\(\int x \csc ^3(a+b x) \sec ^2(a+b x) \, dx\) [287]

Optimal result

Integrand size = 18, antiderivative size = 126 \[ \int x \csc ^3(a+b x) \sec ^2(a+b x) \, dx=-\frac {3 x \text {arctanh}\left (e^{i (a+b x)}\right )}{b}-\frac {\text {arctanh}(\sin (a+b x))}{b^2}-\frac {\csc (a+b x)}{2 b^2}+\frac {3 i \operatorname {PolyLog}\left (2,-e^{i (a+b x)}\right )}{2 b^2}-\frac {3 i \operatorname {PolyLog}\left (2,e^{i (a+b x)}\right )}{2 b^2}+\frac {3 x \sec (a+b x)}{2 b}-\frac {x \csc ^2(a+b x) \sec (a+b x)}{2 b} \]

[Out]

-3*x*arctanh(exp(I*(b*x+a)))/b-arctanh(sin(b*x+a))/b^2-1/2*csc(b*x+a)/b^2+3/2*I*polylog(2,-exp(I*(b*x+a)))/b^2
-3/2*I*polylog(2,exp(I*(b*x+a)))/b^2+3/2*x*sec(b*x+a)/b-1/2*x*csc(b*x+a)^2*sec(b*x+a)/b

Rubi [A] (verified)

Time = 0.18 (sec) , antiderivative size = 126, normalized size of antiderivative = 1.00, number of steps used = 13, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.667, Rules used = {2702, 294, 327, 213, 4505, 6406, 12, 4268, 2317, 2438, 3855, 2701} \[ \int x \csc ^3(a+b x) \sec ^2(a+b x) \, dx=-\frac {\text {arctanh}(\sin (a+b x))}{b^2}-\frac {3 x \text {arctanh}\left (e^{i (a+b x)}\right )}{b}+\frac {3 i \operatorname {PolyLog}\left (2,-e^{i (a+b x)}\right )}{2 b^2}-\frac {3 i \operatorname {PolyLog}\left (2,e^{i (a+b x)}\right )}{2 b^2}-\frac {\csc (a+b x)}{2 b^2}+\frac {3 x \sec (a+b x)}{2 b}-\frac {x \csc ^2(a+b x) \sec (a+b x)}{2 b} \]

[In]

Int[x*Csc[a + b*x]^3*Sec[a + b*x]^2,x]

[Out]

(-3*x*ArcTanh[E^(I*(a + b*x))])/b - ArcTanh[Sin[a + b*x]]/b^2 - Csc[a + b*x]/(2*b^2) + (((3*I)/2)*PolyLog[2, -
E^(I*(a + b*x))])/b^2 - (((3*I)/2)*PolyLog[2, E^(I*(a + b*x))])/b^2 + (3*x*Sec[a + b*x])/(2*b) - (x*Csc[a + b*
x]^2*Sec[a + b*x])/(2*b)

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 213

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

Rule 294

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*n*(p + 1))), x] - Dist[c^n*((m - n + 1)/(b*n*(p + 1))), Int[(c*x)^(m - n)*(a + b*x^n)^(p + 1), x
], x] /; FreeQ[{a, b, c}, x] && IGtQ[n, 0] && LtQ[p, -1] && GtQ[m + 1, n] &&  !ILtQ[(m + n*(p + 1) + 1)/n, 0]
&& IntBinomialQ[a, b, c, n, m, p, x]

Rule 327

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 2317

Int[Log[(a_) + (b_.)*((F_)^((e_.)*((c_.) + (d_.)*(x_))))^(n_.)], x_Symbol] :> Dist[1/(d*e*n*Log[F]), Subst[Int
[Log[a + b*x]/x, x], x, (F^(e*(c + d*x)))^n], x] /; FreeQ[{F, a, b, c, d, e, n}, x] && GtQ[a, 0]

Rule 2438

Int[Log[(c_.)*((d_) + (e_.)*(x_)^(n_.))]/(x_), x_Symbol] :> Simp[-PolyLog[2, (-c)*e*x^n]/n, x] /; FreeQ[{c, d,
e, n}, x] && EqQ[c*d, 1]

