\(\int \csc (c+d x) (a+b \sec (c+d x)) \, dx\) [164]

Optimal result

Integrand size = 17, antiderivative size = 26 \[ \int \csc (c+d x) (a+b \sec (c+d x)) \, dx=-\frac {a \text {arctanh}(\cos (c+d x))}{d}+\frac {b \log (\tan (c+d x))}{d} \]

[Out]

-a*arctanh(cos(d*x+c))/d+b*ln(tan(d*x+c))/d

Rubi [A] (verified)

Time = 0.08 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.294, Rules used = {3957, 2913, 2700, 29, 3855} \[ \int \csc (c+d x) (a+b \sec (c+d x)) \, dx=\frac {b \log (\tan (c+d x))}{d}-\frac {a \text {arctanh}(\cos (c+d x))}{d} \]

[In]

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

[Out]

-((a*ArcTanh[Cos[c + d*x]])/d) + (b*Log[Tan[c + d*x]])/d

Rule 29

Int[(x_)^(-1), x_Symbol] :> Simp[Log[x], x]

Rule 2700

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

Rule 2913

Int[cos[(e_.) + (f_.)*(x_)]^(p_)*((d_.)*sin[(e_.) + (f_.)*(x_)])^(n_.)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]),
x_Symbol] :> Dist[a, Int[Cos[e + f*x]^p*(d*Sin[e + f*x])^n, x], x] + Dist[b/d, Int[Cos[e + f*x]^p*(d*Sin[e +
f*x])^(n + 1), x], x] /; FreeQ[{a, b, d, e, f, n, p}, x] && IntegerQ[(p - 1)/2] && IntegerQ[n] && ((LtQ[p, 0]
&& NeQ[a^2 - b^2, 0]) || LtQ[0, n, p - 1] || LtQ[p + 1, -n, 2*p + 1])

Rule 3855

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

Rule 3957

Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_.)*(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_))^(m_.), x_Symbol] :> Int[(g*Co
s[e + f*x])^p*((b + a*Sin[e + f*x])^m/Sin[e + f*x]^m), x] /; FreeQ[{a, b, e, f, g, p}, x] && IntegerQ[m]

Rubi steps \begin{align*} \text {integral}& = -\int (-b-a \cos (c+d x)) \csc (c+d x) \sec (c+d x) \, dx \\ & = a \int \csc (c+d x) \, dx+b \int \csc (c+d x) \sec (c+d x) \, dx \\ & = -\frac {a \text {arctanh}(\cos (c+d x))}{d}+\frac {b \text {Subst}\left (\int \frac {1}{x} \, dx,x,\tan (c+d x)\right )}{d} \\ & = -\frac {a \text {arctanh}(\cos (c+d x))}{d}+\frac {b \log (\tan (c+d x))}{d} \\ \end{align*}

Mathematica [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(65\) vs. \(2(26)=52\).

Time = 0.02 (sec) , antiderivative size = 65, normalized size of antiderivative = 2.50 \[ \int \csc (c+d x) (a+b \sec (c+d x)) \, dx=-\frac {a \log \left (\cos \left (\frac {c}{2}+\frac {d x}{2}\right )\right )}{d}-\frac {b \log (\cos (c+d x))}{d}+\frac {a \log \left (\sin \left (\frac {c}{2}+\frac {d x}{2}\right )\right )}{d}+\frac {b \log (\sin (c+d x))}{d} \]

[In]

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

[Out]

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

Maple [A] (verified)

Time = 0.57 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.27

method	result	size
derivativedivides	\(\frac {a \ln \left (-\cot \left (d x +c \right )+\csc \left (d x +c \right )\right )+b \ln \left (\tan \left (d x +c \right )\right )}{d}\)	\(33\)
default	\(\frac {a \ln \left (-\cot \left (d x +c \right )+\csc \left (d x +c \right )\right )+b \ln \left (\tan \left (d x +c \right )\right )}{d}\)	\(33\)
parallelrisch	\(\frac {-b \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )-b \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )+\ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \left (a +b \right )}{d}\)	\(50\)
norman	\(\frac {\left (a +b \right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}-\frac {b \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{d}-\frac {b \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{d}\)	\(55\)
risch	\(-\frac {a \ln \left ({\mathrm e}^{i \left (d x +c \right )}+1\right )}{d}+\frac {\ln \left ({\mathrm e}^{i \left (d x +c \right )}+1\right ) b}{d}+\frac {a \ln \left ({\mathrm e}^{i \left (d x +c \right )}-1\right )}{d}+\frac {\ln \left ({\mathrm e}^{i \left (d x +c \right )}-1\right ) b}{d}-\frac {b \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )}{d}\)	\(89\)

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 51, normalized size of antiderivative = 1.96 \[ \int \csc (c+d x) (a+b \sec (c+d x)) \, dx=-\frac {2 \, b \log \left (-\cos \left (d x + c\right )\right ) + {\left (a - b\right )} \log \left (\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) - {\left (a + b\right )} \log \left (-\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right )}{2 \, d} \]

[In]

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

[Out]

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

Sympy [F]

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

[In]

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

[Out]

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

Maxima [A] (verification not implemented)

none

Time = 0.20 (sec) , antiderivative size = 45, normalized size of antiderivative = 1.73 \[ \int \csc (c+d x) (a+b \sec (c+d x)) \, dx=-\frac {{\left (a - b\right )} \log \left (\cos \left (d x + c\right ) + 1\right ) - {\left (a + b\right )} \log \left (\cos \left (d x + c\right ) - 1\right ) + 2 \, b \log \left (\cos \left (d x + c\right )\right )}{2 \, d} \]

[In]

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

[Out]

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

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 61 vs. \(2 (26) = 52\).

Time = 0.28 (sec) , antiderivative size = 61, normalized size of antiderivative = 2.35 \[ \int \csc (c+d x) (a+b \sec (c+d x)) \, dx=\frac {{\left (a + b\right )} \log \left (\frac {{\left | -\cos \left (d x + c\right ) + 1 \right |}}{{\left | \cos \left (d x + c\right ) + 1 \right |}}\right ) - 2 \, b \log \left ({\left | -\frac {\cos \left (d x + c\right ) - 1}{\cos \left (d x + c\right ) + 1} - 1 \right |}\right )}{2 \, d} \]

[In]

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

[Out]

1/2*((a + b)*log(abs(-cos(d*x + c) + 1)/abs(cos(d*x + c) + 1)) - 2*b*log(abs(-(cos(d*x + c) - 1)/(cos(d*x + c)
+ 1) - 1)))/d

Mupad [B] (verification not implemented)

Time = 0.11 (sec) , antiderivative size = 63, normalized size of antiderivative = 2.42 \[ \int \csc (c+d x) (a+b \sec (c+d x)) \, dx=\frac {\frac {a\,\ln \left (\cos \left (c+d\,x\right )-1\right )}{2}-b\,\ln \left (\cos \left (c+d\,x\right )\right )-\frac {a\,\ln \left (\cos \left (c+d\,x\right )+1\right )}{2}+\frac {b\,\ln \left (\cos \left (c+d\,x\right )-1\right )}{2}+\frac {b\,\ln \left (\cos \left (c+d\,x\right )+1\right )}{2}}{d} \]

[In]

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

[Out]

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