3.1.71 \(\int \csc ^3(a+b x) \csc ^3(2 a+2 b x) \, dx\) [71]
Optimal. Leaf size=81 \[ \frac {7 \tanh ^{-1}(\sin (a+b x))}{16 b}-\frac {7 \csc (a+b x)}{16 b}-\frac {7 \csc ^3(a+b x)}{48 b}-\frac {7 \csc ^5(a+b x)}{80 b}+\frac {\csc ^5(a+b x) \sec ^2(a+b x)}{16 b} \]
[Out]
7/16*arctanh(sin(b*x+a))/b-7/16*csc(b*x+a)/b-7/48*csc(b*x+a)^3/b-7/80*csc(b*x+a)^5/b+1/16*csc(b*x+a)^5*sec(b*x
+a)^2/b
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Rubi [A]
time = 0.05, antiderivative size = 81, normalized size of antiderivative = 1.00, number of steps
used = 6, number of rules used = 5, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {4373, 2701,
294, 308, 213} \begin {gather*} -\frac {7 \csc ^5(a+b x)}{80 b}-\frac {7 \csc ^3(a+b x)}{48 b}-\frac {7 \csc (a+b x)}{16 b}+\frac {7 \tanh ^{-1}(\sin (a+b x))}{16 b}+\frac {\csc ^5(a+b x) \sec ^2(a+b x)}{16 b} \end {gather*}
Antiderivative was successfully verified.
[In]
Int[Csc[a + b*x]^3*Csc[2*a + 2*b*x]^3,x]
[Out]
(7*ArcTanh[Sin[a + b*x]])/(16*b) - (7*Csc[a + b*x])/(16*b) - (7*Csc[a + b*x]^3)/(48*b) - (7*Csc[a + b*x]^5)/(8
0*b) + (Csc[a + b*x]^5*Sec[a + b*x]^2)/(16*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 ^3(a+b x) \csc ^3(2 a+2 b x) \, dx &=\frac {1}{8} \int \csc ^6(a+b x) \sec ^3(a+b x) \, dx\\ &=-\frac {\text {Subst}\left (\int \frac {x^8}{\left (-1+x^2\right )^2} \, dx,x,\csc (a+b x)\right )}{8 b}\\ &=\frac {\csc ^5(a+b x) \sec ^2(a+b x)}{16 b}-\frac {7 \text {Subst}\left (\int \frac {x^6}{-1+x^2} \, dx,x,\csc (a+b x)\right )}{16 b}\\ &=\frac {\csc ^5(a+b x) \sec ^2(a+b x)}{16 b}-\frac {7 \text {Subst}\left (\int \left (1+x^2+x^4+\frac {1}{-1+x^2}\right ) \, dx,x,\csc (a+b x)\right )}{16 b}\\ &=-\frac {7 \csc (a+b x)}{16 b}-\frac {7 \csc ^3(a+b x)}{48 b}-\frac {7 \csc ^5(a+b x)}{80 b}+\frac {\csc ^5(a+b x) \sec ^2(a+b x)}{16 b}-\frac {7 \text {Subst}\left (\int \frac {1}{-1+x^2} \, dx,x,\csc (a+b x)\right )}{16 b}\\ &=\frac {7 \tanh ^{-1}(\sin (a+b x))}{16 b}-\frac {7 \csc (a+b x)}{16 b}-\frac {7 \csc ^3(a+b x)}{48 b}-\frac {7 \csc ^5(a+b x)}{80 b}+\frac {\csc ^5(a+b x) \sec ^2(a+b x)}{16 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 = 0.04, size = 31, normalized size = 0.38 \begin {gather*} -\frac {\csc ^5(a+b x) \, _2F_1\left (-\frac {5}{2},2;-\frac {3}{2};\sin ^2(a+b x)\right )}{40 b} \end {gather*}
Antiderivative was successfully verified.
