Optimal. Leaf size=100 \[ \frac {2 \sqrt {a} (B c-A d) \tanh ^{-1}\left (\frac {\sqrt {a} \sqrt {d} \cos (e+f x)}{\sqrt {c+d} \sqrt {a \sin (e+f x)+a}}\right )}{d^{3/2} f \sqrt {c+d}}-\frac {2 a B \cos (e+f x)}{d f \sqrt {a \sin (e+f x)+a}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.25, antiderivative size = 100, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 37, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.081, Rules used = {2981, 2773, 208} \[ \frac {2 \sqrt {a} (B c-A d) \tanh ^{-1}\left (\frac {\sqrt {a} \sqrt {d} \cos (e+f x)}{\sqrt {c+d} \sqrt {a \sin (e+f x)+a}}\right )}{d^{3/2} f \sqrt {c+d}}-\frac {2 a B \cos (e+f x)}{d f \sqrt {a \sin (e+f x)+a}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 208
Rule 2773
Rule 2981
Rubi steps
\begin {align*} \int \frac {\sqrt {a+a \sin (e+f x)} (A+B \sin (e+f x))}{c+d \sin (e+f x)} \, dx &=-\frac {2 a B \cos (e+f x)}{d f \sqrt {a+a \sin (e+f x)}}+\frac {(-a B c+a A d) \int \frac {\sqrt {a+a \sin (e+f x)}}{c+d \sin (e+f x)} \, dx}{a d}\\ &=-\frac {2 a B \cos (e+f x)}{d f \sqrt {a+a \sin (e+f x)}}+\frac {(2 a (B c-A d)) \operatorname {Subst}\left (\int \frac {1}{a c+a d-d x^2} \, dx,x,\frac {a \cos (e+f x)}{\sqrt {a+a \sin (e+f x)}}\right )}{d f}\\ &=\frac {2 \sqrt {a} (B c-A d) \tanh ^{-1}\left (\frac {\sqrt {a} \sqrt {d} \cos (e+f x)}{\sqrt {c+d} \sqrt {a+a \sin (e+f x)}}\right )}{d^{3/2} \sqrt {c+d} f}-\frac {2 a B \cos (e+f x)}{d f \sqrt {a+a \sin (e+f x)}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [C] time = 9.03, size = 903, normalized size = 9.03 \[ \frac {\left (\frac {1}{2}+\frac {i}{2}\right ) \left (-\frac {(2-2 i) B \sqrt {d} \cos \left (\frac {f x}{2}\right ) \left (\cos \left (\frac {e}{2}\right )-\sin \left (\frac {e}{2}\right )\right )}{f}+\frac {(A d-B c) \left (\cos \left (\frac {e}{2}\right )+i \sin \left (\frac {e}{2}\right )\right ) \left ((-1+i) x \cos (e)+(1+i) x \sin (e)+\frac {\text {RootSum}\left [d e^{2 i e} \text {$\#$1}^4+2 i c e^{i e} \text {$\#$1}^2-d\& ,\frac {-\sqrt {d} \sqrt {c+d} e^{i e} f x \text {$\#$1}^3-2 i \sqrt {d} \sqrt {c+d} e^{i e} \log \left (e^{\frac {i f x}{2}}-\text {$\#$1}\right ) \text {$\#$1}^3+\frac {(1-i) c f x \text {$\#$1}^2}{\sqrt {e^{-i e}}}+\frac {(2+2 i) c \log \left (e^{\frac {i f x}{2}}-\text {$\#$1}\right ) \text {$\#$1}^2}{\sqrt {e^{-i e}}}-i \sqrt {d} \sqrt {c+d} f x \text {$\#$1}+2 \sqrt {d} \sqrt {c+d} \log \left (e^{\frac {i f x}{2}}-\text {$\#$1}\right ) \text {$\#$1}+(1+i) d \sqrt {e^{-i e}} f x-(2-2 i) d \sqrt {e^{-i e}} \log \left (e^{\frac {i f x}{2}}-\text {$\#$1}\right )}{d-i c e^{i e} \text {$\#$1}^2}\& \right ] (\cos (e)+i (\sin (e)-1)) \sqrt {\cos (e)-i \sin (e)}}{4 f}\right )}{\sqrt {c+d} (\cos (e)+i (\sin (e)-1)) \sqrt {\cos (e)-i \sin (e)}}+\frac {(A d-B c) \left (\cos \left (\frac {e}{2}\right )+i \sin \left (\frac {e}{2}\right )\right ) \left ((1-i) x \cos (e)-(1+i) x \sin (e)+\frac {\text {RootSum}\left [d e^{2 i e} \text {$\#$1}^4+2 i c e^{i e} \text {$\#$1}^2-d\& ,\frac {-i \sqrt {d} \sqrt {c+d} e^{i e} f x \text {$\#$1}^3+2 \sqrt {d} \sqrt {c+d} e^{i e} \log \left (e^{\frac {i f x}{2}}-\text {$\#$1}\right ) \text {$\#$1}^3-\frac {(1+i) c f x \text {$\#$1}^2}{\sqrt {e^{-i e}}}+\frac {(2-2 i) c \log \left (e^{\frac {i f x}{2}}-\text {$\#$1}\right ) \text {$\#$1}^2}{\sqrt {e^{-i e}}}+\sqrt {d} \sqrt {c+d} f x \text {$\#$1}+2 i \sqrt {d} \sqrt {c+d} \log \left (e^{\frac {i f x}{2}}-\text {$\#$1}\right ) \text {$\#$1}+(1-i) d \sqrt {e^{-i e}} f x+(2+2 i) d \sqrt {e^{-i e}} \log \left (e^{\frac {i f x}{2}}-\text {$\#$1}\right )}{d-i c e^{i e} \text {$\#$1}^2}\& \right ] \sqrt {\cos (e)-i \sin (e)} (-i \cos (e)+\sin (e)-1)}{4 f}\right )}{\sqrt {c+d} (\cos (e)+i (\sin (e)-1)) \sqrt {\cos (e)-i \sin (e)}}+\frac {(2-2 i) B \sqrt {d} \left (\cos \left (\frac {e}{2}\right )+\sin \left (\frac {e}{2}\right )\right ) \sin \left (\frac {f x}{2}\right )}{f}\right ) \sqrt {a (\sin (e+f x)+1)}}{d^{3/2} \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 1.26, size = 651, normalized size = 6.51 \[ \left [-\frac {{\left (B c - A d + {\left (B c - A d\right )} \cos \left (f x + e\right ) + {\left (B c - A d\right )} \sin \left (f x + e\right )\right )} \sqrt {\frac {a}{c d + d^{2}}} \log \left (\frac {a d^{2} \cos \left (f x + e\right )^{3} - a c^{2} - 2 \, a c d - a d^{2} - {\left (6 \, a c d + 7 \, a d^{2}\right )} \cos \left (f x + e\right )^{2} + 4 \, {\left (c^{2} d + 4 \, c d^{2} + 3 \, d^{3} - {\left (c d^{2} + d^{3}\right )} \cos \left (f x + e\right )^{2} + {\left (c^{2} d + 3 \, c d^{2} + 2 \, d^{3}\right )} \cos \left (f x + e\right ) - {\left (c^{2} d + 4 \, c d^{2} + 3 \, d^{3} + {\left (c d^{2} + d^{3}\right )} \cos \left (f x + e\right )\right )} \sin \left (f x + e\right )\right )} \sqrt {a \sin \left (f x + e\right ) + a} \sqrt {\frac {a}{c d + d^{2}}} - {\left (a c^{2} + 8 \, a c d + 9 \, a d^{2}\right )} \cos \left (f x + e\right ) + {\left (a d^{2} \cos \left (f x + e\right )^{2} - a c^{2} - 2 \, a c d - a d^{2} + 2 \, {\left (3 \, a c d + 4 \, a d^{2}\right )} \cos \left (f x + e\right )\right )} \sin \left (f x + e\right )}{d^{2} \cos \left (f x + e\right )^{3} + {\left (2 \, c d + d^{2}\right )} \cos \left (f x + e\right )^{2} - c^{2} - 2 \, c d - d^{2} - {\left (c^{2} + d^{2}\right )} \cos \left (f x + e\right ) + {\left (d^{2} \cos \left (f x + e\right )^{2} - 2 \, c d \cos \left (f x + e\right ) - c^{2} - 2 \, c d - d^{2}\right )} \sin \left (f x + e\right )}\right ) + 4 \, {\left (B \cos \left (f x + e\right ) - B \sin \left (f x + e\right ) + B\right )} \sqrt {a \sin \left (f x + e\right ) + a}}{2 \, {\left (d f \cos \left (f x + e\right ) + d f \sin \left (f x + e\right ) + d f\right )}}, \frac {{\left (B c - A d + {\left (B c - A d\right )} \cos \left (f x + e\right ) + {\left (B c - A d\right )} \sin \left (f x + e\right )\right )} \sqrt {-\frac {a}{c d + d^{2}}} \arctan \left (\frac {\sqrt {a \sin \left (f x + e\right ) + a} {\left (d \sin \left (f x + e\right ) - c - 2 \, d\right )} \sqrt {-\frac {a}{c d + d^{2}}}}{2 \, a \cos \left (f x + e\right )}\right ) - 2 \, {\left (B \cos \left (f x + e\right ) - B \sin \left (f x + e\right ) + B\right )} \sqrt {a \sin \left (f x + e\right ) + a}}{d f \cos \left (f x + e\right ) + d f \sin \left (f x + e\right ) + d f}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [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]
________________________________________________________________________________________
maple [A] time = 1.79, size = 139, normalized size = 1.39 \[ -\frac {2 \left (1+\sin \left (f x +e \right )\right ) \sqrt {-a \left (\sin \left (f x +e \right )-1\right )}\, \left (A \arctanh \left (\frac {\sqrt {-a \left (\sin \left (f x +e \right )-1\right )}\, d}{\sqrt {a \left (c +d \right ) d}}\right ) a d -B \arctanh \left (\frac {\sqrt {-a \left (\sin \left (f x +e \right )-1\right )}\, d}{\sqrt {a \left (c +d \right ) d}}\right ) a c +B \sqrt {-a \left (\sin \left (f x +e \right )-1\right )}\, \sqrt {a \left (c +d \right ) d}\right )}{d \sqrt {a \left (c +d \right ) d}\, \cos \left (f x +e \right ) \sqrt {a +a \sin \left (f x +e \right )}\, f} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (B \sin \left (f x + e\right ) + A\right )} \sqrt {a \sin \left (f x + e\right ) + a}}{d \sin \left (f x + e\right ) + c}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {\left (A+B\,\sin \left (e+f\,x\right )\right )\,\sqrt {a+a\,\sin \left (e+f\,x\right )}}{c+d\,\sin \left (e+f\,x\right )} \,d x \]
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]
________________________________________________________________________________________