\(\int \frac {\sin ^4(c+d x)}{a-b \sin ^4(c+d x)} \, dx\) [205]

Optimal result

Integrand size = 24, antiderivative size = 127 \[ \int \frac {\sin ^4(c+d x)}{a-b \sin ^4(c+d x)} \, dx=-\frac {x}{b}+\frac {\sqrt [4]{a} \arctan \left (\frac {\sqrt {\sqrt {a}-\sqrt {b}} \tan (c+d x)}{\sqrt [4]{a}}\right )}{2 \sqrt {\sqrt {a}-\sqrt {b}} b d}+\frac {\sqrt [4]{a} \arctan \left (\frac {\sqrt {\sqrt {a}+\sqrt {b}} \tan (c+d x)}{\sqrt [4]{a}}\right )}{2 \sqrt {\sqrt {a}+\sqrt {b}} b d} \]

[Out]

-x/b+1/2*a^(1/4)*arctan((a^(1/2)-b^(1/2))^(1/2)*tan(d*x+c)/a^(1/4))/b/d/(a^(1/2)-b^(1/2))^(1/2)+1/2*a^(1/4)*ar
ctan((a^(1/2)+b^(1/2))^(1/2)*tan(d*x+c)/a^(1/4))/b/d/(a^(1/2)+b^(1/2))^(1/2)

Rubi [A] (verified)

Time = 0.12 (sec) , antiderivative size = 127, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.208, Rules used = {3296, 1301, 209, 1180, 211} \[ \int \frac {\sin ^4(c+d x)}{a-b \sin ^4(c+d x)} \, dx=\frac {\sqrt [4]{a} \arctan \left (\frac {\sqrt {\sqrt {a}-\sqrt {b}} \tan (c+d x)}{\sqrt [4]{a}}\right )}{2 b d \sqrt {\sqrt {a}-\sqrt {b}}}+\frac {\sqrt [4]{a} \arctan \left (\frac {\sqrt {\sqrt {a}+\sqrt {b}} \tan (c+d x)}{\sqrt [4]{a}}\right )}{2 b d \sqrt {\sqrt {a}+\sqrt {b}}}-\frac {x}{b} \]

[In]

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

[Out]

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

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^4}{\left (1+x^2\right ) \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 )}+\frac {a \left (1+x^2\right )}{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}{1+x^2} \, dx,x,\tan (c+d x)\right )}{b d}+\frac {a \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 d} \\ & = -\frac {x}{b}+\frac {\left (a \left (1-\frac {\sqrt {b}}{\sqrt {a}}\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 {\left (a \left (1+\frac {\sqrt {b}}{\sqrt {a}}\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}{b}+\frac {\sqrt [4]{a} \arctan \left (\frac {\sqrt {\sqrt {a}-\sqrt {b}} \tan (c+d x)}{\sqrt [4]{a}}\right )}{2 \sqrt {\sqrt {a}-\sqrt {b}} b d}+\frac {\sqrt [4]{a} \arctan \left (\frac {\sqrt {\sqrt {a}+\sqrt {b}} \tan (c+d x)}{\sqrt [4]{a}}\right )}{2 \sqrt {\sqrt {a}+\sqrt {b}} b d} \\ \end{align*}

Mathematica [A] (verified)

Time = 2.23 (sec) , antiderivative size = 143, normalized size of antiderivative = 1.13 \[ \int \frac {\sin ^4(c+d x)}{a-b \sin ^4(c+d x)} \, dx=\frac {-2 (c+d x)+\frac {\sqrt {a} \arctan \left (\frac {\left (\sqrt {a}+\sqrt {b}\right ) \tan (c+d x)}{\sqrt {a+\sqrt {a} \sqrt {b}}}\right )}{\sqrt {a+\sqrt {a} \sqrt {b}}}-\frac {\sqrt {a} \text {arctanh}\left (\frac {\left (\sqrt {a}-\sqrt {b}\right ) \tan (c+d x)}{\sqrt {-a+\sqrt {a} \sqrt {b}}}\right )}{\sqrt {-a+\sqrt {a} \sqrt {b}}}}{2 b d} \]

