Integrand size = 43, antiderivative size = 259 \[ \int \frac {\cos ^4(c+d x) \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right )}{\sqrt {a+a \sec (c+d x)}} \, dx=\frac {(107 A-72 B+112 C) \arctan \left (\frac {\sqrt {a} \tan (c+d x)}{\sqrt {a+a \sec (c+d x)}}\right )}{64 \sqrt {a} d}-\frac {\sqrt {2} (A-B+C) \arctan \left (\frac {\sqrt {a} \tan (c+d x)}{\sqrt {2} \sqrt {a+a \sec (c+d x)}}\right )}{\sqrt {a} d}-\frac {(21 A-56 B+16 C) \sin (c+d x)}{64 d \sqrt {a+a \sec (c+d x)}}+\frac {(43 A-8 B+48 C) \cos (c+d x) \sin (c+d x)}{96 d \sqrt {a+a \sec (c+d x)}}-\frac {(A-8 B) \cos ^2(c+d x) \sin (c+d x)}{24 d \sqrt {a+a \sec (c+d x)}}+\frac {A \cos ^3(c+d x) \sin (c+d x)}{4 d \sqrt {a+a \sec (c+d x)}} \]
1/64*(107*A-72*B+112*C)*arctan(a^(1/2)*tan(d*x+c)/(a+a*sec(d*x+c))^(1/2))/ d/a^(1/2)-(A-B+C)*arctan(1/2*a^(1/2)*tan(d*x+c)*2^(1/2)/(a+a*sec(d*x+c))^( 1/2))*2^(1/2)/d/a^(1/2)-1/64*(21*A-56*B+16*C)*sin(d*x+c)/d/(a+a*sec(d*x+c) )^(1/2)+1/96*(43*A-8*B+48*C)*cos(d*x+c)*sin(d*x+c)/d/(a+a*sec(d*x+c))^(1/2 )-1/24*(A-8*B)*cos(d*x+c)^2*sin(d*x+c)/d/(a+a*sec(d*x+c))^(1/2)+1/4*A*cos( d*x+c)^3*sin(d*x+c)/d/(a+a*sec(d*x+c))^(1/2)
Time = 0.70 (sec) , antiderivative size = 178, normalized size of antiderivative = 0.69 \[ \int \frac {\cos ^4(c+d x) \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right )}{\sqrt {a+a \sec (c+d x)}} \, dx=\frac {\left (3 (107 A-72 B+112 C) \text {arctanh}\left (\sqrt {1-\sec (c+d x)}\right )-192 \sqrt {2} (A-B+C) \text {arctanh}\left (\frac {\sqrt {1-\sec (c+d x)}}{\sqrt {2}}\right )+\cos (c+d x) \left (-63 A+168 B-48 C+2 (43 A-8 B+48 C) \cos (c+d x)-8 (A-8 B) \cos ^2(c+d x)+48 A \cos ^3(c+d x)\right ) \sqrt {1-\sec (c+d x)}\right ) \tan (c+d x)}{192 d \sqrt {1-\sec (c+d x)} \sqrt {a (1+\sec (c+d x))}} \]
Integrate[(Cos[c + d*x]^4*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/Sqrt[a + a*Sec[c + d*x]],x]
((3*(107*A - 72*B + 112*C)*ArcTanh[Sqrt[1 - Sec[c + d*x]]] - 192*Sqrt[2]*( A - B + C)*ArcTanh[Sqrt[1 - Sec[c + d*x]]/Sqrt[2]] + Cos[c + d*x]*(-63*A + 168*B - 48*C + 2*(43*A - 8*B + 48*C)*Cos[c + d*x] - 8*(A - 8*B)*Cos[c + d *x]^2 + 48*A*Cos[c + d*x]^3)*Sqrt[1 - Sec[c + d*x]])*Tan[c + d*x])/(192*d* Sqrt[1 - Sec[c + d*x]]*Sqrt[a*(1 + Sec[c + d*x])])
Time = 1.80 (sec) , antiderivative size = 293, normalized size of antiderivative = 1.13, number of steps used = 20, number of rules used = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.442, Rules used = {3042, 4574, 27, 3042, 4510, 27, 3042, 4510, 27, 3042, 4510, 27, 3042, 4408, 3042, 4261, 216, 4282, 216}
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.
