3.214 \(\int \frac{\sin ^5(c+d x)}{(a-b \sin ^4(c+d x))^2} \, dx\)

Optimal. Leaf size=217 \[ \frac{\left (\sqrt{a}-2 \sqrt{b}\right ) \tan ^{-1}\left (\frac{\sqrt [4]{b} \cos (c+d x)}{\sqrt{\sqrt{a}-\sqrt{b}}}\right )}{8 \sqrt{a} b^{5/4} d \left (\sqrt{a}-\sqrt{b}\right )^{3/2}}+\frac{\left (\sqrt{a}+2 \sqrt{b}\right ) \tanh ^{-1}\left (\frac{\sqrt [4]{b} \cos (c+d x)}{\sqrt{\sqrt{a}+\sqrt{b}}}\right )}{8 \sqrt{a} b^{5/4} d \left (\sqrt{a}+\sqrt{b}\right )^{3/2}}-\frac{\cos (c+d x) \left (a-b \cos ^2(c+d x)+b\right )}{4 b d (a-b) \left (a-b \cos ^4(c+d x)+2 b \cos ^2(c+d x)-b\right )} \]

[Out]

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

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Rubi [A]  time = 0.264157, antiderivative size = 217, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.208, Rules used = {3215, 1205, 1166, 205, 208} \[ \frac{\left (\sqrt{a}-2 \sqrt{b}\right ) \tan ^{-1}\left (\frac{\sqrt [4]{b} \cos (c+d x)}{\sqrt{\sqrt{a}-\sqrt{b}}}\right )}{8 \sqrt{a} b^{5/4} d \left (\sqrt{a}-\sqrt{b}\right )^{3/2}}+\frac{\left (\sqrt{a}+2 \sqrt{b}\right ) \tanh ^{-1}\left (\frac{\sqrt [4]{b} \cos (c+d x)}{\sqrt{\sqrt{a}+\sqrt{b}}}\right )}{8 \sqrt{a} b^{5/4} d \left (\sqrt{a}+\sqrt{b}\right )^{3/2}}-\frac{\cos (c+d x) \left (a-b \cos ^2(c+d x)+b\right )}{4 b d (a-b) \left (a-b \cos ^4(c+d x)+2 b \cos ^2(c+d x)-b\right )} \]

Antiderivative was successfully verified.

[In]

Int[Sin[c + d*x]^5/(a - b*Sin[c + d*x]^4)^2,x]

[Out]

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

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 1205

Int[((d_) + (e_.)*(x_)^2)^(q_)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_), x_Symbol] :> With[{f = Coeff[Polynom
ialRemainder[(d + e*x^2)^q, a + b*x^2 + c*x^4, x], x, 0], g = Coeff[PolynomialRemainder[(d + e*x^2)^q, a + b*x
^2 + c*x^4, x], x, 2]}, Simp[(x*(a + b*x^2 + c*x^4)^(p + 1)*(a*b*g - f*(b^2 - 2*a*c) - c*(b*f - 2*a*g)*x^2))/(
2*a*(p + 1)*(b^2 - 4*a*c)), x] + Dist[1/(2*a*(p + 1)*(b^2 - 4*a*c)), Int[(a + b*x^2 + c*x^4)^(p + 1)*ExpandToS
um[2*a*(p + 1)*(b^2 - 4*a*c)*PolynomialQuotient[(d + e*x^2)^q, a + b*x^2 + c*x^4, x] + b^2*f*(2*p + 3) - 2*a*c
*f*(4*p + 5) - a*b*g + c*(4*p + 7)*(b*f - 2*a*g)*x^2, x], 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] && IGtQ[q, 1] && LtQ[p, -1]

