\(\int \frac {\cos ^7(c+d x)}{a-b \sin ^4(c+d x)} \, dx\) [404]

Optimal result

Integrand size = 24, antiderivative size = 131 \[ \int \frac {\cos ^7(c+d x)}{a-b \sin ^4(c+d x)} \, dx=\frac {\left (\sqrt {a}+\sqrt {b}\right )^3 \arctan \left (\frac {\sqrt [4]{b} \sin (c+d x)}{\sqrt [4]{a}}\right )}{2 a^{3/4} b^{7/4} d}-\frac {\left (\sqrt {a}-\sqrt {b}\right )^3 \text {arctanh}\left (\frac {\sqrt [4]{b} \sin (c+d x)}{\sqrt [4]{a}}\right )}{2 a^{3/4} b^{7/4} d}-\frac {3 \sin (c+d x)}{b d}+\frac {\sin ^3(c+d x)}{3 b d} \]

[Out]

-3*sin(d*x+c)/b/d+1/3*sin(d*x+c)^3/b/d-1/2*arctanh(b^(1/4)*sin(d*x+c)/a^(1/4))*(a^(1/2)-b^(1/2))^3/a^(3/4)/b^(
7/4)/d+1/2*arctan(b^(1/4)*sin(d*x+c)/a^(1/4))*(a^(1/2)+b^(1/2))^3/a^(3/4)/b^(7/4)/d

Rubi [A] (verified)

Time = 0.20 (sec) , antiderivative size = 131, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.208, Rules used = {3302, 1185, 1181, 211, 214} \[ \int \frac {\cos ^7(c+d x)}{a-b \sin ^4(c+d x)} \, dx=\frac {\left (\sqrt {a}+\sqrt {b}\right )^3 \arctan \left (\frac {\sqrt [4]{b} \sin (c+d x)}{\sqrt [4]{a}}\right )}{2 a^{3/4} b^{7/4} d}-\frac {\left (\sqrt {a}-\sqrt {b}\right )^3 \text {arctanh}\left (\frac {\sqrt [4]{b} \sin (c+d x)}{\sqrt [4]{a}}\right )}{2 a^{3/4} b^{7/4} d}+\frac {\sin ^3(c+d x)}{3 b d}-\frac {3 \sin (c+d x)}{b d} \]

[In]

Int[Cos[c + d*x]^7/(a - b*Sin[c + d*x]^4),x]

[Out]

((Sqrt[a] + Sqrt[b])^3*ArcTan[(b^(1/4)*Sin[c + d*x])/a^(1/4)])/(2*a^(3/4)*b^(7/4)*d) - ((Sqrt[a] - Sqrt[b])^3*
ArcTanh[(b^(1/4)*Sin[c + d*x])/a^(1/4)])/(2*a^(3/4)*b^(7/4)*d) - (3*Sin[c + d*x])/(b*d) + Sin[c + d*x]^3/(3*b*
d)

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 214

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

Rule 1181

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

Rule 1185

Int[((d_) + (e_.)*(x_)^2)^(q_)/((a_) + (c_.)*(x_)^4), x_Symbol] :> Int[ExpandIntegrand[(d + e*x^2)^q/(a + c*x^
4), x], x] /; FreeQ[{a, c, d, e}, x] && NeQ[c*d^2 + a*e^2, 0] && IntegerQ[q]

Rule 3302

Int[cos[(e_.) + (f_.)*(x_)]^(m_.)*((a_) + (b_.)*((c_.)*sin[(e_.) + (f_.)*(x_)])^(n_))^(p_.), x_Symbol] :> With
[{ff = FreeFactors[Sin[e + f*x], x]}, Dist[ff/f, Subst[Int[(1 - ff^2*x^2)^((m - 1)/2)*(a + b*(c*ff*x)^n)^p, x]
, x, Sin[e + f*x]/ff], x]] /; FreeQ[{a, b, c, e, f, n, p}, x] && IntegerQ[(m - 1)/2] && (EqQ[n, 4] || GtQ[m, 0
] || IGtQ[p, 0] || IntegersQ[m, p])

Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int \frac {\left (1-x^2\right )^3}{a-b x^4} \, dx,x,\sin (c+d x)\right )}{d} \\ & = \frac {\text {Subst}\left (\int \left (-\frac {3}{b}+\frac {x^2}{b}+\frac {3 a+b-(a+3 b) x^2}{b \left (a-b x^4\right )}\right ) \, dx,x,\sin (c+d x)\right )}{d} \\ & = -\frac {3 \sin (c+d x)}{b d}+\frac {\sin ^3(c+d x)}{3 b d}+\frac {\text {Subst}\left (\int \frac {3 a+b+(-a-3 b) x^2}{a-b x^4} \, dx,x,\sin (c+d x)\right )}{b d} \\ & = -\frac {3 \sin (c+d x)}{b d}+\frac {\sin ^3(c+d x)}{3 b d}-\frac {\left (\sqrt {a}-\sqrt {b}\right )^3 \text {Subst}\left (\int \frac {1}{\sqrt {a} \sqrt {b}-b x^2} \, dx,x,\sin (c+d x)\right )}{2 \sqrt {a} b d}-\frac {\left (\sqrt {a}+\sqrt {b}\right )^3 \text {Subst}\left (\int \frac {1}{-\sqrt {a} \sqrt {b}-b x^2} \, dx,x,\sin (c+d x)\right )}{2 \sqrt {a} b d} \\ & = \frac {\left (\sqrt {a}+\sqrt {b}\right )^3 \arctan \left (\frac {\sqrt [4]{b} \sin (c+d x)}{\sqrt [4]{a}}\right )}{2 a^{3/4} b^{7/4} d}-\frac {\left (\sqrt {a}-\sqrt {b}\right )^3 \text {arctanh}\left (\frac {\sqrt [4]{b} \sin (c+d x)}{\sqrt [4]{a}}\right )}{2 a^{3/4} b^{7/4} d}-\frac {3 \sin (c+d x)}{b d}+\frac {\sin ^3(c+d x)}{3 b d} \\ \end{align*}

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 0.28 (sec) , antiderivative size = 207, normalized size of antiderivative = 1.58 \[ \int \frac {\cos ^7(c+d x)}{a-b \sin ^4(c+d x)} \, dx=\frac {3 \left (\sqrt {a}-\sqrt {b}\right )^3 \log \left (\sqrt [4]{a}-\sqrt [4]{b} \sin (c+d x)\right )+3 i \left (\sqrt {a}+\sqrt {b}\right )^3 \log \left (\sqrt [4]{a}-i \sqrt [4]{b} \sin (c+d x)\right )-3 i \left (\sqrt {a}+\sqrt {b}\right )^3 \log \left (\sqrt [4]{a}+i \sqrt [4]{b} \sin (c+d x)\right )-3 \left (\sqrt {a}-\sqrt {b}\right )^3 \log \left (\sqrt [4]{a}+\sqrt [4]{b} \sin (c+d x)\right )-36 a^{3/4} b^{3/4} \sin (c+d x)+4 a^{3/4} b^{3/4} \sin ^3(c+d x)}{12 a^{3/4} b^{7/4} d} \]

[In]

Integrate[Cos[c + d*x]^7/(a - b*Sin[c + d*x]^4),x]

[Out]

(3*(Sqrt[a] - Sqrt[b])^3*Log[a^(1/4) - b^(1/4)*Sin[c + d*x]] + (3*I)*(Sqrt[a] + Sqrt[b])^3*Log[a^(1/4) - I*b^(
1/4)*Sin[c + d*x]] - (3*I)*(Sqrt[a] + Sqrt[b])^3*Log[a^(1/4) + I*b^(1/4)*Sin[c + d*x]] - 3*(Sqrt[a] - Sqrt[b])
^3*Log[a^(1/4) + b^(1/4)*Sin[c + d*x]] - 36*a^(3/4)*b^(3/4)*Sin[c + d*x] + 4*a^(3/4)*b^(3/4)*Sin[c + d*x]^3)/(
12*a^(3/4)*b^(7/4)*d)

Maple [A] (verified)

Time = 1.97 (sec) , antiderivative size = 178, normalized size of antiderivative = 1.36

