3.43 \(\int \csc (a+b x) \csc ^3(2 a+2 b x) \, dx\)
Optimal. Leaf size=66 \[ -\frac {5 \csc ^3(a+b x)}{48 b}-\frac {5 \csc (a+b x)}{16 b}+\frac {5 \tanh ^{-1}(\sin (a+b x))}{16 b}+\frac {\csc ^3(a+b x) \sec ^2(a+b x)}{16 b} \]
[Out]
5/16*arctanh(sin(b*x+a))/b-5/16*csc(b*x+a)/b-5/48*csc(b*x+a)^3/b+1/16*csc(b*x+a)^3*sec(b*x+a)^2/b
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Rubi [A] time = 0.06, 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 =
{4288, 2621, 288, 302, 207} \[ -\frac {5 \csc ^3(a+b x)}{48 b}-\frac {5 \csc (a+b x)}{16 b}+\frac {5 \tanh ^{-1}(\sin (a+b x))}{16 b}+\frac {\csc ^3(a+b x) \sec ^2(a+b x)}{16 b} \]
Antiderivative was successfully verified.
[In]
Int[Csc[a + b*x]*Csc[2*a + 2*b*x]^3,x]
[Out]
(5*ArcTanh[Sin[a + b*x]])/(16*b) - (5*Csc[a + b*x])/(16*b) - (5*Csc[a + b*x]^3)/(48*b) + (Csc[a + b*x]^3*Sec[a
+ b*x]^2)/(16*b)
Rule 207
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> -Simp[ArcTanh[(Rt[b, 2]*x)/Rt[-a, 2]]/(Rt[-a, 2]*Rt[b, 2]), x] /;
FreeQ[{a, b}, x] && NegQ[a/b] && (LtQ[a, 0] || GtQ[b, 0])
Rule 288
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 302
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 2621
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 4288
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 (a+b x) \csc ^3(2 a+2 b x) \, dx &=\frac {1}{8} \int \csc ^4(a+b x) \sec ^3(a+b x) \, dx\\ &=-\frac {\operatorname {Subst}\left (\int \frac {x^6}{\left (-1+x^2\right )^2} \, dx,x,\csc (a+b x)\right )}{8 b}\\ &=\frac {\csc ^3(a+b x) \sec ^2(a+b x)}{16 b}-\frac {5 \operatorname {Subst}\left (\int \frac {x^4}{-1+x^2} \, dx,x,\csc (a+b x)\right )}{16 b}\\ &=\frac {\csc ^3(a+b x) \sec ^2(a+b x)}{16 b}-\frac {5 \operatorname {Subst}\left (\int \left (1+x^2+\frac {1}{-1+x^2}\right ) \, dx,x,\csc (a+b x)\right )}{16 b}\\ &=-\frac {5 \csc (a+b x)}{16 b}-\frac {5 \csc ^3(a+b x)}{48 b}+\frac {\csc ^3(a+b x) \sec ^2(a+b x)}{16 b}-\frac {5 \operatorname {Subst}\left (\int \frac {1}{-1+x^2} \, dx,x,\csc (a+b x)\right )}{16 b}\\ &=\frac {5 \tanh ^{-1}(\sin (a+b x))}{16 b}-\frac {5 \csc (a+b x)}{16 b}-\frac {5 \csc ^3(a+b x)}{48 b}+\frac {\csc ^3(a+b x) \sec ^2(a+b x)}{16 b}\\ \end {align*}
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Mathematica [C] time = 0.02, size = 31, normalized size = 0.47 \[ -\frac {\csc ^3(a+b x) \, _2F_1\left (-\frac {3}{2},2;-\frac {1}{2};\sin ^2(a+b x)\right )}{24 b} \]
Antiderivative was successfully verified.
