\(\int (c+d x) \cot (a+b x) \csc ^2(a+b x) \, dx\) [49]

Optimal result

Integrand size = 20, antiderivative size = 35 \[ \int (c+d x) \cot (a+b x) \csc ^2(a+b x) \, dx=-\frac {d \cot (a+b x)}{2 b^2}-\frac {(c+d x) \csc ^2(a+b x)}{2 b} \]

[Out]

-1/2*d*cot(b*x+a)/b^2-1/2*(d*x+c)*csc(b*x+a)^2/b

Rubi [A] (verified)

Time = 0.04 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.150, Rules used = {4495, 3852, 8} \[ \int (c+d x) \cot (a+b x) \csc ^2(a+b x) \, dx=-\frac {d \cot (a+b x)}{2 b^2}-\frac {(c+d x) \csc ^2(a+b x)}{2 b} \]

[In]

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

[Out]

-1/2*(d*Cot[a + b*x])/b^2 - ((c + d*x)*Csc[a + b*x]^2)/(2*b)

Rule 8

Int[a_, x_Symbol] :> Simp[a*x, x] /; FreeQ[a, x]

Rule 3852

Int[csc[(c_.) + (d_.)*(x_)]^(n_), x_Symbol] :> Dist[-d^(-1), Subst[Int[ExpandIntegrand[(1 + x^2)^(n/2 - 1), x]
, x], x, Cot[c + d*x]], x] /; FreeQ[{c, d}, x] && IGtQ[n/2, 0]

Rule 4495

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

Rubi steps \begin{align*} \text {integral}& = -\frac {(c+d x) \csc ^2(a+b x)}{2 b}+\frac {d \int \csc ^2(a+b x) \, dx}{2 b} \\ & = -\frac {(c+d x) \csc ^2(a+b x)}{2 b}-\frac {d \text {Subst}(\int 1 \, dx,x,\cot (a+b x))}{2 b^2} \\ & = -\frac {d \cot (a+b x)}{2 b^2}-\frac {(c+d x) \csc ^2(a+b x)}{2 b} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.34 (sec) , antiderivative size = 48, normalized size of antiderivative = 1.37 \[ \int (c+d x) \cot (a+b x) \csc ^2(a+b x) \, dx=-\frac {d \cot (a+b x)}{2 b^2}-\frac {c \csc ^2(a+b x)}{2 b}-\frac {d x \csc ^2(a+b x)}{2 b} \]

[In]

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

[Out]

-1/2*(d*Cot[a + b*x])/b^2 - (c*Csc[a + b*x]^2)/(2*b) - (d*x*Csc[a + b*x]^2)/(2*b)

Maple [A] (verified)

Time = 1.03 (sec) , antiderivative size = 61, normalized size of antiderivative = 1.74

method	result	size
derivativedivides	\(\frac {\frac {d a}{2 b \sin \left (x b +a \right )^{2}}-\frac {c}{2 \sin \left (x b +a \right )^{2}}+\frac {d \left (-\frac {x b +a}{2 \sin \left (x b +a \right )^{2}}-\frac {\cot \left (x b +a \right )}{2}\right )}{b}}{b}\)	\(61\)
default	\(\frac {\frac {d a}{2 b \sin \left (x b +a \right )^{2}}-\frac {c}{2 \sin \left (x b +a \right )^{2}}+\frac {d \left (-\frac {x b +a}{2 \sin \left (x b +a \right )^{2}}-\frac {\cot \left (x b +a \right )}{2}\right )}{b}}{b}\)	\(61\)
parallelrisch	\(-\frac {\sec \left (\frac {a}{2}+\frac {x b}{2}\right )^{2} \csc \left (\frac {a}{2}+\frac {x b}{2}\right )^{2} \left (\cos \left (2 x b +2 a \right ) b c +2 d \sin \left (2 x b +2 a \right )+3 \left (\frac {4 d x}{3}+c \right ) b \right )}{32 b^{2}}\)	\(62\)
risch	\(\frac {2 b d x \,{\mathrm e}^{2 i \left (x b +a \right )}-i d \,{\mathrm e}^{2 i \left (x b +a \right )}+2 b c \,{\mathrm e}^{2 i \left (x b +a \right )}+i d}{b^{2} \left ({\mathrm e}^{2 i \left (x b +a \right )}-1\right )^{2}}\)	\(63\)
norman	\(\frac {-\frac {c}{8 b}-\frac {c \tan \left (\frac {a}{2}+\frac {x b}{2}\right )^{4}}{8 b}-\frac {d \tan \left (\frac {a}{2}+\frac {x b}{2}\right )}{4 b^{2}}+\frac {d \tan \left (\frac {a}{2}+\frac {x b}{2}\right )^{3}}{4 b^{2}}-\frac {d x}{8 b}-\frac {d x \tan \left (\frac {a}{2}+\frac {x b}{2}\right )^{2}}{4 b}-\frac {d x \tan \left (\frac {a}{2}+\frac {x b}{2}\right )^{4}}{8 b}}{\tan \left (\frac {a}{2}+\frac {x b}{2}\right )^{2}}\)	\(112\)

