\(\int (c+d x) \csc ^2(a+b x) \sec ^3(a+b x) \, dx\) [319]

Optimal result

Integrand size = 22, antiderivative size = 162 \[ \int (c+d x) \csc ^2(a+b x) \sec ^3(a+b x) \, dx=-\frac {3 i d x \arctan \left (e^{i (a+b x)}\right )}{b}-\frac {d \text {arctanh}(\cos (a+b x))}{b^2}+\frac {3 c \text {arctanh}(\sin (a+b x))}{2 b}-\frac {3 (c+d x) \csc (a+b x)}{2 b}+\frac {3 i d \operatorname {PolyLog}\left (2,-i e^{i (a+b x)}\right )}{2 b^2}-\frac {3 i d \operatorname {PolyLog}\left (2,i e^{i (a+b x)}\right )}{2 b^2}-\frac {d \sec (a+b x)}{2 b^2}+\frac {(c+d x) \csc (a+b x) \sec ^2(a+b x)}{2 b} \]

[Out]

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

Rubi [A] (verified)

Time = 0.21 (sec) , antiderivative size = 182, normalized size of antiderivative = 1.12, number of steps used = 13, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.545, Rules used = {2701, 294, 327, 213, 4505, 6406, 12, 4266, 2317, 2438, 3855, 2702} \[ \int (c+d x) \csc ^2(a+b x) \sec ^3(a+b x) \, dx=-\frac {3 i d x \arctan \left (e^{i (a+b x)}\right )}{b}-\frac {d \text {arctanh}(\cos (a+b x))}{b^2}+\frac {3 (c+d x) \text {arctanh}(\sin (a+b x))}{2 b}-\frac {3 d x \text {arctanh}(\sin (a+b x))}{2 b}+\frac {3 i d \operatorname {PolyLog}\left (2,-i e^{i (a+b x)}\right )}{2 b^2}-\frac {3 i d \operatorname {PolyLog}\left (2,i e^{i (a+b x)}\right )}{2 b^2}-\frac {d \sec (a+b x)}{2 b^2}-\frac {3 (c+d x) \csc (a+b x)}{2 b}+\frac {(c+d x) \csc (a+b x) \sec ^2(a+b x)}{2 b} \]

[In]

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

[Out]

((-3*I)*d*x*ArcTan[E^(I*(a + b*x))])/b - (d*ArcTanh[Cos[a + b*x]])/b^2 - (3*d*x*ArcTanh[Sin[a + b*x]])/(2*b) +
(3*(c + d*x)*ArcTanh[Sin[a + b*x]])/(2*b) - (3*(c + d*x)*Csc[a + b*x])/(2*b) + (((3*I)/2)*d*PolyLog[2, (-I)*E
^(I*(a + b*x))])/b^2 - (((3*I)/2)*d*PolyLog[2, I*E^(I*(a + b*x))])/b^2 - (d*Sec[a + b*x])/(2*b^2) + ((c + d*x)
*Csc[a + b*x]*Sec[a + b*x]^2)/(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 4266

