Optimal. Leaf size=109 \[ \frac {\tanh ^{-1}(\sin (c+d x))}{b^2 d}-\frac {\sqrt {a-b} (2 a+b) \tanh ^{-1}\left (\frac {\sqrt {a-b} \sin (c+d x)}{\sqrt {a}}\right )}{2 a^{3/2} b^2 d}-\frac {(a-b) \sin (c+d x)}{2 a b d \left (a-(a-b) \sin ^2(c+d x)\right )} \]
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Rubi [A]
time = 0.09, antiderivative size = 109, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 5, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.217, Rules used = {3757, 425, 536,
212, 214} \begin {gather*} -\frac {\sqrt {a-b} (2 a+b) \tanh ^{-1}\left (\frac {\sqrt {a-b} \sin (c+d x)}{\sqrt {a}}\right )}{2 a^{3/2} b^2 d}-\frac {(a-b) \sin (c+d x)}{2 a b d \left (a-(a-b) \sin ^2(c+d x)\right )}+\frac {\tanh ^{-1}(\sin (c+d x))}{b^2 d} \end {gather*}
Antiderivative was successfully verified.
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Rule 212
Rule 214
Rule 425
Rule 536
Rule 3757
Rubi steps
\begin {align*} \int \frac {\sec ^5(c+d x)}{\left (a+b \tan ^2(c+d x)\right )^2} \, dx &=\frac {\text {Subst}\left (\int \frac {1}{\left (1-x^2\right ) \left (a-(a-b) x^2\right )^2} \, dx,x,\sin (c+d x)\right )}{d}\\ &=-\frac {(a-b) \sin (c+d x)}{2 a b d \left (a-(a-b) \sin ^2(c+d x)\right )}-\frac {\text {Subst}\left (\int \frac {-a-b+(-a+b) x^2}{\left (1-x^2\right ) \left (a+(-a+b) x^2\right )} \, dx,x,\sin (c+d x)\right )}{2 a b d}\\ &=-\frac {(a-b) \sin (c+d x)}{2 a b d \left (a-(a-b) \sin ^2(c+d x)\right )}+\frac {\text {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\sin (c+d x)\right )}{b^2 d}-\frac {((a-b) (2 a+b)) \text {Subst}\left (\int \frac {1}{a+(-a+b) x^2} \, dx,x,\sin (c+d x)\right )}{2 a b^2 d}\\ &=\frac {\tanh ^{-1}(\sin (c+d x))}{b^2 d}-\frac {\sqrt {a-b} (2 a+b) \tanh ^{-1}\left (\frac {\sqrt {a-b} \sin (c+d x)}{\sqrt {a}}\right )}{2 a^{3/2} b^2 d}-\frac {(a-b) \sin (c+d x)}{2 a b d \left (a-(a-b) \sin ^2(c+d x)\right )}\\ \end {align*}
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Mathematica [A]
time = 0.55, size = 191, normalized size = 1.75 \begin {gather*} \frac {-4 \log \left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )+4 \log \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )+\frac {\sqrt {a-b} (2 a+b) \log \left (\sqrt {a}-\sqrt {a-b} \sin (c+d x)\right )}{a^{3/2}}+\frac {\left (-2 a^2+a b+b^2\right ) \log \left (\sqrt {a}+\sqrt {a-b} \sin (c+d x)\right )}{a^{3/2} \sqrt {a-b}}+\frac {4 b (-a+b) \sin (c+d x)}{a (a+b+(a-b) \cos (2 (c+d x)))}}{4 b^2 d} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.43, size = 124, normalized size = 1.14
method | result | size |
derivativedivides | \(\frac {-\frac {\ln \left (\sin \left (d x +c \right )-1\right )}{2 b^{2}}+\frac {\ln \left (\sin \left (d x +c \right )+1\right )}{2 b^{2}}+\frac {\left (a -b \right ) \left (\frac {b \sin \left (d x +c \right )}{2 a \left (a \left (\sin ^{2}\left (d x +c \right )\right )-b \left (\sin ^{2}\left (d x +c \right )\right )-a \right )}-\frac {\left (2 a +b \right ) \arctanh \left (\frac {\left (a -b \right ) \sin \left (d x +c \right )}{\sqrt {a \left (a -b \right )}}\right )}{2 a \sqrt {a \left (a -b \right )}}\right )}{b^{2}}}{d}\) | \(124\) |
default | \(\frac {-\frac {\ln \left (\sin \left (d x +c \right )-1\right )}{2 b^{2}}+\frac {\ln \left (\sin \left (d x +c \right )+1\right )}{2 b^{2}}+\frac {\left (a -b \right ) \left (\frac {b \sin \left (d x +c \right )}{2 a \left (a \left (\sin ^{2}\left (d x +c \right )\right )-b \left (\sin ^{2}\left (d x +c \right )\right )-a \right )}-\frac {\left (2 a +b \right ) \arctanh \left (\frac {\left (a -b \right ) \sin \left (d x +c \right )}{\sqrt {a \left (a -b \right )}}\right )}{2 a \sqrt {a \left (a -b \right )}}\right )}{b^{2}}}{d}\) | \(124\) |
