3.1.31 \(\int \csc ^4(2 a+2 b x) \sin ^3(a+b x) \, dx\) [31]

Optimal. Leaf size=43 \[ -\frac {\tanh ^{-1}(\cos (a+b x))}{16 b}+\frac {\sec (a+b x)}{16 b}+\frac {\sec ^3(a+b x)}{48 b} \]

[Out]

-1/16*arctanh(cos(b*x+a))/b+1/16*sec(b*x+a)/b+1/48*sec(b*x+a)^3/b

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Rubi [A]
time = 0.04, antiderivative size = 43, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {4373, 2702, 308, 213} \begin {gather*} \frac {\sec ^3(a+b x)}{48 b}+\frac {\sec (a+b x)}{16 b}-\frac {\tanh ^{-1}(\cos (a+b x))}{16 b} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Csc[2*a + 2*b*x]^4*Sin[a + b*x]^3,x]

[Out]

-1/16*ArcTanh[Cos[a + b*x]]/b + Sec[a + b*x]/(16*b) + Sec[a + b*x]^3/(48*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 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 ^3(a+b x) \, dx &=\frac {1}{16} \int \csc (a+b x) \sec ^4(a+b x) \, dx\\ &=\frac {\text {Subst}\left (\int \frac {x^4}{-1+x^2} \, dx,x,\sec (a+b x)\right )}{16 b}\\ &=\frac {\text {Subst}\left (\int \left (1+x^2+\frac {1}{-1+x^2}\right ) \, dx,x,\sec (a+b x)\right )}{16 b}\\ &=\frac {\sec (a+b x)}{16 b}+\frac {\sec ^3(a+b x)}{48 b}+\frac {\text {Subst}\left (\int \frac {1}{-1+x^2} \, dx,x,\sec (a+b x)\right )}{16 b}\\ &=-\frac {\tanh ^{-1}(\cos (a+b x))}{16 b}+\frac {\sec (a+b x)}{16 b}+\frac {\sec ^3(a+b x)}{48 b}\\ \end {align*}

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Mathematica [A]
time = 0.03, size = 61, normalized size = 1.42 \begin {gather*} \frac {1}{16} \left (-\frac {\log \left (\cos \left (\frac {1}{2} (a+b x)\right )\right )}{b}+\frac {\log \left (\sin \left (\frac {1}{2} (a+b x)\right )\right )}{b}+\frac {\sec (a+b x)}{b}+\frac {\sec ^3(a+b x)}{3 b}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[Csc[2*a + 2*b*x]^4*Sin[a + b*x]^3,x]

[Out]

(-(Log[Cos[(a + b*x)/2]]/b) + Log[Sin[(a + b*x)/2]]/b + Sec[a + b*x]/b + Sec[a + b*x]^3/(3*b))/16

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Maple [A]
time = 0.11, size = 41, normalized size = 0.95

method	result	size
default	\(\frac {\frac {1}{3 \cos \left (x b +a \right )^{3}}+\frac {1}{\cos \left (x b +a \right )}+\ln \left (\csc \left (x b +a \right )-\cot \left (x b +a \right )\right )}{16 b}\)	\(41\)
risch	\(\frac {3 \,{\mathrm e}^{5 i \left (x b +a \right )}+10 \,{\mathrm e}^{3 i \left (x b +a \right )}+3 \,{\mathrm e}^{i \left (x b +a \right )}}{24 b \left ({\mathrm e}^{2 i \left (x b +a \right )}+1\right )^{3}}+\frac {\ln \left ({\mathrm e}^{i \left (x b +a \right )}-1\right )}{16 b}-\frac {\ln \left ({\mathrm e}^{i \left (x b +a \right )}+1\right )}{16 b}\)	\(88\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(csc(2*b*x+2*a)^4*sin(b*x+a)^3,x,method=_RETURNVERBOSE)

[Out]