[In]
Integrate[Csc[a + b*x]*Csc[2*a + 2*b*x]^3,x]
[Out]
-1/24*(Csc[a + b*x]^3*Hypergeometric2F1[-3/2, 2, -1/2, Sin[a + b*x]^2])/b
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fricas [B] time = 0.47, size = 130, normalized size = 1.97 \[ -\frac {30 \, \cos \left (b x + a\right )^{4} - 15 \, {\left (\cos \left (b x + a\right )^{4} - \cos \left (b x + a\right )^{2}\right )} \log \left (\sin \left (b x + a\right ) + 1\right ) \sin \left (b x + a\right ) + 15 \, {\left (\cos \left (b x + a\right )^{4} - \cos \left (b x + a\right )^{2}\right )} \log \left (-\sin \left (b x + a\right ) + 1\right ) \sin \left (b x + a\right ) - 40 \, \cos \left (b x + a\right )^{2} + 6}{96 \, {\left (b \cos \left (b x + a\right )^{4} - b \cos \left (b x + a\right )^{2}\right )} \sin \left (b x + a\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(csc(b*x+a)*csc(2*b*x+2*a)^3,x, algorithm="fricas")
[Out]
-1/96*(30*cos(b*x + a)^4 - 15*(cos(b*x + a)^4 - cos(b*x + a)^2)*log(sin(b*x + a) + 1)*sin(b*x + a) + 15*(cos(b
*x + a)^4 - cos(b*x + a)^2)*log(-sin(b*x + a) + 1)*sin(b*x + a) - 40*cos(b*x + a)^2 + 6)/((b*cos(b*x + a)^4 -
b*cos(b*x + a)^2)*sin(b*x + a))
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giac [B] time = 3.94, size = 3033, normalized size = 45.95 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(csc(b*x+a)*csc(2*b*x+2*a)^3,x, algorithm="giac")
[Out]
-1/192*(24*(tan(1/2*b*x + 2*a)^3*tan(1/2*a)^24 + 30*tan(1/2*b*x + 2*a)^3*tan(1/2*a)^22 - 6*tan(1/2*b*x + 2*a)^
2*tan(1/2*a)^23 + tan(1/2*b*x + 2*a)*tan(1/2*a)^24 - 756*tan(1/2*b*x + 2*a)^3*tan(1/2*a)^20 + 614*tan(1/2*b*x
+ 2*a)^2*tan(1/2*a)^21 - 114*tan(1/2*b*x + 2*a)*tan(1/2*a)^22 + 6*tan(1/2*a)^23 + 2058*tan(1/2*b*x + 2*a)^3*ta
n(1/2*a)^18 - 4578*tan(1/2*b*x + 2*a)^2*tan(1/2*a)^19 + 1932*tan(1/2*b*x + 2*a)*tan(1/2*a)^20 - 182*tan(1/2*a)
^21 - 27*tan(1/2*b*x + 2*a)^3*tan(1/2*a)^16 + 6210*tan(1/2*b*x + 2*a)^2*tan(1/2*a)^17 - 7462*tan(1/2*b*x + 2*a
)*tan(1/2*a)^18 + 1554*tan(1/2*a)^19 - 9396*tan(1/2*b*x + 2*a)^3*tan(1/2*a)^14 + 15588*tan(1/2*b*x + 2*a)^2*ta
n(1/2*a)^15 - 2331*tan(1/2*b*x + 2*a)*tan(1/2*a)^16 - 2178*tan(1/2*a)^17 - 21924*tan(1/2*b*x + 2*a)^2*tan(1/2*
a)^13 + 26028*tan(1/2*b*x + 2*a)*tan(1/2*a)^14 - 5668*tan(1/2*a)^15 + 9396*tan(1/2*b*x + 2*a)^3*tan(1/2*a)^10
- 21924*tan(1/2*b*x + 2*a)^2*tan(1/2*a)^11 + 6468*tan(1/2*a)^13 + 27*tan(1/2*b*x + 2*a)^3*tan(1/2*a)^8 + 15588
