\(\int \frac {\sin ^6(c+d x)}{a-b \sin ^4(c+d x)} \, dx\) [204]

Optimal result

Integrand size = 24, antiderivative size = 155 \[ \int \frac {\sin ^6(c+d x)}{a-b \sin ^4(c+d x)} \, dx=-\frac {x}{2 b}+\frac {a^{3/4} \arctan \left (\frac {\sqrt {\sqrt {a}-\sqrt {b}} \tan (c+d x)}{\sqrt [4]{a}}\right )}{2 \sqrt {\sqrt {a}-\sqrt {b}} b^{3/2} d}-\frac {a^{3/4} \arctan \left (\frac {\sqrt {\sqrt {a}+\sqrt {b}} \tan (c+d x)}{\sqrt [4]{a}}\right )}{2 \sqrt {\sqrt {a}+\sqrt {b}} b^{3/2} d}+\frac {\cos (c+d x) \sin (c+d x)}{2 b d} \]

[Out]

-1/2*x/b+1/2*cos(d*x+c)*sin(d*x+c)/b/d+1/2*a^(3/4)*arctan((a^(1/2)-b^(1/2))^(1/2)*tan(d*x+c)/a^(1/4))/b^(3/2)/
d/(a^(1/2)-b^(1/2))^(1/2)-1/2*a^(3/4)*arctan((a^(1/2)+b^(1/2))^(1/2)*tan(d*x+c)/a^(1/4))/b^(3/2)/d/(a^(1/2)+b^
(1/2))^(1/2)

Rubi [A] (verified)

Time = 0.13 (sec) , antiderivative size = 155, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {3296, 1301, 205, 209, 1144, 211} \[ \int \frac {\sin ^6(c+d x)}{a-b \sin ^4(c+d x)} \, dx=\frac {a^{3/4} \arctan \left (\frac {\sqrt {\sqrt {a}-\sqrt {b}} \tan (c+d x)}{\sqrt [4]{a}}\right )}{2 b^{3/2} d \sqrt {\sqrt {a}-\sqrt {b}}}-\frac {a^{3/4} \arctan \left (\frac {\sqrt {\sqrt {a}+\sqrt {b}} \tan (c+d x)}{\sqrt [4]{a}}\right )}{2 b^{3/2} d \sqrt {\sqrt {a}+\sqrt {b}}}+\frac {\sin (c+d x) \cos (c+d x)}{2 b d}-\frac {x}{2 b} \]

[In]

Int[Sin[c + d*x]^6/(a - b*Sin[c + d*x]^4),x]

[Out]

-1/2*x/b + (a^(3/4)*ArcTan[(Sqrt[Sqrt[a] - Sqrt[b]]*Tan[c + d*x])/a^(1/4)])/(2*Sqrt[Sqrt[a] - Sqrt[b]]*b^(3/2)
*d) - (a^(3/4)*ArcTan[(Sqrt[Sqrt[a] + Sqrt[b]]*Tan[c + d*x])/a^(1/4)])/(2*Sqrt[Sqrt[a] + Sqrt[b]]*b^(3/2)*d) +
(Cos[c + d*x]*Sin[c + d*x])/(2*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 1144

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

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^6}{\left (1+x^2\right )^2 \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 )^2}-\frac {1}{b \left (1+x^2\right )}+\frac {a x^2}{b \left (a+2 a x^2+(a-b) x^4\right )}\right ) \, dx,x,\tan (c+d x)\right )}{d} \\ & = \frac {\text {Subst}\left (\int \frac {1}{\left (1+x^2\right )^2} \, dx,x,\tan (c+d x)\right )}{b d}-\frac {\text {Subst}\left (\int \frac {1}{1+x^2} \, dx,x,\tan (c+d x)\right )}{b d}+\frac {a \text {Subst}\left (\int \frac {x^2}{a+2 a x^2+(a-b) x^4} \, dx,x,\tan (c+d x)\right )}{b d} \\ & = -\frac {x}{b}+\frac {\cos (c+d x) \sin (c+d x)}{2 b d}+\frac {\left (a \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^{3/2} d}+\frac {\text {Subst}\left (\int \frac {1}{1+x^2} \, dx,x,\tan (c+d x)\right )}{2 b d}-\frac {\left (a \left (-1+\frac {\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 d} \\ & = -\frac {x}{2 b}+\frac {a^{3/4} \arctan \left (\frac {\sqrt {\sqrt {a}-\sqrt {b}} \tan (c+d x)}{\sqrt [4]{a}}\right )}{2 \sqrt {\sqrt {a}-\sqrt {b}} b^{3/2} d}-\frac {a^{3/4} \arctan \left (\frac {\sqrt {\sqrt {a}+\sqrt {b}} \tan (c+d x)}{\sqrt [4]{a}}\right )}{2 \sqrt {\sqrt {a}+\sqrt {b}} b^{3/2} d}+\frac {\cos (c+d x) \sin (c+d x)}{2 b d} \\ \end{align*}