\(\displaystyle \int \frac {\cos ^4(c+d x) \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right )}{\sqrt {a \sec (c+d x)+a}} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {A+B \csc \left (c+d x+\frac {\pi }{2}\right )+C \csc \left (c+d x+\frac {\pi }{2}\right )^2}{\csc \left (c+d x+\frac {\pi }{2}\right )^4 \sqrt {a \csc \left (c+d x+\frac {\pi }{2}\right )+a}}dx\) |
\(\Big \downarrow \) 4574 |
\(\displaystyle \frac {\int -\frac {\cos ^3(c+d x) (a (A-8 B)-a (7 A+8 C) \sec (c+d x))}{2 \sqrt {\sec (c+d x) a+a}}dx}{4 a}+\frac {A \sin (c+d x) \cos ^3(c+d x)}{4 d \sqrt {a \sec (c+d x)+a}}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {A \sin (c+d x) \cos ^3(c+d x)}{4 d \sqrt {a \sec (c+d x)+a}}-\frac {\int \frac {\cos ^3(c+d x) (a (A-8 B)-a (7 A+8 C) \sec (c+d x))}{\sqrt {\sec (c+d x) a+a}}dx}{8 a}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {A \sin (c+d x) \cos ^3(c+d x)}{4 d \sqrt {a \sec (c+d x)+a}}-\frac {\int \frac {a (A-8 B)-a (7 A+8 C) \csc \left (c+d x+\frac {\pi }{2}\right )}{\csc \left (c+d x+\frac {\pi }{2}\right )^3 \sqrt {\csc \left (c+d x+\frac {\pi }{2}\right ) a+a}}dx}{8 a}\) |
\(\Big \downarrow \) 4510 |
\(\displaystyle \frac {A \sin (c+d x) \cos ^3(c+d x)}{4 d \sqrt {a \sec (c+d x)+a}}-\frac {\frac {\int -\frac {\cos ^2(c+d x) \left (a^2 (43 A-8 B+48 C)-5 a^2 (A-8 B) \sec (c+d x)\right )}{2 \sqrt {\sec (c+d x) a+a}}dx}{3 a}+\frac {a (A-8 B) \sin (c+d x) \cos ^2(c+d x)}{3 d \sqrt {a \sec (c+d x)+a}}}{8 a}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {A \sin (c+d x) \cos ^3(c+d x)}{4 d \sqrt {a \sec (c+d x)+a}}-\frac {\frac {a (A-8 B) \sin (c+d x) \cos ^2(c+d x)}{3 d \sqrt {a \sec (c+d x)+a}}-\frac {\int \frac {\cos ^2(c+d x) \left (a^2 (43 A-8 B+48 C)-5 a^2 (A-8 B) \sec (c+d x)\right )}{\sqrt {\sec (c+d x) a+a}}dx}{6 a}}{8 a}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {A \sin (c+d x) \cos ^3(c+d x)}{4 d \sqrt {a \sec (c+d x)+a}}-\frac {\frac {a (A-8 B) \sin (c+d x) \cos ^2(c+d x)}{3 d \sqrt {a \sec (c+d x)+a}}-\frac {\int \frac {a^2 (43 A-8 B+48 C)-5 a^2 (A-8 B) \csc \left (c+d x+\frac {\pi }{2}\right )}{\csc \left (c+d x+\frac {\pi }{2}\right )^2 \sqrt {\csc \left (c+d x+\frac {\pi }{2}\right ) a+a}}dx}{6 a}}{8 a}\) |
\(\Big \downarrow \) 4510 |
\(\displaystyle \frac {A \sin (c+d x) \cos ^3(c+d x)}{4 d \sqrt {a \sec (c+d x)+a}}-\frac {\frac {a (A-8 B) \sin (c+d x) \cos ^2(c+d x)}{3 d \sqrt {a \sec (c+d x)+a}}-\frac {\frac {\int -\frac {3 \cos (c+d x) \left (a^3 (21 A-56 B+16 C)-a^3 (43 A-8 B+48 C) \sec (c+d x)\right )}{2 \sqrt {\sec (c+d x) a+a}}dx}{2 a}+\frac {a^2 (43 A-8 B+48 C) \sin (c+d x) \cos (c+d x)}{2 d \sqrt {a \sec (c+d x)+a}}}{6 a}}{8 a}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {A \sin (c+d x) \cos ^3(c+d x)}{4 d \sqrt {a \sec (c+d x)+a}}-\frac {\frac {a (A-8 B) \sin (c+d x) \cos ^2(c+d x)}{3 d \sqrt {a \sec (c+d x)+a}}-\frac {\frac {a^2 (43 A-8 B+48 C) \sin (c+d x) \cos (c+d x)}{2 d \sqrt {a \sec (c+d x)+a}}-\frac {3 \int \frac {\cos (c+d x) \left (a^3 (21 A-56 B+16 C)-a^3 (43 A-8 B+48 C) \sec (c+d x)\right )}{\sqrt {\sec (c+d x) a+a}}dx}{4 a}}{6 a}}{8 a}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {A \sin (c+d x) \cos ^3(c+d x)}{4 d \sqrt {a \sec (c+d x)+a}}-\frac {\frac {a (A-8 B) \sin (c+d x) \cos ^2(c+d x)}{3 d \sqrt {a \sec (c+d x)+a}}-\frac {\frac {a^2 (43 A-8 B+48 C) \sin (c+d x) \cos (c+d x)}{2 d \sqrt {a \sec (c+d x)+a}}-\frac {3 \int \frac {a^3 (21 A-56 B+16 C)-a^3 (43 A-8 B+48 C) \csc \left (c+d x+\frac {\pi }{2}\right )}{\csc \left (c+d x+\frac {\pi }{2}\right ) \sqrt {\csc \left (c+d x+\frac {\pi }{2}\right ) a+a}}dx}{4 a}}{6 a}}{8 a}\) |
\(\Big \downarrow \) 4510 |
\(\displaystyle \frac {A \sin (c+d x) \cos ^3(c+d x)}{4 d \sqrt {a \sec (c+d x)+a}}-\frac {\frac {a (A-8 B) \sin (c+d x) \cos ^2(c+d x)}{3 d \sqrt {a \sec (c+d x)+a}}-\frac {\frac {a^2 (43 A-8 B+48 C) \sin (c+d x) \cos (c+d x)}{2 d \sqrt {a \sec (c+d x)+a}}-\frac {3 \left (\frac {\int -\frac {a^4 (107 A-72 B+112 C)-a^4 (21 A-56 B+16 C) \sec (c+d x)}{2 \sqrt {\sec (c+d x) a+a}}dx}{a}+\frac {a^3 (21 A-56 B+16 C) \sin (c+d x)}{d \sqrt {a \sec (c+d x)+a}}\right )}{4 a}}{6 a}}{8 a}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {A \sin (c+d x) \cos ^3(c+d x)}{4 d \sqrt {a \sec (c+d x)+a}}-\frac {\frac {a (A-8 B) \sin (c+d x) \cos ^2(c+d x)}{3 d \sqrt {a \sec (c+d x)+a}}-\frac {\frac {a^2 (43 A-8 B+48 C) \sin (c+d x) \cos (c+d x)}{2 d \sqrt {a \sec (c+d x)+a}}-\frac {3 \left (\frac {a^3 (21 A-56 B+16 C) \sin (c+d x)}{d \sqrt {a \sec (c+d x)+a}}-\frac {\int \frac {a^4 (107 A-72 B+112 C)-a^4 (21 A-56 B+16 C) \sec (c+d x)}{\sqrt {\sec (c+d x) a+a}}dx}{2 a}\right )}{4 a}}{6 a}}{8 a}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {A \sin (c+d x) \cos ^3(c+d x)}{4 d \sqrt {a \sec (c+d x)+a}}-\frac {\frac {a (A-8 B) \sin (c+d x) \cos ^2(c+d x)}{3 d \sqrt {a \sec (c+d x)+a}}-\frac {\frac {a^2 (43 A-8 B+48 C) \sin (c+d x) \cos (c+d x)}{2 d \sqrt {a \sec (c+d x)+a}}-\frac {3 \left (\frac {a^3 (21 A-56 B+16 C) \sin (c+d x)}{d \sqrt {a \sec (c+d x)+a}}-\frac {\int \frac {a^4 (107 A-72 B+112 C)-a^4 (21 A-56 B+16 C) \csc \left (c+d x+\frac {\pi }{2}\right )}{\sqrt {\csc \left (c+d x+\frac {\pi }{2}\right ) a+a}}dx}{2 a}\right )}{4 a}}{6 a}}{8 a}\) |
\(\Big \downarrow \) 4408 |
\(\displaystyle \frac {A \sin (c+d x) \cos ^3(c+d x)}{4 d \sqrt {a \sec (c+d x)+a}}-\frac {\frac {a (A-8 B) \sin (c+d x) \cos ^2(c+d x)}{3 d \sqrt {a \sec (c+d x)+a}}-\frac {\frac {a^2 (43 A-8 B+48 C) \sin (c+d x) \cos (c+d x)}{2 d \sqrt {a \sec (c+d x)+a}}-\frac {3 \left (\frac {a^3 (21 A-56 B+16 C) \sin (c+d x)}{d \sqrt {a \sec (c+d x)+a}}-\frac {a^3 (107 A-72 B+112 C) \int \sqrt {\sec (c+d x) a+a}dx-128 a^4 (A-B+C) \int \frac {\sec (c+d x)}{\sqrt {\sec (c+d x) a+a}}dx}{2 a}\right )}{4 a}}{6 a}}{8 a}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {A \sin (c+d x) \cos ^3(c+d x)}{4 d \sqrt {a \sec (c+d x)+a}}-\frac {\frac {a (A-8 B) \sin (c+d x) \cos ^2(c+d x)}{3 d \sqrt {a \sec (c+d x)+a}}-\frac {\frac {a^2 (43 A-8 B+48 C) \sin (c+d x) \cos (c+d x)}{2 d \sqrt {a \sec (c+d x)+a}}-\frac {3 \left (\frac {a^3 (21 A-56 B+16 C) \sin (c+d x)}{d \sqrt {a \sec (c+d x)+a}}-\frac {a^3 (107 A-72 B+112 C) \int \sqrt {\csc \left (c+d x+\frac {\pi }{2}\right ) a+a}dx-128 a^4 (A-B+C) \int \frac {\csc \left (c+d x+\frac {\pi }{2}\right )}{\sqrt {\csc \left (c+d x+\frac {\pi }{2}\right ) a+a}}dx}{2 a}\right )}{4 a}}{6 a}}{8 a}\) |
\(\Big \downarrow \) 4261 |
\(\displaystyle \frac {A \sin (c+d x) \cos ^3(c+d x)}{4 d \sqrt {a \sec (c+d x)+a}}-\frac {\frac {a (A-8 B) \sin (c+d x) \cos ^2(c+d x)}{3 d \sqrt {a \sec (c+d x)+a}}-\frac {\frac {a^2 (43 A-8 B+48 C) \sin (c+d x) \cos (c+d x)}{2 d \sqrt {a \sec (c+d x)+a}}-\frac {3 \left (\frac {a^3 (21 A-56 B+16 C) \sin (c+d x)}{d \sqrt {a \sec (c+d x)+a}}-\frac {-128 a^4 (A-B+C) \int \frac {\csc \left (c+d x+\frac {\pi }{2}\right )}{\sqrt {\csc \left (c+d x+\frac {\pi }{2}\right ) a+a}}dx-\frac {2 a^4 (107 A-72 B+112 C) \int \frac {1}{\frac {a^2 \tan ^2(c+d x)}{\sec (c+d x) a+a}+a}d\left (-\frac {a \tan (c+d x)}{\sqrt {\sec (c+d x) a+a}}\right )}{d}}{2 a}\right )}{4 a}}{6 a}}{8 a}\) |
\(\Big \downarrow \) 216 |
\(\displaystyle \frac {A \sin (c+d x) \cos ^3(c+d x)}{4 d \sqrt {a \sec (c+d x)+a}}-\frac {\frac {a (A-8 B) \sin (c+d x) \cos ^2(c+d x)}{3 d \sqrt {a \sec (c+d x)+a}}-\frac {\frac {a^2 (43 A-8 B+48 C) \sin (c+d x) \cos (c+d x)}{2 d \sqrt {a \sec (c+d x)+a}}-\frac {3 \left (\frac {a^3 (21 A-56 B+16 C) \sin (c+d x)}{d \sqrt {a \sec (c+d x)+a}}-\frac {\frac {2 a^{7/2} (107 A-72 B+112 C) \arctan \left (\frac {\sqrt {a} \tan (c+d x)}{\sqrt {a \sec (c+d x)+a}}\right )}{d}-128 a^4 (A-B+C) \int \frac {\csc \left (c+d x+\frac {\pi }{2}\right )}{\sqrt {\csc \left (c+d x+\frac {\pi }{2}\right ) a+a}}dx}{2 a}\right )}{4 a}}{6 a}}{8 a}\) |
\(\Big \downarrow \) 4282 |
\(\displaystyle \frac {A \sin (c+d x) \cos ^3(c+d x)}{4 d \sqrt {a \sec (c+d x)+a}}-\frac {\frac {a (A-8 B) \sin (c+d x) \cos ^2(c+d x)}{3 d \sqrt {a \sec (c+d x)+a}}-\frac {\frac {a^2 (43 A-8 B+48 C) \sin (c+d x) \cos (c+d x)}{2 d \sqrt {a \sec (c+d x)+a}}-\frac {3 \left (\frac {a^3 (21 A-56 B+16 C) \sin (c+d x)}{d \sqrt {a \sec (c+d x)+a}}-\frac {\frac {256 a^4 (A-B+C) \int \frac {1}{\frac {a^2 \tan ^2(c+d x)}{\sec (c+d x) a+a}+2 a}d\left (-\frac {a \tan (c+d x)}{\sqrt {\sec (c+d x) a+a}}\right )}{d}+\frac {2 a^{7/2} (107 A-72 B+112 C) \arctan \left (\frac {\sqrt {a} \tan (c+d x)}{\sqrt {a \sec (c+d x)+a}}\right )}{d}}{2 a}\right )}{4 a}}{6 a}}{8 a}\) |
\(\Big \downarrow \) 216 |
