\(\int \frac {\sin ^8(c+d x)}{a-b \sin ^4(c+d x)} \, dx\) [203]
Optimal result
Integrand size = 24, antiderivative size = 184 \[
\int \frac {\sin ^8(c+d x)}{a-b \sin ^4(c+d x)} \, dx=\frac {5 x}{8 b}-\frac {(a+b) x}{b^2}+\frac {a^{5/4} \arctan \left (\frac {\sqrt {\sqrt {a}-\sqrt {b}} \tan (c+d x)}{\sqrt [4]{a}}\right )}{2 \sqrt {\sqrt {a}-\sqrt {b}} b^2 d}+\frac {a^{5/4} \arctan \left (\frac {\sqrt {\sqrt {a}+\sqrt {b}} \tan (c+d x)}{\sqrt [4]{a}}\right )}{2 \sqrt {\sqrt {a}+\sqrt {b}} b^2 d}+\frac {5 \cos (c+d x) \sin (c+d x)}{8 b d}-\frac {\cos ^3(c+d x) \sin (c+d x)}{4 b d}
\]
[Out]
5/8*x/b-(a+b)*x/b^2+5/8*cos(d*x+c)*sin(d*x+c)/b/d-1/4*cos(d*x+c)^3*sin(d*x+c)/b/d+1/2*a^(5/4)*arctan((a^(1/2)-
b^(1/2))^(1/2)*tan(d*x+c)/a^(1/4))/b^2/d/(a^(1/2)-b^(1/2))^(1/2)+1/2*a^(5/4)*arctan((a^(1/2)+b^(1/2))^(1/2)*ta
n(d*x+c)/a^(1/4))/b^2/d/(a^(1/2)+b^(1/2))^(1/2)
Rubi [A] (verified)
Time = 0.18 (sec) , antiderivative size = 184, normalized size of antiderivative = 1.00, number of
steps used = 12, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {3296, 1301, 205, 209, 1180,
211} \[
\int \frac {\sin ^8(c+d x)}{a-b \sin ^4(c+d x)} \, dx=\frac {a^{5/4} \arctan \left (\frac {\sqrt {\sqrt {a}-\sqrt {b}} \tan (c+d x)}{\sqrt [4]{a}}\right )}{2 b^2 d \sqrt {\sqrt {a}-\sqrt {b}}}+\frac {a^{5/4} \arctan \left (\frac {\sqrt {\sqrt {a}+\sqrt {b}} \tan (c+d x)}{\sqrt [4]{a}}\right )}{2 b^2 d \sqrt {\sqrt {a}+\sqrt {b}}}-\frac {x (a+b)}{b^2}-\frac {\sin (c+d x) \cos ^3(c+d x)}{4 b d}+\frac {5 \sin (c+d x) \cos (c+d x)}{8 b d}+\frac {5 x}{8 b}
\]
[In]
Int[Sin[c + d*x]^8/(a - b*Sin[c + d*x]^4),x]
[Out]
(5*x)/(8*b) - ((a + b)*x)/b^2 + (a^(5/4)*ArcTan[(Sqrt[Sqrt[a] - Sqrt[b]]*Tan[c + d*x])/a^(1/4)])/(2*Sqrt[Sqrt[
a] - Sqrt[b]]*b^2*d) + (a^(5/4)*ArcTan[(Sqrt[Sqrt[a] + Sqrt[b]]*Tan[c + d*x])/a^(1/4)])/(2*Sqrt[Sqrt[a] + Sqrt
[b]]*b^2*d) + (5*Cos[c + d*x]*Sin[c + d*x])/(8*b*d) - (Cos[c + d*x]^3*Sin[c + d*x])/(4*b*d)
Rule 205
Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(-x)*((a + b*x^n)^(p + 1)/(a*n*(p + 1))), x] + Dist[(n*(p
+ 1) + 1)/(a*n*(p + 1)), Int[(a + b*x^n)^(p + 1), x], x] /; FreeQ[{a, b}, x] && IGtQ[n, 0] && LtQ[p, -1] && (
IntegerQ[2*p] || (n == 2 && IntegerQ[4*p]) || (n == 2 && IntegerQ[3*p]) || Denominator[p + 1/n] < Denominator[
p])
Rule 209
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[b, 2]))*ArcTan[Rt[b, 2]*(x/Rt[a, 2])], x] /;
FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a, 0] || GtQ[b, 0])
Rule 211
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[a/b, 2]/a)*ArcTan[x/Rt[a/b, 2]], x] /; FreeQ[{a, b}, x]
&& PosQ[a/b]
Rule 1180
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 1301
Int[(((f_.)*(x_))^(m_.)*((d_) + (e_.)*(x_)^2)^(q_.))/((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4), x_Symbol] :> Int[Ex
pandIntegrand[(f*x)^m*((d + e*x^2)^q/(a + b*x^2 + c*x^4)), x], x] /; FreeQ[{a, b, c, d, e, f, m}, x] && NeQ[b^
2 - 4*a*c, 0] && IntegerQ[q] && IntegerQ[m]
Rule 3296
Int[sin[(e_.) + (f_.)*(x_)]^(m_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]^4)^(p_.), x_Symbol] :> With[{ff = FreeF
actors[Tan[e + f*x], x]}, Dist[ff^(m + 1)/f, Subst[Int[x^m*((a + 2*a*ff^2*x^2 + (a + b)*ff^4*x^4)^p/(1 + ff^2*
x^2)^(m/2 + 2*p + 1)), x], x, Tan[e + f*x]/ff], x]] /; FreeQ[{a, b, e, f}, x] && IntegerQ[m/2] && IntegerQ[p]
Rubi steps \begin{align*}
