Integrand size = 23, antiderivative size = 234 \[ \int (a+b \cos (c+d x))^3 \sec ^{\frac {9}{2}}(c+d x) \, dx=-\frac {2 b \left (9 a^2+5 b^2\right ) \sqrt {\cos (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \sqrt {\sec (c+d x)}}{5 d}+\frac {2 a \left (5 a^2+21 b^2\right ) \sqrt {\cos (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right ) \sqrt {\sec (c+d x)}}{21 d}+\frac {2 b \left (9 a^2+5 b^2\right ) \sqrt {\sec (c+d x)} \sin (c+d x)}{5 d}+\frac {2 a \left (5 a^2+21 b^2\right ) \sec ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{21 d}+\frac {32 a^2 b \sec ^{\frac {5}{2}}(c+d x) \sin (c+d x)}{35 d}+\frac {2 a^2 \sec ^{\frac {5}{2}}(c+d x) (b+a \sec (c+d x)) \sin (c+d x)}{7 d} \] Output:
-2/5*b*(9*a^2+5*b^2)*cos(d*x+c)^(1/2)*EllipticE(sin(1/2*d*x+1/2*c),2^(1/2) )*sec(d*x+c)^(1/2)/d+2/21*a*(5*a^2+21*b^2)*cos(d*x+c)^(1/2)*InverseJacobiA M(1/2*d*x+1/2*c,2^(1/2))*sec(d*x+c)^(1/2)/d+2/5*b*(9*a^2+5*b^2)*sec(d*x+c) ^(1/2)*sin(d*x+c)/d+2/21*a*(5*a^2+21*b^2)*sec(d*x+c)^(3/2)*sin(d*x+c)/d+32 /35*a^2*b*sec(d*x+c)^(5/2)*sin(d*x+c)/d+2/7*a^2*sec(d*x+c)^(5/2)*(b+a*sec( d*x+c))*sin(d*x+c)/d
Time = 1.39 (sec) , antiderivative size = 191, normalized size of antiderivative = 0.82 \[ \int (a+b \cos (c+d x))^3 \sec ^{\frac {9}{2}}(c+d x) \, dx=\frac {\sec ^{\frac {7}{2}}(c+d x) \left (-42 b \left (9 a^2+5 b^2\right ) \cos ^{\frac {7}{2}}(c+d x) E\left (\left .\frac {1}{2} (c+d x)\right |2\right )+10 a \left (5 a^2+21 b^2\right ) \cos ^{\frac {7}{2}}(c+d x) \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )+30 a^3 \sin (c+d x)+50 a^3 \cos ^2(c+d x) \sin (c+d x)+210 a b^2 \cos ^2(c+d x) \sin (c+d x)+378 a^2 b \cos ^3(c+d x) \sin (c+d x)+210 b^3 \cos ^3(c+d x) \sin (c+d x)+63 a^2 b \sin (2 (c+d x))\right )}{105 d} \] Input:
Integrate[(a + b*Cos[c + d*x])^3*Sec[c + d*x]^(9/2),x]
Output:
(Sec[c + d*x]^(7/2)*(-42*b*(9*a^2 + 5*b^2)*Cos[c + d*x]^(7/2)*EllipticE[(c + d*x)/2, 2] + 10*a*(5*a^2 + 21*b^2)*Cos[c + d*x]^(7/2)*EllipticF[(c + d* x)/2, 2] + 30*a^3*Sin[c + d*x] + 50*a^3*Cos[c + d*x]^2*Sin[c + d*x] + 210* a*b^2*Cos[c + d*x]^2*Sin[c + d*x] + 378*a^2*b*Cos[c + d*x]^3*Sin[c + d*x] + 210*b^3*Cos[c + d*x]^3*Sin[c + d*x] + 63*a^2*b*Sin[2*(c + d*x)]))/(105*d )
Time = 1.41 (sec) , antiderivative size = 218, normalized size of antiderivative = 0.93, number of steps used = 20, number of rules used = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.870, Rules used = {3042, 3717, 3042, 4329, 27, 3042, 4535, 3042, 4255, 3042, 4258, 3042, 3120, 4534, 3042, 4255, 3042, 4258, 3042, 3119}
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.
