Integrand size = 37, antiderivative size = 238 \[ \int \frac {(a+a \sec (c+d x))^{5/2} \left (A+C \sec ^2(c+d x)\right )}{\sqrt {\cos (c+d x)}} \, dx=\frac {a^{5/2} (304 A+163 C) \text {arcsinh}\left (\frac {\sqrt {a} \tan (c+d x)}{\sqrt {a+a \sec (c+d x)}}\right ) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}}{64 d}+\frac {a^3 (432 A+299 C) \sin (c+d x)}{192 d \cos ^{\frac {3}{2}}(c+d x) \sqrt {a+a \sec (c+d x)}}+\frac {a^2 (16 A+17 C) \sqrt {a+a \sec (c+d x)} \sin (c+d x)}{32 d \cos ^{\frac {3}{2}}(c+d x)}+\frac {5 a C (a+a \sec (c+d x))^{3/2} \sin (c+d x)}{24 d \cos ^{\frac {3}{2}}(c+d x)}+\frac {C (a+a \sec (c+d x))^{5/2} \sin (c+d x)}{4 d \cos ^{\frac {3}{2}}(c+d x)} \] Output:
1/64*a^(5/2)*(304*A+163*C)*arcsinh(a^(1/2)*tan(d*x+c)/(a+a*sec(d*x+c))^(1/ 2))*cos(d*x+c)^(1/2)*sec(d*x+c)^(1/2)/d+1/192*a^3*(432*A+299*C)*sin(d*x+c) /d/cos(d*x+c)^(3/2)/(a+a*sec(d*x+c))^(1/2)+1/32*a^2*(16*A+17*C)*(a+a*sec(d *x+c))^(1/2)*sin(d*x+c)/d/cos(d*x+c)^(3/2)+5/24*a*C*(a+a*sec(d*x+c))^(3/2) *sin(d*x+c)/d/cos(d*x+c)^(3/2)+1/4*C*(a+a*sec(d*x+c))^(5/2)*sin(d*x+c)/d/c os(d*x+c)^(3/2)
Time = 6.96 (sec) , antiderivative size = 256, normalized size of antiderivative = 1.08 \[ \int \frac {(a+a \sec (c+d x))^{5/2} \left (A+C \sec ^2(c+d x)\right )}{\sqrt {\cos (c+d x)}} \, dx=\frac {a^3 \sqrt {\cos (c+d x)} \sec ^{\frac {3}{2}}(c+d x) \left ((432 A+489 C) \arcsin \left (\sqrt {1-\sec (c+d x)}\right )-480 A \arcsin \left (\sqrt {\sec (c+d x)}\right )+96 A \sqrt {1-\sec (c+d x)} \sec ^{\frac {3}{2}}(c+d x)+326 C \sqrt {1-\sec (c+d x)} \sec ^{\frac {3}{2}}(c+d x)+184 C \sqrt {1-\sec (c+d x)} \sec ^{\frac {5}{2}}(c+d x)+48 C \sqrt {1-\sec (c+d x)} \sec ^{\frac {7}{2}}(c+d x)+528 A \sqrt {-((-1+\sec (c+d x)) \sec (c+d x))}+489 C \sqrt {-((-1+\sec (c+d x)) \sec (c+d x))}\right ) \sin (c+d x)}{192 d \sqrt {1-\sec (c+d x)} \sqrt {a (1+\sec (c+d x))}} \] Input:
Integrate[((a + a*Sec[c + d*x])^(5/2)*(A + C*Sec[c + d*x]^2))/Sqrt[Cos[c + d*x]],x]
Output:
(a^3*Sqrt[Cos[c + d*x]]*Sec[c + d*x]^(3/2)*((432*A + 489*C)*ArcSin[Sqrt[1 - Sec[c + d*x]]] - 480*A*ArcSin[Sqrt[Sec[c + d*x]]] + 96*A*Sqrt[1 - Sec[c + d*x]]*Sec[c + d*x]^(3/2) + 326*C*Sqrt[1 - Sec[c + d*x]]*Sec[c + d*x]^(3/ 2) + 184*C*Sqrt[1 - Sec[c + d*x]]*Sec[c + d*x]^(5/2) + 48*C*Sqrt[1 - Sec[c + d*x]]*Sec[c + d*x]^(7/2) + 528*A*Sqrt[-((-1 + Sec[c + d*x])*Sec[c + d*x ])] + 489*C*Sqrt[-((-1 + Sec[c + d*x])*Sec[c + d*x])])*Sin[c + d*x])/(192* d*Sqrt[1 - Sec[c + d*x]]*Sqrt[a*(1 + Sec[c + d*x])])
Time = 1.63 (sec) , antiderivative size = 254, normalized size of antiderivative = 1.07, number of steps used = 17, number of rules used = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.432, Rules used = {3042, 4753, 3042, 4577, 27, 3042, 4506, 27, 3042, 4506, 27, 3042, 4504, 3042, 4288, 222}
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.
