3.18 \(\int \csc ^3(c+d x) (a+b \tan (c+d x)) \, dx\)

Optimal. Leaf size=60 \[ -\frac{a \tanh ^{-1}(\cos (c+d x))}{2 d}-\frac{a \cot (c+d x) \csc (c+d x)}{2 d}-\frac{b \csc (c+d x)}{d}+\frac{b \tanh ^{-1}(\sin (c+d x))}{d} \]

[Out]

-(a*ArcTanh[Cos[c + d*x]])/(2*d) + (b*ArcTanh[Sin[c + d*x]])/d - (b*Csc[c + d*x])/d - (a*Cot[c + d*x]*Csc[c +
d*x])/(2*d)

________________________________________________________________________________________

Rubi [A]  time = 0.0679689, antiderivative size = 60, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.316, Rules used = {3517, 3768, 3770, 2621, 321, 207} \[ -\frac{a \tanh ^{-1}(\cos (c+d x))}{2 d}-\frac{a \cot (c+d x) \csc (c+d x)}{2 d}-\frac{b \csc (c+d x)}{d}+\frac{b \tanh ^{-1}(\sin (c+d x))}{d} \]

Antiderivative was successfully verified.

[In]

Int[Csc[c + d*x]^3*(a + b*Tan[c + d*x]),x]

[Out]

-(a*ArcTanh[Cos[c + d*x]])/(2*d) + (b*ArcTanh[Sin[c + d*x]])/d - (b*Csc[c + d*x])/d - (a*Cot[c + d*x]*Csc[c +
d*x])/(2*d)

Rule 3517

Int[sin[(e_.) + (f_.)*(x_)]^(m_.)*((a_) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(n_.), x_Symbol] :> Int[Expand[Sin[e
+ f*x]^m*(a + b*Tan[e + f*x])^n, x], x] /; FreeQ[{a, b, e, f}, x] && IntegerQ[(m - 1)/2] && IGtQ[n, 0]

Rule 3768

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 3770

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

Rule 2621

Int[(csc[(e_.) + (f_.)*(x_)]*(a_.))^(m_)*sec[(e_.) + (f_.)*(x_)]^(n_.), x_Symbol] :> -Dist[(f*a^n)^(-1), Subst
[Int[x^(m + n - 1)/(-1 + x^2/a^2)^((n + 1)/2), x], x, a*Csc[e + f*x]], x] /; FreeQ[{a, e, f, m}, x] && Integer
Q[(n + 1)/2] &&  !(IntegerQ[(m + 1)/2] && LtQ[0, m, n])

Rule 321

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(c^(n - 1)*(c*x)^(m - n + 1)*(a + b*x^n
)^(p + 1))/(b*(m + n*p + 1)), x] - Dist[(a*c^n*(m - n + 1))/(b*(m + n*p + 1)), Int[(c*x)^(m - n)*(a + b*x^n)^p
, x], x] /; FreeQ[{a, b, c, p}, x] && IGtQ[n, 0] && GtQ[m, n - 1] && NeQ[m + n*p + 1, 0] && IntBinomialQ[a, b,
c, n, m, p, x]

Rule 207

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> -Simp[ArcTanh[(Rt[b, 2]*x)/Rt[-a, 2]]/(Rt[-a, 2]*Rt[b, 2]), x] /;
FreeQ[{a, b}, x] && NegQ[a/b] && (LtQ[a, 0] || GtQ[b, 0])

Rubi steps

\begin{align*} \int \csc ^3(c+d x) (a+b \tan (c+d x)) \, dx &=\int \left (a \csc ^3(c+d x)+b \csc ^2(c+d x) \sec (c+d x)\right ) \, dx\\ &=a \int \csc ^3(c+d x) \, dx+b \int \csc ^2(c+d x) \sec (c+d x) \, dx\\ &=-\frac{a \cot (c+d x) \csc (c+d x)}{2 d}+\frac{1}{2} a \int \csc (c+d x) \, dx-\frac{b \operatorname{Subst}\left (\int \frac{x^2}{-1+x^2} \, dx,x,\csc (c+d x)\right )}{d}\\ &=-\frac{a \tanh ^{-1}(\cos (c+d x))}{2 d}-\frac{b \csc (c+d x)}{d}-\frac{a \cot (c+d x) \csc (c+d x)}{2 d}-\frac{b \operatorname{Subst}\left (\int \frac{1}{-1+x^2} \, dx,x,\csc (c+d x)\right )}{d}\\ &=-\frac{a \tanh ^{-1}(\cos (c+d x))}{2 d}+\frac{b \tanh ^{-1}(\sin (c+d x))}{d}-\frac{b \csc (c+d x)}{d}-\frac{a \cot (c+d x) \csc (c+d x)}{2 d}\\ \end{align*}

Mathematica [C]  time = 0.0265519, size = 107, normalized size = 1.78 \[ -\frac{a \csc ^2\left (\frac{1}{2} (c+d x)\right )}{8 d}+\frac{a \sec ^2\left (\frac{1}{2} (c+d x)\right )}{8 d}+\frac{a \log \left (\sin \left (\frac{1}{2} (c+d x)\right )\right )}{2 d}-\frac{a \log \left (\cos \left (\frac{1}{2} (c+d x)\right )\right )}{2 d}-\frac{b \csc (c+d x) \, _2F_1\left (-\frac{1}{2},1;\frac{1}{2};\sin ^2(c+d x)\right )}{d} \]

Antiderivative was successfully verified.