1/16/b*(1/3/cos(b*x+a)^3+1/cos(b*x+a)+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. 987 vs. \(2 (37) = 74\).
time = 0.32, size = 987, normalized size = 22.95 \begin {gather*} \frac {4 \, {\left (3 \, \cos \left (5 \, b x + 5 \, a\right ) + 10 \, \cos \left (3 \, b x + 3 \, a\right ) + 3 \, \cos \left (b x + a\right )\right )} \cos \left (6 \, b x + 6 \, a\right ) + 12 \, {\left (3 \, \cos \left (4 \, b x + 4 \, a\right ) + 3 \, \cos \left (2 \, b x + 2 \, a\right ) + 1\right )} \cos \left (5 \, b x + 5 \, a\right ) + 12 \, {\left (10 \, \cos \left (3 \, b x + 3 \, a\right ) + 3 \, \cos \left (b x + a\right )\right )} \cos \left (4 \, b x + 4 \, a\right ) + 40 \, {\left (3 \, \cos \left (2 \, b x + 2 \, a\right ) + 1\right )} \cos \left (3 \, b x + 3 \, a\right ) + 36 \, \cos \left (2 \, b x + 2 \, a\right ) \cos \left (b x + a\right ) - 3 \, {\left (2 \, {\left (3 \, \cos \left (4 \, b x + 4 \, a\right ) + 3 \, \cos \left (2 \, b x + 2 \, a\right ) + 1\right )} \cos \left (6 \, b x + 6 \, a\right ) + \cos \left (6 \, b x + 6 \, a\right )^{2} + 6 \, {\left (3 \, \cos \left (2 \, b x + 2 \, a\right ) + 1\right )} \cos \left (4 \, b x + 4 \, a\right ) + 9 \, \cos \left (4 \, b x + 4 \, a\right )^{2} + 9 \, \cos \left (2 \, b x + 2 \, a\right )^{2} + 6 \, {\left (\sin \left (4 \, b x + 4 \, a\right ) + \sin \left (2 \, b x + 2 \, a\right )\right )} \sin \left (6 \, b x + 6 \, a\right ) + \sin \left (6 \, b x + 6 \, a\right )^{2} + 9 \, \sin \left (4 \, b x + 4 \, a\right )^{2} + 18 \, \sin \left (4 \, b x + 4 \, a\right ) \sin \left (2 \, b x + 2 \, a\right ) + 9 \, \sin \left (2 \, b x + 2 \, a\right )^{2} + 6 \, \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 ) + 3 \, {\left (2 \, {\left (3 \, \cos \left (4 \, b x + 4 \, a\right ) + 3 \, \cos \left (2 \, b x + 2 \, a\right ) + 1\right )} \cos \left (6 \, b x + 6 \, a\right ) + \cos \left (6 \, b x + 6 \, a\right )^{2} + 6 \, {\left (3 \, \cos \left (2 \, b x + 2 \, a\right ) + 1\right )} \cos \left (4 \, b x + 4 \, a\right ) + 9 \, \cos \left (4 \, b x + 4 \, a\right )^{2} + 9 \, \cos \left (2 \, b x + 2 \, a\right )^{2} + 6 \, {\left (\sin \left (4 \, b x + 4 \, a\right ) + \sin \left (2 \, b x + 2 \, a\right )\right )} \sin \left (6 \, b x + 6 \, a\right ) + \sin \left (6 \, b x + 6 \, a\right )^{2} + 9 \, \sin \left (4 \, b x + 4 \, a\right )^{2} + 18 \, \sin \left (4 \, b x + 4 \, a\right ) \sin \left (2 \, b x + 2 \, a\right ) + 9 \, \sin \left (2 \, b x + 2 \, a\right )^{2} + 6 \, \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 \, {\left (3 \, \sin \left (5 \, b x + 5 \, a\right ) + 10 \, \sin \left (3 \, b x + 3 \, a\right ) + 3 \, \sin \left (b x + a\right )\right )} \sin \left (6 \, b x + 6 \, a\right ) + 36 \, {\left (\sin \left (4 \, b x + 4 \, a\right ) + \sin \left (2 \, b x + 2 \, a\right )\right )} \sin \left (5 \, b x + 5 \, a\right ) + 12 \, {\left (10 \, \sin \left (3 \, b x + 3 \, a\right ) + 3 \, \sin \left (b x + a\right )\right )} \sin \left (4 \, b x + 4 \, a\right ) + 120 \, \sin \left (3 \, b x + 3 \, a\right ) \sin \left (2 \, b x + 2 \, a\right ) + 36 \, \sin \left (2 \, b x + 2 \, a\right ) \sin \left (b x + a\right ) + 12 \, \cos \left (b x + a\right )}{96 \, {\left (b \cos \left (6 \, b x + 6 \, a\right )^{2} + 9 \, b \cos \left (4 \, b x + 4 \, a\right )^{2} + 9 \, b \cos \left (2 \, b x + 2 \, a\right )^{2} + b \sin \left (6 \, b x + 6 \, a\right )^{2} + 9 \, b \sin \left (4 \, b x + 4 \, a\right )^{2} + 18 \, b \sin \left (4 \, b x + 4 \, a\right ) \sin \left (2 \, b x + 2 \, a\right ) + 9 \, b \sin \left (2 \, b x + 2 \, a\right )^{2} + 2 \, {\left (3 \, b \cos \left (4 \, b x + 4 \, a\right ) + 3 \, b \cos \left (2 \, b x + 2 \, a\right ) + b\right )} \cos \left (6 \, b x + 6 \, a\right ) + 6 \, {\left (3 \, b \cos \left (2 \, b x + 2 \, a\right ) + b\right )} \cos \left (4 \, b x + 4 \, a\right ) + 6 \, b \cos \left (2 \, b x + 2 \, a\right ) + 6 \, {\left (b \sin \left (4 \, b x + 4 \, a\right ) + b \sin \left (2 \, b x + 2 \, a\right )\right )} \sin \left (6 \, b x + 6 \, a\right ) + b\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)^3,x, algorithm="maxima")

