Integrand size = 49, antiderivative size = 590 \[ \int \frac {(c+d \tan (e+f x))^{5/2} \left (A+B \tan (e+f x)+C \tan ^2(e+f x)\right )}{(a+b \tan (e+f x))^{7/2}} \, dx=-\frac {(i A+B-i C) (c-i d)^{5/2} \text {arctanh}\left (\frac {\sqrt {c-i d} \sqrt {a+b \tan (e+f x)}}{\sqrt {a-i b} \sqrt {c+d \tan (e+f x)}}\right )}{(a-i b)^{7/2} f}-\frac {(B-i (A-C)) (c+i d)^{5/2} \text {arctanh}\left (\frac {\sqrt {c+i d} \sqrt {a+b \tan (e+f x)}}{\sqrt {a+i b} \sqrt {c+d \tan (e+f x)}}\right )}{(a+i b)^{7/2} f}+\frac {2 C d^{5/2} \text {arctanh}\left (\frac {\sqrt {d} \sqrt {a+b \tan (e+f x)}}{\sqrt {b} \sqrt {c+d \tan (e+f x)}}\right )}{b^{7/2} f}-\frac {2 \left (a^6 C d^2+3 a^4 b^2 C d^2-3 a^2 b^4 \left (c^2 C+2 B c d-2 C d^2-A \left (c^2-d^2\right )\right )+b^6 \left (c (c C+2 B d)-A \left (c^2-d^2\right )\right )-a^3 b^3 \left (2 c (A-C) d+B \left (c^2-d^2\right )\right )+3 a b^5 \left (2 c (A-C) d+B \left (c^2-d^2\right )\right )\right ) \sqrt {c+d \tan (e+f x)}}{b^3 \left (a^2+b^2\right )^3 f \sqrt {a+b \tan (e+f x)}}-\frac {2 \left (a^4 C d+b^4 (B c+A d)+2 a b^3 (A c-c C-B d)-a^2 b^2 (B c+(A-3 C) d)\right ) (c+d \tan (e+f x))^{3/2}}{3 b^2 \left (a^2+b^2\right )^2 f (a+b \tan (e+f x))^{3/2}}-\frac {2 \left (A b^2-a (b B-a C)\right ) (c+d \tan (e+f x))^{5/2}}{5 b \left (a^2+b^2\right ) f (a+b \tan (e+f x))^{5/2}} \]
-(I*A+B-I*C)*(c-I*d)^(5/2)*arctanh((c-I*d)^(1/2)*(a+b*tan(f*x+e))^(1/2)/(a -I*b)^(1/2)/(c+d*tan(f*x+e))^(1/2))/(a-I*b)^(7/2)/f-(B-I*(A-C))*(c+I*d)^(5 /2)*arctanh((c+I*d)^(1/2)*(a+b*tan(f*x+e))^(1/2)/(a+I*b)^(1/2)/(c+d*tan(f* x+e))^(1/2))/(a+I*b)^(7/2)/f+2*C*d^(5/2)*arctanh(d^(1/2)*(a+b*tan(f*x+e))^ (1/2)/b^(1/2)/(c+d*tan(f*x+e))^(1/2))/b^(7/2)/f-2*(a^6*C*d^2+3*a^4*b^2*C*d ^2-3*a^2*b^4*(c^2*C+2*B*c*d-2*C*d^2-A*(c^2-d^2))+b^6*(c*(2*B*d+C*c)-A*(c^2 -d^2))-a^3*b^3*(2*c*(A-C)*d+B*(c^2-d^2))+3*a*b^5*(2*c*(A-C)*d+B*(c^2-d^2)) )*(c+d*tan(f*x+e))^(1/2)/b^3/(a^2+b^2)^3/f/(a+b*tan(f*x+e))^(1/2)-2/3*(a^4 *C*d+b^4*(A*d+B*c)+2*a*b^3*(A*c-B*d-C*c)-a^2*b^2*(B*c+(A-3*C)*d))*(c+d*tan (f*x+e))^(3/2)/b^2/(a^2+b^2)^2/f/(a+b*tan(f*x+e))^(3/2)-2/5*(A*b^2-a*(B*b- C*a))*(c+d*tan(f*x+e))^(5/2)/b/(a^2+b^2)/f/(a+b*tan(f*x+e))^(5/2)
Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.