Int[csc[(e_.) + Pi*(k_.) + (f_.)*(x_)]*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> Simp[-2*(c + d*x)^m*(ArcTanh[E
^(I*k*Pi)*E^(I*(e + f*x))]/f), x] + (-Dist[d*(m/f), Int[(c + d*x)^(m - 1)*Log[1 - E^(I*k*Pi)*E^(I*(e + f*x))],
x], x] + Dist[d*(m/f), Int[(c + d*x)^(m - 1)*Log[1 + E^(I*k*Pi)*E^(I*(e + f*x))], x], x]) /; FreeQ[{c, d, e,
f}, x] && IntegerQ[2*k] && 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 (c+d x) \text {arctanh}(\sin (a+b x))}{2 b}-\frac {3 (c+d x) \csc (a+b x)}{2 b}+\frac {(c+d x) \csc (a+b x) \sec ^2(a+b x)}{2 b}-d \int \left (\frac {3 \text {arctanh}(\sin (a+b x))}{2 b}-\frac {3 \csc (a+b x)}{2 b}+\frac {\csc (a+b x) \sec ^2(a+b x)}{2 b}\right ) \, dx \\ & = \frac {3 (c+d x) \text {arctanh}(\sin (a+b x))}{2 b}-\frac {3 (c+d x) \csc (a+b x)}{2 b}+\frac {(c+d x) \csc (a+b x) \sec ^2(a+b x)}{2 b}-\frac {d \int \csc (a+b x) \sec ^2(a+b x) \, dx}{2 b}-\frac {(3 d) \int \text {arctanh}(\sin (a+b x)) \, dx}{2 b}+\frac {(3 d) \int \csc (a+b x) \, dx}{2 b} \\ & = -\frac {3 d \text {arctanh}(\cos (a+b x))}{2 b^2}-\frac {3 d x \text {arctanh}(\sin (a+b x))}{2 b}+\frac {3 (c+d x) \text {arctanh}(\sin (a+b x))}{2 b}-\frac {3 (c+d x) \csc (a+b x)}{2 b}+\frac {(c+d x) \csc (a+b x) \sec ^2(a+b x)}{2 b}-\frac {d \text {Subst}\left (\int \frac {x^2}{-1+x^2} \, dx,x,\sec (a+b x)\right )}{2 b^2}+\frac {(3 d) \int b x \sec (a+b x) \, dx}{2 b} \\ & = -\frac {3 d \text {arctanh}(\cos (a+b x))}{2 b^2}-\frac {3 d x \text {arctanh}(\sin (a+b x))}{2 b}+\frac {3 (c+d x) \text {arctanh}(\sin (a+b x))}{2 b}-\frac {3 (c+d x) \csc (a+b x)}{2 b}-\frac {d \sec (a+b x)}{2 b^2}+\frac {(c+d x) \csc (a+b x) \sec ^2(a+b x)}{2 b}+\frac {1}{2} (3 d) \int x \sec (a+b x) \, dx-\frac {d \text {Subst}\left (\int \frac {1}{-1+x^2} \, dx,x,\sec (a+b x)\right )}{2 b^2} \\ & = -\frac {3 i d x \arctan \left (e^{i (a+b x)}\right )}{b}-\frac {d \text {arctanh}(\cos (a+b x))}{b^2}-\frac {3 d x \text {arctanh}(\sin (a+b x))}{2 b}+\frac {3 (c+d x) \text {arctanh}(\sin (a+b x))}{2 b}-\frac {3 (c+d x) \csc (a+b x)}{2 b}-\frac {d \sec (a+b x)}{2 b^2}+\frac {(c+d x) \csc (a+b x) \sec ^2(a+b x)}{2 b}-\frac {(3 d) \int \log \left (1-i e^{i (a+b x)}\right ) \, dx}{2 b}+\frac {(3 d) \int \log \left (1+i e^{i (a+b x)}\right ) \, dx}{2 b} \\ & = -\frac {3 i d x \arctan \left (e^{i (a+b x)}\right )}{b}-\frac {d \text {arctanh}(\cos (a+b x))}{b^2}-\frac {3 d x \text {arctanh}(\sin (a+b x))}{2 b}+\frac {3 (c+d x) \text {arctanh}(\sin (a+b x))}{2 b}-\frac {3 (c+d x) \csc (a+b x)}{2 b}-\frac {d \sec (a+b x)}{2 b^2}+\frac {(c+d x) \csc (a+b x) \sec ^2(a+b x)}{2 b}+\frac {(3 i d) \text {Subst}\left (\int \frac {\log (1-i x)}{x} \, dx,x,e^{i (a+b x)}\right )}{2 b^2}-\frac {(3 i d) \text {Subst}\left (\int \frac {\log (1+i x)}{x} \, dx,x,e^{i (a+b x)}\right )}{2 b^2} \\ & = -\frac {3 i d x \arctan \left (e^{i (a+b x)}\right )}{b}-\frac {d \text {arctanh}(\cos (a+b x))}{b^2}-\frac {3 d x \text {arctanh}(\sin (a+b x))}{2 b}+\frac {3 (c+d x) \text {arctanh}(\sin (a+b x))}{2 b}-\frac {3 (c+d x) \csc (a+b x)}{2 b}+\frac {3 i d \operatorname {PolyLog}\left (2,-i e^{i (a+b x)}\right )}{2 b^2}-\frac {3 i d \operatorname {PolyLog}\left (2,i e^{i (a+b x)}\right )}{2 b^2}-\frac {d \sec (a+b x)}{2 b^2}+\frac {(c+d x) \csc (a+b x) \sec ^2(a+b x)}{2 b} \\ \end{align*}