[In]

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

[Out]

(-2*(c + d*x) + (Sqrt[a]*ArcTan[((Sqrt[a] + Sqrt[b])*Tan[c + d*x])/Sqrt[a + Sqrt[a]*Sqrt[b]]])/Sqrt[a + Sqrt[a
]*Sqrt[b]] - (Sqrt[a]*ArcTanh[((Sqrt[a] - Sqrt[b])*Tan[c + d*x])/Sqrt[-a + Sqrt[a]*Sqrt[b]]])/Sqrt[-a + Sqrt[a
]*Sqrt[b]])/(2*b*d)

Maple [C] (verified)

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

Time = 0.72 (sec) , antiderivative size = 128, normalized size of antiderivative = 1.01

method	result	size
risch	\(-\frac {x}{b}+\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\left (a \,b^{4} d^{4}-b^{5} d^{4}\right ) \textit {\_Z}^{4}+32 a \,b^{2} d^{2} \textit {\_Z}^{2}+256 a \right )}{\sum }\textit {\_R} \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}+\left (\frac {1}{32} i a \,b^{2} d^{3}-\frac {1}{32} i b^{3} d^{3}\right ) \textit {\_R}^{3}+\left (-\frac {1}{8} b \,d^{2} a +\frac {1}{8} b^{2} d^{2}\right ) \textit {\_R}^{2}+\left (\frac {1}{2} i a d +\frac {1}{2} i d b \right ) \textit {\_R} -\frac {2 a}{b}-1\right )\right )}{16}\)	\(128\)
derivativedivides	\(\frac {-\frac {\arctan \left (\tan \left (d x +c \right )\right )}{b}+\frac {a \left (a -b \right ) \left (\frac {\left (\sqrt {a b}+b \right ) \arctan \left (\frac {\left (a -b \right ) \tan \left (d x +c \right )}{\sqrt {\left (\sqrt {a b}+a \right ) \left (a -b \right )}}\right )}{2 \sqrt {a b}\, \left (a -b \right ) \sqrt {\left (\sqrt {a b}+a \right ) \left (a -b \right )}}+\frac {\left (\sqrt {a b}-b \right ) \operatorname {arctanh}\left (\frac {\left (-a +b \right ) \tan \left (d x +c \right )}{\sqrt {\left (\sqrt {a b}-a \right ) \left (a -b \right )}}\right )}{2 \sqrt {a b}\, \left (a -b \right ) \sqrt {\left (\sqrt {a b}-a \right ) \left (a -b \right )}}\right )}{b}}{d}\)	\(163\)
default	\(\frac {-\frac {\arctan \left (\tan \left (d x +c \right )\right )}{b}+\frac {a \left (a -b \right ) \left (\frac {\left (\sqrt {a b}+b \right ) \arctan \left (\frac {\left (a -b \right ) \tan \left (d x +c \right )}{\sqrt {\left (\sqrt {a b}+a \right ) \left (a -b \right )}}\right )}{2 \sqrt {a b}\, \left (a -b \right ) \sqrt {\left (\sqrt {a b}+a \right ) \left (a -b \right )}}+\frac {\left (\sqrt {a b}-b \right ) \operatorname {arctanh}\left (\frac {\left (-a +b \right ) \tan \left (d x +c \right )}{\sqrt {\left (\sqrt {a b}-a \right ) \left (a -b \right )}}\right )}{2 \sqrt {a b}\, \left (a -b \right ) \sqrt {\left (\sqrt {a b}-a \right ) \left (a -b \right )}}\right )}{b}}{d}\)	\(163\)

[In]

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

[Out]

-x/b+1/16*sum(_R*ln(exp(2*I*(d*x+c))+(1/32*I*a*b^2*d^3-1/32*I*b^3*d^3)*_R^3+(-1/8*b*d^2*a+1/8*b^2*d^2)*_R^2+(1
/2*I*a*d+1/2*I*d*b)*_R-2/b*a-1),_R=RootOf((a*b^4*d^4-b^5*d^4)*_Z^4+32*a*b^2*d^2*_Z^2+256*a))

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1125 vs. \(2 (91) = 182\).

