3.31 \(\int \csc ^4(2 a+2 b x) \sin ^3(a+b x) \, dx\)
Optimal. Leaf size=43 \[ \frac{\sec ^3(a+b x)}{48 b}+\frac{\sec (a+b x)}{16 b}-\frac{\tanh ^{-1}(\cos (a+b x))}{16 b} \]
[Out]
-ArcTanh[Cos[a + b*x]]/(16*b) + Sec[a + b*x]/(16*b) + Sec[a + b*x]^3/(48*b)
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Rubi [A] time = 0.0529464, antiderivative size = 43, normalized size of antiderivative = 1.,
number of steps used = 5, number of rules used = 4, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used =
{4288, 2622, 302, 207} \[ \frac{\sec ^3(a+b x)}{48 b}+\frac{\sec (a+b x)}{16 b}-\frac{\tanh ^{-1}(\cos (a+b x))}{16 b} \]
Antiderivative was successfully verified.
[In]
Int[Csc[2*a + 2*b*x]^4*Sin[a + b*x]^3,x]
[Out]
-ArcTanh[Cos[a + b*x]]/(16*b) + Sec[a + b*x]/(16*b) + Sec[a + b*x]^3/(48*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 2622
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 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 ^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{\operatorname{Subst}\left (\int \frac{x^4}{-1+x^2} \, dx,x,\sec (a+b x)\right )}{16 b}\\ &=\frac{\operatorname{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{\operatorname{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*}
Mathematica [A] time = 0.0263777, size = 61, normalized size = 1.42 \[ \frac{1}{16} \left (\frac{\sec ^3(a+b x)}{3 b}+\frac{\sec (a+b x)}{b}+\frac{\log \left (\sin \left (\frac{1}{2} (a+b x)\right )\right )}{b}-\frac{\log \left (\cos \left (\frac{1}{2} (a+b x)\right )\right )}{b}\right ) \]
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
________________________________________________________________________________________
Maple [A] time = 0.035, size = 49, normalized size = 1.1 \begin{align*}{\frac{1}{48\,b \left ( \cos \left ( bx+a \right ) \right ) ^{3}}}+{\frac{1}{16\,b\cos \left ( bx+a \right ) }}+{\frac{\ln \left ( \csc \left ( bx+a \right ) -\cot \left ( bx+a \right ) \right ) }{16\,b}} \end{align*}
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)
[Out]
1/48/b/cos(b*x+a)^3+1/16/b/cos(b*x+a)+1/16/b*ln(csc(b*x+a)-cot(b*x+a))
________________________________________________________________________________________
Maxima [B] time = 1.29424, size = 1332, normalized size = 30.98 \begin{align*} \text{result too large to display} \end{align*}
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)
________________________________________________________________________________________
Fricas [A] time = 0.501539, size = 194, normalized size = 4.51 \begin{align*} -\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{align*}