Rule 2701

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 2702

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

Rule 3855

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

Rule 4268

Int[csc[(e_.) + (f_.)*(x_)]*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> Simp[-2*(c + d*x)^m*(ArcTanh[E^(I*(e + f*
x))]/f), x] + (-Dist[d*(m/f), Int[(c + d*x)^(m - 1)*Log[1 - E^(I*(e + f*x))], x], x] + Dist[d*(m/f), Int[(c +
d*x)^(m - 1)*Log[1 + E^(I*(e + f*x))], x], x]) /; FreeQ[{c, d, e, f}, x] && IGtQ[m, 0]

Rule 4505

Int[Csc[(a_.) + (b_.)*(x_)]^(n_.)*((c_.) + (d_.)*(x_))^(m_.)*Sec[(a_.) + (b_.)*(x_)]^(p_.), x_Symbol] :> Modul
e[{u = IntHide[Csc[a + b*x]^n*Sec[a + b*x]^p, x]}, Dist[(c + d*x)^m, u, x] - Dist[d*m, Int[(c + d*x)^(m - 1)*u
, x], x]] /; FreeQ[{a, b, c, d}, x] && IntegersQ[n, p] && GtQ[m, 0] && NeQ[n, p]

Rule 6406

Int[ArcTanh[u_], x_Symbol] :> Simp[x*ArcTanh[u], x] - Int[SimplifyIntegrand[x*(D[u, x]/(1 - u^2)), x], x] /; I
nverseFunctionFreeQ[u, x]

Rubi steps \begin{align*} \text {integral}& = -\frac {3 x \text {arctanh}(\cos (a+b x))}{2 b}+\frac {3 x \sec (a+b x)}{2 b}-\frac {x \csc ^2(a+b x) \sec (a+b x)}{2 b}-\int \left (-\frac {3 \text {arctanh}(\cos (a+b x))}{2 b}+\frac {3 \sec (a+b x)}{2 b}-\frac {\csc ^2(a+b x) \sec (a+b x)}{2 b}\right ) \, dx \\ & = -\frac {3 x \text {arctanh}(\cos (a+b x))}{2 b}+\frac {3 x \sec (a+b x)}{2 b}-\frac {x \csc ^2(a+b x) \sec (a+b x)}{2 b}+\frac {\int \csc ^2(a+b x) \sec (a+b x) \, dx}{2 b}+\frac {3 \int \text {arctanh}(\cos (a+b x)) \, dx}{2 b}-\frac {3 \int \sec (a+b x) \, dx}{2 b} \\ & = -\frac {3 \text {arctanh}(\sin (a+b x))}{2 b^2}+\frac {3 x \sec (a+b x)}{2 b}-\frac {x \csc ^2(a+b x) \sec (a+b x)}{2 b}-\frac {\text {Subst}\left (\int \frac {x^2}{-1+x^2} \, dx,x,\csc (a+b x)\right )}{2 b^2}+\frac {3 \int b x \csc (a+b x) \, dx}{2 b} \\ & = -\frac {3 \text {arctanh}(\sin (a+b x))}{2 b^2}-\frac {\csc (a+b x)}{2 b^2}+\frac {3 x \sec (a+b x)}{2 b}-\frac {x \csc ^2(a+b x) \sec (a+b x)}{2 b}+\frac {3}{2} \int x \csc (a+b x) \, dx-\frac {\text {Subst}\left (\int \frac {1}{-1+x^2} \, dx,x,\csc (a+b x)\right )}{2 b^2} \\ & = -\frac {3 x \text {arctanh}\left (e^{i (a+b x)}\right )}{b}-\frac {\text {arctanh}(\sin (a+b x))}{b^2}-\frac {\csc (a+b x)}{2 b^2}+\frac {3 x \sec (a+b x)}{2 b}-\frac {x \csc ^2(a+b x) \sec (a+b x)}{2 b}-\frac {3 \int \log \left (1-e^{i (a+b x)}\right ) \, dx}{2 b}+\frac {3 \int \log \left (1+e^{i (a+b x)}\right ) \, dx}{2 b} \\ & = -\frac {3 x \text {arctanh}\left (e^{i (a+b x)}\right )}{b}-\frac {\text {arctanh}(\sin (a+b x))}{b^2}-\frac {\csc (a+b x)}{2 b^2}+\frac {3 x \sec (a+b x)}{2 b}-\frac {x \csc ^2(a+b x) \sec (a+b x)}{2 b}+\frac {(3 i) \text {Subst}\left (\int \frac {\log (1-x)}{x} \, dx,x,e^{i (a+b x)}\right )}{2 b^2}-\frac {(3 i) \text {Subst}\left (\int \frac {\log (1+x)}{x} \, dx,x,e^{i (a+b x)}\right )}{2 b^2} \\ & = -\frac {3 x \text {arctanh}\left (e^{i (a+b x)}\right )}{b}-\frac {\text {arctanh}(\sin (a+b x))}{b^2}-\frac {\csc (a+b x)}{2 b^2}+\frac {3 i \operatorname {PolyLog}\left (2,-e^{i (a+b x)}\right )}{2 b^2}-\frac {3 i \operatorname {PolyLog}\left (2,e^{i (a+b x)}\right )}{2 b^2}+\frac {3 x \sec (a+b x)}{2 b}-\frac {x \csc ^2(a+b x) \sec (a+b x)}{2 b} \\ \end{align*}