[In]

Integrate[Csc[c + d*x]^3*(a + b*Tan[c + d*x]),x]

[Out]

-(a*Csc[(c + d*x)/2]^2)/(8*d) - (b*Csc[c + d*x]*Hypergeometric2F1[-1/2, 1, 1/2, Sin[c + d*x]^2])/d - (a*Log[Co
s[(c + d*x)/2]])/(2*d) + (a*Log[Sin[(c + d*x)/2]])/(2*d) + (a*Sec[(c + d*x)/2]^2)/(8*d)

________________________________________________________________________________________

Maple [A]  time = 0.082, size = 75, normalized size = 1.3 \begin{align*} -{\frac{b}{d\sin \left ( dx+c \right ) }}+{\frac{b\ln \left ( \sec \left ( dx+c \right ) +\tan \left ( dx+c \right ) \right ) }{d}}-{\frac{\cot \left ( dx+c \right ) a\csc \left ( dx+c \right ) }{2\,d}}+{\frac{a\ln \left ( \csc \left ( dx+c \right ) -\cot \left ( dx+c \right ) \right ) }{2\,d}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(csc(d*x+c)^3*(a+b*tan(d*x+c)),x)

[Out]

-1/d*b/sin(d*x+c)+1/d*b*ln(sec(d*x+c)+tan(d*x+c))-1/2*a*cot(d*x+c)*csc(d*x+c)/d+1/2/d*a*ln(csc(d*x+c)-cot(d*x+
c))

________________________________________________________________________________________

Maxima [A]  time = 1.14193, size = 112, normalized size = 1.87 \begin{align*} \frac{a{\left (\frac{2 \, \cos \left (d x + c\right )}{\cos \left (d x + c\right )^{2} - 1} - \log \left (\cos \left (d x + c\right ) + 1\right ) + \log \left (\cos \left (d x + c\right ) - 1\right )\right )} - 2 \, b{\left (\frac{2}{\sin \left (d x + c\right )} - \log \left (\sin \left (d x + c\right ) + 1\right ) + \log \left (\sin \left (d x + c\right ) - 1\right )\right )}}{4 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csc(d*x+c)^3*(a+b*tan(d*x+c)),x, algorithm="maxima")

[Out]

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

________________________________________________________________________________________

Fricas [B]  time = 2.89013, size = 367, normalized size = 6.12 \begin{align*} \frac{2 \, a \cos \left (d x + c\right ) -{\left (a \cos \left (d x + c\right )^{2} - a\right )} \log \left (\frac{1}{2} \, \cos \left (d x + c\right ) + \frac{1}{2}\right ) +{\left (a \cos \left (d x + c\right )^{2} - a\right )} \log \left (-\frac{1}{2} \, \cos \left (d x + c\right ) + \frac{1}{2}\right ) + 2 \,{\left (b \cos \left (d x + c\right )^{2} - b\right )} \log \left (\sin \left (d x + c\right ) + 1\right ) - 2 \,{\left (b \cos \left (d x + c\right )^{2} - b\right )} \log \left (-\sin \left (d x + c\right ) + 1\right ) + 4 \, b \sin \left (d x + c\right )}{4 \,{\left (d \cos \left (d x + c\right )^{2} - d\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csc(d*x+c)^3*(a+b*tan(d*x+c)),x, algorithm="fricas")

[Out]

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

________________________________________________________________________________________

Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (a + b \tan{\left (c + d x \right )}\right ) \csc ^{3}{\left (c + d x \right )}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csc(d*x+c)**3*(a+b*tan(d*x+c)),x)

[Out]

Integral((a + b*tan(c + d*x))*csc(c + d*x)**3, x)

________________________________________________________________________________________

Giac [B]  time = 1.51231, size = 159, normalized size = 2.65 \begin{align*} \frac{a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} + 8 \, b \log \left ({\left | \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 1 \right |}\right ) - 8 \, b \log \left ({\left | \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) - 1 \right |}\right ) + 4 \, a \log \left ({\left | \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) \right |}\right ) - 4 \, b \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) - \frac{6 \, a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} + 4 \, b \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + a}{\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2}}}{8 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csc(d*x+c)^3*(a+b*tan(d*x+c)),x, algorithm="giac")

[Out]

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