Integrand size = 22, antiderivative size = 117 \[ \int \frac {\sec (c+d x)}{a-b \sin ^4(c+d x)} \, dx=\frac {\sqrt [4]{b} \arctan \left (\frac {\sqrt [4]{b} \sin (c+d x)}{\sqrt [4]{a}}\right )}{2 a^{3/4} \left (\sqrt {a}+\sqrt {b}\right ) d}+\frac {\text {arctanh}(\sin (c+d x))}{(a-b) d}-\frac {\sqrt [4]{b} \text {arctanh}\left (\frac {\sqrt [4]{b} \sin (c+d x)}{\sqrt [4]{a}}\right )}{2 a^{3/4} \left (\sqrt {a}-\sqrt {b}\right ) d} \] Output:
1/2*b^(1/4)*arctan(b^(1/4)*sin(d*x+c)/a^(1/4))/a^(3/4)/(a^(1/2)+b^(1/2))/d +arctanh(sin(d*x+c))/(a-b)/d-1/2*b^(1/4)*arctanh(b^(1/4)*sin(d*x+c)/a^(1/4 ))/a^(3/4)/(a^(1/2)-b^(1/2))/d
Result contains complex when optimal does not.
Time = 0.22 (sec) , antiderivative size = 184, normalized size of antiderivative = 1.57 \[ \int \frac {\sec (c+d x)}{a-b \sin ^4(c+d x)} \, dx=\frac {4 a^{3/4} \text {arctanh}(\sin (c+d x))+\sqrt [4]{b} \left (\left (\sqrt {a}+\sqrt {b}\right ) \log \left (\sqrt [4]{a}-\sqrt [4]{b} \sin (c+d x)\right )+i \left (\left (\sqrt {a}-\sqrt {b}\right ) \log \left (\sqrt [4]{a}-i \sqrt [4]{b} \sin (c+d x)\right )+\left (-\sqrt {a}+\sqrt {b}\right ) \log \left (\sqrt [4]{a}+i \sqrt [4]{b} \sin (c+d x)\right )+i \left (\sqrt {a}+\sqrt {b}\right ) \log \left (\sqrt [4]{a}+\sqrt [4]{b} \sin (c+d x)\right )\right )\right )}{4 a^{3/4} (a-b) d} \] Input:
Integrate[Sec[c + d*x]/(a - b*Sin[c + d*x]^4),x]
Output:
(4*a^(3/4)*ArcTanh[Sin[c + d*x]] + b^(1/4)*((Sqrt[a] + Sqrt[b])*Log[a^(1/4 ) - b^(1/4)*Sin[c + d*x]] + I*((Sqrt[a] - Sqrt[b])*Log[a^(1/4) - I*b^(1/4) *Sin[c + d*x]] + (-Sqrt[a] + Sqrt[b])*Log[a^(1/4) + I*b^(1/4)*Sin[c + d*x] ] + I*(Sqrt[a] + Sqrt[b])*Log[a^(1/4) + b^(1/4)*Sin[c + d*x]])))/(4*a^(3/4 )*(a - b)*d)
Time = 0.34 (sec) , antiderivative size = 112, normalized size of antiderivative = 0.96, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, 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.