\text {integral}& = \frac {\text {Subst}\left (\int \frac {x^8}{\left (1+x^2\right )^3 \left (a+2 a x^2+(a-b) x^4\right )} \, dx,x,\tan (c+d x)\right )}{d} \\ & = \frac {\text {Subst}\left (\int \left (-\frac {1}{b \left (1+x^2\right )^3}+\frac {2}{b \left (1+x^2\right )^2}+\frac {-a-b}{b^2 \left (1+x^2\right )}+\frac {a^2 \left (1+x^2\right )}{b^2 \left (a+2 a x^2+(a-b) x^4\right )}\right ) \, dx,x,\tan (c+d x)\right )}{d} \\ & = \frac {a^2 \text {Subst}\left (\int \frac {1+x^2}{a+2 a x^2+(a-b) x^4} \, dx,x,\tan (c+d x)\right )}{b^2 d}-\frac {\text {Subst}\left (\int \frac {1}{\left (1+x^2\right )^3} \, dx,x,\tan (c+d x)\right )}{b d}+\frac {2 \text {Subst}\left (\int \frac {1}{\left (1+x^2\right )^2} \, dx,x,\tan (c+d x)\right )}{b d}-\frac {(a+b) \text {Subst}\left (\int \frac {1}{1+x^2} \, dx,x,\tan (c+d x)\right )}{b^2 d} \\ & = -\frac {(a+b) x}{b^2}+\frac {\cos (c+d x) \sin (c+d x)}{b d}-\frac {\cos ^3(c+d x) \sin (c+d x)}{4 b d}+\frac {\left (a^{3/2} \left (\sqrt {a}-\sqrt {b}\right )\right ) \text {Subst}\left (\int \frac {1}{a-\sqrt {a} \sqrt {b}+(a-b) x^2} \, dx,x,\tan (c+d x)\right )}{2 b^2 d}+\frac {\left (a^{3/2} \left (\sqrt {a}+\sqrt {b}\right )\right ) \text {Subst}\left (\int \frac {1}{a+\sqrt {a} \sqrt {b}+(a-b) x^2} \, dx,x,\tan (c+d x)\right )}{2 b^2 d}-\frac {3 \text {Subst}\left (\int \frac {1}{\left (1+x^2\right )^2} \, dx,x,\tan (c+d x)\right )}{4 b d}+\frac {\text {Subst}\left (\int \frac {1}{1+x^2} \, dx,x,\tan (c+d x)\right )}{b d} \\ & = \frac {x}{b}-\frac {(a+b) x}{b^2}+\frac {a^{5/4} \arctan \left (\frac {\sqrt {\sqrt {a}-\sqrt {b}} \tan (c+d x)}{\sqrt [4]{a}}\right )}{2 \sqrt {\sqrt {a}-\sqrt {b}} b^2 d}+\frac {a^{5/4} \arctan \left (\frac {\sqrt {\sqrt {a}+\sqrt {b}} \tan (c+d x)}{\sqrt [4]{a}}\right )}{2 \sqrt {\sqrt {a}+\sqrt {b}} b^2 d}+\frac {5 \cos (c+d x) \sin (c+d x)}{8 b d}-\frac {\cos ^3(c+d x) \sin (c+d x)}{4 b d}-\frac {3 \text {Subst}\left (\int \frac {1}{1+x^2} \, dx,x,\tan (c+d x)\right )}{8 b d} \\ & = \frac {5 x}{8 b}-\frac {(a+b) x}{b^2}+\frac {a^{5/4} \arctan \left (\frac {\sqrt {\sqrt {a}-\sqrt {b}} \tan (c+d x)}{\sqrt [4]{a}}\right )}{2 \sqrt {\sqrt {a}-\sqrt {b}} b^2 d}+\frac {a^{5/4} \arctan \left (\frac {\sqrt {\sqrt {a}+\sqrt {b}} \tan (c+d x)}{\sqrt [4]{a}}\right )}{2 \sqrt {\sqrt {a}+\sqrt {b}} b^2 d}+\frac {5 \cos (c+d x) \sin (c+d x)}{8 b d}-\frac {\cos ^3(c+d x) \sin (c+d x)}{4 b d} \\
\end{align*}
Mathematica [A] (verified)
Time = 1.42 (sec) , antiderivative size = 172, normalized size of antiderivative = 0.93
\[
\int \frac {\sin ^8(c+d x)}{a-b \sin ^4(c+d x)} \, dx=-\frac {4 (8 a+3 b) (c+d x)-\frac {16 a^{3/2} \arctan \left (\frac {\left (\sqrt {a}+\sqrt {b}\right ) \tan (c+d x)}{\sqrt {a+\sqrt {a} \sqrt {b}}}\right )}{\sqrt {a+\sqrt {a} \sqrt {b}}}+\frac {16 a^{3/2} \text {arctanh}\left (\frac {\left (\sqrt {a}-\sqrt {b}\right ) \tan (c+d x)}{\sqrt {-a+\sqrt {a} \sqrt {b}}}\right )}{\sqrt {-a+\sqrt {a} \sqrt {b}}}-8 b \sin (2 (c+d x))+b \sin (4 (c+d x))}{32 b^2 d}
\]
[In]
Integrate[Sin[c + d*x]^8/(a - b*Sin[c + d*x]^4),x]
[Out]
-1/32*(4*(8*a + 3*b)*(c + d*x) - (16*a^(3/2)*ArcTan[((Sqrt[a] + Sqrt[b])*Tan[c + d*x])/Sqrt[a + Sqrt[a]*Sqrt[b
]]])/Sqrt[a + Sqrt[a]*Sqrt[b]] + (16*a^(3/2)*ArcTanh[((Sqrt[a] - Sqrt[b])*Tan[c + d*x])/Sqrt[-a + Sqrt[a]*Sqrt
[b]]])/Sqrt[-a + Sqrt[a]*Sqrt[b]] - 8*b*Sin[2*(c + d*x)] + b*Sin[4*(c + d*x)])/(b^2*d)
Maple [A] (verified)
Time = 2.23 (sec) , antiderivative size = 209, normalized size of antiderivative = 1.14