*tan(1/2*b*x + 2*a)^2*tan(1/2*a)^9 - 26028*tan(1/2*b*x + 2*a)*tan(1/2*a)^10 + 6468*tan(1/2*a)^11 - 2058*tan(1/
2*b*x + 2*a)^3*tan(1/2*a)^6 + 6210*tan(1/2*b*x + 2*a)^2*tan(1/2*a)^7 + 2331*tan(1/2*b*x + 2*a)*tan(1/2*a)^8 -
5668*tan(1/2*a)^9 + 756*tan(1/2*b*x + 2*a)^3*tan(1/2*a)^4 - 4578*tan(1/2*b*x + 2*a)^2*tan(1/2*a)^5 + 7462*tan(
1/2*b*x + 2*a)*tan(1/2*a)^6 - 2178*tan(1/2*a)^7 - 30*tan(1/2*b*x + 2*a)^3*tan(1/2*a)^2 + 614*tan(1/2*b*x + 2*a
)^2*tan(1/2*a)^3 - 1932*tan(1/2*b*x + 2*a)*tan(1/2*a)^4 + 1554*tan(1/2*a)^5 - tan(1/2*b*x + 2*a)^3 - 6*tan(1/2
*b*x + 2*a)^2*tan(1/2*a) + 114*tan(1/2*b*x + 2*a)*tan(1/2*a)^2 - 182*tan(1/2*a)^3 - tan(1/2*b*x + 2*a) + 6*tan
(1/2*a))/((tan(1/2*a)^12 - 30*tan(1/2*a)^10 + 255*tan(1/2*a)^8 - 452*tan(1/2*a)^6 + 255*tan(1/2*a)^4 - 30*tan(
1/2*a)^2 + 1)*(tan(1/2*b*x + 2*a)^2*tan(1/2*a)^6 - 15*tan(1/2*b*x + 2*a)^2*tan(1/2*a)^4 + 12*tan(1/2*b*x + 2*a
)*tan(1/2*a)^5 - tan(1/2*a)^6 + 15*tan(1/2*b*x + 2*a)^2*tan(1/2*a)^2 - 40*tan(1/2*b*x + 2*a)*tan(1/2*a)^3 + 15
*tan(1/2*a)^4 - tan(1/2*b*x + 2*a)^2 + 12*tan(1/2*b*x + 2*a)*tan(1/2*a) - 15*tan(1/2*a)^2 + 1)^2) + (27*tan(1/
2*b*x + 2*a)^5*tan(1/2*a)^34 - 9*tan(1/2*b*x + 2*a)^4*tan(1/2*a)^35 + tan(1/2*b*x + 2*a)^3*tan(1/2*a)^36 + 160
2*tan(1/2*b*x + 2*a)^5*tan(1/2*a)^32 - 915*tan(1/2*b*x + 2*a)^4*tan(1/2*a)^33 + 126*tan(1/2*b*x + 2*a)^3*tan(1
/2*a)^34 + 9*tan(1/2*b*x + 2*a)^2*tan(1/2*a)^35 - 50082*tan(1/2*b*x + 2*a)^5*tan(1/2*a)^30 + 58626*tan(1/2*b*x
+ 2*a)^4*tan(1/2*a)^31 - 21141*tan(1/2*b*x + 2*a)^3*tan(1/2*a)^32 + 2373*tan(1/2*b*x + 2*a)^2*tan(1/2*a)^33 +
27*tan(1/2*b*x + 2*a)*tan(1/2*a)^34 + 400050*tan(1/2*b*x + 2*a)^5*tan(1/2*a)^28 - 783810*tan(1/2*b*x + 2*a)^4
*tan(1/2*a)^29 + 487692*tan(1/2*b*x + 2*a)^3*tan(1/2*a)^30 - 118404*tan(1/2*b*x + 2*a)^2*tan(1/2*a)^31 + 9378*
tan(1/2*b*x + 2*a)*tan(1/2*a)^32 + 54*tan(1/2*a)^33 - 1301382*tan(1/2*b*x + 2*a)^5*tan(1/2*a)^26 + 4035870*tan
(1/2*b*x + 2*a)^4*tan(1/2*a)^27 - 3943944*tan(1/2*b*x + 2*a)^3*tan(1/2*a)^28 + 1555092*tan(1/2*b*x + 2*a)^2*ta
n(1/2*a)^29 - 247074*tan(1/2*b*x + 2*a)*tan(1/2*a)^30 + 11610*tan(1/2*a)^31 + 1238250*tan(1/2*b*x + 2*a)^5*tan
(1/2*a)^24 - 8115822*tan(1/2*b*x + 2*a)^4*tan(1/2*a)^25 + 13194468*tan(1/2*b*x + 2*a)^3*tan(1/2*a)^26 - 807523
2*tan(1/2*b*x + 2*a)^2*tan(1/2*a)^27 + 1983762*tan(1/2*b*x + 2*a)*tan(1/2*a)^28 - 160362*tan(1/2*a)^29 + 26423
