3.1.12 \(\int \csc ^5(2 a+2 b x) \sin (a+b x) \, dx\) [12]

Optimal. Leaf size=89 \[ \frac {35 \tanh ^{-1}(\sin (a+b x))}{256 b}-\frac {35 \csc (a+b x)}{256 b}-\frac {35 \csc ^3(a+b x)}{768 b}+\frac {7 \csc ^3(a+b x) \sec ^2(a+b x)}{256 b}+\frac {\csc ^3(a+b x) \sec ^4(a+b x)}{128 b} \]

[Out]

35/256*arctanh(sin(b*x+a))/b-35/256*csc(b*x+a)/b-35/768*csc(b*x+a)^3/b+7/256*csc(b*x+a)^3*sec(b*x+a)^2/b+1/128
*csc(b*x+a)^3*sec(b*x+a)^4/b

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Rubi [A]
time = 0.05, antiderivative size = 89, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 5, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.278, Rules used = {4373, 2701, 294, 308, 213} \begin {gather*} -\frac {35 \csc ^3(a+b x)}{768 b}-\frac {35 \csc (a+b x)}{256 b}+\frac {35 \tanh ^{-1}(\sin (a+b x))}{256 b}+\frac {\csc ^3(a+b x) \sec ^4(a+b x)}{128 b}+\frac {7 \csc ^3(a+b x) \sec ^2(a+b x)}{256 b} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Csc[2*a + 2*b*x]^5*Sin[a + b*x],x]

[Out]

(35*ArcTanh[Sin[a + b*x]])/(256*b) - (35*Csc[a + b*x])/(256*b) - (35*Csc[a + b*x]^3)/(768*b) + (7*Csc[a + b*x]
^3*Sec[a + b*x]^2)/(256*b) + (Csc[a + b*x]^3*Sec[a + b*x]^4)/(128*b)

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 308

Int[(x_)^(m_)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> Int[PolynomialDivide[x^m, a + b*x^n, x], x] /; FreeQ[{a,
b}, x] && IGtQ[m, 0] && IGtQ[n, 0] && GtQ[m, 2*n - 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 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*} \int \csc ^5(2 a+2 b x) \sin (a+b x) \, dx &=\frac {1}{32} \int \csc ^4(a+b x) \sec ^5(a+b x) \, dx\\ &=-\frac {\text {Subst}\left (\int \frac {x^8}{\left (-1+x^2\right )^3} \, dx,x,\csc (a+b x)\right )}{32 b}\\ &=\frac {\csc ^3(a+b x) \sec ^4(a+b x)}{128 b}-\frac {7 \text {Subst}\left (\int \frac {x^6}{\left (-1+x^2\right )^2} \, dx,x,\csc (a+b x)\right )}{128 b}\\ &=\frac {7 \csc ^3(a+b x) \sec ^2(a+b x)}{256 b}+\frac {\csc ^3(a+b x) \sec ^4(a+b x)}{128 b}-\frac {35 \text {Subst}\left (\int \frac {x^4}{-1+x^2} \, dx,x,\csc (a+b x)\right )}{256 b}\\ &=\frac {7 \csc ^3(a+b x) \sec ^2(a+b x)}{256 b}+\frac {\csc ^3(a+b x) \sec ^4(a+b x)}{128 b}-\frac {35 \text {Subst}\left (\int \left (1+x^2+\frac {1}{-1+x^2}\right ) \, dx,x,\csc (a+b x)\right )}{256 b}\\ &=-\frac {35 \csc (a+b x)}{256 b}-\frac {35 \csc ^3(a+b x)}{768 b}+\frac {7 \csc ^3(a+b x) \sec ^2(a+b x)}{256 b}+\frac {\csc ^3(a+b x) \sec ^4(a+b x)}{128 b}-\frac {35 \text {Subst}\left (\int \frac {1}{-1+x^2} \, dx,x,\csc (a+b x)\right )}{256 b}\\ &=\frac {35 \tanh ^{-1}(\sin (a+b x))}{256 b}-\frac {35 \csc (a+b x)}{256 b}-\frac {35 \csc ^3(a+b x)}{768 b}+\frac {7 \csc ^3(a+b x) \sec ^2(a+b x)}{256 b}+\frac {\csc ^3(a+b x) \sec ^4(a+b x)}{128 b}\\ \end {align*}