Time = 0.40 (sec) , antiderivative size = 1125, normalized size of antiderivative = 8.86 \[ \int \frac {\sin ^4(c+d x)}{a-b \sin ^4(c+d x)} \, dx=\text {Too large to display} \]

[In]

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

[Out]

1/8*(b*sqrt(-((a*b^2 - b^3)*d^2*sqrt(a/((a^2*b^3 - 2*a*b^4 + b^5)*d^4)) + a)/((a*b^2 - b^3)*d^2))*log(1/4*cos(
d*x + c)^2 + 1/2*((a*b^2 - b^3)*d^3*sqrt(a/((a^2*b^3 - 2*a*b^4 + b^5)*d^4))*cos(d*x + c)*sin(d*x + c) - b*d*co
s(d*x + c)*sin(d*x + c))*sqrt(-((a*b^2 - b^3)*d^2*sqrt(a/((a^2*b^3 - 2*a*b^4 + b^5)*d^4)) + a)/((a*b^2 - b^3)*
d^2)) - 1/4*(2*(a*b - b^2)*d^2*cos(d*x + c)^2 - (a*b - b^2)*d^2)*sqrt(a/((a^2*b^3 - 2*a*b^4 + b^5)*d^4)) - 1/4
) - b*sqrt(-((a*b^2 - b^3)*d^2*sqrt(a/((a^2*b^3 - 2*a*b^4 + b^5)*d^4)) + a)/((a*b^2 - b^3)*d^2))*log(1/4*cos(d
*x + c)^2 - 1/2*((a*b^2 - b^3)*d^3*sqrt(a/((a^2*b^3 - 2*a*b^4 + b^5)*d^4))*cos(d*x + c)*sin(d*x + c) - b*d*cos
(d*x + c)*sin(d*x + c))*sqrt(-((a*b^2 - b^3)*d^2*sqrt(a/((a^2*b^3 - 2*a*b^4 + b^5)*d^4)) + a)/((a*b^2 - b^3)*d
^2)) - 1/4*(2*(a*b - b^2)*d^2*cos(d*x + c)^2 - (a*b - b^2)*d^2)*sqrt(a/((a^2*b^3 - 2*a*b^4 + b^5)*d^4)) - 1/4)
+ b*sqrt(((a*b^2 - b^3)*d^2*sqrt(a/((a^2*b^3 - 2*a*b^4 + b^5)*d^4)) - a)/((a*b^2 - b^3)*d^2))*log(-1/4*cos(d*
x + c)^2 + 1/2*((a*b^2 - b^3)*d^3*sqrt(a/((a^2*b^3 - 2*a*b^4 + b^5)*d^4))*cos(d*x + c)*sin(d*x + c) + b*d*cos(
d*x + c)*sin(d*x + c))*sqrt(((a*b^2 - b^3)*d^2*sqrt(a/((a^2*b^3 - 2*a*b^4 + b^5)*d^4)) - a)/((a*b^2 - b^3)*d^2
)) - 1/4*(2*(a*b - b^2)*d^2*cos(d*x + c)^2 - (a*b - b^2)*d^2)*sqrt(a/((a^2*b^3 - 2*a*b^4 + b^5)*d^4)) + 1/4) -
b*sqrt(((a*b^2 - b^3)*d^2*sqrt(a/((a^2*b^3 - 2*a*b^4 + b^5)*d^4)) - a)/((a*b^2 - b^3)*d^2))*log(-1/4*cos(d*x
+ c)^2 - 1/2*((a*b^2 - b^3)*d^3*sqrt(a/((a^2*b^3 - 2*a*b^4 + b^5)*d^4))*cos(d*x + c)*sin(d*x + c) + b*d*cos(d*
x + c)*sin(d*x + c))*sqrt(((a*b^2 - b^3)*d^2*sqrt(a/((a^2*b^3 - 2*a*b^4 + b^5)*d^4)) - a)/((a*b^2 - b^3)*d^2))
- 1/4*(2*(a*b - b^2)*d^2*cos(d*x + c)^2 - (a*b - b^2)*d^2)*sqrt(a/((a^2*b^3 - 2*a*b^4 + b^5)*d^4)) + 1/4) - 8
*x)/b

