Integrand size = 21, antiderivative size = 587 \[ \int \frac {\sec (c+d x)}{\left (a+b \sin ^3(c+d x)\right )^2} \, dx=-\frac {\sqrt [3]{b} \left (a^{4/3}-2 b^{4/3}\right ) \arctan \left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} \sin (c+d x)}{\sqrt {3} \sqrt [3]{a}}\right )}{3 \sqrt {3} a^{5/3} \left (a^2-b^2\right ) d}-\frac {\sqrt [3]{b} \left (a^2-2 a^{2/3} b^{4/3}+b^2\right ) \arctan \left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} \sin (c+d x)}{\sqrt {3} \sqrt [3]{a}}\right )}{\sqrt {3} \sqrt [3]{a} \left (a^2-b^2\right )^2 d}-\frac {\log (1-\sin (c+d x))}{2 (a+b)^2 d}+\frac {\log (1+\sin (c+d x))}{2 (a-b)^2 d}-\frac {\sqrt [3]{b} \left (a^{4/3}+2 b^{4/3}\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} \sin (c+d x)\right )}{9 a^{5/3} \left (a^2-b^2\right ) d}-\frac {\sqrt [3]{b} \left (a^2+2 a^{2/3} b^{4/3}+b^2\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} \sin (c+d x)\right )}{3 \sqrt [3]{a} \left (a^2-b^2\right )^2 d}+\frac {\sqrt [3]{b} \left (a^{4/3}+2 b^{4/3}\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} \sin (c+d x)+b^{2/3} \sin ^2(c+d x)\right )}{18 a^{5/3} \left (a^2-b^2\right ) d}+\frac {\sqrt [3]{b} \left (a^2+2 a^{2/3} b^{4/3}+b^2\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} \sin (c+d x)+b^{2/3} \sin ^2(c+d x)\right )}{6 \sqrt [3]{a} \left (a^2-b^2\right )^2 d}-\frac {2 a b \log \left (a+b \sin ^3(c+d x)\right )}{3 \left (a^2-b^2\right )^2 d}+\frac {b (a-\sin (c+d x) (b-a \sin (c+d x)))}{3 a \left (a^2-b^2\right ) d \left (a+b \sin ^3(c+d x)\right )} \]
-1/2*ln(1-sin(d*x+c))/(a+b)^2/d+1/2*ln(1+sin(d*x+c))/(a-b)^2/d-1/9*b^(1/3) *(a^(4/3)+2*b^(4/3))*ln(a^(1/3)+b^(1/3)*sin(d*x+c))/a^(5/3)/(a^2-b^2)/d-1/ 3*b^(1/3)*(a^2+2*a^(2/3)*b^(4/3)+b^2)*ln(a^(1/3)+b^(1/3)*sin(d*x+c))/a^(1/ 3)/(a^2-b^2)^2/d+1/18*b^(1/3)*(a^(4/3)+2*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^(5/3)/(a^2-b^2)/d+1/6*b^(1/3)*(a^2+2 *a^(2/3)*b^(4/3)+b^2)*ln(a^(2/3)-a^(1/3)*b^(1/3)*sin(d*x+c)+b^(2/3)*sin(d* x+c)^2)/a^(1/3)/(a^2-b^2)^2/d-2/3*a*b*ln(a+b*sin(d*x+c)^3)/(a^2-b^2)^2/d+1 /3*b*(a-sin(d*x+c)*(b-a*sin(d*x+c)))/a/(a^2-b^2)/d/(a+b*sin(d*x+c)^3)-1/9* b^(1/3)*(a^(4/3)-2*b^(4/3))*arctan(1/3*(a^(1/3)-2*b^(1/3)*sin(d*x+c))/a^(1 /3)*3^(1/2))/a^(5/3)/(a^2-b^2)/d*3^(1/2)-1/3*b^(1/3)*(a^2-2*a^(2/3)*b^(4/3 )+b^2)*arctan(1/3*(a^(1/3)-2*b^(1/3)*sin(d*x+c))/a^(1/3)*3^(1/2))/a^(1/3)/ (a^2-b^2)^2/d*3^(1/2)
Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.
