\(\int (c+d x) \csc (a+b x) \sec (a+b x) \, dx\) [231]

Optimal result

Integrand size = 18, antiderivative size = 71 \[ \int (c+d x) \csc (a+b x) \sec (a+b x) \, dx=-\frac {2 (c+d x) \text {arctanh}\left (e^{2 i (a+b x)}\right )}{b}+\frac {i d \operatorname {PolyLog}\left (2,-e^{2 i (a+b x)}\right )}{2 b^2}-\frac {i d \operatorname {PolyLog}\left (2,e^{2 i (a+b x)}\right )}{2 b^2} \]

[Out]

-2*(d*x+c)*arctanh(exp(2*I*(b*x+a)))/b+1/2*I*d*polylog(2,-exp(2*I*(b*x+a)))/b^2-1/2*I*d*polylog(2,exp(2*I*(b*x
+a)))/b^2

Rubi [A] (verified)

Time = 0.07 (sec) , antiderivative size = 71, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {4504, 4268, 2317, 2438} \[ \int (c+d x) \csc (a+b x) \sec (a+b x) \, dx=-\frac {2 (c+d x) \text {arctanh}\left (e^{2 i (a+b x)}\right )}{b}+\frac {i d \operatorname {PolyLog}\left (2,-e^{2 i (a+b x)}\right )}{2 b^2}-\frac {i d \operatorname {PolyLog}\left (2,e^{2 i (a+b x)}\right )}{2 b^2} \]

[In]

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

[Out]

(-2*(c + d*x)*ArcTanh[E^((2*I)*(a + b*x))])/b + ((I/2)*d*PolyLog[2, -E^((2*I)*(a + b*x))])/b^2 - ((I/2)*d*Poly
Log[2, E^((2*I)*(a + b*x))])/b^2

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 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 4504

Int[Csc[(a_.) + (b_.)*(x_)]^(n_.)*((c_.) + (d_.)*(x_))^(m_.)*Sec[(a_.) + (b_.)*(x_)]^(n_.), x_Symbol] :> Dist[
2^n, Int[(c + d*x)^m*Csc[2*a + 2*b*x]^n, x], x] /; FreeQ[{a, b, c, d, m}, x] && IntegerQ[n] && RationalQ[m]

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

Mathematica [A] (verified)

Time = 0.21 (sec) , antiderivative size = 141, normalized size of antiderivative = 1.99 \[ \int (c+d x) \csc (a+b x) \sec (a+b x) \, dx=-\frac {c \log (\cos (a+b x))}{b}+\frac {c \log (\sin (a+b x))}{b}+\frac {d \left ((2 a+2 b x) \left (\log \left (1-e^{i (2 a+2 b x)}\right )-\log \left (1+e^{i (2 a+2 b x)}\right )\right )-2 a \log \left (\tan \left (\frac {1}{2} (2 a+2 b x)\right )\right )+i \left (\operatorname {PolyLog}\left (2,-e^{i (2 a+2 b x)}\right )-\operatorname {PolyLog}\left (2,e^{i (2 a+2 b x)}\right )\right )\right )}{2 b^2} \]

[In]

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

[Out]

-((c*Log[Cos[a + b*x]])/b) + (c*Log[Sin[a + b*x]])/b + (d*((2*a + 2*b*x)*(Log[1 - E^(I*(2*a + 2*b*x))] - Log[1
+ E^(I*(2*a + 2*b*x))]) - 2*a*Log[Tan[(2*a + 2*b*x)/2]] + I*(PolyLog[2, -E^(I*(2*a + 2*b*x))] - PolyLog[2, E^
(I*(2*a + 2*b*x))])))/(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. 207 vs. \(2 (59 ) = 118\).

Time = 1.01 (sec) , antiderivative size = 208, normalized size of antiderivative = 2.93

method	result	size
risch	\(\frac {c \ln \left ({\mathrm e}^{i \left (x b +a \right )}+1\right )}{b}-\frac {c \ln \left ({\mathrm e}^{2 i \left (x b +a \right )}+1\right )}{b}+\frac {c \ln \left ({\mathrm e}^{i \left (x b +a \right )}-1\right )}{b}+\frac {d \ln \left ({\mathrm e}^{i \left (x b +a \right )}+1\right ) x}{b}-\frac {i d \operatorname {polylog}\left (2, -{\mathrm e}^{i \left (x b +a \right )}\right )}{b^{2}}-\frac {d \ln \left ({\mathrm e}^{2 i \left (x b +a \right )}+1\right ) x}{b}+\frac {i d \operatorname {polylog}\left (2, -{\mathrm e}^{2 i \left (x b +a \right )}\right )}{2 b^{2}}+\frac {d \ln \left (1-{\mathrm e}^{i \left (x b +a \right )}\right ) x}{b}+\frac {d \ln \left (1-{\mathrm e}^{i \left (x b +a \right )}\right ) a}{b^{2}}-\frac {i d \operatorname {polylog}\left (2, {\mathrm e}^{i \left (x b +a \right )}\right )}{b^{2}}-\frac {d a \ln \left ({\mathrm e}^{i \left (x b +a \right )}-1\right )}{b^{2}}\)	\(208\)

