3.12 \(\int \csc ^5(2 a+2 b x) \sin (a+b x) \, dx\)
Optimal. Leaf size=89 \[ -\frac {35 \csc ^3(a+b x)}{768 b}-\frac {35 \csc (a+b x)}{256 b}+\frac {35 \tanh ^{-1}(\sin (a+b x))}{256 b}+\frac {\csc ^3(a+b x) \sec ^4(a+b x)}{128 b}+\frac {7 \csc ^3(a+b x) \sec ^2(a+b x)}{256 b} \]
[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 = 0.07, 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 =
{4288, 2621, 288, 302, 207} \[ -\frac {35 \csc ^3(a+b x)}{768 b}-\frac {35 \csc (a+b x)}{256 b}+\frac {35 \tanh ^{-1}(\sin (a+b x))}{256 b}+\frac {\csc ^3(a+b x) \sec ^4(a+b x)}{128 b}+\frac {7 \csc ^3(a+b x) \sec ^2(a+b x)}{256 b} \]
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 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 ^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 {\operatorname {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 \operatorname {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 \operatorname {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 \operatorname {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 \operatorname {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] time = 0.03, size = 31, normalized size = 0.35 \[ -\frac {\csc ^3(a+b x) \, _2F_1\left (-\frac {3}{2},3;-\frac {1}{2};\sin ^2(a+b x)\right )}{96 b} \]
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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fricas [A] time = 0.51, size = 140, normalized size = 1.57 \[ -\frac {210 \, \cos \left (b x + a\right )^{6} - 280 \, \cos \left (b x + a\right )^{4} - 105 \, {\left (\cos \left (b x + a\right )^{6} - \cos \left (b x + a\right )^{4}\right )} \log \left (\sin \left (b x + a\right ) + 1\right ) \sin \left (b x + a\right ) + 105 \, {\left (\cos \left (b x + a\right )^{6} - \cos \left (b x + a\right )^{4}\right )} \log \left (-\sin \left (b x + a\right ) + 1\right ) \sin \left (b x + a\right ) + 42 \, \cos \left (b x + a\right )^{2} + 12}{1536 \, {\left (b \cos \left (b x + a\right )^{6} - b \cos \left (b x + a\right )^{4}\right )} \sin \left (b x + a\right )} \]
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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giac [B] time = 12.38, size = 5647, normalized size = 63.45 \[ \text {result too large to display} \]
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]
-1/768*((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*t
an(1/2*a)^36 + 2574*tan(1/2*b*x + 2*a)^5*tan(1/2*a)^32 - 1563*tan(1/2*b*x + 2*a)^4*tan(1/2*a)^33 + 234*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 - 74706*tan(1/2*b*x + 2*a)^5*tan(1/2*a)^30