Time = 4.38 (sec) , antiderivative size = 564, normalized size of antiderivative = 0.96 \[ \int \frac {\sec (c+d x)}{\left (a+b \sin ^3(c+d x)\right )^2} \, dx=\frac {\frac {12 \sqrt {3} \sqrt [3]{a} b^{5/3} \arctan \left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} \sin (c+d x)}{\sqrt {3} \sqrt [3]{a}}\right )}{\left (a^2-b^2\right )^2}+\frac {4 \sqrt {3} b^{5/3} \arctan \left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} \sin (c+d x)}{\sqrt {3} \sqrt [3]{a}}\right )}{a^{5/3} \left (a^2-b^2\right )}-\frac {9 \log (1-\sin (c+d x))}{(a+b)^2}+\frac {9 \log (1+\sin (c+d x))}{(a-b)^2}-\frac {12 \sqrt [3]{a} b^{5/3} \log \left (\sqrt [3]{a}+\sqrt [3]{b} \sin (c+d x)\right )}{\left (a^2-b^2\right )^2}-\frac {4 b^{5/3} \log \left (\sqrt [3]{a}+\sqrt [3]{b} \sin (c+d x)\right )}{a^{5/3} \left (a^2-b^2\right )}+\frac {6 \sqrt [3]{a} b^{5/3} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} \sin (c+d x)+b^{2/3} \sin ^2(c+d x)\right )}{\left (a^2-b^2\right )^2}+\frac {2 b^{5/3} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} \sin (c+d x)+b^{2/3} \sin ^2(c+d x)\right )}{a^{5/3} \left (a^2-b^2\right )}-\frac {12 a b \log \left (a+b \sin ^3(c+d x)\right )}{\left (a^2-b^2\right )^2}+\frac {9 b \left (a^2+b^2\right ) \operatorname {Hypergeometric2F1}\left (\frac {2}{3},1,\frac {5}{3},-\frac {b \sin ^3(c+d x)}{a}\right ) \sin ^2(c+d x)}{a \left (a^2-b^2\right )^2}+\frac {9 b \operatorname {Hypergeometric2F1}\left (\frac {2}{3},2,\frac {5}{3},-\frac {b \sin ^3(c+d x)}{a}\right ) \sin ^2(c+d x)}{a^3-a b^2}+\frac {6 b}{\left (a^2-b^2\right ) \left (a+b \sin ^3(c+d x)\right )}-\frac {6 b^2 \sin (c+d x)}{a \left (a^2-b^2\right ) \left (a+b \sin ^3(c+d x)\right )}}{18 d} \]
((12*Sqrt[3]*a^(1/3)*b^(5/3)*ArcTan[(a^(1/3) - 2*b^(1/3)*Sin[c + d*x])/(Sq rt[3]*a^(1/3))])/(a^2 - b^2)^2 + (4*Sqrt[3]*b^(5/3)*ArcTan[(a^(1/3) - 2*b^ (1/3)*Sin[c + d*x])/(Sqrt[3]*a^(1/3))])/(a^(5/3)*(a^2 - b^2)) - (9*Log[1 - Sin[c + d*x]])/(a + b)^2 + (9*Log[1 + Sin[c + d*x]])/(a - b)^2 - (12*a^(1 /3)*b^(5/3)*Log[a^(1/3) + b^(1/3)*Sin[c + d*x]])/(a^2 - b^2)^2 - (4*b^(5/3 )*Log[a^(1/3) + b^(1/3)*Sin[c + d*x]])/(a^(5/3)*(a^2 - b^2)) + (6*a^(1/3)* 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])/(a^2 - b^2)^2 + (2*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])/(a^(5/3)*(a^2 - b^2)) - (12*a*b*Log[a + b*Sin[c + d*x]^3])/(a^2 - b^2)^2 + (9*b*(a^2 + b^2)*Hypergeometric2F1[2/3, 1, 5/3 , -((b*Sin[c + d*x]^3)/a)]*Sin[c + d*x]^2)/(a*(a^2 - b^2)^2) + (9*b*Hyperg eometric2F1[2/3, 2, 5/3, -((b*Sin[c + d*x]^3)/a)]*Sin[c + d*x]^2)/(a^3 - a *b^2) + (6*b)/((a^2 - b^2)*(a + b*Sin[c + d*x]^3)) - (6*b^2*Sin[c + d*x])/ (a*(a^2 - b^2)*(a + b*Sin[c + d*x]^3)))/(18*d)
Time = 0.91 (sec) , antiderivative size = 561, normalized size of antiderivative = 0.96, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.190, 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.