[In]
Integrate[Csc[a + b*x]^3*Csc[2*a + 2*b*x]^3,x]
[Out]
-1/40*(Csc[a + b*x]^5*Hypergeometric2F1[-5/2, 2, -3/2, Sin[a + b*x]^2])/b
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Maple [A]
time = 0.10, size = 87, normalized size = 1.07
| | |
| method |
result |
size |
| | |
| default |
\(\frac {-\frac {1}{5 \sin \left (x b +a \right )^{5} \cos \left (x b +a \right )^{2}}-\frac {7}{15 \sin \left (x b +a \right )^{3} \cos \left (x b +a \right )^{2}}+\frac {7}{6 \sin \left (x b +a \right ) \cos \left (x b +a \right )^{2}}-\frac {7}{2 \sin \left (x b +a \right )}+\frac {7 \ln \left (\sec \left (x b +a \right )+\tan \left (x b +a \right )\right )}{2}}{8 b}\) |
\(87\) |
| risch |
\(-\frac {i \left (105 \,{\mathrm e}^{13 i \left (x b +a \right )}-350 \,{\mathrm e}^{11 i \left (x b +a \right )}+231 \,{\mathrm e}^{9 i \left (x b +a \right )}+412 \,{\mathrm e}^{7 i \left (x b +a \right )}+231 \,{\mathrm e}^{5 i \left (x b +a \right )}-350 \,{\mathrm e}^{3 i \left (x b +a \right )}+105 \,{\mathrm e}^{i \left (x b +a \right )}\right )}{120 b \left ({\mathrm e}^{2 i \left (x b +a \right )}-1\right )^{5} \left ({\mathrm e}^{2 i \left (x b +a \right )}+1\right )^{2}}+\frac {7 \ln \left (i+{\mathrm e}^{i \left (x b +a \right )}\right )}{16 b}-\frac {7 \ln \left ({\mathrm e}^{i \left (x b +a \right )}-i\right )}{16 b}\) |
\(148\) |
| | |
|
|
|
|
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(csc(b*x+a)^3*csc(2*b*x+2*a)^3,x,method=_RETURNVERBOSE)
[Out]
1/8/b*(-1/5/sin(b*x+a)^5/cos(b*x+a)^2-7/15/sin(b*x+a)^3/cos(b*x+a)^2+7/6/sin(b*x+a)/cos(b*x+a)^2-7/2/sin(b*x+a
)+7/2*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. 3095 vs.
\(2 (71) = 142\).
time = 0.65, size = 3095, normalized size = 38.21 \begin {gather*} \text {Too large to display} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(csc(b*x+a)^3*csc(2*b*x+2*a)^3,x, algorithm="maxima")
[Out]
1/480*(4*(105*sin(13*b*x + 13*a) - 350*sin(11*b*x + 11*a) + 231*sin(9*b*x + 9*a) + 412*sin(7*b*x + 7*a) + 231*
sin(5*b*x + 5*a) - 350*sin(3*b*x + 3*a) + 105*sin(b*x + a))*cos(14*b*x + 14*a) + 420*(3*sin(12*b*x + 12*a) - s
in(10*b*x + 10*a) - 5*sin(8*b*x + 8*a) + 5*sin(6*b*x + 6*a) + sin(4*b*x + 4*a) - 3*sin(2*b*x + 2*a))*cos(13*b*
