Optimal. Leaf size=94 \[ -\frac {2 b \sqrt {a+b \sin (c+d x)}}{d}-\frac {(a-b)^{3/2} \tanh ^{-1}\left (\frac {\sqrt {a+b \sin (c+d x)}}{\sqrt {a-b}}\right )}{d}+\frac {(a+b)^{3/2} \tanh ^{-1}\left (\frac {\sqrt {a+b \sin (c+d x)}}{\sqrt {a+b}}\right )}{d} \]
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Rubi [A] time = 0.17, antiderivative size = 94, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.238, Rules used = {2668, 704, 827, 1166, 206} \[ -\frac {2 b \sqrt {a+b \sin (c+d x)}}{d}-\frac {(a-b)^{3/2} \tanh ^{-1}\left (\frac {\sqrt {a+b \sin (c+d x)}}{\sqrt {a-b}}\right )}{d}+\frac {(a+b)^{3/2} \tanh ^{-1}\left (\frac {\sqrt {a+b \sin (c+d x)}}{\sqrt {a+b}}\right )}{d} \]
Antiderivative was successfully verified.
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Rule 206
Rule 704
Rule 827
Rule 1166
Rule 2668
Rubi steps
\begin {align*} \int \sec (c+d x) (a+b \sin (c+d x))^{3/2} \, dx &=\frac {b \operatorname {Subst}\left (\int \frac {(a+x)^{3/2}}{b^2-x^2} \, dx,x,b \sin (c+d x)\right )}{d}\\ &=-\frac {2 b \sqrt {a+b \sin (c+d x)}}{d}-\frac {b \operatorname {Subst}\left (\int \frac {-a^2-b^2-2 a x}{\sqrt {a+x} \left (b^2-x^2\right )} \, dx,x,b \sin (c+d x)\right )}{d}\\ &=-\frac {2 b \sqrt {a+b \sin (c+d x)}}{d}-\frac {(2 b) \operatorname {Subst}\left (\int \frac {a^2-b^2-2 a x^2}{-a^2+b^2+2 a x^2-x^4} \, dx,x,\sqrt {a+b \sin (c+d x)}\right )}{d}\\ &=-\frac {2 b \sqrt {a+b \sin (c+d x)}}{d}-\frac {(a-b)^2 \operatorname {Subst}\left (\int \frac {1}{a-b-x^2} \, dx,x,\sqrt {a+b \sin (c+d x)}\right )}{d}+\frac {(a+b)^2 \operatorname {Subst}\left (\int \frac {1}{a+b-x^2} \, dx,x,\sqrt {a+b \sin (c+d x)}\right )}{d}\\ &=-\frac {(a-b)^{3/2} \tanh ^{-1}\left (\frac {\sqrt {a+b \sin (c+d x)}}{\sqrt {a-b}}\right )}{d}+\frac {(a+b)^{3/2} \tanh ^{-1}\left (\frac {\sqrt {a+b \sin (c+d x)}}{\sqrt {a+b}}\right )}{d}-\frac {2 b \sqrt {a+b \sin (c+d x)}}{d}\\ \end {align*}
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Mathematica [A] time = 0.10, size = 89, normalized size = 0.95 \[ \frac {-2 b \sqrt {a+b \sin (c+d x)}+(a-b)^{3/2} \left (-\tanh ^{-1}\left (\frac {\sqrt {a+b \sin (c+d x)}}{\sqrt {a-b}}\right )\right )+(a+b)^{3/2} \tanh ^{-1}\left (\frac {\sqrt {a+b \sin (c+d x)}}{\sqrt {a+b}}\right )}{d} \]
Antiderivative was successfully verified.
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fricas [F] time = 1.90, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (b \sec \left (d x + c\right ) \sin \left (d x + c\right ) + a \sec \left (d x + c\right )\right )} \sqrt {b \sin \left (d x + c\right ) + a}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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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.
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maple [B] time = 0.48, size = 218, normalized size = 2.32 \[ -\frac {2 b \sqrt {a +b \sin \left (d x +c \right )}}{d}+\frac {\arctan \left (\frac {\sqrt {a +b \sin \left (d x +c \right )}}{\sqrt {-a +b}}\right ) a^{2}}{d \sqrt {-a +b}}-\frac {2 b \arctan \left (\frac {\sqrt {a +b \sin \left (d x +c \right )}}{\sqrt {-a +b}}\right ) a}{d \sqrt {-a +b}}+\frac {b^{2} \arctan \left (\frac {\sqrt {a +b \sin \left (d x +c \right )}}{\sqrt {-a +b}}\right )}{d \sqrt {-a +b}}+\frac {\arctanh \left (\frac {\sqrt {a +b \sin \left (d x +c \right )}}{\sqrt {a +b}}\right ) a^{2}}{d \sqrt {a +b}}+\frac {2 b \arctanh \left (\frac {\sqrt {a +b \sin \left (d x +c \right )}}{\sqrt {a +b}}\right ) a}{d \sqrt {a +b}}+\frac {b^{2} \arctanh \left (\frac {\sqrt {a +b \sin \left (d x +c \right )}}{\sqrt {a +b}}\right )}{d \sqrt {a +b}} \]
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 \[ \text {Exception raised: ValueError} \]
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 \[ \int \frac {{\left (a+b\,\sin \left (c+d\,x\right )\right )}^{3/2}}{\cos \left (c+d\,x\right )} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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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.
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