3.69 \(\int \csc ^3(a+b x) \csc (2 a+2 b x) \, dx\)
Optimal. Leaf size=43 \[ -\frac{\csc ^3(a+b x)}{6 b}-\frac{\csc (a+b x)}{2 b}+\frac{\tanh ^{-1}(\sin (a+b x))}{2 b} \]
[Out]
ArcTanh[Sin[a + b*x]]/(2*b) - Csc[a + b*x]/(2*b) - Csc[a + b*x]^3/(6*b)
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Rubi [A] time = 0.049419, antiderivative size = 43, normalized size of antiderivative = 1.,
number of steps used = 5, number of rules used = 4, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.222, Rules used =
{4288, 2621, 302, 207} \[ -\frac{\csc ^3(a+b x)}{6 b}-\frac{\csc (a+b x)}{2 b}+\frac{\tanh ^{-1}(\sin (a+b x))}{2 b} \]
Antiderivative was successfully verified.
[In]
Int[Csc[a + b*x]^3*Csc[2*a + 2*b*x],x]
[Out]
ArcTanh[Sin[a + b*x]]/(2*b) - Csc[a + b*x]/(2*b) - Csc[a + b*x]^3/(6*b)
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]
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 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 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])
Rubi steps
\begin{align*} \int \csc ^3(a+b x) \csc (2 a+2 b x) \, dx &=\frac{1}{2} \int \csc ^4(a+b x) \sec (a+b x) \, dx\\ &=-\frac{\operatorname{Subst}\left (\int \frac{x^4}{-1+x^2} \, dx,x,\csc (a+b x)\right )}{2 b}\\ &=-\frac{\operatorname{Subst}\left (\int \left (1+x^2+\frac{1}{-1+x^2}\right ) \, dx,x,\csc (a+b x)\right )}{2 b}\\ &=-\frac{\csc (a+b x)}{2 b}-\frac{\csc ^3(a+b x)}{6 b}-\frac{\operatorname{Subst}\left (\int \frac{1}{-1+x^2} \, dx,x,\csc (a+b x)\right )}{2 b}\\ &=\frac{\tanh ^{-1}(\sin (a+b x))}{2 b}-\frac{\csc (a+b x)}{2 b}-\frac{\csc ^3(a+b x)}{6 b}\\ \end{align*}
Mathematica [C] time = 0.0177232, size = 31, normalized size = 0.72 \[ -\frac{\csc ^3(a+b x) \text{Hypergeometric2F1}\left (-\frac{3}{2},1,-\frac{1}{2},\sin ^2(a+b x)\right )}{6 b} \]
Antiderivative was successfully verified.
[In]
Integrate[Csc[a + b*x]^3*Csc[2*a + 2*b*x],x]
[Out]
-(Csc[a + b*x]^3*Hypergeometric2F1[-3/2, 1, -1/2, Sin[a + b*x]^2])/(6*b)
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Maple [A] time = 0.032, size = 47, normalized size = 1.1 \begin{align*} -{\frac{1}{6\,b \left ( \sin \left ( bx+a \right ) \right ) ^{3}}}-{\frac{1}{2\,b\sin \left ( bx+a \right ) }}+{\frac{\ln \left ( \sec \left ( bx+a \right ) +\tan \left ( bx+a \right ) \right ) }{2\,b}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(csc(b*x+a)^3*csc(2*b*x+2*a),x)
[Out]
-1/6/b/sin(b*x+a)^3-1/2/b/sin(b*x+a)+1/2/b*ln(sec(b*x+a)+tan(b*x+a))
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Maxima [B] time = 1.78185, size = 1126, normalized size = 26.19 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(csc(b*x+a)^3*csc(2*b*x+2*a),x, algorithm="maxima")
[Out]
1/12*(4*(3*sin(5*b*x + 5*a) - 10*sin(3*b*x + 3*a) + 3*sin(b*x + a))*cos(6*b*x + 6*a) + 36*(sin(4*b*x + 4*a) -
sin(2*b*x + 2*a))*cos(5*b*x + 5*a) + 12*(10*sin(3*b*x + 3*a) - 3*sin(b*x + a))*cos(4*b*x + 4*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*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*(3*cos(5*b*x + 5*a) - 10*cos(3*b*x + 3*a) + 3*cos(b*x + a))*sin(6*b