| | |
| method | result | size |
| | |
| derivativedivides |
\(\frac {-\frac {\frac {-\frac {5 \left (\tan ^{3}\left (d x +c \right )\right ) b}{8}-\frac {3 \tan \left (d x +c \right ) b}{8}}{\left (1+\tan ^{2}\left (d x +c \right )\right )^{2}}+\frac {\left (8 a +3 b \right ) \arctan \left (\tan \left (d x +c \right )\right )}{8}}{b^{2}}+\frac {a^{2} \left (a -b \right ) \left (\frac {\left (\sqrt {a b}+b \right ) \arctan \left (\frac {\left (a -b \right ) \tan \left (d x +c \right )}{\sqrt {\left (\sqrt {a b}+a \right ) \left (a -b \right )}}\right )}{2 \sqrt {a b}\, \left (a -b \right ) \sqrt {\left (\sqrt {a b}+a \right ) \left (a -b \right )}}+\frac {\left (\sqrt {a b}-b \right ) \operatorname {arctanh}\left (\frac {\left (-a +b \right ) \tan \left (d x +c \right )}{\sqrt {\left (\sqrt {a b}-a \right ) \left (a -b \right )}}\right )}{2 \sqrt {a b}\, \left (a -b \right ) \sqrt {\left (\sqrt {a b}-a \right ) \left (a -b \right )}}\right )}{b^{2}}}{d}\) |
\(209\) |
| default |
\(\frac {-\frac {\frac {-\frac {5 \left (\tan ^{3}\left (d x +c \right )\right ) b}{8}-\frac {3 \tan \left (d x +c \right ) b}{8}}{\left (1+\tan ^{2}\left (d x +c \right )\right )^{2}}+\frac {\left (8 a +3 b \right ) \arctan \left (\tan \left (d x +c \right )\right )}{8}}{b^{2}}+\frac {a^{2} \left (a -b \right ) \left (\frac {\left (\sqrt {a b}+b \right ) \arctan \left (\frac {\left (a -b \right ) \tan \left (d x +c \right )}{\sqrt {\left (\sqrt {a b}+a \right ) \left (a -b \right )}}\right )}{2 \sqrt {a b}\, \left (a -b \right ) \sqrt {\left (\sqrt {a b}+a \right ) \left (a -b \right )}}+\frac {\left (\sqrt {a b}-b \right ) \operatorname {arctanh}\left (\frac {\left (-a +b \right ) \tan \left (d x +c \right )}{\sqrt {\left (\sqrt {a b}-a \right ) \left (a -b \right )}}\right )}{2 \sqrt {a b}\, \left (a -b \right ) \sqrt {\left (\sqrt {a b}-a \right ) \left (a -b \right )}}\right )}{b^{2}}}{d}\) |
\(209\) |
| risch |
\(-\frac {a x}{b^{2}}-\frac {3 x}{8 b}-\frac {i {\mathrm e}^{2 i \left (d x +c \right )}}{8 b d}+\frac {i {\mathrm e}^{-2 i \left (d x +c \right )}}{8 b d}+\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\left (a \,b^{8} d^{4}-b^{9} d^{4}\right ) \textit {\_Z}^{4}+8192 a^{3} b^{4} d^{2} \textit {\_Z}^{2}+16777216 a^{5}\right )}{\sum }\textit {\_R} \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}+\left (\frac {i b^{5} d^{3}}{131072 a^{2}}-\frac {i b^{6} d^{3}}{131072 a^{3}}\right ) \textit {\_R}^{3}+\left (-\frac {b^{3} d^{2}}{2048 a}+\frac {b^{4} d^{2}}{2048 a^{2}}\right ) \textit {\_R}^{2}+\left (\frac {i d b}{32}+\frac {i b^{2} d}{32 a}\right ) \textit {\_R} -\frac {2 a}{b}-1\right )\right )}{256}-\frac {\sin \left (4 d x +4 c \right )}{32 d b}\) |
\(209\) |
| | |
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[In]
int(sin(d*x+c)^8/(a-b*sin(d*x+c)^4),x,method=_RETURNVERBOSE)
[Out]
1/d*(-1/b^2*((-5/8*tan(d*x+c)^3*b-3/8*tan(d*x+c)*b)/(1+tan(d*x+c)^2)^2+1/8*(8*a+3*b)*arctan(tan(d*x+c)))+a^2/b
^2*(a-b)*(1/2*((a*b)^(1/2)+b)/(a*b)^(1/2)/(a-b)/(((a*b)^(1/2)+a)*(a-b))^(1/2)*arctan((a-b)*tan(d*x+c)/(((a*b)^
(1/2)+a)*(a-b))^(1/2))+1/2*((a*b)^(1/2)-b)/(a*b)^(1/2)/(a-b)/(((a*b)^(1/2)-a)*(a-b))^(1/2)*arctanh((-a+b)*tan(
d*x+c)/(((a*b)^(1/2)-a)*(a-b))^(1/2))))
Fricas [B] (verification not implemented)
Leaf count of result is larger than twice the leaf count of optimal. 1311 vs. \(2 (142) = 284\).