+ 88434*tan(1/2*b*x + 2*a)^4*tan(1/2*a)^31 - 32229*tan(1/2*b*x + 2*a)^3*tan(1/2*a)^32 + 3669*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 + 598014*tan(1/2*b*x + 2*a)^5*tan(1/2*a)^28 - 1170882*
tan(1/2*b*x + 2*a)^4*tan(1/2*a)^29 + 730596*tan(1/2*b*x + 2*a)^3*tan(1/2*a)^30 - 178020*tan(1/2*b*x + 2*a)^2*t
an(1/2*a)^31 + 14238*tan(1/2*b*x + 2*a)*tan(1/2*a)^32 + 54*tan(1/2*a)^33 - 1958598*tan(1/2*b*x + 2*a)^5*tan(1/
2*a)^26 + 6055062*tan(1/2*b*x + 2*a)^4*tan(1/2*a)^27 - 5907096*tan(1/2*b*x + 2*a)^3*tan(1/2*a)^28 + 2329236*ta
n(1/2*b*x + 2*a)^2*tan(1/2*a)^29 - 370194*tan(1/2*b*x + 2*a)*tan(1/2*a)^30 + 17442*tan(1/2*a)^31 + 1859910*tan
(1/2*b*x + 2*a)^5*tan(1/2*a)^24 - 12207078*tan(1/2*b*x + 2*a)^4*tan(1/2*a)^25 + 19818828*tan(1/2*b*x + 2*a)^3*
tan(1/2*a)^26 - 12113616*tan(1/2*b*x + 2*a)^2*tan(1/2*a)^27 + 2973582*tan(1/2*b*x + 2*a)*tan(1/2*a)^28 - 24006
6*tan(1/2*a)^29 + 3993750*tan(1/2*b*x + 2*a)^5*tan(1/2*a)^22 - 1879110*tan(1/2*b*x + 2*a)^4*tan(1/2*a)^23 - 18
713760*tan(1/2*b*x + 2*a)^3*tan(1/2*a)^24 + 24499800*tan(1/2*b*x + 2*a)^2*tan(1/2*a)^25 - 9845190*tan(1/2*b*x
+ 2*a)*tan(1/2*a)^26 + 1208842*tan(1/2*a)^27 - 8882730*tan(1/2*b*x + 2*a)^5*tan(1/2*a)^20 + 38331030*tan(1/2*b
*x + 2*a)^4*tan(1/2*a)^21 - 41093100*tan(1/2*b*x + 2*a)^3*tan(1/2*a)^22 + 3908700*tan(1/2*b*x + 2*a)^2*tan(1/2
*a)^23 + 9319830*tan(1/2*b*x + 2*a)*tan(1/2*a)^24 - 2402946*tan(1/2*a)^25 - 29609100*tan(1/2*b*x + 2*a)^4*tan(
1/2*a)^19 + 87376914*tan(1/2*b*x + 2*a)^3*tan(1/2*a)^20 - 76707084*tan(1/2*b*x + 2*a)^2*tan(1/2*a)^21 + 202110
30*tan(1/2*b*x + 2*a)*tan(1/2*a)^22 - 319590*tan(1/2*a)^23 + 8882730*tan(1/2*b*x + 2*a)^5*tan(1/2*a)^16 - 2960
9100*tan(1/2*b*x + 2*a)^4*tan(1/2*a)^17 + 58847130*tan(1/2*b*x + 2*a)^2*tan(1/2*a)^19 - 44106714*tan(1/2*b*x +
2*a)*tan(1/2*a)^20 + 7594374*tan(1/2*a)^21 - 3993750*tan(1/2*b*x + 2*a)^5*tan(1/2*a)^14 + 38331030*tan(1/2*b*
x + 2*a)^4*tan(1/2*a)^15 - 87376914*tan(1/2*b*x + 2*a)^3*tan(1/2*a)^16 + 58847130*tan(1/2*b*x + 2*a)^2*tan(1/2
*a)^17 - 6185790*tan(1/2*a)^19 - 1859910*tan(1/2*b*x + 2*a)^5*tan(1/2*a)^12 - 1879110*tan(1/2*b*x + 2*a)^4*tan
(1/2*a)^13 + 41093100*tan(1/2*b*x + 2*a)^3*tan(1/2*a)^14 - 76707084*tan(1/2*b*x + 2*a)^2*tan(1/2*a)^15 + 44106
714*tan(1/2*b*x + 2*a)*tan(1/2*a)^16 - 6185790*tan(1/2*a)^17 + 1958598*tan(1/2*b*x + 2*a)^5*tan(1/2*a)^10 - 12
207078*tan(1/2*b*x + 2*a)^4*tan(1/2*a)^11 + 18713760*tan(1/2*b*x + 2*a)^3*tan(1/2*a)^12 + 3908700*tan(1/2*b*x
+ 2*a)^2*tan(1/2*a)^13 - 20211030*tan(1/2*b*x + 2*a)*tan(1/2*a)^14 + 7594374*tan(1/2*a)^15 - 598014*tan(1/2*b*
