3.1.11 \(\int \csc ^4(2 a+2 b x) \sin (a+b x) \, dx\) [11]

Optimal. Leaf size=66 \[ -\frac {5 \tanh ^{-1}(\cos (a+b x))}{32 b}+\frac {5 \sec (a+b x)}{32 b}+\frac {5 \sec ^3(a+b x)}{96 b}-\frac {\csc ^2(a+b x) \sec ^3(a+b x)}{32 b} \]

[Out]

-5/32*arctanh(cos(b*x+a))/b+5/32*sec(b*x+a)/b+5/96*sec(b*x+a)^3/b-1/32*csc(b*x+a)^2*sec(b*x+a)^3/b

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Rubi [A]
time = 0.05, antiderivative size = 66, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.278, Rules used = {4373, 2702, 294, 308, 213} \begin {gather*} \frac {5 \sec ^3(a+b x)}{96 b}+\frac {5 \sec (a+b x)}{32 b}-\frac {5 \tanh ^{-1}(\cos (a+b x))}{32 b}-\frac {\csc ^2(a+b x) \sec ^3(a+b x)}{32 b} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Csc[2*a + 2*b*x]^4*Sin[a + b*x],x]

[Out]

(-5*ArcTanh[Cos[a + b*x]])/(32*b) + (5*Sec[a + b*x])/(32*b) + (5*Sec[a + b*x]^3)/(96*b) - (Csc[a + b*x]^2*Sec[
a + b*x]^3)/(32*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 2702

Int[csc[(e_.) + (f_.)*(x_)]^(n_.)*((a_.)*sec[(e_.) + (f_.)*(x_)])^(m_), x_Symbol] :> Dist[1/(f*a^n), Subst[Int
[x^(m + n - 1)/(-1 + x^2/a^2)^((n + 1)/2), x], x, a*Sec[e + f*x]], x] /; FreeQ[{a, e, f, m}, x] && IntegerQ[(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 ^4(2 a+2 b x) \sin (a+b x) \, dx &=\frac {1}{16} \int \csc ^3(a+b x) \sec ^4(a+b x) \, dx\\ &=\frac {\text {Subst}\left (\int \frac {x^6}{\left (-1+x^2\right )^2} \, dx,x,\sec (a+b x)\right )}{16 b}\\ &=-\frac {\csc ^2(a+b x) \sec ^3(a+b x)}{32 b}+\frac {5 \text {Subst}\left (\int \frac {x^4}{-1+x^2} \, dx,x,\sec (a+b x)\right )}{32 b}\\ &=-\frac {\csc ^2(a+b x) \sec ^3(a+b x)}{32 b}+\frac {5 \text {Subst}\left (\int \left (1+x^2+\frac {1}{-1+x^2}\right ) \, dx,x,\sec (a+b x)\right )}{32 b}\\ &=\frac {5 \sec (a+b x)}{32 b}+\frac {5 \sec ^3(a+b x)}{96 b}-\frac {\csc ^2(a+b x) \sec ^3(a+b x)}{32 b}+\frac {5 \text {Subst}\left (\int \frac {1}{-1+x^2} \, dx,x,\sec (a+b x)\right )}{32 b}\\ &=-\frac {5 \tanh ^{-1}(\cos (a+b x))}{32 b}+\frac {5 \sec (a+b x)}{32 b}+\frac {5 \sec ^3(a+b x)}{96 b}-\frac {\csc ^2(a+b x) \sec ^3(a+b x)}{32 b}\\ \end {align*}

