Integrand size = 23, antiderivative size = 176 \[ \int \frac {1}{\cos ^{\frac {2}{3}}(c+d x) (a+b \cos (c+d x))} \, dx=-\frac {b \operatorname {AppellF1}\left (\frac {1}{2},\frac {1}{3},1,\frac {3}{2},\sin ^2(c+d x),-\frac {b^2 \sin ^2(c+d x)}{a^2-b^2}\right ) \sqrt [3]{\cos ^2(c+d x)} \sin (c+d x)}{\left (a^2-b^2\right ) d \cos ^{\frac {2}{3}}(c+d x)}+\frac {a \operatorname {AppellF1}\left (\frac {1}{2},\frac {5}{6},1,\frac {3}{2},\sin ^2(c+d x),-\frac {b^2 \sin ^2(c+d x)}{a^2-b^2}\right ) \cos ^2(c+d x)^{5/6} \sin (c+d x)}{\left (a^2-b^2\right ) d \cos ^{\frac {5}{3}}(c+d x)} \]
-b*AppellF1(1/2,1/3,1,3/2,sin(d*x+c)^2,-b^2*sin(d*x+c)^2/(a^2-b^2))*(cos(d *x+c)^2)^(1/3)*sin(d*x+c)/(a^2-b^2)/d/cos(d*x+c)^(2/3)+a*AppellF1(1/2,5/6, 1,3/2,sin(d*x+c)^2,-b^2*sin(d*x+c)^2/(a^2-b^2))*(cos(d*x+c)^2)^(5/6)*sin(d *x+c)/(a^2-b^2)/d/cos(d*x+c)^(5/3)
Leaf count is larger than twice the leaf count of optimal. \(4608\) vs. \(2(176)=352\).
Time = 34.39 (sec) , antiderivative size = 4608, normalized size of antiderivative = 26.18 \[ \int \frac {1}{\cos ^{\frac {2}{3}}(c+d x) (a+b \cos (c+d x))} \, dx=\text {Result too large to show} \]
(9*(a^2 - b^2)*Sin[c + d*x]*((a*AppellF1[1/2, -1/3, 1, 3/2, -Tan[c + d*x]^ 2, -((a^2*Tan[c + d*x]^2)/(a^2 - b^2))]*Sqrt[Sec[c + d*x]^2])/(9*(a^2 - b^ 2)*AppellF1[1/2, -1/3, 1, 3/2, -Tan[c + d*x]^2, -((a^2*Tan[c + d*x]^2)/(a^ 2 - b^2))] - 2*(3*a^2*AppellF1[3/2, -1/3, 2, 5/2, -Tan[c + d*x]^2, -((a^2* Tan[c + d*x]^2)/(a^2 - b^2))] + (-a^2 + b^2)*AppellF1[3/2, 2/3, 1, 5/2, -T an[c + d*x]^2, -((a^2*Tan[c + d*x]^2)/(a^2 - b^2))])*Tan[c + d*x]^2) + (b* AppellF1[1/2, 1/6, 1, 3/2, -Tan[c + d*x]^2, -((a^2*Tan[c + d*x]^2)/(a^2 - b^2))])/(-9*(a^2 - b^2)*AppellF1[1/2, 1/6, 1, 3/2, -Tan[c + d*x]^2, -((a^2 *Tan[c + d*x]^2)/(a^2 - b^2))] + (6*a^2*AppellF1[3/2, 1/6, 2, 5/2, -Tan[c + d*x]^2, -((a^2*Tan[c + d*x]^2)/(a^2 - b^2))] + (a^2 - b^2)*AppellF1[3/2, 7/6, 1, 5/2, -Tan[c + d*x]^2, -((a^2*Tan[c + d*x]^2)/(a^2 - b^2))])*Tan[c + d*x]^2)))/(d*Cos[c + d*x]^(5/3)*(a + b*Cos[c + d*x])*(Sec[c + d*x]^2)^( 1/6)*(-b^2 + a^2*Sec[c + d*x]^2)*((9*(a^2 - b^2)*(Sec[c + d*x]^2)^(5/6)*(( a*AppellF1[1/2, -1/3, 1, 3/2, -Tan[c + d*x]^2, -((a^2*Tan[c + d*x]^2)/(a^2 - b^2))]*Sqrt[Sec[c + d*x]^2])/(9*(a^2 - b^2)*AppellF1[1/2, -1/3, 