Mathematica [A] (verified)

Time = 1.31 (sec) , antiderivative size = 157, normalized size of antiderivative = 1.01 \[ \int \frac {\sin ^6(c+d x)}{a-b \sin ^4(c+d x)} \, dx=\frac {-2 \sqrt {b} (c+d x)-\frac {2 a \arctan \left (\frac {\left (\sqrt {a}+\sqrt {b}\right ) \tan (c+d x)}{\sqrt {a+\sqrt {a} \sqrt {b}}}\right )}{\sqrt {a+\sqrt {a} \sqrt {b}}}-\frac {2 a \text {arctanh}\left (\frac {\left (\sqrt {a}-\sqrt {b}\right ) \tan (c+d x)}{\sqrt {-a+\sqrt {a} \sqrt {b}}}\right )}{\sqrt {-a+\sqrt {a} \sqrt {b}}}+\sqrt {b} \sin (2 (c+d x))}{4 b^{3/2} d} \]

[In]

Integrate[Sin[c + d*x]^6/(a - b*Sin[c + d*x]^4),x]

[Out]

(-2*Sqrt[b]*(c + d*x) - (2*a*ArcTan[((Sqrt[a] + Sqrt[b])*Tan[c + d*x])/Sqrt[a + Sqrt[a]*Sqrt[b]]])/Sqrt[a + Sq
rt[a]*Sqrt[b]] - (2*a*ArcTanh[((Sqrt[a] - Sqrt[b])*Tan[c + d*x])/Sqrt[-a + Sqrt[a]*Sqrt[b]]])/Sqrt[-a + Sqrt[a
]*Sqrt[b]] + Sqrt[b]*Sin[2*(c + d*x)])/(4*b^(3/2)*d)

Maple [C] (verified)

Result contains higher order function than in optimal. Order 9 vs. order 3.

Time = 1.31 (sec) , antiderivative size = 170, normalized size of antiderivative = 1.10

method	result	size
risch	\(-\frac {x}{2 b}-\frac {i {\mathrm e}^{2 i \left (d x +c \right )}}{8 b d}+\frac {i {\mathrm e}^{-2 i \left (d x +c \right )}}{8 b d}-\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\left (a \,b^{6} d^{4}-b^{7} d^{4}\right ) \textit {\_Z}^{4}+512 a^{2} b^{3} d^{2} \textit {\_Z}^{2}+65536 a^{3}\right )}{\sum }\textit {\_R} \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}+\left (\frac {i b^{4} d^{3}}{2048 a}-\frac {i b^{5} d^{3}}{2048 a^{2}}\right ) \textit {\_R}^{3}+\left (-\frac {b^{2} d^{2}}{128}+\frac {b^{3} d^{2}}{128 a}\right ) \textit {\_R}^{2}+\frac {i d b \textit {\_R}}{4}-\frac {2 a}{b}-1\right )\right )}{64}\)	\(170\)
derivativedivides	\(\frac {-\frac {-\frac {\tan \left (d x +c \right )}{2 \left (1+\tan ^{2}\left (d x +c \right )\right )}+\frac {\arctan \left (\tan \left (d x +c \right )\right )}{2}}{b}+\frac {a \left (a -b \right ) \left (\frac {\left (\sqrt {a b}+a \right ) \arctan \left (\frac {\left (a -b \right ) \tan \left (d x +c \right )}{\sqrt {\left (\sqrt {a b}+a \right ) \left (a -b \right )}}\right )}{2 \left (a -b \right ) \sqrt {a b}\, \sqrt {\left (\sqrt {a b}+a \right ) \left (a -b \right )}}+\frac {\left (\sqrt {a b}-a \right ) \operatorname {arctanh}\left (\frac {\left (-a +b \right ) \tan \left (d x +c \right )}{\sqrt {\left (\sqrt {a b}-a \right ) \left (a -b \right )}}\right )}{2 \sqrt {a b}\, \left (a -b \right ) \sqrt {\left (\sqrt {a b}-a \right ) \left (a -b \right )}}\right )}{b}}{d}\)	\(186\)
default	\(\frac {-\frac {-\frac {\tan \left (d x +c \right )}{2 \left (1+\tan ^{2}\left (d x +c \right )\right )}+\frac {\arctan \left (\tan \left (d x +c \right )\right )}{2}}{b}+\frac {a \left (a -b \right ) \left (\frac {\left (\sqrt {a b}+a \right ) \arctan \left (\frac {\left (a -b \right ) \tan \left (d x +c \right )}{\sqrt {\left (\sqrt {a b}+a \right ) \left (a -b \right )}}\right )}{2 \left (a -b \right ) \sqrt {a b}\, \sqrt {\left (\sqrt {a b}+a \right ) \left (a -b \right )}}+\frac {\left (\sqrt {a b}-a \right ) \operatorname {arctanh}\left (\frac {\left (-a +b \right ) \tan \left (d x +c \right )}{\sqrt {\left (\sqrt {a b}-a \right ) \left (a -b \right )}}\right )}{2 \sqrt {a b}\, \left (a -b \right ) \sqrt {\left (\sqrt {a b}-a \right ) \left (a -b \right )}}\right )}{b}}{d}\)	\(186\)