[Out]

1/96*(4*(3*cos(5*b*x + 5*a) + 10*cos(3*b*x + 3*a) + 3*cos(b*x + a))*cos(6*b*x + 6*a) + 12*(3*cos(4*b*x + 4*a)
+ 3*cos(2*b*x + 2*a) + 1)*cos(5*b*x + 5*a) + 12*(10*cos(3*b*x + 3*a) + 3*cos(b*x + a))*cos(4*b*x + 4*a) + 40*(
3*cos(2*b*x + 2*a) + 1)*cos(3*b*x + 3*a) + 36*cos(2*b*x + 2*a)*cos(b*x + a) - 3*(2*(3*cos(4*b*x + 4*a) + 3*cos
(2*b*x + 2*a) + 1)*cos(6*b*x + 6*a) + cos(6*b*x + 6*a)^2 + 6*(3*cos(2*b*x + 2*a) + 1)*cos(4*b*x + 4*a) + 9*cos
(4*b*x + 4*a)^2 + 9*cos(2*b*x + 2*a)^2 + 6*(sin(4*b*x + 4*a) + sin(2*b*x + 2*a))*sin(6*b*x + 6*a) + sin(6*b*x
+ 6*a)^2 + 9*sin(4*b*x + 4*a)^2 + 18*sin(4*b*x + 4*a)*sin(2*b*x + 2*a) + 9*sin(2*b*x + 2*a)^2 + 6*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) + 3*(2*(3
*cos(4*b*x + 4*a) + 3*cos(2*b*x + 2*a) + 1)*cos(6*b*x + 6*a) + cos(6*b*x + 6*a)^2 + 6*(3*cos(2*b*x + 2*a) + 1)
*cos(4*b*x + 4*a) + 9*cos(4*b*x + 4*a)^2 + 9*cos(2*b*x + 2*a)^2 + 6*(sin(4*b*x + 4*a) + sin(2*b*x + 2*a))*sin(
6*b*x + 6*a) + sin(6*b*x + 6*a)^2 + 9*sin(4*b*x + 4*a)^2 + 18*sin(4*b*x + 4*a)*sin(2*b*x + 2*a) + 9*sin(2*b*x
+ 2*a)^2 + 6*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*(3*sin(5*b*x + 5*a) + 10*sin(3*b*x + 3*a) + 3*sin(b*x + a))*sin(6*b*x + 6*a) + 36*(sin(4*b
*x + 4*a) + sin(2*b*x + 2*a))*sin(5*b*x + 5*a) + 12*(10*sin(3*b*x + 3*a) + 3*sin(b*x + a))*sin(4*b*x + 4*a) +
120*sin(3*b*x + 3*a)*sin(2*b*x + 2*a) + 36*sin(2*b*x + 2*a)*sin(b*x + a) + 12*cos(b*x + a))/(b*cos(6*b*x + 6*a
)^2 + 9*b*cos(4*b*x + 4*a)^2 + 9*b*cos(2*b*x + 2*a)^2 + b*sin(6*b*x + 6*a)^2 + 9*b*sin(4*b*x + 4*a)^2 + 18*b*s
in(4*b*x + 4*a)*sin(2*b*x + 2*a) + 9*b*sin(2*b*x + 2*a)^2 + 2*(3*b*cos(4*b*x + 4*a) + 3*b*cos(2*b*x + 2*a) + b
)*cos(6*b*x + 6*a) + 6*(3*b*cos(2*b*x + 2*a) + b)*cos(4*b*x + 4*a) + 6*b*cos(2*b*x + 2*a) + 6*(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 = 2.86, size = 67, normalized size = 1.56 \begin {gather*} -\frac {3 \, \cos \left (b x + a\right )^{3} \log \left (\frac {1}{2} \, \cos \left (b x + a\right ) + \frac {1}{2}\right ) - 3 \, \cos \left (b x + a\right )^{3} \log \left (-\frac {1}{2} \, \cos \left (b x + a\right ) + \frac {1}{2}\right ) - 6 \, \cos \left (b x + a\right )^{2} - 2}{96 \, b \cos \left (b x + a\right )^{3}} \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)^3,x, algorithm="fricas")

