Optimal. Leaf size=35 \[ \frac {\sin (c+d x) \sec ^{\frac {3}{2}}(c+d x)}{b d \sqrt {b \sec (c+d x)}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.01, antiderivative size = 35, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.130, Rules used = {17, 3767, 8} \[ \frac {\sin (c+d x) \sec ^{\frac {3}{2}}(c+d x)}{b d \sqrt {b \sec (c+d x)}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 8
Rule 17
Rule 3767
Rubi steps
\begin {align*} \int \frac {\sec ^{\frac {7}{2}}(c+d x)}{(b \sec (c+d x))^{3/2}} \, dx &=\frac {\sqrt {\sec (c+d x)} \int \sec ^2(c+d x) \, dx}{b \sqrt {b \sec (c+d x)}}\\ &=-\frac {\sqrt {\sec (c+d x)} \operatorname {Subst}(\int 1 \, dx,x,-\tan (c+d x))}{b d \sqrt {b \sec (c+d x)}}\\ &=\frac {\sec ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{b d \sqrt {b \sec (c+d x)}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.04, size = 32, normalized size = 0.91 \[ \frac {\sin (c+d x) \sec ^{\frac {5}{2}}(c+d x)}{d (b \sec (c+d x))^{3/2}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.68, size = 33, normalized size = 0.94 \[ \frac {\sqrt {\frac {b}{\cos \left (d x + c\right )}} \sin \left (d x + c\right )}{b^{2} d \sqrt {\cos \left (d x + c\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sec \left (d x + c\right )^{\frac {7}{2}}}{\left (b \sec \left (d x + c\right )\right )^{\frac {3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.82, size = 39, normalized size = 1.11 \[ \frac {\left (\frac {1}{\cos \left (d x +c \right )}\right )^{\frac {7}{2}} \cos \left (d x +c \right ) \sin \left (d x +c \right )}{d \left (\frac {b}{\cos \left (d x +c \right )}\right )^{\frac {3}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [B] time = 0.85, size = 67, normalized size = 1.91 \[ \frac {2 \, \sqrt {b} \sin \left (2 \, d x + 2 \, c\right )}{{\left (b^{2} \cos \left (2 \, d x + 2 \, c\right )^{2} + b^{2} \sin \left (2 \, d x + 2 \, c\right )^{2} + 2 \, b^{2} \cos \left (2 \, d x + 2 \, c\right ) + b^{2}\right )} d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 0.27, size = 51, normalized size = 1.46 \[ \frac {\left (\cos \left (d\,x\right )-\sin \left (d\,x\right )\,1{}\mathrm {i}\right )\,\left (\cos \relax (c)-\sin \relax (c)\,1{}\mathrm {i}\right )\,\sqrt {\frac {b}{\cos \left (c+d\,x\right )}}\,\sqrt {\frac {1}{\cos \left (c+d\,x\right )}}\,1{}\mathrm {i}}{b^2\,d} \]
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]
________________________________________________________________________________________