Mathematica [B] (verified)

Both result and optimal contain complex but leaf count is larger than twice the leaf count of optimal. \(282\) vs. \(2(126)=252\).

Time = 3.69 (sec) , antiderivative size = 282, normalized size of antiderivative = 2.24 \[ \int x \csc ^3(a+b x) \sec ^2(a+b x) \, dx=\frac {8 b x-2 \cot \left (\frac {1}{2} (a+b x)\right )-b x \csc ^2\left (\frac {1}{2} (a+b x)\right )+12 (a+b x) \left (\log \left (1-e^{i (a+b x)}\right )-\log \left (1+e^{i (a+b x)}\right )\right )+8 \log \left (\cos \left (\frac {1}{2} (a+b x)\right )-\sin \left (\frac {1}{2} (a+b x)\right )\right )-8 \log \left (\cos \left (\frac {1}{2} (a+b x)\right )+\sin \left (\frac {1}{2} (a+b x)\right )\right )-12 a \log \left (\tan \left (\frac {1}{2} (a+b x)\right )\right )+12 i \left (\operatorname {PolyLog}\left (2,-e^{i (a+b x)}\right )-\operatorname {PolyLog}\left (2,e^{i (a+b x)}\right )\right )+b x \sec ^2\left (\frac {1}{2} (a+b x)\right )+\frac {8 b x \sin \left (\frac {1}{2} (a+b x)\right )}{\cos \left (\frac {1}{2} (a+b x)\right )-\sin \left (\frac {1}{2} (a+b x)\right )}-\frac {8 b x \sin \left (\frac {1}{2} (a+b x)\right )}{\cos \left (\frac {1}{2} (a+b x)\right )+\sin \left (\frac {1}{2} (a+b x)\right )}-2 \tan \left (\frac {1}{2} (a+b x)\right )}{8 b^2} \]

[In]

Integrate[x*Csc[a + b*x]^3*Sec[a + b*x]^2,x]

[Out]

(8*b*x - 2*Cot[(a + b*x)/2] - b*x*Csc[(a + b*x)/2]^2 + 12*(a + b*x)*(Log[1 - E^(I*(a + b*x))] - Log[1 + E^(I*(
a + b*x))]) + 8*Log[Cos[(a + b*x)/2] - Sin[(a + b*x)/2]] - 8*Log[Cos[(a + b*x)/2] + Sin[(a + b*x)/2]] - 12*a*L
og[Tan[(a + b*x)/2]] + (12*I)*(PolyLog[2, -E^(I*(a + b*x))] - PolyLog[2, E^(I*(a + b*x))]) + b*x*Sec[(a + b*x)
/2]^2 + (8*b*x*Sin[(a + b*x)/2])/(Cos[(a + b*x)/2] - Sin[(a + b*x)/2]) - (8*b*x*Sin[(a + b*x)/2])/(Cos[(a + b*
x)/2] + Sin[(a + b*x)/2]) - 2*Tan[(a + b*x)/2])/(8*b^2)

Maple [B] (verified)

Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 518 vs. \(2 (108 ) = 216\).