*x + 6*a) - 12*(3*cos(4*b*x + 4*a) - 3*cos(2*b*x + 2*a) + 1)*sin(5*b*x + 5*a) - 12*(10*cos(3*b*x + 3*a) - 3*co
s(b*x + a))*sin(4*b*x + 4*a) - 40*(3*cos(2*b*x + 2*a) - 1)*sin(3*b*x + 3*a) + 120*cos(3*b*x + 3*a)*sin(2*b*x +
2*a) - 36*cos(b*x + a)*sin(2*b*x + 2*a) + 36*cos(2*b*x + 2*a)*sin(b*x + a) - 12*sin(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*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(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 [B] time = 0.502252, size = 254, normalized size = 5.91 \begin{align*} \frac{3 \,{\left (\cos \left (b x + a\right )^{2} - 1\right )} \log \left (\sin \left (b x + a\right ) + 1\right ) \sin \left (b x + a\right ) - 3 \,{\left (\cos \left (b x + a\right )^{2} - 1\right )} \log \left (-\sin \left (b x + a\right ) + 1\right ) \sin \left (b x + a\right ) - 6 \, \cos \left (b x + a\right )^{2} + 8}{12 \,{\left (b \cos \left (b x + a\right )^{2} - b\right )} \sin \left (b x + a\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(csc(b*x+a)^3*csc(2*b*x+2*a),x, algorithm="fricas")
[Out]
1/12*(3*(cos(b*x + a)^2 - 1)*log(sin(b*x + a) + 1)*sin(b*x + a) - 3*(cos(b*x + a)^2 - 1)*log(-sin(b*x + a) + 1
)*sin(b*x + a) - 6*cos(b*x + a)^2 + 8)/((b*cos(b*x + a)^2 - b)*sin(b*x + a))
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \csc ^{3}{\left (a + b x \right )} \csc{\left (2 a + 2 b x \right )}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(csc(b*x+a)**3*csc(2*b*x+2*a),x)
[Out]
Integral(csc(a + b*x)**3*csc(2*a + 2*b*x), x)
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Giac [B] time = 2.10202, size = 2830, normalized size = 65.81 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(csc(b*x+a)^3*csc(2*b*x+2*a),x, algorithm="giac")
[Out]
-1/48*((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*ta
n(1/2*a)^36 + 630*tan(1/2*b*x + 2*a)^5*tan(1/2*a)^32 - 267*tan(1/2*b*x + 2*a)^4*tan(1/2*a)^33 + 18*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 - 25458*tan(1/2*b*x + 2*a)^5*tan(1/2*a)^30 + 28
818*tan(1/2*b*x + 2*a)^4*tan(1/2*a)^31 - 10053*tan(1/2*b*x + 2*a)^3*tan(1/2*a)^32 + 1077*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 + 202086*tan(1/2*b*x + 2*a)^5*tan(1/2*a)^28 - 396738*tan(1
/2*b*x + 2*a)^4*tan(1/2*a)^29 + 244788*tan(1/2*b*x + 2*a)^3*tan(1/2*a)^30 - 58788*tan(1/2*b*x + 2*a)^2*tan(1/2
*a)^31 + 4518*tan(1/2*b*x + 2*a)*tan(1/2*a)^32 + 54*tan(1/2*a)^33 - 644166*tan(1/2*b*x + 2*a)^5*tan(1/2*a)^26
+ 2016678*tan(1/2*b*x + 2*a)^4*tan(1/2*a)^27 - 1980792*tan(1/2*b*x + 2*a)^3*tan(1/2*a)^28 + 780948*tan(1/2*b*x
+ 2*a)^2*tan(1/2*a)^29 - 123954*tan(1/2*b*x + 2*a)*tan(1/2*a)^30 + 5778*tan(1/2*a)^31 + 616590*tan(1/2*b*x +
2*a)^5*tan(1/2*a)^24 - 4024566*tan(1/2*b*x + 2*a)^4*tan(1/2*a)^25 + 6570108*tan(1/2*b*x + 2*a)^3*tan(1/2*a)^26
- 4036848*tan(1/2*b*x + 2*a)^2*tan(1/2*a)^27 + 993942*tan(1/2*b*x + 2*a)*tan(1/2*a)^28 - 80658*tan(1/2*a)^29
+ 1290870*tan(1/2*b*x + 2*a)^5*tan(1/2*a)^22 - 545670*tan(1/2*b*x + 2*a)^4*tan(1/2*a)^23 - 6240000*tan(1/2*b*x