| \(\displaystyle \int \frac {\sec (c+d x)}{\left (a+b \sin ^3(c+d x)\right )^2} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {1}{\cos (c+d x) \left (a+b \sin (c+d x)^3\right )^2}dx\) |
| \(\Big \downarrow \) 3702 |
| \(\displaystyle \frac {\int \frac {1}{\left (1-\sin ^2(c+d x)\right ) \left (b \sin ^3(c+d x)+a\right )^2}d\sin (c+d x)}{d}\) |
| \(\Big \downarrow \) 7276 |
| \(\displaystyle \frac {\int \left (\frac {b \left (b \sin ^2(c+d x)-a \sin (c+d x)+b\right )}{\left (b^2-a^2\right ) \left (b \sin ^3(c+d x)+a\right )^2}-\frac {1}{2 (a+b)^2 (\sin (c+d x)-1)}+\frac {1}{2 (a-b)^2 (\sin (c+d x)+1)}+\frac {b \left (-2 a b \sin ^2(c+d x)+\left (a^2+b^2\right ) \sin (c+d x)-2 a b\right )}{\left (a^2-b^2\right )^2 \left (b \sin ^3(c+d x)+a\right )}\right )d\sin (c+d x)}{d}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {\frac {b (a-\sin (c+d x) (b-a \sin (c+d x)))}{3 a \left (a^2-b^2\right ) \left (a+b \sin ^3(c+d x)\right )}-\frac {2 a b \log \left (a+b \sin ^3(c+d x)\right )}{3 \left (a^2-b^2\right )^2}-\frac {\sqrt [3]{b} \left (-2 a^{2/3} b^{4/3}+a^2+b^2\right ) \arctan \left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} \sin (c+d x)}{\sqrt {3} \sqrt [3]{a}}\right )}{\sqrt {3} \sqrt [3]{a} \left (a^2-b^2\right )^2}-\frac {\sqrt [3]{b} \left (a^{4/3}-2 b^{4/3}\right ) \arctan \left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} \sin (c+d x)}{\sqrt {3} \sqrt [3]{a}}\right )}{3 \sqrt {3} a^{5/3} \left (a^2-b^2\right )}+\frac {\sqrt [3]{b} \left (2 a^{2/3} b^{4/3}+a^2+b^2\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} \sin (c+d x)+b^{2/3} \sin ^2(c+d x)\right )}{6 \sqrt [3]{a} \left (a^2-b^2\right )^2}+\frac {\sqrt [3]{b} \left (a^{4/3}+2 b^{4/3}\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} \sin (c+d x)+b^{2/3} \sin ^2(c+d x)\right )}{18 a^{5/3} \left (a^2-b^2\right )}-\frac {\sqrt [3]{b} \left (2 a^{2/3} b^{4/3}+a^2+b^2\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} \sin (c+d x)\right )}{3 \sqrt [3]{a} \left (a^2-b^2\right )^2}-\frac {\sqrt [3]{b} \left (a^{4/3}+2 b^{4/3}\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} \sin (c+d x)\right )}{9 a^{5/3} \left (a^2-b^2\right )}-\frac {\log (1-\sin (c+d x))}{2 (a+b)^2}+\frac {\log (\sin (c+d x)+1)}{2 (a-b)^2}}{d}\) |
(-1/3*(b^(1/3)*(a^(4/3) - 2*b^(4/3))*ArcTan[(a^(1/3) - 2*b^(1/3)*Sin[c + d *x])/(Sqrt[3]*a^(1/3))])/(Sqrt[3]*a^(5/3)*(a^2 - b^2)) - (b^(1/3)*(a^2 - 2 *a^(2/3)*b^(4/3) + b^2)*ArcTan[(a^(1/3) - 2*b^(1/3)*Sin[c + d*x])/(Sqrt[3] *a^(1/3))])/(Sqrt[3]*a^(1/3)*(a^2 - b^2)^2) - Log[1 - Sin[c + d*x]]/(2*(a + b)^2) + Log[1 + Sin[c + d*x]]/(2*(a - b)^2) - (b^(1/3)*(a^(4/3) + 2*b^(4 /3))*Log[a^(1/3) + b^(1/3)*Sin[c + d*x]])/(9*a^(5/3)*(a^2 - b^2)) - (b^(1/ 3)*(a^2 + 2*a^(2/3)*b^(4/3) + b^2)*Log[a^(1/3) + b^(1/3)*Sin[c + d*x]])/(3 *a^(1/3)*(a^2 - b^2)^2) + (b^(1/3)*(a^(4/3) + 2*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])/(18*a^(5/3)*(a^2 - b^ 2)) + (b^(1/3)*(a^2 + 2*a^(2/3)*b^(4/3) + b^2)*Log[a^(2/3) - a^(1/3)*b^(1/ 3)*Sin[c + d*x] + b^(2/3)*Sin[c + d*x]^2])/(6*a^(1/3)*(a^2 - b^2)^2) - (2* a*b*Log[a + b*Sin[c + d*x]^3])/(3*(a^2 - b^2)^2) + (b*(a - Sin[c + d*x]*(b - a*Sin[c + d*x])))/(3*a*(a^2 - b^2)*(a + b*Sin[c + d*x]^3)))/d
3.4.97.3.1 Defintions of rubi rules used
