Optimal. Leaf size=100 \[ -\frac {6 \sqrt {c \cos (a+b x)} E\left (\left .\frac {1}{2} (a+b x)\right |2\right )}{5 b c^4 \sqrt {\cos (a+b x)}}+\frac {2 \sin (a+b x)}{5 b c (c \cos (a+b x))^{5/2}}+\frac {6 \sin (a+b x)}{5 b c^3 \sqrt {c \cos (a+b x)}} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.04, antiderivative size = 100, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 3, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {2716, 2721,
2719} \begin {gather*} -\frac {6 E\left (\left .\frac {1}{2} (a+b x)\right |2\right ) \sqrt {c \cos (a+b x)}}{5 b c^4 \sqrt {\cos (a+b x)}}+\frac {6 \sin (a+b x)}{5 b c^3 \sqrt {c \cos (a+b x)}}+\frac {2 \sin (a+b x)}{5 b c (c \cos (a+b x))^{5/2}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 2716
Rule 2719
Rule 2721
Rubi steps
\begin {align*} \int \frac {1}{(c \cos (a+b x))^{7/2}} \, dx &=\frac {2 \sin (a+b x)}{5 b c (c \cos (a+b x))^{5/2}}+\frac {3 \int \frac {1}{(c \cos (a+b x))^{3/2}} \, dx}{5 c^2}\\ &=\frac {2 \sin (a+b x)}{5 b c (c \cos (a+b x))^{5/2}}+\frac {6 \sin (a+b x)}{5 b c^3 \sqrt {c \cos (a+b x)}}-\frac {3 \int \sqrt {c \cos (a+b x)} \, dx}{5 c^4}\\ &=\frac {2 \sin (a+b x)}{5 b c (c \cos (a+b x))^{5/2}}+\frac {6 \sin (a+b x)}{5 b c^3 \sqrt {c \cos (a+b x)}}-\frac {\left (3 \sqrt {c \cos (a+b x)}\right ) \int \sqrt {\cos (a+b x)} \, dx}{5 c^4 \sqrt {\cos (a+b x)}}\\ &=-\frac {6 \sqrt {c \cos (a+b x)} E\left (\left .\frac {1}{2} (a+b x)\right |2\right )}{5 b c^4 \sqrt {\cos (a+b x)}}+\frac {2 \sin (a+b x)}{5 b c (c \cos (a+b x))^{5/2}}+\frac {6 \sin (a+b x)}{5 b c^3 \sqrt {c \cos (a+b x)}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 0.06, size = 68, normalized size = 0.68 \begin {gather*} \frac {-6 \sqrt {\cos (a+b x)} E\left (\left .\frac {1}{2} (a+b x)\right |2\right )+6 \sin (a+b x)+2 \sec (a+b x) \tan (a+b x)}{5 b c^3 \sqrt {c \cos (a+b x)}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(365\) vs.
\(2(112)=224\).
time = 0.06, size = 366, normalized size = 3.66
method | result | size |
default | \(-\frac {2 \sqrt {c \left (2 \left (\cos ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )-1\right ) \left (\sin ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )}\, \left (24 \cos \left (\frac {b x}{2}+\frac {a}{2}\right ) \left (\sin ^{6}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )-12 \sqrt {2 \left (\sin ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )-1}\, \sqrt {\frac {1}{2}-\frac {\cos \left (b x +a \right )}{2}}\, \EllipticE \left (\cos \left (\frac {b x}{2}+\frac {a}{2}\right ), \sqrt {2}\right ) \left (\sin ^{4}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )-24 \left (\sin ^{4}\left (\frac {b x}{2}+\frac {a}{2}\right )\right ) \cos \left (\frac {b x}{2}+\frac {a}{2}\right )+12 \sqrt {2 \left (\sin ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )-1}\, \sqrt {\frac {1}{2}-\frac {\cos \left (b x +a \right )}{2}}\, \EllipticE \left (\cos \left (\frac {b x}{2}+\frac {a}{2}\right ), \sqrt {2}\right ) \left (\sin ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )+8 \left (\sin ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )\right ) \cos \left (\frac {b x}{2}+\frac {a}{2}\right )-3 \sqrt {\frac {1}{2}-\frac {\cos \left (b x +a \right )}{2}}\, \sqrt {2 \left (\sin ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )-1}\, \EllipticE \left (\cos \left (\frac {b x}{2}+\frac {a}{2}\right ), \sqrt {2}\right )\right ) \sqrt {-2 \left (\sin ^{4}\left (\frac {b x}{2}+\frac {a}{2}\right )\right ) c +c \left (\sin ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )}}{5 c^{4} \sin \left (\frac {b x}{2}+\frac {a}{2}\right )^{3} \left (8 \left (\sin ^{6}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )-12 \left (\sin ^{4}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )+6 \left (\sin ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )-1\right ) \sqrt {c \left (2 \left (\cos ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )-1\right )}\, b}\) | \(366\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [C] Result contains higher order function than in optimal. Order 9 vs. order
4.
time = 0.10, size = 121, normalized size = 1.21 \begin {gather*} \frac {-3 i \, \sqrt {2} \sqrt {c} \cos \left (b x + a\right )^{3} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (b x + a\right ) + i \, \sin \left (b x + a\right )\right )\right ) + 3 i \, \sqrt {2} \sqrt {c} \cos \left (b x + a\right )^{3} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (b x + a\right ) - i \, \sin \left (b x + a\right )\right )\right ) + 2 \, \sqrt {c \cos \left (b x + a\right )} {\left (3 \, \cos \left (b x + a\right )^{2} + 1\right )} \sin \left (b x + a\right )}{5 \, b c^{4} \cos \left (b x + a\right )^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: SystemError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {1}{{\left (c\,\cos \left (a+b\,x\right )\right )}^{7/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________