3.414 \(\int \frac {\cos ^4(c+d x)}{a-b \sin ^4(c+d x)} \, dx\)

Optimal. Leaf size=127 \[ \frac {\left (\sqrt {a}-\sqrt {b}\right )^{3/2} \tan ^{-1}\left (\frac {\sqrt {\sqrt {a}-\sqrt {b}} \tan (c+d x)}{\sqrt [4]{a}}\right )}{2 a^{3/4} b d}+\frac {\left (\sqrt {a}+\sqrt {b}\right )^{3/2} \tan ^{-1}\left (\frac {\sqrt {\sqrt {a}+\sqrt {b}} \tan (c+d x)}{\sqrt [4]{a}}\right )}{2 a^{3/4} b d}-\frac {x}{b} \]

[Out]

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

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Rubi [A]  time = 0.24, antiderivative size = 127, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 5, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.208, Rules used = {3224, 1170, 203, 1166, 205} \[ \frac {\left (\sqrt {a}-\sqrt {b}\right )^{3/2} \tan ^{-1}\left (\frac {\sqrt {\sqrt {a}-\sqrt {b}} \tan (c+d x)}{\sqrt [4]{a}}\right )}{2 a^{3/4} b d}+\frac {\left (\sqrt {a}+\sqrt {b}\right )^{3/2} \tan ^{-1}\left (\frac {\sqrt {\sqrt {a}+\sqrt {b}} \tan (c+d x)}{\sqrt [4]{a}}\right )}{2 a^{3/4} b d}-\frac {x}{b} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 203

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTan[(Rt[b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[b, 2]), x] /;
FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a, 0] || GtQ[b, 0])

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 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 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 3224