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Mathematica [C] Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.
time = 0.03, size = 31, normalized size = 0.35 \begin {gather*} -\frac {\csc ^3(a+b x) \, _2F_1\left (-\frac {3}{2},3;-\frac {1}{2};\sin ^2(a+b x)\right )}{96 b} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[Csc[2*a + 2*b*x]^5*Sin[a + b*x],x]

[Out]

-1/96*(Csc[a + b*x]^3*Hypergeometric2F1[-3/2, 3, -1/2, Sin[a + b*x]^2])/b

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Maple [A]
time = 0.12, size = 87, normalized size = 0.98

method	result	size
default	\(\frac {\frac {1}{4 \sin \left (x b +a \right )^{3} \cos \left (x b +a \right )^{4}}-\frac {7}{12 \sin \left (x b +a \right )^{3} \cos \left (x b +a \right )^{2}}+\frac {35}{24 \sin \left (x b +a \right ) \cos \left (x b +a \right )^{2}}-\frac {35}{8 \sin \left (x b +a \right )}+\frac {35 \ln \left (\sec \left (x b +a \right )+\tan \left (x b +a \right )\right )}{8}}{32 b}\)	\(87\)
risch	\(-\frac {i \left (105 \,{\mathrm e}^{13 i \left (x b +a \right )}+70 \,{\mathrm e}^{11 i \left (x b +a \right )}-329 \,{\mathrm e}^{9 i \left (x b +a \right )}-204 \,{\mathrm e}^{7 i \left (x b +a \right )}-329 \,{\mathrm e}^{5 i \left (x b +a \right )}+70 \,{\mathrm e}^{3 i \left (x b +a \right )}+105 \,{\mathrm e}^{i \left (x b +a \right )}\right )}{384 b \left ({\mathrm e}^{2 i \left (x b +a \right )}+1\right )^{4} \left ({\mathrm e}^{2 i \left (x b +a \right )}-1\right )^{3}}-\frac {35 \ln \left ({\mathrm e}^{i \left (x b +a \right )}-i\right )}{256 b}+\frac {35 \ln \left (i+{\mathrm e}^{i \left (x b +a \right )}\right )}{256 b}\)	\(148\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(csc(2*b*x+2*a)^5*sin(b*x+a),x,method=_RETURNVERBOSE)

[Out]

1/32/b*(1/4/sin(b*x+a)^3/cos(b*x+a)^4-7/12/sin(b*x+a)^3/cos(b*x+a)^2+35/24/sin(b*x+a)/cos(b*x+a)^2-35/8/sin(b*
x+a)+35/8*ln(sec(b*x+a)+tan(b*x+a)))

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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 3088 vs. \(2 (79) = 158\).
time = 0.64, size = 3088, normalized size = 34.70 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csc(2*b*x+2*a)^5*sin(b*x+a),x, algorithm="maxima")

[Out]

