Integrand size = 21, antiderivative size = 290 \[ \int \frac {\sec (c+d x)}{a+b \sin ^3(c+d x)} \, dx=-\frac {\sqrt [3]{b} \left (a^{4/3}-b^{4/3}\right ) \arctan \left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} \sin (c+d x)}{\sqrt {3} \sqrt [3]{a}}\right )}{\sqrt {3} a^{2/3} \left (a^2-b^2\right ) d}-\frac {\log (1-\sin (c+d x))}{2 (a+b) d}+\frac {\log (1+\sin (c+d x))}{2 (a-b) d}-\frac {\sqrt [3]{b} \left (a^{4/3}+b^{4/3}\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} \sin (c+d x)\right )}{3 a^{2/3} \left (a^2-b^2\right ) d}+\frac {\sqrt [3]{b} \left (a^{4/3}+b^{4/3}\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} \sin (c+d x)+b^{2/3} \sin ^2(c+d x)\right )}{6 a^{2/3} \left (a^2-b^2\right ) d}-\frac {b \log \left (a+b \sin ^3(c+d x)\right )}{3 \left (a^2-b^2\right ) d} \] Output:
-1/3*b^(1/3)*(a^(4/3)-b^(4/3))*arctan(1/3*(a^(1/3)-2*b^(1/3)*sin(d*x+c))*3 ^(1/2)/a^(1/3))*3^(1/2)/a^(2/3)/(a^2-b^2)/d-1/2*ln(1-sin(d*x+c))/(a+b)/d+1 /2*ln(1+sin(d*x+c))/(a-b)/d-1/3*b^(1/3)*(a^(4/3)+b^(4/3))*ln(a^(1/3)+b^(1/ 3)*sin(d*x+c))/a^(2/3)/(a^2-b^2)/d+1/6*b^(1/3)*(a^(4/3)+b^(4/3))*ln(a^(2/3 )-a^(1/3)*b^(1/3)*sin(d*x+c)+b^(2/3)*sin(d*x+c)^2)/a^(2/3)/(a^2-b^2)/d-1/3 *b*ln(a+b*sin(d*x+c)^3)/(a^2-b^2)/d
Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.
Time = 0.26 (sec) , antiderivative size = 268, normalized size of antiderivative = 0.92 \[ \int \frac {\sec (c+d x)}{a+b \sin ^3(c+d x)} \, dx=\frac {2 \sqrt {3} b^{5/3} \arctan \left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} \sin (c+d x)}{\sqrt {3} \sqrt [3]{a}}\right )-3 a^{5/3} \log (1-\sin (c+d x))+3 a^{2/3} b \log (1-\sin (c+d x))+3 a^{5/3} \log (1+\sin (c+d x))+3 a^{2/3} b \log (1+\sin (c+d x))-2 b^{5/3} \log \left (\sqrt [3]{a}+\sqrt [3]{b} \sin (c+d x)\right )+b^{5/3} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} \sin (c+d x)+b^{2/3} \sin ^2(c+d x)\right )-2 a^{2/3} b \log \left (a+b \sin ^3(c+d x)\right )+3 a^{2/3} b \operatorname {Hypergeometric2F1}\left (\frac {2}{3},1,\frac {5}{3},-\frac {b \sin ^3(c+d x)}{a}\right ) \sin ^2(c+d x)}{6 a^{2/3} (a-b) (a+b) d} \] Input:
Integrate[Sec[c + d*x]/(a + b*Sin[c + d*x]^3),x]
Output:
(2*Sqrt[3]*b^(5/3)*ArcTan[(a^(1/3) - 2*b^(1/3)*Sin[c + d*x])/(Sqrt[3]*a^(1 /3))] - 3*a^(5/3)*Log[1 - Sin[c + d*x]] + 3*a^(2/3)*b*Log[1 - Sin[c + d*x] ] + 3*a^(5/3)*Log[1 + Sin[c + d*x]] + 3*a^(2/3)*b*Log[1 + Sin[c + d*x]] - 2*b^(5/3)*Log[a^(1/3) + b^(1/3)*Sin[c + d*x]] + b^(5/3)*Log[a^(2/3) - a^(1 /3)*b^(1/3)*Sin[c + d*x] + b^(2/3)*Sin[c + d*x]^2] - 2*a^(2/3)*b*Log[a + b *Sin[c + d*x]^3] + 3*a^(2/3)*b*Hypergeometric2F1[2/3, 1, 5/3, -((b*Sin[c + d*x]^3)/a)]*Sin[c + d*x]^2)/(6*a^(2/3)*(a - b)*(a + b)*d)
Time = 0.61 (sec) , antiderivative size = 276, normalized size of antiderivative = 0.95, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.190, Rules used = {3042, 3702, 7276, 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.