| \(\displaystyle \int \frac {\sec (c+d x)}{a-b \sin ^4(c+d x)} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {1}{\cos (c+d x) \left (a-b \sin (c+d x)^4\right )}dx\) |
| \(\Big \downarrow \) 3702 |
| \(\displaystyle \frac {\int \frac {1}{\left (1-\sin ^2(c+d x)\right ) \left (a-b \sin ^4(c+d x)\right )}d\sin (c+d x)}{d}\) |
| \(\Big \downarrow \) 1485 |
| \(\displaystyle \frac {\int \left (-\frac {b \left (\sin ^2(c+d x)+1\right )}{(a-b) \left (a-b \sin ^4(c+d x)\right )}-\frac {1}{(a-b) \left (\sin ^2(c+d x)-1\right )}\right )d\sin (c+d x)}{d}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {\frac {\sqrt [4]{b} \arctan \left (\frac {\sqrt [4]{b} \sin (c+d x)}{\sqrt [4]{a}}\right )}{2 a^{3/4} \left (\sqrt {a}+\sqrt {b}\right )}-\frac {\sqrt [4]{b} \text {arctanh}\left (\frac {\sqrt [4]{b} \sin (c+d x)}{\sqrt [4]{a}}\right )}{2 a^{3/4} \left (\sqrt {a}-\sqrt {b}\right )}+\frac {\text {arctanh}(\sin (c+d x))}{a-b}}{d}\) |
Input:
Int[Sec[c + d*x]/(a - b*Sin[c + d*x]^4),x]
Output:
((b^(1/4)*ArcTan[(b^(1/4)*Sin[c + d*x])/a^(1/4)])/(2*a^(3/4)*(Sqrt[a] + Sq rt[b])) + ArcTanh[Sin[c + d*x]]/(a - b) - (b^(1/4)*ArcTanh[(b^(1/4)*Sin[c + d*x])/a^(1/4)])/(2*a^(3/4)*(Sqrt[a] - Sqrt[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])
Leaf count of result is larger than twice the leaf count of optimal. \(184\) vs. \(2(89)=178\).
Time = 1.17 (sec) , antiderivative size = 185, normalized size of antiderivative = 1.58
| method | result | size |
| derivativedivides | \(\frac {\frac {b \left (-\frac {\left (\frac {a}{b}\right )^{\frac {1}{4}} \left (\ln \left (\frac {\sin \left (d x +c \right )+\left (\frac {a}{b}\right )^{\frac {1}{4}}}{\sin \left (d x +c \right )-\left (\frac {a}{b}\right )^{\frac {1}{4}}}\right )+2 \arctan \left (\frac {\sin \left (d x +c \right )}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}\right )\right )}{4 a}+\frac {2 \arctan \left (\frac {\sin \left (d x +c \right )}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}\right )-\ln \left (\frac {\sin \left (d x +c \right )+\left (\frac {a}{b}\right )^{\frac {1}{4}}}{\sin \left (d x +c \right )-\left (\frac {a}{b}\right )^{\frac {1}{4}}}\right )}{4 b \left (\frac {a}{b}\right )^{\frac {1}{4}}}\right )}{a -b}+\frac {\ln \left (1+\sin \left (d x +c \right )\right )}{2 a -2 b}-\frac {\ln \left (\sin \left (d x +c \right )-1\right )}{2 a -2 b}}{d}\) | \(185\) |
| default | \(\frac {\frac {b \left (-\frac {\left (\frac {a}{b}\right )^{\frac {1}{4}} \left (\ln \left (\frac {\sin \left (d x +c \right )+\left (\frac {a}{b}\right )^{\frac {1}{4}}}{\sin \left (d x +c \right )-\left (\frac {a}{b}\right )^{\frac {1}{4}}}\right )+2 \arctan \left (\frac {\sin \left (d x +c \right )}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}\right )\right )}{4 a}+\frac {2 \arctan \left (\frac {\sin \left (d x +c \right )}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}\right )-\ln \left (\frac {\sin \left (d x +c \right )+\left (\frac {a}{b}\right )^{\frac {1}{4}}}{\sin \left (d x +c \right )-\left (\frac {a}{b}\right )^{\frac {1}{4}}}\right )}{4 b \left (\frac {a}{b}\right )^{\frac {1}{4}}}\right )}{a -b}+\frac {\ln \left (1+\sin \left (d x +c \right )\right )}{2 a -2 b}-\frac {\ln \left (\sin \left (d x +c \right )-1\right )}{2 a -2 b}}{d}\) | \(185\) |