method	result	size
derivativedivides	\(-\frac {-\frac {\frac {\left (\sin ^{3}\left (d x +c \right )\right )}{3}-3 \sin \left (d x +c \right )}{b}+\frac {\frac {\left (-3 a -b \right ) \left (\frac {a}{b}\right )^{\frac {1}{4}} \left (\ln \left (\frac {\sin \left (d x +c \right )+\left (\frac {a}{b}\right )^{\frac {1}{4}}}{\sin \left (d x +c \right )-\left (\frac {a}{b}\right )^{\frac {1}{4}}}\right )+2 \arctan \left (\frac {\sin \left (d x +c \right )}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}\right )\right )}{4 a}-\frac {\left (a +3 b \right ) \left (2 \arctan \left (\frac {\sin \left (d x +c \right )}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}\right )-\ln \left (\frac {\sin \left (d x +c \right )+\left (\frac {a}{b}\right )^{\frac {1}{4}}}{\sin \left (d x +c \right )-\left (\frac {a}{b}\right )^{\frac {1}{4}}}\right )\right )}{4 b \left (\frac {a}{b}\right )^{\frac {1}{4}}}}{b}}{d}\)	\(178\)
default	\(-\frac {-\frac {\frac {\left (\sin ^{3}\left (d x +c \right )\right )}{3}-3 \sin \left (d x +c \right )}{b}+\frac {\frac {\left (-3 a -b \right ) \left (\frac {a}{b}\right )^{\frac {1}{4}} \left (\ln \left (\frac {\sin \left (d x +c \right )+\left (\frac {a}{b}\right )^{\frac {1}{4}}}{\sin \left (d x +c \right )-\left (\frac {a}{b}\right )^{\frac {1}{4}}}\right )+2 \arctan \left (\frac {\sin \left (d x +c \right )}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}\right )\right )}{4 a}-\frac {\left (a +3 b \right ) \left (2 \arctan \left (\frac {\sin \left (d x +c \right )}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}\right )-\ln \left (\frac {\sin \left (d x +c \right )+\left (\frac {a}{b}\right )^{\frac {1}{4}}}{\sin \left (d x +c \right )-\left (\frac {a}{b}\right )^{\frac {1}{4}}}\right )\right )}{4 b \left (\frac {a}{b}\right )^{\frac {1}{4}}}}{b}}{d}\)	\(178\)
risch	\(\frac {11 i {\mathrm e}^{i \left (d x +c \right )}}{8 b d}-\frac {11 i {\mathrm e}^{-i \left (d x +c \right )}}{8 b d}+\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (256 a^{3} b^{7} d^{4} \textit {\_Z}^{4}+\left (192 a^{4} b^{4} d^{2}+640 a^{3} b^{5} d^{2}+192 a^{2} b^{6} d^{2}\right ) \textit {\_Z}^{2}-a^{6}+6 a^{5} b -15 a^{4} b^{2}+20 a^{3} b^{3}-15 a^{2} b^{4}+6 a \,b^{5}-b^{6}\right )}{\sum }\textit {\_R} \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}+\left (\left (-\frac {128 i a^{4} b^{5} d^{3}}{a^{6}+12 a^{5} b -27 a^{4} b^{2}+27 a^{2} b^{4}-12 a \,b^{5}-b^{6}}-\frac {384 i a^{3} b^{6} d^{3}}{a^{6}+12 a^{5} b -27 a^{4} b^{2}+27 a^{2} b^{4}-12 a \,b^{5}-b^{6}}\right ) \textit {\_R}^{3}+\left (-\frac {72 i a^{5} b^{2} d}{a^{6}+12 a^{5} b -27 a^{4} b^{2}+27 a^{2} b^{4}-12 a \,b^{5}-b^{6}}-\frac {672 i a^{4} b^{3} d}{a^{6}+12 a^{5} b -27 a^{4} b^{2}+27 a^{2} b^{4}-12 a \,b^{5}-b^{6}}-\frac {1008 i a^{3} b^{4} d}{a^{6}+12 a^{5} b -27 a^{4} b^{2}+27 a^{2} b^{4}-12 a \,b^{5}-b^{6}}-\frac {288 i a^{2} b^{5} d}{a^{6}+12 a^{5} b -27 a^{4} b^{2}+27 a^{2} b^{4}-12 a \,b^{5}-b^{6}}-\frac {8 i a \,b^{6} d}{a^{6}+12 a^{5} b -27 a^{4} b^{2}+27 a^{2} b^{4}-12 a \,b^{5}-b^{6}}\right ) \textit {\_R} \right ) {\mathrm e}^{i \left (d x +c \right )}-\frac {a^{6}}{a^{6}+12 a^{5} b -27 a^{4} b^{2}+27 a^{2} b^{4}-12 a \,b^{5}-b^{6}}-\frac {12 a^{5} b}{a^{6}+12 a^{5} b -27 a^{4} b^{2}+27 a^{2} b^{4}-12 a \,b^{5}-b^{6}}+\frac {27 a^{4} b^{2}}{a^{6}+12 a^{5} b -27 a^{4} b^{2}+27 a^{2} b^{4}-12 a \,b^{5}-b^{6}}-\frac {27 a^{2} b^{4}}{a^{6}+12 a^{5} b -27 a^{4} b^{2}+27 a^{2} b^{4}-12 a \,b^{5}-b^{6}}+\frac {12 a \,b^{5}}{a^{6}+12 a^{5} b -27 a^{4} b^{2}+27 a^{2} b^{4}-12 a \,b^{5}-b^{6}}+\frac {b^{6}}{a^{6}+12 a^{5} b -27 a^{4} b^{2}+27 a^{2} b^{4}-12 a \,b^{5}-b^{6}}\right )\right )-\frac {\sin \left (3 d x +3 c \right )}{12 d b}\)	\(806\)