| \(\displaystyle \int \sec ^{\frac {9}{2}}(c+d x) (a+b \cos (c+d x))^3 \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \csc \left (c+d x+\frac {\pi }{2}\right )^{9/2} \left (a+b \sin \left (c+d x+\frac {\pi }{2}\right )\right )^3dx\) |
| \(\Big \downarrow \) 3717 |
| \(\displaystyle \int \sec ^{\frac {3}{2}}(c+d x) (a \sec (c+d x)+b)^3dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \csc \left (c+d x+\frac {\pi }{2}\right )^{3/2} \left (a \csc \left (c+d x+\frac {\pi }{2}\right )+b\right )^3dx\) |
| \(\Big \downarrow \) 4329 |
| \(\displaystyle \frac {2}{7} \int \frac {1}{2} \sec ^{\frac {3}{2}}(c+d x) \left (16 a^2 b \sec ^2(c+d x)+a \left (5 a^2+21 b^2\right ) \sec (c+d x)+b \left (3 a^2+7 b^2\right )\right )dx+\frac {2 a^2 \sin (c+d x) \sec ^{\frac {5}{2}}(c+d x) (a \sec (c+d x)+b)}{7 d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{7} \int \sec ^{\frac {3}{2}}(c+d x) \left (16 a^2 b \sec ^2(c+d x)+a \left (5 a^2+21 b^2\right ) \sec (c+d x)+b \left (3 a^2+7 b^2\right )\right )dx+\frac {2 a^2 \sin (c+d x) \sec ^{\frac {5}{2}}(c+d x) (a \sec (c+d x)+b)}{7 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{7} \int \csc \left (c+d x+\frac {\pi }{2}\right )^{3/2} \left (16 a^2 b \csc \left (c+d x+\frac {\pi }{2}\right )^2+a \left (5 a^2+21 b^2\right ) \csc \left (c+d x+\frac {\pi }{2}\right )+b \left (3 a^2+7 b^2\right )\right )dx+\frac {2 a^2 \sin (c+d x) \sec ^{\frac {5}{2}}(c+d x) (a \sec (c+d x)+b)}{7 d}\) |
| \(\Big \downarrow \) 4535 |
| \(\displaystyle \frac {1}{7} \left (a \left (5 a^2+21 b^2\right ) \int \sec ^{\frac {5}{2}}(c+d x)dx+\int \sec ^{\frac {3}{2}}(c+d x) \left (16 a^2 b \sec ^2(c+d x)+b \left (3 a^2+7 b^2\right )\right )dx\right )+\frac {2 a^2 \sin (c+d x) \sec ^{\frac {5}{2}}(c+d x) (a \sec (c+d x)+b)}{7 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{7} \left (a \left (5 a^2+21 b^2\right ) \int \csc \left (c+d x+\frac {\pi }{2}\right )^{5/2}dx+\int \csc \left (c+d x+\frac {\pi }{2}\right )^{3/2} \left (16 a^2 b \csc \left (c+d x+\frac {\pi }{2}\right )^2+b \left (3 a^2+7 b^2\right )\right )dx\right )+\frac {2 a^2 \sin (c+d x) \sec ^{\frac {5}{2}}(c+d x) (a \sec (c+d x)+b)}{7 d}\) |
| \(\Big \downarrow \) 4255 |
| \(\displaystyle \frac {1}{7} \left (\int \csc \left (c+d x+\frac {\pi }{2}\right )^{3/2} \left (16 a^2 b \csc \left (c+d x+\frac {\pi }{2}\right )^2+b \left (3 a^2+7 b^2\right )\right )dx+a \left (5 a^2+21 b^2\right ) \left (\frac {1}{3} \int \sqrt {\sec (c+d x)}dx+\frac {2 \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x)}{3 d}\right )\right )+\frac {2 a^2 \sin (c+d x) \sec ^{\frac {5}{2}}(c+d x) (a \sec (c+d x)+b)}{7 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{7} \left (\int \csc \left (c+d x+\frac {\pi }{2}\right )^{3/2} \left (16 a^2 b \csc \left (c+d x+\frac {\pi }{2}\right )^2+b \left (3 a^2+7 b^2\right )\right )dx+a \left (5 a^2+21 b^2\right ) \left (\frac {1}{3} \int \sqrt {\csc \left (c+d x+\frac {\pi }{2}\right )}dx+\frac {2 \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x)}{3 d}\right )\right )+\frac {2 a^2 \sin (c+d x) \sec ^{\frac {5}{2}}(c+d x) (a \sec (c+d x)+b)}{7 d}\) |
