3.202 \(\int \frac{\csc ^5(c+d x)}{a-b \sin ^4(c+d x)} \, dx\)

Optimal. Leaf size=229 \[ -\frac{b^{5/4} \tan ^{-1}\left (\frac{\sqrt [4]{b} \cos (c+d x)}{\sqrt{\sqrt{a}-\sqrt{b}}}\right )}{2 a^2 d \sqrt{\sqrt{a}-\sqrt{b}}}+\frac{b^{5/4} \tanh ^{-1}\left (\frac{\sqrt [4]{b} \cos (c+d x)}{\sqrt{\sqrt{a}+\sqrt{b}}}\right )}{2 a^2 d \sqrt{\sqrt{a}+\sqrt{b}}}-\frac{(3 a+8 b) \tanh ^{-1}(\cos (c+d x))}{8 a^2 d}-\frac{3}{16 a d (1-\cos (c+d x))}+\frac{3}{16 a d (\cos (c+d x)+1)}-\frac{1}{16 a d (1-\cos (c+d x))^2}+\frac{1}{16 a d (\cos (c+d x)+1)^2} \]

[Out]

-(b^(5/4)*ArcTan[(b^(1/4)*Cos[c + d*x])/Sqrt[Sqrt[a] - Sqrt[b]]])/(2*a^2*Sqrt[Sqrt[a] - Sqrt[b]]*d) - ((3*a +
8*b)*ArcTanh[Cos[c + d*x]])/(8*a^2*d) + (b^(5/4)*ArcTanh[(b^(1/4)*Cos[c + d*x])/Sqrt[Sqrt[a] + Sqrt[b]]])/(2*a
^2*Sqrt[Sqrt[a] + Sqrt[b]]*d) - 1/(16*a*d*(1 - Cos[c + d*x])^2) - 3/(16*a*d*(1 - Cos[c + d*x])) + 1/(16*a*d*(1
+ Cos[c + d*x])^2) + 3/(16*a*d*(1 + Cos[c + d*x]))

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Rubi [A]  time = 0.246567, antiderivative size = 229, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {3215, 1170, 207, 1166, 205, 208} \[ -\frac{b^{5/4} \tan ^{-1}\left (\frac{\sqrt [4]{b} \cos (c+d x)}{\sqrt{\sqrt{a}-\sqrt{b}}}\right )}{2 a^2 d \sqrt{\sqrt{a}-\sqrt{b}}}+\frac{b^{5/4} \tanh ^{-1}\left (\frac{\sqrt [4]{b} \cos (c+d x)}{\sqrt{\sqrt{a}+\sqrt{b}}}\right )}{2 a^2 d \sqrt{\sqrt{a}+\sqrt{b}}}-\frac{(3 a+8 b) \tanh ^{-1}(\cos (c+d x))}{8 a^2 d}-\frac{3}{16 a d (1-\cos (c+d x))}+\frac{3}{16 a d (\cos (c+d x)+1)}-\frac{1}{16 a d (1-\cos (c+d x))^2}+\frac{1}{16 a d (\cos (c+d x)+1)^2} \]

Antiderivative was successfully verified.

[In]

Int[Csc[c + d*x]^5/(a - b*Sin[c + d*x]^4),x]

[Out]

-(b^(5/4)*ArcTan[(b^(1/4)*Cos[c + d*x])/Sqrt[Sqrt[a] - Sqrt[b]]])/(2*a^2*Sqrt[Sqrt[a] - Sqrt[b]]*d) - ((3*a +
8*b)*ArcTanh[Cos[c + d*x]])/(8*a^2*d) + (b^(5/4)*ArcTanh[(b^(1/4)*Cos[c + d*x])/Sqrt[Sqrt[a] + Sqrt[b]]])/(2*a
^2*Sqrt[Sqrt[a] + Sqrt[b]]*d) - 1/(16*a*d*(1 - Cos[c + d*x])^2) - 3/(16*a*d*(1 - Cos[c + d*x])) + 1/(16*a*d*(1
+ Cos[c + d*x])^2) + 3/(16*a*d*(1 + Cos[c + d*x]))