[In]

int(cos(d*x+c)^7/(a-b*sin(d*x+c)^4),x,method=_RETURNVERBOSE)

[Out]

-1/d*(-1/b*(1/3*sin(d*x+c)^3-3*sin(d*x+c))+1/b*(1/4*(-3*a-b)*(1/b*a)^(1/4)/a*(ln((sin(d*x+c)+(1/b*a)^(1/4))/(s
in(d*x+c)-(1/b*a)^(1/4)))+2*arctan(sin(d*x+c)/(1/b*a)^(1/4)))-1/4*(a+3*b)/b/(1/b*a)^(1/4)*(2*arctan(sin(d*x+c)
/(1/b*a)^(1/4))-ln((sin(d*x+c)+(1/b*a)^(1/4))/(sin(d*x+c)-(1/b*a)^(1/4))))))

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1429 vs. \(2 (101) = 202\).

Time = 0.60 (sec) , antiderivative size = 1429, normalized size of antiderivative = 10.91 \[ \int \frac {\cos ^7(c+d x)}{a-b \sin ^4(c+d x)} \, dx=\text {Too large to display} \]

[In]

integrate(cos(d*x+c)^7/(a-b*sin(d*x+c)^4),x, algorithm="fricas")

[Out]

1/12*(3*b*d*sqrt(-(a*b^3*d^2*sqrt((a^6 + 30*a^5*b + 255*a^4*b^2 + 452*a^3*b^3 + 255*a^2*b^4 + 30*a*b^5 + b^6)/
(a^3*b^7*d^4)) + 6*a^2 + 20*a*b + 6*b^2)/(a*b^3*d^2))*log(1/2*(a^6 + 12*a^5*b - 27*a^4*b^2 + 27*a^2*b^4 - 12*a
*b^5 - b^6)*sin(d*x + c) + 1/2*((a^4*b^5 + 3*a^3*b^6)*d^3*sqrt((a^6 + 30*a^5*b + 255*a^4*b^2 + 452*a^3*b^3 + 2
55*a^2*b^4 + 30*a*b^5 + b^6)/(a^3*b^7*d^4)) - (3*a^5*b^2 + 46*a^4*b^3 + 60*a^3*b^4 + 18*a^2*b^5 + a*b^6)*d)*sq
rt(-(a*b^3*d^2*sqrt((a^6 + 30*a^5*b + 255*a^4*b^2 + 452*a^3*b^3 + 255*a^2*b^4 + 30*a*b^5 + b^6)/(a^3*b^7*d^4))
+ 6*a^2 + 20*a*b + 6*b^2)/(a*b^3*d^2))) - 3*b*d*sqrt((a*b^3*d^2*sqrt((a^6 + 30*a^5*b + 255*a^4*b^2 + 452*a^3*
b^3 + 255*a^2*b^4 + 30*a*b^5 + b^6)/(a^3*b^7*d^4)) - 6*a^2 - 20*a*b - 6*b^2)/(a*b^3*d^2))*log(1/2*(a^6 + 12*a^
5*b - 27*a^4*b^2 + 27*a^2*b^4 - 12*a*b^5 - b^6)*sin(d*x + c) + 1/2*((a^4*b^5 + 3*a^3*b^6)*d^3*sqrt((a^6 + 30*a
^5*b + 255*a^4*b^2 + 452*a^3*b^3 + 255*a^2*b^4 + 30*a*b^5 + b^6)/(a^3*b^7*d^4)) + (3*a^5*b^2 + 46*a^4*b^3 + 60
*a^3*b^4 + 18*a^2*b^5 + a*b^6)*d)*sqrt((a*b^3*d^2*sqrt((a^6 + 30*a^5*b + 255*a^4*b^2 + 452*a^3*b^3 + 255*a^2*b
^4 + 30*a*b^5 + b^6)/(a^3*b^7*d^4)) - 6*a^2 - 20*a*b - 6*b^2)/(a*b^3*d^2))) - 3*b*d*sqrt(-(a*b^3*d^2*sqrt((a^6
+ 30*a^5*b + 255*a^4*b^2 + 452*a^3*b^3 + 255*a^2*b^4 + 30*a*b^5 + b^6)/(a^3*b^7*d^4)) + 6*a^2 + 20*a*b + 6*b^
2)/(a*b^3*d^2))*log(-1/2*(a^6 + 12*a^5*b - 27*a^4*b^2 + 27*a^2*b^4 - 12*a*b^5 - b^6)*sin(d*x + c) + 1/2*((a^4*
b^5 + 3*a^3*b^6)*d^3*sqrt((a^6 + 30*a^5*b + 255*a^4*b^2 + 452*a^3*b^3 + 255*a^2*b^4 + 30*a*b^5 + b^6)/(a^3*b^7
*d^4)) - (3*a^5*b^2 + 46*a^4*b^3 + 60*a^3*b^4 + 18*a^2*b^5 + a*b^6)*d)*sqrt(-(a*b^3*d^2*sqrt((a^6 + 30*a^5*b +
255*a^4*b^2 + 452*a^3*b^3 + 255*a^2*b^4 + 30*a*b^5 + b^6)/(a^3*b^7*d^4)) + 6*a^2 + 20*a*b + 6*b^2)/(a*b^3*d^2
))) + 3*b*d*sqrt((a*b^3*d^2*sqrt((a^6 + 30*a^5*b + 255*a^4*b^2 + 452*a^3*b^3 + 255*a^2*b^4 + 30*a*b^5 + b^6)/(
a^3*b^7*d^4)) - 6*a^2 - 20*a*b - 6*b^2)/(a*b^3*d^2))*log(-1/2*(a^6 + 12*a^5*b - 27*a^4*b^2 + 27*a^2*b^4 - 12*a
*b^5 - b^6)*sin(d*x + c) + 1/2*((a^4*b^5 + 3*a^3*b^6)*d^3*sqrt((a^6 + 30*a^5*b + 255*a^4*b^2 + 452*a^3*b^3 + 2
55*a^2*b^4 + 30*a*b^5 + b^6)/(a^3*b^7*d^4)) + (3*a^5*b^2 + 46*a^4*b^3 + 60*a^3*b^4 + 18*a^2*b^5 + a*b^6)*d)*sq
rt((a*b^3*d^2*sqrt((a^6 + 30*a^5*b + 255*a^4*b^2 + 452*a^3*b^3 + 255*a^2*b^4 + 30*a*b^5 + b^6)/(a^3*b^7*d^4))
- 6*a^2 - 20*a*b - 6*b^2)/(a*b^3*d^2))) - 4*(cos(d*x + c)^2 + 8)*sin(d*x + c))/(b*d)