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

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Mathematica [A]  time = 0.25, size = 171, normalized size = 1.35 \[ \frac {\frac {\left (\sqrt {a}+\sqrt {b}\right )^2 \tan ^{-1}\left (\frac {\left (\sqrt {a}+\sqrt {b}\right ) \tan (c+d x)}{\sqrt {\sqrt {a} \sqrt {b}+a}}\right )}{\sqrt {a} \sqrt {\sqrt {a} \sqrt {b}+a}}-\frac {\left (\sqrt {a}-\sqrt {b}\right )^2 \tanh ^{-1}\left (\frac {\left (\sqrt {a}-\sqrt {b}\right ) \tan (c+d x)}{\sqrt {\sqrt {a} \sqrt {b}-a}}\right )}{\sqrt {a} \sqrt {\sqrt {a} \sqrt {b}-a}}-2 (c+d x)}{2 b d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [B]  time = 0.68, size = 1197, normalized size = 9.43 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/8*(b*sqrt((a*b^2*d^2*sqrt((9*a^2 + 6*a*b + b^2)/(a^3*b^3*d^4)) - a - 3*b)/(a*b^2*d^2))*log(1/4*(3*a^2 - 2*a*
b - b^2)*cos(d*x + c)^2 - 3/4*a^2 + 1/2*a*b + 1/4*b^2 + 1/2*(a^3*b^2*d^3*sqrt((9*a^2 + 6*a*b + b^2)/(a^3*b^3*d
^4))*cos(d*x + c)*sin(d*x + c) + (3*a^2*b + a*b^2)*d*cos(d*x + c)*sin(d*x + c))*sqrt((a*b^2*d^2*sqrt((9*a^2 +
6*a*b + b^2)/(a^3*b^3*d^4)) - a - 3*b)/(a*b^2*d^2)) - 1/4*(2*(a^3*b - a^2*b^2)*d^2*cos(d*x + c)^2 - (a^3*b - a
^2*b^2)*d^2)*sqrt((9*a^2 + 6*a*b + b^2)/(a^3*b^3*d^4))) - b*sqrt((a*b^2*d^2*sqrt((9*a^2 + 6*a*b + b^2)/(a^3*b^
3*d^4)) - a - 3*b)/(a*b^2*d^2))*log(1/4*(3*a^2 - 2*a*b - b^2)*cos(d*x + c)^2 - 3/4*a^2 + 1/2*a*b + 1/4*b^2 - 1
/2*(a^3*b^2*d^3*sqrt((9*a^2 + 6*a*b + b^2)/(a^3*b^3*d^4))*cos(d*x + c)*sin(d*x + c) + (3*a^2*b + a*b^2)*d*cos(
d*x + c)*sin(d*x + c))*sqrt((a*b^2*d^2*sqrt((9*a^2 + 6*a*b + b^2)/(a^3*b^3*d^4)) - a - 3*b)/(a*b^2*d^2)) - 1/4
*(2*(a^3*b - a^2*b^2)*d^2*cos(d*x + c)^2 - (a^3*b - a^2*b^2)*d^2)*sqrt((9*a^2 + 6*a*b + b^2)/(a^3*b^3*d^4))) +
b*sqrt(-(a*b^2*d^2*sqrt((9*a^2 + 6*a*b + b^2)/(a^3*b^3*d^4)) + a + 3*b)/(a*b^2*d^2))*log(-1/4*(3*a^2 - 2*a*b
- b^2)*cos(d*x + c)^2 + 3/4*a^2 - 1/2*a*b - 1/4*b^2 + 1/2*(a^3*b^2*d^3*sqrt((9*a^2 + 6*a*b + b^2)/(a^3*b^3*d^4
))*cos(d*x + c)*sin(d*x + c) - (3*a^2*b + a*b^2)*d*cos(d*x + c)*sin(d*x + c))*sqrt(-(a*b^2*d^2*sqrt((9*a^2 + 6
*a*b + b^2)/(a^3*b^3*d^4)) + a + 3*b)/(a*b^2*d^2)) - 1/4*(2*(a^3*b - a^2*b^2)*d^2*cos(d*x + c)^2 - (a^3*b - a^
2*b^2)*d^2)*sqrt((9*a^2 + 6*a*b + b^2)/(a^3*b^3*d^4))) - b*sqrt(-(a*b^2*d^2*sqrt((9*a^2 + 6*a*b + b^2)/(a^3*b^
3*d^4)) + a + 3*b)/(a*b^2*d^2))*log(-1/4*(3*a^2 - 2*a*b - b^2)*cos(d*x + c)^2 + 3/4*a^2 - 1/2*a*b - 1/4*b^2 -
1/2*(a^3*b^2*d^3*sqrt((9*a^2 + 6*a*b + b^2)/(a^3*b^3*d^4))*cos(d*x + c)*sin(d*x + c) - (3*a^2*b + a*b^2)*d*cos
(d*x + c)*sin(d*x + c))*sqrt(-(a*b^2*d^2*sqrt((9*a^2 + 6*a*b + b^2)/(a^3*b^3*d^4)) + a + 3*b)/(a*b^2*d^2)) - 1
/4*(2*(a^3*b - a^2*b^2)*d^2*cos(d*x + c)^2 - (a^3*b - a^2*b^2)*d^2)*sqrt((9*a^2 + 6*a*b + b^2)/(a^3*b^3*d^4)))
- 8*x)/b

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giac [B]  time = 0.97, size = 906, normalized size = 7.13 \[ -\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 - 3 \, \sqrt {a^{2} - a b + \sqrt {a b} {\left (a - b\right )}} a^{2} b^{2} - 7 \, \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} - 12 \, a^{4} b^{3} + 14 \, a^{3} b^{4} - 4 \, a^{2} b^{5} - a b^{6}\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 - 3 \, \sqrt {a^{2} - a b - \sqrt {a b} {\left (a - b\right )}} a^{2} b^{2} - 7 \, \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} - 12 \, a^{4} b^{3} + 14 \, a^{3} b^{4} - 4 \, a^{2} b^{5} - a b^{6}\right )} {\left | b \right |}}}{2 \, d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(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 - 3*sqrt(a^2 - a*b + sqrt(a*b)*(a - b))*a^2*b^2 - 7*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 - 12*a^4*b^3 + 14*a^3*b^4 - 4*a^2*b^5 - a*b^6)*abs(b)) - ((3*sq
rt(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 -
3*sqrt(a^2 - a*b - sqrt(a*b)*(a - b))*a^2*b^2 - 7*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*sqr
t(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 - 12*a^4*b^3 + 14*a^3*b^4 - 4*a^2*b^5 - a*b^6)*abs(b)))/d