Time = 0.70 (sec) , antiderivative size = 519, normalized size of antiderivative = 4.12

method	result	size
risch	\(\frac {2 i {\mathrm e}^{i \left (x b +a \right )}+6 x b \,{\mathrm e}^{5 i \left (x b +a \right )}-4 x b \,{\mathrm e}^{3 i \left (x b +a \right )}+6 x b \,{\mathrm e}^{i \left (x b +a \right )}-3 i {\mathrm e}^{4 i \left (x b +a \right )} \operatorname {dilog}\left ({\mathrm e}^{i \left (x b +a \right )}+1\right )-4 i \arctan \left ({\mathrm e}^{i \left (x b +a \right )}\right ) {\mathrm e}^{4 i \left (x b +a \right )}-3 i {\mathrm e}^{4 i \left (x b +a \right )} \operatorname {dilog}\left ({\mathrm e}^{i \left (x b +a \right )}\right )+3 i {\mathrm e}^{6 i \left (x b +a \right )} \operatorname {dilog}\left ({\mathrm e}^{i \left (x b +a \right )}\right )+3 i {\mathrm e}^{6 i \left (x b +a \right )} \operatorname {dilog}\left ({\mathrm e}^{i \left (x b +a \right )}+1\right )+4 i \arctan \left ({\mathrm e}^{i \left (x b +a \right )}\right ) {\mathrm e}^{6 i \left (x b +a \right )}+3 \ln \left ({\mathrm e}^{i \left (x b +a \right )}-1\right ) {\mathrm e}^{4 i \left (x b +a \right )} a -3 \ln \left ({\mathrm e}^{i \left (x b +a \right )}-1\right ) {\mathrm e}^{6 i \left (x b +a \right )} a +3 \ln \left ({\mathrm e}^{i \left (x b +a \right )}-1\right ) {\mathrm e}^{2 i \left (x b +a \right )} a -4 i \arctan \left ({\mathrm e}^{i \left (x b +a \right )}\right ) {\mathrm e}^{2 i \left (x b +a \right )}-3 i {\mathrm e}^{2 i \left (x b +a \right )} \operatorname {dilog}\left ({\mathrm e}^{i \left (x b +a \right )}\right )-3 i {\mathrm e}^{2 i \left (x b +a \right )} \operatorname {dilog}\left ({\mathrm e}^{i \left (x b +a \right )}+1\right )-3 \ln \left ({\mathrm e}^{i \left (x b +a \right )}+1\right ) b x +3 \ln \left ({\mathrm e}^{i \left (x b +a \right )}+1\right ) {\mathrm e}^{2 i \left (x b +a \right )} b x -3 \ln \left ({\mathrm e}^{i \left (x b +a \right )}+1\right ) {\mathrm e}^{6 i \left (x b +a \right )} b x +3 \ln \left ({\mathrm e}^{i \left (x b +a \right )}+1\right ) {\mathrm e}^{4 i \left (x b +a \right )} b x -2 i {\mathrm e}^{5 i \left (x b +a \right )}+4 i \arctan \left ({\mathrm e}^{i \left (x b +a \right )}\right )+3 i \operatorname {dilog}\left ({\mathrm e}^{i \left (x b +a \right )}\right )+3 i \operatorname {dilog}\left ({\mathrm e}^{i \left (x b +a \right )}+1\right )-3 a \ln \left ({\mathrm e}^{i \left (x b +a \right )}-1\right )}{2 b^{2} \left ({\mathrm e}^{2 i \left (x b +a \right )}-1\right )^{2} \left ({\mathrm e}^{2 i \left (x b +a \right )}+1\right )}\)	\(519\)

[In]

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

[Out]

