Optimal. Leaf size=349 \[ \frac {b f \tanh ^{-1}(\sin (c+d x))}{\left (a^2-b^2\right ) d^2}+\frac {i b^2 (e+f x) \log \left (1-\frac {i b e^{i (c+d x)}}{a-\sqrt {a^2-b^2}}\right )}{\left (a^2-b^2\right )^{3/2} d}-\frac {i b^2 (e+f x) \log \left (1-\frac {i b e^{i (c+d x)}}{a+\sqrt {a^2-b^2}}\right )}{\left (a^2-b^2\right )^{3/2} d}+\frac {a f \log (\cos (c+d x))}{\left (a^2-b^2\right ) d^2}+\frac {b^2 f \text {Li}_2\left (\frac {i b e^{i (c+d x)}}{a-\sqrt {a^2-b^2}}\right )}{\left (a^2-b^2\right )^{3/2} d^2}-\frac {b^2 f \text {Li}_2\left (\frac {i b e^{i (c+d x)}}{a+\sqrt {a^2-b^2}}\right )}{\left (a^2-b^2\right )^{3/2} d^2}-\frac {b (e+f x) \sec (c+d x)}{\left (a^2-b^2\right ) d}+\frac {a (e+f x) \tan (c+d x)}{\left (a^2-b^2\right ) d} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.56, antiderivative size = 349, normalized size of antiderivative = 1.00, number of steps
used = 15, number of rules used = 11, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.423, Rules used = {4629, 3404,
2296, 2221, 2317, 2438, 6874, 4269, 3556, 4494, 3855} \begin {gather*} \frac {b^2 f \text {PolyLog}\left (2,\frac {i b e^{i (c+d x)}}{a-\sqrt {a^2-b^2}}\right )}{d^2 \left (a^2-b^2\right )^{3/2}}-\frac {b^2 f \text {PolyLog}\left (2,\frac {i b e^{i (c+d x)}}{\sqrt {a^2-b^2}+a}\right )}{d^2 \left (a^2-b^2\right )^{3/2}}+\frac {b f \tanh ^{-1}(\sin (c+d x))}{d^2 \left (a^2-b^2\right )}+\frac {a f \log (\cos (c+d x))}{d^2 \left (a^2-b^2\right )}+\frac {i b^2 (e+f x) \log \left (1-\frac {i b e^{i (c+d x)}}{a-\sqrt {a^2-b^2}}\right )}{d \left (a^2-b^2\right )^{3/2}}-\frac {i b^2 (e+f x) \log \left (1-\frac {i b e^{i (c+d x)}}{\sqrt {a^2-b^2}+a}\right )}{d \left (a^2-b^2\right )^{3/2}}+\frac {a (e+f x) \tan (c+d x)}{d \left (a^2-b^2\right )}-\frac {b (e+f x) \sec (c+d x)}{d \left (a^2-b^2\right )} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 2221
Rule 2296
Rule 2317
Rule 2438
Rule 3404
Rule 3556
Rule 3855
Rule 4269
Rule 4494
Rule 4629
Rule 6874
Rubi steps
\begin {align*} \int \frac {(e+f x) \sec ^2(c+d x)}{a+b \sin (c+d x)} \, dx &=\frac {\int (e+f x) \sec ^2(c+d x) (a-b \sin (c+d x)) \, dx}{a^2-b^2}-\frac {b^2 \int \frac {e+f x}{a+b \sin (c+d x)} \, dx}{a^2-b^2}\\ &=\frac {\int \left (a (e+f x) \sec ^2(c+d x)-b (e+f x) \sec (c+d x) \tan (c+d x)\right ) \, dx}{a^2-b^2}-\frac {\left (2 b^2\right ) \int \frac {e^{i (c+d x)} (e+f x)}{i b+2 a e^{i (c+d x)}-i b e^{2 i (c+d x)}} \, dx}{a^2-b^2}\\ &=\frac {\left (2 i b^3\right ) \int \frac {e^{i (c+d x)} (e+f x)}{2 a-2 \sqrt {a^2-b^2}-2 i b e^{i (c+d x)}} \, dx}{\left (a^2-b^2\right )^{3/2}}-\frac {\left (2 i b^3\right ) \int \frac {e^{i (c+d x)} (e+f x)}{2 a+2 \sqrt {a^2-b^2}-2 i b e^{i (c+d x)}} \, dx}{\left (a^2-b^2\right )^{3/2}}+\frac {a \int (e+f x) \sec ^2(c+d x) \, dx}{a^2-b^2}-\frac {b \int (e+f x) \sec (c+d x) \tan (c+d x) \, dx}{a^2-b^2}\\ &=\frac {i b^2 (e+f x) \log \left (1-\frac {i b e^{i (c+d x)}}{a-\sqrt {a^2-b^2}}\right )}{\left (a^2-b^2\right )^{3/2} d}-\frac {i b^2 (e+f x) \log \left (1-\frac {i b e^{i (c+d x)}}{a+\sqrt {a^2-b^2}}\right )}{\left (a^2-b^2\right )^{3/2} d}-\frac {b (e+f x) \sec (c+d x)}{\left (a^2-b^2\right ) d}+\frac {a (e+f x) \tan (c+d x)}{\left (a^2-b^2\right ) d}-\frac {\left (i b^2 f\right ) \int \log \left (1-\frac {2 i b e^{i (c+d x)}}{2 a-2 \sqrt {a^2-b^2}}\right ) \, dx}{\left (a^2-b^2\right )^{3/2} d}+\frac {\left (i b^2 f\right ) \int \log \left (1-\frac {2 i b e^{i (c+d x)}}{2 a+2 \sqrt {a^2-b^2}}\right ) \, dx}{\left (a^2-b^2\right )^{3/2} d}-\frac {(a f) \int \tan (c+d x) \, dx}{\left (a^2-b^2\right ) d}+\frac {(b f) \int \sec (c+d x) \, dx}{\left (a^2-b^2\right ) d}\\ &=\frac {b f \tanh ^{-1}(\sin (c+d x))}{\left (a^2-b^2\right ) d^2}+\frac {i b^2 (e+f x) \log \left (1-\frac {i b e^{i (c+d x)}}{a-\sqrt {a^2-b^2}}\right )}{\left (a^2-b^2\right )^{3/2} d}-\frac {i b^2 (e+f x) \log \left (1-\frac {i b e^{i (c+d x)}}{a+\sqrt {a^2-b^2}}\right )}{\left (a^2-b^2\right )^{3/2} d}+\frac {a f \log (\cos (c+d x))}{\left (a^2-b^2\right ) d^2}-\frac {b (e+f x) \sec (c+d x)}{\left (a^2-b^2\right ) d}+\frac {a (e+f x) \tan (c+d x)}{\left (a^2-b^2\right ) d}-\frac {\left (b^2 f\right ) \text {Subst}\left (\int \frac {\log \left (1-\frac {2 i b x}{2 a-2 \sqrt {a^2-b^2}}\right )}{x} \, dx,x,e^{i (c+d x)}\right )}{\left (a^2-b^2\right )^{3/2} d^2}+\frac {\left (b^2 f\right ) \text {Subst}\left (\int \frac {\log \left (1-\frac {2 i b x}{2 a+2 \sqrt {a^2-b^2}}\right )}{x} \, dx,x,e^{i (c+d x)}\right )}{\left (a^2-b^2\right )^{3/2} d^2}\\ &=\frac {b f \tanh ^{-1}(\sin (c+d x))}{\left (a^2-b^2\right ) d^2}+\frac {i b^2 (e+f x) \log \left (1-\frac {i b e^{i (c+d x)}}{a-\sqrt {a^2-b^2}}\right )}{\left (a^2-b^2\right )^{3/2} d}-\frac {i b^2 (e+f x) \log \left (1-\frac {i b e^{i (c+d x)}}{a+\sqrt {a^2-b^2}}\right )}{\left (a^2-b^2\right )^{3/2} d}+\frac {a f \log (\cos (c+d x))}{\left (a^2-b^2\right ) d^2}+\frac {b^2 f \text {Li}_2\left (\frac {i b e^{i (c+d x)}}{a-\sqrt {a^2-b^2}}\right )}{\left (a^2-b^2\right )^{3/2} d^2}-\frac {b^2 f \text {Li}_2\left (\frac {i b e^{i (c+d x)}}{a+\sqrt {a^2-b^2}}\right )}{\left (a^2-b^2\right )^{3/2} d^2}-\frac {b (e+f x) \sec (c+d x)}{\left (a^2-b^2\right ) d}+\frac {a (e+f x) \tan (c+d x)}{\left (a^2-b^2\right ) d}\\ \end {align*}
