Integrand size = 24, antiderivative size = 131 \[ \int \frac {\cos ^7(c+d x)}{a-b \sin ^4(c+d x)} \, dx=\frac {\left (\sqrt {a}+\sqrt {b}\right )^3 \arctan \left (\frac {\sqrt [4]{b} \sin (c+d x)}{\sqrt [4]{a}}\right )}{2 a^{3/4} b^{7/4} d}-\frac {\left (\sqrt {a}-\sqrt {b}\right )^3 \text {arctanh}\left (\frac {\sqrt [4]{b} \sin (c+d x)}{\sqrt [4]{a}}\right )}{2 a^{3/4} b^{7/4} d}-\frac {3 \sin (c+d x)}{b d}+\frac {\sin ^3(c+d x)}{3 b d} \] Output:
1/2*(a^(1/2)+b^(1/2))^3*arctan(b^(1/4)*sin(d*x+c)/a^(1/4))/a^(3/4)/b^(7/4) /d-1/2*(a^(1/2)-b^(1/2))^3*arctanh(b^(1/4)*sin(d*x+c)/a^(1/4))/a^(3/4)/b^( 7/4)/d-3*sin(d*x+c)/b/d+1/3*sin(d*x+c)^3/b/d
Result contains complex when optimal does not.
Time = 0.32 (sec) , antiderivative size = 207, normalized size of antiderivative = 1.58 \[ \int \frac {\cos ^7(c+d x)}{a-b \sin ^4(c+d x)} \, dx=\frac {3 \left (\sqrt {a}-\sqrt {b}\right )^3 \log \left (\sqrt [4]{a}-\sqrt [4]{b} \sin (c+d x)\right )+3 i \left (\sqrt {a}+\sqrt {b}\right )^3 \log \left (\sqrt [4]{a}-i \sqrt [4]{b} \sin (c+d x)\right )-3 i \left (\sqrt {a}+\sqrt {b}\right )^3 \log \left (\sqrt [4]{a}+i \sqrt [4]{b} \sin (c+d x)\right )-3 \left (\sqrt {a}-\sqrt {b}\right )^3 \log \left (\sqrt [4]{a}+\sqrt [4]{b} \sin (c+d x)\right )-36 a^{3/4} b^{3/4} \sin (c+d x)+4 a^{3/4} b^{3/4} \sin ^3(c+d x)}{12 a^{3/4} b^{7/4} d} \] Input:
Integrate[Cos[c + d*x]^7/(a - b*Sin[c + d*x]^4),x]
Output:
(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)
Time = 0.37 (sec) , antiderivative size = 123, normalized size of antiderivative = 0.94, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {3042, 3702, 1485, 2009}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int \frac {\cos ^7(c+d x)}{a-b \sin ^4(c+d x)} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\cos (c+d x)^7}{a-b \sin (c+d x)^4}dx\) |
| \(\Big \downarrow \) 3702 |
| \(\displaystyle \frac {\int \frac {\left (1-\sin ^2(c+d x)\right )^3}{a-b \sin ^4(c+d x)}d\sin (c+d x)}{d}\) |
| \(\Big \downarrow \) 1485 |
| \(\displaystyle \frac {\int \left (\frac {\sin ^2(c+d x)}{b}-\frac {3}{b}+\frac {-\left ((a+3 b) \sin ^2(c+d x)\right )+3 a+b}{b \left (a-b \sin ^4(c+d x)\right )}\right )d\sin (c+d x)}{d}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {\frac {\left (\sqrt {a}+\sqrt {b}\right )^3 \arctan \left (\frac {\sqrt [4]{b} \sin (c+d x)}{\sqrt [4]{a}}\right )}{2 a^{3/4} b^{7/4}}-\frac {\left (\sqrt {a}-\sqrt {b}\right )^3 \text {arctanh}\left (\frac {\sqrt [4]{b} \sin (c+d x)}{\sqrt [4]{a}}\right )}{2 a^{3/4} b^{7/4}}+\frac {\sin ^3(c+d x)}{3 b}-\frac {3 \sin (c+d x)}{b}}{d}\) |
Input:
Int[Cos[c + d*x]^7/(a - b*Sin[c + d*x]^4),x]
Output:
(((Sqrt[a] + Sqrt[b])^3*ArcTan[(b^(1/4)*Sin[c + d*x])/a^(1/4)])/(2*a^(3/4) *b^(7/4)) - ((Sqrt[a] - Sqrt[b])^3*ArcTanh[(b^(1/4)*Sin[c + d*x])/a^(1/4)] )/(2*a^(3/4)*b^(7/4)) - (3*Sin[c + d*x])/b + Sin[c + d*x]^3/(3*b))/d
Int[((d_) + (e_.)*(x_)^2)^(q_)/((a_) + (c_.)*(x_)^4), x_Symbol] :> Int[Expa ndIntegrand[(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]