1/2*(4*I*arctan(exp(I*(b*x+a)))+3*I*dilog(exp(I*(b*x+a)))+3*I*dilog(exp(I*(b*x+a))+1)-3*a*ln(exp(I*(b*x+a))-1)
-3*I*exp(4*I*(b*x+a))*dilog(exp(I*(b*x+a))+1)-4*I*arctan(exp(I*(b*x+a)))*exp(4*I*(b*x+a))-3*I*exp(4*I*(b*x+a))
*dilog(exp(I*(b*x+a)))+3*I*exp(6*I*(b*x+a))*dilog(exp(I*(b*x+a)))+3*I*exp(6*I*(b*x+a))*dilog(exp(I*(b*x+a))+1)
+4*I*arctan(exp(I*(b*x+a)))*exp(6*I*(b*x+a))+3*ln(exp(I*(b*x+a))-1)*exp(4*I*(b*x+a))*a-3*ln(exp(I*(b*x+a))-1)*
exp(6*I*(b*x+a))*a+3*ln(exp(I*(b*x+a))-1)*exp(2*I*(b*x+a))*a-4*I*arctan(exp(I*(b*x+a)))*exp(2*I*(b*x+a))-3*I*e
xp(2*I*(b*x+a))*dilog(exp(I*(b*x+a)))-3*I*exp(2*I*(b*x+a))*dilog(exp(I*(b*x+a))+1)-3*ln(exp(I*(b*x+a))+1)*b*x-
2*I*exp(5*I*(b*x+a))+2*I*exp(I*(b*x+a))+3*ln(exp(I*(b*x+a))+1)*exp(2*I*(b*x+a))*b*x-3*ln(exp(I*(b*x+a))+1)*exp
(6*I*(b*x+a))*b*x+3*ln(exp(I*(b*x+a))+1)*exp(4*I*(b*x+a))*b*x+6*x*b*exp(5*I*(b*x+a))-4*x*b*exp(3*I*(b*x+a))+6*
x*b*exp(I*(b*x+a)))/b^2/(exp(2*I*(b*x+a))-1)^2/(exp(2*I*(b*x+a))+1)

Fricas [B] (verification not implemented)

Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 531 vs. \(2 (104) = 208\).

Time = 0.26 (sec) , antiderivative size = 531, normalized size of antiderivative = 4.21 \[ \int x \csc ^3(a+b x) \sec ^2(a+b x) \, dx=\frac {6 \, b x \cos \left (b x + a\right )^{2} - 4 \, b x - 3 \, {\left (i \, \cos \left (b x + a\right )^{3} - i \, \cos \left (b x + a\right )\right )} {\rm Li}_2\left (\cos \left (b x + a\right ) + i \, \sin \left (b x + a\right )\right ) - 3 \, {\left (-i \, \cos \left (b x + a\right )^{3} + i \, \cos \left (b x + a\right )\right )} {\rm Li}_2\left (\cos \left (b x + a\right ) - i \, \sin \left (b x + a\right )\right ) - 3 \, {\left (i \, \cos \left (b x + a\right )^{3} - i \, \cos \left (b x + a\right )\right )} {\rm Li}_2\left (-\cos \left (b x + a\right ) + i \, \sin \left (b x + a\right )\right ) - 3 \, {\left (-i \, \cos \left (b x + a\right )^{3} + i \, \cos \left (b x + a\right )\right )} {\rm Li}_2\left (-\cos \left (b x + a\right ) - i \, \sin \left (b x + a\right )\right ) - 3 \, {\left (b x \cos \left (b x + a\right )^{3} - b x \cos \left (b x + a\right )\right )} \log \left (\cos \left (b x + a\right ) + i \, \sin \left (b x + a\right ) + 1\right ) - 3 \, {\left (b x \cos \left (b x + a\right )^{3} - b x \cos \left (b x + a\right )\right )} \log \left (\cos \left (b x + a\right ) - i \, \sin \left (b x + a\right ) + 1\right ) - 3 \, {\left (a \cos \left (b x + a\right )^{3} - a \cos \left (b x + a\right )\right )} \log \left (-\frac {1}{2} \, \cos \left (b x + a\right ) + \frac {1}{2} i \, \sin \left (b x + a\right ) + \frac {1}{2}\right ) - 3 \, {\left (a \cos \left (b x + a\right )^{3} - a \cos \left (b x + a\right )\right )} \log \left (-\frac {1}{2} \, \cos \left (b x + a\right ) - \frac {1}{2} i \, \sin \left (b x + a\right ) + \frac {1}{2}\right ) + 3 \, {\left ({\left (b x + a\right )} \cos \left (b x + a\right )^{3} - {\left (b x + a\right )} \cos \left (b x + a\right )\right )} \log \left (-\cos \left (b x + a\right ) + i \, \sin \left (b x + a\right ) + 1\right ) + 3 \, {\left ({\left (b x + a\right )} \cos \left (b x + a\right )^{3} - {\left (b x + a\right )} \cos \left (b x + a\right )\right )} \log \left (-\cos \left (b x + a\right ) - i \, \sin \left (b x + a\right ) + 1\right ) - 2 \, {\left (\cos \left (b x + a\right )^{3} - \cos \left (b x + a\right )\right )} \log \left (\sin \left (b x + a\right ) + 1\right ) + 2 \, {\left (\cos \left (b x + a\right )^{3} - \cos \left (b x + a\right )\right )} \log \left (-\sin \left (b x + a\right ) + 1\right ) + 2 \, \cos \left (b x + a\right ) \sin \left (b x + a\right )}{4 \, {\left (b^{2} \cos \left (b x + a\right )^{3} - b^{2} \cos \left (b x + a\right )\right )}} \]