| \(\Big \downarrow \) 4258 |
| \(\displaystyle \frac {1}{7} \left (\int \csc \left (c+d x+\frac {\pi }{2}\right )^{3/2} \left (16 a^2 b \csc \left (c+d x+\frac {\pi }{2}\right )^2+b \left (3 a^2+7 b^2\right )\right )dx+a \left (5 a^2+21 b^2\right ) \left (\frac {1}{3} \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \int \frac {1}{\sqrt {\cos (c+d x)}}dx+\frac {2 \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x)}{3 d}\right )\right )+\frac {2 a^2 \sin (c+d x) \sec ^{\frac {5}{2}}(c+d x) (a \sec (c+d x)+b)}{7 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{7} \left (\int \csc \left (c+d x+\frac {\pi }{2}\right )^{3/2} \left (16 a^2 b \csc \left (c+d x+\frac {\pi }{2}\right )^2+b \left (3 a^2+7 b^2\right )\right )dx+a \left (5 a^2+21 b^2\right ) \left (\frac {1}{3} \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \int \frac {1}{\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}}dx+\frac {2 \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x)}{3 d}\right )\right )+\frac {2 a^2 \sin (c+d x) \sec ^{\frac {5}{2}}(c+d x) (a \sec (c+d x)+b)}{7 d}\) |
| \(\Big \downarrow \) 3120 |
| \(\displaystyle \frac {1}{7} \left (\int \csc \left (c+d x+\frac {\pi }{2}\right )^{3/2} \left (16 a^2 b \csc \left (c+d x+\frac {\pi }{2}\right )^2+b \left (3 a^2+7 b^2\right )\right )dx+a \left (5 a^2+21 b^2\right ) \left (\frac {2 \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x)}{3 d}+\frac {2 \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )}{3 d}\right )\right )+\frac {2 a^2 \sin (c+d x) \sec ^{\frac {5}{2}}(c+d x) (a \sec (c+d x)+b)}{7 d}\) |
| \(\Big \downarrow \) 4534 |
| \(\displaystyle \frac {1}{7} \left (\frac {7}{5} b \left (9 a^2+5 b^2\right ) \int \sec ^{\frac {3}{2}}(c+d x)dx+a \left (5 a^2+21 b^2\right ) \left (\frac {2 \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x)}{3 d}+\frac {2 \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )}{3 d}\right )+\frac {32 a^2 b \sin (c+d x) \sec ^{\frac {5}{2}}(c+d x)}{5 d}\right )+\frac {2 a^2 \sin (c+d x) \sec ^{\frac {5}{2}}(c+d x) (a \sec (c+d x)+b)}{7 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{7} \left (\frac {7}{5} b \left (9 a^2+5 b^2\right ) \int \csc \left (c+d x+\frac {\pi }{2}\right )^{3/2}dx+a \left (5 a^2+21 b^2\right ) \left (\frac {2 \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x)}{3 d}+\frac {2 \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )}{3 d}\right )+\frac {32 a^2 b \sin (c+d x) \sec ^{\frac {5}{2}}(c+d x)}{5 d}\right )+\frac {2 a^2 \sin (c+d x) \sec ^{\frac {5}{2}}(c+d x) (a \sec (c+d x)+b)}{7 d}\) |