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

Mathematica [C]  time = 0.629844, size = 469, normalized size = 2.16 \[ -\frac{\frac{32 \cos (c+d x) (2 a-b \cos (2 (c+d x))+b)}{8 a+4 b \cos (2 (c+d x))-b \cos (4 (c+d x))-3 b}+i \text{RootSum}\left [-16 \text{$\#$1}^4 a+\text{$\#$1}^8 b-4 \text{$\#$1}^6 b+6 \text{$\#$1}^4 b-4 \text{$\#$1}^2 b+b\& ,\frac{-4 i \text{$\#$1}^4 a \log \left (\text{$\#$1}^2-2 \text{$\#$1} \cos (c+d x)+1\right )+4 i \text{$\#$1}^2 a \log \left (\text{$\#$1}^2-2 \text{$\#$1} \cos (c+d x)+1\right )+8 \text{$\#$1}^4 a \tan ^{-1}\left (\frac{\sin (c+d x)}{\cos (c+d x)-\text{$\#$1}}\right )-8 \text{$\#$1}^2 a \tan ^{-1}\left (\frac{\sin (c+d x)}{\cos (c+d x)-\text{$\#$1}}\right )-i \text{$\#$1}^6 b \log \left (\text{$\#$1}^2-2 \text{$\#$1} \cos (c+d x)+1\right )+11 i \text{$\#$1}^4 b \log \left (\text{$\#$1}^2-2 \text{$\#$1} \cos (c+d x)+1\right )-11 i \text{$\#$1}^2 b \log \left (\text{$\#$1}^2-2 \text{$\#$1} \cos (c+d x)+1\right )+i b \log \left (\text{$\#$1}^2-2 \text{$\#$1} \cos (c+d x)+1\right )+2 \text{$\#$1}^6 b \tan ^{-1}\left (\frac{\sin (c+d x)}{\cos (c+d x)-\text{$\#$1}}\right )-22 \text{$\#$1}^4 b \tan ^{-1}\left (\frac{\sin (c+d x)}{\cos (c+d x)-\text{$\#$1}}\right )+22 \text{$\#$1}^2 b \tan ^{-1}\left (\frac{\sin (c+d x)}{\cos (c+d x)-\text{$\#$1}}\right )-2 b \tan ^{-1}\left (\frac{\sin (c+d x)}{\cos (c+d x)-\text{$\#$1}}\right )}{-8 \text{$\#$1}^3 a+\text{$\#$1}^7 b-3 \text{$\#$1}^5 b+3 \text{$\#$1}^3 b-\text{$\#$1} b}\& \right ]}{32 b d (a-b)} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[Sin[c + d*x]^5/(a - b*Sin[c + d*x]^4)^2,x]

[Out]

-((32*Cos[c + d*x]*(2*a + b - b*Cos[2*(c + d*x)]))/(8*a - 3*b + 4*b*Cos[2*(c + d*x)] - b*Cos[4*(c + d*x)]) + I
*RootSum[b - 4*b*#1^2 - 16*a*#1^4 + 6*b*#1^4 - 4*b*#1^6 + b*#1^8 & , (-2*b*ArcTan[Sin[c + d*x]/(Cos[c + d*x] -
#1)] + I*b*Log[1 - 2*Cos[c + d*x]*#1 + #1^2] - 8*a*ArcTan[Sin[c + d*x]/(Cos[c + d*x] - #1)]*#1^2 + 22*b*ArcTa
n[Sin[c + d*x]/(Cos[c + d*x] - #1)]*#1^2 + (4*I)*a*Log[1 - 2*Cos[c + d*x]*#1 + #1^2]*#1^2 - (11*I)*b*Log[1 - 2
*Cos[c + d*x]*#1 + #1^2]*#1^2 + 8*a*ArcTan[Sin[c + d*x]/(Cos[c + d*x] - #1)]*#1^4 - 22*b*ArcTan[Sin[c + d*x]/(
Cos[c + d*x] - #1)]*#1^4 - (4*I)*a*Log[1 - 2*Cos[c + d*x]*#1 + #1^2]*#1^4 + (11*I)*b*Log[1 - 2*Cos[c + d*x]*#1
+ #1^2]*#1^4 + 2*b*ArcTan[Sin[c + d*x]/(Cos[c + d*x] - #1)]*#1^6 - I*b*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) & ])/(32*(a - b)*b*d)

