Optimal. Leaf size=134 \[ \frac{e \left (1-\frac{a+b \sin (c+d x)}{a-b}\right )^{5/4} \left (1-\frac{a+b \sin (c+d x)}{a+b}\right )^{5/4} (a+b \sin (c+d x))^{m+1} F_1\left (m+1;\frac{5}{4},\frac{5}{4};m+2;\frac{a+b \sin (c+d x)}{a-b},\frac{a+b \sin (c+d x)}{a+b}\right )}{b d (m+1) (e \cos (c+d x))^{5/2}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.099244, antiderivative size = 134, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.08, Rules used = {2704, 138} \[ \frac{e \left (1-\frac{a+b \sin (c+d x)}{a-b}\right )^{5/4} \left (1-\frac{a+b \sin (c+d x)}{a+b}\right )^{5/4} (a+b \sin (c+d x))^{m+1} F_1\left (m+1;\frac{5}{4},\frac{5}{4};m+2;\frac{a+b \sin (c+d x)}{a-b},\frac{a+b \sin (c+d x)}{a+b}\right )}{b d (m+1) (e \cos (c+d x))^{5/2}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 2704
Rule 138
Rubi steps
\begin{align*} \int \frac{(a+b \sin (c+d x))^m}{(e \cos (c+d x))^{3/2}} \, dx &=\frac{\left (e \left (1-\frac{a+b \sin (c+d x)}{a-b}\right )^{5/4} \left (1-\frac{a+b \sin (c+d x)}{a+b}\right )^{5/4}\right ) \operatorname{Subst}\left (\int \frac{(a+b x)^m}{\left (-\frac{b}{a-b}-\frac{b x}{a-b}\right )^{5/4} \left (\frac{b}{a+b}-\frac{b x}{a+b}\right )^{5/4}} \, dx,x,\sin (c+d x)\right )}{d (e \cos (c+d x))^{5/2}}\\ &=\frac{e F_1\left (1+m;\frac{5}{4},\frac{5}{4};2+m;\frac{a+b \sin (c+d x)}{a-b},\frac{a+b \sin (c+d x)}{a+b}\right ) (a+b \sin (c+d x))^{1+m} \left (1-\frac{a+b \sin (c+d x)}{a-b}\right )^{5/4} \left (1-\frac{a+b \sin (c+d x)}{a+b}\right )^{5/4}}{b d (1+m) (e \cos (c+d x))^{5/2}}\\ \end{align*}
Mathematica [F] time = 1.9744, size = 0, normalized size = 0. \[ \int \frac{(a+b \sin (c+d x))^m}{(e \cos (c+d x))^{3/2}} \, dx \]
Verification is Not applicable to the result.
[In]
[Out]
________________________________________________________________________________________
Maple [F] time = 0.101, size = 0, normalized size = 0. \begin{align*} \int{ \left ( a+b\sin \left ( dx+c \right ) \right ) ^{m} \left ( e\cos \left ( dx+c \right ) \right ) ^{-{\frac{3}{2}}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (b \sin \left (d x + c\right ) + a\right )}^{m}}{\left (e \cos \left (d x + c\right )\right )^{\frac{3}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{\sqrt{e \cos \left (d x + c\right )}{\left (b \sin \left (d x + c\right ) + a\right )}^{m}}{e^{2} \cos \left (d x + c\right )^{2}}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (b \sin \left (d x + c\right ) + a\right )}^{m}}{\left (e \cos \left (d x + c\right )\right )^{\frac{3}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]