Mathematica [C] (warning: unable to verify)

Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.

Time = 6.96 (sec) , antiderivative size = 669, normalized size of antiderivative = 4.13 \[ \int (c+d x) \csc ^2(a+b x) \sec ^3(a+b x) \, dx=\frac {d \left (a \cos \left (\frac {1}{2} (a+b x)\right )-(a+b x) \cos \left (\frac {1}{2} (a+b x)\right )\right ) \csc \left (\frac {1}{2} (a+b x)\right )}{2 b^2}-\frac {c \csc (a+b x) \operatorname {Hypergeometric2F1}\left (-\frac {1}{2},2,\frac {1}{2},\sin ^2(a+b x)\right )}{b}-\frac {d \log \left (\cos \left (\frac {1}{2} (a+b x)\right )\right )}{b^2}+\frac {d \log \left (\sin \left (\frac {1}{2} (a+b x)\right )\right )}{b^2}-\frac {3 d x \left (a \log \left (1-\tan \left (\frac {1}{2} (a+b x)\right )\right )-a \log \left (1+\tan \left (\frac {1}{2} (a+b x)\right )\right )-i \left (\log \left (1+i \tan \left (\frac {1}{2} (a+b x)\right )\right ) \log \left (\left (\frac {1}{2}-\frac {i}{2}\right ) \left (1+\tan \left (\frac {1}{2} (a+b x)\right )\right )\right )+\operatorname {PolyLog}\left (2,\frac {1}{2} \left ((1+i)-(1-i) \tan \left (\frac {1}{2} (a+b x)\right )\right )\right )\right )+i \left (\log \left (1-i \tan \left (\frac {1}{2} (a+b x)\right )\right ) \log \left (\left (\frac {1}{2}+\frac {i}{2}\right ) \left (1+\tan \left (\frac {1}{2} (a+b x)\right )\right )\right )+\operatorname {PolyLog}\left (2,\left (-\frac {1}{2}-\frac {i}{2}\right ) \left (i+\tan \left (\frac {1}{2} (a+b x)\right )\right )\right )\right )-i \left (\log \left (1-i \tan \left (\frac {1}{2} (a+b x)\right )\right ) \log \left (\left (-\frac {1}{2}+\frac {i}{2}\right ) \left (-1+\tan \left (\frac {1}{2} (a+b x)\right )\right )\right )+\operatorname {PolyLog}\left (2,\frac {1}{2} \left ((1+i)+(1-i) \tan \left (\frac {1}{2} (a+b x)\right )\right )\right )\right )+i \left (\log \left (1+i \tan \left (\frac {1}{2} (a+b x)\right )\right ) \log \left (\left (-\frac {1}{2}-\frac {i}{2}\right ) \left (-1+\tan \left (\frac {1}{2} (a+b x)\right )\right )\right )+\operatorname {PolyLog}\left (2,\frac {1}{2} \left ((1-i)+(1+i) \tan \left (\frac {1}{2} (a+b x)\right )\right )\right )\right )\right )}{2 b \left (a-i \log \left (1-i \tan \left (\frac {1}{2} (a+b x)\right )\right )+i \log \left (1+i \tan \left (\frac {1}{2} (a+b x)\right )\right )\right )}+\frac {d x}{4 b \left (\cos \left (\frac {1}{2} (a+b x)\right )-\sin \left (\frac {1}{2} (a+b x)\right )\right )^2}-\frac {d \sin \left (\frac {1}{2} (a+b x)\right )}{2 b^2 \left (\cos \left (\frac {1}{2} (a+b x)\right )-\sin \left (\frac {1}{2} (a+b x)\right )\right )}-\frac {d x}{4 b \left (\cos \left (\frac {1}{2} (a+b x)\right )+\sin \left (\frac {1}{2} (a+b x)\right )\right )^2}+\frac {d \sin \left (\frac {1}{2} (a+b x)\right )}{2 b^2 \left (\cos \left (\frac {1}{2} (a+b x)\right )+\sin \left (\frac {1}{2} (a+b x)\right )\right )}+\frac {d \sec \left (\frac {1}{2} (a+b x)\right ) \left (a \sin \left (\frac {1}{2} (a+b x)\right )-(a+b x) \sin \left (\frac {1}{2} (a+b x)\right )\right )}{2 b^2} \]

