Optimal. Leaf size=247 \[ \frac {2 b^2 \left (39 a^2+5 b^2\right ) \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x)}{21 d}+\frac {8 a b \left (5 a^2+3 b^2\right ) \sin (c+d x) \sqrt {\sec (c+d x)}}{5 d}-\frac {8 a b \left (5 a^2+3 b^2\right ) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{5 d}+\frac {2 \left (21 a^4+42 a^2 b^2+5 b^4\right ) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} F\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{21 d}+\frac {36 a b^3 \sin (c+d x) \sec ^{\frac {5}{2}}(c+d x)}{35 d}+\frac {2 b^2 \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x) (a+b \sec (c+d x))^2}{7 d} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.37, antiderivative size = 247, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 8, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.348, Rules used = {3842, 4076, 4047, 3768, 3771, 2639, 4046, 2641} \[ \frac {2 b^2 \left (39 a^2+5 b^2\right ) \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x)}{21 d}+\frac {8 a b \left (5 a^2+3 b^2\right ) \sin (c+d x) \sqrt {\sec (c+d x)}}{5 d}+\frac {2 \left (42 a^2 b^2+21 a^4+5 b^4\right ) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} F\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{21 d}-\frac {8 a b \left (5 a^2+3 b^2\right ) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{5 d}+\frac {36 a b^3 \sin (c+d x) \sec ^{\frac {5}{2}}(c+d x)}{35 d}+\frac {2 b^2 \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x) (a+b \sec (c+d x))^2}{7 d} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 2639
Rule 2641
Rule 3768
Rule 3771
Rule 3842
Rule 4046
Rule 4047
Rule 4076
Rubi steps
\begin {align*} \int \sqrt {\sec (c+d x)} (a+b \sec (c+d x))^4 \, dx &=\frac {2 b^2 \sec ^{\frac {3}{2}}(c+d x) (a+b \sec (c+d x))^2 \sin (c+d x)}{7 d}+\frac {2}{7} \int \sqrt {\sec (c+d x)} (a+b \sec (c+d x)) \left (\frac {1}{2} a \left (7 a^2+b^2\right )+\frac {1}{2} b \left (21 a^2+5 b^2\right ) \sec (c+d x)+9 a b^2 \sec ^2(c+d x)\right ) \, dx\\ &=\frac {36 a b^3 \sec ^{\frac {5}{2}}(c+d x) \sin (c+d x)}{35 d}+\frac {2 b^2 \sec ^{\frac {3}{2}}(c+d x) (a+b \sec (c+d x))^2 \sin (c+d x)}{7 d}+\frac {4}{35} \int \sqrt {\sec (c+d x)} \left (\frac {5}{4} a^2 \left (7 a^2+b^2\right )+7 a b \left (5 a^2+3 b^2\right ) \sec (c+d x)+\frac {5}{4} b^2 \left (39 a^2+5 b^2\right ) \sec ^2(c+d x)\right ) \, dx\\ &=\frac {36 a b^3 \sec ^{\frac {5}{2}}(c+d x) \sin (c+d x)}{35 d}+\frac {2 b^2 \sec ^{\frac {3}{2}}(c+d x) (a+b \sec (c+d x))^2 \sin (c+d x)}{7 d}+\frac {4}{35} \int \sqrt {\sec (c+d x)} \left (\frac {5}{4} a^2 \left (7 a^2+b^2\right )+\frac {5}{4} b^2 \left (39 a^2+5 b^2\right ) \sec ^2(c+d x)\right ) \, dx+\frac {1}{5} \left (4 a b \left (5 a^2+3 b^2\right )\right ) \int \sec ^{\frac {3}{2}}(c+d x) \, dx\\ &=\frac {8 a b \left (5 a^2+3 b^2\right ) \sqrt {\sec (c+d x)} \sin (c+d x)}{5 d}+\frac {2 b^2 \left (39 a^2+5 b^2\right ) \sec ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{21 d}+\frac {36 a