x + 2*a)^5*tan(1/2*a)^8 + 6055062*tan(1/2*b*x + 2*a)^4*tan(1/2*a)^9 - 19818828*tan(1/2*b*x + 2*a)^3*tan(1/2*a)
^10 + 24499800*tan(1/2*b*x + 2*a)^2*tan(1/2*a)^11 - 9319830*tan(1/2*b*x + 2*a)*tan(1/2*a)^12 - 319590*tan(1/2*
a)^13 + 74706*tan(1/2*b*x + 2*a)^5*tan(1/2*a)^6 - 1170882*tan(1/2*b*x + 2*a)^4*tan(1/2*a)^7 + 5907096*tan(1/2*
b*x + 2*a)^3*tan(1/2*a)^8 - 12113616*tan(1/2*b*x + 2*a)^2*tan(1/2*a)^9 + 9845190*tan(1/2*b*x + 2*a)*tan(1/2*a)
^10 - 2402946*tan(1/2*a)^11 - 2574*tan(1/2*b*x + 2*a)^5*tan(1/2*a)^4 + 88434*tan(1/2*b*x + 2*a)^4*tan(1/2*a)^5
- 730596*tan(1/2*b*x + 2*a)^3*tan(1/2*a)^6 + 2329236*tan(1/2*b*x + 2*a)^2*tan(1/2*a)^7 - 2973582*tan(1/2*b*x
+ 2*a)*tan(1/2*a)^8 + 1208842*tan(1/2*a)^9 - 27*tan(1/2*b*x + 2*a)^5*tan(1/2*a)^2 - 1563*tan(1/2*b*x + 2*a)^4*
tan(1/2*a)^3 + 32229*tan(1/2*b*x + 2*a)^3*tan(1/2*a)^4 - 178020*tan(1/2*b*x + 2*a)^2*tan(1/2*a)^5 + 370194*tan
(1/2*b*x + 2*a)*tan(1/2*a)^6 - 240066*tan(1/2*a)^7 - 9*tan(1/2*b*x + 2*a)^4*tan(1/2*a) - 234*tan(1/2*b*x + 2*a
)^3*tan(1/2*a)^2 + 3669*tan(1/2*b*x + 2*a)^2*tan(1/2*a)^3 - 14238*tan(1/2*b*x + 2*a)*tan(1/2*a)^4 + 17442*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) + 6*(13*tan(1/2*b*x + 2*a)^7*tan(1/2*a)^48 - 174*tan(1/2*b*x + 2*a)^7*tan(1/2*a)^46 + 162*tan(1/2*b*x + 2
*a)^6*tan(1/2*a)^47 - 5*tan(1/2*b*x + 2*a)^5*tan(1/2*a)^48 - 21522*tan(1/2*b*x + 2*a)^7*tan(1/2*a)^44 + 13014*
tan(1/2*b*x + 2*a)^6*tan(1/2*a)^45 - 2514*tan(1/2*b*x + 2*a)^5*tan(1/2*a)^46 + 234*tan(1/2*b*x + 2*a)^4*tan(1/
2*a)^47 - 5*tan(1/2*b*x + 2*a)^3*tan(1/2*a)^48 + 942746*tan(1/2*b*x + 2*a)^7*tan(1/2*a)^42 - 1100574*tan(1/2*b
*x + 2*a)^6*tan(1/2*a)^43 + 453762*tan(1/2*b*x + 2*a)^5*tan(1/2*a)^44 - 87234*tan(1/2*b*x + 2*a)^4*tan(1/2*a)^
45 + 8718*tan(1/2*b*x + 2*a)^3*tan(1/2*a)^46 - 474*tan(1/2*b*x + 2*a)^2*tan(1/2*a)^47 + 13*tan(1/2*b*x + 2*a)*
tan(1/2*a)^48 - 17981508*tan(1/2*b*x + 2*a)^7*tan(1/2*a)^40 + 30574566*tan(1/2*b*x + 2*a)^6*tan(1/2*a)^41 - 18
985626*tan(1/2*b*x + 2*a)^5*tan(1/2*a)^42 + 5680314*tan(1/2*b*x + 2*a)^4*tan(1/2*a)^43 - 906750*tan(1/2*b*x +
2*a)^3*tan(1/2*a)^44 + 80834*tan(1/2*b*x + 2*a)^2*tan(1/2*a)^45 - 3918*tan(1/2*b*x + 2*a)*tan(1/2*a)^46 + 78*t
an(1/2*a)^47 + 193013886*tan(1/2*b*x + 2*a)^7*tan(1/2*a)^38 - 441566530*tan(1/2*b*x + 2*a)^6*tan(1/2*a)^39 + 3
74555556*tan(1/2*b*x + 2*a)^5*tan(1/2*a)^40 - 153629730*tan(1/2*b*x + 2*a)^4*tan(1/2*a)^41 + 33577926*tan(1/2*
b*x + 2*a)^3*tan(1/2*a)^42 - 4014234*tan(1/2*b*x + 2*a)^2*tan(1/2*a)^43 + 252270*tan(1/2*b*x + 2*a)*tan(1/2*a)