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Maple [B]  time = 0.105, size = 440, normalized size = 2. \begin{align*} -{\frac{ \left ( \cos \left ( dx+c \right ) \right ) ^{3}}{4\,d \left ( b \left ( \cos \left ( dx+c \right ) \right ) ^{4}-2\,b \left ( \cos \left ( dx+c \right ) \right ) ^{2}-a+b \right ) \left ( a-b \right ) }}+{\frac{\cos \left ( dx+c \right ) a}{4\,bd \left ( b \left ( \cos \left ( dx+c \right ) \right ) ^{4}-2\,b \left ( \cos \left ( dx+c \right ) \right ) ^{2}-a+b \right ) \left ( a-b \right ) }}+{\frac{\cos \left ( dx+c \right ) }{4\,d \left ( b \left ( \cos \left ( dx+c \right ) \right ) ^{4}-2\,b \left ( \cos \left ( dx+c \right ) \right ) ^{2}-a+b \right ) \left ( a-b \right ) }}-{\frac{1}{8\,d \left ( a-b \right ) }\arctan \left ({\cos \left ( dx+c \right ) b{\frac{1}{\sqrt{ \left ( \sqrt{ab}-b \right ) b}}}} \right ){\frac{1}{\sqrt{ \left ( \sqrt{ab}-b \right ) b}}}}-{\frac{b}{4\,d \left ( a-b \right ) }\arctan \left ({\cos \left ( dx+c \right ) b{\frac{1}{\sqrt{ \left ( \sqrt{ab}-b \right ) b}}}} \right ){\frac{1}{\sqrt{ab}}}{\frac{1}{\sqrt{ \left ( \sqrt{ab}-b \right ) b}}}}+{\frac{a}{8\,d \left ( a-b \right ) }\arctan \left ({\cos \left ( dx+c \right ) b{\frac{1}{\sqrt{ \left ( \sqrt{ab}-b \right ) b}}}} \right ){\frac{1}{\sqrt{ab}}}{\frac{1}{\sqrt{ \left ( \sqrt{ab}-b \right ) b}}}}+{\frac{1}{8\,d \left ( a-b \right ) }{\it Artanh} \left ({\cos \left ( dx+c \right ) b{\frac{1}{\sqrt{ \left ( \sqrt{ab}+b \right ) b}}}} \right ){\frac{1}{\sqrt{ \left ( \sqrt{ab}+b \right ) b}}}}-{\frac{b}{4\,d \left ( a-b \right ) }{\it Artanh} \left ({\cos \left ( dx+c \right ) b{\frac{1}{\sqrt{ \left ( \sqrt{ab}+b \right ) b}}}} \right ){\frac{1}{\sqrt{ab}}}{\frac{1}{\sqrt{ \left ( \sqrt{ab}+b \right ) b}}}}+{\frac{a}{8\,d \left ( a-b \right ) }{\it Artanh} \left ({\cos \left ( dx+c \right ) b{\frac{1}{\sqrt{ \left ( \sqrt{ab}+b \right ) b}}}} \right ){\frac{1}{\sqrt{ab}}}{\frac{1}{\sqrt{ \left ( \sqrt{ab}+b \right ) b}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sin(d*x+c)^5/(a-b*sin(d*x+c)^4)^2,x)

[Out]

-1/4/d/(b*cos(d*x+c)^4-2*b*cos(d*x+c)^2-a+b)/(a-b)*cos(d*x+c)^3+1/4/d/b*a/(b*cos(d*x+c)^4-2*b*cos(d*x+c)^2-a+b
)/(a-b)*cos(d*x+c)+1/4/d/(b*cos(d*x+c)^4-2*b*cos(d*x+c)^2-a+b)/(a-b)*cos(d*x+c)-1/8/d/(a-b)/(((a*b)^(1/2)-b)*b
)^(1/2)*arctan(cos(d*x+c)*b/(((a*b)^(1/2)-b)*b)^(1/2))-1/4/d/(a-b)/(a*b)^(1/2)/(((a*b)^(1/2)-b)*b)^(1/2)*arcta
n(cos(d*x+c)*b/(((a*b)^(1/2)-b)*b)^(1/2))*b+1/8/d*a/(a-b)/(a*b)^(1/2)/(((a*b)^(1/2)-b)*b)^(1/2)*arctan(cos(d*x
+c)*b/(((a*b)^(1/2)-b)*b)^(1/2))+1/8/d/(a-b)/(((a*b)^(1/2)+b)*b)^(1/2)*arctanh(cos(d*x+c)*b/(((a*b)^(1/2)+b)*b
)^(1/2))-1/4/d/(a-b)/(a*b)^(1/2)/(((a*b)^(1/2)+b)*b)^(1/2)*arctanh(cos(d*x+c)*b/(((a*b)^(1/2)+b)*b)^(1/2))*b+1
/8/d*a/(a-b)/(a*b)^(1/2)/(((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(sin(d*x+c)^5/(a-b*sin(d*x+c)^4)^2,x, algorithm="maxima")

[Out]