| \(\displaystyle \int \frac {(a \sec (c+d x)+a)^{5/2} \left (A+C \sec ^2(c+d x)\right )}{\sqrt {\cos (c+d x)}} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {(a \sec (c+d x)+a)^{5/2} \left (A+C \sec (c+d x)^2\right )}{\sqrt {\cos (c+d x)}}dx\) |
| \(\Big \downarrow \) 4753 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \int \sqrt {\sec (c+d x)} (\sec (c+d x) a+a)^{5/2} \left (C \sec ^2(c+d x)+A\right )dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \int \sqrt {\csc \left (c+d x+\frac {\pi }{2}\right )} \left (\csc \left (c+d x+\frac {\pi }{2}\right ) a+a\right )^{5/2} \left (C \csc \left (c+d x+\frac {\pi }{2}\right )^2+A\right )dx\) |
| \(\Big \downarrow \) 4577 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {\int \frac {1}{2} \sqrt {\sec (c+d x)} (\sec (c+d x) a+a)^{5/2} (a (8 A+C)+5 a C \sec (c+d x))dx}{4 a}+\frac {C \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x) (a \sec (c+d x)+a)^{5/2}}{4 d}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {\int \sqrt {\sec (c+d x)} (\sec (c+d x) a+a)^{5/2} (a (8 A+C)+5 a C \sec (c+d x))dx}{8 a}+\frac {C \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x) (a \sec (c+d x)+a)^{5/2}}{4 d}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {\int \sqrt {\csc \left (c+d x+\frac {\pi }{2}\right )} \left (\csc \left (c+d x+\frac {\pi }{2}\right ) a+a\right )^{5/2} \left (a (8 A+C)+5 a C \csc \left (c+d x+\frac {\pi }{2}\right )\right )dx}{8 a}+\frac {C \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x) (a \sec (c+d x)+a)^{5/2}}{4 d}\right )\) |
| \(\Big \downarrow \) 4506 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {\frac {1}{3} \int \frac {1}{2} \sqrt {\sec (c+d x)} (\sec (c+d x) a+a)^{3/2} \left ((48 A+11 C) a^2+3 (16 A+17 C) \sec (c+d x) a^2\right )dx+\frac {5 a^2 C \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x) (a \sec (c+d x)+a)^{3/2}}{3 d}}{8 a}+\frac {C \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x) (a \sec (c+d x)+a)^{5/2}}{4 d}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {\frac {1}{6} \int \sqrt {\sec (c+d x)} (\sec (c+d x) a+a)^{3/2} \left ((48 A+11 C) a^2+3 (16 A+17 C) \sec (c+d x) a^2\right )dx+\frac {5 a^2 C \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x) (a \sec (c+d x)+a)^{3/2}}{3 d}}{8 a}+\frac {C \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x) (a \sec (c+d x)+a)^{5/2}}{4 d}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {\frac {1}{6} \int \sqrt {\csc \left (c+d x+\frac {\pi }{2}\right )} \left (\csc \left (c+d x+\frac {\pi }{2}\right ) a+a\right )^{3/2} \left ((48 A+11 C) a^2+3 (16 A+17 C) \csc \left (c+d x+\frac {\pi }{2}\right ) a^2\right )dx+\frac {5 a^2 C \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x) (a \sec (c+d x)+a)^{3/2}}{3 d}}{8 a}+\frac {C \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x) (a \sec (c+d x)+a)^{5/2}}{4 d}\right )\) |
| \(\Big \downarrow \) 4506 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {\frac {1}{6} \left (\frac {1}{2} \int \frac {1}{2} \sqrt {\sec (c+d x)} \sqrt {\sec (c+d x) a+a} \left (5 (48 A+19 C) a^3+(432 A+299 C) \sec (c+d x) a^3\right )dx+\frac {3 a^3 (16 A+17 C) \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x) \sqrt {a \sec (c+d x)+a}}{2 d}\right )+\frac {5 a^2 C \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x) (a \sec (c+d x)+a)^{3/2}}{3 d}}{8 a}+\frac {C \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x) (a \sec (c+d x)+a)^{5/2}}{4 d}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {\frac {1}{6} \left (\frac {1}{4} \int \sqrt {\sec (c+d x)} \sqrt {\sec (c+d x) a+a} \left (5 (48 A+19 C) a^3+(432 A+299 C) \sec (c+d x) a^3\right )dx+\frac {3 a^3 (16 A+17 