\(\displaystyle \frac {A \sin (c+d x) \cos ^3(c+d x)}{4 d \sqrt {a \sec (c+d x)+a}}-\frac {\frac {a (A-8 B) \sin (c+d x) \cos ^2(c+d x)}{3 d \sqrt {a \sec (c+d x)+a}}-\frac {\frac {a^2 (43 A-8 B+48 C) \sin (c+d x) \cos (c+d x)}{2 d \sqrt {a \sec (c+d x)+a}}-\frac {3 \left (\frac {a^3 (21 A-56 B+16 C) \sin (c+d x)}{d \sqrt {a \sec (c+d x)+a}}-\frac {\frac {2 a^{7/2} (107 A-72 B+112 C) \arctan \left (\frac {\sqrt {a} \tan (c+d x)}{\sqrt {a \sec (c+d x)+a}}\right )}{d}-\frac {128 \sqrt {2} a^{7/2} (A-B+C) \arctan \left (\frac {\sqrt {a} \tan (c+d x)}{\sqrt {2} \sqrt {a \sec (c+d x)+a}}\right )}{d}}{2 a}\right )}{4 a}}{6 a}}{8 a}\) |
(A*Cos[c + d*x]^3*Sin[c + d*x])/(4*d*Sqrt[a + a*Sec[c + d*x]]) - ((a*(A - 8*B)*Cos[c + d*x]^2*Sin[c + d*x])/(3*d*Sqrt[a + a*Sec[c + d*x]]) - ((a^2*( 43*A - 8*B + 48*C)*Cos[c + d*x]*Sin[c + d*x])/(2*d*Sqrt[a + a*Sec[c + d*x] ]) - (3*(-1/2*((2*a^(7/2)*(107*A - 72*B + 112*C)*ArcTan[(Sqrt[a]*Tan[c + d *x])/Sqrt[a + a*Sec[c + d*x]]])/d - (128*Sqrt[2]*a^(7/2)*(A - B + C)*ArcTa n[(Sqrt[a]*Tan[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sec[c + d*x]])])/d)/a + (a^3* (21*A - 56*B + 16*C)*Sin[c + d*x])/(d*Sqrt[a + a*Sec[c + d*x]])))/(4*a))/( 6*a))/(8*a)
3.6.18.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[b, 2]))*A rcTan[Rt[b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a , 0] || GtQ[b, 0])
Int[Sqrt[csc[(c_.) + (d_.)*(x_)]*(b_.) + (a_)], x_Symbol] :> Simp[-2*(b/d) Subst[Int[1/(a + x^2), x], x, b*(Cot[c + d*x]/Sqrt[a + b*Csc[c + d*x]])], x] /; FreeQ[{a, b, c, d}, x] && EqQ[a^2 - b^2, 0]
Int[csc[(e_.) + (f_.)*(x_)]/Sqrt[csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_)], x_S ymbol] :> Simp[-2/f Subst[Int[1/(2*a + x^2), x], x, b*(Cot[e + f*x]/Sqrt[ a + b*Csc[e + f*x]])], x] /; FreeQ[{a, b, e, f}, x] && EqQ[a^2 - b^2, 0]
Int[(csc[(e_.) + (f_.)*(x_)]*(d_.) + (c_))/Sqrt[csc[(e_.) + (f_.)*(x_)]*(b_ .) + (a_)], x_Symbol] :> Simp[c/a Int[Sqrt[a + b*Csc[e + f*x]], x], x] - Simp[(b*c - a*d)/a Int[Csc[e + f*x]/Sqrt[a + b*Csc[e + f*x]], x], x] /; F reeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 - b^2, 0]
Int[(csc[(e_.) + (f_.)*(x_)]*(d_.))^(n_)*(csc[(e_.) + (f_.)*(x_)]*(b_.) + ( a_))^(m_)*(csc[(e_.) + (f_.)*(x_)]*(B_.) + (A_)), x_Symbol] :> Simp[A*Cot[e + f*x]*(a + b*Csc[e + f*x])^m*((d*Csc[e + f*x])^n/(f*n)), x] - Simp[1/(b*d *n) Int[(a + b*Csc[e + f*x])^m*(d*Csc[e + f*x])^(n + 1)*Simp[a*A*m - b*B* n - A*b*(m + n + 1)*Csc[e + f*x], x], x], x] /; FreeQ[{a, b, d, e, f, A, B, m}, x] && NeQ[A*b - a*B, 0] && EqQ[a^2 - b^2, 0] && LtQ[n, 0]
Int[((A_.) + csc[(e_.) + (f_.)*(x_)]*(B_.) + csc[(e_.) + (f_.)*(x_)]^2*(C_. ))*(csc[(e_.) + (f_.)*(x_)]*(d_.))^(n_)*(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a _))^(m_), x_Symbol] :> Simp[A*Cot[e + f*x]*(a + b*Csc[e + f*x])^m*((d*Csc[e + f*x])^n/(f*n)), x] - Simp[1/(b*d*n) Int[(a + b*Csc[e + f*x])^m*(d*Csc[ e + f*x])^(n + 1)*Simp[a*A*m - b*B*n - b*(A*(m + n + 1) + C*n)*Csc[e + f*x] , x], x], x] /; FreeQ[{a, b, d, e, f, A, B, C, m}, x] && EqQ[a^2 - b^2, 0] && !LtQ[m, -2^(-1)] && (LtQ[n, -2^(-1)] || EqQ[m + n + 1, 0])
Leaf count of result is larger than twice the leaf count of optimal. \(1036\) vs. \(2(226)=452\).