| risch | \(-\frac {\ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right )}{d \left (a -b \right )}+\frac {\ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right )}{d \left (a -b \right )}+2 \left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\left (4096 a^{5} d^{4}-8192 a^{4} b \,d^{4}+4096 a^{3} b^{2} d^{4}\right ) \textit {\_Z}^{4}-256 a^{2} b \,d^{2} \textit {\_Z}^{2}-b \right )}{\sum }\textit {\_R} \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}+\left (\left (-\frac {1024 i d^{3} a^{5}}{a b +b^{2}}+\frac {2048 i d^{3} a^{4} b}{a b +b^{2}}-\frac {1024 i d^{3} a^{3} b^{2}}{a b +b^{2}}\right ) \textit {\_R}^{3}+\left (\frac {48 i d \,a^{2} b}{a b +b^{2}}+\frac {16 i a \,b^{2} d}{a b +b^{2}}\right ) \textit {\_R} \right ) {\mathrm e}^{i \left (d x +c \right )}-\frac {a b}{a b +b^{2}}-\frac {b^{2}}{a b +b^{2}}\right )\right )\) | \(257\) |
Input:
int(sec(d*x+c)/(a-b*sin(d*x+c)^4),x,method=_RETURNVERBOSE)
Output:
1/d*(1/(a-b)*b*(-1/4*(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/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)))))+1/(2*a-2*b)*ln(1+sin(d*x+c))-1/(2*a-2*b)*ln(sin( d*x+c)-1))
Leaf count of result is larger than twice the leaf count of optimal. 1329 vs. \(2 (89) = 178\).
Time = 0.20 (sec) , antiderivative size = 1329, normalized size of antiderivative = 11.36 \[ \int \frac {\sec (c+d x)}{a-b \sin ^4(c+d x)} \, dx=\text {Too large to display} \] Input:
integrate(sec(d*x+c)/(a-b*sin(d*x+c)^4),x, algorithm="fricas")
Output:
-1/4*((a - b)*d*sqrt(((a^3 - 2*a^2*b + a*b^2)*d^2*sqrt((a^2*b + 2*a*b^2 + b^3)/((a^7 - 4*a^6*b + 6*a^5*b^2 - 4*a^4*b^3 + a^3*b^4)*d^4)) + 2*b)/((a^3 - 2*a^2*b + a*b^2)*d^2))*log(1/2*(a*b + b^2)*sin(d*x + c) + 1/2*((a^5 - 2 *a^4*b + a^3*b^2)*d^3*sqrt((a^2*b + 2*a*b^2 + b^3)/((a^7 - 4*a^6*b + 6*a^5 *b^2 - 4*a^4*b^3 + a^3*b^4)*d^4)) - (a^2*b + a*b^2)*d)*sqrt(((a^3 - 2*a^2* b + a*b^2)*d^2*sqrt((a^2*b + 2*a*b^2 + b^3)/((a^7 - 4*a^6*b + 6*a^5*b^2 - 4*a^4*b^3 + a^3*b^4)*d^4)) + 2*b)/((a^3 - 2*a^2*b + a*b^2)*d^2))) - (a - b )*d*sqrt(-((a^3 - 2*a^2*b + a*b^2)*d^2*sqrt((a^2*b + 2*a*b^2 + b^3)/((a^7 - 4*a^6*b + 6*a^5*b^2 - 4*a^4*b^3 + a^3*b^4)*d^4)) - 2*b)/((a^3 - 2*a^2*b + a*b^2)*d^2))*log(1/2*(a*b + b^2)*sin(d*x + c) + 1/2*((a^5 - 2*a^4*b + a^ 3*b^2)*d^3*sqrt((a^2*b + 2*a*b^2 + b^3)/((a^7 - 4*a^6*b + 6*a^5*b^2 - 4*a^ 4*b^3 + a^3*b^4)*d^4)) + (a^2*b + a*b^2)*d)*sqrt(-((a^3 - 2*a^2*b + a*b^2) *d^2*sqrt((a^2*b + 2*a*b^2 + b^3)/((a^7 - 4*a^6*b + 6*a^5*b^2 - 4*a^4*b^3 + a^3*b^4)*d^4)) - 2*b)/((a^3 - 2*a^2*b + a*b^2)*d^2))) - (a - b)*d*sqrt(( (a^3 - 2*a^2*b + a*b^2)*d^2*sqrt((a^2*b + 2*a*b^2 + b^3)/((a^7 - 4*a^6*b + 6*a^5*b^2 - 4*a^4*b^3 + a^3*b^4)*d^4)) + 2*b)/((a^3 - 2*a^2*b + a*b^2)*d^ 2))*log(-1/2*(a*b + b^2)*sin(d*x + c) + 1/2*((a^5 - 2*a^4*b + a^3*b^2)*d^3 *sqrt((a^2*b + 2*a*b^2 + b^3)/((a^7 - 4*a^6*b + 6*a^5*b^2 - 4*a^4*b^3 + a^ 3*b^4)*d^4)) - (a^2*b + a*b^2)*d)*sqrt(((a^3 - 2*a^2*b + a*b^2)*d^2*sqrt(( a^2*b + 2*a*b^2 + b^3)/((a^7 - 4*a^6*b + 6*a^5*b^2 - 4*a^4*b^3 + a^3*b^...