Time = 7.14 (sec) , antiderivative size = 641, normalized size of antiderivative = 1.09 \[ \int \frac {(c+d \tan (e+f x))^{5/2} \left (A+B \tan (e+f x)+C \tan ^2(e+f x)\right )}{(a+b \tan (e+f x))^{7/2}} \, dx=\frac {(B+i (A-C)) (c+d \tan (e+f x))^{5/2}}{5 (a-i b) f (a+b \tan (e+f x))^{5/2}}-\frac {(i A-B-i C) (c+d \tan (e+f x))^{5/2}}{5 (a+i b) f (a+b \tan (e+f x))^{5/2}}-\frac {2 C (b c-a d)^2 \operatorname {Hypergeometric2F1}\left (-\frac {5}{2},-\frac {5}{2},-\frac {3}{2},-\frac {d (a+b \tan (e+f x))}{b c-a d}\right ) \sqrt {c+d \tan (e+f x)}}{5 b^3 f (a+b \tan (e+f x))^{5/2} \sqrt {\frac {b (c+d \tan (e+f x))}{b c-a d}}}+\frac {(A-i B-C) (i c+d) \left (\frac {(c+d \tan (e+f x))^{3/2}}{(a-i b) (a+b \tan (e+f x))^{3/2}}+\frac {3 (c-i d) \left (\frac {\sqrt {-c+i d} \text {arctanh}\left (\frac {\sqrt {-c+i d} \sqrt {a+b \tan (e+f x)}}{\sqrt {-a+i b} \sqrt {c+d \tan (e+f x)}}\right )}{(-a+i b)^{3/2}}+\frac {\sqrt {c+d \tan (e+f x)}}{(a-i b) \sqrt {a+b \tan (e+f x)}}\right )}{a-i b}\right )}{3 (a-i b) f}-\frac {(A+i B-C) (i c-d) \left (\frac {(c+d \tan (e+f x))^{3/2}}{(a+i b) (a+b \tan (e+f x))^{3/2}}-\frac {3 (c+i d) \left (\frac {\sqrt {c+i d} \text {arctanh}\left (\frac {\sqrt {c+i d} \sqrt {a+b \tan (e+f x)}}{\sqrt {a+i b} \sqrt {c+d \tan (e+f x)}}\right )}{(a+i b)^{3/2}}-\frac {\sqrt {c+d \tan (e+f x)}}{(a+i b) \sqrt {a+b \tan (e+f x)}}\right )}{a+i b}\right )}{3 (a+i b) f} \]
Integrate[((c + d*Tan[e + f*x])^(5/2)*(A + B*Tan[e + f*x] + C*Tan[e + f*x] ^2))/(a + b*Tan[e + f*x])^(7/2),x]
((B + I*(A - C))*(c + d*Tan[e + f*x])^(5/2))/(5*(a - I*b)*f*(a + b*Tan[e + f*x])^(5/2)) - ((I*A - B - I*C)*(c + d*Tan[e + f*x])^(5/2))/(5*(a + I*b)* f*(a + b*Tan[e + f*x])^(5/2)) - (2*C*(b*c - a*d)^2*Hypergeometric2F1[-5/2, -5/2, -3/2, -((d*(a + b*Tan[e + f*x]))/(b*c - a*d))]*Sqrt[c + d*Tan[e + f *x]])/(5*b^3*f*(a + b*Tan[e + f*x])^(5/2)*Sqrt[(b*(c + d*Tan[e + f*x]))/(b *c - a*d)]) + ((A - I*B - C)*(I*c + d)*((c + d*Tan[e + f*x])^(3/2)/((a - I *b)*(a + b*Tan[e + f*x])^(3/2)) + (3*(c - I*d)*((Sqrt[-c + I*d]*ArcTanh[(S qrt[-c + I*d]*Sqrt[a + b*Tan[e + f*x]])/(Sqrt[-a + I*b]*Sqrt[c + d*Tan[e + f*x]])])/(-a + I*b)^(3/2) + Sqrt[c + d*Tan[e + f*x]]/((a - I*b)*Sqrt[a + b*Tan[e + f*x]])))/(a - I*b)))/(3*(a - I*b)*f) - ((A + I*B - C)*(I*c - d)* ((c + d*Tan[e + f*x])^(3/2)/((a + I*b)*(a + b*Tan[e + f*x])^(3/2)) - (3*(c + I*d)*((Sqrt[c + I*d]*ArcTanh[(Sqrt[c + I*d]*Sqrt[a + b*Tan[e + f*x]])/( Sqrt[a + I*b]*Sqrt[c + d*Tan[e + f*x]])])/(a + I*b)^(3/2) - Sqrt[c + d*Tan [e + f*x]]/((a + I*b)*Sqrt[a + b*Tan[e + f*x]])))/(a + I*b)))/(3*(a + I*b) *f)
Time = 5.26 (sec) , antiderivative size = 658, normalized size of antiderivative = 1.12, number of steps used = 14, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.265, Rules used = {3042, 4128, 27, 3042, 4128, 27, 3042, 4128, 27, 3042, 4138, 2348, 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.