Sympy [F(-1)]

Timed out. \[ \int \frac {\cos ^7(c+d x)}{a-b \sin ^4(c+d x)} \, dx=\text {Timed out} \]

[In]

integrate(cos(d*x+c)**7/(a-b*sin(d*x+c)**4),x)

[Out]

Timed out

Maxima [A] (verification not implemented)

none

Time = 0.34 (sec) , antiderivative size = 177, normalized size of antiderivative = 1.35 \[ \int \frac {\cos ^7(c+d x)}{a-b \sin ^4(c+d x)} \, dx=\frac {\frac {4 \, {\left (\sin \left (d x + c\right )^{3} - 9 \, \sin \left (d x + c\right )\right )}}{b} + \frac {3 \, {\left (\frac {2 \, {\left (b {\left (3 \, \sqrt {a} + \sqrt {b}\right )} + a^{\frac {3}{2}} + 3 \, a \sqrt {b}\right )} \arctan \left (\frac {\sqrt {b} \sin \left (d x + c\right )}{\sqrt {\sqrt {a} \sqrt {b}}}\right )}{\sqrt {a} \sqrt {\sqrt {a} \sqrt {b}} \sqrt {b}} + \frac {{\left (b {\left (3 \, \sqrt {a} - \sqrt {b}\right )} + a^{\frac {3}{2}} - 3 \, a \sqrt {b}\right )} \log \left (\frac {\sqrt {b} \sin \left (d x + c\right ) - \sqrt {\sqrt {a} \sqrt {b}}}{\sqrt {b} \sin \left (d x + c\right ) + \sqrt {\sqrt {a} \sqrt {b}}}\right )}{\sqrt {a} \sqrt {\sqrt {a} \sqrt {b}} \sqrt {b}}\right )}}{b}}{12 \, d} \]

[In]

integrate(cos(d*x+c)^7/(a-b*sin(d*x+c)^4),x, algorithm="maxima")