Time = 2.44 (sec) , antiderivative size = 1037, normalized size of antiderivative = 4.00
int(cos(d*x+c)^4*(A+B*sec(d*x+c)+C*sec(d*x+c)^2)/(a+a*sec(d*x+c))^(1/2),x, method=_RETURNVERBOSE)
1/192/d/a*(48*A*cos(d*x+c)^4*sin(d*x+c)-8*A*cos(d*x+c)^3*sin(d*x+c)-192*A* 2^(1/2)*ln((cot(d*x+c)^2-2*cot(d*x+c)*csc(d*x+c)+csc(d*x+c)^2-1)^(1/2)-cot (d*x+c)+csc(d*x+c))*(-cos(d*x+c)/(cos(d*x+c)+1))^(1/2)*cos(d*x+c)+64*B*sin (d*x+c)*cos(d*x+c)^3+192*B*2^(1/2)*(-cos(d*x+c)/(cos(d*x+c)+1))^(1/2)*ln(( cot(d*x+c)^2-2*cot(d*x+c)*csc(d*x+c)+csc(d*x+c)^2-1)^(1/2)-cot(d*x+c)+csc( d*x+c))*cos(d*x+c)-192*C*2^(1/2)*ln((cot(d*x+c)^2-2*cot(d*x+c)*csc(d*x+c)+ csc(d*x+c)^2-1)^(1/2)-cot(d*x+c)+csc(d*x+c))*(-cos(d*x+c)/(cos(d*x+c)+1))^ (1/2)*cos(d*x+c)+86*A*cos(d*x+c)^2*sin(d*x+c)+321*A*(-cos(d*x+c)/(cos(d*x+ c)+1))^(1/2)*arctanh(sin(d*x+c)/(cos(d*x+c)+1)/(-cos(d*x+c)/(cos(d*x+c)+1) )^(1/2))*cos(d*x+c)-192*A*2^(1/2)*(-cos(d*x+c)/(cos(d*x+c)+1))^(1/2)*ln((c ot(d*x+c)^2-2*cot(d*x+c)*csc(d*x+c)+csc(d*x+c)^2-1)^(1/2)-cot(d*x+c)+csc(d *x+c))-16*B*sin(d*x+c)*cos(d*x+c)^2-216*B*arctanh(sin(d*x+c)/(cos(d*x+c)+1 )/(-cos(d*x+c)/(cos(d*x+c)+1))^(1/2))*(-cos(d*x+c)/(cos(d*x+c)+1))^(1/2)*c os(d*x+c)+192*B*2^(1/2)*ln((cot(d*x+c)^2-2*cot(d*x+c)*csc(d*x+c)+csc(d*x+c )^2-1)^(1/2)-cot(d*x+c)+csc(d*x+c))*(-cos(d*x+c)/(cos(d*x+c)+1))^(1/2)+96* C*cos(d*x+c)^2*sin(d*x+c)+336*C*(-cos(d*x+c)/(cos(d*x+c)+1))^(1/2)*arctanh (sin(d*x+c)/(cos(d*x+c)+1)/(-cos(d*x+c)/(cos(d*x+c)+1))^(1/2))*cos(d*x+c)- 192*C*2^(1/2)*(-cos(d*x+c)/(cos(d*x+c)+1))^(1/2)*ln((cot(d*x+c)^2-2*cot(d* x+c)*csc(d*x+c)+csc(d*x+c)^2-1)^(1/2)-cot(d*x+c)+csc(d*x+c))-63*A*cos(d*x+ c)*sin(d*x+c)+321*A*(-cos(d*x+c)/(cos(d*x+c)+1))^(1/2)*arctanh(sin(d*x+...
Time = 13.28 (sec) , antiderivative size = 602, normalized size of antiderivative = 2.32 \[ \int \frac {\cos ^4(c+d x) \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right )}{\sqrt {a+a \sec (c+d x)}} \, dx=\left [\frac {192 \, \sqrt {2} {\left ({\left (A - B + C\right )} a \cos \left (d x + c\right ) + {\left (A - B + C\right )} a\right )} \sqrt {-\frac {1}{a}} \log \left (\frac {2 \, \sqrt {2} \sqrt {\frac {a \cos \left (d x + c\right ) + a}{\cos \left (d x + c\right )}} \sqrt {-\frac {1}{a}} \cos \left (d x + c\right ) \sin \left (d x + c\right ) + 3 \, \cos \left (d x + c\right )^{2} + 2 \, \cos \left (d x + c\right ) - 1}{\cos \left (d x + c\right )^{2} + 2 \, \cos \left (d x + c\right ) + 1}\right ) - 3 \, {\left ({\left (107 \, A - 72 \, B + 112 \, C\right )} \cos \left (d x + c\right ) + 107 \, A - 72 \, B + 112 \, C\right )} \sqrt {-a} \log \left (\frac {2 \, a \cos \left (d x + c\right )^{2} + 2 \, \sqrt {-a} \sqrt {\frac {a \cos \left (d x + c\right ) + a}{\cos \left (d x + c\right )}} \cos \left (d x + c\right ) \sin \left (d x + c\right ) + a \cos \left (d x + c\right ) - a}{\cos \left (d x + c\right ) + 1}\right ) + 2 \, {\left (48 \, A \cos \left (d x + c\right )^{4} - 8 \, {\left (A - 8 \, B\right )} \cos \left (d x + c\right )^{3} + 2 \, {\left (43 \, A - 8 \, B + 48 \, C\right )} \cos \left (d x + c\right )^{2} - 3 \, {\left (21 \, A - 56 \, B + 16 \, C\right )} \cos \left (d x + c\right )\right )} \sqrt {\frac {a \cos \left (d x + c\right ) + a}{\cos \left (d x + c\right )}} \sin \left (d x + c\right )}{384 \, {\left (a d \cos \left (d x + c\right ) + a d\right )}}, -\frac {3 \, {\left ({\left (107 \, A - 72 \, B + 112 \, C\right )} \cos \left (d x + c\right ) + 107 \, A - 72 \, B + 112 \, C\right )} \sqrt {a} \arctan \left (\frac {\sqrt {\frac {a \cos \left (d x + c\right ) + a}{\cos \left (d x + c\right )}} \cos \left (d x + c\right )}{\sqrt {a} \sin \left (d x + c\right )}\right ) - {\left (48 \, A \cos \left (d x + c\right )^{4} - 8 \, {\left (A - 8 \, B\right )} \cos \left (d x + c\right )^{3} + 2 \, {\left (43 \, A - 8 \, B + 48 \, C\right )} \cos \left (d x + c\right )^{2} - 3 \, {\left (21 \, A - 56 \, B + 16 \, C\right )} \cos \left (d x + c\right )\right )} \sqrt {\frac {a \cos \left (d x + c\right ) + a}{\cos \left (d x + c\right )}} \sin \left (d x + c\right ) - \frac {192 \, \sqrt {2} {\left ({\left (A - B + C\right )} a \cos \left (d x + c\right ) + {\left (A - B + C\right )} a\right )} \arctan \left (\frac {\sqrt {2} \sqrt {\frac {a \cos \left (d x + c\right ) + a}{\cos \left (d x + c\right )}} \cos \left (d x + c\right )}{\sqrt {a} \sin \left (d x + c\right )}\right )}{\sqrt {a}}}{192 \, {\left (a d \cos \left (d x + c\right ) + a d\right )}}\right ] \]
integrate(cos(d*x+c)^4*(A+B*sec(d*x+c)+C*sec(d*x+c)^2)/(a+a*sec(d*x+c))^(1 /2),x, algorithm="fricas")
[1/384*(192*sqrt(2)*((A - B + C)*a*cos(d*x + c) + (A - B + C)*a)*sqrt(-1/a )*log((2*sqrt(2)*sqrt((a*cos(d*x + c) + a)/cos(d*x + c))*sqrt(-1/a)*cos(d* x + c)*sin(d*x + c) + 3*cos(d*x + c)^2 + 2*cos(d*x + c) - 1)/(cos(d*x + c) ^2 + 2*cos(d*x + c) + 1)) - 3*((107*A - 72*B + 112*C)*cos(d*x + c) + 107*A - 72*B + 112*C)*sqrt(-a)*log((2*a*cos(d*x + c)^2 + 2*sqrt(-a)*sqrt((a*cos (d*x + c) + a)/cos(d*x + c))*cos(d*x + c)*sin(d*x + c) + a*cos(d*x + c) - a)/(cos(d*x + c) + 1)) + 2*(48*A*cos(d*x + c)^4 - 8*(A - 8*B)*cos(d*x + c) ^3 + 2*(43*A - 8*B + 48*C)*cos(d*x + c)^2 - 3*(21*A - 56*B + 16*C)*cos(d*x + c))*sqrt((a*cos(d*x + c) + a)/cos(d*x + c))*sin(d*x + c))/(a*d*cos(d*x + c) + a*d), -1/192*(3*((107*A - 72*B + 112*C)*cos(d*x + c) + 107*A - 72*B + 112*C)*sqrt(a)*arctan(sqrt((a*cos(d*x + c) + a)/cos(d*x + c))*cos(d*x + c)/(sqrt(a)*sin(d*x + c))) - (48*A*cos(d*x + c)^4 - 8*(A - 8*B)*cos(d*x + c)^3 + 2*(43*A - 8*B + 48*C)*cos(d*x + c)^2 - 3*(21*A - 56*B + 16*C)*cos( d*x + c))*sqrt((a*cos(d*x + c) + a)/cos(d*x + c))*sin(d*x + c) - 192*sqrt( 2)*((A - B + C)*a*cos(d*x + c) + (A - B + C)*a)*arctan(sqrt(2)*sqrt((a*cos (d*x + c) + a)/cos(d*x + c))*cos(d*x + c)/(sqrt(a)*sin(d*x + c)))/sqrt(a)) /(a*d*cos(d*x + c) + a*d)]
\[ \int \frac {\cos ^4(c+d x) \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right )}{\sqrt {a+a \sec (c+d x)}} \, dx=\int \frac {\left (A + B \sec {\left (c + d x \right )} + C \sec ^{2}{\left (c + d x \right )}\right ) \cos ^{4}{\left (c + d x \right )}}{\sqrt {a \left (\sec {\left (c + d x \right )} + 1\right )}}\, dx \]
Integral((A + B*sec(c + d*x) + C*sec(c + d*x)**2)*cos(c + d*x)**4/sqrt(a*( sec(c + d*x) + 1)), x)
\[ \int \frac {\cos ^4(c+d x) \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right )}{\sqrt {a+a \sec (c+d x)}} \, dx=\int { \frac {{\left (C \sec \left (d x + c\right )^{2} + B \sec \left (d x + c\right ) + A\right )} \cos \left (d x + c\right )^{4}}{\sqrt {a \sec \left (d x + c\right ) + a}} \,d x } \]
integrate(cos(d*x+c)^4*(A+B*sec(d*x+c)+C*sec(d*x+c)^2)/(a+a*sec(d*x+c))^(1 /2),x, algorithm="maxima")
integrate((C*sec(d*x + c)^2 + B*sec(d*x + c) + A)*cos(d*x + c)^4/sqrt(a*se c(d*x + c) + a), x)
Leaf count of result is larger than twice the leaf count of optimal. 1370 vs. \(2 (226) = 452\).
