Optimal. Leaf size=109 \[ \frac {2 \left (3 a^2+2 b^2\right ) \sqrt {\cos (c+d x)} F\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{3 d \sqrt {e \cos (c+d x)}}-\frac {10 a b \sqrt {e \cos (c+d x)}}{3 d e}-\frac {2 b \sqrt {e \cos (c+d x)} (a+b \sin (c+d x))}{3 d e} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.13, antiderivative size = 109, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.160, Rules used = {2692, 2669, 2642, 2641} \[ \frac {2 \left (3 a^2+2 b^2\right ) \sqrt {\cos (c+d x)} F\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{3 d \sqrt {e \cos (c+d x)}}-\frac {10 a b \sqrt {e \cos (c+d x)}}{3 d e}-\frac {2 b \sqrt {e \cos (c+d x)} (a+b \sin (c+d x))}{3 d e} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 2641
Rule 2642
Rule 2669
Rule 2692
Rubi steps
\begin {align*} \int \frac {(a+b \sin (c+d x))^2}{\sqrt {e \cos (c+d x)}} \, dx &=-\frac {2 b \sqrt {e \cos (c+d x)} (a+b \sin (c+d x))}{3 d e}+\frac {2}{3} \int \frac {\frac {3 a^2}{2}+b^2+\frac {5}{2} a b \sin (c+d x)}{\sqrt {e \cos (c+d x)}} \, dx\\ &=-\frac {10 a b \sqrt {e \cos (c+d x)}}{3 d e}-\frac {2 b \sqrt {e \cos (c+d x)} (a+b \sin (c+d x))}{3 d e}+\frac {1}{3} \left (3 a^2+2 b^2\right ) \int \frac {1}{\sqrt {e \cos (c+d x)}} \, dx\\ &=-\frac {10 a b \sqrt {e \cos (c+d x)}}{3 d e}-\frac {2 b \sqrt {e \cos (c+d x)} (a+b \sin (c+d x))}{3 d e}+\frac {\left (\left (3 a^2+2 b^2\right ) \sqrt {\cos (c+d x)}\right ) \int \frac {1}{\sqrt {\cos (c+d x)}} \, dx}{3 \sqrt {e \cos (c+d x)}}\\ &=-\frac {10 a b \sqrt {e \cos (c+d x)}}{3 d e}+\frac {2 \left (3 a^2+2 b^2\right ) \sqrt {\cos (c+d x)} F\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{3 d \sqrt {e \cos (c+d x)}}-\frac {2 b \sqrt {e \cos (c+d x)} (a+b \sin (c+d x))}{3 d e}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.43, size = 75, normalized size = 0.69 \[ \frac {2 \left (3 a^2+2 b^2\right ) \sqrt {\cos (c+d x)} F\left (\left .\frac {1}{2} (c+d x)\right |2\right )-2 b \cos (c+d x) (6 a+b \sin (c+d x))}{3 d \sqrt {e \cos (c+d x)}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [F] time = 1.09, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-\frac {{\left (b^{2} \cos \left (d x + c\right )^{2} - 2 \, a b \sin \left (d x + c\right ) - a^{2} - b^{2}\right )} \sqrt {e \cos \left (d x + c\right )}}{e \cos \left (d x + c\right )}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (b \sin \left (d x + c\right ) + a\right )}^{2}}{\sqrt {e \cos \left (d x + c\right )}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 1.07, size = 210, normalized size = 1.93 \[ -\frac {2 \left (-4 b^{2} \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) \left (\sin ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+3 \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {2 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1}\, \EllipticF \left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right ) a^{2}+2 \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {2 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1}\, \EllipticF \left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right ) b^{2}-12 a b \left (\sin ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+2 b^{2} \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+6 a b \sin \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3 \sin \left (\frac {d x}{2}+\frac {c}{2}\right ) \sqrt {-2 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) e +e}\, d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (b \sin \left (d x + c\right ) + a\right )}^{2}}{\sqrt {e \cos \left (d x + c\right )}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {{\left (a+b\,\sin \left (c+d\,x\right )\right )}^2}{\sqrt {e\,\cos \left (c+d\,x\right )}} \,d x \]
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]
________________________________________________________________________________________