| \(\displaystyle \int \frac {\sec (c+d x)}{a+b \sin ^3(c+d x)} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {1}{\cos (c+d x) \left (a+b \sin (c+d x)^3\right )}dx\) |
| \(\Big \downarrow \) 3702 |
| \(\displaystyle \frac {\int \frac {1}{\left (1-\sin ^2(c+d x)\right ) \left (b \sin ^3(c+d x)+a\right )}d\sin (c+d x)}{d}\) |
| \(\Big \downarrow \) 7276 |
| \(\displaystyle \frac {\int \left (\frac {b \left (b \sin ^2(c+d x)-a \sin (c+d x)+b\right )}{\left (b^2-a^2\right ) \left (b \sin ^3(c+d x)+a\right )}-\frac {1}{2 (a+b) (\sin (c+d x)-1)}+\frac {1}{2 (a-b) (\sin (c+d x)+1)}\right )d\sin (c+d x)}{d}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {-\frac {b \log \left (a+b \sin ^3(c+d x)\right )}{3 \left (a^2-b^2\right )}-\frac {\sqrt [3]{b} \left (a^{4/3}-b^{4/3}\right ) \arctan \left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} \sin (c+d x)}{\sqrt {3} \sqrt [3]{a}}\right )}{\sqrt {3} a^{2/3} \left (a^2-b^2\right )}+\frac {\sqrt [3]{b} \left (a^{4/3}+b^{4/3}\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} \sin (c+d x)+b^{2/3} \sin ^2(c+d x)\right )}{6 a^{2/3} \left (a^2-b^2\right )}-\frac {\sqrt [3]{b} \left (a^{4/3}+b^{4/3}\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} \sin (c+d x)\right )}{3 a^{2/3} \left (a^2-b^2\right )}-\frac {\log (1-\sin (c+d x))}{2 (a+b)}+\frac {\log (\sin (c+d x)+1)}{2 (a-b)}}{d}\) |
Input:
Int[Sec[c + d*x]/(a + b*Sin[c + d*x]^3),x]
Output:
(-((b^(1/3)*(a^(4/3) - b^(4/3))*ArcTan[(a^(1/3) - 2*b^(1/3)*Sin[c + d*x])/ (Sqrt[3]*a^(1/3))])/(Sqrt[3]*a^(2/3)*(a^2 - b^2))) - Log[1 - Sin[c + d*x]] /(2*(a + b)) + Log[1 + Sin[c + d*x]]/(2*(a - b)) - (b^(1/3)*(a^(4/3) + b^( 4/3))*Log[a^(1/3) + b^(1/3)*Sin[c + d*x]])/(3*a^(2/3)*(a^2 - b^2)) + (b^(1 /3)*(a^(4/3) + b^(4/3))*Log[a^(2/3) - a^(1/3)*b^(1/3)*Sin[c + d*x] + b^(2/ 3)*Sin[c + d*x]^2])/(6*a^(2/3)*(a^2 - b^2)) - (b*Log[a + b*Sin[c + d*x]^3] )/(3*(a^2 - b^2)))/d
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])
Int[(u_)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> With[{v = RationalFunctionE xpand[u/(a + b*x^n), x]}, Int[v, x] /; SumQ[v]] /; FreeQ[{a, b}, x] && IGtQ [n, 0]
Time = 0.96 (sec) , antiderivative size = 300, normalized size of antiderivative = 1.03
| method | result | size |