Time = 0.43 (sec) , antiderivative size = 1311, normalized size of antiderivative = 7.12
\[
\int \frac {\sin ^8(c+d x)}{a-b \sin ^4(c+d x)} \, dx=\text {Too large to display}
\]
[In]
integrate(sin(d*x+c)^8/(a-b*sin(d*x+c)^4),x, algorithm="fricas")
[Out]
-1/8*(b^2*d*sqrt(-((a*b^4 - b^5)*d^2*sqrt(a^5/((a^2*b^7 - 2*a*b^8 + b^9)*d^4)) + a^3)/((a*b^4 - b^5)*d^2))*log
(1/4*a^3*cos(d*x + c)^2 - 1/4*a^3 - 1/4*(2*(a^2*b^3 - a*b^4)*d^2*cos(d*x + c)^2 - (a^2*b^3 - a*b^4)*d^2)*sqrt(
a^5/((a^2*b^7 - 2*a*b^8 + b^9)*d^4)) + 1/2*(a^2*b^2*d*cos(d*x + c)*sin(d*x + c) - (a*b^5 - b^6)*d^3*sqrt(a^5/(
(a^2*b^7 - 2*a*b^8 + b^9)*d^4))*cos(d*x + c)*sin(d*x + c))*sqrt(-((a*b^4 - b^5)*d^2*sqrt(a^5/((a^2*b^7 - 2*a*b
^8 + b^9)*d^4)) + a^3)/((a*b^4 - b^5)*d^2))) - b^2*d*sqrt(-((a*b^4 - b^5)*d^2*sqrt(a^5/((a^2*b^7 - 2*a*b^8 + b
^9)*d^4)) + a^3)/((a*b^4 - b^5)*d^2))*log(1/4*a^3*cos(d*x + c)^2 - 1/4*a^3 - 1/4*(2*(a^2*b^3 - a*b^4)*d^2*cos(
d*x + c)^2 - (a^2*b^3 - a*b^4)*d^2)*sqrt(a^5/((a^2*b^7 - 2*a*b^8 + b^9)*d^4)) - 1/2*(a^2*b^2*d*cos(d*x + c)*si
n(d*x + c) - (a*b^5 - b^6)*d^3*sqrt(a^5/((a^2*b^7 - 2*a*b^8 + b^9)*d^4))*cos(d*x + c)*sin(d*x + c))*sqrt(-((a*
b^4 - b^5)*d^2*sqrt(a^5/((a^2*b^7 - 2*a*b^8 + b^9)*d^4)) + a^3)/((a*b^4 - b^5)*d^2))) - b^2*d*sqrt(((a*b^4 - b
^5)*d^2*sqrt(a^5/((a^2*b^7 - 2*a*b^8 + b^9)*d^4)) - a^3)/((a*b^4 - b^5)*d^2))*log(-1/4*a^3*cos(d*x + c)^2 + 1/
4*a^3 - 1/4*(2*(a^2*b^3 - a*b^4)*d^2*cos(d*x + c)^2 - (a^2*b^3 - a*b^4)*d^2)*sqrt(a^5/((a^2*b^7 - 2*a*b^8 + b^
9)*d^4)) + 1/2*(a^2*b^2*d*cos(d*x + c)*sin(d*x + c) + (a*b^5 - b^6)*d^3*sqrt(a^5/((a^2*b^7 - 2*a*b^8 + b^9)*d^
4))*cos(d*x + c)*sin(d*x + c))*sqrt(((a*b^4 - b^5)*d^2*sqrt(a^5/((a^2*b^7 - 2*a*b^8 + b^9)*d^4)) - a^3)/((a*b^
4 - b^5)*d^2))) + b^2*d*sqrt(((a*b^4 - b^5)*d^2*sqrt(a^5/((a^2*b^7 - 2*a*b^8 + b^9)*d^4)) - a^3)/((a*b^4 - b^5
)*d^2))*log(-1/4*a^3*cos(d*x + c)^2 + 1/4*a^3 - 1/4*(2*(a^2*b^3 - a*b^4)*d^2*cos(d*x + c)^2 - (a^2*b^3 - a*b^4
)*d^2)*sqrt(a^5/((a^2*b^7 - 2*a*b^8 + b^9)*d^4)) - 1/2*(a^2*b^2*d*cos(d*x + c)*sin(d*x + c) + (a*b^5 - b^6)*d^
3*sqrt(a^5/((a^2*b^7 - 2*a*b^8 + b^9)*d^4))*cos(d*x + c)*sin(d*x + c))*sqrt(((a*b^4 - b^5)*d^2*sqrt(a^5/((a^2*
b^7 - 2*a*b^8 + b^9)*d^4)) - a^3)/((a*b^4 - b^5)*d^2))) + (8*a + 3*b)*d*x + (2*b*cos(d*x + c)^3 - 5*b*cos(d*x
+ c))*sin(d*x + c))/(b^2*d)
Sympy [F(-1)]
Timed out. \[
\int \frac {\sin ^8(c+d x)}{a-b \sin ^4(c+d x)} \, dx=\text {Timed out}
\]
[In]
integrate(sin(d*x+c)**8/(a-b*sin(d*x+c)**4),x)
[Out]
Timed out
Maxima [F]
\[
\int \frac {\sin ^8(c+d x)}{a-b \sin ^4(c+d x)} \, dx=\int { -\frac {\sin \left (d x + c\right )^{8}}{b \sin \left (d x + c\right )^{4} - a} \,d x }
\]
[In]
integrate(sin(d*x+c)^8/(a-b*sin(d*x+c)^4),x, algorithm="maxima")
[Out]
-1/32*(512*a^2*b^2*d*integrate((b*cos(8*d*x + 8*c)*cos(4*d*x + 4*c) - 4*b*cos(6*d*x + 6*c)*cos(4*d*x + 4*c) -
2*(8*a - 3*b)*cos(4*d*x + 4*c)^2 + b*sin(8*d*x + 8*c)*sin(4*d*x + 4*c) - 4*b*sin(6*d*x + 6*c)*sin(4*d*x + 4*c)
- 2*(8*a - 3*b)*sin(4*d*x + 4*c)^2 - 4*b*sin(4*d*x + 4*c)*sin(2*d*x + 2*c) - (4*b*cos(2*d*x + 2*c) - b)*cos(4
*d*x + 4*c))/(b^4*cos(8*d*x + 8*c)^2 + 16*b^4*cos(6*d*x + 6*c)^2 + 16*b^4*cos(2*d*x + 2*c)^2 + b^4*sin(8*d*x +
8*c)^2 + 16*b^4*sin(6*d*x + 6*c)^2 + 16*b^4*sin(2*d*x + 2*c)^2 - 8*b^4*cos(2*d*x + 2*c) + b^4 + 4*(64*a^2*b^2
- 48*a*b^3 + 9*b^4)*cos(4*d*x + 4*c)^2 + 4*(64*a^2*b^2 - 48*a*b^3 + 9*b^4)*sin(4*d*x + 4*c)^2 + 16*(8*a*b^3 -
3*b^4)*sin(4*d*x + 4*c)*sin(2*d*x + 2*c) - 2*(4*b^4*cos(6*d*x + 6*c) + 4*b^4*cos(2*d*x + 2*c) - b^4 + 2*(8*a*
b^3 - 3*b^4)*cos(4*d*x + 4*c))*cos(8*d*x + 8*c) + 8*(4*b^4*cos(2*d*x + 2*c) - b^4 + 2*(8*a*b^3 - 3*b^4)*cos(4*
d*x + 4*c))*cos(6*d*x + 6*c) - 4*(8*a*b^3 - 3*b^4 - 4*(8*a*b^3 - 3*b^4)*cos(2*d*x + 2*c))*cos(4*d*x + 4*c) - 4
*(2*b^4*sin(6*d*x + 6*c) + 2*b^4*sin(2*d*x + 2*c) + (8*a*b^3 - 3*b^4)*sin(4*d*x + 4*c))*sin(8*d*x + 8*c) + 16*
(2*b^4*sin(2*d*x + 2*c) + (8*a*b^3 - 3*b^4)*sin(4*d*x + 4*c))*sin(6*d*x + 6*c)), x) + 4*(8*a + 3*b)*d*x + b*si
n(4*d*x + 4*c) - 8*b*sin(2*d*x + 2*c))/(b^2*d)
Giac [B] (verification not implemented)
Leaf count of result is larger than twice the leaf count of optimal. 461 vs. \(2 (142) = 284\).