Sympy [F(-1)]

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

[In]

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

[Out]

Timed out

Maxima [F]

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

[In]

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

[Out]

-(16*a*b*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^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))*sin(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) + x)/b

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 912 vs. \(2 (91) = 182\).

Time = 0.72 (sec) , antiderivative size = 912, normalized size of antiderivative = 7.18 \[ \int \frac {\sin ^4(c+d x)}{a-b \sin ^4(c+d x)} \, dx=-\frac {\frac {2 \, {\left (d x + c\right )}}{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 )}} a^{3} b - 9 \, \sqrt {a^{2} - a b + \sqrt {a b} {\left (a - b\right )}} a^{2} b^{2} + 5 \, \sqrt {a^{2} - a b + \sqrt {a b} {\left (a - b\right )}} a b^{3} + \sqrt {a^{2} - a b + \sqrt {a b} {\left (a - b\right )}} b^{4}\right )} {\left | -a + b \right |} {\left | b \right |} - {\left (3 \, \sqrt {a^{2} - a b + \sqrt {a b} {\left (a - b\right )}} \sqrt {a b} a^{2} b^{2} - 6 \, \sqrt {a^{2} - a b + \sqrt {a b} {\left (a - b\right )}} \sqrt {a b} a b^{3} - \sqrt {a^{2} - a b + \sqrt {a b} {\left (a - b\right )}} \sqrt {a b} b^{4}\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 )}} a^{3} b - 9 \, \sqrt {a^{2} - a b - \sqrt {a b} {\left (a - b\right )}} a^{2} b^{2} + 5 \, \sqrt {a^{2} - a b - \sqrt {a b} {\left (a - b\right )}} a b^{3} + \sqrt {a^{2} - a b - \sqrt {a b} {\left (a - b\right )}} b^{4}\right )} {\left | -a + b \right |} {\left | b \right |} - {\left (3 \, \sqrt {a^{2} - a b - \sqrt {a b} {\left (a - b\right )}} \sqrt {a b} a^{2} b^{2} - 6 \, \sqrt {a^{2} - a b - \sqrt {a b} {\left (a - b\right )}} \sqrt {a b} a b^{3} - \sqrt {a^{2} - a b - \sqrt {a b} {\left (a - b\right )}} \sqrt {a b} b^{4}\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 |}}}{2 \, d} \]

[In]

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

[Out]

-1/2*(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 +
sqrt(a*b)*(a - b))*a^3*b - 9*sqrt(a^2 - a*b + sqrt(a*b)*(a - b))*a^2*b^2 + 5*sqrt(a^2 - a*b + sqrt(a*b)*(a -
b))*a*b^3 + sqrt(a^2 - a*b + sqrt(a*b)*(a - b))*b^4)*abs(-a + b)*abs(b) - (3*sqrt(a^2 - a*b + sqrt(a*b)*(a - b
))*sqrt(a*b)*a^2*b^2 - 6*sqrt(a^2 - a*b + sqrt(a*b)*(a - b))*sqrt(a*b)*a*b^3 - sqrt(a^2 - a*b + sqrt(a*b)*(a -
b))*sqrt(a*b)*b^4)*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 - sqrt(a*b)*(a - b))
*a^3*b - 9*sqrt(a^2 - a*b - sqrt(a*b)*(a - b))*a^2*b^2 + 5*sqrt(a^2 - a*b - sqrt(a*b)*(a - b))*a*b^3 + sqrt(a^
2 - a*b - sqrt(a*b)*(a - b))*b^4)*abs(-a + b)*abs(b) - (3*sqrt(a^2 - a*b - sqrt(a*b)*(a - b))*sqrt(a*b)*a^2*b^
2 - 6*sqrt(a^2 - a*b - sqrt(a*b)*(a - b))*sqrt(a*b)*a*b^3 - sqrt(a^2 - a*b - sqrt(a*b)*(a - b))*sqrt(a*b)*b^4)
*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)))/d

