{\mathrm e}^{4 i \left (d x +c \right )}+5 i {\mathrm e}^{i \left (d x +c \right )}-21 \,{\mathrm e}^{2 i \left (d x +c \right )}+14\right )}{\left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )^{2} \left ({\mathrm e}^{i \left (d x +c \right )}-i\right ) d}+\frac {9 a^{3} \ln \left ({\mathrm e}^{i \left (d x +c \right )}-1\right )}{2 d}-\frac {9 a^{3} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+1\right )}{2 d}\)	\(124\)
derivativedivides	\(\frac {a^{3} \tan \left (d x +c \right )+3 a^{3} \left (\frac {1}{\cos \left (d x +c \right )}+\ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )\right )+3 a^{3} \left (\frac {1}{\sin \left (d x +c \right ) \cos \left (d x +c \right )}-2 \cot \left (d x +c \right )\right )+a^{3} \left (-\frac {1}{2 \sin \left (d x +c \right )^{2} \cos \left (d x +c \right )}+\frac {3}{2 \cos \left (d x +c \right )}+\frac {3 \ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )}{2}\right )}{d}\)	\(128\)
default	\(\frac {a^{3} \tan \left (d x +c \right )+3 a^{3} \left (\frac {1}{\cos \left (d x +c \right )}+\ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )\right )+3 a^{3} \left (\frac {1}{\sin \left (d x +c \right ) \cos \left (d x +c \right )}-2 \cot \left (d x +c \right )\right )+a^{3} \left (-\frac {1}{2 \sin \left (d x +c \right )^{2} \cos \left (d x +c \right )}+\frac {3}{2 \cos \left (d x +c \right )}+\frac {3 \ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )}{2}\right )}{d}\)	\(128\)
norman	\(\frac {\frac {a^{3}}{8 d}-\frac {49 a^{3} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{6}}{2 d}+\frac {3 a^{3} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{2 d}-\frac {13 a^{3} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{2 d}-\frac {27 a^{3} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}{d}-\frac {27 a^{3} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{7}}{d}-\frac {13 a^{3} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{9}}{2 d}+\frac {3 a^{3} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{11}}{2 d}+\frac {a^{3} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{12}}{8 d}-\frac {15 a^{3} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}{2 d}-\frac {69 a^{3} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{8}}{8 d}-\frac {189 a^{3} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}}{8 d}}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-1\right ) \left (1+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right )^{3}}+\frac {9 a^{3} \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2 d}\)	\(278\)

Input:

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

Output:

1/8*(-88+36*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)^3+cot(1/2*d*x+1/2*c)^2+11*tan(1/2*d*x+1/2*c)^2+11*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. 300 vs. \(2 (74) = 148\).

Time = 0.08 (sec) , antiderivative size = 300, normalized size of antiderivative = 3.75 \[ \int \csc ^3(c+d x) \sec ^2(c+d x) (a+a \sin (c+d x))^3 \, dx=\frac {28 \, a^{3} \cos \left (d x + c\right )^{3} + 18 \, a^{3} \cos \left (d x + c\right )^{2} - 26 \, a^{3} \cos \left (d x + c\right ) - 16 \, a^{3} - 9 \, {\left (a^{3} \cos \left (d x + c\right )^{3} + a^{3} \cos \left (d x + c\right )^{2} - a^{3} \cos \left (d x + c\right ) - a^{3} - {\left (a^{3} \cos \left (d x + c\right )^{2} - a^{3}\right )} \sin \left (d x + c\right )\right )} \log \left (\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) + 9 \, {\left (a^{3} \cos \left (d x + c\right )^{3} + a^{3} \cos \left (d x + c\right )^{2} - a^{3} \cos \left (d x + c\right ) - a^{3} - {\left (a^{3} \cos \left (d x + c\right )^{2} - a^{3}\right )} \sin \left (d x + c\right )\right )} \log \left (-\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) + 2 \, {\left (14 \, a^{3} \cos \left (d x + c\right )^{2} + 5 \, a^{3} \cos \left (d x + c\right ) - 8 \, a^{3}\right )} \sin \left (d x + c\right )}{4 \, {\left (d \cos \left (d x + c\right )^{3} + d \cos \left (d x + c\right )^{2} - d \cos \left (d x + c\right ) - {\left (d \cos \left (d x + c\right )^{2} - d\right )} \sin \left (d x + c\right ) - d\right )}} \] Input:

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

Output:

1/4*(28*a^3*cos(d*x + c)^3 + 18*a^3*cos(d*x + c)^2 - 26*a^3*cos(d*x + c) - 
 16*a^3 - 9*(a^3*cos(d*x + c)^3 + a^3*cos(d*x + c)^2 - a^3*cos(d*x + c) - 
a^3 - (a^3*cos(d*x + c)^2 - a^3)*sin(d*x + c))*log(1/2*cos(d*x + c) + 1/2) 
 + 9*(a^3*cos(d*x + c)^3 + a^3*cos(d*x + c)^2 - a^3*cos(d*x + c) - a^3 - ( 
a^3*cos(d*x + c)^2 - a^3)*sin(d*x + c))*log(-1/2*cos(d*x + c) + 1/2) + 2*( 
14*a^3*cos(d*x + c)^2 + 5*a^3*cos(d*x + c) - 8*a^3)*sin(d*x + c))/(d*cos(d 
*x + c)^3 + d*cos(d*x + c)^2 - d*cos(d*x + c) - (d*cos(d*x + c)^2 - d)*sin 
(d*x + c) - d)
 

