\(\int \csc (c+d x) \sec ^2(c+d x) (a+a \sin (c+d x))^3 \, dx\) [768]

Optimal result

Integrand size = 27, antiderivative size = 48 \[ \int \csc (c+d x) \sec ^2(c+d x) (a+a \sin (c+d x))^3 \, dx=-a^3 x-\frac {a^3 \text {arctanh}(\cos (c+d x))}{d}+\frac {4 a^3 \cos (c+d x)}{d (1-\sin (c+d x))} \]

[Out]

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

Rubi [A] (verified)

Time = 0.09 (sec) , antiderivative size = 48, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {2951, 3855, 2727} \[ \int \csc (c+d x) \sec ^2(c+d x) (a+a \sin (c+d x))^3 \, dx=-\frac {a^3 \text {arctanh}(\cos (c+d x))}{d}+\frac {4 a^3 \cos (c+d x)}{d (1-\sin (c+d x))}+a^3 (-x) \]

[In]

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

[Out]

-(a^3*x) - (a^3*ArcTanh[Cos[c + d*x]])/d + (4*a^3*Cos[c + d*x])/(d*(1 - Sin[c + d*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 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 (-a+a \csc (c+d x)-\frac {4 a}{-1+\sin (c+d x)}\right ) \, dx \\ & = -a^3 x+a^3 \int \csc (c+d x) \, dx-\left (4 a^3\right ) \int \frac {1}{-1+\sin (c+d x)} \, dx \\ & = -a^3 x-\frac {a^3 \text {arctanh}(\cos (c+d x))}{d}+\frac {4 a^3 \cos (c+d x)}{d (1-\sin (c+d x))} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.35 (sec) , antiderivative size = 74, normalized size of antiderivative = 1.54 \[ \int \csc (c+d x) \sec ^2(c+d x) (a+a \sin (c+d x))^3 \, dx=-\frac {a^3 \left (c+d x+\log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )-\log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )-\frac {8 \sin \left (\frac {1}{2} (c+d x)\right )}{\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )}\right )}{d} \]

[In]

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

[Out]

-((a^3*(c + d*x + Log[Cos[(c + d*x)/2]] - Log[Sin[(c + d*x)/2]] - (8*Sin[(c + d*x)/2])/(Cos[(c + d*x)/2] - Sin
[(c + d*x)/2])))/d)

Maple [C] (verified)

Result contains complex when optimal does not.

Time = 0.21 (sec) , antiderivative size = 69, normalized size of antiderivative = 1.44

method	result	size
risch	\(-a^{3} x +\frac {8 a^{3}}{d \left ({\mathrm e}^{i \left (d x +c \right )}-i\right )}+\frac {a^{3} \ln \left ({\mathrm e}^{i \left (d x +c \right )}-1\right )}{d}-\frac {a^{3} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+1\right )}{d}\)	\(69\)
parallelrisch	\(-\frac {a^{3} \left (8+\tan \left (\frac {d x}{2}+\frac {c}{2}\right ) d x -\ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )-d x +\ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )\right )}{d \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}\)	\(71\)
derivativedivides	\(\frac {a^{3} \left (\tan \left (d x +c \right )-d x -c \right )+\frac {3 a^{3}}{\cos \left (d x +c \right )}+3 a^{3} \tan \left (d x +c \right )+a^{3} \left (\frac {1}{\cos \left (d x +c \right )}+\ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )\right )}{d}\)	\(77\)
default	\(\frac {a^{3} \left (\tan \left (d x +c \right )-d x -c \right )+\frac {3 a^{3}}{\cos \left (d x +c \right )}+3 a^{3} \tan \left (d x +c \right )+a^{3} \left (\frac {1}{\cos \left (d x +c \right )}+\ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )\right )}{d}\)	\(77\)
norman	\(\frac {a^{3} x -\frac {8 a^{3}}{d}-\frac {8 a^{3} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{d}-\frac {24 a^{3} \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}-\frac {24 a^{3} \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}-\frac {24 a^{3} \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}-\frac {8 a^{3} \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}+2 a^{3} x \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-2 a^{3} x \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-a^{3} x \left (\tan ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-\frac {8 a^{3} \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}-\frac {24 a^{3} \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}}{\left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )-1\right ) \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{3}}+\frac {a^{3} \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}\)	\(246\)

[In]

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

[Out]

-a^3*x+8*a^3/d/(exp(I*(d*x+c))-I)+a^3/d*ln(exp(I*(d*x+c))-1)-a^3/d*ln(exp(I*(d*x+c))+1)

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 151 vs. \(2 (46) = 92\).