+ 2*a)^3*tan(1/2*a)^24 + 8134776*tan(1/2*b*x + 2*a)^2*tan(1/2*a)^25 - 3273030*tan(1/2*b*x + 2*a)*tan(1/2*a)^2
6 + 402730*tan(1/2*a)^27 - 3012066*tan(1/2*b*x + 2*a)^5*tan(1/2*a)^20 + 12765366*tan(1/2*b*x + 2*a)^4*tan(1/2*
a)^21 - 13579740*tan(1/2*b*x + 2*a)^3*tan(1/2*a)^22 + 1241820*tan(1/2*b*x + 2*a)^2*tan(1/2*a)^23 + 3103230*tan
(1/2*b*x + 2*a)*tan(1/2*a)^24 - 796770*tan(1/2*a)^25 - 10040220*tan(1/2*b*x + 2*a)^4*tan(1/2*a)^19 + 29284146*
tan(1/2*b*x + 2*a)^3*tan(1/2*a)^20 - 25575756*tan(1/2*b*x + 2*a)^2*tan(1/2*a)^21 + 6696630*tan(1/2*b*x + 2*a)*
tan(1/2*a)^22 - 99990*tan(1/2*a)^23 + 3012066*tan(1/2*b*x + 2*a)^5*tan(1/2*a)^16 - 10040220*tan(1/2*b*x + 2*a)
^4*tan(1/2*a)^17 + 19709370*tan(1/2*b*x + 2*a)^2*tan(1/2*a)^19 - 14753394*tan(1/2*b*x + 2*a)*tan(1/2*a)^20 + 2
525334*tan(1/2*a)^21 - 1290870*tan(1/2*b*x + 2*a)^5*tan(1/2*a)^14 + 12765366*tan(1/2*b*x + 2*a)^4*tan(1/2*a)^1
5 - 29284146*tan(1/2*b*x + 2*a)^3*tan(1/2*a)^16 + 19709370*tan(1/2*b*x + 2*a)^2*tan(1/2*a)^17 - 2087550*tan(1/
2*a)^19 - 616590*tan(1/2*b*x + 2*a)^5*tan(1/2*a)^12 - 545670*tan(1/2*b*x + 2*a)^4*tan(1/2*a)^13 + 13579740*tan
(1/2*b*x + 2*a)^3*tan(1/2*a)^14 - 25575756*tan(1/2*b*x + 2*a)^2*tan(1/2*a)^15 + 14753394*tan(1/2*b*x + 2*a)*ta
n(1/2*a)^16 - 2087550*tan(1/2*a)^17 + 644166*tan(1/2*b*x + 2*a)^5*tan(1/2*a)^10 - 4024566*tan(1/2*b*x + 2*a)^4
*tan(1/2*a)^11 + 6240000*tan(1/2*b*x + 2*a)^3*tan(1/2*a)^12 + 1241820*tan(1/2*b*x + 2*a)^2*tan(1/2*a)^13 - 669
6630*tan(1/2*b*x + 2*a)*tan(1/2*a)^14 + 2525334*tan(1/2*a)^15 - 202086*tan(1/2*b*x + 2*a)^5*tan(1/2*a)^8 + 201
6678*tan(1/2*b*x + 2*a)^4*tan(1/2*a)^9 - 6570108*tan(1/2*b*x + 2*a)^3*tan(1/2*a)^10 + 8134776*tan(1/2*b*x + 2*
a)^2*tan(1/2*a)^11 - 3103230*tan(1/2*b*x + 2*a)*tan(1/2*a)^12 - 99990*tan(1/2*a)^13 + 25458*tan(1/2*b*x + 2*a)
^5*tan(1/2*a)^6 - 396738*tan(1/2*b*x + 2*a)^4*tan(1/2*a)^7 + 1980792*tan(1/2*b*x + 2*a)^3*tan(1/2*a)^8 - 40368
48*tan(1/2*b*x + 2*a)^2*tan(1/2*a)^9 + 3273030*tan(1/2*b*x + 2*a)*tan(1/2*a)^10 - 796770*tan(1/2*a)^11 - 630*t
an(1/2*b*x + 2*a)^5*tan(1/2*a)^4 + 28818*tan(1/2*b*x + 2*a)^4*tan(1/2*a)^5 - 244788*tan(1/2*b*x + 2*a)^3*tan(1
/2*a)^6 + 780948*tan(1/2*b*x + 2*a)^2*tan(1/2*a)^7 - 993942*tan(1/2*b*x + 2*a)*tan(1/2*a)^8 + 402730*tan(1/2*a
)^9 - 27*tan(1/2*b*x + 2*a)^5*tan(1/2*a)^2 - 267*tan(1/2*b*x + 2*a)^4*tan(1/2*a)^3 + 10053*tan(1/2*b*x + 2*a)^
3*tan(1/2*a)^4 - 58788*tan(1/2*b*x + 2*a)^2*tan(1/2*a)^5 + 123954*tan(1/2*b*x + 2*a)*tan(1/2*a)^6 - 80658*tan(
1/2*a)^7 - 9*tan(1/2*b*x + 2*a)^4*tan(1/2*a) - 18*tan(1/2*b*x + 2*a)^3*tan(1/2*a)^2 + 1077*tan(1/2*b*x + 2*a)^
2*tan(1/2*a)^3 - 4518*tan(1/2*b*x + 2*a)*tan(1/2*a)^4 + 5778*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*ta
n(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)*t
an(1/2*a)^2 + 10*tan(1/2*a)^3 + tan(1/2*b*x + 2*a) - 3*tan(1/2*a))^3) - 24*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 - t
an(1/2*b*x + 2*a) + 3*tan(1/2*a) - 1)) + 24*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