^44 - 6614*tan(1/2*a)^45 - 1202084806*tan(1/2*b*x + 2*a)^7*tan(1/2*a)^36 + 3626590890*tan(1/2*b*x + 2*a)^6*tan
(1/2*a)^37 - 4076849790*tan(1/2*b*x + 2*a)^5*tan(1/2*a)^38 + 2214510870*tan(1/2*b*x + 2*a)^4*tan(1/2*a)^39 - 6
33112668*tan(1/2*b*x + 2*a)^3*tan(1/2*a)^40 + 97266546*tan(1/2*b*x + 2*a)^2*tan(1/2*a)^41 - 7629702*tan(1/2*b*
x + 2*a)*tan(1/2*a)^42 + 243198*tan(1/2*a)^43 + 4119245478*tan(1/2*b*x + 2*a)^7*tan(1/2*a)^34 - 16824081618*ta
n(1/2*b*x + 2*a)^6*tan(1/2*a)^35 + 25189696662*tan(1/2*b*x + 2*a)^5*tan(1/2*a)^36 - 18102012318*tan(1/2*b*x +
2*a)^4*tan(1/2*a)^37 + 6776146338*tan(1/2*b*x + 2*a)^3*tan(1/2*a)^38 - 1334518022*tan(1/2*b*x + 2*a)^2*tan(1/2
*a)^39 + 130749372*tan(1/2*b*x + 2*a)*tan(1/2*a)^40 - 5045814*tan(1/2*a)^41 - 6836731551*tan(1/2*b*x + 2*a)^7*
tan(1/2*a)^32 + 41573675178*tan(1/2*b*x + 2*a)^6*tan(1/2*a)^33 - 86629166118*tan(1/2*b*x + 2*a)^5*tan(1/2*a)^3
4 + 84079436982*tan(1/2*b*x + 2*a)^4*tan(1/2*a)^35 - 41858006762*tan(1/2*b*x + 2*a)^3*tan(1/2*a)^36 + 10786903
998*tan(1/2*b*x + 2*a)^2*tan(1/2*a)^37 - 1346794722*tan(1/2*b*x + 2*a)*tan(1/2*a)^38 + 64179762*tan(1/2*a)^39
+ 267313972*tan(1/2*b*x + 2*a)^7*tan(1/2*a)^30 - 37264510188*tan(1/2*b*x + 2*a)^6*tan(1/2*a)^31 + 143316996039
*tan(1/2*b*x + 2*a)^5*tan(1/2*a)^32 - 208138130542*tan(1/2*b*x + 2*a)^4*tan(1/2*a)^33 + 144563703930*tan(1/2*b
*x + 2*a)^3*tan(1/2*a)^34 - 50381554230*tan(1/2*b*x + 2*a)^2*tan(1/2*a)^35 + 8311517370*tan(1/2*b*x + 2*a)*tan
(1/2*a)^36 - 507068874*tan(1/2*a)^37 + 17351751084*tan(1/2*b*x + 2*a)^7*tan(1/2*a)^28 - 56502600420*tan(1/2*b*
x + 2*a)^6*tan(1/2*a)^29 - 5710349876*tan(1/2*b*x + 2*a)^5*tan(1/2*a)^30 + 186100588644*tan(1/2*b*x + 2*a)^4*t
an(1/2*a)^31 - 239405365305*tan(1/2*b*x + 2*a)^3*tan(1/2*a)^32 + 125386031934*tan(1/2*b*x + 2*a)^2*tan(1/2*a)^
33 - 29001178170*tan(1/2*b*x + 2*a)*tan(1/2*a)^34 + 2392888914*tan(1/2*a)^35 - 22757749308*tan(1/2*b*x + 2*a)^
7*tan(1/2*a)^26 + 164419919188*tan(1/2*b*x + 2*a)^6*tan(1/2*a)^27 - 363804236364*tan(1/2*b*x + 2*a)^5*tan(1/2*
a)^28 + 282793219500*tan(1/2*b*x + 2*a)^4*tan(1/2*a)^29 + 8521880716*tan(1/2*b*x + 2*a)^3*tan(1/2*a)^30 - 1116
37405764*tan(1/2*b*x + 2*a)^2*tan(1/2*a)^31 + 48133667169*tan(1/2*b*x + 2*a)*tan(1/2*a)^32 - 6017063802*tan(1/
2*a)^33 - 98616913668*tan(1/2*b*x + 2*a)^6*tan(1/2*a)^25 + 478784400444*tan(1/2*b*x + 2*a)^5*tan(1/2*a)^26 - 8
21763522748*tan(1/2*b*x + 2*a)^4*tan(1/2*a)^27 + 607022398644*tan(1/2*b*x + 2*a)^3*tan(1/2*a)^28 - 17113731793
2*tan(1/2*b*x + 2*a)^2*tan(1/2*a)^29 - 1237805836*tan(1/2*b*x + 2*a)*tan(1/2*a)^30 + 5308170444*tan(1/2*a)^31
