\(\int \csc ^3(2 a+2 b x) \sin ^2(a+b x) \, dx\) [20]

Optimal result

Integrand size = 20, antiderivative size = 30 \[ \int \csc ^3(2 a+2 b x) \sin ^2(a+b x) \, dx=\frac {\log (\tan (a+b x))}{8 b}+\frac {\tan ^2(a+b x)}{16 b} \]

[Out]

1/8*ln(tan(b*x+a))/b+1/16*tan(b*x+a)^2/b

Rubi [A] (verified)

Time = 0.06 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.150, Rules used = {4373, 2700, 14} \[ \int \csc ^3(2 a+2 b x) \sin ^2(a+b x) \, dx=\frac {\tan ^2(a+b x)}{16 b}+\frac {\log (\tan (a+b x))}{8 b} \]

[In]

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

[Out]

Log[Tan[a + b*x]]/(8*b) + Tan[a + b*x]^2/(16*b)

Rule 14

Int[(u_)*((c_.)*(x_))^(m_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*u, x], x] /; FreeQ[{c, m}, x] && SumQ[u]
&&  !LinearQ[u, x] &&  !MatchQ[u, (a_) + (b_.)*(v_) /; FreeQ[{a, b}, x] && InverseFunctionQ[v]]

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 4373

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

Rubi steps \begin{align*} \text {integral}& = \frac {1}{8} \int \csc (a+b x) \sec ^3(a+b x) \, dx \\ & = \frac {\text {Subst}\left (\int \frac {1+x^2}{x} \, dx,x,\tan (a+b x)\right )}{8 b} \\ & = \frac {\text {Subst}\left (\int \left (\frac {1}{x}+x\right ) \, dx,x,\tan (a+b x)\right )}{8 b} \\ & = \frac {\log (\tan (a+b x))}{8 b}+\frac {\tan ^2(a+b x)}{16 b} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.03 (sec) , antiderivative size = 43, normalized size of antiderivative = 1.43 \[ \int \csc ^3(2 a+2 b x) \sin ^2(a+b x) \, dx=\frac {1}{8} \left (-\frac {\log (\cos (a+b x))}{b}+\frac {\log (\sin (a+b x))}{b}+\frac {\sec ^2(a+b x)}{2 b}\right ) \]

[In]

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

[Out]

(-(Log[Cos[a + b*x]]/b) + Log[Sin[a + b*x]]/b + Sec[a + b*x]^2/(2*b))/8

Maple [A] (verified)

Time = 0.99 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.80

method	result	size
default	\(\frac {\frac {1}{2 \cos \left (x b +a \right )^{2}}+\ln \left (\tan \left (x b +a \right )\right )}{8 b}\)	\(24\)
risch	\(\frac {{\mathrm e}^{2 i \left (x b +a \right )}}{4 b \left ({\mathrm e}^{2 i \left (x b +a \right )}+1\right )^{2}}-\frac {\ln \left ({\mathrm e}^{2 i \left (x b +a \right )}+1\right )}{8 b}+\frac {\ln \left ({\mathrm e}^{2 i \left (x b +a \right )}-1\right )}{8 b}\)	\(63\)
parallelrisch	\(\frac {-\csc \left (x b +a \right )^{2} \left (\sec \left (x b +a \right )^{2}-2\right ) \cos \left (2 x b +2 a \right )-2 \csc \left (x b +a \right ) \sec \left (x b +a \right ) \sin \left (2 x b +2 a \right )+16 \ln \left (\tan \left (x b +a \right )^{\frac {1}{4}}\right )+\csc \left (x b +a \right )^{2} \left (\sec \left (x b +a \right )^{2}-2\right )}{32 b}\)	\(89\)

[In]

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

[Out]

1/8/b*(1/2/cos(b*x+a)^2+ln(tan(b*x+a)))

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 56 vs. \(2 (26) = 52\).

Time = 0.25 (sec) , antiderivative size = 56, normalized size of antiderivative = 1.87 \[ \int \csc ^3(2 a+2 b x) \sin ^2(a+b x) \, dx=-\frac {\cos \left (b x + a\right )^{2} \log \left (\cos \left (b x + a\right )^{2}\right ) - \cos \left (b x + a\right )^{2} \log \left (-\frac {1}{4} \, \cos \left (b x + a\right )^{2} + \frac {1}{4}\right ) - 1}{16 \, b \cos \left (b x + a\right )^{2}} \]

[In]

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

[Out]

-1/16*(cos(b*x + a)^2*log(cos(b*x + a)^2) - cos(b*x + a)^2*log(-1/4*cos(b*x + a)^2 + 1/4) - 1)/(b*cos(b*x + a)
^2)

Sympy [F(-1)]

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

[In]

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

[Out]

Timed out

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 641 vs. \(2 (26) = 52\).