Rule 3215

Int[sin[(e_.) + (f_.)*(x_)]^(m_.)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]^4)^(p_.), x_Symbol] :> With[{ff = Free
Factors[Cos[e + f*x], x]}, -Dist[ff/f, Subst[Int[(1 - ff^2*x^2)^((m - 1)/2)*(a + b - 2*b*ff^2*x^2 + b*ff^4*x^4
)^p, x], x, Cos[e + f*x]/ff], x]] /; FreeQ[{a, b, e, f, p}, x] && IntegerQ[(m - 1)/2]

Rule 1170

Int[((d_) + (e_.)*(x_)^2)^(q_)/((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4), x_Symbol] :> Int[ExpandIntegrand[(d + e*x
^2)^q/(a + b*x^2 + c*x^4), x], x] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a
*e^2, 0] && IntegerQ[q]

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 1166

Int[((d_) + (e_.)*(x_)^2)/((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[b^2 - 4*a*c, 2]}, Di
st[e/2 + (2*c*d - b*e)/(2*q), Int[1/(b/2 - q/2 + c*x^2), x], x] + Dist[e/2 - (2*c*d - b*e)/(2*q), Int[1/(b/2 +
q/2 + c*x^2), x], x]] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - a*e^2, 0] && PosQ[b^
2 - 4*a*c]

Rule 205

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[a/b, 2]*ArcTan[x/Rt[a/b, 2]])/a, x] /; FreeQ[{a, b}, x]
&& PosQ[a/b]

Rule 208

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-(a/b), 2]*ArcTanh[x/Rt[-(a/b), 2]])/a, x] /; FreeQ[{a,
b}, x] && NegQ[a/b]

Rubi steps

\begin{align*} \int \frac{\csc ^5(c+d x)}{a-b \sin ^4(c+d x)} \, dx &=-\frac{\operatorname{Subst}\left (\int \frac{1}{\left (1-x^2\right )^3 \left (a-b+2 b x^2-b x^4\right )} \, dx,x,\cos (c+d x)\right )}{d}\\ &=-\frac{\operatorname{Subst}\left (\int \left (-\frac{1}{8 a (-1+x)^3}+\frac{3}{16 a (-1+x)^2}+\frac{1}{8 a (1+x)^3}+\frac{3}{16 a (1+x)^2}+\frac{-3 a-8 b}{8 a^2 \left (-1+x^2\right )}-\frac{b^2 \left (-1+x^2\right )}{a^2 \left (a-b+2 b x^2-b x^4\right )}\right ) \, dx,x,\cos (c+d x)\right )}{d}\\ &=-\frac{1}{16 a d (1-\cos (c+d x))^2}-\frac{3}{16 a d (1-\cos (c+d x))}+\frac{1}{16 a d (1+\cos (c+d x))^2}+\frac{3}{16 a d (1+\cos (c+d x))}+\frac{b^2 \operatorname{Subst}\left (\int \frac{-1+x^2}{a-b+2 b x^2-b x^4} \, dx,x,\cos (c+d x)\right )}{a^2 d}+\frac{(3 a+8 b) \operatorname{Subst}\left (\int \frac{1}{-1+x^2} \, dx,x,\cos (c+d x)\right )}{8 a^2 d}\\ &=-\frac{(3 a+8 b) \tanh ^{-1}(\cos (c+d x))}{8 a^2 d}-\frac{1}{16 a d (1-\cos (c+d x))^2}-\frac{3}{16 a d (1-\cos (c+d x))}+\frac{1}{16 a d (1+\cos (c+d x))^2}+\frac{3}{16 a d (1+\cos (c+d x))}+\frac{b^2 \operatorname{Subst}\left (\int \frac{1}{-\sqrt{a} \sqrt{b}+b-b x^2} \, dx,x,\cos (c+d x)\right )}{2 a^2 d}+\frac{b^2 \operatorname{Subst}\left (\int \frac{1}{\sqrt{a} \sqrt{b}+b-b x^2} \, dx,x,\cos (c+d x)\right )}{2 a^2 d}\\ &=-\frac{b^{5/4} \tan ^{-1}\left (\frac{\sqrt [4]{b} \cos (c+d x)}{\sqrt{\sqrt{a}-\sqrt{b}}}\right )}{2 a^2 \sqrt{\sqrt{a}-\sqrt{b}} d}-\frac{(3 a+8 b) \tanh ^{-1}(\cos (c+d x))}{8 a^2 d}+\frac{b^{5/4} \tanh ^{-1}\left (\frac{\sqrt [4]{b} \cos (c+d x)}{\sqrt{\sqrt{a}+\sqrt{b}}}\right )}{2 a^2 \sqrt{\sqrt{a}+\sqrt{b}} d}-\frac{1}{16 a d (1-\cos (c+d x))^2}-\frac{3}{16 a d (1-\cos (c+d x))}+\frac{1}{16 a d (1+\cos (c+d x))^2}+\frac{3}{16 a d (1+\cos (c+d x))}\\ \end{align*}