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]}, Si mp[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])
Time = 3.14 (sec) , antiderivative size = 178, normalized size of antiderivative = 1.36
| method | result | size |
| derivativedivides | \(-\frac {-\frac {\frac {\sin \left (d x +c \right )^{3}}{3}-3 \sin \left (d x +c \right )}{b}+\frac {\frac {\left (-3 a -b \right ) \left (\frac {a}{b}\right )^{\frac {1}{4}} \left (\ln \left (\frac {\sin \left (d x +c \right )+\left (\frac {a}{b}\right )^{\frac {1}{4}}}{\sin \left (d x +c \right )-\left (\frac {a}{b}\right )^{\frac {1}{4}}}\right )+2 \arctan \left (\frac {\sin \left (d x +c \right )}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}\right )\right )}{4 a}-\frac {\left (a +3 b \right ) \left (2 \arctan \left (\frac {\sin \left (d x +c \right )}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}\right )-\ln \left (\frac {\sin \left (d x +c \right )+\left (\frac {a}{b}\right )^{\frac {1}{4}}}{\sin \left (d x +c \right )-\left (\frac {a}{b}\right )^{\frac {1}{4}}}\right )\right )}{4 b \left (\frac {a}{b}\right )^{\frac {1}{4}}}}{b}}{d}\) | \(178\) |
| default | \(-\frac {-\frac {\frac {\sin \left (d x +c \right )^{3}}{3}-3 \sin \left (d x +c \right )}{b}+\frac {\frac {\left (-3 a -b \right ) \left (\frac {a}{b}\right )^{\frac {1}{4}} \left (\ln \left (\frac {\sin \left (d x +c \right )+\left (\frac {a}{b}\right )^{\frac {1}{4}}}{\sin \left (d x +c \right )-\left (\frac {a}{b}\right )^{\frac {1}{4}}}\right )+2 \arctan \left (\frac {\sin \left (d x +c \right )}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}\right )\right )}{4 a}-\frac {\left (a +3 b \right ) \left (2 \arctan \left (\frac {\sin \left (d x +c \right )}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}\right )-\ln \left (\frac {\sin \left (d x +c \right )+\left (\frac {a}{b}\right )^{\frac {1}{4}}}{\sin \left (d x +c \right )-\left (\frac {a}{b}\right )^{\frac {1}{4}}}\right )\right )}{4 b \left (\frac {a}{b}\right )^{\frac {1}{4}}}}{b}}{d}\) | \(178\) |
| risch | \(\frac {11 i {\mathrm e}^{i \left (d x +c \right )}}{8 b d}-\frac {11 i {\mathrm e}^{-i \left (d x +c \right )}}{8 b d}+\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (256 a^{3} b^{7} d^{4} \textit {\_Z}^{4}+\left (192 a^{4} b^{4} d^{2}+640 a^{3} b^{5} d^{2}+192 a^{2} b^{6} d^{2}\right ) \textit {\_Z}^{2}-a^{6}+6 a^{5} b -15 a^{4} b^{2}+20 a^{3} b^{3}-15 b^{4} a^{2}+6 b^{5} a -b^{6}\right )}{\sum }\textit {\_R} \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}+\left (\left (-\frac {128 i a^{4} b^{5} d^{3}}{a^{6}+12 a^{5} b -27 a^{4} b^{2}+27 b^{4} a^{2}-12 b^{5} a -b^{6}}-\frac {384 i a^{3} b^{6} d^{3}}{a^{6}+12 a^{5} b -27 a^{4} b^{2}+27 b^{4} a^{2}-12 b^{5} a -b^{6}}\right ) \textit {\_R}^{3}+\left (-\frac {72 i a^{5} b^{2} d}{a^{6}+12 a^{5} b -27 a^{4} b^{2}+27 b^{4} a^{2}-12 b^{5} a -b^{6}}-\frac {672 i a^{4} b^{3} d}{a^{6}+12 a^{5} b -27 a^{4} b^{2}+27 b^{4} a^{2}-12 b^{5} a -b^{6}}-\frac {1008 i a^{3} b^{4} d}{a^{6}+12 a^{5} b -27 a^{4} b^{2}+27 