Mupad [B] (verification not implemented)

Time = 16.32 (sec) , antiderivative size = 2991, normalized size of antiderivative = 23.55 \[ \int \frac {\sin ^4(c+d x)}{a-b \sin ^4(c+d x)} \, dx=\text {Too large to display} \]

[In]

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

[Out]

- atan((18*a^5*tan(c + d*x))/(18*a^5 - 50*a^4*b + 32*a^3*b^2) - (50*a^4*tan(c + d*x))/(32*a^3*b - 50*a^4 + (18
*a^5)/b) + (32*a^3*b*tan(c + d*x))/(32*a^3*b - 50*a^4 + (18*a^5)/b))/(b*d) - (atan((((-(a*b^2 - (a*b^5)^(1/2))
/(16*(a*b^4 - b^5)))^(1/2)*(((-(a*b^2 - (a*b^5)^(1/2))/(16*(a*b^4 - b^5)))^(1/2)*(320*a^3*b^5 - 64*a^2*b^6 - 4
48*a^4*b^4 + 192*a^5*b^3 + tan(c + d*x)*(-(a*b^2 - (a*b^5)^(1/2))/(16*(a*b^4 - b^5)))^(1/2)*(768*a^2*b^7 - 768
*a^3*b^6 - 768*a^4*b^5 + 768*a^5*b^4)) + tan(c + d*x)*(176*a^2*b^5 - 400*a^3*b^4 + 80*a^4*b^3 + 144*a^5*b^2))*
(-(a*b^2 - (a*b^5)^(1/2))/(16*(a*b^4 - b^5)))^(1/2) + 12*a^5*b - 16*a^2*b^4 + 28*a^3*b^3 - 24*a^4*b^2) + tan(c
+ d*x)*(18*a^4*b + 6*a^5 - 4*a^2*b^3 - 20*a^3*b^2))*(-(a*b^2 - (a*b^5)^(1/2))/(16*(a*b^4 - b^5)))^(1/2)*1i +
((-(a*b^2 - (a*b^5)^(1/2))/(16*(a*b^4 - b^5)))^(1/2)*(((-(a*b^2 - (a*b^5)^(1/2))/(16*(a*b^4 - b^5)))^(1/2)*(64
*a^2*b^6 - 320*a^3*b^5 + 448*a^4*b^4 - 192*a^5*b^3 + tan(c + d*x)*(-(a*b^2 - (a*b^5)^(1/2))/(16*(a*b^4 - b^5))
)^(1/2)*(768*a^2*b^7 - 768*a^3*b^6 - 768*a^4*b^5 + 768*a^5*b^4)) + tan(c + d*x)*(176*a^2*b^5 - 400*a^3*b^4 + 8
0*a^4*b^3 + 144*a^5*b^2))*(-(a*b^2 - (a*b^5)^(1/2))/(16*(a*b^4 - b^5)))^(1/2) - 12*a^5*b + 16*a^2*b^4 - 28*a^3
*b^3 + 24*a^4*b^2) + tan(c + d*x)*(18*a^4*b + 6*a^5 - 4*a^2*b^3 - 20*a^3*b^2))*(-(a*b^2 - (a*b^5)^(1/2))/(16*(
a*b^4 - b^5)))^(1/2)*1i)/(6*a^3*b - 6*a^4 + ((-(a*b^2 - (a*b^5)^(1/2))/(16*(a*b^4 - b^5)))^(1/2)*(((-(a*b^2 -