Time = 0.21 (sec) , antiderivative size = 641, normalized size of antiderivative = 21.37 \[ \int \csc ^3(2 a+2 b x) \sin ^2(a+b x) \, dx=\frac {4 \, \cos \left (4 \, b x + 4 \, a\right ) \cos \left (2 \, b x + 2 \, a\right ) + 8 \, \cos \left (2 \, b x + 2 \, a\right )^{2} - {\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 )} \log \left (\cos \left (2 \, b x\right )^{2} + 2 \, \cos \left (2 \, b x\right ) \cos \left (2 \, a\right ) + \cos \left (2 \, a\right )^{2} + \sin \left (2 \, b x\right )^{2} - 2 \, \sin \left (2 \, b x\right ) \sin \left (2 \, a\right ) + \sin \left (2 \, a\right )^{2}\right ) + {\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 )} \log \left (\cos \left (b x\right )^{2} + 2 \, \cos \left (b x\right ) \cos \left (a\right ) + \cos \left (a\right )^{2} + \sin \left (b x\right )^{2} - 2 \, \sin \left (b x\right ) \sin \left (a\right ) + \sin \left (a\right )^{2}\right ) + {\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 )} \log \left (\cos \left (b x\right )^{2} - 2 \, \cos \left (b x\right ) \cos \left (a\right ) + \cos \left (a\right )^{2} + \sin \left (b x\right )^{2} + 2 \, \sin \left (b x\right ) \sin \left (a\right ) + \sin \left (a\right )^{2}\right ) + 4 \, \sin \left (4 \, b x + 4 \, a\right ) \sin \left (2 \, b x + 2 \, a\right ) + 8 \, \sin \left (2 \, b x + 2 \, a\right )^{2} + 4 \, \cos \left (2 \, b x + 2 \, a\right )}{16 \, {\left (b \cos \left (4 \, b x + 4 \, a\right )^{2} + 4 \, b \cos \left (2 \, b x + 2 \, a\right )^{2} + b \sin \left (4 \, b x + 4 \, a\right )^{2} + 4 \, b \sin \left (4 \, b x + 4 \, a\right ) \sin \left (2 \, b x + 2 \, a\right ) + 4 \, b \sin \left (2 \, b x + 2 \, a\right )^{2} + 2 \, {\left (2 \, b \cos \left (2 \, b x + 2 \, a\right ) + b\right )} \cos \left (4 \, b x + 4 \, a\right ) + 4 \, b \cos \left (2 \, b x + 2 \, a\right ) + b\right )}} \]

[In]

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

[Out]

1/16*(4*cos(4*b*x + 4*a)*cos(2*b*x + 2*a) + 8*cos(2*b*x + 2*a)^2 - (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*s
in(2*b*x + 2*a)^2 + 4*cos(2*b*x + 2*a) + 1)*log(cos(2*b*x)^2 + 2*cos(2*b*x)*cos(2*a) + cos(2*a)^2 + sin(2*b*x)
^2 - 2*sin(2*b*x)*sin(2*a) + sin(2*a)^2) + (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*co
s(2*b*x + 2*a) + 1)*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*(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)*log(cos(b*x)^2 - 2*c
os(b*x)*cos(a) + cos(a)^2 + sin(b*x)^2 + 2*sin(b*x)*sin(a) + sin(a)^2) + 4*sin(4*b*x + 4*a)*sin(2*b*x + 2*a) +
8*sin(2*b*x + 2*a)^2 + 4*cos(2*b*x + 2*a))/(b*cos(4*b*x + 4*a)^2 + 4*b*cos(2*b*x + 2*a)^2 + b*sin(4*b*x + 4*a
)^2 + 4*b*sin(4*b*x + 4*a)*sin(2*b*x + 2*a) + 4*b*sin(2*b*x + 2*a)^2 + 2*(2*b*cos(2*b*x + 2*a) + b)*cos(4*b*x
+ 4*a) + 4*b*cos(2*b*x + 2*a) + b)

Giac [A] (verification not implemented)

none

Time = 0.34 (sec) , antiderivative size = 41, normalized size of antiderivative = 1.37 \[ \int \csc ^3(2 a+2 b x) \sin ^2(a+b x) \, dx=-\frac {\frac {1}{\sin \left (b x + a\right )^{2} - 1} + \log \left (-\sin \left (b x + a\right )^{2} + 1\right ) - 2 \, \log \left ({\left | \sin \left (b x + a\right ) \right |}\right )}{16 \, b} \]

[In]

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

[Out]

-1/16*(1/(sin(b*x + a)^2 - 1) + log(-sin(b*x + a)^2 + 1) - 2*log(abs(sin(b*x + a))))/b

Mupad [B] (verification not implemented)

Time = 0.11 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.17 \[ \int \csc ^3(2 a+2 b x) \sin ^2(a+b x) \, dx=\frac {\frac {\ln \left ({\sin \left (a+b\,x\right )}^2\right )}{16}-\frac {\ln \left (\cos \left (a+b\,x\right )\right )}{8}+\frac {1}{16\,{\cos \left (a+b\,x\right )}^2}}{b} \]

[In]

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

[Out]

(log(sin(a + b*x)^2)/16 - log(cos(a + b*x))/8 + 1/(16*cos(a + b*x)^2))/b