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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(205\) vs. \(2(66)=132\).
time = 0.48, size = 205, normalized size = 3.11 \begin {gather*} \frac {\csc ^8(a+b x) \left (22-40 \cos (2 (a+b x))+13 \cos (3 (a+b x))-30 \cos (4 (a+b x))+13 \cos (5 (a+b x))+15 \cos (3 (a+b x)) \log \left (\cos \left (\frac {1}{2} (a+b x)\right )\right )+15 \cos (5 (a+b x)) \log \left (\cos \left (\frac {1}{2} (a+b x)\right )\right )-15 \cos (3 (a+b x)) \log \left (\sin \left (\frac {1}{2} (a+b x)\right )\right )-15 \cos (5 (a+b x)) \log \left (\sin \left (\frac {1}{2} (a+b x)\right )\right )+\cos (a+b x) \left (-26-30 \log \left (\cos \left (\frac {1}{2} (a+b x)\right )\right )+30 \log \left (\sin \left (\frac {1}{2} (a+b x)\right )\right )\right )\right )}{24 b \left (\csc ^2\left (\frac {1}{2} (a+b x)\right )-\sec ^2\left (\frac {1}{2} (a+b x)\right )\right )^3} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[Csc[2*a + 2*b*x]^4*Sin[a + b*x],x]

[Out]

(Csc[a + b*x]^8*(22 - 40*Cos[2*(a + b*x)] + 13*Cos[3*(a + b*x)] - 30*Cos[4*(a + b*x)] + 13*Cos[5*(a + b*x)] +
15*Cos[3*(a + b*x)]*Log[Cos[(a + b*x)/2]] + 15*Cos[5*(a + b*x)]*Log[Cos[(a + b*x)/2]] - 15*Cos[3*(a + b*x)]*Lo
g[Sin[(a + b*x)/2]] - 15*Cos[5*(a + b*x)]*Log[Sin[(a + b*x)/2]] + Cos[a + b*x]*(-26 - 30*Log[Cos[(a + b*x)/2]]
+ 30*Log[Sin[(a + b*x)/2]])))/(24*b*(Csc[(a + b*x)/2]^2 - Sec[(a + b*x)/2]^2)^3)

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Maple [A]
time = 0.13, size = 71, normalized size = 1.08

method	result	size
default	\(\frac {\frac {1}{3 \sin \left (x b +a \right )^{2} \cos \left (x b +a \right )^{3}}-\frac {5}{6 \sin \left (x b +a \right )^{2} \cos \left (x b +a \right )}+\frac {5}{2 \cos \left (x b +a \right )}+\frac {5 \ln \left (\csc \left (x b +a \right )-\cot \left (x b +a \right )\right )}{2}}{16 b}\)	\(71\)
risch	\(\frac {15 \,{\mathrm e}^{9 i \left (x b +a \right )}+20 \,{\mathrm e}^{7 i \left (x b +a \right )}-22 \,{\mathrm e}^{5 i \left (x b +a \right )}+20 \,{\mathrm e}^{3 i \left (x b +a \right )}+15 \,{\mathrm e}^{i \left (x b +a \right )}}{48 b \left ({\mathrm e}^{2 i \left (x b +a \right )}+1\right )^{3} \left ({\mathrm e}^{2 i \left (x b +a \right )}-1\right )^{2}}+\frac {5 \ln \left ({\mathrm e}^{i \left (x b +a \right )}-1\right )}{32 b}-\frac {5 \ln \left ({\mathrm e}^{i \left (x b +a \right )}+1\right )}{32 b}\)	\(123\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(csc(2*b*x+2*a)^4*sin(b*x+a),x,method=_RETURNVERBOSE)

[Out]

1/16/b*(1/3/sin(b*x+a)^2/cos(b*x+a)^3-5/6/sin(b*x+a)^2/cos(b*x+a)+5/2/cos(b*x+a)+5/2*ln(csc(b*x+a)-cot(b*x+a))
)

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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 2174 vs. \(2 (58) = 116\).
time = 0.35, size = 2174, normalized size = 32.94 \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)^4*sin(b*x+a),x, algorithm="maxima")

[Out]