[In]

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

[Out]

(d*(a*Cos[(a + b*x)/2] - (a + b*x)*Cos[(a + b*x)/2])*Csc[(a + b*x)/2])/(2*b^2) - (c*Csc[a + b*x]*Hypergeometri
c2F1[-1/2, 2, 1/2, Sin[a + b*x]^2])/b - (d*Log[Cos[(a + b*x)/2]])/b^2 + (d*Log[Sin[(a + b*x)/2]])/b^2 - (3*d*x
*(a*Log[1 - Tan[(a + b*x)/2]] - a*Log[1 + Tan[(a + b*x)/2]] - I*(Log[1 + I*Tan[(a + b*x)/2]]*Log[(1/2 - I/2)*(
1 + Tan[(a + b*x)/2])] + PolyLog[2, ((1 + I) - (1 - I)*Tan[(a + b*x)/2])/2]) + I*(Log[1 - I*Tan[(a + b*x)/2]]*
Log[(1/2 + I/2)*(1 + Tan[(a + b*x)/2])] + PolyLog[2, (-1/2 - I/2)*(I + Tan[(a + b*x)/2])]) - I*(Log[1 - I*Tan[
(a + b*x)/2]]*Log[(-1/2 + I/2)*(-1 + Tan[(a + b*x)/2])] + PolyLog[2, ((1 + I) + (1 - I)*Tan[(a + b*x)/2])/2])
+ I*(Log[1 + I*Tan[(a + b*x)/2]]*Log[(-1/2 - I/2)*(-1 + Tan[(a + b*x)/2])] + PolyLog[2, ((1 - I) + (1 + I)*Tan
[(a + b*x)/2])/2])))/(2*b*(a - I*Log[1 - I*Tan[(a + b*x)/2]] + I*Log[1 + I*Tan[(a + b*x)/2]])) + (d*x)/(4*b*(C
os[(a + b*x)/2] - Sin[(a + b*x)/2])^2) - (d*Sin[(a + b*x)/2])/(2*b^2*(Cos[(a + b*x)/2] - Sin[(a + b*x)/2])) -
(d*x)/(4*b*(Cos[(a + b*x)/2] + Sin[(a + b*x)/2])^2) + (d*Sin[(a + b*x)/2])/(2*b^2*(Cos[(a + b*x)/2] + Sin[(a +
b*x)/2])) + (d*Sec[(a + b*x)/2]*(a*Sin[(a + b*x)/2] - (a + b*x)*Sin[(a + b*x)/2]))/(2*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. 343 vs. \(2 (139 ) = 278\).