b^{4} a^{2}-12 b^{5} a -b^{6}}-\frac {288 i a^{2} b^{5} d}{a^{6}+12 a^{5} b -27 a^{4} b^{2}+27 b^{4} a^{2}-12 b^{5} a -b^{6}}-\frac {8 i a \,b^{6} d}{a^{6}+12 a^{5} b -27 a^{4} b^{2}+27 b^{4} a^{2}-12 b^{5} a -b^{6}}\right ) \textit {\_R} \right ) {\mathrm e}^{i \left (d x +c \right )}-\frac {a^{6}}{a^{6}+12 a^{5} b -27 a^{4} b^{2}+27 b^{4} a^{2}-12 b^{5} a -b^{6}}-\frac {12 a^{5} b}{a^{6}+12 a^{5} b -27 a^{4} b^{2}+27 b^{4} a^{2}-12 b^{5} a -b^{6}}+\frac {27 a^{4} b^{2}}{a^{6}+12 a^{5} b -27 a^{4} b^{2}+27 b^{4} a^{2}-12 b^{5} a -b^{6}}-\frac {27 b^{4} a^{2}}{a^{6}+12 a^{5} b -27 a^{4} b^{2}+27 b^{4} a^{2}-12 b^{5} a -b^{6}}+\frac {12 b^{5} a}{a^{6}+12 a^{5} b -27 a^{4} b^{2}+27 b^{4} a^{2}-12 b^{5} a -b^{6}}+\frac {b^{6}}{a^{6}+12 a^{5} b -27 a^{4} b^{2}+27 b^{4} a^{2}-12 b^{5} a -b^{6}}\right )\right )-\frac {\sin \left (3 d x +3 c \right )}{12 b d}\) | \(806\) |
Input:
int(cos(d*x+c)^7/(a-b*sin(d*x+c)^4),x,method=_RETURNVERBOSE)
Output:
-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))/(sin(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))))))
Leaf count of result is larger than twice the leaf count of optimal. 1429 vs. \(2 (101) = 202\).
Time = 0.30 (sec) , antiderivative size = 1429, normalized size of antiderivative = 10.91 \[ \int \frac {\cos ^7(c+d x)}{a-b \sin ^4(c+d x)} \, dx=\text {Too large to display} \] Input:
integrate(cos(d*x+c)^7/(a-b*sin(d*x+c)^4),x, algorithm="fricas")
Output:
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 + 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 + 2 55*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)*s in(d*x + c) + 1/2*((a^4*b^5 + 3*a^3*b^6)*d^3*sqrt((a^6 + 30*a^5*b + 255...
Timed out. \[ \int \frac {\cos ^7(c+d x)}{a-b \sin ^4(c+d x)} \, dx=\text {Timed out} \] Input:
integrate(cos(d*x+c)**7/(a-b*sin(d*x+c)**4),x)
Output:
Timed out
Time = 0.11 (sec) , antiderivative size = 177, normalized size of antiderivative = 1.35 \[ \int \frac {\cos ^7(c+d x)}{a-b \sin ^4(c+d x)} \, dx=\frac {\frac {4 \, {\left (\sin \left (d x + c\right )^{3} - 9 \, \sin \left (d x + c\right )\right )}}{b} + \frac {3 \, {\left (\frac {2 \, {\left (b {\left (3 \, \sqrt {a} + \sqrt {b}\right )} + a^{\frac {3}{2}} + 3 \, a \sqrt {b}\right )} \arctan \left (\frac {\sqrt {b} \sin \left (d x + c\right )}{\sqrt {\sqrt {a} \sqrt {b}}}\right )}{\sqrt {a} \sqrt {\sqrt {a} \sqrt {b}} \sqrt {b}} + \frac {{\left (b {\left (3 \, \sqrt {a} - \sqrt {b}\right )} + a^{\frac {3}{2}} - 3 \, a \sqrt {b}\right )} \log \left (\frac {\sqrt {b} \sin \left (d x + c\right ) - \sqrt {\sqrt {a} \sqrt {b}}}{\sqrt {b} \sin \left (d x + c\right ) + \sqrt {\sqrt {a} \sqrt {b}}}\right )}{\sqrt {a} \sqrt {\sqrt {a} \sqrt {b}} \sqrt {b}}\right )}}{b}}{12 \, d} \] Input:
integrate(cos(d*x+c)^7/(a-b*sin(d*x+c)^4),x, algorithm="maxima")
Output:
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(sqrt(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
Leaf count of result is larger than twice the leaf count of optimal. 376 vs. \(2 (101) = 202\).