[In]

int(sin(d*x+c)^6/(a-b*sin(d*x+c)^4),x,method=_RETURNVERBOSE)

[Out]

-1/2*x/b-1/8*I/b/d*exp(2*I*(d*x+c))+1/8*I/b/d*exp(-2*I*(d*x+c))-1/64*sum(_R*ln(exp(2*I*(d*x+c))+(1/2048*I/a*b^
4*d^3-1/2048*I/a^2*b^5*d^3)*_R^3+(-1/128*b^2*d^2+1/128/a*b^3*d^2)*_R^2+1/4*I*d*b*_R-2/b*a-1),_R=RootOf((a*b^6*
d^4-b^7*d^4)*_Z^4+512*a^2*b^3*d^2*_Z^2+65536*a^3))

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1275 vs. \(2 (111) = 222\).

Time = 0.45 (sec) , antiderivative size = 1275, normalized size of antiderivative = 8.23 \[ \int \frac {\sin ^6(c+d x)}{a-b \sin ^4(c+d x)} \, dx=\text {Too large to display} \]

[In]

integrate(sin(d*x+c)^6/(a-b*sin(d*x+c)^4),x, algorithm="fricas")

[Out]

-1/8*(b*d*sqrt(-((a*b^3 - b^4)*d^2*sqrt(a^3/((a^2*b^5 - 2*a*b^6 + b^7)*d^4)) + a^2)/((a*b^3 - b^4)*d^2))*log(1
/4*a^2*cos(d*x + c)^2 - 1/4*a^2 - 1/4*(2*(a^2*b^2 - a*b^3)*d^2*cos(d*x + c)^2 - (a^2*b^2 - a*b^3)*d^2)*sqrt(a^
3/((a^2*b^5 - 2*a*b^6 + b^7)*d^4)) + 1/2*((a*b^4 - b^5)*d^3*sqrt(a^3/((a^2*b^5 - 2*a*b^6 + b^7)*d^4))*cos(d*x
+ c)*sin(d*x + c) - a^2*b*d*cos(d*x + c)*sin(d*x + c))*sqrt(-((a*b^3 - b^4)*d^2*sqrt(a^3/((a^2*b^5 - 2*a*b^6 +
b^7)*d^4)) + a^2)/((a*b^3 - b^4)*d^2))) - b*d*sqrt(-((a*b^3 - b^4)*d^2*sqrt(a^3/((a^2*b^5 - 2*a*b^6 + b^7)*d^
4)) + a^2)/((a*b^3 - b^4)*d^2))*log(1/4*a^2*cos(d*x + c)^2 - 1/4*a^2 - 1/4*(2*(a^2*b^2 - a*b^3)*d^2*cos(d*x +
c)^2 - (a^2*b^2 - a*b^3)*d^2)*sqrt(a^3/((a^2*b^5 - 2*a*b^6 + b^7)*d^4)) - 1/2*((a*b^4 - b^5)*d^3*sqrt(a^3/((a^
2*b^5 - 2*a*b^6 + b^7)*d^4))*cos(d*x + c)*sin(d*x + c) - a^2*b*d*cos(d*x + c)*sin(d*x + c))*sqrt(-((a*b^3 - b^