Time = 0.74 (sec) , antiderivative size = 461, normalized size of antiderivative = 2.51
\[
\int \frac {\sin ^8(c+d x)}{a-b \sin ^4(c+d x)} \, dx=\frac {\frac {4 \, {\left (3 \, \sqrt {a^{2} - a b + \sqrt {a b} {\left (a - b\right )}} a^{3} - 6 \, \sqrt {a^{2} - a b + \sqrt {a b} {\left (a - b\right )}} a^{2} b - \sqrt {a^{2} - a b + \sqrt {a b} {\left (a - b\right )}} a b^{2}\right )} {\left (\pi \left \lfloor \frac {d x + c}{\pi } + \frac {1}{2} \right \rfloor + \arctan \left (\frac {\tan \left (d x + c\right )}{\sqrt {\frac {a b^{2} + \sqrt {a^{2} b^{4} - {\left (a b^{2} - b^{3}\right )} a b^{2}}}{a b^{2} - b^{3}}}}\right )\right )} {\left | -a + b \right |}}{3 \, a^{4} b^{2} - 12 \, a^{3} b^{3} + 14 \, a^{2} b^{4} - 4 \, a b^{5} - b^{6}} + \frac {4 \, {\left (3 \, \sqrt {a^{2} - a b - \sqrt {a b} {\left (a - b\right )}} a^{3} - 6 \, \sqrt {a^{2} - a b - \sqrt {a b} {\left (a - b\right )}} a^{2} b - \sqrt {a^{2} - a b - \sqrt {a b} {\left (a - b\right )}} a b^{2}\right )} {\left (\pi \left \lfloor \frac {d x + c}{\pi } + \frac {1}{2} \right \rfloor + \arctan \left (\frac {\tan \left (d x + c\right )}{\sqrt {\frac {a b^{2} - \sqrt {a^{2} b^{4} - {\left (a b^{2} - b^{3}\right )} a b^{2}}}{a b^{2} - b^{3}}}}\right )\right )} {\left | -a + b \right |}}{3 \, a^{4} b^{2} - 12 \, a^{3} b^{3} + 14 \, a^{2} b^{4} - 4 \, a b^{5} - b^{6}} - \frac {{\left (d x + c\right )} {\left (8 \, a + 3 \, b\right )}}{b^{2}} + \frac {5 \, \tan \left (d x + c\right )^{3} + 3 \, \tan \left (d x + c\right )}{{\left (\tan \left (d x + c\right )^{2} + 1\right )}^{2} b}}{8 \, d}
\]
[In]
integrate(sin(d*x+c)^8/(a-b*sin(d*x+c)^4),x, algorithm="giac")
[Out]
1/8*(4*(3*sqrt(a^2 - a*b + sqrt(a*b)*(a - b))*a^3 - 6*sqrt(a^2 - a*b + sqrt(a*b)*(a - b))*a^2*b - sqrt(a^2 - a
*b + sqrt(a*b)*(a - b))*a*b^2)*(pi*floor((d*x + c)/pi + 1/2) + arctan(tan(d*x + c)/sqrt((a*b^2 + sqrt(a^2*b^4
- (a*b^2 - b^3)*a*b^2))/(a*b^2 - b^3))))*abs(-a + b)/(3*a^4*b^2 - 12*a^3*b^3 + 14*a^2*b^4 - 4*a*b^5 - b^6) + 4
*(3*sqrt(a^2 - a*b - sqrt(a*b)*(a - b))*a^3 - 6*sqrt(a^2 - a*b - sqrt(a*b)*(a - b))*a^2*b - sqrt(a^2 - a*b - s
qrt(a*b)*(a - b))*a*b^2)*(pi*floor((d*x + c)/pi + 1/2) + arctan(tan(d*x + c)/sqrt((a*b^2 - sqrt(a^2*b^4 - (a*b
^2 - b^3)*a*b^2))/(a*b^2 - b^3))))*abs(-a + b)/(3*a^4*b^2 - 12*a^3*b^3 + 14*a^2*b^4 - 4*a*b^5 - b^6) - (d*x +
c)*(8*a + 3*b)/b^2 + (5*tan(d*x + c)^3 + 3*tan(d*x + c))/((tan(d*x + c)^2 + 1)^2*b))/d
Mupad [B] (verification not implemented)
Time = 17.02 (sec) , antiderivative size = 5022, normalized size of antiderivative = 27.29
\[
\int \frac {\sin ^8(c+d x)}{a-b \sin ^4(c+d x)} \, dx=\text {Too large to display}
\]
[In]
int(sin(c + d*x)^8/(a - b*sin(c + d*x)^4),x)
[Out]
(atan(((((((2048*a^3*b^10 + 8192*a^4*b^9 - 22528*a^5*b^8 + 12288*a^6*b^7)/(64*b^5) - (tan(c + d*x)*(-((a^5*b^9
)^(1/2) + a^3*b^4)/(16*(a*b^8 - b^9)))^(1/2)*(12288*a^2*b^11 - 12288*a^3*b^10 - 12288*a^4*b^9 + 12288*a^5*b^8)