| derivativedivides | \(\frac {\frac {\left (-b \left (\frac {\ln \left (\sin \left (d x +c \right )+\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {2}{3}}}-\frac {\ln \left (\sin \left (d x +c \right )^{2}-\left (\frac {a}{b}\right )^{\frac {1}{3}} \sin \left (d x +c \right )+\left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{6 b \left (\frac {a}{b}\right )^{\frac {2}{3}}}+\frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 \sin \left (d x +c \right )}{\left (\frac {a}{b}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {2}{3}}}\right )+a \left (-\frac {\ln \left (\sin \left (d x +c \right )+\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {1}{3}}}+\frac {\ln \left (\sin \left (d x +c \right )^{2}-\left (\frac {a}{b}\right )^{\frac {1}{3}} \sin \left (d x +c \right )+\left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{6 b \left (\frac {a}{b}\right )^{\frac {1}{3}}}+\frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 \sin \left (d x +c \right )}{\left (\frac {a}{b}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {1}{3}}}\right )-\frac {\ln \left (a +b \sin \left (d x +c \right )^{3}\right )}{3}\right ) b}{\left (a -b \right ) \left (a +b \right )}-\frac {\ln \left (\sin \left (d x +c \right )-1\right )}{2 a +2 b}+\frac {\ln \left (1+\sin \left (d x +c \right )\right )}{2 a -2 b}}{d}\) | \(300\) |
| default | \(\frac {\frac {\left (-b \left (\frac {\ln \left (\sin \left (d x +c \right )+\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {2}{3}}}-\frac {\ln \left (\sin \left (d x +c \right )^{2}-\left (\frac {a}{b}\right )^{\frac {1}{3}} \sin \left (d x +c \right )+\left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{6 b \left (\frac {a}{b}\right )^{\frac {2}{3}}}+\frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 \sin \left (d x +c \right )}{\left (\frac {a}{b}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {2}{3}}}\right )+a \left (-\frac {\ln \left (\sin \left (d x +c \right )+\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {1}{3}}}+\frac {\ln \left (\sin \left (d x +c \right )^{2}-\left (\frac {a}{b}\right )^{\frac {1}{3}} \sin \left (d x +c \right )+\left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{6 b \left (\frac {a}{b}\right )^{\frac {1}{3}}}+\frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 \sin \left (d x +c \right )}{\left (\frac {a}{b}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {1}{3}}}\right )-\frac {\ln \left (a +b \sin \left (d x +c \right )^{3}\right )}{3}\right ) b}{\left (a -b \right ) \left (a +b \right )}-\frac {\ln \left (\sin \left (d x +c \right )-1\right )}{2 a +2 b}+\frac {\ln \left (1+\sin \left (d x +c \right )\right )}{2 a -2 b}}{d}\) | \(300\) |