| \(\Big \downarrow \) 4255 |
| \(\displaystyle \frac {1}{7} \left (\frac {7}{5} b \left (9 a^2+5 b^2\right ) \left (\frac {2 \sin (c+d x) \sqrt {\sec (c+d x)}}{d}-\int \frac {1}{\sqrt {\sec (c+d x)}}dx\right )+a \left (5 a^2+21 b^2\right ) \left (\frac {2 \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x)}{3 d}+\frac {2 \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )}{3 d}\right )+\frac {32 a^2 b \sin (c+d x) \sec ^{\frac {5}{2}}(c+d x)}{5 d}\right )+\frac {2 a^2 \sin (c+d x) \sec ^{\frac {5}{2}}(c+d x) (a \sec (c+d x)+b)}{7 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{7} \left (\frac {7}{5} b \left (9 a^2+5 b^2\right ) \left (\frac {2 \sin (c+d x) \sqrt {\sec (c+d x)}}{d}-\int \frac {1}{\sqrt {\csc \left (c+d x+\frac {\pi }{2}\right )}}dx\right )+a \left (5 a^2+21 b^2\right ) \left (\frac {2 \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x)}{3 d}+\frac {2 \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )}{3 d}\right )+\frac {32 a^2 b \sin (c+d x) \sec ^{\frac {5}{2}}(c+d x)}{5 d}\right )+\frac {2 a^2 \sin (c+d x) \sec ^{\frac {5}{2}}(c+d x) (a \sec (c+d x)+b)}{7 d}\) |
| \(\Big \downarrow \) 4258 |
| \(\displaystyle \frac {1}{7} \left (\frac {7}{5} b \left (9 a^2+5 b^2\right ) \left (\frac {2 \sin (c+d x) \sqrt {\sec (c+d x)}}{d}-\sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \int \sqrt {\cos (c+d x)}dx\right )+a \left (5 a^2+21 b^2\right ) \left (\frac {2 \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x)}{3 d}+\frac {2 \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )}{3 d}\right )+\frac {32 a^2 b \sin (c+d x) \sec ^{\frac {5}{2}}(c+d x)}{5 d}\right )+\frac {2 a^2 \sin (c+d x) \sec ^{\frac {5}{2}}(c+d x) (a \sec (c+d x)+b)}{7 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{7} \left (\frac {7}{5} b \left (9 a^2+5 b^2\right ) \left (\frac {2 \sin (c+d x) \sqrt {\sec (c+d x)}}{d}-\sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \int \sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}dx\right )+a \left (5 a^2+21 b^2\right ) \left (\frac {2 \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x)}{3 d}+\frac {2 \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )}{3 d}\right )+\frac {32 a^2 b \sin (c+d x) \sec ^{\frac {5}{2}}(c+d x)}{5 d}\right )+\frac {2 a^2 \sin (c+d x) \sec ^{\frac {5}{2}}(c+d x) (a \sec (c+d x)+b)}{7 d}\) |
| \(\Big \downarrow \) 3119 |
| \(\displaystyle \frac {1}{7} \left (a \left (5 a^2+21 b^2\right ) \left (\frac {2 \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x)}{3 d}+\frac {2 \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )}{3 d}\right )+\frac {7}{5} b \left (9 a^2+5 b^2\right ) \left (\frac {2 \sin (c+d x) \sqrt {\sec (c+d x)}}{d}-\frac {2 \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{d}\right )+\frac {32 a^2 b \sin (c+d x) \sec ^{\frac {5}{2}}(c+d x)}{5 d}\right )+\frac {2 a^2 \sin (c+d x) \sec ^{\frac {5}{2}}(c+d x) (a \sec (c+d x)+b)}{7 d}\) |