1/192*(4*(15*cos(9*b*x + 9*a) + 20*cos(7*b*x + 7*a) - 22*cos(5*b*x + 5*a) + 20*cos(3*b*x + 3*a) + 15*cos(b*x +
a))*cos(10*b*x + 10*a) + 60*(cos(8*b*x + 8*a) - 2*cos(6*b*x + 6*a) - 2*cos(4*b*x + 4*a) + cos(2*b*x + 2*a) +
1)*cos(9*b*x + 9*a) + 4*(20*cos(7*b*x + 7*a) - 22*cos(5*b*x + 5*a) + 20*cos(3*b*x + 3*a) + 15*cos(b*x + a))*co
s(8*b*x + 8*a) - 80*(2*cos(6*b*x + 6*a) + 2*cos(4*b*x + 4*a) - cos(2*b*x + 2*a) - 1)*cos(7*b*x + 7*a) + 8*(22*
cos(5*b*x + 5*a) - 20*cos(3*b*x + 3*a) - 15*cos(b*x + a))*cos(6*b*x + 6*a) + 88*(2*cos(4*b*x + 4*a) - cos(2*b*
x + 2*a) - 1)*cos(5*b*x + 5*a) - 40*(4*cos(3*b*x + 3*a) + 3*cos(b*x + a))*cos(4*b*x + 4*a) + 80*(cos(2*b*x + 2
*a) + 1)*cos(3*b*x + 3*a) + 60*cos(2*b*x + 2*a)*cos(b*x + a) - 15*(2*(cos(8*b*x + 8*a) - 2*cos(6*b*x + 6*a) -
2*cos(4*b*x + 4*a) + cos(2*b*x + 2*a) + 1)*cos(10*b*x + 10*a) + cos(10*b*x + 10*a)^2 - 2*(2*cos(6*b*x + 6*a) +
2*cos(4*b*x + 4*a) - cos(2*b*x + 2*a) - 1)*cos(8*b*x + 8*a) + cos(8*b*x + 8*a)^2 + 4*(2*cos(4*b*x + 4*a) - co
s(2*b*x + 2*a) - 1)*cos(6*b*x + 6*a) + 4*cos(6*b*x + 6*a)^2 - 4*(cos(2*b*x + 2*a) + 1)*cos(4*b*x + 4*a) + 4*co
s(4*b*x + 4*a)^2 + cos(2*b*x + 2*a)^2 + 2*(sin(8*b*x + 8*a) - 2*sin(6*b*x + 6*a) - 2*sin(4*b*x + 4*a) + sin(2*
b*x + 2*a))*sin(10*b*x + 10*a) + sin(10*b*x + 10*a)^2 - 2*(2*sin(6*b*x + 6*a) + 2*sin(4*b*x + 4*a) - sin(2*b*x
+ 2*a))*sin(8*b*x + 8*a) + sin(8*b*x + 8*a)^2 + 4*(2*sin(4*b*x + 4*a) - sin(2*b*x + 2*a))*sin(6*b*x + 6*a) +
4*sin(6*b*x + 6*a)^2 + 4*sin(4*b*x + 4*a)^2 - 4*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 + 2*cos(b*x)*cos(a) + cos(a)^2 + sin(b*x)^2 - 2*sin(b*x)*sin(a) + sin(a)^2)
+ 15*(2*(cos(8*b*x + 8*a) - 2*cos(6*b*x + 6*a) - 2*cos(4*b*x + 4*a) + cos(2*b*x + 2*a) + 1)*cos(10*b*x + 10*a)
+ cos(10*b*x + 10*a)^2 - 2*(2*cos(6*b*x + 6*a) + 2*cos(4*b*x + 4*a) - cos(2*b*x + 2*a) - 1)*cos(8*b*x + 8*a)
+ cos(8*b*x + 8*a)^2 + 4*(2*cos(4*b*x + 4*a) - cos(2*b*x + 2*a) - 1)*cos(6*b*x + 6*a) + 4*cos(6*b*x + 6*a)^2 -
4*(cos(2*b*x + 2*a) + 1)*cos(4*b*x + 4*a) + 4*cos(4*b*x + 4*a)^2 + cos(2*b*x + 2*a)^2 + 2*(sin(8*b*x + 8*a) -