[Out]

-1/96*(3*cos(b*x + a)^3*log(1/2*cos(b*x + a) + 1/2) - 3*cos(b*x + a)^3*log(-1/2*cos(b*x + a) + 1/2) - 6*cos(b*
x + a)^2 - 2)/(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)**3,x)

[Out]

Timed out

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 98 vs. \(2 (37) = 74\).
time = 0.47, size = 98, normalized size = 2.28 \begin {gather*} \frac {\frac {8 \, {\left (\frac {3 \, {\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 )}^{2}}{{\left (\cos \left (b x + a\right ) + 1\right )}^{2}} + 2\right )}}{{\left (\frac {\cos \left (b x + a\right ) - 1}{\cos \left (b x + a\right ) + 1} + 1\right )}^{3}} + 3 \, \log \left (-\frac {\cos \left (b x + a\right ) - 1}{\cos \left (b x + a\right ) + 1}\right )}{96 \, 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)^3,x, algorithm="giac")

[Out]

1/96*(8*(3*(cos(b*x + a) - 1)/(cos(b*x + a) + 1) + 3*(cos(b*x + a) - 1)^2/(cos(b*x + a) + 1)^2 + 2)/((cos(b*x
+ a) - 1)/(cos(b*x + a) + 1) + 1)^3 + 3*log(-(cos(b*x + a) - 1)/(cos(b*x + a) + 1)))/b

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Mupad [B]
time = 0.07, size = 37, normalized size = 0.86 \begin {gather*} \frac {\frac {{\cos \left (a+b\,x\right )}^2}{16}+\frac {1}{48}}{b\,{\cos \left (a+b\,x\right )}^3}-\frac {\mathrm {atanh}\left (\cos \left (a+b\,x\right )\right )}{16\,b} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sin(a + b*x)^3/sin(2*a + 2*b*x)^4,x)

[Out]

(cos(a + b*x)^2/16 + 1/48)/(b*cos(a + b*x)^3) - atanh(cos(a + b*x))/(16*b)

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