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maple [B]  time = 0.62, size = 449, normalized size = 3.54 \[ -\frac {\arctan \left (\tan \left (d x +c \right )\right )}{d b}+\frac {a \arctanh \left (\frac {\left (-a +b \right ) \tan \left (d x +c \right )}{\sqrt {\left (\sqrt {a b}-a \right ) \left (a -b \right )}}\right )}{2 d b \sqrt {\left (\sqrt {a b}-a \right ) \left (a -b \right )}}+\frac {a \arctanh \left (\frac {\left (-a +b \right ) \tan \left (d x +c \right )}{\sqrt {\left (\sqrt {a b}-a \right ) \left (a -b \right )}}\right )}{2 d \sqrt {a b}\, \sqrt {\left (\sqrt {a b}-a \right ) \left (a -b \right )}}+\frac {a \arctan \left (\frac {\left (a -b \right ) \tan \left (d x +c \right )}{\sqrt {\left (\sqrt {a b}+a \right ) \left (a -b \right )}}\right )}{2 d b \sqrt {\left (\sqrt {a b}+a \right ) \left (a -b \right )}}-\frac {a \arctan \left (\frac {\left (a -b \right ) \tan \left (d x +c \right )}{\sqrt {\left (\sqrt {a b}+a \right ) \left (a -b \right )}}\right )}{2 d \sqrt {a b}\, \sqrt {\left (\sqrt {a b}+a \right ) \left (a -b \right )}}-\frac {\arctanh \left (\frac {\left (-a +b \right ) \tan \left (d x +c \right )}{\sqrt {\left (\sqrt {a b}-a \right ) \left (a -b \right )}}\right )}{2 d \sqrt {\left (\sqrt {a b}-a \right ) \left (a -b \right )}}-\frac {b \arctanh \left (\frac {\left (-a +b \right ) \tan \left (d x +c \right )}{\sqrt {\left (\sqrt {a b}-a \right ) \left (a -b \right )}}\right )}{2 d \sqrt {a b}\, \sqrt {\left (\sqrt {a b}-a \right ) \left (a -b \right )}}-\frac {\arctan \left (\frac {\left (a -b \right ) \tan \left (d x +c \right )}{\sqrt {\left (\sqrt {a b}+a \right ) \left (a -b \right )}}\right )}{2 d \sqrt {\left (\sqrt {a b}+a \right ) \left (a -b \right )}}+\frac {b \arctan \left (\frac {\left (a -b \right ) \tan \left (d x +c \right )}{\sqrt {\left (\sqrt {a b}+a \right ) \left (a -b \right )}}\right )}{2 d \sqrt {a b}\, \sqrt {\left (\sqrt {a b}+a \right ) \left (a -b \right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(b*integrate(-8*(4*b^2*cos(6*d*x + 6*c)^2 + 4*b^2*cos(2*d*x + 2*c)^2 + 4*b^2*sin(6*d*x + 6*c)^2 + 4*b^2*sin(2
*d*x + 2*c)^2 + 4*(8*a^2 - 3*a*b)*cos(4*d*x + 4*c)^2 - b^2*cos(2*d*x + 2*c) + 4*(8*a^2 - 3*a*b)*sin(4*d*x + 4*
c)^2 + 6*(4*a*b - b^2)*sin(4*d*x + 4*c)*sin(2*d*x + 2*c) - (b^2*cos(6*d*x + 6*c) + 2*a*b*cos(4*d*x + 4*c) + b^
2*cos(2*d*x + 2*c))*cos(8*d*x + 8*c) + (8*b^2*cos(2*d*x + 2*c) - b^2 + 6*(4*a*b - b^2)*cos(4*d*x + 4*c))*cos(6
*d*x + 6*c) - 2*(a*b - 3*(4*a*b - b^2)*cos(2*d*x + 2*c))*cos(4*d*x + 4*c) - (b^2*sin(6*d*x + 6*c) + 2*a*b*sin(
4*d*x + 4*c) + b^2*sin(2*d*x + 2*c))*sin(8*d*x + 8*c) + 2*(4*b^2*sin(2*d*x + 2*c) + 3*(4*a*b - b^2)*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))*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