\[ \int \frac {\sec (c+d x)}{a-b \sin ^4(c+d x)} \, dx=\int \frac {\sec {\left (c + d x \right )}}{a - b \sin ^{4}{\left (c + d x \right )}}\, dx \] Input:
integrate(sec(d*x+c)/(a-b*sin(d*x+c)**4),x)
Output:
Integral(sec(c + d*x)/(a - b*sin(c + d*x)**4), x)
Time = 0.11 (sec) , antiderivative size = 167, normalized size of antiderivative = 1.43 \[ \int \frac {\sec (c+d x)}{a-b \sin ^4(c+d x)} \, dx=\frac {\frac {b {\left (\frac {2 \, {\left (\sqrt {a} - \sqrt {b}\right )} \arctan \left (\frac {\sqrt {b} \sin \left (d x + c\right )}{\sqrt {\sqrt {a} \sqrt {b}}}\right )}{\sqrt {a} \sqrt {\sqrt {a} \sqrt {b}} \sqrt {b}} + \frac {{\left (\sqrt {a} + \sqrt {b}\right )} \log \left (\frac {\sqrt {b} \sin \left (d x + c\right ) - \sqrt {\sqrt {a} \sqrt {b}}}{\sqrt {b} \sin \left (d x + c\right ) + \sqrt {\sqrt {a} \sqrt {b}}}\right )}{\sqrt {a} \sqrt {\sqrt {a} \sqrt {b}} \sqrt {b}}\right )}}{a - b} + \frac {2 \, \log \left (\sin \left (d x + c\right ) + 1\right )}{a - b} - \frac {2 \, \log \left (\sin \left (d x + c\right ) - 1\right )}{a - b}}{4 \, d} \] Input:
integrate(sec(d*x+c)/(a-b*sin(d*x+c)^4),x, algorithm="maxima")
Output:
1/4*(b*(2*(sqrt(a) - sqrt(b))*arctan(sqrt(b)*sin(d*x + c)/sqrt(sqrt(a)*sqr t(b)))/(sqrt(a)*sqrt(sqrt(a)*sqrt(b))*sqrt(b)) + (sqrt(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)))/(a - b) + 2*l og(sin(d*x + c) + 1)/(a - b) - 2*log(sin(d*x + c) - 1)/(a - b))/d
Leaf count of result is larger than twice the leaf count of optimal. 384 vs. \(2 (89) = 178\).