1/2*(4*b^2*cos(2*d*x + 2*c)*cos(d*x + c) + 4*b^2*sin(2*d*x + 2*c)*sin(d*x + c) - b^2*cos(d*x + c) - 4*(4*a*b +
b^2)*sin(3*d*x + 3*c)*sin(2*d*x + 2*c) - (b^2*cos(7*d*x + 7*c) + b^2*cos(d*x + c) - (4*a*b + b^2)*cos(5*d*x +
5*c) - (4*a*b + b^2)*cos(3*d*x + 3*c))*cos(8*d*x + 8*c) + (4*b^2*cos(6*d*x + 6*c) + 4*b^2*cos(2*d*x + 2*c) -
b^2 + 2*(8*a*b - 3*b^2)*cos(4*d*x + 4*c))*cos(7*d*x + 7*c) + 4*(b^2*cos(d*x + c) - (4*a*b + b^2)*cos(5*d*x + 5
*c) - (4*a*b + b^2)*cos(3*d*x + 3*c))*cos(6*d*x + 6*c) + (4*a*b + b^2 - 2*(32*a^2 - 4*a*b - 3*b^2)*cos(4*d*x +
4*c) - 4*(4*a*b + b^2)*cos(2*d*x + 2*c))*cos(5*d*x + 5*c) - 2*((32*a^2 - 4*a*b - 3*b^2)*cos(3*d*x + 3*c) - (8
*a*b - 3*b^2)*cos(d*x + c))*cos(4*d*x + 4*c) + (4*a*b + b^2 - 4*(4*a*b + b^2)*cos(2*d*x + 2*c))*cos(3*d*x + 3*
c) + 2*((a*b^3 - b^4)*d*cos(8*d*x + 8*c)^2 + 16*(a*b^3 - b^4)*d*cos(6*d*x + 6*c)^2 + 4*(64*a^3*b - 112*a^2*b^2
+ 57*a*b^3 - 9*b^4)*d*cos(4*d*x + 4*c)^2 + 16*(a*b^3 - b^4)*d*cos(2*d*x + 2*c)^2 + (a*b^3 - b^4)*d*sin(8*d*x
+ 8*c)^2 + 16*(a*b^3 - b^4)*d*sin(6*d*x + 6*c)^2 + 4*(64*a^3*b - 112*a^2*b^2 + 57*a*b^3 - 9*b^4)*d*sin(4*d*x +
4*c)^2 + 16*(8*a^2*b^2 - 11*a*b^3 + 3*b^4)*d*sin(4*d*x + 4*c)*sin(2*d*x + 2*c) + 16*(a*b^3 - b^4)*d*sin(2*d*x
+ 2*c)^2 - 8*(a*b^3 - b^4)*d*cos(2*d*x + 2*c) + (a*b^3 - b^4)*d - 2*(4*(a*b^3 - b^4)*d*cos(6*d*x + 6*c) + 2*(
8*a^2*b^2 - 11*a*b^3 + 3*b^4)*d*cos(4*d*x + 4*c) + 4*(a*b^3 - b^4)*d*cos(2*d*x + 2*c) - (a*b^3 - b^4)*d)*cos(8
*d*x + 8*c) + 8*(2*(8*a^2*b^2 - 11*a*b^3 + 3*b^4)*d*cos(4*d*x + 4*c) + 4*(a*b^3 - b^4)*d*cos(2*d*x + 2*c) - (a
*b^3 - b^4)*d)*cos(6*d*x + 6*c) + 4*(4*(8*a^2*b^2 - 11*a*b^3 + 3*b^4)*d*cos(2*d*x + 2*c) - (8*a^2*b^2 - 11*a*b
^3 + 3*b^4)*d)*cos(4*d*x + 4*c) - 4*(2*(a*b^3 - b^4)*d*sin(6*d*x + 6*c) + (8*a^2*b^2 - 11*a*b^3 + 3*b^4)*d*sin
(4*d*x + 4*c) + 2*(a*b^3 - b^4)*d*sin(2*d*x + 2*c))*sin(8*d*x + 8*c) + 16*((8*a^2*b^2 - 11*a*b^3 + 3*b^4)*d*si
n(4*d*x + 4*c) + 2*(a*b^3 - b^4)*d*sin(2*d*x + 2*c))*sin(6*d*x + 6*c))*integrate(-1/2*(4*b^2*cos(d*x + c)*sin(
2*d*x + 2*c) - 4*b^2*cos(2*d*x + 2*c)*sin(d*x + c) + 4*(4*a*b - 11*b^2)*cos(3*d*x + 3*c)*sin(2*d*x + 2*c) + b^