C) \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x) \sqrt {a \sec (c+d x)+a}}{2 d}\right )+\frac {5 a^2 C \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x) (a \sec (c+d x)+a)^{3/2}}{3 d}}{8 a}+\frac {C \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x) (a \sec (c+d x)+a)^{5/2}}{4 d}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {\frac {1}{6} \left (\frac {1}{4} \int \sqrt {\csc \left (c+d x+\frac {\pi }{2}\right )} \sqrt {\csc \left (c+d x+\frac {\pi }{2}\right ) a+a} \left (5 (48 A+19 C) a^3+(432 A+299 C) \csc \left (c+d x+\frac {\pi }{2}\right ) a^3\right )dx+\frac {3 a^3 (16 A+17 C) \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x) \sqrt {a \sec (c+d x)+a}}{2 d}\right )+\frac {5 a^2 C \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x) (a \sec (c+d x)+a)^{3/2}}{3 d}}{8 a}+\frac {C \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x) (a \sec (c+d x)+a)^{5/2}}{4 d}\right )\) |
| \(\Big \downarrow \) 4504 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {\frac {1}{6} \left (\frac {1}{4} \left (\frac {3}{2} a^3 (304 A+163 C) \int \sqrt {\sec (c+d x)} \sqrt {\sec (c+d x) a+a}dx+\frac {a^4 (432 A+299 C) \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x)}{d \sqrt {a \sec (c+d x)+a}}\right )+\frac {3 a^3 (16 A+17 C) \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x) \sqrt {a \sec (c+d x)+a}}{2 d}\right )+\frac {5 a^2 C \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x) (a \sec (c+d x)+a)^{3/2}}{3 d}}{8 a}+\frac {C \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x) (a \sec (c+d x)+a)^{5/2}}{4 d}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {\frac {1}{6} \left (\frac {1}{4} \left (\frac {3}{2} a^3 (304 A+163 C) \int \sqrt {\csc \left (c+d x+\frac {\pi }{2}\right )} \sqrt {\csc \left (c+d x+\frac {\pi }{2}\right ) a+a}dx+\frac {a^4 (432 A+299 C) \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x)}{d \sqrt {a \sec (c+d x)+a}}\right )+\frac {3 a^3 (16 A+17 C) \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x) \sqrt {a \sec (c+d x)+a}}{2 d}\right )+\frac {5 a^2 C \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x) (a \sec (c+d x)+a)^{3/2}}{3 d}}{8 a}+\frac {C \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x) (a \sec (c+d x)+a)^{5/2}}{4 d}\right )\) |
| \(\Big \downarrow \) 4288 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {\frac {1}{6} \left (\frac {1}{4} \left (\frac {a^4 (432 A+299 C) \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x)}{d \sqrt {a \sec (c+d x)+a}}-\frac {3 a^3 (304 A+163 C) \int \frac {1}{\sqrt {\frac {a \tan ^2(c+d x)}{\sec (c+d x) a+a}+1}}d\left (-\frac {a \tan (c+d x)}{\sqrt {\sec (c+d x) a+a}}\right )}{d}\right )+\frac {3 a^3 (16 A+17 C) \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x) \sqrt {a \sec (c+d x)+a}}{2 d}\right )+\frac {5 a^2 C \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x) (a \sec (c+d x)+a)^{3/2}}{3 d}}{8 a}+\frac {C \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x) (a \sec (c+d x)+a)^{5/2}}{4 d}\right )\) |
| \(\Big \downarrow \) 222 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {\frac {5 a^2 C \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x) (a \sec (c+d x)+a)^{3/2}}{3 d}+\frac {1}{6} \left (\frac {3 a^3 (16 A+17 C) \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x) \sqrt {a \sec (c+d x)+a}}{2 d}+\frac {1}{4} \left (\frac {3 a^{7/2} (304 A+163 C) \text {arcsinh}\left (\frac {\sqrt {a} \tan (c+d x)}{\sqrt {a \sec (c+d x)+a}}\right )}{d}+\frac {a^4 (432 A+299 C) \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x)}{d \sqrt {a \sec (c+d x)+a}}\right )\right )}{8 a}+\frac {C \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x) (a \sec (c+d x)+a)^{5/2}}{4 d}\right )\) |