[In]

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

[Out]

1/4*(6*b*x*cos(b*x + a)^2 - 4*b*x - 3*(I*cos(b*x + a)^3 - I*cos(b*x + a))*dilog(cos(b*x + a) + I*sin(b*x + a))
- 3*(-I*cos(b*x + a)^3 + I*cos(b*x + a))*dilog(cos(b*x + a) - I*sin(b*x + a)) - 3*(I*cos(b*x + a)^3 - I*cos(b
*x + a))*dilog(-cos(b*x + a) + I*sin(b*x + a)) - 3*(-I*cos(b*x + a)^3 + I*cos(b*x + a))*dilog(-cos(b*x + a) -
I*sin(b*x + a)) - 3*(b*x*cos(b*x + a)^3 - b*x*cos(b*x + a))*log(cos(b*x + a) + I*sin(b*x + a) + 1) - 3*(b*x*co
s(b*x + a)^3 - b*x*cos(b*x + a))*log(cos(b*x + a) - I*sin(b*x + a) + 1) - 3*(a*cos(b*x + a)^3 - a*cos(b*x + a)
)*log(-1/2*cos(b*x + a) + 1/2*I*sin(b*x + a) + 1/2) - 3*(a*cos(b*x + a)^3 - a*cos(b*x + a))*log(-1/2*cos(b*x +
a) - 1/2*I*sin(b*x + a) + 1/2) + 3*((b*x + a)*cos(b*x + a)^3 - (b*x + a)*cos(b*x + a))*log(-cos(b*x + a) + I*
sin(b*x + a) + 1) + 3*((b*x + a)*cos(b*x + a)^3 - (b*x + a)*cos(b*x + a))*log(-cos(b*x + a) - I*sin(b*x + a) +
1) - 2*(cos(b*x + a)^3 - cos(b*x + a))*log(sin(b*x + a) + 1) + 2*(cos(b*x + a)^3 - cos(b*x + a))*log(-sin(b*x
+ a) + 1) + 2*cos(b*x + a)*sin(b*x + a))/(b^2*cos(b*x + a)^3 - b^2*cos(b*x + a))

Sympy [F]

\[ \int x \csc ^3(a+b x) \sec ^2(a+b x) \, dx=\int x \csc ^{3}{\left (a + b x \right )} \sec ^{2}{\left (a + b x \right )}\, dx \]

[In]

integrate(x*csc(b*x+a)**3*sec(b*x+a)**2,x)

[Out]

Integral(x*csc(a + b*x)**3*sec(a + b*x)**2, x)

Maxima [B] (verification not implemented)

Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 1173 vs. \(2 (104) = 208\).

Time = 0.49 (sec) , antiderivative size = 1173, normalized size of antiderivative = 9.31 \[ \int x \csc ^3(a+b x) \sec ^2(a+b x) \, dx=\text {Too large to display} \]

[In]

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

[Out]