)/(16*b^4))*(-((a^5*b^9)^(1/2) + a^3*b^4)/(16*(a*b^8 - b^9)))^(1/2) - (tan(c + d*x)*(432*a^2*b^9 + 1584*a^3*b^
8 - 880*a^4*b^7 - 5488*a^5*b^6 + 2048*a^6*b^5 + 2304*a^7*b^4))/(16*b^4))*(-((a^5*b^9)^(1/2) + a^3*b^4)/(16*(a*
b^8 - b^9)))^(1/2) - (144*a^3*b^8 + 624*a^4*b^7 + 112*a^5*b^6 - 1648*a^6*b^5 + 1536*a^7*b^4 - 768*a^8*b^3)/(64
*b^5))*(-((a^5*b^9)^(1/2) + a^3*b^4)/(16*(a*b^8 - b^9)))^(1/2) + (tan(c + d*x)*(9*a^4*b^5 - 96*a^9 - 336*a^8*b
+ 93*a^5*b^4 + 259*a^6*b^3 + 71*a^7*b^2))/(16*b^4))*(-((a^5*b^9)^(1/2) + a^3*b^4)/(16*(a*b^8 - b^9)))^(1/2)*1
i - (((((2048*a^3*b^10 + 8192*a^4*b^9 - 22528*a^5*b^8 + 12288*a^6*b^7)/(64*b^5) + (tan(c + d*x)*(-((a^5*b^9)^(
1/2) + a^3*b^4)/(16*(a*b^8 - b^9)))^(1/2)*(12288*a^2*b^11 - 12288*a^3*b^10 - 12288*a^4*b^9 + 12288*a^5*b^8))/(
16*b^4))*(-((a^5*b^9)^(1/2) + a^3*b^4)/(16*(a*b^8 - b^9)))^(1/2) + (tan(c + d*x)*(432*a^2*b^9 + 1584*a^3*b^8 -
880*a^4*b^7 - 5488*a^5*b^6 + 2048*a^6*b^5 + 2304*a^7*b^4))/(16*b^4))*(-((a^5*b^9)^(1/2) + a^3*b^4)/(16*(a*b^8
- b^9)))^(1/2) - (144*a^3*b^8 + 624*a^4*b^7 + 112*a^5*b^6 - 1648*a^6*b^5 + 1536*a^7*b^4 - 768*a^8*b^3)/(64*b^
5))*(-((a^5*b^9)^(1/2) + a^3*b^4)/(16*(a*b^8 - b^9)))^(1/2) - (tan(c + d*x)*(9*a^4*b^5 - 96*a^9 - 336*a^8*b +
93*a^5*b^4 + 259*a^6*b^3 + 71*a^7*b^2))/(16*b^4))*(-((a^5*b^9)^(1/2) + a^3*b^4)/(16*(a*b^8 - b^9)))^(1/2)*1i)/
((((((2048*a^3*b^10 + 8192*a^4*b^9 - 22528*a^5*b^8 + 12288*a^6*b^7)/(64*b^5) - (tan(c + d*x)*(-((a^5*b^9)^(1/2
) + a^3*b^4)/(16*(a*b^8 - b^9)))^(1/2)*(12288*a^2*b^11 - 12288*a^3*b^10 - 12288*a^4*b^9 + 12288*a^5*b^8))/(16*
b^4))*(-((a^5*b^9)^(1/2) + a^3*b^4)/(16*(a*b^8 - b^9)))^(1/2) - (tan(c + d*x)*(432*a^2*b^9 + 1584*a^3*b^8 - 88
0*a^4*b^7 - 5488*a^5*b^6 + 2048*a^6*b^5 + 2304*a^7*b^4))/(16*b^4))*(-((a^5*b^9)^(1/2) + a^3*b^4)/(16*(a*b^8 -
b^9)))^(1/2) - (144*a^3*b^8 + 624*a^4*b^7 + 112*a^5*b^6 - 1648*a^6*b^5 + 1536*a^7*b^4 - 768*a^8*b^3)/(64*b^5))
*(-((a^5*b^9)^(1/2) + a^3*b^4)/(16*(a*b^8 - b^9)))^(1/2) + (tan(c + d*x)*(9*a^4*b^5 - 96*a^9 - 336*a^8*b + 93*
a^5*b^4 + 259*a^6*b^3 + 71*a^7*b^2))/(16*b^4))*(-((a^5*b^9)^(1/2) + a^3*b^4)/(16*(a*b^8 - b^9)))^(1/2) + (((((
2048*a^3*b^10 + 8192*a^4*b^9 - 22528*a^5*b^8 + 12288*a^6*b^7)/(64*b^5) + (tan(c + d*x)*(-((a^5*b^9)^(1/2) + a^
3*b^4)/(16*(a*b^8 - b^9)))^(1/2)*(12288*a^2*b^11 - 12288*a^3*b^10 - 12288*a^4*b^9 + 12288*a^5*b^8))/(16*b^4))*
(-((a^5*b^9)^(1/2) + a^3*b^4)/(16*(a*b^8 - b^9)))^(1/2) + (tan(c + d*x)*(432*a^2*b^9 + 1584*a^3*b^8 - 880*a^4*
b^7 - 5488*a^5*b^6 + 2048*a^6*b^5 + 2304*a^7*b^4))/(16*b^4))*(-((a^5*b^9)^(1/2) + a^3*b^4)/(16*(a*b^8 - b^9)))
^(1/2) - (144*a^3*b^8 + 624*a^4*b^7 + 112*a^5*b^6 - 1648*a^6*b^5 + 1536*a^7*b^4 - 768*a^8*b^3)/(64*b^5))*(-((a
^5*b^9)^(1/2) + a^3*b^4)/(16*(a*b^8 - b^9)))^(1/2) - (tan(c + d*x)*(9*a^4*b^5 - 96*a^9 - 336*a^8*b + 93*a^5*b^
4 + 259*a^6*b^3 + 71*a^7*b^2))/(16*b^4))*(-((a^5*b^9)^(1/2) + a^3*b^4)/(16*(a*b^8 - b^9)))^(1/2) + (63*a^8*b -