2*sin(d*x + c) - (b^2*sin(7*d*x + 7*c) - b^2*sin(d*x + c) + (4*a*b - 11*b^2)*sin(5*d*x + 5*c) - (4*a*b - 11*b^
2)*sin(3*d*x + 3*c))*cos(8*d*x + 8*c) - 2*(2*b^2*sin(6*d*x + 6*c) + 2*b^2*sin(2*d*x + 2*c) + (8*a*b - 3*b^2)*s
in(4*d*x + 4*c))*cos(7*d*x + 7*c) - 4*(b^2*sin(d*x + c) - (4*a*b - 11*b^2)*sin(5*d*x + 5*c) + (4*a*b - 11*b^2)
*sin(3*d*x + 3*c))*cos(6*d*x + 6*c) - 2*((32*a^2 - 100*a*b + 33*b^2)*sin(4*d*x + 4*c) + 2*(4*a*b - 11*b^2)*sin
(2*d*x + 2*c))*cos(5*d*x + 5*c) - 2*((32*a^2 - 100*a*b + 33*b^2)*sin(3*d*x + 3*c) + (8*a*b - 3*b^2)*sin(d*x +
c))*cos(4*d*x + 4*c) + (b^2*cos(7*d*x + 7*c) - b^2*cos(d*x + c) + (4*a*b - 11*b^2)*cos(5*d*x + 5*c) - (4*a*b -
11*b^2)*cos(3*d*x + 3*c))*sin(8*d*x + 8*c) + (4*b^2*cos(6*d*x + 6*c) + 4*b^2*cos(2*d*x + 2*c) - b^2 + 2*(8*a*
b - 3*b^2)*cos(4*d*x + 4*c))*sin(7*d*x + 7*c) + 4*(b^2*cos(d*x + c) - (4*a*b - 11*b^2)*cos(5*d*x + 5*c) + (4*a
*b - 11*b^2)*cos(3*d*x + 3*c))*sin(6*d*x + 6*c) - (4*a*b - 11*b^2 - 2*(32*a^2 - 100*a*b + 33*b^2)*cos(4*d*x +
4*c) - 4*(4*a*b - 11*b^2)*cos(2*d*x + 2*c))*sin(5*d*x + 5*c) + 2*((32*a^2 - 100*a*b + 33*b^2)*cos(3*d*x + 3*c)
+ (8*a*b - 3*b^2)*cos(d*x + c))*sin(4*d*x + 4*c) + (4*a*b - 11*b^2 - 4*(4*a*b - 11*b^2)*cos(2*d*x + 2*c))*sin
(3*d*x + 3*c))/(a*b^3 - b^4 + (a*b^3 - b^4)*cos(8*d*x + 8*c)^2 + 16*(a*b^3 - b^4)*cos(6*d*x + 6*c)^2 + 4*(64*a
^3*b - 112*a^2*b^2 + 57*a*b^3 - 9*b^4)*cos(4*d*x + 4*c)^2 + 16*(a*b^3 - b^4)*cos(2*d*x + 2*c)^2 + (a*b^3 - b^4
)*sin(8*d*x + 8*c)^2 + 16*(a*b^3 - b^4)*sin(6*d*x + 6*c)^2 + 4*(64*a^3*b - 112*a^2*b^2 + 57*a*b^3 - 9*b^4)*sin
(4*d*x + 4*c)^2 + 16*(8*a^2*b^2 - 11*a*b^3 + 3*b^4)*sin(4*d*x + 4*c)*sin(2*d*x + 2*c) + 16*(a*b^3 - b^4)*sin(2
*d*x + 2*c)^2 + 2*(a*b^3 - b^4 - 4*(a*b^3 - b^4)*cos(6*d*x + 6*c) - 2*(8*a^2*b^2 - 11*a*b^3 + 3*b^4)*cos(4*d*x
+ 4*c) - 4*(a*b^3 - b^4)*cos(2*d*x + 2*c))*cos(8*d*x + 8*c) - 8*(a*b^3 - b^4 - 2*(8*a^2*b^2 - 11*a*b^3 + 3*b^
4)*cos(4*d*x + 4*c) - 4*(a*b^3 - b^4)*cos(2*d*x + 2*c))*cos(6*d*x + 6*c) - 4*(8*a^2*b^2 - 11*a*b^3 + 3*b^4 - 4