| \(\displaystyle \int \frac {(c+d \tan (e+f x))^{5/2} \left (A+B \tan (e+f x)+C \tan ^2(e+f x)\right )}{(a+b \tan (e+f x))^{7/2}} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {(c+d \tan (e+f x))^{5/2} \left (A+B \tan (e+f x)+C \tan (e+f x)^2\right )}{(a+b \tan (e+f x))^{7/2}}dx\) |
| \(\Big \downarrow \) 4128 |
| \(\displaystyle \frac {2 \int \frac {5 (c+d \tan (e+f x))^{3/2} \left (\left (a^2+b^2\right ) C d \tan ^2(e+f x)-b ((A-C) (b c-a d)-B (a c+b d)) \tan (e+f x)+(b B-a C) (b c-a d)+A b (a c+b d)\right )}{2 (a+b \tan (e+f x))^{5/2}}dx}{5 b \left (a^2+b^2\right )}-\frac {2 \left (A b^2-a (b B-a C)\right ) (c+d \tan (e+f x))^{5/2}}{5 b f \left (a^2+b^2\right ) (a+b \tan (e+f x))^{5/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int \frac {(c+d \tan (e+f x))^{3/2} \left (\left (a^2+b^2\right ) C d \tan ^2(e+f x)-b ((A-C) (b c-a d)-B (a c+b d)) \tan (e+f x)+(b B-a C) (b c-a d)+A b (a c+b d)\right )}{(a+b \tan (e+f x))^{5/2}}dx}{b \left (a^2+b^2\right )}-\frac {2 \left (A b^2-a (b B-a C)\right ) (c+d \tan (e+f x))^{5/2}}{5 b f \left (a^2+b^2\right ) (a+b \tan (e+f x))^{5/2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\int \frac {(c+d \tan (e+f x))^{3/2} \left (\left (a^2+b^2\right ) C d \tan (e+f x)^2-b ((A-C) (b c-a d)-B (a c+b d)) \tan (e+f x)+(b B-a C) (b c-a d)+A b (a c+b d)\right )}{(a+b \tan (e+f x))^{5/2}}dx}{b \left (a^2+b^2\right )}-\frac {2 \left (A b^2-a (b B-a C)\right ) (c+d \tan (e+f x))^{5/2}}{5 b f \left (a^2+b^2\right ) (a+b \tan (e+f x))^{5/2}}\) |
| \(\Big \downarrow \) 4128 |
| \(\displaystyle \frac {\frac {2 \int \frac {3 \sqrt {c+d \tan (e+f x)} \left (-\left (((a c+b d) ((A-C) (b c-a d)-B (a c+b d))+(b c-a d) (b B c+b (A-C) d+a (A c-C c-B d))) \tan (e+f x) b^2\right )+(a c+b d) ((b B-a C) (b c-a d)+A b (a c+b d)) b+\left (a^2+b^2\right )^2 C d^2 \tan ^2(e+f x)-(b c-a d) \left (((A-C) (b c-a d)-B (a c+b d)) b^2+a \left (a^2+b^2\right ) C d\right )\right )}{2 (a+b \tan (e+f x))^{3/2}}dx}{3 b \left (a^2+b^2\right )}-\frac {2 (c+d \tan (e+f x))^{3/2} \left (a^4 C d-a^2 b^2 (d (A-3 C)+B c)+2 a b^3 (A c-B d-c C)+b^4 (A d+B c)\right )}{3 b f \left (a^2+b^2\right ) (a+b \tan (e+f x))^{3/2}}}{b \left (a^2+b^2\right )}-\frac {2 \left (A b^2-a (b B-a C)\right ) (c+d \tan (e+f x))^{5/2}}{5 b f \left (a^2+b^2\right ) (a+b \tan (e+f x))^{5/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {\int \frac {\sqrt {c+d \tan (e+f x)} \left (-\left (((a c+b d) ((A-C) (b c-a d)-B (a c+b d))+(b c-a d) (b B c+b (A-C) d+a (A c-C c-B d))) \tan (e+f x) b^2\right )+(a c+b d) ((b B-a C) (b c-a d)+A b (a c+b d)) b+\left (a^2+b^2\right )^2 C d^2 \tan ^2(e+f x)-(b c-a d) \left (((A-C) (b c-a d)-B (a c+b d)) b^2+a \left (a^2+b^2\right ) C d\right )\right )}{(a+b \tan (e+f x))^{3/2}}dx}{b \left (a^2+b^2\right )}-\frac {2 (c+d \tan (e+f x))^{3/2} \left (a^4 C d-a^2 b^2 (d (A-3 C)+B c)+2 a b^3 (A c-B d-c C)+b^4 (A d+B c)\right )}{3 b f \left (a^2+b^2\right ) (a+b \tan (e+f x))^{3/2}}}{b \left (a^2+b^2\right )}-\frac {2 \left (A b^2-a (b B-a C)\right ) (c+d \tan (e+f x))^{5/2}}{5 b f \left (a^2+b^2\right ) (a+b \tan (e+f x))^{5/2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {\int \frac {\sqrt {c+d \tan (e+f x)} \left (-\left (((a c+b d) ((A-C) (b c-a d)-B (a c+b d))+(b c-a d) (b B c+b (A-C) d+a (A c-C c-B d))) \tan (e+f x) b^2\right )+(a c+b d) ((b B-a C) (b c-a d)+A b (a c+b d)) b+\left (a^2+b^2\right )^2 C d^2 \tan (e+f x)^2-(b c-a d) \left (((A-C) (b c-a d)-B (a c+b d)) b^2+a \left (a^2+b^2\right ) C d\right )\right )}{(a+b \tan (e+f x))^{3/2}}dx}{b \left (a^2+b^2\right )}-\frac {2 (c+d \tan (e+f x))^{3/2} \left (a^4 C d-a^2 b^2 (d (A-3 C)+B c)+2 a b^3 (A c-B d-c C)+b^4 (A d+B c)\right )}{3 b f \left (a^2+b^2\right ) (a+b \tan (e+f x))^{3/2}}}{b \left (a^2+b^2\right )}-\frac {2 \left (A b^2-a (b B-a C)\right ) (c+d \tan (e+f x))^{5/2}}{5 b f \left (a^2+b^2\right ) (a+b \tan (e+f x))^{5/2}}\) |
| \(\Big \downarrow \) 4128 |
| \(\displaystyle \frac {\frac {\frac {2 \int \frac {C d^3 a^6+3 b^2 C d^3 a^4-b^3 \left (C c^3+3 B d c^2-3 C d^2 c-B d^3-A \left (c^3-3 c d^2\right )\right ) a^3+3 b^4 \left (B c^3+3 A d c^2-3 C d c^2-3 B d^2 c-A d^3+2 C d^3\right ) a^2-3 b^5 \left (A c^3-C c^3-3 B d c^2-3 A d^2 c+3 C d^2 c+B d^3\right ) a+\left (a^2+b^2\right )^3 C d^3 \tan ^2(e+f x)-b^6 \left (B c^3+3 A d c^2-3 C d c^2-3 B d^2 c-A d^3\right )+b^3 \left (\left ((A-C) d \left (3 c^2-d^2\right )+B \left (c^3-3 c d^2\right )\right ) a^3+3 b \left (C c^3+3 B d c^2-3 C d^2 c-B d^3-A \left (c^3-3 c d^2\right )\right ) a^2-3 b^2 \left ((A-C) d \left (3 c^2-d^2\right )+B \left (c^3-3 c d^2\right )\right ) a+b^3 \left (A c^3-C c^3-3 B d c^2-3 A d^2 c+3 C d^2 c+B d^3\right )\right ) \tan (e+f x)}{2 \sqrt {a+b \tan (e+f x)} \sqrt {c+d \tan (e+f x)}}dx}{b \left (a^2+b^2\right )}-\frac {2 \left (C d^2 a^6+3 b^2 C d^2 a^4-b^3 \left (2 c (A-C) d+B \left (c^2-d^2\right )\right ) a^3-3 b^4 \left (C c^2+2 B d c-2 C d^2-A \left (c^2-d^2\right )\right ) a^2+3 b^5 \left (2 c (A-C) d+B \left (c^2-d^2\right )\right ) a+b^6 \left (c (c C+2 B d)-A \left (c^2-d^2\right )\right )\right ) \sqrt {c+d \tan (e+f x)}}{b \left (a^2+b^2\right ) f \sqrt {a+b \tan (e+f x)}}}{b \left (a^2+b^2\right )}-\frac {2 \left (C d a^4-b^2 (B c+(A-3 C) d) a^2+2 b^3 (A c-C c-B d) a+b^4 (B c+A d)\right ) (c+d \tan (e+f x))^{3/2}}{3 b \left (a^2+b^2\right ) f (a+b \tan (e+f x))^{3/2}}}{b \left (a^2+b^2\right )}-\frac {2 \left (A b^2-a (b B-a C)\right ) (c+d \tan (e+f x))^{5/2}}{5 b \left (a^2+b^2\right ) f (a+b \tan (e+f x))^{5/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {\frac {\int \frac {C d^3 a^6+3 b^2 C d^3 a^4-b^3 \left (C c^3+3 B d c^2-3 C d^2 c-B d^3-A \left (c^3-3 c d^2\right )\right ) a^3+3 b^4 \left (B c^3+3 A d c^2-3 C d c^2-3 B d^2 c-A d^3+2 C d^3\right ) a^2-3 b^5 \left (A c^3-C c^3-3 B d c^2-3 A d^2 c+3 C d^2 c+B d^3\right ) a+\left (a^2+b^2\right )^3 C d^3 \tan ^2(e+f x)-b^6 \left (B c^3+3 A d c^2-3 C d c^2-3 B d^2 c-A d^3\right )-b^3 \left (-\left (\left ((A-C) d \left (3 c^2-d^2\right )+B \left (c^3-3 c d^2\right )\right ) a^3\right )+3 b \left (A c^3-C c^3-3 B d c^2-3 A d^2 c+3 C d^2 c+B d^3\right ) a^2+3 b^2 \left ((A-C) d \left (3 c^2-d^2\right )+B \left (c^3-3 c d^2\right )\right ) a+b^3 \left (C c^3+3 B d c^2-3 C d^2 c-B d^3-A \left (c^3-3 c d^2\right )\right )\right ) \tan (e+f x)}{\sqrt {a+b \tan (e+f x)} \sqrt {c+d \tan (e+f x)}}dx}{b \left (a^2+b^2\right )}-\frac {2 \left (C d^2 a^6+3 b^2 C d^2 a^4-b^3 \left (2 c (A-C) d+B \left (c^2-d^2\right )\right ) a^3-3 b^4 \left (C c^2+2 B d c-2 C d^2-A \left (c^2-d^2\right )\right ) a^2+3 b^5 \left (2 c (A-C) d+B \left (c^2-d^2\right )\right ) a+b^6 \left (c (c C+2 B d)-A \left (c^2-d^2\right )\right )\right ) \sqrt {c+d \tan (e+f x)}}{b \left (a^2+b^2\right ) f \sqrt {a+b \tan (e+f x)}}}{b \left (a^2+b^2\right )}-\frac {2 \left (C d a^4-b^2 (B c+(A-3 C) d) a^2+2 b^3 (A c-C c-B d) a+b^4 (B c+A d)\right ) (c+d \tan (e+f x))^{3/2}}{3 b \left (a^2+b^2\right ) f (a+b \tan (e+f x))^{3/2}}}{b \left (a^2+b^2\right )}-\frac {2 \left (A b^2-a (b B-a C)\right ) (c+d \tan (e+f x))^{5/2}}{5 b \left (a^2+b^2\right ) f (a+b \tan (e+f x))^{5/2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {\frac {\int \frac {C d^3 a^6+3 b^2 C d^3 a^4-b^3 \left (C c^3+3 B d c^2-3 C d^2 c-B d^3-A \left (c^3-3 c d^2\right )\right ) a^3+3 b^4 \left (B c^3+3 A d c^2-3 C d c^2-3 B d^2 c-A d^3+2 C d^3\right ) a^2-3 b^5 \left (A c^3-C c^3-3 B d c^2-3 A d^2 c+3 C d^2 c+B d^3\right ) a+\left (a^2+b^2\right )^3 C d^3 \tan (e+f x)^2-b^6 \left (B c^3+3 A d c^2-3 C d c^2-3 B d^2 c-A d^3\right )-b^3 \left (-\left (\left ((A-C) d \left (3 c^2-d^2\right )+B \left (c^3-3 c d^2\right )\right ) a^3\right )+3 b \left (A c^3-C c^3-3 B d c^2-3 A d^2 c+3 C d^2 c+B d^3\right ) a^2+3 b^2 \left ((A-C) d \left (3 c^2-d^2\right )+B \left (c^3-3 c d^2\right )\right ) a+b^3 \left (C c^3+3 B d c^2-3 C d^2 c-B d^3-A \left (c^3-3 c d^2\right )\right )\right ) \tan (e+f x)}{\sqrt {a+b \tan (e+f x)} \sqrt {c+d \tan (e+f x)}}dx}{b \left (a^2+b^2\right )}-\frac {2 \left (C d^2 a^6+3 b^2 C d^2 a^4-b^3 \left (2 c (A-C) d+B \left (c^2-d^2\right )\right ) a^3-3 b^4 \left (C c^2+2 B d c-2 C d^2-A \left (c^2-d^2\right )\right ) a^2+3 b^5 \left (2 c (A-C) d+B \left (c^2-d^2\right )\right ) a+b^6 \left (c (c C+2 B d)-A \left (c^2-d^2\right )\right )\right ) \sqrt {c+d \tan (e+f x)}}{b \left (a^2+b^2\right ) f \sqrt {a+b \tan (e+f x)}}}{b \left (a^2+b^2\right )}-\frac {2 \left (C d a^4-b^2 (B c+(A-3 C) d) a^2+2 b^3 (A c-C c-B d) a+b^4 (B c+A d)\right ) (c+d \tan (e+f x))^{3/2}}{3 b \left (a^2+b^2\right ) f (a+b \tan (e+f x))^{3/2}}}{b \left (a^2+b^2\right )}-\frac {2 \left (A b^2-a (b B-a C)\right ) (c+d \tan (e+f x))^{5/2}}{5 b \left (a^2+b^2\right ) f (a+b \tan (e+f x))^{5/2}}\) |
| \(\Big \downarrow \) 4138 |
| \(\displaystyle \frac {\frac {\frac {\int \frac {C d^3 a^6+3 b^2 C d^3 a^4-b^3 \left (C c^3+3 B d c^2-3 C d^2 c-B d^3-A \left (c^3-3 c d^2\right )\right ) a^3+3 b^4 \left (B c^3+3 A d c^2-3 C d c^2-3 B d^2 c-A d^3+2 C d^3\right ) a^2-3 b^5 \left (A c^3-C c^3-3 B d c^2-3 A d^2 c+3 C d^2 c+B d^3\right ) a+\left (a^2+b^2\right )^3 C d^3 \tan ^2(e+f x)-b^6 \left (B c^3+3 A d c^2-3 C d c^2-3 B d^2 c-A d^3\right )+b^3 \left (\left ((A-C) d \left (3 c^2-d^2\right )+B \left (c^3-3 c d^2\right )\right ) a^3+3 b \left (C c^3+3 B d c^2-3 C d^2 c-B d^3-A \left (c^3-3 c d^2\right )\right ) a^2-3 b^2 \left ((A-C) d \left (3 c^2-d^2\right )+B \left (c^3-3 