Input:
Int[(a + b*Cos[c + d*x])^3*Sec[c + d*x]^(9/2),x]
Output:
(2*a^2*Sec[c + d*x]^(5/2)*(b + a*Sec[c + d*x])*Sin[c + d*x])/(7*d) + ((32* a^2*b*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(5*d) + (7*b*(9*a^2 + 5*b^2)*((-2*S qrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/d + (2*Sqr t[Sec[c + d*x]]*Sin[c + d*x])/d))/5 + a*(5*a^2 + 21*b^2)*((2*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(3*d) + (2*Sec[c + d*x ]^(3/2)*Sin[c + d*x])/(3*d)))/7
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[Sqrt[sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[(2/d)*EllipticE[(1/2)* (c - Pi/2 + d*x), 2], x] /; FreeQ[{c, d}, x]
Int[1/Sqrt[sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[(2/d)*EllipticF[(1/2 )*(c - Pi/2 + d*x), 2], x] /; FreeQ[{c, d}, x]
Int[(csc[(e_.) + (f_.)*(x_)]*(d_.))^(m_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x _)]^(n_.))^(p_.), x_Symbol] :> Simp[d^(n*p) Int[(d*Csc[e + f*x])^(m - n*p )*(b + a*Csc[e + f*x]^n)^p, x], x] /; FreeQ[{a, b, d, e, f, m, n, p}, x] && !IntegerQ[m] && IntegersQ[n, p]
Int[(csc[(c_.) + (d_.)*(x_)]*(b_.))^(n_), x_Symbol] :> Simp[(-b)*Cos[c + d* x]*((b*Csc[c + d*x])^(n - 1)/(d*(n - 1))), x] + Simp[b^2*((n - 2)/(n - 1)) Int[(b*Csc[c + d*x])^(n - 2), x], x] /; FreeQ[{b, c, d}, x] && GtQ[n, 1] && IntegerQ[2*n]
Int[(csc[(c_.) + (d_.)*(x_)]*(b_.))^(n_), x_Symbol] :> Simp[(b*Csc[c + d*x] )^n*Sin[c + d*x]^n Int[1/Sin[c + d*x]^n, x], x] /; FreeQ[{b, c, d}, x] && EqQ[n^2, 1/4]
Int[(csc[(e_.) + (f_.)*(x_)]*(d_.))^(n_)*(csc[(e_.) + (f_.)*(x_)]*(b_.) + ( a_))^(m_), x_Symbol] :> Simp[(-b^2)*Cot[e + f*x]*(a + b*Csc[e + f*x])^(m - 2)*((d*Csc[e + f*x])^n/(f*(m + n - 1))), x] + Simp[1/(d*(m + n - 1)) Int[ (a + b*Csc[e + f*x])^(m - 3)*(d*Csc[e + f*x])^n*Simp[a^3*d*(m + n - 1) + a* b^2*d*n + b*(b^2*d*(m + n - 2) + 3*a^2*d*(m + n - 1))*Csc[e + f*x] + a*b^2* d*(3*m + 2*n - 4)*Csc[e + f*x]^2, x], x], x] /; FreeQ[{a, b, d, e, f, n}, x ] && NeQ[a^2 - b^2, 0] && GtQ[m, 2] && (IntegerQ[m] || IntegersQ[2*m, 2*n]) && !(IGtQ[n, 2] && !IntegerQ[m])
Int[(csc[(e_.) + (f_.)*(x_)]*(b_.))^(m_.)*(csc[(e_.) + (f_.)*(x_)]^2*(C_.) + (A_)), x_Symbol] :> Simp[(-C)*Cot[e + f*x]*((b*Csc[e + f*x])^m/(f*(m + 1) )), x] + Simp[(C*m + A*(m + 1))/(m + 1) Int[(b*Csc[e + f*x])^m, x], x] /; FreeQ[{b, e, f, A, C, m}, x] && NeQ[C*m + A*(m + 1), 0] && !LeQ[m, -1]
Int[(csc[(e_.) + (f_.)*(x_)]*(b_.))^(m_.)*((A_.) + csc[(e_.) + (f_.)*(x_)]* (B_.) + csc[(e_.) + (f_.)*(x_)]^2*(C_.)), x_Symbol] :> Simp[B/b Int[(b*Cs c[e + f*x])^(m + 1), x], x] + Int[(b*Csc[e + f*x])^m*(A + C*Csc[e + f*x]^2) , x] /; FreeQ[{b, e, f, A, B, C, m}, x]
Leaf count of result is larger than twice the leaf count of optimal. \(819\) vs. \(2(209)=418\).
Time = 965.34 (sec) , antiderivative size = 820, normalized size of antiderivative = 3.50