2*sin(6*b*x + 6*a) - 2*sin(4*b*x + 4*a) + sin(2*b*x + 2*a))*sin(10*b*x + 10*a) + sin(10*b*x + 10*a)^2 - 2*(2*
sin(6*b*x + 6*a) + 2*sin(4*b*x + 4*a) - sin(2*b*x + 2*a))*sin(8*b*x + 8*a) + sin(8*b*x + 8*a)^2 + 4*(2*sin(4*b
*x + 4*a) - sin(2*b*x + 2*a))*sin(6*b*x + 6*a) + 4*sin(6*b*x + 6*a)^2 + 4*sin(4*b*x + 4*a)^2 - 4*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 - 2*cos(b*x)*cos(a) + cos(a
)^2 + sin(b*x)^2 + 2*sin(b*x)*sin(a) + sin(a)^2) + 4*(15*sin(9*b*x + 9*a) + 20*sin(7*b*x + 7*a) - 22*sin(5*b*x
+ 5*a) + 20*sin(3*b*x + 3*a) + 15*sin(b*x + a))*sin(10*b*x + 10*a) + 60*(sin(8*b*x + 8*a) - 2*sin(6*b*x + 6*a
) - 2*sin(4*b*x + 4*a) + sin(2*b*x + 2*a))*sin(9*b*x + 9*a) + 4*(20*sin(7*b*x + 7*a) - 22*sin(5*b*x + 5*a) + 2
0*sin(3*b*x + 3*a) + 15*sin(b*x + a))*sin(8*b*x + 8*a) - 80*(2*sin(6*b*x + 6*a) + 2*sin(4*b*x + 4*a) - sin(2*b
*x + 2*a))*sin(7*b*x + 7*a) + 8*(22*sin(5*b*x + 5*a) - 20*sin(3*b*x + 3*a) - 15*sin(b*x + a))*sin(6*b*x + 6*a)
+ 88*(2*sin(4*b*x + 4*a) - sin(2*b*x + 2*a))*sin(5*b*x + 5*a) - 40*(4*sin(3*b*x + 3*a) + 3*sin(b*x + a))*sin(
4*b*x + 4*a) + 80*sin(3*b*x + 3*a)*sin(2*b*x + 2*a) + 60*sin(2*b*x + 2*a)*sin(b*x + a) + 60*cos(b*x + a))/(b*c
os(10*b*x + 10*a)^2 + b*cos(8*b*x + 8*a)^2 + 4*b*cos(6*b*x + 6*a)^2 + 4*b*cos(4*b*x + 4*a)^2 + b*cos(2*b*x + 2
*a)^2 + b*sin(10*b*x + 10*a)^2 + b*sin(8*b*x + 8*a)^2 + 4*b*sin(6*b*x + 6*a)^2 + 4*b*sin(4*b*x + 4*a)^2 - 4*b*
sin(4*b*x + 4*a)*sin(2*b*x + 2*a) + b*sin(2*b*x + 2*a)^2 + 2*(b*cos(8*b*x + 8*a) - 2*b*cos(6*b*x + 6*a) - 2*b*
cos(4*b*x + 4*a) + b*cos(2*b*x + 2*a) + b)*cos(10*b*x + 10*a) - 2*(2*b*cos(6*b*x + 6*a) + 2*b*cos(4*b*x + 4*a)
- b*cos(2*b*x + 2*a) - b)*cos(8*b*x + 8*a) + 4*(2*b*cos(4*b*x + 4*a) - b*cos(2*b*x + 2*a) - b)*cos(6*b*x + 6*
a) - 4*(b*cos(2*b*x + 2*a) + b)*cos(4*b*x + 4*a) + 2*b*cos(2*b*x + 2*a) + 2*(b*sin(8*b*x + 8*a) - 2*b*sin(6*b*
x + 6*a) - 2*b*sin(4*b*x + 4*a) + b*sin(2*b*x + 2*a))*sin(10*b*x + 10*a) - 2*(2*b*sin(6*b*x + 6*a) + 2*b*sin(4
*b*x + 4*a) - b*sin(2*b*x + 2*a))*sin(8*b*x + 8*a) + 4*(2*b*sin(4*b*x + 4*a) - b*sin(2*b*x + 2*a))*sin(6*b*x +
6*a) + b)