[Out]

1/12*(4*(sin(d*x + c)^3 - 9*sin(d*x + c))/b + 3*(2*(b*(3*sqrt(a) + sqrt(b)) + a^(3/2) + 3*a*sqrt(b))*arctan(sq
rt(b)*sin(d*x + c)/sqrt(sqrt(a)*sqrt(b)))/(sqrt(a)*sqrt(sqrt(a)*sqrt(b))*sqrt(b)) + (b*(3*sqrt(a) - sqrt(b)) +
a^(3/2) - 3*a*sqrt(b))*log((sqrt(b)*sin(d*x + c) - sqrt(sqrt(a)*sqrt(b)))/(sqrt(b)*sin(d*x + c) + sqrt(sqrt(a
)*sqrt(b))))/(sqrt(a)*sqrt(sqrt(a)*sqrt(b))*sqrt(b)))/b)/d

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 360 vs. \(2 (101) = 202\).

Time = 0.72 (sec) , antiderivative size = 360, normalized size of antiderivative = 2.75 \[ \int \frac {\cos ^7(c+d x)}{a-b \sin ^4(c+d x)} \, dx=\frac {\frac {8 \, {\left (b^{2} \sin \left (d x + c\right )^{3} - 9 \, b^{2} \sin \left (d x + c\right )\right )}}{b^{3}} - \frac {6 \, \sqrt {2} {\left (\left (-a b^{3}\right )^{\frac {3}{4}} {\left (a + 3 \, b\right )} - \left (-a b^{3}\right )^{\frac {1}{4}} {\left (3 \, a b^{2} + b^{3}\right )}\right )} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} \left (-\frac {a}{b}\right )^{\frac {1}{4}} + 2 \, \sin \left (d x + c\right )\right )}}{2 \, \left (-\frac {a}{b}\right )^{\frac {1}{4}}}\right )}{a b^{4}} - \frac {6 \, \sqrt {2} {\left (\left (-a b^{3}\right )^{\frac {3}{4}} {\left (a + 3 \, b\right )} - \left (-a b^{3}\right )^{\frac {1}{4}} {\left (3 \, a b^{2} + b^{3}\right )}\right )} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} \left (-\frac {a}{b}\right )^{\frac {1}{4}} - 2 \, \sin \left (d x + c\right )\right )}}{2 \, \left (-\frac {a}{b}\right )^{\frac {1}{4}}}\right )}{a b^{4}} + \frac {3 \, \sqrt {2} {\left (\left (-a b^{3}\right )^{\frac {3}{4}} {\left (a + 3 \, b\right )} + \left (-a b^{3}\right )^{\frac {1}{4}} {\left (3 \, a b^{2} + b^{3}\right )}\right )} \log \left (\sin \left (d x + c\right )^{2} + \sqrt {2} \left (-\frac {a}{b}\right )^{\frac {1}{4}} \sin \left (d x + c\right ) + \sqrt {-\frac {a}{b}}\right )}{a b^{4}} - \frac {3 \, \sqrt {2} {\left (\left (-a b^{3}\right )^{\frac {3}{4}} {\left (a + 3 \, b\right )} + \left (-a b^{3}\right )^{\frac {1}{4}} {\left (3 \, a b^{2} + b^{3}\right )}\right )} \log \left (\sin \left (d x + c\right )^{2} - \sqrt {2} \left (-\frac {a}{b}\right )^{\frac {1}{4}} \sin \left (d x + c\right ) + \sqrt {-\frac {a}{b}}\right )}{a b^{4}}}{24 \, d} \]

[In]

integrate(cos(d*x+c)^7/(a-b*sin(d*x+c)^4),x, algorithm="giac")

[Out]

1/24*(8*(b^2*sin(d*x + c)^3 - 9*b^2*sin(d*x + c))/b^3 - 6*sqrt(2)*((-a*b^3)^(3/4)*(a + 3*b) - (-a*b^3)^(1/4)*(
3*a*b^2 + b^3))*arctan(1/2*sqrt(2)*(sqrt(2)*(-a/b)^(1/4) + 2*sin(d*x + c))/(-a/b)^(1/4))/(a*b^4) - 6*sqrt(2)*(
(-a*b^3)^(3/4)*(a + 3*b) - (-a*b^3)^(1/4)*(3*a*b^2 + b^3))*arctan(-1/2*sqrt(2)*(sqrt(2)*(-a/b)^(1/4) - 2*sin(d
*x + c))/(-a/b)^(1/4))/(a*b^4) + 3*sqrt(2)*((-a*b^3)^(3/4)*(a + 3*b) + (-a*b^3)^(1/4)*(3*a*b^2 + b^3))*log(sin
(d*x + c)^2 + sqrt(2)*(-a/b)^(1/4)*sin(d*x + c) + sqrt(-a/b))/(a*b^4) - 3*sqrt(2)*((-a*b^3)^(3/4)*(a + 3*b) +
(-a*b^3)^(1/4)*(3*a*b^2 + b^3))*log(sin(d*x + c)^2 - sqrt(2)*(-a/b)^(1/4)*sin(d*x + c) + sqrt(-a/b))/(a*b^4))/
d