Time = 0.90 (sec) , antiderivative size = 344, normalized size of antiderivative = 2.12

method	result	size
risch	\(-\frac {i \left (3 d x b \,{\mathrm e}^{5 i \left (x b +a \right )}+3 c b \,{\mathrm e}^{5 i \left (x b +a \right )}+2 d x b \,{\mathrm e}^{3 i \left (x b +a \right )}+2 c b \,{\mathrm e}^{3 i \left (x b +a \right )}-i d \,{\mathrm e}^{5 i \left (x b +a \right )}+3 d x b \,{\mathrm e}^{i \left (x b +a \right )}+3 c b \,{\mathrm e}^{i \left (x b +a \right )}+i d \,{\mathrm e}^{i \left (x b +a \right )}\right )}{b^{2} \left ({\mathrm e}^{2 i \left (x b +a \right )}+1\right )^{2} \left ({\mathrm e}^{2 i \left (x b +a \right )}-1\right )}-\frac {3 i c \arctan \left ({\mathrm e}^{i \left (x b +a \right )}\right )}{b}+\frac {3 i d a \arctan \left ({\mathrm e}^{i \left (x b +a \right )}\right )}{b^{2}}-\frac {d \ln \left ({\mathrm e}^{i \left (x b +a \right )}+1\right )}{b^{2}}+\frac {d \ln \left ({\mathrm e}^{i \left (x b +a \right )}-1\right )}{b^{2}}-\frac {3 d \ln \left (1+i {\mathrm e}^{i \left (x b +a \right )}\right ) a}{2 b^{2}}-\frac {3 d \ln \left (1+i {\mathrm e}^{i \left (x b +a \right )}\right ) x}{2 b}+\frac {3 d \ln \left (1-i {\mathrm e}^{i \left (x b +a \right )}\right ) x}{2 b}+\frac {3 d \ln \left (1-i {\mathrm e}^{i \left (x b +a \right )}\right ) a}{2 b^{2}}+\frac {3 i d \operatorname {dilog}\left (1+i {\mathrm e}^{i \left (x b +a \right )}\right )}{2 b^{2}}-\frac {3 i d \operatorname {dilog}\left (1-i {\mathrm e}^{i \left (x b +a \right )}\right )}{2 b^{2}}\)	\(344\)

[In]

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

[Out]

-I/b^2/(exp(2*I*(b*x+a))+1)^2/(exp(2*I*(b*x+a))-1)*(3*d*x*b*exp(5*I*(b*x+a))+3*c*b*exp(5*I*(b*x+a))+2*d*x*b*ex
p(3*I*(b*x+a))+2*c*b*exp(3*I*(b*x+a))-I*d*exp(5*I*(b*x+a))+3*d*x*b*exp(I*(b*x+a))+3*c*b*exp(I*(b*x+a))+I*d*exp
(I*(b*x+a)))-3*I/b*c*arctan(exp(I*(b*x+a)))+3*I/b^2*a*d*arctan(exp(I*(b*x+a)))-d/b^2*ln(exp(I*(b*x+a))+1)+d/b^
2*ln(exp(I*(b*x+a))-1)-3/2/b^2*d*ln(1+I*exp(I*(b*x+a)))*a-3/2/b*d*ln(1+I*exp(I*(b*x+a)))*x+3/2/b*d*ln(1-I*exp(
I*(b*x+a)))*x+3/2/b^2*d*ln(1-I*exp(I*(b*x+a)))*a+3/2*I/b^2*d*dilog(1+I*exp(I*(b*x+a)))-3/2*I/b^2*d*dilog(1-I*e
xp(I*(b*x+a)))

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. 592 vs. \(2 (132) = 264\).