c d^2\right )\right ) a+b^3 \left (A c^3-C c^3-3 B d c^2-3 A d^2 c+3 C d^2 c+B d^3\right )\right ) \tan (e+f x)}{\sqrt {a+b \tan (e+f x)} \sqrt {c+d \tan (e+f x)} \left (\tan ^2(e+f x)+1\right )}d\tan (e+f x)}{b \left (a^2+b^2\right ) f}-\frac {2 \left (C d^2 a^6+3 b^2 C d^2 a^4-b^3 \left (2 c (A-C) d+B \left (c^2-d^2\right )\right ) a^3-3 b^4 \left (C c^2+2 B d c-2 C d^2-A \left (c^2-d^2\right )\right ) a^2+3 b^5 \left (2 c (A-C) d+B \left (c^2-d^2\right )\right ) a+b^6 \left (c (c C+2 B d)-A \left (c^2-d^2\right )\right )\right ) \sqrt {c+d \tan (e+f x)}}{b \left (a^2+b^2\right ) f \sqrt {a+b \tan (e+f x)}}}{b \left (a^2+b^2\right )}-\frac {2 \left (C d a^4-b^2 (B c+(A-3 C) d) a^2+2 b^3 (A c-C c-B d) a+b^4 (B c+A d)\right ) (c+d \tan (e+f x))^{3/2}}{3 b \left (a^2+b^2\right ) f (a+b \tan (e+f x))^{3/2}}}{b \left (a^2+b^2\right )}-\frac {2 \left (A b^2-a (b B-a C)\right ) (c+d \tan (e+f x))^{5/2}}{5 b \left (a^2+b^2\right ) f (a+b \tan (e+f x))^{5/2}}\) |
| \(\Big \downarrow \) 2348 |
| \(\displaystyle \frac {\frac {\frac {\int \left (\frac {\left (a^2+b^2\right )^3 C d^3}{\sqrt {a+b \tan (e+f x)} \sqrt {c+d \tan (e+f x)}}+\frac {-A c^3 b^6-B d^3 b^6+3 A c d^2 b^6-3 c C d^2 b^6+c^3 C b^6+3 B c^2 d b^6+3 a B c^3 b^5-3 a A d^3 b^5+3 a C d^3 b^5-9 a B c d^2 b^5+9 a A c^2 d b^5-9 a c^2 C d b^5+3 a^2 A c^3 b^4+3 a^2 B d^3 b^4-9 a^2 A c d^2 b^4+9 a^2 c C d^2 b^4-3 a^2 c^3 C b^4-9 a^2 B c^2 d b^4-a^3 B c^3 b^3+a^3 A d^3 b^3-a^3 C d^3 b^3+3 a^3 B c d^2 b^3-3 a^3 A c^2 d b^3+3 a^3 c^2 C d b^3+i \left (-B c^3 b^6+A d^3 b^6-C d^3 b^6+3 B c d^2 b^6-3 A c^2 d b^6+3 c^2 C d b^6-3 a A c^3 b^5-3 a B d^3 b^5+9 a A c d^2 b^5-9 a c C d^2 b^5+3 a c^3 C b^5+9 a B c^2 d b^5+3 a^2 B c^3 b^4-3 a^2 A d^3 b^4+3 a^2 C d^3 b^4-9 a^2 B c d^2 b^4+9 a^2 A c^2 d b^4-9 a^2 c^2 C d b^4+a^3 A c^3 b^3+a^3 B d^3 b^3-3 a^3 A c d^2 b^3+3 a^3 c C d^2 b^3-a^3 c^3 C b^3-3 a^3 B c^2 d b^3\right )}{2 (i-\tan (e+f x)) \sqrt {a+b \tan (e+f x)} \sqrt {c+d \tan (e+f x)}}+\frac {A c^3 b^6+B d^3 b^6-3 A c d^2 b^6+3 c C d^2 b^6-c^3 C b^6-3 B c^2 d b^6-3 a B c^3 b^5+3 a A d^3 b^5-3 a C d^3 b^5+9 a B c d^2 b^5-9 a A c^2 d b^5+9 a c^2 C d b^5-3 a^2 A c^3 b^4-3 a^2 B d^3 b^4+9 a^2 A c d^2 b^4-9 a^2 c C d^2 b^4+3 a^2 c^3 C b^4+9 a^2 B c^2 d b^4+a^3 B c^3 b^3-a^3 A d^3 b^3+a^3 C d^3 b^3-3 a^3 B c d^2 b^3+3 a^3 A c^2 d b^3-3 a^3 c^2 C d b^3+i \left (-B c^3 b^6+A d^3 b^6-C d^3 b^6+3 B c d^2 b^6-3 A c^2 d b^6+3 c^2 C d b^6-3 a A c^3 b^5-3 a B d^3 b^5+9 a A c d^2 b^5-9 a c C d^2 b^5+3 a c^3 C b^5+9 a B c^2 d b^5+3 a^2 B c^3 b^4-3 a^2 A d^3 b^4+3 a^2 C d^3 b^4-9 a^2 B c d^2 b^4+9 a^2 A c^2 d b^4-9 a^2 c^2 C d b^4+a^3 A c^3 b^3+a^3 B d^3 b^3-3 a^3 A c d^2 b^3+3 a^3 c C d^2 b^3-a^3 c^3 C b^3-3 a^3 B c^2 d b^3\right )}{2 (\tan (e+f x)+i) \sqrt {a+b \tan (e+f x)} \sqrt {c+d \tan (e+f x)}}\right )d\tan (e+f x)}{b \left (a^2+b^2\right ) f}-\frac {2 \left (C d^2 a^6+3 b^2 C d^2 a^4-b^3 \left (2 c (A-C) d+B \left (c^2-d^2\right )\right ) a^3-3 b^4 \left (C c^2+2 B d c-2 C d^2-A \left (c^2-d^2\right )\right ) a^2+3 b^5 \left (2 c (A-C) d+B \left (c^2-d^2\right )\right ) a+b^6 \left (c (c C+2 B d)-A \left (c^2-d^2\right )\right )\right ) \sqrt {c+d \tan (e+f x)}}{b \left (a^2+b^2\right ) f \sqrt {a+b \tan (e+f x)}}}{b \left (a^2+b^2\right )}-\frac {2 \left (C d a^4-b^2 (B c+(A-3 C) d) a^2+2 b^3 (A c-C c-B d) a+b^4 (B c+A d)\right ) (c+d \tan (e+f x))^{3/2}}{3 b \left (a^2+b^2\right ) f (a+b \tan (e+f x))^{3/2}}}{b \left (a^2+b^2\right )}-\frac {2 \left (A b^2-a (b B-a C)\right ) (c+d \tan (e+f x))^{5/2}}{5 b \left (a^2+b^2\right ) f (a+b \tan (e+f x))^{5/2}}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {2 \left (A b^2-a (b B-a C)\right ) (c+d \tan (e+f x))^{5/2}}{5 b f \left (a^2+b^2\right ) (a+b \tan (e+f x))^{5/2}}+\frac {-\frac {2 (c+d \tan (e+f x))^{3/2} \left (a^4 C d-a^2 b^2 (d (A-3 C)+B c)+2 a b^3 (A c-B d-c C)+b^4 (A d+B c)\right )}{3 b f \left (a^2+b^2\right ) (a+b \tan (e+f x))^{3/2}}+\frac {-\frac {2 \sqrt {c+d \tan (e+f x)} \left (a^6 C d^2+3 a^4 b^2 C d^2-a^3 b^3 \left (2 c d (A-C)+B \left (c^2-d^2\right )\right )-3 a^2 b^4 \left (-A \left (c^2-d^2\right )+2 B c d+c^2 C-2 C d^2\right )+3 a b^5 \left (2 c d (A-C)+B \left (c^2-d^2\right )\right )+b^6 \left (c (2 B d+c C)-A \left (c^2-d^2\right )\right )\right )}{b f \left (a^2+b^2\right ) \sqrt {a+b \tan (e+f x)}}+\frac {\frac {2 C d^{5/2} \left (a^2+b^2\right )^3 \text {arctanh}\left (\frac {\sqrt {d} \sqrt {a+b \tan (e+f x)}}{\sqrt {b} \sqrt {c+d \tan (e+f x)}}\right )}{\sqrt {b}}-\frac {b^3 (a+i b)^3 (c-i d)^{5/2} (i A+B-i C) \text {arctanh}\left (\frac {\sqrt {c-i d} \sqrt {a+b \tan (e+f x)}}{\sqrt {a-i b} \sqrt {c+d \tan (e+f x)}}\right )}{\sqrt {a-i b}}-\frac {b^3 (b+i a)^3 (c+i d)^{5/2} (A+i B-C) \text {arctanh}\left (\frac {\sqrt {c+i d} \sqrt {a+b \tan (e+f x)}}{\sqrt {a+i b} \sqrt {c+d \tan (e+f x)}}\right )}{\sqrt {a+i b}}}{b f \left (a^2+b^2\right )}}{b \left (a^2+b^2\right )}}{b \left (a^2+b^2\right )}\) |
Int[((c + d*Tan[e + f*x])^(5/2)*(A + B*Tan[e + f*x] + C*Tan[e + f*x]^2))/( a + b*Tan[e + f*x])^(7/2),x]
(-2*(A*b^2 - a*(b*B - a*C))*(c + d*Tan[e + f*x])^(5/2))/(5*b*(a^2 + b^2)*f *(a + b*Tan[e + f*x])^(5/2)) + ((-2*(a^4*C*d + b^4*(B*c + A*d) + 2*a*b^3*( A*c - c*C - B*d) - a^2*b^2*(B*c + (A - 3*C)*d))*(c + d*Tan[e + f*x])^(3/2) )/(3*b*(a^2 + b^2)*f*(a + b*Tan[e + f*x])^(3/2)) + ((-(((a + I*b)^3*b^3*(I *A + B - I*C)*(c - I*d)^(5/2)*ArcTanh[(Sqrt[c - I*d]*Sqrt[a + b*Tan[e + f* x]])/(Sqrt[a - I*b]*Sqrt[c + d*Tan[e + f*x]])])/Sqrt[a - I*b]) - (b^3*(I*a + b)^3*(A + I*B - C)*(c + I*d)^(5/2)*ArcTanh[(Sqrt[c + I*d]*Sqrt[a + b*Ta n[e + f*x]])/(Sqrt[a + I*b]*Sqrt[c + d*Tan[e + f*x]])])/Sqrt[a + I*b] + (2 *(a^2 + b^2)^3*C*d^(5/2)*ArcTanh[(Sqrt[d]*Sqrt[a + b*Tan[e + f*x]])/(Sqrt[ b]*Sqrt[c + d*Tan[e + f*x]])])/Sqrt[b])/(b*(a^2 + b^2)*f) - (2*(a^6*C*d^2 + 3*a^4*b^2*C*d^2 - 3*a^2*b^4*(c^2*C + 2*B*c*d - 2*C*d^2 - A*(c^2 - d^2)) + b^6*(c*(c*C + 2*B*d) - A*(c^2 - d^2)) - a^3*b^3*(2*c*(A - C)*d + B*(c^2 - d^2)) + 3*a*b^5*(2*c*(A - C)*d + B*(c^2 - d^2)))*Sqrt[c + d*Tan[e + f*x] ])/(b*(a^2 + b^2)*f*Sqrt[a + b*Tan[e + f*x]]))/(b*(a^2 + b^2)))/(b*(a^2 + b^2))