x + 13*a) + 12*(350*sin(11*b*x + 11*a) - 231*sin(9*b*x + 9*a) - 412*sin(7*b*x + 7*a) - 231*sin(5*b*x + 5*a) +
350*sin(3*b*x + 3*a) - 105*sin(b*x + a))*cos(12*b*x + 12*a) + 1400*(sin(10*b*x + 10*a) + 5*sin(8*b*x + 8*a) -
5*sin(6*b*x + 6*a) - sin(4*b*x + 4*a) + 3*sin(2*b*x + 2*a))*cos(11*b*x + 11*a) + 4*(231*sin(9*b*x + 9*a) + 412
*sin(7*b*x + 7*a) + 231*sin(5*b*x + 5*a) - 350*sin(3*b*x + 3*a) + 105*sin(b*x + a))*cos(10*b*x + 10*a) - 924*(
5*sin(8*b*x + 8*a) - 5*sin(6*b*x + 6*a) - sin(4*b*x + 4*a) + 3*sin(2*b*x + 2*a))*cos(9*b*x + 9*a) + 20*(412*si
n(7*b*x + 7*a) + 231*sin(5*b*x + 5*a) - 350*sin(3*b*x + 3*a) + 105*sin(b*x + a))*cos(8*b*x + 8*a) + 1648*(5*si
n(6*b*x + 6*a) + sin(4*b*x + 4*a) - 3*sin(2*b*x + 2*a))*cos(7*b*x + 7*a) - 140*(33*sin(5*b*x + 5*a) - 50*sin(3
*b*x + 3*a) + 15*sin(b*x + a))*cos(6*b*x + 6*a) + 924*(sin(4*b*x + 4*a) - 3*sin(2*b*x + 2*a))*cos(5*b*x + 5*a)
+ 140*(10*sin(3*b*x + 3*a) - 3*sin(b*x + a))*cos(4*b*x + 4*a) + 105*(2*(3*cos(12*b*x + 12*a) - cos(10*b*x + 1
0*a) - 5*cos(8*b*x + 8*a) + 5*cos(6*b*x + 6*a) + cos(4*b*x + 4*a) - 3*cos(2*b*x + 2*a) + 1)*cos(14*b*x + 14*a)
- cos(14*b*x + 14*a)^2 + 6*(cos(10*b*x + 10*a) + 5*cos(8*b*x + 8*a) - 5*cos(6*b*x + 6*a) - cos(4*b*x + 4*a) +
3*cos(2*b*x + 2*a) - 1)*cos(12*b*x + 12*a) - 9*cos(12*b*x + 12*a)^2 - 2*(5*cos(8*b*x + 8*a) - 5*cos(6*b*x + 6
*a) - cos(4*b*x + 4*a) + 3*cos(2*b*x + 2*a) - 1)*cos(10*b*x + 10*a) - cos(10*b*x + 10*a)^2 + 10*(5*cos(6*b*x +
6*a) + cos(4*b*x + 4*a) - 3*cos(2*b*x + 2*a) + 1)*cos(8*b*x + 8*a) - 25*cos(8*b*x + 8*a)^2 - 10*(cos(4*b*x +
4*a) - 3*cos(2*b*x + 2*a) + 1)*cos(6*b*x + 6*a) - 25*cos(6*b*x + 6*a)^2 + 2*(3*cos(2*b*x + 2*a) - 1)*cos(4*b*x
+ 4*a) - cos(4*b*x + 4*a)^2 - 9*cos(2*b*x + 2*a)^2 + 2*(3*sin(12*b*x + 12*a) - sin(10*b*x + 10*a) - 5*sin(8*b
*x + 8*a) + 5*sin(6*b*x + 6*a) + sin(4*b*x + 4*a) - 3*sin(2*b*x + 2*a))*sin(14*b*x + 14*a) - sin(14*b*x + 14*a
)^2 + 6*(sin(10*b*x + 10*a) + 5*sin(8*b*x + 8*a) - 5*sin(6*b*x + 6*a) - sin(4*b*x + 4*a) + 3*sin(2*b*x + 2*a))
*sin(12*b*x + 12*a) - 9*sin(12*b*x + 12*a)^2 - 2*(5*sin(8*b*x + 8*a) - 5*sin(6*b*x + 6*a) - sin(4*b*x + 4*a) +
3*sin(2*b*x + 2*a))*sin(10*b*x + 10*a) - sin(10*b*x + 10*a)^2 + 10*(5*sin(6*b*x + 6*a) + sin(4*b*x + 4*a) - 3
*sin(2*b*x + 2*a))*sin(8*b*x + 8*a) - 25*sin(8*b*x + 8*a)^2 - 10*(sin(4*b*x + 4*a) - 3*sin(2*b*x + 2*a))*sin(6
