Optimal. Leaf size=277 \[ \frac {4 a^3 (24 A+23 B) \sin (c+d x) \sec ^{\frac {5}{2}}(c+d x)}{105 d}+\frac {4 a^3 (13 A+11 B) \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x)}{21 d}+\frac {2 (9 A+13 B) \sin (c+d x) \sec ^{\frac {5}{2}}(c+d x) \left (a^3 \sec (c+d x)+a^3\right )}{63 d}+\frac {4 a^3 (21 A+17 B) \sin (c+d x) \sqrt {\sec (c+d x)}}{15 d}+\frac {4 a^3 (13 A+11 B) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} F\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{21 d}-\frac {4 a^3 (21 A+17 B) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{15 d}+\frac {2 a B \sin (c+d x) \sec ^{\frac {5}{2}}(c+d x) (a \sec (c+d x)+a)^2}{9 d} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.54, antiderivative size = 277, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 7, integrand size = 33, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.212, Rules used = {4018, 3997, 3787, 3768, 3771, 2639, 2641} \[ \frac {4 a^3 (24 A+23 B) \sin (c+d x) \sec ^{\frac {5}{2}}(c+d x)}{105 d}+\frac {4 a^3 (13 A+11 B) \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x)}{21 d}+\frac {2 (9 A+13 B) \sin (c+d x) \sec ^{\frac {5}{2}}(c+d x) \left (a^3 \sec (c+d x)+a^3\right )}{63 d}+\frac {4 a^3 (21 A+17 B) \sin (c+d x) \sqrt {\sec (c+d x)}}{15 d}+\frac {4 a^3 (13 A+11 B) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} F\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{21 d}-\frac {4 a^3 (21 A+17 B) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{15 d}+\frac {2 a B \sin (c+d x) \sec ^{\frac {5}{2}}(c+d x) (a \sec (c+d x)+a)^2}{9 d} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 2639
Rule 2641
Rule 3768
Rule 3771
Rule 3787
Rule 3997
Rule 4018
Rubi steps
\begin {align*} \int \sec ^{\frac {3}{2}}(c+d x) (a+a \sec (c+d x))^3 (A+B \sec (c+d x)) \, dx &=\frac {2 a B \sec ^{\frac {5}{2}}(c+d x) (a+a \sec (c+d x))^2 \sin (c+d x)}{9 d}+\frac {2}{9} \int \sec ^{\frac {3}{2}}(c+d x) (a+a \sec (c+d x))^2 \left (\frac {3}{2} a (3 A+B)+\frac {1}{2} a (9 A+13 B) \sec (c+d x)\right ) \, dx\\ &=\frac {2 a B \sec ^{\frac {5}{2}}(c+d x) (a+a \sec (c+d x))^2 \sin (c+d x)}{9 d}+\frac {2 (9 A+13 B) \sec ^{\frac {5}{2}}(c+d x) \left (a^3+a^3 \sec (c+d x)\right ) \sin (c+d x)}{63 d}+\frac {4}{63} \int \sec ^{\frac {3}{2}}(c+d x) (a+a \sec (c+d x)) \left (\frac {15}{2} a^2 (3 A+2 B)+\frac {3}{2} a^2 (24 A+23 B) \sec (c+d x)\right ) \, dx\\ &=\frac {4 a^3 (24 A+23 B) \sec ^{\frac {5}{2}}(c+d x) \sin (c+d x)}{105 d}+\frac {2 a B \sec ^{\frac {5}{2}}(c+d x) (a+a \sec (c+d x))^2 \sin (c+d x)}{9 d}+\frac {2 (9 A+13 B) \sec ^{\frac {5}{2}}(c+d x) \left (a^3+a^3 \sec (c+d x)\right ) \sin (c+d x)}{63 d}+\frac {8}{315} \int \sec ^{\frac {3}{2}}(c+d x) \left (\frac {21}{4} a^3 (21 A+17 B)+\frac {45}{4} a^3 (13 A+11 B) \sec (c+d x)\right ) \, dx\\ &=\frac {4 a^3 (24 A+23 