1, 3/2, -Tan[c + d*x]^2, -((a^2*Tan[c + d*x]^2)/(a^2 - b^2))] - 2*(3*a^2*AppellF1 [3/2, -1/3, 2, 5/2, -Tan[c + d*x]^2, -((a^2*Tan[c + d*x]^2)/(a^2 - b^2))] + (-a^2 + b^2)*AppellF1[3/2, 2/3, 1, 5/2, -Tan[c + d*x]^2, -((a^2*Tan[c + d*x]^2)/(a^2 - b^2))])*Tan[c + d*x]^2) + (b*AppellF1[1/2, 1/6, 1, 3/2, -Ta n[c + d*x]^2, -((a^2*Tan[c + d*x]^2)/(a^2 - b^2))])/(-9*(a^2 - b^2)*App...
Time = 0.46 (sec) , antiderivative size = 176, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.217, Rules used = {3042, 3302, 3042, 3668, 333}
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.
\(\displaystyle \int \frac {1}{\cos ^{\frac {2}{3}}(c+d x) (a+b \cos (c+d x))} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {1}{\sin \left (c+d x+\frac {\pi }{2}\right )^{2/3} \left (a+b \sin \left (c+d x+\frac {\pi }{2}\right )\right )}dx\) |
\(\Big \downarrow \) 3302 |
\(\displaystyle a \int \frac {1}{\cos ^{\frac {2}{3}}(c+d x) \left (a^2-b^2 \cos ^2(c+d x)\right )}dx-b \int \frac {\sqrt [3]{\cos (c+d x)}}{a^2-b^2 \cos ^2(c+d x)}dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle a \int \frac {1}{\sin \left (c+d x+\frac {\pi }{2}\right )^{2/3} \left (a^2-b^2 \sin \left (c+d x+\frac {\pi }{2}\right )^2\right )}dx-b \int \frac {\sqrt [3]{\sin \left (c+d x+\frac {\pi }{2}\right )}}{a^2-b^2 \sin \left (c+d x+\frac {\pi }{2}\right )^2}dx\) |
\(\Big \downarrow \) 3668 |
\(\displaystyle \frac {a \cos ^2(c+d x)^{5/6} \int \frac {1}{\left (1-\sin ^2(c+d x)\right )^{5/6} \left (a^2-b^2+b^2 \sin ^2(c+d x)\right )}d\sin (c+d x)}{d \cos ^{\frac {5}{3}}(c+d x)}-\frac {b \sqrt [3]{\cos ^2(c+d x)} \int \frac {1}{\sqrt [3]{1-\sin ^2(c+d x)} \left (a^2-b^2+b^2 \sin ^2(c+d x)\right )}d\sin (c+d x)}{d \cos ^{\frac {2}{3}}(c+d x)}\) |
\(\Big \downarrow \) 333 |
\(\displaystyle \frac {a \sin (c+d x) \cos ^2(c+d x)^{5/6} \operatorname {AppellF1}\left (\frac {1}{2},\frac {5}{6},1,\frac {3}{2},\sin ^2(c+d x),-\frac {b^2 \sin ^2(c+d x)}{a^2-b^2}\right )}{d \left (a^2-b^2\right ) \cos ^{\frac {5}{3}}(c+d x)}-\frac {b \sin (c+d x) \sqrt [3]{\cos ^2(c+d x)} \operatorname {AppellF1}\left (\frac {1}{2},\frac {1}{3},1,\frac {3}{2},\sin ^2(c+d x),-\frac {b^2 \sin ^2(c+d x)}{a^2-b^2}\right )}{d \left (a^2-b^2\right ) \cos ^{\frac {2}{3}}(c+d x)}\) |