216*a^9 + 27*a^6*b^3 + 126*a^7*b^2)/(32*b^5)))*(-((a^5*b^9)^(1/2) + a^3*b^4)/(16*(a*b^8 - b^9)))^(1/2)*2i)/d
+ (atan(((((((2048*a^3*b^10 + 8192*a^4*b^9 - 22528*a^5*b^8 + 12288*a^6*b^7)/(64*b^5) - (tan(c + d*x)*(((a^5*b^
9)^(1/2) - a^3*b^4)/(16*(a*b^8 - b^9)))^(1/2)*(12288*a^2*b^11 - 12288*a^3*b^10 - 12288*a^4*b^9 + 12288*a^5*b^8
))/(16*b^4))*(((a^5*b^9)^(1/2) - a^3*b^4)/(16*(a*b^8 - b^9)))^(1/2) - (tan(c + d*x)*(432*a^2*b^9 + 1584*a^3*b^
8 - 880*a^4*b^7 - 5488*a^5*b^6 + 2048*a^6*b^5 + 2304*a^7*b^4))/(16*b^4))*(((a^5*b^9)^(1/2) - a^3*b^4)/(16*(a*b
^8 - b^9)))^(1/2) - (144*a^3*b^8 + 624*a^4*b^7 + 112*a^5*b^6 - 1648*a^6*b^5 + 1536*a^7*b^4 - 768*a^8*b^3)/(64*
b^5))*(((a^5*b^9)^(1/2) - a^3*b^4)/(16*(a*b^8 - b^9)))^(1/2) + (tan(c + d*x)*(9*a^4*b^5 - 96*a^9 - 336*a^8*b +
93*a^5*b^4 + 259*a^6*b^3 + 71*a^7*b^2))/(16*b^4))*(((a^5*b^9)^(1/2) - a^3*b^4)/(16*(a*b^8 - b^9)))^(1/2)*1i -
(((((2048*a^3*b^10 + 8192*a^4*b^9 - 22528*a^5*b^8 + 12288*a^6*b^7)/(64*b^5) + (tan(c + d*x)*(((a^5*b^9)^(1/2)
- a^3*b^4)/(16*(a*b^8 - b^9)))^(1/2)*(12288*a^2*b^11 - 12288*a^3*b^10 - 12288*a^4*b^9 + 12288*a^5*b^8))/(16*b
^4))*(((a^5*b^9)^(1/2) - a^3*b^4)/(16*(a*b^8 - b^9)))^(1/2) + (tan(c + d*x)*(432*a^2*b^9 + 1584*a^3*b^8 - 880*
a^4*b^7 - 5488*a^5*b^6 + 2048*a^6*b^5 + 2304*a^7*b^4))/(16*b^4))*(((a^5*b^9)^(1/2) - a^3*b^4)/(16*(a*b^8 - b^9
)))^(1/2) - (144*a^3*b^8 + 624*a^4*b^7 + 112*a^5*b^6 - 1648*a^6*b^5 + 1536*a^7*b^4 - 768*a^8*b^3)/(64*b^5))*((
(a^5*b^9)^(1/2) - a^3*b^4)/(16*(a*b^8 - b^9)))^(1/2) - (tan(c + d*x)*(9*a^4*b^5 - 96*a^9 - 336*a^8*b + 93*a^5*
b^4 + 259*a^6*b^3 + 71*a^7*b^2))/(16*b^4))*(((a^5*b^9)^(1/2) - a^3*b^4)/(16*(a*b^8 - b^9)))^(1/2)*1i)/((((((20
48*a^3*b^10 + 8192*a^4*b^9 - 22528*a^5*b^8 + 12288*a^6*b^7)/(64*b^5) - (tan(c + d*x)*(((a^5*b^9)^(1/2) - a^3*b
^4)/(16*(a*b^8 - b^9)))^(1/2)*(12288*a^2*b^11 - 12288*a^3*b^10 - 12288*a^4*b^9 + 12288*a^5*b^8))/(16*b^4))*(((
a^5*b^9)^(1/2) - a^3*b^4)/(16*(a*b^8 - b^9)))^(1/2) - (tan(c + d*x)*(432*a^2*b^9 + 1584*a^3*b^8 - 880*a^4*b^7
- 5488*a^5*b^6 + 2048*a^6*b^5 + 2304*a^7*b^4))/(16*b^4))*(((a^5*b^9)^(1/2) - a^3*b^4)/(16*(a*b^8 - b^9)))^(1/2
) - (144*a^3*b^8 + 624*a^4*b^7 + 112*a^5*b^6 - 1648*a^6*b^5 + 1536*a^7*b^4 - 768*a^8*b^3)/(64*b^5))*(((a^5*b^9
)^(1/2) - a^3*b^4)/(16*(a*b^8 - b^9)))^(1/2) + (tan(c + d*x)*(9*a^4*b^5 - 96*a^9 - 336*a^8*b + 93*a^5*b^4 + 25
9*a^6*b^3 + 71*a^7*b^2))/(16*b^4))*(((a^5*b^9)^(1/2) - a^3*b^4)/(16*(a*b^8 - b^9)))^(1/2) + (((((2048*a^3*b^10
+ 8192*a^4*b^9 - 22528*a^5*b^8 + 12288*a^6*b^7)/(64*b^5) + (tan(c + d*x)*(((a^5*b^9)^(1/2) - a^3*b^4)/(16*(a*
b^8 - b^9)))^(1/2)*(12288*a^2*b^11 - 12288*a^3*b^10 - 12288*a^4*b^9 + 12288*a^5*b^8))/(16*b^4))*(((a^5*b^9)^(1
/2) - a^3*b^4)/(16*(a*b^8 - b^9)))^(1/2) + (tan(c + d*x)*(432*a^2*b^9 + 1584*a^3*b^8 - 880*a^4*b^7 - 5488*a^5*
b^6 + 2048*a^6*b^5 + 2304*a^7*b^4))/(16*b^4))*(((a^5*b^9)^(1/2) - a^3*b^4)/(16*(a*b^8 - b^9)))^(1/2) - (144*a^
3*b^8 + 624*a^4*b^7 + 112*a^5*b^6 - 1648*a^6*b^5 + 1536*a^7*b^4 - 768*a^8*b^3)/(64*b^5))*(((a^5*b^9)^(1/2) - a