1/1536*(4*(105*sin(13*b*x + 13*a) + 70*sin(11*b*x + 11*a) - 329*sin(9*b*x + 9*a) - 204*sin(7*b*x + 7*a) - 329*
sin(5*b*x + 5*a) + 70*sin(3*b*x + 3*a) + 105*sin(b*x + a))*cos(14*b*x + 14*a) - 420*(sin(12*b*x + 12*a) - 3*si
n(10*b*x + 10*a) - 3*sin(8*b*x + 8*a) + 3*sin(6*b*x + 6*a) + 3*sin(4*b*x + 4*a) - sin(2*b*x + 2*a))*cos(13*b*x
+ 13*a) + 4*(70*sin(11*b*x + 11*a) - 329*sin(9*b*x + 9*a) - 204*sin(7*b*x + 7*a) - 329*sin(5*b*x + 5*a) + 70*
sin(3*b*x + 3*a) + 105*sin(b*x + a))*cos(12*b*x + 12*a) + 280*(3*sin(10*b*x + 10*a) + 3*sin(8*b*x + 8*a) - 3*s
in(6*b*x + 6*a) - 3*sin(4*b*x + 4*a) + sin(2*b*x + 2*a))*cos(11*b*x + 11*a) + 12*(329*sin(9*b*x + 9*a) + 204*s
in(7*b*x + 7*a) + 329*sin(5*b*x + 5*a) - 70*sin(3*b*x + 3*a) - 105*sin(b*x + a))*cos(10*b*x + 10*a) - 1316*(3*
sin(8*b*x + 8*a) - 3*sin(6*b*x + 6*a) - 3*sin(4*b*x + 4*a) + sin(2*b*x + 2*a))*cos(9*b*x + 9*a) + 12*(204*sin(
7*b*x + 7*a) + 329*sin(5*b*x + 5*a) - 70*sin(3*b*x + 3*a) - 105*sin(b*x + a))*cos(8*b*x + 8*a) + 816*(3*sin(6*
b*x + 6*a) + 3*sin(4*b*x + 4*a) - sin(2*b*x + 2*a))*cos(7*b*x + 7*a) - 84*(47*sin(5*b*x + 5*a) - 10*sin(3*b*x
+ 3*a) - 15*sin(b*x + a))*cos(6*b*x + 6*a) + 1316*(3*sin(4*b*x + 4*a) - sin(2*b*x + 2*a))*cos(5*b*x + 5*a) + 4
20*(2*sin(3*b*x + 3*a) + 3*sin(b*x + a))*cos(4*b*x + 4*a) - 105*(2*(cos(12*b*x + 12*a) - 3*cos(10*b*x + 10*a)
- 3*cos(8*b*x + 8*a) + 3*cos(6*b*x + 6*a) + 3*cos(4*b*x + 4*a) - cos(2*b*x + 2*a) - 1)*cos(14*b*x + 14*a) + co
s(14*b*x + 14*a)^2 - 2*(3*cos(10*b*x + 10*a) + 3*cos(8*b*x + 8*a) - 3*cos(6*b*x + 6*a) - 3*cos(4*b*x + 4*a) +
cos(2*b*x + 2*a) + 1)*cos(12*b*x + 12*a) + cos(12*b*x + 12*a)^2 + 6*(3*cos(8*b*x + 8*a) - 3*cos(6*b*x + 6*a) -
3*cos(4*b*x + 4*a) + cos(2*b*x + 2*a) + 1)*cos(10*b*x + 10*a) + 9*cos(10*b*x + 10*a)^2 - 6*(3*cos(6*b*x + 6*a
) + 3*cos(4*b*x + 4*a) - cos(2*b*x + 2*a) - 1)*cos(8*b*x + 8*a) + 9*cos(8*b*x + 8*a)^2 + 6*(3*cos(4*b*x + 4*a)
- cos(2*b*x + 2*a) - 1)*cos(6*b*x + 6*a) + 9*cos(6*b*x + 6*a)^2 - 6*(cos(2*b*x + 2*a) + 1)*cos(4*b*x + 4*a) +
9*cos(4*b*x + 4*a)^2 + cos(2*b*x + 2*a)^2 + 2*(sin(12*b*x + 12*a) - 3*sin(10*b*x + 10*a) - 3*sin(8*b*x + 8*a)
+ 3*sin(6*b*x + 6*a) + 3*sin(4*b*x + 4*a) - sin(2*b*x + 2*a))*sin(14*b*x + 14*a) + sin(14*b*x + 14*a)^2 - 2*(
3*sin(10*b*x + 10*a) + 3*sin(8*b*x + 8*a) - 3*sin(6*b*x + 6*a) - 3*sin(4*b*x + 4*a) + sin(2*b*x + 2*a))*sin(12
*b*x + 12*a) + sin(12*b*x + 12*a)^2 + 6*(3*sin(8*b*x + 8*a) - 3*sin(6*b*x + 6*a) - 3*sin(4*b*x + 4*a) + sin(2*
b*x + 2*a))*sin(10*b*x + 10*a) + 9*sin(10*b*x + 10*a)^2 - 6*(3*sin(6*b*x + 6*a) + 3*sin(4*b*x + 4*a) - sin(2*b