Mupad [B] (verification not implemented)

Time = 0.76 (sec) , antiderivative size = 1931, normalized size of antiderivative = 14.74 \[ \int \frac {\cos ^7(c+d x)}{a-b \sin ^4(c+d x)} \, dx=\text {Too large to display} \]

[In]

int(cos(c + d*x)^7/(a - b*sin(c + d*x)^4),x)

[Out]

(atan((a^3*sin(c + d*x)*(- (a^3*b^7)^(1/2)/(16*b^7) - (3*a)/(8*b^3) - 5/(4*b^2) - 3/(8*a*b) - (15*(a^3*b^7)^(1
/2))/(16*a*b^6) - (15*(a^3*b^7)^(1/2))/(16*a^2*b^5) - (a^3*b^7)^(1/2)/(16*a^3*b^4))^(1/2)*8i)/(92*a*b + (120*(
a^3*b^7)^(1/2))/b^3 + 120*a^2 + 6*b^2 + (36*a^3)/b + (2*a^4)/b^2 + (36*(a^3*b^7)^(1/2))/(a*b^2) + (2*(a^3*b^7)
^(1/2))/(a^2*b) + (6*a^2*(a^3*b^7)^(1/2))/b^5 + (92*a*(a^3*b^7)^(1/2))/b^4) + (b^3*sin(c + d*x)*(- (a^3*b^7)^(
1/2)/(16*b^7) - (3*a)/(8*b^3) - 5/(4*b^2) - 3/(8*a*b) - (15*(a^3*b^7)^(1/2))/(16*a*b^6) - (15*(a^3*b^7)^(1/2))
/(16*a^2*b^5) - (a^3*b^7)^(1/2)/(16*a^3*b^4))^(1/2)*8i)/(92*a*b + (120*(a^3*b^7)^(1/2))/b^3 + 120*a^2 + 6*b^2
+ (36*a^3)/b + (2*a^4)/b^2 + (36*(a^3*b^7)^(1/2))/(a*b^2) + (2*(a^3*b^7)^(1/2))/(a^2*b) + (6*a^2*(a^3*b^7)^(1/
2))/b^5 + (92*a*(a^3*b^7)^(1/2))/b^4) + (a*b^2*sin(c + d*x)*(- (a^3*b^7)^(1/2)/(16*b^7) - (3*a)/(8*b^3) - 5/(4
*b^2) - 3/(8*a*b) - (15*(a^3*b^7)^(1/2))/(16*a*b^6) - (15*(a^3*b^7)^(1/2))/(16*a^2*b^5) - (a^3*b^7)^(1/2)/(16*
a^3*b^4))^(1/2)*120i)/(92*a*b + (120*(a^3*b^7)^(1/2))/b^3 + 120*a^2 + 6*b^2 + (36*a^3)/b + (2*a^4)/b^2 + (36*(
a^3*b^7)^(1/2))/(a*b^2) + (2*(a^3*b^7)^(1/2))/(a^2*b) + (6*a^2*(a^3*b^7)^(1/2))/b^5 + (92*a*(a^3*b^7)^(1/2))/b
^4) + (a^2*b*sin(c + d*x)*(- (a^3*b^7)^(1/2)/(16*b^7) - (3*a)/(8*b^3) - 5/(4*b^2) - 3/(8*a*b) - (15*(a^3*b^7)^
(1/2))/(16*a*b^6) - (15*(a^3*b^7)^(1/2))/(16*a^2*b^5) - (a^3*b^7)^(1/2)/(16*a^3*b^4))^(1/2)*120i)/(92*a*b + (1
20*(a^3*b^7)^(1/2))/b^3 + 120*a^2 + 6*b^2 + (36*a^3)/b + (2*a^4)/b^2 + (36*(a^3*b^7)^(1/2))/(a*b^2) + (2*(a^3*
b^7)^(1/2))/(a^2*b) + (6*a^2*(a^3*b^7)^(1/2))/b^5 + (92*a*(a^3*b^7)^(1/2))/b^4))*(-(a^3*(a^3*b^7)^(1/2) + b^3*