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mupad [B]  time = 16.40, size = 4299, normalized size = 33.85 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

atan((90*a^4*tan(c + d*x))/(10*a*b^3 + 132*a^3*b - 90*a^4 - 2*b^4 - 68*a^2*b^2 + (18*a^5)/b) - (18*a^5*tan(c +
d*x))/(10*a*b^4 - 90*a^4*b + 18*a^5 - 2*b^5 - 68*a^2*b^3 + 132*a^3*b^2) + (2*b^4*tan(c + d*x))/(10*a*b^3 + 13
2*a^3*b - 90*a^4 - 2*b^4 - 68*a^2*b^2 + (18*a^5)/b) + (68*a^2*b^2*tan(c + d*x))/(10*a*b^3 + 132*a^3*b - 90*a^4
- 2*b^4 - 68*a^2*b^2 + (18*a^5)/b) - (10*a*b^3*tan(c + d*x))/(10*a*b^3 + 132*a^3*b - 90*a^4 - 2*b^4 - 68*a^2*
b^2 + (18*a^5)/b) - (132*a^3*b*tan(c + d*x))/(10*a*b^3 + 132*a^3*b - 90*a^4 - 2*b^4 - 68*a^2*b^2 + (18*a^5)/b)
)/(b*d) + (atan(((tan(c + d*x)*(30*a*b^4 - 30*a^4*b + 6*a^5 - 6*b^5 - 60*a^2*b^3 + 60*a^3*b^2) + (-(3*a*(a^3*b
^5)^(1/2) + b*(a^3*b^5)^(1/2) + 3*a^2*b^3 + a^3*b^2)/(16*a^3*b^4))^(1/2)*(36*a*b^5 - 12*a^5*b - 4*b^6 + ((-(3*
a*(a^3*b^5)^(1/2) + b*(a^3*b^5)^(1/2) + 3*a^2*b^3 + a^3*b^2)/(16*a^3*b^4))^(1/2)*(64*a*b^7 + 256*a^2*b^6 - 896
*a^3*b^5 + 768*a^4*b^4 - 192*a^5*b^3 + tan(c + d*x)*(-(3*a*(a^3*b^5)^(1/2) + b*(a^3*b^5)^(1/2) + 3*a^2*b^3 + a
^3*b^2)/(16*a^3*b^4))^(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)*(80*a*b^6
- 16*b^7 + 224*a^2*b^5 - 480*a^3*b^4 + 48*a^4*b^3 + 144*a^5*b^2))*(-(3*a*(a^3*b^5)^(1/2) + b*(a^3*b^5)^(1/2) +
3*a^2*b^3 + a^3*b^2)/(16*a^3*b^4))^(1/2) - 72*a^2*b^4 + 40*a^3*b^3 + 12*a^4*b^2))*(-(3*a*(a^3*b^5)^(1/2) + b*
(a^3*b^5)^(1/2) + 3*a^2*b^3 + a^3*b^2)/(16*a^3*b^4))^(1/2)*1i + (tan(c + d*x)*(30*a*b^4 - 30*a^4*b + 6*a^5 - 6
*b^5 - 60*a^2*b^3 + 60*a^3*b^2) - (-(3*a*(a^3*b^5)^(1/2) + b*(a^3*b^5)^(1/2) + 3*a^2*b^3 + a^3*b^2)/(16*a^3*b^
4))^(1/2)*(36*a*b^5 - 12*a^5*b - 4*b^6 + ((-(3*a*(a^3*b^5)^(1/2) + b*(a^3*b^5)^(1/2) + 3*a^2*b^3 + a^3*b^2)/(1