| risch | \(-\frac {i x}{a -b}-\frac {i c}{d \left (a -b \right )}+\frac {i x}{a +b}+\frac {i c}{d \left (a +b \right )}+\frac {2 i a^{2} b \,d^{3} x}{a^{4} d^{3}-a^{2} b^{2} d^{3}}+\frac {2 i a^{2} b \,d^{2} c}{a^{4} d^{3}-a^{2} b^{2} d^{3}}+\frac {\ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right )}{d \left (a -b \right )}-\frac {\ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right )}{d \left (a +b \right )}+2 \left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\left (216 a^{4} d^{3}-216 a^{2} b^{2} d^{3}\right ) \textit {\_Z}^{3}+108 a^{2} b \,d^{2} \textit {\_Z}^{2}+b \right )}{\sum }\textit {\_R} \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}+\left (\left (\frac {72 i a^{5} d^{2}}{a^{2} b +b^{3}}-\frac {72 i a^{3} b^{2} d^{2}}{a^{2} b +b^{3}}\right ) \textit {\_R}^{2}+\left (\frac {24 i a^{3} b d}{a^{2} b +b^{3}}+\frac {12 i a \,b^{3} d}{a^{2} b +b^{3}}\right ) \textit {\_R} -\frac {2 i a \,b^{2}}{a^{2} b +b^{3}}\right ) {\mathrm e}^{i \left (d x +c \right )}-\frac {a^{2} b}{a^{2} b +b^{3}}-\frac {b^{3}}{a^{2} b +b^{3}}\right )\right )\) | \(367\) |
Input:
int(sec(d*x+c)/(a+b*sin(d*x+c)^3),x,method=_RETURNVERBOSE)
Output:
1/d*((-b*(1/3/b/(1/b*a)^(2/3)*ln(sin(d*x+c)+(1/b*a)^(1/3))-1/6/b/(1/b*a)^( 2/3)*ln(sin(d*x+c)^2-(1/b*a)^(1/3)*sin(d*x+c)+(1/b*a)^(2/3))+1/3/b/(1/b*a) ^(2/3)*3^(1/2)*arctan(1/3*3^(1/2)*(2/(1/b*a)^(1/3)*sin(d*x+c)-1)))+a*(-1/3 /b/(1/b*a)^(1/3)*ln(sin(d*x+c)+(1/b*a)^(1/3))+1/6/b/(1/b*a)^(1/3)*ln(sin(d *x+c)^2-(1/b*a)^(1/3)*sin(d*x+c)+(1/b*a)^(2/3))+1/3*3^(1/2)/b/(1/b*a)^(1/3 )*arctan(1/3*3^(1/2)*(2/(1/b*a)^(1/3)*sin(d*x+c)-1)))-1/3*ln(a+b*sin(d*x+c )^3))*b/(a-b)/(a+b)-1/(2*a+2*b)*ln(sin(d*x+c)-1)+1/(2*a-2*b)*ln(1+sin(d*x+ c)))
Result contains complex when optimal does not.
Time = 0.88 (sec) , antiderivative size = 4396, normalized size of antiderivative = 15.16 \[ \int \frac {\sec (c+d x)}{a+b \sin ^3(c+d x)} \, dx=\text {Too large to display} \] Input:
integrate(sec(d*x+c)/(a+b*sin(d*x+c)^3),x, algorithm="fricas")
Output:
Too large to include
\[ \int \frac {\sec (c+d x)}{a+b \sin ^3(c+d x)} \, dx=\int \frac {\sec {\left (c + d x \right )}}{a + b \sin ^{3}{\left (c + d x \right )}}\, dx \] Input:
integrate(sec(d*x+c)/(a+b*sin(d*x+c)**3),x)
Output:
Integral(sec(c + d*x)/(a + b*sin(c + d*x)**3), x)