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Fricas [A]
time = 3.08, size = 112, normalized size = 1.70 \begin {gather*} \frac {30 \, \cos \left (b x + a\right )^{4} - 20 \, \cos \left (b x + a\right )^{2} - 15 \, {\left (\cos \left (b x + a\right )^{5} - \cos \left (b x + a\right )^{3}\right )} \log \left (\frac {1}{2} \, \cos \left (b x + a\right ) + \frac {1}{2}\right ) + 15 \, {\left (\cos \left (b x + a\right )^{5} - \cos \left (b x + a\right )^{3}\right )} \log \left (-\frac {1}{2} \, \cos \left (b x + a\right ) + \frac {1}{2}\right ) - 4}{192 \, {\left (b \cos \left (b x + a\right )^{5} - b \cos \left (b x + a\right )^{3}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csc(2*b*x+2*a)^4*sin(b*x+a),x, algorithm="fricas")

[Out]

1/192*(30*cos(b*x + a)^4 - 20*cos(b*x + a)^2 - 15*(cos(b*x + a)^5 - cos(b*x + a)^3)*log(1/2*cos(b*x + a) + 1/2
) + 15*(cos(b*x + a)^5 - cos(b*x + a)^3)*log(-1/2*cos(b*x + a) + 1/2) - 4)/(b*cos(b*x + a)^5 - b*cos(b*x + a)^
3)

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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)**4*sin(b*x+a),x)

[Out]

Timed out

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 160 vs. \(2 (58) = 116\).
time = 0.47, size = 160, normalized size = 2.42 \begin {gather*} -\frac {\frac {3 \, {\left (\frac {10 \, {\left (\cos \left (b x + a\right ) - 1\right )}}{\cos \left (b x + a\right ) + 1} - 1\right )} {\left (\cos \left (b x + a\right ) + 1\right )}}{\cos \left (b x + a\right ) - 1} + \frac {3 \, {\left (\cos \left (b x + a\right ) - 1\right )}}{\cos \left (b x + a\right ) + 1} - \frac {16 \, {\left (\frac {12 \, {\left (\cos \left (b x + a\right ) - 1\right )}}{\cos \left (b x + a\right ) + 1} + \frac {9 \, {\left (\cos \left (b x + a\right ) - 1\right )}^{2}}{{\left (\cos \left (b x + a\right ) + 1\right )}^{2}} + 7\right )}}{{\left (\frac {\cos \left (b x + a\right ) - 1}{\cos \left (b x + a\right ) + 1} + 1\right )}^{3}} - 30 \, \log \left (-\frac {\cos \left (b x + a\right ) - 1}{\cos \left (b x + a\right ) + 1}\right )}{384 \, b} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csc(2*b*x+2*a)^4*sin(b*x+a),x, algorithm="giac")

[Out]

-1/384*(3*(10*(cos(b*x + a) - 1)/(cos(b*x + a) + 1) - 1)*(cos(b*x + a) + 1)/(cos(b*x + a) - 1) + 3*(cos(b*x +
a) - 1)/(cos(b*x + a) + 1) - 16*(12*(cos(b*x + a) - 1)/(cos(b*x + a) + 1) + 9*(cos(b*x + a) - 1)^2/(cos(b*x +
a) + 1)^2 + 7)/((cos(b*x + a) - 1)/(cos(b*x + a) + 1) + 1)^3 - 30*log(-(cos(b*x + a) - 1)/(cos(b*x + a) + 1)))
/b

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Mupad [B]
time = 0.10, size = 60, normalized size = 0.91 \begin {gather*} \frac {-\frac {5\,{\cos \left (a+b\,x\right )}^4}{32}+\frac {5\,{\cos \left (a+b\,x\right )}^2}{48}+\frac {1}{48}}{b\,\left ({\cos \left (a+b\,x\right )}^3-{\cos \left (a+b\,x\right )}^5\right )}-\frac {5\,\mathrm {atanh}\left (\cos \left (a+b\,x\right )\right )}{32\,b} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sin(a + b*x)/sin(2*a + 2*b*x)^4,x)

[Out]

((5*cos(a + b*x)^2)/48 - (5*cos(a + b*x)^4)/32 + 1/48)/(b*(cos(a + b*x)^3 - cos(a + b*x)^5)) - (5*atanh(cos(a
+ b*x)))/(32*b)

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