[In]

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

[Out]

1/b*c*ln(exp(I*(b*x+a))+1)-1/b*c*ln(exp(2*I*(b*x+a))+1)+1/b*c*ln(exp(I*(b*x+a))-1)+1/b*d*ln(exp(I*(b*x+a))+1)*
x-I*d*polylog(2,-exp(I*(b*x+a)))/b^2-1/b*d*ln(exp(2*I*(b*x+a))+1)*x+1/2*I*d*polylog(2,-exp(2*I*(b*x+a)))/b^2+1
/b*d*ln(1-exp(I*(b*x+a)))*x+1/b^2*d*ln(1-exp(I*(b*x+a)))*a-I*d*polylog(2,exp(I*(b*x+a)))/b^2-1/b^2*d*a*ln(exp(
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. 554 vs. \(2 (55) = 110\).

Time = 0.32 (sec) , antiderivative size = 554, normalized size of antiderivative = 7.80 \[ \int (c+d x) \csc (a+b x) \sec (a+b x) \, dx=\frac {-i \, d {\rm Li}_2\left (\cos \left (b x + a\right ) + i \, \sin \left (b x + a\right )\right ) + i \, d {\rm Li}_2\left (\cos \left (b x + a\right ) - i \, \sin \left (b x + a\right )\right ) - i \, d {\rm Li}_2\left (i \, \cos \left (b x + a\right ) + \sin \left (b x + a\right )\right ) + i \, d {\rm Li}_2\left (i \, \cos \left (b x + a\right ) - \sin \left (b x + a\right )\right ) + i \, d {\rm Li}_2\left (-i \, \cos \left (b x + a\right ) + \sin \left (b x + a\right )\right ) - i \, d {\rm Li}_2\left (-i \, \cos \left (b x + a\right ) - \sin \left (b x + a\right )\right ) + i \, d {\rm Li}_2\left (-\cos \left (b x + a\right ) + i \, \sin \left (b x + a\right )\right ) - i \, d {\rm Li}_2\left (-\cos \left (b x + a\right ) - i \, \sin \left (b x + a\right )\right ) + {\left (b d x + b c\right )} \log \left (\cos \left (b x + a\right ) + i \, \sin \left (b x + a\right ) + 1\right ) - {\left (b c - a d\right )} \log \left (\cos \left (b x + a\right ) + i \, \sin \left (b x + a\right ) + i\right ) + {\left (b d x + b c\right )} \log \left (\cos \left (b x + a\right ) - i \, \sin \left (b x + a\right ) + 1\right ) - {\left (b c - a d\right )} \log \left (\cos \left (b x + a\right ) - i \, \sin \left (b x + a\right ) + i\right ) - {\left (b d x + a d\right )} \log \left (i \, \cos \left (b x + a\right ) + \sin \left (b x + a\right ) + 1\right ) - {\left (b d x + a d\right )} \log \left (i \, \cos \left (b x + a\right ) - \sin \left (b x + a\right ) + 1\right ) - {\left (b d x + a d\right )} \log \left (-i \, \cos \left (b x + a\right ) + \sin \left (b x + a\right ) + 1\right ) - {\left (b d x + a d\right )} \log \left (-i \, \cos \left (b x + a\right ) - \sin \left (b x + a\right ) + 1\right ) + {\left (b c - a d\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 ) + {\left (b c - a d\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 ) + {\left (b d x + a d\right )} \log \left (-\cos \left (b x + a\right ) + i \, \sin \left (b x + a\right ) + 1\right ) - {\left (b c - a d\right )} \log \left (-\cos \left (b x + a\right ) + i \, \sin \left (b x + a\right ) + i\right ) + {\left (b d x + a d\right )} \log \left (-\cos \left (b x + a\right ) - i \, \sin \left (b x + a\right ) + 1\right ) - {\left (b c - a d\right )} \log \left (-\cos \left (b x + a\right ) - i \, \sin \left (b x + a\right ) + i\right )}{2 \, b^{2}} \]

[In]

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

[Out]