Time = 0.12 (sec) , antiderivative size = 288, normalized size of antiderivative = 0.99 \[ \int \frac {\sec (c+d x)}{a+b \sin ^3(c+d x)} \, dx=\frac {\frac {2 \, \sqrt {3} {\left (a {\left (3 \, \left (\frac {a}{b}\right )^{\frac {2}{3}} + 2\right )} - b {\left (3 \, \left (\frac {a}{b}\right )^{\frac {1}{3}} + \frac {2 \, a}{b}\right )}\right )} \arctan \left (-\frac {\sqrt {3} {\left (\left (\frac {a}{b}\right )^{\frac {1}{3}} - 2 \, \sin \left (d x + c\right )\right )}}{3 \, \left (\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{{\left (a^{2} \left (\frac {a}{b}\right )^{\frac {2}{3}} - b^{2} \left (\frac {a}{b}\right )^{\frac {2}{3}}\right )} \left (\frac {a}{b}\right )^{\frac {1}{3}}} - \frac {3 \, {\left (b {\left (2 \, \left (\frac {a}{b}\right )^{\frac {2}{3}} - 1\right )} - a \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )} \log \left (\sin \left (d x + c\right )^{2} - \left (\frac {a}{b}\right )^{\frac {1}{3}} \sin \left (d x + c\right ) + \left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{a^{2} \left (\frac {a}{b}\right )^{\frac {2}{3}} - b^{2} \left (\frac {a}{b}\right )^{\frac {2}{3}}} - \frac {6 \, {\left (b {\left (\left (\frac {a}{b}\right )^{\frac {2}{3}} + 1\right )} + a \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )} \log \left (\left (\frac {a}{b}\right )^{\frac {1}{3}} + \sin \left (d x + c\right )\right )}{a^{2} \left (\frac {a}{b}\right )^{\frac {2}{3}} - b^{2} \left (\frac {a}{b}\right )^{\frac {2}{3}}} + \frac {9 \, \log \left (\sin \left (d x + c\right ) + 1\right )}{a - b} - \frac {9 \, \log \left (\sin \left (d x + c\right ) - 1\right )}{a + b}}{18 \, d} \] Input:
integrate(sec(d*x+c)/(a+b*sin(d*x+c)^3),x, algorithm="maxima")
Output:
1/18*(2*sqrt(3)*(a*(3*(a/b)^(2/3) + 2) - b*(3*(a/b)^(1/3) + 2*a/b))*arctan (-1/3*sqrt(3)*((a/b)^(1/3) - 2*sin(d*x + c))/(a/b)^(1/3))/((a^2*(a/b)^(2/3 ) - b^2*(a/b)^(2/3))*(a/b)^(1/3)) - 3*(b*(2*(a/b)^(2/3) - 1) - a*(a/b)^(1/ 3))*log(sin(d*x + c)^2 - (a/b)^(1/3)*sin(d*x + c) + (a/b)^(2/3))/(a^2*(a/b )^(2/3) - b^2*(a/b)^(2/3)) - 6*(b*((a/b)^(2/3) + 1) + a*(a/b)^(1/3))*log(( a/b)^(1/3) + sin(d*x + c))/(a^2*(a/b)^(2/3) - b^2*(a/b)^(2/3)) + 9*log(sin (d*x + c) + 1)/(a - b) - 9*log(sin(d*x + c) - 1)/(a + b))/d
Time = 0.42 (sec) , antiderivative size = 335, normalized size of antiderivative = 1.16 \[ \int \frac {\sec (c+d x)}{a+b \sin ^3(c+d x)} \, dx=-\frac {{\left (a^{3} b^{2} d \left (-\frac {a}{b}\right )^{\frac {1}{3}} - a b^{4} d \left (-\frac {a}{b}\right )^{\frac {1}{3}} - a^{2} b^{3} d + b^{5} d\right )} \left (-\frac {a}{b}\right )^{\frac {1}{3}} \log \left ({\left | -\left (-\frac {a}{b}\right )^{\frac {1}{3}} + \sin \left (d x + c\right ) \right |}\right )}{3 \, {\left (a^{5} b d^{2} - 2 \, a^{3} b^{3} d^{2} + a