[In]

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

[Out]

1/b*(1/2/b*d*a/sin(b*x+a)^2-1/2*c/sin(b*x+a)^2+1/b*d*(-1/2*(b*x+a)/sin(b*x+a)^2-1/2*cot(b*x+a)))

Fricas [A] (verification not implemented)

none

Time = 0.24 (sec) , antiderivative size = 44, normalized size of antiderivative = 1.26 \[ \int (c+d x) \cot (a+b x) \csc ^2(a+b x) \, dx=\frac {b d x + d \cos \left (b x + a\right ) \sin \left (b x + a\right ) + b c}{2 \, {\left (b^{2} \cos \left (b x + a\right )^{2} - b^{2}\right )}} \]

[In]

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

[Out]

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

Sympy [F]

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

[In]

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

[Out]

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

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 287 vs. \(2 (31) = 62\).

Time = 0.25 (sec) , antiderivative size = 287, normalized size of antiderivative = 8.20 \[ \int (c+d x) \cot (a+b x) \csc ^2(a+b x) \, dx=\frac {\frac {2 \, {\left (4 \, {\left (b x + a\right )} \cos \left (2 \, b x + 2 \, a\right )^{2} + 4 \, {\left (b x + a\right )} \sin \left (2 \, b x + 2 \, a\right )^{2} - {\left (2 \, {\left (b x + a\right )} \cos \left (2 \, b x + 2 \, a\right ) + \sin \left (2 \, b x + 2 \, a\right )\right )} \cos \left (4 \, b x + 4 \, a\right ) - 2 \, {\left (b x + a\right )} \cos \left (2 \, b x + 2 \, a\right ) - {\left (2 \, {\left (b x + a\right )} \sin \left (2 \, b x + 2 \, a\right ) - \cos \left (2 \, b x + 2 \, a\right ) + 1\right )} \sin \left (4 \, b x + 4 \, a\right ) + \sin \left (2 \, b x + 2 \, a\right )\right )} d}{{\left (2 \, {\left (2 \, \cos \left (2 \, b x + 2 \, a\right ) - 1\right )} \cos \left (4 \, b x + 4 \, a\right ) - \cos \left (4 \, b x + 4 \, a\right )^{2} - 4 \, \cos \left (2 \, b x + 2 \, a\right )^{2} - \sin \left (4 \, b x + 4 \, a\right )^{2} + 4 \, \sin \left (4 \, b x + 4 \, a\right ) \sin \left (2 \, b x + 2 \, a\right ) - 4 \, \sin \left (2 \, b x + 2 \, a\right )^{2} + 4 \, \cos \left (2 \, b x + 2 \, a\right ) - 1\right )} b} - \frac {c}{\sin \left (b x + a\right )^{2}} + \frac {a d}{b \sin \left (b x + a\right )^{2}}}{2 \, b} \]

[In]

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

[Out]

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

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 526 vs. \(2 (31) = 62\).