*(8*a^2*b^2 - 11*a*b^3 + 3*b^4)*cos(2*d*x + 2*c))*cos(4*d*x + 4*c) - 8*(a*b^3 - b^4)*cos(2*d*x + 2*c) - 4*(2*(
a*b^3 - b^4)*sin(6*d*x + 6*c) + (8*a^2*b^2 - 11*a*b^3 + 3*b^4)*sin(4*d*x + 4*c) + 2*(a*b^3 - b^4)*sin(2*d*x +
2*c))*sin(8*d*x + 8*c) + 16*((8*a^2*b^2 - 11*a*b^3 + 3*b^4)*sin(4*d*x + 4*c) + 2*(a*b^3 - b^4)*sin(2*d*x + 2*c
))*sin(6*d*x + 6*c)), x) - (b^2*sin(7*d*x + 7*c) + b^2*sin(d*x + c) - (4*a*b + b^2)*sin(5*d*x + 5*c) - (4*a*b
+ b^2)*sin(3*d*x + 3*c))*sin(8*d*x + 8*c) + 2*(2*b^2*sin(6*d*x + 6*c) + 2*b^2*sin(2*d*x + 2*c) + (8*a*b - 3*b^
2)*sin(4*d*x + 4*c))*sin(7*d*x + 7*c) + 4*(b^2*sin(d*x + c) - (4*a*b + b^2)*sin(5*d*x + 5*c) - (4*a*b + b^2)*s
in(3*d*x + 3*c))*sin(6*d*x + 6*c) - 2*((32*a^2 - 4*a*b - 3*b^2)*sin(4*d*x + 4*c) + 2*(4*a*b + b^2)*sin(2*d*x +
2*c))*sin(5*d*x + 5*c) - 2*((32*a^2 - 4*a*b - 3*b^2)*sin(3*d*x + 3*c) - (8*a*b - 3*b^2)*sin(d*x + c))*sin(4*d
*x + 4*c))/((a*b^3 - b^4)*d*cos(8*d*x + 8*c)^2 + 16*(a*b^3 - b^4)*d*cos(6*d*x + 6*c)^2 + 4*(64*a^3*b - 112*a^2
*b^2 + 57*a*b^3 - 9*b^4)*d*cos(4*d*x + 4*c)^2 + 16*(a*b^3 - b^4)*d*cos(2*d*x + 2*c)^2 + (a*b^3 - b^4)*d*sin(8*
d*x + 8*c)^2 + 16*(a*b^3 - b^4)*d*sin(6*d*x + 6*c)^2 + 4*(64*a^3*b - 112*a^2*b^2 + 57*a*b^3 - 9*b^4)*d*sin(4*d
*x + 4*c)^2 + 16*(8*a^2*b^2 - 11*a*b^3 + 3*b^4)*d*sin(4*d*x + 4*c)*sin(2*d*x + 2*c) + 16*(a*b^3 - b^4)*d*sin(2
*d*x + 2*c)^2 - 8*(a*b^3 - b^4)*d*cos(2*d*x + 2*c) + (a*b^3 - b^4)*d - 2*(4*(a*b^3 - b^4)*d*cos(6*d*x + 6*c) +
2*(8*a^2*b^2 - 11*a*b^3 + 3*b^4)*d*cos(4*d*x + 4*c) + 4*(a*b^3 - b^4)*d*cos(2*d*x + 2*c) - (a*b^3 - b^4)*d)*c
os(8*d*x + 8*c) + 8*(2*(8*a^2*b^2 - 11*a*b^3 + 3*b^4)*d*cos(4*d*x + 4*c) + 4*(a*b^3 - b^4)*d*cos(2*d*x + 2*c)
- (a*b^3 - b^4)*d)*cos(6*d*x + 6*c) + 4*(4*(8*a^2*b^2 - 11*a*b^3 + 3*b^4)*d*cos(2*d*x + 2*c) - (8*a^2*b^2 - 11
*a*b^3 + 3*b^4)*d)*cos(4*d*x + 4*c) - 4*(2*(a*b^3 - b^4)*d*sin(6*d*x + 6*c) + (8*a^2*b^2 - 11*a*b^3 + 3*b^4)*d
*sin(4*d*x + 4*c) + 2*(a*b^3 - b^4)*d*sin(2*d*x + 2*c))*sin(8*d*x + 8*c) + 16*((8*a^2*b^2 - 11*a*b^3 + 3*b^4)*
d*sin(4*d*x + 4*c) + 2*(a*b^3 - b^4)*d*sin(2*d*x + 2*c))*sin(6*d*x + 6*c))