b^{5} d^{2}\right )}} - \frac {b \log \left ({\left | b \sin \left (d x + c\right )^{3} + a \right |}\right )}{3 \, {\left (a^{2} d - b^{2} d\right )}} - \frac {{\left (\left (-a b^{2}\right )^{\frac {1}{3}} b^{2} + \left (-a b^{2}\right )^{\frac {2}{3}} a\right )} \arctan \left (\frac {\sqrt {3} {\left (\left (-\frac {a}{b}\right )^{\frac {1}{3}} + 2 \, \sin \left (d x + c\right )\right )}}{3 \, \left (-\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{{\left (\sqrt {3} a^{3} b - \sqrt {3} a b^{3}\right )} d} - \frac {{\left (\left (-a b^{2}\right )^{\frac {1}{3}} b^{2} - \left (-a b^{2}\right )^{\frac {2}{3}} a\right )} \log \left (\sin \left (d x + c\right )^{2} + \left (-\frac {a}{b}\right )^{\frac {1}{3}} \sin \left (d x + c\right ) + \left (-\frac {a}{b}\right )^{\frac {2}{3}}\right )}{6 \, {\left (a^{3} b - a b^{3}\right )} d} + \frac {\log \left ({\left | \sin \left (d x + c\right ) + 1 \right |}\right )}{2 \, {\left (a d - b d\right )}} - \frac {\log \left ({\left | \sin \left (d x + c\right ) - 1 \right |}\right )}{2 \, {\left (a d + b d\right )}} \] Input:
integrate(sec(d*x+c)/(a+b*sin(d*x+c)^3),x, algorithm="giac")
Output:
-1/3*(a^3*b^2*d*(-a/b)^(1/3) - a*b^4*d*(-a/b)^(1/3) - a^2*b^3*d + b^5*d)*( -a/b)^(1/3)*log(abs(-(-a/b)^(1/3) + sin(d*x + c)))/(a^5*b*d^2 - 2*a^3*b^3* d^2 + a*b^5*d^2) - 1/3*b*log(abs(b*sin(d*x + c)^3 + a))/(a^2*d - b^2*d) - ((-a*b^2)^(1/3)*b^2 + (-a*b^2)^(2/3)*a)*arctan(1/3*sqrt(3)*((-a/b)^(1/3) + 2*sin(d*x + c))/(-a/b)^(1/3))/((sqrt(3)*a^3*b - sqrt(3)*a*b^3)*d) - 1/6*( (-a*b^2)^(1/3)*b^2 - (-a*b^2)^(2/3)*a)*log(sin(d*x + c)^2 + (-a/b)^(1/3)*s in(d*x + c) + (-a/b)^(2/3))/((a^3*b - 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 = 35.64 (sec) , antiderivative size = 600, normalized size of antiderivative = 2.07 \[ \int \frac {\sec (c+d x)}{a+b \sin ^3(c+d x)} \, dx=\frac {\left (\sum _{k=1}^3\ln \left (-{\mathrm {root}\left (27\,a^2\,b^2\,z^3-27\,a^4\,z^3-27\,a^2\,b\,z^2-b,z,k\right )}^2\,a\,b^4\,13-{\mathrm {root}\left (27\,a^2\,b^2\,z^3-27\,a^4\,z^3-27\,a^2\,b\,z^2-b,z,k\right )}^3\,a\,b^5\,36-{\mathrm {root}\left (27\,a^2\,b^2\,z^3-27\,a^4\,z^3-27\,a^2\,b\,z^2-b,z,k\right )}^4\,a\,b^6\,36-{\mathrm {root}\left (27\,a^2\,b^2\,z^3-27\,a^4\,z^3-27\,a^2\,b\,z^2-b,z,k\right )}^2\,b^5\,\sin \left (c+d\,x\right )\,16-{\mathrm {root}\left (27\,a^2\,b^2\,z^3-27\,a^4\,z^3-27\,a^2\,b\,z^2-b,z,k\right )}^3\,b^6\,\sin \left (c+d\,x\right )\,12-{\mathrm {root}\left (27\,a^2\,b^2\,z^3-27\,a^4\,z^3-27\,a^2\,b\,z^2-b,z,k\right )}^3\,a^3\,b^3\,27-{\mathrm {root}\left (27\,a^2\,b^2\,z^3-27\,a^4\,z^3-27\,a^2\,b\,z^2-b,z,k\right )}^4\,a^3\,b^4\,180-\mathrm {root}\left (27\,a^2\,b^2\,z^3-27\,a^4\,z^3-27\,a^2\,b\,z^2-b,z,k\right )\,b^4\,\sin \left (c+d\,x\right )\,5-{\mathrm {root}\left (27\,a^2\,b^2\,z^3-27\,a^4\,z^3-27\,a^2\,b\,z^2-b,z,k\right )}^3\,a^2\,b^4\,\sin \left (c+d\,x\right )\,69-{\mathrm {root}\left (27\,a^2\,b^2\,z^3-27\,a^4\,z^3-27\,a^2\,b\,z^2-b,z,k\right )}^4\,a^2\,b^5\,\sin \left (c+d\,x\right )\,162-{\mathrm {root}\left (27\,a^2\,b^2\,z^3-27\,a^4\,z^3-27\,a^2\,b\,z^2-b,z,k\right )}^4\,a^4\,b^3\,\sin \left (c+d\,x\right )\,54\right )\,\mathrm {root}\left (27\,a^2\,b^2\,z^3-27\,a^4\,z^3-27\,a^2\,b\,z^2-b,z,k\right )\right )-\frac {\ln \left (\sin \left (c+d\,x\right )-1\right )}{2\,a+2\,b}+\frac {\ln \left (\sin \left (c+d\,x\right )+1\right )}{2\,a-2\,b}}{d} \] Input:
int(1/(cos(c + d*x)*(a + b*sin(c + d*x)^3)),x)
Output:
(symsum(log(- 13*root(27*a^2*b^2*z^3 - 27*a^4*z^3 - 27*a^2*b*z^2 - b, z, k )^2*a*b^4 - 36*root(27*a^2*b^2*z^3 - 27*a^4*z^3 - 27*a^2*b*z^2 - b, z, k)^ 3*a*b^5 - 36*root(27*a^2*b^2*z^3 - 27*a^4*z^3 - 27*a^2*b*z^2 - b, z, k)^4* a*b^6 - 16*root(27*a^2*b^2*z^3 - 27*a^4*z^3 - 27*a^2*b*z^2 - b, z, k)^2*b^ 5*sin(c + d*x) - 12*root(27*a^2*b^2*z^3 - 27*a^4*z^3 - 27*a^2*b*z^2 - b, z , k)^3*b^6*sin(c + d*x) - 27*root(27*a^2*b^2*z^3 - 27*a^4*z^3 - 27*a^2*b*z ^2 - b, z, k)^3*a^3*b^3 - 180*root(27*a^2*b^2*z^3 - 27*a^4*z^3 - 27*a^2*b* z^2 - b, z, k)^4*a^3*b^4 - 5*root(27*a^2*b^2*z^3 - 27*a^4*z^3 - 27*a^2*b*z ^2 - b, z, k)*b^4*sin(c + d*x) - 69*root(27*a^2*b^2*z^3 - 27*a^4*z^3 - 27* a^2*b*z^2 - b, z, k)^3*a^2*b^4*sin(c + d*x) - 162*root(27*a^2*b^2*z^3 - 27 *a^4*z^3 - 27*a^2*b*z^2 - b, z, k)^4*a^2*b^5*sin(c + d*x) - 54*root(27*a^2 *b^2*z^3 - 27*a^4*z^3 - 27*a^2*b*z^2 - b, z, k)^4*a^4*b^3*sin(c + d*x))*ro ot(27*a^2*b^2*z^3 - 27*a^4*z^3 - 27*a^2*b*z^2 - b, z, k), k, 1, 3) - log(s in(c + d*x) - 1)/(2*a + 2*b) + log(sin(c + d*x) + 1)/(2*a - 2*b))/d
\[ \int \frac {\sec (c+d x)}{a+b \sin ^3(c+d x)} \, dx=\frac {-\left (\int \frac {\sin \left (d x +c \right )^{3}}{\cos \left (d x +c \right ) \sin \left (d x +c \right )^{3} b +\cos \left (d x +c \right ) a}d x \right ) b d -\mathrm {log}\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )+\mathrm {log}\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{a d} \] Input:
int(sec(d*x+c)/(a+b*sin(d*x+c)^3),x)
Output:
( - int(sin(c + d*x)**3/(cos(c + d*x)*sin(c + d*x)**3*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)