Time = 0.42 (sec) , antiderivative size = 384, normalized size of antiderivative = 3.28 \[ \int \frac {\sec (c+d x)}{a-b \sin ^4(c+d x)} \, dx=-\frac {{\left (\left (-a b^{3}\right )^{\frac {1}{4}} b^{2} + \left (-a b^{3}\right )^{\frac {3}{4}}\right )} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} \left (-\frac {a}{b}\right )^{\frac {1}{4}} + 2 \, \sin \left (d x + c\right )\right )}}{2 \, \left (-\frac {a}{b}\right )^{\frac {1}{4}}}\right )}{2 \, {\left (\sqrt {2} a^{2} b^{2} - \sqrt {2} a b^{3}\right )} d} - \frac {{\left (\left (-a b^{3}\right )^{\frac {1}{4}} b^{2} + \left (-a b^{3}\right )^{\frac {3}{4}}\right )} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} \left (-\frac {a}{b}\right )^{\frac {1}{4}} - 2 \, \sin \left (d x + c\right )\right )}}{2 \, \left (-\frac {a}{b}\right )^{\frac {1}{4}}}\right )}{2 \, {\left (\sqrt {2} a^{2} b^{2} - \sqrt {2} a b^{3}\right )} d} - \frac {{\left (\left (-a b^{3}\right )^{\frac {1}{4}} b^{2} - \left (-a b^{3}\right )^{\frac {3}{4}}\right )} \log \left (\sin \left (d x + c\right )^{2} + \sqrt {2} \left (-\frac {a}{b}\right )^{\frac {1}{4}} \sin \left (d x + c\right ) + \sqrt {-\frac {a}{b}}\right )}{4 \, {\left (\sqrt {2} a^{2} b^{2} - \sqrt {2} a b^{3}\right )} d} + \frac {{\left (\left (-a b^{3}\right )^{\frac {1}{4}} b^{2} - \left (-a b^{3}\right )^{\frac {3}{4}}\right )} \log \left (\sin \left (d x + c\right )^{2} - \sqrt {2} \left (-\frac {a}{b}\right )^{\frac {1}{4}} \sin \left (d x + c\right ) + \sqrt {-\frac {a}{b}}\right )}{4 \, {\left (\sqrt {2} a^{2} b^{2} - \sqrt {2} a b^{3}\right )} d} + \frac {\log \left ({\left | \sin \left (d x + c\right ) + 1 \right |}\right )}{2 \, {\left (a d - b d\right )}} - \frac {\log \left ({\left | \sin \left (d x + c\right ) - 1 \right |}\right )}{2 \, {\left (a d - b d\right )}} \] Input:
integrate(sec(d*x+c)/(a-b*sin(d*x+c)^4),x, algorithm="giac")
Output:
-1/2*((-a*b^3)^(1/4)*b^2 + (-a*b^3)^(3/4))*arctan(1/2*sqrt(2)*(sqrt(2)*(-a /b)^(1/4) + 2*sin(d*x + c))/(-a/b)^(1/4))/((sqrt(2)*a^2*b^2 - sqrt(2)*a*b^ 3)*d) - 1/2*((-a*b^3)^(1/4)*b^2 + (-a*b^3)^(3/4))*arctan(-1/2*sqrt(2)*(sqr t(2)*(-a/b)^(1/4) - 2*sin(d*x + c))/(-a/b)^(1/4))/((sqrt(2)*a^2*b^2 - sqrt (2)*a*b^3)*d) - 1/4*((-a*b^3)^(1/4)*b^2 - (-a*b^3)^(3/4))*log(sin(d*x + c) ^2 + sqrt(2)*(-a/b)^(1/4)*sin(d*x + c) + sqrt(-a/b))/((sqrt(2)*a^2*b^2 - s qrt(2)*a*b^3)*d) + 1/4*((-a*b^3)^(1/4)*b^2 - (-a*b^3)^(3/4))*log(sin(d*x + c)^2 - sqrt(2)*(-a/b)^(1/4)*sin(d*x + c) + sqrt(-a/b))/((sqrt(2)*a^2*b^2 - sqrt(2)*a*b^3)*d) + 1/2*log(abs(sin(d*x + c) + 1))/(a*d - b*d) - 1/2*log (abs(sin(d*x + c) - 1))/(a*d - b*d)