*x + 2*a))*sin(8*b*x + 8*a) + 9*sin(8*b*x + 8*a)^2 + 6*(3*sin(4*b*x + 4*a) - sin(2*b*x + 2*a))*sin(6*b*x + 6*a
) + 9*sin(6*b*x + 6*a)^2 + 9*sin(4*b*x + 4*a)^2 - 6*sin(4*b*x + 4*a)*sin(2*b*x + 2*a) + sin(2*b*x + 2*a)^2 + 2
*cos(2*b*x + 2*a) + 1)*log((cos(b*x + 2*a)^2 + cos(a)^2 - 2*cos(a)*sin(b*x + 2*a) + sin(b*x + 2*a)^2 + 2*cos(b
*x + 2*a)*sin(a) + sin(a)^2)/(cos(b*x + 2*a)^2 + cos(a)^2 + 2*cos(a)*sin(b*x + 2*a) + sin(b*x + 2*a)^2 - 2*cos
(b*x + 2*a)*sin(a) + sin(a)^2)) - 4*(105*cos(13*b*x + 13*a) + 70*cos(11*b*x + 11*a) - 329*cos(9*b*x + 9*a) - 2
04*cos(7*b*x + 7*a) - 329*cos(5*b*x + 5*a) + 70*cos(3*b*x + 3*a) + 105*cos(b*x + a))*sin(14*b*x + 14*a) + 420*
(cos(12*b*x + 12*a) - 3*cos(10*b*x + 10*a) - 3*cos(8*b*x + 8*a) + 3*cos(6*b*x + 6*a) + 3*cos(4*b*x + 4*a) - co
s(2*b*x + 2*a) - 1)*sin(13*b*x + 13*a) - 4*(70*cos(11*b*x + 11*a) - 329*cos(9*b*x + 9*a) - 204*cos(7*b*x + 7*a
) - 329*cos(5*b*x + 5*a) + 70*cos(3*b*x + 3*a) + 105*cos(b*x + a))*sin(12*b*x + 12*a) - 280*(3*cos(10*b*x + 10
*a) + 3*cos(8*b*x + 8*a) - 3*cos(6*b*x + 6*a) - 3*cos(4*b*x + 4*a) + cos(2*b*x + 2*a) + 1)*sin(11*b*x + 11*a)
- 12*(329*cos(9*b*x + 9*a) + 204*cos(7*b*x + 7*a) + 329*cos(5*b*x + 5*a) - 70*cos(3*b*x + 3*a) - 105*cos(b*x +
a))*sin(10*b*x + 10*a) + 1316*(3*cos(8*b*x + 8*a) - 3*cos(6*b*x + 6*a) - 3*cos(4*b*x + 4*a) + cos(2*b*x + 2*a
) + 1)*sin(9*b*x + 9*a) - 12*(204*cos(7*b*x + 7*a) + 329*cos(5*b*x + 5*a) - 70*cos(3*b*x + 3*a) - 105*cos(b*x
+ a))*sin(8*b*x + 8*a) - 816*(3*cos(6*b*x + 6*a) + 3*cos(4*b*x + 4*a) - cos(2*b*x + 2*a) - 1)*sin(7*b*x + 7*a)
+ 84*(47*cos(5*b*x + 5*a) - 10*cos(3*b*x + 3*a) - 15*cos(b*x + a))*sin(6*b*x + 6*a) - 1316*(3*cos(4*b*x + 4*a
) - cos(2*b*x + 2*a) - 1)*sin(5*b*x + 5*a) - 420*(2*cos(3*b*x + 3*a) + 3*cos(b*x + a))*sin(4*b*x + 4*a) - 280*
(cos(2*b*x + 2*a) + 1)*sin(3*b*x + 3*a) + 280*cos(3*b*x + 3*a)*sin(2*b*x + 2*a) + 420*cos(b*x + a)*sin(2*b*x +
2*a) - 420*cos(2*b*x + 2*a)*sin(b*x + a) - 420*sin(b*x + a))/(b*cos(14*b*x + 14*a)^2 + b*cos(12*b*x + 12*a)^2
+ 9*b*cos(10*b*x + 10*a)^2 + 9*b*cos(8*b*x + 8*a)^2 + 9*b*cos(6*b*x + 6*a)^2 + 9*b*cos(4*b*x + 4*a)^2 + b*cos
(2*b*x + 2*a)^2 + b*sin(14*b*x + 14*a)^2 + b*sin(12*b*x + 12*a)^2 + 9*b*sin(10*b*x + 10*a)^2 + 9*b*sin(8*b*x +
8*a)^2 + 9*b*sin(6*b*x + 6*a)^2 + 9*b*sin(4*b*x + 4*a)^2 - 6*b*sin(4*b*x + 4*a)*sin(2*b*x + 2*a) + b*sin(2*b*
x + 2*a)^2 + 2*(b*cos(12*b*x + 12*a) - 3*b*cos(10*b*x + 10*a) - 3*b*cos(8*b*x + 8*a) + 3*b*cos(6*b*x + 6*a) +
3*b*cos(4*b*x + 4*a) - b*cos(2*b*x + 2*a) - b)*...