Time = 2.29 (sec) , antiderivative size = 1370, normalized size of antiderivative = 5.29 \[ \int \frac {\cos ^4(c+d x) \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right )}{\sqrt {a+a \sec (c+d x)}} \, dx=\text {Too large to display} \]
integrate(cos(d*x+c)^4*(A+B*sec(d*x+c)+C*sec(d*x+c)^2)/(a+a*sec(d*x+c))^(1 /2),x, algorithm="giac")
1/384*(192*sqrt(2)*(A - B + C)*log((sqrt(-a)*tan(1/2*d*x + 1/2*c) - sqrt(- a*tan(1/2*d*x + 1/2*c)^2 + a))^2)/(sqrt(-a)*sgn(cos(d*x + c))) + 3*(107*A - 72*B + 112*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)))/(sqrt(-a)*sgn(cos(d*x + c))) - 3*(107*A - 72*B + 112*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)))/(sqrt(-a)*sgn(cos(d* x + c))) + 4*sqrt(2)*(1599*(sqrt(-a)*tan(1/2*d*x + 1/2*c) - sqrt(-a*tan(1/ 2*d*x + 1/2*c)^2 + a))^14*A*sqrt(-a) - 1320*(sqrt(-a)*tan(1/2*d*x + 1/2*c) - sqrt(-a*tan(1/2*d*x + 1/2*c)^2 + a))^14*B*sqrt(-a) + 816*(sqrt(-a)*tan( 1/2*d*x + 1/2*c) - sqrt(-a*tan(1/2*d*x + 1/2*c)^2 + a))^14*C*sqrt(-a) - 18 219*(sqrt(-a)*tan(1/2*d*x + 1/2*c) - sqrt(-a*tan(1/2*d*x + 1/2*c)^2 + a))^ 12*A*sqrt(-a)*a + 18504*(sqrt(-a)*tan(1/2*d*x + 1/2*c) - sqrt(-a*tan(1/2*d *x + 1/2*c)^2 + a))^12*B*sqrt(-a)*a - 12528*(sqrt(-a)*tan(1/2*d*x + 1/2*c) - sqrt(-a*tan(1/2*d*x + 1/2*c)^2 + a))^12*C*sqrt(-a)*a + 91467*(sqrt(-a)* tan(1/2*d*x + 1/2*c) - sqrt(-a*tan(1/2*d*x + 1/2*c)^2 + a))^10*A*sqrt(-a)* a^2 - 96072*(sqrt(-a)*tan(1/2*d*x + 1/2*c) - sqrt(-a*tan(1/2*d*x + 1/2*c)^ 2 + a))^10*B*sqrt(-a)*a^2 + 64752*(sqrt(-a)*tan(1/2*d*x + 1/2*c) - sqrt(-a *tan(1/2*d*x + 1/2*c)^2 + a))^10*C*sqrt(-a)*a^2 - 177735*(sqrt(-a)*tan(1/2 *d*x + 1/2*c) - sqrt(-a*tan(1/2*d*x + 1/2*c)^2 + a))^8*A*sqrt(-a)*a^3 + 21 5016*(sqrt(-a)*tan(1/2*d*x + 1/2*c) - sqrt(-a*tan(1/2*d*x + 1/2*c)^2 + ...
Timed out. \[ \int \frac {\cos ^4(c+d x) \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right )}{\sqrt {a+a \sec (c+d x)}} \, dx=\int \frac {{\cos \left (c+d\,x\right )}^4\,\left (A+\frac {B}{\cos \left (c+d\,x\right )}+\frac {C}{{\cos \left (c+d\,x\right )}^2}\right )}{\sqrt {a+\frac {a}{\cos \left (c+d\,x\right )}}} \,d x \]