Time = 37.94 (sec) , antiderivative size = 3891, normalized size of antiderivative = 33.26 \[ \int \frac {\sec (c+d x)}{a-b \sin ^4(c+d x)} \, dx=\text {Too large to display} \] Input:
int(1/(cos(c + d*x)*(a - b*sin(c + d*x)^4)),x)
Output:
(atan(((b^5*sin(c + d*x)*3i + ((((32*a*b^7 + 64*a^2*b^6 - 224*a^3*b^5 + 12 8*a^4*b^4 - (sin(c + d*x)*(512*a^2*b^7 - 512*a^3*b^6 - 512*a^4*b^5 + 512*a ^5*b^4))/(4*(a - b)))/(2*(a - b)) + (sin(c + d*x)*(32*a*b^6 - 16*b^7 + 240 *a^2*b^5))/2)/(2*(a - b)) - 10*a*b^5 + 2*b^6)*1i)/(2*(a - b)))/(a - b) + ( b^5*sin(c + d*x)*3i - ((((32*a*b^7 + 64*a^2*b^6 - 224*a^3*b^5 + 128*a^4*b^ 4 + (sin(c + d*x)*(512*a^2*b^7 - 512*a^3*b^6 - 512*a^4*b^5 + 512*a^5*b^4)) /(4*(a - b)))/(2*(a - b)) - (sin(c + d*x)*(32*a*b^6 - 16*b^7 + 240*a^2*b^5 ))/2)/(2*(a - b)) - 10*a*b^5 + 2*b^6)*1i)/(2*(a - b)))/(a - b))/((3*b^5*si n(c + d*x) + (((32*a*b^7 + 64*a^2*b^6 - 224*a^3*b^5 + 128*a^4*b^4 - (sin(c + d*x)*(512*a^2*b^7 - 512*a^3*b^6 - 512*a^4*b^5 + 512*a^5*b^4))/(4*(a - b )))/(2*(a - b)) + (sin(c + d*x)*(32*a*b^6 - 16*b^7 + 240*a^2*b^5))/2)/(2*( a - b)) - 10*a*b^5 + 2*b^6)/(2*(a - b)))/(a - b) - (3*b^5*sin(c + d*x) - ( ((32*a*b^7 + 64*a^2*b^6 - 224*a^3*b^5 + 128*a^4*b^4 + (sin(c + d*x)*(512*a ^2*b^7 - 512*a^3*b^6 - 512*a^4*b^5 + 512*a^5*b^4))/(4*(a - b)))/(2*(a - b) ) - (sin(c + d*x)*(32*a*b^6 - 16*b^7 + 240*a^2*b^5))/2)/(2*(a - b)) - 10*a *b^5 + 2*b^6)/(2*(a - b)))/(a - b)))*1i)/(d*(a - b)) - (atan(((((((2*a^2*b + a*(a^3*b)^(1/2) + b*(a^3*b)^(1/2))/(16*(a^5 - 2*a^4*b + a^3*b^2)))^(1/2 )*(64*a*b^7 + 128*a^2*b^6 - 448*a^3*b^5 + 256*a^4*b^4 + sin(c + d*x)*((2*a ^2*b + a*(a^3*b)^(1/2) + b*(a^3*b)^(1/2))/(16*(a^5 - 2*a^4*b + a^3*b^2)))^ (1/2)*(512*a^2*b^7 - 512*a^3*b^6 - 512*a^4*b^5 + 512*a^5*b^4)) - sin(c ...
\[ \int \frac {\sec (c+d x)}{a-b \sin ^4(c+d x)} \, dx=\frac {-\left (\int \frac {\sin \left (d x +c \right )^{4}}{\cos \left (d x +c \right ) \sin \left (d x +c \right )^{4} b -\cos \left (d x +c \right ) a}d x \right ) b d -\mathrm {log}\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )+\mathrm {log}\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{a d} \] Input:
int(sec(d*x+c)/(a-b*sin(d*x+c)^4),x)
Output:
( - int(sin(c + d*x)**4/(cos(c + d*x)*sin(c + d*x)**4*b - cos(c + d*x)*a), x)*b*d - log(tan((c + d*x)/2) - 1) + log(tan((c + d*x)/2) + 1))/(a*d)