________________________________________________________________________________________
Mathematica [B] Both result and optimal contain complex but leaf count is larger than twice
the leaf count of optimal. \(842\) vs. \(2(349)=698\).
time = 5.57, size = 842, normalized size = 2.41 \begin {gather*} \frac {\frac {b d (e+f x)}{-a^2+b^2}+\frac {f \log \left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )}{a+b}+\frac {f \log \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )}{a-b}+\frac {b^2 d (e+f x) \left (\frac {2 (d e-c f) \tan ^{-1}\left (\frac {b+a \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a^2-b^2}}\right )}{\sqrt {a^2-b^2}}-\frac {i f \left (\log \left (1-i \tan \left (\frac {1}{2} (c+d x)\right )\right ) \log \left (\frac {b+\sqrt {-a^2+b^2}+a \tan \left (\frac {1}{2} (c+d x)\right )}{-i a+b+\sqrt {-a^2+b^2}}\right )+\text {Li}_2\left (\frac {a \left (1-i \tan \left (\frac {1}{2} (c+d x)\right )\right )}{a+i \left (b+\sqrt {-a^2+b^2}\right )}\right )\right )}{\sqrt {-a^2+b^2}}+\frac {i f \left (\log \left (1+i \tan \left (\frac {1}{2} (c+d x)\right )\right ) \log \left (\frac {b+\sqrt {-a^2+b^2}+a \tan \left (\frac {1}{2} (c+d x)\right )}{i a+b+\sqrt {-a^2+b^2}}\right )+\text {Li}_2\left (\frac {a \left (1+i \tan \left (\frac {1}{2} (c+d x)\right )\right )}{a-i \left (b+\sqrt {-a^2+b^2}\right )}\right )\right )}{\sqrt {-a^2+b^2}}+\frac {i f \left (\log \left (1-i \tan \left (\frac {1}{2} (c+d x)\right )\right ) \log \left (\frac {-b+\sqrt {-a^2+b^2}-a \tan \left (\frac {1}{2} (c+d x)\right )}{i a-b+\sqrt {-a^2+b^2}}\right )+\text {Li}_2\left (\frac {a \left (i+\tan \left (\frac {1}{2} (c+d x)\right )\right )}{i a-b+\sqrt {-a^2+b^2}}\right )\right )}{\sqrt {-a^2+b^2}}-\frac {i f \left (\log \left (1+i \tan \left (\frac {1}{2} (c+d x)\right )\right ) \log \left (\frac {b-\sqrt {-a^2+b^2}+a \tan \left (\frac {1}{2} (c+d x)\right )}{i a+b-\sqrt {-a^2+b^2}}\right )+\text {Li}_2\left (\frac {a+i a \tan \left (\frac {1}{2} (c+d x)\right )}{a+i \left (-b+\sqrt {-a^2+b^2}\right )}\right )\right )}{\sqrt {-a^2+b^2}}\right )}{\left (-a^2+b^2\right ) \left (d e-c f+i f \log \left (1-i \tan \left (\frac {1}{2} (c+d x)\right )\right )-i f \log \left (1+i \tan \left (\frac {1}{2} (c+d x)\right )\right )\right )}+\frac {d (e+f x) \sin \left (\frac {1}{2} (c+d x)\right )}{(a+b) \left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )}+\frac {d (e+f x) \sin \left (\frac {1}{2} (c+d x)\right )}{(a-b) \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )}}{d^2} \end {gather*}
Warning: Unable to verify antiderivative.