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)
________________________________________________________________________________________
Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
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
________________________________________________________________________________________
Giac [B] time = 2.82518, size = 2854, normalized size = 66.37 \begin{align*} \text{result too large to display} \end{align*}
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]
-1/48*(4*(18*tan(1/2*b*x + 2*a)^5*tan(1/2*a)^35 - 3*tan(1/2*b*x + 2*a)^4*tan(1/2*a)^36 - 762*tan(1/2*b*x + 2*a
)^5*tan(1/2*a)^33 + 486*tan(1/2*b*x + 2*a)^4*tan(1/2*a)^34 - 72*tan(1/2*b*x + 2*a)^3*tan(1/2*a)^35 + 3*tan(1/2
*b*x + 2*a)^2*tan(1/2*a)^36 + 13644*tan(1/2*b*x + 2*a)^5*tan(1/2*a)^31 - 15561*tan(1/2*b*x + 2*a)^4*tan(1/2*a)
^32 + 5424*tan(1/2*b*x + 2*a)^3*tan(1/2*a)^33 - 756*tan(1/2*b*x + 2*a)^2*tan(1/2*a)^34 + 54*tan(1/2*b*x + 2*a)
*tan(1/2*a)^35 - 2*tan(1/2*a)^36 - 140076*tan(1/2*b*x + 2*a)^5*tan(1/2*a)^29 + 233916*tan(1/2*b*x + 2*a)^4*tan
(1/2*a)^30 - 126936*tan(1/2*b*x + 2*a)^3*tan(1/2*a)^31 + 28701*tan(1/2*b*x + 2*a)^2*tan(1/2*a)^32 - 2934*tan(1
/2*b*x + 2*a)*tan(1/2*a)^33 + 126*tan(1/2*a)^34 + 811140*tan(1/2*b*x + 2*a)^5*tan(1/2*a)^27 - 1893744*tan(1/2*
b*x + 2*a)^4*tan(1/2*a)^28 + 1474776*tan(1/2*b*x + 2*a)^3*tan(1/2*a)^29 - 471816*tan(1/2*b*x + 2*a)^2*tan(1/2*
a)^30 + 64908*tan(1/2*b*x + 2*a)*tan(1/2*a)^31 - 3330*tan(1/2*a)^32 - 1739556*tan(1/2*b*x + 2*a)^5*tan(1/2*a)^
25 + 6839316*tan(1/2*b*x + 2*a)^4*tan(1/2*a)^26 - 8095576*tan(1/2*b*x + 2*a)^3*tan(1/2*a)^27 + 3832164*tan(1/2
*b*x + 2*a)^2*tan(1/2*a)^28 - 727596*tan(1/2*b*x + 2*a)*tan(1/2*a)^29 + 47148*tan(1/2*a)^30 - 461700*tan(1/2*b
*x + 2*a)^5*tan(1/2*a)^23 - 6247800*tan(1/2*b*x + 2*a)^4*tan(1/2*a)^24 + 16873272*tan(1/2*b*x + 2*a)^3*tan(1/2
*a)^25 - 13541976*tan(1/2*b*x + 2*a)^2*tan(1/2*a)^26 + 4049556*tan(1/2*b*x + 2*a)*tan(1/2*a)^27 - 368892*tan(1
/2*a)^28 + 5103972*tan(1/2*b*x + 2*a)^5*tan(1/2*a)^21 - 14529780*tan(1/2*b*x + 2*a)^4*tan(1/2*a)^22 + 3596040*
tan(1/2*b*x + 2*a)^3*tan(1/2*a)^23 + 12515100*tan(1/2*b*x + 2*a)^2*tan(1/2*a)^24 - 8506836*tan(1/2*b*x + 2*a)*
tan(1/2*a)^25 + 1396836*tan(1/2*a)^26 - 3586680*tan(1/2*b*x + 2*a)^5*tan(1/2*a)^19 + 28026162*tan(1/2*b*x + 2*
a)^4*tan(1/2*a)^20 - 51234312*tan(1/2*b*x + 2*a)^3*tan(1/2*a)^21 + 28649880*tan(1/2*b*x + 2*a)^2*tan(1/2*a)^22
- 1942020*tan(1/2*b*x + 2*a)*tan(1/2*a)^23 - 1226940*tan(1/2*a)^24 - 3586680*tan(1/2*b*x + 2*a)^5*tan(1/2*a)^
17 + 37245240*tan(1/2*b*x + 2*a)^3*tan(1/2*a)^19 - 56612142*tan(1/2*b*x + 2*a)^2*tan(1/2*a)^20 + 25560228*tan(