Time = 0.35 (sec) , antiderivative size = 526, normalized size of antiderivative = 15.03 \[ \int (c+d x) \cot (a+b x) \csc ^2(a+b x) \, dx=-\frac {b d x \tan \left (\frac {1}{2} \, b x\right )^{4} \tan \left (\frac {1}{2} \, a\right )^{4} + b c \tan \left (\frac {1}{2} \, b x\right )^{4} \tan \left (\frac {1}{2} \, a\right )^{4} + 2 \, b d x \tan \left (\frac {1}{2} \, b x\right )^{4} \tan \left (\frac {1}{2} \, a\right )^{2} + 2 \, b d x \tan \left (\frac {1}{2} \, b x\right )^{2} \tan \left (\frac {1}{2} \, a\right )^{4} + 2 \, b c \tan \left (\frac {1}{2} \, b x\right )^{4} \tan \left (\frac {1}{2} \, a\right )^{2} - 2 \, d \tan \left (\frac {1}{2} \, b x\right )^{4} \tan \left (\frac {1}{2} \, a\right )^{3} + 2 \, b c \tan \left (\frac {1}{2} \, b x\right )^{2} \tan \left (\frac {1}{2} \, a\right )^{4} - 2 \, d \tan \left (\frac {1}{2} \, b x\right )^{3} \tan \left (\frac {1}{2} \, a\right )^{4} + b d x \tan \left (\frac {1}{2} \, b x\right )^{4} + 4 \, b d x \tan \left (\frac {1}{2} \, b x\right )^{2} \tan \left (\frac {1}{2} \, a\right )^{2} + b d x \tan \left (\frac {1}{2} \, a\right )^{4} + b c \tan \left (\frac {1}{2} \, b x\right )^{4} + 2 \, d \tan \left (\frac {1}{2} \, b x\right )^{4} \tan \left (\frac {1}{2} \, a\right ) + 4 \, b c \tan \left (\frac {1}{2} \, b x\right )^{2} \tan \left (\frac {1}{2} \, a\right )^{2} + 12 \, d \tan \left (\frac {1}{2} \, b x\right )^{3} \tan \left (\frac {1}{2} \, a\right )^{2} + 12 \, d \tan \left (\frac {1}{2} \, b x\right )^{2} \tan \left (\frac {1}{2} \, a\right )^{3} + b c \tan \left (\frac {1}{2} \, a\right )^{4} + 2 \, d \tan \left (\frac {1}{2} \, b x\right ) \tan \left (\frac {1}{2} \, a\right )^{4} + 2 \, b d x \tan \left (\frac {1}{2} \, b x\right )^{2} + 2 \, b d x \tan \left (\frac {1}{2} \, a\right )^{2} + 2 \, b c \tan \left (\frac {1}{2} \, b x\right )^{2} - 2 \, d \tan \left (\frac {1}{2} \, b x\right )^{3} - 12 \, d \tan \left (\frac {1}{2} \, b x\right )^{2} \tan \left (\frac {1}{2} \, a\right ) + 2 \, b c \tan \left (\frac {1}{2} \, a\right )^{2} - 12 \, d \tan \left (\frac {1}{2} \, b x\right ) \tan \left (\frac {1}{2} \, a\right )^{2} - 2 \, d \tan \left (\frac {1}{2} \, a\right )^{3} + b d x + b c + 2 \, d \tan \left (\frac {1}{2} \, b x\right ) + 2 \, d \tan \left (\frac {1}{2} \, a\right )}{8 \, {\left (b^{2} \tan \left (\frac {1}{2} \, b x\right )^{4} \tan \left (\frac {1}{2} \, a\right )^{2} + 2 \, b^{2} \tan \left (\frac {1}{2} \, b x\right )^{3} \tan \left (\frac {1}{2} \, a\right )^{3} + b^{2} \tan \left (\frac {1}{2} \, b x\right )^{2} \tan \left (\frac {1}{2} \, a\right )^{4} - 2 \, b^{2} \tan \left (\frac {1}{2} \, b x\right )^{3} \tan \left (\frac {1}{2} \, a\right ) - 4 \, b^{2} \tan \left (\frac {1}{2} \, b x\right )^{2} \tan \left (\frac {1}{2} \, a\right )^{2} - 2 \, b^{2} \tan \left (\frac {1}{2} \, b x\right ) \tan \left (\frac {1}{2} \, a\right )^{3} + b^{2} \tan \left (\frac {1}{2} \, b x\right )^{2} + 2 \, b^{2} \tan \left (\frac {1}{2} \, b x\right ) \tan \left (\frac {1}{2} \, a\right ) + b^{2} \tan \left (\frac {1}{2} \, a\right )^{2}\right )}} \]