6*a^3*b^4))^(1/2)*(64*a*b^7 + 256*a^2*b^6 - 896*a^3*b^5 + 768*a^4*b^4 - 192*a^5*b^3 - tan(c + d*x)*(-(3*a*(a^3
*b^5)^(1/2) + b*(a^3*b^5)^(1/2) + 3*a^2*b^3 + a^3*b^2)/(16*a^3*b^4))^(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)*(80*a*b^6 - 16*b^7 + 224*a^2*b^5 - 480*a^3*b^4 + 48*a^4*b^3 + 144*a^5*b^2
))*(-(3*a*(a^3*b^5)^(1/2) + b*(a^3*b^5)^(1/2) + 3*a^2*b^3 + a^3*b^2)/(16*a^3*b^4))^(1/2) - 72*a^2*b^4 + 40*a^3
*b^3 + 12*a^4*b^2))*(-(3*a*(a^3*b^5)^(1/2) + b*(a^3*b^5)^(1/2) + 3*a^2*b^3 + a^3*b^2)/(16*a^3*b^4))^(1/2)*1i)/
((tan(c + d*x)*(30*a*b^4 - 30*a^4*b + 6*a^5 - 6*b^5 - 60*a^2*b^3 + 60*a^3*b^2) + (-(3*a*(a^3*b^5)^(1/2) + b*(a
^3*b^5)^(1/2) + 3*a^2*b^3 + a^3*b^2)/(16*a^3*b^4))^(1/2)*(36*a*b^5 - 12*a^5*b - 4*b^6 + ((-(3*a*(a^3*b^5)^(1/2
) + b*(a^3*b^5)^(1/2) + 3*a^2*b^3 + a^3*b^2)/(16*a^3*b^4))^(1/2)*(64*a*b^7 + 256*a^2*b^6 - 896*a^3*b^5 + 768*a
^4*b^4 - 192*a^5*b^3 + tan(c + d*x)*(-(3*a*(a^3*b^5)^(1/2) + b*(a^3*b^5)^(1/2) + 3*a^2*b^3 + a^3*b^2)/(16*a^3*
b^4))^(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)*(80*a*b^6 - 16*b^7 + 224*a
^2*b^5 - 480*a^3*b^4 + 48*a^4*b^3 + 144*a^5*b^2))*(-(3*a*(a^3*b^5)^(1/2) + b*(a^3*b^5)^(1/2) + 3*a^2*b^3 + a^3
*b^2)/(16*a^3*b^4))^(1/2) - 72*a^2*b^4 + 40*a^3*b^3 + 12*a^4*b^2))*(-(3*a*(a^3*b^5)^(1/2) + b*(a^3*b^5)^(1/2)
+ 3*a^2*b^3 + a^3*b^2)/(16*a^3*b^4))^(1/2) - (tan(c + d*x)*(30*a*b^4 - 30*a^4*b + 6*a^5 - 6*b^5 - 60*a^2*b^3 +
60*a^3*b^2) - (-(3*a*(a^3*b^5)^(1/2) + b*(a^3*b^5)^(1/2) + 3*a^2*b^3 + a^3*b^2)/(16*a^3*b^4))^(1/2)*(36*a*b^5
- 12*a^5*b - 4*b^6 + ((-(3*a*(a^3*b^5)^(1/2) + b*(a^3*b^5)^(1/2) + 3*a^2*b^3 + a^3*b^2)/(16*a^3*b^4))^(1/2)*(
64*a*b^7 + 256*a^2*b^6 - 896*a^3*b^5 + 768*a^4*b^4 - 192*a^5*b^3 - tan(c + d*x)*(-(3*a*(a^3*b^5)^(1/2) + b*(a^
3*b^5)^(1/2) + 3*a^2*b^3 + a^3*b^2)/(16*a^3*b^4))^(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)*(80*a*b^6 - 16*b^7 + 224*a^2*b^5 - 480*a^3*b^4 + 48*a^4*b^3 + 144*a^5*b^2))*(-(3*a*(a^3*b^5)