[In]
[Out]
________________________________________________________________________________________
Maple [B] Both result and optimal contain complex but leaf count of result is larger than twice
the leaf count of optimal. 1541 vs. \(2 (319 ) = 638\).
time = 0.33, size = 1542, normalized size = 4.42
method | result | size |
risch | \(\text {Expression too large to display}\) | \(1542\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: ValueError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [B] Both result and optimal contain complex but leaf count of result is larger than twice
the leaf count of optimal. 1273 vs. \(2 (315) = 630\).
time = 0.58, size = 1273, normalized size = 3.65 \begin {gather*} -\frac {i \, b^{3} f \sqrt {-\frac {a^{2} - b^{2}}{b^{2}}} \cos \left (d x + c\right ) {\rm Li}_2\left (\frac {i \, a \cos \left (d x + c\right ) - a \sin \left (d x + c\right ) + {\left (b \cos \left (d x + c\right ) + i \, b \sin \left (d x + c\right )\right )} \sqrt {-\frac {a^{2} - b^{2}}{b^{2}}} - b}{b} + 1\right ) - i \, b^{3} f \sqrt {-\frac {a^{2} - b^{2}}{b^{2}}} \cos \left (d x + c\right ) {\rm Li}_2\left (\frac {i \, a \cos \left (d x + c\right ) - a \sin \left (d x + c\right ) - {\left (b \cos \left (d x + c\right ) + i \, b \sin \left (d x + c\right )\right )} \sqrt {-\frac {a^{2} - b^{2}}{b^{2}}} - b}{b} + 1\right ) - i \, b^{3} f \sqrt {-\frac {a^{2} - b^{2}}{b^{2}}} \cos \left (d x + c\right ) {\rm Li}_2\left (\frac {-i \, a \cos \left (d x + c\right ) - a \sin \left (d x + c\right ) + {\left (b \cos \left (d x + c\right ) - i \, b \sin \left (d x + c\right )\right )} \sqrt {-\frac {a^{2} - b^{2}}{b^{2}}} - b}{b} + 1\right ) + i \, b^{3} f \sqrt {-\frac {a^{2} - b^{2}}{b^{2}}} \cos \left (d x + c\right ) {\rm Li}_2\left (\frac {-i \, a \cos \left (d x + c\right ) - a \sin \left (d x + c\right ) - {\left (b \cos \left (d x + c\right ) - i \, b \sin \left (d x + c\right )\right )} \sqrt {-\frac {a^{2} - b^{2}}{b^{2}}} - b}{b} + 1\right ) + 2 \, {\left (a^{2} b - b^{3}\right )} d f x - {\left (a^{3} + a^{2} b - a b^{2} - b^{3}\right )} f \cos \left (d x + c\right ) \log \left (\sin \left (d x + c\right ) + 1\right ) - {\left (a^{3} - a^{2} b - a b^{2} + b^{3}\right )} f \cos \left (d x + c\right ) \log \left (-\sin \left (d x + c\right ) + 1\right ) - {\left (b^{3} c f - b^{3} d e\right )} \sqrt {-\frac {a^{2} - b^{2}}{b^{2}}} \cos \left (d x + c\right ) \log \left (2 \, b \cos \left (d x + c\right ) + 2 i \, b \sin \left (d x + c\right ) + 2 \, b \sqrt {-\frac {a^{2} - b^{2}}{b^{2}}} + 2 i \, a\right ) - {\left (b^{3} c f - b^{3} d e\right )} \sqrt {-\frac {a^{2} - b^{2}}{b^{2}}} \cos \left (d x + c\right ) \log \left (2 \, b \cos \left (d x + c\right ) - 2 i \, b \sin \left (d x + c\right ) + 2 \, b \sqrt {-\frac {a^{2} - b^{2}}{b^{2}}} - 2 i \, a\right ) + {\left (b^{3} c f - b^{3} d e\right )} \sqrt {-\frac {a^{2} - b^{2}}{b^{2}}} \cos \left (d x + c\right ) \log \left (-2 \, b \cos \left (d x + c\right ) + 2 i \, b \sin \left (d x + c\right ) + 2 \, b \sqrt {-\frac {a^{2} - b^{2}}{b^{2}}} + 2 i \, a\right ) + {\left (b^{3} c f - b^{3} d e\right )} \sqrt {-\frac {a^{2} - b^{2}}{b^{2}}} \cos \left (d x + c\right ) \log \left (-2 \, b \cos \left (d x + c\right ) - 2 i \, b \sin \left (d x + c\right ) + 2 \, b \sqrt {-\frac {a^{2} - b^{2}}{b^{2}}} - 2 i \, a\right ) - {\left (b^{3} d f x + b^{3} c f\right )} \sqrt {-\frac {a^{2} - b^{2}}{b^{2}}} \cos \left (d x + c\right ) \log \left (-\frac {i \, a \cos \left (d x + c\right ) - a \sin \left (d x + c\right ) + {\left (b \cos \left (d x + c\right ) + i \, b \sin \left (d x + c\right )\right )} \sqrt {-\frac {a^{2} - b^{2}}{b^{2}}} - b}{b}\right ) + {\left (b^{3} d f x + b^{3} c f\right )} \sqrt {-\frac {a^{2} - b^{2}}{b^{2}}} \cos \left (d x + c\right ) \log \left (-\frac {i \, a \cos \left (d x + c\right ) - a \sin \left (d x + c\right ) - {\left (b \cos \left (d x + c\right ) + i \, b \sin \left (d x + c\right )\right )} \sqrt {-\frac {a^{2} - b^{2}}{b^{2}}} - b}{b}\right ) - {\left (b^{3} d f x + b^{3} c f\right )} \sqrt {-\frac {a^{2} - b^{2}}{b^{2}}} \cos \left (d x + c\right ) \log \left (-\frac {-i \, a \cos \left (d x + c\right ) - a \sin \left (d x + c\right ) + {\left (b \cos \left (d x + c\right ) - i \, b \sin \left (d x + c\right )\right )} \sqrt {-\frac {a^{2} - b^{2}}{b^{2}}} - b}{b}\right ) + {\left (b^{3} d f x + b^{3} c f\right )} \sqrt {-\frac {a^{2} - b^{2}}{b^{2}}} \cos \left (d x + c\right ) \log \left (-\frac {-i \, a \cos \left (d x + c\right ) - a \sin \left (d x + c\right ) - {\left (b \cos \left (d x + c\right ) - i \, b \sin \left (d x + c\right )\right )} \sqrt {-\frac {a^{2} - b^{2}}{b^{2}}} - b}{b}\right ) + 2 \, {\left (a^{2} b - b^{3}\right )} d e - 2 \, {\left ({\left (a^{3} - a b^{2}\right )} d f x + {\left (a^{3} - a b^{2}\right )} d e\right )} \sin \left (d x + c\right )}{2 \, {\left (a^{4} - 2 \, a^{2} b^{2} + b^{4}\right )} d^{2} \cos \left (d x + c\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (e + f x\right ) \sec ^{2}{\left (c + d x \right )}}{a + b \sin {\left (c + d x \right )}}\, dx \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(-1)]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \text {Hanged} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________