b^3 \sec ^{\frac {5}{2}}(c+d x) \sin (c+d x)}{35 d}+\frac {2 b^2 \sec ^{\frac {3}{2}}(c+d x) (a+b \sec (c+d x))^2 \sin (c+d x)}{7 d}-\frac {1}{5} \left (4 a b \left (5 a^2+3 b^2\right )\right ) \int \frac {1}{\sqrt {\sec (c+d x)}} \, dx+\frac {1}{21} \left (21 a^4+42 a^2 b^2+5 b^4\right ) \int \sqrt {\sec (c+d x)} \, dx\\ &=\frac {8 a b \left (5 a^2+3 b^2\right ) \sqrt {\sec (c+d x)} \sin (c+d x)}{5 d}+\frac {2 b^2 \left (39 a^2+5 b^2\right ) \sec ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{21 d}+\frac {36 a b^3 \sec ^{\frac {5}{2}}(c+d x) \sin (c+d x)}{35 d}+\frac {2 b^2 \sec ^{\frac {3}{2}}(c+d x) (a+b \sec (c+d x))^2 \sin (c+d x)}{7 d}-\frac {1}{5} \left (4 a b \left (5 a^2+3 b^2\right ) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \sqrt {\cos (c+d x)} \, dx+\frac {1}{21} \left (\left (21 a^4+42 a^2 b^2+5 b^4\right ) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \frac {1}{\sqrt {\cos (c+d x)}} \, dx\\ &=-\frac {8 a b \left (5 a^2+3 b^2\right ) \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 {2 \left (21 a^4+42 a^2 b^2+5 b^4\right ) \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 {8 a b \left (5 a^2+3 b^2\right ) \sqrt {\sec (c+d x)} \sin (c+d x)}{5 d}+\frac {2 b^2 \left (39 a^2+5 b^2\right ) \sec ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{21 d}+\frac {36 a b^3 \sec ^{\frac {5}{2}}(c+d x) \sin (c+d x)}{35 d}+\frac {2 b^2 \sec ^{\frac {3}{2}}(c+d x) (a+b \sec (c+d x))^2 \sin (c+d x)}{7 d}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 1.72, size = 168, normalized size = 0.68 \[ \frac {2 \sec ^{\frac {7}{2}}(c+d x) \left (-84 a b \left (5 a^2+3 b^2\right ) \cos ^{\frac {7}{2}}(c+d x) E\left (\left .\frac {1}{2} (c+d x)\right |2\right )+b \sin (c+d x) \left (84 a \left (5 a^2+3 b^2\right ) \cos ^3(c+d x)+5 b \left (42 a^2+5 b^2\right ) \cos ^2(c+d x)+15 b^3\right )+5 \left (21 a^4+42 a^2 b^2+5 b^4\right ) \cos ^{\frac {7}{2}}(c+d x) F\left (\left .\frac {1}{2} (c+d x)\right |2\right )+42 a b^3 \sin (2 (c+d x))\right )}{105 d} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [F] time = 0.52, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (b^{4} \sec \left (d x + c\right )^{4} + 4 \, a b^{3} \sec \left (d x + c\right )^{3} + 6 \, a^{2} b^{2} \sec \left (d x + c\right )^{2} + 4 \, a^{3} b \sec \left (d x + c\right ) + a^{4}\right )} \sqrt {\sec \left (d x + c\right )}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (b \sec \left (d x + c\right ) + a\right )}^{4} \sqrt {\sec \left (d x + c\right )}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [B] time = 12.66, size = 925, normalized size = 3.74 \[ \text {result too large to display} \]
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 \[ \int {\left (a+\frac {b}{\cos \left (c+d\,x\right )}\right )}^4\,\sqrt {\frac {1}{\cos \left (c+d\,x\right )}} \,d x \]
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]
________________________________________________________________________________________