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 = 2.21 (sec) , antiderivative size = 389, normalized size of antiderivative = 0.66
| method | result | size |
| derivativedivides | \(\frac {-\frac {\ln \left (\sin \left (d x +c \right )-1\right )}{2 \left (a +b \right )^{2}}+\frac {\ln \left (1+\sin \left (d x +c \right )\right )}{2 \left (a -b \right )^{2}}+\frac {b \left (\frac {\left (\frac {a^{2}}{3}-\frac {b^{2}}{3}\right ) \left (\sin ^{2}\left (d x +c \right )\right )-\frac {b \left (a^{2}-b^{2}\right ) \sin \left (d x +c \right )}{3 a}+\frac {a^{2}}{3}-\frac {b^{2}}{3}}{a +b \left (\sin ^{3}\left (d x +c \right )\right )}+\frac {\frac {2 \left (-4 a^{2} b +b^{3}\right ) \left (\frac {\ln \left (\sin \left (d x +c \right )+\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {2}{3}}}-\frac {\ln \left (\sin ^{2}\left (d x +c \right )-\left (\frac {a}{b}\right )^{\frac {1}{3}} \sin \left (d x +c \right )+\left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{6 b \left (\frac {a}{b}\right )^{\frac {2}{3}}}+\frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 \sin \left (d x +c \right )}{\left (\frac {a}{b}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {2}{3}}}\right )}{3}+\frac {2 \left (2 a^{3}+a \,b^{2}\right ) \left (-\frac {\ln \left (\sin \left (d x +c \right )+\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {1}{3}}}+\frac {\ln \left (\sin ^{2}\left (d x +c \right )-\left (\frac {a}{b}\right )^{\frac {1}{3}} \sin \left (d x +c \right )+\left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{6 b \left (\frac {a}{b}\right )^{\frac {1}{3}}}+\frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 \sin \left (d x +c \right )}{\left (\frac {a}{b}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{3}-\frac {2 a^{2} \ln \left (a +b \left (\sin ^{3}\left (d x +c \right )\right )\right )}{3}}{a}\right )}{\left (a -b \right )^{2} \left (a +b \right )^{2}}}{d}\) | \(389\) |
| default | \(\frac {-\frac {\ln \left (\sin \left (d x +c \right )-1\right )}{2 \left (a +b \right )^{2}}+\frac {\ln \left (1+\sin \left (d x +c \right )\right )}{2 \left (a -b \right )^{2}}+\frac {b \left (\frac {\left (\frac {a^{2}}{3}-\frac {b^{2}}{3}\right ) \left (\sin ^{2}\left (d x +c \right )\right )-\frac {b \left (a^{2}-b^{2}\right ) \sin \left (d x +c \right )}{3 a}+\frac {a^{2}}{3}-\frac {b^{2}}{3}}{a +b \left (\sin ^{3}\left (d x +c \right )\right )}+\frac {\frac {2 \left (-4 a^{2} b +b^{3}\right ) \left (\frac {\ln \left (\sin \left (d x +c \right )+\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {2}{3}}}-\frac {\ln \left (\sin ^{2}\left (d x +c \right )-\left (\frac {a}{b}\right )^{\frac {1}{3}} \sin \left (d x +c \right )+\left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{6 b \left (\frac {a}{b}\right )^{\frac {2}{3}}}+\frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 \sin \left (d x +c \right )}{\left (\frac {a}{b}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {2}{3}}}\right )}{3}+\frac {2 \left (2 a^{3}+a \,b^{2}\right ) \left (-\frac {\ln \left (\sin \left (d x +c \right )+\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {1}{3}}}+\frac {\ln \left (\sin ^{2}\left (d x +c \right )-\left (\frac {a}{b}\right )^{\frac {1}{3}} \sin \left (d x +c \right )+\left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{6 b \left (\frac {a}{b}\right )^{\frac {1}{3}}}+\frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 \sin \left (d x +c \right )}{\left (\frac {a}{b}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{3}-\frac {2 a^{2} \ln \left (a +b \left (\sin ^{3}\left (d x +c \right )\right )\right )}{3}}{a}\right )}{\left (a -b \right )^{2} \left (a +b \right )^{2}}}{d}\) | \(389\) |
