Optimal. Leaf size=144 \[ -\frac {(2 a-3 b) \tanh ^{-1}\left (\frac {\sqrt {a+b \sin (c+d x)}}{\sqrt {a-b}}\right )}{4 (a-b)^{3/2} d}+\frac {(2 a+3 b) \tanh ^{-1}\left (\frac {\sqrt {a+b \sin (c+d x)}}{\sqrt {a+b}}\right )}{4 (a+b)^{3/2} d}-\frac {\sec ^2(c+d x) (b-a \sin (c+d x)) \sqrt {a+b \sin (c+d x)}}{2 \left (a^2-b^2\right ) d} \]
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Rubi [A]
time = 0.20, antiderivative size = 144, normalized size of antiderivative = 1.00, number of steps
used = 6, number of rules used = 5, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.217, Rules used = {2747, 755, 841,
1180, 212} \begin {gather*} -\frac {\sec ^2(c+d x) (b-a \sin (c+d x)) \sqrt {a+b \sin (c+d x)}}{2 d \left (a^2-b^2\right )}-\frac {(2 a-3 b) \tanh ^{-1}\left (\frac {\sqrt {a+b \sin (c+d x)}}{\sqrt {a-b}}\right )}{4 d (a-b)^{3/2}}+\frac {(2 a+3 b) \tanh ^{-1}\left (\frac {\sqrt {a+b \sin (c+d x)}}{\sqrt {a+b}}\right )}{4 d (a+b)^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 212
Rule 755
Rule 841
Rule 1180
Rule 2747
Rubi steps
\begin {align*} \int \frac {\sec ^3(c+d x)}{\sqrt {a+b \sin (c+d x)}} \, dx &=\frac {b^3 \text {Subst}\left (\int \frac {1}{\sqrt {a+x} \left (b^2-x^2\right )^2} \, dx,x,b \sin (c+d x)\right )}{d}\\ &=-\frac {\sec ^2(c+d x) (b-a \sin (c+d x)) \sqrt {a+b \sin (c+d x)}}{2 \left (a^2-b^2\right ) d}+\frac {b \text {Subst}\left (\int \frac {\frac {1}{2} \left (2 a^2-3 b^2\right )+\frac {a x}{2}}{\sqrt {a+x} \left (b^2-x^2\right )} \, dx,x,b \sin (c+d x)\right )}{2 \left (a^2-b^2\right ) d}\\ &=-\frac {\sec ^2(c+d x) (b-a \sin (c+d x)) \sqrt {a+b \sin (c+d x)}}{2 \left (a^2-b^2\right ) d}+\frac {b \text {Subst}\left (\int \frac {-\frac {a^2}{2}+\frac {1}{2} \left (2 a^2-3 b^2\right )+\frac {a x^2}{2}}{-a^2+b^2+2 a x^2-x^4} \, dx,x,\sqrt {a+b \sin (c+d x)}\right )}{\left (a^2-b^2\right ) d}\\ &=-\frac {\sec ^2(c+d x) (b-a \sin (c+d x)) \sqrt {a+b \sin (c+d x)}}{2 \left (a^2-b^2\right ) d}-\frac {(2 a-3 b) \text {Subst}\left (\int \frac {1}{a-b-x^2} \, dx,x,\sqrt {a+b \sin (c+d x)}\right )}{4 (a-b) d}+\frac {(2 a+3 b) \text {Subst}\left (\int \frac {1}{a+b-x^2} \, dx,x,\sqrt {a+b \sin (c+d x)}\right )}{4 (a+b) d}\\ &=-\frac {(2 a-3 b) \tanh ^{-1}\left (\frac {\sqrt {a+b \sin (c+d x)}}{\sqrt {a-b}}\right )}{4 (a-b)^{3/2} d}+\frac {(2 a+3 b) \tanh ^{-1}\left (\frac {\sqrt {a+b \sin (c+d x)}}{\sqrt {a+b}}\right )}{4 (a+b)^{3/2} d}-\frac {\sec ^2(c+d x) (b-a \sin (c+d x)) \sqrt {a+b \sin (c+d x)}}{2 \left (a^2-b^2\right ) d}\\ \end {align*}
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Mathematica [A]