| method | result | size |
| default | \(\text {Expression too large to display}\) | \(820\) |
| parts | \(\text {Expression too large to display}\) | \(1008\) |
Input:
int((a+cos(d*x+c)*b)^3*sec(d*x+c)^(9/2),x,method=_RETURNVERBOSE)
Output:
-(-(-2*cos(1/2*d*x+1/2*c)^2+1)*sin(1/2*d*x+1/2*c)^2)^(1/2)*(2*a^3*(-1/56*c os(1/2*d*x+1/2*c)*(-2*sin(1/2*d*x+1/2*c)^4+sin(1/2*d*x+1/2*c)^2)^(1/2)/(co s(1/2*d*x+1/2*c)^2-1/2)^4-5/42*cos(1/2*d*x+1/2*c)*(-2*sin(1/2*d*x+1/2*c)^4 +sin(1/2*d*x+1/2*c)^2)^(1/2)/(cos(1/2*d*x+1/2*c)^2-1/2)^2+5/21*(sin(1/2*d* x+1/2*c)^2)^(1/2)*(-2*cos(1/2*d*x+1/2*c)^2+1)^(1/2)/(-2*sin(1/2*d*x+1/2*c) ^4+sin(1/2*d*x+1/2*c)^2)^(1/2)*EllipticF(cos(1/2*d*x+1/2*c),2^(1/2)))+2*b^ 3/sin(1/2*d*x+1/2*c)^2/(2*sin(1/2*d*x+1/2*c)^2-1)*(-2*sin(1/2*d*x+1/2*c)^4 +sin(1/2*d*x+1/2*c)^2)^(1/2)*(2*sin(1/2*d*x+1/2*c)^2*cos(1/2*d*x+1/2*c)-(s in(1/2*d*x+1/2*c)^2)^(1/2)*(2*sin(1/2*d*x+1/2*c)^2-1)^(1/2)*EllipticE(cos( 1/2*d*x+1/2*c),2^(1/2)))+6/5*a^2*b/sin(1/2*d*x+1/2*c)^2/(8*sin(1/2*d*x+1/2 *c)^6-12*sin(1/2*d*x+1/2*c)^4+6*sin(1/2*d*x+1/2*c)^2-1)*(24*cos(1/2*d*x+1/ 2*c)*sin(1/2*d*x+1/2*c)^6-12*EllipticE(cos(1/2*d*x+1/2*c),2^(1/2))*(sin(1/ 2*d*x+1/2*c)^2)^(1/2)*(2*sin(1/2*d*x+1/2*c)^2-1)^(1/2)*sin(1/2*d*x+1/2*c)^ 4-24*sin(1/2*d*x+1/2*c)^4*cos(1/2*d*x+1/2*c)+12*EllipticE(cos(1/2*d*x+1/2* c),2^(1/2))*(sin(1/2*d*x+1/2*c)^2)^(1/2)*(2*sin(1/2*d*x+1/2*c)^2-1)^(1/2)* sin(1/2*d*x+1/2*c)^2+8*sin(1/2*d*x+1/2*c)^2*cos(1/2*d*x+1/2*c)-3*(sin(1/2* d*x+1/2*c)^2)^(1/2)*(2*sin(1/2*d*x+1/2*c)^2-1)^(1/2)*EllipticE(cos(1/2*d*x +1/2*c),2^(1/2)))*(-2*sin(1/2*d*x+1/2*c)^4+sin(1/2*d*x+1/2*c)^2)^(1/2)+6*b ^2*a*(-1/6*cos(1/2*d*x+1/2*c)*(-2*sin(1/2*d*x+1/2*c)^4+sin(1/2*d*x+1/2*c)^ 2)^(1/2)/(cos(1/2*d*x+1/2*c)^2-1/2)^2+1/3*(sin(1/2*d*x+1/2*c)^2)^(1/2)*...
Result contains complex when optimal does not.
Time = 0.10 (sec) , antiderivative size = 270, normalized size of antiderivative = 1.15 \[ \int (a+b \cos (c+d x))^3 \sec ^{\frac {9}{2}}(c+d x) \, dx=-\frac {5 \, \sqrt {2} {\left (5 i \, a^{3} + 21 i \, a b^{2}\right )} \cos \left (d x + c\right )^{3} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right ) + 5 \, \sqrt {2} {\left (-5 i \, a^{3} - 21 i \, a b^{2}\right )} \cos \left (d x + c\right )^{3} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right ) + 21 \, \sqrt {2} {\left (9 i \, a^{2} b + 5 i \, b^{3}\right )} \cos \left (d x + c\right )^{3} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right )\right ) + 21 \, \sqrt {2} {\left (-9 i \, a^{2} b - 5 i \, b^{3}\right )} \cos \left (d x + c\right )^{3} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right )\right ) - \frac {2 \, {\left (63 \, a^{2} b \cos \left (d x + c\right ) + 21 \, {\left (9 \, a^{2} b + 5 \, b^{3}\right )} \cos \left (d x + c\right )^{3} + 15 \, a^{3} + 5 \, {\left (5 \, a^{3} + 21 \, a b^{2}\right )} \cos \left (d x + c\right )^{2}\right )} \sin \left (d x + c\right )}{\sqrt {\cos \left (d x + c\right )}}}{105 \, d \cos \left (d x + c\right )^{3}} \] Input:
