Optimal. Leaf size=70 \[ \frac {\sqrt {\sec (c+d x)} \sqrt {b \sec (c+d x)} \sin (c+d x)}{d}+\frac {\sec ^{\frac {5}{2}}(c+d x) \sqrt {b \sec (c+d x)} \sin ^3(c+d x)}{3 d} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.01, antiderivative size = 70, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 2, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.087, Rules used = {17, 3852}
\begin {gather*} \frac {\sin ^3(c+d x) \sec ^{\frac {5}{2}}(c+d x) \sqrt {b \sec (c+d x)}}{3 d}+\frac {\sin (c+d x) \sqrt {\sec (c+d x)} \sqrt {b \sec (c+d x)}}{d} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 17
Rule 3852
Rubi steps
\begin {align*} \int \sec ^{\frac {7}{2}}(c+d x) \sqrt {b \sec (c+d x)} \, dx &=\frac {\sqrt {b \sec (c+d x)} \int \sec ^4(c+d x) \, dx}{\sqrt {\sec (c+d x)}}\\ &=-\frac {\sqrt {b \sec (c+d x)} \text {Subst}\left (\int \left (1+x^2\right ) \, dx,x,-\tan (c+d x)\right )}{d \sqrt {\sec (c+d x)}}\\ &=\frac {\sqrt {\sec (c+d x)} \sqrt {b \sec (c+d x)} \sin (c+d x)}{d}+\frac {\sec ^{\frac {5}{2}}(c+d x) \sqrt {b \sec (c+d x)} \sin ^3(c+d x)}{3 d}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 0.09, size = 45, normalized size = 0.64 \begin {gather*} \frac {\sqrt {b \sec (c+d x)} \left (\tan (c+d x)+\frac {1}{3} \tan ^3(c+d x)\right )}{d \sqrt {\sec (c+d x)}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A]
time = 33.39, size = 52, normalized size = 0.74
method | result | size |
default | \(\frac {\left (2 \left (\cos ^{2}\left (d x +c \right )\right )+1\right ) \cos \left (d x +c \right ) \sin \left (d x +c \right ) \left (\frac {1}{\cos \left (d x +c \right )}\right )^{\frac {7}{2}} \sqrt {\frac {b}{\cos \left (d x +c \right )}}}{3 d}\) | \(52\) |
risch | \(\frac {4 i \sqrt {\frac {{\mathrm e}^{i \left (d x +c \right )}}{{\mathrm e}^{2 i \left (d x +c \right )}+1}}\, \sqrt {\frac {b \,{\mathrm e}^{i \left (d x +c \right )}}{{\mathrm e}^{2 i \left (d x +c \right )}+1}}\, \left (4 \cos \left (d x +c \right )+2 i \sin \left (d x +c \right )\right )}{3 \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )^{2} d}\) | \(89\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 294 vs.
\(2 (60) = 120\).
time = 0.60, size = 294, normalized size = 4.20 \begin {gather*} \frac {4 \, {\left ({\left (3 \, \cos \left (2 \, d x + 2 \, c\right ) + 1\right )} \sin \left (6 \, d x + 6 \, c\right ) + 3 \, {\left (3 \, \cos \left (2 \, d x + 2 \, c\right ) + 1\right )} \sin \left (4 \, d x + 4 \, c\right ) - 3 \, \cos \left (6 \, d x + 6 \, c\right ) \sin \left (2 \, d x + 2 \, c\right ) - 9 \, \cos \left (4 \, d x + 4 \, c\right ) \sin \left (2 \, d x + 2 \, c\right )\right )} \sqrt {b}}{3 \, {\left (2 \, {\left (3 \, \cos \left (4 \, d x + 4 \, c\right ) + 3 \, \cos \left (2 \, d x + 2 \, c\right ) + 1\right )} \cos \left (6 \, d x + 6 \, c\right ) + \cos \left (6 \, d x + 6 \, c\right )^{2} + 6 \, {\left (3 \, \cos \left (2 \, d x + 2 \, c\right ) + 1\right )} \cos \left (4 \, d x + 4 \, c\right ) + 9 \, \cos \left (4 \, d x + 4 \, c\right )^{2} + 9 \, \cos \left (2 \, d x + 2 \, c\right )^{2} + 6 \, {\left (\sin \left (4 \, d x + 4 \, c\right ) + \sin \left (2 \, d x + 2 \, c\right )\right )} \sin \left (6 \, d x + 6 \, c\right ) + \sin \left (6 \, d x + 6 \, c\right )^{2} + 9 \, \sin \left (4 \, d x + 4 \, c\right )^{2} + 18 \, \sin \left (4 \, d x + 4 \, c\right ) \sin \left (2 \, d x + 2 \, c\right ) + 9 \, \sin \left (2 \, d x + 2 \, c\right )^{2} + 6 \, \cos \left (2 \, d x + 2 \, c\right ) + 1\right )} d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A]
time = 2.76, size = 43, normalized size = 0.61 \begin {gather*} \frac {{\left (2 \, \cos \left (d x + c\right )^{2} + 1\right )} \sqrt {\frac {b}{\cos \left (d x + c\right )}} \sin \left (d x + c\right )}{3 \, d \cos \left (d x + c\right )^{\frac {5}{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \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 [B]
time = 2.55, size = 126, normalized size = 1.80 \begin {gather*} \frac {2\,\cos \left (c+d\,x\right )\,\sqrt {\frac {b}{\cos \left (c+d\,x\right )}}\,\sqrt {\frac {1}{\cos \left (c+d\,x\right )}}\,\left (4\,\sin \left (c+d\,x\right )+5\,\sin \left (3\,c+3\,d\,x\right )+\sin \left (5\,c+5\,d\,x\right )+\cos \left (c+d\,x\right )\,10{}\mathrm {i}+\cos \left (3\,c+3\,d\,x\right )\,5{}\mathrm {i}+\cos \left (5\,c+5\,d\,x\right )\,1{}\mathrm {i}\right )}{3\,d\,\left (10\,\cos \left (c+d\,x\right )+5\,\cos \left (3\,c+3\,d\,x\right )+\cos \left (5\,c+5\,d\,x\right )\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________