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Fricas [B]  time = 5.03459, size = 5237, normalized size = 24.13 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sin(d*x+c)^5/(a-b*sin(d*x+c)^4)^2,x, algorithm="fricas")

[Out]

-1/16*(4*b*cos(d*x + c)^3 - ((a*b^2 - b^3)*d*cos(d*x + c)^4 - 2*(a*b^2 - b^3)*d*cos(d*x + c)^2 - (a^2*b - 2*a*
b^2 + b^3)*d)*sqrt(((a^4*b^2 - 3*a^3*b^3 + 3*a^2*b^4 - a*b^5)*d^2*sqrt((a^4 - 10*a^3*b + 41*a^2*b^2 - 80*a*b^3
+ 64*b^4)/((a^7*b^5 - 6*a^6*b^6 + 15*a^5*b^7 - 20*a^4*b^8 + 15*a^3*b^9 - 6*a^2*b^10 + a*b^11)*d^4)) + a^2 - a
*b - 4*b^2)/((a^4*b^2 - 3*a^3*b^3 + 3*a^2*b^4 - a*b^5)*d^2))*log((a^3 - 9*a^2*b + 28*a*b^2 - 32*b^3)*cos(d*x +
c) - (2*(a^4*b^5 - 3*a^3*b^6 + 3*a^2*b^7 - a*b^8)*d^3*sqrt((a^4 - 10*a^3*b + 41*a^2*b^2 - 80*a*b^3 + 64*b^4)/
((a^7*b^5 - 6*a^6*b^6 + 15*a^5*b^7 - 20*a^4*b^8 + 15*a^3*b^9 - 6*a^2*b^10 + a*b^11)*d^4)) - (a^4*b - 8*a^3*b^2
+ 23*a^2*b^3 - 24*a*b^4)*d)*sqrt(((a^4*b^2 - 3*a^3*b^3 + 3*a^2*b^4 - a*b^5)*d^2*sqrt((a^4 - 10*a^3*b + 41*a^2
*b^2 - 80*a*b^3 + 64*b^4)/((a^7*b^5 - 6*a^6*b^6 + 15*a^5*b^7 - 20*a^4*b^8 + 15*a^3*b^9 - 6*a^2*b^10 + a*b^11)*
d^4)) + a^2 - a*b - 4*b^2)/((a^4*b^2 - 3*a^3*b^3 + 3*a^2*b^4 - a*b^5)*d^2))) + ((a*b^2 - b^3)*d*cos(d*x + c)^4
- 2*(a*b^2 - b^3)*d*cos(d*x + c)^2 - (a^2*b - 2*a*b^2 + b^3)*d)*sqrt(-((a^4*b^2 - 3*a^3*b^3 + 3*a^2*b^4 - a*b
^5)*d^2*sqrt((a^4 - 10*a^3*b + 41*a^2*b^2 - 80*a*b^3 + 64*b^4)/((a^7*b^5 - 6*a^6*b^6 + 15*a^5*b^7 - 20*a^4*b^8
+ 15*a^3*b^9 - 6*a^2*b^10 + a*b^11)*d^4)) - a^2 + a*b + 4*b^2)/((a^4*b^2 - 3*a^3*b^3 + 3*a^2*b^4 - a*b^5)*d^2
))*log((a^3 - 9*a^2*b + 28*a*b^2 - 32*b^3)*cos(d*x + c) - (2*(a^4*b^5 - 3*a^3*b^6 + 3*a^2*b^7 - a*b^8)*d^3*sqr
t((a^4 - 10*a^3*b + 41*a^2*b^2 - 80*a*b^3 + 64*b^4)/((a^7*b^5 - 6*a^6*b^6 + 15*a^5*b^7 - 20*a^4*b^8 + 15*a^3*b
^9 - 6*a^2*b^10 + a*b^11)*d^4)) + (a^4*b - 8*a^3*b^2 + 23*a^2*b^3 - 24*a*b^4)*d)*sqrt(-((a^4*b^2 - 3*a^3*b^3 +
3*a^2*b^4 - a*b^5)*d^2*sqrt((a^4 - 10*a^3*b + 41*a^2*b^2 - 80*a*b^3 + 64*b^4)/((a^7*b^5 - 6*a^6*b^6 + 15*a^5*
b^7 - 20*a^4*b^8 + 15*a^3*b^9 - 6*a^2*b^10 + a*b^11)*d^4)) - a^2 + a*b + 4*b^2)/((a^4*b^2 - 3*a^3*b^3 + 3*a^2*
b^4 - a*b^5)*d^2))) + ((a*b^2 - b^3)*d*cos(d*x + c)^4 - 2*(a*b^2 - b^3)*d*cos(d*x + c)^2 - (a^2*b - 2*a*b^2 +