Mathematica [C]  time = 1.12617, size = 409, normalized size = 1.79 \[ \frac{-8 i b^2 \text{RootSum}\left [-16 \text{$\#$1}^4 a+\text{$\#$1}^8 b-4 \text{$\#$1}^6 b+6 \text{$\#$1}^4 b-4 \text{$\#$1}^2 b+b\& ,\frac{-i \text{$\#$1}^6 \log \left (\text{$\#$1}^2-2 \text{$\#$1} \cos (c+d x)+1\right )+3 i \text{$\#$1}^4 \log \left (\text{$\#$1}^2-2 \text{$\#$1} \cos (c+d x)+1\right )-3 i \text{$\#$1}^2 \log \left (\text{$\#$1}^2-2 \text{$\#$1} \cos (c+d x)+1\right )+i \log \left (\text{$\#$1}^2-2 \text{$\#$1} \cos (c+d x)+1\right )+2 \text{$\#$1}^6 \tan ^{-1}\left (\frac{\sin (c+d x)}{\cos (c+d x)-\text{$\#$1}}\right )-6 \text{$\#$1}^4 \tan ^{-1}\left (\frac{\sin (c+d x)}{\cos (c+d x)-\text{$\#$1}}\right )+6 \text{$\#$1}^2 \tan ^{-1}\left (\frac{\sin (c+d x)}{\cos (c+d x)-\text{$\#$1}}\right )-2 \tan ^{-1}\left (\frac{\sin (c+d x)}{\cos (c+d x)-\text{$\#$1}}\right )}{-8 \text{$\#$1}^3 a+\text{$\#$1}^7 b-3 \text{$\#$1}^5 b+3 \text{$\#$1}^3 b-\text{$\#$1} b}\& \right ]-a \csc ^4\left (\frac{1}{2} (c+d x)\right )-6 a \csc ^2\left (\frac{1}{2} (c+d x)\right )+a \sec ^4\left (\frac{1}{2} (c+d x)\right )+6 a \sec ^2\left (\frac{1}{2} (c+d x)\right )+24 a \log \left (\sin \left (\frac{1}{2} (c+d x)\right )\right )-24 a \log \left (\cos \left (\frac{1}{2} (c+d x)\right )\right )+64 b \log \left (\sin \left (\frac{1}{2} (c+d x)\right )\right )-64 b \log \left (\cos \left (\frac{1}{2} (c+d x)\right )\right )}{64 a^2 d} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[Csc[c + d*x]^5/(a - b*Sin[c + d*x]^4),x]

[Out]

(-6*a*Csc[(c + d*x)/2]^2 - a*Csc[(c + d*x)/2]^4 - 24*a*Log[Cos[(c + d*x)/2]] - 64*b*Log[Cos[(c + d*x)/2]] + 24
*a*Log[Sin[(c + d*x)/2]] + 64*b*Log[Sin[(c + d*x)/2]] - (8*I)*b^2*RootSum[b - 4*b*#1^2 - 16*a*#1^4 + 6*b*#1^4
- 4*b*#1^6 + b*#1^8 & , (-2*ArcTan[Sin[c + d*x]/(Cos[c + d*x] - #1)] + I*Log[1 - 2*Cos[c + d*x]*#1 + #1^2] + 6
*ArcTan[Sin[c + d*x]/(Cos[c + d*x] - #1)]*#1^2 - (3*I)*Log[1 - 2*Cos[c + d*x]*#1 + #1^2]*#1^2 - 6*ArcTan[Sin[c
+ d*x]/(Cos[c + d*x] - #1)]*#1^4 + (3*I)*Log[1 - 2*Cos[c + d*x]*#1 + #1^2]*#1^4 + 2*ArcTan[Sin[c + d*x]/(Cos[
c + d*x] - #1)]*#1^6 - I*Log[1 - 2*Cos[c + d*x]*#1 + #1^2]*#1^6)/(-(b*#1) - 8*a*#1^3 + 3*b*#1^3 - 3*b*#1^5 + b
*#1^7) & ] + 6*a*Sec[(c + d*x)/2]^2 + a*Sec[(c + d*x)/2]^4)/(64*a^2*d)

