Optimal. Leaf size=255 \[ \frac {2 a^2 (35 A+49 B+33 C) \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x)}{105 d}+\frac {4 a^2 (5 A+4 B+3 C) \sin (c+d x) \sqrt {\sec (c+d x)}}{5 d}+\frac {4 a^2 (14 A+7 B+6 C) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} F\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{21 d}-\frac {4 a^2 (5 A+4 B+3 C) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{5 d}+\frac {2 (7 B+4 C) \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x) \left (a^2 \sec (c+d x)+a^2\right )}{35 d}+\frac {2 C \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x) (a \sec (c+d x)+a)^2}{7 d} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.51, antiderivative size = 255, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 8, integrand size = 43, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.186, Rules used = {4088, 4018, 3997, 3787, 3771, 2641, 3768, 2639} \[ \frac {2 a^2 (35 A+49 B+33 C) \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x)}{105 d}+\frac {4 a^2 (5 A+4 B+3 C) \sin (c+d x) \sqrt {\sec (c+d x)}}{5 d}+\frac {4 a^2 (14 A+7 B+6 C) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} F\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{21 d}-\frac {4 a^2 (5 A+4 B+3 C) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{5 d}+\frac {2 (7 B+4 C) \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x) \left (a^2 \sec (c+d x)+a^2\right )}{35 d}+\frac {2 C \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x) (a \sec (c+d x)+a)^2}{7 d} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 2639
Rule 2641
Rule 3768
Rule 3771
Rule 3787
Rule 3997
Rule 4018
Rule 4088
Rubi steps
\begin {align*} \int \sqrt {\sec (c+d x)} (a+a \sec (c+d x))^2 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx &=\frac {2 C \sec ^{\frac {3}{2}}(c+d x) (a+a \sec (c+d x))^2 \sin (c+d x)}{7 d}+\frac {2 \int \sqrt {\sec (c+d x)} (a+a \sec (c+d x))^2 \left (\frac {1}{2} a (7 A+C)+\frac {1}{2} a (7 B+4 C) \sec (c+d x)\right ) \, dx}{7 a}\\ &=\frac {2 C \sec ^{\frac {3}{2}}(c+d x) (a+a \sec (c+d x))^2 \sin (c+d x)}{7 d}+\frac {2 (7 B+4 C) \sec ^{\frac {3}{2}}(c+d x) \left (a^2+a^2 \sec (c+d x)\right ) \sin (c+d x)}{35 d}+\frac {4 \int \sqrt {\sec (c+d x)} (a+a \sec (c+d x)) \left (\frac {1}{4} a^2 (35 A+7 B+9 C)+\frac {1}{4} a^2 (35 A+49 B+33 C) \sec (c+d x)\right ) \, dx}{35 a}\\ &=\frac {2 a^2 (35 A+49 B+33 C) \sec ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{105 d}+\frac {2 C \sec ^{\frac {3}{2}}(c+d x) (a+a \sec (c+d x))^2 \sin (c+d x)}{7 d}+\frac {2 (7 B+4 C) \sec ^{\frac {3}{2}}(c+d x) \left (a^2+a^2 \sec (c+d x)\right ) \sin (c+d x)}{35 d}+\frac {8 \int \sqrt {\sec (c+d x)} \left (\frac {5}{4} a^3 (14 A+7 B+6 C)+\frac {21}{4} a^3 (5 A+4 B+3 C) \sec (c+d x)\right ) \, dx}{105 a}\\ &=\frac {2 a^2 (35 A+49 B+33 C) \sec ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{105 d}+\frac {2 C \sec ^{\frac {3}{2}}(c+d x) (a+a \sec (c+d x))^2 \sin (c+d x)}{7 d}+\frac {2 (7 B+4 C) \sec ^{\frac {3}{2}}(c+d x) \left (a^2+a^2 \sec (c+d x)\right ) \sin (c+d x)}{35 d}+\frac {1}{5} \left (2 a^2 (5 A+4 B+3 C)\right ) \int \sec ^{\frac {3}{2}}(c+d x) \, dx+\frac {1}{21} \left (2 a^2 (14 A+7 B+6 C)\right ) \int \sqrt {\sec (c+d x)} \, dx\\ &=\frac {4 a^2 (5 A+4 B+3 C) \sqrt {\sec (c+d x)} \sin (c+d x)}{5 d}+\frac {2 a^2 (35 A+49 B+33 C) \sec ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{105 d}+\frac {2 C \sec ^{\frac {3}{2}}(c+d x) (a+a \sec (c+d x))^2 \sin (c+d x)}{7 d}+\frac {2 (7 B+4 C) \sec ^{\frac {3}{2}}(c+d x) \left (a^2+a^2 \sec (c+d x)\right ) \sin (c+d x)}{35 d}-\frac {1}{5} \left (2 a^2 (5 A+4 B+3 C)\right ) \int \frac {1}{\sqrt {\sec (c+d x)}} \, dx+\frac {1}{21} \left (2 a^2 (14 A+7 B+6 C) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \frac {1}{\sqrt {\cos (c+d x)}} \, dx\\ &=\frac {4 a^2 (14 A+7 B+6 C) \sqrt {\cos (c+d x)} F\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \sqrt {\sec (c+d x)}}{21 d}+\frac {4 a^2 (5 A+4 B+3 C) \sqrt {\sec (c+d x)} \sin (c+d x)}{5 d}+\frac {2 a^2 (35 A+49 B+33 C) \sec ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{105 d}+\frac {2 C \sec ^{\frac {3}{2}}(c+d x) (a+a \sec (c+d x))^2 \sin (c+d x)}{7 d}+\frac {2 (7 B+4 C) \sec ^{\frac {3}{2}}(c+d x) \left (a^2+a^2 \sec (c+d x)\right ) \sin (c+d x)}{35 d}-\frac {1}{5} \left (2 a^2 (5 A+4 B+3 C) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \sqrt {\cos (c+d x)} \, dx\\ &=-\frac {4 a^2 (5 A+4 B+3 C) \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 {4 a^2 (14 A+7 B+6 C) \sqrt {\cos (c+d x)} F\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \sqrt {\sec (c+d x)}}{21 d}+\frac {4 a^2 (5 A+4 B+3 C) \sqrt {\sec (c+d x)} \sin (c+d x)}{5 d}+\frac {2 a^2 (35 A+49 B+33 C) \sec ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{105 d}+\frac {2 C \sec ^{\frac {3}{2}}(c+d x) (a+a \sec (c+d x))^2 \sin (c+d x)}{7 d}+\frac {2 (7 B+4 C) \sec ^{\frac {3}{2}}(c+d x) \left (a^2+a^2 \sec (c+d x)\right ) \sin (c+d x)}{35 d}\\ \end {align*}
________________________________________________________________________________________
Mathematica [C] time = 7.05, size = 1216, normalized size = 4.77 \[ \frac {\sqrt {2} A e^{-i d x} \sqrt {\frac {e^{i (c+d x)}}{1+e^{2 i (c+d x)}}} \sqrt {1+e^{2 i (c+d x)}} \cos ^4(c+d x) \csc (c) \left (e^{2 i d x} \left (-1+e^{2 i c}\right ) \, _2F_1\left (\frac {1}{2},\frac {3}{4};\frac {7}{4};-e^{2 i (c+d x)}\right )-3 \sqrt {1+e^{2 i (c+d x)}}\right ) (\sec (c+d x) a+a)^2 \left (C \sec ^2(c+d x)+B \sec (c+d x)+A\right ) \sec ^4\left (\frac {c}{2}+\frac {d x}{2}\right )}{3 d (\cos (2 c+2 d x) A+A+2 C+2 B \cos (c+d x))}+\frac {4 \sqrt {2} B e^{-i d x} \sqrt {\frac {e^{i (c+d x)}}{1+e^{2 i (c+d x)}}} \sqrt {1+e^{2 i (c+d x)}} \cos ^4(c+d x) \csc (c) \left (e^{2 i d x} \left (-1+e^{2 i c}\right ) \, _2F_1\left (\frac {1}{2},\frac {3}{4};\frac {7}{4};-e^{2 i (c+d x)}\right )-3 \sqrt {1+e^{2 i (c+d x)}}\right ) (\sec (c+d x) a+a)^2 \left (C \sec ^2(c+d x)+B \sec (c+d x)+A\right ) \sec ^4\left (\frac {c}{2}+\frac {d x}{2}\right )}{15 d (\cos (2 c+2 d x) A+A+2 C+2 B \cos (c+d x))}+\frac {\sqrt {2} C e^{-i d x} \sqrt {\frac {e^{i (c+d x)}}{1+e^{2 i (c+d x)}}} \sqrt {1+e^{2 i (c+d x)}} \cos ^4(c+d x) \csc (c) \left (e^{2 i d x} \left (-1+e^{2 i c}\right ) \, _2F_1\left (\frac {1}{2},\frac {3}{4};\frac {7}{4};-e^{2 i (c+d x)}\right )-3 \sqrt {1+e^{2 i (c+d x)}}\right ) (\sec (c+d x) a+a)^2 \left (C \sec ^2(c+d x)+B \sec (c+d x)+A\right ) \sec ^4\left (\frac {c}{2}+\frac {d x}{2}\right )}{5 d (\cos (2 c+2 d x) A+A+2 C+2 B \cos (c+d x))}+\frac {4 A \sqrt {\cos (c+d x)} F\left (\left .