Time = 0.40 (sec) , antiderivative size = 376, normalized size of antiderivative = 2.87 \[ \int \frac {\cos ^7(c+d x)}{a-b \sin ^4(c+d x)} \, dx=-\frac {\sqrt {2} {\left (\left (-a b^{3}\right )^{\frac {3}{4}} {\left (a + 3 \, b\right )} - \left (-a b^{3}\right )^{\frac {1}{4}} {\left (3 \, a b^{2} + b^{3}\right )}\right )} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} \left (-\frac {a}{b}\right )^{\frac {1}{4}} + 2 \, \sin \left (d x + c\right )\right )}}{2 \, \left (-\frac {a}{b}\right )^{\frac {1}{4}}}\right )}{4 \, a b^{4} d} - \frac {\sqrt {2} {\left (\left (-a b^{3}\right )^{\frac {3}{4}} {\left (a + 3 \, b\right )} - \left (-a b^{3}\right )^{\frac {1}{4}} {\left (3 \, a b^{2} + b^{3}\right )}\right )} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} \left (-\frac {a}{b}\right )^{\frac {1}{4}} - 2 \, \sin \left (d x + c\right )\right )}}{2 \, \left (-\frac {a}{b}\right )^{\frac {1}{4}}}\right )}{4 \, a b^{4} d} + \frac {\sqrt {2} {\left (\left (-a b^{3}\right )^{\frac {3}{4}} {\left (a + 3 \, b\right )} + \left (-a b^{3}\right )^{\frac {1}{4}} {\left (3 \, a b^{2} + b^{3}\right )}\right )} \log \left (\sin \left (d x + c\right )^{2} + \sqrt {2} \left (-\frac {a}{b}\right )^{\frac {1}{4}} \sin \left (d x + c\right ) + \sqrt {-\frac {a}{b}}\right )}{8 \, a b^{4} d} - \frac {\sqrt {2} {\left (\left (-a b^{3}\right )^{\frac {3}{4}} {\left (a + 3 \, b\right )} + \left (-a b^{3}\right )^{\frac {1}{4}} {\left (3 \, a b^{2} + b^{3}\right )}\right )} \log \left (\sin \left (d x + c\right )^{2} - \sqrt {2} \left (-\frac {a}{b}\right )^{\frac {1}{4}} \sin \left (d x + c\right ) + \sqrt {-\frac {a}{b}}\right )}{8 \, a b^{4} d} + \frac {b^{2} d^{2} \sin \left (d x + c\right )^{3} - 9 \, b^{2} d^{2} \sin \left (d x + c\right )}{3 \, b^{3} d^{3}} \] Input:
integrate(cos(d*x+c)^7/(a-b*sin(d*x+c)^4),x, algorithm="giac")
Output:
-1/4*sqrt(2)*((-a*b^3)^(3/4)*(a + 3*b) - (-a*b^3)^(1/4)*(3*a*b^2 + b^3))*a rctan(1/2*sqrt(2)*(sqrt(2)*(-a/b)^(1/4) + 2*sin(d*x + c))/(-a/b)^(1/4))/(a *b^4*d) - 1/4*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*d) + 1/8*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) - 1/8*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) + 1/3*(b^2*d^2*sin(d*x + c)^3 - 9*b^2*d^2*sin( d*x + c))/(b^3*d^3)
Time = 0.75 (sec) , antiderivative size = 1931, normalized size of antiderivative = 14.74 \[ \int \frac {\cos ^7(c+d x)}{a-b \sin ^4(c+d x)} \, dx=\text {Too large to display} \] Input:
int(cos(c + d*x)^7/(a - b*sin(c + d*x)^4),x)
Output:
(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...
\[ \int \frac {\cos ^7(c+d x)}{a-b \sin ^4(c+d x)} \, dx=-\left (\int \frac {\cos \left (d x +c \right )^{7}}{\sin \left (d x +c \right )^{4} b -a}d x \right ) \] Input:
int(cos(d*x+c)^7/(a-b*sin(d*x+c)^4),x)
Output:
- int(cos(c + d*x)**7/(sin(c + d*x)**4*b - a),x)