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Maple [A]  time = 0.142, size = 232, normalized size = 1. \begin{align*} -{\frac{1}{16\,da \left ( -1+\cos \left ( dx+c \right ) \right ) ^{2}}}+{\frac{3}{16\,da \left ( -1+\cos \left ( dx+c \right ) \right ) }}+{\frac{3\,\ln \left ( -1+\cos \left ( dx+c \right ) \right ) }{16\,da}}+{\frac{\ln \left ( -1+\cos \left ( dx+c \right ) \right ) b}{2\,{a}^{2}d}}+{\frac{1}{16\,da \left ( 1+\cos \left ( dx+c \right ) \right ) ^{2}}}+{\frac{3}{16\,da \left ( 1+\cos \left ( dx+c \right ) \right ) }}-{\frac{3\,\ln \left ( 1+\cos \left ( dx+c \right ) \right ) }{16\,da}}-{\frac{\ln \left ( 1+\cos \left ( dx+c \right ) \right ) b}{2\,{a}^{2}d}}-{\frac{{b}^{2}}{2\,{a}^{2}d}\arctan \left ({b\cos \left ( dx+c \right ){\frac{1}{\sqrt{ \left ( \sqrt{ab}-b \right ) b}}}} \right ){\frac{1}{\sqrt{ \left ( \sqrt{ab}-b \right ) b}}}}+{\frac{{b}^{2}}{2\,{a}^{2}d}{\it Artanh} \left ({b\cos \left ( dx+c \right ){\frac{1}{\sqrt{ \left ( \sqrt{ab}+b \right ) b}}}} \right ){\frac{1}{\sqrt{ \left ( \sqrt{ab}+b \right ) b}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(csc(d*x+c)^5/(a-b*sin(d*x+c)^4),x)

[Out]

-1/16/d/a/(-1+cos(d*x+c))^2+3/16/d/a/(-1+cos(d*x+c))+3/16/d/a*ln(-1+cos(d*x+c))+1/2/d/a^2*ln(-1+cos(d*x+c))*b+
1/16/a/d/(1+cos(d*x+c))^2+3/16/a/d/(1+cos(d*x+c))-3/16/d/a*ln(1+cos(d*x+c))-1/2/d/a^2*ln(1+cos(d*x+c))*b-1/2/d
/a^2*b^2/(((a*b)^(1/2)-b)*b)^(1/2)*arctan(cos(d*x+c)*b/(((a*b)^(1/2)-b)*b)^(1/2))+1/2/d/a^2*b^2/(((a*b)^(1/2)+
b)*b)^(1/2)*arctanh(cos(d*x+c)*b/(((a*b)^(1/2)+b)*b)^(1/2))

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csc(d*x+c)^5/(a-b*sin(d*x+c)^4),x, algorithm="maxima")

[Out]