[In]

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

[Out]

-1/8*(b*d*x*tan(1/2*b*x)^4*tan(1/2*a)^4 + b*c*tan(1/2*b*x)^4*tan(1/2*a)^4 + 2*b*d*x*tan(1/2*b*x)^4*tan(1/2*a)^
2 + 2*b*d*x*tan(1/2*b*x)^2*tan(1/2*a)^4 + 2*b*c*tan(1/2*b*x)^4*tan(1/2*a)^2 - 2*d*tan(1/2*b*x)^4*tan(1/2*a)^3
+ 2*b*c*tan(1/2*b*x)^2*tan(1/2*a)^4 - 2*d*tan(1/2*b*x)^3*tan(1/2*a)^4 + b*d*x*tan(1/2*b*x)^4 + 4*b*d*x*tan(1/2
*b*x)^2*tan(1/2*a)^2 + b*d*x*tan(1/2*a)^4 + b*c*tan(1/2*b*x)^4 + 2*d*tan(1/2*b*x)^4*tan(1/2*a) + 4*b*c*tan(1/2
*b*x)^2*tan(1/2*a)^2 + 12*d*tan(1/2*b*x)^3*tan(1/2*a)^2 + 12*d*tan(1/2*b*x)^2*tan(1/2*a)^3 + b*c*tan(1/2*a)^4
+ 2*d*tan(1/2*b*x)*tan(1/2*a)^4 + 2*b*d*x*tan(1/2*b*x)^2 + 2*b*d*x*tan(1/2*a)^2 + 2*b*c*tan(1/2*b*x)^2 - 2*d*t
an(1/2*b*x)^3 - 12*d*tan(1/2*b*x)^2*tan(1/2*a) + 2*b*c*tan(1/2*a)^2 - 12*d*tan(1/2*b*x)*tan(1/2*a)^2 - 2*d*tan
(1/2*a)^3 + b*d*x + b*c + 2*d*tan(1/2*b*x) + 2*d*tan(1/2*a))/(b^2*tan(1/2*b*x)^4*tan(1/2*a)^2 + 2*b^2*tan(1/2*
b*x)^3*tan(1/2*a)^3 + b^2*tan(1/2*b*x)^2*tan(1/2*a)^4 - 2*b^2*tan(1/2*b*x)^3*tan(1/2*a) - 4*b^2*tan(1/2*b*x)^2
*tan(1/2*a)^2 - 2*b^2*tan(1/2*b*x)*tan(1/2*a)^3 + b^2*tan(1/2*b*x)^2 + 2*b^2*tan(1/2*b*x)*tan(1/2*a) + b^2*tan
(1/2*a)^2)

Mupad [B] (verification not implemented)

Time = 24.14 (sec) , antiderivative size = 53, normalized size of antiderivative = 1.51 \[ \int (c+d x) \cot (a+b x) \csc ^2(a+b x) \, dx=\frac {d\,1{}\mathrm {i}-{\mathrm {e}}^{a\,2{}\mathrm {i}+b\,x\,2{}\mathrm {i}}\,\left (-b\,\left (2\,c+2\,d\,x\right )+d\,1{}\mathrm {i}\right )}{b^2\,{\left ({\mathrm {e}}^{a\,2{}\mathrm {i}+b\,x\,2{}\mathrm {i}}-1\right )}^2} \]

[In]

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

[Out]

(d*1i - exp(a*2i + b*x*2i)*(d*1i - b*(2*c + 2*d*x)))/(b^2*(exp(a*2i + b*x*2i) - 1)^2)