1/2*(-I*d*dilog(cos(b*x + a) + I*sin(b*x + a)) + I*d*dilog(cos(b*x + a) - I*sin(b*x + a)) - I*d*dilog(I*cos(b*
x + a) + sin(b*x + a)) + I*d*dilog(I*cos(b*x + a) - sin(b*x + a)) + I*d*dilog(-I*cos(b*x + a) + sin(b*x + a))
- I*d*dilog(-I*cos(b*x + a) - sin(b*x + a)) + I*d*dilog(-cos(b*x + a) + I*sin(b*x + a)) - I*d*dilog(-cos(b*x +
a) - I*sin(b*x + a)) + (b*d*x + b*c)*log(cos(b*x + a) + I*sin(b*x + a) + 1) - (b*c - a*d)*log(cos(b*x + a) +
I*sin(b*x + a) + I) + (b*d*x + b*c)*log(cos(b*x + a) - I*sin(b*x + a) + 1) - (b*c - a*d)*log(cos(b*x + a) - I*
sin(b*x + a) + I) - (b*d*x + a*d)*log(I*cos(b*x + a) + sin(b*x + a) + 1) - (b*d*x + a*d)*log(I*cos(b*x + a) -
sin(b*x + a) + 1) - (b*d*x + a*d)*log(-I*cos(b*x + a) + sin(b*x + a) + 1) - (b*d*x + a*d)*log(-I*cos(b*x + a)
- sin(b*x + a) + 1) + (b*c - a*d)*log(-1/2*cos(b*x + a) + 1/2*I*sin(b*x + a) + 1/2) + (b*c - a*d)*log(-1/2*cos
(b*x + a) - 1/2*I*sin(b*x + a) + 1/2) + (b*d*x + a*d)*log(-cos(b*x + a) + I*sin(b*x + a) + 1) - (b*c - a*d)*lo
g(-cos(b*x + a) + I*sin(b*x + a) + I) + (b*d*x + a*d)*log(-cos(b*x + a) - I*sin(b*x + a) + 1) - (b*c - a*d)*lo
g(-cos(b*x + a) - I*sin(b*x + a) + I))/b^2

Sympy [F]

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

[In]

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

[Out]

Integral((c + d*x)*csc(a + b*x)*sec(a + b*x), 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. 269 vs. \(2 (55) = 110\).

Time = 0.41 (sec) , antiderivative size = 269, normalized size of antiderivative = 3.79 \[ \int (c+d x) \csc (a+b x) \sec (a+b x) \, dx=-\frac {2 i \, b d x \arctan \left (\sin \left (b x + a\right ), -\cos \left (b x + a\right ) + 1\right ) - 2 i \, b c \arctan \left (\sin \left (b x + a\right ), \cos \left (b x + a\right ) - 1\right ) - 2 \, {\left (-i \, b d x - i \, b c\right )} \arctan \left (\sin \left (2 \, b x + 2 \, a\right ), \cos \left (2 \, b x + 2 \, a\right ) + 1\right ) - 2 \, {\left (i \, b d x + i \, b c\right )} \arctan \left (\sin \left (b x + a\right ), \cos \left (b x + a\right ) + 1\right ) - i \, d {\rm Li}_2\left (-e^{\left (2 i \, b x + 2 i \, a\right )}\right ) + 2 i \, d {\rm Li}_2\left (-e^{\left (i \, b x + i \, a\right )}\right ) + 2 i \, d {\rm Li}_2\left (e^{\left (i \, b x + i \, a\right )}\right ) + {\left (b d x + b c\right )} \log \left (\cos \left (2 \, b x + 2 \, a\right )^{2} + \sin \left (2 \, b x + 2 \, a\right )^{2} + 2 \, \cos \left (2 \, b x + 2 \, a\right ) + 1\right ) - {\left (b d x + b c\right )} \log \left (\cos \left (b x + a\right )^{2} + \sin \left (b x + a\right )^{2} + 2 \, \cos \left (b x + a\right ) + 1\right ) - {\left (b d x + b c\right )} \log \left (\cos \left (b x + a\right )^{2} + \sin \left (b x + a\right )^{2} - 2 \, \cos \left (b x + a\right ) + 1\right )}{2 \, b^{2}} \]

[In]

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

[Out]

-1/2*(2*I*b*d*x*arctan2(sin(b*x + a), -cos(b*x + a) + 1) - 2*I*b*c*arctan2(sin(b*x + a), cos(b*x + a) - 1) - 2
*(-I*b*d*x - I*b*c)*arctan2(sin(2*b*x + 2*a), cos(2*b*x + 2*a) + 1) - 2*(I*b*d*x + I*b*c)*arctan2(sin(b*x + a)
, cos(b*x + a) + 1) - I*d*dilog(-e^(2*I*b*x + 2*I*a)) + 2*I*d*dilog(-e^(I*b*x + I*a)) + 2*I*d*dilog(e^(I*b*x +
I*a)) + (b*d*x + b*c)*log(cos(2*b*x + 2*a)^2 + sin(2*b*x + 2*a)^2 + 2*cos(2*b*x + 2*a) + 1) - (b*d*x + b*c)*l
og(cos(b*x + a)^2 + sin(b*x + a)^2 + 2*cos(b*x + a) + 1) - (b*d*x + b*c)*log(cos(b*x + a)^2 + sin(b*x + a)^2 -
2*cos(b*x + a) + 1))/b^2

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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