4)*d^2*sqrt(a^3/((a^2*b^5 - 2*a*b^6 + b^7)*d^4)) + a^2)/((a*b^3 - b^4)*d^2))) + b*d*sqrt(((a*b^3 - b^4)*d^2*sq
rt(a^3/((a^2*b^5 - 2*a*b^6 + b^7)*d^4)) - a^2)/((a*b^3 - b^4)*d^2))*log(-1/4*a^2*cos(d*x + c)^2 + 1/4*a^2 - 1/
4*(2*(a^2*b^2 - a*b^3)*d^2*cos(d*x + c)^2 - (a^2*b^2 - a*b^3)*d^2)*sqrt(a^3/((a^2*b^5 - 2*a*b^6 + b^7)*d^4)) +
1/2*((a*b^4 - b^5)*d^3*sqrt(a^3/((a^2*b^5 - 2*a*b^6 + b^7)*d^4))*cos(d*x + c)*sin(d*x + c) + a^2*b*d*cos(d*x
+ c)*sin(d*x + c))*sqrt(((a*b^3 - b^4)*d^2*sqrt(a^3/((a^2*b^5 - 2*a*b^6 + b^7)*d^4)) - a^2)/((a*b^3 - b^4)*d^2
))) - b*d*sqrt(((a*b^3 - b^4)*d^2*sqrt(a^3/((a^2*b^5 - 2*a*b^6 + b^7)*d^4)) - a^2)/((a*b^3 - b^4)*d^2))*log(-1
/4*a^2*cos(d*x + c)^2 + 1/4*a^2 - 1/4*(2*(a^2*b^2 - a*b^3)*d^2*cos(d*x + c)^2 - (a^2*b^2 - a*b^3)*d^2)*sqrt(a^
3/((a^2*b^5 - 2*a*b^6 + b^7)*d^4)) - 1/2*((a*b^4 - b^5)*d^3*sqrt(a^3/((a^2*b^5 - 2*a*b^6 + b^7)*d^4))*cos(d*x
+ c)*sin(d*x + c) + a^2*b*d*cos(d*x + c)*sin(d*x + c))*sqrt(((a*b^3 - b^4)*d^2*sqrt(a^3/((a^2*b^5 - 2*a*b^6 +
b^7)*d^4)) - a^2)/((a*b^3 - b^4)*d^2))) + 4*d*x - 4*cos(d*x + c)*sin(d*x + c))/(b*d)

Sympy [F(-1)]

Timed out. \[ \int \frac {\sin ^6(c+d x)}{a-b \sin ^4(c+d x)} \, dx=\text {Timed out} \]

[In]

integrate(sin(d*x+c)**6/(a-b*sin(d*x+c)**4),x)

[Out]

Timed out

Maxima [F]

\[ \int \frac {\sin ^6(c+d x)}{a-b \sin ^4(c+d x)} \, dx=\int { -\frac {\sin \left (d x + c\right )^{6}}{b \sin \left (d x + c\right )^{4} - a} \,d x } \]

[In]

integrate(sin(d*x+c)^6/(a-b*sin(d*x+c)^4),x, algorithm="maxima")

[Out]