B) \sec ^{\frac {5}{2}}(c+d x) \sin (c+d x)}{105 d}+\frac {2 a B \sec ^{\frac {5}{2}}(c+d x) (a+a \sec (c+d x))^2 \sin (c+d x)}{9 d}+\frac {2 (9 A+13 B) \sec ^{\frac {5}{2}}(c+d x) \left (a^3+a^3 \sec (c+d x)\right ) \sin (c+d x)}{63 d}+\frac {1}{7} \left (2 a^3 (13 A+11 B)\right ) \int \sec ^{\frac {5}{2}}(c+d x) \, dx+\frac {1}{15} \left (2 a^3 (21 A+17 B)\right ) \int \sec ^{\frac {3}{2}}(c+d x) \, dx\\ &=\frac {4 a^3 (21 A+17 B) \sqrt {\sec (c+d x)} \sin (c+d x)}{15 d}+\frac {4 a^3 (13 A+11 B) \sec ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{21 d}+\frac {4 a^3 (24 A+23 B) \sec ^{\frac {5}{2}}(c+d x) \sin (c+d x)}{105 d}+\frac {2 a B \sec ^{\frac {5}{2}}(c+d x) (a+a \sec (c+d x))^2 \sin (c+d x)}{9 d}+\frac {2 (9 A+13 B) \sec ^{\frac {5}{2}}(c+d x) \left (a^3+a^3 \sec (c+d x)\right ) \sin (c+d x)}{63 d}+\frac {1}{21} \left (2 a^3 (13 A+11 B)\right ) \int \sqrt {\sec (c+d x)} \, dx-\frac {1}{15} \left (2 a^3 (21 A+17 B)\right ) \int \frac {1}{\sqrt {\sec (c+d x)}} \, dx\\ &=\frac {4 a^3 (21 A+17 B) \sqrt {\sec (c+d x)} \sin (c+d x)}{15 d}+\frac {4 a^3 (13 A+11 B) \sec ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{21 d}+\frac {4 a^3 (24 A+23 B) \sec ^{\frac {5}{2}}(c+d x) \sin (c+d x)}{105 d}+\frac {2 a B \sec ^{\frac {5}{2}}(c+d x) (a+a \sec (c+d x))^2 \sin (c+d x)}{9 d}+\frac {2 (9 A+13 B) \sec ^{\frac {5}{2}}(c+d x) \left (a^3+a^3 \sec (c+d x)\right ) \sin (c+d x)}{63 d}+\frac {1}{21} \left (2 a^3 (13 A+11 B) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \frac {1}{\sqrt {\cos (c+d x)}} \, dx-\frac {1}{15} \left (2 a^3 (21 A+17 B) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \sqrt {\cos (c+d x)} \, dx\\ &=-\frac {4 a^3 (21 A+17 B) \sqrt {\cos (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \sqrt {\sec (c+d x)}}{15 d}+\frac {4 a^3 (13 A+11 B) \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^3 (21 A+17 B) \sqrt {\sec (c+d x)} \sin (c+d x)}{15 d}+\frac {4 a^3 (13 A+11 B) \sec ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{21 d}+\frac {4 a^3 (24 A+23 B) \sec ^{\frac {5}{2}}(c+d x) \sin (c+d x)}{105 d}+\frac {2 a B \sec ^{\frac {5}{2}}(c+d x) (a+a \sec (c+d x))^2 \sin (c+d x)}{9 d}+\frac {2 (9 A+13 B) \sec ^{\frac {5}{2}}(c+d x) \left (a^3+a^3 \sec (c+d x)\right ) \sin (c+d x)}{63 d}\\ \end {align*}
________________________________________________________________________________________
Mathematica [C] time = 6.96, size = 793, normalized size = 2.86 \[ \frac {7 A \csc (c) e^{-i d x} \sqrt {\frac {e^{i (c+d x)}}{1+e^{2 i (c+d x)}}} \sqrt {1+e^{2 i (c+d x)}} \left (\left (-1+e^{2 i c}\right ) e^{2 i d x} \, _2F_1\left (\frac {1}{2},\frac {3}{4};\frac {7}{4};-e^{2 i (c+d x)}\right )-3 \sqrt {1+e^{2 i (c+d x)}}\right ) \cos ^4(c+d x) \sec ^6\left (\frac {c}{2}+\frac {d x}{2}\right ) (a \sec (c+d x)+a)^3 (A+B \sec (c+d x))}{30 \sqrt {2} d (A \cos (c+d x)+B)}+\frac {17 B \csc (c) e^{-i d x} \sqrt {\frac {e^{i (c+d x)}}{1+e^{2 i (c+d x)}}} \sqrt {1+e^{2 i (c+d x)}} \left (\left (-1+e^{2 i c}\right ) e^{2 i d x} \, _2F_1\left (\frac {1}{2},\frac {3}{4};\frac {7}{4};-e^{2 i (c+d x)}\right )-3 \sqrt {1+e^{2 i (c+d x)}}\right ) \cos ^4(c+d x) \sec ^6\left (\frac {c}{2}+\frac {d x}{2}\right ) (a \sec (c+d x)+a)^3 (A+B \sec (c+d x))}{90 \sqrt {2} d (A \cos (c+d x)+B)}+\frac {13 A \sqrt {\cos (c+d x)} \sec ^6\left (\frac {c}{2}+\frac {d x}{2}\right ) F\left (\left .\frac {1}{2} (c+d x)\right |2\right ) (a \sec (c+d x)+a)^3 (A+B \sec (c+d x))}{42 d \sec ^{\frac {7}{2}}(c+d x) (A \cos (c+d x)+B)}+\frac {11 B \sqrt {\cos (c+d x)} \sec ^6\left (\frac {c}{2}+\frac {d x}{2}\right ) F\left (\left .\frac {1}{2} (c+d x)\right |2\right ) (a \sec (c+d x)+a)^3 (A+B \sec (c+d x))}{42 d \sec ^{\frac {7}{2}}(c+d x) (A \cos (c+d x)+B)}+\frac {\sec ^6\left (\frac {c}{2}+\frac {d x}{2}\right ) (a \sec (c+d x)+a)^3 (A+B \sec (c+d x)) \left (\frac {(21 A+17 B) \csc (c) \cos (d x)}{30 d}+\frac {\sec (c) \sec ^3(c+d x) (9 A \sin (d x)+7 B \sin (c)+27 B \sin (d x))}{252 d}+\frac {\sec (c) \sec ^2(c+d x) (45 A \sin (c)+189 A \sin (d x)+135 B \sin (c)+238 B \sin (d x))}{1260 d}+\frac {\sec (c) \sec (c+d x) (189 A \sin (c)+390 A \sin (d x)+238 B \sin (c)+330 B \sin (d x))}{1260 d}+\frac {(13 A+11 B) \tan (c)}{42 d}+\frac {B \sec (c) \sin (d x) \sec ^4(c+d x)}{36 d}\right )}{\sec ^{\frac {7}{2}}(c+d x) (A \cos (c+d x)+B)} \]
Warning: Unable to verify antiderivative.
[In]
[Out]
________________________________________________________________________________________
fricas [F] time = 0.45, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (B a^{3} \sec \left (d x + c\right )^{5} + {\left (A + 3 \, B\right )} a^{3} \sec \left (d x + c\right )^{4} + 3 \, {\left (A + B\right )} a^{3} \sec \left (d x + c\right )^{3} + {\left (3 \, A + B\right )} a^{3} \sec \left (d x + c\right )^{2} + A a^{3} \sec \left (d x + c\right )\right )} \sqrt {\sec \left (d x + c\right )}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (B \sec \left (d x + c\right ) + A\right )} {\left (a \sec \left (d x + c\right ) + a\right )}^{3} \sec \left (d x + c\right )^{\frac {3}{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [B] time = 18.52, size = 1180, normalized size = 4.26 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [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]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \left (A+\frac {B}{\cos \left (c+d\,x\right )}\right )\,{\left (a+\frac {a}{\cos \left (c+d\,x\right )}\right )}^3\,{\left (\frac {1}{\cos \left (c+d\,x\right )}\right )}^{3/2} \,d x \]
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]
________________________________________________________________________________________