| risch | \(-\frac {i x}{a^{2}-2 a b +b^{2}}-\frac {i c}{d \left (a^{2}-2 a b +b^{2}\right )}+\frac {i x}{a^{2}+2 a b +b^{2}}+\frac {i c}{d \left (a^{2}+2 a b +b^{2}\right )}+\frac {4 i a^{6} b \,d^{3} x}{a^{9} d^{3}-2 a^{7} b^{2} d^{3}+a^{5} b^{4} d^{3}}+\frac {4 i a^{6} b \,d^{2} c}{a^{9} d^{3}-2 a^{7} b^{2} d^{3}+a^{5} b^{4} d^{3}}-\frac {2 b \left (i a \,{\mathrm e}^{5 i \left (d x +c \right )}-6 i a \,{\mathrm e}^{3 i \left (d x +c \right )}+2 b \,{\mathrm e}^{4 i \left (d x +c \right )}+i a \,{\mathrm e}^{i \left (d x +c \right )}-2 b \,{\mathrm e}^{2 i \left (d x +c \right )}\right )}{3 d \left (-a^{2}+b^{2}\right ) a \left (b \,{\mathrm e}^{6 i \left (d x +c \right )}-3 b \,{\mathrm e}^{4 i \left (d x +c \right )}+3 b \,{\mathrm e}^{2 i \left (d x +c \right )}-8 i a \,{\mathrm e}^{3 i \left (d x +c \right )}-b \right )}+\frac {\ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right )}{d \left (a^{2}-2 a b +b^{2}\right )}-\frac {\ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right )}{d \left (a^{2}+2 a b +b^{2}\right )}+2 \left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\left (729 a^{9} d^{3}-1458 a^{7} b^{2} d^{3}+729 a^{5} b^{4} d^{3}\right ) \textit {\_Z}^{3}+729 a^{6} b \,d^{2} \textit {\_Z}^{2}+27 a^{3} b^{2} d \textit {\_Z} +8 a^{2} b -b^{3}\right )}{\sum }\textit {\_R} \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}+\left (\left (\frac {324 i a^{11} d^{2}}{8 a^{6} b +28 a^{4} b^{3}-10 a^{2} b^{5}+b^{7}}-\frac {486 i a^{9} b^{2} d^{2}}{8 a^{6} b +28 a^{4} b^{3}-10 a^{2} b^{5}+b^{7}}+\frac {162 i a^{5} b^{6} d^{2}}{8 a^{6} b +28 a^{4} b^{3}-10 a^{2} b^{5}+b^{7}}\right ) \textit {\_R}^{2}+\left (\frac {216 i a^{8} b d}{8 a^{6} b +28 a^{4} b^{3}-10 a^{2} b^{5}+b^{7}}+\frac {396 i a^{6} b^{3} d}{8 a^{6} b +28 a^{4} b^{3}-10 a^{2} b^{5}+b^{7}}-\frac {144 i a^{4} b^{5} d}{8 a^{6} b +28 a^{4} b^{3}-10 a^{2} b^{5}+b^{7}}+\frac {18 i a^{2} b^{7} d}{8 a^{6} b +28 a^{4} b^{3}-10 a^{2} b^{5}+b^{7}}\right ) \textit {\_R} -\frac {28 i a^{5} b^{2}}{8 a^{6} b +28 a^{4} b^{3}-10 a^{2} b^{5}+b^{7}}+\frac {10 i a^{3} b^{4}}{8 a^{6} b +28 a^{4} b^{3}-10 a^{2} b^{5}+b^{7}}\right ) {\mathrm e}^{i \left (d x +c \right )}-\frac {8 a^{6} b}{8 a^{6} b +28 a^{4} b^{3}-10 a^{2} b^{5}+b^{7}}-\frac {28 a^{4} b^{3}}{8 a^{6} b +28 a^{4} b^{3}-10 a^{2} b^{5}+b^{7}}+\frac {10 a^{2} b^{5}}{8 a^{6} b +28 a^{4} b^{3}-10 a^{2} b^{5}+b^{7}}-\frac {b^{7}}{8 a^{6} b +28 a^{4} b^{3}-10 a^{2} b^{5}+b^{7}}\right )\right )\) | \(946\) |
1/d*(-1/2/(a+b)^2*ln(sin(d*x+c)-1)+1/2/(a-b)^2*ln(1+sin(d*x+c))+b/(a-b)^2/ (a+b)^2*(((1/3*a^2-1/3*b^2)*sin(d*x+c)^2-1/3*b*(a^2-b^2)/a*sin(d*x+c)+1/3* a^2-1/3*b^2)/(a+b*sin(d*x+c)^3)+2/3/a*((-4*a^2*b+b^3)*(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)))+(2*a^3+a*b^2)*(-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)))-a^2*ln(a+b*sin(d*x+c)^3))))
Result contains complex when optimal does not.