1/4*(4*b*d*integrate(-4*(4*a*b*cos(6*d*x + 6*c)^2 + 4*a*b*cos(2*d*x + 2*c)^2 + 4*a*b*sin(6*d*x + 6*c)^2 + 4*a*
b*sin(2*d*x + 2*c)^2 - 4*(8*a^2 - 3*a*b)*cos(4*d*x + 4*c)^2 - a*b*cos(2*d*x + 2*c) - 4*(8*a^2 - 3*a*b)*sin(4*d
*x + 4*c)^2 + 2*(8*a^2 - 7*a*b)*sin(4*d*x + 4*c)*sin(2*d*x + 2*c) - (a*b*cos(6*d*x + 6*c) - 2*a*b*cos(4*d*x +
4*c) + a*b*cos(2*d*x + 2*c))*cos(8*d*x + 8*c) + (8*a*b*cos(2*d*x + 2*c) - a*b + 2*(8*a^2 - 7*a*b)*cos(4*d*x +
4*c))*cos(6*d*x + 6*c) + 2*(a*b + (8*a^2 - 7*a*b)*cos(2*d*x + 2*c))*cos(4*d*x + 4*c) - (a*b*sin(6*d*x + 6*c) -
2*a*b*sin(4*d*x + 4*c) + a*b*sin(2*d*x + 2*c))*sin(8*d*x + 8*c) + 2*(4*a*b*sin(2*d*x + 2*c) + (8*a^2 - 7*a*b)
*sin(4*d*x + 4*c))*sin(6*d*x + 6*c))/(b^3*cos(8*d*x + 8*c)^2 + 16*b^3*cos(6*d*x + 6*c)^2 + 16*b^3*cos(2*d*x +
2*c)^2 + b^3*sin(8*d*x + 8*c)^2 + 16*b^3*sin(6*d*x + 6*c)^2 + 16*b^3*sin(2*d*x + 2*c)^2 - 8*b^3*cos(2*d*x + 2*
c) + b^3 + 4*(64*a^2*b - 48*a*b^2 + 9*b^3)*cos(4*d*x + 4*c)^2 + 4*(64*a^2*b - 48*a*b^2 + 9*b^3)*sin(4*d*x + 4*
c)^2 + 16*(8*a*b^2 - 3*b^3)*sin(4*d*x + 4*c)*sin(2*d*x + 2*c) - 2*(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))*cos(8*d*x + 8*c) + 8*(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))*cos(6*d*x + 6*c) - 4*(8*a*b^2 - 3*b^3 - 4*(8*a*b^2 - 3*b^3)*cos(2*d*x + 2*c))*
cos(4*d*x + 4*c) - 4*(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))*si
n(8*d*x + 8*c) + 16*(2*b^3*sin(2*d*x + 2*c) + (8*a*b^2 - 3*b^3)*sin(4*d*x + 4*c))*sin(6*d*x + 6*c)), x) - 2*d*
x + sin(2*d*x + 2*c))/(b*d)

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 695 vs. \(2 (111) = 222\).

Time = 0.78 (sec) , antiderivative size = 695, normalized size of antiderivative = 4.48 \[ \int \frac {\sin ^6(c+d x)}{a-b \sin ^4(c+d x)} \, dx=-\frac {\frac {d x + c}{b} + \frac {{\left ({\left (3 \, \sqrt {a^{2} - a b + \sqrt {a b} {\left (a - b\right )}} \sqrt {a b} a^{2} - 6 \, \sqrt {a^{2} - a b + \sqrt {a b} {\left (a - b\right )}} \sqrt {a b} a b - \sqrt {a^{2} - a b + \sqrt {a b} {\left (a - b\right )}} \sqrt {a b} b^{2}\right )} b^{2} {\left | -a + b \right |} - {\left (3 \, \sqrt {a^{2} - a b + \sqrt {a b} {\left (a - b\right )}} \sqrt {a b} a^{3} b - 6 \, \sqrt {a^{2} - a b + \sqrt {a b} {\left (a - b\right )}} \sqrt {a b} a^{2} b^{2} - \sqrt {a^{2} - a b + \sqrt {a b} {\left (a - b\right )}} \sqrt {a b} a b^{3}\right )} {\left | -a + b \right |}\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 + \sqrt {a^{2} b^{2} - {\left (a b - b^{2}\right )} a b}}{a b - b^{2}}}}\right )\right )}}{{\left (3 \, a^{5} b^{2} - 15 \, a^{4} b^{3} + 26 \, a^{3} b^{4} - 18 \, a^{2} b^{5} + 3 \, a b^{6} + b^{7}\right )} {\left | b \right |}} - \frac {{\left ({\left (3 \, \sqrt {a^{2} - a b - \sqrt {a b} {\left (a - b\right )}} \sqrt {a b} a^{2} - 6 \, \sqrt {a^{2} - a b - \sqrt {a b} {\left (a - b\right )}} \sqrt {a b} a b - \sqrt {a^{2} - a b - \sqrt {a b} {\left (a - b\right )}} \sqrt {a b} b^{2}\right )} b^{2} {\left | -a + b \right |} - {\left (3 \, \sqrt {a^{2} - a b - \sqrt {a b} {\left (a - b\right )}} \sqrt {a b} a^{3} b - 6 \, \sqrt {a^{2} - a b - \sqrt {a b} {\left (a - b\right )}} \sqrt {a b} a^{2} b^{2} - \sqrt {a^{2} - a b - \sqrt {a b} {\left (a - b\right )}} \sqrt {a b} a b^{3}\right )} {\left | -a + b \right |}\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 - \sqrt {a^{2} b^{2} - {\left (a b - b^{2}\right )} a b}}{a b - b^{2}}}}\right )\right )}}{{\left (3 \, a^{5} b^{2} - 15 \, a^{4} b^{3} + 26 \, a^{3} b^{4} - 18 \, a^{2} b^{5} + 3 \, a b^{6} + b^{7}\right )} {\left | b \right |}} - \frac {\tan \left (d x + c\right )}{{\left (\tan \left (d x + c\right )^{2} + 1\right )} b}}{2 \, d} \]