-1/16*(48*a*cos(2*d*x + 2*c)*cos(d*x + c) - 176*a*sin(3*d*x + 3*c)*sin(2*d*x + 2*c) + 48*a*sin(2*d*x + 2*c)*si
n(d*x + c) - 4*(3*a*cos(7*d*x + 7*c) - 11*a*cos(5*d*x + 5*c) - 11*a*cos(3*d*x + 3*c) + 3*a*cos(d*x + c))*cos(8
*d*x + 8*c) + 12*(4*a*cos(6*d*x + 6*c) - 6*a*cos(4*d*x + 4*c) + 4*a*cos(2*d*x + 2*c) - a)*cos(7*d*x + 7*c) - 1
6*(11*a*cos(5*d*x + 5*c) + 11*a*cos(3*d*x + 3*c) - 3*a*cos(d*x + c))*cos(6*d*x + 6*c) + 44*(6*a*cos(4*d*x + 4*
c) - 4*a*cos(2*d*x + 2*c) + a)*cos(5*d*x + 5*c) + 24*(11*a*cos(3*d*x + 3*c) - 3*a*cos(d*x + c))*cos(4*d*x + 4*
c) - 44*(4*a*cos(2*d*x + 2*c) - a)*cos(3*d*x + 3*c) - 12*a*cos(d*x + c) + 16*(a^2*d*cos(8*d*x + 8*c)^2 + 16*a^
2*d*cos(6*d*x + 6*c)^2 + 36*a^2*d*cos(4*d*x + 4*c)^2 + 16*a^2*d*cos(2*d*x + 2*c)^2 + a^2*d*sin(8*d*x + 8*c)^2
+ 16*a^2*d*sin(6*d*x + 6*c)^2 + 36*a^2*d*sin(4*d*x + 4*c)^2 - 48*a^2*d*sin(4*d*x + 4*c)*sin(2*d*x + 2*c) + 16*
a^2*d*sin(2*d*x + 2*c)^2 - 8*a^2*d*cos(2*d*x + 2*c) + a^2*d - 2*(4*a^2*d*cos(6*d*x + 6*c) - 6*a^2*d*cos(4*d*x
+ 4*c) + 4*a^2*d*cos(2*d*x + 2*c) - a^2*d)*cos(8*d*x + 8*c) - 8*(6*a^2*d*cos(4*d*x + 4*c) - 4*a^2*d*cos(2*d*x
+ 2*c) + a^2*d)*cos(6*d*x + 6*c) - 12*(4*a^2*d*cos(2*d*x + 2*c) - a^2*d)*cos(4*d*x + 4*c) - 4*(2*a^2*d*sin(6*d
*x + 6*c) - 3*a^2*d*sin(4*d*x + 4*c) + 2*a^2*d*sin(2*d*x + 2*c))*sin(8*d*x + 8*c) - 16*(3*a^2*d*sin(4*d*x + 4*
c) - 2*a^2*d*sin(2*d*x + 2*c))*sin(6*d*x + 6*c))*integrate(-2*(12*b^3*cos(3*d*x + 3*c)*sin(2*d*x + 2*c) - 4*b^
3*cos(d*x + c)*sin(2*d*x + 2*c) + 4*b^3*cos(2*d*x + 2*c)*sin(d*x + c) - b^3*sin(d*x + c) + (b^3*sin(7*d*x + 7*
c) - 3*b^3*sin(5*d*x + 5*c) + 3*b^3*sin(3*d*x + 3*c) - b^3*sin(d*x + c))*cos(8*d*x + 8*c) + 2*(2*b^3*sin(6*d*x
+ 6*c) + 2*b^3*sin(2*d*x + 2*c) + (8*a*b^2 - 3*b^3)*sin(4*d*x + 4*c))*cos(7*d*x + 7*c) + 4*(3*b^3*sin(5*d*x +
5*c) - 3*b^3*sin(3*d*x + 3*c) + b^3*sin(d*x + c))*cos(6*d*x + 6*c) - 6*(2*b^3*sin(2*d*x + 2*c) + (8*a*b^2 - 3
*b^3)*sin(4*d*x + 4*c))*cos(5*d*x + 5*c) - 2*(3*(8*a*b^2 - 3*b^3)*sin(3*d*x + 3*c) - (8*a*b^2 - 3*b^3)*sin(d*x
+ c))*cos(4*d*x + 4*c) - (b^3*cos(7*d*x + 7*c) - 3*b^3*cos(5*d*x + 5*c) + 3*b^3*cos(3*d*x + 3*c) - b^3*cos(d*
x + c))*sin(8*d*x + 8*c) - (4*b^3*cos(6*d*x + 6*c) + 4*b^3*cos(2*d*x + 2*c) - b^3 + 2*(8*a*b^2 - 3*b^3)*cos(4*