1/2*b*x + 2*a)*tan(1/2*a)^21 - 2935620*tan(1/2*a)^22 + 5103972*tan(1/2*b*x + 2*a)^5*tan(1/2*a)^15 - 28026162*t
an(1/2*b*x + 2*a)^4*tan(1/2*a)^16 + 37245240*tan(1/2*b*x + 2*a)^3*tan(1/2*a)^17 - 18495360*tan(1/2*b*x + 2*a)*
tan(1/2*a)^19 + 5548608*tan(1/2*a)^20 - 461700*tan(1/2*b*x + 2*a)^5*tan(1/2*a)^13 + 14529780*tan(1/2*b*x + 2*a
)^4*tan(1/2*a)^14 - 51234312*tan(1/2*b*x + 2*a)^3*tan(1/2*a)^15 + 56612142*tan(1/2*b*x + 2*a)^2*tan(1/2*a)^16
- 18495360*tan(1/2*b*x + 2*a)*tan(1/2*a)^17 - 1739556*tan(1/2*b*x + 2*a)^5*tan(1/2*a)^11 + 6247800*tan(1/2*b*x
+ 2*a)^4*tan(1/2*a)^12 + 3596040*tan(1/2*b*x + 2*a)^3*tan(1/2*a)^13 - 28649880*tan(1/2*b*x + 2*a)^2*tan(1/2*a
)^14 + 25560228*tan(1/2*b*x + 2*a)*tan(1/2*a)^15 - 5548608*tan(1/2*a)^16 + 811140*tan(1/2*b*x + 2*a)^5*tan(1/2
*a)^9 - 6839316*tan(1/2*b*x + 2*a)^4*tan(1/2*a)^10 + 16873272*tan(1/2*b*x + 2*a)^3*tan(1/2*a)^11 - 12515100*ta
n(1/2*b*x + 2*a)^2*tan(1/2*a)^12 - 1942020*tan(1/2*b*x + 2*a)*tan(1/2*a)^13 + 2935620*tan(1/2*a)^14 - 140076*t
an(1/2*b*x + 2*a)^5*tan(1/2*a)^7 + 1893744*tan(1/2*b*x + 2*a)^4*tan(1/2*a)^8 - 8095576*tan(1/2*b*x + 2*a)^3*ta
n(1/2*a)^9 + 13541976*tan(1/2*b*x + 2*a)^2*tan(1/2*a)^10 - 8506836*tan(1/2*b*x + 2*a)*tan(1/2*a)^11 + 1226940*
tan(1/2*a)^12 + 13644*tan(1/2*b*x + 2*a)^5*tan(1/2*a)^5 - 233916*tan(1/2*b*x + 2*a)^4*tan(1/2*a)^6 + 1474776*t
an(1/2*b*x + 2*a)^3*tan(1/2*a)^7 - 3832164*tan(1/2*b*x + 2*a)^2*tan(1/2*a)^8 + 4049556*tan(1/2*b*x + 2*a)*tan(
1/2*a)^9 - 1396836*tan(1/2*a)^10 - 762*tan(1/2*b*x + 2*a)^5*tan(1/2*a)^3 + 15561*tan(1/2*b*x + 2*a)^4*tan(1/2*
a)^4 - 126936*tan(1/2*b*x + 2*a)^3*tan(1/2*a)^5 + 471816*tan(1/2*b*x + 2*a)^2*tan(1/2*a)^6 - 727596*tan(1/2*b*
x + 2*a)*tan(1/2*a)^7 + 368892*tan(1/2*a)^8 + 18*tan(1/2*b*x + 2*a)^5*tan(1/2*a) - 486*tan(1/2*b*x + 2*a)^4*ta
n(1/2*a)^2 + 5424*tan(1/2*b*x + 2*a)^3*tan(1/2*a)^3 - 28701*tan(1/2*b*x + 2*a)^2*tan(1/2*a)^4 + 64908*tan(1/2*
b*x + 2*a)*tan(1/2*a)^5 - 47148*tan(1/2*a)^6 + 3*tan(1/2*b*x + 2*a)^4 - 72*tan(1/2*b*x + 2*a)^3*tan(1/2*a) + 7
56*tan(1/2*b*x + 2*a)^2*tan(1/2*a)^2 - 2934*tan(1/2*b*x + 2*a)*tan(1/2*a)^3 + 3330*tan(1/2*a)^4 - 3*tan(1/2*b*
x + 2*a)^2 + 54*tan(1/2*b*x + 2*a)*tan(1/2*a) - 126*tan(1/2*a)^2 + 2)/((tan(1/2*a)^18 - 45*tan(1/2*a)^16 + 720
*tan(1/2*a)^14 - 4728*tan(1/2*a)^12 + 10890*tan(1/2*a)^10 - 10890*tan(1/2*a)^8 + 4728*tan(1/2*a)^6 - 720*tan(1
/2*a)^4 + 45*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
)^3) + 3*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) + 3*tan(1/2*a)^2 - 1)) - 3*
log(abs(3*tan(1/2*b*x + 2*a)*tan(1/2*a)^2 - tan(1/2*a)^3 - tan(1/2*b*x + 2*a) + 3*tan(1/2*a))))/b