Optimal. Leaf size=26 \[ \text {Int}\left (\frac {(a+b \sec (c+d x))^n}{\sin ^{\frac {3}{2}}(c+d x)},x\right ) \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.04, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \[ \int \frac {(a+b \sec (c+d x))^n}{\sin ^{\frac {3}{2}}(c+d x)} \, dx \]
Verification is Not applicable to the result.
[In]
[Out]
Rubi steps
\begin {align*} \int \frac {(a+b \sec (c+d x))^n}{\sin ^{\frac {3}{2}}(c+d x)} \, dx &=\int \frac {(a+b \sec (c+d x))^n}{\sin ^{\frac {3}{2}}(c+d x)} \, dx\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 2.91, size = 0, normalized size = 0.00 \[ \int \frac {(a+b \sec (c+d x))^n}{\sin ^{\frac {3}{2}}(c+d x)} \, dx \]
Verification is Not applicable to the result.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 1.05, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-\frac {{\left (b \sec \left (d x + c\right ) + a\right )}^{n} \sqrt {\sin \left (d x + c\right )}}{\cos \left (d x + c\right )^{2} - 1}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (b \sec \left (d x + c\right ) + a\right )}^{n}}{\sin \left (d x + c\right )^{\frac {3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.83, size = 0, normalized size = 0.00 \[ \int \frac {\left (a +b \sec \left (d x +c \right )\right )^{n}}{\sin \left (d x +c \right )^{\frac {3}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (b \sec \left (d x + c\right ) + a\right )}^{n}}{\sin \left (d x + c\right )^{\frac {3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [A] time = 0.00, size = -1, normalized size = -0.04 \[ \int \frac {{\left (a+\frac {b}{\cos \left (c+d\,x\right )}\right )}^n}{{\sin \left (c+d\,x\right )}^{3/2}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [A] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\left (a + b \sec {\left (c + d x \right )}\right )^{n}}{\sin ^{\frac {3}{2}}{\left (c + d x \right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________