Time = 0.32 (sec) , antiderivative size = 592, normalized size of antiderivative = 3.65 \[ \int (c+d x) \csc ^2(a+b x) \sec ^3(a+b x) \, dx=\frac {-3 i \, d \cos \left (b x + a\right )^{2} {\rm Li}_2\left (i \, \cos \left (b x + a\right ) + \sin \left (b x + a\right )\right ) \sin \left (b x + a\right ) - 3 i \, d \cos \left (b x + a\right )^{2} {\rm Li}_2\left (i \, \cos \left (b x + a\right ) - \sin \left (b x + a\right )\right ) \sin \left (b x + a\right ) + 3 i \, d \cos \left (b x + a\right )^{2} {\rm Li}_2\left (-i \, \cos \left (b x + a\right ) + \sin \left (b x + a\right )\right ) \sin \left (b x + a\right ) + 3 i \, d \cos \left (b x + a\right )^{2} {\rm Li}_2\left (-i \, \cos \left (b x + a\right ) - \sin \left (b x + a\right )\right ) \sin \left (b x + a\right ) + 3 \, {\left (b c - a d\right )} \cos \left (b x + a\right )^{2} \log \left (\cos \left (b x + a\right ) + i \, \sin \left (b x + a\right ) + i\right ) \sin \left (b x + a\right ) - 3 \, {\left (b c - a d\right )} \cos \left (b x + a\right )^{2} \log \left (\cos \left (b x + a\right ) - i \, \sin \left (b x + a\right ) + i\right ) \sin \left (b x + a\right ) - 2 \, d \cos \left (b x + a\right )^{2} \log \left (\frac {1}{2} \, \cos \left (b x + a\right ) + \frac {1}{2}\right ) \sin \left (b x + a\right ) + 3 \, {\left (b d x + a d\right )} \cos \left (b x + a\right )^{2} \log \left (i \, \cos \left (b x + a\right ) + \sin \left (b x + a\right ) + 1\right ) \sin \left (b x + a\right ) - 3 \, {\left (b d x + a d\right )} \cos \left (b x + a\right )^{2} \log \left (i \, \cos \left (b x + a\right ) - \sin \left (b x + a\right ) + 1\right ) \sin \left (b x + a\right ) + 3 \, {\left (b d x + a d\right )} \cos \left (b x + a\right )^{2} \log \left (-i \, \cos \left (b x + a\right ) + \sin \left (b x + a\right ) + 1\right ) \sin \left (b x + a\right ) - 3 \, {\left (b d x + a d\right )} \cos \left (b x + a\right )^{2} \log \left (-i \, \cos \left (b x + a\right ) - \sin \left (b x + a\right ) + 1\right ) \sin \left (b x + a\right ) + 2 \, d \cos \left (b x + a\right )^{2} \log \left (-\frac {1}{2} \, \cos \left (b x + a\right ) + \frac {1}{2}\right ) \sin \left (b x + a\right ) + 3 \, {\left (b c - a d\right )} \cos \left (b x + a\right )^{2} \log \left (-\cos \left (b x + a\right ) + i \, \sin \left (b x + a\right ) + i\right ) \sin \left (b x + a\right ) - 3 \, {\left (b c - a d\right )} \cos \left (b x + a\right )^{2} \log \left (-\cos \left (b x + a\right ) - i \, \sin \left (b x + a\right ) + i\right ) \sin \left (b x + a\right ) + 2 \, b d x - 6 \, {\left (b d x + b c\right )} \cos \left (b x + a\right )^{2} - 2 \, d \cos \left (b x + a\right ) \sin \left (b x + a\right ) + 2 \, b c}{4 \, b^{2} \cos \left (b x + a\right )^{2} \sin \left (b x + a\right )} \]

[In]

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

[Out]

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

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F]

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

[In]

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

[Out]