(a*b^5)^(1/2))/(16*(a*b^4 - b^5)))^(1/2)*(320*a^3*b^5 - 64*a^2*b^6 - 448*a^4*b^4 + 192*a^5*b^3 + tan(c + d*x)*
(-(a*b^2 - (a*b^5)^(1/2))/(16*(a*b^4 - b^5)))^(1/2)*(768*a^2*b^7 - 768*a^3*b^6 - 768*a^4*b^5 + 768*a^5*b^4)) +
tan(c + d*x)*(176*a^2*b^5 - 400*a^3*b^4 + 80*a^4*b^3 + 144*a^5*b^2))*(-(a*b^2 - (a*b^5)^(1/2))/(16*(a*b^4 - b
^5)))^(1/2) + 12*a^5*b - 16*a^2*b^4 + 28*a^3*b^3 - 24*a^4*b^2) + tan(c + d*x)*(18*a^4*b + 6*a^5 - 4*a^2*b^3 -
20*a^3*b^2))*(-(a*b^2 - (a*b^5)^(1/2))/(16*(a*b^4 - b^5)))^(1/2) - ((-(a*b^2 - (a*b^5)^(1/2))/(16*(a*b^4 - b^5
)))^(1/2)*(((-(a*b^2 - (a*b^5)^(1/2))/(16*(a*b^4 - b^5)))^(1/2)*(64*a^2*b^6 - 320*a^3*b^5 + 448*a^4*b^4 - 192*
a^5*b^3 + tan(c + d*x)*(-(a*b^2 - (a*b^5)^(1/2))/(16*(a*b^4 - b^5)))^(1/2)*(768*a^2*b^7 - 768*a^3*b^6 - 768*a^
4*b^5 + 768*a^5*b^4)) + tan(c + d*x)*(176*a^2*b^5 - 400*a^3*b^4 + 80*a^4*b^3 + 144*a^5*b^2))*(-(a*b^2 - (a*b^5
)^(1/2))/(16*(a*b^4 - b^5)))^(1/2) - 12*a^5*b + 16*a^2*b^4 - 28*a^3*b^3 + 24*a^4*b^2) + tan(c + d*x)*(18*a^4*b
+ 6*a^5 - 4*a^2*b^3 - 20*a^3*b^2))*(-(a*b^2 - (a*b^5)^(1/2))/(16*(a*b^4 - b^5)))^(1/2)))*(-(a*b^2 - (a*b^5)^(
1/2))/(16*(a*b^4 - b^5)))^(1/2)*2i)/d - (atan(((tan(c + d*x)*(18*a^4*b + 6*a^5 - 4*a^2*b^3 - 20*a^3*b^2) + (-(
a*b^2 + (a*b^5)^(1/2))/(16*(a*b^4 - b^5)))^(1/2)*((tan(c + d*x)*(176*a^2*b^5 - 400*a^3*b^4 + 80*a^4*b^3 + 144*
a^5*b^2) + (-(a*b^2 + (a*b^5)^(1/2))/(16*(a*b^4 - b^5)))^(1/2)*(320*a^3*b^5 - 64*a^2*b^6 - 448*a^4*b^4 + 192*a
^5*b^3 + tan(c + d*x)*(-(a*b^2 + (a*b^5)^(1/2))/(16*(a*b^4 - b^5)))^(1/2)*(768*a^2*b^7 - 768*a^3*b^6 - 768*a^4
*b^5 + 768*a^5*b^4)))*(-(a*b^2 + (a*b^5)^(1/2))/(16*(a*b^4 - b^5)))^(1/2) + 12*a^5*b - 16*a^2*b^4 + 28*a^3*b^3
- 24*a^4*b^2))*(-(a*b^2 + (a*b^5)^(1/2))/(16*(a*b^4 - b^5)))^(1/2)*1i + (tan(c + d*x)*(18*a^4*b + 6*a^5 - 4*a