^(1/2) + b*(a^3*b^5)^(1/2) + 3*a^2*b^3 + a^3*b^2)/(16*a^3*b^4))^(1/2) - 72*a^2*b^4 + 40*a^3*b^3 + 12*a^4*b^2))
*(-(3*a*(a^3*b^5)^(1/2) + b*(a^3*b^5)^(1/2) + 3*a^2*b^3 + a^3*b^2)/(16*a^3*b^4))^(1/2)))*(-(3*a*(a^3*b^5)^(1/2
) + b*(a^3*b^5)^(1/2) + 3*a^2*b^3 + a^3*b^2)/(16*a^3*b^4))^(1/2)*2i)/d + (atan(((tan(c + d*x)*(30*a*b^4 - 30*a
^4*b + 6*a^5 - 6*b^5 - 60*a^2*b^3 + 60*a^3*b^2) + ((3*a*(a^3*b^5)^(1/2) + b*(a^3*b^5)^(1/2) - 3*a^2*b^3 - a^3*
b^2)/(16*a^3*b^4))^(1/2)*(36*a*b^5 - 12*a^5*b - 4*b^6 + (((3*a*(a^3*b^5)^(1/2) + b*(a^3*b^5)^(1/2) - 3*a^2*b^3
- a^3*b^2)/(16*a^3*b^4))^(1/2)*(64*a*b^7 + 256*a^2*b^6 - 896*a^3*b^5 + 768*a^4*b^4 - 192*a^5*b^3 + tan(c + d*
x)*((3*a*(a^3*b^5)^(1/2) + b*(a^3*b^5)^(1/2) - 3*a^2*b^3 - a^3*b^2)/(16*a^3*b^4))^(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)*(80*a*b^6 - 16*b^7 + 224*a^2*b^5 - 480*a^3*b^4 + 48*a^4*b^3
+ 144*a^5*b^2))*((3*a*(a^3*b^5)^(1/2) + b*(a^3*b^5)^(1/2) - 3*a^2*b^3 - a^3*b^2)/(16*a^3*b^4))^(1/2) - 72*a^2*
b^4 + 40*a^3*b^3 + 12*a^4*b^2))*((3*a*(a^3*b^5)^(1/2) + b*(a^3*b^5)^(1/2) - 3*a^2*b^3 - a^3*b^2)/(16*a^3*b^4))
^(1/2)*1i + (tan(c + d*x)*(30*a*b^4 - 30*a^4*b + 6*a^5 - 6*b^5 - 60*a^2*b^3 + 60*a^3*b^2) - ((3*a*(a^3*b^5)^(1
/2) + b*(a^3*b^5)^(1/2) - 3*a^2*b^3 - a^3*b^2)/(16*a^3*b^4))^(1/2)*(36*a*b^5 - 12*a^5*b - 4*b^6 + (((3*a*(a^3*
b^5)^(1/2) + b*(a^3*b^5)^(1/2) - 3*a^2*b^3 - a^3*b^2)/(16*a^3*b^4))^(1/2)*(64*a*b^7 + 256*a^2*b^6 - 896*a^3*b^
5 + 768*a^4*b^4 - 192*a^5*b^3 - tan(c + d*x)*((3*a*(a^3*b^5)^(1/2) + b*(a^3*b^5)^(1/2) - 3*a^2*b^3 - a^3*b^2)/
(16*a^3*b^4))^(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)*(80*a*b^6 - 16*b^7
+ 224*a^2*b^5 - 480*a^3*b^4 + 48*a^4*b^3 + 144*a^5*b^2))*((3*a*(a^3*b^5)^(1/2) + b*(a^3*b^5)^(1/2) - 3*a^2*b^
3 - a^3*b^2)/(16*a^3*b^4))^(1/2) - 72*a^2*b^4 + 40*a^3*b^3 + 12*a^4*b^2))*((3*a*(a^3*b^5)^(1/2) + b*(a^3*b^5)^