Input:
Int[((a + a*Sec[c + d*x])^(5/2)*(A + C*Sec[c + d*x]^2))/Sqrt[Cos[c + d*x]] ,x]
Output:
Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]]*((C*Sec[c + d*x]^(3/2)*(a + a*Sec[c + d*x])^(5/2)*Sin[c + d*x])/(4*d) + ((5*a^2*C*Sec[c + d*x]^(3/2)*(a + a*Se c[c + d*x])^(3/2)*Sin[c + d*x])/(3*d) + ((3*a^3*(16*A + 17*C)*Sec[c + d*x] ^(3/2)*Sqrt[a + a*Sec[c + d*x]]*Sin[c + d*x])/(2*d) + ((3*a^(7/2)*(304*A + 163*C)*ArcSinh[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]])/d + (a^4 *(432*A + 299*C)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(d*Sqrt[a + a*Sec[c + d* x]]))/4)/6)/(8*a))
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Simp[ArcSinh[Rt[b, 2]*(x/Sqrt [a])]/Rt[b, 2], x] /; FreeQ[{a, b}, x] && GtQ[a, 0] && PosQ[b]
Int[Sqrt[csc[(e_.) + (f_.)*(x_)]*(d_.)]*Sqrt[csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_)], x_Symbol] :> Simp[-2*(a/(b*f))*Sqrt[a*(d/b)] Subst[Int[1/Sqrt[1 + x^2/a], x], x, b*(Cot[e + f*x]/Sqrt[a + b*Csc[e + f*x]])], x] /; FreeQ[{a , b, d, e, f}, x] && EqQ[a^2 - b^2, 0] && GtQ[a*(d/b), 0]
Int[(csc[(e_.) + (f_.)*(x_)]*(d_.))^(n_)*Sqrt[csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_)]*(csc[(e_.) + (f_.)*(x_)]*(B_.) + (A_)), x_Symbol] :> Simp[-2*b*B*C ot[e + f*x]*((d*Csc[e + f*x])^n/(f*(2*n + 1)*Sqrt[a + b*Csc[e + f*x]])), x] + Simp[(A*b*(2*n + 1) + 2*a*B*n)/(b*(2*n + 1)) Int[Sqrt[a + b*Csc[e + f* x]]*(d*Csc[e + f*x])^n, x], x] /; FreeQ[{a, b, d, e, f, A, B, n}, x] && NeQ [A*b - a*B, 0] && EqQ[a^2 - b^2, 0] && NeQ[A*b*(2*n + 1) + 2*a*B*n, 0] && !LtQ[n, 0]
Int[(csc[(e_.) + (f_.)*(x_)]*(d_.))^(n_)*(csc[(e_.) + (f_.)*(x_)]*(b_.) + ( a_))^(m_)*(csc[(e_.) + (f_.)*(x_)]*(B_.) + (A_)), x_Symbol] :> Simp[(-b)*B* Cot[e + f*x]*(a + b*Csc[e + f*x])^(m - 1)*((d*Csc[e + f*x])^n/(f*(m + n))), x] + Simp[1/(d*(m + n)) Int[(a + b*Csc[e + f*x])^(m - 1)*(d*Csc[e + f*x] )^n*Simp[a*A*d*(m + n) + B*(b*d*n) + (A*b*d*(m + n) + a*B*d*(2*m + n - 1))* Csc[e + f*x], x], x], x] /; FreeQ[{a, b, d, e, f, A, B, n}, x] && NeQ[A*b - a*B, 0] && EqQ[a^2 - b^2, 0] && GtQ[m, 1/2] && !LtQ[n, -1]
Int[((A_.) + csc[(e_.) + (f_.)*(x_)]^2*(C_.))*(csc[(e_.) + (f_.)*(x_)]*(d_. ))^(n_)*(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_))^(m_), x_Symbol] :> Simp[(-C) *Cot[e + f*x]*(a + b*Csc[e + f*x])^m*((d*Csc[e + f*x])^n/(f*(m + n + 1))), x] + Simp[1/(b*(m + n + 1)) Int[(a + b*Csc[e + f*x])^m*(d*Csc[e + f*x])^n *Simp[A*b*(m + n + 1) + b*C*n + a*C*m*Csc[e + f*x], x], x], x] /; FreeQ[{a, b, d, e, f, A, C, m, n}, x] && EqQ[a^2 - b^2, 0] && !LtQ[m, -2^(-1)] && !LtQ[n, -2^(-1)] && NeQ[m + n + 1, 0]
Int[(cos[(a_.) + (b_.)*(x_)]*(c_.))^(m_.)*(u_), x_Symbol] :> Simp[(c*Cos[a + b*x])^m*(c*Sec[a + b*x])^m Int[ActivateTrig[u]/(c*Sec[a + b*x])^m, x], x] /; FreeQ[{a, b, c, m}, x] && !IntegerQ[m] && KnownSecantIntegrandQ[u, x ]
Leaf count of result is larger than twice the leaf count of optimal. \(602\) vs. \(2(202)=404\).
Time = 4.03 (sec) , antiderivative size = 603, normalized size of antiderivative = 2.53
| method | result | size |
| default | \(4 \sqrt {2}\, \left (-\frac {C \sqrt {\frac {a \cos \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}{2 \cos \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-1}}\, a^{2} \left (\left (-15648 \cos \left (\frac {d x}{2}+\frac {c}{2}\right )^{8}+26080 \cos \left (\frac {d x}{2}+\frac {c}{2}\right )^{6}-17120 \cos \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}+5192 \cos \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-598\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+\sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {2}\, \left (\cot \left (\frac {d x}{2}+\frac {c}{2}\right )-\csc \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{2}\right ) \left (15648 \cos \left (\frac {d x}{2}+\frac {c}{2}\right )^{9}-39120 \cos \left (\frac {d x}{2}+\frac {c}{2}\right )^{7}+39120 \cos \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}-19560 \cos \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}+4890 \cos \left (\frac {d x}{2}+\frac {c}{2}\right )-489 \sec \left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {2}\, \left (\cot \left (\frac {d x}{2}+\frac {c}{2}\right )-\csc \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{2}\right ) \left (15648 \cos \left (\frac {d x}{2}+\frac {c}{2}\right )^{9}-39120 \cos \left (\frac {d x}{2}+\frac {c}{2}\right )^{7}+39120 \cos \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}-19560 \cos \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}+4890 \cos \left (\frac {d x}{2}+\frac {c}{2}\right )-489 \sec \left (\frac {d x}{2}+\frac {c}{2}\right )\right )\right )}{1536 d \left (2 \cos \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-1\right )^{\frac {9}{2}}}-\frac {A \sqrt {\frac {a \cos \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}{2 \cos \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-1}}\, a^{2} \left (\left (-88 \cos \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}+80 \cos \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-18\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+\sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {2}\, \left (\cot \left (\frac {d x}{2}+\frac {c}{2}\right )-\csc \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{2}\right ) \left (152 \cos \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}-228 \cos \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}+114 \cos \left (\frac {d x}{2}+\frac {c}{2}\right )-19 \sec \left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {2}\, \left (\cot \left (\frac {d x}{2}+\frac {c}{2}\right )-\csc \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{2}\right ) \left (152 \cos \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}-228 \cos \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}+114 \cos \left (\frac {d x}{2}+\frac {c}{2}\right )-19 \sec \left (\frac {d x}{2}+\frac {c}{2}\right )\right )\right )}{32 d \left (2 \cos \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-1\right )^{\frac {5}{2}}}\right )\) | \(603\) |
Input:
int((a+a*sec(d*x+c))^(5/2)*(A+C*sec(d*x+c)^2)/cos(d*x+c)^(1/2),x,method=_R ETURNVERBOSE)
Output:
4*2^(1/2)*(-1/1536*C/d*(a/(2*cos(1/2*d*x+1/2*c)^2-1)*cos(1/2*d*x+1/2*c)^2) ^(1/2)*a^2/(2*cos(1/2*d*x+1/2*c)^2-1)^(9/2)*((-15648*cos(1/2*d*x+1/2*c)^8+ 26080*cos(1/2*d*x+1/2*c)^6-17120*cos(1/2*d*x+1/2*c)^4+5192*cos(1/2*d*x+1/2 *c)^2-598)*tan(1/2*d*x+1/2*c)+2^(1/2)*arctanh(1/2*2^(1/2)*(cot(1/2*d*x+1/2 *c)-csc(1/2*d*x+1/2*c)-1))*(15648*cos(1/2*d*x+1/2*c)^9-39120*cos(1/2*d*x+1 /2*c)^7+39120*cos(1/2*d*x+1/2*c)^5-19560*cos(1/2*d*x+1/2*c)^3+4890*cos(1/2 *d*x+1/2*c)-489*sec(1/2*d*x+1/2*c))+2^(1/2)*arctanh(1/2*2^(1/2)*(cot(1/2*d *x+1/2*c)-csc(1/2*d*x+1/2*c)+1))*(15648*cos(1/2*d*x+1/2*c)^9-39120*cos(1/2 *d*x+1/2*c)^7+39120*cos(1/2*d*x+1/2*c)^5-19560*cos(1/2*d*x+1/2*c)^3+4890*c os(1/2*d*x+1/2*c)-489*sec(1/2*d*x+1/2*c)))-1/32*A/d*(a/(2*cos(1/2*d*x+1/2* c)^2-1)*cos(1/2*d*x+1/2*c)^2)^(1/2)*a^2/(2*cos(1/2*d*x+1/2*c)^2-1)^(5/2)*( (-88*cos(1/2*d*x+1/2*c)^4+80*cos(1/2*d*x+1/2*c)^2-18)*tan(1/2*d*x+1/2*c)+2 ^(1/2)*arctanh(1/2*2^(1/2)*(cot(1/2*d*x+1/2*c)-csc(1/2*d*x+1/2*c)+1))*(152 *cos(1/2*d*x+1/2*c)^5-228*cos(1/2*d*x+1/2*c)^3+114*cos(1/2*d*x+1/2*c)-19*s ec(1/2*d*x+1/2*c))+2^(1/2)*arctanh(1/2*2^(1/2)*(cot(1/2*d*x+1/2*c)-csc(1/2 *d*x+1/2*c)-1))*(152*cos(1/2*d*x+1/2*c)^5-228*cos(1/2*d*x+1/2*c)^3+114*cos (1/2*d*x+1/2*c)-19*sec(1/2*d*x+1/2*c))))