d*x + 4*c))*sin(7*d*x + 7*c) - 4*(3*b^3*cos(5*d*x + 5*c) - 3*b^3*cos(3*d*x + 3*c) + b^3*cos(d*x + c))*sin(6*d*
x + 6*c) + 3*(4*b^3*cos(2*d*x + 2*c) - b^3 + 2*(8*a*b^2 - 3*b^3)*cos(4*d*x + 4*c))*sin(5*d*x + 5*c) + 2*(3*(8*
a*b^2 - 3*b^3)*cos(3*d*x + 3*c) - (8*a*b^2 - 3*b^3)*cos(d*x + c))*sin(4*d*x + 4*c) - 3*(4*b^3*cos(2*d*x + 2*c)
- b^3)*sin(3*d*x + 3*c))/(a^2*b^2*cos(8*d*x + 8*c)^2 + 16*a^2*b^2*cos(6*d*x + 6*c)^2 + 16*a^2*b^2*cos(2*d*x +
2*c)^2 + a^2*b^2*sin(8*d*x + 8*c)^2 + 16*a^2*b^2*sin(6*d*x + 6*c)^2 + 16*a^2*b^2*sin(2*d*x + 2*c)^2 - 8*a^2*b
^2*cos(2*d*x + 2*c) + a^2*b^2 + 4*(64*a^4 - 48*a^3*b + 9*a^2*b^2)*cos(4*d*x + 4*c)^2 + 4*(64*a^4 - 48*a^3*b +
9*a^2*b^2)*sin(4*d*x + 4*c)^2 + 16*(8*a^3*b - 3*a^2*b^2)*sin(4*d*x + 4*c)*sin(2*d*x + 2*c) - 2*(4*a^2*b^2*cos(
6*d*x + 6*c) + 4*a^2*b^2*cos(2*d*x + 2*c) - a^2*b^2 + 2*(8*a^3*b - 3*a^2*b^2)*cos(4*d*x + 4*c))*cos(8*d*x + 8*
c) + 8*(4*a^2*b^2*cos(2*d*x + 2*c) - a^2*b^2 + 2*(8*a^3*b - 3*a^2*b^2)*cos(4*d*x + 4*c))*cos(6*d*x + 6*c) - 4*
(8*a^3*b - 3*a^2*b^2 - 4*(8*a^3*b - 3*a^2*b^2)*cos(2*d*x + 2*c))*cos(4*d*x + 4*c) - 4*(2*a^2*b^2*sin(6*d*x + 6
*c) + 2*a^2*b^2*sin(2*d*x + 2*c) + (8*a^3*b - 3*a^2*b^2)*sin(4*d*x + 4*c))*sin(8*d*x + 8*c) + 16*(2*a^2*b^2*si
n(2*d*x + 2*c) + (8*a^3*b - 3*a^2*b^2)*sin(4*d*x + 4*c))*sin(6*d*x + 6*c)), x) + ((3*a + 8*b)*cos(8*d*x + 8*c)
^2 + 16*(3*a + 8*b)*cos(6*d*x + 6*c)^2 + 36*(3*a + 8*b)*cos(4*d*x + 4*c)^2 + 16*(3*a + 8*b)*cos(2*d*x + 2*c)^2
+ (3*a + 8*b)*sin(8*d*x + 8*c)^2 + 16*(3*a + 8*b)*sin(6*d*x + 6*c)^2 + 36*(3*a + 8*b)*sin(4*d*x + 4*c)^2 - 48
*(3*a + 8*b)*sin(4*d*x + 4*c)*sin(2*d*x + 2*c) + 16*(3*a + 8*b)*sin(2*d*x + 2*c)^2 - 2*(4*(3*a + 8*b)*cos(6*d*
x + 6*c) - 6*(3*a + 8*b)*cos(4*d*x + 4*c) + 4*(3*a + 8*b)*cos(2*d*x + 2*c) - 3*a - 8*b)*cos(8*d*x + 8*c) - 8*(
6*(3*a + 8*b)*cos(4*d*x + 4*c) - 4*(3*a + 8*b)*cos(2*d*x + 2*c) + 3*a + 8*b)*cos(6*d*x + 6*c) - 12*(4*(3*a + 8
*b)*cos(2*d*x + 2*c) - 3*a - 8*b)*cos(4*d*x + 4*c) - 8*(3*a + 8*b)*cos(2*d*x + 2*c) - 4*(2*(3*a + 8*b)*sin(6*d
*x + 6*c) - 3*(3*a + 8*b)*sin(4*d*x + 4*c) + 2*(3*a + 8*b)*sin(2*d*x + 2*c))*sin(8*d*x + 8*c) - 16*(3*(3*a + 8
*b)*sin(4*d*x + 4*c) - 2*(3*a + 8*b)*sin(2*d*x + 2*c))*sin(6*d*x + 6*c) + 3*a + 8*b)*log(cos(d*x)^2 + 2*cos(d*