Time = 2.60 (sec) , antiderivative size = 10855, normalized size of antiderivative = 18.49 \[ \int \frac {\sec (c+d x)}{\left (a+b \sin ^3(c+d x)\right )^2} \, dx=\text {Too large to display} \]
Timed out. \[ \int \frac {\sec (c+d x)}{\left (a+b \sin ^3(c+d x)\right )^2} \, dx=\text {Timed out} \]
Time = 0.32 (sec) , antiderivative size = 483, normalized size of antiderivative = 0.82 \[ \int \frac {\sec (c+d x)}{\left (a+b \sin ^3(c+d x)\right )^2} \, dx=\frac {\frac {4 \, \sqrt {3} {\left (2 \, a^{3} {\left (\left (\frac {a}{b}\right )^{\frac {2}{3}} + 1\right )} - 2 \, a^{2} b {\left (2 \, \left (\frac {a}{b}\right )^{\frac {1}{3}} + \frac {a}{b}\right )} + a b^{2} \left (\frac {a}{b}\right )^{\frac {2}{3}} + b^{3} \left (\frac {a}{b}\right )^{\frac {1}{3}}\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^{5} \left (\frac {a}{b}\right )^{\frac {2}{3}} - 2 \, a^{3} b^{2} \left (\frac {a}{b}\right )^{\frac {2}{3}} + a b^{4} \left (\frac {a}{b}\right )^{\frac {2}{3}}\right )} \left (\frac {a}{b}\right )^{\frac {1}{3}}} - \frac {2 \, {\left (2 \, a^{2} b {\left (3 \, \left (\frac {a}{b}\right )^{\frac {2}{3}} - 2\right )} - 2 \, a^{3} \left (\frac {a}{b}\right )^{\frac {1}{3}} - a b^{2} \left (\frac {a}{b}\right )^{\frac {1}{3}} + b^{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^{5} \left (\frac {a}{b}\right )^{\frac {2}{3}} - 2 \, a^{3} b^{2} \left (\frac {a}{b}\right )^{\frac {2}{3}} + a b^{4} \left (\frac {a}{b}\right )^{\frac {2}{3}}} - \frac {4 \, {\left (a^{2} b {\left (3 \, \left (\frac {a}{b}\right )^{\frac {2}{3}} + 4\right )} + 2 \, a^{3} \left (\frac {a}{b}\right )^{\frac {1}{3}} + a b^{2} \left (\frac {a}{b}\right )^{\frac {1}{3}} - b^{3}\right )} \log \left (\left (\frac {a}{b}\right )^{\frac {1}{3}} + \sin \left (d x + c\right )\right )}{a^{5} \left (\frac {a}{b}\right )^{\frac {2}{3}} - 2 \, a^{3} b^{2} \left (\frac {a}{b}\right )^{\frac {2}{3}} + a b^{4} \left (\frac {a}{b}\right )^{\frac {2}{3}}} + \frac {6 \, {\left (a b \sin \left (d x + c\right )^{2} - b^{2} \sin \left (d x + c\right ) + a b\right )}}{a^{4} - a^{2} b^{2} + {\left (a^{3} b - a b^{3}\right )} \sin \left (d x + c\right )^{3}} + \frac {9 \, \log \left (\sin \left (d x + c\right ) + 1\right )}{a^{2} - 2 \, a b + b^{2}} - \frac {9 \, \log \left (\sin \left (d x + c\right ) - 1\right )}{a^{2} + 2 \, a b + b^{2}}}{18 \, d} \]