10*tan(1/2*b*x + 2*a)^5*tan(1/2*a)^22 - 1212390*tan(1/2*b*x + 2*a)^4*tan(1/2*a)^23 - 12476880*tan(1/2*b*x + 2*
a)^3*tan(1/2*a)^24 + 16317288*tan(1/2*b*x + 2*a)^2*tan(1/2*a)^25 - 6559110*tan(1/2*b*x + 2*a)*tan(1/2*a)^26 +
805786*tan(1/2*a)^27 - 5947398*tan(1/2*b*x + 2*a)^5*tan(1/2*a)^20 + 25548198*tan(1/2*b*x + 2*a)^4*tan(1/2*a)^2
1 - 27336420*tan(1/2*b*x + 2*a)^3*tan(1/2*a)^22 + 2575260*tan(1/2*b*x + 2*a)^2*tan(1/2*a)^23 + 6211530*tan(1/2
*b*x + 2*a)*tan(1/2*a)^24 - 1599858*tan(1/2*a)^25 - 19824660*tan(1/2*b*x + 2*a)^4*tan(1/2*a)^19 + 58330530*tan
(1/2*b*x + 2*a)^3*tan(1/2*a)^20 - 51141420*tan(1/2*b*x + 2*a)^2*tan(1/2*a)^21 + 13453830*tan(1/2*b*x + 2*a)*ta
n(1/2*a)^22 - 209790*tan(1/2*a)^23 + 5947398*tan(1/2*b*x + 2*a)^5*tan(1/2*a)^16 - 19824660*tan(1/2*b*x + 2*a)^
4*tan(1/2*a)^17 + 39278250*tan(1/2*b*x + 2*a)^2*tan(1/2*a)^19 - 29430054*tan(1/2*b*x + 2*a)*tan(1/2*a)^20 + 50
59854*tan(1/2*a)^21 - 2642310*tan(1/2*b*x + 2*a)^5*tan(1/2*a)^14 + 25548198*tan(1/2*b*x + 2*a)^4*tan(1/2*a)^15
- 58330530*tan(1/2*b*x + 2*a)^3*tan(1/2*a)^16 + 39278250*tan(1/2*b*x + 2*a)^2*tan(1/2*a)^17 - 4136670*tan(1/2
*a)^19 - 1238250*tan(1/2*b*x + 2*a)^5*tan(1/2*a)^12 - 1212390*tan(1/2*b*x + 2*a)^4*tan(1/2*a)^13 + 27336420*ta
n(1/2*b*x + 2*a)^3*tan(1/2*a)^14 - 51141420*tan(1/2*b*x + 2*a)^2*tan(1/2*a)^15 + 29430054*tan(1/2*b*x + 2*a)*t
an(1/2*a)^16 - 4136670*tan(1/2*a)^17 + 1301382*tan(1/2*b*x + 2*a)^5*tan(1/2*a)^10 - 8115822*tan(1/2*b*x + 2*a)
^4*tan(1/2*a)^11 + 12476880*tan(1/2*b*x + 2*a)^3*tan(1/2*a)^12 + 2575260*tan(1/2*b*x + 2*a)^2*tan(1/2*a)^13 -
13453830*tan(1/2*b*x + 2*a)*tan(1/2*a)^14 + 5059854*tan(1/2*a)^15 - 400050*tan(1/2*b*x + 2*a)^5*tan(1/2*a)^8 +
4035870*tan(1/2*b*x + 2*a)^4*tan(1/2*a)^9 - 13194468*tan(1/2*b*x + 2*a)^3*tan(1/2*a)^10 + 16317288*tan(1/2*b*
x + 2*a)^2*tan(1/2*a)^11 - 6211530*tan(1/2*b*x + 2*a)*tan(1/2*a)^12 - 209790*tan(1/2*a)^13 + 50082*tan(1/2*b*x
+ 2*a)^5*tan(1/2*a)^6 - 783810*tan(1/2*b*x + 2*a)^4*tan(1/2*a)^7 + 3943944*tan(1/2*b*x + 2*a)^3*tan(1/2*a)^8
- 8075232*tan(1/2*b*x + 2*a)^2*tan(1/2*a)^9 + 6559110*tan(1/2*b*x + 2*a)*tan(1/2*a)^10 - 1599858*tan(1/2*a)^11
- 1602*tan(1/2*b*x + 2*a)^5*tan(1/2*a)^4 + 58626*tan(1/2*b*x + 2*a)^4*tan(1/2*a)^5 - 487692*tan(1/2*b*x + 2*a
)^3*tan(1/2*a)^6 + 1555092*tan(1/2*b*x + 2*a)^2*tan(1/2*a)^7 - 1983762*tan(1/2*b*x + 2*a)*tan(1/2*a)^8 + 80578