Time = 0.16 (sec) , antiderivative size = 497, normalized size of antiderivative = 2.09 \[ \int \frac {(a+a \sec (c+d x))^{5/2} \left (A+C \sec ^2(c+d x)\right )}{\sqrt {\cos (c+d x)}} \, dx=\left [\frac {4 \, {\left (3 \, {\left (176 \, A + 163 \, C\right )} a^{2} \cos \left (d x + c\right )^{3} + 2 \, {\left (48 \, A + 163 \, C\right )} a^{2} \cos \left (d x + c\right )^{2} + 184 \, C a^{2} \cos \left (d x + c\right ) + 48 \, C a^{2}\right )} \sqrt {\frac {a \cos \left (d x + c\right ) + a}{\cos \left (d x + c\right )}} \sqrt {\cos \left (d x + c\right )} \sin \left (d x + c\right ) + 3 \, {\left ({\left (304 \, A + 163 \, C\right )} a^{2} \cos \left (d x + c\right )^{5} + {\left (304 \, A + 163 \, C\right )} a^{2} \cos \left (d x + c\right )^{4}\right )} \sqrt {a} \log \left (\frac {a \cos \left (d x + c\right )^{3} - 4 \, \sqrt {a} \sqrt {\frac {a \cos \left (d x + c\right ) + a}{\cos \left (d x + c\right )}} {\left (\cos \left (d x + c\right ) - 2\right )} \sqrt {\cos \left (d x + c\right )} \sin \left (d x + c\right ) - 7 \, a \cos \left (d x + c\right )^{2} + 8 \, a}{\cos \left (d x + c\right )^{3} + \cos \left (d x + c\right )^{2}}\right )}{768 \, {\left (d \cos \left (d x + c\right )^{5} + d \cos \left (d x + c\right )^{4}\right )}}, \frac {2 \, {\left (3 \, {\left (176 \, A + 163 \, C\right )} a^{2} \cos \left (d x + c\right )^{3} + 2 \, {\left (48 \, A + 163 \, C\right )} a^{2} \cos \left (d x + c\right )^{2} + 184 \, C a^{2} \cos \left (d x + c\right ) + 48 \, C a^{2}\right )} \sqrt {\frac {a \cos \left (d x + c\right ) + a}{\cos \left (d x + c\right )}} \sqrt {\cos \left (d x + c\right )} \sin \left (d x + c\right ) + 3 \, {\left ({\left (304 \, A + 163 \, C\right )} a^{2} \cos \left (d x + c\right )^{5} + {\left (304 \, A + 163 \, C\right )} a^{2} \cos \left (d x + c\right )^{4}\right )} \sqrt {-a} \arctan \left (\frac {2 \, \sqrt {-a} \sqrt {\frac {a \cos \left (d x + c\right ) + a}{\cos \left (d x + c\right )}} \sqrt {\cos \left (d x + c\right )} \sin \left (d x + c\right )}{a \cos \left (d x + c\right )^{2} - a \cos \left (d x + c\right ) - 2 \, a}\right )}{384 \, {\left (d \cos \left (d x + c\right )^{5} + d \cos \left (d x + c\right )^{4}\right )}}\right ] \] Input:
integrate((a+a*sec(d*x+c))^(5/2)*(A+C*sec(d*x+c)^2)/cos(d*x+c)^(1/2),x, al gorithm="fricas")
Output:
[1/768*(4*(3*(176*A + 163*C)*a^2*cos(d*x + c)^3 + 2*(48*A + 163*C)*a^2*cos (d*x + c)^2 + 184*C*a^2*cos(d*x + c) + 48*C*a^2)*sqrt((a*cos(d*x + c) + a) /cos(d*x + c))*sqrt(cos(d*x + c))*sin(d*x + c) + 3*((304*A + 163*C)*a^2*co s(d*x + c)^5 + (304*A + 163*C)*a^2*cos(d*x + c)^4)*sqrt(a)*log((a*cos(d*x + c)^3 - 4*sqrt(a)*sqrt((a*cos(d*x + c) + a)/cos(d*x + c))*(cos(d*x + c) - 2)*sqrt(cos(d*x + c))*sin(d*x + c) - 7*a*cos(d*x + c)^2 + 8*a)/(cos(d*x + c)^3 + cos(d*x + c)^2)))/(d*cos(d*x + c)^5 + d*cos(d*x + c)^4), 1/384*(2* (3*(176*A + 163*C)*a^2*cos(d*x + c)^3 + 2*(48*A + 163*C)*a^2*cos(d*x + c)^ 2 + 184*C*a^2*cos(d*x + c) + 48*C*a^2)*sqrt((a*cos(d*x + c) + a)/cos(d*x + c))*sqrt(cos(d*x + c))*sin(d*x + c) + 3*((304*A + 163*C)*a^2*cos(d*x + c) ^5 + (304*A + 163*C)*a^2*cos(d*x + c)^4)*sqrt(-a)*arctan(2*sqrt(-a)*sqrt(( a*cos(d*x + c) + a)/cos(d*x + c))*sqrt(cos(d*x + c))*sin(d*x + c)/(a*cos(d *x + c)^2 - a*cos(d*x + c) - 2*a)))/(d*cos(d*x + c)^5 + d*cos(d*x + c)^4)]
Timed out. \[ \int \frac {(a+a \sec (c+d x))^{5/2} \left (A+C \sec ^2(c+d x)\right )}{\sqrt {\cos (c+d x)}} \, dx=\text {Timed out} \] Input:
integrate((a+a*sec(d*x+c))**(5/2)*(A+C*sec(d*x+c)**2)/cos(d*x+c)**(1/2),x)
Output:
Timed out
Leaf count of result is larger than twice the leaf count of optimal. 6687 vs. \(2 (202) = 404\).
