Integrand size = 23, antiderivative size = 72 \[ \int \frac {(b \cos (c+d x))^{3/2}}{\cos ^{\frac {11}{2}}(c+d x)} \, dx=\frac {b \sqrt {b \cos (c+d x)} \sin (c+d x)}{d \cos ^{\frac {3}{2}}(c+d x)}+\frac {b \sqrt {b \cos (c+d x)} \sin ^3(c+d x)}{3 d \cos ^{\frac {7}{2}}(c+d x)} \]
[Out]
Time = 0.02 (sec) , antiderivative size = 72, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.087, Rules used = {17, 3852} \[ \int \frac {(b \cos (c+d x))^{3/2}}{\cos ^{\frac {11}{2}}(c+d x)} \, dx=\frac {b \sin ^3(c+d x) \sqrt {b \cos (c+d x)}}{3 d \cos ^{\frac {7}{2}}(c+d x)}+\frac {b \sin (c+d x) \sqrt {b \cos (c+d x)}}{d \cos ^{\frac {3}{2}}(c+d x)} \]
[In]
[Out]
Rule 17
Rule 3852
Rubi steps \begin{align*} \text {integral}& = \frac {\left (b \sqrt {b \cos (c+d x)}\right ) \int \sec ^4(c+d x) \, dx}{\sqrt {\cos (c+d x)}} \\ & = -\frac {\left (b \sqrt {b \cos (c+d x)}\right ) \text {Subst}\left (\int \left (1+x^2\right ) \, dx,x,-\tan (c+d x)\right )}{d \sqrt {\cos (c+d x)}} \\ & = \frac {b \sqrt {b \cos (c+d x)} \sin (c+d x)}{d \cos ^{\frac {3}{2}}(c+d x)}+\frac {b \sqrt {b \cos (c+d x)} \sin ^3(c+d x)}{3 d \cos ^{\frac {7}{2}}(c+d x)} \\ \end{align*}
Time = 0.07 (sec) , antiderivative size = 45, normalized size of antiderivative = 0.62 \[ \int \frac {(b \cos (c+d x))^{3/2}}{\cos ^{\frac {11}{2}}(c+d x)} \, dx=\frac {(b \cos (c+d x))^{3/2} \left (\tan (c+d x)+\frac {1}{3} \tan ^3(c+d x)\right )}{d \cos ^{\frac {3}{2}}(c+d x)} \]
[In]
[Out]
Time = 2.77 (sec) , antiderivative size = 43, normalized size of antiderivative = 0.60
method | result | size |
default | \(\frac {b \left (2 \left (\cos ^{2}\left (d x +c \right )\right )+1\right ) \sqrt {\cos \left (d x +c \right ) b}\, \sin \left (d x +c \right )}{3 d \cos \left (d x +c \right )^{\frac {7}{2}}}\) | \(43\) |
risch | \(\frac {4 i b \sqrt {\cos \left (d x +c \right ) b}\, \left (3 \,{\mathrm e}^{2 i \left (d x +c \right )}+1\right )}{3 \sqrt {\cos \left (d x +c \right )}\, d \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )^{3}}\) | \(52\) |
[In]
[Out]
none
Time = 0.28 (sec) , antiderivative size = 42, normalized size of antiderivative = 0.58 \[ \int \frac {(b \cos (c+d x))^{3/2}}{\cos ^{\frac {11}{2}}(c+d x)} \, dx=\frac {{\left (2 \, b \cos \left (d x + c\right )^{2} + b\right )} \sqrt {b \cos \left (d x + c\right )} \sin \left (d x + c\right )}{3 \, d \cos \left (d x + c\right )^{\frac {7}{2}}} \]
[In]
[Out]
Timed out. \[ \int \frac {(b \cos (c+d x))^{3/2}}{\cos ^{\frac {11}{2}}(c+d x)} \, dx=\text {Timed out} \]
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. 299 vs. \(2 (62) = 124\).
Time = 0.41 (sec) , antiderivative size = 299, normalized size of antiderivative = 4.15 \[ \int \frac {(b \cos (c+d x))^{3/2}}{\cos ^{\frac {11}{2}}(c+d x)} \, dx=-\frac {4 \, {\left (3 \, b \cos \left (6 \, d x + 6 \, c\right ) \sin \left (2 \, d x + 2 \, c\right ) + 9 \, b \cos \left (4 \, d x + 4 \, c\right ) \sin \left (2 \, d x + 2 \, c\right ) - {\left (3 \, b \cos \left (2 \, d x + 2 \, c\right ) + b\right )} \sin \left (6 \, d x + 6 \, c\right ) - 3 \, {\left (3 \, b \cos \left (2 \, d x + 2 \, c\right ) + b\right )} \sin \left (4 \, d x + 4 \, c\right )\right )} \sqrt {b}}{3 \, {\left (2 \, {\left (3 \, \cos \left (4 \, d x + 4 \, c\right ) + 3 \, \cos \left (2 \, d x + 2 \, c\right ) + 1\right )} \cos \left (6 \, d x + 6 \, c\right ) + \cos \left (6 \, d x + 6 \, c\right )^{2} + 6 \, {\left (3 \, \cos \left (2 \, d x + 2 \, c\right ) + 1\right )} \cos \left (4 \, d x + 4 \, c\right ) + 9 \, \cos \left (4 \, d x + 4 \, c\right )^{2} + 9 \, \cos \left (2 \, d x + 2 \, c\right )^{2} + 6 \, {\left (\sin \left (4 \, d x + 4 \, c\right ) + \sin \left (2 \, d x + 2 \, c\right )\right )} \sin \left (6 \, d x + 6 \, c\right ) + \sin \left (6 \, d x + 6 \, c\right )^{2} + 9 \, \sin \left (4 \, d x + 4 \, c\right )^{2} + 18 \, \sin \left (4 \, d x + 4 \, c\right ) \sin \left (2 \, d x + 2 \, c\right ) + 9 \, \sin \left (2 \, d x + 2 \, c\right )^{2} + 6 \, \cos \left (2 \, d x + 2 \, c\right ) + 1\right )} d} \]
[In]
[Out]
\[ \int \frac {(b \cos (c+d x))^{3/2}}{\cos ^{\frac {11}{2}}(c+d x)} \, dx=\int { \frac {\left (b \cos \left (d x + c\right )\right )^{\frac {3}{2}}}{\cos \left (d x + c\right )^{\frac {11}{2}}} \,d x } \]
[In]
[Out]
Time = 15.35 (sec) , antiderivative size = 129, normalized size of antiderivative = 1.79 \[ \int \frac {(b \cos (c+d x))^{3/2}}{\cos ^{\frac {11}{2}}(c+d x)} \, dx=\frac {2\,b\,\sqrt {b\,\cos \left (c+d\,x\right )}\,\left (\cos \left (2\,c+2\,d\,x\right )\,15{}\mathrm {i}+\cos \left (4\,c+4\,d\,x\right )\,6{}\mathrm {i}+\cos \left (6\,c+6\,d\,x\right )\,1{}\mathrm {i}+9\,\sin \left (2\,c+2\,d\,x\right )+6\,\sin \left (4\,c+4\,d\,x\right )+\sin \left (6\,c+6\,d\,x\right )+10{}\mathrm {i}\right )}{3\,d\,\sqrt {\cos \left (c+d\,x\right )}\,\left (15\,\cos \left (2\,c+2\,d\,x\right )+6\,\cos \left (4\,c+4\,d\,x\right )+\cos \left (6\,c+6\,d\,x\right )+10\right )} \]
[In]
[Out]