Optimal. Leaf size=85 \[ \frac {2^{\frac {1}{4}+m} \, _2F_1\left (-\frac {3}{4},\frac {7}{4}-m;\frac {1}{4};\frac {1}{2} (1-\sin (c+d x))\right ) (1+\sin (c+d x))^{\frac {3}{4}-m} (a+a \sin (c+d x))^m}{3 d e (e \cos (c+d x))^{3/2}} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.07, antiderivative size = 85, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 3, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.120, Rules used = {2768, 72, 71}
\begin {gather*} \frac {2^{m+\frac {1}{4}} (\sin (c+d x)+1)^{\frac {3}{4}-m} (a \sin (c+d x)+a)^m \, _2F_1\left (-\frac {3}{4},\frac {7}{4}-m;\frac {1}{4};\frac {1}{2} (1-\sin (c+d x))\right )}{3 d e (e \cos (c+d x))^{3/2}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 71
Rule 72
Rule 2768
Rubi steps
\begin {align*} \int \frac {(a+a \sin (c+d x))^m}{(e \cos (c+d x))^{5/2}} \, dx &=\frac {\left (a^2 (a-a \sin (c+d x))^{3/4} (a+a \sin (c+d x))^{3/4}\right ) \text {Subst}\left (\int \frac {(a+a x)^{-\frac {7}{4}+m}}{(a-a x)^{7/4}} \, dx,x,\sin (c+d x)\right )}{d e (e \cos (c+d x))^{3/2}}\\ &=\frac {\left (2^{-\frac {7}{4}+m} a (a-a \sin (c+d x))^{3/4} (a+a \sin (c+d x))^m \left (\frac {a+a \sin (c+d x)}{a}\right )^{\frac {3}{4}-m}\right ) \text {Subst}\left (\int \frac {\left (\frac {1}{2}+\frac {x}{2}\right )^{-\frac {7}{4}+m}}{(a-a x)^{7/4}} \, dx,x,\sin (c+d x)\right )}{d e (e \cos (c+d x))^{3/2}}\\ &=\frac {2^{\frac {1}{4}+m} \, _2F_1\left (-\frac {3}{4},\frac {7}{4}-m;\frac {1}{4};\frac {1}{2} (1-\sin (c+d x))\right ) (1+\sin (c+d x))^{\frac {3}{4}-m} (a+a \sin (c+d x))^m}{3 d e (e \cos (c+d x))^{3/2}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 0.12, size = 85, normalized size = 1.00 \begin {gather*} \frac {2^{\frac {1}{4}+m} \, _2F_1\left (-\frac {3}{4},\frac {7}{4}-m;\frac {1}{4};\frac {1}{2} (1-\sin (c+d x))\right ) (1+\sin (c+d x))^{\frac {3}{4}-m} (a (1+\sin (c+d x)))^m}{3 d e (e \cos (c+d x))^{3/2}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [F]
time = 0.13, size = 0, normalized size = 0.00 \[\int \frac {\left (a +a \sin \left (d x +c \right )\right )^{m}}{\left (e \cos \left (d x +c \right )\right )^{\frac {5}{2}}}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [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]
________________________________________________________________________________________
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 [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {{\left (a+a\,\sin \left (c+d\,x\right )\right )}^m}{{\left (e\,\cos \left (c+d\,x\right )\right )}^{5/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________