Time = 3.07 (sec) , antiderivative size = 6687, normalized size of antiderivative = 28.10 \[ \int \frac {(a+a \sec (c+d x))^{5/2} \left (A+C \sec ^2(c+d x)\right )}{\sqrt {\cos (c+d x)}} \, dx=\text {Too large to display} \] Input:
integrate((a+a*sec(d*x+c))^(5/2)*(A+C*sec(d*x+c)^2)/cos(d*x+c)^(1/2),x, al gorithm="maxima")
Output:
-1/768*(48*(88*sqrt(2)*a^2*cos(7/2*d*x + 7/2*c)*sin(2*d*x + 2*c) - 56*sqrt (2)*a^2*cos(5/2*d*x + 5/2*c)*sin(2*d*x + 2*c) - 28*sqrt(2)*a^2*sin(3/2*d*x + 3/2*c) + 44*sqrt(2)*a^2*sin(1/2*d*x + 1/2*c) - 19*(a^2*log(2*cos(1/2*d* x + 1/2*c)^2 + 2*sin(1/2*d*x + 1/2*c)^2 + 2*sqrt(2)*cos(1/2*d*x + 1/2*c) + 2*sqrt(2)*sin(1/2*d*x + 1/2*c) + 2) - a^2*log(2*cos(1/2*d*x + 1/2*c)^2 + 2*sin(1/2*d*x + 1/2*c)^2 + 2*sqrt(2)*cos(1/2*d*x + 1/2*c) - 2*sqrt(2)*sin( 1/2*d*x + 1/2*c) + 2) + a^2*log(2*cos(1/2*d*x + 1/2*c)^2 + 2*sin(1/2*d*x + 1/2*c)^2 - 2*sqrt(2)*cos(1/2*d*x + 1/2*c) + 2*sqrt(2)*sin(1/2*d*x + 1/2*c ) + 2) - a^2*log(2*cos(1/2*d*x + 1/2*c)^2 + 2*sin(1/2*d*x + 1/2*c)^2 - 2*s qrt(2)*cos(1/2*d*x + 1/2*c) - 2*sqrt(2)*sin(1/2*d*x + 1/2*c) + 2))*cos(4*d *x + 4*c)^2 - 76*(a^2*log(2*cos(1/2*d*x + 1/2*c)^2 + 2*sin(1/2*d*x + 1/2*c )^2 + 2*sqrt(2)*cos(1/2*d*x + 1/2*c) + 2*sqrt(2)*sin(1/2*d*x + 1/2*c) + 2) - a^2*log(2*cos(1/2*d*x + 1/2*c)^2 + 2*sin(1/2*d*x + 1/2*c)^2 + 2*sqrt(2) *cos(1/2*d*x + 1/2*c) - 2*sqrt(2)*sin(1/2*d*x + 1/2*c) + 2) + a^2*log(2*co s(1/2*d*x + 1/2*c)^2 + 2*sin(1/2*d*x + 1/2*c)^2 - 2*sqrt(2)*cos(1/2*d*x + 1/2*c) + 2*sqrt(2)*sin(1/2*d*x + 1/2*c) + 2) - a^2*log(2*cos(1/2*d*x + 1/2 *c)^2 + 2*sin(1/2*d*x + 1/2*c)^2 - 2*sqrt(2)*cos(1/2*d*x + 1/2*c) - 2*sqrt (2)*sin(1/2*d*x + 1/2*c) + 2))*cos(2*d*x + 2*c)^2 - 19*a^2*log(2*cos(1/2*d *x + 1/2*c)^2 + 2*sin(1/2*d*x + 1/2*c)^2 + 2*sqrt(2)*cos(1/2*d*x + 1/2*c) + 2*sqrt(2)*sin(1/2*d*x + 1/2*c) + 2) + 19*a^2*log(2*cos(1/2*d*x + 1/2*...
Leaf count of result is larger than twice the leaf count of optimal. 942 vs. \(2 (202) = 404\).
Time = 0.97 (sec) , antiderivative size = 942, normalized size of antiderivative = 3.96 \[ \int \frac {(a+a \sec (c+d x))^{5/2} \left (A+C \sec ^2(c+d x)\right )}{\sqrt {\cos (c+d x)}} \, dx=\text {Too large to display} \] Input:
integrate((a+a*sec(d*x+c))^(5/2)*(A+C*sec(d*x+c)^2)/cos(d*x+c)^(1/2),x, al gorithm="giac")
Output:
1/384*(3*(304*A*a^(5/2)*sgn(cos(d*x + c)) + 163*C*a^(5/2)*sgn(cos(d*x + c) ))*log(abs((sqrt(a)*tan(1/2*d*x + 1/2*c) - sqrt(a*tan(1/2*d*x + 1/2*c)^2 + a))^2 - a*(2*sqrt(2) + 3))) - 3*(304*A*a^(5/2)*sgn(cos(d*x + c)) + 163*C* a^(5/2)*sgn(cos(d*x + c)))*log(abs((sqrt(a)*tan(1/2*d*x + 1/2*c) - sqrt(a* tan(1/2*d*x + 1/2*c)^2 + a))^2 + a*(2*sqrt(2) - 3))) + 4*sqrt(2)*(912*(sqr t(a)*tan(1/2*d*x + 1/2*c) - sqrt(a*tan(1/2*d*x + 1/2*c)^2 + a))^14*A*a^(7/ 2)*sgn(cos(d*x + c)) + 489*(sqrt(a)*tan(1/2*d*x + 1/2*c) - sqrt(a*tan(1/2* d*x + 1/2*c)^2 + a))^14*C*a^(7/2)*sgn(cos(d*x + c)) - 19152*(sqrt(a)*tan(1 /2*d*x + 1/2*c) - sqrt(a*tan(1/2*d*x + 1/2*c)^2 + a))^12*A*a^(9/2)*sgn(cos (d*x + c)) - 10269*(sqrt(a)*tan(1/2*d*x + 