time = 0.54, size = 176, normalized size = 1.22 \begin {gather*} \frac {\sqrt {a+b} \left (2 a^2-a b-3 b^2\right ) \tanh ^{-1}\left (\frac {\sqrt {a+b \sin (c+d x)}}{\sqrt {a-b}}\right )-\sqrt {a-b} \left (\left (2 a^2+a b-3 b^2\right ) \tanh ^{-1}\left (\frac {\sqrt {a+b \sin (c+d x)}}{\sqrt {a+b}}\right )+2 \sqrt {a+b} \sec ^2(c+d x) (-b+a \sin (c+d x)) \sqrt {a+b \sin (c+d x)}\right )}{4 \sqrt {a-b} \sqrt {a+b} \left (-a^2+b^2\right ) d} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 2.31, size = 204, normalized size = 1.42
method | result | size |
default | \(\frac {-\frac {b \sqrt {a +b \sin \left (d x +c \right )}}{4 \left (a +b \right ) \left (b \sin \left (d x +c \right )-b \right )}+\frac {\arctanh \left (\frac {\sqrt {a +b \sin \left (d x +c \right )}}{\sqrt {a +b}}\right ) a}{2 \left (a +b \right )^{\frac {3}{2}}}+\frac {3 \arctanh \left (\frac {\sqrt {a +b \sin \left (d x +c \right )}}{\sqrt {a +b}}\right ) b}{4 \left (a +b \right )^{\frac {3}{2}}}-\frac {b \sqrt {a +b \sin \left (d x +c \right )}}{4 \left (a -b \right ) \left (b \sin \left (d x +c \right )+b \right )}+\frac {\arctan \left (\frac {\sqrt {a +b \sin \left (d x +c \right )}}{\sqrt {-a +b}}\right ) a}{2 \left (a -b \right ) \sqrt {-a +b}}-\frac {3 \arctan \left (\frac {\sqrt {a +b \sin \left (d x +c \right )}}{\sqrt {-a +b}}\right ) b}{4 \left (a -b \right ) \sqrt {-a +b}}}{d}\) | \(204\) |
Verification of antiderivative is not currently implemented for this CAS.
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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.
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Fricas [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sec ^{3}{\left (c + d x \right )}}{\sqrt {a + b \sin {\left (c + d x \right )}}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 2.21, size = 212, normalized size = 1.47 \begin {gather*} \frac {b^{3} {\left (\frac {{\left (2 \, a - 3 \, b\right )} \arctan \left (\frac {\sqrt {b \sin \left (d x + c\right ) + a}}{\sqrt {-a + b}}\right )}{{\left (a b^{3} - b^{4}\right )} \sqrt {-a + b}} - \frac {{\left (2 \, a + 3 \, b\right )} \arctan \left (\frac {\sqrt {b \sin \left (d x + c\right ) + a}}{\sqrt {-a - b}}\right )}{{\left (a b^{3} + b^{4}\right )} \sqrt {-a - b}} - \frac {2 \, {\left ({\left (b \sin \left (d x + c\right ) + a\right )}^{\frac {3}{2}} a - \sqrt {b \sin \left (d x + c\right ) + a} a^{2} - \sqrt {b \sin \left (d x + c\right ) + a} b^{2}\right )}}{{\left (a^{2} b^{2} - b^{4}\right )} {\left ({\left (b \sin \left (d x + c\right ) + a\right )}^{2} - 2 \, {\left (b \sin \left (d x + c\right ) + a\right )} a + a^{2} - b^{2}\right )}}\right )}}{4 \, d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {1}{{\cos \left (c+d\,x\right )}^3\,\sqrt {a+b\,\sin \left (c+d\,x\right )}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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