1/4*(8*(b*d*x + b*c)*cos(3*b*x + 3*a)*sin(2*b*x + 2*a) - 4*(d*cos(5*b*x + 5*a) - d*cos(b*x + a) - 3*(b*d*x + b
*c)*sin(5*b*x + 5*a) - 2*(b*d*x + b*c)*sin(3*b*x + 3*a) - 3*(b*d*x + b*c)*sin(b*x + a))*cos(6*b*x + 6*a) - 4*(
d*cos(4*b*x + 4*a) - d*cos(2*b*x + 2*a) + 3*(b*d*x + b*c)*sin(4*b*x + 4*a) - 3*(b*d*x + b*c)*sin(2*b*x + 2*a)
- d)*cos(5*b*x + 5*a) + 4*(d*cos(b*x + a) + 2*(b*d*x + b*c)*sin(3*b*x + 3*a) + 3*(b*d*x + b*c)*sin(b*x + a))*c
os(4*b*x + 4*a) - 4*(d*cos(b*x + a) + 3*(b*d*x + b*c)*sin(b*x + a))*cos(2*b*x + 2*a) - 4*d*cos(b*x + a) + 12*(
b^2*d*cos(6*b*x + 6*a)^2 + b^2*d*cos(4*b*x + 4*a)^2 + b^2*d*cos(2*b*x + 2*a)^2 + b^2*d*sin(6*b*x + 6*a)^2 + b^
2*d*sin(4*b*x + 4*a)^2 - 2*b^2*d*sin(4*b*x + 4*a)*sin(2*b*x + 2*a) + b^2*d*sin(2*b*x + 2*a)^2 + 2*b^2*d*cos(2*
b*x + 2*a) + b^2*d + 2*(b^2*d*cos(4*b*x + 4*a) - b^2*d*cos(2*b*x + 2*a) - b^2*d)*cos(6*b*x + 6*a) - 2*(b^2*d*c
os(2*b*x + 2*a) + b^2*d)*cos(4*b*x + 4*a) + 2*(b^2*d*sin(4*b*x + 4*a) - b^2*d*sin(2*b*x + 2*a))*sin(6*b*x + 6*
a))*integrate((x*cos(2*b*x + 2*a)*cos(b*x + a) + x*sin(2*b*x + 2*a)*sin(b*x + a) + x*cos(b*x + a))/(cos(2*b*x
+ 2*a)^2 + sin(2*b*x + 2*a)^2 + 2*cos(2*b*x + 2*a) + 1), x) + 3*(b*c*cos(6*b*x + 6*a)^2 + b*c*cos(4*b*x + 4*a)
^2 + b*c*cos(2*b*x + 2*a)^2 + b*c*sin(6*b*x + 6*a)^2 + b*c*sin(4*b*x + 4*a)^2 - 2*b*c*sin(4*b*x + 4*a)*sin(2*b
*x + 2*a) + b*c*sin(2*b*x + 2*a)^2 + 2*b*c*cos(2*b*x + 2*a) + b*c + 2*(b*c*cos(4*b*x + 4*a) - b*c*cos(2*b*x +
2*a) - b*c)*cos(6*b*x + 6*a) - 2*(b*c*cos(2*b*x + 2*a) + b*c)*cos(4*b*x + 4*a) + 2*(b*c*sin(4*b*x + 4*a) - b*c
*sin(2*b*x + 2*a))*sin(6*b*x + 6*a))*log(cos(b*x + a)^2 + sin(b*x + a)^2 + 2*sin(b*x + a) + 1) - 3*(b*c*cos(6*
b*x + 6*a)^2 + b*c*cos(4*b*x + 4*a)^2 + b*c*cos(2*b*x + 2*a)^2 + b*c*sin(6*b*x + 6*a)^2 + b*c*sin(4*b*x + 4*a)
^2 - 2*b*c*sin(4*b*x + 4*a)*sin(2*b*x + 2*a) + b*c*sin(2*b*x + 2*a)^2 + 2*b*c*cos(2*b*x + 2*a) + b*c + 2*(b*c*
cos(4*b*x + 4*a) - b*c*cos(2*b*x + 2*a) - b*c)*cos(6*b*x + 6*a) - 2*(b*c*cos(2*b*x + 2*a) + b*c)*cos(4*b*x + 4
*a) + 2*(b*c*sin(4*b*x + 4*a) - b*c*sin(2*b*x + 2*a))*sin(6*b*x + 6*a))*log(cos(b*x + a)^2 + sin(b*x + a)^2 -