x)*cos(c) + cos(c)^2 + sin(d*x)^2 - 2*sin(d*x)*sin(c) + sin(c)^2) - ((3*a + 8*b)*cos(8*d*x + 8*c)^2 + 16*(3*a
+ 8*b)*cos(6*d*x + 6*c)^2 + 36*(3*a + 8*b)*cos(4*d*x + 4*c)^2 + 16*(3*a + 8*b)*cos(2*d*x + 2*c)^2 + (3*a + 8*b
)*sin(8*d*x + 8*c)^2 + 16*(3*a + 8*b)*sin(6*d*x + 6*c)^2 + 36*(3*a + 8*b)*sin(4*d*x + 4*c)^2 - 48*(3*a + 8*b)*
sin(4*d*x + 4*c)*sin(2*d*x + 2*c) + 16*(3*a + 8*b)*sin(2*d*x + 2*c)^2 - 2*(4*(3*a + 8*b)*cos(6*d*x + 6*c) - 6*
(3*a + 8*b)*cos(4*d*x + 4*c) + 4*(3*a + 8*b)*cos(2*d*x + 2*c) - 3*a - 8*b)*cos(8*d*x + 8*c) - 8*(6*(3*a + 8*b)
*cos(4*d*x + 4*c) - 4*(3*a + 8*b)*cos(2*d*x + 2*c) + 3*a + 8*b)*cos(6*d*x + 6*c) - 12*(4*(3*a + 8*b)*cos(2*d*x
+ 2*c) - 3*a - 8*b)*cos(4*d*x + 4*c) - 8*(3*a + 8*b)*cos(2*d*x + 2*c) - 4*(2*(3*a + 8*b)*sin(6*d*x + 6*c) - 3
*(3*a + 8*b)*sin(4*d*x + 4*c) + 2*(3*a + 8*b)*sin(2*d*x + 2*c))*sin(8*d*x + 8*c) - 16*(3*(3*a + 8*b)*sin(4*d*x
+ 4*c) - 2*(3*a + 8*b)*sin(2*d*x + 2*c))*sin(6*d*x + 6*c) + 3*a + 8*b)*log(cos(d*x)^2 - 2*cos(d*x)*cos(c) + c
os(c)^2 + sin(d*x)^2 + 2*sin(d*x)*sin(c) + sin(c)^2) - 4*(3*a*sin(7*d*x + 7*c) - 11*a*sin(5*d*x + 5*c) - 11*a*
sin(3*d*x + 3*c) + 3*a*sin(d*x + c))*sin(8*d*x + 8*c) + 24*(2*a*sin(6*d*x + 6*c) - 3*a*sin(4*d*x + 4*c) + 2*a*
sin(2*d*x + 2*c))*sin(7*d*x + 7*c) - 16*(11*a*sin(5*d*x + 5*c) + 11*a*sin(3*d*x + 3*c) - 3*a*sin(d*x + c))*sin
(6*d*x + 6*c) + 88*(3*a*sin(4*d*x + 4*c) - 2*a*sin(2*d*x + 2*c))*sin(5*d*x + 5*c) + 24*(11*a*sin(3*d*x + 3*c)
- 3*a*sin(d*x + c))*sin(4*d*x + 4*c))/(a^2*d*cos(8*d*x + 8*c)^2 + 16*a^2*d*cos(6*d*x + 6*c)^2 + 36*a^2*d*cos(4
*d*x + 4*c)^2 + 16*a^2*d*cos(2*d*x + 2*c)^2 + a^2*d*sin(8*d*x + 8*c)^2 + 16*a^2*d*sin(6*d*x + 6*c)^2 + 36*a^2*
d*sin(4*d*x + 4*c)^2 - 48*a^2*d*sin(4*d*x + 4*c)*sin(2*d*x + 2*c) + 16*a^2*d*sin(2*d*x + 2*c)^2 - 8*a^2*d*cos(
2*d*x + 2*c) + a^2*d - 2*(4*a^2*d*cos(6*d*x + 6*c) - 6*a^2*d*cos(4*d*x + 4*c) + 4*a^2*d*cos(2*d*x + 2*c) - a^2
*d)*cos(8*d*x + 8*c) - 8*(6*a^2*d*cos(4*d*x + 4*c) - 4*a^2*d*cos(2*d*x + 2*c) + a^2*d)*cos(6*d*x + 6*c) - 12*(
4*a^2*d*cos(2*d*x + 2*c) - a^2*d)*cos(4*d*x + 4*c) - 4*(2*a^2*d*sin(6*d*x + 6*c) - 3*a^2*d*sin(4*d*x + 4*c) +
2*a^2*d*sin(2*d*x + 2*c))*sin(8*d*x + 8*c) - 16*(3*a^2*d*sin(4*d*x + 4*c) - 2*a^2*d*sin(2*d*x + 2*c))*sin(6*d*
x + 6*c))