3.2.45.3.1 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[(Px_)*((c_) + (d_.)*(x_))^(m_.)*((e_) + (f_.)*(x_))^(n_.)*((a_.) + (b_. )*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegrand[Px*(c + d*x)^m*(e + f*x)^ n*(a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, f, m, n, p}, x] && PolyQ[P x, x] && (IntegerQ[p] || (IntegerQ[2*p] && IntegerQ[m] && ILtQ[n, 0])) && !(IGtQ[m, 0] && IGtQ[n, 0])
Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)])^(n_)*((A_.) + (B_.)*tan[(e_.) + (f_.)*(x_)] + (C_.)*tan[(e_.) + (f_.)*(x_)]^2), x_Symbol] :> Simp[(A*d^2 + c*(c*C - B*d))*(a + b*Tan[e + f*x])^m*((c + d*Tan[e + f*x])^(n + 1)/(d*f*(n + 1)*(c^2 + d^2))), x] - Sim p[1/(d*(n + 1)*(c^2 + d^2)) Int[(a + b*Tan[e + f*x])^(m - 1)*(c + d*Tan[e + f*x])^(n + 1)*Simp[A*d*(b*d*m - a*c*(n + 1)) + (c*C - B*d)*(b*c*m + a*d* (n + 1)) - d*(n + 1)*((A - C)*(b*c - a*d) + B*(a*c + b*d))*Tan[e + f*x] - b *(d*(B*c - A*d)*(m + n + 1) - C*(c^2*m - d^2*(n + 1)))*Tan[e + f*x]^2, x], x], x] /; FreeQ[{a, b, c, d, e, f, A, B, C}, x] && NeQ[b*c - a*d, 0] && NeQ [a^2 + b^2, 0] && NeQ[c^2 + d^2, 0] && GtQ[m, 0] && LtQ[n, -1]
Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)])^(n_)*((A_.) + (B_.)*tan[(e_.) + (f_.)*(x_)] + (C_.)*tan[(e_.) + (f_.)*(x_)]^2), x_Symbol] :> With[{ff = FreeFactors[Tan[e + f*x], x]}, S imp[ff/f Subst[Int[(a + b*ff*x)^m*(c + d*ff*x)^n*((A + B*ff*x + C*ff^2*x^ 2)/(1 + ff^2*x^2)), x], x, Tan[e + f*x]/ff], x]] /; FreeQ[{a, b, c, d, e, f , A, B, C, m, n}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 + b^2, 0] && NeQ[c^2 + d^2, 0]
Timed out.
\[\int \frac {\left (c +d \tan \left (f x +e \right )\right )^{\frac {5}{2}} \left (A +B \tan \left (f x +e \right )+C \tan \left (f x +e \right )^{2}\right )}{\left (a +b \tan \left (f x +e \right )\right )^{\frac {7}{2}}}d x\]
Timed out. \[ \int \frac {(c+d \tan (e+f x))^{5/2} \left (A+B \tan (e+f x)+C \tan ^2(e+f x)\right )}{(a+b \tan (e+f x))^{7/2}} \, dx=\text {Timed out} \]
integrate((c+d*tan(f*x+e))^(5/2)*(A+B*tan(f*x+e)+C*tan(f*x+e)^2)/(a+b*tan( f*x+e))^(7/2),x, algorithm="fricas")
\[ \int \frac {(c+d \tan (e+f x))^{5/2} \left (A+B \tan (e+f x)+C \tan ^2(e+f x)\right )}{(a+b \tan (e+f x))^{7/2}} \, dx=\int \frac {\left (c + d \tan {\left (e + f x \right )}\right )^{\frac {5}{2}} \left (A + B \tan {\left (e + f x \right )} + C \tan ^{2}{\left (e + f x \right )}\right )}{\left (a + b \tan {\left (e + f x \right )}\right )^{\frac {7}{2}}}\, dx \]
integrate((c+d*tan(f*x+e))**(5/2)*(A+B*tan(f*x+e)+C*tan(f*x+e)**2)/(a+b*ta n(f*x+e))**(7/2),x)
Integral((c + d*tan(e + f*x))**(5/2)*(A + B*tan(e + f*x) + C*tan(e + f*x)* *2)/(a + b*tan(e + f*x))**(7/2), x)
Timed out. \[ \int \frac {(c+d \tan (e+f x))^{5/2} \left (A+B \tan (e+f x)+C \tan ^2(e+f x)\right )}{(a+b \tan (e+f x))^{7/2}} \, dx=\text {Timed out} \]
integrate((c+d*tan(f*x+e))^(5/2)*(A+B*tan(f*x+e)+C*tan(f*x+e)^2)/(a+b*tan( f*x+e))^(7/2),x, algorithm="maxima")
Timed out. \[ \int \frac {(c+d \tan (e+f x))^{5/2} \left (A+B \tan (e+f x)+C \tan ^2(e+f x)\right )}{(a+b \tan (e+f x))^{7/2}} \, dx=\text {Timed out} \]
integrate((c+d*tan(f*x+e))^(5/2)*(A+B*tan(f*x+e)+C*tan(f*x+e)^2)/(a+b*tan( f*x+e))^(7/2),x, algorithm="giac")
Timed out. \[ \int \frac {(c+d \tan (e+f x))^{5/2} \left (A+B \tan (e+f x)+C \tan ^2(e+f x)\right )}{(a+b \tan (e+f x))^{7/2}} \, dx=\text {Hanged} \]