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Fricas [A]
time = 2.81, size = 140, normalized size = 1.57 \begin {gather*} -\frac {210 \, \cos \left (b x + a\right )^{6} - 280 \, \cos \left (b x + a\right )^{4} - 105 \, {\left (\cos \left (b x + a\right )^{6} - \cos \left (b x + a\right )^{4}\right )} \log \left (\sin \left (b x + a\right ) + 1\right ) \sin \left (b x + a\right ) + 105 \, {\left (\cos \left (b x + a\right )^{6} - \cos \left (b x + a\right )^{4}\right )} \log \left (-\sin \left (b x + a\right ) + 1\right ) \sin \left (b x + a\right ) + 42 \, \cos \left (b x + a\right )^{2} + 12}{1536 \, {\left (b \cos \left (b x + a\right )^{6} - b \cos \left (b x + a\right )^{4}\right )} \sin \left (b x + a\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csc(2*b*x+2*a)^5*sin(b*x+a),x, algorithm="fricas")

[Out]

-1/1536*(210*cos(b*x + a)^6 - 280*cos(b*x + a)^4 - 105*(cos(b*x + a)^6 - cos(b*x + a)^4)*log(sin(b*x + a) + 1)
*sin(b*x + a) + 105*(cos(b*x + a)^6 - cos(b*x + a)^4)*log(-sin(b*x + a) + 1)*sin(b*x + a) + 42*cos(b*x + a)^2
+ 12)/((b*cos(b*x + a)^6 - b*cos(b*x + a)^4)*sin(b*x + a))

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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csc(2*b*x+2*a)**5*sin(b*x+a),x)

[Out]

Timed out

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Giac [A]
time = 0.44, size = 85, normalized size = 0.96 \begin {gather*} -\frac {\frac {6 \, {\left (11 \, \sin \left (b x + a\right )^{3} - 13 \, \sin \left (b x + a\right )\right )}}{{\left (\sin \left (b x + a\right )^{2} - 1\right )}^{2}} + \frac {16 \, {\left (9 \, \sin \left (b x + a\right )^{2} + 1\right )}}{\sin \left (b x + a\right )^{3}} - 105 \, \log \left (\sin \left (b x + a\right ) + 1\right ) + 105 \, \log \left (-\sin \left (b x + a\right ) + 1\right )}{1536 \, b} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csc(2*b*x+2*a)^5*sin(b*x+a),x, algorithm="giac")

[Out]

-1/1536*(6*(11*sin(b*x + a)^3 - 13*sin(b*x + a))/(sin(b*x + a)^2 - 1)^2 + 16*(9*sin(b*x + a)^2 + 1)/sin(b*x +
a)^3 - 105*log(sin(b*x + a) + 1) + 105*log(-sin(b*x + a) + 1))/b

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Mupad [B]
time = 0.17, size = 79, normalized size = 0.89 \begin {gather*} \frac {35\,\mathrm {atanh}\left (\sin \left (a+b\,x\right )\right )}{256\,b}-\frac {\frac {35\,{\sin \left (a+b\,x\right )}^6}{256}-\frac {175\,{\sin \left (a+b\,x\right )}^4}{768}+\frac {7\,{\sin \left (a+b\,x\right )}^2}{96}+\frac {1}{96}}{b\,\left ({\sin \left (a+b\,x\right )}^7-2\,{\sin \left (a+b\,x\right )}^5+{\sin \left (a+b\,x\right )}^3\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sin(a + b*x)/sin(2*a + 2*b*x)^5,x)

[Out]

(35*atanh(sin(a + b*x)))/(256*b) - ((7*sin(a + b*x)^2)/96 - (175*sin(a + b*x)^4)/768 + (35*sin(a + b*x)^6)/256
+ 1/96)/(b*(sin(a + b*x)^3 - 2*sin(a + b*x)^5 + sin(a + b*x)^7))

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