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Fricas [B]  time = 4.91023, size = 2292, normalized size = 10.01 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csc(d*x+c)^5/(a-b*sin(d*x+c)^4),x, algorithm="fricas")

[Out]

1/16*(6*a*cos(d*x + c)^3 + 4*(a^2*d*cos(d*x + c)^4 - 2*a^2*d*cos(d*x + c)^2 + a^2*d)*sqrt(-((a^5 - a^4*b)*d^2*
sqrt(b^5/((a^9 - 2*a^8*b + a^7*b^2)*d^4)) + b^3)/((a^5 - a^4*b)*d^2))*log(b^4*cos(d*x + c) + (a^2*b^3*d - (a^7
- a^6*b)*d^3*sqrt(b^5/((a^9 - 2*a^8*b + a^7*b^2)*d^4)))*sqrt(-((a^5 - a^4*b)*d^2*sqrt(b^5/((a^9 - 2*a^8*b + a
^7*b^2)*d^4)) + b^3)/((a^5 - a^4*b)*d^2))) - 4*(a^2*d*cos(d*x + c)^4 - 2*a^2*d*cos(d*x + c)^2 + a^2*d)*sqrt(((
a^5 - a^4*b)*d^2*sqrt(b^5/((a^9 - 2*a^8*b + a^7*b^2)*d^4)) - b^3)/((a^5 - a^4*b)*d^2))*log(b^4*cos(d*x + c) -
(a^2*b^3*d + (a^7 - a^6*b)*d^3*sqrt(b^5/((a^9 - 2*a^8*b + a^7*b^2)*d^4)))*sqrt(((a^5 - a^4*b)*d^2*sqrt(b^5/((a
^9 - 2*a^8*b + a^7*b^2)*d^4)) - b^3)/((a^5 - a^4*b)*d^2))) - 4*(a^2*d*cos(d*x + c)^4 - 2*a^2*d*cos(d*x + c)^2
+ a^2*d)*sqrt(-((a^5 - a^4*b)*d^2*sqrt(b^5/((a^9 - 2*a^8*b + a^7*b^2)*d^4)) + b^3)/((a^5 - a^4*b)*d^2))*log(-b
^4*cos(d*x + c) + (a^2*b^3*d - (a^7 - a^6*b)*d^3*sqrt(b^5/((a^9 - 2*a^8*b + a^7*b^2)*d^4)))*sqrt(-((a^5 - a^4*
b)*d^2*sqrt(b^5/((a^9 - 2*a^8*b + a^7*b^2)*d^4)) + b^3)/((a^5 - a^4*b)*d^2))) + 4*(a^2*d*cos(d*x + c)^4 - 2*a^
2*d*cos(d*x + c)^2 + a^2*d)*sqrt(((a^5 - a^4*b)*d^2*sqrt(b^5/((a^9 - 2*a^8*b + a^7*b^2)*d^4)) - b^3)/((a^5 - a
^4*b)*d^2))*log(-b^4*cos(d*x + c) - (a^2*b^3*d + (a^7 - a^6*b)*d^3*sqrt(b^5/((a^9 - 2*a^8*b + a^7*b^2)*d^4)))*
sqrt(((a^5 - a^4*b)*d^2*sqrt(b^5/((a^9 - 2*a^8*b + a^7*b^2)*d^4)) - b^3)/((a^5 - a^4*b)*d^2))) - 10*a*cos(d*x
+ c) - ((3*a + 8*b)*cos(d*x + c)^4 - 2*(3*a + 8*b)*cos(d*x + c)^2 + 3*a + 8*b)*log(1/2*cos(d*x + c) + 1/2) + (
(3*a + 8*b)*cos(d*x + c)^4 - 2*(3*a + 8*b)*cos(d*x + c)^2 + 3*a + 8*b)*log(-1/2*cos(d*x + c) + 1/2))/(a^2*d*co
s(d*x + c)^4 - 2*a^2*d*cos(d*x + c)^2 + a^2*d)

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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(d*x+c)**5/(a-b*sin(d*x+c)**4),x)

[Out]

Timed out

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Giac [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: NotImplementedError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csc(d*x+c)^5/(a-b*sin(d*x+c)^4),x, algorithm="giac")

[Out]

Exception raised: NotImplementedError