+ 22757749308*tan(1/2*b*x + 2*a)^7*tan(1/2*a)^22 - 98616913668*tan(1/2*b*x + 2*a)^6*tan(1/2*a)^23 + 4928297283
00*tan(1/2*b*x + 2*a)^4*tan(1/2*a)^25 - 795918156804*tan(1/2*b*x + 2*a)^3*tan(1/2*a)^26 + 492162548604*tan(1/2
*b*x + 2*a)^2*tan(1/2*a)^27 - 121726145364*tan(1/2*b*x + 2*a)*tan(1/2*a)^28 + 8269813092*tan(1/2*a)^29 - 17351
751084*tan(1/2*b*x + 2*a)^7*tan(1/2*a)^20 + 164419919188*tan(1/2*b*x + 2*a)^6*tan(1/2*a)^21 - 478784400444*tan
(1/2*b*x + 2*a)^5*tan(1/2*a)^22 + 492829728300*tan(1/2*b*x + 2*a)^4*tan(1/2*a)^23 - 293938021260*tan(1/2*b*x +
2*a)^2*tan(1/2*a)^25 + 158215581828*tan(1/2*b*x + 2*a)*tan(1/2*a)^26 - 23358680596*tan(1/2*a)^27 - 267313972*
tan(1/2*b*x + 2*a)^7*tan(1/2*a)^18 - 56502600420*tan(1/2*b*x + 2*a)^6*tan(1/2*a)^19 + 363804236364*tan(1/2*b*x
+ 2*a)^5*tan(1/2*a)^20 - 821763522748*tan(1/2*b*x + 2*a)^4*tan(1/2*a)^21 + 795918156804*tan(1/2*b*x + 2*a)^3*
tan(1/2*a)^22 - 293938021260*tan(1/2*b*x + 2*a)^2*tan(1/2*a)^23 + 13852570212*tan(1/2*a)^25 + 6836731551*tan(1
/2*b*x + 2*a)^7*tan(1/2*a)^16 - 37264510188*tan(1/2*b*x + 2*a)^6*tan(1/2*a)^17 + 5710349876*tan(1/2*b*x + 2*a)
^5*tan(1/2*a)^18 + 282793219500*tan(1/2*b*x + 2*a)^4*tan(1/2*a)^19 - 607022398644*tan(1/2*b*x + 2*a)^3*tan(1/2
*a)^20 + 492162548604*tan(1/2*b*x + 2*a)^2*tan(1/2*a)^21 - 158215581828*tan(1/2*b*x + 2*a)*tan(1/2*a)^22 + 138
52570212*tan(1/2*a)^23 - 4119245478*tan(1/2*b*x + 2*a)^7*tan(1/2*a)^14 + 41573675178*tan(1/2*b*x + 2*a)^6*tan(
1/2*a)^15 - 143316996039*tan(1/2*b*x + 2*a)^5*tan(1/2*a)^16 + 186100588644*tan(1/2*b*x + 2*a)^4*tan(1/2*a)^17
- 8521880716*tan(1/2*b*x + 2*a)^3*tan(1/2*a)^18 - 171137317932*tan(1/2*b*x + 2*a)^2*tan(1/2*a)^19 + 1217261453
64*tan(1/2*b*x + 2*a)*tan(1/2*a)^20 - 23358680596*tan(1/2*a)^21 + 1202084806*tan(1/2*b*x + 2*a)^7*tan(1/2*a)^1
2 - 16824081618*tan(1/2*b*x + 2*a)^6*tan(1/2*a)^13 + 86629166118*tan(1/2*b*x + 2*a)^5*tan(1/2*a)^14 - 20813813
0542*tan(1/2*b*x + 2*a)^4*tan(1/2*a)^15 + 239405365305*tan(1/2*b*x + 2*a)^3*tan(1/2*a)^16 - 111637405764*tan(1
/2*b*x + 2*a)^2*tan(1/2*a)^17 + 1237805836*tan(1/2*b*x + 2*a)*tan(1/2*a)^18 + 8269813092*tan(1/2*a)^19 - 19301
3886*tan(1/2*b*x + 2*a)^7*tan(1/2*a)^10 + 3626590890*tan(1/2*b*x + 2*a)^6*tan(1/2*a)^11 - 25189696662*tan(1/2*
b*x + 2*a)^5*tan(1/2*a)^12 + 84079436982*tan(1/2*b*x + 2*a)^4*tan(1/2*a)^13 - 144563703930*tan(1/2*b*x + 2*a)^
3*tan(1/2*a)^14 + 125386031934*tan(1/2*b*x + 2*a)^2*tan(1/2*a)^15 - 48133667169*tan(1/2*b*x + 2*a)*tan(1/2*a)^
16 + 5308170444*tan(1/2*a)^17 + 17981508*tan(1/2*b*x + 2*a)^7*tan(1/2*a)^8 - 441566530*tan(1/2*b*x + 2*a)^6*ta
