Optimal. Leaf size=209 \[ \frac {2 A b^3 \sin (c+d x)}{7 d (b \cos (c+d x))^{7/2}}+\frac {2 b (5 A+7 C) \sin (c+d x)}{21 d (b \cos (c+d x))^{3/2}}+\frac {2 (5 A+7 C) \sqrt {\cos (c+d x)} F\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{21 d \sqrt {b \cos (c+d x)}}+\frac {2 b^2 B \sin (c+d x)}{5 d (b \cos (c+d x))^{5/2}}+\frac {6 B \sin (c+d x)}{5 d \sqrt {b \cos (c+d x)}}-\frac {6 B E\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \sqrt {b \cos (c+d x)}}{5 b d \sqrt {\cos (c+d x)}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.26, antiderivative size = 209, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 8, integrand size = 41, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.195, Rules used = {16, 3021, 2748, 2636, 2640, 2639, 2642, 2641} \[ \frac {2 A b^3 \sin (c+d x)}{7 d (b \cos (c+d x))^{7/2}}+\frac {2 b (5 A+7 C) \sin (c+d x)}{21 d (b \cos (c+d x))^{3/2}}+\frac {2 (5 A+7 C) \sqrt {\cos (c+d x)} F\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{21 d \sqrt {b \cos (c+d x)}}+\frac {2 b^2 B \sin (c+d x)}{5 d (b \cos (c+d x))^{5/2}}+\frac {6 B \sin (c+d x)}{5 d \sqrt {b \cos (c+d x)}}-\frac {6 B E\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \sqrt {b \cos (c+d x)}}{5 b d \sqrt {\cos (c+d x)}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 16
Rule 2636
Rule 2639
Rule 2640
Rule 2641
Rule 2642
Rule 2748
Rule 3021
Rubi steps
\begin {align*} \int \frac {\left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec ^4(c+d x)}{\sqrt {b \cos (c+d x)}} \, dx &=b^4 \int \frac {A+B \cos (c+d x)+C \cos ^2(c+d x)}{(b \cos (c+d x))^{9/2}} \, dx\\ &=\frac {2 A b^3 \sin (c+d x)}{7 d (b \cos (c+d x))^{7/2}}+\frac {1}{7} (2 b) \int \frac {\frac {7 b^2 B}{2}+\frac {1}{2} b^2 (5 A+7 C) \cos (c+d x)}{(b \cos (c+d x))^{7/2}} \, dx\\ &=\frac {2 A b^3 \sin (c+d x)}{7 d (b \cos (c+d x))^{7/2}}+\left (b^3 B\right ) \int \frac {1}{(b \cos (c+d x))^{7/2}} \, dx+\frac {1}{7} \left (b^2 (5 A+7 C)\right ) \int \frac {1}{(b \cos (c+d x))^{5/2}} \, dx\\ &=\frac {2 A b^3 \sin (c+d x)}{7 d (b \cos (c+d x))^{7/2}}+\frac {2 b^2 B \sin (c+d x)}{5 d (b \cos (c+d x))^{5/2}}+\frac {2 b (5 A+7 C) \sin (c+d x)}{21 d (b \cos (c+d x))^{3/2}}+\frac {1}{5} (3 b B) \int \frac {1}{(b \cos (c+d x))^{3/2}} \, dx+\frac {1}{21} (5 A+7 C) \int \frac {1}{\sqrt {b \cos (c+d x)}} \, dx\\ &=\frac {2 A b^3 \sin (c+d x)}{7 d (b \cos (c+d x))^{7/2}}+\frac {2 b^2 B \sin (c+d x)}{5 d (b \cos (c+d x))^{5/2}}+\frac {2 b (5 A+7 C) \sin (c+d x)}{21 d (b \cos (c+d x))^{3/2}}+\frac {6 B \sin (c+d x)}{5 d \sqrt {b \cos (c+d x)}}-\frac {(3 B) \int \sqrt {b \cos (c+d x)} \, dx}{5 b}+\frac {\left ((5 A+7 C) \sqrt {\cos (c+d x)}\right ) \int \frac {1}{\sqrt {\cos (c+d x)}} \, dx}{21 \sqrt {b \cos (c+d x)}}\\ &=\frac {2 (5 A+7 C) \sqrt {\cos (c+d x)} F\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{21 d \sqrt {b \cos (c+d x)}}+\frac {2 A b^3 \sin (c+d x)}{7 d (b \cos (c+d x))^{7/2}}+\frac {2 b^2 B \sin (c+d x)}{5 d (b \cos (c+d x))^{5/2}}+\frac {2 b (5 A+7 C) \sin (c+d x)}{21 d (b \cos (c+d x))^{3/2}}+\frac {6 B \sin (c+d x)}{5 d \sqrt {b \cos (c+d x)}}-\frac {\left (3 B \sqrt {b \cos (c+d x)}\right ) \int \sqrt {\cos (c+d x)} \, dx}{5 b \sqrt {\cos (c+d x)}}\\ &=-\frac {6 B \sqrt {b \cos (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{5 b d \sqrt {\cos (c+d x)}}+\frac {2 (5 A+7 C) \sqrt {\cos (c+d x)} F\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{21 d \sqrt {b \cos (c+d x)}}+\frac {2 A b^3 \sin (c+d x)}{7 d (b \cos (c+d x))^{7/2}}+\frac {2 b^2 B \sin (c+d x)}{5 d (b \cos (c+d x))^{5/2}}+\frac {2 b (5 A+7 C) \sin (c+d x)}{21 d (b \cos (c+d x))^{3/2}}+\frac {6 B \sin (c+d x)}{5 d \sqrt {b \cos (c+d x)}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.69, size = 133, normalized size = 0.64 \[ \frac {2 \left (5 (5 A+7 C) \sqrt {\cos (c+d x)} F\left (\left .\frac {1}{2} (c+d x)\right |2\right )+25 A \tan (c+d x)+15 A \tan (c+d x) \sec ^2(c+d x)+63 B \sin (c+d x)-63 B \sqrt {\cos (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right )+21 B \tan (c+d x) \sec (c+d x)+35 C \tan (c+d x)\right )}{105 d \sqrt {b \cos (c+d x)}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [F] time = 0.52, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (C \cos \left (d x + c\right )^{2} + B \cos \left (d x + c\right ) + A\right )} \sqrt {b \cos \left (d x + c\right )} \sec \left (d x + c\right )^{4}}{b \cos \left (d x + c\right )}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (C \cos \left (d x + c\right )^{2} + B \cos \left (d x + c\right ) + A\right )} \sec \left (d x + c\right )^{4}}{\sqrt {b \cos \left (d x + c\right )}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [B] time = 5.06, size = 726, normalized size = 3.47 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (C \cos \left (d x + c\right )^{2} + B \cos \left (d x + c\right ) + A\right )} \sec \left (d x + c\right )^{4}}{\sqrt {b \cos \left (d x + c\right )}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {C\,{\cos \left (c+d\,x\right )}^2+B\,\cos \left (c+d\,x\right )+A}{{\cos \left (c+d\,x\right )}^4\,\sqrt {b\,\cos \left (c+d\,x\right )}} \,d x \]
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]
________________________________________________________________________________________