6*tan(1/2*a)^9 - 27*tan(1/2*b*x + 2*a)^5*tan(1/2*a)^2 - 915*tan(1/2*b*x + 2*a)^4*tan(1/2*a)^3 + 21141*tan(1/2*
b*x + 2*a)^3*tan(1/2*a)^4 - 118404*tan(1/2*b*x + 2*a)^2*tan(1/2*a)^5 + 247074*tan(1/2*b*x + 2*a)*tan(1/2*a)^6
- 160362*tan(1/2*a)^7 - 9*tan(1/2*b*x + 2*a)^4*tan(1/2*a) - 126*tan(1/2*b*x + 2*a)^3*tan(1/2*a)^2 + 2373*tan(1
/2*b*x + 2*a)^2*tan(1/2*a)^3 - 9378*tan(1/2*b*x + 2*a)*tan(1/2*a)^4 + 11610*tan(1/2*a)^5 - tan(1/2*b*x + 2*a)^
3 + 9*tan(1/2*b*x + 2*a)^2*tan(1/2*a) - 27*tan(1/2*b*x + 2*a)*tan(1/2*a)^2 + 54*tan(1/2*a)^3)/((27*tan(1/2*a)^
15 - 270*tan(1/2*a)^13 + 981*tan(1/2*a)^11 - 1540*tan(1/2*a)^9 + 981*tan(1/2*a)^7 - 270*tan(1/2*a)^5 + 27*tan(
1/2*a)^3)*(3*tan(1/2*b*x + 2*a)^2*tan(1/2*a)^5 - tan(1/2*b*x + 2*a)*tan(1/2*a)^6 - 10*tan(1/2*b*x + 2*a)^2*tan
(1/2*a)^3 + 15*tan(1/2*b*x + 2*a)*tan(1/2*a)^4 - 3*tan(1/2*a)^5 + 3*tan(1/2*b*x + 2*a)^2*tan(1/2*a) - 15*tan(1
/2*b*x + 2*a)*tan(1/2*a)^2 + 10*tan(1/2*a)^3 + tan(1/2*b*x + 2*a) - 3*tan(1/2*a))^3) - 60*log(abs(tan(1/2*b*x
+ 2*a)*tan(1/2*a)^3 + 3*tan(1/2*b*x + 2*a)*tan(1/2*a)^2 - tan(1/2*a)^3 - 3*tan(1/2*b*x + 2*a)*tan(1/2*a) + 3*t
an(1/2*a)^2 - tan(1/2*b*x + 2*a) + 3*tan(1/2*a) - 1)) + 60*log(abs(tan(1/2*b*x + 2*a)*tan(1/2*a)^3 - 3*tan(1/2
*b*x + 2*a)*tan(1/2*a)^2 + tan(1/2*a)^3 - 3*tan(1/2*b*x + 2*a)*tan(1/2*a) + 3*tan(1/2*a)^2 + tan(1/2*b*x + 2*a
) - 3*tan(1/2*a) - 1)))/b
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maple [A] time = 0.76, size = 76, normalized size = 1.15 \[ -\frac {1}{24 b \sin \left (b x +a \right )^{3} \cos \left (b x +a \right )^{2}}+\frac {5}{48 b \sin \left (b x +a \right ) \cos \left (b x +a \right )^{2}}-\frac {5}{16 b \sin \left (b x +a \right )}+\frac {5 \ln \left (\sec \left (b x +a \right )+\tan \left (b x +a \right )\right )}{16 b} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(csc(b*x+a)*csc(2*b*x+2*a)^3,x)
[Out]
-1/24/b/sin(b*x+a)^3/cos(b*x+a)^2+5/48/b/sin(b*x+a)/cos(b*x+a)^2-5/16/b/sin(b*x+a)+5/16/b*ln(sec(b*x+a)+tan(b*
x+a))
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maxima [B] time = 0.53, size = 1780, normalized size = 26.97 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(csc(b*x+a)*csc(2*b*x+2*a)^3,x, algorithm="maxima")
[Out]
1/96*(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))*cos(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))*co