n(1/2*a)^9 + 4076849790*tan(1/2*b*x + 2*a)^5*tan(1/2*a)^10 - 18102012318*tan(1/2*b*x + 2*a)^4*tan(1/2*a)^11 +
41858006762*tan(1/2*b*x + 2*a)^3*tan(1/2*a)^12 - 50381554230*tan(1/2*b*x + 2*a)^2*tan(1/2*a)^13 + 29001178170*
tan(1/2*b*x + 2*a)*tan(1/2*a)^14 - 6017063802*tan(1/2*a)^15 - 942746*tan(1/2*b*x + 2*a)^7*tan(1/2*a)^6 + 30574
566*tan(1/2*b*x + 2*a)^6*tan(1/2*a)^7 - 374555556*tan(1/2*b*x + 2*a)^5*tan(1/2*a)^8 + 2214510870*tan(1/2*b*x +
2*a)^4*tan(1/2*a)^9 - 6776146338*tan(1/2*b*x + 2*a)^3*tan(1/2*a)^10 + 10786903998*tan(1/2*b*x + 2*a)^2*tan(1/
2*a)^11 - 8311517370*tan(1/2*b*x + 2*a)*tan(1/2*a)^12 + 2392888914*tan(1/2*a)^13 + 21522*tan(1/2*b*x + 2*a)^7*
tan(1/2*a)^4 - 1100574*tan(1/2*b*x + 2*a)^6*tan(1/2*a)^5 + 18985626*tan(1/2*b*x + 2*a)^5*tan(1/2*a)^6 - 153629
730*tan(1/2*b*x + 2*a)^4*tan(1/2*a)^7 + 633112668*tan(1/2*b*x + 2*a)^3*tan(1/2*a)^8 - 1334518022*tan(1/2*b*x +
2*a)^2*tan(1/2*a)^9 + 1346794722*tan(1/2*b*x + 2*a)*tan(1/2*a)^10 - 507068874*tan(1/2*a)^11 + 174*tan(1/2*b*x
+ 2*a)^7*tan(1/2*a)^2 + 13014*tan(1/2*b*x + 2*a)^6*tan(1/2*a)^3 - 453762*tan(1/2*b*x + 2*a)^5*tan(1/2*a)^4 +
5680314*tan(1/2*b*x + 2*a)^4*tan(1/2*a)^5 - 33577926*tan(1/2*b*x + 2*a)^3*tan(1/2*a)^6 + 97266546*tan(1/2*b*x
+ 2*a)^2*tan(1/2*a)^7 - 130749372*tan(1/2*b*x + 2*a)*tan(1/2*a)^8 + 64179762*tan(1/2*a)^9 - 13*tan(1/2*b*x + 2
*a)^7 + 162*tan(1/2*b*x + 2*a)^6*tan(1/2*a) + 2514*tan(1/2*b*x + 2*a)^5*tan(1/2*a)^2 - 87234*tan(1/2*b*x + 2*a
)^4*tan(1/2*a)^3 + 906750*tan(1/2*b*x + 2*a)^3*tan(1/2*a)^4 - 4014234*tan(1/2*b*x + 2*a)^2*tan(1/2*a)^5 + 7629
702*tan(1/2*b*x + 2*a)*tan(1/2*a)^6 - 5045814*tan(1/2*a)^7 + 5*tan(1/2*b*x + 2*a)^5 + 234*tan(1/2*b*x + 2*a)^4
*tan(1/2*a) - 8718*tan(1/2*b*x + 2*a)^3*tan(1/2*a)^2 + 80834*tan(1/2*b*x + 2*a)^2*tan(1/2*a)^3 - 252270*tan(1/
2*b*x + 2*a)*tan(1/2*a)^4 + 243198*tan(1/2*a)^5 + 5*tan(1/2*b*x + 2*a)^3 - 474*tan(1/2*b*x + 2*a)^2*tan(1/2*a)
+ 3918*tan(1/2*b*x + 2*a)*tan(1/2*a)^2 - 6614*tan(1/2*a)^3 - 13*tan(1/2*b*x + 2*a) + 78*tan(1/2*a))/((tan(1/2
*a)^24 - 60*tan(1/2*a)^22 + 1410*tan(1/2*a)^20 - 16204*tan(1/2*a)^18 + 92655*tan(1/2*a)^16 - 245880*tan(1/2*a)
^14 + 336156*tan(1/2*a)^12 - 245880*tan(1/2*a)^10 + 92655*tan(1/2*a)^8 - 16204*tan(1/2*a)^6 + 1410*tan(1/2*a)^
4 - 60*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)^4) -
105*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)) + 105*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