(8*I*b*x*cos(3*b*x + 3*a) - 8*b*x*sin(3*b*x + 3*a) - 4*(cos(6*b*x + 6*a) - cos(4*b*x + 4*a) - cos(2*b*x + 2*a)
+ I*sin(6*b*x + 6*a) - I*sin(4*b*x + 4*a) - I*sin(2*b*x + 2*a) + 1)*arctan2(2*(cos(b*x + 2*a)*cos(a) + sin(b*
x + 2*a)*sin(a))/(cos(b*x + 2*a)^2 + cos(a)^2 + 2*cos(a)*sin(b*x + 2*a) + sin(b*x + 2*a)^2 - 2*cos(b*x + 2*a)*
sin(a) + sin(a)^2), (cos(b*x + 2*a)^2 - cos(a)^2 + sin(b*x + 2*a)^2 - sin(a)^2)/(cos(b*x + 2*a)^2 + cos(a)^2 +
2*cos(a)*sin(b*x + 2*a) + sin(b*x + 2*a)^2 - 2*cos(b*x + 2*a)*sin(a) + sin(a)^2)) - 6*(b*x*cos(6*b*x + 6*a) -
b*x*cos(4*b*x + 4*a) - b*x*cos(2*b*x + 2*a) + I*b*x*sin(6*b*x + 6*a) - I*b*x*sin(4*b*x + 4*a) - I*b*x*sin(2*b
*x + 2*a) + b*x)*arctan2(sin(b*x + a), cos(b*x + a) + 1) - 6*(b*x*cos(6*b*x + 6*a) - b*x*cos(4*b*x + 4*a) - b*
x*cos(2*b*x + 2*a) + I*b*x*sin(6*b*x + 6*a) - I*b*x*sin(4*b*x + 4*a) - I*b*x*sin(2*b*x + 2*a) + b*x)*arctan2(s
in(b*x + a), -cos(b*x + a) + 1) - 4*(3*I*b*x + 1)*cos(5*b*x + 5*a) - 4*(3*I*b*x - 1)*cos(b*x + a) + 6*(cos(6*b
*x + 6*a) - cos(4*b*x + 4*a) - cos(2*b*x + 2*a) + I*sin(6*b*x + 6*a) - I*sin(4*b*x + 4*a) - I*sin(2*b*x + 2*a)
+ 1)*dilog(-e^(I*b*x + I*a)) - 6*(cos(6*b*x + 6*a) - cos(4*b*x + 4*a) - cos(2*b*x + 2*a) + I*sin(6*b*x + 6*a)
- I*sin(4*b*x + 4*a) - I*sin(2*b*x + 2*a) + 1)*dilog(e^(I*b*x + I*a)) - 3*(-I*b*x*cos(6*b*x + 6*a) + I*b*x*co
s(4*b*x + 4*a) + I*b*x*cos(2*b*x + 2*a) + b*x*sin(6*b*x + 6*a) - b*x*sin(4*b*x + 4*a) - b*x*sin(2*b*x + 2*a) -
I*b*x)*log(cos(b*x + a)^2 + sin(b*x + a)^2 + 2*cos(b*x + a) + 1) - 3*(I*b*x*cos(6*b*x + 6*a) - I*b*x*cos(4*b*
x + 4*a) - I*b*x*cos(2*b*x + 2*a) - b*x*sin(6*b*x + 6*a) + b*x*sin(4*b*x + 4*a) + b*x*sin(2*b*x + 2*a) + I*b*x
)*log(cos(b*x + a)^2 + sin(b*x + a)^2 - 2*cos(b*x + a) + 1) - 2*(I*cos(6*b*x + 6*a) - I*cos(4*b*x + 4*a) - I*c
os(2*b*x + 2*a) - sin(6*b*x + 6*a) + sin(4*b*x + 4*a) + sin(2*b*x + 2*a) + I)*log((cos(b*x + 2*a)^2 + cos(a)^2
- 2*cos(a)*sin(b*x + 2*a) + sin(b*x + 2*a)^2 + 2*cos(b*x + 2*a)*sin(a) + sin(a)^2)/(cos(b*x + 2*a)^2 + cos(a)
^2 + 2*cos(a)*sin(b*x + 2*a) + sin(b*x + 2*a)^2 - 2*cos(b*x + 2*a)*sin(a) + sin(a)^2)) + 4*(3*b*x - I)*sin(5*b
*x + 5*a) + 4*(3*b*x + I)*sin(b*x + a))/(-4*I*b^2*cos(6*b*x + 6*a) + 4*I*b^2*cos(4*b*x + 4*a) + 4*I*b^2*cos(2*
b*x + 2*a) + 4*b^2*sin(6*b*x + 6*a) - 4*b^2*sin(4*b*x + 4*a) - 4*b^2*sin(2*b*x + 2*a) - 4*I*b^2)

Giac [F]

\[ \int x \csc ^3(a+b x) \sec ^2(a+b x) \, dx=\int { x \csc \left (b x + a\right )^{3} \sec \left (b x + a\right )^{2} \,d x } \]

[In]

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

[Out]

integrate(x*csc(b*x + a)^3*sec(b*x + a)^2, x)

Mupad [F(-1)]

Timed out. \[ \int x \csc ^3(a+b x) \sec ^2(a+b x) \, dx=\int \frac {x}{{\cos \left (a+b\,x\right )}^2\,{\sin \left (a+b\,x\right )}^3} \,d x \]

[In]

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

[Out]

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