s(9*b*x + 9*a) + 4*(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))*cos(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))*cos(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))*cos(6*b*x + 6*a) + 88*(2*sin(4*b*x + 4*a) + sin(2*b*x + 2*a))
*cos(5*b*x + 5*a) - 40*(4*sin(3*b*x + 3*a) - 3*sin(b*x + a))*cos(4*b*x + 4*a) + 15*(2*(cos(8*b*x + 8*a) + 2*co
s(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*c
os(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))*si
n(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*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*(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))*sin(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)*sin(9*b*x + 9*a) - 4*(20*cos(7*b*x + 7*a) + 22*cos(5*b*x + 5*a) + 2
0*cos(3*b*x + 3*a) - 15*cos(b*x + a))*sin(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)*sin(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))*sin(6*b*x +
6*a) - 88*(2*cos(4*b*x + 4*a) + cos(2*b*x + 2*a) - 1)*sin(5*b*x + 5*a) + 40*(4*cos(3*b*x + 3*a) - 3*cos(b*x +
a))*sin(4*b*x + 4*a) - 80*(cos(2*b*x + 2*a) - 1)*sin(3*b*x + 3*a) + 80*cos(3*b*x + 3*a)*sin(2*b*x + 2*a) - 60*
cos(b*x + a)*sin(2*b*x + 2*a) + 60*cos(2*b*x + 2*a)*sin(b*x + a) - 60*sin(b*x + a))/(b*cos(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*c
os(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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mupad [B] time = 0.19, size = 61, normalized size = 0.92 \[ \frac {5\,\mathrm {atanh}\left (\sin \left (a+b\,x\right )\right )}{16\,b}-\frac {-\frac {5\,{\sin \left (a+b\,x\right )}^4}{16}+\frac {5\,{\sin \left (a+b\,x\right )}^2}{24}+\frac {1}{24}}{b\,\left ({\sin \left (a+b\,x\right )}^3-{\sin \left (a+b\,x\right )}^5\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(1/(sin(a + b*x)*sin(2*a + 2*b*x)^3),x)
[Out]
(5*atanh(sin(a + b*x)))/(16*b) - ((5*sin(a + b*x)^2)/24 - (5*sin(a + b*x)^4)/16 + 1/24)/(b*(sin(a + b*x)^3 - s
in(a + b*x)^5))
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \csc {\left (a + b x \right )} \csc ^{3}{\left (2 a + 2 b x \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(csc(b*x+a)*csc(2*b*x+2*a)**3,x)
[Out]
Integral(csc(a + b*x)*csc(2*a + 2*b*x)**3, x)
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