\frac {1}{2} (c+d x)\right |2\right ) (\sec (c+d x) a+a)^2 \left (C \sec ^2(c+d x)+B \sec (c+d x)+A\right ) \sec ^4\left (\frac {c}{2}+\frac {d x}{2}\right )}{3 d (\cos (2 c+2 d x) A+A+2 C+2 B \cos (c+d x)) \sec ^{\frac {7}{2}}(c+d x)}+\frac {2 B \sqrt {\cos (c+d x)} F\left (\left .\frac {1}{2} (c+d x)\right |2\right ) (\sec (c+d x) a+a)^2 \left (C \sec ^2(c+d x)+B \sec (c+d x)+A\right ) \sec ^4\left (\frac {c}{2}+\frac {d x}{2}\right )}{3 d (\cos (2 c+2 d x) A+A+2 C+2 B \cos (c+d x)) \sec ^{\frac {7}{2}}(c+d x)}+\frac {4 C \sqrt {\cos (c+d x)} F\left (\left .\frac {1}{2} (c+d x)\right |2\right ) (\sec (c+d x) a+a)^2 \left (C \sec ^2(c+d x)+B \sec (c+d x)+A\right ) \sec ^4\left (\frac {c}{2}+\frac {d x}{2}\right )}{7 d (\cos (2 c+2 d x) A+A+2 C+2 B \cos (c+d x)) \sec ^{\frac {7}{2}}(c+d x)}+\frac {(\sec (c+d x) a+a)^2 \left (C \sec ^2(c+d x)+B \sec (c+d x)+A\right ) \left (\frac {C \sec (c) \sin (d x) \sec ^3(c+d x)}{7 d}+\frac {\sec (c) (5 C \sin (c)+7 B \sin (d x)+14 C \sin (d x)) \sec ^2(c+d x)}{35 d}+\frac {\sec (c) (21 B \sin (c)+42 C \sin (c)+35 A \sin (d x)+70 B \sin (d x)+60 C \sin (d x)) \sec (c+d x)}{105 d}+\frac {2 (5 A+4 B+3 C) \cos (d x) \csc (c)}{5 d}+\frac {(7 A+14 B+12 C) \tan (c)}{21 d}\right ) \sec ^4\left (\frac {c}{2}+\frac {d x}{2}\right )}{(\cos (2 c+2 d x) A+A+2 C+2 B \cos (c+d x)) \sec ^{\frac {7}{2}}(c+d x)} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [F] time = 0.46, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (C a^{2} \sec \left (d x + c\right )^{4} + {\left (B + 2 \, C\right )} a^{2} \sec \left (d x + c\right )^{3} + {\left (A + 2 \, B + C\right )} a^{2} \sec \left (d x + c\right )^{2} + {\left (2 \, A + B\right )} a^{2} \sec \left (d x + c\right ) + A a^{2}\right )} \sqrt {\sec \left (d x + c\right )}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (C \sec \left (d x + c\right )^{2} + B \sec \left (d x + c\right ) + A\right )} {\left (a \sec \left (d x + c\right ) + a\right )}^{2} \sqrt {\sec \left (d x + c\right )}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [B] time = 17.16, size = 934, normalized size = 3.66 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (C \sec \left (d x + c\right )^{2} + B \sec \left (d x + c\right ) + A\right )} {\left (a \sec \left (d x + c\right ) + a\right )}^{2} \sqrt {\sec \left (d x + c\right )}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int {\left (a+\frac {a}{\cos \left (c+d\,x\right )}\right )}^2\,\sqrt {\frac {1}{\cos \left (c+d\,x\right )}}\,\left (A+\frac {B}{\cos \left (c+d\,x\right )}+\frac {C}{{\cos \left (c+d\,x\right )}^2}\right ) \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________