2*sin(b*x + a) + 1) - 2*(d*cos(6*b*x + 6*a)^2 + d*cos(4*b*x + 4*a)^2 + d*cos(2*b*x + 2*a)^2 + d*sin(6*b*x + 6*
a)^2 + d*sin(4*b*x + 4*a)^2 - 2*d*sin(4*b*x + 4*a)*sin(2*b*x + 2*a) + d*sin(2*b*x + 2*a)^2 + 2*(d*cos(4*b*x +
4*a) - d*cos(2*b*x + 2*a) - d)*cos(6*b*x + 6*a) - 2*(d*cos(2*b*x + 2*a) + d)*cos(4*b*x + 4*a) + 2*d*cos(2*b*x
+ 2*a) + 2*(d*sin(4*b*x + 4*a) - d*sin(2*b*x + 2*a))*sin(6*b*x + 6*a) + d)*log(cos(b*x)^2 + 2*cos(b*x)*cos(a)
+ cos(a)^2 + sin(b*x)^2 - 2*sin(b*x)*sin(a) + sin(a)^2) + 2*(d*cos(6*b*x + 6*a)^2 + d*cos(4*b*x + 4*a)^2 + d*c
os(2*b*x + 2*a)^2 + d*sin(6*b*x + 6*a)^2 + d*sin(4*b*x + 4*a)^2 - 2*d*sin(4*b*x + 4*a)*sin(2*b*x + 2*a) + d*si
n(2*b*x + 2*a)^2 + 2*(d*cos(4*b*x + 4*a) - d*cos(2*b*x + 2*a) - d)*cos(6*b*x + 6*a) - 2*(d*cos(2*b*x + 2*a) +
d)*cos(4*b*x + 4*a) + 2*d*cos(2*b*x + 2*a) + 2*(d*sin(4*b*x + 4*a) - d*sin(2*b*x + 2*a))*sin(6*b*x + 6*a) + d)
*log(cos(b*x)^2 - 2*cos(b*x)*cos(a) + cos(a)^2 + sin(b*x)^2 + 2*sin(b*x)*sin(a) + sin(a)^2) - 4*(3*(b*d*x + b*
c)*cos(5*b*x + 5*a) + 2*(b*d*x + b*c)*cos(3*b*x + 3*a) + 3*(b*d*x + b*c)*cos(b*x + a) + d*sin(5*b*x + 5*a) - d
*sin(b*x + a))*sin(6*b*x + 6*a) - 4*(3*b*d*x + 3*b*c - 3*(b*d*x + b*c)*cos(4*b*x + 4*a) + 3*(b*d*x + b*c)*cos(
2*b*x + 2*a) + d*sin(4*b*x + 4*a) - d*sin(2*b*x + 2*a))*sin(5*b*x + 5*a) - 4*(2*(b*d*x + b*c)*cos(3*b*x + 3*a)
+ 3*(b*d*x + b*c)*cos(b*x + a) - d*sin(b*x + a))*sin(4*b*x + 4*a) - 8*(b*d*x + b*c + (b*d*x + b*c)*cos(2*b*x
+ 2*a))*sin(3*b*x + 3*a) + 4*(3*(b*d*x + b*c)*cos(b*x + a) - d*sin(b*x + a))*sin(2*b*x + 2*a) - 12*(b*d*x + b*
c)*sin(b*x + a))/(b^2*cos(6*b*x + 6*a)^2 + b^2*cos(4*b*x + 4*a)^2 + b^2*cos(2*b*x + 2*a)^2 + b^2*sin(6*b*x + 6
*a)^2 + b^2*sin(4*b*x + 4*a)^2 - 2*b^2*sin(4*b*x + 4*a)*sin(2*b*x + 2*a) + b^2*sin(2*b*x + 2*a)^2 + 2*b^2*cos(
2*b*x + 2*a) + b^2 + 2*(b^2*cos(4*b*x + 4*a) - b^2*cos(2*b*x + 2*a) - b^2)*cos(6*b*x + 6*a) - 2*(b^2*cos(2*b*x
+ 2*a) + b^2)*cos(4*b*x + 4*a) + 2*(b^2*sin(4*b*x + 4*a) - b^2*sin(2*b*x + 2*a))*sin(6*b*x + 6*a))

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

\text{Hanged}