risch | \(\frac {i \left (a -b \right ) \left ({\mathrm e}^{3 i \left (d x +c \right )}-{\mathrm e}^{i \left (d x +c \right )}\right )}{b d a \left (a \,{\mathrm e}^{4 i \left (d x +c \right )}-b \,{\mathrm e}^{4 i \left (d x +c \right )}+2 a \,{\mathrm e}^{2 i \left (d x +c \right )}+2 b \,{\mathrm e}^{2 i \left (d x +c \right )}+a -b \right )}-\frac {\ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right )}{d \,b^{2}}+\frac {\ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right )}{d \,b^{2}}+\frac {\sqrt {a \left (a -b \right )}\, \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}-\frac {2 i \sqrt {a \left (a -b \right )}\, {\mathrm e}^{i \left (d x +c \right )}}{a -b}-1\right )}{2 a d \,b^{2}}+\frac {\sqrt {a \left (a -b \right )}\, \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}-\frac {2 i \sqrt {a \left (a -b \right )}\, {\mathrm e}^{i \left (d x +c \right )}}{a -b}-1\right )}{4 a^{2} d b}-\frac {\sqrt {a \left (a -b \right )}\, \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}+\frac {2 i \sqrt {a \left (a -b \right )}\, {\mathrm e}^{i \left (d x +c \right )}}{a -b}-1\right )}{2 a d \,b^{2}}-\frac {\sqrt {a \left (a -b \right )}\, \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}+\frac {2 i \sqrt {a \left (a -b \right )}\, {\mathrm e}^{i \left (d x +c \right )}}{a -b}-1\right )}{4 a^{2} d b}\) | \(375\) |
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 [A]
time = 2.54, size = 407, normalized size = 3.73 \begin {gather*} \left [\frac {{\left ({\left (2 \, a^{2} - a b - b^{2}\right )} \cos \left (d x + c\right )^{2} + 2 \, a b + b^{2}\right )} \sqrt {\frac {a - b}{a}} \log \left (-\frac {{\left (a - b\right )} \cos \left (d x + c\right )^{2} + 2 \, a \sqrt {\frac {a - b}{a}} \sin \left (d x + c\right ) - 2 \, a + b}{{\left (a - b\right )} \cos \left (d x + c\right )^{2} + b}\right ) + 2 \, {\left ({\left (a^{2} - a b\right )} \cos \left (d x + c\right )^{2} + a b\right )} \log \left (\sin \left (d x + c\right ) + 1\right ) - 2 \, {\left ({\left (a^{2} - a b\right )} \cos \left (d x + c\right )^{2} + a b\right )} \log \left (-\sin \left (d x + c\right ) + 1\right ) - 2 \, {\left (a b - b^{2}\right )} \sin \left (d x + c\right )}{4 \, {\left (a b^{3} d + {\left (a^{2} b^{2} - a b^{3}\right )} d \cos \left (d x + c\right )^{2}\right )}}, \frac {{\left ({\left (2 \, a^{2} - a b - b^{2}\right )} \cos \left (d x + c\right )^{2} + 2 \, a b + b^{2}\right )} \sqrt {-\frac {a - b}{a}} \arctan \left (\sqrt {-\frac {a - b}{a}} \sin \left (d x + c\right )\right ) + {\left ({\left (a^{2} - a b\right )} \cos \left (d x + c\right )^{2} + a b\right )} \log \left (\sin \left (d x + c\right ) + 1\right ) - {\left ({\left (a^{2} - a b\right )} \cos \left (d x + c\right )^{2} + a b\right )} \log \left (-\sin \left (d x + c\right ) + 1\right ) - {\left (a b - b^{2}\right )} \sin \left (d x + c\right )}{2 \, {\left (a b^{3} d + {\left (a^{2} b^{2} - a b^{3}\right )} d \cos \left (d x + c\right )^{2}\right )}}\right ] \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 ^{5}{\left (c + d x \right )}}{\left (a + b \tan ^{2}{\left (c + d x \right )}\right )^{2}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.74, size = 153, normalized size = 1.40 \begin {gather*} \frac {\frac {\log \left ({\left | \sin \left (d x + c\right ) + 1 \right |}\right )}{b^{2}} - \frac {\log \left ({\left | \sin \left (d x + c\right ) - 1 \right |}\right )}{b^{2}} - \frac {{\left (2 \, a^{2} - a b - b^{2}\right )} \arctan \left (-\frac {a \sin \left (d x + c\right ) - b \sin \left (d x + c\right )}{\sqrt {-a^{2} + a b}}\right )}{\sqrt {-a^{2} + a b} a b^{2}} + \frac {a \sin \left (d x + c\right ) - b \sin \left (d x + c\right )}{{\left (a \sin \left (d x + c\right )^{2} - b \sin \left (d x + c\right )^{2} - a\right )} a b}}{2 \, d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 14.37, size = 946, normalized size = 8.68 \begin {gather*} \frac {\left (-a^{5/2}\,\mathrm {atanh}\left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )\,2{}\mathrm {i}+a^2\,\mathrm {atan}\left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (a-2\,b+2\,a\,\cos \left (c+d\,x\right )-2\,b\,\cos \left (c+d\,x\right )\right )}{2\,\sqrt {a}\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3\,\sqrt {b-a}}\right )\,\sqrt {b-a}\,1{}\mathrm {i}+\frac {b^2\,\mathrm {atan}\left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (a-2\,b+2\,a\,\cos \left (c+d\,x\right )-2\,b\,\cos \left (c+d\,x\right )\right )}{2\,\sqrt {a}\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3\,\sqrt {b-a}}\right )\,\sqrt {b-a}\,1{}\mathrm {i}}{2}-a^{3/2}\,b\,\mathrm {atanh}\left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )\,2{}\mathrm {i}-\sqrt {a}\,b^2\,\sin \left (c+d\,x\right )\,1{}\mathrm {i}+a^2\,\mathrm {atan}\left (\frac {\sqrt {a}\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{2\,\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\sqrt {b-a}}\right )\,\sqrt {b-a}\,1{}\mathrm {i}+\frac {b^2\,\mathrm {atan}\left (\frac {\sqrt {a}\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{2\,\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\sqrt {b-a}}\right )\,\sqrt {b-a}\,1{}\mathrm {i}}{2}-a^{5/2}\,\mathrm {atanh}\left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )\,\cos \left (2\,c+2\,d\,x\right )\,2{}\mathrm {i}+a^{3/2}\,b\,\sin \left (c+d\,x\right )\,1{}\mathrm {i}+a^2\,\mathrm {atan}\left (\frac {\sqrt {a}\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{2\,\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\sqrt {b-a}}\right )\,\cos \left (2\,c+2\,d\,x\right )\,\sqrt {b-a}\,1{}\mathrm {i}-\frac {b^2\,\mathrm {atan}\left (\frac {\sqrt {a}\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{2\,\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\sqrt {b-a}}\right )\,\cos \left (2\,c+2\,d\,x\right )\,\sqrt {b-a}\,1{}\mathrm {i}}{2}+\frac {a\,b\,\mathrm {atan}\left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (a-2\,b+2\,a\,\cos \left (c+d\,x\right )-2\,b\,\cos \left (c+d\,x\right )\right )}{2\,\sqrt {a}\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3\,\sqrt {b-a}}\right )\,\sqrt {b-a}\,3{}\mathrm {i}}{2}+\frac {a\,b\,\mathrm {atan}\left (\frac {\sqrt {a}\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{2\,\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\sqrt {b-a}}\right )\,\sqrt {b-a}\,3{}\mathrm {i}}{2}+a^2\,\mathrm {atan}\left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (a-2\,b+2\,a\,\cos \left (c+d\,x\right )-2\,b\,\cos \left (c+d\,x\right )\right )}{2\,\sqrt {a}\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3\,\sqrt {b-a}}\right )\,\cos \left (2\,c+2\,d\,x\right )\,\sqrt {b-a}\,1{}\mathrm {i}-\frac {b^2\,\mathrm {atan}\left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (a-2\,b+2\,a\,\cos \left (c+d\,x\right )-2\,b\,\cos \left (c+d\,x\right )\right )}{2\,\sqrt {a}\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3\,\sqrt {b-a}}\right )\,\cos \left (2\,c+2\,d\,x\right )\,\sqrt {b-a}\,1{}\mathrm {i}}{2}+a^{3/2}\,b\,\mathrm {atanh}\left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )\,\cos \left (2\,c+2\,d\,x\right )\,2{}\mathrm {i}-\frac {a\,b\,\mathrm {atan}\left (\frac {\sqrt {a}\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{2\,\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\sqrt {b-a}}\right )\,\cos \left (2\,c+2\,d\,x\right )\,\sqrt {b-a}\,1{}\mathrm {i}}{2}-\frac {a\,b\,\mathrm {atan}\left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (a-2\,b+2\,a\,\cos \left (c+d\,x\right )-2\,b\,\cos \left (c+d\,x\right )\right )}{2\,\sqrt {a}\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3\,\sqrt {b-a}}\right )\,\cos \left (2\,c+2\,d\,x\right )\,\sqrt {b-a}\,1{}\mathrm {i}}{2}\right )\,1{}\mathrm {i}}{2\,a^{3/2}\,b^2\,d\,\left (\frac {a}{2}+\frac {b}{2}+\frac {a\,\cos \left (2\,c+2\,d\,x\right )}{2}-\frac {b\,\cos \left (2\,c+2\,d\,x\right )}{2}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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