1/18*(4*sqrt(3)*(2*a^3*((a/b)^(2/3) + 1) - 2*a^2*b*(2*(a/b)^(1/3) + a/b) + a*b^2*(a/b)^(2/3) + b^3*(a/b)^(1/3))*arctan(-1/3*sqrt(3)*((a/b)^(1/3) - 2 *sin(d*x + c))/(a/b)^(1/3))/((a^5*(a/b)^(2/3) - 2*a^3*b^2*(a/b)^(2/3) + a* b^4*(a/b)^(2/3))*(a/b)^(1/3)) - 2*(2*a^2*b*(3*(a/b)^(2/3) - 2) - 2*a^3*(a/ b)^(1/3) - a*b^2*(a/b)^(1/3) + b^3)*log(sin(d*x + c)^2 - (a/b)^(1/3)*sin(d *x + c) + (a/b)^(2/3))/(a^5*(a/b)^(2/3) - 2*a^3*b^2*(a/b)^(2/3) + a*b^4*(a /b)^(2/3)) - 4*(a^2*b*(3*(a/b)^(2/3) + 4) + 2*a^3*(a/b)^(1/3) + a*b^2*(a/b )^(1/3) - b^3)*log((a/b)^(1/3) + sin(d*x + c))/(a^5*(a/b)^(2/3) - 2*a^3*b^ 2*(a/b)^(2/3) + a*b^4*(a/b)^(2/3)) + 6*(a*b*sin(d*x + c)^2 - b^2*sin(d*x + c) + a*b)/(a^4 - a^2*b^2 + (a^3*b - a*b^3)*sin(d*x + c)^3) + 9*log(sin(d* x + c) + 1)/(a^2 - 2*a*b + b^2) - 9*log(sin(d*x + c) - 1)/(a^2 + 2*a*b + b ^2))/d
\[ \int \frac {\sec (c+d x)}{\left (a+b \sin ^3(c+d x)\right )^2} \, dx=\int { \frac {\sec \left (d x + c\right )}{{\left (b \sin \left (d x + c\right )^{3} + a\right )}^{2}} \,d x } \]
Time = 14.82 (sec) , antiderivative size = 980, normalized size of antiderivative = 1.67 \[ \int \frac {\sec (c+d x)}{\left (a+b \sin ^3(c+d x)\right )^2} \, dx=\frac {\sum _{k=1}^3\ln \left (\frac {\frac {8\,b^6}{27}-\frac {16\,a^2\,b^4}{27}}{a^7-2\,a^5\,b^2+a^3\,b^4}+\mathrm {root}\left (1458\,a^7\,b^2\,z^3-729\,a^5\,b^4\,z^3-729\,a^9\,z^3-1458\,a^6\,b\,z^2-108\,a^3\,b^2\,z-64\,a^2\,b+8\,b^3,z,k\right )\,\left (\frac {\frac {128\,a^3\,b^5}{27}+\frac {32\,a\,b^7}{27}}{a^7-2\,a^5\,b^2+a^3\,b^4}-\mathrm {root}\left (1458\,a^7\,b^2\,z^3-729\,a^5\,b^4\,z^3-729\,a^9\,z^3-1458\,a^6\,b\,z^2-108\,a^3\,b^2\,z-64\,a^2\,b+8\,b^3,z,k\right )\,\left (\mathrm {root}\left (1458\,a^7\,b^2\,z^3-729\,a^5\,b^4\,z^3-729\,a^9\,z^3-1458\,a^6\,b\,z^2-108\,a^3\,b^2\,z-64\,a^2\,b+8\,b^3,z,k\right )\,\left (\frac {27\,a^9\,b^3+34\,a^7\,b^5-77\,a^5\,b^7+16\,a^3\,b^9}{a^7-2\,a^5\,b^2+a^3\,b^4}+\mathrm {root}\left (1458\,a^7\,b^2\,z^3-729\,a^5\,b^4\,z^3-729\,a^9\,z^3-1458\,a^6\,b\,z^2-108\,a^3\,b^2\,z-64\,a^2\,b+8\,b^3,z,k\right )\,\left (\frac {180\,a^{10}\,b^4-324\,a^8\,b^6+108\,a^6\,b^8+36\,a^4\,b^{10}}{a^7-2\,a^5\,b^2+a^3\,b^4}+\frac {\sin \left (c+d\,x\right )\,\left (1458\,a^{11}\,b^3+1458\,a^9\,b^5-7290\,a^7\,b^7+4374\,a^5\,b^9\right )}{27\,\left (a^7-2\,a^5\,b^2+a^3\,b^4\right )}\right )+\frac {\sin \left (c+d\,x\right )\,\left (2484\,a^8\,b^4-1836\,a^6\,b^6-864\,a^4\,b^8+216\,a^2\,b^{10}\right )}{27\,\left (a^7-2\,a^5\,b^2+a^3\,b^4\right )}\right )+\frac {\frac {388\,a^6\,b^4}{9}-\frac {353\,a^4\,b^6}{9}+\frac {64\,a^2\,b^8}{9}}{a^7-2\,a^5\,b^2+a^3\,b^4}+\frac {\sin \left (c+d\,x\right )\,\left (447\,a^5\,b^5-408\,a^3\,b^7+96\,a\,b^9\right )}{27\,\left (a^7-2\,a^5\,b^2+a^3\,b^4\right )}\right )+\frac {\sin \left (c+d\,x\right )\,\left (-236\,a^4\,b^4+134\,a^2\,b^6+16\,b^8\right )}{27\,\left (a^7-2\,a^5\,b^2+a^3\,b^4\right )}\right )+\frac {8\,a\,b^5\,\sin \left (c+d\,x\right )}{9\,\left (a^7-2\,a^5\,b^2+a^3\,b^4\right )}\right )\,\mathrm {root}\left (1458\,a^7\,b^2\,z^3-729\,a^5\,b^4\,z^3-729\,a^9\,z^3-1458\,a^6\,b\,z^2-108\,a^3\,b^2\,z-64\,a^2\,b+8\,b^3,z,k\right )}{d}-\frac {\ln \left (\sin \left (c+d\,x\right )-1\right )}{d\,\left (2\,a^2+4\,a\,b+2\,b^2\right )}+\frac {\ln \left (\sin \left (c+d\,x\right )+1\right )}{d\,\left (2\,a^2-4\,a\,b+2\,b^2\right )}+\frac {\frac {b}{3\,\left (a^2-b^2\right )}+\frac {b\,{\sin \left (c+d\,x\right )}^2}{3\,\left (a^2-b^2\right )}-\frac {b^2\,\sin \left (c+d\,x\right )}{3\,a\,\left (a^2-b^2\right )}}{d\,\left (b\,{\sin \left (c+d\,x\right )}^3+a\right )} \]
symsum(log(((8*b^6)/27 - (16*a^2*b^4)/27)/(a^7 + a^3*b^4 - 2*a^5*b^2) + ro ot(1458*a^7*b^2*z^3 - 729*a^5*b^4*z^3 - 729*a^9*z^3 - 1458*a^6*b*z^2 - 108 *a^3*b^2*z - 64*a^2*b + 8*b^3, z, k)*(((32*a*b^7)/27 + (128*a^3*b^5)/27)/( a^7 + a^3*b^4 - 2*a^5*b^2) - root(1458*a^7*b^2*z^3 - 729*a^5*b^4*z^3 - 729 *a^9*z^3 - 1458*a^6*b*z^2 - 108*a^3*b^2*z - 64*a^2*b + 8*b^3, z, k)*(root( 1458*a^7*b^2*z^3 - 729*a^5*b^4*z^3 - 729*a^9*z^3 - 1458*a^6*b*z^2 - 108*a^ 3*b^2*z - 64*a^2*b + 8*b^3, z, k)*((16*a^3*b^9 - 77*a^5*b^7 + 34*a^7*b^5 + 27*a^9*b^3)/(a^7 + a^3*b^4 - 2*a^5*b^2) + root(1458*a^7*b^2*z^3 - 729*a^5 *b^4*z^3 - 729*a^9*z^3 - 1458*a^6*b*z^2 - 108*a^3*b^2*z - 64*a^2*b + 8*b^3 , z, k)*((36*a^4*b^10 + 108*a^6*b^8 - 324*a^8*b^6 + 180*a^10*b^4)/(a^7 + a ^3*b^4 - 2*a^5*b^2) + (sin(c + d*x)*(4374*a^5*b^9 - 7290*a^7*b^7 + 1458*a^ 9*b^5 + 1458*a^11*b^3))/(27*(a^7 + a^3*b^4 - 2*a^5*b^2))) + (sin(c + d*x)* (216*a^2*b^10 - 864*a^4*b^8 - 1836*a^6*b^6 + 2484*a^8*b^4))/(27*(a^7 + a^3 *b^4 - 2*a^5*b^2))) + ((64*a^2*b^8)/9 - (353*a^4*b^6)/9 + (388*a^6*b^4)/9) /(a^7 + a^3*b^4 - 2*a^5*b^2) + (sin(c + d*x)*(96*a*b^9 - 408*a^3*b^7 + 447 *a^5*b^5))/(27*(a^7 + a^3*b^4 - 2*a^5*b^2))) + (sin(c + d*x)*(16*b^8 + 134 *a^2*b^6 - 236*a^4*b^4))/(27*(a^7 + a^3*b^4 - 2*a^5*b^2))) + (8*a*b^5*sin( c + d*x))/(9*(a^7 + a^3*b^4 - 2*a^5*b^2)))*root(1458*a^7*b^2*z^3 - 729*a^5 *b^4*z^3 - 729*a^9*z^3 - 1458*a^6*b*z^2 - 108*a^3*b^2*z - 64*a^2*b + 8*b^3 , z, k), k, 1, 3)/d - log(sin(c + d*x) - 1)/(d*(4*a*b + 2*a^2 + 2*b^2))...