*b*x + 6*a) - 25*sin(6*b*x + 6*a)^2 - sin(4*b*x + 4*a)^2 + 6*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*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) - 350*cos(11*b*x + 11*a) + 231*cos(9*b*
x + 9*a) + 412*cos(7*b*x + 7*a) + 231*cos(5*b*x + 5*a) - 350*cos(3*b*x + 3*a) + 105*cos(b*x + a))*sin(14*b*x +
14*a) - 420*(3*cos(12*b*x + 12*a) - cos(10*b*x + 10*a) - 5*cos(8*b*x + 8*a) + 5*cos(6*b*x + 6*a) + cos(4*b*x
+ 4*a) - 3*cos(2*b*x + 2*a) + 1)*sin(13*b*x + 13*a) - 12*(350*cos(11*b*x + 11*a) - 231*cos(9*b*x + 9*a) - 412*
cos(7*b*x + 7*a) - 231*cos(5*b*x + 5*a) + 350*cos(3*b*x + 3*a) - 105*cos(b*x + a))*sin(12*b*x + 12*a) - 1400*(
cos(10*b*x + 10*a) + 5*cos(8*b*x + 8*a) - 5*cos(6*b*x + 6*a) - cos(4*b*x + 4*a) + 3*cos(2*b*x + 2*a) - 1)*sin(
11*b*x + 11*a) - 4*(231*cos(9*b*x + 9*a) + 412*cos(7*b*x + 7*a) + 231*cos(5*b*x + 5*a) - 350*cos(3*b*x + 3*a)
+ 105*cos(b*x + a))*sin(10*b*x + 10*a) + 924*(5*cos(8*b*x + 8*a) - 5*cos(6*b*x + 6*a) - cos(4*b*x + 4*a) + 3*c
os(2*b*x + 2*a) - 1)*sin(9*b*x + 9*a) - 20*(412*cos(7*b*x + 7*a) + 231*cos(5*b*x + 5*a) - 350*cos(3*b*x + 3*a)
+ 105*cos(b*x + a))*sin(8*b*x + 8*a) - 1648*(5*cos(6*b*x + 6*a) + cos(4*b*x + 4*a) - 3*cos(2*b*x + 2*a) + 1)*
sin(7*b*x + 7*a) + 140*(33*cos(5*b*x + 5*a) - 50*cos(3*b*x + 3*a) + 15*cos(b*x + a))*sin(6*b*x + 6*a) - 924*(c
os(4*b*x + 4*a) - 3*cos(2*b*x + 2*a) + 1)*sin(5*b*x + 5*a) - 140*(10*cos(3*b*x + 3*a) - 3*cos(b*x + a))*sin(4*
b*x + 4*a) - 1400*(3*cos(2*b*x + 2*a) - 1)*sin(3*b*x + 3*a) + 4200*cos(3*b*x + 3*a)*sin(2*b*x + 2*a) - 1260*co
s(b*x + a)*sin(2*b*x + 2*a) + 1260*cos(2*b*x + 2*a)*sin(b*x + a) - 420*sin(b*x + a))/(b*cos(14*b*x + 14*a)^2 +
9*b*cos(12*b*x + 12*a)^2 + b*cos(10*b*x + 10*a)^2 + 25*b*cos(8*b*x + 8*a)^2 + 25*b*cos(6*b*x + 6*a)^2 + b*cos
(4*b*x + 4*a)^2 + 9*b*cos(2*b*x + 2*a)^2 + b*sin(14*b*x + 14*a)^2 + 9*b*sin(12*b*x + 12*a)^2 + b*sin(10*b*x +
10*a)^2 + 25*b*sin(8*b*x + 8*a)^2 + 25*b*sin(6*b*x + 6*a)^2 + b*sin(4*b*x + 4*a)^2 - 6*b*sin(4*b*x + 4*a)*sin(
2*b*x + 2*a) + 9*b*sin(2*b*x + 2*a)^2 - 2*(3*b*cos(12*b*x + 12*a) - b*cos(10*b*x + 10*a) - 5*b*cos(8*b*x + 8*a
) + 5*b*cos(6*b*x + 6*a) + b*cos(4*b*x + 4*a) -...
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 166 vs.