[In]

integrate(sin(d*x+c)^6/(a-b*sin(d*x+c)^4),x, algorithm="giac")

[Out]

-1/2*((d*x + c)/b + ((3*sqrt(a^2 - a*b + sqrt(a*b)*(a - b))*sqrt(a*b)*a^2 - 6*sqrt(a^2 - a*b + sqrt(a*b)*(a -
b))*sqrt(a*b)*a*b - sqrt(a^2 - a*b + sqrt(a*b)*(a - b))*sqrt(a*b)*b^2)*b^2*abs(-a + b) - (3*sqrt(a^2 - a*b + s
qrt(a*b)*(a - b))*sqrt(a*b)*a^3*b - 6*sqrt(a^2 - a*b + sqrt(a*b)*(a - b))*sqrt(a*b)*a^2*b^2 - sqrt(a^2 - a*b +
sqrt(a*b)*(a - b))*sqrt(a*b)*a*b^3)*abs(-a + b))*(pi*floor((d*x + c)/pi + 1/2) + arctan(tan(d*x + c)/sqrt((a*
b + sqrt(a^2*b^2 - (a*b - b^2)*a*b))/(a*b - b^2))))/((3*a^5*b^2 - 15*a^4*b^3 + 26*a^3*b^4 - 18*a^2*b^5 + 3*a*b
^6 + b^7)*abs(b)) - ((3*sqrt(a^2 - a*b - sqrt(a*b)*(a - b))*sqrt(a*b)*a^2 - 6*sqrt(a^2 - a*b - sqrt(a*b)*(a -
b))*sqrt(a*b)*a*b - sqrt(a^2 - a*b - sqrt(a*b)*(a - b))*sqrt(a*b)*b^2)*b^2*abs(-a + b) - (3*sqrt(a^2 - a*b - s
qrt(a*b)*(a - b))*sqrt(a*b)*a^3*b - 6*sqrt(a^2 - a*b - sqrt(a*b)*(a - b))*sqrt(a*b)*a^2*b^2 - sqrt(a^2 - a*b -
sqrt(a*b)*(a - b))*sqrt(a*b)*a*b^3)*abs(-a + b))*(pi*floor((d*x + c)/pi + 1/2) + arctan(tan(d*x + c)/sqrt((a*
b - sqrt(a^2*b^2 - (a*b - b^2)*a*b))/(a*b - b^2))))/((3*a^5*b^2 - 15*a^4*b^3 + 26*a^3*b^4 - 18*a^2*b^5 + 3*a*b
^6 + b^7)*abs(b)) - tan(d*x + c)/((tan(d*x + c)^2 + 1)*b))/d

Mupad [B] (verification not implemented)

Time = 16.09 (sec) , antiderivative size = 1273, normalized size of antiderivative = 8.21 \[ \int \frac {\sin ^6(c+d x)}{a-b \sin ^4(c+d x)} \, dx=\text {Too large to display} \]

[In]

int(sin(c + d*x)^6/(a - b*sin(c + d*x)^4),x)