(a^3*b^7)^(1/2) + 6*a^2*b^6 + 20*a^3*b^5 + 6*a^4*b^4 + 15*a*b^2*(a^3*b^7)^(1/2) + 15*a^2*b*(a^3*b^7)^(1/2))/(1
6*a^3*b^7))^(1/2)*2i)/d - (3*sin(c + d*x))/(b*d) + (atan((a^3*sin(c + d*x)*((a^3*b^7)^(1/2)/(16*b^7) - (3*a)/(
8*b^3) - 5/(4*b^2) - 3/(8*a*b) + (15*(a^3*b^7)^(1/2))/(16*a*b^6) + (15*(a^3*b^7)^(1/2))/(16*a^2*b^5) + (a^3*b^
7)^(1/2)/(16*a^3*b^4))^(1/2)*8i)/(92*a*b - (120*(a^3*b^7)^(1/2))/b^3 + 120*a^2 + 6*b^2 + (36*a^3)/b + (2*a^4)/
b^2 - (36*(a^3*b^7)^(1/2))/(a*b^2) - (2*(a^3*b^7)^(1/2))/(a^2*b) - (6*a^2*(a^3*b^7)^(1/2))/b^5 - (92*a*(a^3*b^
7)^(1/2))/b^4) + (b^3*sin(c + d*x)*((a^3*b^7)^(1/2)/(16*b^7) - (3*a)/(8*b^3) - 5/(4*b^2) - 3/(8*a*b) + (15*(a^
3*b^7)^(1/2))/(16*a*b^6) + (15*(a^3*b^7)^(1/2))/(16*a^2*b^5) + (a^3*b^7)^(1/2)/(16*a^3*b^4))^(1/2)*8i)/(92*a*b
- (120*(a^3*b^7)^(1/2))/b^3 + 120*a^2 + 6*b^2 + (36*a^3)/b + (2*a^4)/b^2 - (36*(a^3*b^7)^(1/2))/(a*b^2) - (2*
(a^3*b^7)^(1/2))/(a^2*b) - (6*a^2*(a^3*b^7)^(1/2))/b^5 - (92*a*(a^3*b^7)^(1/2))/b^4) + (a*b^2*sin(c + d*x)*((a
^3*b^7)^(1/2)/(16*b^7) - (3*a)/(8*b^3) - 5/(4*b^2) - 3/(8*a*b) + (15*(a^3*b^7)^(1/2))/(16*a*b^6) + (15*(a^3*b^
7)^(1/2))/(16*a^2*b^5) + (a^3*b^7)^(1/2)/(16*a^3*b^4))^(1/2)*120i)/(92*a*b - (120*(a^3*b^7)^(1/2))/b^3 + 120*a
^2 + 6*b^2 + (36*a^3)/b + (2*a^4)/b^2 - (36*(a^3*b^7)^(1/2))/(a*b^2) - (2*(a^3*b^7)^(1/2))/(a^2*b) - (6*a^2*(a
^3*b^7)^(1/2))/b^5 - (92*a*(a^3*b^7)^(1/2))/b^4) + (a^2*b*sin(c + d*x)*((a^3*b^7)^(1/2)/(16*b^7) - (3*a)/(8*b^
3) - 5/(4*b^2) - 3/(8*a*b) + (15*(a^3*b^7)^(1/2))/(16*a*b^6) + (15*(a^3*b^7)^(1/2))/(16*a^2*b^5) + (a^3*b^7)^(
1/2)/(16*a^3*b^4))^(1/2)*120i)/(92*a*b - (120*(a^3*b^7)^(1/2))/b^3 + 120*a^2 + 6*b^2 + (36*a^3)/b + (2*a^4)/b^
2 - (36*(a^3*b^7)^(1/2))/(a*b^2) - (2*(a^3*b^7)^(1/2))/(a^2*b) - (6*a^2*(a^3*b^7)^(1/2))/b^5 - (92*a*(a^3*b^7)
^(1/2))/b^4))*((a^3*(a^3*b^7)^(1/2) + b^3*(a^3*b^7)^(1/2) - 6*a^2*b^6 - 20*a^3*b^5 - 6*a^4*b^4 + 15*a*b^2*(a^3
*b^7)^(1/2) + 15*a^2*b*(a^3*b^7)^(1/2))/(16*a^3*b^7))^(1/2)*2i)/d + sin(c + d*x)^3/(3*b*d)