-((b*AppellF1[1/2, 1/3, 1, 3/2, Sin[c + d*x]^2, -((b^2*Sin[c + d*x]^2)/(a^ 2 - b^2))]*(Cos[c + d*x]^2)^(1/3)*Sin[c + d*x])/((a^2 - b^2)*d*Cos[c + d*x ]^(2/3))) + (a*AppellF1[1/2, 5/6, 1, 3/2, Sin[c + d*x]^2, -((b^2*Sin[c + d *x]^2)/(a^2 - b^2))]*(Cos[c + d*x]^2)^(5/6)*Sin[c + d*x])/((a^2 - b^2)*d*C os[c + d*x]^(5/3))
3.7.79.3.1 Defintions of rubi rules used
Int[((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2)^(q_), x_Symbol] :> Sim p[a^p*c^q*x*AppellF1[1/2, -p, -q, 3/2, (-b)*(x^2/a), (-d)*(x^2/c)], x] /; F reeQ[{a, b, c, d, p, q}, x] && NeQ[b*c - a*d, 0] && (IntegerQ[p] || GtQ[a, 0]) && (IntegerQ[q] || GtQ[c, 0])
Int[((d_.)*sin[(e_.) + (f_.)*(x_)])^(n_.)/((a_) + (b_.)*sin[(e_.) + (f_.)*( x_)]), x_Symbol] :> Simp[a Int[(d*Sin[e + f*x])^n/(a^2 - b^2*Sin[e + f*x] ^2), x], x] - Simp[b/d Int[(d*Sin[e + f*x])^(n + 1)/(a^2 - b^2*Sin[e + f* x]^2), x], x] /; FreeQ[{a, b, d, e, f, n}, x] && NeQ[a^2 - b^2, 0]
Int[((d_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x _)]^2)^(p_.), x_Symbol] :> With[{ff = FreeFactors[Cos[e + f*x], x]}, Simp[( -ff)*d^(2*IntPart[(m - 1)/2] + 1)*((d*Sin[e + f*x])^(2*FracPart[(m - 1)/2]) /(f*(Sin[e + f*x]^2)^FracPart[(m - 1)/2])) Subst[Int[(1 - ff^2*x^2)^((m - 1)/2)*(a + b - b*ff^2*x^2)^p, x], x, Cos[e + f*x]/ff], x]] /; FreeQ[{a, b, d, e, f, m, p}, x] && !IntegerQ[m]
\[\int \frac {1}{\cos \left (d x +c \right )^{\frac {2}{3}} \left (a +\cos \left (d x +c \right ) b \right )}d x\]
Timed out. \[ \int \frac {1}{\cos ^{\frac {2}{3}}(c+d x) (a+b \cos (c+d x))} \, dx=\text {Timed out} \]
Timed out. \[ \int \frac {1}{\cos ^{\frac {2}{3}}(c+d x) (a+b \cos (c+d x))} \, dx=\text {Timed out} \]
\[ \int \frac {1}{\cos ^{\frac {2}{3}}(c+d x) (a+b \cos (c+d x))} \, dx=\int { \frac {1}{{\left (b \cos \left (d x + c\right ) + a\right )} \cos \left (d x + c\right )^{\frac {2}{3}}} \,d x } \]
\[ \int \frac {1}{\cos ^{\frac {2}{3}}(c+d x) (a+b \cos (c+d x))} \, dx=\int { \frac {1}{{\left (b \cos \left (d x + c\right ) + a\right )} \cos \left (d x + c\right )^{\frac {2}{3}}} \,d x } \]
Timed out. \[ \int \frac {1}{\cos ^{\frac {2}{3}}(c+d x) (a+b \cos (c+d x))} \, dx=\int \frac {1}{{\cos \left (c+d\,x\right )}^{2/3}\,\left (a+b\,\cos \left (c+d\,x\right )\right )} \,d x \]