[Out]

sin(2*c + 2*d*x)/(4*b*d) - (atan((a*b^7*sin(c + d*x)*(((a^3*b^7)^(1/2) - a^2*b^3)/(16*a*b^6 - 16*b^7))^(1/2)*4
i - b^12*sin(c + d*x)*(((a^3*b^7)^(1/2) - a^2*b^3)/(16*a*b^6 - 16*b^7))^(5/2)*3072i - b^10*sin(c + d*x)*(((a^3
*b^7)^(1/2) - a^2*b^3)/(16*a*b^6 - 16*b^7))^(3/2)*192i + a*b^9*sin(c + d*x)*(((a^3*b^7)^(1/2) - a^2*b^3)/(16*a
*b^6 - 16*b^7))^(3/2)*192i + a^2*b^6*sin(c + d*x)*(((a^3*b^7)^(1/2) - a^2*b^3)/(16*a*b^6 - 16*b^7))^(1/2)*24i
+ a^3*b^5*sin(c + d*x)*(((a^3*b^7)^(1/2) - a^2*b^3)/(16*a*b^6 - 16*b^7))^(1/2)*4i + a^4*b^4*sin(c + d*x)*(((a^
3*b^7)^(1/2) - a^2*b^3)/(16*a*b^6 - 16*b^7))^(1/2)*8i + a^2*b^8*sin(c + d*x)*(((a^3*b^7)^(1/2) - a^2*b^3)/(16*
a*b^6 - 16*b^7))^(3/2)*448i + a^3*b^7*sin(c + d*x)*(((a^3*b^7)^(1/2) - a^2*b^3)/(16*a*b^6 - 16*b^7))^(3/2)*320
i + a^2*b^10*sin(c + d*x)*(((a^3*b^7)^(1/2) - a^2*b^3)/(16*a*b^6 - 16*b^7))^(5/2)*3072i)/(a^2*b^5*cos(c + d*x)
+ a^3*b^4*cos(c + d*x) - a^4*b^3*cos(c + d*x) - a^2*cos(c + d*x)*(a^3*b^7)^(1/2) + 2*a*b*cos(c + d*x)*(a^3*b^
7)^(1/2)))*(((a^3*b^7)^(1/2) - a^2*b^3)/(16*a*b^6 - 16*b^7))^(1/2)*2i)/d - (atan((a*b^7*sin(c + d*x)*(-((a^3*b
^7)^(1/2) + a^2*b^3)/(16*a*b^6 - 16*b^7))^(1/2)*4i - b^12*sin(c + d*x)*(-((a^3*b^7)^(1/2) + a^2*b^3)/(16*a*b^6
- 16*b^7))^(5/2)*3072i - b^10*sin(c + d*x)*(-((a^3*b^7)^(1/2) + a^2*b^3)/(16*a*b^6 - 16*b^7))^(3/2)*192i + a*
b^9*sin(c + d*x)*(-((a^3*b^7)^(1/2) + a^2*b^3)/(16*a*b^6 - 16*b^7))^(3/2)*192i + a^2*b^6*sin(c + d*x)*(-((a^3*
b^7)^(1/2) + a^2*b^3)/(16*a*b^6 - 16*b^7))^(1/2)*24i + a^3*b^5*sin(c + d*x)*(-((a^3*b^7)^(1/2) + a^2*b^3)/(16*
a*b^6 - 16*b^7))^(1/2)*4i + a^4*b^4*sin(c + d*x)*(-((a^3*b^7)^(1/2) + a^2*b^3)/(16*a*b^6 - 16*b^7))^(1/2)*8i +
a^2*b^8*sin(c + d*x)*(-((a^3*b^7)^(1/2) + a^2*b^3)/(16*a*b^6 - 16*b^7))^(3/2)*448i + a^3*b^7*sin(c + d*x)*(-(
(a^3*b^7)^(1/2) + a^2*b^3)/(16*a*b^6 - 16*b^7))^(3/2)*320i + a^2*b^10*sin(c + d*x)*(-((a^3*b^7)^(1/2) + a^2*b^
3)/(16*a*b^6 - 16*b^7))^(5/2)*3072i)/(a^2*b^5*cos(c + d*x) + a^3*b^4*cos(c + d*x) - a^4*b^3*cos(c + d*x) + a^2
*cos(c + d*x)*(a^3*b^7)^(1/2) - 2*a*b*cos(c + d*x)*(a^3*b^7)^(1/2)))*(-((a^3*b^7)^(1/2) + a^2*b^3)/(16*a*b^6 -
16*b^7))^(1/2)*2i)/d - atan(sin(c + d*x)/cos(c + d*x))/(2*b*d)