^2*b^3 - 20*a^3*b^2) + (-(a*b^2 + (a*b^5)^(1/2))/(16*(a*b^4 - b^5)))^(1/2)*((tan(c + d*x)*(176*a^2*b^5 - 400*a
^3*b^4 + 80*a^4*b^3 + 144*a^5*b^2) + (-(a*b^2 + (a*b^5)^(1/2))/(16*(a*b^4 - b^5)))^(1/2)*(64*a^2*b^6 - 320*a^3
*b^5 + 448*a^4*b^4 - 192*a^5*b^3 + tan(c + d*x)*(-(a*b^2 + (a*b^5)^(1/2))/(16*(a*b^4 - b^5)))^(1/2)*(768*a^2*b
^7 - 768*a^3*b^6 - 768*a^4*b^5 + 768*a^5*b^4)))*(-(a*b^2 + (a*b^5)^(1/2))/(16*(a*b^4 - b^5)))^(1/2) - 12*a^5*b
+ 16*a^2*b^4 - 28*a^3*b^3 + 24*a^4*b^2))*(-(a*b^2 + (a*b^5)^(1/2))/(16*(a*b^4 - b^5)))^(1/2)*1i)/((tan(c + d*
x)*(18*a^4*b + 6*a^5 - 4*a^2*b^3 - 20*a^3*b^2) + (-(a*b^2 + (a*b^5)^(1/2))/(16*(a*b^4 - b^5)))^(1/2)*((tan(c +
d*x)*(176*a^2*b^5 - 400*a^3*b^4 + 80*a^4*b^3 + 144*a^5*b^2) + (-(a*b^2 + (a*b^5)^(1/2))/(16*(a*b^4 - b^5)))^(
1/2)*(320*a^3*b^5 - 64*a^2*b^6 - 448*a^4*b^4 + 192*a^5*b^3 + tan(c + d*x)*(-(a*b^2 + (a*b^5)^(1/2))/(16*(a*b^4
- b^5)))^(1/2)*(768*a^2*b^7 - 768*a^3*b^6 - 768*a^4*b^5 + 768*a^5*b^4)))*(-(a*b^2 + (a*b^5)^(1/2))/(16*(a*b^4
- b^5)))^(1/2) + 12*a^5*b - 16*a^2*b^4 + 28*a^3*b^3 - 24*a^4*b^2))*(-(a*b^2 + (a*b^5)^(1/2))/(16*(a*b^4 - b^5
)))^(1/2) - (tan(c + d*x)*(18*a^4*b + 6*a^5 - 4*a^2*b^3 - 20*a^3*b^2) + (-(a*b^2 + (a*b^5)^(1/2))/(16*(a*b^4 -
b^5)))^(1/2)*((tan(c + d*x)*(176*a^2*b^5 - 400*a^3*b^4 + 80*a^4*b^3 + 144*a^5*b^2) + (-(a*b^2 + (a*b^5)^(1/2)
)/(16*(a*b^4 - b^5)))^(1/2)*(64*a^2*b^6 - 320*a^3*b^5 + 448*a^4*b^4 - 192*a^5*b^3 + tan(c + d*x)*(-(a*b^2 + (a
*b^5)^(1/2))/(16*(a*b^4 - b^5)))^(1/2)*(768*a^2*b^7 - 768*a^3*b^6 - 768*a^4*b^5 + 768*a^5*b^4)))*(-(a*b^2 + (a
*b^5)^(1/2))/(16*(a*b^4 - b^5)))^(1/2) - 12*a^5*b + 16*a^2*b^4 - 28*a^3*b^3 + 24*a^4*b^2))*(-(a*b^2 + (a*b^5)^
(1/2))/(16*(a*b^4 - b^5)))^(1/2) + 6*a^3*b - 6*a^4))*(-(a*b^2 + (a*b^5)^(1/2))/(16*(a*b^4 - b^5)))^(1/2)*2i)/d