Sympy [F(-1)]

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

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

Output:

Timed out
 

Maxima [A] (verification not implemented)

Time = 0.03 (sec) , antiderivative size = 135, normalized size of antiderivative = 1.69 \[ \int \csc ^3(c+d x) \sec ^2(c+d x) (a+a \sin (c+d x))^3 \, dx=\frac {a^{3} {\left (\frac {2 \, {\left (3 \, \cos \left (d x + c\right )^{2} - 2\right )}}{\cos \left (d x + c\right )^{3} - \cos \left (d x + c\right )} - 3 \, \log \left (\cos \left (d x + c\right ) + 1\right ) + 3 \, \log \left (\cos \left (d x + c\right ) - 1\right )\right )} + 6 \, a^{3} {\left (\frac {2}{\cos \left (d x + c\right )} - \log \left (\cos \left (d x + c\right ) + 1\right ) + \log \left (\cos \left (d x + c\right ) - 1\right )\right )} - 12 \, a^{3} {\left (\frac {1}{\tan \left (d x + c\right )} - \tan \left (d x + c\right )\right )} + 4 \, a^{3} \tan \left (d x + c\right )}{4 \, d} \] Input:

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

Output:

1/4*(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) - lo 
g(cos(d*x + c) + 1) + log(cos(d*x + c) - 1)) - 12*a^3*(1/tan(d*x + c) - ta 
n(d*x + c)) + 4*a^3*tan(d*x + c))/d
 

Giac [A] (verification not implemented)

Time = 0.19 (sec) , antiderivative size = 116, normalized size of antiderivative = 1.45 \[ \int \csc ^3(c+d x) \sec ^2(c+d x) (a+a \sin (c+d x))^3 \, dx=\frac {a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 36 \, a^{3} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) \right |}\right ) + 12 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - \frac {64 \, a^{3}}{\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1} - \frac {54 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 12 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + a^{3}}{\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2}}}{8 \, d} \] Input:

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

Output:

1/8*(a^3*tan(1/2*d*x + 1/2*c)^2 + 36*a^3*log(abs(tan(1/2*d*x + 1/2*c))) + 
12*a^3*tan(1/2*d*x + 1/2*c) - 64*a^3/(tan(1/2*d*x + 1/2*c) - 1) - (54*a^3* 
tan(1/2*d*x + 1/2*c)^2 + 12*a^3*tan(1/2*d*x + 1/2*c) + a^3)/tan(1/2*d*x + 
1/2*c)^2)/d
 

Mupad [B] (verification not implemented)

Time = 32.09 (sec) , antiderivative size = 125, normalized size of antiderivative = 1.56 \[ \int \csc ^3(c+d x) \sec ^2(c+d x) (a+a \sin (c+d x))^3 \, dx=\frac {a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{8\,d}+\frac {9\,a^3\,\ln \left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )}{2\,d}+\frac {3\,a^3\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{2\,d}-\frac {-38\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+\frac {11\,a^3\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{2}+\frac {a^3}{2}}{d\,\left (4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2-4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3\right )} \] Input:

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

Output:

(a^3*tan(c/2 + (d*x)/2)^2)/(8*d) + (9*a^3*log(tan(c/2 + (d*x)/2)))/(2*d) + 
 (3*a^3*tan(c/2 + (d*x)/2))/(2*d) - (a^3/2 - 38*a^3*tan(c/2 + (d*x)/2)^2 + 
 (11*a^3*tan(c/2 + (d*x)/2))/2)/(d*(4*tan(c/2 + (d*x)/2)^2 - 4*tan(c/2 + ( 
d*x)/2)^3))
 

Reduce [B] (verification not implemented)

Time = 0.16 (sec) , antiderivative size = 128, normalized size of antiderivative = 1.60 \[ \int \csc ^3(c+d x) \sec ^2(c+d x) (a+a \sin (c+d x))^3 \, dx=\frac {a^{3} \left (36 \,\mathrm {log}\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}-36 \,\mathrm {log}\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}+11 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}-88 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}+11 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{8 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} d \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )} \] Input:

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

Output:

(a**3*(36*log(tan((c + d*x)/2))*tan((c + d*x)/2)**3 - 36*log(tan((c + d*x) 
/2))*tan((c + d*x)/2)**2 + tan((c + d*x)/2)**5 + 11*tan((c + d*x)/2)**4 - 
88*tan((c + d*x)/2)**3 + 11*tan((c + d*x)/2) + 1))/(8*tan((c + d*x)/2)**2* 
d*(tan((c + d*x)/2) - 1))