integrate((a+b*cos(d*x+c))^3*sec(d*x+c)^(9/2),x, algorithm="fricas")
Output:
-1/105*(5*sqrt(2)*(5*I*a^3 + 21*I*a*b^2)*cos(d*x + c)^3*weierstrassPInvers e(-4, 0, cos(d*x + c) + I*sin(d*x + c)) + 5*sqrt(2)*(-5*I*a^3 - 21*I*a*b^2 )*cos(d*x + c)^3*weierstrassPInverse(-4, 0, cos(d*x + c) - I*sin(d*x + c)) + 21*sqrt(2)*(9*I*a^2*b + 5*I*b^3)*cos(d*x + c)^3*weierstrassZeta(-4, 0, weierstrassPInverse(-4, 0, cos(d*x + c) + I*sin(d*x + c))) + 21*sqrt(2)*(- 9*I*a^2*b - 5*I*b^3)*cos(d*x + c)^3*weierstrassZeta(-4, 0, weierstrassPInv erse(-4, 0, cos(d*x + c) - I*sin(d*x + c))) - 2*(63*a^2*b*cos(d*x + c) + 2 1*(9*a^2*b + 5*b^3)*cos(d*x + c)^3 + 15*a^3 + 5*(5*a^3 + 21*a*b^2)*cos(d*x + c)^2)*sin(d*x + c)/sqrt(cos(d*x + c)))/(d*cos(d*x + c)^3)
Timed out. \[ \int (a+b \cos (c+d x))^3 \sec ^{\frac {9}{2}}(c+d x) \, dx=\text {Timed out} \] Input:
integrate((a+b*cos(d*x+c))**3*sec(d*x+c)**(9/2),x)
Output:
Timed out
\[ \int (a+b \cos (c+d x))^3 \sec ^{\frac {9}{2}}(c+d x) \, dx=\int { {\left (b \cos \left (d x + c\right ) + a\right )}^{3} \sec \left (d x + c\right )^{\frac {9}{2}} \,d x } \] Input:
integrate((a+b*cos(d*x+c))^3*sec(d*x+c)^(9/2),x, algorithm="maxima")
Output:
integrate((b*cos(d*x + c) + a)^3*sec(d*x + c)^(9/2), x)
\[ \int (a+b \cos (c+d x))^3 \sec ^{\frac {9}{2}}(c+d x) \, dx=\int { {\left (b \cos \left (d x + c\right ) + a\right )}^{3} \sec \left (d x + c\right )^{\frac {9}{2}} \,d x } \] Input:
integrate((a+b*cos(d*x+c))^3*sec(d*x+c)^(9/2),x, algorithm="giac")
Output:
integrate((b*cos(d*x + c) + a)^3*sec(d*x + c)^(9/2), x)
Timed out. \[ \int (a+b \cos (c+d x))^3 \sec ^{\frac {9}{2}}(c+d x) \, dx=\int {\left (\frac {1}{\cos \left (c+d\,x\right )}\right )}^{9/2}\,{\left (a+b\,\cos \left (c+d\,x\right )\right )}^3 \,d x \] Input:
int((1/cos(c + d*x))^(9/2)*(a + b*cos(c + d*x))^3,x)
Output:
int((1/cos(c + d*x))^(9/2)*(a + b*cos(c + d*x))^3, x)
\[ \int (a+b \cos (c+d x))^3 \sec ^{\frac {9}{2}}(c+d x) \, dx=3 \left (\int \sqrt {\sec \left (d x +c \right )}\, \cos \left (d x +c \right ) \sec \left (d x +c \right )^{4}d x \right ) a^{2} b +\left (\int \sqrt {\sec \left (d x +c \right )}\, \cos \left (d x +c \right )^{3} \sec \left (d x +c \right )^{4}d x \right ) b^{3}+3 \left (\int \sqrt {\sec \left (d x +c \right )}\, \cos \left (d x +c \right )^{2} \sec \left (d x +c \right )^{4}d x \right ) a \,b^{2}+\left (\int \sqrt {\sec \left (d x +c \right )}\, \sec \left (d x +c \right )^{4}d x \right ) a^{3} \] Input:
int((a+b*cos(d*x+c))^3*sec(d*x+c)^(9/2),x)
Output:
3*int(sqrt(sec(c + d*x))*cos(c + d*x)*sec(c + d*x)**4,x)*a**2*b + int(sqrt (sec(c + d*x))*cos(c + d*x)**3*sec(c + d*x)**4,x)*b**3 + 3*int(sqrt(sec(c + d*x))*cos(c + d*x)**2*sec(c + d*x)**4,x)*a*b**2 + int(sqrt(sec(c + d*x)) *sec(c + d*x)**4,x)*a**3