Time = 0.27 (sec) , antiderivative size = 151, normalized size of antiderivative = 3.15 \[ \int \csc (c+d x) \sec ^2(c+d x) (a+a \sin (c+d x))^3 \, dx=-\frac {2 \, a^{3} d x - 8 \, a^{3} + 2 \, {\left (a^{3} d x - 4 \, a^{3}\right )} \cos \left (d x + c\right ) + {\left (a^{3} \cos \left (d x + c\right ) - a^{3} \sin \left (d x + c\right ) + a^{3}\right )} \log \left (\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) - {\left (a^{3} \cos \left (d x + c\right ) - a^{3} \sin \left (d x + c\right ) + a^{3}\right )} \log \left (-\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) - 2 \, {\left (a^{3} d x + 4 \, a^{3}\right )} \sin \left (d x + c\right )}{2 \, {\left (d \cos \left (d x + c\right ) - d \sin \left (d x + c\right ) + d\right )}} \]

[In]

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

[Out]

-1/2*(2*a^3*d*x - 8*a^3 + 2*(a^3*d*x - 4*a^3)*cos(d*x + c) + (a^3*cos(d*x + c) - a^3*sin(d*x + c) + a^3)*log(1
/2*cos(d*x + c) + 1/2) - (a^3*cos(d*x + c) - a^3*sin(d*x + c) + a^3)*log(-1/2*cos(d*x + c) + 1/2) - 2*(a^3*d*x
+ 4*a^3)*sin(d*x + c))/(d*cos(d*x + c) - d*sin(d*x + c) + d)

Sympy [F(-1)]

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

[In]

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

[Out]

Timed out

Maxima [A] (verification not implemented)

none

Time = 0.33 (sec) , antiderivative size = 84, normalized size of antiderivative = 1.75 \[ \int \csc (c+d x) \sec ^2(c+d x) (a+a \sin (c+d x))^3 \, dx=-\frac {2 \, {\left (d x + c - \tan \left (d x + c\right )\right )} a^{3} - a^{3} {\left (\frac {2}{\cos \left (d x + c\right )} - \log \left (\cos \left (d x + c\right ) + 1\right ) + \log \left (\cos \left (d x + c\right ) - 1\right )\right )} - 6 \, a^{3} \tan \left (d x + c\right ) - \frac {6 \, a^{3}}{\cos \left (d x + c\right )}}{2 \, d} \]

[In]

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

[Out]

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

Giac [A] (verification not implemented)

none

Time = 0.38 (sec) , antiderivative size = 49, normalized size of antiderivative = 1.02 \[ \int \csc (c+d x) \sec ^2(c+d x) (a+a \sin (c+d x))^3 \, dx=-\frac {{\left (d x + c\right )} a^{3} - a^{3} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) \right |}\right ) + \frac {8 \, a^{3}}{\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1}}{d} \]

[In]

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

[Out]

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

Mupad [B] (verification not implemented)

Time = 9.94 (sec) , antiderivative size = 112, normalized size of antiderivative = 2.33 \[ \int \csc (c+d x) \sec ^2(c+d x) (a+a \sin (c+d x))^3 \, dx=\frac {a^3\,\ln \left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )}{d}+\frac {2\,a^3\,\mathrm {atan}\left (\frac {4\,a^6}{4\,a^6+4\,a^6\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}-\frac {4\,a^6\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{4\,a^6+4\,a^6\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )}{d}-\frac {8\,a^3}{d\,\left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )-1\right )} \]

[In]

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

[Out]

(a^3*log(tan(c/2 + (d*x)/2)))/d + (2*a^3*atan((4*a^6)/(4*a^6 + 4*a^6*tan(c/2 + (d*x)/2)) - (4*a^6*tan(c/2 + (d
*x)/2))/(4*a^6 + 4*a^6*tan(c/2 + (d*x)/2))))/d - (8*a^3)/(d*(tan(c/2 + (d*x)/2) - 1))