(1/2) - 3*a^2*b^3 - a^3*b^2)/(16*a^3*b^4))^(1/2)*1i)/((tan(c + d*x)*(30*a*b^4 - 30*a^4*b + 6*a^5 - 6*b^5 - 60*
a^2*b^3 + 60*a^3*b^2) + ((3*a*(a^3*b^5)^(1/2) + b*(a^3*b^5)^(1/2) - 3*a^2*b^3 - a^3*b^2)/(16*a^3*b^4))^(1/2)*(
36*a*b^5 - 12*a^5*b - 4*b^6 + (((3*a*(a^3*b^5)^(1/2) + b*(a^3*b^5)^(1/2) - 3*a^2*b^3 - a^3*b^2)/(16*a^3*b^4))^
(1/2)*(64*a*b^7 + 256*a^2*b^6 - 896*a^3*b^5 + 768*a^4*b^4 - 192*a^5*b^3 + tan(c + d*x)*((3*a*(a^3*b^5)^(1/2) +
b*(a^3*b^5)^(1/2) - 3*a^2*b^3 - a^3*b^2)/(16*a^3*b^4))^(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)*(80*a*b^6 - 16*b^7 + 224*a^2*b^5 - 480*a^3*b^4 + 48*a^4*b^3 + 144*a^5*b^2))*((3*a*(a^3
*b^5)^(1/2) + b*(a^3*b^5)^(1/2) - 3*a^2*b^3 - a^3*b^2)/(16*a^3*b^4))^(1/2) - 72*a^2*b^4 + 40*a^3*b^3 + 12*a^4*
b^2))*((3*a*(a^3*b^5)^(1/2) + b*(a^3*b^5)^(1/2) - 3*a^2*b^3 - a^3*b^2)/(16*a^3*b^4))^(1/2) - (tan(c + d*x)*(30
*a*b^4 - 30*a^4*b + 6*a^5 - 6*b^5 - 60*a^2*b^3 + 60*a^3*b^2) - ((3*a*(a^3*b^5)^(1/2) + b*(a^3*b^5)^(1/2) - 3*a
^2*b^3 - a^3*b^2)/(16*a^3*b^4))^(1/2)*(36*a*b^5 - 12*a^5*b - 4*b^6 + (((3*a*(a^3*b^5)^(1/2) + b*(a^3*b^5)^(1/2
) - 3*a^2*b^3 - a^3*b^2)/(16*a^3*b^4))^(1/2)*(64*a*b^7 + 256*a^2*b^6 - 896*a^3*b^5 + 768*a^4*b^4 - 192*a^5*b^3
- tan(c + d*x)*((3*a*(a^3*b^5)^(1/2) + b*(a^3*b^5)^(1/2) - 3*a^2*b^3 - a^3*b^2)/(16*a^3*b^4))^(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)*(80*a*b^6 - 16*b^7 + 224*a^2*b^5 - 480*a^3*b^4
+ 48*a^4*b^3 + 144*a^5*b^2))*((3*a*(a^3*b^5)^(1/2) + b*(a^3*b^5)^(1/2) - 3*a^2*b^3 - a^3*b^2)/(16*a^3*b^4))^(1
/2) - 72*a^2*b^4 + 40*a^3*b^3 + 12*a^4*b^2))*((3*a*(a^3*b^5)^(1/2) + b*(a^3*b^5)^(1/2) - 3*a^2*b^3 - a^3*b^2)/
(16*a^3*b^4))^(1/2)))*((3*a*(a^3*b^5)^(1/2) + b*(a^3*b^5)^(1/2) - 3*a^2*b^3 - a^3*b^2)/(16*a^3*b^4))^(1/2)*2i)
/d

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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