________________________________________________________________________________________
maple [A] time = 0.98, size = 97, normalized size = 1.09 \[ \frac {1}{128 b \sin \left (b x +a \right )^{3} \cos \left (b x +a \right )^{4}}-\frac {7}{384 b \sin \left (b x +a \right )^{3} \cos \left (b x +a \right )^{2}}+\frac {35}{768 b \sin \left (b x +a \right ) \cos \left (b x +a \right )^{2}}-\frac {35}{256 b \sin \left (b x +a \right )}+\frac {35 \ln \left (\sec \left (b x +a \right )+\tan \left (b x +a \right )\right )}{256 b} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(csc(2*b*x+2*a)^5*sin(b*x+a),x)
[Out]
1/128/b/sin(b*x+a)^3/cos(b*x+a)^4-7/384/b/sin(b*x+a)^3/cos(b*x+a)^2+35/768/b/sin(b*x+a)/cos(b*x+a)^2-35/256/b/
sin(b*x+a)+35/256/b*ln(sec(b*x+a)+tan(b*x+a))
________________________________________________________________________________________
maxima [B] time = 0.62, size = 3088, normalized size = 34.70 \[ \text {result too large to display} \]
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)*cos(14*b*x + 14*a) - 2*(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)*cos(12*b*x + 12*a) + 6*(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)*cos(10*b*x + 10*a) - 6*(
3*b*cos(6*b*x + 6*a) + 3*b*cos(4*b*x + 4*a) - b*cos(2*b*x + 2*a) - b)*cos(8*b*x + 8*a) + 6*(3*b*cos(4*b*x + 4*
a) - b*cos(2*b*x + 2*a) - b)*cos(6*b*x + 6*a) - 6*(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(12*b*x + 12*a) - 3*b*sin(10*b*x + 10*a) - 3*b*sin(8*b*x + 8*a) + 3*b*sin(6*b*x + 6*a) + 3*b*si
n(4*b*x + 4*a) - b*sin(2*b*x + 2*a))*sin(14*b*x + 14*a) - 2*(3*b*sin(10*b*x + 10*a) + 3*b*sin(8*b*x + 8*a) - 3
*b*sin(6*b*x + 6*a) - 3*b*sin(4*b*x + 4*a) + b*sin(2*b*x + 2*a))*sin(12*b*x + 12*a) + 6*(3*b*sin(8*b*x + 8*a)
- 3*b*sin(6*b*x + 6*a) - 3*b*sin(4*b*x + 4*a) + b*sin(2*b*x + 2*a))*sin(10*b*x + 10*a) - 6*(3*b*sin(6*b*x + 6*
a) + 3*b*sin(4*b*x + 4*a) - b*sin(2*b*x + 2*a))*sin(8*b*x + 8*a) + 6*(3*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 = 0.17, size = 79, normalized size = 0.89 \[ \frac {35\,\mathrm {atanh}\left (\sin \left (a+b\,x\right )\right )}{256\,b}-\frac {\frac {35\,{\sin \left (a+b\,x\right )}^6}{256}-\frac {175\,{\sin \left (a+b\,x\right )}^4}{768}+\frac {7\,{\sin \left (a+b\,x\right )}^2}{96}+\frac {1}{96}}{b\,\left ({\sin \left (a+b\,x\right )}^7-2\,{\sin \left (a+b\,x\right )}^5+{\sin \left (a+b\,x\right )}^3\right )} \]
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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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
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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