\(2 (71) = 142\).
time = 2.83, size = 166, normalized size = 2.05 \begin {gather*} -\frac {210 \, \cos \left (b x + a\right )^{6} - 490 \, \cos \left (b x + a\right )^{4} - 105 \, {\left (\cos \left (b x + a\right )^{6} - 2 \, \cos \left (b x + a\right )^{4} + \cos \left (b x + a\right )^{2}\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} - 2 \, \cos \left (b x + a\right )^{4} + \cos \left (b x + a\right )^{2}\right )} \log \left (-\sin \left (b x + a\right ) + 1\right ) \sin \left (b x + a\right ) + 322 \, \cos \left (b x + a\right )^{2} - 30}{480 \, {\left (b \cos \left (b x + a\right )^{6} - 2 \, b \cos \left (b x + a\right )^{4} + b \cos \left (b x + a\right )^{2}\right )} \sin \left (b x + a\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(csc(b*x+a)^3*csc(2*b*x+2*a)^3,x, algorithm="fricas")
[Out]
-1/480*(210*cos(b*x + a)^6 - 490*cos(b*x + a)^4 - 105*(cos(b*x + a)^6 - 2*cos(b*x + a)^4 + cos(b*x + a)^2)*log
(sin(b*x + a) + 1)*sin(b*x + a) + 105*(cos(b*x + a)^6 - 2*cos(b*x + a)^4 + cos(b*x + a)^2)*log(-sin(b*x + a) +
1)*sin(b*x + a) + 322*cos(b*x + a)^2 - 30)/((b*cos(b*x + a)^6 - 2*b*cos(b*x + a)^4 + b*cos(b*x + a)^2)*sin(b*
x + a))
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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \csc ^{3}{\left (a + b x \right )} \csc ^{3}{\left (2 a + 2 b x \right )}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(csc(b*x+a)**3*csc(2*b*x+2*a)**3,x)
[Out]
Integral(csc(a + b*x)**3*csc(2*a + 2*b*x)**3, x)
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Giac [A]
time = 0.48, size = 82, normalized size = 1.01 \begin {gather*} -\frac {\frac {30 \, \sin \left (b x + a\right )}{\sin \left (b x + a\right )^{2} - 1} + \frac {4 \, {\left (45 \, \sin \left (b x + a\right )^{4} + 10 \, \sin \left (b x + a\right )^{2} + 3\right )}}{\sin \left (b x + a\right )^{5}} - 105 \, \log \left (\sin \left (b x + a\right ) + 1\right ) + 105 \, \log \left (-\sin \left (b x + a\right ) + 1\right )}{480 \, b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(csc(b*x+a)^3*csc(2*b*x+2*a)^3,x, algorithm="giac")
[Out]
-1/480*(30*sin(b*x + a)/(sin(b*x + a)^2 - 1) + 4*(45*sin(b*x + a)^4 + 10*sin(b*x + a)^2 + 3)/sin(b*x + a)^5 -
105*log(sin(b*x + a) + 1) + 105*log(-sin(b*x + a) + 1))/b
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Mupad [B]
time = 0.18, size = 71, normalized size = 0.88 \begin {gather*} \frac {7\,\mathrm {atanh}\left (\sin \left (a+b\,x\right )\right )}{16\,b}-\frac {-\frac {7\,{\sin \left (a+b\,x\right )}^6}{16}+\frac {7\,{\sin \left (a+b\,x\right )}^4}{24}+\frac {7\,{\sin \left (a+b\,x\right )}^2}{120}+\frac {1}{40}}{b\,\left ({\sin \left (a+b\,x\right )}^5-{\sin \left (a+b\,x\right )}^7\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(1/(sin(a + b*x)^3*sin(2*a + 2*b*x)^3),x)
[Out]
(7*atanh(sin(a + b*x)))/(16*b) - ((7*sin(a + b*x)^2)/120 + (7*sin(a + b*x)^4)/24 - (7*sin(a + b*x)^6)/16 + 1/4
0)/(b*(sin(a + b*x)^5 - sin(a + b*x)^7))
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