b^3)*d)*sqrt(((a^4*b^2 - 3*a^3*b^3 + 3*a^2*b^4 - a*b^5)*d^2*sqrt((a^4 - 10*a^3*b + 41*a^2*b^2 - 80*a*b^3 + 64*
b^4)/((a^7*b^5 - 6*a^6*b^6 + 15*a^5*b^7 - 20*a^4*b^8 + 15*a^3*b^9 - 6*a^2*b^10 + a*b^11)*d^4)) + a^2 - a*b - 4
*b^2)/((a^4*b^2 - 3*a^3*b^3 + 3*a^2*b^4 - a*b^5)*d^2))*log(-(a^3 - 9*a^2*b + 28*a*b^2 - 32*b^3)*cos(d*x + c) -
(2*(a^4*b^5 - 3*a^3*b^6 + 3*a^2*b^7 - a*b^8)*d^3*sqrt((a^4 - 10*a^3*b + 41*a^2*b^2 - 80*a*b^3 + 64*b^4)/((a^7
*b^5 - 6*a^6*b^6 + 15*a^5*b^7 - 20*a^4*b^8 + 15*a^3*b^9 - 6*a^2*b^10 + a*b^11)*d^4)) - (a^4*b - 8*a^3*b^2 + 23
*a^2*b^3 - 24*a*b^4)*d)*sqrt(((a^4*b^2 - 3*a^3*b^3 + 3*a^2*b^4 - a*b^5)*d^2*sqrt((a^4 - 10*a^3*b + 41*a^2*b^2
- 80*a*b^3 + 64*b^4)/((a^7*b^5 - 6*a^6*b^6 + 15*a^5*b^7 - 20*a^4*b^8 + 15*a^3*b^9 - 6*a^2*b^10 + a*b^11)*d^4))
+ a^2 - a*b - 4*b^2)/((a^4*b^2 - 3*a^3*b^3 + 3*a^2*b^4 - a*b^5)*d^2))) - ((a*b^2 - b^3)*d*cos(d*x + c)^4 - 2*
(a*b^2 - b^3)*d*cos(d*x + c)^2 - (a^2*b - 2*a*b^2 + b^3)*d)*sqrt(-((a^4*b^2 - 3*a^3*b^3 + 3*a^2*b^4 - a*b^5)*d
^2*sqrt((a^4 - 10*a^3*b + 41*a^2*b^2 - 80*a*b^3 + 64*b^4)/((a^7*b^5 - 6*a^6*b^6 + 15*a^5*b^7 - 20*a^4*b^8 + 15
*a^3*b^9 - 6*a^2*b^10 + a*b^11)*d^4)) - a^2 + a*b + 4*b^2)/((a^4*b^2 - 3*a^3*b^3 + 3*a^2*b^4 - a*b^5)*d^2))*lo
g(-(a^3 - 9*a^2*b + 28*a*b^2 - 32*b^3)*cos(d*x + c) - (2*(a^4*b^5 - 3*a^3*b^6 + 3*a^2*b^7 - a*b^8)*d^3*sqrt((a
^4 - 10*a^3*b + 41*a^2*b^2 - 80*a*b^3 + 64*b^4)/((a^7*b^5 - 6*a^6*b^6 + 15*a^5*b^7 - 20*a^4*b^8 + 15*a^3*b^9 -
6*a^2*b^10 + a*b^11)*d^4)) + (a^4*b - 8*a^3*b^2 + 23*a^2*b^3 - 24*a*b^4)*d)*sqrt(-((a^4*b^2 - 3*a^3*b^3 + 3*a
^2*b^4 - a*b^5)*d^2*sqrt((a^4 - 10*a^3*b + 41*a^2*b^2 - 80*a*b^3 + 64*b^4)/((a^7*b^5 - 6*a^6*b^6 + 15*a^5*b^7
- 20*a^4*b^8 + 15*a^3*b^9 - 6*a^2*b^10 + a*b^11)*d^4)) - a^2 + a*b + 4*b^2)/((a^4*b^2 - 3*a^3*b^3 + 3*a^2*b^4
- a*b^5)*d^2))) - 4*(a + b)*cos(d*x + c))/((a*b^2 - b^3)*d*cos(d*x + c)^4 - 2*(a*b^2 - b^3)*d*cos(d*x + c)^2 -
(a^2*b - 2*a*b^2 + b^3)*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(sin(d*x+c)**5/(a-b*sin(d*x+c)**4)**2,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(sin(d*x+c)^5/(a-b*sin(d*x+c)^4)^2,x, algorithm="giac")

[Out]

Exception raised: NotImplementedError