1/2*c) - sqrt(a*tan(1/2*d*x + 1/ 2*c)^2 + a))^12*C*a^(9/2)*sgn(cos(d*x + c)) + 137424*(sqrt(a)*tan(1/2*d*x + 1/2*c) - sqrt(a*tan(1/2*d*x + 1/2*c)^2 + a))^10*A*a^(11/2)*sgn(cos(d*x + c)) + 69885*(sqrt(a)*tan(1/2*d*x + 1/2*c) - sqrt(a*tan(1/2*d*x + 1/2*c)^2 + a))^10*C*a^(11/2)*sgn(cos(d*x + c)) - 374544*(sqrt(a)*tan(1/2*d*x + 1/2 *c) - sqrt(a*tan(1/2*d*x + 1/2*c)^2 + a))^8*A*a^(13/2)*sgn(cos(d*x + c)) - 259233*(sqrt(a)*tan(1/2*d*x + 1/2*c) - sqrt(a*tan(1/2*d*x + 1/2*c)^2 + a) )^8*C*a^(13/2)*sgn(cos(d*x + c)) + 266928*(sqrt(a)*tan(1/2*d*x + 1/2*c) - sqrt(a*tan(1/2*d*x + 1/2*c)^2 + a))^6*A*a^(15/2)*sgn(cos(d*x + c)) + 20997 9*(sqrt(a)*tan(1/2*d*x + 1/2*c) - sqrt(a*tan(1/2*d*x + 1/2*c)^2 + a))^6*C* a^(15/2)*sgn(cos(d*x + c)) - 75888*(sqrt(a)*tan(1/2*d*x + 1/2*c) - sqrt...
Timed out. \[ \int \frac {(a+a \sec (c+d x))^{5/2} \left (A+C \sec ^2(c+d x)\right )}{\sqrt {\cos (c+d x)}} \, dx=\int \frac {\left (A+\frac {C}{{\cos \left (c+d\,x\right )}^2}\right )\,{\left (a+\frac {a}{\cos \left (c+d\,x\right )}\right )}^{5/2}}{\sqrt {\cos \left (c+d\,x\right )}} \,d x \] Input:
int(((A + C/cos(c + d*x)^2)*(a + a/cos(c + d*x))^(5/2))/cos(c + d*x)^(1/2) ,x)
Output:
int(((A + C/cos(c + d*x)^2)*(a + a/cos(c + d*x))^(5/2))/cos(c + d*x)^(1/2) , x)
\[ \int \frac {(a+a \sec (c+d x))^{5/2} \left (A+C \sec ^2(c+d x)\right )}{\sqrt {\cos (c+d x)}} \, dx=\sqrt {a}\, a^{2} \left (\left (\int \frac {\sqrt {\sec \left (d x +c \right )+1}\, \sqrt {\cos \left (d x +c \right )}\, \sec \left (d x +c \right )^{4}}{\cos \left (d x +c \right )}d x \right ) c +2 \left (\int \frac {\sqrt {\sec \left (d x +c \right )+1}\, \sqrt {\cos \left (d x +c \right )}\, \sec \left (d x +c \right )^{3}}{\cos \left (d x +c \right )}d x \right ) c +\left (\int \frac {\sqrt {\sec \left (d x +c \right )+1}\, \sqrt {\cos \left (d x +c \right )}\, \sec \left (d x +c \right )^{2}}{\cos \left (d x +c \right )}d x \right ) a +\left (\int \frac {\sqrt {\sec \left (d x +c \right )+1}\, \sqrt {\cos \left (d x +c \right )}\, \sec \left (d x +c \right )^{2}}{\cos \left (d x +c \right )}d x \right ) c +2 \left (\int \frac {\sqrt {\sec \left (d x +c \right )+1}\, \sqrt {\cos \left (d x +c \right )}\, \sec \left (d x +c \right )}{\cos \left (d x +c \right )}d x \right ) a +\left (\int \frac {\sqrt {\sec \left (d x +c \right )+1}\, \sqrt {\cos \left (d x +c \right )}}{\cos \left (d x +c \right )}d x \right ) a \right ) \] Input:
int((a+a*sec(d*x+c))^(5/2)*(A+C*sec(d*x+c)^2)/cos(d*x+c)^(1/2),x)
Output:
sqrt(a)*a**2*(int((sqrt(sec(c + d*x) + 1)*sqrt(cos(c + d*x))*sec(c + d*x)* *4)/cos(c + d*x),x)*c + 2*int((sqrt(sec(c + d*x) + 1)*sqrt(cos(c + d*x))*s ec(c + d*x)**3)/cos(c + d*x),x)*c + int((sqrt(sec(c + d*x) + 1)*sqrt(cos(c + d*x))*sec(c + d*x)**2)/cos(c + d*x),x)*a + int((sqrt(sec(c + d*x) + 1)* sqrt(cos(c + d*x))*sec(c + d*x)**2)/cos(c + d*x),x)*c + 2*int((sqrt(sec(c + d*x) + 1)*sqrt(cos(c + d*x))*sec(c + d*x))/cos(c + d*x),x)*a + int((sqrt (sec(c + d*x) + 1)*sqrt(cos(c + d*x)))/cos(c + d*x),x)*a)