^3*b^4)/(16*(a*b^8 - b^9)))^(1/2) - (tan(c + d*x)*(9*a^4*b^5 - 96*a^9 - 336*a^8*b + 93*a^5*b^4 + 259*a^6*b^3 +
71*a^7*b^2))/(16*b^4))*(((a^5*b^9)^(1/2) - a^3*b^4)/(16*(a*b^8 - b^9)))^(1/2) + (63*a^8*b - 216*a^9 + 27*a^6*
b^3 + 126*a^7*b^2)/(32*b^5)))*(((a^5*b^9)^(1/2) - a^3*b^4)/(16*(a*b^8 - b^9)))^(1/2)*2i)/d + ((3*tan(c + d*x))
/(8*b) + (5*tan(c + d*x)^3)/(8*b))/(d*(2*tan(c + d*x)^2 + tan(c + d*x)^4 + 1)) + (atan((((a*8i + b*3i)*((tan(c
+ d*x)*(9*a^4*b^5 - 96*a^9 - 336*a^8*b + 93*a^5*b^4 + 259*a^6*b^3 + 71*a^7*b^2))/(16*b^4) - ((((9*a^3*b^8)/4
+ (39*a^4*b^7)/4 + (7*a^5*b^6)/4 - (103*a^6*b^5)/4 + 24*a^7*b^4 - 12*a^8*b^3)/b^5 + (((tan(c + d*x)*(432*a^2*b
^9 + 1584*a^3*b^8 - 880*a^4*b^7 - 5488*a^5*b^6 + 2048*a^6*b^5 + 2304*a^7*b^4))/(16*b^4) - (((32*a^3*b^10 + 128
*a^4*b^9 - 352*a^5*b^8 + 192*a^6*b^7)/b^5 - (tan(c + d*x)*(a*8i + b*3i)*(12288*a^2*b^11 - 12288*a^3*b^10 - 122
88*a^4*b^9 + 12288*a^5*b^8))/(256*b^6))*(a*8i + b*3i))/(16*b^2))*(a*8i + b*3i))/(16*b^2))*(a*8i + b*3i))/(16*b
^2))*1i)/(16*b^2) + ((a*8i + b*3i)*((tan(c + d*x)*(9*a^4*b^5 - 96*a^9 - 336*a^8*b + 93*a^5*b^4 + 259*a^6*b^3 +
71*a^7*b^2))/(16*b^4) + ((((9*a^3*b^8)/4 + (39*a^4*b^7)/4 + (7*a^5*b^6)/4 - (103*a^6*b^5)/4 + 24*a^7*b^4 - 12
*a^8*b^3)/b^5 - (((tan(c + d*x)*(432*a^2*b^9 + 1584*a^3*b^8 - 880*a^4*b^7 - 5488*a^5*b^6 + 2048*a^6*b^5 + 2304
*a^7*b^4))/(16*b^4) + (((32*a^3*b^10 + 128*a^4*b^9 - 352*a^5*b^8 + 192*a^6*b^7)/b^5 + (tan(c + d*x)*(a*8i + b*
3i)*(12288*a^2*b^11 - 12288*a^3*b^10 - 12288*a^4*b^9 + 12288*a^5*b^8))/(256*b^6))*(a*8i + b*3i))/(16*b^2))*(a*
8i + b*3i))/(16*b^2))*(a*8i + b*3i))/(16*b^2))*1i)/(16*b^2))/(((63*a^8*b)/32 - (27*a^9)/4 + (27*a^6*b^3)/32 +
(63*a^7*b^2)/16)/b^5 + ((a*8i + b*3i)*((tan(c + d*x)*(9*a^4*b^5 - 96*a^9 - 336*a^8*b + 93*a^5*b^4 + 259*a^6*b^
3 + 71*a^7*b^2))/(16*b^4) - ((((9*a^3*b^8)/4 + (39*a^4*b^7)/4 + (7*a^5*b^6)/4 - (103*a^6*b^5)/4 + 24*a^7*b^4 -
12*a^8*b^3)/b^5 + (((tan(c + d*x)*(432*a^2*b^9 + 1584*a^3*b^8 - 880*a^4*b^7 - 5488*a^5*b^6 + 2048*a^6*b^5 + 2
304*a^7*b^4))/(16*b^4) - (((32*a^3*b^10 + 128*a^4*b^9 - 352*a^5*b^8 + 192*a^6*b^7)/b^5 - (tan(c + d*x)*(a*8i +
b*3i)*(12288*a^2*b^11 - 12288*a^3*b^10 - 12288*a^4*b^9 + 12288*a^5*b^8))/(256*b^6))*(a*8i + b*3i))/(16*b^2))*
(a*8i + b*3i))/(16*b^2))*(a*8i + b*3i))/(16*b^2)))/(16*b^2) - ((a*8i + b*3i)*((tan(c + d*x)*(9*a^4*b^5 - 96*a^
9 - 336*a^8*b + 93*a^5*b^4 + 259*a^6*b^3 + 71*a^7*b^2))/(16*b^4) + ((((9*a^3*b^8)/4 + (39*a^4*b^7)/4 + (7*a^5*
b^6)/4 - (103*a^6*b^5)/4 + 24*a^7*b^4 - 12*a^8*b^3)/b^5 - (((tan(c + d*x)*(432*a^2*b^9 + 1584*a^3*b^8 - 880*a^
4*b^7 - 5488*a^5*b^6 + 2048*a^6*b^5 + 2304*a^7*b^4))/(16*b^4) + (((32*a^3*b^10 + 128*a^4*b^9 - 352*a^5*b^8 + 1
92*a^6*b^7)/b^5 + (tan(c + d*x)*(a*8i + b*3i)*(12288*a^2*b^11 - 12288*a^3*b^10 - 12288*a^4*b^9 + 12288*a^5*b^8
))/(256*b^6))*(a*8i + b*3i))/(16*b^2))*(a*8i + b*3i))/(16*b^2))*(a*8i + b*3i))/(16*b^2)))/(16*b^2)))*(a*8i + b
*3i)*1i)/(8*b^2*d)