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

Optimal. Leaf size=48 \[ \frac{4 a^3 \cos (c+d x)}{d (1-\sin (c+d x))}-\frac{a^3 \tanh ^{-1}(\cos (c+d x))}{d}+a^3 (-x) \]

[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]))

________________________________________________________________________________________

Rubi [A]  time = 0.104201, antiderivative size = 48, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.111, Rules used = {2872, 3770, 2648} \[ \frac{4 a^3 \cos (c+d x)}{d (1-\sin (c+d x))}-\frac{a^3 \tanh ^{-1}(\cos (c+d x))}{d}+a^3 (-x) \]

Antiderivative was successfully verified.

[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 2872

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 3770

Int[csc[(c_.) + (d_.)*(x_)], x_Symbol] :> -Simp[ArcTanh[Cos[c + d*x]]/d, x] /; FreeQ[{c, d}, x]

Rule 2648

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]

Rubi steps

\begin{align*} \int \csc (c+d x) \sec ^2(c+d x) (a+a \sin (c+d x))^3 \, dx &=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 \tanh ^{-1}(\cos (c+d x))}{d}+\frac{4 a^3 \cos (c+d x)}{d (1-\sin (c+d x))}\\ \end{align*}

Mathematica [A]  time = 0.133245, size = 74, normalized size = 1.54 \[ -\frac{a^3 \left (-\log \left (\sin \left (\frac{1}{2} (c+d x)\right )\right )+\log \left (\cos \left (\frac{1}{2} (c+d x)\right )\right )-\frac{8 \sin \left (\frac{1}{2} (c+d x)\right )}{\cos \left (\frac{1}{2} (c+d x)\right )-\sin \left (\frac{1}{2} (c+d x)\right )}+c+d x\right )}{d} \]

Antiderivative was successfully verified.

[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 [A]  time = 0.096, size = 70, normalized size = 1.5 \begin{align*} -{a}^{3}x+4\,{\frac{{a}^{3}\tan \left ( dx+c \right ) }{d}}-{\frac{{a}^{3}c}{d}}+4\,{\frac{{a}^{3}}{d\cos \left ( dx+c \right ) }}+{\frac{{a}^{3}\ln \left ( \csc \left ( dx+c \right ) -\cot \left ( dx+c \right ) \right ) }{d}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-a^3*x+4*a^3*tan(d*x+c)/d-1/d*a^3*c+4/d*a^3/cos(d*x+c)+1/d*a^3*ln(csc(d*x+c)-cot(d*x+c))

________________________________________________________________________________________

Maxima [A]  time = 1.47459, size = 113, normalized size = 2.35 \begin{align*} -\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} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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

________________________________________________________________________________________

Fricas [B]  time = 1.12461, size = 382, normalized size = 7.96 \begin{align*} -\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 )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

________________________________________________________________________________________

Giac [A]  time = 1.27733, size = 66, normalized size = 1.38 \begin{align*} -\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} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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